LMFDB - Embedded newform 450.2.p.e.443.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [450,2,Mod(257,450)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(450, base_ring=CyclotomicField(12))

chi = DirichletCharacter(H, H._module([10, 3]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("450.257");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 450 = 2 \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	450.p (of order $12$, degree $4$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$3.59326809096$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$2$ over $\Q(\zeta_{12})$
Coefficient field:	$\Q(\zeta_{24})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			443.2
Root			$-0.258819 - 0.965926i$ of defining polynomial
Character	$\chi$	$=$	450.443
Dual form			450.2.p.e.257.2

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.965926 - 0.258819i) q^{2} +(1.22474 - 1.22474i) q^{3} +(0.866025 - 0.500000i) q^{4} +(0.866025 - 1.50000i) q^{6} +(0.328169 + 1.22474i) q^{7} +(0.707107 - 0.707107i) q^{8} -3.00000i q^{9} +O(q^{10})$	\(q+(0.965926 - 0.258819i) q^{2} +(1.22474 - 1.22474i) q^{3} +(0.866025 - 0.500000i) q^{4} +(0.866025 - 1.50000i) q^{6} +(0.328169 + 1.22474i) q^{7} +(0.707107 - 0.707107i) q^{8} -3.00000i q^{9} +(3.00000 + 1.73205i) q^{11} +(0.448288 - 1.67303i) q^{12} +(-0.328169 + 1.22474i) q^{13} +(0.633975 + 1.09808i) q^{14} +(0.500000 - 0.866025i) q^{16} +(-0.776457 - 2.89778i) q^{18} -7.19615i q^{19} +(1.90192 + 1.09808i) q^{21} +(3.34607 + 0.896575i) q^{22} +(-7.91688 - 2.12132i) q^{23} -1.73205i q^{24} +1.26795i q^{26} +(-3.67423 - 3.67423i) q^{27} +(0.896575 + 0.896575i) q^{28} +(-3.63397 + 6.29423i) q^{29} +(5.09808 + 8.83013i) q^{31} +(0.258819 - 0.965926i) q^{32} +(5.79555 - 1.55291i) q^{33} +(-1.50000 - 2.59808i) q^{36} +(-1.55291 + 1.55291i) q^{37} +(-1.86250 - 6.95095i) q^{38} +(1.09808 + 1.90192i) q^{39} +(-1.50000 + 0.866025i) q^{41} +(2.12132 + 0.568406i) q^{42} +(-6.24384 + 1.67303i) q^{43} +3.46410 q^{44} -8.19615 q^{46} +(-5.79555 + 1.55291i) q^{47} +(-0.448288 - 1.67303i) q^{48} +(4.66987 - 2.69615i) q^{49} +(0.328169 + 1.22474i) q^{52} +(-1.55291 + 1.55291i) q^{53} +(-4.50000 - 2.59808i) q^{54} +(1.09808 + 0.633975i) q^{56} +(-8.81345 - 8.81345i) q^{57} +(-1.88108 + 7.02030i) q^{58} +(6.23205 + 10.7942i) q^{59} +(2.00000 - 3.46410i) q^{61} +(7.20977 + 7.20977i) q^{62} +(3.67423 - 0.984508i) q^{63} -1.00000i q^{64} +(5.19615 - 3.00000i) q^{66} +(12.0394 + 3.22595i) q^{67} +(-12.2942 + 7.09808i) q^{69} -10.7321i q^{71} +(-2.12132 - 2.12132i) q^{72} +(3.67423 + 3.67423i) q^{73} +(-1.09808 + 1.90192i) q^{74} +(-3.59808 - 6.23205i) q^{76} +(-1.13681 + 4.24264i) q^{77} +(1.55291 + 1.55291i) q^{78} +(8.66025 + 5.00000i) q^{79} -9.00000 q^{81} +(-1.22474 + 1.22474i) q^{82} +(1.76097 + 6.57201i) q^{83} +2.19615 q^{84} +(-5.59808 + 3.23205i) q^{86} +(3.25813 + 12.1595i) q^{87} +(3.34607 - 0.896575i) q^{88} -8.66025 q^{89} -1.60770 q^{91} +(-7.91688 + 2.12132i) q^{92} +(17.0585 + 4.57081i) q^{93} +(-5.19615 + 3.00000i) q^{94} +(-0.866025 - 1.50000i) q^{96} +(-3.79435 - 14.1607i) q^{97} +(3.81294 - 3.81294i) q^{98} +(5.19615 - 9.00000i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q + 24 q^{11} + 12 q^{14} + 4 q^{16} + 36 q^{21} - 36 q^{29} + 20 q^{31} - 12 q^{36} - 12 q^{39} - 12 q^{41} - 24 q^{46} + 72 q^{49} - 36 q^{54} - 12 q^{56} + 36 q^{59} + 16 q^{61} - 36 q^{69} + 12 q^{74} - 8 q^{76} - 72 q^{81} - 24 q^{84} - 24 q^{86} - 96 q^{91}+O(q^{100}) $ 8 * q + 24 * q^11 + 12 * q^14 + 4 * q^16 + 36 * q^21 - 36 * q^29 + 20 * q^31 - 12 * q^36 - 12 * q^39 - 12 * q^41 - 24 * q^46 + 72 * q^49 - 36 * q^54 - 12 * q^56 + 36 * q^59 + 16 * q^61 - 36 * q^69 + 12 * q^74 - 8 * q^76 - 72 * q^81 - 24 * q^84 - 24 * q^86 - 96 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/450\mathbb{Z}\right)^\times$.

$n$	$101$	$127$
$\chi(n)$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{3}{4}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	0.965926	−	0.258819i	0.683013	−	0.183013i
$2$	0.965926	−	0.258819i	0.683013	−	0.183013i
$3$	1.22474	−	1.22474i	0.707107	−	0.707107i
$3$	1.22474	−	1.22474i	0.707107	−	0.707107i
$4$	0.866025	−	0.500000i	0.433013	−	0.250000i
$5$		0			0
$5$		0			0
$6$	0.866025	−	1.50000i	0.353553	−	0.612372i
$7$	0.328169	+	1.22474i	0.124036	+	0.462910i	0.999803	−	0.0198238i	$-0.00631052\pi$
$7$	0.328169	+	1.22474i	0.124036	+	0.462910i	−0.875767	+	0.482734i	$0.839644\pi$
$8$	0.707107	−	0.707107i	0.250000	−	0.250000i
$9$		−	3.00000i		−	1.00000i
$10$		0			0
$11$	3.00000	+	1.73205i	0.904534	+	0.522233i	0.878668	−	0.477432i	$-0.158432\pi$
$11$	3.00000	+	1.73205i	0.904534	+	0.522233i	0.0258656	+	0.999665i	$0.491766\pi$
$12$	0.448288	−	1.67303i	0.129410	−	0.482963i
$13$	−0.328169	+	1.22474i	−0.0910178	+	0.339683i	−0.996386	−	0.0849451i	$-0.972929\pi$
$13$	−0.328169	+	1.22474i	−0.0910178	+	0.339683i	0.905368	+	0.424628i	$0.139595\pi$
$14$	0.633975	+	1.09808i	0.169437	+	0.293473i
$15$		0			0
$16$	0.500000	−	0.866025i	0.125000	−	0.216506i
$17$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$17$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$18$	−0.776457	−	2.89778i	−0.183013	−	0.683013i
$19$		−	7.19615i		−	1.65091i	−0.564467	−	0.825455i	$-0.690918\pi$
$19$		−	7.19615i		−	1.65091i	0.564467	−	0.825455i	$-0.309082\pi$
$20$		0			0
$21$	1.90192	+	1.09808i	0.415034	+	0.239620i
$22$	3.34607	+	0.896575i	0.713384	+	0.191151i
$23$	−7.91688	−	2.12132i	−1.65078	−	0.442326i	−0.690951	−	0.722902i	$-0.742808\pi$
$23$	−7.91688	−	2.12132i	−1.65078	−	0.442326i	−0.959832	+	0.280576i	$0.909475\pi$
$24$		−	1.73205i		−	0.353553i
$25$		0			0
$26$			1.26795i			0.248665i
$27$	−3.67423	−	3.67423i	−0.707107	−	0.707107i
$28$	0.896575	+	0.896575i	0.169437	+	0.169437i
$29$	−3.63397	+	6.29423i	−0.674812	+	1.16881i	0.301712	+	0.953399i	$0.402442\pi$
$29$	−3.63397	+	6.29423i	−0.674812	+	1.16881i	−0.976524	+	0.215410i	$0.930891\pi$
$30$		0			0
$31$	5.09808	+	8.83013i	0.915642	+	1.58594i	0.805959	+	0.591971i	$0.201650\pi$
$31$	5.09808	+	8.83013i	0.915642	+	1.58594i	0.109682	+	0.993967i	$0.465017\pi$
$32$	0.258819	−	0.965926i	0.0457532	−	0.170753i
$33$	5.79555	−	1.55291i	1.00888	−	0.270328i
$34$		0			0
$35$		0			0
$36$	−1.50000	−	2.59808i	−0.250000	−	0.433013i
$37$	−1.55291	+	1.55291i	−0.255298	+	0.255298i	−0.823138	−	0.567841i	$-0.807779\pi$
$37$	−1.55291	+	1.55291i	−0.255298	+	0.255298i	0.567841	+	0.823138i	$0.307779\pi$
$38$	−1.86250	−	6.95095i	−0.302138	−	1.12759i
$39$	1.09808	+	1.90192i	0.175833	+	0.304552i
$40$		0			0
$41$	−1.50000	+	0.866025i	−0.234261	+	0.135250i	−0.612536	−	0.790443i	$-0.709851\pi$
$41$	−1.50000	+	0.866025i	−0.234261	+	0.135250i	0.378275	+	0.925693i	$0.376517\pi$
$42$	2.12132	+	0.568406i	0.327327	+	0.0877070i
$43$	−6.24384	+	1.67303i	−0.952177	+	0.255135i	−0.701286	−	0.712880i	$-0.747390\pi$
$43$	−6.24384	+	1.67303i	−0.952177	+	0.255135i	−0.250891	+	0.968015i	$0.580724\pi$
$44$	3.46410			0.522233
$45$		0			0
$46$	−8.19615			−1.20846
$47$	−5.79555	+	1.55291i	−0.845369	+	0.226516i	−0.655407	−	0.755276i	$-0.727503\pi$
$47$	−5.79555	+	1.55291i	−0.845369	+	0.226516i	−0.189961	+	0.981792i	$0.560836\pi$
$48$	−0.448288	−	1.67303i	−0.0647048	−	0.241481i
$49$	4.66987	−	2.69615i	0.667125	−	0.385165i
$50$		0			0
$51$		0			0
$52$	0.328169	+	1.22474i	0.0455089	+	0.169842i
$53$	−1.55291	+	1.55291i	−0.213309	+	0.213309i	−0.805672	−	0.592362i	$-0.798195\pi$
$53$	−1.55291	+	1.55291i	−0.213309	+	0.213309i	0.592362	+	0.805672i	$0.298195\pi$
$54$	−4.50000	−	2.59808i	−0.612372	−	0.353553i
$55$		0			0
$56$	1.09808	+	0.633975i	0.146737	+	0.0847184i
$57$	−8.81345	−	8.81345i	−1.16737	−	1.16737i
$58$	−1.88108	+	7.02030i	−0.246998	+	0.921811i
$59$	6.23205	+	10.7942i	0.811344	+	1.40529i	0.911924	+	0.410360i	$0.134597\pi$
$59$	6.23205	+	10.7942i	0.811344	+	1.40529i	−0.100580	+	0.994929i	$0.532070\pi$
$60$		0			0
$61$	2.00000	−	3.46410i	0.256074	−	0.443533i	−0.709113	−	0.705095i	$-0.750904\pi$
$61$	2.00000	−	3.46410i	0.256074	−	0.443533i	0.965187	+	0.261562i	$0.0842377\pi$
$62$	7.20977	+	7.20977i	0.915642	+	0.915642i
$63$	3.67423	−	0.984508i	0.462910	−	0.124036i
$64$		−	1.00000i		−	0.125000i
$65$		0			0
$66$	5.19615	−	3.00000i	0.639602	−	0.369274i
$67$	12.0394	+	3.22595i	1.47085	+	0.394112i	0.903221	−	0.429175i	$-0.141196\pi$
$67$	12.0394	+	3.22595i	1.47085	+	0.394112i	0.567625	+	0.823287i	$0.307862\pi$
$68$		0			0
$69$	−12.2942	+	7.09808i	−1.48005	+	0.854508i
$70$		0			0
$71$		−	10.7321i		−	1.27366i	−0.771004	−	0.636830i	$-0.780245\pi$
$71$		−	10.7321i		−	1.27366i	0.771004	−	0.636830i	$-0.219755\pi$
$72$	−2.12132	−	2.12132i	−0.250000	−	0.250000i
$73$	3.67423	+	3.67423i	0.430037	+	0.430037i	0.888641	−	0.458604i	$-0.151650\pi$
$73$	3.67423	+	3.67423i	0.430037	+	0.430037i	−0.458604	+	0.888641i	$0.651650\pi$
$74$	−1.09808	+	1.90192i	−0.127649	+	0.221094i
$75$		0			0
$76$	−3.59808	−	6.23205i	−0.412728	−	0.714865i
$77$	−1.13681	+	4.24264i	−0.129552	+	0.483494i
$78$	1.55291	+	1.55291i	0.175833	+	0.175833i
$79$	8.66025	+	5.00000i	0.974355	+	0.562544i	0.900561	−	0.434730i	$-0.143156\pi$
$79$	8.66025	+	5.00000i	0.974355	+	0.562544i	0.0737937	+	0.997274i	$0.476489\pi$
$80$		0			0
$81$	−9.00000			−1.00000
$82$	−1.22474	+	1.22474i	−0.135250	+	0.135250i
$83$	1.76097	+	6.57201i	0.193291	+	0.721372i	0.992703	+	0.120588i	$0.0384781\pi$
$83$	1.76097	+	6.57201i	0.193291	+	0.721372i	−0.799412	+	0.600784i	$0.794855\pi$
$84$	2.19615			0.239620
$85$		0			0
$86$	−5.59808	+	3.23205i	−0.603656	+	0.348521i
$87$	3.25813	+	12.1595i	0.349308	+	1.30364i
$88$	3.34607	−	0.896575i	0.356692	−	0.0955753i
$89$	−8.66025			−0.917985			−0.458993	−	0.888440i	$-0.651790\pi$
$89$	−8.66025			−0.917985			−0.458993	+	0.888440i	$0.651790\pi$
$90$		0			0
$91$	−1.60770			−0.168532
$92$	−7.91688	+	2.12132i	−0.825391	+	0.221163i
$93$	17.0585	+	4.57081i	1.76888	+	0.473971i
$94$	−5.19615	+	3.00000i	−0.535942	+	0.309426i
$95$		0			0
$96$	−0.866025	−	1.50000i	−0.0883883	−	0.153093i
$97$	−3.79435	−	14.1607i	−0.385258	−	1.43780i	−0.837760	−	0.546039i	$-0.816135\pi$
$97$	−3.79435	−	14.1607i	−0.385258	−	1.43780i	0.452501	−	0.891764i	$-0.350532\pi$
$98$	3.81294	−	3.81294i	0.385165	−	0.385165i
$99$	5.19615	−	9.00000i	0.522233	−	0.904534i
$100$		0			0
$101$	−9.29423	−	5.36603i	−0.924810	−	0.533939i	−0.0396438	−	0.999214i	$-0.512622\pi$
$101$	−9.29423	−	5.36603i	−0.924810	−	0.533939i	−0.885167	+	0.465274i	$0.845956\pi$
$102$		0			0
$103$	−2.68973	+	10.0382i	−0.265027	+	0.989093i	0.697207	+	0.716869i	$0.254426\pi$
$103$	−2.68973	+	10.0382i	−0.265027	+	0.989093i	−0.962234	+	0.272223i	$0.912241\pi$
$104$	0.633975	+	1.09808i	0.0621663	+	0.107675i
$105$		0			0
$106$	−1.09808	+	1.90192i	−0.106655	+	0.184731i
$107$	0.568406	+	0.568406i	0.0549499	+	0.0549499i	0.734048	−	0.679098i	$-0.237629\pi$
$107$	0.568406	+	0.568406i	0.0549499	+	0.0549499i	−0.679098	+	0.734048i	$0.737629\pi$
$108$	−5.01910	−	1.34486i	−0.482963	−	0.129410i
$109$		−	12.1962i		−	1.16818i	−0.811689	−	0.584090i	$-0.801452\pi$
$109$		−	12.1962i		−	1.16818i	0.811689	−	0.584090i	$-0.198548\pi$
$110$		0			0
$111$			3.80385i			0.361045i
$112$	1.22474	+	0.328169i	0.115728	+	0.0310091i
$113$	−7.14042	−	1.91327i	−0.671714	−	0.179985i	−0.0931872	−	0.995649i	$-0.529706\pi$
$113$	−7.14042	−	1.91327i	−0.671714	−	0.179985i	−0.578527	+	0.815663i	$0.696372\pi$
$114$	−10.7942	−	6.23205i	−1.01097	−	0.583685i
$115$		0			0
$116$			7.26795i			0.674812i
$117$	3.67423	+	0.984508i	0.339683	+	0.0910178i
$118$	8.81345	+	8.81345i	0.811344	+	0.811344i
$119$		0			0
$120$		0			0
$121$	0.500000	+	0.866025i	0.0454545	+	0.0787296i
$122$	1.03528	−	3.86370i	0.0937295	−	0.349803i
$123$	−0.776457	+	2.89778i	−0.0700108	+	0.261284i
$124$	8.83013	+	5.09808i	0.792969	+	0.457821i
$125$		0			0
$126$	3.29423	−	1.90192i	0.293473	−	0.169437i
$127$	−1.55291	+	1.55291i	−0.137799	+	0.137799i	−0.772641	−	0.634843i	$-0.781065\pi$
$127$	−1.55291	+	1.55291i	−0.137799	+	0.137799i	0.634843	+	0.772641i	$0.281065\pi$
$128$	−0.258819	−	0.965926i	−0.0228766	−	0.0853766i
$129$	−5.59808	+	9.69615i	−0.492883	+	0.853699i
$130$		0			0
$131$	−6.00000	+	3.46410i	−0.524222	+	0.302660i	−0.738661	−	0.674078i	$-0.764541\pi$
$131$	−6.00000	+	3.46410i	−0.524222	+	0.302660i	0.214438	+	0.976738i	$0.431208\pi$
$132$	4.24264	−	4.24264i	0.369274	−	0.369274i
$133$	8.81345	−	2.36156i	0.764223	−	0.204773i
$134$	12.4641			1.07673
$135$		0			0
$136$		0			0
$137$	−7.14042	+	1.91327i	−0.610047	+	0.163462i	−0.550599	−	0.834770i	$-0.685601\pi$
$137$	−7.14042	+	1.91327i	−0.610047	+	0.163462i	−0.0594480	+	0.998231i	$0.518934\pi$
$138$	−10.0382	+	10.0382i	−0.854508	+	0.854508i
$139$	−6.92820	+	4.00000i	−0.587643	+	0.339276i	−0.764165	−	0.645021i	$-0.776849\pi$
$139$	−6.92820	+	4.00000i	−0.587643	+	0.339276i	0.176522	+	0.984297i	$0.443515\pi$
$140$		0			0
$141$	−5.19615	+	9.00000i	−0.437595	+	0.757937i
$142$	−2.77766	−	10.3664i	−0.233096	−	0.869926i
$143$	−3.10583	+	3.10583i	−0.259722	+	0.259722i
$144$	−2.59808	−	1.50000i	−0.216506	−	0.125000i
$145$		0			0
$146$	4.50000	+	2.59808i	0.372423	+	0.215018i
$147$	2.41730	−	9.02150i	0.199376	−	0.744081i
$148$	−0.568406	+	2.12132i	−0.0467227	+	0.174371i
$149$	−9.00000	−	15.5885i	−0.737309	−	1.27706i	−0.953703	−	0.300750i	$-0.902763\pi$
$149$	−9.00000	−	15.5885i	−0.737309	−	1.27706i	0.216394	−	0.976306i	$-0.430570\pi$
$150$		0			0
$151$	4.00000	−	6.92820i	0.325515	−	0.563809i	−0.656101	−	0.754673i	$-0.727796\pi$
$151$	4.00000	−	6.92820i	0.325515	−	0.563809i	0.981617	+	0.190864i	$0.0611289\pi$
$152$	−5.08845	−	5.08845i	−0.412728	−	0.412728i
$153$		0			0
$154$			4.39230i			0.353942i
$155$		0			0
$156$	1.90192	+	1.09808i	0.152276	+	0.0879165i
$157$	7.58871	+	2.03339i	0.605645	+	0.162282i	0.548593	−	0.836089i	$-0.315163\pi$
$157$	7.58871	+	2.03339i	0.605645	+	0.162282i	0.0570512	+	0.998371i	$0.481830\pi$
$158$	9.65926	+	2.58819i	0.768449	+	0.205905i
$159$			3.80385i			0.301665i
$160$		0			0
$161$		−	10.3923i		−	0.819028i
$162$	−8.69333	+	2.32937i	−0.683013	+	0.183013i
$163$	−14.3688	−	14.3688i	−1.12545	−	1.12545i	−0.990908	−	0.134541i	$-0.957044\pi$
$163$	−14.3688	−	14.3688i	−1.12545	−	1.12545i	−0.134541	−	0.990908i	$-0.542956\pi$
$164$	−0.866025	+	1.50000i	−0.0676252	+	0.117130i
$165$		0			0
$166$	3.40192	+	5.89230i	0.264040	+	0.457332i
$167$	1.13681	−	4.24264i	0.0879692	−	0.328305i	−0.907891	−	0.419207i	$-0.862308\pi$
$167$	1.13681	−	4.24264i	0.0879692	−	0.328305i	0.995860	+	0.0909015i	$0.0289749\pi$
$168$	2.12132	−	0.568406i	0.163663	−	0.0438535i
$169$	9.86603	+	5.69615i	0.758925	+	0.438166i
$170$		0			0
$171$	−21.5885			−1.65091
$172$	−4.57081	+	4.57081i	−0.348521	+	0.348521i
$173$	1.13681	+	4.24264i	0.0864302	+	0.322562i	0.995581	−	0.0939047i	$-0.0299349\pi$
$173$	1.13681	+	4.24264i	0.0864302	+	0.322562i	−0.909151	+	0.416467i	$0.863268\pi$
$174$	6.29423	+	10.9019i	0.477164	+	0.826473i
$175$		0			0
$176$	3.00000	−	1.73205i	0.226134	−	0.130558i
$177$	20.8528	+	5.58750i	1.56740	+	0.419983i
$178$	−8.36516	+	2.24144i	−0.626995	+	0.168003i
$179$	4.85641			0.362985			0.181492	−	0.983392i	$-0.441907\pi$
$179$	4.85641			0.362985			0.181492	+	0.983392i	$0.441907\pi$
$180$		0			0
$181$	8.39230			0.623795			0.311898	−	0.950116i	$-0.399035\pi$
$181$	8.39230			0.623795			0.311898	+	0.950116i	$0.399035\pi$
$182$	−1.55291	+	0.416102i	−0.115110	+	0.0308435i
$183$	−1.79315	−	6.69213i	−0.132554	−	0.494697i
$184$	−7.09808	+	4.09808i	−0.523277	+	0.302114i
$185$		0			0
$186$	17.6603			1.29491
$187$		0			0
$188$	−4.24264	+	4.24264i	−0.309426	+	0.309426i
$189$	3.29423	−	5.70577i	0.239620	−	0.415034i
$190$		0			0
$191$	3.00000	+	1.73205i	0.217072	+	0.125327i	0.604594	−	0.796534i	$-0.293335\pi$
$191$	3.00000	+	1.73205i	0.217072	+	0.125327i	−0.387522	+	0.921861i	$0.626669\pi$
$192$	−1.22474	−	1.22474i	−0.0883883	−	0.0883883i
$193$	6.45189	−	24.0788i	0.464417	−	1.73323i	−0.194396	−	0.980923i	$-0.562275\pi$
$193$	6.45189	−	24.0788i	0.464417	−	1.73323i	0.658813	−	0.752306i	$-0.271059\pi$
$194$	−7.33013	−	12.6962i	−0.526272	−	0.911531i
$195$		0			0
$196$	2.69615	−	4.66987i	0.192582	−	0.333562i
$197$	−2.68973	−	2.68973i	−0.191635	−	0.191635i	0.604767	−	0.796402i	$-0.293266\pi$
$197$	−2.68973	−	2.68973i	−0.191635	−	0.191635i	−0.796402	+	0.604767i	$0.793266\pi$
$198$	2.68973	−	10.0382i	0.191151	−	0.713384i
$199$			0.392305i			0.0278098i	0.999903	+	0.0139049i	$0.00442620\pi$
$199$			0.392305i			0.0278098i	−0.999903	+	0.0139049i	$0.995574\pi$
$200$		0			0
$201$	18.6962	−	10.7942i	1.31872	−	0.761366i
$202$	−10.3664	−	2.77766i	−0.729375	−	0.195435i
$203$	−8.90138	−	2.38512i	−0.624755	−	0.167403i
$204$		0			0
$205$		0			0
$206$			10.3923i			0.724066i
$207$	−6.36396	+	23.7506i	−0.442326	+	1.65078i
$208$	0.896575	+	0.896575i	0.0621663	+	0.0621663i
$209$	12.4641	−	21.5885i	0.862160	−	1.49330i
$210$		0			0
$211$	4.59808	+	7.96410i	0.316545	+	0.548271i	0.979765	−	0.200153i	$-0.0641440\pi$
$211$	4.59808	+	7.96410i	0.316545	+	0.548271i	−0.663220	+	0.748424i	$0.730811\pi$
$212$	−0.568406	+	2.12132i	−0.0390383	+	0.145693i
$213$	−13.1440	−	13.1440i	−0.900614	−	0.900614i
$214$	0.696152	+	0.401924i	0.0475880	+	0.0274749i
$215$		0			0
$216$	−5.19615			−0.353553
$217$	−9.14162	+	9.14162i	−0.620574	+	0.620574i
$218$	−3.15660	−	11.7806i	−0.213792	−	0.797881i
$219$	9.00000			0.608164
$220$		0			0
$221$		0			0
$222$	0.984508	+	3.67423i	0.0660759	+	0.246598i
$223$	9.14162	−	2.44949i	0.612168	−	0.164030i	0.0606032	−	0.998162i	$-0.480698\pi$
$223$	9.14162	−	2.44949i	0.612168	−	0.164030i	0.551565	+	0.834132i	$0.314031\pi$
$224$	1.26795			0.0847184
$225$		0			0
$226$	−7.39230			−0.491729
$227$	22.4058	−	6.00361i	1.48712	−	0.398473i	0.578358	−	0.815783i	$-0.303694\pi$
$227$	22.4058	−	6.00361i	1.48712	−	0.398473i	0.908764	+	0.417310i	$0.137027\pi$
$228$	−12.0394	−	3.22595i	−0.797329	−	0.213644i
$229$	−14.0263	+	8.09808i	−0.926883	+	0.535136i	−0.885824	−	0.464021i	$-0.846406\pi$
$229$	−14.0263	+	8.09808i	−0.926883	+	0.535136i	−0.0410583	+	0.999157i	$0.513073\pi$
$230$		0			0
$231$	3.80385	+	6.58846i	0.250275	+	0.433489i
$232$	1.88108	+	7.02030i	0.123499	+	0.460905i
$233$	17.9551	−	17.9551i	1.17628	−	1.17628i	0.195590	−	0.980686i	$-0.437338\pi$
$233$	17.9551	−	17.9551i	1.17628	−	1.17628i	0.980686	−	0.195590i	$-0.0626622\pi$
$234$	3.80385			0.248665
$235$		0			0
$236$	10.7942	+	6.23205i	0.702644	+	0.405672i
$237$	16.7303	−	4.48288i	1.08675	−	0.291194i
$238$		0			0
$239$	7.09808	+	12.2942i	0.459136	+	0.795248i	0.998916	−	0.0465591i	$-0.0148256\pi$
$239$	7.09808	+	12.2942i	0.459136	+	0.795248i	−0.539779	+	0.841807i	$0.681492\pi$
$240$		0			0
$241$	5.69615	−	9.86603i	0.366921	−	0.635527i	−0.622161	−	0.782889i	$-0.713745\pi$
$241$	5.69615	−	9.86603i	0.366921	−	0.635527i	0.989083	+	0.147363i	$0.0470785\pi$
$242$	0.707107	+	0.707107i	0.0454545	+	0.0454545i
$243$	−11.0227	+	11.0227i	−0.707107	+	0.707107i
$244$		−	4.00000i		−	0.256074i
$245$		0			0
$246$			3.00000i			0.191273i
$247$	8.81345	+	2.36156i	0.560786	+	0.150262i
$248$	9.84873	+	2.63896i	0.625395	+	0.167574i
$249$	10.2058	+	5.89230i	0.646764	+	0.373410i
$250$		0			0
$251$		−	9.00000i		−	0.568075i	−0.958813	−	0.284037i	$-0.908326\pi$
$251$		−	9.00000i		−	0.568075i	0.958813	−	0.284037i	$-0.0916740\pi$
$252$	2.68973	−	2.68973i	0.169437	−	0.169437i
$253$	−20.0764	−	20.0764i	−1.26219	−	1.26219i
$254$	−1.09808	+	1.90192i	−0.0688994	+	0.119337i
$255$		0			0
$256$	−0.500000	−	0.866025i	−0.0312500	−	0.0541266i
$257$	−1.19256	+	4.45069i	−0.0743898	+	0.277627i	−0.993094	−	0.117319i	$-0.962570\pi$
$257$	−1.19256	+	4.45069i	−0.0743898	+	0.277627i	0.918704	+	0.394946i	$0.129237\pi$
$258$	−2.89778	+	10.8147i	−0.180408	+	0.673291i
$259$	−2.41154	−	1.39230i	−0.149846	−	0.0865136i
$260$		0			0
$261$	18.8827	+	10.9019i	1.16881	+	0.674812i
$262$	−4.89898	+	4.89898i	−0.302660	+	0.302660i
$263$	4.81105	+	17.9551i	0.296662	+	1.10716i	0.939889	+	0.341481i	$0.110929\pi$
$263$	4.81105	+	17.9551i	0.296662	+	1.10716i	−0.643227	+	0.765676i	$0.722405\pi$
$264$	3.00000	−	5.19615i	0.184637	−	0.319801i
$265$		0			0
$266$	7.90192	−	4.56218i	0.484498	−	0.279725i
$267$	−10.6066	+	10.6066i	−0.649113	+	0.649113i
$268$	12.0394	−	3.22595i	0.735423	−	0.197056i
$269$	−28.0526			−1.71039			−0.855197	−	0.518303i	$-0.826564\pi$
$269$	−28.0526			−1.71039			−0.855197	+	0.518303i	$0.826564\pi$
$270$		0			0
$271$	−4.58846			−0.278729			−0.139364	−	0.990241i	$-0.544506\pi$
$271$	−4.58846			−0.278729			−0.139364	+	0.990241i	$0.544506\pi$
$272$		0			0
$273$	−1.96902	+	1.96902i	−0.119170	+	0.119170i
$274$	−6.40192	+	3.69615i	−0.386754	+	0.223293i
$275$		0			0
$276$	−7.09808	+	12.2942i	−0.427254	+	0.740026i
$277$	−6.60420	−	24.6472i	−0.396808	−	1.48091i	−0.818679	−	0.574251i	$-0.805293\pi$
$277$	−6.60420	−	24.6472i	−0.396808	−	1.48091i	0.421871	−	0.906656i	$-0.361373\pi$
$278$	−5.65685	+	5.65685i	−0.339276	+	0.339276i
$279$	26.4904	−	15.2942i	1.58594	−	0.915642i
$280$		0			0
$281$	9.00000	+	5.19615i	0.536895	+	0.309976i	0.743820	−	0.668380i	$-0.233012\pi$
$281$	9.00000	+	5.19615i	0.536895	+	0.309976i	−0.206925	+	0.978357i	$0.566345\pi$
$282$	−2.68973	+	10.0382i	−0.160171	+	0.597766i
$283$	2.56961	−	9.58991i	0.152747	−	0.570061i	−0.846540	−	0.532325i	$-0.821319\pi$
$283$	2.56961	−	9.58991i	0.152747	−	0.570061i	0.999288	−	0.0377364i	$-0.0120147\pi$
$284$	−5.36603	−	9.29423i	−0.318415	−	0.551511i
$285$		0			0
$286$	−2.19615	+	3.80385i	−0.129861	+	0.224926i
$287$	−1.55291	−	1.55291i	−0.0916656	−	0.0916656i
$288$	−2.89778	−	0.776457i	−0.170753	−	0.0457532i
$289$		−	17.0000i		−	1.00000i
$290$		0			0
$291$	−21.9904	−	12.6962i	−1.28910	−	0.744262i
$292$	5.01910	+	1.34486i	0.293720	+	0.0787022i
$293$	13.7124	+	3.67423i	0.801089	+	0.214651i	0.636062	−	0.771638i	$-0.280562\pi$
$293$	13.7124	+	3.67423i	0.801089	+	0.214651i	0.165027	+	0.986289i	$0.447229\pi$
$294$		−	9.33975i		−	0.544705i
$295$		0			0
$296$			2.19615i			0.127649i
$297$	−4.65874	−	17.3867i	−0.270328	−	1.00888i
$298$	−12.7279	−	12.7279i	−0.737309	−	0.737309i
$299$	5.19615	−	9.00000i	0.300501	−	0.520483i
$300$		0			0
$301$	−4.09808	−	7.09808i	−0.236209	−	0.409126i
$302$	2.07055	−	7.72741i	0.119147	−	0.444662i
$303$	−17.9551	+	4.81105i	−1.03149	+	0.276387i
$304$	−6.23205	−	3.59808i	−0.357433	−	0.206364i
$305$		0			0
$306$		0			0
$307$	−2.44949	+	2.44949i	−0.139800	+	0.139800i	−0.773543	−	0.633743i	$-0.781517\pi$
$307$	−2.44949	+	2.44949i	−0.139800	+	0.139800i	0.633743	+	0.773543i	$0.281517\pi$
$308$	1.13681	+	4.24264i	0.0647759	+	0.241747i
$309$	9.00000	+	15.5885i	0.511992	+	0.886796i
$310$		0			0
$311$	21.5885	−	12.4641i	1.22417	−	0.706774i	0.258365	−	0.966047i	$-0.416816\pi$
$311$	21.5885	−	12.4641i	1.22417	−	0.706774i	0.965804	+	0.259273i	$0.0834829\pi$
$312$	2.12132	+	0.568406i	0.120096	+	0.0321797i
$313$	3.22595	−	0.864390i	0.182341	−	0.0488582i	−0.166493	−	0.986043i	$-0.553244\pi$
$313$	3.22595	−	0.864390i	0.182341	−	0.0488582i	0.348834	+	0.937184i	$0.386578\pi$
$314$	7.85641			0.443363
$315$		0			0
$316$	10.0000			0.562544
$317$	23.7506	−	6.36396i	1.33397	−	0.357436i	0.479775	−	0.877392i	$-0.340718\pi$
$317$	23.7506	−	6.36396i	1.33397	−	0.357436i	0.854193	+	0.519956i	$0.174052\pi$
$318$	0.984508	+	3.67423i	0.0552085	+	0.206041i
$319$	−21.8038	+	12.5885i	−1.22078	+	0.704818i
$320$		0			0
$321$	1.39230			0.0777109
$322$	−2.68973	−	10.0382i	−0.149893	−	0.559407i
$323$		0			0
$324$	−7.79423	+	4.50000i	−0.433013	+	0.250000i
$325$		0			0
$326$	−17.5981	−	10.1603i	−0.974667	−	0.562724i
$327$	−14.9372	−	14.9372i	−0.826028	−	0.826028i
$328$	−0.448288	+	1.67303i	−0.0247525	+	0.0923778i
$329$	−3.80385	−	6.58846i	−0.209713	−	0.363233i
$330$		0			0
$331$	−8.79423	+	15.2321i	−0.483375	+	0.837229i	−0.999818	−	0.0190922i	$-0.993922\pi$
$331$	−8.79423	+	15.2321i	−0.483375	+	0.837229i	0.516443	+	0.856321i	$0.327256\pi$
$332$	4.81105	+	4.81105i	0.264040	+	0.264040i
$333$	4.65874	+	4.65874i	0.255298	+	0.255298i
$334$		−	4.39230i		−	0.240336i
$335$		0			0
$336$	1.90192	−	1.09808i	0.103758	−	0.0599050i
$337$	−26.5283	−	7.10823i	−1.44509	−	0.387210i	−0.550775	−	0.834653i	$-0.685668\pi$
$337$	−26.5283	−	7.10823i	−1.44509	−	0.387210i	−0.894312	+	0.447443i	$0.852335\pi$
$338$	11.0041	+	2.94855i	0.598545	+	0.160380i
$339$	−11.0885	+	6.40192i	−0.602242	+	0.347705i
$340$		0			0
$341$			35.3205i			1.91271i
$342$	−20.8528	+	5.58750i	−1.12759	+	0.302138i
$343$	11.1106	+	11.1106i	0.599918	+	0.599918i
$344$	−3.23205	+	5.59808i	−0.174261	+	0.301828i
$345$		0			0
$346$	2.19615	+	3.80385i	0.118066	+	0.204496i
$347$	8.48528	−	31.6675i	0.455514	−	1.70000i	−0.231059	−	0.972940i	$-0.574219\pi$
$347$	8.48528	−	31.6675i	0.455514	−	1.70000i	0.686573	−	0.727061i	$-0.259114\pi$
$348$	8.90138	+	8.90138i	0.477164	+	0.477164i
$349$	1.73205	+	1.00000i	0.0927146	+	0.0535288i	0.545640	−	0.838019i	$-0.316286\pi$
$349$	1.73205	+	1.00000i	0.0927146	+	0.0535288i	−0.452926	+	0.891548i	$0.649620\pi$
$350$		0			0
$351$	5.70577	−	3.29423i	0.304552	−	0.175833i
$352$	2.44949	−	2.44949i	0.130558	−	0.130558i
$353$	3.88229	+	14.4889i	0.206633	+	0.771166i	0.988946	+	0.148279i	$0.0473735\pi$
$353$	3.88229	+	14.4889i	0.206633	+	0.771166i	−0.782312	+	0.622886i	$0.785960\pi$
$354$	21.5885			1.14741
$355$		0			0
$356$	−7.50000	+	4.33013i	−0.397499	+	0.229496i
$357$		0			0
$358$	4.69093	−	1.25693i	0.247923	−	0.0664308i
$359$	0.679492			0.0358622			0.0179311	−	0.999839i	$-0.494292\pi$
$359$	0.679492			0.0358622			0.0179311	+	0.999839i	$0.494292\pi$
$360$		0			0
$361$	−32.7846			−1.72551
$362$	8.10634	−	2.17209i	0.426060	−	0.114162i
$363$	1.67303	+	0.448288i	0.0878114	+	0.0235290i
$364$	−1.39230	+	0.803848i	−0.0729766	+	0.0421331i
$365$		0			0
$366$	−3.46410	−	6.00000i	−0.181071	−	0.313625i
$367$	8.24504	+	30.7709i	0.430388	+	1.60623i	0.751868	+	0.659313i	$0.229153\pi$
$367$	8.24504	+	30.7709i	0.430388	+	1.60623i	−0.321481	+	0.946916i	$0.604181\pi$
$368$	−5.79555	+	5.79555i	−0.302114	+	0.302114i
$369$	2.59808	+	4.50000i	0.135250	+	0.234261i
$370$		0			0
$371$	−2.41154	−	1.39230i	−0.125201	−	0.0722849i
$372$	17.0585	−	4.57081i	0.884442	−	0.236985i
$373$	−2.92996	+	10.9348i	−0.151708	+	0.566181i	0.847657	+	0.530545i	$0.178013\pi$
$373$	−2.92996	+	10.9348i	−0.151708	+	0.566181i	−0.999365	+	0.0356365i	$0.988654\pi$
$374$		0			0
$375$		0			0
$376$	−3.00000	+	5.19615i	−0.154713	+	0.267971i
$377$	−6.51626	−	6.51626i	−0.335605	−	0.335605i
$378$	1.70522	−	6.36396i	0.0877070	−	0.327327i
$379$			20.3923i			1.04748i	0.851877	+	0.523741i	$0.175464\pi$
$379$			20.3923i			1.04748i	−0.851877	+	0.523741i	$0.824536\pi$
$380$		0			0
$381$			3.80385i			0.194877i
$382$	3.34607	+	0.896575i	0.171200	+	0.0458728i
$383$	35.3417	+	9.46979i	1.80588	+	0.483884i	0.994871	−	0.101152i	$-0.0322529\pi$
$383$	35.3417	+	9.46979i	1.80588	+	0.483884i	0.811007	+	0.585036i	$0.198920\pi$
$384$	−1.50000	−	0.866025i	−0.0765466	−	0.0441942i
$385$		0			0
$386$		−	24.9282i		−	1.26881i
$387$	5.01910	+	18.7315i	0.255135	+	0.952177i
$388$	−10.3664	−	10.3664i	−0.526272	−	0.526272i
$389$	5.19615	−	9.00000i	0.263455	−	0.456318i	−0.703702	−	0.710495i	$-0.748471\pi$
$389$	5.19615	−	9.00000i	0.263455	−	0.456318i	0.967158	+	0.254177i	$0.0818045\pi$
$390$		0			0
$391$		0			0
$392$	1.39563	−	5.20857i	0.0704900	−	0.263072i
$393$	−3.10583	+	11.5911i	−0.156668	+	0.584694i
$394$	−3.29423	−	1.90192i	−0.165961	−	0.0958175i
$395$		0			0
$396$		−	10.3923i		−	0.522233i
$397$	−6.03579	+	6.03579i	−0.302928	+	0.302928i	−0.842158	−	0.539231i	$-0.818715\pi$
$397$	−6.03579	+	6.03579i	−0.302928	+	0.302928i	0.539231	+	0.842158i	$0.318715\pi$
$398$	0.101536	+	0.378937i	0.00508954	+	0.0189944i
$399$	7.90192	−	13.6865i	0.395591	−	0.685184i
$400$		0			0
$401$	9.00000	−	5.19615i	0.449439	−	0.259483i	−0.258154	−	0.966104i	$-0.583114\pi$
$401$	9.00000	−	5.19615i	0.449439	−	0.259483i	0.707593	+	0.706620i	$0.249781\pi$
$402$	15.2653	−	15.2653i	0.761366	−	0.761366i
$403$	−12.4877	+	3.34607i	−0.622056	+	0.166679i
$404$	−10.7321			−0.533939
$405$		0			0
$406$	−9.21539			−0.457352
$407$	−7.34847	+	1.96902i	−0.364250	+	0.0976005i
$408$		0			0
$409$	−0.526279	+	0.303848i	−0.0260228	+	0.0150243i	−0.512955	−	0.858416i	$-0.671449\pi$
$409$	−0.526279	+	0.303848i	−0.0260228	+	0.0150243i	0.486932	+	0.873440i	$0.338116\pi$
$410$		0			0
$411$	−6.40192	+	11.0885i	−0.315784	+	0.546953i
$412$	2.68973	+	10.0382i	0.132513	+	0.494546i
$413$	−11.1750	+	11.1750i	−0.549886	+	0.549886i
$414$			24.5885i			1.20846i
$415$		0			0
$416$	1.09808	+	0.633975i	0.0538376	+	0.0310832i
$417$	−3.58630	+	13.3843i	−0.175622	+	0.655430i
$418$	6.45189	−	24.0788i	0.315572	−	1.17773i
$419$	4.50000	+	7.79423i	0.219839	+	0.380773i	0.954759	−	0.297382i	$-0.0961133\pi$
$419$	4.50000	+	7.79423i	0.219839	+	0.380773i	−0.734919	+	0.678155i	$0.762780\pi$
$420$		0			0
$421$	4.29423	−	7.43782i	0.209288	−	0.362497i	−0.742203	−	0.670176i	$-0.766219\pi$
$421$	4.29423	−	7.43782i	0.209288	−	0.362497i	0.951490	+	0.307678i	$0.0995521\pi$
$422$	6.50266	+	6.50266i	0.316545	+	0.316545i
$423$	4.65874	+	17.3867i	0.226516	+	0.845369i
$424$			2.19615i			0.106655i
$425$		0			0
$426$	−16.0981	−	9.29423i	−0.779954	−	0.450307i
$427$	4.89898	+	1.31268i	0.237078	+	0.0635249i
$428$	0.776457	+	0.208051i	0.0375315	+	0.0100565i
$429$			7.60770i			0.367303i
$430$		0			0
$431$		−	3.12436i		−	0.150495i	−0.997165	−	0.0752475i	$-0.976025\pi$
$431$		−	3.12436i		−	0.150495i	0.997165	−	0.0752475i	$-0.0239747\pi$
$432$	−5.01910	+	1.34486i	−0.241481	+	0.0647048i
$433$	21.8695	+	21.8695i	1.05098	+	1.05098i	0.998629	+	0.0523546i	$0.0166726\pi$
$433$	21.8695	+	21.8695i	1.05098	+	1.05098i	0.0523546	+	0.998629i	$0.483327\pi$
$434$	−6.46410	+	11.1962i	−0.310287	+	0.537433i
$435$		0			0
$436$	−6.09808	−	10.5622i	−0.292045	−	0.505837i
$437$	−15.2653	+	56.9710i	−0.730240	+	2.72529i
$438$	8.69333	−	2.32937i	0.415383	−	0.111302i
$439$	−28.2224	−	16.2942i	−1.34698	−	0.777681i	−0.359162	−	0.933275i	$-0.616937\pi$
$439$	−28.2224	−	16.2942i	−1.34698	−	0.777681i	−0.987821	+	0.155594i	$0.950271\pi$
$440$		0			0
$441$	−8.08846	−	14.0096i	−0.385165	−	0.667125i
$442$		0			0
$443$	−2.68973	−	10.0382i	−0.127793	−	0.476929i	0.872131	−	0.489272i	$-0.162738\pi$
$443$	−2.68973	−	10.0382i	−0.127793	−	0.476929i	−0.999924	+	0.0123433i	$0.996071\pi$
$444$	1.90192	+	3.29423i	0.0902613	+	0.156337i
$445$		0			0
$446$	8.19615	−	4.73205i	0.388099	−	0.224069i
$447$	−30.1146	−	8.06918i	−1.42437	−	0.381659i
$448$	1.22474	−	0.328169i	0.0578638	−	0.0155045i
$449$	−5.87564			−0.277289			−0.138644	−	0.990342i	$-0.544275\pi$
$449$	−5.87564			−0.277289			−0.138644	+	0.990342i	$0.544275\pi$
$450$		0			0
$451$	−6.00000			−0.282529
$452$	−7.14042	+	1.91327i	−0.335857	+	0.0899926i
$453$	−3.58630	−	13.3843i	−0.168499	−	0.628847i
$454$	20.0885	−	11.5981i	0.942798	−	0.544325i
$455$		0			0
$456$	−12.4641			−0.583685
$457$	0.928761	+	3.46618i	0.0434456	+	0.162141i	0.984240	−	0.176835i	$-0.0565860\pi$
$457$	0.928761	+	3.46618i	0.0434456	+	0.162141i	−0.940795	+	0.338976i	$0.889919\pi$
$458$	−11.4524	+	11.4524i	−0.535136	+	0.535136i
$459$		0			0
$460$		0			0
$461$	−24.2942	−	14.0263i	−1.13150	−	0.653269i	−0.187185	−	0.982325i	$-0.559936\pi$
$461$	−24.2942	−	14.0263i	−1.13150	−	0.653269i	−0.944310	+	0.329056i	$0.893270\pi$
$462$	5.37945	+	5.37945i	0.250275	+	0.250275i
$463$	−2.12132	+	7.91688i	−0.0985861	+	0.367928i	−0.997539	−	0.0701175i	$-0.977663\pi$
$463$	−2.12132	+	7.91688i	−0.0985861	+	0.367928i	0.898953	+	0.438046i	$0.144329\pi$
$464$	3.63397	+	6.29423i	0.168703	+	0.292202i
$465$		0			0
$466$	12.6962	−	21.9904i	0.588138	−	1.01868i
$467$	−15.2653	−	15.2653i	−0.706396	−	0.706396i	0.259380	−	0.965775i	$-0.416482\pi$
$467$	−15.2653	−	15.2653i	−0.706396	−	0.706396i	−0.965775	+	0.259380i	$0.916482\pi$
$468$	3.67423	−	0.984508i	0.169842	−	0.0455089i
$469$			15.8038i			0.729754i
$470$		0			0
$471$	11.7846	−	6.80385i	0.543006	−	0.313505i
$472$	12.0394	+	3.22595i	0.554158	+	0.148486i
$473$	−21.6293	−	5.79555i	−0.994517	−	0.266480i
$474$	15.0000	−	8.66025i	0.688973	−	0.397779i
$475$		0			0
$476$		0			0
$477$	4.65874	+	4.65874i	0.213309	+	0.213309i
$478$	10.0382	+	10.0382i	0.459136	+	0.459136i
$479$	−10.5622	+	18.2942i	−0.482598	+	0.835885i	−0.999800	−	0.0199786i	$-0.993640\pi$
$479$	−10.5622	+	18.2942i	−0.482598	+	0.835885i	0.517202	+	0.855863i	$0.326974\pi$
$480$		0			0
$481$	−1.39230	−	2.41154i	−0.0634836	−	0.109957i
$482$	2.94855	−	11.0041i	0.134303	−	0.501224i
$483$	−12.7279	−	12.7279i	−0.579141	−	0.579141i
$484$	0.866025	+	0.500000i	0.0393648	+	0.0227273i
$485$		0			0
$486$	−7.79423	+	13.5000i	−0.353553	+	0.612372i
$487$	−15.5935	+	15.5935i	−0.706610	+	0.706610i	−0.965821	−	0.259211i	$-0.916537\pi$
$487$	−15.5935	+	15.5935i	−0.706610	+	0.706610i	0.259211	+	0.965821i	$0.416537\pi$
$488$	−1.03528	−	3.86370i	−0.0468648	−	0.174902i
$489$	−35.1962			−1.59163
$490$		0			0
$491$	−31.7942	+	18.3564i	−1.43485	+	0.828413i	−0.997486	−	0.0708697i	$-0.977423\pi$
$491$	−31.7942	+	18.3564i	−1.43485	+	0.828413i	−0.437368	+	0.899283i	$0.644089\pi$
$492$	0.776457	+	2.89778i	0.0350054	+	0.130642i
$493$		0			0
$494$	9.12436			0.410524
$495$		0			0
$496$	10.1962			0.457821
$497$	13.1440	−	3.52193i	0.589590	−	0.157980i
$498$	11.3831	+	3.05008i	0.510087	+	0.136677i
$499$	−16.9641	+	9.79423i	−0.759417	+	0.438450i	−0.829087	−	0.559120i	$-0.811139\pi$
$499$	−16.9641	+	9.79423i	−0.759417	+	0.438450i	0.0696691	+	0.997570i	$0.477806\pi$
$500$		0			0
$501$	−3.80385	−	6.58846i	−0.169943	−	0.294351i
$502$	−2.32937	−	8.69333i	−0.103965	−	0.388002i
$503$	25.8719	−	25.8719i	1.15357	−	1.15357i	0.167742	−	0.985831i	$-0.446352\pi$
$503$	25.8719	−	25.8719i	1.15357	−	1.15357i	0.985831	−	0.167742i	$-0.0536476\pi$
$504$	1.90192	−	3.29423i	0.0847184	−	0.146737i
$505$		0			0
$506$	−24.5885	−	14.1962i	−1.09309	−	0.631096i
$507$	19.0597	−	5.10703i	0.846471	−	0.226811i
$508$	−0.568406	+	2.12132i	−0.0252189	+	0.0941184i
$509$	10.3923	+	18.0000i	0.460631	+	0.797836i	0.998992	−	0.0448779i	$-0.0142899\pi$
$509$	10.3923	+	18.0000i	0.460631	+	0.797836i	−0.538362	+	0.842714i	$0.680957\pi$
$510$		0			0
$511$	−3.29423	+	5.70577i	−0.145728	+	0.252408i
$512$	−0.707107	−	0.707107i	−0.0312500	−	0.0312500i
$513$	−26.4404	+	26.4404i	−1.16737	+	1.16737i
$514$			4.60770i			0.203237i
$515$		0			0
$516$			11.1962i			0.492883i
$517$	−20.0764	−	5.37945i	−0.882959	−	0.236588i
$518$	−2.68973	−	0.720710i	−0.118180	−	0.0316662i
$519$	6.58846	+	3.80385i	0.289201	+	0.166970i
$520$		0			0
$521$			38.7846i			1.69918i	0.527440	+	0.849592i	$0.323152\pi$
$521$			38.7846i			1.69918i	−0.527440	+	0.849592i	$0.676848\pi$
$522$	21.0609	+	5.64325i	0.921811	+	0.246998i
$523$	14.1929	+	14.1929i	0.620612	+	0.620612i	0.945688	−	0.325076i	$-0.105390\pi$
$523$	14.1929	+	14.1929i	0.620612	+	0.620612i	−0.325076	+	0.945688i	$0.605390\pi$
$524$	−3.46410	+	6.00000i	−0.151330	+	0.262111i
$525$		0			0
$526$	9.29423	+	16.0981i	0.405248	+	0.701909i
$527$		0			0
$528$	1.55291	−	5.79555i	0.0675819	−	0.252219i
$529$	38.2583	+	22.0885i	1.66341	+	0.960368i
$530$		0			0
$531$	32.3827	−	18.6962i	1.40529	−	0.811344i
$532$	6.45189	−	6.45189i	0.279725	−	0.279725i
$533$	−0.568406	−	2.12132i	−0.0246204	−	0.0918846i
$534$	−7.50000	+	12.9904i	−0.324557	+	0.562149i
$535$		0			0
$536$	10.7942	−	6.23205i	0.466240	−	0.269184i
$537$	5.94786	−	5.94786i	0.256669	−	0.256669i
$538$	−27.0967	+	7.26054i	−1.16822	+	0.313024i
$539$	18.6795			0.804583
$540$		0			0
$541$	12.3923			0.532787			0.266393	−	0.963864i	$-0.414168\pi$
$541$	12.3923			0.532787			0.266393	+	0.963864i	$0.414168\pi$
$542$	−4.43211	+	1.18758i	−0.190375	+	0.0510109i
$543$	10.2784	−	10.2784i	0.441090	−	0.441090i
$544$		0			0
$545$		0			0
$546$	−1.39230	+	2.41154i	−0.0595851	+	0.103205i
$547$	−5.73981	−	21.4213i	−0.245416	−	0.915907i	−0.973174	−	0.230072i	$-0.926104\pi$
$547$	−5.73981	−	21.4213i	−0.245416	−	0.915907i	0.727757	−	0.685835i	$-0.240563\pi$
$548$	−5.22715	+	5.22715i	−0.223293	+	0.223293i
$549$	−10.3923	−	6.00000i	−0.443533	−	0.256074i
$550$		0			0
$551$	45.2942	+	26.1506i	1.92960	+	1.11405i
$552$	−3.67423	+	13.7124i	−0.156386	+	0.583640i
$553$	−3.28169	+	12.2474i	−0.139552	+	0.520814i
$554$	−12.7583	−	22.0981i	−0.542050	−	0.938857i
$555$		0			0
$556$	−4.00000	+	6.92820i	−0.169638	+	0.293821i
$557$	−11.5911	−	11.5911i	−0.491131	−	0.491131i	0.417531	−	0.908662i	$-0.362895\pi$
$557$	−11.5911	−	11.5911i	−0.491131	−	0.491131i	−0.908662	+	0.417531i	$0.862895\pi$
$558$	21.6293	−	21.6293i	0.915642	−	0.915642i
$559$		−	8.19615i		−	0.346660i
$560$		0			0
$561$		0			0
$562$	10.0382	+	2.68973i	0.423436	+	0.113459i
$563$	−32.4440	−	8.69333i	−1.36735	−	0.366380i	−0.500840	−	0.865540i	$-0.666975\pi$
$563$	−32.4440	−	8.69333i	−1.36735	−	0.366380i	−0.866510	+	0.499160i	$0.833642\pi$
$564$			10.3923i			0.437595i
$565$		0			0
$566$		−	9.92820i		−	0.417314i
$567$	−2.95352	−	11.0227i	−0.124036	−	0.462910i
$568$	−7.58871	−	7.58871i	−0.318415	−	0.318415i
$569$	5.53590	−	9.58846i	0.232077	−	0.401969i	−0.726342	−	0.687333i	$-0.758781\pi$
$569$	5.53590	−	9.58846i	0.232077	−	0.401969i	0.958419	+	0.285364i	$0.0921146\pi$
$570$		0			0
$571$	−4.40192	−	7.62436i	−0.184215	−	0.319069i	0.759097	−	0.650978i	$-0.225641\pi$
$571$	−4.40192	−	7.62436i	−0.184215	−	0.319069i	−0.943312	+	0.331908i	$0.892308\pi$
$572$	−1.13681	+	4.24264i	−0.0475325	+	0.177394i
$573$	5.79555	−	1.55291i	0.242113	−	0.0648739i
$574$	−1.90192	−	1.09808i	−0.0793848	−	0.0458328i
$575$		0			0
$576$	−3.00000			−0.125000
$577$	15.9217	−	15.9217i	0.662828	−	0.662828i	−0.293217	−	0.956046i	$-0.594726\pi$
$577$	15.9217	−	15.9217i	0.662828	−	0.662828i	0.956046	+	0.293217i	$0.0947260\pi$
$578$	−4.39992	−	16.4207i	−0.183013	−	0.683013i
$579$	−21.5885	−	37.3923i	−0.897186	−	1.55397i
$580$		0			0
$581$	−7.47114	+	4.31347i	−0.309955	+	0.178953i
$582$	−24.5271	−	6.57201i	−1.01668	−	0.272419i
$583$	−7.34847	+	1.96902i	−0.304342	+	0.0815483i
$584$	5.19615			0.215018
$585$		0			0
$586$	14.1962			0.586438
$587$	−15.8338	+	4.24264i	−0.653529	+	0.175113i	−0.570324	−	0.821420i	$-0.693182\pi$
$587$	−15.8338	+	4.24264i	−0.653529	+	0.175113i	−0.0832050	+	0.996532i	$0.526516\pi$
$588$	−2.41730	−	9.02150i	−0.0996879	−	0.372040i
$589$	63.5429	−	36.6865i	2.61824	−	1.51164i
$590$		0			0
$591$	−6.58846			−0.271013
$592$	0.568406	+	2.12132i	0.0233613	+	0.0871857i
$593$	−14.8492	+	14.8492i	−0.609785	+	0.609785i	−0.942890	−	0.333105i	$-0.891904\pi$
$593$	−14.8492	+	14.8492i	−0.609785	+	0.609785i	0.333105	+	0.942890i	$0.391904\pi$
$594$	−9.00000	−	15.5885i	−0.369274	−	0.639602i
$595$		0			0
$596$	−15.5885	−	9.00000i	−0.638528	−	0.368654i
$597$	0.480473	+	0.480473i	0.0196645	+	0.0196645i
$598$	2.68973	−	10.0382i	0.109991	−	0.410492i
$599$	5.36603	+	9.29423i	0.219250	+	0.379752i	0.954579	−	0.297958i	$-0.0963057\pi$
$599$	5.36603	+	9.29423i	0.219250	+	0.379752i	−0.735329	+	0.677710i	$0.762972\pi$
$600$		0			0
$601$	−6.39230	+	11.0718i	−0.260748	+	0.451628i	−0.966441	−	0.256890i	$-0.917302\pi$
$601$	−6.39230	+	11.0718i	−0.260748	+	0.451628i	0.705693	+	0.708518i	$0.250636\pi$
$602$	−5.79555	−	5.79555i	−0.236209	−	0.236209i
$603$	9.67784	−	36.1182i	0.394112	−	1.47085i
$604$		−	8.00000i		−	0.325515i
$605$		0			0
$606$	−16.0981	+	9.29423i	−0.653940	+	0.377552i
$607$	16.8183	+	4.50644i	0.682632	+	0.182911i	0.583438	−	0.812157i	$-0.301707\pi$
$607$	16.8183	+	4.50644i	0.682632	+	0.182911i	0.0991937	+	0.995068i	$0.468374\pi$
$608$	−6.95095	−	1.86250i	−0.281898	−	0.0755344i
$609$	−13.8231	+	7.98076i	−0.560140	+	0.323397i
$610$		0			0
$611$		−	7.60770i		−	0.307774i
$612$		0			0
$613$	2.68973	+	2.68973i	0.108637	+	0.108637i	0.759336	−	0.650699i	$-0.225524\pi$
$613$	2.68973	+	2.68973i	0.108637	+	0.108637i	−0.650699	+	0.759336i	$0.725524\pi$
$614$	−1.73205	+	3.00000i	−0.0698999	+	0.121070i
$615$		0			0
$616$	2.19615	+	3.80385i	0.0884855	+	0.153261i
$617$	−7.70882	+	28.7697i	−0.310346	+	1.15823i	0.617900	+	0.786257i	$0.287984\pi$
$617$	−7.70882	+	28.7697i	−0.310346	+	1.15823i	−0.928245	+	0.371969i	$0.878683\pi$
$618$	12.7279	+	12.7279i	0.511992	+	0.511992i
$619$	−14.2128	−	8.20577i	−0.571261	−	0.329818i	0.186392	−	0.982476i	$-0.440321\pi$
$619$	−14.2128	−	8.20577i	−0.571261	−	0.329818i	−0.757653	+	0.652658i	$0.773654\pi$
$620$		0			0
$621$	21.2942	+	36.8827i	0.854508	+	1.48005i
$622$	17.6269	−	17.6269i	0.706774	−	0.706774i
$623$	−2.84203	−	10.6066i	−0.113864	−	0.424945i
$624$	2.19615			0.0879165
$625$		0			0
$626$	2.89230	−	1.66987i	0.115600	−	0.0667415i
$627$	−11.1750	−	41.7057i	−0.446287	−	1.66557i
$628$	7.58871	−	2.03339i	0.302822	−	0.0811410i
$629$		0			0
$630$		0			0
$631$	40.7846			1.62361			0.811805	−	0.583929i	$-0.198485\pi$
$631$	40.7846			1.62361			0.811805	+	0.583929i	$0.198485\pi$
$632$	9.65926	−	2.58819i	0.384225	−	0.102953i
$633$	15.3855	+	4.12252i	0.611517	+	0.163856i
$634$	21.2942	−	12.2942i	0.845702	−	0.488266i
$635$		0			0
$636$	1.90192	+	3.29423i	0.0754162	+	0.130625i
$637$	1.76959	+	6.60420i	0.0701137	+	0.261668i
$638$	−17.8028	+	17.8028i	−0.704818	+	0.704818i
$639$	−32.1962			−1.27366
$640$		0			0
$641$	−0.911543	−	0.526279i	−0.0360038	−	0.0207868i	0.481890	−	0.876232i	$-0.339950\pi$
$641$	−0.911543	−	0.526279i	−0.0360038	−	0.0207868i	−0.517894	+	0.855445i	$0.673284\pi$
$642$	1.34486	−	0.360355i	0.0530775	−	0.0142221i
$643$	−4.29839	+	16.0418i	−0.169512	+	0.632627i	0.827910	+	0.560861i	$0.189530\pi$
$643$	−4.29839	+	16.0418i	−0.169512	+	0.632627i	−0.997422	+	0.0717654i	$0.977137\pi$
$644$	−5.19615	−	9.00000i	−0.204757	−	0.354650i
$645$		0			0
$646$		0			0
$647$	2.68973	+	2.68973i	0.105744	+	0.105744i	0.757999	−	0.652255i	$-0.226177\pi$
$647$	2.68973	+	2.68973i	0.105744	+	0.105744i	−0.652255	+	0.757999i	$0.726177\pi$
$648$	−6.36396	+	6.36396i	−0.250000	+	0.250000i
$649$			43.1769i			1.69484i
$650$		0			0
$651$			22.3923i			0.877624i
$652$	−19.6281	−	5.25933i	−0.768696	−	0.205971i
$653$	2.12132	+	0.568406i	0.0830137	+	0.0222434i	0.300087	−	0.953912i	$-0.402984\pi$
$653$	2.12132	+	0.568406i	0.0830137	+	0.0222434i	−0.217073	+	0.976155i	$0.569651\pi$
$654$	−18.2942	−	10.5622i	−0.715361	−	0.413014i
$655$		0			0
$656$			1.73205i			0.0676252i
$657$	11.0227	−	11.0227i	0.430037	−	0.430037i
$658$	−5.37945	−	5.37945i	−0.209713	−	0.209713i
$659$	11.0885	−	19.2058i	0.431945	−	0.748151i	−0.565096	−	0.825025i	$-0.691161\pi$
$659$	11.0885	−	19.2058i	0.431945	−	0.748151i	0.997041	+	0.0768747i	$0.0244941\pi$
$660$		0			0
$661$	−11.5885	−	20.0718i	−0.450739	−	0.780702i	0.547693	−	0.836679i	$-0.315506\pi$
$661$	−11.5885	−	20.0718i	−0.450739	−	0.780702i	−0.998432	+	0.0559768i	$0.982173\pi$
$662$	−4.55223	+	16.9891i	−0.176927	+	0.660302i
$663$		0			0
$664$	5.89230	+	3.40192i	0.228666	+	0.132020i
$665$		0			0
$666$	5.70577	+	3.29423i	0.221094	+	0.127649i
$667$	42.1218	−	42.1218i	1.63096	−	1.63096i
$668$	−1.13681	−	4.24264i	−0.0439846	−	0.164153i
$669$	8.19615	−	14.1962i	0.316882	−	0.548855i
$670$		0			0
$671$	12.0000	−	6.92820i	0.463255	−	0.267460i
$672$	1.55291	−	1.55291i	0.0599050	−	0.0599050i
$673$	1.55291	−	0.416102i	0.0598604	−	0.0160396i	−0.228765	−	0.973482i	$-0.573469\pi$
$673$	1.55291	−	0.416102i	0.0598604	−	0.0160396i	0.288625	+	0.957442i	$0.406802\pi$
$674$	−27.4641			−1.05788
$675$		0			0
$676$	11.3923			0.438166
$677$	−6.36396	+	1.70522i	−0.244587	+	0.0655369i	−0.379030	−	0.925384i	$-0.623742\pi$
$677$	−6.36396	+	1.70522i	−0.244587	+	0.0655369i	0.134443	+	0.990921i	$0.457076\pi$
$678$	−9.05369	+	9.05369i	−0.347705	+	0.347705i
$679$	16.0981	−	9.29423i	0.617787	−	0.356680i
$680$		0			0
$681$	20.0885	−	34.7942i	0.769791	−	1.33332i
$682$	9.14162	+	34.1170i	0.350051	+	1.30641i
$683$	27.9933	−	27.9933i	1.07113	−	1.07113i	0.0738643	−	0.997268i	$-0.476467\pi$
$683$	27.9933	−	27.9933i	1.07113	−	1.07113i	0.997268	−	0.0738643i	$-0.0235332\pi$
$684$	−18.6962	+	10.7942i	−0.714865	+	0.412728i
$685$		0			0
$686$	13.6077	+	7.85641i	0.519544	+	0.299959i
$687$	−7.26054	+	27.0967i	−0.277007	+	1.03380i
$688$	−1.67303	+	6.24384i	−0.0637838	+	0.238044i
$689$	−1.39230	−	2.41154i	−0.0530426	−	0.0918725i
$690$		0			0
$691$	−6.20577	+	10.7487i	−0.236079	+	0.408900i	−0.959586	−	0.281417i	$-0.909196\pi$
$691$	−6.20577	+	10.7487i	−0.236079	+	0.408900i	0.723507	+	0.690317i	$0.242529\pi$
$692$	3.10583	+	3.10583i	0.118066	+	0.118066i
$693$	12.7279	+	3.41044i	0.483494	+	0.129552i
$694$		−	32.7846i		−	1.24449i
$695$		0			0
$696$	10.9019	+	6.29423i	0.413236	+	0.238582i
$697$		0			0
$698$	1.93185	+	0.517638i	0.0731217	+	0.0195929i
$699$		−	43.9808i		−	1.66351i
$700$		0			0
$701$		−	21.4641i		−	0.810688i	−0.914164	−	0.405344i	$-0.867152\pi$
$701$		−	21.4641i		−	0.810688i	0.914164	−	0.405344i	$-0.132848\pi$
$702$	4.65874	−	4.65874i	0.175833	−	0.175833i
$703$	11.1750	+	11.1750i	0.421473	+	0.421473i
$704$	1.73205	−	3.00000i	0.0652791	−	0.113067i
$705$		0			0
$706$	7.50000	+	12.9904i	0.282266	+	0.488899i
$707$	3.52193	−	13.1440i	0.132456	−	0.494332i
$708$	20.8528	−	5.58750i	0.783698	−	0.209991i
$709$	−22.3468	−	12.9019i	−0.839251	−	0.484542i	0.0177584	−	0.999842i	$-0.494347\pi$
$709$	−22.3468	−	12.9019i	−0.839251	−	0.484542i	−0.857010	+	0.515300i	$0.827680\pi$
$710$		0			0
$711$	15.0000	−	25.9808i	0.562544	−	0.974355i
$712$	−6.12372	+	6.12372i	−0.229496	+	0.229496i
$713$	−21.6293	−	80.7217i	−0.810024	−	3.02305i
$714$		0			0
$715$		0			0
$716$	4.20577	−	2.42820i	0.157177	−	0.0907462i
$717$	23.7506	+	6.36396i	0.886983	+	0.237666i
$718$	0.656339	−	0.175865i	0.0244943	−	0.00656324i
$719$	39.1244			1.45909			0.729546	−	0.683932i	$-0.239731\pi$
$719$	39.1244			1.45909			0.729546	+	0.683932i	$0.239731\pi$
$720$		0			0
$721$	−13.1769			−0.490734
$722$	−31.6675	+	8.48528i	−1.17854	+	0.315789i
$723$	−5.10703	−	19.0597i	−0.189933	−	0.708838i
$724$	7.26795	−	4.19615i	0.270111	−	0.155949i
$725$		0			0
$726$	1.73205			0.0642824
$727$	1.79315	+	6.69213i	0.0665043	+	0.248197i	0.991173	−	0.132575i	$-0.0423247\pi$
$727$	1.79315	+	6.69213i	0.0665043	+	0.248197i	−0.924669	+	0.380773i	$0.875658\pi$
$728$	−1.13681	+	1.13681i	−0.0421331	+	0.0421331i
$729$			27.0000i			1.00000i
$730$		0			0
$731$		0			0
$732$	−4.89898	−	4.89898i	−0.181071	−	0.181071i
$733$	−8.90138	+	33.2204i	−0.328780	+	1.22702i	0.581677	+	0.813420i	$0.302397\pi$
$733$	−8.90138	+	33.2204i	−0.328780	+	1.22702i	−0.910457	+	0.413604i	$0.864270\pi$
$734$	15.9282	+	27.5885i	0.587921	+	1.01831i
$735$		0			0
$736$	−4.09808	+	7.09808i	−0.151057	+	0.261639i
$737$	30.5307	+	30.5307i	1.12461	+	1.12461i
$738$	3.67423	+	3.67423i	0.135250	+	0.135250i
$739$		−	11.5885i		−	0.426288i	−0.977021	−	0.213144i	$-0.931630\pi$
$739$		−	11.5885i		−	0.426288i	0.977021	−	0.213144i	$-0.0683704\pi$
$740$		0			0
$741$	13.6865	−	7.90192i	0.502787	−	0.290284i
$742$	−2.68973	−	0.720710i	−0.0987430	−	0.0264581i
$743$	−38.0315	−	10.1905i	−1.39524	−	0.373853i	−0.518606	−	0.855013i	$-0.673549\pi$
$743$	−38.0315	−	10.1905i	−1.39524	−	0.373853i	−0.876633	+	0.481160i	$0.840216\pi$
$744$	15.2942	−	8.83013i	0.560714	−	0.323728i
$745$		0			0
$746$			11.3205i			0.414473i
$747$	19.7160	−	5.28290i	0.721372	−	0.193291i
$748$		0			0
$749$	−0.509619	+	0.882686i	−0.0186211	+	0.0322526i
$750$		0			0
$751$	10.2942	+	17.8301i	0.375642	+	0.650631i	0.990423	−	0.138067i	$-0.0440890\pi$
$751$	10.2942	+	17.8301i	0.375642	+	0.650631i	−0.614781	+	0.788698i	$0.710756\pi$
$752$	−1.55291	+	5.79555i	−0.0566290	+	0.211342i
$753$	−11.0227	−	11.0227i	−0.401690	−	0.401690i
$754$	−7.98076	−	4.60770i	−0.290642	−	0.167802i
$755$		0			0
$756$		−	6.58846i		−	0.239620i
$757$	10.5187	−	10.5187i	0.382308	−	0.382308i	−0.489625	−	0.871933i	$-0.662866\pi$
$757$	10.5187	−	10.5187i	0.382308	−	0.382308i	0.871933	+	0.489625i	$0.162866\pi$
$758$	5.27792	+	19.6975i	0.191703	+	0.715444i
$759$	−49.1769			−1.78501
$760$		0			0
$761$	−9.91154	+	5.72243i	−0.359293	+	0.207438i	−0.668771	−	0.743469i	$-0.733179\pi$
$761$	−9.91154	+	5.72243i	−0.359293	+	0.207438i	0.309478	+	0.950907i	$0.399846\pi$
$762$	0.984508	+	3.67423i	0.0356650	+	0.133103i
$763$	14.9372	−	4.00240i	0.540762	−	0.144897i
$764$	3.46410			0.125327
$765$		0			0
$766$	36.5885			1.32199
$767$	−15.2653	+	4.09034i	−0.551200	+	0.147693i
$768$	−1.67303	−	0.448288i	−0.0603704	−	0.0161762i
$769$	28.9186	−	16.6962i	1.04283	−	0.602079i	0.122197	−	0.992506i	$-0.461006\pi$
$769$	28.9186	−	16.6962i	1.04283	−	0.602079i	0.920634	+	0.390427i	$0.127673\pi$
$770$		0			0
$771$	3.99038	+	6.91154i	0.143710	+	0.248913i
$772$	−6.45189	−	24.0788i	−0.232209	−	0.866615i
$773$	6.51626	−	6.51626i	0.234374	−	0.234374i	−0.580142	−	0.814516i	$-0.697003\pi$
$773$	6.51626	−	6.51626i	0.234374	−	0.234374i	0.814516	+	0.580142i	$0.197003\pi$
$774$	9.69615	+	16.7942i	0.348521	+	0.603656i
$775$		0			0
$776$	−12.6962	−	7.33013i	−0.455765	−	0.263136i
$777$	−4.65874	+	1.24831i	−0.167131	+	0.0447827i
$778$	2.68973	−	10.0382i	0.0964314	−	0.359887i
$779$	6.23205	+	10.7942i	0.223286	+	0.386743i
$780$		0			0
$781$	18.5885	−	32.1962i	0.665147	−	1.15207i
$782$		0			0
$783$	36.4785	−	9.77440i	1.30364	−	0.349308i
$784$		−	5.39230i		−	0.192582i
$785$		0			0
$786$			12.0000i			0.428026i
$787$	16.7303	+	4.48288i	0.596372	+	0.159797i	0.544364	−	0.838849i	$-0.316771\pi$
$787$	16.7303	+	4.48288i	0.596372	+	0.159797i	0.0520081	+	0.998647i	$0.483438\pi$
$788$	−3.67423	−	0.984508i	−0.130889	−	0.0350717i
$789$	27.8827	+	16.0981i	0.992650	+	0.573107i
$790$		0			0
$791$		−	9.37307i		−	0.333268i
$792$	−2.68973	−	10.0382i	−0.0955753	−	0.356692i
$793$	3.58630	+	3.58630i	0.127353	+	0.127353i
$794$	−4.26795	+	7.39230i	−0.151464	+	0.262343i
$795$		0			0
$796$	0.196152	+	0.339746i	0.00695244	+	0.0120420i
$797$	−2.84203	+	10.6066i	−0.100670	+	0.375705i	−0.997818	−	0.0660248i	$-0.978968\pi$
$797$	−2.84203	+	10.6066i	−0.100670	+	0.375705i	0.897148	+	0.441730i	$0.145635\pi$
$798$	4.09034	−	15.2653i	0.144796	−	0.540387i
$799$		0			0
$800$		0			0
$801$			25.9808i			0.917985i
$802$	7.34847	−	7.34847i	0.259483	−	0.259483i
$803$	4.65874	+	17.3867i	0.164403	+	0.613562i
$804$	10.7942	−	18.6962i	0.380683	−	0.659362i
$805$		0			0
$806$	−11.1962	+	6.46410i	−0.394368	+	0.227688i
$807$	−34.3572	+	34.3572i	−1.20943	+	1.20943i
$808$	−10.3664	+	2.77766i	−0.364687	+	0.0977177i
$809$	−1.73205			−0.0608957			−0.0304478	−	0.999536i	$-0.509693\pi$
$809$	−1.73205			−0.0608957			−0.0304478	+	0.999536i	$0.509693\pi$
$810$		0			0
$811$	−38.3731			−1.34746			−0.673730	−	0.738977i	$-0.735309\pi$
$811$	−38.3731			−1.34746			−0.673730	+	0.738977i	$0.735309\pi$
$812$	−8.90138	+	2.38512i	−0.312377	+	0.0837013i
$813$	−5.61969	+	5.61969i	−0.197091	+	0.197091i
$814$	−6.58846	+	3.80385i	−0.230925	+	0.133325i
$815$		0			0
$816$		0			0
$817$	12.0394	+	44.9316i	0.421205	+	1.57196i
$818$	−0.429705	+	0.429705i	−0.0150243	+	0.0150243i
$819$			4.82309i			0.168532i
$820$		0			0
$821$	−18.5885	−	10.7321i	−0.648742	−	0.374551i	0.139232	−	0.990260i	$-0.455537\pi$
$821$	−18.5885	−	10.7321i	−0.648742	−	0.374551i	−0.787974	+	0.615709i	$0.788870\pi$
$822$	−3.31388	+	12.3676i	−0.115585	+	0.431368i
$823$	7.41284	−	27.6651i	0.258395	−	0.964345i	−0.707775	−	0.706438i	$-0.750301\pi$
$823$	7.41284	−	27.6651i	0.258395	−	0.964345i	0.966170	−	0.257906i	$-0.0830326\pi$
$824$	5.19615	+	9.00000i	0.181017	+	0.313530i
$825$		0			0
$826$	−7.90192	+	13.6865i	−0.274943	+	0.476215i
$827$	7.91688	+	7.91688i	0.275297	+	0.275297i	0.831228	−	0.555931i	$-0.187638\pi$
$827$	7.91688	+	7.91688i	0.275297	+	0.275297i	−0.555931	+	0.831228i	$0.687638\pi$
$828$	6.36396	+	23.7506i	0.221163	+	0.825391i
$829$		−	2.58846i		−	0.0899008i	−0.998989	−	0.0449504i	$-0.985687\pi$
$829$		−	2.58846i		−	0.0899008i	0.998989	−	0.0449504i	$-0.0143130\pi$
$830$		0			0
$831$	−38.2750	−	22.0981i	−1.32774	−	0.766574i
$832$	1.22474	+	0.328169i	0.0424604	+	0.0113772i
$833$		0			0
$834$			13.8564i			0.479808i
$835$		0			0
$836$		−	24.9282i		−	0.862160i
$837$	13.7124	−	51.1755i	0.473971	−	1.76888i
$838$	6.36396	+	6.36396i	0.219839	+	0.219839i
$839$	−15.7583	+	27.2942i	−0.544038	+	0.942301i	0.454629	+	0.890681i	$0.349772\pi$
$839$	−15.7583	+	27.2942i	−0.544038	+	0.942301i	−0.998667	+	0.0516204i	$0.983561\pi$
$840$		0			0
$841$	−11.9115	−	20.6314i	−0.410743	−	0.711427i
$842$	2.22286	−	8.29581i	0.0766047	−	0.285893i
$843$	17.3867	−	4.65874i	0.598829	−	0.160456i
$844$	7.96410	+	4.59808i	0.274136	+	0.158272i
$845$		0			0
$846$	9.00000	+	15.5885i	0.309426	+	0.535942i
$847$	−0.896575	+	0.896575i	−0.0308067	+	0.0308067i
$848$	0.568406	+	2.12132i	0.0195191	+	0.0728464i
$849$	−8.59808	−	14.8923i	−0.295085	−	0.511103i
$850$		0			0
$851$	15.5885	−	9.00000i	0.534365	−	0.308516i
$852$	−17.9551	−	4.81105i	−0.615131	−	0.164824i
$853$	−51.5037	+	13.8004i	−1.76345	+	0.472515i	−0.987412	−	0.158169i	$-0.949441\pi$
$853$	−51.5037	+	13.8004i	−1.76345	+	0.472515i	−0.776040	+	0.630684i	$0.782774\pi$
$854$	5.07180			0.173553
$855$		0			0
$856$	0.803848			0.0274749
$857$	0.208051	−	0.0557471i	0.00710689	−	0.00190429i	−0.255264	−	0.966871i	$-0.582162\pi$
$857$	0.208051	−	0.0557471i	0.00710689	−	0.00190429i	0.262371	+	0.964967i	$0.415496\pi$
$858$	1.96902	+	7.34847i	0.0672211	+	0.250873i
$859$	9.01666	−	5.20577i	0.307644	−	0.177619i	−0.338227	−	0.941064i	$-0.609827\pi$
$859$	9.01666	−	5.20577i	0.307644	−	0.177619i	0.645872	+	0.763446i	$0.276494\pi$
$860$		0			0
$861$	−3.80385			−0.129635
$862$	−0.808643	−	3.01790i	−0.0275425	−	0.102790i
$863$	3.52193	−	3.52193i	0.119888	−	0.119888i	−0.644617	−	0.764505i	$-0.722983\pi$
$863$	3.52193	−	3.52193i	0.119888	−	0.119888i	0.764505	+	0.644617i	$0.222983\pi$
$864$	−4.50000	+	2.59808i	−0.153093	+	0.0883883i
$865$		0			0
$866$	26.7846	+	15.4641i	0.910178	+	0.525492i
$867$	−20.8207	−	20.8207i	−0.707107	−	0.707107i
$868$	−3.34607	+	12.4877i	−0.113573	+	0.423860i
$869$	17.3205	+	30.0000i	0.587558	+	1.01768i
$870$		0			0
$871$	−7.90192	+	13.6865i	−0.267746	+	0.463750i
$872$	−8.62398	−	8.62398i	−0.292045	−	0.292045i
$873$	−42.4822	+	11.3831i	−1.43780	+	0.385258i
$874$			58.9808i			1.99505i
$875$		0			0
$876$	7.79423	−	4.50000i	0.263343	−	0.152041i
$877$	−22.2856	−	5.97142i	−0.752533	−	0.201641i	−0.137892	−	0.990447i	$-0.544033\pi$
$877$	−22.2856	−	5.97142i	−0.752533	−	0.201641i	−0.614641	+	0.788807i	$0.710699\pi$
$878$	−31.4780	−	8.43451i	−1.06233	−	0.284651i
$879$	21.2942	−	12.2942i	0.718237	−	0.414674i
$880$		0			0
$881$			53.3205i			1.79641i	0.439573	+	0.898207i	$0.355130\pi$
$881$			53.3205i			1.79641i	−0.439573	+	0.898207i	$0.644870\pi$
$882$	−11.4388	−	11.4388i	−0.385165	−	0.385165i
$883$	1.31268	+	1.31268i	0.0441751	+	0.0441751i	0.728849	−	0.684674i	$-0.240055\pi$
$883$	1.31268	+	1.31268i	0.0441751	+	0.0441751i	−0.684674	+	0.728849i	$0.740055\pi$
$884$		0			0
$885$		0			0
$886$	−5.19615	−	9.00000i	−0.174568	−	0.302361i
$887$	2.68973	−	10.0382i	0.0903122	−	0.337050i	−0.905955	−	0.423374i	$-0.860846\pi$
$887$	2.68973	−	10.0382i	0.0903122	−	0.337050i	0.996267	+	0.0863246i	$0.0275122\pi$
$888$	2.68973	+	2.68973i	0.0902613	+	0.0902613i
$889$	−2.41154	−	1.39230i	−0.0808805	−	0.0466964i
$890$		0			0
$891$	−27.0000	−	15.5885i	−0.904534	−	0.522233i
$892$	6.69213	−	6.69213i	0.224069	−	0.224069i
$893$	11.1750	+	41.7057i	0.373957	+	1.39563i
$894$	−31.1769			−1.04271
$895$		0			0
$896$	1.09808	−	0.633975i	0.0366842	−	0.0211796i
$897$	−4.65874	−	17.3867i	−0.155551	−	0.580524i
$898$	−5.67544	+	1.52073i	−0.189392	+	0.0507474i
$899$	−74.1051			−2.47154
$900$		0			0
$901$		0			0
$902$	−5.79555	+	1.55291i	−0.192971	+	0.0517064i
$903$	−13.7124	−	3.67423i	−0.456321	−	0.122271i
$904$	−6.40192	+	3.69615i	−0.212925	+	0.122932i
$905$		0			0
$906$	−6.92820	−	12.0000i	−0.230174	−	0.398673i
$907$	8.54103	+	31.8756i	0.283600	+	1.05841i	0.949856	+	0.312687i	$0.101229\pi$
$907$	8.54103	+	31.8756i	0.283600	+	1.05841i	−0.666256	+	0.745723i	$0.732104\pi$
$908$	16.4022	−	16.4022i	0.544325	−	0.544325i
$909$	−16.0981	+	27.8827i	−0.533939	+	0.924810i
$910$		0			0
$911$	−12.8827	−	7.43782i	−0.426822	−	0.246426i	0.271170	−	0.962532i	$-0.412590\pi$
$911$	−12.8827	−	7.43782i	−0.426822	−	0.246426i	−0.697992	+	0.716106i	$0.745923\pi$
$912$	−12.0394	+	3.22595i	−0.398664	+	0.106822i
$913$	−6.10016	+	22.7661i	−0.201886	+	0.753449i
$914$	1.79423	+	3.10770i	0.0593478	+	0.102793i
$915$		0			0
$916$	−8.09808	+	14.0263i	−0.267568	+	0.463441i
$917$	−6.21166	−	6.21166i	−0.205127	−	0.205127i
$918$		0			0
$919$			27.4115i			0.904223i	0.891961	+	0.452112i	$0.149329\pi$
$919$			27.4115i			0.904223i	−0.891961	+	0.452112i	$0.850671\pi$
$920$		0			0
$921$			6.00000i			0.197707i
$922$	−27.0967	−	7.26054i	−0.892382	−	0.239113i
$923$	13.1440	+	3.52193i	0.432641	+	0.115926i
$924$	6.58846	+	3.80385i	0.216744	+	0.125137i
$925$		0			0
$926$			8.19615i			0.269342i
$927$	30.1146	+	8.06918i	0.989093	+	0.265027i
$928$	5.13922	+	5.13922i	0.168703	+	0.168703i
$929$	2.07180	−	3.58846i	0.0679734	−	0.117733i	−0.830036	−	0.557710i	$-0.811680\pi$
$929$	2.07180	−	3.58846i	0.0679734	−	0.117733i	0.898009	+	0.439977i	$0.145013\pi$
$930$		0			0
$931$	−19.4019	−	33.6051i	−0.635872	−	1.10136i
$932$	6.57201	−	24.5271i	0.215273	−	0.803411i
$933$	11.1750	−	41.7057i	0.365853	−	1.36538i
$934$	−18.6962	−	10.7942i	−0.611757	−	0.353198i
$935$		0			0
$936$	3.29423	−	1.90192i	0.107675	−	0.0621663i
$937$	9.22955	−	9.22955i	0.301516	−	0.301516i	−0.540091	−	0.841607i	$-0.681610\pi$
$937$	9.22955	−	9.22955i	0.301516	−	0.301516i	0.841607	+	0.540091i	$0.181610\pi$
$938$	4.09034	+	15.2653i	0.133554	+	0.498431i
$939$	2.89230	−	5.00962i	0.0943868	−	0.163483i
$940$		0			0
$941$	−36.8827	+	21.2942i	−1.20234	+	0.694172i	−0.961075	−	0.276287i	$-0.910896\pi$
$941$	−36.8827	+	21.2942i	−1.20234	+	0.694172i	−0.241266	+	0.970459i	$0.577563\pi$
$942$	9.62209	−	9.62209i	0.313505	−	0.313505i
$943$	13.7124	−	3.67423i	0.446538	−	0.119650i
$944$	12.4641			0.405672
$945$		0			0
$946$	−22.3923			−0.728037
$947$	29.3381	−	7.86113i	0.953361	−	0.255452i	0.251573	−	0.967838i	$-0.419052\pi$
$947$	29.3381	−	7.86113i	0.953361	−	0.255452i	0.701788	+	0.712386i	$0.252385\pi$
$948$	12.2474	−	12.2474i	0.397779	−	0.397779i
$949$	−5.70577	+	3.29423i	−0.185217	+	0.106935i
$950$		0			0
$951$	21.2942	−	36.8827i	0.690513	−	1.19600i
$952$		0			0
$953$	−28.5617	+	28.5617i	−0.925203	+	0.925203i	−0.997391	−	0.0721877i	$-0.977002\pi$
$953$	−28.5617	+	28.5617i	−0.925203	+	0.925203i	0.0721877	+	0.997391i	$0.477002\pi$
$954$	5.70577	+	3.29423i	0.184731	+	0.106655i
$955$		0			0
$956$	12.2942	+	7.09808i	0.397624	+	0.229568i
$957$	−11.2865	+	42.1218i	−0.364841	+	1.36160i
$958$	−5.46739	+	20.4046i	−0.176643	+	0.659241i
$959$	−4.68653	−	8.11731i	−0.151336	−	0.262122i
$960$		0			0
$961$	−36.4808	+	63.1865i	−1.17680	+	2.03828i
$962$	−1.96902	−	1.96902i	−0.0634836	−	0.0634836i
$963$	1.70522	−	1.70522i	0.0549499	−	0.0549499i
$964$		−	11.3923i		−	0.366921i
$965$		0			0
$966$	−15.5885	−	9.00000i	−0.501550	−	0.289570i
$967$	−40.1528	−	10.7589i	−1.29123	−	0.345983i	−0.453100	−	0.891460i	$-0.649682\pi$
$967$	−40.1528	−	10.7589i	−1.29123	−	0.345983i	−0.838126	+	0.545476i	$0.816349\pi$
$968$	0.965926	+	0.258819i	0.0310460	+	0.00831876i
$969$		0			0
$970$		0			0
$971$		−	36.0333i		−	1.15636i	−0.815908	−	0.578182i	$-0.803762\pi$
$971$		−	36.0333i		−	1.15636i	0.815908	−	0.578182i	$-0.196238\pi$
$972$	−4.03459	+	15.0573i	−0.129410	+	0.482963i
$973$	−7.17260	−	7.17260i	−0.229943	−	0.229943i
$974$	−11.0263	+	19.0981i	−0.353305	+	0.611942i
$975$		0			0
$976$	−2.00000	−	3.46410i	−0.0640184	−	0.110883i
$977$	4.18689	−	15.6257i	0.133951	−	0.499910i	−0.866049	−	0.499958i	$-0.833349\pi$
$977$	4.18689	−	15.6257i	0.133951	−	0.499910i	1.00000		4.80066e-5i	$1.52810e-5\pi$
$978$	−33.9969	+	9.10943i	−1.08710	+	0.291288i
$979$	−25.9808	−	15.0000i	−0.830349	−	0.479402i
$980$		0			0
$981$	−36.5885			−1.16818
$982$	−25.9599	+	25.9599i	−0.828413	+	0.828413i
$983$	4.96335	+	18.5235i	0.158306	+	0.590807i	0.998800	+	0.0489851i	$0.0155987\pi$
$983$	4.96335	+	18.5235i	0.158306	+	0.590807i	−0.840493	+	0.541822i	$0.817735\pi$
$984$	1.50000	+	2.59808i	0.0478183	+	0.0828236i
$985$		0			0
$986$		0			0
$987$	−12.7279	−	3.41044i	−0.405134	−	0.108555i
$988$	8.81345	−	2.36156i	0.280393	−	0.0751311i
$989$	52.9808			1.68469
$990$		0			0
$991$	33.1769			1.05390			0.526950	−	0.849896i	$-0.323336\pi$
$991$	33.1769			1.05390			0.526950	+	0.849896i	$0.323336\pi$
$992$	9.84873	−	2.63896i	0.312697	−	0.0837870i
$993$	7.88469	+	29.4261i	0.250213	+	0.933808i
$994$	11.7846	−	6.80385i	0.373785	−	0.215805i
$995$		0			0
$996$	11.7846			0.373410
$997$	8.33298	+	31.0991i	0.263908	+	0.984918i	0.962915	+	0.269804i	$0.0869588\pi$
$997$	8.33298	+	31.0991i	0.263908	+	0.984918i	−0.699007	+	0.715115i	$0.746375\pi$
$998$	−13.8511	+	13.8511i	−0.438450	+	0.438450i
$999$	11.4115			0.361045

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	450.2.p.e.443.2	yes	8
3.2	odd	2		1350.2.q.a.143.1		8
5.2	odd	4		450.2.p.c.407.2	yes	8
5.3	odd	4		450.2.p.c.407.1	yes	8
5.4	even	2	inner	450.2.p.e.443.1	yes	8
9.4	even	3		1350.2.q.d.1043.1		8
9.5	odd	6		450.2.p.c.293.2	yes	8
15.2	even	4		1350.2.q.d.1007.1		8
15.8	even	4		1350.2.q.d.1007.2		8
15.14	odd	2		1350.2.q.a.143.2		8
45.4	even	6		1350.2.q.d.1043.2		8
45.13	odd	12		1350.2.q.a.557.2		8
45.14	odd	6		450.2.p.c.293.1	✓	8
45.22	odd	12		1350.2.q.a.557.1		8
45.23	even	12	inner	450.2.p.e.257.1	yes	8
45.32	even	12	inner	450.2.p.e.257.2	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
450.2.p.c.293.1	✓	8	45.14	odd	6
450.2.p.c.293.2	yes	8	9.5	odd	6
450.2.p.c.407.1	yes	8	5.3	odd	4
450.2.p.c.407.2	yes	8	5.2	odd	4
450.2.p.e.257.1	yes	8	45.23	even	12	inner
450.2.p.e.257.2	yes	8	45.32	even	12	inner
450.2.p.e.443.1	yes	8	5.4	even	2	inner
450.2.p.e.443.2	yes	8	1.1	even	1	trivial
1350.2.q.a.143.1		8	3.2	odd	2
1350.2.q.a.143.2		8	15.14	odd	2
1350.2.q.a.557.1		8	45.22	odd	12
1350.2.q.a.557.2		8	45.13	odd	12
1350.2.q.d.1007.1		8	15.2	even	4
1350.2.q.d.1007.2		8	15.8	even	4
1350.2.q.d.1043.1		8	9.4	even	3
1350.2.q.d.1043.2		8	45.4	even	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	450.p (of order \(12\), degree \(4\), minimal)

\(f(q)\)	\(=\)	\(q+(0.965926 - 0.258819i) q^{2} +(1.22474 - 1.22474i) q^{3} +(0.866025 - 0.500000i) q^{4} +(0.866025 - 1.50000i) q^{6} +(0.328169 + 1.22474i) q^{7} +(0.707107 - 0.707107i) q^{8} -3.00000i q^{9} +O(q^{10})\)	\(q+(0.965926 - 0.258819i) q^{2} +(1.22474 - 1.22474i) q^{3} +(0.866025 - 0.500000i) q^{4} +(0.866025 - 1.50000i) q^{6} +(0.328169 + 1.22474i) q^{7} +(0.707107 - 0.707107i) q^{8} -3.00000i q^{9} +(3.00000 + 1.73205i) q^{11} +(0.448288 - 1.67303i) q^{12} +(-0.328169 + 1.22474i) q^{13} +(0.633975 + 1.09808i) q^{14} +(0.500000 - 0.866025i) q^{16} +(-0.776457 - 2.89778i) q^{18} -7.19615i q^{19} +(1.90192 + 1.09808i) q^{21} +(3.34607 + 0.896575i) q^{22} +(-7.91688 - 2.12132i) q^{23} -1.73205i q^{24} +1.26795i q^{26} +(-3.67423 - 3.67423i) q^{27} +(0.896575 + 0.896575i) q^{28} +(-3.63397 + 6.29423i) q^{29} +(5.09808 + 8.83013i) q^{31} +(0.258819 - 0.965926i) q^{32} +(5.79555 - 1.55291i) q^{33} +(-1.50000 - 2.59808i) q^{36} +(-1.55291 + 1.55291i) q^{37} +(-1.86250 - 6.95095i) q^{38} +(1.09808 + 1.90192i) q^{39} +(-1.50000 + 0.866025i) q^{41} +(2.12132 + 0.568406i) q^{42} +(-6.24384 + 1.67303i) q^{43} +3.46410 q^{44} -8.19615 q^{46} +(-5.79555 + 1.55291i) q^{47} +(-0.448288 - 1.67303i) q^{48} +(4.66987 - 2.69615i) q^{49} +(0.328169 + 1.22474i) q^{52} +(-1.55291 + 1.55291i) q^{53} +(-4.50000 - 2.59808i) q^{54} +(1.09808 + 0.633975i) q^{56} +(-8.81345 - 8.81345i) q^{57} +(-1.88108 + 7.02030i) q^{58} +(6.23205 + 10.7942i) q^{59} +(2.00000 - 3.46410i) q^{61} +(7.20977 + 7.20977i) q^{62} +(3.67423 - 0.984508i) q^{63} -1.00000i q^{64} +(5.19615 - 3.00000i) q^{66} +(12.0394 + 3.22595i) q^{67} +(-12.2942 + 7.09808i) q^{69} -10.7321i q^{71} +(-2.12132 - 2.12132i) q^{72} +(3.67423 + 3.67423i) q^{73} +(-1.09808 + 1.90192i) q^{74} +(-3.59808 - 6.23205i) q^{76} +(-1.13681 + 4.24264i) q^{77} +(1.55291 + 1.55291i) q^{78} +(8.66025 + 5.00000i) q^{79} -9.00000 q^{81} +(-1.22474 + 1.22474i) q^{82} +(1.76097 + 6.57201i) q^{83} +2.19615 q^{84} +(-5.59808 + 3.23205i) q^{86} +(3.25813 + 12.1595i) q^{87} +(3.34607 - 0.896575i) q^{88} -8.66025 q^{89} -1.60770 q^{91} +(-7.91688 + 2.12132i) q^{92} +(17.0585 + 4.57081i) q^{93} +(-5.19615 + 3.00000i) q^{94} +(-0.866025 - 1.50000i) q^{96} +(-3.79435 - 14.1607i) q^{97} +(3.81294 - 3.81294i) q^{98} +(5.19615 - 9.00000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q + 24 q^{11} + 12 q^{14} + 4 q^{16} + 36 q^{21} - 36 q^{29} + 20 q^{31} - 12 q^{36} - 12 q^{39} - 12 q^{41} - 24 q^{46} + 72 q^{49} - 36 q^{54} - 12 q^{56} + 36 q^{59} + 16 q^{61} - 36 q^{69} + 12 q^{74} - 8 q^{76} - 72 q^{81} - 24 q^{84} - 24 q^{86} - 96 q^{91}+O(q^{100}) \) 8 * q + 24 * q^11 + 12 * q^14 + 4 * q^16 + 36 * q^21 - 36 * q^29 + 20 * q^31 - 12 * q^36 - 12 * q^39 - 12 * q^41 - 24 * q^46 + 72 * q^49 - 36 * q^54 - 12 * q^56 + 36 * q^59 + 16 * q^61 - 36 * q^69 + 12 * q^74 - 8 * q^76 - 72 * q^81 - 24 * q^84 - 24 * q^86 - 96 * q^91

Introduction

L-functions

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