LMFDB - Embedded newform 168.2.i.e.125.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [168,2,Mod(125,168)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(168, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 1, 1]))

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

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

chi := DirichletCharacter("168.125");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 168 = 2^{3} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	168.i (of order $2$, degree $1$, 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:	$1.34148675396$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.3317760000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 8x^{6} + 13x^{4} + 12x^{2} + 36 $ x^8 - 8x^6 + 13x^4 + 12*x^2 + 36
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			125.5
Root			$-2.15988 - 0.258819i$ of defining polynomial
Character	$\chi$	$=$	168.125
Dual form			168.2.i.e.125.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.866025 - 1.11803i) q^{2} +(-1.58114 + 0.707107i) q^{3} +(-0.500000 - 1.93649i) q^{4} -1.41421i q^{5} +(-0.578737 + 2.38014i) q^{6} +(1.00000 - 2.44949i) q^{7} +(-2.59808 - 1.11803i) q^{8} +(2.00000 - 2.23607i) q^{9} +O(q^{10})$	\(q+(0.866025 - 1.11803i) q^{2} +(-1.58114 + 0.707107i) q^{3} +(-0.500000 - 1.93649i) q^{4} -1.41421i q^{5} +(-0.578737 + 2.38014i) q^{6} +(1.00000 - 2.44949i) q^{7} +(-2.59808 - 1.11803i) q^{8} +(2.00000 - 2.23607i) q^{9} +(-1.58114 - 1.22474i) q^{10} -3.46410 q^{11} +(2.15988 + 2.70831i) q^{12} +3.16228 q^{13} +(-1.87259 - 3.23935i) q^{14} +(1.00000 + 2.23607i) q^{15} +(-3.50000 + 1.93649i) q^{16} +(-0.767949 - 4.17256i) q^{18} +3.16228 q^{19} +(-2.73861 + 0.707107i) q^{20} +(0.150912 + 4.58009i) q^{21} +(-3.00000 + 3.87298i) q^{22} +4.47214i q^{23} +(4.89849 - 0.0693504i) q^{24} +3.00000 q^{25} +(2.73861 - 3.53553i) q^{26} +(-1.58114 + 4.94975i) q^{27} +(-5.24342 - 0.711747i) q^{28} +6.92820 q^{29} +(3.36603 + 0.818458i) q^{30} +4.89898i q^{31} +(-0.866025 + 5.59017i) q^{32} +(5.47723 - 2.44949i) q^{33} +(-3.46410 - 1.41421i) q^{35} +(-5.33013 - 2.75495i) q^{36} +(2.73861 - 3.53553i) q^{38} +(-5.00000 + 2.23607i) q^{39} +(-1.58114 + 3.67423i) q^{40} -10.9545 q^{41} +(5.25139 + 3.79775i) q^{42} -7.74597i q^{43} +(1.73205 + 6.70820i) q^{44} +(-3.16228 - 2.82843i) q^{45} +(5.00000 + 3.87298i) q^{46} +10.9545 q^{47} +(4.16468 - 5.53674i) q^{48} +(-5.00000 - 4.89898i) q^{49} +(2.59808 - 3.35410i) q^{50} +(-1.58114 - 6.12372i) q^{52} +(4.16468 + 6.05437i) q^{54} +4.89898i q^{55} +(-5.33669 + 5.24593i) q^{56} +(-5.00000 + 2.23607i) q^{57} +(6.00000 - 7.74597i) q^{58} -9.89949i q^{59} +(3.83013 - 3.05453i) q^{60} +3.16228 q^{61} +(5.47723 + 4.24264i) q^{62} +(-3.47723 - 7.13505i) q^{63} +(5.50000 + 5.80948i) q^{64} -4.47214i q^{65} +(2.00480 - 8.24504i) q^{66} +7.74597i q^{67} +(-3.16228 - 7.07107i) q^{69} +(-4.58114 + 2.64824i) q^{70} -8.94427i q^{71} +(-7.69615 + 3.57341i) q^{72} +14.6969i q^{73} +(-4.74342 + 2.12132i) q^{75} +(-1.58114 - 6.12372i) q^{76} +(-3.46410 + 8.48528i) q^{77} +(-1.83013 + 7.52666i) q^{78} -10.0000 q^{79} +(2.73861 + 4.94975i) q^{80} +(-1.00000 - 8.94427i) q^{81} +(-9.48683 + 12.2474i) q^{82} +7.07107i q^{83} +(8.79385 - 2.58228i) q^{84} +(-8.66025 - 6.70820i) q^{86} +(-10.9545 + 4.89898i) q^{87} +(9.00000 + 3.87298i) q^{88} +(-5.90089 + 1.08604i) q^{90} +(3.16228 - 7.74597i) q^{91} +(8.66025 - 2.23607i) q^{92} +(-3.46410 - 7.74597i) q^{93} +(9.48683 - 12.2474i) q^{94} -4.47214i q^{95} +(-2.58354 - 9.45121i) q^{96} -4.89898i q^{97} +(-9.80735 + 1.34753i) q^{98} +(-6.92820 + 7.74597i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 4 q^{4} + 8 q^{7} + 16 q^{9}+O(q^{10}) $ 8 * q - 4 * q^4 + 8 * q^7 + 16 * q^9	$ 8 q - 4 q^{4} + 8 q^{7} + 16 q^{9} + 8 q^{15} - 28 q^{16} - 20 q^{18} - 24 q^{22} + 24 q^{25} - 4 q^{28} + 20 q^{30} - 8 q^{36} - 40 q^{39} + 12 q^{42} + 40 q^{46} - 40 q^{49} - 40 q^{57} + 48 q^{58} - 4 q^{60} + 16 q^{63} + 44 q^{64} - 24 q^{70} - 20 q^{72} + 20 q^{78} - 80 q^{79} - 8 q^{81} + 60 q^{84} + 72 q^{88}+O(q^{100}) $ 8 * q - 4 * q^4 + 8 * q^7 + 16 * q^9 + 8 * q^15 - 28 * q^16 - 20 * q^18 - 24 * q^22 + 24 * q^25 - 4 * q^28 + 20 * q^30 - 8 * q^36 - 40 * q^39 + 12 * q^42 + 40 * q^46 - 40 * q^49 - 40 * q^57 + 48 * q^58 - 4 * q^60 + 16 * q^63 + 44 * q^64 - 24 * q^70 - 20 * q^72 + 20 * q^78 - 80 * q^79 - 8 * q^81 + 60 * q^84 + 72 * q^88

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/168\mathbb{Z}\right)^\times$.

$n$	$73$	$85$	$113$	$127$
$\chi(n)$	$-1$	$-1$	$-1$	$1$

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 $

$2$	0.866025	−	1.11803i	0.612372	−	0.790569i
$2$	0.866025	−	1.11803i	0.612372	−	0.790569i
$3$	−1.58114	+	0.707107i	−0.912871	+	0.408248i
$3$	−1.58114	+	0.707107i	−0.912871	+	0.408248i
$4$	−0.500000	−	1.93649i	−0.250000	−	0.968246i
$5$		−	1.41421i		−	0.632456i	−0.948683	−	0.316228i	$-0.897584\pi$
$5$		−	1.41421i		−	0.632456i	0.948683	−	0.316228i	$-0.102416\pi$
$6$	−0.578737	+	2.38014i	−0.236268	+	0.971688i
$7$	1.00000	−	2.44949i	0.377964	−	0.925820i
$7$	1.00000	−	2.44949i	0.377964	−	0.925820i
$8$	−2.59808	−	1.11803i	−0.918559	−	0.395285i
$9$	2.00000	−	2.23607i	0.666667	−	0.745356i
$10$	−1.58114	−	1.22474i	−0.500000	−	0.387298i
$11$	−3.46410			−1.04447			−0.522233	−	0.852803i	$-0.674901\pi$
$11$	−3.46410			−1.04447			−0.522233	+	0.852803i	$0.674901\pi$
$12$	2.15988	+	2.70831i	0.623502	+	0.781821i
$13$	3.16228			0.877058			0.438529	−	0.898717i	$-0.355500\pi$
$13$	3.16228			0.877058			0.438529	+	0.898717i	$0.355500\pi$
$14$	−1.87259	−	3.23935i	−0.500470	−	0.865754i
$15$	1.00000	+	2.23607i	0.258199	+	0.577350i
$16$	−3.50000	+	1.93649i	−0.875000	+	0.484123i
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$	−0.767949	−	4.17256i	−0.181007	−	0.983482i
$19$	3.16228			0.725476			0.362738	−	0.931891i	$-0.381842\pi$
$19$	3.16228			0.725476			0.362738	+	0.931891i	$0.381842\pi$
$20$	−2.73861	+	0.707107i	−0.612372	+	0.158114i
$21$	0.150912	+	4.58009i	0.0329317	+	0.999458i
$22$	−3.00000	+	3.87298i	−0.639602	+	0.825723i
$23$			4.47214i			0.932505i	0.884652	+	0.466252i	$0.154396\pi$
$23$			4.47214i			0.932505i	−0.884652	+	0.466252i	$0.845604\pi$
$24$	4.89849	−	0.0693504i	0.999900	−	0.0141561i
$25$	3.00000			0.600000
$26$	2.73861	−	3.53553i	0.537086	−	0.693375i
$27$	−1.58114	+	4.94975i	−0.304290	+	0.952579i
$28$	−5.24342	−	0.711747i	−0.990913	−	0.134508i
$29$	6.92820			1.28654			0.643268	−	0.765641i	$-0.277578\pi$
$29$	6.92820			1.28654			0.643268	+	0.765641i	$0.277578\pi$
$30$	3.36603	+	0.818458i	0.614549	+	0.149429i
$31$			4.89898i			0.879883i	0.898027	+	0.439941i	$0.145001\pi$
$31$			4.89898i			0.879883i	−0.898027	+	0.439941i	$0.854999\pi$
$32$	−0.866025	+	5.59017i	−0.153093	+	0.988212i
$33$	5.47723	−	2.44949i	0.953463	−	0.426401i
$34$		0			0
$35$	−3.46410	−	1.41421i	−0.585540	−	0.239046i
$36$	−5.33013	−	2.75495i	−0.888355	−	0.459158i
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$	2.73861	−	3.53553i	0.444262	−	0.573539i
$39$	−5.00000	+	2.23607i	−0.800641	+	0.358057i
$40$	−1.58114	+	3.67423i	−0.250000	+	0.580948i
$41$	−10.9545			−1.71080			−0.855399	−	0.517970i	$-0.826688\pi$
$41$	−10.9545			−1.71080			−0.855399	+	0.517970i	$0.826688\pi$
$42$	5.25139	+	3.79775i	0.810307	+	0.586005i
$43$		−	7.74597i		−	1.18125i	−0.806947	−	0.590624i	$-0.798881\pi$
$43$		−	7.74597i		−	1.18125i	0.806947	−	0.590624i	$-0.201119\pi$
$44$	1.73205	+	6.70820i	0.261116	+	1.01130i
$45$	−3.16228	−	2.82843i	−0.471405	−	0.421637i
$46$	5.00000	+	3.87298i	0.737210	+	0.571040i
$47$	10.9545			1.59787			0.798935	−	0.601417i	$-0.205397\pi$
$47$	10.9545			1.59787			0.798935	+	0.601417i	$0.205397\pi$
$48$	4.16468	−	5.53674i	0.601120	−	0.799159i
$49$	−5.00000	−	4.89898i	−0.714286	−	0.699854i
$50$	2.59808	−	3.35410i	0.367423	−	0.474342i
$51$		0			0
$52$	−1.58114	−	6.12372i	−0.219265	−	0.849208i
$53$		0			0			−	1.00000i	$-0.5\pi$
$53$		0			0				1.00000i	$0.5\pi$
$54$	4.16468	+	6.05437i	0.566741	+	0.823896i
$55$			4.89898i			0.660578i
$56$	−5.33669	+	5.24593i	−0.713145	+	0.701016i
$57$	−5.00000	+	2.23607i	−0.662266	+	0.296174i
$58$	6.00000	−	7.74597i	0.787839	−	1.01710i
$59$		−	9.89949i		−	1.28880i	−0.764687	−	0.644402i	$-0.777106\pi$
$59$		−	9.89949i		−	1.28880i	0.764687	−	0.644402i	$-0.222894\pi$
$60$	3.83013	−	3.05453i	0.494467	−	0.394338i
$61$	3.16228			0.404888			0.202444	−	0.979294i	$-0.435112\pi$
$61$	3.16228			0.404888			0.202444	+	0.979294i	$0.435112\pi$
$62$	5.47723	+	4.24264i	0.695608	+	0.538816i
$63$	−3.47723	−	7.13505i	−0.438089	−	0.898931i
$64$	5.50000	+	5.80948i	0.687500	+	0.726184i
$65$		−	4.47214i		−	0.554700i
$66$	2.00480	−	8.24504i	0.246774	−	1.01489i
$67$			7.74597i			0.946320i	0.880976	+	0.473160i	$0.156887\pi$
$67$			7.74597i			0.946320i	−0.880976	+	0.473160i	$0.843113\pi$
$68$		0			0
$69$	−3.16228	−	7.07107i	−0.380693	−	0.851257i
$70$	−4.58114	+	2.64824i	−0.547551	+	0.316525i
$71$		−	8.94427i		−	1.06149i	−0.847532	−	0.530745i	$-0.821912\pi$
$71$		−	8.94427i		−	1.06149i	0.847532	−	0.530745i	$-0.178088\pi$
$72$	−7.69615	+	3.57341i	−0.907000	+	0.421130i
$73$			14.6969i			1.72015i	0.510171	+	0.860073i	$0.329582\pi$
$73$			14.6969i			1.72015i	−0.510171	+	0.860073i	$0.670418\pi$
$74$		0			0
$75$	−4.74342	+	2.12132i	−0.547723	+	0.244949i
$76$	−1.58114	−	6.12372i	−0.181369	−	0.702439i
$77$	−3.46410	+	8.48528i	−0.394771	+	0.966988i
$78$	−1.83013	+	7.52666i	−0.207221	+	0.852227i
$79$	−10.0000			−1.12509			−0.562544	−	0.826767i	$-0.690177\pi$
$79$	−10.0000			−1.12509			−0.562544	+	0.826767i	$0.690177\pi$
$80$	2.73861	+	4.94975i	0.306186	+	0.553399i
$81$	−1.00000	−	8.94427i	−0.111111	−	0.993808i
$82$	−9.48683	+	12.2474i	−1.04765	+	1.35250i
$83$			7.07107i			0.776151i	0.921628	+	0.388075i	$0.126860\pi$
$83$			7.07107i			0.776151i	−0.921628	+	0.388075i	$0.873140\pi$
$84$	8.79385	−	2.58228i	0.959488	−	0.281750i
$85$		0			0
$86$	−8.66025	−	6.70820i	−0.933859	−	0.723364i
$87$	−10.9545	+	4.89898i	−1.17444	+	0.525226i
$88$	9.00000	+	3.87298i	0.959403	+	0.412861i
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$	−5.90089	+	1.08604i	−0.622008	+	0.114479i
$91$	3.16228	−	7.74597i	0.331497	−	0.811998i
$92$	8.66025	−	2.23607i	0.902894	−	0.233126i
$93$	−3.46410	−	7.74597i	−0.359211	−	0.803219i
$94$	9.48683	−	12.2474i	0.978492	−	1.26323i
$95$		−	4.47214i		−	0.458831i
$96$	−2.58354	−	9.45121i	−0.263682	−	0.964610i
$97$		−	4.89898i		−	0.497416i	−0.968579	−	0.248708i	$-0.919994\pi$
$97$		−	4.89898i		−	0.497416i	0.968579	−	0.248708i	$-0.0800060\pi$
$98$	−9.80735	+	1.34753i	−0.990692	+	0.136121i
$99$	−6.92820	+	7.74597i	−0.696311	+	0.778499i
$100$	−1.50000	−	5.80948i	−0.150000	−	0.580948i
$101$			15.5563i			1.54791i	0.633238	+	0.773957i	$0.281726\pi$
$101$			15.5563i			1.54791i	−0.633238	+	0.773957i	$0.718274\pi$
$102$		0			0
$103$		−	9.79796i		−	0.965422i	−0.875780	−	0.482711i	$-0.839652\pi$
$103$		−	9.79796i		−	0.965422i	0.875780	−	0.482711i	$-0.160348\pi$
$104$	−8.21584	−	3.53553i	−0.805629	−	0.346688i
$105$	6.47723	−	0.213422i	0.632112	−	0.0208278i
$106$		0			0
$107$	10.3923			1.00466			0.502331	−	0.864675i	$-0.332476\pi$
$107$	10.3923			1.00466			0.502331	+	0.864675i	$0.332476\pi$
$108$	10.3757	+	0.586988i	0.998404	+	0.0564830i
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$	5.47723	+	4.24264i	0.522233	+	0.404520i
$111$		0			0
$112$	1.24342	+	10.5097i	0.117492	+	0.993074i
$113$			4.47214i			0.420703i	0.977626	+	0.210352i	$0.0674609\pi$
$113$			4.47214i			0.420703i	−0.977626	+	0.210352i	$0.932539\pi$
$114$	−1.83013	+	7.52666i	−0.171407	+	0.704936i
$115$	6.32456			0.589768
$116$	−3.46410	−	13.4164i	−0.321634	−	1.24568i
$117$	6.32456	−	7.07107i	0.584705	−	0.653720i
$118$	−11.0680	−	8.57321i	−1.01889	−	0.789228i
$119$		0			0
$120$	−0.0980762	−	6.92751i	−0.00895309	−	0.632392i
$121$	1.00000			0.0909091
$122$	2.73861	−	3.53553i	0.247942	−	0.320092i
$123$	17.3205	−	7.74597i	1.56174	−	0.698430i
$124$	9.48683	−	2.44949i	0.851943	−	0.219971i
$125$		−	11.3137i		−	1.01193i
$126$	−10.9886	−	2.29148i	−0.978942	−	0.204141i
$127$	8.00000			0.709885			0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000			0.709885			0.354943	+	0.934888i	$0.384500\pi$
$128$	11.2583	−	1.11803i	0.995105	−	0.0988212i
$129$	5.47723	+	12.2474i	0.482243	+	1.07833i
$130$	−5.00000	−	3.87298i	−0.438529	−	0.339683i
$131$		−	1.41421i		−	0.123560i	−0.998090	−	0.0617802i	$-0.980322\pi$
$131$		−	1.41421i		−	0.123560i	0.998090	−	0.0617802i	$-0.0196778\pi$
$132$	−7.48203	−	9.38186i	−0.651227	−	0.816586i
$133$	3.16228	−	7.74597i	0.274204	−	0.671660i
$134$	8.66025	+	6.70820i	0.748132	+	0.579501i
$135$	7.00000	+	2.23607i	0.602464	+	0.192450i
$136$		0			0
$137$			17.8885i			1.52832i	0.645026	+	0.764161i	$0.276847\pi$
$137$			17.8885i			1.52832i	−0.645026	+	0.764161i	$0.723153\pi$
$138$	−10.6443	−	2.58819i	−0.906104	−	0.220321i
$139$	−15.8114			−1.34110			−0.670552	−	0.741862i	$-0.733943\pi$
$139$	−15.8114			−1.34110			−0.670552	+	0.741862i	$0.733943\pi$
$140$	−1.00656	+	7.41531i	−0.0850700	+	0.626708i
$141$	−17.3205	+	7.74597i	−1.45865	+	0.652328i
$142$	−10.0000	−	7.74597i	−0.839181	−	0.650027i
$143$	−10.9545			−0.916057
$144$	−2.66987	+	11.6992i	−0.222489	+	0.974935i
$145$		−	9.79796i		−	0.813676i
$146$	16.4317	+	12.7279i	1.35990	+	1.05337i
$147$	11.3698	+	4.21043i	0.937765	+	0.347271i
$148$		0			0
$149$	−6.92820			−0.567581			−0.283790	−	0.958886i	$-0.591592\pi$
$149$	−6.92820			−0.567581			−0.283790	+	0.958886i	$0.591592\pi$
$150$	−1.73621	+	7.14042i	−0.141761	+	0.583013i
$151$	8.00000			0.651031			0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000			0.651031			0.325515	+	0.945537i	$0.394462\pi$
$152$	−8.21584	−	3.53553i	−0.666392	−	0.286770i
$153$		0			0
$154$	6.48683	+	11.2215i	0.522724	+	0.904250i
$155$	6.92820			0.556487
$156$	6.83013	+	8.56442i	0.546848	+	0.685703i
$157$	−15.8114			−1.26189			−0.630943	−	0.775829i	$-0.717332\pi$
$157$	−15.8114			−1.26189			−0.630943	+	0.775829i	$0.717332\pi$
$158$	−8.66025	+	11.1803i	−0.688973	+	0.889460i
$159$		0			0
$160$	7.90569	+	1.22474i	0.625000	+	0.0968246i
$161$	10.9545	+	4.47214i	0.863332	+	0.352454i
$162$	−10.8660	−	6.62793i	−0.853716	−	0.520740i
$163$			23.2379i			1.82013i	0.414462	+	0.910066i	$0.363970\pi$
$163$			23.2379i			1.82013i	−0.414462	+	0.910066i	$0.636030\pi$
$164$	5.47723	+	21.2132i	0.427699	+	1.65647i
$165$	−3.46410	−	7.74597i	−0.269680	−	0.603023i
$166$	7.90569	+	6.12372i	0.613601	+	0.475293i
$167$	21.9089			1.69536			0.847681	−	0.530506i	$-0.177998\pi$
$167$	21.9089			1.69536			0.847681	+	0.530506i	$0.177998\pi$
$168$	4.72862	−	12.0681i	0.364821	−	0.931078i
$169$	−3.00000			−0.230769
$170$		0			0
$171$	6.32456	−	7.07107i	0.483651	−	0.540738i
$172$	−15.0000	+	3.87298i	−1.14374	+	0.295312i
$173$			7.07107i			0.537603i	0.963196	+	0.268802i	$0.0866276\pi$
$173$			7.07107i			0.537603i	−0.963196	+	0.268802i	$0.913372\pi$
$174$	−4.00961	+	16.4901i	−0.303968	+	1.25011i
$175$	3.00000	−	7.34847i	0.226779	−	0.555492i
$176$	12.1244	−	6.70820i	0.913908	−	0.505650i
$177$	7.00000	+	15.6525i	0.526152	+	1.17651i
$178$		0			0
$179$	−10.3923			−0.776757			−0.388379	−	0.921500i	$-0.626965\pi$
$179$	−10.3923			−0.776757			−0.388379	+	0.921500i	$0.626965\pi$
$180$	−3.89609	+	7.53794i	−0.290397	+	0.561845i
$181$	3.16228			0.235050			0.117525	−	0.993070i	$-0.462504\pi$
$181$	3.16228			0.235050			0.117525	+	0.993070i	$0.462504\pi$
$182$	−5.92164	−	10.2437i	−0.438941	−	0.759316i
$183$	−5.00000	+	2.23607i	−0.369611	+	0.165295i
$184$	5.00000	−	11.6190i	0.368605	−	0.856560i
$185$		0			0
$186$	−11.6603	−	2.83522i	−0.854971	−	0.207888i
$187$		0			0
$188$	−5.47723	−	21.2132i	−0.399468	−	1.54713i
$189$	10.5432	+	8.82273i	0.766906	+	0.641759i
$190$	−5.00000	−	3.87298i	−0.362738	−	0.280976i
$191$		−	8.94427i		−	0.647185i	−0.946197	−	0.323592i	$-0.895109\pi$
$191$		−	8.94427i		−	0.647185i	0.946197	−	0.323592i	$-0.104891\pi$
$192$	−12.8042	−	5.29650i	−0.924062	−	0.382242i
$193$	−4.00000			−0.287926			−0.143963	−	0.989583i	$-0.545985\pi$
$193$	−4.00000			−0.287926			−0.143963	+	0.989583i	$0.545985\pi$
$194$	−5.47723	−	4.24264i	−0.393242	−	0.304604i
$195$	3.16228	+	7.07107i	0.226455	+	0.506370i
$196$	−6.98683	+	12.1319i	−0.499059	+	0.866568i
$197$		0			0			−	1.00000i	$-0.5\pi$
$197$		0			0				1.00000i	$0.5\pi$
$198$	2.66025	+	14.4542i	0.189056	+	1.02721i
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$	−7.79423	−	3.35410i	−0.551135	−	0.237171i
$201$	−5.47723	−	12.2474i	−0.386334	−	0.863868i
$202$	17.3925	+	13.4722i	1.22373	+	0.947900i
$203$	6.92820	−	16.9706i	0.486265	−	1.19110i
$204$		0			0
$205$			15.4919i			1.08200i
$206$	−10.9545	−	8.48528i	−0.763233	−	0.591198i
$207$	10.0000	+	8.94427i	0.695048	+	0.621670i
$208$	−11.0680	+	6.12372i	−0.767426	+	0.424604i
$209$	−10.9545			−0.757735
$210$	5.37083	−	7.42659i	0.370622	−	0.512483i
$211$			7.74597i			0.533254i	0.963800	+	0.266627i	$0.0859092\pi$
$211$			7.74597i			0.533254i	−0.963800	+	0.266627i	$0.914091\pi$
$212$		0			0
$213$	6.32456	+	14.1421i	0.433351	+	0.969003i
$214$	9.00000	−	11.6190i	0.615227	−	0.794255i
$215$	−10.9545			−0.747087
$216$	9.64191	−	11.0921i	0.656049	−	0.754719i
$217$	12.0000	+	4.89898i	0.814613	+	0.332564i
$218$		0			0
$219$	−10.3923	−	23.2379i	−0.702247	−	1.57027i
$220$	9.48683	−	2.44949i	0.639602	−	0.165145i
$221$		0			0
$222$		0			0
$223$			9.79796i			0.656120i	0.944657	+	0.328060i	$0.106395\pi$
$223$			9.79796i			0.656120i	−0.944657	+	0.328060i	$0.893605\pi$
$224$	12.8270	+	7.71149i	0.857043	+	0.515246i
$225$	6.00000	−	6.70820i	0.400000	−	0.447214i
$226$	5.00000	+	3.87298i	0.332595	+	0.257627i
$227$		−	9.89949i		−	0.657053i	−0.944495	−	0.328526i	$-0.893448\pi$
$227$		−	9.89949i		−	0.657053i	0.944495	−	0.328526i	$-0.106552\pi$
$228$	6.83013	+	8.56442i	0.452336	+	0.567193i
$229$	−15.8114			−1.04485			−0.522423	−	0.852686i	$-0.674972\pi$
$229$	−15.8114			−1.04485			−0.522423	+	0.852686i	$0.674972\pi$
$230$	5.47723	−	7.07107i	0.361158	−	0.466252i
$231$	−0.522774	−	15.8659i	−0.0343960	−	1.04390i
$232$	−18.0000	−	7.74597i	−1.18176	−	0.508548i
$233$		−	8.94427i		−	0.585959i	−0.956119	−	0.292979i	$-0.905353\pi$
$233$		−	8.94427i		−	0.585959i	0.956119	−	0.292979i	$-0.0946467\pi$
$234$	−2.42847	−	13.1948i	−0.158754	−	0.862571i
$235$		−	15.4919i		−	1.01058i
$236$	−19.1703	+	4.94975i	−1.24788	+	0.322201i
$237$	15.8114	−	7.07107i	1.02706	−	0.459315i
$238$		0			0
$239$			4.47214i			0.289278i	0.989484	+	0.144639i	$0.0462022\pi$
$239$			4.47214i			0.289278i	−0.989484	+	0.144639i	$0.953798\pi$
$240$	−7.83013	−	5.88975i	−0.505433	−	0.380181i
$241$		−	4.89898i		−	0.315571i	−0.987473	−	0.157786i	$-0.949565\pi$
$241$		−	4.89898i		−	0.315571i	0.987473	−	0.157786i	$-0.0504355\pi$
$242$	0.866025	−	1.11803i	0.0556702	−	0.0718699i
$243$	7.90569	+	13.4350i	0.507151	+	0.861858i
$244$	−1.58114	−	6.12372i	−0.101222	−	0.392031i
$245$	−6.92820	+	7.07107i	−0.442627	+	0.451754i
$246$	6.33975	−	26.0731i	0.404207	−	1.66236i
$247$	10.0000			0.636285
$248$	5.47723	−	12.7279i	0.347804	−	0.808224i
$249$	−5.00000	−	11.1803i	−0.316862	−	0.708525i
$250$	−12.6491	−	9.79796i	−0.800000	−	0.619677i
$251$			15.5563i			0.981908i	0.871185	+	0.490954i	$0.163352\pi$
$251$			15.5563i			0.981908i	−0.871185	+	0.490954i	$0.836648\pi$
$252$	−12.0783	+	10.3011i	−0.760864	+	0.648911i
$253$		−	15.4919i		−	0.973970i
$254$	6.92820	−	8.94427i	0.434714	−	0.561214i
$255$		0			0
$256$	8.50000	−	13.5554i	0.531250	−	0.847215i
$257$	−10.9545			−0.683320			−0.341660	−	0.939824i	$-0.610989\pi$
$257$	−10.9545			−0.683320			−0.341660	+	0.939824i	$0.610989\pi$
$258$	18.4365	+	4.48288i	1.14781	+	0.279092i
$259$		0			0
$260$	−8.66025	+	2.23607i	−0.537086	+	0.138675i
$261$	13.8564	−	15.4919i	0.857690	−	0.958927i
$262$	−1.58114	−	1.22474i	−0.0976831	−	0.0756650i
$263$		−	8.94427i		−	0.551527i	−0.961225	−	0.275764i	$-0.911069\pi$
$263$		−	8.94427i		−	0.551527i	0.961225	−	0.275764i	$-0.0889307\pi$
$264$	−16.9689	+	0.240237i	−1.04436	+	0.0147855i
$265$		0			0
$266$	−5.92164	−	10.2437i	−0.363079	−	0.628084i
$267$		0			0
$268$	15.0000	−	3.87298i	0.916271	−	0.236580i
$269$		−	9.89949i		−	0.603583i	−0.953374	−	0.301791i	$-0.902415\pi$
$269$		−	9.89949i		−	0.603583i	0.953374	−	0.301791i	$-0.0975846\pi$
$270$	8.56218	−	5.88975i	0.521078	−	0.358439i
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$		0			0
$273$	0.477226	+	14.4835i	0.0288830	+	0.876582i
$274$	20.0000	+	15.4919i	1.20824	+	0.935902i
$275$	−10.3923			−0.626680
$276$	−12.1119	+	9.65926i	−0.729052	+	0.581419i
$277$			15.4919i			0.930820i	0.885095	+	0.465410i	$0.154093\pi$
$277$			15.4919i			0.930820i	−0.885095	+	0.465410i	$0.845907\pi$
$278$	−13.6931	+	17.6777i	−0.821255	+	1.06024i
$279$	10.9545	+	9.79796i	0.655826	+	0.586588i
$280$	7.41886	+	7.54722i	0.443362	+	0.451033i
$281$		−	8.94427i		−	0.533571i	−0.963756	−	0.266785i	$-0.914039\pi$
$281$		−	8.94427i		−	0.533571i	0.963756	−	0.266785i	$-0.0859614\pi$
$282$	−6.33975	+	26.0731i	−0.377526	+	1.55263i
$283$	−15.8114			−0.939889			−0.469945	−	0.882696i	$-0.655726\pi$
$283$	−15.8114			−0.939889			−0.469945	+	0.882696i	$0.655726\pi$
$284$	−17.3205	+	4.47214i	−1.02778	+	0.265372i
$285$	3.16228	+	7.07107i	0.187317	+	0.418854i
$286$	−9.48683	+	12.2474i	−0.560968	+	0.724207i
$287$	−10.9545	+	26.8328i	−0.646621	+	1.58389i
$288$	10.7679	+	13.1168i	0.634507	+	0.772917i
$289$	−17.0000			−1.00000
$290$	−10.9545	−	8.48528i	−0.643268	−	0.498273i
$291$	3.46410	+	7.74597i	0.203069	+	0.454077i
$292$	28.4605	−	7.34847i	1.66552	−	0.430037i
$293$		−	26.8701i		−	1.56977i	−0.619644	−	0.784883i	$-0.712723\pi$
$293$		−	26.8701i		−	1.56977i	0.619644	−	0.784883i	$-0.287277\pi$
$294$	14.5539	−	9.06548i	0.848803	−	0.528709i
$295$	−14.0000			−0.815112
$296$		0			0
$297$	5.47723	−	17.1464i	0.317821	−	0.994937i
$298$	−6.00000	+	7.74597i	−0.347571	+	0.448712i
$299$			14.1421i			0.817861i
$300$	6.47963	+	8.12493i	0.374101	+	0.469093i
$301$	−18.9737	−	7.74597i	−1.09362	−	0.446470i
$302$	6.92820	−	8.94427i	0.398673	−	0.514685i
$303$	−11.0000	−	24.5967i	−0.631933	−	1.41305i
$304$	−11.0680	+	6.12372i	−0.634792	+	0.351220i
$305$		−	4.47214i		−	0.256074i
$306$		0			0
$307$	3.16228			0.180481			0.0902404	−	0.995920i	$-0.471236\pi$
$307$	3.16228			0.180481			0.0902404	+	0.995920i	$0.471236\pi$
$308$	18.1637	+	2.46556i	1.03497	+	0.140489i
$309$	6.92820	+	15.4919i	0.394132	+	0.881305i
$310$	6.00000	−	7.74597i	0.340777	−	0.439941i
$311$	−10.9545			−0.621170			−0.310585	−	0.950546i	$-0.600525\pi$
$311$	−10.9545			−0.621170			−0.310585	+	0.950546i	$0.600525\pi$
$312$	15.4904	−	0.219305i	0.876970	−	0.0124157i
$313$			9.79796i			0.553813i	0.960897	+	0.276907i	$0.0893093\pi$
$313$			9.79796i			0.553813i	−0.960897	+	0.276907i	$0.910691\pi$
$314$	−13.6931	+	17.6777i	−0.772744	+	0.997609i
$315$	−10.0905	+	4.91754i	−0.568534	+	0.277072i
$316$	5.00000	+	19.3649i	0.281272	+	1.08936i
$317$	6.92820			0.389127			0.194563	−	0.980890i	$-0.437671\pi$
$317$	6.92820			0.389127			0.194563	+	0.980890i	$0.437671\pi$
$318$		0			0
$319$	−24.0000			−1.34374
$320$	8.21584	−	7.77817i	0.459279	−	0.434813i
$321$	−16.4317	+	7.34847i	−0.917127	+	0.410152i
$322$	14.4868	−	8.37447i	0.807320	−	0.466691i
$323$		0			0
$324$	−16.8205	+	6.40863i	−0.934473	+	0.356035i
$325$	9.48683			0.526235
$326$	25.9808	+	20.1246i	1.43894	+	1.11460i
$327$		0			0
$328$	28.4605	+	12.2474i	1.57147	+	0.676252i
$329$	10.9545	−	26.8328i	0.603938	−	1.47934i
$330$	−11.6603	−	2.83522i	−0.641876	−	0.156074i
$331$		−	7.74597i		−	0.425757i	−0.977079	−	0.212878i	$-0.931716\pi$
$331$		−	7.74597i		−	0.425757i	0.977079	−	0.212878i	$-0.0682838\pi$
$332$	13.6931	−	3.53553i	0.751505	−	0.194038i
$333$		0			0
$334$	18.9737	−	24.4949i	1.03819	−	1.34030i
$335$	10.9545			0.598506
$336$	−9.39750	−	15.7381i	−0.512676	−	0.858582i
$337$	8.00000			0.435788			0.217894	−	0.975972i	$-0.430081\pi$
$337$	8.00000			0.435788			0.217894	+	0.975972i	$0.430081\pi$
$338$	−2.59808	+	3.35410i	−0.141317	+	0.182439i
$339$	−3.16228	−	7.07107i	−0.171751	−	0.384048i
$340$		0			0
$341$		−	16.9706i		−	0.919007i
$342$	−2.42847	−	13.1948i	−0.131317	−	0.713493i
$343$	−17.0000	+	7.34847i	−0.917914	+	0.396780i
$344$	−8.66025	+	20.1246i	−0.466930	+	1.08505i
$345$	−10.0000	+	4.47214i	−0.538382	+	0.240772i
$346$	7.90569	+	6.12372i	0.425013	+	0.329213i
$347$	−17.3205			−0.929814			−0.464907	−	0.885360i	$-0.653912\pi$
$347$	−17.3205			−0.929814			−0.464907	+	0.885360i	$0.653912\pi$
$348$	14.9641	+	18.7637i	0.802158	+	1.00584i
$349$	22.1359			1.18491			0.592455	−	0.805604i	$-0.298159\pi$
$349$	22.1359			1.18491			0.592455	+	0.805604i	$0.298159\pi$
$350$	−5.61776	−	9.71806i	−0.300282	−	0.519452i
$351$	−5.00000	+	15.6525i	−0.266880	+	0.835467i
$352$	3.00000	−	19.3649i	0.159901	−	1.03215i
$353$	10.9545			0.583047			0.291523	−	0.956564i	$-0.405838\pi$
$353$	10.9545			0.583047			0.291523	+	0.956564i	$0.405838\pi$
$354$	23.5622	+	5.72920i	1.25232	+	0.304504i
$355$	−12.6491			−0.671345
$356$		0			0
$357$		0			0
$358$	−9.00000	+	11.6190i	−0.475665	+	0.614081i
$359$			31.3050i			1.65221i	0.563515	+	0.826106i	$0.309449\pi$
$359$			31.3050i			1.65221i	−0.563515	+	0.826106i	$0.690551\pi$
$360$	5.05356	+	10.8840i	0.266346	+	0.573637i
$361$	−9.00000			−0.473684
$362$	2.73861	−	3.53553i	0.143938	−	0.185824i
$363$	−1.58114	+	0.707107i	−0.0829883	+	0.0371135i
$364$	−16.5811	−	2.25074i	−0.869088	−	0.117971i
$365$	20.7846			1.08792
$366$	−1.83013	+	7.52666i	−0.0956623	+	0.393425i
$367$		−	19.5959i		−	1.02290i	−0.859313	−	0.511449i	$-0.829109\pi$
$367$		−	19.5959i		−	1.02290i	0.859313	−	0.511449i	$-0.170891\pi$
$368$	−8.66025	−	15.6525i	−0.451447	−	0.815942i
$369$	−21.9089	+	24.4949i	−1.14053	+	1.27515i
$370$		0			0
$371$		0			0
$372$	−13.2679	+	10.5812i	−0.687911	+	0.548609i
$373$			30.9839i			1.60428i	0.597133	+	0.802142i	$0.296306\pi$
$373$			30.9839i			1.60428i	−0.597133	+	0.802142i	$0.703694\pi$
$374$		0			0
$375$	8.00000	+	17.8885i	0.413118	+	0.923760i
$376$	−28.4605	−	12.2474i	−1.46774	−	0.631614i
$377$	21.9089			1.12837
$378$	18.9948	−	4.14697i	0.976987	−	0.213297i
$379$		−	23.2379i		−	1.19365i	−0.802371	−	0.596825i	$-0.796429\pi$
$379$		−	23.2379i		−	1.19365i	0.802371	−	0.596825i	$-0.203571\pi$
$380$	−8.66025	+	2.23607i	−0.444262	+	0.114708i
$381$	−12.6491	+	5.65685i	−0.648034	+	0.289809i
$382$	−10.0000	−	7.74597i	−0.511645	−	0.396318i
$383$	−10.9545			−0.559746			−0.279873	−	0.960037i	$-0.590292\pi$
$383$	−10.9545			−0.559746			−0.279873	+	0.960037i	$0.590292\pi$
$384$	−17.0104	+	9.72861i	−0.868059	+	0.496461i
$385$	12.0000	+	4.89898i	0.611577	+	0.249675i
$386$	−3.46410	+	4.47214i	−0.176318	+	0.227626i
$387$	−17.3205	−	15.4919i	−0.880451	−	0.787499i
$388$	−9.48683	+	2.44949i	−0.481621	+	0.124354i
$389$	27.7128			1.40510			0.702548	−	0.711637i	$-0.252046\pi$
$389$	27.7128			1.40510			0.702548	+	0.711637i	$0.252046\pi$
$390$	10.6443	+	2.58819i	0.538995	+	0.131058i
$391$		0			0
$392$	7.51316	+	18.3181i	0.379472	+	0.925203i
$393$	1.00000	+	2.23607i	0.0504433	+	0.112795i
$394$		0			0
$395$			14.1421i			0.711568i
$396$	18.4641	+	9.54342i	0.927856	+	0.479575i
$397$	22.1359			1.11097			0.555486	−	0.831526i	$-0.312532\pi$
$397$	22.1359			1.11097			0.555486	+	0.831526i	$0.312532\pi$
$398$		0			0
$399$	0.477226	+	14.4835i	0.0238912	+	0.725083i
$400$	−10.5000	+	5.80948i	−0.525000	+	0.290474i
$401$			4.47214i			0.223328i	0.993746	+	0.111664i	$0.0356180\pi$
$401$			4.47214i			0.223328i	−0.993746	+	0.111664i	$0.964382\pi$
$402$	−18.4365	−	4.48288i	−0.919528	−	0.223586i
$403$			15.4919i			0.771708i
$404$	30.1247	−	7.77817i	1.49876	−	0.386979i
$405$	−12.6491	+	1.41421i	−0.628539	+	0.0702728i
$406$	−12.9737	−	22.4429i	−0.643872	−	1.11382i
$407$		0			0
$408$		0			0
$409$		−	9.79796i		−	0.484478i	−0.970217	−	0.242239i	$-0.922118\pi$
$409$		−	9.79796i		−	0.484478i	0.970217	−	0.242239i	$-0.0778818\pi$
$410$	17.3205	+	13.4164i	0.855399	+	0.662589i
$411$	−12.6491	−	28.2843i	−0.623935	−	1.39516i
$412$	−18.9737	+	4.89898i	−0.934765	+	0.241355i
$413$	−24.2487	−	9.89949i	−1.19320	−	0.487122i
$414$	18.6603	−	3.43437i	0.917101	−	0.168790i
$415$	10.0000			0.490881
$416$	−2.73861	+	17.6777i	−0.134272	+	0.866719i
$417$	25.0000	−	11.1803i	1.22426	−	0.547504i
$418$	−9.48683	+	12.2474i	−0.464016	+	0.599042i
$419$			24.0416i			1.17451i	0.809402	+	0.587255i	$0.199792\pi$
$419$			24.0416i			1.17451i	−0.809402	+	0.587255i	$0.800208\pi$
$420$	−3.65190	−	12.4364i	−0.178195	−	0.606833i
$421$		−	30.9839i		−	1.51006i	−0.655690	−	0.755031i	$-0.727622\pi$
$421$		−	30.9839i		−	1.51006i	0.655690	−	0.755031i	$-0.272378\pi$
$422$	8.66025	+	6.70820i	0.421575	+	0.326550i
$423$	21.9089	−	24.4949i	1.06525	−	1.19098i
$424$		0			0
$425$		0			0
$426$	21.2886	+	5.17638i	1.03144	+	0.250796i
$427$	3.16228	−	7.74597i	0.153033	−	0.374854i
$428$	−5.19615	−	20.1246i	−0.251166	−	0.972760i
$429$	17.3205	−	7.74597i	0.836242	−	0.373979i
$430$	−9.48683	+	12.2474i	−0.457496	+	0.590624i
$431$		−	22.3607i		−	1.07708i	−0.842601	−	0.538538i	$-0.818977\pi$
$431$		−	22.3607i		−	1.07708i	0.842601	−	0.538538i	$-0.181023\pi$
$432$	−4.05116	−	20.3860i	−0.194911	−	0.980821i
$433$		−	14.6969i		−	0.706290i	−0.935569	−	0.353145i	$-0.885112\pi$
$433$		−	14.6969i		−	0.706290i	0.935569	−	0.353145i	$-0.114888\pi$
$434$	15.8695	−	9.17377i	0.761762	−	0.440355i
$435$	6.92820	+	15.4919i	0.332182	+	0.742781i
$436$		0			0
$437$			14.1421i			0.676510i
$438$	−34.9808	−	8.50566i	−1.67145	−	0.406416i
$439$			24.4949i			1.16908i	0.811366	+	0.584539i	$0.198725\pi$
$439$			24.4949i			1.16908i	−0.811366	+	0.584539i	$0.801275\pi$
$440$	5.47723	−	12.7279i	0.261116	−	0.606780i
$441$	−20.9545	+	1.38238i	−0.997831	+	0.0658277i
$442$		0			0
$443$	17.3205			0.822922			0.411461	−	0.911427i	$-0.365019\pi$
$443$	17.3205			0.822922			0.411461	+	0.911427i	$0.365019\pi$
$444$		0			0
$445$		0			0
$446$	10.9545	+	8.48528i	0.518708	+	0.401790i
$447$	10.9545	−	4.89898i	0.518128	−	0.231714i
$448$	19.7302	−	7.66272i	0.932167	−	0.362029i
$449$		−	8.94427i		−	0.422106i	−0.977475	−	0.211053i	$-0.932311\pi$
$449$		−	8.94427i		−	0.422106i	0.977475	−	0.211053i	$-0.0676893\pi$
$450$	−2.30385	−	12.5177i	−0.108604	−	0.590089i
$451$	37.9473			1.78687
$452$	8.66025	−	2.23607i	0.407344	−	0.105176i
$453$	−12.6491	+	5.65685i	−0.594307	+	0.265782i
$454$	−11.0680	−	8.57321i	−0.519446	−	0.402361i
$455$	−10.9545	−	4.47214i	−0.513553	−	0.209657i
$456$	15.4904	−	0.219305i	0.725404	−	0.0102699i
$457$	8.00000			0.374224			0.187112	−	0.982339i	$-0.440087\pi$
$457$	8.00000			0.374224			0.187112	+	0.982339i	$0.440087\pi$
$458$	−13.6931	+	17.6777i	−0.639835	+	0.826023i
$459$		0			0
$460$	−3.16228	−	12.2474i	−0.147442	−	0.571040i
$461$			15.5563i			0.724531i	0.932075	+	0.362266i	$0.117997\pi$
$461$			15.5563i			0.724531i	−0.932075	+	0.362266i	$0.882003\pi$
$462$	−18.1913	−	13.1558i	−0.846338	−	0.612063i
$463$	14.0000			0.650635			0.325318	−	0.945605i	$-0.394529\pi$
$463$	14.0000			0.650635			0.325318	+	0.945605i	$0.394529\pi$
$464$	−24.2487	+	13.4164i	−1.12572	+	0.622841i
$465$	−10.9545	+	4.89898i	−0.508001	+	0.227185i
$466$	−10.0000	−	7.74597i	−0.463241	−	0.358825i
$467$			7.07107i			0.327210i	0.986526	+	0.163605i	$0.0523123\pi$
$467$			7.07107i			0.327210i	−0.986526	+	0.163605i	$0.947688\pi$
$468$	−16.8553	−	8.71191i	−0.779138	−	0.402708i
$469$	18.9737	+	7.74597i	0.876122	+	0.357676i
$470$	−17.3205	−	13.4164i	−0.798935	−	0.618853i
$471$	25.0000	−	11.1803i	1.15194	−	0.515163i
$472$	−11.0680	+	25.7196i	−0.509445	+	1.18384i
$473$			26.8328i			1.23377i
$474$	5.78737	−	23.8014i	0.265823	−	1.09323i
$475$	9.48683			0.435286
$476$		0			0
$477$		0			0
$478$	5.00000	+	3.87298i	0.228695	+	0.177146i
$479$	10.9545			0.500522			0.250261	−	0.968178i	$-0.419484\pi$
$479$	10.9545			0.500522			0.250261	+	0.968178i	$0.419484\pi$
$480$	−13.3660	+	3.65368i	−0.610073	+	0.166767i
$481$		0			0
$482$	−5.47723	−	4.24264i	−0.249481	−	0.193247i
$483$	−20.4828	+	0.674899i	−0.931999	+	0.0307090i
$484$	−0.500000	−	1.93649i	−0.0227273	−	0.0880223i
$485$	−6.92820			−0.314594
$486$	21.8674	+	2.79624i	0.991923	+	0.126840i
$487$	8.00000			0.362515			0.181257	−	0.983436i	$-0.441983\pi$
$487$	8.00000			0.362515			0.181257	+	0.983436i	$0.441983\pi$
$488$	−8.21584	−	3.53553i	−0.371914	−	0.160046i
$489$	−16.4317	−	36.7423i	−0.743066	−	1.66155i
$490$	1.90569	+	13.8697i	0.0860905	+	0.626569i
$491$	3.46410			0.156333			0.0781664	−	0.996940i	$-0.475093\pi$
$491$	3.46410			0.156333			0.0781664	+	0.996940i	$0.475093\pi$
$492$	−23.6603	−	29.6680i	−1.06669	−	1.33754i
$493$		0			0
$494$	8.66025	−	11.1803i	0.389643	−	0.503027i
$495$	10.9545	+	9.79796i	0.492366	+	0.440386i
$496$	−9.48683	−	17.1464i	−0.425971	−	0.769897i
$497$	−21.9089	−	8.94427i	−0.982749	−	0.401205i
$498$	−16.8301	−	4.09229i	−0.754176	−	0.183380i
$499$			7.74597i			0.346757i	0.984855	+	0.173379i	$0.0554684\pi$
$499$			7.74597i			0.346757i	−0.984855	+	0.173379i	$0.944532\pi$
$500$	−21.9089	+	5.65685i	−0.979796	+	0.252982i
$501$	−34.6410	+	15.4919i	−1.54765	+	0.692129i
$502$	17.3925	+	13.4722i	0.776266	+	0.601293i
$503$	−32.8634			−1.46530			−0.732652	−	0.680603i	$-0.761718\pi$
$503$	−32.8634			−1.46530			−0.732652	+	0.680603i	$0.761718\pi$
$504$	1.05687	+	22.4251i	0.0470768	+	0.998891i
$505$	22.0000			0.978987
$506$	−17.3205	−	13.4164i	−0.769991	−	0.596432i
$507$	4.74342	−	2.12132i	0.210663	−	0.0942111i
$508$	−4.00000	−	15.4919i	−0.177471	−	0.687343i
$509$		−	9.89949i		−	0.438787i	−0.975636	−	0.219394i	$-0.929592\pi$
$509$		−	9.89949i		−	0.438787i	0.975636	−	0.219394i	$-0.0704079\pi$
$510$		0			0
$511$	36.0000	+	14.6969i	1.59255	+	0.650154i
$512$	−7.79423	−	21.2426i	−0.344459	−	0.938801i
$513$	−5.00000	+	15.6525i	−0.220755	+	0.691074i
$514$	−9.48683	+	12.2474i	−0.418446	+	0.540212i
$515$	−13.8564			−0.610586
$516$	20.9785	−	16.7303i	0.923526	−	0.736512i
$517$	−37.9473			−1.66892
$518$		0			0
$519$	−5.00000	−	11.1803i	−0.219476	−	0.490762i
$520$	−5.00000	+	11.6190i	−0.219265	+	0.509525i
$521$	32.8634			1.43977			0.719885	−	0.694094i	$-0.244195\pi$
$521$	32.8634			1.43977			0.719885	+	0.694094i	$0.244195\pi$
$522$	−5.32051	−	28.9083i	−0.232872	−	1.26528i
$523$	3.16228			0.138277			0.0691384	−	0.997607i	$-0.477975\pi$
$523$	3.16228			0.138277			0.0691384	+	0.997607i	$0.477975\pi$
$524$	−2.73861	+	0.707107i	−0.119637	+	0.0308901i
$525$	0.452736	+	13.7403i	0.0197590	+	0.599675i
$526$	−10.0000	−	7.74597i	−0.436021	−	0.337740i
$527$		0			0
$528$	−14.4269	+	19.1798i	−0.627849	+	0.834694i
$529$	3.00000			0.130435
$530$		0			0
$531$	−22.1359	−	19.7990i	−0.960618	−	0.859203i
$532$	−16.5811	−	2.25074i	−0.718884	−	0.0975820i
$533$	−34.6410			−1.50047
$534$		0			0
$535$		−	14.6969i		−	0.635404i
$536$	8.66025	−	20.1246i	0.374066	−	0.869251i
$537$	16.4317	−	7.34847i	0.709079	−	0.317110i
$538$	−11.0680	−	8.57321i	−0.477174	−	0.369618i
$539$	17.3205	+	16.9706i	0.746047	+	0.730974i
$540$	0.830127	−	14.6735i	0.0357230	−	0.631446i
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$		0			0
$543$	−5.00000	+	2.23607i	−0.214571	+	0.0959589i
$544$		0			0
$545$		0			0
$546$	16.6064	+	12.0095i	0.710686	+	0.513961i
$547$		−	7.74597i		−	0.331194i	−0.986194	−	0.165597i	$-0.947045\pi$
$547$		−	7.74597i		−	0.331194i	0.986194	−	0.165597i	$-0.0529550\pi$
$548$	34.6410	−	8.94427i	1.47979	−	0.382080i
$549$	6.32456	−	7.07107i	0.269925	−	0.301786i
$550$	−9.00000	+	11.6190i	−0.383761	+	0.495434i
$551$	21.9089			0.933351
$552$	0.310144	+	21.9067i	0.0132006	+	0.932411i
$553$	−10.0000	+	24.4949i	−0.425243	+	1.04163i
$554$	17.3205	+	13.4164i	0.735878	+	0.570009i
$555$		0			0
$556$	7.90569	+	30.6186i	0.335276	+	1.29852i
$557$		0			0			−	1.00000i	$-0.5\pi$
$557$		0			0				1.00000i	$0.5\pi$
$558$	20.4413	−	3.76217i	0.865349	−	0.159265i
$559$		−	24.4949i		−	1.03602i
$560$	14.8630	−	1.75846i	0.628075	−	0.0743083i
$561$		0			0
$562$	−10.0000	−	7.74597i	−0.421825	−	0.326744i
$563$		−	1.41421i		−	0.0596020i	−0.999556	−	0.0298010i	$-0.990513\pi$
$563$		−	1.41421i		−	0.0596020i	0.999556	−	0.0298010i	$-0.00948736\pi$
$564$	23.6603	+	29.6680i	0.996276	+	1.24925i
$565$	6.32456			0.266076
$566$	−13.6931	+	17.6777i	−0.575562	+	0.743048i
$567$	−22.9089	−	6.49478i	−0.962083	−	0.272755i
$568$	−10.0000	+	23.2379i	−0.419591	+	0.975041i
$569$		−	22.3607i		−	0.937408i	−0.883355	−	0.468704i	$-0.844721\pi$
$569$		−	22.3607i		−	0.937408i	0.883355	−	0.468704i	$-0.155279\pi$
$570$	10.6443	+	2.58819i	0.445841	+	0.108407i
$571$			7.74597i			0.324159i	0.986778	+	0.162079i	$0.0518200\pi$
$571$			7.74597i			0.324159i	−0.986778	+	0.162079i	$0.948180\pi$
$572$	5.47723	+	21.2132i	0.229014	+	0.886969i
$573$	6.32456	+	14.1421i	0.264212	+	0.590796i
$574$	20.5132	+	35.4853i	0.856203	+	1.48113i
$575$			13.4164i			0.559503i
$576$	23.9904	−	0.679424i	0.999599	−	0.0283093i
$577$		−	29.3939i		−	1.22368i	−0.790980	−	0.611842i	$-0.790429\pi$
$577$		−	29.3939i		−	1.22368i	0.790980	−	0.611842i	$-0.209571\pi$
$578$	−14.7224	+	19.0066i	−0.612372	+	0.790569i
$579$	6.32456	−	2.82843i	0.262840	−	0.117545i
$580$	−18.9737	+	4.89898i	−0.787839	+	0.203419i
$581$	17.3205	+	7.07107i	0.718576	+	0.293357i
$582$	11.6603	+	2.83522i	0.483333	+	0.117524i
$583$		0			0
$584$	16.4317	−	38.1838i	0.679948	−	1.58006i
$585$	−10.0000	−	8.94427i	−0.413449	−	0.369800i
$586$	−30.0416	−	23.2702i	−1.24101	−	0.961281i
$587$		−	35.3553i		−	1.45927i	−0.683836	−	0.729636i	$-0.739690\pi$
$587$		−	35.3553i		−	1.45927i	0.683836	−	0.729636i	$-0.260310\pi$
$588$	2.46857	−	24.1227i	0.101802	−	0.994805i
$589$			15.4919i			0.638334i
$590$	−12.1244	+	15.6525i	−0.499152	+	0.644402i
$591$		0			0
$592$		0			0
$593$	32.8634			1.34954			0.674768	−	0.738030i	$-0.264244\pi$
$593$	32.8634			1.34954			0.674768	+	0.738030i	$0.264244\pi$
$594$	−14.4269	−	20.9730i	−0.591942	−	0.860531i
$595$		0			0
$596$	3.46410	+	13.4164i	0.141895	+	0.549557i
$597$		0			0
$598$	15.8114	+	12.2474i	0.646576	+	0.500835i
$599$		−	8.94427i		−	0.365453i	−0.983164	−	0.182727i	$-0.941508\pi$
$599$		−	8.94427i		−	0.365453i	0.983164	−	0.182727i	$-0.0584923\pi$
$600$	14.6955	−	0.208051i	0.599940	−	0.00849365i
$601$			24.4949i			0.999168i	0.866266	+	0.499584i	$0.166514\pi$
$601$			24.4949i			0.999168i	−0.866266	+	0.499584i	$0.833486\pi$
$602$	−25.0919	+	14.5050i	−1.02267	+	0.591180i
$603$	17.3205	+	15.4919i	0.705346	+	0.630880i
$604$	−4.00000	−	15.4919i	−0.162758	−	0.630358i
$605$		−	1.41421i		−	0.0574960i
$606$	−37.0263	−	9.00303i	−1.50409	−	0.365723i
$607$			19.5959i			0.795374i	0.917521	+	0.397687i	$0.130187\pi$
$607$			19.5959i			0.795374i	−0.917521	+	0.397687i	$0.869813\pi$
$608$	−2.73861	+	17.6777i	−0.111065	+	0.716924i
$609$	1.04555	+	31.7318i	0.0423678	+	1.28584i
$610$	−5.00000	−	3.87298i	−0.202444	−	0.156813i
$611$	34.6410			1.40143
$612$		0			0
$613$		−	46.4758i		−	1.87714i	−0.345089	−	0.938570i	$-0.612151\pi$
$613$		−	46.4758i		−	1.87714i	0.345089	−	0.938570i	$-0.387849\pi$
$614$	2.73861	−	3.53553i	0.110521	−	0.142683i
$615$	−10.9545	−	24.4949i	−0.441726	−	0.987730i
$616$	18.4868	−	18.1724i	0.744856	−	0.732188i
$617$			31.3050i			1.26029i	0.776478	+	0.630145i	$0.217005\pi$
$617$			31.3050i			1.26029i	−0.776478	+	0.630145i	$0.782995\pi$
$618$	23.3205	+	5.67044i	0.938088	+	0.228099i
$619$	3.16228			0.127103			0.0635513	−	0.997979i	$-0.479757\pi$
$619$	3.16228			0.127103			0.0635513	+	0.997979i	$0.479757\pi$
$620$	−3.46410	−	13.4164i	−0.139122	−	0.538816i
$621$	−22.1359	−	7.07107i	−0.888285	−	0.283752i
$622$	−9.48683	+	12.2474i	−0.380387	+	0.491078i
$623$		0			0
$624$	13.1699	−	17.5087i	0.527217	−	0.700909i
$625$	−1.00000			−0.0400000
$626$	10.9545	+	8.48528i	0.437828	+	0.339140i
$627$	17.3205	−	7.74597i	0.691714	−	0.309344i
$628$	7.90569	+	30.6186i	0.315472	+	1.22182i
$629$		0			0
$630$	−3.24064	+	15.5402i	−0.129110	+	0.619137i
$631$	−10.0000			−0.398094			−0.199047	−	0.979990i	$-0.563785\pi$
$631$	−10.0000			−0.398094			−0.199047	+	0.979990i	$0.563785\pi$
$632$	25.9808	+	11.1803i	1.03346	+	0.444730i
$633$	−5.47723	−	12.2474i	−0.217700	−	0.486792i
$634$	6.00000	−	7.74597i	0.238290	−	0.307632i
$635$		−	11.3137i		−	0.448971i
$636$		0			0
$637$	−15.8114	−	15.4919i	−0.626470	−	0.613813i
$638$	−20.7846	+	26.8328i	−0.822871	+	1.06232i
$639$	−20.0000	−	17.8885i	−0.791188	−	0.707660i
$640$	−1.58114	−	15.9217i	−0.0625000	−	0.629360i
$641$			31.3050i			1.23647i	0.785993	+	0.618236i	$0.212152\pi$
$641$			31.3050i			1.23647i	−0.785993	+	0.618236i	$0.787848\pi$
$642$	−6.01441	+	24.7351i	−0.237370	+	0.976218i
$643$	3.16228			0.124708			0.0623540	−	0.998054i	$-0.480139\pi$
$643$	3.16228			0.124708			0.0623540	+	0.998054i	$0.480139\pi$
$644$	3.18303	−	23.4493i	0.125429	−	0.924031i
$645$	17.3205	−	7.74597i	0.681994	−	0.304997i
$646$		0			0
$647$		0			0			−	1.00000i	$-0.5\pi$
$647$		0			0				1.00000i	$0.5\pi$
$648$	−7.40192	+	24.3559i	−0.290775	+	0.956791i
$649$			34.2929i			1.34611i
$650$	8.21584	−	10.6066i	0.322252	−	0.416025i
$651$	−22.4378	+	0.739315i	−0.879405	+	0.0289760i
$652$	45.0000	−	11.6190i	1.76234	−	0.455033i
$653$	−48.4974			−1.89785			−0.948925	−	0.315501i	$-0.897828\pi$
$653$	−48.4974			−1.89785			−0.948925	+	0.315501i	$0.897828\pi$
$654$		0			0
$655$	−2.00000			−0.0781465
$656$	38.3406	−	21.2132i	1.49695	−	0.828236i
$657$	32.8634	+	29.3939i	1.28212	+	1.14676i
$658$	−20.5132	−	35.4853i	−0.799687	−	1.38336i
$659$	−24.2487			−0.944596			−0.472298	−	0.881439i	$-0.656575\pi$
$659$	−24.2487			−0.944596			−0.472298	+	0.881439i	$0.656575\pi$
$660$	−13.2679	+	10.5812i	−0.516454	+	0.411872i
$661$	41.1096			1.59898			0.799489	−	0.600680i	$-0.205104\pi$
$661$	41.1096			1.59898			0.799489	+	0.600680i	$0.205104\pi$
$662$	−8.66025	−	6.70820i	−0.336590	−	0.260722i
$663$		0			0
$664$	7.90569	−	18.3712i	0.306800	−	0.712940i
$665$	−10.9545	−	4.47214i	−0.424795	−	0.173422i
$666$		0			0
$667$			30.9839i			1.19970i
$668$	−10.9545	−	42.4264i	−0.423840	−	1.64153i
$669$	−6.92820	−	15.4919i	−0.267860	−	0.598953i
$670$	9.48683	−	12.2474i	0.366508	−	0.473160i
$671$	−10.9545			−0.422892
$672$	−25.7342	−	3.12285i	−0.992717	−	0.120467i
$673$	−46.0000			−1.77317			−0.886585	−	0.462566i	$-0.846929\pi$
$673$	−46.0000			−1.77317			−0.886585	+	0.462566i	$0.846929\pi$
$674$	6.92820	−	8.94427i	0.266864	−	0.344520i
$675$	−4.74342	+	14.8492i	−0.182574	+	0.571548i
$676$	1.50000	+	5.80948i	0.0576923	+	0.223441i
$677$			7.07107i			0.271763i	0.990725	+	0.135882i	$0.0433867\pi$
$677$			7.07107i			0.271763i	−0.990725	+	0.135882i	$0.956613\pi$
$678$	−10.6443	−	2.58819i	−0.408792	−	0.0993989i
$679$	−12.0000	−	4.89898i	−0.460518	−	0.188006i
$680$		0			0
$681$	7.00000	+	15.6525i	0.268241	+	0.599804i
$682$	−18.9737	−	14.6969i	−0.726539	−	0.562775i
$683$	31.1769			1.19295			0.596476	−	0.802631i	$-0.296567\pi$
$683$	31.1769			1.19295			0.596476	+	0.802631i	$0.296567\pi$
$684$	−16.8553	−	8.71191i	−0.644480	−	0.333108i
$685$	25.2982			0.966595
$686$	−6.50659	+	25.3705i	−0.248423	+	0.968652i
$687$	25.0000	−	11.1803i	0.953809	−	0.426557i
$688$	15.0000	+	27.1109i	0.571870	+	1.03359i
$689$		0			0
$690$	−3.66025	+	15.0533i	−0.139343	+	0.573070i
$691$	3.16228			0.120299			0.0601494	−	0.998189i	$-0.480842\pi$
$691$	3.16228			0.120299			0.0601494	+	0.998189i	$0.480842\pi$
$692$	13.6931	−	3.53553i	0.520532	−	0.134401i
$693$	12.0455	+	24.7165i	0.457569	+	0.938903i
$694$	−15.0000	+	19.3649i	−0.569392	+	0.735082i
$695$			22.3607i			0.848189i
$696$	33.9377	−	0.480473i	1.28641	−	0.0182123i
$697$		0			0
$698$	19.1703	−	24.7487i	0.725606	−	0.936754i
$699$	6.32456	+	14.1421i	0.239217	+	0.534905i
$700$	−15.7302	−	2.13524i	−0.594548	−	0.0807045i
$701$	−20.7846			−0.785024			−0.392512	−	0.919747i	$-0.628394\pi$
$701$	−20.7846			−0.785024			−0.392512	+	0.919747i	$0.628394\pi$
$702$	13.1699	+	19.1456i	0.497065	+	0.722605i
$703$		0			0
$704$	−19.0526	−	20.1246i	−0.718070	−	0.758475i
$705$	10.9545	+	24.4949i	0.412568	+	0.922531i
$706$	9.48683	−	12.2474i	0.357042	−	0.460939i
$707$	38.1051	+	15.5563i	1.43309	+	0.585057i
$708$	26.8109	−	21.3817i	1.00761	−	0.803573i
$709$		−	30.9839i		−	1.16362i	−0.813323	−	0.581812i	$-0.802344\pi$
$709$		−	30.9839i		−	1.16362i	0.813323	−	0.581812i	$-0.197656\pi$
$710$	−10.9545	+	14.1421i	−0.411113	+	0.530745i
$711$	−20.0000	+	22.3607i	−0.750059	+	0.838591i
$712$		0			0
$713$	−21.9089			−0.820495
$714$		0			0
$715$			15.4919i			0.579365i
$716$	5.19615	+	20.1246i	0.194189	+	0.752092i
$717$	−3.16228	−	7.07107i	−0.118097	−	0.264074i
$718$	35.0000	+	27.1109i	1.30619	+	1.01177i
$719$	−32.8634			−1.22560			−0.612798	−	0.790239i	$-0.709956\pi$
$719$	−32.8634			−1.22560			−0.612798	+	0.790239i	$0.709956\pi$
$720$	16.5452	+	3.77577i	0.616603	+	0.140715i
$721$	−24.0000	−	9.79796i	−0.893807	−	0.364895i
$722$	−7.79423	+	10.0623i	−0.290071	+	0.374480i
$723$	3.46410	+	7.74597i	0.128831	+	0.288076i
$724$	−1.58114	−	6.12372i	−0.0587626	−	0.227586i
$725$	20.7846			0.771921
$726$	−0.578737	+	2.38014i	−0.0214789	+	0.0883353i
$727$		−	19.5959i		−	0.726772i	−0.931639	−	0.363386i	$-0.881621\pi$
$727$		−	19.5959i		−	0.726772i	0.931639	−	0.363386i	$-0.118379\pi$
$728$	−16.8761	+	16.5891i	−0.625470	+	0.614832i
$729$	−22.0000	−	15.6525i	−0.814815	−	0.579721i
$730$	18.0000	−	23.2379i	0.666210	−	0.860073i
$731$		0			0
$732$	6.83013	+	8.56442i	0.252449	+	0.316550i
$733$	−15.8114			−0.584007			−0.292003	−	0.956417i	$-0.594322\pi$
$733$	−15.8114			−0.584007			−0.292003	+	0.956417i	$0.594322\pi$
$734$	−21.9089	−	16.9706i	−0.808672	−	0.626395i
$735$	5.95445	−	16.0793i	0.219633	−	0.593095i
$736$	−25.0000	−	3.87298i	−0.921512	−	0.142760i
$737$		−	26.8328i		−	0.988399i
$738$	8.41246	+	45.7081i	0.309667	+	1.68254i
$739$		−	23.2379i		−	0.854820i	−0.904058	−	0.427410i	$-0.859426\pi$
$739$		−	23.2379i		−	0.854820i	0.904058	−	0.427410i	$-0.140574\pi$
$740$		0			0
$741$	−15.8114	+	7.07107i	−0.580846	+	0.259762i
$742$		0			0
$743$		−	49.1935i		−	1.80473i	−0.430968	−	0.902367i	$-0.641828\pi$
$743$		−	49.1935i		−	1.80473i	0.430968	−	0.902367i	$-0.358172\pi$
$744$	0.339746	+	23.9976i	0.0124557	+	0.879795i
$745$			9.79796i			0.358969i
$746$	34.6410	+	26.8328i	1.26830	+	0.982419i
$747$	15.8114	+	14.1421i	0.578508	+	0.517434i
$748$		0			0
$749$	10.3923	−	25.4558i	0.379727	−	0.930136i
$750$	26.9282	+	6.54766i	0.983279	+	0.239087i
$751$	−40.0000			−1.45962			−0.729810	−	0.683650i	$-0.760392\pi$
$751$	−40.0000			−1.45962			−0.729810	+	0.683650i	$0.760392\pi$
$752$	−38.3406	+	21.2132i	−1.39814	+	0.773566i
$753$	−11.0000	−	24.5967i	−0.400862	−	0.896355i
$754$	18.9737	−	24.4949i	0.690980	−	0.892052i
$755$		−	11.3137i		−	0.411748i
$756$	11.8135	−	24.8282i	0.429654	−	0.902994i
$757$			46.4758i			1.68919i	0.535404	+	0.844596i	$0.320159\pi$
$757$			46.4758i			1.68919i	−0.535404	+	0.844596i	$0.679841\pi$
$758$	−25.9808	−	20.1246i	−0.943664	−	0.730959i
$759$	10.9545	+	24.4949i	0.397621	+	0.889108i
$760$	−5.00000	+	11.6190i	−0.181369	+	0.421464i
$761$	−10.9545			−0.397099			−0.198549	−	0.980091i	$-0.563623\pi$
$761$	−10.9545			−0.397099			−0.198549	+	0.980091i	$0.563623\pi$
$762$	−4.62990	+	19.0411i	−0.167723	+	0.689787i
$763$		0			0
$764$	−17.3205	+	4.47214i	−0.626634	+	0.161796i
$765$		0			0
$766$	−9.48683	+	12.2474i	−0.342773	+	0.442518i
$767$		−	31.3050i		−	1.13036i
$768$	−3.85454	+	27.4434i	−0.139089	+	0.990280i
$769$		−	24.4949i		−	0.883309i	−0.897185	−	0.441654i	$-0.854392\pi$
$769$		−	24.4949i		−	0.883309i	0.897185	−	0.441654i	$-0.145608\pi$
$770$	15.8695	−	9.17377i	0.571898	−	0.330600i
$771$	17.3205	−	7.74597i	0.623783	−	0.278964i
$772$	2.00000	+	7.74597i	0.0719816	+	0.278783i
$773$		−	35.3553i		−	1.27164i	−0.771836	−	0.635822i	$-0.780661\pi$
$773$		−	35.3553i		−	1.27164i	0.771836	−	0.635822i	$-0.219339\pi$
$774$	−32.3205	+	5.94851i	−1.16174	+	0.213815i
$775$			14.6969i			0.527930i
$776$	−5.47723	+	12.7279i	−0.196621	+	0.456906i
$777$		0			0
$778$	24.0000	−	30.9839i	0.860442	−	1.11083i
$779$	−34.6410			−1.24114
$780$	12.1119	−	9.65926i	0.433676	−	0.345857i
$781$			30.9839i			1.10869i
$782$		0			0
$783$	−10.9545	+	34.2929i	−0.391480	+	1.22553i
$784$	26.9868	+	7.46397i	0.963815	+	0.266570i
$785$			22.3607i			0.798087i
$786$	3.36603	+	0.818458i	0.120062	+	0.0291934i
$787$	−53.7587			−1.91629			−0.958146	−	0.286281i	$-0.907581\pi$
$787$	−53.7587			−1.91629			−0.958146	+	0.286281i	$0.907581\pi$
$788$		0			0
$789$	6.32456	+	14.1421i	0.225160	+	0.503473i
$790$	15.8114	+	12.2474i	0.562544	+	0.435745i
$791$	10.9545	+	4.47214i	0.389495	+	0.159011i
$792$	26.6603	−	12.3786i	0.947331	−	0.439856i
$793$	10.0000			0.355110
$794$	19.1703	−	24.7487i	0.680328	−	0.878300i
$795$		0			0
$796$		0			0
$797$			49.4975i			1.75329i	0.481137	+	0.876645i	$0.340224\pi$
$797$			49.4975i			1.75329i	−0.481137	+	0.876645i	$0.659776\pi$
$798$	16.6064	+	12.0095i	0.587859	+	0.425133i
$799$		0			0
$800$	−2.59808	+	16.7705i	−0.0918559	+	0.592927i
$801$		0			0
$802$	5.00000	+	3.87298i	0.176556	+	0.136760i
$803$		−	50.9117i		−	1.79663i
$804$	−20.9785	+	16.7303i	−0.739854	+	0.590033i
$805$	6.32456	−	15.4919i	0.222911	−	0.546019i
$806$	17.3205	+	13.4164i	0.610089	+	0.472573i
$807$	7.00000	+	15.6525i	0.246412	+	0.550993i
$808$	17.3925	−	40.4166i	0.611867	−	1.42185i
$809$		−	22.3607i		−	0.786160i	−0.919504	−	0.393080i	$-0.871410\pi$
$809$		−	22.3607i		−	0.786160i	0.919504	−	0.393080i	$-0.128590\pi$
$810$	−9.37331	+	15.3669i	−0.329345	+	0.539937i
$811$	22.1359			0.777298			0.388649	−	0.921386i	$-0.372942\pi$
$811$	22.1359			0.777298			0.388649	+	0.921386i	$0.372942\pi$
$812$	−36.3275	−	4.93113i	−1.27484	−	0.173049i
$813$		0			0
$814$		0			0
$815$	32.8634			1.15115
$816$		0			0
$817$		−	24.4949i		−	0.856968i
$818$	−10.9545	−	8.48528i	−0.383013	−	0.296681i
$819$	−10.9960	−	22.5630i	−0.384230	−	0.788415i
$820$	30.0000	−	7.74597i	1.04765	−	0.270501i
$821$	−13.8564			−0.483592			−0.241796	−	0.970327i	$-0.577736\pi$
$821$	−13.8564			−0.483592			−0.241796	+	0.970327i	$0.577736\pi$
$822$	−42.5772	−	10.3528i	−1.48505	−	0.361094i
$823$	26.0000			0.906303			0.453152	−	0.891434i	$-0.350300\pi$
$823$	26.0000			0.906303			0.453152	+	0.891434i	$0.350300\pi$
$824$	−10.9545	+	25.4558i	−0.381616	+	0.886796i
$825$	16.4317	−	7.34847i	0.572078	−	0.255841i
$826$	−32.0680	+	18.5377i	−1.11579	+	0.645008i
$827$	−51.9615			−1.80688			−0.903440	−	0.428715i	$-0.858966\pi$
$827$	−51.9615			−1.80688			−0.903440	+	0.428715i	$0.858966\pi$
$828$	12.3205	−	23.8371i	0.428167	−	0.828395i
$829$	−34.7851			−1.20813			−0.604067	−	0.796933i	$-0.706454\pi$
$829$	−34.7851			−1.20813			−0.604067	+	0.796933i	$0.706454\pi$
$830$	8.66025	−	11.1803i	0.300602	−	0.388075i
$831$	−10.9545	−	24.4949i	−0.380006	−	0.849719i
$832$	17.3925	+	18.3712i	0.602977	+	0.636906i
$833$		0			0
$834$	9.15064	−	37.6333i	0.316861	−	1.30313i
$835$		−	30.9839i		−	1.07224i
$836$	5.47723	+	21.2132i	0.189434	+	0.733674i
$837$	−24.2487	−	7.74597i	−0.838158	−	0.267740i
$838$	26.8794	+	20.8207i	0.928532	+	0.719238i
$839$	−21.9089			−0.756379			−0.378190	−	0.925728i	$-0.623453\pi$
$839$	−21.9089			−0.756379			−0.378190	+	0.925728i	$0.623453\pi$
$840$	−17.0669	−	6.68727i	−0.588865	−	0.230733i
$841$	19.0000			0.655172
$842$	−34.6410	−	26.8328i	−1.19381	−	0.924720i
$843$	6.32456	+	14.1421i	0.217829	+	0.487081i
$844$	15.0000	−	3.87298i	0.516321	−	0.133314i
$845$			4.24264i			0.145951i
$846$	−8.41246	−	45.7081i	−0.289226	−	1.57148i
$847$	1.00000	−	2.44949i	0.0343604	−	0.0841655i
$848$		0			0
$849$	25.0000	−	11.1803i	0.857998	−	0.383708i
$850$		0			0
$851$		0			0
$852$	24.2239	−	19.3185i	0.829895	−	0.661841i
$853$	−53.7587			−1.84066			−0.920332	−	0.391139i	$-0.872081\pi$
$853$	−53.7587			−1.84066			−0.920332	+	0.391139i	$0.872081\pi$
$854$	−5.92164	−	10.2437i	−0.202634	−	0.350533i
$855$	−10.0000	−	8.94427i	−0.341993	−	0.305888i
$856$	−27.0000	−	11.6190i	−0.922841	−	0.397128i
$857$	−54.7723			−1.87098			−0.935492	−	0.353347i	$-0.885043\pi$
$857$	−54.7723			−1.87098			−0.935492	+	0.353347i	$0.885043\pi$
$858$	6.33975	−	26.0731i	0.216435	−	0.890122i
$859$	3.16228			0.107896			0.0539478	−	0.998544i	$-0.482820\pi$
$859$	3.16228			0.107896			0.0539478	+	0.998544i	$0.482820\pi$
$860$	5.47723	+	21.2132i	0.186772	+	0.723364i
$861$	−1.65316	−	50.1724i	−0.0563395	−	1.70987i
$862$	−25.0000	−	19.3649i	−0.851503	−	0.659572i
$863$		−	8.94427i		−	0.304467i	−0.988345	−	0.152233i	$-0.951353\pi$
$863$		−	8.94427i		−	0.304467i	0.988345	−	0.152233i	$-0.0486465\pi$
$864$	−26.3006	−	13.1254i	−0.894765	−	0.446537i
$865$	10.0000			0.340010
$866$	−16.4317	−	12.7279i	−0.558371	−	0.432512i
$867$	26.8794	−	12.0208i	0.912871	−	0.408248i
$868$	3.48683	−	25.6874i	0.118351	−	0.871887i
$869$	34.6410			1.17512
$870$	23.3205	+	5.67044i	0.790639	+	0.192246i
$871$			24.4949i			0.829978i
$872$		0			0
$873$	−10.9545	−	9.79796i	−0.370752	−	0.331611i
$874$	15.8114	+	12.2474i	0.534828	+	0.414276i
$875$	−27.7128	−	11.3137i	−0.936864	−	0.382473i
$876$	−39.8038	+	31.7436i	−1.34485	+	1.07252i
$877$		−	15.4919i		−	0.523125i	−0.965186	−	0.261563i	$-0.915762\pi$
$877$		−	15.4919i		−	0.523125i	0.965186	−	0.261563i	$-0.0842378\pi$
$878$	27.3861	+	21.2132i	0.924237	+	0.715911i
$879$	19.0000	+	42.4853i	0.640854	+	1.43299i
$880$	−9.48683	−	17.1464i	−0.319801	−	0.578006i
$881$	32.8634			1.10719			0.553597	−	0.832785i	$-0.313255\pi$
$881$	32.8634			1.10719			0.553597	+	0.832785i	$0.313255\pi$
$882$	−16.6015	+	24.6250i	−0.559003	+	0.829166i
$883$		−	23.2379i		−	0.782018i	−0.920387	−	0.391009i	$-0.872126\pi$
$883$		−	23.2379i		−	0.782018i	0.920387	−	0.391009i	$-0.127874\pi$
$884$		0			0
$885$	22.1359	−	9.89949i	0.744092	−	0.332768i
$886$	15.0000	−	19.3649i	0.503935	−	0.650577i
$887$	21.9089			0.735629			0.367814	−	0.929899i	$-0.380106\pi$
$887$	21.9089			0.735629			0.367814	+	0.929899i	$0.380106\pi$
$888$		0			0
$889$	8.00000	−	19.5959i	0.268311	−	0.657226i
$890$		0			0
$891$	3.46410	+	30.9839i	0.116052	+	1.03800i
$892$	18.9737	−	4.89898i	0.635285	−	0.164030i
$893$	34.6410			1.15922
$894$	4.00961	−	16.4901i	0.134101	−	0.551511i
$895$			14.6969i			0.491264i
$896$	8.51972	−	28.6952i	0.284624	−	0.958639i
$897$	−10.0000	−	22.3607i	−0.333890	−	0.746601i
$898$	−10.0000	−	7.74597i	−0.333704	−	0.258486i
$899$			33.9411i			1.13200i
$900$	−15.9904	−	8.26485i	−0.533013	−	0.275495i
$901$		0			0
$902$	32.8634	−	42.4264i	1.09423	−	1.41264i
$903$	35.4772	−	1.16896i	1.18061	−	0.0389005i
$904$	5.00000	−	11.6190i	0.166298	−	0.386441i
$905$		−	4.47214i		−	0.148659i
$906$	−4.62990	+	19.0411i	−0.153818	+	0.632599i
$907$			38.7298i			1.28600i	0.765865	+	0.643002i	$0.222311\pi$
$907$			38.7298i			1.28600i	−0.765865	+	0.643002i	$0.777689\pi$
$908$	−19.1703	+	4.94975i	−0.636188	+	0.164263i
$909$	34.7851	+	31.1127i	1.15375	+	1.03194i
$910$	−14.4868	+	8.37447i	−0.480234	+	0.277611i
$911$			31.3050i			1.03718i	0.855023	+	0.518590i	$0.173543\pi$
$911$			31.3050i			1.03718i	−0.855023	+	0.518590i	$0.826457\pi$
$912$	13.1699	−	17.5087i	0.436098	−	0.579771i
$913$		−	24.4949i		−	0.810663i
$914$	6.92820	−	8.94427i	0.229165	−	0.295850i
$915$	3.16228	+	7.07107i	0.104542	+	0.233762i
$916$	7.90569	+	30.6186i	0.261211	+	1.01167i
$917$	−3.46410	−	1.41421i	−0.114395	−	0.0467014i
$918$		0			0
$919$	−10.0000			−0.329870			−0.164935	−	0.986304i	$-0.552741\pi$
$919$	−10.0000			−0.329870			−0.164935	+	0.986304i	$0.552741\pi$
$920$	−16.4317	−	7.07107i	−0.541736	−	0.233126i
$921$	−5.00000	+	2.23607i	−0.164756	+	0.0736809i
$922$	17.3925	+	13.4722i	0.572792	+	0.443683i
$923$		−	28.2843i		−	0.930988i
$924$	−30.4628	+	8.94530i	−1.00215	+	0.294279i
$925$		0			0
$926$	12.1244	−	15.6525i	0.398431	−	0.514372i
$927$	−21.9089	−	19.5959i	−0.719583	−	0.643614i
$928$	−6.00000	+	38.7298i	−0.196960	+	1.27137i
$929$	−21.9089			−0.718808			−0.359404	−	0.933182i	$-0.617020\pi$
$929$	−21.9089			−0.718808			−0.359404	+	0.933182i	$0.617020\pi$
$930$	−4.00961	+	16.4901i	−0.131480	+	0.540731i
$931$	−15.8114	−	15.4919i	−0.518197	−	0.507728i
$932$	−17.3205	+	4.47214i	−0.567352	+	0.146490i
$933$	17.3205	−	7.74597i	0.567048	−	0.253592i
$934$	7.90569	+	6.12372i	0.258682	+	0.200374i
$935$		0			0
$936$	−24.3374	+	11.3001i	−0.795492	+	0.369355i
$937$			44.0908i			1.44038i	0.693775	+	0.720192i	$0.255946\pi$
$937$			44.0908i			1.44038i	−0.693775	+	0.720192i	$0.744054\pi$
$938$	25.0919	−	14.5050i	0.819281	−	0.473605i
$939$	−6.92820	−	15.4919i	−0.226093	−	0.505560i
$940$	−30.0000	+	7.74597i	−0.978492	+	0.252646i
$941$		−	1.41421i		−	0.0461020i	−0.999734	−	0.0230510i	$-0.992662\pi$
$941$		−	1.41421i		−	0.0461020i	0.999734	−	0.0230510i	$-0.00733802\pi$
$942$	9.15064	−	37.6333i	0.298144	−	1.22616i
$943$		−	48.9898i		−	1.59533i
$944$	19.1703	+	34.6482i	0.623940	+	1.12770i
$945$	12.4772	−	14.9104i	0.405884	−	0.485034i
$946$	30.0000	+	23.2379i	0.975384	+	0.755529i
$947$	17.3205			0.562841			0.281420	−	0.959585i	$-0.409194\pi$
$947$	17.3205			0.562841			0.281420	+	0.959585i	$0.409194\pi$
$948$	−21.5988	−	27.0831i	−0.701495	−	0.879618i
$949$			46.4758i			1.50867i
$950$	8.21584	−	10.6066i	0.266557	−	0.344124i
$951$	−10.9545	+	4.89898i	−0.355222	+	0.158860i
$952$		0			0
$953$			17.8885i			0.579467i	0.957107	+	0.289733i	$0.0935666\pi$
$953$			17.8885i			0.579467i	−0.957107	+	0.289733i	$0.906433\pi$
$954$		0			0
$955$	−12.6491			−0.409316
$956$	8.66025	−	2.23607i	0.280093	−	0.0723196i
$957$	37.9473	−	16.9706i	1.22666	−	0.548580i
$958$	9.48683	−	12.2474i	0.306506	−	0.395697i
$959$	43.8178	+	17.8885i	1.41495	+	0.577651i
$960$	−7.49038	+	18.1078i	−0.241751	+	0.584428i
$961$	7.00000			0.225806
$962$		0			0
$963$	20.7846	−	23.2379i	0.669775	−	0.748831i
$964$	−9.48683	+	2.44949i	−0.305550	+	0.0788928i
$965$			5.65685i			0.182101i
$966$	−16.9841	+	23.4849i	−0.546453	+	0.755615i
$967$	32.0000			1.02905			0.514525	−	0.857475i	$-0.327968\pi$
$967$	32.0000			1.02905			0.514525	+	0.857475i	$0.327968\pi$
$968$	−2.59808	−	1.11803i	−0.0835053	−	0.0359350i
$969$		0			0
$970$	−6.00000	+	7.74597i	−0.192648	+	0.248708i
$971$			41.0122i			1.31614i	0.752955	+	0.658072i	$0.228628\pi$
$971$			41.0122i			1.31614i	−0.752955	+	0.658072i	$0.771372\pi$
$972$	22.0640	−	22.0268i	0.707702	−	0.706511i
$973$	−15.8114	+	38.7298i	−0.506890	+	1.24162i
$974$	6.92820	−	8.94427i	0.221994	−	0.286593i
$975$	−15.0000	+	6.70820i	−0.480384	+	0.214834i
$976$	−11.0680	+	6.12372i	−0.354277	+	0.196016i
$977$		−	8.94427i		−	0.286153i	−0.989712	−	0.143076i	$-0.954301\pi$
$977$		−	8.94427i		−	0.286153i	0.989712	−	0.143076i	$-0.0456994\pi$
$978$	−55.3094	−	13.4486i	−1.76860	−	0.430040i
$979$		0			0
$980$	17.1572	+	9.88087i	0.548066	+	0.315633i
$981$		0			0
$982$	3.00000	−	3.87298i	0.0957338	−	0.123592i
$983$	43.8178			1.39757			0.698785	−	0.715331i	$-0.253724\pi$
$983$	43.8178			1.39757			0.698785	+	0.715331i	$0.253724\pi$
$984$	−53.6603	+	0.759695i	−1.71063	+	0.0242182i
$985$		0			0
$986$		0			0
$987$	1.65316	+	50.1724i	0.0526206	+	1.59700i
$988$	−5.00000	−	19.3649i	−0.159071	−	0.616080i
$989$	34.6410			1.10152
$990$	20.4413	−	3.76217i	0.649667	−	0.119570i
$991$	2.00000			0.0635321			0.0317660	−	0.999495i	$-0.489887\pi$
$991$	2.00000			0.0635321			0.0317660	+	0.999495i	$0.489887\pi$
$992$	−27.3861	−	4.24264i	−0.869510	−	0.134704i
$993$	5.47723	+	12.2474i	0.173814	+	0.388661i
$994$	−28.9737	+	16.7489i	−0.918989	+	0.531244i
$995$		0			0
$996$	−19.1506	+	15.2726i	−0.606811	+	0.483932i
$997$	3.16228			0.100150			0.0500752	−	0.998745i	$-0.484054\pi$
$997$	3.16228			0.100150			0.0500752	+	0.998745i	$0.484054\pi$
$998$	8.66025	+	6.70820i	0.274136	+	0.212344i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	168.2.i.e.125.5	yes	8
3.2	odd	2	inner	168.2.i.e.125.3	yes	8
4.3	odd	2		672.2.i.d.209.6		8
7.6	odd	2	inner	168.2.i.e.125.6	yes	8
8.3	odd	2		672.2.i.d.209.4		8
8.5	even	2	inner	168.2.i.e.125.2	yes	8
12.11	even	2		672.2.i.d.209.8		8
21.20	even	2	inner	168.2.i.e.125.4	yes	8
24.5	odd	2	inner	168.2.i.e.125.8	yes	8
24.11	even	2		672.2.i.d.209.2		8
28.27	even	2		672.2.i.d.209.3		8
56.13	odd	2	inner	168.2.i.e.125.1	✓	8
56.27	even	2		672.2.i.d.209.5		8
84.83	odd	2		672.2.i.d.209.1		8
168.83	odd	2		672.2.i.d.209.7		8
168.125	even	2	inner	168.2.i.e.125.7	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.2.i.e.125.1	✓	8	56.13	odd	2	inner
168.2.i.e.125.2	yes	8	8.5	even	2	inner
168.2.i.e.125.3	yes	8	3.2	odd	2	inner
168.2.i.e.125.4	yes	8	21.20	even	2	inner
168.2.i.e.125.5	yes	8	1.1	even	1	trivial
168.2.i.e.125.6	yes	8	7.6	odd	2	inner
168.2.i.e.125.7	yes	8	168.125	even	2	inner
168.2.i.e.125.8	yes	8	24.5	odd	2	inner
672.2.i.d.209.1		8	84.83	odd	2
672.2.i.d.209.2		8	24.11	even	2
672.2.i.d.209.3		8	28.27	even	2
672.2.i.d.209.4		8	8.3	odd	2
672.2.i.d.209.5		8	56.27	even	2
672.2.i.d.209.6		8	4.3	odd	2
672.2.i.d.209.7		8	168.83	odd	2
672.2.i.d.209.8		8	12.11	even	2

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:	\( N \)	\(=\)	\( 168 = 2^{3} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	168.i (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(0.866025 - 1.11803i) q^{2} +(-1.58114 + 0.707107i) q^{3} +(-0.500000 - 1.93649i) q^{4} -1.41421i q^{5} +(-0.578737 + 2.38014i) q^{6} +(1.00000 - 2.44949i) q^{7} +(-2.59808 - 1.11803i) q^{8} +(2.00000 - 2.23607i) q^{9} +O(q^{10})\)	\(q+(0.866025 - 1.11803i) q^{2} +(-1.58114 + 0.707107i) q^{3} +(-0.500000 - 1.93649i) q^{4} -1.41421i q^{5} +(-0.578737 + 2.38014i) q^{6} +(1.00000 - 2.44949i) q^{7} +(-2.59808 - 1.11803i) q^{8} +(2.00000 - 2.23607i) q^{9} +(-1.58114 - 1.22474i) q^{10} -3.46410 q^{11} +(2.15988 + 2.70831i) q^{12} +3.16228 q^{13} +(-1.87259 - 3.23935i) q^{14} +(1.00000 + 2.23607i) q^{15} +(-3.50000 + 1.93649i) q^{16} +(-0.767949 - 4.17256i) q^{18} +3.16228 q^{19} +(-2.73861 + 0.707107i) q^{20} +(0.150912 + 4.58009i) q^{21} +(-3.00000 + 3.87298i) q^{22} +4.47214i q^{23} +(4.89849 - 0.0693504i) q^{24} +3.00000 q^{25} +(2.73861 - 3.53553i) q^{26} +(-1.58114 + 4.94975i) q^{27} +(-5.24342 - 0.711747i) q^{28} +6.92820 q^{29} +(3.36603 + 0.818458i) q^{30} +4.89898i q^{31} +(-0.866025 + 5.59017i) q^{32} +(5.47723 - 2.44949i) q^{33} +(-3.46410 - 1.41421i) q^{35} +(-5.33013 - 2.75495i) q^{36} +(2.73861 - 3.53553i) q^{38} +(-5.00000 + 2.23607i) q^{39} +(-1.58114 + 3.67423i) q^{40} -10.9545 q^{41} +(5.25139 + 3.79775i) q^{42} -7.74597i q^{43} +(1.73205 + 6.70820i) q^{44} +(-3.16228 - 2.82843i) q^{45} +(5.00000 + 3.87298i) q^{46} +10.9545 q^{47} +(4.16468 - 5.53674i) q^{48} +(-5.00000 - 4.89898i) q^{49} +(2.59808 - 3.35410i) q^{50} +(-1.58114 - 6.12372i) q^{52} +(4.16468 + 6.05437i) q^{54} +4.89898i q^{55} +(-5.33669 + 5.24593i) q^{56} +(-5.00000 + 2.23607i) q^{57} +(6.00000 - 7.74597i) q^{58} -9.89949i q^{59} +(3.83013 - 3.05453i) q^{60} +3.16228 q^{61} +(5.47723 + 4.24264i) q^{62} +(-3.47723 - 7.13505i) q^{63} +(5.50000 + 5.80948i) q^{64} -4.47214i q^{65} +(2.00480 - 8.24504i) q^{66} +7.74597i q^{67} +(-3.16228 - 7.07107i) q^{69} +(-4.58114 + 2.64824i) q^{70} -8.94427i q^{71} +(-7.69615 + 3.57341i) q^{72} +14.6969i q^{73} +(-4.74342 + 2.12132i) q^{75} +(-1.58114 - 6.12372i) q^{76} +(-3.46410 + 8.48528i) q^{77} +(-1.83013 + 7.52666i) q^{78} -10.0000 q^{79} +(2.73861 + 4.94975i) q^{80} +(-1.00000 - 8.94427i) q^{81} +(-9.48683 + 12.2474i) q^{82} +7.07107i q^{83} +(8.79385 - 2.58228i) q^{84} +(-8.66025 - 6.70820i) q^{86} +(-10.9545 + 4.89898i) q^{87} +(9.00000 + 3.87298i) q^{88} +(-5.90089 + 1.08604i) q^{90} +(3.16228 - 7.74597i) q^{91} +(8.66025 - 2.23607i) q^{92} +(-3.46410 - 7.74597i) q^{93} +(9.48683 - 12.2474i) q^{94} -4.47214i q^{95} +(-2.58354 - 9.45121i) q^{96} -4.89898i q^{97} +(-9.80735 + 1.34753i) q^{98} +(-6.92820 + 7.74597i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{4} + 8 q^{7} + 16 q^{9}+O(q^{10}) \) 8 * q - 4 * q^4 + 8 * q^7 + 16 * q^9	\( 8 q - 4 q^{4} + 8 q^{7} + 16 q^{9} + 8 q^{15} - 28 q^{16} - 20 q^{18} - 24 q^{22} + 24 q^{25} - 4 q^{28} + 20 q^{30} - 8 q^{36} - 40 q^{39} + 12 q^{42} + 40 q^{46} - 40 q^{49} - 40 q^{57} + 48 q^{58} - 4 q^{60} + 16 q^{63} + 44 q^{64} - 24 q^{70} - 20 q^{72} + 20 q^{78} - 80 q^{79} - 8 q^{81} + 60 q^{84} + 72 q^{88}+O(q^{100}) \) 8 * q - 4 * q^4 + 8 * q^7 + 16 * q^9 + 8 * q^15 - 28 * q^16 - 20 * q^18 - 24 * q^22 + 24 * q^25 - 4 * q^28 + 20 * q^30 - 8 * q^36 - 40 * q^39 + 12 * q^42 + 40 * q^46 - 40 * q^49 - 40 * q^57 + 48 * q^58 - 4 * q^60 + 16 * q^63 + 44 * q^64 - 24 * q^70 - 20 * q^72 + 20 * q^78 - 80 * q^79 - 8 * q^81 + 60 * q^84 + 72 * q^88

Introduction

L-functions

Modular forms

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