LMFDB - Embedded newform 448.2.ba.a.81.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [448,2,Mod(81,448)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(448, base_ring=CyclotomicField(12))

chi = DirichletCharacter(H, H._module([0, 9, 8]))

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

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

chi := DirichletCharacter("448.81");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 448 = 2^{6} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	448.ba (of order $12$, degree $4$, not 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:	$3.57729801055$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 112)
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			81.1
Root			$0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	448.81
Dual form			448.2.ba.a.177.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.500000 + 0.133975i) q^{3} +(-0.866025 - 0.232051i) q^{5} +(-1.73205 - 2.00000i) q^{7} +(-2.36603 + 1.36603i) q^{9} +O(q^{10})$	\(q+(-0.500000 + 0.133975i) q^{3} +(-0.866025 - 0.232051i) q^{5} +(-1.73205 - 2.00000i) q^{7} +(-2.36603 + 1.36603i) q^{9} +(0.767949 + 2.86603i) q^{11} +(3.73205 + 3.73205i) q^{13} +0.464102 q^{15} +(-3.23205 + 5.59808i) q^{17} +(-0.767949 + 2.86603i) q^{19} +(1.13397 + 0.767949i) q^{21} +(-3.86603 + 2.23205i) q^{23} +(-3.63397 - 2.09808i) q^{25} +(2.09808 - 2.09808i) q^{27} +(-0.267949 - 0.267949i) q^{29} +(1.86603 - 3.23205i) q^{31} +(-0.767949 - 1.33013i) q^{33} +(1.03590 + 2.13397i) q^{35} +(-1.13397 - 0.303848i) q^{37} +(-2.36603 - 1.36603i) q^{39} -4.92820i q^{41} +(-6.46410 + 6.46410i) q^{43} +(2.36603 - 0.633975i) q^{45} +(2.13397 + 3.69615i) q^{47} +(-1.00000 + 6.92820i) q^{49} +(0.866025 - 3.23205i) q^{51} +(-1.06218 - 3.96410i) q^{53} -2.66025i q^{55} -1.53590i q^{57} +(3.03590 + 11.3301i) q^{59} +(1.86603 - 6.96410i) q^{61} +(6.83013 + 2.36603i) q^{63} +(-2.36603 - 4.09808i) q^{65} +(4.96410 - 1.33013i) q^{67} +(1.63397 - 1.63397i) q^{69} -0.535898i q^{71} +(6.23205 + 3.59808i) q^{73} +(2.09808 + 0.562178i) q^{75} +(4.40192 - 6.50000i) q^{77} +(-8.33013 - 14.4282i) q^{79} +(3.33013 - 5.76795i) q^{81} +(-1.53590 - 1.53590i) q^{83} +(4.09808 - 4.09808i) q^{85} +(0.169873 + 0.0980762i) q^{87} +(-4.50000 + 2.59808i) q^{89} +(1.00000 - 13.9282i) q^{91} +(-0.500000 + 1.86603i) q^{93} +(1.33013 - 2.30385i) q^{95} -2.92820 q^{97} +(-5.73205 - 5.73205i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{3} - 6 q^{9}+O(q^{10}) $ 4 * q - 2 * q^3 - 6 * q^9	$ 4 q - 2 q^{3} - 6 q^{9} + 10 q^{11} + 8 q^{13} - 12 q^{15} - 6 q^{17} - 10 q^{19} + 8 q^{21} - 12 q^{23} - 18 q^{25} - 2 q^{27} - 8 q^{29} + 4 q^{31} - 10 q^{33} + 18 q^{35} - 8 q^{37} - 6 q^{39} - 12 q^{43} + 6 q^{45} + 12 q^{47} - 4 q^{49} + 20 q^{53} + 26 q^{59} + 4 q^{61} + 10 q^{63} - 6 q^{65} + 6 q^{67} + 10 q^{69} + 18 q^{73} - 2 q^{75} + 28 q^{77} - 16 q^{79} - 4 q^{81} - 20 q^{83} + 6 q^{85} + 18 q^{87} - 18 q^{89} + 4 q^{91} - 2 q^{93} - 12 q^{95} + 16 q^{97} - 16 q^{99}+O(q^{100}) $ 4 * q - 2 * q^3 - 6 * q^9 + 10 * q^11 + 8 * q^13 - 12 * q^15 - 6 * q^17 - 10 * q^19 + 8 * q^21 - 12 * q^23 - 18 * q^25 - 2 * q^27 - 8 * q^29 + 4 * q^31 - 10 * q^33 + 18 * q^35 - 8 * q^37 - 6 * q^39 - 12 * q^43 + 6 * q^45 + 12 * q^47 - 4 * q^49 + 20 * q^53 + 26 * q^59 + 4 * q^61 + 10 * q^63 - 6 * q^65 + 6 * q^67 + 10 * q^69 + 18 * q^73 - 2 * q^75 + 28 * q^77 - 16 * q^79 - 4 * q^81 - 20 * q^83 + 6 * q^85 + 18 * q^87 - 18 * q^89 + 4 * q^91 - 2 * q^93 - 12 * q^95 + 16 * q^97 - 16 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/448\mathbb{Z}\right)^\times$.

$n$	$127$	$129$	$197$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{3}{4}\right)$

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			0
$2$		0			0
$3$	−0.500000	+	0.133975i	−0.288675	+	0.0773503i	−0.400251	−	0.916406i	$-0.631077\pi$
$3$	−0.500000	+	0.133975i	−0.288675	+	0.0773503i	0.111576	+	0.993756i	$0.464410\pi$
$4$		0			0
$5$	−0.866025	−	0.232051i	−0.387298	−	0.103776i	0.0599153	−	0.998203i	$-0.480917\pi$
$5$	−0.866025	−	0.232051i	−0.387298	−	0.103776i	−0.447214	+	0.894427i	$0.647584\pi$
$6$		0			0
$7$	−1.73205	−	2.00000i	−0.654654	−	0.755929i
$7$	−1.73205	−	2.00000i	−0.654654	−	0.755929i
$8$		0			0
$9$	−2.36603	+	1.36603i	−0.788675	+	0.455342i
$10$		0			0
$11$	0.767949	+	2.86603i	0.231545	+	0.864139i	0.979676	+	0.200587i	$0.0642851\pi$
$11$	0.767949	+	2.86603i	0.231545	+	0.864139i	−0.748130	+	0.663552i	$0.769048\pi$
$12$		0			0
$13$	3.73205	+	3.73205i	1.03508	+	1.03508i	0.999362	+	0.0357229i	$0.0113734\pi$
$13$	3.73205	+	3.73205i	1.03508	+	1.03508i	0.0357229	+	0.999362i	$0.488627\pi$
$14$		0			0
$15$	0.464102			0.119831
$16$		0			0
$17$	−3.23205	+	5.59808i	−0.783887	+	1.35773i	0.145774	+	0.989318i	$0.453433\pi$
$17$	−3.23205	+	5.59808i	−0.783887	+	1.35773i	−0.929661	+	0.368415i	$0.879901\pi$
$18$		0			0
$19$	−0.767949	+	2.86603i	−0.176180	+	0.657511i	0.820168	+	0.572123i	$0.193880\pi$
$19$	−0.767949	+	2.86603i	−0.176180	+	0.657511i	−0.996348	+	0.0853887i	$0.972787\pi$
$20$		0			0
$21$	1.13397	+	0.767949i	0.247454	+	0.167580i
$22$		0			0
$23$	−3.86603	+	2.23205i	−0.806122	+	0.465415i	−0.845607	−	0.533805i	$-0.820761\pi$
$23$	−3.86603	+	2.23205i	−0.806122	+	0.465415i	0.0394853	+	0.999220i	$0.487428\pi$
$24$		0			0
$25$	−3.63397	−	2.09808i	−0.726795	−	0.419615i
$26$		0			0
$27$	2.09808	−	2.09808i	0.403775	−	0.403775i
$28$		0			0
$29$	−0.267949	−	0.267949i	−0.0497569	−	0.0497569i	0.681791	−	0.731547i	$-0.261202\pi$
$29$	−0.267949	−	0.267949i	−0.0497569	−	0.0497569i	−0.731547	+	0.681791i	$0.761202\pi$
$30$		0			0
$31$	1.86603	−	3.23205i	0.335148	−	0.580493i	−0.648365	−	0.761329i	$-0.724547\pi$
$31$	1.86603	−	3.23205i	0.335148	−	0.580493i	0.983513	+	0.180836i	$0.0578803\pi$
$32$		0			0
$33$	−0.767949	−	1.33013i	−0.133683	−	0.231545i
$34$		0			0
$35$	1.03590	+	2.13397i	0.175099	+	0.360708i
$36$		0			0
$37$	−1.13397	−	0.303848i	−0.186424	−	0.0499522i	0.164399	−	0.986394i	$-0.447432\pi$
$37$	−1.13397	−	0.303848i	−0.186424	−	0.0499522i	−0.350823	+	0.936442i	$0.614098\pi$
$38$		0			0
$39$	−2.36603	−	1.36603i	−0.378867	−	0.218739i
$40$		0			0
$41$		−	4.92820i		−	0.769656i	−0.922988	−	0.384828i	$-0.874261\pi$
$41$		−	4.92820i		−	0.769656i	0.922988	−	0.384828i	$-0.125739\pi$
$42$		0			0
$43$	−6.46410	+	6.46410i	−0.985766	+	0.985766i	−0.999900	−	0.0141339i	$-0.995501\pi$
$43$	−6.46410	+	6.46410i	−0.985766	+	0.985766i	0.0141339	+	0.999900i	$0.495501\pi$
$44$		0			0
$45$	2.36603	−	0.633975i	0.352706	−	0.0945074i
$46$		0			0
$47$	2.13397	+	3.69615i	0.311272	+	0.539139i	0.978638	−	0.205591i	$-0.0659116\pi$
$47$	2.13397	+	3.69615i	0.311272	+	0.539139i	−0.667366	+	0.744730i	$0.732578\pi$
$48$		0			0
$49$	−1.00000	+	6.92820i	−0.142857	+	0.989743i
$50$		0			0
$51$	0.866025	−	3.23205i	0.121268	−	0.452578i
$52$		0			0
$53$	−1.06218	−	3.96410i	−0.145901	−	0.544511i	−0.999714	−	0.0239302i	$-0.992382\pi$
$53$	−1.06218	−	3.96410i	−0.145901	−	0.544511i	0.853812	−	0.520581i	$-0.174285\pi$
$54$		0			0
$55$		−	2.66025i		−	0.358709i
$56$		0			0
$57$		−	1.53590i		−	0.203435i
$58$		0			0
$59$	3.03590	+	11.3301i	0.395240	+	1.47506i	0.821370	+	0.570395i	$0.193210\pi$
$59$	3.03590	+	11.3301i	0.395240	+	1.47506i	−0.426130	+	0.904662i	$0.640123\pi$
$60$		0			0
$61$	1.86603	−	6.96410i	0.238920	−	0.891662i	−0.737422	−	0.675432i	$-0.763957\pi$
$61$	1.86603	−	6.96410i	0.238920	−	0.891662i	0.976342	−	0.216230i	$-0.0693761\pi$
$62$		0			0
$63$	6.83013	+	2.36603i	0.860515	+	0.298091i
$64$		0			0
$65$	−2.36603	−	4.09808i	−0.293469	−	0.508304i
$66$		0			0
$67$	4.96410	−	1.33013i	0.606462	−	0.162501i	0.0574958	−	0.998346i	$-0.481688\pi$
$67$	4.96410	−	1.33013i	0.606462	−	0.162501i	0.548966	+	0.835845i	$0.315022\pi$
$68$		0			0
$69$	1.63397	−	1.63397i	0.196707	−	0.196707i
$70$		0			0
$71$		−	0.535898i		−	0.0635994i	−0.999494	−	0.0317997i	$-0.989876\pi$
$71$		−	0.535898i		−	0.0635994i	0.999494	−	0.0317997i	$-0.0101239\pi$
$72$		0			0
$73$	6.23205	+	3.59808i	0.729406	+	0.421123i	0.818205	−	0.574927i	$-0.194969\pi$
$73$	6.23205	+	3.59808i	0.729406	+	0.421123i	−0.0887986	+	0.996050i	$0.528303\pi$
$74$		0			0
$75$	2.09808	+	0.562178i	0.242265	+	0.0649147i
$76$		0			0
$77$	4.40192	−	6.50000i	0.501646	−	0.740744i
$78$		0			0
$79$	−8.33013	−	14.4282i	−0.937213	−	1.62330i	−0.770640	−	0.637270i	$-0.780063\pi$
$79$	−8.33013	−	14.4282i	−0.937213	−	1.62330i	−0.166572	−	0.986029i	$-0.553270\pi$
$80$		0			0
$81$	3.33013	−	5.76795i	0.370014	−	0.640883i
$82$		0			0
$83$	−1.53590	−	1.53590i	−0.168587	−	0.168587i	0.617771	−	0.786358i	$-0.288036\pi$
$83$	−1.53590	−	1.53590i	−0.168587	−	0.168587i	−0.786358	+	0.617771i	$0.788036\pi$
$84$		0			0
$85$	4.09808	−	4.09808i	0.444499	−	0.444499i
$86$		0			0
$87$	0.169873	+	0.0980762i	0.0182123	+	0.0105149i
$88$		0			0
$89$	−4.50000	+	2.59808i	−0.476999	+	0.275396i	−0.719165	−	0.694839i	$-0.755475\pi$
$89$	−4.50000	+	2.59808i	−0.476999	+	0.275396i	0.242166	+	0.970235i	$0.422142\pi$
$90$		0			0
$91$	1.00000	−	13.9282i	0.104828	−	1.46007i
$92$		0			0
$93$	−0.500000	+	1.86603i	−0.0518476	+	0.193498i
$94$		0			0
$95$	1.33013	−	2.30385i	0.136468	−	0.236370i
$96$		0			0
$97$	−2.92820			−0.297314			−0.148657	−	0.988889i	$-0.547495\pi$
$97$	−2.92820			−0.297314			−0.148657	+	0.988889i	$0.547495\pi$
$98$		0			0
$99$	−5.73205	−	5.73205i	−0.576093	−	0.576093i
$100$		0			0
$101$	5.06218	+	18.8923i	0.503706	+	1.87985i	0.474450	+	0.880283i	$0.342647\pi$
$101$	5.06218	+	18.8923i	0.503706	+	1.87985i	0.0292559	+	0.999572i	$0.490686\pi$
$102$		0			0
$103$	5.59808	−	3.23205i	0.551595	−	0.318463i	−0.198170	−	0.980168i	$-0.563500\pi$
$103$	5.59808	−	3.23205i	0.551595	−	0.318463i	0.749765	+	0.661704i	$0.230167\pi$
$104$		0			0
$105$	−0.803848	−	0.928203i	−0.0784475	−	0.0905834i
$106$		0			0
$107$	−3.23205	−	0.866025i	−0.312454	−	0.0837218i	0.0991843	−	0.995069i	$-0.468377\pi$
$107$	−3.23205	−	0.866025i	−0.312454	−	0.0837218i	−0.411638	+	0.911347i	$0.635043\pi$
$108$		0			0
$109$	−2.13397	+	0.571797i	−0.204398	+	0.0547682i	−0.359565	−	0.933120i	$-0.617075\pi$
$109$	−2.13397	+	0.571797i	−0.204398	+	0.0547682i	0.155167	+	0.987888i	$0.450408\pi$
$110$		0			0
$111$	0.607695			0.0576799
$112$		0			0
$113$	−1.46410			−0.137731			−0.0688655	−	0.997626i	$-0.521938\pi$
$113$	−1.46410			−0.137731			−0.0688655	+	0.997626i	$0.521938\pi$
$114$		0			0
$115$	3.86603	−	1.03590i	0.360509	−	0.0965980i
$116$		0			0
$117$	−13.9282	−	3.73205i	−1.28766	−	0.345028i
$118$		0			0
$119$	16.7942	−	3.23205i	1.53952	−	0.296282i
$120$		0			0
$121$	1.90192	−	1.09808i	0.172902	−	0.0998251i
$122$		0			0
$123$	0.660254	+	2.46410i	0.0595331	+	0.222181i
$124$		0			0
$125$	5.83013	+	5.83013i	0.521462	+	0.521462i
$126$		0			0
$127$	−9.46410			−0.839803			−0.419902	−	0.907570i	$-0.637935\pi$
$127$	−9.46410			−0.839803			−0.419902	+	0.907570i	$0.637935\pi$
$128$		0			0
$129$	2.36603	−	4.09808i	0.208317	−	0.360815i
$130$		0			0
$131$	2.03590	−	7.59808i	0.177877	−	0.663847i	−0.818166	−	0.574982i	$-0.805009\pi$
$131$	2.03590	−	7.59808i	0.177877	−	0.663847i	0.996044	−	0.0888654i	$-0.0283241\pi$
$132$		0			0
$133$	7.06218	−	3.42820i	0.612368	−	0.297263i
$134$		0			0
$135$	−2.30385	+	1.33013i	−0.198284	+	0.114479i
$136$		0			0
$137$	15.2321	+	8.79423i	1.30136	+	0.751342i	0.980638	−	0.195831i	$-0.0627404\pi$
$137$	15.2321	+	8.79423i	1.30136	+	0.751342i	0.320724	+	0.947173i	$0.396074\pi$
$138$		0			0
$139$	11.9282	−	11.9282i	1.01174	−	1.01174i	0.0118067	−	0.999930i	$-0.496242\pi$
$139$	11.9282	−	11.9282i	1.01174	−	1.01174i	0.999930	−	0.0118067i	$-0.00375827\pi$
$140$		0			0
$141$	−1.56218	−	1.56218i	−0.131559	−	0.131559i
$142$		0			0
$143$	−7.83013	+	13.5622i	−0.654788	+	1.13413i
$144$		0			0
$145$	0.169873	+	0.294229i	0.0141072	+	0.0244344i
$146$		0			0
$147$	−0.428203	−	3.59808i	−0.0353176	−	0.296764i
$148$		0			0
$149$	−7.59808	−	2.03590i	−0.622459	−	0.166787i	−0.0662134	−	0.997805i	$-0.521092\pi$
$149$	−7.59808	−	2.03590i	−0.622459	−	0.166787i	−0.556245	+	0.831018i	$0.687758\pi$
$150$		0			0
$151$	−9.86603	−	5.69615i	−0.802886	−	0.463546i	0.0415935	−	0.999135i	$-0.486757\pi$
$151$	−9.86603	−	5.69615i	−0.802886	−	0.463546i	−0.844479	+	0.535588i	$0.820090\pi$
$152$		0			0
$153$		−	17.6603i		−	1.42775i
$154$		0			0
$155$	−2.36603	+	2.36603i	−0.190044	+	0.190044i
$156$		0			0
$157$	−18.2583	+	4.89230i	−1.45717	+	0.390448i	−0.898513	−	0.438948i	$-0.855351\pi$
$157$	−18.2583	+	4.89230i	−1.45717	+	0.390448i	−0.558661	+	0.829396i	$0.688685\pi$
$158$		0			0
$159$	1.06218	+	1.83975i	0.0842362	+	0.145901i
$160$		0			0
$161$	11.1603	+	3.86603i	0.879551	+	0.304685i
$162$		0			0
$163$	−3.23205	+	12.0622i	−0.253154	+	0.944783i	0.715954	+	0.698147i	$0.245992\pi$
$163$	−3.23205	+	12.0622i	−0.253154	+	0.944783i	−0.969108	+	0.246636i	$0.920675\pi$
$164$		0			0
$165$	0.356406	+	1.33013i	0.0277462	+	0.103550i
$166$		0			0
$167$			21.8564i			1.69130i	0.533738	+	0.845650i	$0.320787\pi$
$167$			21.8564i			1.69130i	−0.533738	+	0.845650i	$0.679213\pi$
$168$		0			0
$169$			14.8564i			1.14280i
$170$		0			0
$171$	−2.09808	−	7.83013i	−0.160444	−	0.598785i
$172$		0			0
$173$	0.598076	−	2.23205i	0.0454709	−	0.169700i	−0.939456	−	0.342668i	$-0.888669\pi$
$173$	0.598076	−	2.23205i	0.0454709	−	0.169700i	0.984927	+	0.172969i	$0.0553359\pi$
$174$		0			0
$175$	2.09808	+	10.9019i	0.158600	+	0.824108i
$176$		0			0
$177$	−3.03590	−	5.25833i	−0.228192	−	0.395240i
$178$		0			0
$179$	−8.96410	+	2.40192i	−0.670008	+	0.179528i	−0.577759	−	0.816208i	$-0.696072\pi$
$179$	−8.96410	+	2.40192i	−0.670008	+	0.179528i	−0.0922498	+	0.995736i	$0.529406\pi$
$180$		0			0
$181$	13.3923	−	13.3923i	0.995442	−	0.995442i	−0.00454748	−	0.999990i	$-0.501448\pi$
$181$	13.3923	−	13.3923i	0.995442	−	0.995442i	0.999990	+	0.00454748i	$0.00144751\pi$
$182$		0			0
$183$			3.73205i			0.275881i
$184$		0			0
$185$	0.911543	+	0.526279i	0.0670180	+	0.0386928i
$186$		0			0
$187$	−18.5263	−	4.96410i	−1.35478	−	0.363011i
$188$		0			0
$189$	−7.83013	−	0.562178i	−0.569558	−	0.0408924i
$190$		0			0
$191$	4.33013	+	7.50000i	0.313317	+	0.542681i	0.979078	−	0.203484i	$-0.0652264\pi$
$191$	4.33013	+	7.50000i	0.313317	+	0.542681i	−0.665761	+	0.746165i	$0.731893\pi$
$192$		0			0
$193$	11.5000	−	19.9186i	0.827788	−	1.43377i	−0.0719816	−	0.997406i	$-0.522932\pi$
$193$	11.5000	−	19.9186i	0.827788	−	1.43377i	0.899770	−	0.436365i	$-0.143734\pi$
$194$		0			0
$195$	1.73205	+	1.73205i	0.124035	+	0.124035i
$196$		0			0
$197$	−16.6603	+	16.6603i	−1.18699	+	1.18699i	−0.209100	+	0.977894i	$0.567053\pi$
$197$	−16.6603	+	16.6603i	−1.18699	+	1.18699i	−0.977894	+	0.209100i	$0.932947\pi$
$198$		0			0
$199$	10.3301	+	5.96410i	0.732283	+	0.422784i	0.819257	−	0.573427i	$-0.194386\pi$
$199$	10.3301	+	5.96410i	0.732283	+	0.422784i	−0.0869736	+	0.996211i	$0.527720\pi$
$200$		0			0
$201$	−2.30385	+	1.33013i	−0.162501	+	0.0938199i
$202$		0			0
$203$	−0.0717968	+	1.00000i	−0.00503915	+	0.0701862i
$204$		0			0
$205$	−1.14359	+	4.26795i	−0.0798720	+	0.298087i
$206$		0			0
$207$	6.09808	−	10.5622i	0.423846	−	0.734122i
$208$		0			0
$209$	−8.80385			−0.608975
$210$		0			0
$211$	15.9282	+	15.9282i	1.09654	+	1.09654i	0.994812	+	0.101731i	$0.0324380\pi$
$211$	15.9282	+	15.9282i	1.09654	+	1.09654i	0.101731	+	0.994812i	$0.467562\pi$
$212$		0			0
$213$	0.0717968	+	0.267949i	0.00491943	+	0.0183596i
$214$		0			0
$215$	7.09808	−	4.09808i	0.484085	−	0.279486i
$216$		0			0
$217$	−9.69615	+	1.86603i	−0.658218	+	0.126674i
$218$		0			0
$219$	−3.59808	−	0.964102i	−0.243135	−	0.0651479i
$220$		0			0
$221$	−32.9545	+	8.83013i	−2.21676	+	0.593979i
$222$		0			0
$223$	−9.85641			−0.660034			−0.330017	−	0.943975i	$-0.607054\pi$
$223$	−9.85641			−0.660034			−0.330017	+	0.943975i	$0.607054\pi$
$224$		0			0
$225$	11.4641			0.764273
$226$		0			0
$227$	−11.1603	+	2.99038i	−0.740732	+	0.198479i	−0.609403	−	0.792860i	$-0.708591\pi$
$227$	−11.1603	+	2.99038i	−0.740732	+	0.198479i	−0.131329	+	0.991339i	$0.541924\pi$
$228$		0			0
$229$	13.3301	+	3.57180i	0.880880	+	0.236031i	0.670787	−	0.741650i	$-0.265957\pi$
$229$	13.3301	+	3.57180i	0.880880	+	0.236031i	0.210093	+	0.977681i	$0.432623\pi$
$230$		0			0
$231$	−1.33013	+	3.83975i	−0.0875159	+	0.252637i
$232$		0			0
$233$	0.696152	−	0.401924i	0.0456065	−	0.0263309i	−0.477023	−	0.878891i	$-0.658284\pi$
$233$	0.696152	−	0.401924i	0.0456065	−	0.0263309i	0.522630	+	0.852560i	$0.324951\pi$
$234$		0			0
$235$	−0.990381	−	3.69615i	−0.0646053	−	0.241110i
$236$		0			0
$237$	6.09808	+	6.09808i	0.396113	+	0.396113i
$238$		0			0
$239$	8.53590			0.552141			0.276071	−	0.961137i	$-0.410968\pi$
$239$	8.53590			0.552141			0.276071	+	0.961137i	$0.410968\pi$
$240$		0			0
$241$	−6.96410	+	12.0622i	−0.448597	+	0.776993i	−0.998295	−	0.0583704i	$-0.981410\pi$
$241$	−6.96410	+	12.0622i	−0.448597	+	0.776993i	0.549698	+	0.835364i	$0.314743\pi$
$242$		0			0
$243$	−3.19615	+	11.9282i	−0.205033	+	0.765195i
$244$		0			0
$245$	2.47372	−	5.76795i	0.158040	−	0.368501i
$246$		0			0
$247$	−13.5622	+	7.83013i	−0.862941	+	0.498219i
$248$		0			0
$249$	0.973721	+	0.562178i	0.0617070	+	0.0356266i
$250$		0			0
$251$	17.5885	−	17.5885i	1.11017	−	1.11017i	0.117047	−	0.993126i	$-0.462657\pi$
$251$	17.5885	−	17.5885i	1.11017	−	1.11017i	0.993126	−	0.117047i	$-0.0373429\pi$
$252$		0			0
$253$	−9.36603	−	9.36603i	−0.588837	−	0.588837i
$254$		0			0
$255$	−1.50000	+	2.59808i	−0.0939336	+	0.162698i
$256$		0			0
$257$	9.69615	+	16.7942i	0.604829	+	1.04760i	0.992078	+	0.125620i	$0.0400920\pi$
$257$	9.69615	+	16.7942i	0.604829	+	1.04760i	−0.387249	+	0.921975i	$0.626575\pi$
$258$		0			0
$259$	1.35641	+	2.79423i	0.0842830	+	0.173625i
$260$		0			0
$261$	1.00000	+	0.267949i	0.0618984	+	0.0165856i
$262$		0			0
$263$	−3.99038	−	2.30385i	−0.246057	−	0.142061i	0.371900	−	0.928273i	$-0.378706\pi$
$263$	−3.99038	−	2.30385i	−0.246057	−	0.142061i	−0.617958	+	0.786211i	$0.712040\pi$
$264$		0			0
$265$			3.67949i			0.226029i
$266$		0			0
$267$	1.90192	−	1.90192i	0.116396	−	0.116396i
$268$		0			0
$269$	9.79423	−	2.62436i	0.597165	−	0.160010i	0.0524390	−	0.998624i	$-0.483300\pi$
$269$	9.79423	−	2.62436i	0.597165	−	0.160010i	0.544726	+	0.838614i	$0.316634\pi$
$270$		0			0
$271$	6.06218	+	10.5000i	0.368251	+	0.637830i	0.989292	−	0.145948i	$-0.0466233\pi$
$271$	6.06218	+	10.5000i	0.368251	+	0.637830i	−0.621041	+	0.783778i	$0.713290\pi$
$272$		0			0
$273$	1.36603	+	7.09808i	0.0826756	+	0.429595i
$274$		0			0
$275$	3.22243	−	12.0263i	0.194320	−	0.725212i
$276$		0			0
$277$	−1.20577	−	4.50000i	−0.0724478	−	0.270379i	0.920195	−	0.391461i	$-0.128030\pi$
$277$	−1.20577	−	4.50000i	−0.0724478	−	0.270379i	−0.992642	+	0.121082i	$0.961364\pi$
$278$		0			0
$279$			10.1962i			0.610428i
$280$		0			0
$281$		−	0.928203i		−	0.0553720i	−0.999617	−	0.0276860i	$-0.991186\pi$
$281$		−	0.928203i		−	0.0553720i	0.999617	−	0.0276860i	$-0.00881385\pi$
$282$		0			0
$283$	3.89230	+	14.5263i	0.231374	+	0.863498i	0.979750	+	0.200224i	$0.0641669\pi$
$283$	3.89230	+	14.5263i	0.231374	+	0.863498i	−0.748377	+	0.663274i	$0.769166\pi$
$284$		0			0
$285$	−0.356406	+	1.33013i	−0.0211117	+	0.0787899i
$286$		0			0
$287$	−9.85641	+	8.53590i	−0.581805	+	0.503858i
$288$		0			0
$289$	−12.3923	−	21.4641i	−0.728959	−	1.26259i
$290$		0			0
$291$	1.46410	−	0.392305i	0.0858272	−	0.0229973i
$292$		0			0
$293$	−7.92820	+	7.92820i	−0.463171	+	0.463171i	−0.899693	−	0.436523i	$-0.856210\pi$
$293$	−7.92820	+	7.92820i	−0.463171	+	0.463171i	0.436523	+	0.899693i	$0.356210\pi$
$294$		0			0
$295$		−	10.5167i		−	0.612304i
$296$		0			0
$297$	7.62436	+	4.40192i	0.442410	+	0.255426i
$298$		0			0
$299$	−22.7583	−	6.09808i	−1.31615	−	0.352661i
$300$		0			0
$301$	24.1244	+	1.73205i	1.39050	+	0.0998337i
$302$		0			0
$303$	−5.06218	−	8.76795i	−0.290815	−	0.503706i
$304$		0			0
$305$	−3.23205	+	5.59808i	−0.185067	+	0.320545i
$306$		0			0
$307$	−9.00000	−	9.00000i	−0.513657	−	0.513657i	0.401988	−	0.915645i	$-0.368319\pi$
$307$	−9.00000	−	9.00000i	−0.513657	−	0.513657i	−0.915645	+	0.401988i	$0.868319\pi$
$308$		0			0
$309$	−2.36603	+	2.36603i	−0.134598	+	0.134598i
$310$		0			0
$311$	−5.59808	−	3.23205i	−0.317438	−	0.183273i	0.332812	−	0.942993i	$-0.392002\pi$
$311$	−5.59808	−	3.23205i	−0.317438	−	0.183273i	−0.650250	+	0.759720i	$0.725336\pi$
$312$		0			0
$313$	−18.6962	+	10.7942i	−1.05677	+	0.610126i	−0.924537	−	0.381093i	$-0.875548\pi$
$313$	−18.6962	+	10.7942i	−1.05677	+	0.610126i	−0.132232	+	0.991219i	$0.542214\pi$
$314$		0			0
$315$	−5.36603	−	3.63397i	−0.302341	−	0.204751i
$316$		0			0
$317$	−3.79423	+	14.1603i	−0.213105	+	0.795319i	0.773720	+	0.633528i	$0.218394\pi$
$317$	−3.79423	+	14.1603i	−0.213105	+	0.795319i	−0.986825	+	0.161791i	$0.948273\pi$
$318$		0			0
$319$	0.562178	−	0.973721i	0.0314759	−	0.0545179i
$320$		0			0
$321$	1.73205			0.0966736
$322$		0			0
$323$	−13.5622	−	13.5622i	−0.754620	−	0.754620i
$324$		0			0
$325$	−5.73205	−	21.3923i	−0.317957	−	1.18663i
$326$		0			0
$327$	0.990381	−	0.571797i	0.0547682	−	0.0316204i
$328$		0			0
$329$	3.69615	−	10.6699i	0.203775	−	0.588249i
$330$		0			0
$331$	19.6244	+	5.25833i	1.07865	+	0.289024i	0.754044	−	0.656824i	$-0.228101\pi$
$331$	19.6244	+	5.25833i	1.07865	+	0.289024i	0.324609	+	0.945848i	$0.394767\pi$
$332$		0			0
$333$	3.09808	−	0.830127i	0.169774	−	0.0454907i
$334$		0			0
$335$	−4.60770			−0.251745
$336$		0			0
$337$	−6.14359			−0.334663			−0.167331	−	0.985901i	$-0.553515\pi$
$337$	−6.14359			−0.334663			−0.167331	+	0.985901i	$0.553515\pi$
$338$		0			0
$339$	0.732051	−	0.196152i	0.0397595	−	0.0106535i
$340$		0			0
$341$	10.6962	+	2.86603i	0.579229	+	0.155204i
$342$		0			0
$343$	15.5885	−	10.0000i	0.841698	−	0.539949i
$344$		0			0
$345$	−1.79423	+	1.03590i	−0.0965980	+	0.0557709i
$346$		0			0
$347$	6.30385	+	23.5263i	0.338408	+	1.26296i	0.900127	+	0.435628i	$0.143474\pi$
$347$	6.30385	+	23.5263i	0.338408	+	1.26296i	−0.561718	+	0.827329i	$0.689860\pi$
$348$		0			0
$349$	6.12436	+	6.12436i	0.327829	+	0.327829i	0.851761	−	0.523931i	$-0.175535\pi$
$349$	6.12436	+	6.12436i	0.327829	+	0.327829i	−0.523931	+	0.851761i	$0.675535\pi$
$350$		0			0
$351$	15.6603			0.835883
$352$		0			0
$353$	13.8923	−	24.0622i	0.739413	−	1.28070i	−0.213347	−	0.976976i	$-0.568437\pi$
$353$	13.8923	−	24.0622i	0.739413	−	1.28070i	0.952760	−	0.303724i	$-0.0982301\pi$
$354$		0			0
$355$	−0.124356	+	0.464102i	−0.00660011	+	0.0246320i
$356$		0			0
$357$	−7.96410	+	3.86603i	−0.421505	+	0.204612i
$358$		0			0
$359$	3.86603	−	2.23205i	0.204041	−	0.117803i	−0.394498	−	0.918897i	$-0.629081\pi$
$359$	3.86603	−	2.23205i	0.204041	−	0.117803i	0.598539	+	0.801094i	$0.295748\pi$
$360$		0			0
$361$	8.83013	+	5.09808i	0.464744	+	0.268320i
$362$		0			0
$363$	−0.803848	+	0.803848i	−0.0421911	+	0.0421911i
$364$		0			0
$365$	−4.56218	−	4.56218i	−0.238795	−	0.238795i
$366$		0			0
$367$	3.06218	−	5.30385i	0.159844	−	0.276859i	−0.774968	−	0.632000i	$-0.782234\pi$
$367$	3.06218	−	5.30385i	0.159844	−	0.276859i	0.934812	+	0.355142i	$0.115567\pi$
$368$		0			0
$369$	6.73205	+	11.6603i	0.350457	+	0.607009i
$370$		0			0
$371$	−6.08846	+	8.99038i	−0.316097	+	0.466757i
$372$		0			0
$373$	0.866025	+	0.232051i	0.0448411	+	0.0120151i	0.281170	−	0.959658i	$-0.409278\pi$
$373$	0.866025	+	0.232051i	0.0448411	+	0.0120151i	−0.236329	+	0.971673i	$0.575944\pi$
$374$		0			0
$375$	−3.69615	−	2.13397i	−0.190868	−	0.110198i
$376$		0			0
$377$		−	2.00000i		−	0.103005i
$378$		0			0
$379$	−0.124356	+	0.124356i	−0.00638772	+	0.00638772i	−0.710293	−	0.703906i	$-0.751438\pi$
$379$	−0.124356	+	0.124356i	−0.00638772	+	0.00638772i	0.703906	+	0.710293i	$0.251438\pi$
$380$		0			0
$381$	4.73205	−	1.26795i	0.242430	−	0.0649590i
$382$		0			0
$383$	−12.5981	−	21.8205i	−0.643732	−	1.11498i	−0.984593	−	0.174862i	$-0.944052\pi$
$383$	−12.5981	−	21.8205i	−0.643732	−	1.11498i	0.340861	−	0.940114i	$-0.389281\pi$
$384$		0			0
$385$	−5.32051	+	4.60770i	−0.271158	+	0.234830i
$386$		0			0
$387$	6.46410	−	24.1244i	0.328589	−	1.22631i
$388$		0			0
$389$	5.99038	+	22.3564i	0.303724	+	1.13351i	0.934038	+	0.357174i	$0.116260\pi$
$389$	5.99038	+	22.3564i	0.303724	+	1.13351i	−0.630313	+	0.776341i	$0.717074\pi$
$390$		0			0
$391$		−	28.8564i		−	1.45933i
$392$		0			0
$393$			4.07180i			0.205395i
$394$		0			0
$395$	3.86603	+	14.4282i	0.194521	+	0.725962i
$396$		0			0
$397$	6.86603	−	25.6244i	0.344596	−	1.28605i	−0.548488	−	0.836159i	$-0.684796\pi$
$397$	6.86603	−	25.6244i	0.344596	−	1.28605i	0.893084	−	0.449891i	$-0.148537\pi$
$398$		0			0
$399$	−3.07180	+	2.66025i	−0.153782	+	0.133179i
$400$		0			0
$401$	7.50000	+	12.9904i	0.374532	+	0.648709i	0.990257	−	0.139253i	$-0.0444700\pi$
$401$	7.50000	+	12.9904i	0.374532	+	0.648709i	−0.615725	+	0.787961i	$0.711137\pi$
$402$		0			0
$403$	19.0263	−	5.09808i	0.947766	−	0.253953i
$404$		0			0
$405$	−4.22243	+	4.22243i	−0.209814	+	0.209814i
$406$		0			0
$407$		−	3.48334i		−	0.172663i
$408$		0			0
$409$	−7.50000	−	4.33013i	−0.370851	−	0.214111i	0.302979	−	0.952997i	$-0.402019\pi$
$409$	−7.50000	−	4.33013i	−0.370851	−	0.214111i	−0.673830	+	0.738886i	$0.735352\pi$
$410$		0			0
$411$	−8.79423	−	2.35641i	−0.433787	−	0.116233i
$412$		0			0
$413$	17.4019	−	25.6962i	0.856293	−	1.26442i
$414$		0			0
$415$	0.973721	+	1.68653i	0.0477981	+	0.0827887i
$416$		0			0
$417$	−4.36603	+	7.56218i	−0.213805	+	0.370321i
$418$		0			0
$419$	19.0000	+	19.0000i	0.928211	+	0.928211i	0.997590	−	0.0693796i	$-0.0221020\pi$
$419$	19.0000	+	19.0000i	0.928211	+	0.928211i	−0.0693796	+	0.997590i	$0.522102\pi$
$420$		0			0
$421$	8.66025	−	8.66025i	0.422075	−	0.422075i	−0.463843	−	0.885918i	$-0.653530\pi$
$421$	8.66025	−	8.66025i	0.422075	−	0.422075i	0.885918	+	0.463843i	$0.153530\pi$
$422$		0			0
$423$	−10.0981	−	5.83013i	−0.490985	−	0.283470i
$424$		0			0
$425$	23.4904	−	13.5622i	1.13945	−	0.657862i
$426$		0			0
$427$	−17.1603	+	8.33013i	−0.830443	+	0.403123i
$428$		0			0
$429$	2.09808	−	7.83013i	0.101296	−	0.378042i
$430$		0			0
$431$	15.3301	−	26.5526i	0.738426	−	1.27899i	−0.214777	−	0.976663i	$-0.568903\pi$
$431$	15.3301	−	26.5526i	0.738426	−	1.27899i	0.953204	−	0.302329i	$-0.0977640\pi$
$432$		0			0
$433$	−33.1769			−1.59438			−0.797190	−	0.603728i	$-0.793681\pi$
$433$	−33.1769			−1.59438			−0.797190	+	0.603728i	$0.793681\pi$
$434$		0			0
$435$	−0.124356	−	0.124356i	−0.00596240	−	0.00596240i
$436$		0			0
$437$	−3.42820	−	12.7942i	−0.163993	−	0.612031i
$438$		0			0
$439$	−6.52628	+	3.76795i	−0.311482	+	0.179834i	−0.647590	−	0.761989i	$-0.724223\pi$
$439$	−6.52628	+	3.76795i	−0.311482	+	0.179834i	0.336107	+	0.941824i	$0.390890\pi$
$440$		0			0
$441$	−7.09808	−	17.7583i	−0.338004	−	0.845635i
$442$		0			0
$443$	10.4282	+	2.79423i	0.495459	+	0.132758i	0.497892	−	0.867239i	$-0.334108\pi$
$443$	10.4282	+	2.79423i	0.495459	+	0.132758i	−0.00243278	+	0.999997i	$0.500774\pi$
$444$		0			0
$445$	4.50000	−	1.20577i	0.213320	−	0.0571590i
$446$		0			0
$447$	4.07180			0.192589
$448$		0			0
$449$	−23.3205			−1.10056			−0.550281	−	0.834979i	$-0.685480\pi$
$449$	−23.3205			−1.10056			−0.550281	+	0.834979i	$0.685480\pi$
$450$		0			0
$451$	14.1244	−	3.78461i	0.665090	−	0.178210i
$452$		0			0
$453$	5.69615	+	1.52628i	0.267629	+	0.0717109i
$454$		0			0
$455$	−4.09808	+	11.8301i	−0.192121	+	0.554605i
$456$		0			0
$457$	16.2846	−	9.40192i	0.761762	−	0.439803i	−0.0681661	−	0.997674i	$-0.521715\pi$
$457$	16.2846	−	9.40192i	0.761762	−	0.439803i	0.829928	+	0.557871i	$0.188381\pi$
$458$		0			0
$459$	4.96410	+	18.5263i	0.231704	+	0.864733i
$460$		0			0
$461$	18.6603	+	18.6603i	0.869095	+	0.869095i	0.992372	−	0.123278i	$-0.0393405\pi$
$461$	18.6603	+	18.6603i	0.869095	+	0.869095i	−0.123278	+	0.992372i	$0.539341\pi$
$462$		0			0
$463$	−2.14359			−0.0996212			−0.0498106	−	0.998759i	$-0.515862\pi$
$463$	−2.14359			−0.0996212			−0.0498106	+	0.998759i	$0.515862\pi$
$464$		0			0
$465$	0.866025	−	1.50000i	0.0401610	−	0.0695608i
$466$		0			0
$467$	6.96410	−	25.9904i	0.322260	−	1.20269i	−0.594777	−	0.803890i	$-0.702760\pi$
$467$	6.96410	−	25.9904i	0.322260	−	1.20269i	0.917038	−	0.398801i	$-0.130574\pi$
$468$		0			0
$469$	−11.2583	−	7.62436i	−0.519861	−	0.352060i
$470$		0			0
$471$	8.47372	−	4.89230i	0.390448	−	0.225426i
$472$		0			0
$473$	−23.4904	−	13.5622i	−1.08009	−	0.623590i
$474$		0			0
$475$	8.80385	−	8.80385i	0.403948	−	0.403948i
$476$		0			0
$477$	7.92820	+	7.92820i	0.363007	+	0.363007i
$478$		0			0
$479$	7.79423	−	13.5000i	0.356127	−	0.616831i	−0.631183	−	0.775634i	$-0.717430\pi$
$479$	7.79423	−	13.5000i	0.356127	−	0.616831i	0.987310	+	0.158803i	$0.0507636\pi$
$480$		0			0
$481$	−3.09808	−	5.36603i	−0.141260	−	0.244670i
$482$		0			0
$483$	−6.09808	−	0.437822i	−0.277472	−	0.0199216i
$484$		0			0
$485$	2.53590	+	0.679492i	0.115149	+	0.0308541i
$486$		0			0
$487$	−10.6699	−	6.16025i	−0.483498	−	0.279148i	0.238375	−	0.971173i	$-0.423385\pi$
$487$	−10.6699	−	6.16025i	−0.483498	−	0.279148i	−0.721873	+	0.692025i	$0.756719\pi$
$488$		0			0
$489$		−	6.46410i		−	0.292317i
$490$		0			0
$491$	3.58846	−	3.58846i	0.161945	−	0.161945i	−0.621483	−	0.783428i	$-0.713469\pi$
$491$	3.58846	−	3.58846i	0.161945	−	0.161945i	0.783428	+	0.621483i	$0.213469\pi$
$492$		0			0
$493$	2.36603	−	0.633975i	0.106560	−	0.0285528i
$494$		0			0
$495$	3.63397	+	6.29423i	0.163335	+	0.282905i
$496$		0			0
$497$	−1.07180	+	0.928203i	−0.0480767	+	0.0416356i
$498$		0			0
$499$	−6.69615	+	24.9904i	−0.299761	+	1.11872i	0.637601	+	0.770367i	$0.279927\pi$
$499$	−6.69615	+	24.9904i	−0.299761	+	1.11872i	−0.937362	+	0.348356i	$0.886740\pi$
$500$		0			0
$501$	−2.92820	−	10.9282i	−0.130822	−	0.488236i
$502$		0			0
$503$			31.8564i			1.42041i	0.703996	+	0.710203i	$0.251397\pi$
$503$			31.8564i			1.42041i	−0.703996	+	0.710203i	$0.748603\pi$
$504$		0			0
$505$		−	17.5359i		−	0.780337i
$506$		0			0
$507$	−1.99038	−	7.42820i	−0.0883959	−	0.329898i
$508$		0			0
$509$	2.52628	−	9.42820i	0.111975	−	0.417898i	−0.887067	−	0.461640i	$-0.847261\pi$
$509$	2.52628	−	9.42820i	0.111975	−	0.417898i	0.999043	+	0.0437420i	$0.0139280\pi$
$510$		0			0
$511$	−3.59808	−	18.6962i	−0.159170	−	0.827069i
$512$		0			0
$513$	4.40192	+	7.62436i	0.194350	+	0.336624i
$514$		0			0
$515$	−5.59808	+	1.50000i	−0.246681	+	0.0660979i
$516$		0			0
$517$	−8.95448	+	8.95448i	−0.393818	+	0.393818i
$518$		0			0
$519$			1.19615i			0.0525053i
$520$		0			0
$521$	−13.3756	−	7.72243i	−0.585998	−	0.338326i	0.177516	−	0.984118i	$-0.443194\pi$
$521$	−13.3756	−	7.72243i	−0.585998	−	0.338326i	−0.763513	+	0.645792i	$0.776527\pi$
$522$		0			0
$523$	32.2846	+	8.65064i	1.41171	+	0.378266i	0.882534	−	0.470248i	$-0.155836\pi$
$523$	32.2846	+	8.65064i	1.41171	+	0.378266i	0.529173	+	0.848514i	$0.322502\pi$
$524$		0			0
$525$	−2.50962	−	5.16987i	−0.109529	−	0.225632i
$526$		0			0
$527$	12.0622	+	20.8923i	0.525437	+	0.910083i
$528$		0			0
$529$	−1.53590	+	2.66025i	−0.0667782	+	0.115663i
$530$		0			0
$531$	−22.6603	−	22.6603i	−0.983371	−	0.983371i
$532$		0			0
$533$	18.3923	−	18.3923i	0.796659	−	0.796659i
$534$		0			0
$535$	2.59808	+	1.50000i	0.112325	+	0.0648507i
$536$		0			0
$537$	4.16025	−	2.40192i	0.179528	−	0.103651i
$538$		0			0
$539$	−20.6244	+	2.45448i	−0.888354	+	0.105722i
$540$		0			0
$541$	2.47372	−	9.23205i	0.106354	−	0.396917i	−0.892142	−	0.451756i	$-0.850798\pi$
$541$	2.47372	−	9.23205i	0.106354	−	0.396917i	0.998495	+	0.0548389i	$0.0174645\pi$
$542$		0			0
$543$	−4.90192	+	8.49038i	−0.210362	+	0.364357i
$544$		0			0
$545$	1.98076			0.0848465
$546$		0			0
$547$	23.0526	+	23.0526i	0.985656	+	0.985656i	0.999899	−	0.0142423i	$-0.00453363\pi$
$547$	23.0526	+	23.0526i	0.985656	+	0.985656i	−0.0142423	+	0.999899i	$0.504534\pi$
$548$		0			0
$549$	5.09808	+	19.0263i	0.217581	+	0.812022i
$550$		0			0
$551$	0.973721	−	0.562178i	0.0414819	−	0.0239496i
$552$		0			0
$553$	−14.4282	+	41.6506i	−0.613550	+	1.77117i
$554$		0			0
$555$	−0.526279	−	0.141016i	−0.0223393	−	0.00598580i
$556$		0			0
$557$	6.59808	−	1.76795i	0.279569	−	0.0749104i	−0.116310	−	0.993213i	$-0.537107\pi$
$557$	6.59808	−	1.76795i	0.279569	−	0.0749104i	0.395879	+	0.918303i	$0.370440\pi$
$558$		0			0
$559$	−48.2487			−2.04070
$560$		0			0
$561$	9.92820			0.419169
$562$		0			0
$563$	2.42820	−	0.650635i	0.102337	−	0.0274210i	−0.207287	−	0.978280i	$-0.566464\pi$
$563$	2.42820	−	0.650635i	0.102337	−	0.0274210i	0.309624	+	0.950859i	$0.399797\pi$
$564$		0			0
$565$	1.26795	+	0.339746i	0.0533430	+	0.0142932i
$566$		0			0
$567$	−17.3038	+	3.33013i	−0.726693	+	0.139852i
$568$		0			0
$569$	−5.08846	+	2.93782i	−0.213319	+	0.123160i	−0.602853	−	0.797852i	$-0.705969\pi$
$569$	−5.08846	+	2.93782i	−0.213319	+	0.123160i	0.389534	+	0.921012i	$0.372636\pi$
$570$		0			0
$571$	7.16025	+	26.7224i	0.299647	+	1.11830i	0.937456	+	0.348104i	$0.113174\pi$
$571$	7.16025	+	26.7224i	0.299647	+	1.11830i	−0.637809	+	0.770195i	$0.720159\pi$
$572$		0			0
$573$	−3.16987	−	3.16987i	−0.132423	−	0.132423i
$574$		0			0
$575$	18.7321			0.781181
$576$		0			0
$577$	−6.62436	+	11.4737i	−0.275776	+	0.477657i	−0.970330	−	0.241782i	$-0.922268\pi$
$577$	−6.62436	+	11.4737i	−0.275776	+	0.477657i	0.694555	+	0.719440i	$0.255601\pi$
$578$		0			0
$579$	−3.08142	+	11.5000i	−0.128059	+	0.477924i
$580$		0			0
$581$	−0.411543	+	5.73205i	−0.0170737	+	0.237806i
$582$		0			0
$583$	10.5455	−	6.08846i	0.436751	−	0.252158i
$584$		0			0
$585$	11.1962	+	6.46410i	0.462904	+	0.267258i
$586$		0			0
$587$	−21.9282	+	21.9282i	−0.905074	+	0.905074i	−0.995870	−	0.0907957i	$-0.971059\pi$
$587$	−21.9282	+	21.9282i	−0.905074	+	0.905074i	0.0907957	+	0.995870i	$0.471059\pi$
$588$		0			0
$589$	7.83013	+	7.83013i	0.322635	+	0.322635i
$590$		0			0
$591$	6.09808	−	10.5622i	0.250841	−	0.434470i
$592$		0			0
$593$	5.69615	+	9.86603i	0.233913	+	0.405149i	0.958956	−	0.283554i	$-0.0915136\pi$
$593$	5.69615	+	9.86603i	0.233913	+	0.405149i	−0.725043	+	0.688703i	$0.758180\pi$
$594$		0			0
$595$	−15.2942	−	1.09808i	−0.627002	−	0.0450167i
$596$		0			0
$597$	−5.96410	−	1.59808i	−0.244094	−	0.0654049i
$598$		0			0
$599$	−16.6699	−	9.62436i	−0.681113	−	0.393241i	0.119162	−	0.992875i	$-0.461979\pi$
$599$	−16.6699	−	9.62436i	−0.681113	−	0.393241i	−0.800274	+	0.599634i	$0.795313\pi$
$600$		0			0
$601$			10.0000i			0.407909i	0.978980	+	0.203954i	$0.0653794\pi$
$601$			10.0000i			0.407909i	−0.978980	+	0.203954i	$0.934621\pi$
$602$		0			0
$603$	−9.92820	+	9.92820i	−0.404308	+	0.404308i
$604$		0			0
$605$	−1.90192	+	0.509619i	−0.0773242	+	0.0207190i
$606$		0			0
$607$	8.52628	+	14.7679i	0.346071	+	0.599413i	0.985548	−	0.169398i	$-0.0541823\pi$
$607$	8.52628	+	14.7679i	0.346071	+	0.599413i	−0.639477	+	0.768810i	$0.720849\pi$
$608$		0			0
$609$	−0.0980762	−	0.509619i	−0.00397425	−	0.0206508i
$610$		0			0
$611$	−5.83013	+	21.7583i	−0.235862	+	0.880248i
$612$		0			0
$613$	2.47372	+	9.23205i	0.0999126	+	0.372879i	0.997718	−	0.0675126i	$-0.0215063\pi$
$613$	2.47372	+	9.23205i	0.0999126	+	0.372879i	−0.897806	+	0.440392i	$0.854840\pi$
$614$		0			0
$615$		−	2.28719i		−	0.0922283i
$616$		0			0
$617$			0.535898i			0.0215745i	0.999942	+	0.0107872i	$0.00343375\pi$
$617$			0.535898i			0.0215745i	−0.999942	+	0.0107872i	$0.996566\pi$
$618$		0			0
$619$	−5.10770	−	19.0622i	−0.205296	−	0.766174i	−0.989359	−	0.145493i	$-0.953523\pi$
$619$	−5.10770	−	19.0622i	−0.205296	−	0.766174i	0.784064	−	0.620680i	$-0.213144\pi$
$620$		0			0
$621$	−3.42820	+	12.7942i	−0.137569	+	0.513415i
$622$		0			0
$623$	12.9904	+	4.50000i	0.520449	+	0.180289i
$624$		0			0
$625$	6.79423	+	11.7679i	0.271769	+	0.470718i
$626$		0			0
$627$	4.40192	−	1.17949i	0.175796	−	0.0471044i
$628$		0			0
$629$	5.36603	−	5.36603i	0.213957	−	0.213957i
$630$		0			0
$631$		−	32.2487i		−	1.28380i	−0.766788	−	0.641900i	$-0.778146\pi$
$631$		−	32.2487i		−	1.28380i	0.766788	−	0.641900i	$-0.221854\pi$
$632$		0			0
$633$	−10.0981	−	5.83013i	−0.401362	−	0.231727i
$634$		0			0
$635$	8.19615	+	2.19615i	0.325254	+	0.0871517i
$636$		0			0
$637$	−29.5885	+	22.1244i	−1.17234	+	0.876599i
$638$		0			0
$639$	0.732051	+	1.26795i	0.0289595	+	0.0501593i
$640$		0			0
$641$	5.57180	−	9.65064i	0.220073	−	0.381177i	−0.734757	−	0.678330i	$-0.762704\pi$
$641$	5.57180	−	9.65064i	0.220073	−	0.381177i	0.954830	+	0.297153i	$0.0960372\pi$
$642$		0			0
$643$	5.39230	+	5.39230i	0.212652	+	0.212652i	0.805393	−	0.592741i	$-0.201954\pi$
$643$	5.39230	+	5.39230i	0.212652	+	0.212652i	−0.592741	+	0.805393i	$0.701954\pi$
$644$		0			0
$645$	−3.00000	+	3.00000i	−0.118125	+	0.118125i
$646$		0			0
$647$	−30.8660	−	17.8205i	−1.21347	−	0.700596i	−0.249955	−	0.968257i	$-0.580416\pi$
$647$	−30.8660	−	17.8205i	−1.21347	−	0.700596i	−0.963513	+	0.267661i	$0.913749\pi$
$648$		0			0
$649$	−30.1410	+	17.4019i	−1.18314	+	0.683085i
$650$		0			0
$651$	4.59808	−	2.23205i	0.180213	−	0.0874810i
$652$		0			0
$653$	−8.72243	+	32.5526i	−0.341335	+	1.27388i	0.555501	+	0.831516i	$0.312527\pi$
$653$	−8.72243	+	32.5526i	−0.341335	+	1.27388i	−0.896836	+	0.442364i	$0.854140\pi$
$654$		0			0
$655$	−3.52628	+	6.10770i	−0.137783	+	0.238647i
$656$		0			0
$657$	−19.6603			−0.767020
$658$		0			0
$659$	−18.8564	−	18.8564i	−0.734541	−	0.734541i	0.236975	−	0.971516i	$-0.423844\pi$
$659$	−18.8564	−	18.8564i	−0.734541	−	0.734541i	−0.971516	+	0.236975i	$0.923844\pi$
$660$		0			0
$661$	−5.93782	−	22.1603i	−0.230955	−	0.861934i	−0.979931	−	0.199337i	$-0.936121\pi$
$661$	−5.93782	−	22.1603i	−0.230955	−	0.861934i	0.748976	−	0.662597i	$-0.230546\pi$
$662$		0			0
$663$	15.2942	−	8.83013i	0.593979	−	0.342934i
$664$		0			0
$665$	−6.91154	+	1.33013i	−0.268018	+	0.0515801i
$666$		0			0
$667$	1.63397	+	0.437822i	0.0632677	+	0.0169525i
$668$		0			0
$669$	4.92820	−	1.32051i	0.190535	−	0.0510538i
$670$		0			0
$671$	21.3923			0.825841
$672$		0			0
$673$	40.7846			1.57213			0.786066	−	0.618143i	$-0.212115\pi$
$673$	40.7846			1.57213			0.786066	+	0.618143i	$0.212115\pi$
$674$		0			0
$675$	−12.0263	+	3.22243i	−0.462892	+	0.124031i
$676$		0			0
$677$	−42.7224	−	11.4474i	−1.64196	−	0.439961i	−0.684611	−	0.728908i	$-0.740028\pi$
$677$	−42.7224	−	11.4474i	−1.64196	−	0.439961i	−0.957345	+	0.288947i	$0.906695\pi$
$678$		0			0
$679$	5.07180	+	5.85641i	0.194638	+	0.224748i
$680$		0			0
$681$	5.17949	−	2.99038i	0.198479	−	0.114592i
$682$		0			0
$683$	−12.2128	−	45.5788i	−0.467310	−	1.74403i	−0.649114	−	0.760691i	$-0.724860\pi$
$683$	−12.2128	−	45.5788i	−0.467310	−	1.74403i	0.181804	−	0.983335i	$-0.441806\pi$
$684$		0			0
$685$	−11.1506	−	11.1506i	−0.426044	−	0.426044i
$686$		0			0
$687$	−7.14359			−0.272545
$688$		0			0
$689$	10.8301	−	18.7583i	0.412595	−	0.714635i
$690$		0			0
$691$	−6.16025	+	22.9904i	−0.234347	+	0.874595i	0.744095	+	0.668074i	$0.232881\pi$
$691$	−6.16025	+	22.9904i	−0.234347	+	0.874595i	−0.978442	+	0.206521i	$0.933786\pi$
$692$		0			0
$693$	−1.53590	+	21.3923i	−0.0583440	+	0.812626i
$694$		0			0
$695$	−13.0981	+	7.56218i	−0.496838	+	0.286850i
$696$		0			0
$697$	27.5885	+	15.9282i	1.04499	+	0.603324i
$698$		0			0
$699$	−0.294229	+	0.294229i	−0.0111287	+	0.0111287i
$700$		0			0
$701$	−13.3923	−	13.3923i	−0.505820	−	0.505820i	0.407420	−	0.913241i	$-0.366428\pi$
$701$	−13.3923	−	13.3923i	−0.505820	−	0.505820i	−0.913241	+	0.407420i	$0.866428\pi$
$702$		0			0
$703$	1.74167	−	3.01666i	0.0656883	−	0.113776i
$704$		0			0
$705$	0.990381	+	1.71539i	0.0372999	+	0.0646053i
$706$		0			0
$707$	29.0167	−	42.8468i	1.09128	−	1.61142i
$708$		0			0
$709$	45.6506	+	12.2321i	1.71445	+	0.459384i	0.976507	−	0.215484i	$-0.0691329\pi$
$709$	45.6506	+	12.2321i	1.71445	+	0.459384i	0.737938	+	0.674868i	$0.235800\pi$
$710$		0			0
$711$	39.4186	+	22.7583i	1.47831	+	0.853504i
$712$		0			0
$713$			16.6603i			0.623931i
$714$		0			0
$715$	9.92820	−	9.92820i	0.371294	−	0.371294i
$716$		0			0
$717$	−4.26795	+	1.14359i	−0.159389	+	0.0427083i
$718$		0			0
$719$	0.205771	+	0.356406i	0.00767398	+	0.0132917i	0.869837	−	0.493339i	$-0.164224\pi$
$719$	0.205771	+	0.356406i	0.00767398	+	0.0132917i	−0.862163	+	0.506631i	$0.830891\pi$
$720$		0			0
$721$	−16.1603	−	5.59808i	−0.601839	−	0.208483i
$722$		0			0
$723$	1.86603	−	6.96410i	0.0693982	−	0.258998i
$724$		0			0
$725$	0.411543	+	1.53590i	0.0152843	+	0.0570418i
$726$		0			0
$727$			41.3205i			1.53249i	0.642547	+	0.766246i	$0.277878\pi$
$727$			41.3205i			1.53249i	−0.642547	+	0.766246i	$0.722122\pi$
$728$		0			0
$729$			13.5885i			0.503276i
$730$		0			0
$731$	−15.2942	−	57.0788i	−0.565677	−	2.11114i
$732$		0			0
$733$	−8.81347	+	32.8923i	−0.325533	+	1.21490i	0.588242	+	0.808685i	$0.299820\pi$
$733$	−8.81347	+	32.8923i	−0.325533	+	1.21490i	−0.913775	+	0.406220i	$0.866847\pi$
$734$		0			0
$735$	−0.464102	+	3.21539i	−0.0171186	+	0.118601i
$736$		0			0
$737$	7.62436	+	13.2058i	0.280847	+	0.486441i
$738$		0			0
$739$	−38.4808	+	10.3109i	−1.41554	+	0.379292i	−0.883899	−	0.467679i	$-0.845090\pi$
$739$	−38.4808	+	10.3109i	−1.41554	+	0.379292i	−0.531639	+	0.846971i	$0.678424\pi$
$740$		0			0
$741$	5.73205	−	5.73205i	0.210572	−	0.210572i
$742$		0			0
$743$			11.0718i			0.406185i	0.979160	+	0.203092i	$0.0650992\pi$
$743$			11.0718i			0.406185i	−0.979160	+	0.203092i	$0.934901\pi$
$744$		0			0
$745$	6.10770	+	3.52628i	0.223769	+	0.129193i
$746$		0			0
$747$	5.73205	+	1.53590i	0.209725	+	0.0561956i
$748$		0			0
$749$	3.86603	+	7.96410i	0.141261	+	0.291002i
$750$		0			0
$751$	6.52628	+	11.3038i	0.238147	+	0.412483i	0.960183	−	0.279373i	$-0.0901266\pi$
$751$	6.52628	+	11.3038i	0.238147	+	0.412483i	−0.722035	+	0.691856i	$0.756793\pi$
$752$		0			0
$753$	−6.43782	+	11.1506i	−0.234607	+	0.406352i
$754$		0			0
$755$	7.22243	+	7.22243i	0.262851	+	0.262851i
$756$		0			0
$757$	−18.6603	+	18.6603i	−0.678218	+	0.678218i	−0.959597	−	0.281378i	$-0.909208\pi$
$757$	−18.6603	+	18.6603i	−0.678218	+	0.678218i	0.281378	+	0.959597i	$0.409208\pi$
$758$		0			0
$759$	5.93782	+	3.42820i	0.215529	+	0.124436i
$760$		0			0
$761$	11.7679	−	6.79423i	0.426588	−	0.246291i	−0.271304	−	0.962494i	$-0.587455\pi$
$761$	11.7679	−	6.79423i	0.426588	−	0.246291i	0.697892	+	0.716203i	$0.254122\pi$
$762$		0			0
$763$	4.83975	+	3.27757i	0.175211	+	0.118656i
$764$		0			0
$765$	−4.09808	+	15.2942i	−0.148166	+	0.552964i
$766$		0			0
$767$	−30.9545	+	53.6147i	−1.11770	+	1.93592i
$768$		0			0
$769$	9.85641			0.355431			0.177716	−	0.984082i	$-0.443129\pi$
$769$	9.85641			0.355431			0.177716	+	0.984082i	$0.443129\pi$
$770$		0			0
$771$	−7.09808	−	7.09808i	−0.255631	−	0.255631i
$772$		0			0
$773$	−2.93782	−	10.9641i	−0.105666	−	0.394351i	0.892754	−	0.450545i	$-0.148770\pi$
$773$	−2.93782	−	10.9641i	−0.105666	−	0.394351i	−0.998420	+	0.0561936i	$0.982104\pi$
$774$		0			0
$775$	−13.5622	+	7.83013i	−0.487168	+	0.281266i
$776$		0			0
$777$	−1.05256	−	1.21539i	−0.0377603	−	0.0436019i
$778$		0			0
$779$	14.1244	+	3.78461i	0.506058	+	0.135598i
$780$		0			0
$781$	1.53590	−	0.411543i	0.0549588	−	0.0147262i
$782$		0			0
$783$	−1.12436			−0.0401812
$784$		0			0
$785$	16.9474			0.604880
$786$		0			0
$787$	13.1603	−	3.52628i	0.469112	−	0.125698i	−0.0165161	−	0.999864i	$-0.505257\pi$
$787$	13.1603	−	3.52628i	0.469112	−	0.125698i	0.485628	+	0.874165i	$0.338591\pi$
$788$		0			0
$789$	2.30385	+	0.617314i	0.0820191	+	0.0219770i
$790$		0			0
$791$	2.53590	+	2.92820i	0.0901662	+	0.104115i
$792$		0			0
$793$	32.9545	−	19.0263i	1.17025	−	0.675643i
$794$		0			0
$795$	−0.492958	−	1.83975i	−0.0174834	−	0.0652491i
$796$		0			0
$797$	−13.3397	−	13.3397i	−0.472518	−	0.472518i	0.430211	−	0.902729i	$-0.358439\pi$
$797$	−13.3397	−	13.3397i	−0.472518	−	0.472518i	−0.902729	+	0.430211i	$0.858439\pi$
$798$		0			0
$799$	−27.5885			−0.976009
$800$		0			0
$801$	7.09808	−	12.2942i	0.250798	−	0.434395i
$802$		0			0
$803$	−5.52628	+	20.6244i	−0.195018	+	0.727818i
$804$		0			0
$805$	−8.76795	−	5.93782i	−0.309030	−	0.209281i
$806$		0			0
$807$	−4.54552	+	2.62436i	−0.160010	+	0.0923817i
$808$		0			0
$809$	−29.4282	−	16.9904i	−1.03464	−	0.597350i	−0.116330	−	0.993211i	$-0.537113\pi$
$809$	−29.4282	−	16.9904i	−1.03464	−	0.597350i	−0.918311	+	0.395861i	$0.870446\pi$
$810$		0			0
$811$	−24.3205	+	24.3205i	−0.854009	+	0.854009i	−0.990624	−	0.136616i	$-0.956377\pi$
$811$	−24.3205	+	24.3205i	−0.854009	+	0.854009i	0.136616	+	0.990624i	$0.456377\pi$
$812$		0			0
$813$	−4.43782	−	4.43782i	−0.155641	−	0.155641i
$814$		0			0
$815$	5.59808	−	9.69615i	0.196092	−	0.339641i
$816$		0			0
$817$	−13.5622	−	23.4904i	−0.474481	−	0.821824i
$818$		0			0
$819$	16.6603	+	34.3205i	0.582156	+	1.19926i
$820$		0			0
$821$	0.598076	+	0.160254i	0.0208730	+	0.00559290i	0.269240	−	0.963073i	$-0.413227\pi$
$821$	0.598076	+	0.160254i	0.0208730	+	0.00559290i	−0.248367	+	0.968666i	$0.579894\pi$
$822$		0			0
$823$	14.9378	+	8.62436i	0.520700	+	0.300626i	0.737221	−	0.675652i	$-0.236138\pi$
$823$	14.9378	+	8.62436i	0.520700	+	0.300626i	−0.216521	+	0.976278i	$0.569471\pi$
$824$		0			0
$825$			6.44486i			0.224381i
$826$		0			0
$827$	3.78461	−	3.78461i	0.131604	−	0.131604i	−0.638237	−	0.769840i	$-0.720336\pi$
$827$	3.78461	−	3.78461i	0.131604	−	0.131604i	0.769840	+	0.638237i	$0.220336\pi$
$828$		0			0
$829$	−3.06218	+	0.820508i	−0.106354	+	0.0284974i	−0.311603	−	0.950212i	$-0.600866\pi$
$829$	−3.06218	+	0.820508i	−0.106354	+	0.0284974i	0.205250	+	0.978710i	$0.434199\pi$
$830$		0			0
$831$	1.20577	+	2.08846i	0.0418277	+	0.0724478i
$832$		0			0
$833$	−35.5526	−	27.9904i	−1.23182	−	0.969809i
$834$		0			0
$835$	5.07180	−	18.9282i	0.175517	−	0.655037i
$836$		0			0
$837$	−2.86603	−	10.6962i	−0.0990643	−	0.369713i
$838$		0			0
$839$		−	17.7128i		−	0.611514i	−0.952110	−	0.305757i	$-0.901090\pi$
$839$		−	17.7128i		−	0.611514i	0.952110	−	0.305757i	$-0.0989096\pi$
$840$		0			0
$841$		−	28.8564i		−	0.995048i
$842$		0			0
$843$	0.124356	+	0.464102i	0.00428304	+	0.0159845i
$844$		0			0
$845$	3.44744	−	12.8660i	0.118596	−	0.442605i
$846$		0			0
$847$	−5.49038	−	1.90192i	−0.188652	−	0.0653509i
$848$		0			0
$849$	−3.89230	−	6.74167i	−0.133584	−	0.231374i
$850$		0			0
$851$	5.06218	−	1.35641i	0.173529	−	0.0464970i
$852$		0			0
$853$	−11.8756	+	11.8756i	−0.406614	+	0.406614i	−0.880556	−	0.473942i	$-0.842831\pi$
$853$	−11.8756	+	11.8756i	−0.406614	+	0.406614i	0.473942	+	0.880556i	$0.342831\pi$
$854$		0			0
$855$			7.26795i			0.248559i
$856$		0			0
$857$	11.8923	+	6.86603i	0.406233	+	0.234539i	0.689170	−	0.724600i	$-0.257975\pi$
$857$	11.8923	+	6.86603i	0.406233	+	0.234539i	−0.282937	+	0.959139i	$0.591309\pi$
$858$		0			0
$859$	−13.6244	−	3.65064i	−0.464857	−	0.124558i	0.0187858	−	0.999824i	$-0.494020\pi$
$859$	−13.6244	−	3.65064i	−0.464857	−	0.124558i	−0.483643	+	0.875265i	$0.660687\pi$
$860$		0			0
$861$	3.78461	−	5.58846i	0.128979	−	0.190454i
$862$		0			0
$863$	−16.3301	−	28.2846i	−0.555884	−	0.962819i	−0.997834	−	0.0657797i	$-0.979047\pi$
$863$	−16.3301	−	28.2846i	−0.555884	−	0.962819i	0.441950	−	0.897040i	$-0.354287\pi$
$864$		0			0
$865$	−1.03590	+	1.79423i	−0.0352216	+	0.0610056i
$866$		0			0
$867$	9.07180	+	9.07180i	0.308094	+	0.308094i
$868$		0			0
$869$	34.9545	−	34.9545i	1.18575	−	1.18575i
$870$		0			0
$871$	23.4904	+	13.5622i	0.795941	+	0.459537i
$872$		0			0
$873$	6.92820	−	4.00000i	0.234484	−	0.135379i
$874$		0			0
$875$	1.56218	−	21.7583i	0.0528112	−	0.735566i
$876$		0			0
$877$	8.18653	−	30.5526i	0.276440	−	1.03169i	−0.678431	−	0.734664i	$-0.737340\pi$
$877$	8.18653	−	30.5526i	0.276440	−	1.03169i	0.954871	−	0.297022i	$-0.0959936\pi$
$878$		0			0
$879$	2.90192	−	5.02628i	0.0978795	−	0.169532i
$880$		0			0
$881$	50.0000			1.68454			0.842271	−	0.539054i	$-0.181218\pi$
$881$	50.0000			1.68454			0.842271	+	0.539054i	$0.181218\pi$
$882$		0			0
$883$	5.00000	+	5.00000i	0.168263	+	0.168263i	0.786216	−	0.617952i	$-0.212037\pi$
$883$	5.00000	+	5.00000i	0.168263	+	0.168263i	−0.617952	+	0.786216i	$0.712037\pi$
$884$		0			0
$885$	1.40897	+	5.25833i	0.0473619	+	0.176757i
$886$		0			0
$887$	39.7750	−	22.9641i	1.33551	−	0.771059i	0.349375	−	0.936983i	$-0.386394\pi$
$887$	39.7750	−	22.9641i	1.33551	−	0.771059i	0.986139	+	0.165924i	$0.0530606\pi$
$888$		0			0
$889$	16.3923	+	18.9282i	0.549780	+	0.634832i
$890$		0			0
$891$	19.0885	+	5.11474i	0.639487	+	0.171350i
$892$		0			0
$893$	−12.2321	+	3.27757i	−0.409330	+	0.109680i
$894$		0			0
$895$	8.32051			0.278124
$896$		0			0
$897$	12.1962			0.407218
$898$		0			0
$899$	−1.36603	+	0.366025i	−0.0455595	+	0.0122076i
$900$		0			0
$901$	25.6244	+	6.86603i	0.853671	+	0.228740i
$902$		0			0
$903$	−12.2942	+	2.36603i	−0.409126	+	0.0787364i
$904$		0			0
$905$	−14.7058	+	8.49038i	−0.488836	+	0.282230i
$906$		0			0
$907$	−7.16025	−	26.7224i	−0.237752	−	0.887304i	−0.976889	−	0.213748i	$-0.931433\pi$
$907$	−7.16025	−	26.7224i	−0.237752	−	0.887304i	0.739136	−	0.673556i	$-0.235234\pi$
$908$		0			0
$909$	−37.7846	−	37.7846i	−1.25324	−	1.25324i
$910$		0			0
$911$	−7.32051			−0.242539			−0.121270	−	0.992620i	$-0.538697\pi$
$911$	−7.32051			−0.242539			−0.121270	+	0.992620i	$0.538697\pi$
$912$		0			0
$913$	3.22243	−	5.58142i	0.106647	−	0.184718i
$914$		0			0
$915$	0.866025	−	3.23205i	0.0286299	−	0.106848i
$916$		0			0
$917$	−18.7224	+	9.08846i	−0.618269	+	0.300127i
$918$		0			0
$919$	15.8660	−	9.16025i	0.523372	−	0.302169i	−0.214941	−	0.976627i	$-0.568956\pi$
$919$	15.8660	−	9.16025i	0.523372	−	0.302169i	0.738313	+	0.674458i	$0.235623\pi$
$920$		0			0
$921$	5.70577	+	3.29423i	0.188012	+	0.108549i
$922$		0			0
$923$	2.00000	−	2.00000i	0.0658308	−	0.0658308i
$924$		0			0
$925$	3.48334	+	3.48334i	0.114531	+	0.114531i
$926$		0			0
$927$	−8.83013	+	15.2942i	−0.290019	+	0.502328i
$928$		0			0
$929$	−25.0167	−	43.3301i	−0.820770	−	1.42162i	−0.905110	−	0.425178i	$-0.860211\pi$
$929$	−25.0167	−	43.3301i	−0.820770	−	1.42162i	0.0843396	−	0.996437i	$-0.473122\pi$
$930$		0			0
$931$	−19.0885	−	8.18653i	−0.625599	−	0.268303i
$932$		0			0
$933$	3.23205	+	0.866025i	0.105813	+	0.0283524i
$934$		0			0
$935$	14.8923	+	8.59808i	0.487030	+	0.281187i
$936$		0			0
$937$		−	29.0718i		−	0.949734i	−0.880058	−	0.474867i	$-0.842496\pi$
$937$		−	29.0718i		−	0.949734i	0.880058	−	0.474867i	$-0.157504\pi$
$938$		0			0
$939$	7.90192	−	7.90192i	0.257870	−	0.257870i
$940$		0			0
$941$	−23.5263	+	6.30385i	−0.766935	+	0.205500i	−0.621017	−	0.783797i	$-0.713280\pi$
$941$	−23.5263	+	6.30385i	−0.766935	+	0.205500i	−0.145918	+	0.989297i	$0.546614\pi$
$942$		0			0
$943$	11.0000	+	19.0526i	0.358209	+	0.620437i
$944$		0			0
$945$	6.65064	+	2.30385i	0.216345	+	0.0749442i
$946$		0			0
$947$	1.44744	−	5.40192i	0.0470355	−	0.175539i	−0.938412	−	0.345518i	$-0.887703\pi$
$947$	1.44744	−	5.40192i	0.0470355	−	0.175539i	0.985448	+	0.169979i	$0.0543700\pi$
$948$		0			0
$949$	9.83013	+	36.6865i	0.319099	+	1.19090i
$950$		0			0
$951$		−	7.58846i		−	0.246073i
$952$		0			0
$953$			16.5359i			0.535650i	0.963468	+	0.267825i	$0.0863050\pi$
$953$			16.5359i			0.535650i	−0.963468	+	0.267825i	$0.913695\pi$
$954$		0			0
$955$	−2.00962	−	7.50000i	−0.0650297	−	0.242694i
$956$		0			0
$957$	−0.150635	+	0.562178i	−0.00486934	+	0.0181726i
$958$		0			0
$959$	−8.79423	−	45.6962i	−0.283980	−	1.47561i
$960$		0			0
$961$	8.53590	+	14.7846i	0.275352	+	0.476923i
$962$		0			0
$963$	8.83013	−	2.36603i	0.284547	−	0.0762441i
$964$		0			0
$965$	−14.5814	+	14.5814i	−0.469392	+	0.469392i
$966$		0			0
$967$			60.2487i			1.93747i	0.248102	+	0.968734i	$0.420193\pi$
$967$			60.2487i			1.93747i	−0.248102	+	0.968734i	$0.579807\pi$
$968$		0			0
$969$	8.59808	+	4.96410i	0.276210	+	0.159470i
$970$		0			0
$971$	52.4090	+	14.0429i	1.68188	+	0.450659i	0.968275	−	0.249885i	$-0.0803930\pi$
$971$	52.4090	+	14.0429i	1.68188	+	0.450659i	0.713608	+	0.700545i	$0.247060\pi$
$972$		0			0
$973$	−44.5167	−	3.19615i	−1.42714	−	0.102464i
$974$		0			0
$975$	5.73205	+	9.92820i	0.183573	+	0.317957i
$976$		0			0
$977$	8.57180	−	14.8468i	0.274236	−	0.474991i	−0.695706	−	0.718327i	$-0.744908\pi$
$977$	8.57180	−	14.8468i	0.274236	−	0.474991i	0.969942	+	0.243336i	$0.0782417\pi$
$978$		0			0
$979$	−10.9019	−	10.9019i	−0.348427	−	0.348427i
$980$		0			0
$981$	4.26795	−	4.26795i	0.136265	−	0.136265i
$982$		0			0
$983$	−17.1340	−	9.89230i	−0.546489	−	0.315516i	0.201216	−	0.979547i	$-0.435511\pi$
$983$	−17.1340	−	9.89230i	−0.546489	−	0.315516i	−0.747705	+	0.664031i	$0.768844\pi$
$984$		0			0
$985$	18.2942	−	10.5622i	0.582903	−	0.336539i
$986$		0			0
$987$	−0.418584	+	5.83013i	−0.0133237	+	0.185575i
$988$		0			0
$989$	10.5622	−	39.4186i	0.335858	−	1.25344i
$990$		0			0
$991$	−4.20577	+	7.28461i	−0.133601	+	0.231403i	−0.925062	−	0.379816i	$-0.875987\pi$
$991$	−4.20577	+	7.28461i	−0.133601	+	0.231403i	0.791461	+	0.611219i	$0.209321\pi$
$992$		0			0
$993$	−10.5167			−0.333736
$994$		0			0
$995$	−7.56218	−	7.56218i	−0.239737	−	0.239737i
$996$		0			0
$997$	0.401924	+	1.50000i	0.0127291	+	0.0475055i	0.971998	−	0.234988i	$-0.0755051\pi$
$997$	0.401924	+	1.50000i	0.0127291	+	0.0475055i	−0.959269	+	0.282494i	$0.908838\pi$
$998$		0			0
$999$	−3.01666	+	1.74167i	−0.0954429	+	0.0551040i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	448.2.ba.a.81.1		4
4.3	odd	2		112.2.w.b.109.1	yes	4
7.2	even	3		448.2.ba.b.401.1		4
8.3	odd	2		896.2.ba.b.417.1		4
8.5	even	2		896.2.ba.c.417.1		4
16.3	odd	4		896.2.ba.d.865.1		4
16.5	even	4		448.2.ba.b.305.1		4
16.11	odd	4		112.2.w.a.53.1	✓	4
16.13	even	4		896.2.ba.a.865.1		4
28.3	even	6		784.2.m.d.589.2		4
28.11	odd	6		784.2.m.e.589.2		4
28.19	even	6		784.2.x.a.765.1		4
28.23	odd	6		112.2.w.a.93.1	yes	4
28.27	even	2		784.2.x.h.557.1		4
56.37	even	6		896.2.ba.a.289.1		4
56.51	odd	6		896.2.ba.d.289.1		4
112.11	odd	12		784.2.m.e.197.2		4
112.27	even	4		784.2.x.a.165.1		4
112.37	even	12	inner	448.2.ba.a.177.1		4
112.51	odd	12		896.2.ba.b.737.1		4
112.59	even	12		784.2.m.d.197.2		4
112.75	even	12		784.2.x.h.373.1		4
112.93	even	12		896.2.ba.c.737.1		4
112.107	odd	12		112.2.w.b.37.1	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
112.2.w.a.53.1	✓	4	16.11	odd	4
112.2.w.a.93.1	yes	4	28.23	odd	6
112.2.w.b.37.1	yes	4	112.107	odd	12
112.2.w.b.109.1	yes	4	4.3	odd	2
448.2.ba.a.81.1		4	1.1	even	1	trivial
448.2.ba.a.177.1		4	112.37	even	12	inner
448.2.ba.b.305.1		4	16.5	even	4
448.2.ba.b.401.1		4	7.2	even	3
784.2.m.d.197.2		4	112.59	even	12
784.2.m.d.589.2		4	28.3	even	6
784.2.m.e.197.2		4	112.11	odd	12
784.2.m.e.589.2		4	28.11	odd	6
784.2.x.a.165.1		4	112.27	even	4
784.2.x.a.765.1		4	28.19	even	6
784.2.x.h.373.1		4	112.75	even	12
784.2.x.h.557.1		4	28.27	even	2
896.2.ba.a.289.1		4	56.37	even	6
896.2.ba.a.865.1		4	16.13	even	4
896.2.ba.b.417.1		4	8.3	odd	2
896.2.ba.b.737.1		4	112.51	odd	12
896.2.ba.c.417.1		4	8.5	even	2
896.2.ba.c.737.1		4	112.93	even	12
896.2.ba.d.289.1		4	56.51	odd	6
896.2.ba.d.865.1		4	16.3	odd	4

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:	\( N \)	\(=\)	\( 448 = 2^{6} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	448.ba (of order \(12\), degree \(4\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.500000 + 0.133975i) q^{3} +(-0.866025 - 0.232051i) q^{5} +(-1.73205 - 2.00000i) q^{7} +(-2.36603 + 1.36603i) q^{9} +O(q^{10})\)	\(q+(-0.500000 + 0.133975i) q^{3} +(-0.866025 - 0.232051i) q^{5} +(-1.73205 - 2.00000i) q^{7} +(-2.36603 + 1.36603i) q^{9} +(0.767949 + 2.86603i) q^{11} +(3.73205 + 3.73205i) q^{13} +0.464102 q^{15} +(-3.23205 + 5.59808i) q^{17} +(-0.767949 + 2.86603i) q^{19} +(1.13397 + 0.767949i) q^{21} +(-3.86603 + 2.23205i) q^{23} +(-3.63397 - 2.09808i) q^{25} +(2.09808 - 2.09808i) q^{27} +(-0.267949 - 0.267949i) q^{29} +(1.86603 - 3.23205i) q^{31} +(-0.767949 - 1.33013i) q^{33} +(1.03590 + 2.13397i) q^{35} +(-1.13397 - 0.303848i) q^{37} +(-2.36603 - 1.36603i) q^{39} -4.92820i q^{41} +(-6.46410 + 6.46410i) q^{43} +(2.36603 - 0.633975i) q^{45} +(2.13397 + 3.69615i) q^{47} +(-1.00000 + 6.92820i) q^{49} +(0.866025 - 3.23205i) q^{51} +(-1.06218 - 3.96410i) q^{53} -2.66025i q^{55} -1.53590i q^{57} +(3.03590 + 11.3301i) q^{59} +(1.86603 - 6.96410i) q^{61} +(6.83013 + 2.36603i) q^{63} +(-2.36603 - 4.09808i) q^{65} +(4.96410 - 1.33013i) q^{67} +(1.63397 - 1.63397i) q^{69} -0.535898i q^{71} +(6.23205 + 3.59808i) q^{73} +(2.09808 + 0.562178i) q^{75} +(4.40192 - 6.50000i) q^{77} +(-8.33013 - 14.4282i) q^{79} +(3.33013 - 5.76795i) q^{81} +(-1.53590 - 1.53590i) q^{83} +(4.09808 - 4.09808i) q^{85} +(0.169873 + 0.0980762i) q^{87} +(-4.50000 + 2.59808i) q^{89} +(1.00000 - 13.9282i) q^{91} +(-0.500000 + 1.86603i) q^{93} +(1.33013 - 2.30385i) q^{95} -2.92820 q^{97} +(-5.73205 - 5.73205i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{3} - 6 q^{9}+O(q^{10}) \) 4 * q - 2 * q^3 - 6 * q^9	\( 4 q - 2 q^{3} - 6 q^{9} + 10 q^{11} + 8 q^{13} - 12 q^{15} - 6 q^{17} - 10 q^{19} + 8 q^{21} - 12 q^{23} - 18 q^{25} - 2 q^{27} - 8 q^{29} + 4 q^{31} - 10 q^{33} + 18 q^{35} - 8 q^{37} - 6 q^{39} - 12 q^{43} + 6 q^{45} + 12 q^{47} - 4 q^{49} + 20 q^{53} + 26 q^{59} + 4 q^{61} + 10 q^{63} - 6 q^{65} + 6 q^{67} + 10 q^{69} + 18 q^{73} - 2 q^{75} + 28 q^{77} - 16 q^{79} - 4 q^{81} - 20 q^{83} + 6 q^{85} + 18 q^{87} - 18 q^{89} + 4 q^{91} - 2 q^{93} - 12 q^{95} + 16 q^{97} - 16 q^{99}+O(q^{100}) \) 4 * q - 2 * q^3 - 6 * q^9 + 10 * q^11 + 8 * q^13 - 12 * q^15 - 6 * q^17 - 10 * q^19 + 8 * q^21 - 12 * q^23 - 18 * q^25 - 2 * q^27 - 8 * q^29 + 4 * q^31 - 10 * q^33 + 18 * q^35 - 8 * q^37 - 6 * q^39 - 12 * q^43 + 6 * q^45 + 12 * q^47 - 4 * q^49 + 20 * q^53 + 26 * q^59 + 4 * q^61 + 10 * q^63 - 6 * q^65 + 6 * q^67 + 10 * q^69 + 18 * q^73 - 2 * q^75 + 28 * q^77 - 16 * q^79 - 4 * q^81 - 20 * q^83 + 6 * q^85 + 18 * q^87 - 18 * q^89 + 4 * q^91 - 2 * q^93 - 12 * q^95 + 16 * q^97 - 16 * q^99

Introduction

L-functions

Modular forms

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Database

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