LMFDB - Embedded newform 2576.2.a.v.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2576,2,Mod(1,2576)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2576.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2576, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2576 = 2^{4} \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2576.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,-2,0,2,0,3,0,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

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

Self dual:	yes
Analytic conductor:	$20.5694635607$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 161)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.17009$ of defining polynomial
Character	$\chi$	$=$	2576.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.17009 q^{3} -1.17009 q^{5} +1.00000 q^{7} -1.63090 q^{9} -3.70928 q^{11} +4.34017 q^{13} -1.36910 q^{15} +3.17009 q^{17} -5.26180 q^{19} +1.17009 q^{21} -1.00000 q^{23} -3.63090 q^{25} -5.41855 q^{27} +0.630898 q^{29} -9.32684 q^{31} -4.34017 q^{33} -1.17009 q^{35} -3.07838 q^{37} +5.07838 q^{39} +2.68035 q^{41} +8.49693 q^{43} +1.90829 q^{45} -6.09171 q^{47} +1.00000 q^{49} +3.70928 q^{51} +4.15676 q^{53} +4.34017 q^{55} -6.15676 q^{57} +8.40522 q^{59} -6.92881 q^{61} -1.63090 q^{63} -5.07838 q^{65} -9.86603 q^{67} -1.17009 q^{69} -10.8371 q^{71} -13.0205 q^{73} -4.24846 q^{75} -3.70928 q^{77} +4.38962 q^{79} -1.44748 q^{81} -14.6537 q^{83} -3.70928 q^{85} +0.738205 q^{87} -6.77205 q^{89} +4.34017 q^{91} -10.9132 q^{93} +6.15676 q^{95} +8.58864 q^{97} +6.04945 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 2 q^{3} + 2 q^{5} + 3 q^{7} - q^{9} - 4 q^{11} + 2 q^{13} - 8 q^{15} + 4 q^{17} - 8 q^{19} - 2 q^{21} - 3 q^{23} - 7 q^{25} - 2 q^{27} - 2 q^{29} - 16 q^{31} - 2 q^{33} + 2 q^{35} - 6 q^{37} + 12 q^{39}+ \cdots + 6 q^{97}+O(q^{100}) $ 3 * q - 2 * q^3 + 2 * q^5 + 3 * q^7 - q^9 - 4 * q^11 + 2 * q^13 - 8 * q^15 + 4 * q^17 - 8 * q^19 - 2 * q^21 - 3 * q^23 - 7 * q^25 - 2 * q^27 - 2 * q^29 - 16 * q^31 - 2 * q^33 + 2 * q^35 - 6 * q^37 + 12 * q^39 - 14 * q^41 + 8 * q^43 + 8 * q^45 - 16 * q^47 + 3 * q^49 + 4 * q^51 + 6 * q^53 + 2 * q^55 - 12 * q^57 + 10 * q^59 + 10 * q^61 - q^63 - 12 * q^65 - 16 * q^67 + 2 * q^69 - 4 * q^71 - 6 * q^73 - 4 * q^75 - 4 * q^77 - 16 * q^79 - 5 * q^81 - 20 * q^83 - 4 * q^85 + 10 * q^87 + 4 * q^89 + 2 * q^91 + 12 * q^93 + 12 * q^95 + 6 * q^97

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
$3$	1.17009	0.675550	0.337775	−	0.941227i	$-0.390326\pi$
$3$	1.17009	0.675550	0.337775	+	0.941227i	$0.390326\pi$
$4$	0	0
$5$	−1.17009	−0.523279	−0.261639	−	0.965166i	$-0.584263\pi$
$5$	−1.17009	−0.523279	−0.261639	+	0.965166i	$0.584263\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−1.63090	−0.543633
$10$	0	0
$11$	−3.70928	−1.11839	−0.559194	−	0.829037i	$-0.688889\pi$
$11$	−3.70928	−1.11839	−0.559194	+	0.829037i	$0.688889\pi$
$12$	0	0
$13$	4.34017	1.20375	0.601874	−	0.798591i	$-0.294421\pi$
$13$	4.34017	1.20375	0.601874	+	0.798591i	$0.294421\pi$
$14$	0	0
$15$	−1.36910	−0.353501
$16$	0	0
$17$	3.17009	0.768859	0.384429	−	0.923154i	$-0.374398\pi$
$17$	3.17009	0.768859	0.384429	+	0.923154i	$0.374398\pi$
$18$	0	0
$19$	−5.26180	−1.20714	−0.603569	−	0.797311i	$-0.706255\pi$
$19$	−5.26180	−1.20714	−0.603569	+	0.797311i	$0.706255\pi$
$20$	0	0
$21$	1.17009	0.255334
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	−3.63090	−0.726180
$26$	0	0
$27$	−5.41855	−1.04280
$28$	0	0
$29$	0.630898	0.117155	0.0585774	−	0.998283i	$-0.481344\pi$
$29$	0.630898	0.117155	0.0585774	+	0.998283i	$0.481344\pi$
$30$	0	0
$31$	−9.32684	−1.67515	−0.837575	−	0.546322i	$-0.816027\pi$
$31$	−9.32684	−1.67515	−0.837575	+	0.546322i	$0.816027\pi$
$32$	0	0
$33$	−4.34017	−0.755527
$34$	0	0
$35$	−1.17009	−0.197781
$36$	0	0
$37$	−3.07838	−0.506082	−0.253041	−	0.967456i	$-0.581431\pi$
$37$	−3.07838	−0.506082	−0.253041	+	0.967456i	$0.581431\pi$
$38$	0	0
$39$	5.07838	0.813191
$40$	0	0
$41$	2.68035	0.418600	0.209300	−	0.977852i	$-0.432882\pi$
$41$	2.68035	0.418600	0.209300	+	0.977852i	$0.432882\pi$
$42$	0	0
$43$	8.49693	1.29577	0.647885	−	0.761738i	$-0.275654\pi$
$43$	8.49693	1.29577	0.647885	+	0.761738i	$0.275654\pi$
$44$	0	0
$45$	1.90829	0.284471
$46$	0	0
$47$	−6.09171	−0.888567	−0.444284	−	0.895886i	$-0.646542\pi$
$47$	−6.09171	−0.888567	−0.444284	+	0.895886i	$0.646542\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	3.70928	0.519402
$52$	0	0
$53$	4.15676	0.570974	0.285487	−	0.958383i	$-0.407845\pi$
$53$	4.15676	0.570974	0.285487	+	0.958383i	$0.407845\pi$
$54$	0	0
$55$	4.34017	0.585229
$56$	0	0
$57$	−6.15676	−0.815482
$58$	0	0
$59$	8.40522	1.09427	0.547133	−	0.837046i	$-0.315719\pi$
$59$	8.40522	1.09427	0.547133	+	0.837046i	$0.315719\pi$
$60$	0	0
$61$	−6.92881	−0.887143	−0.443572	−	0.896239i	$-0.646289\pi$
$61$	−6.92881	−0.887143	−0.443572	+	0.896239i	$0.646289\pi$
$62$	0	0
$63$	−1.63090	−0.205474
$64$	0	0
$65$	−5.07838	−0.629895
$66$	0	0
$67$	−9.86603	−1.20533	−0.602664	−	0.797995i	$-0.705894\pi$
$67$	−9.86603	−1.20533	−0.602664	+	0.797995i	$0.705894\pi$
$68$	0	0
$69$	−1.17009	−0.140862
$70$	0	0
$71$	−10.8371	−1.28613	−0.643064	−	0.765813i	$-0.722337\pi$
$71$	−10.8371	−1.28613	−0.643064	+	0.765813i	$0.722337\pi$
$72$	0	0
$73$	−13.0205	−1.52394	−0.761968	−	0.647614i	$-0.775767\pi$
$73$	−13.0205	−1.52394	−0.761968	+	0.647614i	$0.775767\pi$
$74$	0	0
$75$	−4.24846	−0.490570
$76$	0	0
$77$	−3.70928	−0.422711
$78$	0	0
$79$	4.38962	0.493871	0.246935	−	0.969032i	$-0.420576\pi$
$79$	4.38962	0.493871	0.246935	+	0.969032i	$0.420576\pi$
$80$	0	0
$81$	−1.44748	−0.160831
$82$	0	0
$83$	−14.6537	−1.60845	−0.804225	−	0.594324i	$-0.797420\pi$
$83$	−14.6537	−1.60845	−0.804225	+	0.594324i	$0.797420\pi$
$84$	0	0
$85$	−3.70928	−0.402327
$86$	0	0
$87$	0.738205	0.0791439
$88$	0	0
$89$	−6.77205	−0.717836	−0.358918	−	0.933369i	$-0.616854\pi$
$89$	−6.77205	−0.717836	−0.358918	+	0.933369i	$0.616854\pi$
$90$	0	0
$91$	4.34017	0.454974
$92$	0	0
$93$	−10.9132	−1.13165
$94$	0	0
$95$	6.15676	0.631670
$96$	0	0
$97$	8.58864	0.872044	0.436022	−	0.899936i	$-0.356387\pi$
$97$	8.58864	0.872044	0.436022	+	0.899936i	$0.356387\pi$
$98$	0	0
$99$	6.04945	0.607992
$100$	0	0
$101$	−4.92162	−0.489720	−0.244860	−	0.969558i	$-0.578742\pi$
$101$	−4.92162	−0.489720	−0.244860	+	0.969558i	$0.578742\pi$
$102$	0	0
$103$	−16.6803	−1.64356	−0.821782	−	0.569803i	$-0.807020\pi$
$103$	−16.6803	−1.64356	−0.821782	+	0.569803i	$0.807020\pi$
$104$	0	0
$105$	−1.36910	−0.133611
$106$	0	0
$107$	15.3340	1.48240	0.741198	−	0.671286i	$-0.234258\pi$
$107$	15.3340	1.48240	0.741198	+	0.671286i	$0.234258\pi$
$108$	0	0
$109$	−11.1773	−1.07059	−0.535294	−	0.844666i	$-0.679799\pi$
$109$	−11.1773	−1.07059	−0.535294	+	0.844666i	$0.679799\pi$
$110$	0	0
$111$	−3.60197	−0.341884
$112$	0	0
$113$	13.5174	1.27161	0.635807	−	0.771848i	$-0.280667\pi$
$113$	13.5174	1.27161	0.635807	+	0.771848i	$0.280667\pi$
$114$	0	0
$115$	1.17009	0.109111
$116$	0	0
$117$	−7.07838	−0.654396
$118$	0	0
$119$	3.17009	0.290601
$120$	0	0
$121$	2.75872	0.250793
$122$	0	0
$123$	3.13624	0.282785
$124$	0	0
$125$	10.0989	0.903273
$126$	0	0
$127$	−2.92162	−0.259252	−0.129626	−	0.991563i	$-0.541378\pi$
$127$	−2.92162	−0.259252	−0.129626	+	0.991563i	$0.541378\pi$
$128$	0	0
$129$	9.94214	0.875357
$130$	0	0
$131$	6.27513	0.548260	0.274130	−	0.961693i	$-0.411610\pi$
$131$	6.27513	0.548260	0.274130	+	0.961693i	$0.411610\pi$
$132$	0	0
$133$	−5.26180	−0.456256
$134$	0	0
$135$	6.34017	0.545675
$136$	0	0
$137$	7.75872	0.662873	0.331436	−	0.943478i	$-0.392467\pi$
$137$	7.75872	0.662873	0.331436	+	0.943478i	$0.392467\pi$
$138$	0	0
$139$	2.92881	0.248418	0.124209	−	0.992256i	$-0.460361\pi$
$139$	2.92881	0.248418	0.124209	+	0.992256i	$0.460361\pi$
$140$	0	0
$141$	−7.12783	−0.600271
$142$	0	0
$143$	−16.0989	−1.34626
$144$	0	0
$145$	−0.738205	−0.0613046
$146$	0	0
$147$	1.17009	0.0965071
$148$	0	0
$149$	0.837101	0.0685780	0.0342890	−	0.999412i	$-0.489083\pi$
$149$	0.837101	0.0685780	0.0342890	+	0.999412i	$0.489083\pi$
$150$	0	0
$151$	−15.9155	−1.29518	−0.647592	−	0.761988i	$-0.724224\pi$
$151$	−15.9155	−1.29518	−0.647592	+	0.761988i	$0.724224\pi$
$152$	0	0
$153$	−5.17009	−0.417977
$154$	0	0
$155$	10.9132	0.876570
$156$	0	0
$157$	−11.1701	−0.891470	−0.445735	−	0.895165i	$-0.647058\pi$
$157$	−11.1701	−0.891470	−0.445735	+	0.895165i	$0.647058\pi$
$158$	0	0
$159$	4.86376	0.385722
$160$	0	0
$161$	−1.00000	−0.0788110
$162$	0	0
$163$	−14.6225	−1.14532	−0.572661	−	0.819792i	$-0.694089\pi$
$163$	−14.6225	−1.14532	−0.572661	+	0.819792i	$0.694089\pi$
$164$	0	0
$165$	5.07838	0.395351
$166$	0	0
$167$	−6.00719	−0.464850	−0.232425	−	0.972614i	$-0.574666\pi$
$167$	−6.00719	−0.464850	−0.232425	+	0.972614i	$0.574666\pi$
$168$	0	0
$169$	5.83710	0.449008
$170$	0	0
$171$	8.58145	0.656240
$172$	0	0
$173$	14.0989	1.07192	0.535960	−	0.844244i	$-0.319950\pi$
$173$	14.0989	1.07192	0.535960	+	0.844244i	$0.319950\pi$
$174$	0	0
$175$	−3.63090	−0.274470
$176$	0	0
$177$	9.83483	0.739231
$178$	0	0
$179$	18.1256	1.35477	0.677384	−	0.735630i	$-0.263114\pi$
$179$	18.1256	1.35477	0.677384	+	0.735630i	$0.263114\pi$
$180$	0	0
$181$	12.4319	0.924054	0.462027	−	0.886866i	$-0.347122\pi$
$181$	12.4319	0.924054	0.462027	+	0.886866i	$0.347122\pi$
$182$	0	0
$183$	−8.10731	−0.599309
$184$	0	0
$185$	3.60197	0.264822
$186$	0	0
$187$	−11.7587	−0.859883
$188$	0	0
$189$	−5.41855	−0.394142
$190$	0	0
$191$	4.86376	0.351930	0.175965	−	0.984396i	$-0.443696\pi$
$191$	4.86376	0.351930	0.175965	+	0.984396i	$0.443696\pi$
$192$	0	0
$193$	−17.2351	−1.24061	−0.620306	−	0.784360i	$-0.712992\pi$
$193$	−17.2351	−1.24061	−0.620306	+	0.784360i	$0.712992\pi$
$194$	0	0
$195$	−5.94214	−0.425526
$196$	0	0
$197$	−12.5236	−0.892269	−0.446134	−	0.894966i	$-0.647200\pi$
$197$	−12.5236	−0.892269	−0.446134	+	0.894966i	$0.647200\pi$
$198$	0	0
$199$	−1.16290	−0.0824357	−0.0412178	−	0.999150i	$-0.513124\pi$
$199$	−1.16290	−0.0824357	−0.0412178	+	0.999150i	$0.513124\pi$
$200$	0	0
$201$	−11.5441	−0.814259
$202$	0	0
$203$	0.630898	0.0442803
$204$	0	0
$205$	−3.13624	−0.219044
$206$	0	0
$207$	1.63090	0.113355
$208$	0	0
$209$	19.5174	1.35005
$210$	0	0
$211$	−4.76487	−0.328027	−0.164013	−	0.986458i	$-0.552444\pi$
$211$	−4.76487	−0.328027	−0.164013	+	0.986458i	$0.552444\pi$
$212$	0	0
$213$	−12.6803	−0.868843
$214$	0	0
$215$	−9.94214	−0.678048
$216$	0	0
$217$	−9.32684	−0.633147
$218$	0	0
$219$	−15.2351	−1.02949
$220$	0	0
$221$	13.7587	0.925512
$222$	0	0
$223$	21.3679	1.43090	0.715450	−	0.698664i	$-0.246222\pi$
$223$	21.3679	1.43090	0.715450	+	0.698664i	$0.246222\pi$
$224$	0	0
$225$	5.92162	0.394775
$226$	0	0
$227$	5.84324	0.387830	0.193915	−	0.981018i	$-0.437881\pi$
$227$	5.84324	0.387830	0.193915	+	0.981018i	$0.437881\pi$
$228$	0	0
$229$	2.85658	0.188768	0.0943839	−	0.995536i	$-0.469912\pi$
$229$	2.85658	0.188768	0.0943839	+	0.995536i	$0.469912\pi$
$230$	0	0
$231$	−4.34017	−0.285562
$232$	0	0
$233$	−13.5259	−0.886108	−0.443054	−	0.896495i	$-0.646105\pi$
$233$	−13.5259	−0.886108	−0.443054	+	0.896495i	$0.646105\pi$
$234$	0	0
$235$	7.12783	0.464968
$236$	0	0
$237$	5.13624	0.333634
$238$	0	0
$239$	−4.76487	−0.308214	−0.154107	−	0.988054i	$-0.549250\pi$
$239$	−4.76487	−0.308214	−0.154107	+	0.988054i	$0.549250\pi$
$240$	0	0
$241$	24.4775	1.57673	0.788366	−	0.615207i	$-0.210928\pi$
$241$	24.4775	1.57673	0.788366	+	0.615207i	$0.210928\pi$
$242$	0	0
$243$	14.5620	0.934151
$244$	0	0
$245$	−1.17009	−0.0747541
$246$	0	0
$247$	−22.8371	−1.45309
$248$	0	0
$249$	−17.1461	−1.08659
$250$	0	0
$251$	26.2557	1.65724	0.828621	−	0.559810i	$-0.189126\pi$
$251$	26.2557	1.65724	0.828621	+	0.559810i	$0.189126\pi$
$252$	0	0
$253$	3.70928	0.233200
$254$	0	0
$255$	−4.34017	−0.271792
$256$	0	0
$257$	7.84324	0.489248	0.244624	−	0.969618i	$-0.421335\pi$
$257$	7.84324	0.489248	0.244624	+	0.969618i	$0.421335\pi$
$258$	0	0
$259$	−3.07838	−0.191281
$260$	0	0
$261$	−1.02893	−0.0636891
$262$	0	0
$263$	−10.9132	−0.672937	−0.336469	−	0.941695i	$-0.609233\pi$
$263$	−10.9132	−0.672937	−0.336469	+	0.941695i	$0.609233\pi$
$264$	0	0
$265$	−4.86376	−0.298779
$266$	0	0
$267$	−7.92389	−0.484934
$268$	0	0
$269$	−3.57531	−0.217990	−0.108995	−	0.994042i	$-0.534763\pi$
$269$	−3.57531	−0.217990	−0.108995	+	0.994042i	$0.534763\pi$
$270$	0	0
$271$	5.98053	0.363291	0.181646	−	0.983364i	$-0.441858\pi$
$271$	5.98053	0.363291	0.181646	+	0.983364i	$0.441858\pi$
$272$	0	0
$273$	5.07838	0.307357
$274$	0	0
$275$	13.4680	0.812151
$276$	0	0
$277$	−13.3112	−0.799795	−0.399898	−	0.916560i	$-0.630954\pi$
$277$	−13.3112	−0.799795	−0.399898	+	0.916560i	$0.630954\pi$
$278$	0	0
$279$	15.2111	0.910666
$280$	0	0
$281$	−2.13009	−0.127071	−0.0635354	−	0.997980i	$-0.520238\pi$
$281$	−2.13009	−0.127071	−0.0635354	+	0.997980i	$0.520238\pi$
$282$	0	0
$283$	−13.6742	−0.812847	−0.406423	−	0.913685i	$-0.633224\pi$
$283$	−13.6742	−0.812847	−0.406423	+	0.913685i	$0.633224\pi$
$284$	0	0
$285$	7.20394	0.426724
$286$	0	0
$287$	2.68035	0.158216
$288$	0	0
$289$	−6.95055	−0.408856
$290$	0	0
$291$	10.0494	0.589109
$292$	0	0
$293$	−13.2690	−0.775182	−0.387591	−	0.921831i	$-0.626693\pi$
$293$	−13.2690	−0.775182	−0.387591	+	0.921831i	$0.626693\pi$
$294$	0	0
$295$	−9.83483	−0.572606
$296$	0	0
$297$	20.0989	1.16626
$298$	0	0
$299$	−4.34017	−0.250999
$300$	0	0
$301$	8.49693	0.489755
$302$	0	0
$303$	−5.75872	−0.330830
$304$	0	0
$305$	8.10731	0.464223
$306$	0	0
$307$	17.5103	0.999363	0.499682	−	0.866209i	$-0.333450\pi$
$307$	17.5103	0.999363	0.499682	+	0.866209i	$0.333450\pi$
$308$	0	0
$309$	−19.5174	−1.11031
$310$	0	0
$311$	−2.03385	−0.115329	−0.0576645	−	0.998336i	$-0.518365\pi$
$311$	−2.03385	−0.115329	−0.0576645	+	0.998336i	$0.518365\pi$
$312$	0	0
$313$	12.3786	0.699677	0.349839	−	0.936810i	$-0.386236\pi$
$313$	12.3786	0.699677	0.349839	+	0.936810i	$0.386236\pi$
$314$	0	0
$315$	1.90829	0.107520
$316$	0	0
$317$	−20.1483	−1.13164	−0.565822	−	0.824527i	$-0.691441\pi$
$317$	−20.1483	−1.13164	−0.565822	+	0.824527i	$0.691441\pi$
$318$	0	0
$319$	−2.34017	−0.131025
$320$	0	0
$321$	17.9421	1.00143
$322$	0	0
$323$	−16.6803	−0.928119
$324$	0	0
$325$	−15.7587	−0.874137
$326$	0	0
$327$	−13.0784	−0.723236
$328$	0	0
$329$	−6.09171	−0.335847
$330$	0	0
$331$	6.70701	0.368650	0.184325	−	0.982865i	$-0.440990\pi$
$331$	6.70701	0.368650	0.184325	+	0.982865i	$0.440990\pi$
$332$	0	0
$333$	5.02052	0.275123
$334$	0	0
$335$	11.5441	0.630722
$336$	0	0
$337$	−29.0205	−1.58085	−0.790424	−	0.612560i	$-0.790140\pi$
$337$	−29.0205	−1.58085	−0.790424	+	0.612560i	$0.790140\pi$
$338$	0	0
$339$	15.8166	0.859039
$340$	0	0
$341$	34.5958	1.87347
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	1.36910	0.0737100
$346$	0	0
$347$	18.2413	0.979243	0.489622	−	0.871935i	$-0.337135\pi$
$347$	18.2413	0.979243	0.489622	+	0.871935i	$0.337135\pi$
$348$	0	0
$349$	−8.35455	−0.447209	−0.223604	−	0.974680i	$-0.571782\pi$
$349$	−8.35455	−0.447209	−0.223604	+	0.974680i	$0.571782\pi$
$350$	0	0
$351$	−23.5174	−1.25527
$352$	0	0
$353$	−31.5897	−1.68135	−0.840675	−	0.541541i	$-0.817841\pi$
$353$	−31.5897	−1.68135	−0.840675	+	0.541541i	$0.817841\pi$
$354$	0	0
$355$	12.6803	0.673003
$356$	0	0
$357$	3.70928	0.196316
$358$	0	0
$359$	4.48852	0.236895	0.118447	−	0.992960i	$-0.462208\pi$
$359$	4.48852	0.236895	0.118447	+	0.992960i	$0.462208\pi$
$360$	0	0
$361$	8.68649	0.457184
$362$	0	0
$363$	3.22795	0.169423
$364$	0	0
$365$	15.2351	0.797443
$366$	0	0
$367$	18.7526	0.978877	0.489438	−	0.872038i	$-0.337202\pi$
$367$	18.7526	0.978877	0.489438	+	0.872038i	$0.337202\pi$
$368$	0	0
$369$	−4.37137	−0.227564
$370$	0	0
$371$	4.15676	0.215808
$372$	0	0
$373$	22.5113	1.16559	0.582796	−	0.812619i	$-0.301959\pi$
$373$	22.5113	1.16559	0.582796	+	0.812619i	$0.301959\pi$
$374$	0	0
$375$	11.8166	0.610206
$376$	0	0
$377$	2.73820	0.141025
$378$	0	0
$379$	22.3318	1.14711	0.573553	−	0.819169i	$-0.305565\pi$
$379$	22.3318	1.14711	0.573553	+	0.819169i	$0.305565\pi$
$380$	0	0
$381$	−3.41855	−0.175138
$382$	0	0
$383$	−15.2351	−0.778479	−0.389239	−	0.921137i	$-0.627262\pi$
$383$	−15.2351	−0.778479	−0.389239	+	0.921137i	$0.627262\pi$
$384$	0	0
$385$	4.34017	0.221196
$386$	0	0
$387$	−13.8576	−0.704422
$388$	0	0
$389$	21.3874	1.08438	0.542191	−	0.840255i	$-0.317595\pi$
$389$	21.3874	1.08438	0.542191	+	0.840255i	$0.317595\pi$
$390$	0	0
$391$	−3.17009	−0.160318
$392$	0	0
$393$	7.34244	0.370377
$394$	0	0
$395$	−5.13624	−0.258432
$396$	0	0
$397$	20.6537	1.03658	0.518289	−	0.855205i	$-0.326569\pi$
$397$	20.6537	1.03658	0.518289	+	0.855205i	$0.326569\pi$
$398$	0	0
$399$	−6.15676	−0.308223
$400$	0	0
$401$	24.3402	1.21549	0.607745	−	0.794132i	$-0.292074\pi$
$401$	24.3402	1.21549	0.607745	+	0.794132i	$0.292074\pi$
$402$	0	0
$403$	−40.4801	−2.01646
$404$	0	0
$405$	1.69368	0.0841595
$406$	0	0
$407$	11.4186	0.565997
$408$	0	0
$409$	17.2885	0.854859	0.427430	−	0.904049i	$-0.359419\pi$
$409$	17.2885	0.854859	0.427430	+	0.904049i	$0.359419\pi$
$410$	0	0
$411$	9.07838	0.447803
$412$	0	0
$413$	8.40522	0.413594
$414$	0	0
$415$	17.1461	0.841668
$416$	0	0
$417$	3.42696	0.167819
$418$	0	0
$419$	3.05172	0.149086	0.0745430	−	0.997218i	$-0.476250\pi$
$419$	3.05172	0.149086	0.0745430	+	0.997218i	$0.476250\pi$
$420$	0	0
$421$	13.8843	0.676679	0.338339	−	0.941024i	$-0.390135\pi$
$421$	13.8843	0.676679	0.338339	+	0.941024i	$0.390135\pi$
$422$	0	0
$423$	9.93495	0.483054
$424$	0	0
$425$	−11.5103	−0.558330
$426$	0	0
$427$	−6.92881	−0.335309
$428$	0	0
$429$	−18.8371	−0.909464
$430$	0	0
$431$	−15.2039	−0.732348	−0.366174	−	0.930546i	$-0.619332\pi$
$431$	−15.2039	−0.732348	−0.366174	+	0.930546i	$0.619332\pi$
$432$	0	0
$433$	12.8566	0.617848	0.308924	−	0.951087i	$-0.400031\pi$
$433$	12.8566	0.617848	0.308924	+	0.951087i	$0.400031\pi$
$434$	0	0
$435$	−0.863763	−0.0414143
$436$	0	0
$437$	5.26180	0.251706
$438$	0	0
$439$	−14.5620	−0.695005	−0.347503	−	0.937679i	$-0.612970\pi$
$439$	−14.5620	−0.695005	−0.347503	+	0.937679i	$0.612970\pi$
$440$	0	0
$441$	−1.63090	−0.0776618
$442$	0	0
$443$	30.2557	1.43749	0.718745	−	0.695274i	$-0.244717\pi$
$443$	30.2557	1.43749	0.718745	+	0.695274i	$0.244717\pi$
$444$	0	0
$445$	7.92389	0.375628
$446$	0	0
$447$	0.979481	0.0463279
$448$	0	0
$449$	−4.68422	−0.221062	−0.110531	−	0.993873i	$-0.535255\pi$
$449$	−4.68422	−0.221062	−0.110531	+	0.993873i	$0.535255\pi$
$450$	0	0
$451$	−9.94214	−0.468157
$452$	0	0
$453$	−18.6225	−0.874961
$454$	0	0
$455$	−5.07838	−0.238078
$456$	0	0
$457$	23.4596	1.09739	0.548697	−	0.836022i	$-0.315124\pi$
$457$	23.4596	1.09739	0.548697	+	0.836022i	$0.315124\pi$
$458$	0	0
$459$	−17.1773	−0.801767
$460$	0	0
$461$	6.58145	0.306529	0.153264	−	0.988185i	$-0.451021\pi$
$461$	6.58145	0.306529	0.153264	+	0.988185i	$0.451021\pi$
$462$	0	0
$463$	−20.0144	−0.930147	−0.465073	−	0.885272i	$-0.653972\pi$
$463$	−20.0144	−0.930147	−0.465073	+	0.885272i	$0.653972\pi$
$464$	0	0
$465$	12.7694	0.592167
$466$	0	0
$467$	−21.4596	−0.993031	−0.496516	−	0.868028i	$-0.665387\pi$
$467$	−21.4596	−0.993031	−0.496516	+	0.868028i	$0.665387\pi$
$468$	0	0
$469$	−9.86603	−0.455571
$470$	0	0
$471$	−13.0700	−0.602232
$472$	0	0
$473$	−31.5174	−1.44917
$474$	0	0
$475$	19.1050	0.876599
$476$	0	0
$477$	−6.77924	−0.310400
$478$	0	0
$479$	14.3090	0.653794	0.326897	−	0.945060i	$-0.393997\pi$
$479$	14.3090	0.653794	0.326897	+	0.945060i	$0.393997\pi$
$480$	0	0
$481$	−13.3607	−0.609195
$482$	0	0
$483$	−1.17009	−0.0532408
$484$	0	0
$485$	−10.0494	−0.456322
$486$	0	0
$487$	5.36069	0.242916	0.121458	−	0.992597i	$-0.461243\pi$
$487$	5.36069	0.242916	0.121458	+	0.992597i	$0.461243\pi$
$488$	0	0
$489$	−17.1096	−0.773722
$490$	0	0
$491$	−8.99386	−0.405887	−0.202944	−	0.979190i	$-0.565051\pi$
$491$	−8.99386	−0.405887	−0.202944	+	0.979190i	$0.565051\pi$
$492$	0	0
$493$	2.00000	0.0900755
$494$	0	0
$495$	−7.07838	−0.318149
$496$	0	0
$497$	−10.8371	−0.486110
$498$	0	0
$499$	−2.02666	−0.0907259	−0.0453629	−	0.998971i	$-0.514444\pi$
$499$	−2.02666	−0.0907259	−0.0453629	+	0.998971i	$0.514444\pi$
$500$	0	0
$501$	−7.02893	−0.314029
$502$	0	0
$503$	32.4124	1.44520	0.722599	−	0.691268i	$-0.242947\pi$
$503$	32.4124	1.44520	0.722599	+	0.691268i	$0.242947\pi$
$504$	0	0
$505$	5.75872	0.256260
$506$	0	0
$507$	6.82991	0.303327
$508$	0	0
$509$	−10.4826	−0.464631	−0.232315	−	0.972640i	$-0.574630\pi$
$509$	−10.4826	−0.464631	−0.232315	+	0.972640i	$0.574630\pi$
$510$	0	0
$511$	−13.0205	−0.575994
$512$	0	0
$513$	28.5113	1.25880
$514$	0	0
$515$	19.5174	0.860041
$516$	0	0
$517$	22.5958	0.993763
$518$	0	0
$519$	16.4969	0.724135
$520$	0	0
$521$	−24.6030	−1.07788	−0.538939	−	0.842345i	$-0.681175\pi$
$521$	−24.6030	−1.07788	−0.538939	+	0.842345i	$0.681175\pi$
$522$	0	0
$523$	−12.1978	−0.533372	−0.266686	−	0.963783i	$-0.585929\pi$
$523$	−12.1978	−0.533372	−0.266686	+	0.963783i	$0.585929\pi$
$524$	0	0
$525$	−4.24846	−0.185418
$526$	0	0
$527$	−29.5669	−1.28795
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−13.7081	−0.594879
$532$	0	0
$533$	11.6332	0.503888
$534$	0	0
$535$	−17.9421	−0.775706
$536$	0	0
$537$	21.2085	0.915213
$538$	0	0
$539$	−3.70928	−0.159770
$540$	0	0
$541$	17.5259	0.753495	0.376748	−	0.926316i	$-0.377042\pi$
$541$	17.5259	0.753495	0.376748	+	0.926316i	$0.377042\pi$
$542$	0	0
$543$	14.5464	0.624245
$544$	0	0
$545$	13.0784	0.560216
$546$	0	0
$547$	−12.5958	−0.538559	−0.269279	−	0.963062i	$-0.586785\pi$
$547$	−12.5958	−0.538559	−0.269279	+	0.963062i	$0.586785\pi$
$548$	0	0
$549$	11.3002	0.482280
$550$	0	0
$551$	−3.31965	−0.141422
$552$	0	0
$553$	4.38962	0.186666
$554$	0	0
$555$	4.21461	0.178900
$556$	0	0
$557$	−26.5113	−1.12332	−0.561660	−	0.827368i	$-0.689837\pi$
$557$	−26.5113	−1.12332	−0.561660	+	0.827368i	$0.689837\pi$
$558$	0	0
$559$	36.8781	1.55978
$560$	0	0
$561$	−13.7587	−0.580894
$562$	0	0
$563$	17.0472	0.718453	0.359227	−	0.933250i	$-0.383041\pi$
$563$	17.0472	0.718453	0.359227	+	0.933250i	$0.383041\pi$
$564$	0	0
$565$	−15.8166	−0.665409
$566$	0	0
$567$	−1.44748	−0.0607885
$568$	0	0
$569$	−25.1194	−1.05306	−0.526530	−	0.850156i	$-0.676507\pi$
$569$	−25.1194	−1.05306	−0.526530	+	0.850156i	$0.676507\pi$
$570$	0	0
$571$	12.0144	0.502786	0.251393	−	0.967885i	$-0.419111\pi$
$571$	12.0144	0.502786	0.251393	+	0.967885i	$0.419111\pi$
$572$	0	0
$573$	5.69102	0.237746
$574$	0	0
$575$	3.63090	0.151419
$576$	0	0
$577$	−10.1133	−0.421021	−0.210511	−	0.977592i	$-0.567513\pi$
$577$	−10.1133	−0.421021	−0.210511	+	0.977592i	$0.567513\pi$
$578$	0	0
$579$	−20.1666	−0.838095
$580$	0	0
$581$	−14.6537	−0.607937
$582$	0	0
$583$	−15.4186	−0.638571
$584$	0	0
$585$	8.28231	0.342432
$586$	0	0
$587$	14.2485	0.588097	0.294049	−	0.955790i	$-0.404997\pi$
$587$	14.2485	0.588097	0.294049	+	0.955790i	$0.404997\pi$
$588$	0	0
$589$	49.0759	2.02214
$590$	0	0
$591$	−14.6537	−0.602772
$592$	0	0
$593$	−20.2557	−0.831800	−0.415900	−	0.909410i	$-0.636533\pi$
$593$	−20.2557	−0.831800	−0.415900	+	0.909410i	$0.636533\pi$
$594$	0	0
$595$	−3.70928	−0.152065
$596$	0	0
$597$	−1.36069	−0.0556894
$598$	0	0
$599$	19.2351	0.785926	0.392963	−	0.919554i	$-0.371450\pi$
$599$	19.2351	0.785926	0.392963	+	0.919554i	$0.371450\pi$
$600$	0	0
$601$	−21.4329	−0.874267	−0.437134	−	0.899397i	$-0.644006\pi$
$601$	−21.4329	−0.874267	−0.437134	+	0.899397i	$0.644006\pi$
$602$	0	0
$603$	16.0905	0.655255
$604$	0	0
$605$	−3.22795	−0.131235
$606$	0	0
$607$	37.7926	1.53395	0.766977	−	0.641675i	$-0.221760\pi$
$607$	37.7926	1.53395	0.766977	+	0.641675i	$0.221760\pi$
$608$	0	0
$609$	0.738205	0.0299136
$610$	0	0
$611$	−26.4391	−1.06961
$612$	0	0
$613$	14.6947	0.593514	0.296757	−	0.954953i	$-0.404095\pi$
$613$	14.6947	0.593514	0.296757	+	0.954953i	$0.404095\pi$
$614$	0	0
$615$	−3.66967	−0.147975
$616$	0	0
$617$	12.1256	0.488157	0.244078	−	0.969756i	$-0.421515\pi$
$617$	12.1256	0.488157	0.244078	+	0.969756i	$0.421515\pi$
$618$	0	0
$619$	12.0144	0.482899	0.241449	−	0.970413i	$-0.422377\pi$
$619$	12.0144	0.482899	0.241449	+	0.970413i	$0.422377\pi$
$620$	0	0
$621$	5.41855	0.217439
$622$	0	0
$623$	−6.77205	−0.271317
$624$	0	0
$625$	6.33791	0.253516
$626$	0	0
$627$	22.8371	0.912026
$628$	0	0
$629$	−9.75872	−0.389106
$630$	0	0
$631$	−10.9300	−0.435118	−0.217559	−	0.976047i	$-0.569809\pi$
$631$	−10.9300	−0.435118	−0.217559	+	0.976047i	$0.569809\pi$
$632$	0	0
$633$	−5.57531	−0.221599
$634$	0	0
$635$	3.41855	0.135661
$636$	0	0
$637$	4.34017	0.171964
$638$	0	0
$639$	17.6742	0.699181
$640$	0	0
$641$	−33.1194	−1.30814	−0.654069	−	0.756435i	$-0.726939\pi$
$641$	−33.1194	−1.30814	−0.654069	+	0.756435i	$0.726939\pi$
$642$	0	0
$643$	17.5297	0.691305	0.345653	−	0.938363i	$-0.387658\pi$
$643$	17.5297	0.691305	0.345653	+	0.938363i	$0.387658\pi$
$644$	0	0
$645$	−11.6332	−0.458055
$646$	0	0
$647$	6.88777	0.270786	0.135393	−	0.990792i	$-0.456770\pi$
$647$	6.88777	0.270786	0.135393	+	0.990792i	$0.456770\pi$
$648$	0	0
$649$	−31.1773	−1.22382
$650$	0	0
$651$	−10.9132	−0.427722
$652$	0	0
$653$	−36.3545	−1.42266	−0.711332	−	0.702856i	$-0.751908\pi$
$653$	−36.3545	−1.42266	−0.711332	+	0.702856i	$0.751908\pi$
$654$	0	0
$655$	−7.34244	−0.286893
$656$	0	0
$657$	21.2351	0.828461
$658$	0	0
$659$	2.63931	0.102813	0.0514064	−	0.998678i	$-0.483630\pi$
$659$	2.63931	0.102813	0.0514064	+	0.998678i	$0.483630\pi$
$660$	0	0
$661$	21.0010	0.816846	0.408423	−	0.912793i	$-0.366079\pi$
$661$	21.0010	0.816846	0.408423	+	0.912793i	$0.366079\pi$
$662$	0	0
$663$	16.0989	0.625229
$664$	0	0
$665$	6.15676	0.238749
$666$	0	0
$667$	−0.630898	−0.0244285
$668$	0	0
$669$	25.0023	0.966644
$670$	0	0
$671$	25.7009	0.992171
$672$	0	0
$673$	−51.2534	−1.97567	−0.987836	−	0.155497i	$-0.950302\pi$
$673$	−51.2534	−1.97567	−0.987836	+	0.155497i	$0.950302\pi$
$674$	0	0
$675$	19.6742	0.757260
$676$	0	0
$677$	41.4668	1.59370	0.796849	−	0.604179i	$-0.206499\pi$
$677$	41.4668	1.59370	0.796849	+	0.604179i	$0.206499\pi$
$678$	0	0
$679$	8.58864	0.329602
$680$	0	0
$681$	6.83710	0.261998
$682$	0	0
$683$	26.2557	1.00464	0.502322	−	0.864680i	$-0.332479\pi$
$683$	26.2557	1.00464	0.502322	+	0.864680i	$0.332479\pi$
$684$	0	0
$685$	−9.07838	−0.346867
$686$	0	0
$687$	3.34244	0.127522
$688$	0	0
$689$	18.0410	0.687309
$690$	0	0
$691$	12.0338	0.457789	0.228895	−	0.973451i	$-0.426489\pi$
$691$	12.0338	0.457789	0.228895	+	0.973451i	$0.426489\pi$
$692$	0	0
$693$	6.04945	0.229800
$694$	0	0
$695$	−3.42696	−0.129992
$696$	0	0
$697$	8.49693	0.321844
$698$	0	0
$699$	−15.8264	−0.598610
$700$	0	0
$701$	34.6491	1.30868	0.654340	−	0.756200i	$-0.272946\pi$
$701$	34.6491	1.30868	0.654340	+	0.756200i	$0.272946\pi$
$702$	0	0
$703$	16.1978	0.610911
$704$	0	0
$705$	8.34017	0.314109
$706$	0	0
$707$	−4.92162	−0.185097
$708$	0	0
$709$	7.65983	0.287671	0.143835	−	0.989602i	$-0.454056\pi$
$709$	7.65983	0.287671	0.143835	+	0.989602i	$0.454056\pi$
$710$	0	0
$711$	−7.15902	−0.268484
$712$	0	0
$713$	9.32684	0.349293
$714$	0	0
$715$	18.8371	0.704468
$716$	0	0
$717$	−5.57531	−0.208214
$718$	0	0
$719$	−22.4196	−0.836110	−0.418055	−	0.908422i	$-0.637288\pi$
$719$	−22.4196	−0.836110	−0.418055	+	0.908422i	$0.637288\pi$
$720$	0	0
$721$	−16.6803	−0.621209
$722$	0	0
$723$	28.6407	1.06516
$724$	0	0
$725$	−2.29072	−0.0850754
$726$	0	0
$727$	−34.8371	−1.29204	−0.646018	−	0.763322i	$-0.723567\pi$
$727$	−34.8371	−1.29204	−0.646018	+	0.763322i	$0.723567\pi$
$728$	0	0
$729$	21.3812	0.791897
$730$	0	0
$731$	26.9360	0.996264
$732$	0	0
$733$	−38.6752	−1.42850	−0.714251	−	0.699889i	$-0.753233\pi$
$733$	−38.6752	−1.42850	−0.714251	+	0.699889i	$0.753233\pi$
$734$	0	0
$735$	−1.36910	−0.0505001
$736$	0	0
$737$	36.5958	1.34802
$738$	0	0
$739$	51.3484	1.88888	0.944441	−	0.328682i	$-0.106604\pi$
$739$	51.3484	1.88888	0.944441	+	0.328682i	$0.106604\pi$
$740$	0	0
$741$	−26.7214	−0.981635
$742$	0	0
$743$	−8.00597	−0.293710	−0.146855	−	0.989158i	$-0.546915\pi$
$743$	−8.00597	−0.293710	−0.146855	+	0.989158i	$0.546915\pi$
$744$	0	0
$745$	−0.979481	−0.0358854
$746$	0	0
$747$	23.8987	0.874406
$748$	0	0
$749$	15.3340	0.560293
$750$	0	0
$751$	−23.3958	−0.853724	−0.426862	−	0.904317i	$-0.640381\pi$
$751$	−23.3958	−0.853724	−0.426862	+	0.904317i	$0.640381\pi$
$752$	0	0
$753$	30.7214	1.11955
$754$	0	0
$755$	18.6225	0.677742
$756$	0	0
$757$	−29.1194	−1.05836	−0.529182	−	0.848509i	$-0.677501\pi$
$757$	−29.1194	−1.05836	−0.529182	+	0.848509i	$0.677501\pi$
$758$	0	0
$759$	4.34017	0.157538
$760$	0	0
$761$	−10.1834	−0.369149	−0.184574	−	0.982819i	$-0.559091\pi$
$761$	−10.1834	−0.369149	−0.184574	+	0.982819i	$0.559091\pi$
$762$	0	0
$763$	−11.1773	−0.404645
$764$	0	0
$765$	6.04945	0.218718
$766$	0	0
$767$	36.4801	1.31722
$768$	0	0
$769$	0.431882	0.0155741	0.00778703	−	0.999970i	$-0.497521\pi$
$769$	0.431882	0.0155741	0.00778703	+	0.999970i	$0.497521\pi$
$770$	0	0
$771$	9.17727	0.330511
$772$	0	0
$773$	−0.0650468	−0.00233957	−0.00116978	−	0.999999i	$-0.500372\pi$
$773$	−0.0650468	−0.00233957	−0.00116978	+	0.999999i	$0.500372\pi$
$774$	0	0
$775$	33.8648	1.21646
$776$	0	0
$777$	−3.60197	−0.129220
$778$	0	0
$779$	−14.1034	−0.505308
$780$	0	0
$781$	40.1978	1.43839
$782$	0	0
$783$	−3.41855	−0.122169
$784$	0	0
$785$	13.0700	0.466487
$786$	0	0
$787$	−31.5486	−1.12459	−0.562294	−	0.826937i	$-0.690081\pi$
$787$	−31.5486	−1.12459	−0.562294	+	0.826937i	$0.690081\pi$
$788$	0	0
$789$	−12.7694	−0.454603
$790$	0	0
$791$	13.5174	0.480625
$792$	0	0
$793$	−30.0722	−1.06790
$794$	0	0
$795$	−5.69102	−0.201840
$796$	0	0
$797$	51.8336	1.83604	0.918020	−	0.396533i	$-0.129787\pi$
$797$	51.8336	1.83604	0.918020	+	0.396533i	$0.129787\pi$
$798$	0	0
$799$	−19.3112	−0.683183
$800$	0	0
$801$	11.0445	0.390239
$802$	0	0
$803$	48.2967	1.70435
$804$	0	0
$805$	1.17009	0.0412401
$806$	0	0
$807$	−4.18342	−0.147263
$808$	0	0
$809$	44.4391	1.56239	0.781197	−	0.624284i	$-0.214609\pi$
$809$	44.4391	1.56239	0.781197	+	0.624284i	$0.214609\pi$
$810$	0	0
$811$	23.6358	0.829966	0.414983	−	0.909829i	$-0.363788\pi$
$811$	23.6358	0.829966	0.414983	+	0.909829i	$0.363788\pi$
$812$	0	0
$813$	6.99773	0.245421
$814$	0	0
$815$	17.1096	0.599322
$816$	0	0
$817$	−44.7091	−1.56417
$818$	0	0
$819$	−7.07838	−0.247339
$820$	0	0
$821$	−14.9399	−0.521405	−0.260703	−	0.965419i	$-0.583954\pi$
$821$	−14.9399	−0.521405	−0.260703	+	0.965419i	$0.583954\pi$
$822$	0	0
$823$	−30.7670	−1.07247	−0.536234	−	0.844069i	$-0.680154\pi$
$823$	−30.7670	−1.07247	−0.536234	+	0.844069i	$0.680154\pi$
$824$	0	0
$825$	15.7587	0.548648
$826$	0	0
$827$	24.5113	0.852342	0.426171	−	0.904643i	$-0.359862\pi$
$827$	24.5113	0.852342	0.426171	+	0.904643i	$0.359862\pi$
$828$	0	0
$829$	43.8720	1.52374	0.761869	−	0.647731i	$-0.224282\pi$
$829$	43.8720	1.52374	0.761869	+	0.647731i	$0.224282\pi$
$830$	0	0
$831$	−15.5753	−0.540301
$832$	0	0
$833$	3.17009	0.109837
$834$	0	0
$835$	7.02893	0.243246
$836$	0	0
$837$	50.5380	1.74685
$838$	0	0
$839$	5.39189	0.186149	0.0930743	−	0.995659i	$-0.470331\pi$
$839$	5.39189	0.186149	0.0930743	+	0.995659i	$0.470331\pi$
$840$	0	0
$841$	−28.6020	−0.986275
$842$	0	0
$843$	−2.49239	−0.0858426
$844$	0	0
$845$	−6.82991	−0.234956
$846$	0	0
$847$	2.75872	0.0947909
$848$	0	0
$849$	−16.0000	−0.549119
$850$	0	0
$851$	3.07838	0.105525
$852$	0	0
$853$	−1.33403	−0.0456763	−0.0228382	−	0.999739i	$-0.507270\pi$
$853$	−1.33403	−0.0456763	−0.0228382	+	0.999739i	$0.507270\pi$
$854$	0	0
$855$	−10.0410	−0.343396
$856$	0	0
$857$	12.1034	0.413445	0.206723	−	0.978400i	$-0.433720\pi$
$857$	12.1034	0.413445	0.206723	+	0.978400i	$0.433720\pi$
$858$	0	0
$859$	−40.5718	−1.38429	−0.692146	−	0.721757i	$-0.743335\pi$
$859$	−40.5718	−1.38429	−0.692146	+	0.721757i	$0.743335\pi$
$860$	0	0
$861$	3.13624	0.106883
$862$	0	0
$863$	−34.0944	−1.16059	−0.580293	−	0.814408i	$-0.697062\pi$
$863$	−34.0944	−1.16059	−0.580293	+	0.814408i	$0.697062\pi$
$864$	0	0
$865$	−16.4969	−0.560912
$866$	0	0
$867$	−8.13275	−0.276203
$868$	0	0
$869$	−16.2823	−0.552340
$870$	0	0
$871$	−42.8203	−1.45091
$872$	0	0
$873$	−14.0072	−0.474071
$874$	0	0
$875$	10.0989	0.341405
$876$	0	0
$877$	−20.4163	−0.689409	−0.344704	−	0.938711i	$-0.612021\pi$
$877$	−20.4163	−0.689409	−0.344704	+	0.938711i	$0.612021\pi$
$878$	0	0
$879$	−15.5259	−0.523674
$880$	0	0
$881$	−35.4668	−1.19491	−0.597453	−	0.801904i	$-0.703821\pi$
$881$	−35.4668	−1.19491	−0.597453	+	0.801904i	$0.703821\pi$
$882$	0	0
$883$	−7.91548	−0.266377	−0.133189	−	0.991091i	$-0.542522\pi$
$883$	−7.91548	−0.266377	−0.133189	+	0.991091i	$0.542522\pi$
$884$	0	0
$885$	−11.5076	−0.386824
$886$	0	0
$887$	8.10608	0.272176	0.136088	−	0.990697i	$-0.456547\pi$
$887$	8.10608	0.272176	0.136088	+	0.990697i	$0.456547\pi$
$888$	0	0
$889$	−2.92162	−0.0979881
$890$	0	0
$891$	5.36910	0.179872
$892$	0	0
$893$	32.0533	1.07262
$894$	0	0
$895$	−21.2085	−0.708921
$896$	0	0
$897$	−5.07838	−0.169562
$898$	0	0
$899$	−5.88428	−0.196252
$900$	0	0
$901$	13.1773	0.438999
$902$	0	0
$903$	9.94214	0.330854
$904$	0	0
$905$	−14.5464	−0.483538
$906$	0	0
$907$	18.8371	0.625476	0.312738	−	0.949839i	$-0.398754\pi$
$907$	18.8371	0.625476	0.312738	+	0.949839i	$0.398754\pi$
$908$	0	0
$909$	8.02666	0.266228
$910$	0	0
$911$	−33.6514	−1.11492	−0.557461	−	0.830203i	$-0.688224\pi$
$911$	−33.6514	−1.11492	−0.557461	+	0.830203i	$0.688224\pi$
$912$	0	0
$913$	54.3545	1.79887
$914$	0	0
$915$	9.48625	0.313606
$916$	0	0
$917$	6.27513	0.207223
$918$	0	0
$919$	−42.5464	−1.40348	−0.701738	−	0.712435i	$-0.747592\pi$
$919$	−42.5464	−1.40348	−0.701738	+	0.712435i	$0.747592\pi$
$920$	0	0
$921$	20.4885	0.675120
$922$	0	0
$923$	−47.0349	−1.54817
$924$	0	0
$925$	11.1773	0.367507
$926$	0	0
$927$	27.2039	0.893495
$928$	0	0
$929$	36.3012	1.19100	0.595502	−	0.803354i	$-0.296953\pi$
$929$	36.3012	1.19100	0.595502	+	0.803354i	$0.296953\pi$
$930$	0	0
$931$	−5.26180	−0.172448
$932$	0	0
$933$	−2.37978	−0.0779105
$934$	0	0
$935$	13.7587	0.449958
$936$	0	0
$937$	6.50412	0.212480	0.106240	−	0.994341i	$-0.466119\pi$
$937$	6.50412	0.212480	0.106240	+	0.994341i	$0.466119\pi$
$938$	0	0
$939$	14.4840	0.472667
$940$	0	0
$941$	−2.35965	−0.0769223	−0.0384611	−	0.999260i	$-0.512246\pi$
$941$	−2.35965	−0.0769223	−0.0384611	+	0.999260i	$0.512246\pi$
$942$	0	0
$943$	−2.68035	−0.0872841
$944$	0	0
$945$	6.34017	0.206246
$946$	0	0
$947$	−1.17727	−0.0382563	−0.0191281	−	0.999817i	$-0.506089\pi$
$947$	−1.17727	−0.0382563	−0.0191281	+	0.999817i	$0.506089\pi$
$948$	0	0
$949$	−56.5113	−1.83443
$950$	0	0
$951$	−23.5753	−0.764482
$952$	0	0
$953$	−31.8264	−1.03096	−0.515479	−	0.856902i	$-0.672386\pi$
$953$	−31.8264	−1.03096	−0.515479	+	0.856902i	$0.672386\pi$
$954$	0	0
$955$	−5.69102	−0.184157
$956$	0	0
$957$	−2.73820	−0.0885136
$958$	0	0
$959$	7.75872	0.250542
$960$	0	0
$961$	55.9900	1.80613
$962$	0	0
$963$	−25.0082	−0.805879
$964$	0	0
$965$	20.1666	0.649186
$966$	0	0
$967$	−42.6225	−1.37065	−0.685323	−	0.728239i	$-0.740339\pi$
$967$	−42.6225	−1.37065	−0.685323	+	0.728239i	$0.740339\pi$
$968$	0	0
$969$	−19.5174	−0.626991
$970$	0	0
$971$	−55.7998	−1.79070	−0.895350	−	0.445364i	$-0.853074\pi$
$971$	−55.7998	−1.79070	−0.895350	+	0.445364i	$0.853074\pi$
$972$	0	0
$973$	2.92881	0.0938933
$974$	0	0
$975$	−18.4391	−0.590523
$976$	0	0
$977$	52.2245	1.67081	0.835404	−	0.549636i	$-0.185234\pi$
$977$	52.2245	1.67081	0.835404	+	0.549636i	$0.185234\pi$
$978$	0	0
$979$	25.1194	0.802820
$980$	0	0
$981$	18.2290	0.582007
$982$	0	0
$983$	16.7649	0.534716	0.267358	−	0.963597i	$-0.413849\pi$
$983$	16.7649	0.534716	0.267358	+	0.963597i	$0.413849\pi$
$984$	0	0
$985$	14.6537	0.466905
$986$	0	0
$987$	−7.12783	−0.226881
$988$	0	0
$989$	−8.49693	−0.270187
$990$	0	0
$991$	39.0349	1.23998	0.619992	−	0.784608i	$-0.287136\pi$
$991$	39.0349	1.23998	0.619992	+	0.784608i	$0.287136\pi$
$992$	0	0
$993$	7.84778	0.249042
$994$	0	0
$995$	1.36069	0.0431368
$996$	0	0
$997$	−11.2618	−0.356665	−0.178332	−	0.983970i	$-0.557070\pi$
$997$	−11.2618	−0.356665	−0.178332	+	0.983970i	$0.557070\pi$
$998$	0	0
$999$	16.6803	0.527743

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2576.2.a.v.1.3		3
4.3	odd	2		161.2.a.c.1.1	✓	3
12.11	even	2		1449.2.a.m.1.3		3
20.19	odd	2		4025.2.a.j.1.3		3
28.27	even	2		1127.2.a.f.1.1		3
92.91	even	2		3703.2.a.c.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
161.2.a.c.1.1	✓	3	4.3	odd	2
1127.2.a.f.1.1		3	28.27	even	2
1449.2.a.m.1.3		3	12.11	even	2
2576.2.a.v.1.3		3	1.1	even	1	trivial
3703.2.a.c.1.1		3	92.91	even	2
4025.2.a.j.1.3		3	20.19	odd	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2576 = 2^{4} \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2576.a (trivial)

\(f(q)\)	\(=\)	\(q+1.17009 q^{3} -1.17009 q^{5} +1.00000 q^{7} -1.63090 q^{9} -3.70928 q^{11} +4.34017 q^{13} -1.36910 q^{15} +3.17009 q^{17} -5.26180 q^{19} +1.17009 q^{21} -1.00000 q^{23} -3.63090 q^{25} -5.41855 q^{27} +0.630898 q^{29} -9.32684 q^{31} -4.34017 q^{33} -1.17009 q^{35} -3.07838 q^{37} +5.07838 q^{39} +2.68035 q^{41} +8.49693 q^{43} +1.90829 q^{45} -6.09171 q^{47} +1.00000 q^{49} +3.70928 q^{51} +4.15676 q^{53} +4.34017 q^{55} -6.15676 q^{57} +8.40522 q^{59} -6.92881 q^{61} -1.63090 q^{63} -5.07838 q^{65} -9.86603 q^{67} -1.17009 q^{69} -10.8371 q^{71} -13.0205 q^{73} -4.24846 q^{75} -3.70928 q^{77} +4.38962 q^{79} -1.44748 q^{81} -14.6537 q^{83} -3.70928 q^{85} +0.738205 q^{87} -6.77205 q^{89} +4.34017 q^{91} -10.9132 q^{93} +6.15676 q^{95} +8.58864 q^{97} +6.04945 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 2 q^{3} + 2 q^{5} + 3 q^{7} - q^{9} - 4 q^{11} + 2 q^{13} - 8 q^{15} + 4 q^{17} - 8 q^{19} - 2 q^{21} - 3 q^{23} - 7 q^{25} - 2 q^{27} - 2 q^{29} - 16 q^{31} - 2 q^{33} + 2 q^{35} - 6 q^{37} + 12 q^{39}+ \cdots + 6 q^{97}+O(q^{100}) \) 3 * q - 2 * q^3 + 2 * q^5 + 3 * q^7 - q^9 - 4 * q^11 + 2 * q^13 - 8 * q^15 + 4 * q^17 - 8 * q^19 - 2 * q^21 - 3 * q^23 - 7 * q^25 - 2 * q^27 - 2 * q^29 - 16 * q^31 - 2 * q^33 + 2 * q^35 - 6 * q^37 + 12 * q^39 - 14 * q^41 + 8 * q^43 + 8 * q^45 - 16 * q^47 + 3 * q^49 + 4 * q^51 + 6 * q^53 + 2 * q^55 - 12 * q^57 + 10 * q^59 + 10 * q^61 - q^63 - 12 * q^65 - 16 * q^67 + 2 * q^69 - 4 * q^71 - 6 * q^73 - 4 * q^75 - 4 * q^77 - 16 * q^79 - 5 * q^81 - 20 * q^83 - 4 * q^85 + 10 * q^87 + 4 * q^89 + 2 * q^91 + 12 * q^93 + 12 * q^95 + 6 * q^97

Introduction

L-functions

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