LMFDB - Embedded newform 324.4.a.e.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [324,4,Mod(1,324)] 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("324.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(324, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 324 = 2^{2} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	324.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$19.1166188419$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{3}, \sqrt{7})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 5x^{2} + 1 $ x^4 - 5*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{4} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.18890$ of defining polynomial
Character	$\chi$	$=$	324.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-10.6784 q^{5} +23.4955 q^{7} +4.91010 q^{11} +1.50455 q^{13} -99.2990 q^{17} +28.5045 q^{19} +152.691 q^{23} -10.9727 q^{25} +240.453 q^{29} +256.973 q^{31} -250.893 q^{35} +359.468 q^{37} -108.500 q^{41} +411.459 q^{43} +311.769 q^{47} +209.036 q^{49} -702.578 q^{53} -52.4318 q^{55} +479.190 q^{59} +498.459 q^{61} -16.0661 q^{65} -332.432 q^{67} -884.729 q^{71} +305.000 q^{73} +115.365 q^{77} -703.386 q^{79} +420.747 q^{83} +1060.35 q^{85} +1610.00 q^{89} +35.3500 q^{91} -304.382 q^{95} +396.018 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 16 q^{7} + 116 q^{13} + 224 q^{19} + 616 q^{25} + 368 q^{31} + 668 q^{37} + 656 q^{43} + 1716 q^{49} + 1440 q^{55} + 1004 q^{61} + 320 q^{67} + 1220 q^{73} - 64 q^{79} + 612 q^{85} - 3488 q^{91} + 2024 q^{97}+O(q^{100}) $ 4 * q - 16 * q^7 + 116 * q^13 + 224 * q^19 + 616 * q^25 + 368 * q^31 + 668 * q^37 + 656 * q^43 + 1716 * q^49 + 1440 * q^55 + 1004 * q^61 + 320 * q^67 + 1220 * q^73 - 64 * q^79 + 612 * q^85 - 3488 * q^91 + 2024 * 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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−10.6784	−0.955101	−0.477551	−	0.878604i	$-0.658475\pi$
$5$	−10.6784	−0.955101	−0.477551	+	0.878604i	$0.658475\pi$
$6$	0	0
$7$	23.4955	1.26864	0.634318	−	0.773073i	$-0.281281\pi$
$7$	23.4955	1.26864	0.634318	+	0.773073i	$0.281281\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	4.91010	0.134586	0.0672932	−	0.997733i	$-0.478564\pi$
$11$	4.91010	0.134586	0.0672932	+	0.997733i	$0.478564\pi$
$12$	0	0
$13$	1.50455	0.0320989	0.0160495	−	0.999871i	$-0.494891\pi$
$13$	1.50455	0.0320989	0.0160495	+	0.999871i	$0.494891\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−99.2990	−1.41668	−0.708340	−	0.705872i	$-0.750555\pi$
$17$	−99.2990	−1.41668	−0.708340	+	0.705872i	$0.750555\pi$
$18$	0	0
$19$	28.5045	0.344178	0.172089	−	0.985081i	$-0.444948\pi$
$19$	28.5045	0.344178	0.172089	+	0.985081i	$0.444948\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	152.691	1.38427	0.692135	−	0.721768i	$-0.256670\pi$
$23$	152.691	1.38427	0.692135	+	0.721768i	$0.256670\pi$
$24$	0	0
$25$	−10.9727	−0.0877818
$26$	0	0
$27$	0	0
$28$	0	0
$29$	240.453	1.53969	0.769846	−	0.638230i	$-0.220333\pi$
$29$	240.453	1.53969	0.769846	+	0.638230i	$0.220333\pi$
$30$	0	0
$31$	256.973	1.48883	0.744414	−	0.667718i	$-0.232729\pi$
$31$	256.973	1.48883	0.744414	+	0.667718i	$0.232729\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−250.893	−1.21168
$36$	0	0
$37$	359.468	1.59719	0.798597	−	0.601866i	$-0.205576\pi$
$37$	359.468	1.59719	0.798597	+	0.601866i	$0.205576\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−108.500	−0.413289	−0.206644	−	0.978416i	$-0.566254\pi$
$41$	−108.500	−0.413289	−0.206644	+	0.978416i	$0.566254\pi$
$42$	0	0
$43$	411.459	1.45923	0.729615	−	0.683858i	$-0.239699\pi$
$43$	411.459	1.45923	0.729615	+	0.683858i	$0.239699\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	311.769	0.967579	0.483789	−	0.875184i	$-0.339260\pi$
$47$	311.769	0.967579	0.483789	+	0.875184i	$0.339260\pi$
$48$	0	0
$49$	209.036	0.609435
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−702.578	−1.82088	−0.910438	−	0.413645i	$-0.864255\pi$
$53$	−702.578	−1.82088	−0.910438	+	0.413645i	$0.864255\pi$
$54$	0	0
$55$	−52.4318	−0.128544
$56$	0	0
$57$	0	0
$58$	0	0
$59$	479.190	1.05738	0.528689	−	0.848816i	$-0.322684\pi$
$59$	479.190	1.05738	0.528689	+	0.848816i	$0.322684\pi$
$60$	0	0
$61$	498.459	1.04625	0.523124	−	0.852256i	$-0.324766\pi$
$61$	498.459	1.04625	0.523124	+	0.852256i	$0.324766\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−16.0661	−0.0306577
$66$	0	0
$67$	−332.432	−0.606164	−0.303082	−	0.952964i	$-0.598016\pi$
$67$	−332.432	−0.606164	−0.303082	+	0.952964i	$0.598016\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−884.729	−1.47885	−0.739423	−	0.673242i	$-0.764901\pi$
$71$	−884.729	−1.47885	−0.739423	+	0.673242i	$0.764901\pi$
$72$	0	0
$73$	305.000	0.489008	0.244504	−	0.969648i	$-0.421375\pi$
$73$	305.000	0.489008	0.244504	+	0.969648i	$0.421375\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	115.365	0.170741
$78$	0	0
$79$	−703.386	−1.00174	−0.500868	−	0.865524i	$-0.666986\pi$
$79$	−703.386	−1.00174	−0.500868	+	0.865524i	$0.666986\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	420.747	0.556421	0.278210	−	0.960520i	$-0.410259\pi$
$83$	420.747	0.556421	0.278210	+	0.960520i	$0.410259\pi$
$84$	0	0
$85$	1060.35	1.35307
$86$	0	0
$87$	0	0
$88$	0	0
$89$	1610.00	1.91752	0.958761	−	0.284212i	$-0.0917319\pi$
$89$	1610.00	1.91752	0.958761	+	0.284212i	$0.0917319\pi$
$90$	0	0
$91$	35.3500	0.0407218
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−304.382	−0.328725
$96$	0	0
$97$	396.018	0.414531	0.207266	−	0.978285i	$-0.433544\pi$
$97$	396.018	0.414531	0.207266	+	0.978285i	$0.433544\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	272.966	0.268922	0.134461	−	0.990919i	$-0.457070\pi$
$101$	272.966	0.268922	0.134461	+	0.990919i	$0.457070\pi$
$102$	0	0
$103$	−397.082	−0.379861	−0.189930	−	0.981798i	$-0.560826\pi$
$103$	−397.082	−0.379861	−0.189930	+	0.981798i	$0.560826\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	360.870	0.326043	0.163022	−	0.986622i	$-0.447876\pi$
$107$	360.870	0.326043	0.163022	+	0.986622i	$0.447876\pi$
$108$	0	0
$109$	120.568	0.105948	0.0529740	−	0.998596i	$-0.483130\pi$
$109$	120.568	0.105948	0.0529740	+	0.998596i	$0.483130\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−1224.62	−1.01949	−0.509747	−	0.860324i	$-0.670261\pi$
$113$	−1224.62	−1.01949	−0.509747	+	0.860324i	$0.670261\pi$
$114$	0	0
$115$	−1630.49	−1.32212
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−2333.08	−1.79725
$120$	0	0
$121$	−1306.89	−0.981886
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1451.97	1.03894
$126$	0	0
$127$	−1991.46	−1.39144	−0.695722	−	0.718311i	$-0.744915\pi$
$127$	−1991.46	−1.39144	−0.695722	+	0.718311i	$0.744915\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	1577.49	1.05210	0.526052	−	0.850452i	$-0.323672\pi$
$131$	1577.49	1.05210	0.526052	+	0.850452i	$0.323672\pi$
$132$	0	0
$133$	669.727	0.436637
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−836.008	−0.521351	−0.260675	−	0.965427i	$-0.583945\pi$
$137$	−836.008	−0.521351	−0.260675	+	0.965427i	$0.583945\pi$
$138$	0	0
$139$	−2728.61	−1.66502	−0.832509	−	0.554011i	$-0.813097\pi$
$139$	−2728.61	−1.66502	−0.832509	+	0.554011i	$0.813097\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	7.38747	0.00432008
$144$	0	0
$145$	−2567.65	−1.47056
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−2776.08	−1.52635	−0.763173	−	0.646195i	$-0.776359\pi$
$149$	−2776.08	−1.52635	−0.763173	+	0.646195i	$0.776359\pi$
$150$	0	0
$151$	−763.027	−0.411220	−0.205610	−	0.978634i	$-0.565918\pi$
$151$	−763.027	−0.411220	−0.205610	+	0.978634i	$0.565918\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−2744.05	−1.42198
$156$	0	0
$157$	705.486	0.358624	0.179312	−	0.983792i	$-0.442613\pi$
$157$	705.486	0.358624	0.179312	+	0.983792i	$0.442613\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	3587.54	1.75613
$162$	0	0
$163$	1587.60	0.762886	0.381443	−	0.924392i	$-0.375427\pi$
$163$	1587.60	0.762886	0.381443	+	0.924392i	$0.375427\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	660.864	0.306223	0.153111	−	0.988209i	$-0.451071\pi$
$167$	660.864	0.306223	0.153111	+	0.988209i	$0.451071\pi$
$168$	0	0
$169$	−2194.74	−0.998970
$170$	0	0
$171$	0	0
$172$	0	0
$173$	2037.37	0.895367	0.447684	−	0.894192i	$-0.352249\pi$
$173$	2037.37	0.895367	0.447684	+	0.894192i	$0.352249\pi$
$174$	0	0
$175$	−257.809	−0.111363
$176$	0	0
$177$	0	0
$178$	0	0
$179$	2579.01	1.07690	0.538448	−	0.842659i	$-0.319011\pi$
$179$	2579.01	1.07690	0.538448	+	0.842659i	$0.319011\pi$
$180$	0	0
$181$	1851.65	0.760400	0.380200	−	0.924904i	$-0.375855\pi$
$181$	1851.65	0.760400	0.380200	+	0.924904i	$0.375855\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−3838.53	−1.52548
$186$	0	0
$187$	−487.568	−0.190666
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−56.9661	−0.0215807	−0.0107904	−	0.999942i	$-0.503435\pi$
$191$	−56.9661	−0.0215807	−0.0107904	+	0.999942i	$0.503435\pi$
$192$	0	0
$193$	1814.03	0.676562	0.338281	−	0.941045i	$-0.390155\pi$
$193$	1814.03	0.676562	0.338281	+	0.941045i	$0.390155\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	873.334	0.315850	0.157925	−	0.987451i	$-0.449520\pi$
$197$	873.334	0.315850	0.157925	+	0.987451i	$0.449520\pi$
$198$	0	0
$199$	−5063.60	−1.80376	−0.901882	−	0.431982i	$-0.857814\pi$
$199$	−5063.60	−1.80376	−0.901882	+	0.431982i	$0.857814\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	5649.56	1.95331
$204$	0	0
$205$	1158.60	0.394732
$206$	0	0
$207$	0	0
$208$	0	0
$209$	139.960	0.0463218
$210$	0	0
$211$	430.777	0.140549	0.0702747	−	0.997528i	$-0.477612\pi$
$211$	430.777	0.140549	0.0702747	+	0.997528i	$0.477612\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−4393.71	−1.39371
$216$	0	0
$217$	6037.69	1.88878
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−149.400	−0.0454739
$222$	0	0
$223$	−3263.68	−0.980054	−0.490027	−	0.871707i	$-0.663013\pi$
$223$	−3263.68	−0.980054	−0.490027	+	0.871707i	$0.663013\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	748.679	0.218906	0.109453	−	0.993992i	$-0.465090\pi$
$227$	748.679	0.218906	0.109453	+	0.993992i	$0.465090\pi$
$228$	0	0
$229$	−2987.00	−0.861949	−0.430974	−	0.902364i	$-0.641830\pi$
$229$	−2987.00	−0.861949	−0.430974	+	0.902364i	$0.641830\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−3785.23	−1.06429	−0.532144	−	0.846654i	$-0.678613\pi$
$233$	−3785.23	−1.06429	−0.532144	+	0.846654i	$0.678613\pi$
$234$	0	0
$235$	−3329.18	−0.924136
$236$	0	0
$237$	0	0
$238$	0	0
$239$	608.375	0.164655	0.0823274	−	0.996605i	$-0.473765\pi$
$239$	608.375	0.164655	0.0823274	+	0.996605i	$0.473765\pi$
$240$	0	0
$241$	5343.41	1.42821	0.714106	−	0.700037i	$-0.246833\pi$
$241$	5343.41	1.42821	0.714106	+	0.700037i	$0.246833\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−2232.16	−0.582073
$246$	0	0
$247$	42.8864	0.0110478
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−6475.89	−1.62850	−0.814252	−	0.580512i	$-0.802853\pi$
$251$	−6475.89	−1.62850	−0.814252	+	0.580512i	$0.802853\pi$
$252$	0	0
$253$	749.727	0.186304
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−5.29064	−0.00128413	−0.000642064	−	1.00000i	$-0.500204\pi$
$257$	−5.29064	−0.00128413	−0.000642064		1.00000i	$0.500204\pi$
$258$	0	0
$259$	8445.87	2.02626
$260$	0	0
$261$	0	0
$262$	0	0
$263$	7046.94	1.65222	0.826108	−	0.563512i	$-0.190550\pi$
$263$	7046.94	1.65222	0.826108	+	0.563512i	$0.190550\pi$
$264$	0	0
$265$	7502.37	1.73912
$266$	0	0
$267$	0	0
$268$	0	0
$269$	787.057	0.178393	0.0891965	−	0.996014i	$-0.471570\pi$
$269$	787.057	0.178393	0.0891965	+	0.996014i	$0.471570\pi$
$270$	0	0
$271$	5076.65	1.13795	0.568975	−	0.822355i	$-0.307340\pi$
$271$	5076.65	1.13795	0.568975	+	0.822355i	$0.307340\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−53.8772	−0.0118142
$276$	0	0
$277$	817.782	0.177385	0.0886927	−	0.996059i	$-0.471731\pi$
$277$	817.782	0.177385	0.0886927	+	0.996059i	$0.471731\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	5237.49	1.11190	0.555948	−	0.831217i	$-0.312356\pi$
$281$	5237.49	1.11190	0.555948	+	0.831217i	$0.312356\pi$
$282$	0	0
$283$	−2098.05	−0.440694	−0.220347	−	0.975422i	$-0.570719\pi$
$283$	−2098.05	−0.440694	−0.220347	+	0.975422i	$0.570719\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−2549.25	−0.524312
$288$	0	0
$289$	4947.29	1.00698
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−2490.06	−0.496487	−0.248243	−	0.968698i	$-0.579853\pi$
$293$	−2490.06	−0.496487	−0.248243	+	0.968698i	$0.579853\pi$
$294$	0	0
$295$	−5116.96	−1.00990
$296$	0	0
$297$	0	0
$298$	0	0
$299$	229.730	0.0444336
$300$	0	0
$301$	9667.42	1.85123
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−5322.72	−0.999273
$306$	0	0
$307$	2068.21	0.384492	0.192246	−	0.981347i	$-0.438423\pi$
$307$	2068.21	0.384492	0.192246	+	0.981347i	$0.438423\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	4911.57	0.895528	0.447764	−	0.894152i	$-0.352220\pi$
$311$	4911.57	0.895528	0.447764	+	0.894152i	$0.352220\pi$
$312$	0	0
$313$	5296.34	0.956443	0.478221	−	0.878239i	$-0.341282\pi$
$313$	5296.34	0.956443	0.478221	+	0.878239i	$0.341282\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1430.80	0.253507	0.126754	−	0.991934i	$-0.459544\pi$
$317$	1430.80	0.253507	0.126754	+	0.991934i	$0.459544\pi$
$318$	0	0
$319$	1180.65	0.207222
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−2830.47	−0.487590
$324$	0	0
$325$	−16.5090	−0.00281770
$326$	0	0
$327$	0	0
$328$	0	0
$329$	7325.16	1.22750
$330$	0	0
$331$	−5334.51	−0.885835	−0.442917	−	0.896562i	$-0.646057\pi$
$331$	−5334.51	−0.885835	−0.442917	+	0.896562i	$0.646057\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	3549.83	0.578948
$336$	0	0
$337$	−8671.58	−1.40169	−0.700847	−	0.713311i	$-0.747195\pi$
$337$	−8671.58	−1.40169	−0.700847	+	0.713311i	$0.747195\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	1261.76	0.200376
$342$	0	0
$343$	−3147.54	−0.495484
$344$	0	0
$345$	0	0
$346$	0	0
$347$	5371.73	0.831036	0.415518	−	0.909585i	$-0.363600\pi$
$347$	5371.73	0.831036	0.415518	+	0.909585i	$0.363600\pi$
$348$	0	0
$349$	−5041.53	−0.773257	−0.386629	−	0.922236i	$-0.626360\pi$
$349$	−5041.53	−0.773257	−0.386629	+	0.922236i	$0.626360\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	4851.78	0.731541	0.365771	−	0.930705i	$-0.380805\pi$
$353$	4851.78	0.731541	0.365771	+	0.930705i	$0.380805\pi$
$354$	0	0
$355$	9447.45	1.41245
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−3143.59	−0.462151	−0.231075	−	0.972936i	$-0.574224\pi$
$359$	−3143.59	−0.462151	−0.231075	+	0.972936i	$0.574224\pi$
$360$	0	0
$361$	−6046.49	−0.881541
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−3256.90	−0.467052
$366$	0	0
$367$	9981.83	1.41975	0.709873	−	0.704329i	$-0.248752\pi$
$367$	9981.83	1.41975	0.709873	+	0.704329i	$0.248752\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−16507.4	−2.31003
$372$	0	0
$373$	−6064.84	−0.841891	−0.420945	−	0.907086i	$-0.638302\pi$
$373$	−6064.84	−0.841891	−0.420945	+	0.907086i	$0.638302\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	361.773	0.0494224
$378$	0	0
$379$	11238.7	1.52321	0.761603	−	0.648044i	$-0.224413\pi$
$379$	11238.7	1.52321	0.761603	+	0.648044i	$0.224413\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−14688.9	−1.95970	−0.979851	−	0.199730i	$-0.935994\pi$
$383$	−14688.9	−1.95970	−0.979851	+	0.199730i	$0.935994\pi$
$384$	0	0
$385$	−1231.91	−0.163075
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−1355.67	−0.176697	−0.0883483	−	0.996090i	$-0.528159\pi$
$389$	−1355.67	−0.176697	−0.0883483	+	0.996090i	$0.528159\pi$
$390$	0	0
$391$	−15162.0	−1.96107
$392$	0	0
$393$	0	0
$394$	0	0
$395$	7511.01	0.956759
$396$	0	0
$397$	−9070.50	−1.14669	−0.573345	−	0.819314i	$-0.694354\pi$
$397$	−9070.50	−1.14669	−0.573345	+	0.819314i	$0.694354\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−3283.84	−0.408945	−0.204473	−	0.978872i	$-0.565548\pi$
$401$	−3283.84	−0.408945	−0.204473	+	0.978872i	$0.565548\pi$
$402$	0	0
$403$	386.627	0.0477898
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1765.03	0.214961
$408$	0	0
$409$	6219.30	0.751894	0.375947	−	0.926641i	$-0.377317\pi$
$409$	6219.30	0.751894	0.375947	+	0.926641i	$0.377317\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	11258.8	1.34143
$414$	0	0
$415$	−4492.88	−0.531438
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−5969.10	−0.695966	−0.347983	−	0.937501i	$-0.613133\pi$
$419$	−5969.10	−0.695966	−0.347983	+	0.937501i	$0.613133\pi$
$420$	0	0
$421$	6539.09	0.756997	0.378498	−	0.925602i	$-0.376441\pi$
$421$	6539.09	0.756997	0.378498	+	0.925602i	$0.376441\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	1089.58	0.124359
$426$	0	0
$427$	11711.5	1.32731
$428$	0	0
$429$	0	0
$430$	0	0
$431$	11929.1	1.33319	0.666596	−	0.745419i	$-0.267750\pi$
$431$	11929.1	1.33319	0.666596	+	0.745419i	$0.267750\pi$
$432$	0	0
$433$	−4047.92	−0.449262	−0.224631	−	0.974444i	$-0.572118\pi$
$433$	−4047.92	−0.449262	−0.224631	+	0.974444i	$0.572118\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	4352.38	0.476436
$438$	0	0
$439$	−12124.2	−1.31813	−0.659063	−	0.752088i	$-0.729047\pi$
$439$	−12124.2	−1.31813	−0.659063	+	0.752088i	$0.729047\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−12437.6	−1.33392	−0.666962	−	0.745092i	$-0.732406\pi$
$443$	−12437.6	−1.33392	−0.666962	+	0.745092i	$0.732406\pi$
$444$	0	0
$445$	−17192.1	−1.83143
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−2925.50	−0.307489	−0.153745	−	0.988111i	$-0.549133\pi$
$449$	−2925.50	−0.307489	−0.153745	+	0.988111i	$0.549133\pi$
$450$	0	0
$451$	−532.745	−0.0556231
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−377.480	−0.0388935
$456$	0	0
$457$	−17323.5	−1.77322	−0.886608	−	0.462521i	$-0.846945\pi$
$457$	−17323.5	−1.77322	−0.886608	+	0.462521i	$0.846945\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−13388.4	−1.35263	−0.676314	−	0.736613i	$-0.736424\pi$
$461$	−13388.4	−1.35263	−0.676314	+	0.736613i	$0.736424\pi$
$462$	0	0
$463$	−11483.2	−1.15263	−0.576315	−	0.817228i	$-0.695510\pi$
$463$	−11483.2	−1.15263	−0.576315	+	0.817228i	$0.695510\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	18016.7	1.78526	0.892629	−	0.450793i	$-0.148859\pi$
$467$	18016.7	1.78526	0.892629	+	0.450793i	$0.148859\pi$
$468$	0	0
$469$	−7810.64	−0.769001
$470$	0	0
$471$	0	0
$472$	0	0
$473$	2020.31	0.196393
$474$	0	0
$475$	−312.773	−0.0302126
$476$	0	0
$477$	0	0
$478$	0	0
$479$	17052.2	1.62659	0.813294	−	0.581853i	$-0.197672\pi$
$479$	17052.2	1.62659	0.813294	+	0.581853i	$0.197672\pi$
$480$	0	0
$481$	540.836	0.0512682
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−4228.82	−0.395919
$486$	0	0
$487$	3327.54	0.309620	0.154810	−	0.987944i	$-0.450523\pi$
$487$	3327.54	0.309620	0.154810	+	0.987944i	$0.450523\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	5972.19	0.548923	0.274462	−	0.961598i	$-0.411500\pi$
$491$	5972.19	0.548923	0.274462	+	0.961598i	$0.411500\pi$
$492$	0	0
$493$	−23876.8	−2.18125
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−20787.1	−1.87612
$498$	0	0
$499$	6934.90	0.622142	0.311071	−	0.950387i	$-0.399312\pi$
$499$	6934.90	0.622142	0.311071	+	0.950387i	$0.399312\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	5751.15	0.509803	0.254902	−	0.966967i	$-0.417957\pi$
$503$	5751.15	0.509803	0.254902	+	0.966967i	$0.417957\pi$
$504$	0	0
$505$	−2914.83	−0.256848
$506$	0	0
$507$	0	0
$508$	0	0
$509$	12012.5	1.04606	0.523029	−	0.852315i	$-0.324802\pi$
$509$	12012.5	1.04606	0.523029	+	0.852315i	$0.324802\pi$
$510$	0	0
$511$	7166.11	0.620372
$512$	0	0
$513$	0	0
$514$	0	0
$515$	4240.18	0.362805
$516$	0	0
$517$	1530.82	0.130223
$518$	0	0
$519$	0	0
$520$	0	0
$521$	13667.9	1.14933	0.574666	−	0.818388i	$-0.305132\pi$
$521$	13667.9	1.14933	0.574666	+	0.818388i	$0.305132\pi$
$522$	0	0
$523$	−22339.9	−1.86779	−0.933895	−	0.357548i	$-0.883613\pi$
$523$	−22339.9	−1.86779	−0.933895	+	0.357548i	$0.883613\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−25517.1	−2.10919
$528$	0	0
$529$	11147.5	0.916206
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−163.243	−0.0132661
$534$	0	0
$535$	−3853.50	−0.311404
$536$	0	0
$537$	0	0
$538$	0	0
$539$	1026.39	0.0820218
$540$	0	0
$541$	22676.0	1.80207	0.901034	−	0.433749i	$-0.142809\pi$
$541$	22676.0	1.80207	0.901034	+	0.433749i	$0.142809\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−1287.47	−0.101191
$546$	0	0
$547$	10032.5	0.784205	0.392102	−	0.919922i	$-0.371748\pi$
$547$	10032.5	0.784205	0.392102	+	0.919922i	$0.371748\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	6854.01	0.529929
$552$	0	0
$553$	−16526.4	−1.27084
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−12557.1	−0.955224	−0.477612	−	0.878571i	$-0.658498\pi$
$557$	−12557.1	−0.955224	−0.477612	+	0.878571i	$0.658498\pi$
$558$	0	0
$559$	619.059	0.0468397
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−20107.7	−1.50522	−0.752609	−	0.658467i	$-0.771205\pi$
$563$	−20107.7	−1.50522	−0.752609	+	0.658467i	$0.771205\pi$
$564$	0	0
$565$	13077.0	0.973720
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−5390.81	−0.397178	−0.198589	−	0.980083i	$-0.563636\pi$
$569$	−5390.81	−0.397178	−0.198589	+	0.980083i	$0.563636\pi$
$570$	0	0
$571$	14230.2	1.04293	0.521467	−	0.853271i	$-0.325385\pi$
$571$	14230.2	1.04293	0.521467	+	0.853271i	$0.325385\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−1675.43	−0.121514
$576$	0	0
$577$	−4781.42	−0.344979	−0.172490	−	0.985011i	$-0.555181\pi$
$577$	−4781.42	−0.344979	−0.172490	+	0.985011i	$0.555181\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	9885.63	0.705895
$582$	0	0
$583$	−3449.73	−0.245065
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−12736.8	−0.895579	−0.447790	−	0.894139i	$-0.647789\pi$
$587$	−12736.8	−0.895579	−0.447790	+	0.894139i	$0.647789\pi$
$588$	0	0
$589$	7324.89	0.512423
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−11262.9	−0.779954	−0.389977	−	0.920825i	$-0.627517\pi$
$593$	−11262.9	−0.779954	−0.389977	+	0.920825i	$0.627517\pi$
$594$	0	0
$595$	24913.4	1.71655
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−3683.92	−0.251287	−0.125643	−	0.992075i	$-0.540100\pi$
$599$	−3683.92	−0.251287	−0.125643	+	0.992075i	$0.540100\pi$
$600$	0	0
$601$	−9424.33	−0.639644	−0.319822	−	0.947478i	$-0.603623\pi$
$601$	−9424.33	−0.639644	−0.319822	+	0.947478i	$0.603623\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	13955.4	0.937801
$606$	0	0
$607$	2711.28	0.181297	0.0906486	−	0.995883i	$-0.471106\pi$
$607$	2711.28	0.181297	0.0906486	+	0.995883i	$0.471106\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	469.071	0.0310582
$612$	0	0
$613$	−15014.7	−0.989298	−0.494649	−	0.869093i	$-0.664703\pi$
$613$	−15014.7	−0.989298	−0.494649	+	0.869093i	$0.664703\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	25910.7	1.69064	0.845319	−	0.534262i	$-0.179410\pi$
$617$	25910.7	1.69064	0.845319	+	0.534262i	$0.179410\pi$
$618$	0	0
$619$	29146.4	1.89256	0.946278	−	0.323355i	$-0.104811\pi$
$619$	29146.4	1.89256	0.946278	+	0.323355i	$0.104811\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	37827.7	2.43264
$624$	0	0
$625$	−14133.0	−0.904513
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−35694.8	−2.26271
$630$	0	0
$631$	−10221.9	−0.644893	−0.322446	−	0.946588i	$-0.604505\pi$
$631$	−10221.9	−0.644893	−0.322446	+	0.946588i	$0.604505\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	21265.5	1.32897
$636$	0	0
$637$	314.505	0.0195622
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−11741.1	−0.723470	−0.361735	−	0.932281i	$-0.617815\pi$
$641$	−11741.1	−0.723470	−0.361735	+	0.932281i	$0.617815\pi$
$642$	0	0
$643$	15723.8	0.964362	0.482181	−	0.876071i	$-0.339845\pi$
$643$	15723.8	0.964362	0.482181	+	0.876071i	$0.339845\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−6458.34	−0.392432	−0.196216	−	0.980561i	$-0.562865\pi$
$647$	−6458.34	−0.392432	−0.196216	+	0.980561i	$0.562865\pi$
$648$	0	0
$649$	2352.87	0.142309
$650$	0	0
$651$	0	0
$652$	0	0
$653$	3658.97	0.219275	0.109637	−	0.993972i	$-0.465031\pi$
$653$	3658.97	0.219275	0.109637	+	0.993972i	$0.465031\pi$
$654$	0	0
$655$	−16845.0	−1.00487
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−10479.8	−0.619476	−0.309738	−	0.950822i	$-0.600241\pi$
$659$	−10479.8	−0.619476	−0.309738	+	0.950822i	$0.600241\pi$
$660$	0	0
$661$	19818.8	1.16620	0.583102	−	0.812399i	$-0.301839\pi$
$661$	19818.8	1.16620	0.583102	+	0.812399i	$0.301839\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−7151.59	−0.417032
$666$	0	0
$667$	36715.0	2.13135
$668$	0	0
$669$	0	0
$670$	0	0
$671$	2447.48	0.140811
$672$	0	0
$673$	−27868.0	−1.59619	−0.798094	−	0.602533i	$-0.794158\pi$
$673$	−27868.0	−1.59619	−0.798094	+	0.602533i	$0.794158\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	14400.4	0.817508	0.408754	−	0.912645i	$-0.365963\pi$
$677$	14400.4	0.817508	0.408754	+	0.912645i	$0.365963\pi$
$678$	0	0
$679$	9304.63	0.525889
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−8387.44	−0.469892	−0.234946	−	0.972008i	$-0.575491\pi$
$683$	−8387.44	−0.469892	−0.234946	+	0.972008i	$0.575491\pi$
$684$	0	0
$685$	8927.20	0.497942
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−1057.06	−0.0584482
$690$	0	0
$691$	21175.7	1.16579	0.582895	−	0.812547i	$-0.301920\pi$
$691$	21175.7	1.16579	0.582895	+	0.812547i	$0.301920\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	29137.1	1.59026
$696$	0	0
$697$	10773.9	0.585497
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−7467.68	−0.402354	−0.201177	−	0.979555i	$-0.564477\pi$
$701$	−7467.68	−0.402354	−0.201177	+	0.979555i	$0.564477\pi$
$702$	0	0
$703$	10246.5	0.549720
$704$	0	0
$705$	0	0
$706$	0	0
$707$	6413.46	0.341164
$708$	0	0
$709$	19065.2	1.00988	0.504942	−	0.863153i	$-0.331514\pi$
$709$	19065.2	1.00988	0.504942	+	0.863153i	$0.331514\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	39237.4	2.06094
$714$	0	0
$715$	−78.8861	−0.00412611
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−6863.35	−0.355994	−0.177997	−	0.984031i	$-0.556962\pi$
$719$	−6863.35	−0.355994	−0.177997	+	0.984031i	$0.556962\pi$
$720$	0	0
$721$	−9329.62	−0.481904
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−2638.43	−0.135157
$726$	0	0
$727$	14783.6	0.754184	0.377092	−	0.926176i	$-0.376924\pi$
$727$	14783.6	0.754184	0.377092	+	0.926176i	$0.376924\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−40857.5	−2.06726
$732$	0	0
$733$	4001.98	0.201660	0.100830	−	0.994904i	$-0.467850\pi$
$733$	4001.98	0.201660	0.100830	+	0.994904i	$0.467850\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−1632.27	−0.0815815
$738$	0	0
$739$	16889.8	0.840733	0.420367	−	0.907354i	$-0.361901\pi$
$739$	16889.8	0.840733	0.420367	+	0.907354i	$0.361901\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−30516.2	−1.50677	−0.753386	−	0.657579i	$-0.771581\pi$
$743$	−30516.2	−1.50677	−0.753386	+	0.657579i	$0.771581\pi$
$744$	0	0
$745$	29644.0	1.45781
$746$	0	0
$747$	0	0
$748$	0	0
$749$	8478.81	0.413630
$750$	0	0
$751$	13862.2	0.673554	0.336777	−	0.941585i	$-0.390663\pi$
$751$	13862.2	0.673554	0.336777	+	0.941585i	$0.390663\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	8147.88	0.392757
$756$	0	0
$757$	−15205.3	−0.730049	−0.365024	−	0.930998i	$-0.618939\pi$
$757$	−15205.3	−0.730049	−0.365024	+	0.930998i	$0.618939\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	4840.85	0.230592	0.115296	−	0.993331i	$-0.463218\pi$
$761$	4840.85	0.230592	0.115296	+	0.993331i	$0.463218\pi$
$762$	0	0
$763$	2832.80	0.134409
$764$	0	0
$765$	0	0
$766$	0	0
$767$	720.964	0.0339407
$768$	0	0
$769$	−17056.2	−0.799819	−0.399909	−	0.916555i	$-0.630958\pi$
$769$	−17056.2	−0.799819	−0.399909	+	0.916555i	$0.630958\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−10350.7	−0.481618	−0.240809	−	0.970573i	$-0.577413\pi$
$773$	−10350.7	−0.481618	−0.240809	+	0.970573i	$0.577413\pi$
$774$	0	0
$775$	−2819.69	−0.130692
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−3092.74	−0.142245
$780$	0	0
$781$	−4344.11	−0.199033
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−7533.43	−0.342522
$786$	0	0
$787$	6400.88	0.289919	0.144960	−	0.989438i	$-0.453695\pi$
$787$	6400.88	0.289919	0.144960	+	0.989438i	$0.453695\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−28773.1	−1.29337
$792$	0	0
$793$	749.955	0.0335834
$794$	0	0
$795$	0	0
$796$	0	0
$797$	31581.1	1.40359	0.701795	−	0.712379i	$-0.252382\pi$
$797$	31581.1	1.40359	0.701795	+	0.712379i	$0.252382\pi$
$798$	0	0
$799$	−30958.4	−1.37075
$800$	0	0
$801$	0	0
$802$	0	0
$803$	1497.58	0.0658138
$804$	0	0
$805$	−38309.0	−1.67729
$806$	0	0
$807$	0	0
$808$	0	0
$809$	3349.34	0.145558	0.0727791	−	0.997348i	$-0.476813\pi$
$809$	3349.34	0.145558	0.0727791	+	0.997348i	$0.476813\pi$
$810$	0	0
$811$	5186.82	0.224579	0.112290	−	0.993676i	$-0.464182\pi$
$811$	5186.82	0.224579	0.112290	+	0.993676i	$0.464182\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−16953.0	−0.728633
$816$	0	0
$817$	11728.5	0.502236
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−26665.0	−1.13352	−0.566758	−	0.823884i	$-0.691803\pi$
$821$	−26665.0	−1.13352	−0.566758	+	0.823884i	$0.691803\pi$
$822$	0	0
$823$	−26304.3	−1.11411	−0.557053	−	0.830477i	$-0.688068\pi$
$823$	−26304.3	−1.11411	−0.557053	+	0.830477i	$0.688068\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	10768.0	0.452769	0.226385	−	0.974038i	$-0.427309\pi$
$827$	10768.0	0.452769	0.226385	+	0.974038i	$0.427309\pi$
$828$	0	0
$829$	−11960.9	−0.501110	−0.250555	−	0.968102i	$-0.580613\pi$
$829$	−11960.9	−0.501110	−0.250555	+	0.968102i	$0.580613\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−20757.1	−0.863374
$834$	0	0
$835$	−7056.94	−0.292474
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−14354.1	−0.590656	−0.295328	−	0.955396i	$-0.595429\pi$
$839$	−14354.1	−0.590656	−0.295328	+	0.955396i	$0.595429\pi$
$840$	0	0
$841$	33428.8	1.37065
$842$	0	0
$843$	0	0
$844$	0	0
$845$	23436.2	0.954117
$846$	0	0
$847$	−30706.0	−1.24566
$848$	0	0
$849$	0	0
$850$	0	0
$851$	54887.5	2.21095
$852$	0	0
$853$	−12131.1	−0.486941	−0.243470	−	0.969908i	$-0.578286\pi$
$853$	−12131.1	−0.486941	−0.243470	+	0.969908i	$0.578286\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−18752.9	−0.747477	−0.373739	−	0.927534i	$-0.621924\pi$
$857$	−18752.9	−0.747477	−0.373739	+	0.927534i	$0.621924\pi$
$858$	0	0
$859$	−11292.5	−0.448539	−0.224269	−	0.974527i	$-0.572000\pi$
$859$	−11292.5	−0.448539	−0.224269	+	0.974527i	$0.572000\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−5959.72	−0.235077	−0.117538	−	0.993068i	$-0.537500\pi$
$863$	−5959.72	−0.235077	−0.117538	+	0.993068i	$0.537500\pi$
$864$	0	0
$865$	−21755.8	−0.855166
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−3453.70	−0.134820
$870$	0	0
$871$	−500.159	−0.0194572
$872$	0	0
$873$	0	0
$874$	0	0
$875$	34114.6	1.31804
$876$	0	0
$877$	17824.9	0.686324	0.343162	−	0.939276i	$-0.388502\pi$
$877$	17824.9	0.686324	0.343162	+	0.939276i	$0.388502\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	44756.5	1.71156	0.855781	−	0.517339i	$-0.173077\pi$
$881$	44756.5	1.71156	0.855781	+	0.517339i	$0.173077\pi$
$882$	0	0
$883$	−7241.57	−0.275989	−0.137995	−	0.990433i	$-0.544066\pi$
$883$	−7241.57	−0.275989	−0.137995	+	0.990433i	$0.544066\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−33337.9	−1.26198	−0.630990	−	0.775791i	$-0.717351\pi$
$887$	−33337.9	−1.26198	−0.630990	+	0.775791i	$0.717351\pi$
$888$	0	0
$889$	−46790.2	−1.76524
$890$	0	0
$891$	0	0
$892$	0	0
$893$	8886.84	0.333020
$894$	0	0
$895$	−27539.6	−1.02855
$896$	0	0
$897$	0	0
$898$	0	0
$899$	61789.9	2.29234
$900$	0	0
$901$	69765.2	2.57960
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−19772.6	−0.726259
$906$	0	0
$907$	−18508.3	−0.677571	−0.338785	−	0.940864i	$-0.610016\pi$
$907$	−18508.3	−0.677571	−0.338785	+	0.940864i	$0.610016\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−14705.6	−0.534817	−0.267409	−	0.963583i	$-0.586167\pi$
$911$	−14705.6	−0.534817	−0.267409	+	0.963583i	$0.586167\pi$
$912$	0	0
$913$	2065.91	0.0748867
$914$	0	0
$915$	0	0
$916$	0	0
$917$	37063.8	1.33474
$918$	0	0
$919$	−44333.5	−1.59132	−0.795662	−	0.605741i	$-0.792877\pi$
$919$	−44333.5	−1.59132	−0.795662	+	0.605741i	$0.792877\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−1331.12	−0.0474693
$924$	0	0
$925$	−3944.35	−0.140205
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−30228.0	−1.06755	−0.533773	−	0.845628i	$-0.679226\pi$
$929$	−30228.0	−1.06755	−0.533773	+	0.845628i	$0.679226\pi$
$930$	0	0
$931$	5958.49	0.209755
$932$	0	0
$933$	0	0
$934$	0	0
$935$	5206.43	0.182105
$936$	0	0
$937$	22568.0	0.786834	0.393417	−	0.919360i	$-0.371293\pi$
$937$	22568.0	0.786834	0.393417	+	0.919360i	$0.371293\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−35085.1	−1.21545	−0.607727	−	0.794146i	$-0.707918\pi$
$941$	−35085.1	−1.21545	−0.607727	+	0.794146i	$0.707918\pi$
$942$	0	0
$943$	−16566.9	−0.572103
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−34248.1	−1.17520	−0.587600	−	0.809152i	$-0.699927\pi$
$947$	−34248.1	−1.17520	−0.587600	+	0.809152i	$0.699927\pi$
$948$	0	0
$949$	458.886	0.0156966
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−38627.8	−1.31299	−0.656494	−	0.754331i	$-0.727961\pi$
$953$	−38627.8	−1.31299	−0.656494	+	0.754331i	$0.727961\pi$
$954$	0	0
$955$	608.304	0.0206118
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−19642.4	−0.661404
$960$	0	0
$961$	36244.0	1.21661
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−19370.8	−0.646186
$966$	0	0
$967$	−15628.8	−0.519739	−0.259869	−	0.965644i	$-0.583680\pi$
$967$	−15628.8	−0.519739	−0.259869	+	0.965644i	$0.583680\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	26637.8	0.880379	0.440190	−	0.897905i	$-0.354911\pi$
$971$	26637.8	0.880379	0.440190	+	0.897905i	$0.354911\pi$
$972$	0	0
$973$	−64109.9	−2.11230
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−1777.58	−0.0582085	−0.0291043	−	0.999576i	$-0.509265\pi$
$977$	−1777.58	−0.0582085	−0.0291043	+	0.999576i	$0.509265\pi$
$978$	0	0
$979$	7905.26	0.258073
$980$	0	0
$981$	0	0
$982$	0	0
$983$	47634.4	1.54558	0.772788	−	0.634664i	$-0.218861\pi$
$983$	47634.4	1.54558	0.772788	+	0.634664i	$0.218861\pi$
$984$	0	0
$985$	−9325.77	−0.301669
$986$	0	0
$987$	0	0
$988$	0	0
$989$	62826.0	2.01997
$990$	0	0
$991$	−33719.5	−1.08086	−0.540432	−	0.841388i	$-0.681739\pi$
$991$	−33719.5	−1.08086	−0.540432	+	0.841388i	$0.681739\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	54070.9	1.72278
$996$	0	0
$997$	−400.777	−0.0127309	−0.00636547	−	0.999980i	$-0.502026\pi$
$997$	−400.777	−0.0127309	−0.00636547	+	0.999980i	$0.502026\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	324.4.a.e.1.2	✓	4
3.2	odd	2	inner	324.4.a.e.1.3	yes	4
4.3	odd	2		1296.4.a.z.1.2		4
9.2	odd	6		324.4.e.i.109.2		8
9.4	even	3		324.4.e.i.217.3		8
9.5	odd	6		324.4.e.i.217.2		8
9.7	even	3		324.4.e.i.109.3		8
12.11	even	2		1296.4.a.z.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
324.4.a.e.1.2	✓	4	1.1	even	1	trivial
324.4.a.e.1.3	yes	4	3.2	odd	2	inner
324.4.e.i.109.2		8	9.2	odd	6
324.4.e.i.109.3		8	9.7	even	3
324.4.e.i.217.2		8	9.5	odd	6
324.4.e.i.217.3		8	9.4	even	3
1296.4.a.z.1.2		4	4.3	odd	2
1296.4.a.z.1.3		4	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 324 = 2^{2} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	324.a (trivial)

\(f(q)\)	\(=\)	\(q-10.6784 q^{5} +23.4955 q^{7} +4.91010 q^{11} +1.50455 q^{13} -99.2990 q^{17} +28.5045 q^{19} +152.691 q^{23} -10.9727 q^{25} +240.453 q^{29} +256.973 q^{31} -250.893 q^{35} +359.468 q^{37} -108.500 q^{41} +411.459 q^{43} +311.769 q^{47} +209.036 q^{49} -702.578 q^{53} -52.4318 q^{55} +479.190 q^{59} +498.459 q^{61} -16.0661 q^{65} -332.432 q^{67} -884.729 q^{71} +305.000 q^{73} +115.365 q^{77} -703.386 q^{79} +420.747 q^{83} +1060.35 q^{85} +1610.00 q^{89} +35.3500 q^{91} -304.382 q^{95} +396.018 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 16 q^{7} + 116 q^{13} + 224 q^{19} + 616 q^{25} + 368 q^{31} + 668 q^{37} + 656 q^{43} + 1716 q^{49} + 1440 q^{55} + 1004 q^{61} + 320 q^{67} + 1220 q^{73} - 64 q^{79} + 612 q^{85} - 3488 q^{91} + 2024 q^{97}+O(q^{100}) \) 4 * q - 16 * q^7 + 116 * q^13 + 224 * q^19 + 616 * q^25 + 368 * q^31 + 668 * q^37 + 656 * q^43 + 1716 * q^49 + 1440 * q^55 + 1004 * q^61 + 320 * q^67 + 1220 * q^73 - 64 * q^79 + 612 * q^85 - 3488 * q^91 + 2024 * q^97

Introduction

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