LMFDB - Embedded newform 1296.4.a.z.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1296,4,Mod(1,1296)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1296, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1296.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1296 = 2^{4} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1296.a (trivial)

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:	yes
Analytic conductor:	$76.4664753674$
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:	no (minimal twist has level 324)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.18890$ of defining polynomial
Character	$\chi$	$=$	1296.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-10.6784 q^{5} -23.4955 q^{7} +O(q^{10})$	$q-10.6784 q^{5} -23.4955 q^{7} -4.91010 q^{11} +1.50455 q^{13} -99.2990 q^{17} -28.5045 q^{19} -152.691 q^{23} -10.9727 q^{25} +240.453 q^{29} -256.973 q^{31} +250.893 q^{35} +359.468 q^{37} -108.500 q^{41} -411.459 q^{43} -311.769 q^{47} +209.036 q^{49} -702.578 q^{53} +52.4318 q^{55} -479.190 q^{59} +498.459 q^{61} -16.0661 q^{65} +332.432 q^{67} +884.729 q^{71} +305.000 q^{73} +115.365 q^{77} +703.386 q^{79} -420.747 q^{83} +1060.35 q^{85} +1610.00 q^{89} -35.3500 q^{91} +304.382 q^{95} +396.018 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 16 q^{7}+O(q^{10}) $ 4 * q + 16 * q^7	$ 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

Display 100 coefficients 10 coefficients

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	$-0.718719\pi$
$7$	−23.4955	−1.26864	−0.634318	+	0.773073i	$0.718719\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−4.91010	−0.134586	−0.0672932	−	0.997733i	$-0.521436\pi$
$11$	−4.91010	−0.134586	−0.0672932	+	0.997733i	$0.521436\pi$
$12$	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	$-0.555052\pi$
$19$	−28.5045	−0.344178	−0.172089	+	0.985081i	$0.555052\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−152.691	−1.38427	−0.692135	−	0.721768i	$-0.743330\pi$
$23$	−152.691	−1.38427	−0.692135	+	0.721768i	$0.743330\pi$
$24$	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	$-0.767271\pi$
$31$	−256.973	−1.48883	−0.744414	+	0.667718i	$0.767271\pi$
$32$	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	$-0.760301\pi$
$43$	−411.459	−1.45923	−0.729615	+	0.683858i	$0.760301\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−311.769	−0.967579	−0.483789	−	0.875184i	$-0.660740\pi$
$47$	−311.769	−0.967579	−0.483789	+	0.875184i	$0.660740\pi$
$48$	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	$-0.677316\pi$
$59$	−479.190	−1.05738	−0.528689	+	0.848816i	$0.677316\pi$
$60$	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	$-0.401984\pi$
$67$	332.432	0.606164	0.303082	+	0.952964i	$0.401984\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	884.729	1.47885	0.739423	−	0.673242i	$-0.235099\pi$
$71$	884.729	1.47885	0.739423	+	0.673242i	$0.235099\pi$
$72$	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	$-0.333014\pi$
$79$	703.386	1.00174	0.500868	+	0.865524i	$0.333014\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−420.747	−0.556421	−0.278210	−	0.960520i	$-0.589741\pi$
$83$	−420.747	−0.556421	−0.278210	+	0.960520i	$0.589741\pi$
$84$	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	$-0.439174\pi$
$103$	397.082	0.379861	0.189930	+	0.981798i	$0.439174\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−360.870	−0.326043	−0.163022	−	0.986622i	$-0.552124\pi$
$107$	−360.870	−0.326043	−0.163022	+	0.986622i	$0.552124\pi$
$108$	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	$-0.255085\pi$
$127$	1991.46	1.39144	0.695722	+	0.718311i	$0.255085\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−1577.49	−1.05210	−0.526052	−	0.850452i	$-0.676328\pi$
$131$	−1577.49	−1.05210	−0.526052	+	0.850452i	$0.676328\pi$
$132$	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	$-0.186903\pi$
$139$	2728.61	1.66502	0.832509	+	0.554011i	$0.186903\pi$
$140$	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	$-0.434082\pi$
$151$	763.027	0.411220	0.205610	+	0.978634i	$0.434082\pi$
$152$	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	$-0.624573\pi$
$163$	−1587.60	−0.762886	−0.381443	+	0.924392i	$0.624573\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−660.864	−0.306223	−0.153111	−	0.988209i	$-0.548929\pi$
$167$	−660.864	−0.306223	−0.153111	+	0.988209i	$0.548929\pi$
$168$	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	$-0.680989\pi$
$179$	−2579.01	−1.07690	−0.538448	+	0.842659i	$0.680989\pi$
$180$	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	$-0.496565\pi$
$191$	56.9661	0.0215807	0.0107904	+	0.999942i	$0.496565\pi$
$192$	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	$-0.142186\pi$
$199$	5063.60	1.80376	0.901882	+	0.431982i	$0.142186\pi$
$200$	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	$-0.522388\pi$
$211$	−430.777	−0.140549	−0.0702747	+	0.997528i	$0.522388\pi$
$212$	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	$-0.336987\pi$
$223$	3263.68	0.980054	0.490027	+	0.871707i	$0.336987\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−748.679	−0.218906	−0.109453	−	0.993992i	$-0.534910\pi$
$227$	−748.679	−0.218906	−0.109453	+	0.993992i	$0.534910\pi$
$228$	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	$-0.526235\pi$
$239$	−608.375	−0.164655	−0.0823274	+	0.996605i	$0.526235\pi$
$240$	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	$-0.197147\pi$
$251$	6475.89	1.62850	0.814252	+	0.580512i	$0.197147\pi$
$252$	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	$-0.809450\pi$
$263$	−7046.94	−1.65222	−0.826108	+	0.563512i	$0.809450\pi$
$264$	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	$-0.692660\pi$
$271$	−5076.65	−1.13795	−0.568975	+	0.822355i	$0.692660\pi$
$272$	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	$-0.429281\pi$
$283$	2098.05	0.440694	0.220347	+	0.975422i	$0.429281\pi$
$284$	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	$-0.561577\pi$
$307$	−2068.21	−0.384492	−0.192246	+	0.981347i	$0.561577\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−4911.57	−0.895528	−0.447764	−	0.894152i	$-0.647780\pi$
$311$	−4911.57	−0.895528	−0.447764	+	0.894152i	$0.647780\pi$
$312$	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	$-0.353943\pi$
$331$	5334.51	0.885835	0.442917	+	0.896562i	$0.353943\pi$
$332$	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	$-0.636400\pi$
$347$	−5371.73	−0.831036	−0.415518	+	0.909585i	$0.636400\pi$
$348$	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	$-0.425776\pi$
$359$	3143.59	0.462151	0.231075	+	0.972936i	$0.425776\pi$
$360$	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	$-0.751248\pi$
$367$	−9981.83	−1.41975	−0.709873	+	0.704329i	$0.751248\pi$
$368$	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	$-0.775587\pi$
$379$	−11238.7	−1.52321	−0.761603	+	0.648044i	$0.775587\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	14688.9	1.95970	0.979851	−	0.199730i	$-0.0640064\pi$
$383$	14688.9	1.95970	0.979851	+	0.199730i	$0.0640064\pi$
$384$	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	$-0.386867\pi$
$419$	5969.10	0.695966	0.347983	+	0.937501i	$0.386867\pi$
$420$	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	$-0.732250\pi$
$431$	−11929.1	−1.33319	−0.666596	+	0.745419i	$0.732250\pi$
$432$	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	$-0.270953\pi$
$439$	12124.2	1.31813	0.659063	+	0.752088i	$0.270953\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	12437.6	1.33392	0.666962	−	0.745092i	$-0.267594\pi$
$443$	12437.6	1.33392	0.666962	+	0.745092i	$0.267594\pi$
$444$	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	$-0.304490\pi$
$463$	11483.2	1.15263	0.576315	+	0.817228i	$0.304490\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−18016.7	−1.78526	−0.892629	−	0.450793i	$-0.851141\pi$
$467$	−18016.7	−1.78526	−0.892629	+	0.450793i	$0.851141\pi$
$468$	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	$-0.802328\pi$
$479$	−17052.2	−1.62659	−0.813294	+	0.581853i	$0.802328\pi$
$480$	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	$-0.549477\pi$
$487$	−3327.54	−0.309620	−0.154810	+	0.987944i	$0.549477\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−5972.19	−0.548923	−0.274462	−	0.961598i	$-0.588500\pi$
$491$	−5972.19	−0.548923	−0.274462	+	0.961598i	$0.588500\pi$
$492$	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	$-0.600688\pi$
$499$	−6934.90	−0.622142	−0.311071	+	0.950387i	$0.600688\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−5751.15	−0.509803	−0.254902	−	0.966967i	$-0.582043\pi$
$503$	−5751.15	−0.509803	−0.254902	+	0.966967i	$0.582043\pi$
$504$	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	$-0.116387\pi$
$523$	22339.9	1.86779	0.933895	+	0.357548i	$0.116387\pi$
$524$	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	$-0.628252\pi$
$547$	−10032.5	−0.784205	−0.392102	+	0.919922i	$0.628252\pi$
$548$	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	$-0.228795\pi$
$563$	20107.7	1.50522	0.752609	+	0.658467i	$0.228795\pi$
$564$	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	$-0.674615\pi$
$571$	−14230.2	−1.04293	−0.521467	+	0.853271i	$0.674615\pi$
$572$	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	$-0.352211\pi$
$587$	12736.8	0.895579	0.447790	+	0.894139i	$0.352211\pi$
$588$	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	$-0.459900\pi$
$599$	3683.92	0.251287	0.125643	+	0.992075i	$0.459900\pi$
$600$	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	$-0.528894\pi$
$607$	−2711.28	−0.181297	−0.0906486	+	0.995883i	$0.528894\pi$
$608$	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	$-0.895189\pi$
$619$	−29146.4	−1.89256	−0.946278	+	0.323355i	$0.895189\pi$
$620$	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	$-0.395495\pi$
$631$	10221.9	0.644893	0.322446	+	0.946588i	$0.395495\pi$
$632$	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	$-0.660155\pi$
$643$	−15723.8	−0.964362	−0.482181	+	0.876071i	$0.660155\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	6458.34	0.392432	0.196216	−	0.980561i	$-0.437135\pi$
$647$	6458.34	0.392432	0.196216	+	0.980561i	$0.437135\pi$
$648$	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	$-0.399759\pi$
$659$	10479.8	0.619476	0.309738	+	0.950822i	$0.399759\pi$
$660$	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	$-0.424509\pi$
$683$	8387.44	0.469892	0.234946	+	0.972008i	$0.424509\pi$
$684$	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	$-0.698080\pi$
$691$	−21175.7	−1.16579	−0.582895	+	0.812547i	$0.698080\pi$
$692$	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	$-0.443038\pi$
$719$	6863.35	0.355994	0.177997	+	0.984031i	$0.443038\pi$
$720$	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	$-0.623076\pi$
$727$	−14783.6	−0.754184	−0.377092	+	0.926176i	$0.623076\pi$
$728$	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	$-0.638099\pi$
$739$	−16889.8	−0.840733	−0.420367	+	0.907354i	$0.638099\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	30516.2	1.50677	0.753386	−	0.657579i	$-0.228419\pi$
$743$	30516.2	1.50677	0.753386	+	0.657579i	$0.228419\pi$
$744$	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	$-0.609337\pi$
$751$	−13862.2	−0.673554	−0.336777	+	0.941585i	$0.609337\pi$
$752$	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	$-0.546305\pi$
$787$	−6400.88	−0.289919	−0.144960	+	0.989438i	$0.546305\pi$
$788$	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	$-0.535818\pi$
$811$	−5186.82	−0.224579	−0.112290	+	0.993676i	$0.535818\pi$
$812$	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	$-0.311932\pi$
$823$	26304.3	1.11411	0.557053	+	0.830477i	$0.311932\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−10768.0	−0.452769	−0.226385	−	0.974038i	$-0.572691\pi$
$827$	−10768.0	−0.452769	−0.226385	+	0.974038i	$0.572691\pi$
$828$	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	$-0.404571\pi$
$839$	14354.1	0.590656	0.295328	+	0.955396i	$0.404571\pi$
$840$	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	$-0.428000\pi$
$859$	11292.5	0.448539	0.224269	+	0.974527i	$0.428000\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	5959.72	0.235077	0.117538	−	0.993068i	$-0.462500\pi$
$863$	5959.72	0.235077	0.117538	+	0.993068i	$0.462500\pi$
$864$	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	$-0.455934\pi$
$883$	7241.57	0.275989	0.137995	+	0.990433i	$0.455934\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	33337.9	1.26198	0.630990	−	0.775791i	$-0.282649\pi$
$887$	33337.9	1.26198	0.630990	+	0.775791i	$0.282649\pi$
$888$	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	$-0.389984\pi$
$907$	18508.3	0.677571	0.338785	+	0.940864i	$0.389984\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	14705.6	0.534817	0.267409	−	0.963583i	$-0.413833\pi$
$911$	14705.6	0.534817	0.267409	+	0.963583i	$0.413833\pi$
$912$	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	$-0.207123\pi$
$919$	44333.5	1.59132	0.795662	+	0.605741i	$0.207123\pi$
$920$	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	$-0.300073\pi$
$947$	34248.1	1.17520	0.587600	+	0.809152i	$0.300073\pi$
$948$	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	$-0.416320\pi$
$967$	15628.8	0.519739	0.259869	+	0.965644i	$0.416320\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−26637.8	−0.880379	−0.440190	−	0.897905i	$-0.645089\pi$
$971$	−26637.8	−0.880379	−0.440190	+	0.897905i	$0.645089\pi$
$972$	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	$-0.781139\pi$
$983$	−47634.4	−1.54558	−0.772788	+	0.634664i	$0.781139\pi$
$984$	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	$-0.318261\pi$
$991$	33719.5	1.08086	0.540432	+	0.841388i	$0.318261\pi$
$992$	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	1296.4.a.z.1.2		4
3.2	odd	2	inner	1296.4.a.z.1.3		4
4.3	odd	2		324.4.a.e.1.2	✓	4
12.11	even	2		324.4.a.e.1.3	yes	4
36.7	odd	6		324.4.e.i.109.3		8
36.11	even	6		324.4.e.i.109.2		8
36.23	even	6		324.4.e.i.217.2		8
36.31	odd	6		324.4.e.i.217.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
324.4.a.e.1.2	✓	4	4.3	odd	2
324.4.a.e.1.3	yes	4	12.11	even	2
324.4.e.i.109.2		8	36.11	even	6
324.4.e.i.109.3		8	36.7	odd	6
324.4.e.i.217.2		8	36.23	even	6
324.4.e.i.217.3		8	36.31	odd	6
1296.4.a.z.1.2		4	1.1	even	1	trivial
1296.4.a.z.1.3		4	3.2	odd	2	inner

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:	\( N \)	\(=\)	\( 1296 = 2^{4} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1296.a (trivial)

\(f(q)\)	\(=\)	\(q-10.6784 q^{5} -23.4955 q^{7} +O(q^{10})\)	\(q-10.6784 q^{5} -23.4955 q^{7} -4.91010 q^{11} +1.50455 q^{13} -99.2990 q^{17} -28.5045 q^{19} -152.691 q^{23} -10.9727 q^{25} +240.453 q^{29} -256.973 q^{31} +250.893 q^{35} +359.468 q^{37} -108.500 q^{41} -411.459 q^{43} -311.769 q^{47} +209.036 q^{49} -702.578 q^{53} +52.4318 q^{55} -479.190 q^{59} +498.459 q^{61} -16.0661 q^{65} +332.432 q^{67} +884.729 q^{71} +305.000 q^{73} +115.365 q^{77} +703.386 q^{79} -420.747 q^{83} +1060.35 q^{85} +1610.00 q^{89} -35.3500 q^{91} +304.382 q^{95} +396.018 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 16 q^{7}+O(q^{10}) \) 4 * q + 16 * q^7	\( 4 q + 16 q^{7} + 116 q^{13} - 224 q^{19} + 616 q^{25} - 368 q^{31} + 668 q^{37} - 656 q^{43} + 1716 q^{49} - 1440 q^{55} + 1004 q^{61} - 320 q^{67} + 1220 q^{73} + 64 q^{79} + 612 q^{85} + 3488 q^{91} + 2024 q^{97}+O(q^{100}) \) 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

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