LMFDB - Embedded newform 1152.3.h.d.449.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1152,3,Mod(449,1152)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1152, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1152.449");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1152 = 2^{7} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1152.h (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$31.3897264543$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.2
Root			$-0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1152.449
Dual form			1152.3.h.d.449.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.24264 q^{5} -8.48528 q^{7} +O(q^{10})$	$q-4.24264 q^{5} -8.48528 q^{7} +4.00000 q^{11} +18.0000i q^{13} -4.24264i q^{17} -16.9706i q^{19} +36.0000i q^{23} -7.00000 q^{25} -12.7279 q^{29} +8.48528 q^{31} +36.0000 q^{35} -36.0000i q^{37} +29.6985i q^{41} -67.8823i q^{43} -36.0000i q^{47} +23.0000 q^{49} +80.6102 q^{53} -16.9706 q^{55} +80.0000 q^{59} -36.0000i q^{61} -76.3675i q^{65} -118.794i q^{67} -108.000i q^{71} -56.0000 q^{73} -33.9411 q^{77} +25.4558 q^{79} +76.0000 q^{83} +18.0000i q^{85} +89.0955i q^{89} -152.735i q^{91} +72.0000i q^{95} +104.000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q + 16 q^{11} - 28 q^{25} + 144 q^{35} + 92 q^{49} + 320 q^{59} - 224 q^{73} + 304 q^{83} + 416 q^{97}+O(q^{100}) $ 4 * q + 16 * q^11 - 28 * q^25 + 144 * q^35 + 92 * q^49 + 320 * q^59 - 224 * q^73 + 304 * q^83 + 416 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1152\mathbb{Z}\right)^\times$.

$n$	$127$	$641$	$901$
$\chi(n)$	$1$	$-1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−4.24264	−0.848528	−0.424264	−	0.905539i	$-0.639467\pi$
$5$	−4.24264	−0.848528	−0.424264	+	0.905539i	$0.639467\pi$
$6$	0	0
$7$	−8.48528	−1.21218	−0.606092	−	0.795395i	$-0.707263\pi$
$7$	−8.48528	−1.21218	−0.606092	+	0.795395i	$0.707263\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	4.00000	0.363636	0.181818	−	0.983332i	$-0.441802\pi$
$11$	4.00000	0.363636	0.181818	+	0.983332i	$0.441802\pi$
$12$	0	0
$13$	18.0000i	1.38462i	0.721602	+	0.692308i	$0.243406\pi$
$13$	18.0000i	1.38462i	−0.721602	+	0.692308i	$0.756594\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 4.24264i	− 0.249567i	−0.992184	−	0.124784i	$-0.960176\pi$
$17$	− 4.24264i	− 0.249567i	0.992184	−	0.124784i	$-0.0398236\pi$
$18$	0	0
$19$	− 16.9706i	− 0.893188i	−0.894737	−	0.446594i	$-0.852637\pi$
$19$	− 16.9706i	− 0.893188i	0.894737	−	0.446594i	$-0.147363\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	36.0000i	1.56522i	0.622514	+	0.782609i	$0.286111\pi$
$23$	36.0000i	1.56522i	−0.622514	+	0.782609i	$0.713889\pi$
$24$	0	0
$25$	−7.00000	−0.280000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−12.7279	−0.438894	−0.219447	−	0.975624i	$-0.570425\pi$
$29$	−12.7279	−0.438894	−0.219447	+	0.975624i	$0.570425\pi$
$30$	0	0
$31$	8.48528	0.273719	0.136859	−	0.990590i	$-0.456299\pi$
$31$	8.48528	0.273719	0.136859	+	0.990590i	$0.456299\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	36.0000	1.02857
$36$	0	0
$37$	− 36.0000i	− 0.972973i	−0.873688	−	0.486486i	$-0.838278\pi$
$37$	− 36.0000i	− 0.972973i	0.873688	−	0.486486i	$-0.161722\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	29.6985i	0.724353i	0.932109	+	0.362177i	$0.117966\pi$
$41$	29.6985i	0.724353i	−0.932109	+	0.362177i	$0.882034\pi$
$42$	0	0
$43$	− 67.8823i	− 1.57866i	−0.613971	−	0.789328i	$-0.710429\pi$
$43$	− 67.8823i	− 1.57866i	0.613971	−	0.789328i	$-0.289571\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 36.0000i	− 0.765957i	−0.923757	−	0.382979i	$-0.874898\pi$
$47$	− 36.0000i	− 0.765957i	0.923757	−	0.382979i	$-0.125102\pi$
$48$	0	0
$49$	23.0000	0.469388
$50$	0	0
$51$	0	0
$52$	0	0
$53$	80.6102	1.52095	0.760473	−	0.649369i	$-0.224967\pi$
$53$	80.6102	1.52095	0.760473	+	0.649369i	$0.224967\pi$
$54$	0	0
$55$	−16.9706	−0.308556
$56$	0	0
$57$	0	0
$58$	0	0
$59$	80.0000	1.35593	0.677966	−	0.735093i	$-0.262862\pi$
$59$	80.0000	1.35593	0.677966	+	0.735093i	$0.262862\pi$
$60$	0	0
$61$	− 36.0000i	− 0.590164i	−0.955472	−	0.295082i	$-0.904653\pi$
$61$	− 36.0000i	− 0.590164i	0.955472	−	0.295082i	$-0.0953470\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 76.3675i	− 1.17489i
$66$	0	0
$67$	− 118.794i	− 1.77304i	−0.462687	−	0.886522i	$-0.653114\pi$
$67$	− 118.794i	− 1.77304i	0.462687	−	0.886522i	$-0.346886\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 108.000i	− 1.52113i	−0.649264	−	0.760563i	$-0.724923\pi$
$71$	− 108.000i	− 1.52113i	0.649264	−	0.760563i	$-0.275077\pi$
$72$	0	0
$73$	−56.0000	−0.767123	−0.383562	−	0.923515i	$-0.625303\pi$
$73$	−56.0000	−0.767123	−0.383562	+	0.923515i	$0.625303\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−33.9411	−0.440794
$78$	0	0
$79$	25.4558	0.322226	0.161113	−	0.986936i	$-0.448492\pi$
$79$	25.4558	0.322226	0.161113	+	0.986936i	$0.448492\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	76.0000	0.915663	0.457831	−	0.889039i	$-0.348626\pi$
$83$	76.0000	0.915663	0.457831	+	0.889039i	$0.348626\pi$
$84$	0	0
$85$	18.0000i	0.211765i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	89.0955i	1.00107i	0.865716	+	0.500536i	$0.166864\pi$
$89$	89.0955i	1.00107i	−0.865716	+	0.500536i	$0.833136\pi$
$90$	0	0
$91$	− 152.735i	− 1.67841i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	72.0000i	0.757895i
$96$	0	0
$97$	104.000	1.07216	0.536082	−	0.844166i	$-0.319904\pi$
$97$	104.000	1.07216	0.536082	+	0.844166i	$0.319904\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−148.492	−1.47022	−0.735111	−	0.677947i	$-0.762870\pi$
$101$	−148.492	−1.47022	−0.735111	+	0.677947i	$0.762870\pi$
$102$	0	0
$103$	178.191	1.73001	0.865004	−	0.501764i	$-0.167316\pi$
$103$	178.191	1.73001	0.865004	+	0.501764i	$0.167316\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	128.000	1.19626	0.598131	−	0.801398i	$-0.295910\pi$
$107$	128.000	1.19626	0.598131	+	0.801398i	$0.295910\pi$
$108$	0	0
$109$	126.000i	1.15596i	0.816050	+	0.577982i	$0.196159\pi$
$109$	126.000i	1.15596i	−0.816050	+	0.577982i	$0.803841\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 89.0955i	− 0.788455i	−0.919013	−	0.394228i	$-0.871012\pi$
$113$	− 89.0955i	− 0.788455i	0.919013	−	0.394228i	$-0.128988\pi$
$114$	0	0
$115$	− 152.735i	− 1.32813i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	36.0000i	0.302521i
$120$	0	0
$121$	−105.000	−0.867769
$122$	0	0
$123$	0	0
$124$	0	0
$125$	135.765	1.08612
$126$	0	0
$127$	−161.220	−1.26945	−0.634726	−	0.772737i	$-0.718887\pi$
$127$	−161.220	−1.26945	−0.634726	+	0.772737i	$0.718887\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	152.000	1.16031	0.580153	−	0.814508i	$-0.302993\pi$
$131$	152.000	1.16031	0.580153	+	0.814508i	$0.302993\pi$
$132$	0	0
$133$	144.000i	1.08271i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	207.889i	1.51744i	0.651416	+	0.758720i	$0.274175\pi$
$137$	207.889i	1.51744i	−0.651416	+	0.758720i	$0.725825\pi$
$138$	0	0
$139$	− 118.794i	− 0.854633i	−0.904102	−	0.427316i	$-0.859459\pi$
$139$	− 118.794i	− 0.854633i	0.904102	−	0.427316i	$-0.140541\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	72.0000i	0.503497i
$144$	0	0
$145$	54.0000	0.372414
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−97.5807	−0.654904	−0.327452	−	0.944868i	$-0.606190\pi$
$149$	−97.5807	−0.654904	−0.327452	+	0.944868i	$0.606190\pi$
$150$	0	0
$151$	−246.073	−1.62962	−0.814812	−	0.579726i	$-0.803160\pi$
$151$	−246.073	−1.62962	−0.814812	+	0.579726i	$0.803160\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−36.0000	−0.232258
$156$	0	0
$157$	− 252.000i	− 1.60510i	−0.596588	−	0.802548i	$-0.703477\pi$
$157$	− 252.000i	− 1.60510i	0.596588	−	0.802548i	$-0.296523\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 305.470i	− 1.89733i
$162$	0	0
$163$	− 101.823i	− 0.624683i	−0.949970	−	0.312342i	$-0.898887\pi$
$163$	− 101.823i	− 0.624683i	0.949970	−	0.312342i	$-0.101113\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	0	0
$169$	−155.000	−0.917160
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−29.6985	−0.171668	−0.0858338	−	0.996309i	$-0.527355\pi$
$173$	−29.6985	−0.171668	−0.0858338	+	0.996309i	$0.527355\pi$
$174$	0	0
$175$	59.3970	0.339411
$176$	0	0
$177$	0	0
$178$	0	0
$179$	56.0000	0.312849	0.156425	−	0.987690i	$-0.450003\pi$
$179$	56.0000	0.312849	0.156425	+	0.987690i	$0.450003\pi$
$180$	0	0
$181$	− 126.000i	− 0.696133i	−0.937470	−	0.348066i	$-0.886838\pi$
$181$	− 126.000i	− 0.696133i	0.937470	−	0.348066i	$-0.113162\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	152.735i	0.825595i
$186$	0	0
$187$	− 16.9706i	− 0.0907517i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	0	0
$193$	298.000	1.54404	0.772021	−	0.635597i	$-0.219246\pi$
$193$	298.000	1.54404	0.772021	+	0.635597i	$0.219246\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	326.683	1.65829	0.829146	−	0.559033i	$-0.188827\pi$
$197$	326.683	1.65829	0.829146	+	0.559033i	$0.188827\pi$
$198$	0	0
$199$	−59.3970	−0.298477	−0.149239	−	0.988801i	$-0.547682\pi$
$199$	−59.3970	−0.298477	−0.149239	+	0.988801i	$0.547682\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	108.000	0.532020
$204$	0	0
$205$	− 126.000i	− 0.614634i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 67.8823i	− 0.324795i
$210$	0	0
$211$	50.9117i	0.241288i	0.992696	+	0.120644i	$0.0384959\pi$
$211$	50.9117i	0.241288i	−0.992696	+	0.120644i	$0.961504\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	288.000i	1.33953i
$216$	0	0
$217$	−72.0000	−0.331797
$218$	0	0
$219$	0	0
$220$	0	0
$221$	76.3675	0.345554
$222$	0	0
$223$	280.014	1.25567	0.627835	−	0.778347i	$-0.283941\pi$
$223$	280.014	1.25567	0.627835	+	0.778347i	$0.283941\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−284.000	−1.25110	−0.625551	−	0.780184i	$-0.715126\pi$
$227$	−284.000	−1.25110	−0.625551	+	0.780184i	$0.715126\pi$
$228$	0	0
$229$	126.000i	0.550218i	0.961413	+	0.275109i	$0.0887140\pi$
$229$	126.000i	0.550218i	−0.961413	+	0.275109i	$0.911286\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	284.257i	1.21999i	0.792406	+	0.609993i	$0.208828\pi$
$233$	284.257i	1.21999i	−0.792406	+	0.609993i	$0.791172\pi$
$234$	0	0
$235$	152.735i	0.649936i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 360.000i	− 1.50628i	−0.657862	−	0.753138i	$-0.728539\pi$
$239$	− 360.000i	− 1.50628i	0.657862	−	0.753138i	$-0.271461\pi$
$240$	0	0
$241$	−32.0000	−0.132780	−0.0663900	−	0.997794i	$-0.521148\pi$
$241$	−32.0000	−0.132780	−0.0663900	+	0.997794i	$0.521148\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−97.5807	−0.398289
$246$	0	0
$247$	305.470	1.23672
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−20.0000	−0.0796813	−0.0398406	−	0.999206i	$-0.512685\pi$
$251$	−20.0000	−0.0796813	−0.0398406	+	0.999206i	$0.512685\pi$
$252$	0	0
$253$	144.000i	0.569170i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 386.080i	− 1.50226i	−0.660155	−	0.751129i	$-0.729510\pi$
$257$	− 386.080i	− 1.50226i	0.660155	−	0.751129i	$-0.270490\pi$
$258$	0	0
$259$	305.470i	1.17942i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 72.0000i	− 0.273764i	−0.990587	−	0.136882i	$-0.956292\pi$
$263$	− 72.0000i	− 0.273764i	0.990587	−	0.136882i	$-0.0437082\pi$
$264$	0	0
$265$	−342.000	−1.29057
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−394.566	−1.46679	−0.733393	−	0.679805i	$-0.762065\pi$
$269$	−394.566	−1.46679	−0.733393	+	0.679805i	$0.762065\pi$
$270$	0	0
$271$	−93.3381	−0.344421	−0.172211	−	0.985060i	$-0.555091\pi$
$271$	−93.3381	−0.344421	−0.172211	+	0.985060i	$0.555091\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−28.0000	−0.101818
$276$	0	0
$277$	− 126.000i	− 0.454874i	−0.973793	−	0.227437i	$-0.926965\pi$
$277$	− 126.000i	− 0.454874i	0.973793	−	0.227437i	$-0.0730345\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	89.0955i	0.317066i	0.987354	+	0.158533i	$0.0506764\pi$
$281$	89.0955i	0.317066i	−0.987354	+	0.158533i	$0.949324\pi$
$282$	0	0
$283$	203.647i	0.719600i	0.933029	+	0.359800i	$0.117155\pi$
$283$	203.647i	0.719600i	−0.933029	+	0.359800i	$0.882845\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 252.000i	− 0.878049i
$288$	0	0
$289$	271.000	0.937716
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−190.919	−0.651600	−0.325800	−	0.945439i	$-0.605634\pi$
$293$	−190.919	−0.651600	−0.325800	+	0.945439i	$0.605634\pi$
$294$	0	0
$295$	−339.411	−1.15055
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−648.000	−2.16722
$300$	0	0
$301$	576.000i	1.91362i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	152.735i	0.500771i
$306$	0	0
$307$	288.500i	0.939738i	0.882736	+	0.469869i	$0.155699\pi$
$307$	288.500i	0.939738i	−0.882736	+	0.469869i	$0.844301\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 504.000i	− 1.62058i	−0.586030	−	0.810289i	$-0.699310\pi$
$311$	− 504.000i	− 1.62058i	0.586030	−	0.810289i	$-0.300690\pi$
$312$	0	0
$313$	58.0000	0.185304	0.0926518	−	0.995699i	$-0.470466\pi$
$313$	58.0000	0.185304	0.0926518	+	0.995699i	$0.470466\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	182.434	0.575500	0.287750	−	0.957706i	$-0.407093\pi$
$317$	182.434	0.575500	0.287750	+	0.957706i	$0.407093\pi$
$318$	0	0
$319$	−50.9117	−0.159598
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−72.0000	−0.222910
$324$	0	0
$325$	− 126.000i	− 0.387692i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	305.470i	0.928481i
$330$	0	0
$331$	356.382i	1.07668i	0.842727	+	0.538341i	$0.180949\pi$
$331$	356.382i	1.07668i	−0.842727	+	0.538341i	$0.819051\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	504.000i	1.50448i
$336$	0	0
$337$	−8.00000	−0.0237389	−0.0118694	−	0.999930i	$-0.503778\pi$
$337$	−8.00000	−0.0237389	−0.0118694	+	0.999930i	$0.503778\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	33.9411	0.0995341
$342$	0	0
$343$	220.617	0.643199
$344$	0	0
$345$	0	0
$346$	0	0
$347$	28.0000	0.0806916	0.0403458	−	0.999186i	$-0.487154\pi$
$347$	28.0000	0.0806916	0.0403458	+	0.999186i	$0.487154\pi$
$348$	0	0
$349$	252.000i	0.722063i	0.932554	+	0.361032i	$0.117575\pi$
$349$	252.000i	0.722063i	−0.932554	+	0.361032i	$0.882425\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 564.271i	− 1.59850i	−0.600997	−	0.799251i	$-0.705230\pi$
$353$	− 564.271i	− 1.59850i	0.600997	−	0.799251i	$-0.294770\pi$
$354$	0	0
$355$	458.205i	1.29072i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	252.000i	0.701950i	0.936385	+	0.350975i	$0.114150\pi$
$359$	252.000i	0.701950i	−0.936385	+	0.350975i	$0.885850\pi$
$360$	0	0
$361$	73.0000	0.202216
$362$	0	0
$363$	0	0
$364$	0	0
$365$	237.588	0.650926
$366$	0	0
$367$	8.48528	0.0231207	0.0115603	−	0.999933i	$-0.496320\pi$
$367$	8.48528	0.0231207	0.0115603	+	0.999933i	$0.496320\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−684.000	−1.84367
$372$	0	0
$373$	− 252.000i	− 0.675603i	−0.941217	−	0.337802i	$-0.890317\pi$
$373$	− 252.000i	− 0.675603i	0.941217	−	0.337802i	$-0.109683\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 229.103i	− 0.607699i
$378$	0	0
$379$	593.970i	1.56720i	0.621264	+	0.783601i	$0.286619\pi$
$379$	593.970i	1.56720i	−0.621264	+	0.783601i	$0.713381\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	504.000i	1.31593i	0.753050	+	0.657963i	$0.228582\pi$
$383$	504.000i	1.31593i	−0.753050	+	0.657963i	$0.771418\pi$
$384$	0	0
$385$	144.000	0.374026
$386$	0	0
$387$	0	0
$388$	0	0
$389$	207.889	0.534420	0.267210	−	0.963638i	$-0.413898\pi$
$389$	207.889	0.534420	0.267210	+	0.963638i	$0.413898\pi$
$390$	0	0
$391$	152.735	0.390627
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−108.000	−0.273418
$396$	0	0
$397$	− 756.000i	− 1.90428i	−0.305659	−	0.952141i	$-0.598877\pi$
$397$	− 756.000i	− 1.90428i	0.305659	−	0.952141i	$-0.401123\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	504.874i	1.25904i	0.776985	+	0.629519i	$0.216748\pi$
$401$	504.874i	1.25904i	−0.776985	+	0.629519i	$0.783252\pi$
$402$	0	0
$403$	152.735i	0.378995i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 144.000i	− 0.353808i
$408$	0	0
$409$	−136.000	−0.332518	−0.166259	−	0.986082i	$-0.553169\pi$
$409$	−136.000	−0.332518	−0.166259	+	0.986082i	$0.553169\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−678.823	−1.64364
$414$	0	0
$415$	−322.441	−0.776966
$416$	0	0
$417$	0	0
$418$	0	0
$419$	28.0000	0.0668258	0.0334129	−	0.999442i	$-0.489362\pi$
$419$	28.0000	0.0668258	0.0334129	+	0.999442i	$0.489362\pi$
$420$	0	0
$421$	− 270.000i	− 0.641330i	−0.947193	−	0.320665i	$-0.896094\pi$
$421$	− 270.000i	− 0.641330i	0.947193	−	0.320665i	$-0.103906\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	29.6985i	0.0698788i
$426$	0	0
$427$	305.470i	0.715387i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	252.000i	0.584687i	0.956313	+	0.292343i	$0.0944350\pi$
$431$	252.000i	0.584687i	−0.956313	+	0.292343i	$0.905565\pi$
$432$	0	0
$433$	−574.000	−1.32564	−0.662818	−	0.748781i	$-0.730640\pi$
$433$	−574.000	−1.32564	−0.662818	+	0.748781i	$0.730640\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	610.940	1.39803
$438$	0	0
$439$	93.3381	0.212615	0.106308	−	0.994333i	$-0.466097\pi$
$439$	93.3381	0.212615	0.106308	+	0.994333i	$0.466097\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−716.000	−1.61625	−0.808126	−	0.589009i	$-0.799518\pi$
$443$	−716.000	−1.61625	−0.808126	+	0.589009i	$0.799518\pi$
$444$	0	0
$445$	− 378.000i	− 0.849438i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 199.404i	− 0.444107i	−0.975034	−	0.222054i	$-0.928724\pi$
$449$	− 199.404i	− 0.444107i	0.975034	−	0.222054i	$-0.0712760\pi$
$450$	0	0
$451$	118.794i	0.263401i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	648.000i	1.42418i
$456$	0	0
$457$	−40.0000	−0.0875274	−0.0437637	−	0.999042i	$-0.513935\pi$
$457$	−40.0000	−0.0875274	−0.0437637	+	0.999042i	$0.513935\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	742.462	1.61055	0.805273	−	0.592904i	$-0.202018\pi$
$461$	742.462	1.61055	0.805273	+	0.592904i	$0.202018\pi$
$462$	0	0
$463$	−653.367	−1.41116	−0.705580	−	0.708631i	$-0.749313\pi$
$463$	−653.367	−1.41116	−0.705580	+	0.708631i	$0.749313\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	52.0000	0.111349	0.0556745	−	0.998449i	$-0.482269\pi$
$467$	52.0000	0.111349	0.0556745	+	0.998449i	$0.482269\pi$
$468$	0	0
$469$	1008.00i	2.14925i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 271.529i	− 0.574057i
$474$	0	0
$475$	118.794i	0.250093i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 684.000i	− 1.42797i	−0.700158	−	0.713987i	$-0.746887\pi$
$479$	− 684.000i	− 1.42797i	0.700158	−	0.713987i	$-0.253113\pi$
$480$	0	0
$481$	648.000	1.34719
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−441.235	−0.909762
$486$	0	0
$487$	−229.103	−0.470437	−0.235218	−	0.971943i	$-0.575580\pi$
$487$	−229.103	−0.470437	−0.235218	+	0.971943i	$0.575580\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	440.000	0.896130	0.448065	−	0.894001i	$-0.352113\pi$
$491$	440.000	0.896130	0.448065	+	0.894001i	$0.352113\pi$
$492$	0	0
$493$	54.0000i	0.109533i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	916.410i	1.84388i
$498$	0	0
$499$	475.176i	0.952256i	0.879376	+	0.476128i	$0.157960\pi$
$499$	475.176i	0.952256i	−0.879376	+	0.476128i	$0.842040\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 900.000i	− 1.78926i	−0.446803	−	0.894632i	$-0.647438\pi$
$503$	− 900.000i	− 1.78926i	0.446803	−	0.894632i	$-0.352562\pi$
$504$	0	0
$505$	630.000	1.24752
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−182.434	−0.358416	−0.179208	−	0.983811i	$-0.557353\pi$
$509$	−182.434	−0.358416	−0.179208	+	0.983811i	$0.557353\pi$
$510$	0	0
$511$	475.176	0.929894
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−756.000	−1.46796
$516$	0	0
$517$	− 144.000i	− 0.278530i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	4.24264i	0.00814326i	0.999992	+	0.00407163i	$0.00129604\pi$
$521$	4.24264i	0.00814326i	−0.999992	+	0.00407163i	$0.998704\pi$
$522$	0	0
$523$	− 254.558i	− 0.486727i	−0.969935	−	0.243364i	$-0.921749\pi$
$523$	− 254.558i	− 0.486727i	0.969935	−	0.243364i	$-0.0782509\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 36.0000i	− 0.0683112i
$528$	0	0
$529$	−767.000	−1.44991
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−534.573	−1.00295
$534$	0	0
$535$	−543.058	−1.01506
$536$	0	0
$537$	0	0
$538$	0	0
$539$	92.0000	0.170686
$540$	0	0
$541$	414.000i	0.765250i	0.923904	+	0.382625i	$0.124980\pi$
$541$	414.000i	0.765250i	−0.923904	+	0.382625i	$0.875020\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 534.573i	− 0.980867i
$546$	0	0
$547$	− 67.8823i	− 0.124099i	−0.998073	−	0.0620496i	$-0.980236\pi$
$547$	− 67.8823i	− 0.124099i	0.998073	−	0.0620496i	$-0.0197637\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	216.000i	0.392015i
$552$	0	0
$553$	−216.000	−0.390597
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−275.772	−0.495102	−0.247551	−	0.968875i	$-0.579626\pi$
$557$	−275.772	−0.495102	−0.247551	+	0.968875i	$0.579626\pi$
$558$	0	0
$559$	1221.88	2.18583
$560$	0	0
$561$	0	0
$562$	0	0
$563$	892.000	1.58437	0.792185	−	0.610281i	$-0.208944\pi$
$563$	892.000	1.58437	0.792185	+	0.610281i	$0.208944\pi$
$564$	0	0
$565$	378.000i	0.669027i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 844.285i	− 1.48381i	−0.670507	−	0.741903i	$-0.733924\pi$
$569$	− 844.285i	− 1.48381i	0.670507	−	0.741903i	$-0.266076\pi$
$570$	0	0
$571$	− 441.235i	− 0.772740i	−0.922344	−	0.386370i	$-0.873729\pi$
$571$	− 441.235i	− 0.772740i	0.922344	−	0.386370i	$-0.126271\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 252.000i	− 0.438261i
$576$	0	0
$577$	−326.000	−0.564991	−0.282496	−	0.959269i	$-0.591162\pi$
$577$	−326.000	−0.564991	−0.282496	+	0.959269i	$0.591162\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−644.881	−1.10995
$582$	0	0
$583$	322.441	0.553072
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−400.000	−0.681431	−0.340716	−	0.940166i	$-0.610669\pi$
$587$	−400.000	−0.681431	−0.340716	+	0.940166i	$0.610669\pi$
$588$	0	0
$589$	− 144.000i	− 0.244482i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	903.682i	1.52392i	0.647626	+	0.761958i	$0.275762\pi$
$593$	903.682i	1.52392i	−0.647626	+	0.761958i	$0.724238\pi$
$594$	0	0
$595$	− 152.735i	− 0.256698i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	756.000i	1.26210i	0.775741	+	0.631052i	$0.217376\pi$
$599$	756.000i	1.26210i	−0.775741	+	0.631052i	$0.782624\pi$
$600$	0	0
$601$	778.000	1.29451	0.647255	−	0.762274i	$-0.275917\pi$
$601$	778.000	1.29451	0.647255	+	0.762274i	$0.275917\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	445.477	0.736326
$606$	0	0
$607$	381.838	0.629057	0.314529	−	0.949248i	$-0.398154\pi$
$607$	381.838	0.629057	0.314529	+	0.949248i	$0.398154\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	648.000	1.06056
$612$	0	0
$613$	180.000i	0.293638i	0.989163	+	0.146819i	$0.0469035\pi$
$613$	180.000i	0.293638i	−0.989163	+	0.146819i	$0.953097\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 453.963i	− 0.735758i	−0.929874	−	0.367879i	$-0.880084\pi$
$617$	− 453.963i	− 0.735758i	0.929874	−	0.367879i	$-0.119916\pi$
$618$	0	0
$619$	− 33.9411i	− 0.0548322i	−0.999624	−	0.0274161i	$-0.991272\pi$
$619$	− 33.9411i	− 0.0548322i	0.999624	−	0.0274161i	$-0.00872791\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 756.000i	− 1.21348i
$624$	0	0
$625$	−401.000	−0.641600
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−152.735	−0.242822
$630$	0	0
$631$	755.190	1.19681	0.598407	−	0.801192i	$-0.295800\pi$
$631$	755.190	1.19681	0.598407	+	0.801192i	$0.295800\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	684.000	1.07717
$636$	0	0
$637$	414.000i	0.649922i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	233.345i	0.364033i	0.983295	+	0.182017i	$0.0582625\pi$
$641$	233.345i	0.364033i	−0.983295	+	0.182017i	$0.941738\pi$
$642$	0	0
$643$	− 1187.94i	− 1.84750i	−0.383002	−	0.923748i	$-0.625110\pi$
$643$	− 1187.94i	− 1.84750i	0.383002	−	0.923748i	$-0.374890\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 324.000i	− 0.500773i	−0.968146	−	0.250386i	$-0.919442\pi$
$647$	− 324.000i	− 0.500773i	0.968146	−	0.250386i	$-0.0805577\pi$
$648$	0	0
$649$	320.000	0.493066
$650$	0	0
$651$	0	0
$652$	0	0
$653$	886.712	1.35790	0.678952	−	0.734182i	$-0.262434\pi$
$653$	886.712	1.35790	0.678952	+	0.734182i	$0.262434\pi$
$654$	0	0
$655$	−644.881	−0.984552
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−1120.00	−1.69954	−0.849772	−	0.527150i	$-0.823261\pi$
$659$	−1120.00	−1.69954	−0.849772	+	0.527150i	$0.823261\pi$
$660$	0	0
$661$	− 180.000i	− 0.272315i	−0.990687	−	0.136157i	$-0.956525\pi$
$661$	− 180.000i	− 0.272315i	0.990687	−	0.136157i	$-0.0434753\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 610.940i	− 0.918707i
$666$	0	0
$667$	− 458.205i	− 0.686964i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 144.000i	− 0.214605i
$672$	0	0
$673$	302.000	0.448737	0.224368	−	0.974504i	$-0.427968\pi$
$673$	302.000	0.448737	0.224368	+	0.974504i	$0.427968\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	776.403	1.14683	0.573415	−	0.819265i	$-0.305618\pi$
$677$	776.403	1.14683	0.573415	+	0.819265i	$0.305618\pi$
$678$	0	0
$679$	−882.469	−1.29966
$680$	0	0
$681$	0	0
$682$	0	0
$683$	1060.00	1.55198	0.775988	−	0.630747i	$-0.217252\pi$
$683$	1060.00	1.55198	0.775988	+	0.630747i	$0.217252\pi$
$684$	0	0
$685$	− 882.000i	− 1.28759i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	1450.98i	2.10593i
$690$	0	0
$691$	− 712.764i	− 1.03150i	−0.856740	−	0.515748i	$-0.827514\pi$
$691$	− 712.764i	− 1.03150i	0.856740	−	0.515748i	$-0.172486\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	504.000i	0.725180i
$696$	0	0
$697$	126.000	0.180775
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−284.257	−0.405502	−0.202751	−	0.979230i	$-0.564988\pi$
$701$	−284.257	−0.405502	−0.202751	+	0.979230i	$0.564988\pi$
$702$	0	0
$703$	−610.940	−0.869047
$704$	0	0
$705$	0	0
$706$	0	0
$707$	1260.00	1.78218
$708$	0	0
$709$	− 882.000i	− 1.24401i	−0.783015	−	0.622003i	$-0.786319\pi$
$709$	− 882.000i	− 1.24401i	0.783015	−	0.622003i	$-0.213681\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	305.470i	0.428429i
$714$	0	0
$715$	− 305.470i	− 0.427231i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	252.000i	0.350487i	0.984525	+	0.175243i	$0.0560712\pi$
$719$	252.000i	0.350487i	−0.984525	+	0.175243i	$0.943929\pi$
$720$	0	0
$721$	−1512.00	−2.09709
$722$	0	0
$723$	0	0
$724$	0	0
$725$	89.0955	0.122890
$726$	0	0
$727$	42.4264	0.0583582	0.0291791	−	0.999574i	$-0.490711\pi$
$727$	42.4264	0.0583582	0.0291791	+	0.999574i	$0.490711\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−288.000	−0.393981
$732$	0	0
$733$	162.000i	0.221010i	0.993876	+	0.110505i	$0.0352467\pi$
$733$	162.000i	0.221010i	−0.993876	+	0.110505i	$0.964753\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 475.176i	− 0.644743i
$738$	0	0
$739$	− 237.588i	− 0.321499i	−0.986995	−	0.160750i	$-0.948609\pi$
$739$	− 237.588i	− 0.321499i	0.986995	−	0.160750i	$-0.0513912\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 1008.00i	− 1.35666i	−0.734756	−	0.678331i	$-0.762704\pi$
$743$	− 1008.00i	− 1.35666i	0.734756	−	0.678331i	$-0.237296\pi$
$744$	0	0
$745$	414.000	0.555705
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−1086.12	−1.45009
$750$	0	0
$751$	−500.632	−0.666620	−0.333310	−	0.942817i	$-0.608166\pi$
$751$	−500.632	−0.666620	−0.333310	+	0.942817i	$0.608166\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	1044.00	1.38278
$756$	0	0
$757$	− 1026.00i	− 1.35535i	−0.735362	−	0.677675i	$-0.762988\pi$
$757$	− 1026.00i	− 1.35535i	0.735362	−	0.677675i	$-0.237012\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 1260.06i	− 1.65580i	−0.560875	−	0.827900i	$-0.689535\pi$
$761$	− 1260.06i	− 1.65580i	0.560875	−	0.827900i	$-0.310465\pi$
$762$	0	0
$763$	− 1069.15i	− 1.40124i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1440.00i	1.87744i
$768$	0	0
$769$	694.000	0.902471	0.451235	−	0.892405i	$-0.350983\pi$
$769$	694.000	0.902471	0.451235	+	0.892405i	$0.350983\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	1438.26	1.86061	0.930307	−	0.366781i	$-0.119540\pi$
$773$	1438.26	1.86061	0.930307	+	0.366781i	$0.119540\pi$
$774$	0	0
$775$	−59.3970	−0.0766413
$776$	0	0
$777$	0	0
$778$	0	0
$779$	504.000	0.646983
$780$	0	0
$781$	− 432.000i	− 0.553137i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	1069.15i	1.36197i
$786$	0	0
$787$	− 865.499i	− 1.09974i	−0.835249	−	0.549872i	$-0.814676\pi$
$787$	− 865.499i	− 1.09974i	0.835249	−	0.549872i	$-0.185324\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	756.000i	0.955752i
$792$	0	0
$793$	648.000	0.817150
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−640.639	−0.803813	−0.401906	−	0.915681i	$-0.631652\pi$
$797$	−640.639	−0.803813	−0.401906	+	0.915681i	$0.631652\pi$
$798$	0	0
$799$	−152.735	−0.191158
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−224.000	−0.278954
$804$	0	0
$805$	1296.00i	1.60994i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 683.065i	− 0.844333i	−0.906518	−	0.422166i	$-0.861270\pi$
$809$	− 683.065i	− 0.844333i	0.906518	−	0.422166i	$-0.138730\pi$
$810$	0	0
$811$	− 1306.73i	− 1.61126i	−0.592418	−	0.805631i	$-0.701827\pi$
$811$	− 1306.73i	− 1.61126i	0.592418	−	0.805631i	$-0.298173\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	432.000i	0.530061i
$816$	0	0
$817$	−1152.00	−1.41004
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−114.551	−0.139527	−0.0697633	−	0.997564i	$-0.522224\pi$
$821$	−114.551	−0.139527	−0.0697633	+	0.997564i	$0.522224\pi$
$822$	0	0
$823$	−653.367	−0.793884	−0.396942	−	0.917844i	$-0.629929\pi$
$823$	−653.367	−0.793884	−0.396942	+	0.917844i	$0.629929\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	776.000	0.938331	0.469166	−	0.883110i	$-0.344555\pi$
$827$	776.000	0.938331	0.469166	+	0.883110i	$0.344555\pi$
$828$	0	0
$829$	702.000i	0.846803i	0.905942	+	0.423402i	$0.139164\pi$
$829$	702.000i	0.846803i	−0.905942	+	0.423402i	$0.860836\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 97.5807i	− 0.117144i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 612.000i	− 0.729440i	−0.931117	−	0.364720i	$-0.881165\pi$
$839$	− 612.000i	− 0.729440i	0.931117	−	0.364720i	$-0.118835\pi$
$840$	0	0
$841$	−679.000	−0.807372
$842$	0	0
$843$	0	0
$844$	0	0
$845$	657.609	0.778236
$846$	0	0
$847$	890.955	1.05189
$848$	0	0
$849$	0	0
$850$	0	0
$851$	1296.00	1.52291
$852$	0	0
$853$	252.000i	0.295428i	0.989030	+	0.147714i	$0.0471915\pi$
$853$	252.000i	0.295428i	−0.989030	+	0.147714i	$0.952809\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 649.124i	− 0.757438i	−0.925512	−	0.378719i	$-0.876365\pi$
$857$	− 649.124i	− 0.757438i	0.925512	−	0.378719i	$-0.123635\pi$
$858$	0	0
$859$	339.411i	0.395124i	0.980290	+	0.197562i	$0.0633023\pi$
$859$	339.411i	0.395124i	−0.980290	+	0.197562i	$0.936698\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	1008.00i	1.16802i	0.811747	+	0.584009i	$0.198517\pi$
$863$	1008.00i	1.16802i	−0.811747	+	0.584009i	$0.801483\pi$
$864$	0	0
$865$	126.000	0.145665
$866$	0	0
$867$	0	0
$868$	0	0
$869$	101.823	0.117173
$870$	0	0
$871$	2138.29	2.45498
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−1152.00	−1.31657
$876$	0	0
$877$	1260.00i	1.43672i	0.695674	+	0.718358i	$0.255106\pi$
$877$	1260.00i	1.43672i	−0.695674	+	0.718358i	$0.744894\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	1514.62i	1.71921i	0.510960	+	0.859604i	$0.329290\pi$
$881$	1514.62i	1.71921i	−0.510960	+	0.859604i	$0.670710\pi$
$882$	0	0
$883$	− 950.352i	− 1.07628i	−0.842857	−	0.538138i	$-0.819128\pi$
$883$	− 950.352i	− 1.07628i	0.842857	−	0.538138i	$-0.180872\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 216.000i	− 0.243517i	−0.992560	−	0.121759i	$-0.961147\pi$
$887$	− 216.000i	− 0.243517i	0.992560	−	0.121759i	$-0.0388534\pi$
$888$	0	0
$889$	1368.00	1.53881
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−610.940	−0.684144
$894$	0	0
$895$	−237.588	−0.265461
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−108.000	−0.120133
$900$	0	0
$901$	− 342.000i	− 0.379578i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	534.573i	0.590688i
$906$	0	0
$907$	− 746.705i	− 0.823269i	−0.911349	−	0.411634i	$-0.864958\pi$
$907$	− 746.705i	− 0.823269i	0.911349	−	0.411634i	$-0.135042\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 144.000i	− 0.158068i	−0.996872	−	0.0790340i	$-0.974816\pi$
$911$	− 144.000i	− 0.158068i	0.996872	−	0.0790340i	$-0.0251836\pi$
$912$	0	0
$913$	304.000	0.332968
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−1289.76	−1.40650
$918$	0	0
$919$	−653.367	−0.710954	−0.355477	−	0.934685i	$-0.615682\pi$
$919$	−653.367	−0.710954	−0.355477	+	0.934685i	$0.615682\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	1944.00	2.10618
$924$	0	0
$925$	252.000i	0.272432i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 1260.06i	− 1.35637i	−0.734893	−	0.678183i	$-0.762768\pi$
$929$	− 1260.06i	− 1.35637i	0.734893	−	0.678183i	$-0.237232\pi$
$930$	0	0
$931$	− 390.323i	− 0.419251i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	72.0000i	0.0770053i
$936$	0	0
$937$	1154.00	1.23159	0.615795	−	0.787906i	$-0.288835\pi$
$937$	1154.00	1.23159	0.615795	+	0.787906i	$0.288835\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−1090.36	−1.15872	−0.579362	−	0.815071i	$-0.696698\pi$
$941$	−1090.36	−1.15872	−0.579362	+	0.815071i	$0.696698\pi$
$942$	0	0
$943$	−1069.15	−1.13377
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−1120.00	−1.18268	−0.591341	−	0.806422i	$-0.701401\pi$
$947$	−1120.00	−1.18268	−0.591341	+	0.806422i	$0.701401\pi$
$948$	0	0
$949$	− 1008.00i	− 1.06217i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 335.169i	− 0.351698i	−0.984417	−	0.175849i	$-0.943733\pi$
$953$	− 335.169i	− 0.351698i	0.984417	−	0.175849i	$-0.0562671\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 1764.00i	− 1.83942i
$960$	0	0
$961$	−889.000	−0.925078
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−1264.31	−1.31016
$966$	0	0
$967$	313.955	0.324670	0.162335	−	0.986736i	$-0.448098\pi$
$967$	313.955	0.324670	0.162335	+	0.986736i	$0.448098\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	772.000	0.795057	0.397528	−	0.917590i	$-0.369868\pi$
$971$	772.000	0.795057	0.397528	+	0.917590i	$0.369868\pi$
$972$	0	0
$973$	1008.00i	1.03597i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1192.18i	1.22025i	0.792306	+	0.610124i	$0.208880\pi$
$977$	1192.18i	1.22025i	−0.792306	+	0.610124i	$0.791120\pi$
$978$	0	0
$979$	356.382i	0.364026i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 504.000i	− 0.512716i	−0.966582	−	0.256358i	$-0.917477\pi$
$983$	− 504.000i	− 0.512716i	0.966582	−	0.256358i	$-0.0825226\pi$
$984$	0	0
$985$	−1386.00	−1.40711
$986$	0	0
$987$	0	0
$988$	0	0
$989$	2443.76	2.47094
$990$	0	0
$991$	−364.867	−0.368181	−0.184090	−	0.982909i	$-0.558934\pi$
$991$	−364.867	−0.368181	−0.184090	+	0.982909i	$0.558934\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	252.000	0.253266
$996$	0	0
$997$	756.000i	0.758275i	0.925340	+	0.379137i	$0.123779\pi$
$997$	756.000i	0.758275i	−0.925340	+	0.379137i	$0.876221\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1152.3.h.d.449.2	yes	4
3.2	odd	2		1152.3.h.a.449.4	yes	4
4.3	odd	2		1152.3.h.a.449.2	yes	4
8.3	odd	2	inner	1152.3.h.d.449.3	yes	4
8.5	even	2		1152.3.h.a.449.3	yes	4
12.11	even	2	inner	1152.3.h.d.449.4	yes	4
16.3	odd	4		2304.3.e.e.1025.3		4
16.5	even	4		2304.3.e.l.1025.2		4
16.11	odd	4		2304.3.e.l.1025.1		4
16.13	even	4		2304.3.e.e.1025.4		4
24.5	odd	2	inner	1152.3.h.d.449.1	yes	4
24.11	even	2		1152.3.h.a.449.1	✓	4
48.5	odd	4		2304.3.e.l.1025.4		4
48.11	even	4		2304.3.e.l.1025.3		4
48.29	odd	4		2304.3.e.e.1025.2		4
48.35	even	4		2304.3.e.e.1025.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1152.3.h.a.449.1	✓	4	24.11	even	2
1152.3.h.a.449.2	yes	4	4.3	odd	2
1152.3.h.a.449.3	yes	4	8.5	even	2
1152.3.h.a.449.4	yes	4	3.2	odd	2
1152.3.h.d.449.1	yes	4	24.5	odd	2	inner
1152.3.h.d.449.2	yes	4	1.1	even	1	trivial
1152.3.h.d.449.3	yes	4	8.3	odd	2	inner
1152.3.h.d.449.4	yes	4	12.11	even	2	inner
2304.3.e.e.1025.1		4	48.35	even	4
2304.3.e.e.1025.2		4	48.29	odd	4
2304.3.e.e.1025.3		4	16.3	odd	4
2304.3.e.e.1025.4		4	16.13	even	4
2304.3.e.l.1025.1		4	16.11	odd	4
2304.3.e.l.1025.2		4	16.5	even	4
2304.3.e.l.1025.3		4	48.11	even	4
2304.3.e.l.1025.4		4	48.5	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1152 = 2^{7} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1152.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-4.24264 q^{5} -8.48528 q^{7} +O(q^{10})\)	\(q-4.24264 q^{5} -8.48528 q^{7} +4.00000 q^{11} +18.0000i q^{13} -4.24264i q^{17} -16.9706i q^{19} +36.0000i q^{23} -7.00000 q^{25} -12.7279 q^{29} +8.48528 q^{31} +36.0000 q^{35} -36.0000i q^{37} +29.6985i q^{41} -67.8823i q^{43} -36.0000i q^{47} +23.0000 q^{49} +80.6102 q^{53} -16.9706 q^{55} +80.0000 q^{59} -36.0000i q^{61} -76.3675i q^{65} -118.794i q^{67} -108.000i q^{71} -56.0000 q^{73} -33.9411 q^{77} +25.4558 q^{79} +76.0000 q^{83} +18.0000i q^{85} +89.0955i q^{89} -152.735i q^{91} +72.0000i q^{95} +104.000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q + 16 q^{11} - 28 q^{25} + 144 q^{35} + 92 q^{49} + 320 q^{59} - 224 q^{73} + 304 q^{83} + 416 q^{97}+O(q^{100}) \) 4 * q + 16 * q^11 - 28 * q^25 + 144 * q^35 + 92 * q^49 + 320 * q^59 - 224 * q^73 + 304 * q^83 + 416 * q^97

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