LMFDB - Embedded newform 4896.2.f.c.2449.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4896,2,Mod(2449,4896)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4896, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4896.2449");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4896 = 2^{5} \cdot 3^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4896.f (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$39.0947568296$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.1649659456.5
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 2x^{5} + 4x^{4} - 4x^{3} - 8x + 16 $ x^8 - x^7 - 2x^5 + 4x^4 - 4x^3 - 8x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 136)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2449.5
Root			$-1.13622 + 0.842022i$ of defining polynomial
Character	$\chi$	$=$	4896.2449
Dual form			4896.2.f.c.2449.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.436910i q^{5} +1.90455 q^{7} +O(q^{10})$	$q+0.436910i q^{5} +1.90455 q^{7} -2.45665i q^{11} -2.93118i q^{13} +1.00000 q^{17} +0.713086i q^{19} -0.640331 q^{23} +4.80911 q^{25} +7.72544i q^{29} +6.23255 q^{31} +0.832119i q^{35} +1.93832i q^{37} +1.06377 q^{41} -7.65379i q^{43} -2.73578 q^{47} -3.37267 q^{49} -4.91329i q^{53} +1.07333 q^{55} -6.77997i q^{59} +13.9968i q^{61} +1.28066 q^{65} -13.2364i q^{67} +14.0581 q^{71} -9.52886 q^{73} -4.67882i q^{77} +3.35967 q^{79} -12.6423i q^{83} +0.436910i q^{85} -0.155777 q^{89} -5.58259i q^{91} -0.311555 q^{95} +3.59222 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{7}+O(q^{10}) $ 8 * q - 8 * q^7	$ 8 q - 8 q^{7} + 8 q^{17} + 12 q^{23} - 8 q^{25} + 12 q^{31} - 28 q^{47} - 8 q^{49} - 44 q^{55} - 24 q^{65} + 8 q^{73} + 44 q^{79} - 8 q^{89} - 16 q^{95} + 8 q^{97}+O(q^{100}) $ 8 * q - 8 * q^7 + 8 * q^17 + 12 * q^23 - 8 * q^25 + 12 * q^31 - 28 * q^47 - 8 * q^49 - 44 * q^55 - 24 * q^65 + 8 * q^73 + 44 * q^79 - 8 * q^89 - 16 * q^95 + 8 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4896\mathbb{Z}\right)^\times$.

$n$	$613$	$2143$	$3809$	$4321$
$\chi(n)$	$-1$	$1$	$1$	$1$

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
$5$	0.436910i	0.195392i	0.995216	+	0.0976961i	$0.0311473\pi$
$5$	0.436910i	0.195392i	−0.995216	+	0.0976961i	$0.968853\pi$
$6$	0	0
$7$	1.90455	0.719854	0.359927	−	0.932980i	$-0.382802\pi$
$7$	1.90455	0.719854	0.359927	+	0.932980i	$0.382802\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 2.45665i	− 0.740707i	−0.928891	−	0.370353i	$-0.879237\pi$
$11$	− 2.45665i	− 0.740707i	0.928891	−	0.370353i	$-0.120763\pi$
$12$	0	0
$13$	− 2.93118i	− 0.812962i	−0.913659	−	0.406481i	$-0.866756\pi$
$13$	− 2.93118i	− 0.812962i	0.913659	−	0.406481i	$-0.133244\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	0.713086i	0.163593i	0.996649	+	0.0817966i	$0.0260658\pi$
$19$	0.713086i	0.163593i	−0.996649	+	0.0817966i	$0.973934\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−0.640331	−0.133518	−0.0667591	−	0.997769i	$-0.521266\pi$
$23$	−0.640331	−0.133518	−0.0667591	+	0.997769i	$0.521266\pi$
$24$	0	0
$25$	4.80911	0.961822
$26$	0	0
$27$	0	0
$28$	0	0
$29$	7.72544i	1.43458i	0.696776	+	0.717289i	$0.254617\pi$
$29$	7.72544i	1.43458i	−0.696776	+	0.717289i	$0.745383\pi$
$30$	0	0
$31$	6.23255	1.11940	0.559699	−	0.828696i	$-0.310917\pi$
$31$	6.23255	1.11940	0.559699	+	0.828696i	$0.310917\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0.832119i	0.140654i
$36$	0	0
$37$	1.93832i	0.318659i	0.987225	+	0.159329i	$0.0509332\pi$
$37$	1.93832i	0.318659i	−0.987225	+	0.159329i	$0.949067\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	1.06377	0.166133	0.0830663	−	0.996544i	$-0.473529\pi$
$41$	1.06377	0.166133	0.0830663	+	0.996544i	$0.473529\pi$
$42$	0	0
$43$	− 7.65379i	− 1.16719i	−0.812044	−	0.583596i	$-0.801645\pi$
$43$	− 7.65379i	− 1.16719i	0.812044	−	0.583596i	$-0.198355\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−2.73578	−0.399054	−0.199527	−	0.979892i	$-0.563941\pi$
$47$	−2.73578	−0.399054	−0.199527	+	0.979892i	$0.563941\pi$
$48$	0	0
$49$	−3.37267	−0.481810
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 4.91329i	− 0.674893i	−0.941345	−	0.337446i	$-0.890437\pi$
$53$	− 4.91329i	− 0.674893i	0.941345	−	0.337446i	$-0.109563\pi$
$54$	0	0
$55$	1.07333	0.144728
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 6.77997i	− 0.882676i	−0.897341	−	0.441338i	$-0.854504\pi$
$59$	− 6.77997i	− 0.882676i	0.897341	−	0.441338i	$-0.145496\pi$
$60$	0	0
$61$	13.9968i	1.79211i	0.443941	+	0.896056i	$0.353580\pi$
$61$	13.9968i	1.79211i	−0.443941	+	0.896056i	$0.646420\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	1.28066	0.158846
$66$	0	0
$67$	− 13.2364i	− 1.61708i	−0.588441	−	0.808540i	$-0.700258\pi$
$67$	− 13.2364i	− 1.61708i	0.588441	−	0.808540i	$-0.299742\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	14.0581	1.66839	0.834194	−	0.551471i	$-0.185933\pi$
$71$	14.0581	1.66839	0.834194	+	0.551471i	$0.185933\pi$
$72$	0	0
$73$	−9.52886	−1.11527	−0.557634	−	0.830087i	$-0.688291\pi$
$73$	−9.52886	−1.11527	−0.557634	+	0.830087i	$0.688291\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 4.67882i	− 0.533201i
$78$	0	0
$79$	3.35967	0.377992	0.188996	−	0.981978i	$-0.439477\pi$
$79$	3.35967	0.377992	0.188996	+	0.981978i	$0.439477\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 12.6423i	− 1.38768i	−0.720132	−	0.693838i	$-0.755919\pi$
$83$	− 12.6423i	− 1.38768i	0.720132	−	0.693838i	$-0.244081\pi$
$84$	0	0
$85$	0.436910i	0.0473896i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−0.155777	−0.0165124	−0.00825618	−	0.999966i	$-0.502628\pi$
$89$	−0.155777	−0.0165124	−0.00825618	+	0.999966i	$0.502628\pi$
$90$	0	0
$91$	− 5.58259i	− 0.585214i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−0.311555	−0.0319648
$96$	0	0
$97$	3.59222	0.364734	0.182367	−	0.983231i	$-0.441624\pi$
$97$	3.59222	0.364734	0.182367	+	0.983231i	$0.441624\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 11.0935i	− 1.10385i	−0.833895	−	0.551924i	$-0.813894\pi$
$101$	− 11.0935i	− 1.10385i	0.833895	−	0.551924i	$-0.186106\pi$
$102$	0	0
$103$	−8.74534	−0.861704	−0.430852	−	0.902423i	$-0.641787\pi$
$103$	−8.74534	−0.861704	−0.430852	+	0.902423i	$0.641787\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 9.86212i	− 0.953407i	−0.879064	−	0.476703i	$-0.841832\pi$
$107$	− 9.86212i	− 0.953407i	0.879064	−	0.476703i	$-0.158168\pi$
$108$	0	0
$109$	− 0.512152i	− 0.0490553i	−0.999699	−	0.0245276i	$-0.992192\pi$
$109$	− 0.512152i	− 0.0490553i	0.999699	−	0.0245276i	$-0.00780818\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	7.28066	0.684907	0.342453	−	0.939535i	$-0.388742\pi$
$113$	7.28066	0.684907	0.342453	+	0.939535i	$0.388742\pi$
$114$	0	0
$115$	− 0.279767i	− 0.0260884i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	1.90455	0.174590
$120$	0	0
$121$	4.96489	0.451353
$122$	0	0
$123$	0	0
$124$	0	0
$125$	4.28570i	0.383325i
$126$	0	0
$127$	2.40778	0.213656	0.106828	−	0.994277i	$-0.465931\pi$
$127$	2.40778	0.213656	0.106828	+	0.994277i	$0.465931\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 16.0583i	− 1.40302i	−0.712660	−	0.701509i	$-0.752510\pi$
$131$	− 16.0583i	− 1.40302i	0.712660	−	0.701509i	$-0.247490\pi$
$132$	0	0
$133$	1.35811i	0.117763i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	17.8000	1.52076	0.760378	−	0.649480i	$-0.225014\pi$
$137$	17.8000	1.52076	0.760378	+	0.649480i	$0.225014\pi$
$138$	0	0
$139$	21.7161i	1.84194i	0.389637	+	0.920968i	$0.372600\pi$
$139$	21.7161i	1.84194i	−0.389637	+	0.920968i	$0.627400\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−7.20087	−0.602167
$144$	0	0
$145$	−3.37532	−0.280305
$146$	0	0
$147$	0	0
$148$	0	0
$149$	21.7223i	1.77956i	0.456391	+	0.889779i	$0.349142\pi$
$149$	21.7223i	1.77956i	−0.456391	+	0.889779i	$0.650858\pi$
$150$	0	0
$151$	−18.6820	−1.52032	−0.760159	−	0.649737i	$-0.774879\pi$
$151$	−18.6820	−1.52032	−0.760159	+	0.649737i	$0.774879\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	2.72306i	0.218722i
$156$	0	0
$157$	0.627594i	0.0500875i	0.999686	+	0.0250437i	$0.00797251\pi$
$157$	0.627594i	0.0500875i	−0.999686	+	0.0250437i	$0.992027\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−1.21954	−0.0961136
$162$	0	0
$163$	− 8.56523i	− 0.670880i	−0.942061	−	0.335440i	$-0.891115\pi$
$163$	− 8.56523i	− 0.670880i	0.942061	−	0.335440i	$-0.108885\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	21.8672	1.69213	0.846067	−	0.533076i	$-0.178964\pi$
$167$	21.8672	1.69213	0.846067	+	0.533076i	$0.178964\pi$
$168$	0	0
$169$	4.40820	0.339092
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 1.38597i	− 0.105374i	−0.998611	−	0.0526868i	$-0.983222\pi$
$173$	− 1.38597i	− 0.105374i	0.998611	−	0.0526868i	$-0.0167785\pi$
$174$	0	0
$175$	9.15921	0.692371
$176$	0	0
$177$	0	0
$178$	0	0
$179$	10.8611i	0.811800i	0.913918	+	0.405900i	$0.133042\pi$
$179$	10.8611i	0.811800i	−0.913918	+	0.405900i	$0.866958\pi$
$180$	0	0
$181$	14.6244i	1.08703i	0.839401	+	0.543513i	$0.182906\pi$
$181$	14.6244i	1.08703i	−0.839401	+	0.543513i	$0.817094\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−0.846874	−0.0622634
$186$	0	0
$187$	− 2.45665i	− 0.179648i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	2.26464	0.163863	0.0819317	−	0.996638i	$-0.473891\pi$
$191$	2.26464	0.163863	0.0819317	+	0.996638i	$0.473891\pi$
$192$	0	0
$193$	17.3067	1.24576	0.622880	−	0.782317i	$-0.285962\pi$
$193$	17.3067	1.24576	0.622880	+	0.782317i	$0.285962\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 23.9821i	− 1.70865i	−0.519737	−	0.854326i	$-0.673970\pi$
$197$	− 23.9821i	− 1.70865i	0.519737	−	0.854326i	$-0.326030\pi$
$198$	0	0
$199$	−2.42344	−0.171793	−0.0858964	−	0.996304i	$-0.527375\pi$
$199$	−2.42344	−0.171793	−0.0858964	+	0.996304i	$0.527375\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	14.7135i	1.03269i
$204$	0	0
$205$	0.464771i	0.0324610i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	1.75180	0.121175
$210$	0	0
$211$	− 13.5202i	− 0.930771i	−0.885108	−	0.465385i	$-0.845916\pi$
$211$	− 13.5202i	− 0.930771i	0.885108	−	0.465385i	$-0.154084\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	3.34402	0.228060
$216$	0	0
$217$	11.8702	0.805803
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 2.93118i	− 0.197172i
$222$	0	0
$223$	23.9462	1.60356	0.801778	−	0.597621i	$-0.203887\pi$
$223$	23.9462	1.60356	0.801778	+	0.597621i	$0.203887\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	10.2295i	0.678954i	0.940614	+	0.339477i	$0.110250\pi$
$227$	10.2295i	0.678954i	−0.940614	+	0.339477i	$0.889750\pi$
$228$	0	0
$229$	− 27.2472i	− 1.80055i	−0.435323	−	0.900274i	$-0.643366\pi$
$229$	− 27.2472i	− 1.80055i	0.435323	−	0.900274i	$-0.356634\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−8.02600	−0.525801	−0.262900	−	0.964823i	$-0.584679\pi$
$233$	−8.02600	−0.525801	−0.262900	+	0.964823i	$0.584679\pi$
$234$	0	0
$235$	− 1.19529i	− 0.0779720i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	17.8829	1.15675	0.578373	−	0.815773i	$-0.303688\pi$
$239$	17.8829	1.15675	0.578373	+	0.815773i	$0.303688\pi$
$240$	0	0
$241$	−23.9887	−1.54524	−0.772622	−	0.634866i	$-0.781055\pi$
$241$	−23.9887	−1.54524	−0.772622	+	0.634866i	$0.781055\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 1.47355i	− 0.0941419i
$246$	0	0
$247$	2.09018	0.132995
$248$	0	0
$249$	0	0
$250$	0	0
$251$	3.40979i	0.215224i	0.994193	+	0.107612i	$0.0343204\pi$
$251$	3.40979i	0.215224i	−0.994193	+	0.107612i	$0.965680\pi$
$252$	0	0
$253$	1.57307i	0.0988978i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−4.03511	−0.251703	−0.125852	−	0.992049i	$-0.540166\pi$
$257$	−4.03511	−0.251703	−0.125852	+	0.992049i	$0.540166\pi$
$258$	0	0
$259$	3.69165i	0.229388i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−17.6442	−1.08799	−0.543995	−	0.839089i	$-0.683089\pi$
$263$	−17.6442	−1.08799	−0.543995	+	0.839089i	$0.683089\pi$
$264$	0	0
$265$	2.14667	0.131869
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 27.2910i	− 1.66396i	−0.554803	−	0.831982i	$-0.687206\pi$
$269$	− 27.2910i	− 1.66396i	0.554803	−	0.831982i	$-0.312794\pi$
$270$	0	0
$271$	30.2273	1.83618	0.918088	−	0.396376i	$-0.129732\pi$
$271$	30.2273	1.83618	0.918088	+	0.396376i	$0.129732\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 11.8143i	− 0.712428i
$276$	0	0
$277$	7.97166i	0.478971i	0.970900	+	0.239485i	$0.0769787\pi$
$277$	7.97166i	0.478971i	−0.970900	+	0.239485i	$0.923021\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	14.0833	0.840140	0.420070	−	0.907492i	$-0.362006\pi$
$281$	14.0833	0.840140	0.420070	+	0.907492i	$0.362006\pi$
$282$	0	0
$283$	− 19.0947i	− 1.13506i	−0.823353	−	0.567530i	$-0.807899\pi$
$283$	− 19.0947i	− 1.13506i	0.823353	−	0.567530i	$-0.192101\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	2.02600	0.119591
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	0	0
$292$	0	0
$293$	16.1537i	0.943710i	0.881676	+	0.471855i	$0.156415\pi$
$293$	16.1537i	0.943710i	−0.881676	+	0.471855i	$0.843585\pi$
$294$	0	0
$295$	2.96224	0.172468
$296$	0	0
$297$	0	0
$298$	0	0
$299$	1.87692i	0.108545i
$300$	0	0
$301$	− 14.5771i	− 0.840208i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−6.11536	−0.350165
$306$	0	0
$307$	20.1771i	1.15157i	0.817602	+	0.575783i	$0.195303\pi$
$307$	20.1771i	1.15157i	−0.817602	+	0.575783i	$0.804697\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	11.7205	0.664611	0.332305	−	0.943172i	$-0.392174\pi$
$311$	11.7205	0.664611	0.332305	+	0.943172i	$0.392174\pi$
$312$	0	0
$313$	20.1795	1.14062	0.570308	−	0.821431i	$-0.306824\pi$
$313$	20.1795	1.14062	0.570308	+	0.821431i	$0.306824\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	3.84879i	0.216170i	0.994142	+	0.108085i	$0.0344718\pi$
$317$	3.84879i	0.216170i	−0.994142	+	0.108085i	$0.965528\pi$
$318$	0	0
$319$	18.9787	1.06260
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0.713086i	0.0396772i
$324$	0	0
$325$	− 14.0964i	− 0.781925i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−5.21043	−0.287261
$330$	0	0
$331$	8.11856i	0.446236i	0.974791	+	0.223118i	$0.0716236\pi$
$331$	8.11856i	0.446236i	−0.974791	+	0.223118i	$0.928376\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	5.78311	0.315965
$336$	0	0
$337$	−18.5545	−1.01073	−0.505363	−	0.862907i	$-0.668641\pi$
$337$	−18.5545	−1.01073	−0.505363	+	0.862907i	$0.668641\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 15.3112i	− 0.829146i
$342$	0	0
$343$	−19.7553	−1.06669
$344$	0	0
$345$	0	0
$346$	0	0
$347$	18.6104i	0.999056i	0.866298	+	0.499528i	$0.166493\pi$
$347$	18.6104i	0.999056i	−0.866298	+	0.499528i	$0.833507\pi$
$348$	0	0
$349$	− 5.40574i	− 0.289363i	−0.989478	−	0.144681i	$-0.953784\pi$
$349$	− 5.40574i	− 0.289363i	0.989478	−	0.144681i	$-0.0462157\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−18.4980	−0.984547	−0.492274	−	0.870440i	$-0.663834\pi$
$353$	−18.4980	−0.984547	−0.492274	+	0.870440i	$0.663834\pi$
$354$	0	0
$355$	6.14212i	0.325990i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	11.6018	0.612319	0.306159	−	0.951980i	$-0.400956\pi$
$359$	11.6018	0.612319	0.306159	+	0.951980i	$0.400956\pi$
$360$	0	0
$361$	18.4915	0.973237
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 4.16326i	− 0.217915i
$366$	0	0
$367$	−13.2425	−0.691254	−0.345627	−	0.938372i	$-0.612334\pi$
$367$	−13.2425	−0.691254	−0.345627	+	0.938372i	$0.612334\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 9.35764i	− 0.485824i
$372$	0	0
$373$	− 31.3896i	− 1.62529i	−0.582756	−	0.812647i	$-0.698026\pi$
$373$	− 31.3896i	− 1.62529i	0.582756	−	0.812647i	$-0.301974\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	22.6446	1.16626
$378$	0	0
$379$	− 9.54065i	− 0.490070i	−0.969514	−	0.245035i	$-0.921200\pi$
$379$	− 9.54065i	− 0.490070i	0.969514	−	0.245035i	$-0.0787995\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−16.1535	−0.825407	−0.412704	−	0.910865i	$-0.635415\pi$
$383$	−16.1535	−0.825407	−0.412704	+	0.910865i	$0.635415\pi$
$384$	0	0
$385$	2.04422	0.104183
$386$	0	0
$387$	0	0
$388$	0	0
$389$	5.46924i	0.277301i	0.990341	+	0.138651i	$0.0442765\pi$
$389$	5.46924i	0.277301i	−0.990341	+	0.138651i	$0.955723\pi$
$390$	0	0
$391$	−0.640331	−0.0323829
$392$	0	0
$393$	0	0
$394$	0	0
$395$	1.46787i	0.0738567i
$396$	0	0
$397$	− 4.65455i	− 0.233605i	−0.993155	−	0.116803i	$-0.962736\pi$
$397$	− 4.65455i	− 0.233605i	0.993155	−	0.116803i	$-0.0372645\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	35.1731	1.75646	0.878230	−	0.478239i	$-0.158725\pi$
$401$	35.1731	1.75646	0.878230	+	0.478239i	$0.158725\pi$
$402$	0	0
$403$	− 18.2687i	− 0.910029i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	4.76178	0.236033
$408$	0	0
$409$	−9.65109	−0.477216	−0.238608	−	0.971116i	$-0.576691\pi$
$409$	−9.65109	−0.477216	−0.238608	+	0.971116i	$0.576691\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 12.9128i	− 0.635398i
$414$	0	0
$415$	5.52356	0.271141
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 23.8116i	− 1.16327i	−0.813449	−	0.581636i	$-0.802413\pi$
$419$	− 23.8116i	− 1.16327i	0.813449	−	0.581636i	$-0.197587\pi$
$420$	0	0
$421$	8.23400i	0.401301i	0.979663	+	0.200650i	$0.0643055\pi$
$421$	8.23400i	0.401301i	−0.979663	+	0.200650i	$0.935695\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	4.80911	0.233276
$426$	0	0
$427$	26.6578i	1.29006i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−17.3796	−0.837147	−0.418574	−	0.908183i	$-0.637470\pi$
$431$	−17.3796	−0.837147	−0.418574	+	0.908183i	$0.637470\pi$
$432$	0	0
$433$	4.50021	0.216266	0.108133	−	0.994136i	$-0.465513\pi$
$433$	4.50021	0.216266	0.108133	+	0.994136i	$0.465513\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 0.456611i	− 0.0218427i
$438$	0	0
$439$	−27.0585	−1.29143	−0.645716	−	0.763578i	$-0.723441\pi$
$439$	−27.0585	−1.29143	−0.645716	+	0.763578i	$0.723441\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 26.3039i	− 1.24973i	−0.780731	−	0.624867i	$-0.785153\pi$
$443$	− 26.3039i	− 1.24973i	0.780731	−	0.624867i	$-0.214847\pi$
$444$	0	0
$445$	− 0.0680607i	− 0.00322639i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−11.4716	−0.541376	−0.270688	−	0.962667i	$-0.587251\pi$
$449$	−11.4716	−0.541376	−0.270688	+	0.962667i	$0.587251\pi$
$450$	0	0
$451$	− 2.61330i	− 0.123056i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	2.43909	0.114346
$456$	0	0
$457$	−28.5522	−1.33562	−0.667808	−	0.744334i	$-0.732767\pi$
$457$	−28.5522	−1.33562	−0.667808	+	0.744334i	$0.732767\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 5.47283i	− 0.254895i	−0.991845	−	0.127447i	$-0.959322\pi$
$461$	− 5.47283i	− 0.254895i	0.991845	−	0.127447i	$-0.0406784\pi$
$462$	0	0
$463$	0.381781	0.0177429	0.00887143	−	0.999961i	$-0.497176\pi$
$463$	0.381781	0.0177429	0.00887143	+	0.999961i	$0.497176\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	25.9977i	1.20303i	0.798861	+	0.601516i	$0.205436\pi$
$467$	25.9977i	1.20303i	−0.798861	+	0.601516i	$0.794564\pi$
$468$	0	0
$469$	− 25.2094i	− 1.16406i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−18.8027	−0.864547
$474$	0	0
$475$	3.42931i	0.157348i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	31.5010	1.43932	0.719660	−	0.694327i	$-0.244298\pi$
$479$	31.5010	1.43932	0.719660	+	0.694327i	$0.244298\pi$
$480$	0	0
$481$	5.68157	0.259058
$482$	0	0
$483$	0	0
$484$	0	0
$485$	1.56948i	0.0712662i
$486$	0	0
$487$	0.586541	0.0265787	0.0132894	−	0.999912i	$-0.495770\pi$
$487$	0.586541	0.0265787	0.0132894	+	0.999912i	$0.495770\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 16.0764i	− 0.725517i	−0.931883	−	0.362758i	$-0.881835\pi$
$491$	− 16.0764i	− 0.725517i	0.931883	−	0.362758i	$-0.118165\pi$
$492$	0	0
$493$	7.72544i	0.347936i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	26.7744	1.20100
$498$	0	0
$499$	25.8096i	1.15540i	0.816250	+	0.577699i	$0.196049\pi$
$499$	25.8096i	1.15540i	−0.816250	+	0.577699i	$0.803951\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−0.604761	−0.0269650	−0.0134825	−	0.999909i	$-0.504292\pi$
$503$	−0.604761	−0.0269650	−0.0134825	+	0.999909i	$0.504292\pi$
$504$	0	0
$505$	4.84687	0.215683
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 1.83522i	− 0.0813448i	−0.999173	−	0.0406724i	$-0.987050\pi$
$509$	− 1.83522i	− 0.0813448i	0.999173	−	0.0406724i	$-0.0129500\pi$
$510$	0	0
$511$	−18.1482	−0.802831
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 3.82093i	− 0.168370i
$516$	0	0
$517$	6.72083i	0.295582i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	32.3002	1.41510	0.707549	−	0.706664i	$-0.249801\pi$
$521$	32.3002	1.41510	0.707549	+	0.706664i	$0.249801\pi$
$522$	0	0
$523$	− 25.5968i	− 1.11927i	−0.828739	−	0.559636i	$-0.810941\pi$
$523$	− 25.5968i	− 1.11927i	0.828739	−	0.559636i	$-0.189059\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	6.23255	0.271494
$528$	0	0
$529$	−22.5900	−0.982173
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 3.11809i	− 0.135060i
$534$	0	0
$535$	4.30886	0.186288
$536$	0	0
$537$	0	0
$538$	0	0
$539$	8.28546i	0.356880i
$540$	0	0
$541$	8.26545i	0.355360i	0.984088	+	0.177680i	$0.0568591\pi$
$541$	8.26545i	0.355360i	−0.984088	+	0.177680i	$0.943141\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0.223764	0.00958502
$546$	0	0
$547$	32.5335i	1.39103i	0.718511	+	0.695515i	$0.244824\pi$
$547$	32.5335i	1.39103i	−0.718511	+	0.695515i	$0.755176\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−5.50890	−0.234687
$552$	0	0
$553$	6.39867	0.272099
$554$	0	0
$555$	0	0
$556$	0	0
$557$	7.82495i	0.331554i	0.986163	+	0.165777i	$0.0530132\pi$
$557$	7.82495i	0.331554i	−0.986163	+	0.165777i	$0.946987\pi$
$558$	0	0
$559$	−22.4346	−0.948883
$560$	0	0
$561$	0	0
$562$	0	0
$563$	24.5308i	1.03385i	0.856030	+	0.516926i	$0.172924\pi$
$563$	24.5308i	1.03385i	−0.856030	+	0.516926i	$0.827076\pi$
$564$	0	0
$565$	3.18099i	0.133825i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−22.1162	−0.927159	−0.463579	−	0.886055i	$-0.653435\pi$
$569$	−22.1162	−0.927159	−0.463579	+	0.886055i	$0.653435\pi$
$570$	0	0
$571$	20.8768i	0.873667i	0.899542	+	0.436834i	$0.143900\pi$
$571$	20.8768i	0.873667i	−0.899542	+	0.436834i	$0.856100\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−3.07942	−0.128421
$576$	0	0
$577$	31.8573	1.32624	0.663119	−	0.748514i	$-0.269233\pi$
$577$	31.8573	1.32624	0.663119	+	0.748514i	$0.269233\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 24.0780i	− 0.998924i
$582$	0	0
$583$	−12.0702	−0.499898
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 8.88262i	− 0.366625i	−0.983055	−	0.183312i	$-0.941318\pi$
$587$	− 8.88262i	− 0.366625i	0.983055	−	0.183312i	$-0.0586820\pi$
$588$	0	0
$589$	4.44434i	0.183126i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	18.8469	0.773948	0.386974	−	0.922091i	$-0.373520\pi$
$593$	18.8469	0.773948	0.386974	+	0.922091i	$0.373520\pi$
$594$	0	0
$595$	0.832119i	0.0341136i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−36.9822	−1.51105	−0.755526	−	0.655119i	$-0.772618\pi$
$599$	−36.9822	−1.51105	−0.755526	+	0.655119i	$0.772618\pi$
$600$	0	0
$601$	−8.64463	−0.352622	−0.176311	−	0.984335i	$-0.556416\pi$
$601$	−8.64463	−0.352622	−0.176311	+	0.984335i	$0.556416\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	2.16921i	0.0881909i
$606$	0	0
$607$	10.1623	0.412476	0.206238	−	0.978502i	$-0.433878\pi$
$607$	10.1623	0.412476	0.206238	+	0.978502i	$0.433878\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	8.01905i	0.324416i
$612$	0	0
$613$	− 1.66006i	− 0.0670491i	−0.999438	−	0.0335246i	$-0.989327\pi$
$613$	− 1.66006i	− 0.0670491i	0.999438	−	0.0335246i	$-0.0106732\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	4.05888	0.163404	0.0817021	−	0.996657i	$-0.473964\pi$
$617$	4.05888	0.163404	0.0817021	+	0.996657i	$0.473964\pi$
$618$	0	0
$619$	5.38423i	0.216411i	0.994129	+	0.108205i	$0.0345104\pi$
$619$	5.38423i	0.216411i	−0.994129	+	0.108205i	$0.965490\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−0.296686	−0.0118865
$624$	0	0
$625$	22.1731	0.886923
$626$	0	0
$627$	0	0
$628$	0	0
$629$	1.93832i	0.0772861i
$630$	0	0
$631$	−18.8564	−0.750663	−0.375332	−	0.926891i	$-0.622471\pi$
$631$	−18.8564	−0.750663	−0.375332	+	0.926891i	$0.622471\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	1.05199i	0.0417468i
$636$	0	0
$637$	9.88590i	0.391694i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−16.4078	−0.648069	−0.324034	−	0.946045i	$-0.605039\pi$
$641$	−16.4078	−0.648069	−0.324034	+	0.946045i	$0.605039\pi$
$642$	0	0
$643$	− 2.41495i	− 0.0952362i	−0.998866	−	0.0476181i	$-0.984837\pi$
$643$	− 2.41495i	− 0.0952362i	0.998866	−	0.0476181i	$-0.0151630\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−30.1944	−1.18706	−0.593532	−	0.804810i	$-0.702267\pi$
$647$	−30.1944	−1.18706	−0.593532	+	0.804810i	$0.702267\pi$
$648$	0	0
$649$	−16.6560	−0.653805
$650$	0	0
$651$	0	0
$652$	0	0
$653$	40.7911i	1.59628i	0.602474	+	0.798139i	$0.294182\pi$
$653$	40.7911i	1.59628i	−0.602474	+	0.798139i	$0.705818\pi$
$654$	0	0
$655$	7.01603	0.274139
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 0.290016i	− 0.0112974i	−0.999984	−	0.00564872i	$-0.998202\pi$
$659$	− 0.290016i	− 0.0112974i	0.999984	−	0.00564872i	$-0.00179805\pi$
$660$	0	0
$661$	42.0741i	1.63649i	0.574867	+	0.818247i	$0.305054\pi$
$661$	42.0741i	1.63649i	−0.574867	+	0.818247i	$0.694946\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−0.593373	−0.0230100
$666$	0	0
$667$	− 4.94683i	− 0.191542i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	34.3853	1.32743
$672$	0	0
$673$	15.3448	0.591500	0.295750	−	0.955265i	$-0.404430\pi$
$673$	15.3448	0.591500	0.295750	+	0.955265i	$0.404430\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	24.3563i	0.936087i	0.883705	+	0.468044i	$0.155041\pi$
$677$	24.3563i	0.936087i	−0.883705	+	0.468044i	$0.844959\pi$
$678$	0	0
$679$	6.84157	0.262555
$680$	0	0
$681$	0	0
$682$	0	0
$683$	38.9677i	1.49106i	0.666474	+	0.745529i	$0.267803\pi$
$683$	38.9677i	1.49106i	−0.666474	+	0.745529i	$0.732197\pi$
$684$	0	0
$685$	7.77700i	0.297144i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−14.4017	−0.548662
$690$	0	0
$691$	− 35.8965i	− 1.36557i	−0.730621	−	0.682783i	$-0.760770\pi$
$691$	− 35.8965i	− 1.36557i	0.730621	−	0.682783i	$-0.239230\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−9.48799	−0.359900
$696$	0	0
$697$	1.06377	0.0402931
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 3.41645i	− 0.129037i	−0.997916	−	0.0645187i	$-0.979449\pi$
$701$	− 3.41645i	− 0.129037i	0.997916	−	0.0645187i	$-0.0205512\pi$
$702$	0	0
$703$	−1.38219	−0.0521304
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 21.1282i	− 0.794609i
$708$	0	0
$709$	29.5034i	1.10803i	0.832508	+	0.554013i	$0.186904\pi$
$709$	29.5034i	1.10803i	−0.832508	+	0.554013i	$0.813096\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−3.99089	−0.149460
$714$	0	0
$715$	− 3.14613i	− 0.117659i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−1.45859	−0.0543964	−0.0271982	−	0.999630i	$-0.508659\pi$
$719$	−1.45859	−0.0543964	−0.0271982	+	0.999630i	$0.508659\pi$
$720$	0	0
$721$	−16.6560	−0.620301
$722$	0	0
$723$	0	0
$724$	0	0
$725$	37.1525i	1.37981i
$726$	0	0
$727$	−8.29669	−0.307707	−0.153854	−	0.988094i	$-0.549168\pi$
$727$	−8.29669	−0.307707	−0.153854	+	0.988094i	$0.549168\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 7.65379i	− 0.283086i
$732$	0	0
$733$	− 11.9905i	− 0.442878i	−0.975174	−	0.221439i	$-0.928925\pi$
$733$	− 11.9905i	− 0.442878i	0.975174	−	0.221439i	$-0.0710753\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−32.5171	−1.19778
$738$	0	0
$739$	− 5.49124i	− 0.201998i	−0.994887	−	0.100999i	$-0.967796\pi$
$739$	− 5.49124i	− 0.201998i	0.994887	−	0.100999i	$-0.0322040\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−38.7305	−1.42088	−0.710442	−	0.703755i	$-0.751505\pi$
$743$	−38.7305	−1.42088	−0.710442	+	0.703755i	$0.751505\pi$
$744$	0	0
$745$	−9.49068	−0.347712
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 18.7829i	− 0.686314i
$750$	0	0
$751$	−24.6699	−0.900216	−0.450108	−	0.892974i	$-0.648614\pi$
$751$	−24.6699	−0.900216	−0.450108	+	0.892974i	$0.648614\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 8.16235i	− 0.297058i
$756$	0	0
$757$	− 35.8258i	− 1.30211i	−0.759030	−	0.651056i	$-0.774326\pi$
$757$	− 35.8258i	− 1.30211i	0.759030	−	0.651056i	$-0.225674\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−39.3297	−1.42570	−0.712850	−	0.701317i	$-0.752596\pi$
$761$	−39.3297	−1.42570	−0.712850	+	0.701317i	$0.752596\pi$
$762$	0	0
$763$	− 0.975422i	− 0.0353126i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−19.8733	−0.717583
$768$	0	0
$769$	12.6582	0.456467	0.228234	−	0.973606i	$-0.426705\pi$
$769$	12.6582	0.456467	0.228234	+	0.973606i	$0.426705\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	31.5053i	1.13317i	0.824005	+	0.566583i	$0.191735\pi$
$773$	31.5053i	1.13317i	−0.824005	+	0.566583i	$0.808265\pi$
$774$	0	0
$775$	29.9730	1.07666
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0.758558i	0.0271782i
$780$	0	0
$781$	− 34.5358i	− 1.23579i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−0.274202	−0.00978670
$786$	0	0
$787$	33.8345i	1.20607i	0.797714	+	0.603035i	$0.206042\pi$
$787$	33.8345i	1.20607i	−0.797714	+	0.603035i	$0.793958\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	13.8664	0.493033
$792$	0	0
$793$	41.0272	1.45692
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 8.42810i	− 0.298538i	−0.988797	−	0.149269i	$-0.952308\pi$
$797$	− 8.42810i	− 0.298538i	0.988797	−	0.149269i	$-0.0476921\pi$
$798$	0	0
$799$	−2.73578	−0.0967848
$800$	0	0
$801$	0	0
$802$	0	0
$803$	23.4090i	0.826087i
$804$	0	0
$805$	− 0.532831i	− 0.0187798i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−20.0902	−0.706333	−0.353167	−	0.935560i	$-0.614895\pi$
$809$	−20.0902	−0.706333	−0.353167	+	0.935560i	$0.614895\pi$
$810$	0	0
$811$	1.90429i	0.0668688i	0.999441	+	0.0334344i	$0.0106445\pi$
$811$	1.90429i	0.0668688i	−0.999441	+	0.0334344i	$0.989356\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	3.74223	0.131085
$816$	0	0
$817$	5.45781	0.190945
$818$	0	0
$819$	0	0
$820$	0	0
$821$	32.8749i	1.14734i	0.819085	+	0.573672i	$0.194481\pi$
$821$	32.8749i	1.14734i	−0.819085	+	0.573672i	$0.805519\pi$
$822$	0	0
$823$	45.7119	1.59342	0.796708	−	0.604364i	$-0.206573\pi$
$823$	45.7119	1.59342	0.796708	+	0.604364i	$0.206573\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 27.5172i	− 0.956868i	−0.878123	−	0.478434i	$-0.841205\pi$
$827$	− 27.5172i	− 0.956868i	0.878123	−	0.478434i	$-0.158795\pi$
$828$	0	0
$829$	− 19.8717i	− 0.690173i	−0.938571	−	0.345087i	$-0.887850\pi$
$829$	− 19.8717i	− 0.690173i	0.938571	−	0.345087i	$-0.112150\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−3.37267	−0.116856
$834$	0	0
$835$	9.55400i	0.330630i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−49.0516	−1.69345	−0.846725	−	0.532030i	$-0.821429\pi$
$839$	−49.0516	−1.69345	−0.846725	+	0.532030i	$0.821429\pi$
$840$	0	0
$841$	−30.6824	−1.05801
$842$	0	0
$843$	0	0
$844$	0	0
$845$	1.92599i	0.0662559i
$846$	0	0
$847$	9.45590	0.324909
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 1.24117i	− 0.0425467i
$852$	0	0
$853$	21.3483i	0.730951i	0.930821	+	0.365476i	$0.119094\pi$
$853$	21.3483i	0.730951i	−0.930821	+	0.365476i	$0.880906\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−17.2036	−0.587663	−0.293831	−	0.955857i	$-0.594930\pi$
$857$	−17.2036	−0.587663	−0.293831	+	0.955857i	$0.594930\pi$
$858$	0	0
$859$	5.99373i	0.204503i	0.994759	+	0.102252i	$0.0326047\pi$
$859$	5.99373i	0.204503i	−0.994759	+	0.102252i	$0.967395\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−13.6503	−0.464661	−0.232330	−	0.972637i	$-0.574635\pi$
$863$	−13.6503	−0.464661	−0.232330	+	0.972637i	$0.574635\pi$
$864$	0	0
$865$	0.605545	0.0205892
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 8.25352i	− 0.279982i
$870$	0	0
$871$	−38.7982	−1.31463
$872$	0	0
$873$	0	0
$874$	0	0
$875$	8.16235i	0.275938i
$876$	0	0
$877$	− 54.0172i	− 1.82403i	−0.410156	−	0.912015i	$-0.634526\pi$
$877$	− 54.0172i	− 1.82403i	0.410156	−	0.912015i	$-0.365474\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	9.71975	0.327467	0.163733	−	0.986505i	$-0.447646\pi$
$881$	9.71975	0.327467	0.163733	+	0.986505i	$0.447646\pi$
$882$	0	0
$883$	26.1534i	0.880131i	0.897966	+	0.440066i	$0.145045\pi$
$883$	26.1534i	0.880131i	−0.897966	+	0.440066i	$0.854955\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−46.9597	−1.57675	−0.788376	−	0.615194i	$-0.789078\pi$
$887$	−46.9597	−1.57675	−0.788376	+	0.615194i	$0.789078\pi$
$888$	0	0
$889$	4.58576	0.153801
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 1.95084i	− 0.0652825i
$894$	0	0
$895$	−4.74534	−0.158619
$896$	0	0
$897$	0	0
$898$	0	0
$899$	48.1492i	1.60586i
$900$	0	0
$901$	− 4.91329i	− 0.163686i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−6.38956	−0.212396
$906$	0	0
$907$	− 33.7479i	− 1.12058i	−0.828296	−	0.560291i	$-0.810689\pi$
$907$	− 33.7479i	− 1.12058i	0.828296	−	0.560291i	$-0.189311\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	29.8152	0.987822	0.493911	−	0.869513i	$-0.335567\pi$
$911$	29.8152	0.987822	0.493911	+	0.869513i	$0.335567\pi$
$912$	0	0
$913$	−31.0577	−1.02786
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 30.5839i	− 1.00997i
$918$	0	0
$919$	32.0669	1.05779	0.528894	−	0.848688i	$-0.322607\pi$
$919$	32.0669	1.05779	0.528894	+	0.848688i	$0.322607\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 41.2068i	− 1.35634i
$924$	0	0
$925$	9.32162i	0.306493i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−21.5540	−0.707165	−0.353583	−	0.935403i	$-0.615037\pi$
$929$	−21.5540	−0.707165	−0.353583	+	0.935403i	$0.615037\pi$
$930$	0	0
$931$	− 2.40501i	− 0.0788209i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	1.07333	0.0351018
$936$	0	0
$937$	−2.68886	−0.0878411	−0.0439206	−	0.999035i	$-0.513985\pi$
$937$	−2.68886	−0.0878411	−0.0439206	+	0.999035i	$0.513985\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 28.4782i	− 0.928361i	−0.885741	−	0.464181i	$-0.846349\pi$
$941$	− 28.4782i	− 0.928361i	0.885741	−	0.464181i	$-0.153651\pi$
$942$	0	0
$943$	−0.681163	−0.0221817
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 8.02522i	− 0.260784i	−0.991462	−	0.130392i	$-0.958376\pi$
$947$	− 8.02522i	− 0.260784i	0.991462	−	0.130392i	$-0.0416237\pi$
$948$	0	0
$949$	27.9308i	0.906672i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	5.46774	0.177118	0.0885588	−	0.996071i	$-0.471774\pi$
$953$	5.46774	0.177118	0.0885588	+	0.996071i	$0.471774\pi$
$954$	0	0
$955$	0.989442i	0.0320176i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	33.9011	1.09472
$960$	0	0
$961$	7.84463	0.253053
$962$	0	0
$963$	0	0
$964$	0	0
$965$	7.56146i	0.243412i
$966$	0	0
$967$	−20.7947	−0.668711	−0.334355	−	0.942447i	$-0.608519\pi$
$967$	−20.7947	−0.668711	−0.334355	+	0.942447i	$0.608519\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	38.0104i	1.21981i	0.792474	+	0.609905i	$0.208793\pi$
$971$	38.0104i	1.21981i	−0.792474	+	0.609905i	$0.791207\pi$
$972$	0	0
$973$	41.3595i	1.32593i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−3.31354	−0.106009	−0.0530047	−	0.998594i	$-0.516880\pi$
$977$	−3.31354	−0.106009	−0.0530047	+	0.998594i	$0.516880\pi$
$978$	0	0
$979$	0.382690i	0.0122308i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	12.2421	0.390463	0.195231	−	0.980757i	$-0.437454\pi$
$983$	12.2421	0.390463	0.195231	+	0.980757i	$0.437454\pi$
$984$	0	0
$985$	10.4780	0.333857
$986$	0	0
$987$	0	0
$988$	0	0
$989$	4.90095i	0.155841i
$990$	0	0
$991$	42.6576	1.35506	0.677532	−	0.735494i	$-0.263050\pi$
$991$	42.6576	1.35506	0.677532	+	0.735494i	$0.263050\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 1.05882i	− 0.0335670i
$996$	0	0
$997$	13.6630i	0.432713i	0.976314	+	0.216356i	$0.0694173\pi$
$997$	13.6630i	0.432713i	−0.976314	+	0.216356i	$0.930583\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4896.2.f.c.2449.5		8
3.2	odd	2		544.2.c.a.273.4		8
4.3	odd	2		1224.2.f.d.613.1		8
8.3	odd	2		1224.2.f.d.613.2		8
8.5	even	2	inner	4896.2.f.c.2449.4		8
12.11	even	2		136.2.c.a.69.8	yes	8
24.5	odd	2		544.2.c.a.273.5		8
24.11	even	2		136.2.c.a.69.7	✓	8
48.5	odd	4		4352.2.a.be.1.4		8
48.11	even	4		4352.2.a.bc.1.5		8
48.29	odd	4		4352.2.a.be.1.5		8
48.35	even	4		4352.2.a.bc.1.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
136.2.c.a.69.7	✓	8	24.11	even	2
136.2.c.a.69.8	yes	8	12.11	even	2
544.2.c.a.273.4		8	3.2	odd	2
544.2.c.a.273.5		8	24.5	odd	2
1224.2.f.d.613.1		8	4.3	odd	2
1224.2.f.d.613.2		8	8.3	odd	2
4352.2.a.bc.1.4		8	48.35	even	4
4352.2.a.bc.1.5		8	48.11	even	4
4352.2.a.be.1.4		8	48.5	odd	4
4352.2.a.be.1.5		8	48.29	odd	4
4896.2.f.c.2449.4		8	8.5	even	2	inner
4896.2.f.c.2449.5		8	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 4896 = 2^{5} \cdot 3^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4896.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+0.436910i q^{5} +1.90455 q^{7} +O(q^{10})\)	\(q+0.436910i q^{5} +1.90455 q^{7} -2.45665i q^{11} -2.93118i q^{13} +1.00000 q^{17} +0.713086i q^{19} -0.640331 q^{23} +4.80911 q^{25} +7.72544i q^{29} +6.23255 q^{31} +0.832119i q^{35} +1.93832i q^{37} +1.06377 q^{41} -7.65379i q^{43} -2.73578 q^{47} -3.37267 q^{49} -4.91329i q^{53} +1.07333 q^{55} -6.77997i q^{59} +13.9968i q^{61} +1.28066 q^{65} -13.2364i q^{67} +14.0581 q^{71} -9.52886 q^{73} -4.67882i q^{77} +3.35967 q^{79} -12.6423i q^{83} +0.436910i q^{85} -0.155777 q^{89} -5.58259i q^{91} -0.311555 q^{95} +3.59222 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{7}+O(q^{10}) \) 8 * q - 8 * q^7	\( 8 q - 8 q^{7} + 8 q^{17} + 12 q^{23} - 8 q^{25} + 12 q^{31} - 28 q^{47} - 8 q^{49} - 44 q^{55} - 24 q^{65} + 8 q^{73} + 44 q^{79} - 8 q^{89} - 16 q^{95} + 8 q^{97}+O(q^{100}) \) 8 * q - 8 * q^7 + 8 * q^17 + 12 * q^23 - 8 * q^25 + 12 * q^31 - 28 * q^47 - 8 * q^49 - 44 * q^55 - 24 * q^65 + 8 * q^73 + 44 * q^79 - 8 * q^89 - 16 * q^95 + 8 * q^97

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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