LMFDB - Embedded newform 4232.2.a.x.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4232,2,Mod(1,4232)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4232, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4232.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 4232 = 2^{3} \cdot 23^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4232.a (trivial)

Newform invariants

comment:select newform

sage:traces = [12,0,8,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$33.7926901354$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 20x^{10} + 157x^{8} - 616x^{6} + 1264x^{4} - 1272x^{2} + 484 $ x^12 - 20x^10 + 157x^8 - 616x^6 + 1264x^4 - 1272*x^2 + 484
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$1.90564$ of defining polynomial
Character	$\chi$	$=$	4232.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+0.368525 q^{3} -1.72190 q^{5} -0.634565 q^{7} -2.86419 q^{9} -6.09935 q^{11} +6.49865 q^{13} -0.634565 q^{15} -2.10696 q^{17} +4.13463 q^{19} -0.233853 q^{21} -2.03505 q^{25} -2.16110 q^{27} +0.137641 q^{29} +6.17402 q^{31} -2.24777 q^{33} +1.09266 q^{35} -9.22178 q^{37} +2.39492 q^{39} -3.58758 q^{41} -5.59880 q^{43} +4.93186 q^{45} +7.61453 q^{47} -6.59733 q^{49} -0.776469 q^{51} +7.23472 q^{53} +10.5025 q^{55} +1.52371 q^{57} -9.94415 q^{59} +1.63915 q^{61} +1.81751 q^{63} -11.1900 q^{65} +13.6926 q^{67} +4.13695 q^{71} +4.17219 q^{73} -0.749968 q^{75} +3.87043 q^{77} -2.88084 q^{79} +7.79615 q^{81} -10.3441 q^{83} +3.62799 q^{85} +0.0507241 q^{87} -1.91162 q^{89} -4.12381 q^{91} +2.27528 q^{93} -7.11942 q^{95} +6.18953 q^{97} +17.4697 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 8 q^{3} + 8 q^{9} + 16 q^{13} + 4 q^{25} + 8 q^{27} + 8 q^{31} + 56 q^{35} + 64 q^{39} - 40 q^{41} + 32 q^{47} + 28 q^{49} + 64 q^{55} + 60 q^{59} - 32 q^{71} + 28 q^{73} + 16 q^{75} + 24 q^{77}+ \cdots + 16 q^{95}+O(q^{100}) $ 12 * q + 8 * q^3 + 8 * q^9 + 16 * q^13 + 4 * q^25 + 8 * q^27 + 8 * q^31 + 56 * q^35 + 64 * q^39 - 40 * q^41 + 32 * q^47 + 28 * q^49 + 64 * q^55 + 60 * q^59 - 32 * q^71 + 28 * q^73 + 16 * q^75 + 24 * q^77 + 4 * q^81 + 64 * q^85 - 16 * q^87 + 24 * q^93 + 16 * q^95

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
$3$	0.368525	0.212768	0.106384	−	0.994325i	$-0.466073\pi$
$3$	0.368525	0.212768	0.106384	+	0.994325i	$0.466073\pi$
$4$	0	0
$5$	−1.72190	−0.770058	−0.385029	−	0.922904i	$-0.625809\pi$
$5$	−1.72190	−0.770058	−0.385029	+	0.922904i	$0.625809\pi$
$6$	0	0
$7$	−0.634565	−0.239843	−0.119921	−	0.992783i	$-0.538264\pi$
$7$	−0.634565	−0.239843	−0.119921	+	0.992783i	$0.538264\pi$
$8$	0	0
$9$	−2.86419	−0.954730
$10$	0	0
$11$	−6.09935	−1.83902	−0.919512	−	0.393062i	$-0.871416\pi$
$11$	−6.09935	−1.83902	−0.919512	+	0.393062i	$0.871416\pi$
$12$	0	0
$13$	6.49865	1.80240	0.901201	−	0.433402i	$-0.142687\pi$
$13$	6.49865	1.80240	0.901201	+	0.433402i	$0.142687\pi$
$14$	0	0
$15$	−0.634565	−0.163844
$16$	0	0
$17$	−2.10696	−0.511014	−0.255507	−	0.966807i	$-0.582242\pi$
$17$	−2.10696	−0.511014	−0.255507	+	0.966807i	$0.582242\pi$
$18$	0	0
$19$	4.13463	0.948548	0.474274	−	0.880377i	$-0.342711\pi$
$19$	4.13463	0.948548	0.474274	+	0.880377i	$0.342711\pi$
$20$	0	0
$21$	−0.233853	−0.0510309
$22$	0	0
$23$	0	0
$23$	0	0
$24$	0	0
$25$	−2.03505	−0.407010
$26$	0	0
$27$	−2.16110	−0.415904
$28$	0	0
$29$	0.137641	0.0255593	0.0127796	−	0.999918i	$-0.495932\pi$
$29$	0.137641	0.0255593	0.0127796	+	0.999918i	$0.495932\pi$
$30$	0	0
$31$	6.17402	1.10889	0.554443	−	0.832222i	$-0.312931\pi$
$31$	6.17402	1.10889	0.554443	+	0.832222i	$0.312931\pi$
$32$	0	0
$33$	−2.24777	−0.391286
$34$	0	0
$35$	1.09266	0.184693
$36$	0	0
$37$	−9.22178	−1.51605	−0.758026	−	0.652225i	$-0.773836\pi$
$37$	−9.22178	−1.51605	−0.758026	+	0.652225i	$0.773836\pi$
$38$	0	0
$39$	2.39492	0.383494
$40$	0	0
$41$	−3.58758	−0.560286	−0.280143	−	0.959958i	$-0.590382\pi$
$41$	−3.58758	−0.560286	−0.280143	+	0.959958i	$0.590382\pi$
$42$	0	0
$43$	−5.59880	−0.853809	−0.426904	−	0.904297i	$-0.640396\pi$
$43$	−5.59880	−0.853809	−0.426904	+	0.904297i	$0.640396\pi$
$44$	0	0
$45$	4.93186	0.735198
$46$	0	0
$47$	7.61453	1.11069	0.555347	−	0.831619i	$-0.312586\pi$
$47$	7.61453	1.11069	0.555347	+	0.831619i	$0.312586\pi$
$48$	0	0
$49$	−6.59733	−0.942475
$50$	0	0
$51$	−0.776469	−0.108727
$52$	0	0
$53$	7.23472	0.993766	0.496883	−	0.867818i	$-0.334478\pi$
$53$	7.23472	0.993766	0.496883	+	0.867818i	$0.334478\pi$
$54$	0	0
$55$	10.5025	1.41616
$56$	0	0
$57$	1.52371	0.201821
$58$	0	0
$59$	−9.94415	−1.29462	−0.647309	−	0.762228i	$-0.724106\pi$
$59$	−9.94415	−1.29462	−0.647309	+	0.762228i	$0.724106\pi$
$60$	0	0
$61$	1.63915	0.209872	0.104936	−	0.994479i	$-0.466536\pi$
$61$	1.63915	0.209872	0.104936	+	0.994479i	$0.466536\pi$
$62$	0	0
$63$	1.81751	0.228985
$64$	0	0
$65$	−11.1900	−1.38795
$66$	0	0
$67$	13.6926	1.67282	0.836411	−	0.548103i	$-0.184650\pi$
$67$	13.6926	1.67282	0.836411	+	0.548103i	$0.184650\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	4.13695	0.490966	0.245483	−	0.969401i	$-0.421053\pi$
$71$	4.13695	0.490966	0.245483	+	0.969401i	$0.421053\pi$
$72$	0	0
$73$	4.17219	0.488318	0.244159	−	0.969735i	$-0.421488\pi$
$73$	4.17219	0.488318	0.244159	+	0.969735i	$0.421488\pi$
$74$	0	0
$75$	−0.749968	−0.0865988
$76$	0	0
$77$	3.87043	0.441077
$78$	0	0
$79$	−2.88084	−0.324120	−0.162060	−	0.986781i	$-0.551814\pi$
$79$	−2.88084	−0.324120	−0.162060	+	0.986781i	$0.551814\pi$
$80$	0	0
$81$	7.79615	0.866239
$82$	0	0
$83$	−10.3441	−1.13541	−0.567705	−	0.823232i	$-0.692169\pi$
$83$	−10.3441	−1.13541	−0.567705	+	0.823232i	$0.692169\pi$
$84$	0	0
$85$	3.62799	0.393510
$86$	0	0
$87$	0.0507241	0.00543820
$88$	0	0
$89$	−1.91162	−0.202632	−0.101316	−	0.994854i	$-0.532305\pi$
$89$	−1.91162	−0.202632	−0.101316	+	0.994854i	$0.532305\pi$
$90$	0	0
$91$	−4.12381	−0.432293
$92$	0	0
$93$	2.27528	0.235936
$94$	0	0
$95$	−7.11942	−0.730437
$96$	0	0
$97$	6.18953	0.628451	0.314226	−	0.949348i	$-0.398255\pi$
$97$	6.18953	0.628451	0.314226	+	0.949348i	$0.398255\pi$
$98$	0	0
$99$	17.4697	1.75577
$100$	0	0
$101$	11.6775	1.16195	0.580976	−	0.813921i	$-0.302671\pi$
$101$	11.6775	1.16195	0.580976	+	0.813921i	$0.302671\pi$
$102$	0	0
$103$	10.7801	1.06220	0.531098	−	0.847311i	$-0.321780\pi$
$103$	10.7801	1.06220	0.531098	+	0.847311i	$0.321780\pi$
$104$	0	0
$105$	0.402672	0.0392968
$106$	0	0
$107$	12.0266	1.16266	0.581329	−	0.813668i	$-0.302533\pi$
$107$	12.0266	1.16266	0.581329	+	0.813668i	$0.302533\pi$
$108$	0	0
$109$	12.6322	1.20994	0.604971	−	0.796248i	$-0.293185\pi$
$109$	12.6322	1.20994	0.604971	+	0.796248i	$0.293185\pi$
$110$	0	0
$111$	−3.39846	−0.322567
$112$	0	0
$113$	11.9070	1.12012	0.560059	−	0.828453i	$-0.310778\pi$
$113$	11.9070	1.12012	0.560059	+	0.828453i	$0.310778\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−18.6134	−1.72081
$118$	0	0
$119$	1.33700	0.122563
$120$	0	0
$121$	26.2021	2.38201
$122$	0	0
$123$	−1.32211	−0.119211
$124$	0	0
$125$	12.1137	1.08348
$126$	0	0
$127$	4.21736	0.374230	0.187115	−	0.982338i	$-0.440086\pi$
$127$	4.21736	0.374230	0.187115	+	0.982338i	$0.440086\pi$
$128$	0	0
$129$	−2.06330	−0.181663
$130$	0	0
$131$	13.4556	1.17562	0.587811	−	0.808998i	$-0.299990\pi$
$131$	13.4556	1.17562	0.587811	+	0.808998i	$0.299990\pi$
$132$	0	0
$133$	−2.62369	−0.227503
$134$	0	0
$135$	3.72121	0.320271
$136$	0	0
$137$	0.974858	0.0832878	0.0416439	−	0.999133i	$-0.486741\pi$
$137$	0.974858	0.0832878	0.0416439	+	0.999133i	$0.486741\pi$
$138$	0	0
$139$	11.4887	0.974455	0.487227	−	0.873275i	$-0.338008\pi$
$139$	11.4887	0.974455	0.487227	+	0.873275i	$0.338008\pi$
$140$	0	0
$141$	2.80615	0.236320
$142$	0	0
$143$	−39.6376	−3.31466
$144$	0	0
$145$	−0.237004	−0.0196821
$146$	0	0
$147$	−2.43128	−0.200529
$148$	0	0
$149$	−21.9309	−1.79665	−0.898327	−	0.439328i	$-0.855217\pi$
$149$	−21.9309	−1.79665	−0.898327	+	0.439328i	$0.855217\pi$
$150$	0	0
$151$	21.3858	1.74035	0.870176	−	0.492741i	$-0.164005\pi$
$151$	21.3858	1.74035	0.870176	+	0.492741i	$0.164005\pi$
$152$	0	0
$153$	6.03474	0.487880
$154$	0	0
$155$	−10.6311	−0.853907
$156$	0	0
$157$	4.03077	0.321691	0.160845	−	0.986980i	$-0.448578\pi$
$157$	4.03077	0.321691	0.160845	+	0.986980i	$0.448578\pi$
$158$	0	0
$159$	2.66618	0.211442
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−2.07630	−0.162628	−0.0813141	−	0.996689i	$-0.525912\pi$
$163$	−2.07630	−0.162628	−0.0813141	+	0.996689i	$0.525912\pi$
$164$	0	0
$165$	3.87043	0.301313
$166$	0	0
$167$	6.50877	0.503663	0.251832	−	0.967771i	$-0.418967\pi$
$167$	6.50877	0.503663	0.251832	+	0.967771i	$0.418967\pi$
$168$	0	0
$169$	29.2324	2.24865
$170$	0	0
$171$	−11.8424	−0.905607
$172$	0	0
$173$	−23.5705	−1.79203	−0.896016	−	0.444021i	$-0.853551\pi$
$173$	−23.5705	−1.79203	−0.896016	+	0.444021i	$0.853551\pi$
$174$	0	0
$175$	1.29137	0.0976185
$176$	0	0
$177$	−3.66467	−0.275453
$178$	0	0
$179$	19.5642	1.46230	0.731148	−	0.682219i	$-0.238985\pi$
$179$	19.5642	1.46230	0.731148	+	0.682219i	$0.238985\pi$
$180$	0	0
$181$	12.0695	0.897121	0.448560	−	0.893753i	$-0.351937\pi$
$181$	12.0695	0.897121	0.448560	+	0.893753i	$0.351937\pi$
$182$	0	0
$183$	0.604070	0.0446541
$184$	0	0
$185$	15.8790	1.16745
$186$	0	0
$187$	12.8511	0.939767
$188$	0	0
$189$	1.37136	0.0997517
$190$	0	0
$191$	−3.99790	−0.289278	−0.144639	−	0.989485i	$-0.546202\pi$
$191$	−3.99790	−0.289278	−0.144639	+	0.989485i	$0.546202\pi$
$192$	0	0
$193$	−8.97997	−0.646392	−0.323196	−	0.946332i	$-0.604757\pi$
$193$	−8.97997	−0.646392	−0.323196	+	0.946332i	$0.604757\pi$
$194$	0	0
$195$	−4.12381	−0.295312
$196$	0	0
$197$	4.28855	0.305547	0.152773	−	0.988261i	$-0.451180\pi$
$197$	4.28855	0.305547	0.152773	+	0.988261i	$0.451180\pi$
$198$	0	0
$199$	−1.36952	−0.0970830	−0.0485415	−	0.998821i	$-0.515457\pi$
$199$	−1.36952	−0.0970830	−0.0485415	+	0.998821i	$0.515457\pi$
$200$	0	0
$201$	5.04608	0.355923
$202$	0	0
$203$	−0.0873420	−0.00613021
$204$	0	0
$205$	6.17746	0.431453
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−25.2185	−1.74440
$210$	0	0
$211$	−26.0920	−1.79624	−0.898122	−	0.439745i	$-0.855069\pi$
$211$	−26.0920	−1.79624	−0.898122	+	0.439745i	$0.855069\pi$
$212$	0	0
$213$	1.52457	0.104462
$214$	0	0
$215$	9.64059	0.657483
$216$	0	0
$217$	−3.91781	−0.265959
$218$	0	0
$219$	1.53756	0.103898
$220$	0	0
$221$	−13.6924	−0.921052
$222$	0	0
$223$	8.99643	0.602445	0.301223	−	0.953554i	$-0.402605\pi$
$223$	8.99643	0.602445	0.301223	+	0.953554i	$0.402605\pi$
$224$	0	0
$225$	5.82877	0.388585
$226$	0	0
$227$	−3.51671	−0.233412	−0.116706	−	0.993166i	$-0.537234\pi$
$227$	−3.51671	−0.233412	−0.116706	+	0.993166i	$0.537234\pi$
$228$	0	0
$229$	11.0578	0.730718	0.365359	−	0.930867i	$-0.380946\pi$
$229$	11.0578	0.730718	0.365359	+	0.930867i	$0.380946\pi$
$230$	0	0
$231$	1.42635	0.0938471
$232$	0	0
$233$	−14.4616	−0.947408	−0.473704	−	0.880684i	$-0.657083\pi$
$233$	−14.4616	−0.947408	−0.473704	+	0.880684i	$0.657083\pi$
$234$	0	0
$235$	−13.1115	−0.855299
$236$	0	0
$237$	−1.06166	−0.0689625
$238$	0	0
$239$	20.6516	1.33584	0.667921	−	0.744232i	$-0.267184\pi$
$239$	20.6516	1.33584	0.667921	+	0.744232i	$0.267184\pi$
$240$	0	0
$241$	−12.9423	−0.833687	−0.416843	−	0.908978i	$-0.636864\pi$
$241$	−12.9423	−0.833687	−0.416843	+	0.908978i	$0.636864\pi$
$242$	0	0
$243$	9.35638	0.600212
$244$	0	0
$245$	11.3600	0.725761
$246$	0	0
$247$	26.8695	1.70966
$248$	0	0
$249$	−3.81206	−0.241579
$250$	0	0
$251$	12.0115	0.758159	0.379080	−	0.925364i	$-0.376241\pi$
$251$	12.0115	0.758159	0.379080	+	0.925364i	$0.376241\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	1.33700	0.0837265
$256$	0	0
$257$	−2.03916	−0.127199	−0.0635997	−	0.997975i	$-0.520258\pi$
$257$	−2.03916	−0.127199	−0.0635997	+	0.997975i	$0.520258\pi$
$258$	0	0
$259$	5.85182	0.363614
$260$	0	0
$261$	−0.394229	−0.0244022
$262$	0	0
$263$	−22.9911	−1.41769	−0.708847	−	0.705362i	$-0.750784\pi$
$263$	−22.9911	−1.41769	−0.708847	+	0.705362i	$0.750784\pi$
$264$	0	0
$265$	−12.4575	−0.765257
$266$	0	0
$267$	−0.704481	−0.0431136
$268$	0	0
$269$	−12.7407	−0.776815	−0.388407	−	0.921488i	$-0.626975\pi$
$269$	−12.7407	−0.776815	−0.388407	+	0.921488i	$0.626975\pi$
$270$	0	0
$271$	−16.9416	−1.02913	−0.514563	−	0.857452i	$-0.672046\pi$
$271$	−16.9416	−1.02913	−0.514563	+	0.857452i	$0.672046\pi$
$272$	0	0
$273$	−1.51973	−0.0919782
$274$	0	0
$275$	12.4125	0.748502
$276$	0	0
$277$	−0.609022	−0.0365926	−0.0182963	−	0.999833i	$-0.505824\pi$
$277$	−0.609022	−0.0365926	−0.0182963	+	0.999833i	$0.505824\pi$
$278$	0	0
$279$	−17.6836	−1.05869
$280$	0	0
$281$	−7.85105	−0.468354	−0.234177	−	0.972194i	$-0.575240\pi$
$281$	−7.85105	−0.468354	−0.234177	+	0.972194i	$0.575240\pi$
$282$	0	0
$283$	21.7802	1.29470	0.647348	−	0.762194i	$-0.275878\pi$
$283$	21.7802	1.29470	0.647348	+	0.762194i	$0.275878\pi$
$284$	0	0
$285$	−2.62369	−0.155414
$286$	0	0
$287$	2.27655	0.134381
$288$	0	0
$289$	−12.5607	−0.738865
$290$	0	0
$291$	2.28100	0.133714
$292$	0	0
$293$	−9.72236	−0.567986	−0.283993	−	0.958826i	$-0.591659\pi$
$293$	−9.72236	−0.567986	−0.283993	+	0.958826i	$0.591659\pi$
$294$	0	0
$295$	17.1229	0.996931
$296$	0	0
$297$	13.1813	0.764858
$298$	0	0
$299$	0	0
$300$	0	0
$301$	3.55280	0.204780
$302$	0	0
$303$	4.30344	0.247226
$304$	0	0
$305$	−2.82247	−0.161614
$306$	0	0
$307$	14.6370	0.835381	0.417690	−	0.908589i	$-0.362840\pi$
$307$	14.6370	0.835381	0.417690	+	0.908589i	$0.362840\pi$
$308$	0	0
$309$	3.97274	0.226001
$310$	0	0
$311$	19.2886	1.09376	0.546879	−	0.837211i	$-0.315816\pi$
$311$	19.2886	1.09376	0.546879	+	0.837211i	$0.315816\pi$
$312$	0	0
$313$	−31.9483	−1.80582	−0.902912	−	0.429825i	$-0.858575\pi$
$313$	−31.9483	−1.80582	−0.902912	+	0.429825i	$0.858575\pi$
$314$	0	0
$315$	−3.12958	−0.176332
$316$	0	0
$317$	−25.6140	−1.43863	−0.719314	−	0.694686i	$-0.755544\pi$
$317$	−25.6140	−1.43863	−0.719314	+	0.694686i	$0.755544\pi$
$318$	0	0
$319$	−0.839520	−0.0470041
$320$	0	0
$321$	4.43212	0.247377
$322$	0	0
$323$	−8.71151	−0.484721
$324$	0	0
$325$	−13.2251	−0.733596
$326$	0	0
$327$	4.65527	0.257437
$328$	0	0
$329$	−4.83191	−0.266392
$330$	0	0
$331$	31.2416	1.71719	0.858597	−	0.512651i	$-0.171336\pi$
$331$	31.2416	1.71719	0.858597	+	0.512651i	$0.171336\pi$
$332$	0	0
$333$	26.4129	1.44742
$334$	0	0
$335$	−23.5774	−1.28817
$336$	0	0
$337$	14.3237	0.780260	0.390130	−	0.920760i	$-0.372430\pi$
$337$	14.3237	0.780260	0.390130	+	0.920760i	$0.372430\pi$
$338$	0	0
$339$	4.38804	0.238326
$340$	0	0
$341$	−37.6575	−2.03927
$342$	0	0
$343$	8.62838	0.465889
$344$	0	0
$345$	0	0
$346$	0	0
$347$	23.5062	1.26188	0.630939	−	0.775832i	$-0.282670\pi$
$347$	23.5062	1.26188	0.630939	+	0.775832i	$0.282670\pi$
$348$	0	0
$349$	22.5045	1.20464	0.602320	−	0.798255i	$-0.294243\pi$
$349$	22.5045	1.20464	0.602320	+	0.798255i	$0.294243\pi$
$350$	0	0
$351$	−14.0442	−0.749626
$352$	0	0
$353$	−32.7316	−1.74213	−0.871065	−	0.491168i	$-0.836570\pi$
$353$	−32.7316	−1.74213	−0.871065	+	0.491168i	$0.836570\pi$
$354$	0	0
$355$	−7.12343	−0.378073
$356$	0	0
$357$	0.492720	0.0260775
$358$	0	0
$359$	9.50246	0.501521	0.250760	−	0.968049i	$-0.419319\pi$
$359$	9.50246	0.501521	0.250760	+	0.968049i	$0.419319\pi$
$360$	0	0
$361$	−1.90487	−0.100256
$362$	0	0
$363$	9.65614	0.506816
$364$	0	0
$365$	−7.18410	−0.376033
$366$	0	0
$367$	−27.1399	−1.41669	−0.708346	−	0.705865i	$-0.750558\pi$
$367$	−27.1399	−1.41669	−0.708346	+	0.705865i	$0.750558\pi$
$368$	0	0
$369$	10.2755	0.534922
$370$	0	0
$371$	−4.59090	−0.238348
$372$	0	0
$373$	6.57493	0.340437	0.170219	−	0.985406i	$-0.445553\pi$
$373$	6.57493	0.340437	0.170219	+	0.985406i	$0.445553\pi$
$374$	0	0
$375$	4.46419	0.230530
$376$	0	0
$377$	0.894480	0.0460680
$378$	0	0
$379$	−11.6777	−0.599846	−0.299923	−	0.953963i	$-0.596961\pi$
$379$	−11.6777	−0.599846	−0.299923	+	0.953963i	$0.596961\pi$
$380$	0	0
$381$	1.55420	0.0796242
$382$	0	0
$383$	−15.6271	−0.798507	−0.399254	−	0.916841i	$-0.630731\pi$
$383$	−15.6271	−0.798507	−0.399254	+	0.916841i	$0.630731\pi$
$384$	0	0
$385$	−6.66451	−0.339655
$386$	0	0
$387$	16.0360	0.815157
$388$	0	0
$389$	8.36477	0.424111	0.212055	−	0.977258i	$-0.431984\pi$
$389$	8.36477	0.424111	0.212055	+	0.977258i	$0.431984\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	4.95873	0.250135
$394$	0	0
$395$	4.96053	0.249592
$396$	0	0
$397$	14.1204	0.708682	0.354341	−	0.935116i	$-0.384705\pi$
$397$	14.1204	0.708682	0.354341	+	0.935116i	$0.384705\pi$
$398$	0	0
$399$	−0.966895	−0.0484053
$400$	0	0
$401$	21.6128	1.07929	0.539645	−	0.841893i	$-0.318559\pi$
$401$	21.6128	1.07929	0.539645	+	0.841893i	$0.318559\pi$
$402$	0	0
$403$	40.1228	1.99866
$404$	0	0
$405$	−13.4242	−0.667054
$406$	0	0
$407$	56.2469	2.78806
$408$	0	0
$409$	−22.4656	−1.11085	−0.555427	−	0.831566i	$-0.687445\pi$
$409$	−22.4656	−1.11085	−0.555427	+	0.831566i	$0.687445\pi$
$410$	0	0
$411$	0.359260	0.0177210
$412$	0	0
$413$	6.31020	0.310505
$414$	0	0
$415$	17.8115	0.874333
$416$	0	0
$417$	4.23386	0.207333
$418$	0	0
$419$	22.4591	1.09720	0.548598	−	0.836086i	$-0.315162\pi$
$419$	22.4591	1.09720	0.548598	+	0.836086i	$0.315162\pi$
$420$	0	0
$421$	0.226618	0.0110447	0.00552235	−	0.999985i	$-0.498242\pi$
$421$	0.226618	0.0110447	0.00552235	+	0.999985i	$0.498242\pi$
$422$	0	0
$423$	−21.8095	−1.06041
$424$	0	0
$425$	4.28778	0.207988
$426$	0	0
$427$	−1.04015	−0.0503364
$428$	0	0
$429$	−14.6074	−0.705254
$430$	0	0
$431$	6.59735	0.317783	0.158892	−	0.987296i	$-0.449208\pi$
$431$	6.59735	0.317783	0.158892	+	0.987296i	$0.449208\pi$
$432$	0	0
$433$	−22.2825	−1.07083	−0.535414	−	0.844590i	$-0.679844\pi$
$433$	−22.2825	−1.07083	−0.535414	+	0.844590i	$0.679844\pi$
$434$	0	0
$435$	−0.0873420	−0.00418773
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−37.5746	−1.79334	−0.896668	−	0.442703i	$-0.854020\pi$
$439$	−37.5746	−1.79334	−0.896668	+	0.442703i	$0.854020\pi$
$440$	0	0
$441$	18.8960	0.899809
$442$	0	0
$443$	7.82901	0.371967	0.185984	−	0.982553i	$-0.440453\pi$
$443$	7.82901	0.371967	0.185984	+	0.982553i	$0.440453\pi$
$444$	0	0
$445$	3.29163	0.156038
$446$	0	0
$447$	−8.08211	−0.382271
$448$	0	0
$449$	−10.7712	−0.508327	−0.254163	−	0.967161i	$-0.581800\pi$
$449$	−10.7712	−0.508327	−0.254163	+	0.967161i	$0.581800\pi$
$450$	0	0
$451$	21.8819	1.03038
$452$	0	0
$453$	7.88121	0.370291
$454$	0	0
$455$	7.10081	0.332891
$456$	0	0
$457$	16.5736	0.775281	0.387641	−	0.921811i	$-0.373290\pi$
$457$	16.5736	0.775281	0.387641	+	0.921811i	$0.373290\pi$
$458$	0	0
$459$	4.55336	0.212533
$460$	0	0
$461$	2.45504	0.114343	0.0571714	−	0.998364i	$-0.481792\pi$
$461$	2.45504	0.114343	0.0571714	+	0.998364i	$0.481792\pi$
$462$	0	0
$463$	19.9722	0.928189	0.464094	−	0.885786i	$-0.346380\pi$
$463$	19.9722	0.928189	0.464094	+	0.885786i	$0.346380\pi$
$464$	0	0
$465$	−3.91781	−0.181684
$466$	0	0
$467$	22.6227	1.04685	0.523426	−	0.852071i	$-0.324654\pi$
$467$	22.6227	1.04685	0.523426	+	0.852071i	$0.324654\pi$
$468$	0	0
$469$	−8.68886	−0.401214
$470$	0	0
$471$	1.48544	0.0684455
$472$	0	0
$473$	34.1491	1.57018
$474$	0	0
$475$	−8.41417	−0.386069
$476$	0	0
$477$	−20.7216	−0.948778
$478$	0	0
$479$	40.8007	1.86423	0.932116	−	0.362160i	$-0.117961\pi$
$479$	40.8007	1.86423	0.932116	+	0.362160i	$0.117961\pi$
$480$	0	0
$481$	−59.9291	−2.73253
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−10.6578	−0.483944
$486$	0	0
$487$	−13.8661	−0.628332	−0.314166	−	0.949368i	$-0.601725\pi$
$487$	−13.8661	−0.628332	−0.314166	+	0.949368i	$0.601725\pi$
$488$	0	0
$489$	−0.765168	−0.0346021
$490$	0	0
$491$	4.34220	0.195961	0.0979804	−	0.995188i	$-0.468762\pi$
$491$	4.34220	0.195961	0.0979804	+	0.995188i	$0.468762\pi$
$492$	0	0
$493$	−0.290004	−0.0130611
$494$	0	0
$495$	−30.0811	−1.35205
$496$	0	0
$497$	−2.62516	−0.117755
$498$	0	0
$499$	−11.9929	−0.536877	−0.268438	−	0.963297i	$-0.586508\pi$
$499$	−11.9929	−0.536877	−0.268438	+	0.963297i	$0.586508\pi$
$500$	0	0
$501$	2.39864	0.107164
$502$	0	0
$503$	−23.0497	−1.02774	−0.513868	−	0.857869i	$-0.671788\pi$
$503$	−23.0497	−1.02774	−0.513868	+	0.857869i	$0.671788\pi$
$504$	0	0
$505$	−20.1075	−0.894771
$506$	0	0
$507$	10.7729	0.478441
$508$	0	0
$509$	−9.74900	−0.432117	−0.216058	−	0.976380i	$-0.569320\pi$
$509$	−9.74900	−0.432117	−0.216058	+	0.976380i	$0.569320\pi$
$510$	0	0
$511$	−2.64752	−0.117120
$512$	0	0
$513$	−8.93535	−0.394505
$514$	0	0
$515$	−18.5623	−0.817952
$516$	0	0
$517$	−46.4437	−2.04259
$518$	0	0
$519$	−8.68633	−0.381287
$520$	0	0
$521$	−16.6549	−0.729664	−0.364832	−	0.931073i	$-0.618874\pi$
$521$	−16.6549	−0.729664	−0.364832	+	0.931073i	$0.618874\pi$
$522$	0	0
$523$	−35.7730	−1.56425	−0.782123	−	0.623124i	$-0.785863\pi$
$523$	−35.7730	−1.56425	−0.782123	+	0.623124i	$0.785863\pi$
$524$	0	0
$525$	0.475903	0.0207701
$526$	0	0
$527$	−13.0084	−0.566656
$528$	0	0
$529$	0	0
$530$	0	0
$531$	28.4819	1.23601
$532$	0	0
$533$	−23.3144	−1.00986
$534$	0	0
$535$	−20.7087	−0.895315
$536$	0	0
$537$	7.20989	0.311130
$538$	0	0
$539$	40.2394	1.73324
$540$	0	0
$541$	23.0468	0.990859	0.495429	−	0.868648i	$-0.335011\pi$
$541$	23.0468	0.990859	0.495429	+	0.868648i	$0.335011\pi$
$542$	0	0
$543$	4.44792	0.190879
$544$	0	0
$545$	−21.7514	−0.931726
$546$	0	0
$547$	31.3085	1.33865	0.669327	−	0.742968i	$-0.266582\pi$
$547$	31.3085	1.33865	0.669327	+	0.742968i	$0.266582\pi$
$548$	0	0
$549$	−4.69485	−0.200371
$550$	0	0
$551$	0.569093	0.0242442
$552$	0	0
$553$	1.82808	0.0777380
$554$	0	0
$555$	5.85182	0.248396
$556$	0	0
$557$	14.4758	0.613358	0.306679	−	0.951813i	$-0.400782\pi$
$557$	14.4758	0.613358	0.306679	+	0.951813i	$0.400782\pi$
$558$	0	0
$559$	−36.3846	−1.53891
$560$	0	0
$561$	4.73596	0.199952
$562$	0	0
$563$	30.7739	1.29696	0.648482	−	0.761230i	$-0.275404\pi$
$563$	30.7739	1.29696	0.648482	+	0.761230i	$0.275404\pi$
$564$	0	0
$565$	−20.5027	−0.862557
$566$	0	0
$567$	−4.94716	−0.207761
$568$	0	0
$569$	22.5834	0.946745	0.473372	−	0.880862i	$-0.343037\pi$
$569$	22.5834	0.946745	0.473372	+	0.880862i	$0.343037\pi$
$570$	0	0
$571$	2.71458	0.113601	0.0568007	−	0.998386i	$-0.481910\pi$
$571$	2.71458	0.113601	0.0568007	+	0.998386i	$0.481910\pi$
$572$	0	0
$573$	−1.47333	−0.0615491
$574$	0	0
$575$	0	0
$576$	0	0
$577$	3.48252	0.144979	0.0724896	−	0.997369i	$-0.476906\pi$
$577$	3.48252	0.144979	0.0724896	+	0.997369i	$0.476906\pi$
$578$	0	0
$579$	−3.30934	−0.137532
$580$	0	0
$581$	6.56399	0.272320
$582$	0	0
$583$	−44.1271	−1.82756
$584$	0	0
$585$	32.0504	1.32512
$586$	0	0
$587$	−6.02649	−0.248740	−0.124370	−	0.992236i	$-0.539691\pi$
$587$	−6.02649	−0.248740	−0.124370	+	0.992236i	$0.539691\pi$
$588$	0	0
$589$	25.5273	1.05183
$590$	0	0
$591$	1.58044	0.0650106
$592$	0	0
$593$	12.3095	0.505489	0.252745	−	0.967533i	$-0.418667\pi$
$593$	12.3095	0.505489	0.252745	+	0.967533i	$0.418667\pi$
$594$	0	0
$595$	−2.30219	−0.0943807
$596$	0	0
$597$	−0.504704	−0.0206562
$598$	0	0
$599$	−27.2651	−1.11402	−0.557011	−	0.830505i	$-0.688052\pi$
$599$	−27.2651	−1.11402	−0.557011	+	0.830505i	$0.688052\pi$
$600$	0	0
$601$	32.7110	1.33431	0.667155	−	0.744919i	$-0.267512\pi$
$601$	32.7110	1.33431	0.667155	+	0.744919i	$0.267512\pi$
$602$	0	0
$603$	−39.2183	−1.59709
$604$	0	0
$605$	−45.1175	−1.83429
$606$	0	0
$607$	−7.17263	−0.291128	−0.145564	−	0.989349i	$-0.546500\pi$
$607$	−7.17263	−0.291128	−0.145564	+	0.989349i	$0.546500\pi$
$608$	0	0
$609$	−0.0321877	−0.00130431
$610$	0	0
$611$	49.4842	2.00191
$612$	0	0
$613$	33.4238	1.34997	0.674987	−	0.737830i	$-0.264149\pi$
$613$	33.4238	1.34997	0.674987	+	0.737830i	$0.264149\pi$
$614$	0	0
$615$	2.27655	0.0917994
$616$	0	0
$617$	28.2378	1.13681	0.568405	−	0.822749i	$-0.307561\pi$
$617$	28.2378	1.13681	0.568405	+	0.822749i	$0.307561\pi$
$618$	0	0
$619$	−3.01950	−0.121364	−0.0606820	−	0.998157i	$-0.519328\pi$
$619$	−3.01950	−0.121364	−0.0606820	+	0.998157i	$0.519328\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	1.21305	0.0485998
$624$	0	0
$625$	−10.6833	−0.427333
$626$	0	0
$627$	−9.29367	−0.371153
$628$	0	0
$629$	19.4300	0.774723
$630$	0	0
$631$	−42.3604	−1.68634	−0.843171	−	0.537646i	$-0.819314\pi$
$631$	−42.3604	−1.68634	−0.843171	+	0.537646i	$0.819314\pi$
$632$	0	0
$633$	−9.61555	−0.382184
$634$	0	0
$635$	−7.26188	−0.288179
$636$	0	0
$637$	−42.8737	−1.69872
$638$	0	0
$639$	−11.8490	−0.468740
$640$	0	0
$641$	1.22041	0.0482034	0.0241017	−	0.999710i	$-0.492327\pi$
$641$	1.22041	0.0482034	0.0241017	+	0.999710i	$0.492327\pi$
$642$	0	0
$643$	−4.17771	−0.164753	−0.0823763	−	0.996601i	$-0.526251\pi$
$643$	−4.17771	−0.164753	−0.0823763	+	0.996601i	$0.526251\pi$
$644$	0	0
$645$	3.55280	0.139891
$646$	0	0
$647$	−1.43130	−0.0562701	−0.0281351	−	0.999604i	$-0.508957\pi$
$647$	−1.43130	−0.0562701	−0.0281351	+	0.999604i	$0.508957\pi$
$648$	0	0
$649$	60.6529	2.38083
$650$	0	0
$651$	−1.44381	−0.0565875
$652$	0	0
$653$	−31.9104	−1.24875	−0.624375	−	0.781125i	$-0.714646\pi$
$653$	−31.9104	−1.24875	−0.624375	+	0.781125i	$0.714646\pi$
$654$	0	0
$655$	−23.1693	−0.905298
$656$	0	0
$657$	−11.9499	−0.466211
$658$	0	0
$659$	−17.5556	−0.683870	−0.341935	−	0.939724i	$-0.611082\pi$
$659$	−17.5556	−0.683870	−0.341935	+	0.939724i	$0.611082\pi$
$660$	0	0
$661$	33.8621	1.31708	0.658542	−	0.752544i	$-0.271173\pi$
$661$	33.8621	1.31708	0.658542	+	0.752544i	$0.271173\pi$
$662$	0	0
$663$	−5.04600	−0.195970
$664$	0	0
$665$	4.51773	0.175190
$666$	0	0
$667$	0	0
$668$	0	0
$669$	3.31541	0.128181
$670$	0	0
$671$	−9.99779	−0.385960
$672$	0	0
$673$	−39.2280	−1.51213	−0.756064	−	0.654498i	$-0.772880\pi$
$673$	−39.2280	−1.51213	−0.756064	+	0.654498i	$0.772880\pi$
$674$	0	0
$675$	4.39795	0.169277
$676$	0	0
$677$	−34.1775	−1.31355	−0.656773	−	0.754088i	$-0.728079\pi$
$677$	−34.1775	−1.31355	−0.656773	+	0.754088i	$0.728079\pi$
$678$	0	0
$679$	−3.92766	−0.150730
$680$	0	0
$681$	−1.29600	−0.0496627
$682$	0	0
$683$	−28.6245	−1.09528	−0.547642	−	0.836713i	$-0.684474\pi$
$683$	−28.6245	−1.09528	−0.547642	+	0.836713i	$0.684474\pi$
$684$	0	0
$685$	−1.67861	−0.0641364
$686$	0	0
$687$	4.07507	0.155473
$688$	0	0
$689$	47.0159	1.79116
$690$	0	0
$691$	−37.0757	−1.41042	−0.705212	−	0.708996i	$-0.749148\pi$
$691$	−37.0757	−1.41042	−0.705212	+	0.708996i	$0.749148\pi$
$692$	0	0
$693$	−11.0857	−0.421109
$694$	0	0
$695$	−19.7823	−0.750387
$696$	0	0
$697$	7.55890	0.286314
$698$	0	0
$699$	−5.32945	−0.201578
$700$	0	0
$701$	41.6704	1.57387	0.786935	−	0.617035i	$-0.211667\pi$
$701$	41.6704	1.57387	0.786935	+	0.617035i	$0.211667\pi$
$702$	0	0
$703$	−38.1286	−1.43805
$704$	0	0
$705$	−4.83191	−0.181980
$706$	0	0
$707$	−7.41011	−0.278686
$708$	0	0
$709$	−17.7695	−0.667349	−0.333674	−	0.942688i	$-0.608289\pi$
$709$	−17.7695	−0.667349	−0.333674	+	0.942688i	$0.608289\pi$
$710$	0	0
$711$	8.25128	0.309447
$712$	0	0
$713$	0	0
$714$	0	0
$715$	68.2520	2.55248
$716$	0	0
$717$	7.61064	0.284225
$718$	0	0
$719$	−4.89622	−0.182598	−0.0912991	−	0.995824i	$-0.529102\pi$
$719$	−4.89622	−0.182598	−0.0912991	+	0.995824i	$0.529102\pi$
$720$	0	0
$721$	−6.84067	−0.254760
$722$	0	0
$723$	−4.76956	−0.177382
$724$	0	0
$725$	−0.280106	−0.0104029
$726$	0	0
$727$	21.5740	0.800134	0.400067	−	0.916486i	$-0.368987\pi$
$727$	21.5740	0.800134	0.400067	+	0.916486i	$0.368987\pi$
$728$	0	0
$729$	−19.9404	−0.738533
$730$	0	0
$731$	11.7965	0.436308
$732$	0	0
$733$	0.713744	0.0263627	0.0131814	−	0.999913i	$-0.495804\pi$
$733$	0.713744	0.0263627	0.0131814	+	0.999913i	$0.495804\pi$
$734$	0	0
$735$	4.18643	0.154419
$736$	0	0
$737$	−83.5162	−3.07636
$738$	0	0
$739$	35.3971	1.30210	0.651051	−	0.759034i	$-0.274328\pi$
$739$	35.3971	1.30210	0.651051	+	0.759034i	$0.274328\pi$
$740$	0	0
$741$	9.90208	0.363762
$742$	0	0
$743$	12.9756	0.476029	0.238015	−	0.971262i	$-0.423503\pi$
$743$	12.9756	0.476029	0.238015	+	0.971262i	$0.423503\pi$
$744$	0	0
$745$	37.7630	1.38353
$746$	0	0
$747$	29.6274	1.08401
$748$	0	0
$749$	−7.63168	−0.278855
$750$	0	0
$751$	39.4606	1.43994	0.719968	−	0.694007i	$-0.244156\pi$
$751$	39.4606	1.43994	0.719968	+	0.694007i	$0.244156\pi$
$752$	0	0
$753$	4.42654	0.161312
$754$	0	0
$755$	−36.8243	−1.34017
$756$	0	0
$757$	−6.95737	−0.252870	−0.126435	−	0.991975i	$-0.540353\pi$
$757$	−6.95737	−0.252870	−0.126435	+	0.991975i	$0.540353\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−7.72553	−0.280050	−0.140025	−	0.990148i	$-0.544718\pi$
$761$	−7.72553	−0.280050	−0.140025	+	0.990148i	$0.544718\pi$
$762$	0	0
$763$	−8.01592	−0.290196
$764$	0	0
$765$	−10.3912	−0.375696
$766$	0	0
$767$	−64.6235	−2.33342
$768$	0	0
$769$	25.6432	0.924719	0.462360	−	0.886693i	$-0.347003\pi$
$769$	25.6432	0.924719	0.462360	+	0.886693i	$0.347003\pi$
$770$	0	0
$771$	−0.751482	−0.0270640
$772$	0	0
$773$	39.1386	1.40772	0.703859	−	0.710340i	$-0.251459\pi$
$773$	39.1386	1.40772	0.703859	+	0.710340i	$0.251459\pi$
$774$	0	0
$775$	−12.5644	−0.451328
$776$	0	0
$777$	2.15654	0.0773655
$778$	0	0
$779$	−14.8333	−0.531458
$780$	0	0
$781$	−25.2327	−0.902899
$782$	0	0
$783$	−0.297456	−0.0106302
$784$	0	0
$785$	−6.94060	−0.247721
$786$	0	0
$787$	37.3144	1.33012	0.665058	−	0.746792i	$-0.268407\pi$
$787$	37.3144	1.33012	0.665058	+	0.746792i	$0.268407\pi$
$788$	0	0
$789$	−8.47281	−0.301640
$790$	0	0
$791$	−7.55578	−0.268652
$792$	0	0
$793$	10.6523	0.378274
$794$	0	0
$795$	−4.59090	−0.162822
$796$	0	0
$797$	−34.6928	−1.22888	−0.614441	−	0.788963i	$-0.710618\pi$
$797$	−34.6928	−1.22888	−0.614441	+	0.788963i	$0.710618\pi$
$798$	0	0
$799$	−16.0435	−0.567579
$800$	0	0
$801$	5.47525	0.193458
$802$	0	0
$803$	−25.4477	−0.898028
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−4.69527	−0.165281
$808$	0	0
$809$	34.6506	1.21825	0.609125	−	0.793074i	$-0.291521\pi$
$809$	34.6506	1.21825	0.609125	+	0.793074i	$0.291521\pi$
$810$	0	0
$811$	−19.3338	−0.678901	−0.339451	−	0.940624i	$-0.610241\pi$
$811$	−19.3338	−0.678901	−0.339451	+	0.940624i	$0.610241\pi$
$812$	0	0
$813$	−6.24339	−0.218965
$814$	0	0
$815$	3.57518	0.125233
$816$	0	0
$817$	−23.1489	−0.809879
$818$	0	0
$819$	11.8114	0.412723
$820$	0	0
$821$	−2.71040	−0.0945937	−0.0472968	−	0.998881i	$-0.515061\pi$
$821$	−2.71040	−0.0945937	−0.0472968	+	0.998881i	$0.515061\pi$
$822$	0	0
$823$	−35.9917	−1.25459	−0.627295	−	0.778781i	$-0.715838\pi$
$823$	−35.9917	−1.25459	−0.627295	+	0.778781i	$0.715838\pi$
$824$	0	0
$825$	4.57432	0.159257
$826$	0	0
$827$	−23.7293	−0.825149	−0.412575	−	0.910924i	$-0.635370\pi$
$827$	−23.7293	−0.825149	−0.412575	+	0.910924i	$0.635370\pi$
$828$	0	0
$829$	−25.2855	−0.878201	−0.439101	−	0.898438i	$-0.644703\pi$
$829$	−25.2855	−0.878201	−0.439101	+	0.898438i	$0.644703\pi$
$830$	0	0
$831$	−0.224440	−0.00778574
$832$	0	0
$833$	13.9003	0.481618
$834$	0	0
$835$	−11.2075	−0.387850
$836$	0	0
$837$	−13.3427	−0.461191
$838$	0	0
$839$	35.2390	1.21658	0.608292	−	0.793713i	$-0.291855\pi$
$839$	35.2390	1.21658	0.608292	+	0.793713i	$0.291855\pi$
$840$	0	0
$841$	−28.9811	−0.999347
$842$	0	0
$843$	−2.89331	−0.0996509
$844$	0	0
$845$	−50.3354	−1.73159
$846$	0	0
$847$	−16.6269	−0.571308
$848$	0	0
$849$	8.02654	0.275470
$850$	0	0
$851$	0	0
$852$	0	0
$853$	17.8585	0.611463	0.305732	−	0.952118i	$-0.401099\pi$
$853$	17.8585	0.611463	0.305732	+	0.952118i	$0.401099\pi$
$854$	0	0
$855$	20.3914	0.697370
$856$	0	0
$857$	−5.03092	−0.171853	−0.0859265	−	0.996301i	$-0.527385\pi$
$857$	−5.03092	−0.171853	−0.0859265	+	0.996301i	$0.527385\pi$
$858$	0	0
$859$	46.2075	1.57658	0.788290	−	0.615304i	$-0.210967\pi$
$859$	46.2075	1.57658	0.788290	+	0.615304i	$0.210967\pi$
$860$	0	0
$861$	0.838967	0.0285919
$862$	0	0
$863$	2.72309	0.0926951	0.0463475	−	0.998925i	$-0.485242\pi$
$863$	2.72309	0.0926951	0.0463475	+	0.998925i	$0.485242\pi$
$864$	0	0
$865$	40.5861	1.37997
$866$	0	0
$867$	−4.62894	−0.157207
$868$	0	0
$869$	17.5713	0.596065
$870$	0	0
$871$	88.9836	3.01510
$872$	0	0
$873$	−17.7280	−0.600001
$874$	0	0
$875$	−7.68691	−0.259865
$876$	0	0
$877$	11.4314	0.386009	0.193005	−	0.981198i	$-0.438177\pi$
$877$	11.4314	0.386009	0.193005	+	0.981198i	$0.438177\pi$
$878$	0	0
$879$	−3.58294	−0.120849
$880$	0	0
$881$	32.8953	1.10827	0.554136	−	0.832426i	$-0.313049\pi$
$881$	32.8953	1.10827	0.554136	+	0.832426i	$0.313049\pi$
$882$	0	0
$883$	32.7067	1.10067	0.550333	−	0.834945i	$-0.314501\pi$
$883$	32.7067	1.10067	0.550333	+	0.834945i	$0.314501\pi$
$884$	0	0
$885$	6.31020	0.212115
$886$	0	0
$887$	−20.2976	−0.681527	−0.340764	−	0.940149i	$-0.610686\pi$
$887$	−20.2976	−0.681527	−0.340764	+	0.940149i	$0.610686\pi$
$888$	0	0
$889$	−2.67618	−0.0897563
$890$	0	0
$891$	−47.5515	−1.59303
$892$	0	0
$893$	31.4832	1.05355
$894$	0	0
$895$	−33.6876	−1.12605
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0.849797	0.0283423
$900$	0	0
$901$	−15.2433	−0.507828
$902$	0	0
$903$	1.30930	0.0435707
$904$	0	0
$905$	−20.7825	−0.690835
$906$	0	0
$907$	34.0919	1.13200	0.566001	−	0.824404i	$-0.308490\pi$
$907$	34.0919	1.13200	0.566001	+	0.824404i	$0.308490\pi$
$908$	0	0
$909$	−33.4465	−1.10935
$910$	0	0
$911$	52.5516	1.74111	0.870557	−	0.492068i	$-0.163759\pi$
$911$	52.5516	1.74111	0.870557	+	0.492068i	$0.163759\pi$
$912$	0	0
$913$	63.0922	2.08805
$914$	0	0
$915$	−1.04015	−0.0343863
$916$	0	0
$917$	−8.53846	−0.281965
$918$	0	0
$919$	−20.9862	−0.692269	−0.346135	−	0.938185i	$-0.612506\pi$
$919$	−20.9862	−0.692269	−0.346135	+	0.938185i	$0.612506\pi$
$920$	0	0
$921$	5.39412	0.177742
$922$	0	0
$923$	26.8846	0.884918
$924$	0	0
$925$	18.7668	0.617048
$926$	0	0
$927$	−30.8763	−1.01411
$928$	0	0
$929$	−48.6762	−1.59701	−0.798507	−	0.601985i	$-0.794377\pi$
$929$	−48.6762	−1.59701	−0.798507	+	0.601985i	$0.794377\pi$
$930$	0	0
$931$	−27.2775	−0.893983
$932$	0	0
$933$	7.10835	0.232717
$934$	0	0
$935$	−22.1284	−0.723675
$936$	0	0
$937$	9.68130	0.316274	0.158137	−	0.987417i	$-0.449451\pi$
$937$	9.68130	0.316274	0.158137	+	0.987417i	$0.449451\pi$
$938$	0	0
$939$	−11.7738	−0.384222
$940$	0	0
$941$	−15.8327	−0.516130	−0.258065	−	0.966128i	$-0.583085\pi$
$941$	−15.8327	−0.516130	−0.258065	+	0.966128i	$0.583085\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	−2.36135	−0.0768146
$946$	0	0
$947$	37.6683	1.22405	0.612027	−	0.790837i	$-0.290354\pi$
$947$	37.6683	1.22405	0.612027	+	0.790837i	$0.290354\pi$
$948$	0	0
$949$	27.1136	0.880144
$950$	0	0
$951$	−9.43942	−0.306094
$952$	0	0
$953$	−32.4286	−1.05047	−0.525233	−	0.850958i	$-0.676022\pi$
$953$	−32.4286	−1.05047	−0.525233	+	0.850958i	$0.676022\pi$
$954$	0	0
$955$	6.88399	0.222761
$956$	0	0
$957$	−0.309384	−0.0100010
$958$	0	0
$959$	−0.618611	−0.0199760
$960$	0	0
$961$	7.11851	0.229629
$962$	0	0
$963$	−34.4466	−1.11002
$964$	0	0
$965$	15.4626	0.497760
$966$	0	0
$967$	2.57278	0.0827350	0.0413675	−	0.999144i	$-0.486829\pi$
$967$	2.57278	0.0827350	0.0413675	+	0.999144i	$0.486829\pi$
$968$	0	0
$969$	−3.21041	−0.103133
$970$	0	0
$971$	41.4181	1.32917	0.664585	−	0.747213i	$-0.268608\pi$
$971$	41.4181	1.32917	0.664585	+	0.747213i	$0.268608\pi$
$972$	0	0
$973$	−7.29029	−0.233716
$974$	0	0
$975$	−4.87378	−0.156086
$976$	0	0
$977$	10.2799	0.328883	0.164442	−	0.986387i	$-0.447418\pi$
$977$	10.2799	0.328883	0.164442	+	0.986387i	$0.447418\pi$
$978$	0	0
$979$	11.6597	0.372644
$980$	0	0
$981$	−36.1809	−1.15517
$982$	0	0
$983$	46.5963	1.48619	0.743096	−	0.669185i	$-0.233357\pi$
$983$	46.5963	1.48619	0.743096	+	0.669185i	$0.233357\pi$
$984$	0	0
$985$	−7.38447	−0.235289
$986$	0	0
$987$	−1.78068	−0.0566797
$988$	0	0
$989$	0	0
$990$	0	0
$991$	19.9332	0.633200	0.316600	−	0.948559i	$-0.397459\pi$
$991$	19.9332	0.633200	0.316600	+	0.948559i	$0.397459\pi$
$992$	0	0
$993$	11.5133	0.365364
$994$	0	0
$995$	2.35819	0.0747596
$996$	0	0
$997$	−33.8980	−1.07356	−0.536780	−	0.843722i	$-0.680360\pi$
$997$	−33.8980	−1.07356	−0.536780	+	0.843722i	$0.680360\pi$
$998$	0	0
$999$	19.9292	0.630532

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4232.2.a.x.1.5	✓	12
4.3	odd	2		8464.2.a.cf.1.7		12
23.22	odd	2	inner	4232.2.a.x.1.6	yes	12
92.91	even	2		8464.2.a.cf.1.8		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4232.2.a.x.1.5	✓	12	1.1	even	1	trivial
4232.2.a.x.1.6	yes	12	23.22	odd	2	inner
8464.2.a.cf.1.7		12	4.3	odd	2
8464.2.a.cf.1.8		12	92.91	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4232 = 2^{3} \cdot 23^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4232.a (trivial)

\(f(q)\)	\(=\)	\(q+0.368525 q^{3} -1.72190 q^{5} -0.634565 q^{7} -2.86419 q^{9} -6.09935 q^{11} +6.49865 q^{13} -0.634565 q^{15} -2.10696 q^{17} +4.13463 q^{19} -0.233853 q^{21} -2.03505 q^{25} -2.16110 q^{27} +0.137641 q^{29} +6.17402 q^{31} -2.24777 q^{33} +1.09266 q^{35} -9.22178 q^{37} +2.39492 q^{39} -3.58758 q^{41} -5.59880 q^{43} +4.93186 q^{45} +7.61453 q^{47} -6.59733 q^{49} -0.776469 q^{51} +7.23472 q^{53} +10.5025 q^{55} +1.52371 q^{57} -9.94415 q^{59} +1.63915 q^{61} +1.81751 q^{63} -11.1900 q^{65} +13.6926 q^{67} +4.13695 q^{71} +4.17219 q^{73} -0.749968 q^{75} +3.87043 q^{77} -2.88084 q^{79} +7.79615 q^{81} -10.3441 q^{83} +3.62799 q^{85} +0.0507241 q^{87} -1.91162 q^{89} -4.12381 q^{91} +2.27528 q^{93} -7.11942 q^{95} +6.18953 q^{97} +17.4697 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 8 q^{3} + 8 q^{9} + 16 q^{13} + 4 q^{25} + 8 q^{27} + 8 q^{31} + 56 q^{35} + 64 q^{39} - 40 q^{41} + 32 q^{47} + 28 q^{49} + 64 q^{55} + 60 q^{59} - 32 q^{71} + 28 q^{73} + 16 q^{75} + 24 q^{77}+ \cdots + 16 q^{95}+O(q^{100}) \) 12 * q + 8 * q^3 + 8 * q^9 + 16 * q^13 + 4 * q^25 + 8 * q^27 + 8 * q^31 + 56 * q^35 + 64 * q^39 - 40 * q^41 + 32 * q^47 + 28 * q^49 + 64 * q^55 + 60 * q^59 - 32 * q^71 + 28 * q^73 + 16 * q^75 + 24 * q^77 + 4 * q^81 + 64 * q^85 - 16 * q^87 + 24 * q^93 + 16 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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