LMFDB - Embedded newform 5292.2.a.bb.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5292,2,Mod(1,5292)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5292, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("5292.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5292 = 2^{2} \cdot 3^{3} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5292.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$42.2568327497$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 6x^{2} + 4 $ x^4 - 6*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$0.874032$ of defining polynomial
Character	$\chi$	$=$	5292.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.23607 q^{5} +O(q^{10})$	$q+2.23607 q^{5} +0.540182 q^{11} -2.62210 q^{13} -1.47214 q^{17} +4.03631 q^{19} -3.16228 q^{23} -0.540182 q^{29} -2.62210 q^{31} +4.70820 q^{37} +2.23607 q^{41} +8.70820 q^{43} +4.52786 q^{47} +11.6476 q^{53} +1.20788 q^{55} +11.9443 q^{59} +9.69316 q^{61} -5.86319 q^{65} +3.70820 q^{67} -13.7295 q^{71} -1.41421 q^{73} +5.00000 q^{79} +3.76393 q^{83} -3.29180 q^{85} +6.76393 q^{89} +9.02546 q^{95} -9.69316 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q + 12 q^{17} - 8 q^{37} + 8 q^{43} + 36 q^{47} + 12 q^{59} - 12 q^{67} + 20 q^{79} + 24 q^{83} - 40 q^{85} + 36 q^{89}+O(q^{100}) $ 4 * q + 12 * q^17 - 8 * q^37 + 8 * q^43 + 36 * q^47 + 12 * q^59 - 12 * q^67 + 20 * q^79 + 24 * q^83 - 40 * q^85 + 36 * q^89

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	2.23607	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$5$	2.23607	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	0	0
$10$	0	0
$11$	0.540182	0.162871	0.0814354	−	0.996679i	$-0.474050\pi$
$11$	0.540182	0.162871	0.0814354	+	0.996679i	$0.474050\pi$
$12$	0	0
$13$	−2.62210	−0.727239	−0.363619	−	0.931548i	$-0.618459\pi$
$13$	−2.62210	−0.727239	−0.363619	+	0.931548i	$0.618459\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−1.47214	−0.357045	−0.178523	−	0.983936i	$-0.557132\pi$
$17$	−1.47214	−0.357045	−0.178523	+	0.983936i	$0.557132\pi$
$18$	0	0
$19$	4.03631	0.925993	0.462996	−	0.886360i	$-0.346774\pi$
$19$	4.03631	0.925993	0.462996	+	0.886360i	$0.346774\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−3.16228	−0.659380	−0.329690	−	0.944089i	$-0.606944\pi$
$23$	−3.16228	−0.659380	−0.329690	+	0.944089i	$0.606944\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−0.540182	−0.100309	−0.0501546	−	0.998741i	$-0.515971\pi$
$29$	−0.540182	−0.100309	−0.0501546	+	0.998741i	$0.515971\pi$
$30$	0	0
$31$	−2.62210	−0.470942	−0.235471	−	0.971881i	$-0.575663\pi$
$31$	−2.62210	−0.470942	−0.235471	+	0.971881i	$0.575663\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	4.70820	0.774024	0.387012	−	0.922075i	$-0.373507\pi$
$37$	4.70820	0.774024	0.387012	+	0.922075i	$0.373507\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	2.23607	0.349215	0.174608	−	0.984638i	$-0.444134\pi$
$41$	2.23607	0.349215	0.174608	+	0.984638i	$0.444134\pi$
$42$	0	0
$43$	8.70820	1.32799	0.663994	−	0.747738i	$-0.268860\pi$
$43$	8.70820	1.32799	0.663994	+	0.747738i	$0.268860\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	4.52786	0.660457	0.330228	−	0.943901i	$-0.392874\pi$
$47$	4.52786	0.660457	0.330228	+	0.943901i	$0.392874\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	0	0
$52$	0	0
$53$	11.6476	1.59992	0.799958	−	0.600056i	$-0.204855\pi$
$53$	11.6476	1.59992	0.799958	+	0.600056i	$0.204855\pi$
$54$	0	0
$55$	1.20788	0.162871
$56$	0	0
$57$	0	0
$58$	0	0
$59$	11.9443	1.55501	0.777506	−	0.628876i	$-0.216485\pi$
$59$	11.9443	1.55501	0.777506	+	0.628876i	$0.216485\pi$
$60$	0	0
$61$	9.69316	1.24108	0.620541	−	0.784174i	$-0.286913\pi$
$61$	9.69316	1.24108	0.620541	+	0.784174i	$0.286913\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−5.86319	−0.727239
$66$	0	0
$67$	3.70820	0.453029	0.226515	−	0.974008i	$-0.427267\pi$
$67$	3.70820	0.453029	0.226515	+	0.974008i	$0.427267\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−13.7295	−1.62939	−0.814694	−	0.579891i	$-0.803095\pi$
$71$	−13.7295	−1.62939	−0.814694	+	0.579891i	$0.803095\pi$
$72$	0	0
$73$	−1.41421	−0.165521	−0.0827606	−	0.996569i	$-0.526374\pi$
$73$	−1.41421	−0.165521	−0.0827606	+	0.996569i	$0.526374\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	5.00000	0.562544	0.281272	−	0.959628i	$-0.409244\pi$
$79$	5.00000	0.562544	0.281272	+	0.959628i	$0.409244\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	3.76393	0.413145	0.206573	−	0.978431i	$-0.433769\pi$
$83$	3.76393	0.413145	0.206573	+	0.978431i	$0.433769\pi$
$84$	0	0
$85$	−3.29180	−0.357045
$86$	0	0
$87$	0	0
$88$	0	0
$89$	6.76393	0.716975	0.358488	−	0.933534i	$-0.383293\pi$
$89$	6.76393	0.716975	0.358488	+	0.933534i	$0.383293\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	9.02546	0.925993
$96$	0	0
$97$	−9.69316	−0.984192	−0.492096	−	0.870541i	$-0.663769\pi$
$97$	−9.69316	−0.984192	−0.492096	+	0.870541i	$0.663769\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	9.70820	0.966002	0.483001	−	0.875620i	$-0.339547\pi$
$101$	9.70820	0.966002	0.483001	+	0.875620i	$0.339547\pi$
$102$	0	0
$103$	−9.89949	−0.975426	−0.487713	−	0.873004i	$-0.662169\pi$
$103$	−9.89949	−0.975426	−0.487713	+	0.873004i	$0.662169\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	2.08191	0.201266	0.100633	−	0.994924i	$-0.467913\pi$
$107$	2.08191	0.201266	0.100633	+	0.994924i	$0.467913\pi$
$108$	0	0
$109$	−2.70820	−0.259399	−0.129699	−	0.991553i	$-0.541401\pi$
$109$	−2.70820	−0.259399	−0.129699	+	0.991553i	$0.541401\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	7.86629	0.739998	0.369999	−	0.929032i	$-0.379358\pi$
$113$	7.86629	0.739998	0.369999	+	0.929032i	$0.379358\pi$
$114$	0	0
$115$	−7.07107	−0.659380
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−10.7082	−0.973473
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−11.1803	−1.00000
$126$	0	0
$127$	1.00000	0.0887357	0.0443678	−	0.999015i	$-0.485873\pi$
$127$	1.00000	0.0887357	0.0443678	+	0.999015i	$0.485873\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	18.0000	1.57267	0.786334	−	0.617802i	$-0.211977\pi$
$131$	18.0000	1.57267	0.786334	+	0.617802i	$0.211977\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	19.5927	1.67391	0.836957	−	0.547269i	$-0.184332\pi$
$137$	19.5927	1.67391	0.836957	+	0.547269i	$0.184332\pi$
$138$	0	0
$139$	12.5216	1.06207	0.531034	−	0.847351i	$-0.321804\pi$
$139$	12.5216	1.06207	0.531034	+	0.847351i	$0.321804\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−1.41641	−0.118446
$144$	0	0
$145$	−1.20788	−0.100309
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−16.4304	−1.34603	−0.673015	−	0.739629i	$-0.735001\pi$
$149$	−16.4304	−1.34603	−0.673015	+	0.739629i	$0.735001\pi$
$150$	0	0
$151$	5.00000	0.406894	0.203447	−	0.979086i	$-0.434786\pi$
$151$	5.00000	0.406894	0.203447	+	0.979086i	$0.434786\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−5.86319	−0.470942
$156$	0	0
$157$	17.7658	1.41786	0.708932	−	0.705277i	$-0.249177\pi$
$157$	17.7658	1.41786	0.708932	+	0.705277i	$0.249177\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−2.70820	−0.212123	−0.106061	−	0.994360i	$-0.533824\pi$
$163$	−2.70820	−0.212123	−0.106061	+	0.994360i	$0.533824\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	17.1803	1.32945	0.664727	−	0.747086i	$-0.268548\pi$
$167$	17.1803	1.32945	0.664727	+	0.747086i	$0.268548\pi$
$168$	0	0
$169$	−6.12461	−0.471124
$170$	0	0
$171$	0	0
$172$	0	0
$173$	19.4164	1.47620	0.738101	−	0.674690i	$-0.235723\pi$
$173$	19.4164	1.47620	0.738101	+	0.674690i	$0.235723\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	19.5927	1.46442	0.732212	−	0.681077i	$-0.238488\pi$
$179$	19.5927	1.46442	0.732212	+	0.681077i	$0.238488\pi$
$180$	0	0
$181$	−5.24419	−0.389798	−0.194899	−	0.980823i	$-0.562438\pi$
$181$	−5.24419	−0.389798	−0.194899	+	0.980823i	$0.562438\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	10.5279	0.774024
$186$	0	0
$187$	−0.795221	−0.0581523
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−8.40647	−0.608271	−0.304135	−	0.952629i	$-0.598368\pi$
$191$	−8.40647	−0.608271	−0.304135	+	0.952629i	$0.598368\pi$
$192$	0	0
$193$	−18.1246	−1.30464	−0.652319	−	0.757944i	$-0.726204\pi$
$193$	−18.1246	−1.30464	−0.652319	+	0.757944i	$0.726204\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−19.5138	−1.39030	−0.695152	−	0.718863i	$-0.744663\pi$
$197$	−19.5138	−1.39030	−0.695152	+	0.718863i	$0.744663\pi$
$198$	0	0
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	5.00000	0.349215
$206$	0	0
$207$	0	0
$208$	0	0
$209$	2.18034	0.150817
$210$	0	0
$211$	−25.4164	−1.74974	−0.874869	−	0.484360i	$-0.839053\pi$
$211$	−25.4164	−1.74974	−0.874869	+	0.484360i	$0.839053\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	19.4721	1.32799
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	3.86008	0.259657
$222$	0	0
$223$	7.86629	0.526766	0.263383	−	0.964691i	$-0.415162\pi$
$223$	7.86629	0.526766	0.263383	+	0.964691i	$0.415162\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	8.29180	0.550346	0.275173	−	0.961395i	$-0.411265\pi$
$227$	8.29180	0.550346	0.275173	+	0.961395i	$0.411265\pi$
$228$	0	0
$229$	−25.4558	−1.68217	−0.841085	−	0.540903i	$-0.818082\pi$
$229$	−25.4558	−1.68217	−0.841085	+	0.540903i	$0.818082\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−19.5927	−1.28356	−0.641779	−	0.766890i	$-0.721803\pi$
$233$	−19.5927	−1.28356	−0.641779	+	0.766890i	$0.721803\pi$
$234$	0	0
$235$	10.1246	0.660457
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−5.86319	−0.379258	−0.189629	−	0.981856i	$-0.560729\pi$
$239$	−5.86319	−0.379258	−0.189629	+	0.981856i	$0.560729\pi$
$240$	0	0
$241$	−2.62210	−0.168904	−0.0844520	−	0.996428i	$-0.526914\pi$
$241$	−2.62210	−0.168904	−0.0844520	+	0.996428i	$0.526914\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−10.5836	−0.673418
$248$	0	0
$249$	0	0
$250$	0	0
$251$	9.76393	0.616294	0.308147	−	0.951339i	$-0.400291\pi$
$251$	9.76393	0.616294	0.308147	+	0.951339i	$0.400291\pi$
$252$	0	0
$253$	−1.70820	−0.107394
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−1.41641	−0.0883531	−0.0441765	−	0.999024i	$-0.514066\pi$
$257$	−1.41641	−0.0883531	−0.0441765	+	0.999024i	$0.514066\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	5.86319	0.361539	0.180770	−	0.983525i	$-0.442141\pi$
$263$	5.86319	0.361539	0.180770	+	0.983525i	$0.442141\pi$
$264$	0	0
$265$	26.0447	1.59992
$266$	0	0
$267$	0	0
$268$	0	0
$269$	1.47214	0.0897577	0.0448789	−	0.998992i	$-0.485710\pi$
$269$	1.47214	0.0897577	0.0448789	+	0.998992i	$0.485710\pi$
$270$	0	0
$271$	−9.07417	−0.551217	−0.275608	−	0.961270i	$-0.588879\pi$
$271$	−9.07417	−0.551217	−0.275608	+	0.961270i	$0.588879\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−1.29180	−0.0776165	−0.0388083	−	0.999247i	$-0.512356\pi$
$277$	−1.29180	−0.0776165	−0.0388083	+	0.999247i	$0.512356\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−7.32611	−0.437039	−0.218519	−	0.975833i	$-0.570123\pi$
$281$	−7.32611	−0.437039	−0.218519	+	0.975833i	$0.570123\pi$
$282$	0	0
$283$	12.9343	0.768862	0.384431	−	0.923154i	$-0.374398\pi$
$283$	12.9343	0.768862	0.384431	+	0.923154i	$0.374398\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−14.8328	−0.872519
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−11.9443	−0.697792	−0.348896	−	0.937161i	$-0.613443\pi$
$293$	−11.9443	−0.697792	−0.348896	+	0.937161i	$0.613443\pi$
$294$	0	0
$295$	26.7082	1.55501
$296$	0	0
$297$	0	0
$298$	0	0
$299$	8.29180	0.479527
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	21.6746	1.24108
$306$	0	0
$307$	−1.41421	−0.0807134	−0.0403567	−	0.999185i	$-0.512849\pi$
$307$	−1.41421	−0.0807134	−0.0403567	+	0.999185i	$0.512849\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	17.1803	0.974208	0.487104	−	0.873344i	$-0.338053\pi$
$311$	17.1803	0.974208	0.487104	+	0.873344i	$0.338053\pi$
$312$	0	0
$313$	25.4558	1.43885	0.719425	−	0.694570i	$-0.244406\pi$
$313$	25.4558	1.43885	0.719425	+	0.694570i	$0.244406\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−19.0525	−1.07009	−0.535047	−	0.844822i	$-0.679706\pi$
$317$	−19.0525	−1.07009	−0.535047	+	0.844822i	$0.679706\pi$
$318$	0	0
$319$	−0.291796	−0.0163374
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−5.94200	−0.330622
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	13.0000	0.714545	0.357272	−	0.934000i	$-0.383707\pi$
$331$	13.0000	0.714545	0.357272	+	0.934000i	$0.383707\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	8.29180	0.453029
$336$	0	0
$337$	−28.7082	−1.56384	−0.781918	−	0.623382i	$-0.785758\pi$
$337$	−28.7082	−1.56384	−0.781918	+	0.623382i	$0.785758\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−1.41641	−0.0767028
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	21.5958	1.15932	0.579661	−	0.814858i	$-0.303185\pi$
$347$	21.5958	1.15932	0.579661	+	0.814858i	$0.303185\pi$
$348$	0	0
$349$	−30.7000	−1.64334	−0.821668	−	0.569967i	$-0.806956\pi$
$349$	−30.7000	−1.64334	−0.821668	+	0.569967i	$0.806956\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	14.2361	0.757709	0.378855	−	0.925456i	$-0.376318\pi$
$353$	14.2361	0.757709	0.378855	+	0.925456i	$0.376318\pi$
$354$	0	0
$355$	−30.7000	−1.62939
$356$	0	0
$357$	0	0
$358$	0	0
$359$	13.6507	0.720454	0.360227	−	0.932865i	$-0.382699\pi$
$359$	13.6507	0.720454	0.360227	+	0.932865i	$0.382699\pi$
$360$	0	0
$361$	−2.70820	−0.142537
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−3.16228	−0.165521
$366$	0	0
$367$	−13.1406	−0.685933	−0.342966	−	0.939348i	$-0.611432\pi$
$367$	−13.1406	−0.685933	−0.342966	+	0.939348i	$0.611432\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−7.29180	−0.377555	−0.188777	−	0.982020i	$-0.560452\pi$
$373$	−7.29180	−0.377555	−0.188777	+	0.982020i	$0.560452\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	1.41641	0.0729487
$378$	0	0
$379$	8.41641	0.432322	0.216161	−	0.976358i	$-0.430646\pi$
$379$	8.41641	0.432322	0.216161	+	0.976358i	$0.430646\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	36.5967	1.87001	0.935003	−	0.354639i	$-0.115396\pi$
$383$	36.5967	1.87001	0.935003	+	0.354639i	$0.115396\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	24.9157	1.26327	0.631637	−	0.775264i	$-0.282383\pi$
$389$	24.9157	1.26327	0.631637	+	0.775264i	$0.282383\pi$
$390$	0	0
$391$	4.65530	0.235429
$392$	0	0
$393$	0	0
$394$	0	0
$395$	11.1803	0.562544
$396$	0	0
$397$	6.65841	0.334176	0.167088	−	0.985942i	$-0.446564\pi$
$397$	6.65841	0.334176	0.167088	+	0.985942i	$0.446564\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−11.7264	−0.585587	−0.292793	−	0.956176i	$-0.594585\pi$
$401$	−11.7264	−0.585587	−0.292793	+	0.956176i	$0.594585\pi$
$402$	0	0
$403$	6.87539	0.342487
$404$	0	0
$405$	0	0
$406$	0	0
$407$	2.54328	0.126066
$408$	0	0
$409$	−5.24419	−0.259309	−0.129654	−	0.991559i	$-0.541387\pi$
$409$	−5.24419	−0.259309	−0.129654	+	0.991559i	$0.541387\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	8.41641	0.413145
$416$	0	0
$417$	0	0
$418$	0	0
$419$	0.819660	0.0400430	0.0200215	−	0.999800i	$-0.493627\pi$
$419$	0.819660	0.0400430	0.0200215	+	0.999800i	$0.493627\pi$
$420$	0	0
$421$	25.4164	1.23872	0.619360	−	0.785107i	$-0.287392\pi$
$421$	25.4164	1.23872	0.619360	+	0.785107i	$0.287392\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−13.7295	−0.661325	−0.330663	−	0.943749i	$-0.607272\pi$
$431$	−13.7295	−0.661325	−0.330663	+	0.943749i	$0.607272\pi$
$432$	0	0
$433$	−2.62210	−0.126010	−0.0630049	−	0.998013i	$-0.520068\pi$
$433$	−2.62210	−0.126010	−0.0630049	+	0.998013i	$0.520068\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−12.7639	−0.610582
$438$	0	0
$439$	41.1884	1.96582	0.982908	−	0.184097i	$-0.0589361\pi$
$439$	41.1884	1.96582	0.982908	+	0.184097i	$0.0589361\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	40.6482	1.93126	0.965628	−	0.259928i	$-0.0836987\pi$
$443$	40.6482	1.93126	0.965628	+	0.259928i	$0.0836987\pi$
$444$	0	0
$445$	15.1246	0.716975
$446$	0	0
$447$	0	0
$448$	0	0
$449$	33.3221	1.57257	0.786284	−	0.617865i	$-0.212002\pi$
$449$	33.3221	1.57257	0.786284	+	0.617865i	$0.212002\pi$
$450$	0	0
$451$	1.20788	0.0568770
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	18.0000	0.842004	0.421002	−	0.907060i	$-0.361678\pi$
$457$	18.0000	0.842004	0.421002	+	0.907060i	$0.361678\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−23.9443	−1.11520	−0.557598	−	0.830111i	$-0.688277\pi$
$461$	−23.9443	−1.11520	−0.557598	+	0.830111i	$0.688277\pi$
$462$	0	0
$463$	−25.2918	−1.17541	−0.587705	−	0.809075i	$-0.699968\pi$
$463$	−25.2918	−1.17541	−0.587705	+	0.809075i	$0.699968\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	11.2361	0.519943	0.259972	−	0.965616i	$-0.416287\pi$
$467$	11.2361	0.519943	0.259972	+	0.965616i	$0.416287\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	4.70401	0.216291
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−5.18034	−0.236696	−0.118348	−	0.992972i	$-0.537760\pi$
$479$	−5.18034	−0.236696	−0.118348	+	0.992972i	$0.537760\pi$
$480$	0	0
$481$	−12.3454	−0.562900
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−21.6746	−0.984192
$486$	0	0
$487$	−14.8328	−0.672139	−0.336070	−	0.941837i	$-0.609098\pi$
$487$	−14.8328	−0.672139	−0.336070	+	0.941837i	$0.609098\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−35.3252	−1.59421	−0.797103	−	0.603844i	$-0.793635\pi$
$491$	−35.3252	−1.59421	−0.797103	+	0.603844i	$0.793635\pi$
$492$	0	0
$493$	0.795221	0.0358149
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−14.4164	−0.645367	−0.322684	−	0.946507i	$-0.604585\pi$
$499$	−14.4164	−0.645367	−0.322684	+	0.946507i	$0.604585\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	18.0557	0.805065	0.402533	−	0.915406i	$-0.368130\pi$
$503$	18.0557	0.805065	0.402533	+	0.915406i	$0.368130\pi$
$504$	0	0
$505$	21.7082	0.966002
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−29.9443	−1.32726	−0.663628	−	0.748063i	$-0.730984\pi$
$509$	−29.9443	−1.32726	−0.663628	+	0.748063i	$0.730984\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−22.1359	−0.975426
$516$	0	0
$517$	2.44587	0.107569
$518$	0	0
$519$	0	0
$520$	0	0
$521$	17.9443	0.786153	0.393076	−	0.919506i	$-0.371411\pi$
$521$	17.9443	0.786153	0.393076	+	0.919506i	$0.371411\pi$
$522$	0	0
$523$	38.5663	1.68639	0.843194	−	0.537610i	$-0.180673\pi$
$523$	38.5663	1.68639	0.843194	+	0.537610i	$0.180673\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	3.86008	0.168148
$528$	0	0
$529$	−13.0000	−0.565217
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−5.86319	−0.253963
$534$	0	0
$535$	4.65530	0.201266
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	41.2492	1.77344	0.886721	−	0.462304i	$-0.152977\pi$
$541$	41.2492	1.77344	0.886721	+	0.462304i	$0.152977\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−6.05573	−0.259399
$546$	0	0
$547$	34.7082	1.48402	0.742008	−	0.670391i	$-0.233874\pi$
$547$	34.7082	1.48402	0.742008	+	0.670391i	$0.233874\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−2.18034	−0.0928856
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−2.00310	−0.0848742	−0.0424371	−	0.999099i	$-0.513512\pi$
$557$	−2.00310	−0.0848742	−0.0424371	+	0.999099i	$0.513512\pi$
$558$	0	0
$559$	−22.8337	−0.965765
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−15.5967	−0.657325	−0.328662	−	0.944448i	$-0.606598\pi$
$563$	−15.5967	−0.657325	−0.328662	+	0.944448i	$0.606598\pi$
$564$	0	0
$565$	17.5896	0.739998
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−46.5114	−1.94986	−0.974930	−	0.222511i	$-0.928575\pi$
$569$	−46.5114	−1.94986	−0.974930	+	0.222511i	$0.928575\pi$
$570$	0	0
$571$	−14.7082	−0.615519	−0.307760	−	0.951464i	$-0.599579\pi$
$571$	−14.7082	−0.615519	−0.307760	+	0.951464i	$0.599579\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−30.7000	−1.27806	−0.639030	−	0.769182i	$-0.720664\pi$
$577$	−30.7000	−1.27806	−0.639030	+	0.769182i	$0.720664\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	6.29180	0.260580
$584$	0	0
$585$	0	0
$586$	0	0
$587$	19.4164	0.801401	0.400700	−	0.916209i	$-0.368767\pi$
$587$	19.4164	0.801401	0.400700	+	0.916209i	$0.368767\pi$
$588$	0	0
$589$	−10.5836	−0.436089
$590$	0	0
$591$	0	0
$592$	0	0
$593$	2.88854	0.118618	0.0593091	−	0.998240i	$-0.481110\pi$
$593$	2.88854	0.118618	0.0593091	+	0.998240i	$0.481110\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	4.70401	0.192201	0.0961003	−	0.995372i	$-0.469363\pi$
$599$	4.70401	0.192201	0.0961003	+	0.995372i	$0.469363\pi$
$600$	0	0
$601$	−22.8337	−0.931408	−0.465704	−	0.884941i	$-0.654199\pi$
$601$	−22.8337	−0.931408	−0.465704	+	0.884941i	$0.654199\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−23.9443	−0.973473
$606$	0	0
$607$	−22.8337	−0.926793	−0.463397	−	0.886151i	$-0.653369\pi$
$607$	−22.8337	−0.926793	−0.463397	+	0.886151i	$0.653369\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−11.8725	−0.480310
$612$	0	0
$613$	32.8328	1.32610	0.663052	−	0.748573i	$-0.269261\pi$
$613$	32.8328	1.32610	0.663052	+	0.748573i	$0.269261\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	14.8886	0.599394	0.299697	−	0.954034i	$-0.403114\pi$
$617$	14.8886	0.599394	0.299697	+	0.954034i	$0.403114\pi$
$618$	0	0
$619$	−20.1815	−0.811165	−0.405582	−	0.914058i	$-0.632931\pi$
$619$	−20.1815	−0.811165	−0.405582	+	0.914058i	$0.632931\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−25.0000	−1.00000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−6.93112	−0.276362
$630$	0	0
$631$	−4.70820	−0.187431	−0.0937153	−	0.995599i	$-0.529874\pi$
$631$	−4.70820	−0.187431	−0.0937153	+	0.995599i	$0.529874\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	2.23607	0.0887357
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	1.46292	0.0577819	0.0288910	−	0.999583i	$-0.490802\pi$
$641$	1.46292	0.0577819	0.0288910	+	0.999583i	$0.490802\pi$
$642$	0	0
$643$	−0.588890	−0.0232235	−0.0116118	−	0.999933i	$-0.503696\pi$
$643$	−0.588890	−0.0232235	−0.0116118	+	0.999933i	$0.503696\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	19.4164	0.763338	0.381669	−	0.924299i	$-0.375349\pi$
$647$	19.4164	0.763338	0.381669	+	0.924299i	$0.375349\pi$
$648$	0	0
$649$	6.45207	0.253266
$650$	0	0
$651$	0	0
$652$	0	0
$653$	35.3252	1.38238	0.691192	−	0.722672i	$-0.257086\pi$
$653$	35.3252	1.38238	0.691192	+	0.722672i	$0.257086\pi$
$654$	0	0
$655$	40.2492	1.57267
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−15.7326	−0.612854	−0.306427	−	0.951894i	$-0.599134\pi$
$659$	−15.7326	−0.612854	−0.306427	+	0.951894i	$0.599134\pi$
$660$	0	0
$661$	28.0779	1.09211	0.546053	−	0.837751i	$-0.316130\pi$
$661$	28.0779	1.09211	0.546053	+	0.837751i	$0.316130\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	1.70820	0.0661419
$668$	0	0
$669$	0	0
$670$	0	0
$671$	5.23607	0.202136
$672$	0	0
$673$	29.1246	1.12267	0.561336	−	0.827588i	$-0.310288\pi$
$673$	29.1246	1.12267	0.561336	+	0.827588i	$0.310288\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−18.0000	−0.691796	−0.345898	−	0.938272i	$-0.612426\pi$
$677$	−18.0000	−0.691796	−0.345898	+	0.938272i	$0.612426\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−47.0516	−1.80038	−0.900190	−	0.435498i	$-0.856572\pi$
$683$	−47.0516	−1.80038	−0.900190	+	0.435498i	$0.856572\pi$
$684$	0	0
$685$	43.8105	1.67391
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−30.5410	−1.16352
$690$	0	0
$691$	22.8337	0.868637	0.434318	−	0.900759i	$-0.356989\pi$
$691$	22.8337	0.868637	0.434318	+	0.900759i	$0.356989\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	27.9991	1.06207
$696$	0	0
$697$	−3.29180	−0.124686
$698$	0	0
$699$	0	0
$700$	0	0
$701$	21.5958	0.815661	0.407830	−	0.913058i	$-0.366285\pi$
$701$	21.5958	0.815661	0.407830	+	0.913058i	$0.366285\pi$
$702$	0	0
$703$	19.0038	0.716741
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	20.4164	0.766754	0.383377	−	0.923592i	$-0.374761\pi$
$709$	20.4164	0.766754	0.383377	+	0.923592i	$0.374761\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	8.29180	0.310530
$714$	0	0
$715$	−3.16718	−0.118446
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−19.4721	−0.726188	−0.363094	−	0.931752i	$-0.618280\pi$
$719$	−19.4721	−0.726188	−0.363094	+	0.931752i	$0.618280\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−20.1815	−0.748492	−0.374246	−	0.927329i	$-0.622098\pi$
$727$	−20.1815	−0.748492	−0.374246	+	0.927329i	$0.622098\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−12.8197	−0.474152
$732$	0	0
$733$	−46.4326	−1.71503	−0.857514	−	0.514461i	$-0.827992\pi$
$733$	−46.4326	−1.71503	−0.857514	+	0.514461i	$0.827992\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	2.00310	0.0737853
$738$	0	0
$739$	−43.4164	−1.59710	−0.798549	−	0.601930i	$-0.794399\pi$
$739$	−43.4164	−1.59710	−0.798549	+	0.601930i	$0.794399\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−16.4304	−0.602772	−0.301386	−	0.953502i	$-0.597449\pi$
$743$	−16.4304	−0.602772	−0.301386	+	0.953502i	$0.597449\pi$
$744$	0	0
$745$	−36.7394	−1.34603
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−33.1246	−1.20873	−0.604367	−	0.796706i	$-0.706574\pi$
$751$	−33.1246	−1.20873	−0.604367	+	0.796706i	$0.706574\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	11.1803	0.406894
$756$	0	0
$757$	9.58359	0.348322	0.174161	−	0.984717i	$-0.444279\pi$
$757$	9.58359	0.348322	0.174161	+	0.984717i	$0.444279\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−29.9443	−1.08548	−0.542740	−	0.839901i	$-0.682613\pi$
$761$	−29.9443	−1.08548	−0.542740	+	0.839901i	$0.682613\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−31.3190	−1.13086
$768$	0	0
$769$	30.7000	1.10707	0.553536	−	0.832825i	$-0.313278\pi$
$769$	30.7000	1.10707	0.553536	+	0.832825i	$0.313278\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−6.81966	−0.245286	−0.122643	−	0.992451i	$-0.539137\pi$
$773$	−6.81966	−0.245286	−0.122643	+	0.992451i	$0.539137\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	9.02546	0.323371
$780$	0	0
$781$	−7.41641	−0.265380
$782$	0	0
$783$	0	0
$784$	0	0
$785$	39.7255	1.41786
$786$	0	0
$787$	26.0447	0.928394	0.464197	−	0.885732i	$-0.346343\pi$
$787$	26.0447	0.928394	0.464197	+	0.885732i	$0.346343\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−25.4164	−0.902563
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−48.5410	−1.71941	−0.859706	−	0.510790i	$-0.829353\pi$
$797$	−48.5410	−1.71941	−0.859706	+	0.510790i	$0.829353\pi$
$798$	0	0
$799$	−6.66563	−0.235813
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−0.763932	−0.0269586
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−25.3770	−0.892209	−0.446104	−	0.894981i	$-0.647189\pi$
$809$	−25.3770	−0.892209	−0.446104	+	0.894981i	$0.647189\pi$
$810$	0	0
$811$	−33.3221	−1.17010	−0.585049	−	0.810998i	$-0.698925\pi$
$811$	−33.3221	−1.17010	−0.585049	+	0.810998i	$0.698925\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−6.05573	−0.212123
$816$	0	0
$817$	35.1490	1.22971
$818$	0	0
$819$	0	0
$820$	0	0
$821$	13.7295	0.479162	0.239581	−	0.970876i	$-0.422990\pi$
$821$	13.7295	0.479162	0.239581	+	0.970876i	$0.422990\pi$
$822$	0	0
$823$	−25.8328	−0.900475	−0.450238	−	0.892909i	$-0.648661\pi$
$823$	−25.8328	−0.900475	−0.450238	+	0.892909i	$0.648661\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	35.3252	1.22838	0.614189	−	0.789159i	$-0.289483\pi$
$827$	35.3252	1.22838	0.614189	+	0.789159i	$0.289483\pi$
$828$	0	0
$829$	16.9405	0.588366	0.294183	−	0.955749i	$-0.404952\pi$
$829$	16.9405	0.588366	0.294183	+	0.955749i	$0.404952\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	38.4164	1.32945
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−24.5967	−0.849174	−0.424587	−	0.905387i	$-0.639581\pi$
$839$	−24.5967	−0.849174	−0.424587	+	0.905387i	$0.639581\pi$
$840$	0	0
$841$	−28.7082	−0.989938
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−13.6950	−0.471124
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−14.8886	−0.510376
$852$	0	0
$853$	−20.2117	−0.692034	−0.346017	−	0.938228i	$-0.612466\pi$
$853$	−20.2117	−0.692034	−0.346017	+	0.938228i	$0.612466\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	6.59675	0.225341	0.112670	−	0.993632i	$-0.464060\pi$
$857$	6.59675	0.225341	0.112670	+	0.993632i	$0.464060\pi$
$858$	0	0
$859$	−30.1111	−1.02738	−0.513690	−	0.857976i	$-0.671722\pi$
$859$	−30.1111	−1.02738	−0.513690	+	0.857976i	$0.671722\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	52.9936	1.80392	0.901962	−	0.431816i	$-0.142127\pi$
$863$	52.9936	1.80392	0.901962	+	0.431816i	$0.142127\pi$
$864$	0	0
$865$	43.4164	1.47620
$866$	0	0
$867$	0	0
$868$	0	0
$869$	2.70091	0.0916220
$870$	0	0
$871$	−9.72327	−0.329460
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−10.7082	−0.361590	−0.180795	−	0.983521i	$-0.557867\pi$
$877$	−10.7082	−0.361590	−0.180795	+	0.983521i	$0.557867\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−20.8328	−0.701875	−0.350938	−	0.936399i	$-0.614137\pi$
$881$	−20.8328	−0.701875	−0.350938	+	0.936399i	$0.614137\pi$
$882$	0	0
$883$	−41.2492	−1.38815	−0.694073	−	0.719904i	$-0.744186\pi$
$883$	−41.2492	−1.38815	−0.694073	+	0.719904i	$0.744186\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−13.3607	−0.448608	−0.224304	−	0.974519i	$-0.572011\pi$
$887$	−13.3607	−0.448608	−0.224304	+	0.974519i	$0.572011\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	18.2759	0.611578
$894$	0	0
$895$	43.8105	1.46442
$896$	0	0
$897$	0	0
$898$	0	0
$899$	1.41641	0.0472398
$900$	0	0
$901$	−17.1468	−0.571242
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−11.7264	−0.389798
$906$	0	0
$907$	5.58359	0.185400	0.0927001	−	0.995694i	$-0.470450\pi$
$907$	5.58359	0.185400	0.0927001	+	0.995694i	$0.470450\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−0.540182	−0.0178970	−0.00894851	−	0.999960i	$-0.502848\pi$
$911$	−0.540182	−0.0178970	−0.00894851	+	0.999960i	$0.502848\pi$
$912$	0	0
$913$	2.03321	0.0672893
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	11.5836	0.382107	0.191054	−	0.981580i	$-0.438810\pi$
$919$	11.5836	0.382107	0.191054	+	0.981580i	$0.438810\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	36.0000	1.18495
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−10.6393	−0.349065	−0.174532	−	0.984651i	$-0.555841\pi$
$929$	−10.6393	−0.349065	−0.174532	+	0.984651i	$0.555841\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−1.77817	−0.0581523
$936$	0	0
$937$	29.4922	0.963467	0.481733	−	0.876318i	$-0.340007\pi$
$937$	29.4922	0.963467	0.481733	+	0.876318i	$0.340007\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−35.1803	−1.14685	−0.573423	−	0.819259i	$-0.694385\pi$
$941$	−35.1803	−1.14685	−0.573423	+	0.819259i	$0.694385\pi$
$942$	0	0
$943$	−7.07107	−0.230266
$944$	0	0
$945$	0	0
$946$	0	0
$947$	25.4558	0.827204	0.413602	−	0.910458i	$-0.364271\pi$
$947$	25.4558	0.827204	0.413602	+	0.910458i	$0.364271\pi$
$948$	0	0
$949$	3.70820	0.120373
$950$	0	0
$951$	0	0
$952$	0	0
$953$	23.3739	0.757156	0.378578	−	0.925569i	$-0.376413\pi$
$953$	23.3739	0.757156	0.378578	+	0.925569i	$0.376413\pi$
$954$	0	0
$955$	−18.7974	−0.608271
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−24.1246	−0.778213
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−40.5279	−1.30464
$966$	0	0
$967$	−43.4164	−1.39618	−0.698089	−	0.716011i	$-0.745966\pi$
$967$	−43.4164	−1.39618	−0.698089	+	0.716011i	$0.745966\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−22.3050	−0.715800	−0.357900	−	0.933760i	$-0.616507\pi$
$971$	−22.3050	−0.715800	−0.357900	+	0.933760i	$0.616507\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−54.9179	−1.75698	−0.878490	−	0.477762i	$-0.841448\pi$
$977$	−54.9179	−1.75698	−0.878490	+	0.477762i	$0.841448\pi$
$978$	0	0
$979$	3.65375	0.116774
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−3.11146	−0.0992400	−0.0496200	−	0.998768i	$-0.515801\pi$
$983$	−3.11146	−0.0992400	−0.0496200	+	0.998768i	$0.515801\pi$
$984$	0	0
$985$	−43.6343	−1.39030
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−27.5378	−0.875650
$990$	0	0
$991$	−30.1246	−0.956940	−0.478470	−	0.878104i	$-0.658808\pi$
$991$	−30.1246	−0.956940	−0.478470	+	0.878104i	$0.658808\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	45.6374	1.44535	0.722675	−	0.691188i	$-0.242912\pi$
$997$	45.6374	1.44535	0.722675	+	0.691188i	$0.242912\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5292.2.a.bb.1.4	yes	4
3.2	odd	2		5292.2.a.y.1.1	✓	4
7.6	odd	2		5292.2.a.y.1.2	yes	4
21.20	even	2	inner	5292.2.a.bb.1.3	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5292.2.a.y.1.1	✓	4	3.2	odd	2
5292.2.a.y.1.2	yes	4	7.6	odd	2
5292.2.a.bb.1.3	yes	4	21.20	even	2	inner
5292.2.a.bb.1.4	yes	4	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:	\( N \)	\(=\)	\( 5292 = 2^{2} \cdot 3^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5292.a (trivial)

\(f(q)\)	\(=\)	\(q+2.23607 q^{5} +O(q^{10})\)	\(q+2.23607 q^{5} +0.540182 q^{11} -2.62210 q^{13} -1.47214 q^{17} +4.03631 q^{19} -3.16228 q^{23} -0.540182 q^{29} -2.62210 q^{31} +4.70820 q^{37} +2.23607 q^{41} +8.70820 q^{43} +4.52786 q^{47} +11.6476 q^{53} +1.20788 q^{55} +11.9443 q^{59} +9.69316 q^{61} -5.86319 q^{65} +3.70820 q^{67} -13.7295 q^{71} -1.41421 q^{73} +5.00000 q^{79} +3.76393 q^{83} -3.29180 q^{85} +6.76393 q^{89} +9.02546 q^{95} -9.69316 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q + 12 q^{17} - 8 q^{37} + 8 q^{43} + 36 q^{47} + 12 q^{59} - 12 q^{67} + 20 q^{79} + 24 q^{83} - 40 q^{85} + 36 q^{89}+O(q^{100}) \) 4 * q + 12 * q^17 - 8 * q^37 + 8 * q^43 + 36 * q^47 + 12 * q^59 - 12 * q^67 + 20 * q^79 + 24 * q^83 - 40 * q^85 + 36 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more