LMFDB - Embedded newform 6525.2.a.bt.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6525,2,Mod(1,6525)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("6525.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6525, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 6525 = 3^{2} \cdot 5^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6525.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,0,14,0,0,0,0,0,0,10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$52.1023873189$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.337383424.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 13x^{4} + 41x^{2} - 1 $ x^6 - 13x^4 + 41x^2 - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 145)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$2.30229$ of defining polynomial
Character	$\chi$	$=$	6525.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.30229 q^{2} +3.30056 q^{4} +3.91261 q^{7} +2.99427 q^{8} -2.65427 q^{11} +5.62692 q^{13} +9.00799 q^{14} +0.292570 q^{16} +1.86794 q^{17} +1.69944 q^{19} -6.11092 q^{22} -0.691975 q^{23} +12.9548 q^{26} +12.9138 q^{28} +1.00000 q^{29} -0.654273 q^{31} -5.31495 q^{32} +4.30056 q^{34} -3.91261 q^{37} +3.91261 q^{38} +10.7155 q^{43} -8.76059 q^{44} -1.59313 q^{46} +4.93495 q^{47} +8.30855 q^{49} +18.5720 q^{52} +7.67159 q^{53} +11.7154 q^{56} +2.30229 q^{58} -10.0000 q^{59} -4.70743 q^{61} -1.50633 q^{62} -12.8217 q^{64} +6.47253 q^{67} +6.16526 q^{68} -2.00000 q^{71} +10.5619 q^{73} -9.00799 q^{74} +5.60911 q^{76} -10.3851 q^{77} -2.05316 q^{79} +1.86794 q^{83} +24.6703 q^{86} -7.94761 q^{88} -3.30855 q^{89} +22.0160 q^{91} -2.28390 q^{92} +11.3617 q^{94} -0.384703 q^{97} +19.1287 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 14 q^{4} + 10 q^{11} + 8 q^{14} + 42 q^{16} + 16 q^{19} + 46 q^{26} + 6 q^{29} + 22 q^{31} + 20 q^{34} + 2 q^{44} - 44 q^{46} - 2 q^{49} - 16 q^{56} - 60 q^{59} + 12 q^{61} + 38 q^{64} - 12 q^{71}+ \cdots + 2 q^{94}+O(q^{100}) $ 6 * q + 14 * q^4 + 10 * q^11 + 8 * q^14 + 42 * q^16 + 16 * q^19 + 46 * q^26 + 6 * q^29 + 22 * q^31 + 20 * q^34 + 2 * q^44 - 44 * q^46 - 2 * q^49 - 16 * q^56 - 60 * q^59 + 12 * q^61 + 38 * q^64 - 12 * q^71 - 8 * q^74 - 24 * q^76 + 2 * q^79 + 30 * q^86 + 32 * q^89 + 40 * q^91 + 2 * q^94

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 $

$2$	2.30229	1.62797	0.813984	−	0.580887i	$-0.197294\pi$
$2$	2.30229	1.62797	0.813984	+	0.580887i	$0.197294\pi$
$3$	0	0
$3$	0	0
$4$	3.30056	1.65028
$5$	0	0
$5$	0	0
$6$	0	0
$7$	3.91261	1.47883	0.739415	−	0.673250i	$-0.235102\pi$
$7$	3.91261	1.47883	0.739415	+	0.673250i	$0.235102\pi$
$8$	2.99427	1.05863
$9$	0	0
$10$	0	0
$11$	−2.65427	−0.800294	−0.400147	−	0.916451i	$-0.631041\pi$
$11$	−2.65427	−0.800294	−0.400147	+	0.916451i	$0.631041\pi$
$12$	0	0
$13$	5.62692	1.56063	0.780314	−	0.625388i	$-0.215059\pi$
$13$	5.62692	1.56063	0.780314	+	0.625388i	$0.215059\pi$
$14$	9.00799	2.40749
$15$	0	0
$16$	0.292570	0.0731426
$17$	1.86794	0.453043	0.226522	−	0.974006i	$-0.427265\pi$
$17$	1.86794	0.453043	0.226522	+	0.974006i	$0.427265\pi$
$18$	0	0
$19$	1.69944	0.389879	0.194939	−	0.980815i	$-0.437549\pi$
$19$	1.69944	0.389879	0.194939	+	0.980815i	$0.437549\pi$
$20$	0	0
$21$	0	0
$22$	−6.11092	−1.30285
$23$	−0.691975	−0.144287	−0.0721433	−	0.997394i	$-0.522984\pi$
$23$	−0.691975	−0.144287	−0.0721433	+	0.997394i	$0.522984\pi$
$24$	0	0
$25$	0	0
$26$	12.9548	2.54065
$27$	0	0
$28$	12.9138	2.44048
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	−0.654273	−0.117511	−0.0587555	−	0.998272i	$-0.518713\pi$
$31$	−0.654273	−0.117511	−0.0587555	+	0.998272i	$0.518713\pi$
$32$	−5.31495	−0.939560
$33$	0	0
$34$	4.30056	0.737540
$35$	0	0
$36$	0	0
$37$	−3.91261	−0.643230	−0.321615	−	0.946871i	$-0.604226\pi$
$37$	−3.91261	−0.643230	−0.321615	+	0.946871i	$0.604226\pi$
$38$	3.91261	0.634710
$39$	0	0
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	10.7155	1.63410	0.817050	−	0.576567i	$-0.195608\pi$
$43$	10.7155	1.63410	0.817050	+	0.576567i	$0.195608\pi$
$44$	−8.76059	−1.32071
$45$	0	0
$46$	−1.59313	−0.234894
$47$	4.93495	0.719836	0.359918	−	0.932984i	$-0.382805\pi$
$47$	4.93495	0.719836	0.359918	+	0.932984i	$0.382805\pi$
$48$	0	0
$49$	8.30855	1.18694
$50$	0	0
$51$	0	0
$52$	18.5720	2.57547
$53$	7.67159	1.05377	0.526887	−	0.849935i	$-0.323359\pi$
$53$	7.67159	1.05377	0.526887	+	0.849935i	$0.323359\pi$
$54$	0	0
$55$	0	0
$56$	11.7154	1.56554
$57$	0	0
$58$	2.30229	0.302306
$59$	−10.0000	−1.30189	−0.650945	−	0.759125i	$-0.725627\pi$
$59$	−10.0000	−1.30189	−0.650945	+	0.759125i	$0.725627\pi$
$60$	0	0
$61$	−4.70743	−0.602725	−0.301362	−	0.953510i	$-0.597441\pi$
$61$	−4.70743	−0.602725	−0.301362	+	0.953510i	$0.597441\pi$
$62$	−1.50633	−0.191304
$63$	0	0
$64$	−12.8217	−1.60272
$65$	0	0
$66$	0	0
$67$	6.47253	0.790746	0.395373	−	0.918521i	$-0.370615\pi$
$67$	6.47253	0.790746	0.395373	+	0.918521i	$0.370615\pi$
$68$	6.16526	0.747648
$69$	0	0
$70$	0	0
$71$	−2.00000	−0.237356	−0.118678	−	0.992933i	$-0.537866\pi$
$71$	−2.00000	−0.237356	−0.118678	+	0.992933i	$0.537866\pi$
$72$	0	0
$73$	10.5619	1.23617	0.618087	−	0.786110i	$-0.287908\pi$
$73$	10.5619	1.23617	0.618087	+	0.786110i	$0.287908\pi$
$74$	−9.00799	−1.04716
$75$	0	0
$76$	5.60911	0.643409
$77$	−10.3851	−1.18350
$78$	0	0
$79$	−2.05316	−0.230998	−0.115499	−	0.993308i	$-0.536847\pi$
$79$	−2.05316	−0.230998	−0.115499	+	0.993308i	$0.536847\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	1.86794	0.205034	0.102517	−	0.994731i	$-0.467310\pi$
$83$	1.86794	0.205034	0.102517	+	0.994731i	$0.467310\pi$
$84$	0	0
$85$	0	0
$86$	24.6703	2.66026
$87$	0	0
$88$	−7.94761	−0.847218
$89$	−3.30855	−0.350705	−0.175353	−	0.984506i	$-0.556107\pi$
$89$	−3.30855	−0.350705	−0.175353	+	0.984506i	$0.556107\pi$
$90$	0	0
$91$	22.0160	2.30790
$92$	−2.28390	−0.238113
$93$	0	0
$94$	11.3617	1.17187
$95$	0	0
$96$	0	0
$97$	−0.384703	−0.0390607	−0.0195303	−	0.999809i	$-0.506217\pi$
$97$	−0.384703	−0.0390607	−0.0195303	+	0.999809i	$0.506217\pi$
$98$	19.1287	1.93229
$99$	0	0
$100$	0	0
$101$	−11.9097	−1.18506	−0.592528	−	0.805550i	$-0.701870\pi$
$101$	−11.9097	−1.18506	−0.592528	+	0.805550i	$0.701870\pi$
$102$	0	0
$103$	−14.9585	−1.47390	−0.736951	−	0.675946i	$-0.763735\pi$
$103$	−14.9585	−1.47390	−0.736951	+	0.675946i	$0.763735\pi$
$104$	16.8485	1.65213
$105$	0	0
$106$	17.6623	1.71551
$107$	3.91261	0.378247	0.189123	−	0.981953i	$-0.439435\pi$
$107$	3.91261	0.378247	0.189123	+	0.981953i	$0.439435\pi$
$108$	0	0
$109$	−7.55595	−0.723729	−0.361864	−	0.932231i	$-0.617860\pi$
$109$	−7.55595	−0.723729	−0.361864	+	0.932231i	$0.617860\pi$
$110$	0	0
$111$	0	0
$112$	1.14472	0.108165
$113$	18.6944	1.75862	0.879309	−	0.476251i	$-0.158005\pi$
$113$	18.6944	1.75862	0.879309	+	0.476251i	$0.158005\pi$
$114$	0	0
$115$	0	0
$116$	3.30056	0.306449
$117$	0	0
$118$	−23.0229	−2.11943
$119$	7.30855	0.669973
$120$	0	0
$121$	−3.95483	−0.359530
$122$	−10.8379	−0.981216
$123$	0	0
$124$	−2.15947	−0.193926
$125$	0	0
$126$	0	0
$127$	4.98929	0.442728	0.221364	−	0.975191i	$-0.428949\pi$
$127$	4.98929	0.442728	0.221364	+	0.975191i	$0.428949\pi$
$128$	−18.8895	−1.66961
$129$	0	0
$130$	0	0
$131$	−19.7154	−1.72254	−0.861272	−	0.508144i	$-0.830332\pi$
$131$	−19.7154	−1.72254	−0.861272	+	0.508144i	$0.830332\pi$
$132$	0	0
$133$	6.64926	0.576564
$134$	14.9017	1.28731
$135$	0	0
$136$	5.59313	0.479607
$137$	6.26455	0.535217	0.267608	−	0.963528i	$-0.413767\pi$
$137$	6.26455	0.535217	0.267608	+	0.963528i	$0.413767\pi$
$138$	0	0
$139$	21.9097	1.85835	0.929177	−	0.369636i	$-0.120518\pi$
$139$	21.9097	1.85835	0.929177	+	0.369636i	$0.120518\pi$
$140$	0	0
$141$	0	0
$142$	−4.60459	−0.386408
$143$	−14.9354	−1.24896
$144$	0	0
$145$	0	0
$146$	24.3165	2.01245
$147$	0	0
$148$	−12.9138	−1.06151
$149$	−4.24740	−0.347961	−0.173980	−	0.984749i	$-0.555663\pi$
$149$	−4.24740	−0.347961	−0.173980	+	0.984749i	$0.555663\pi$
$150$	0	0
$151$	−11.3085	−0.920277	−0.460138	−	0.887847i	$-0.652200\pi$
$151$	−11.3085	−0.920277	−0.460138	+	0.887847i	$0.652200\pi$
$152$	5.08858	0.412739
$153$	0	0
$154$	−23.9097	−1.92670
$155$	0	0
$156$	0	0
$157$	−4.12059	−0.328859	−0.164430	−	0.986389i	$-0.552578\pi$
$157$	−4.12059	−0.328859	−0.164430	+	0.986389i	$0.552578\pi$
$158$	−4.72697	−0.376057
$159$	0	0
$160$	0	0
$161$	−2.70743	−0.213375
$162$	0	0
$163$	3.19755	0.250452	0.125226	−	0.992128i	$-0.460034\pi$
$163$	3.19755	0.250452	0.125226	+	0.992128i	$0.460034\pi$
$164$	0	0
$165$	0	0
$166$	4.30056	0.333788
$167$	−14.6050	−1.13017	−0.565086	−	0.825032i	$-0.691157\pi$
$167$	−14.6050	−1.13017	−0.565086	+	0.825032i	$0.691157\pi$
$168$	0	0
$169$	18.6623	1.43556
$170$	0	0
$171$	0	0
$172$	35.3672	2.69672
$173$	9.00120	0.684348	0.342174	−	0.939637i	$-0.388837\pi$
$173$	9.00120	0.684348	0.342174	+	0.939637i	$0.388837\pi$
$174$	0	0
$175$	0	0
$176$	−0.776562	−0.0585356
$177$	0	0
$178$	−7.61725	−0.570937
$179$	21.3085	1.59268	0.796338	−	0.604852i	$-0.206768\pi$
$179$	21.3085	1.59268	0.796338	+	0.604852i	$0.206768\pi$
$180$	0	0
$181$	−16.9708	−1.26143	−0.630715	−	0.776014i	$-0.717238\pi$
$181$	−16.9708	−1.26143	−0.630715	+	0.776014i	$0.717238\pi$
$182$	50.6873	3.75719
$183$	0	0
$184$	−2.07196	−0.152747
$185$	0	0
$186$	0	0
$187$	−4.95804	−0.362568
$188$	16.2881	1.18793
$189$	0	0
$190$	0	0
$191$	8.40687	0.608300	0.304150	−	0.952624i	$-0.401628\pi$
$191$	8.40687	0.608300	0.304150	+	0.952624i	$0.401628\pi$
$192$	0	0
$193$	0.791267	0.0569567	0.0284783	−	0.999594i	$-0.490934\pi$
$193$	0.791267	0.0569567	0.0284783	+	0.999594i	$0.490934\pi$
$194$	−0.885700	−0.0635895
$195$	0	0
$196$	27.4228	1.95877
$197$	−24.5062	−1.74599	−0.872997	−	0.487726i	$-0.837826\pi$
$197$	−24.5062	−1.74599	−0.872997	+	0.487726i	$0.837826\pi$
$198$	0	0
$199$	19.3085	1.36875	0.684373	−	0.729132i	$-0.260076\pi$
$199$	19.3085	1.36875	0.684373	+	0.729132i	$0.260076\pi$
$200$	0	0
$201$	0	0
$202$	−27.4196	−1.92923
$203$	3.91261	0.274612
$204$	0	0
$205$	0	0
$206$	−34.4388	−2.39947
$207$	0	0
$208$	1.64627	0.114148
$209$	−4.51078	−0.312017
$210$	0	0
$211$	19.2554	1.32560	0.662798	−	0.748798i	$-0.269369\pi$
$211$	19.2554	1.32560	0.662798	+	0.748798i	$0.269369\pi$
$212$	25.3205	1.73902
$213$	0	0
$214$	9.00799	0.615773
$215$	0	0
$216$	0	0
$217$	−2.55992	−0.173779
$218$	−17.3960	−1.17821
$219$	0	0
$220$	0	0
$221$	10.5108	0.707032
$222$	0	0
$223$	10.8691	0.727852	0.363926	−	0.931428i	$-0.381436\pi$
$223$	10.8691	0.727852	0.363926	+	0.931428i	$0.381436\pi$
$224$	−20.7954	−1.38945
$225$	0	0
$226$	43.0399	2.86297
$227$	−17.3104	−1.14893	−0.574467	−	0.818528i	$-0.694790\pi$
$227$	−17.3104	−1.14893	−0.574467	+	0.818528i	$0.694790\pi$
$228$	0	0
$229$	18.7074	1.23622	0.618111	−	0.786091i	$-0.287898\pi$
$229$	18.7074	1.23622	0.618111	+	0.786091i	$0.287898\pi$
$230$	0	0
$231$	0	0
$232$	2.99427	0.196583
$233$	−14.6281	−0.958320	−0.479160	−	0.877728i	$-0.659059\pi$
$233$	−14.6281	−0.958320	−0.479160	+	0.877728i	$0.659059\pi$
$234$	0	0
$235$	0	0
$236$	−33.0056	−2.14848
$237$	0	0
$238$	16.8264	1.09070
$239$	2.00000	0.129369	0.0646846	−	0.997906i	$-0.479396\pi$
$239$	2.00000	0.129369	0.0646846	+	0.997906i	$0.479396\pi$
$240$	0	0
$241$	−5.55595	−0.357890	−0.178945	−	0.983859i	$-0.557268\pi$
$241$	−5.55595	−0.357890	−0.178945	+	0.983859i	$0.557268\pi$
$242$	−9.10519	−0.585304
$243$	0	0
$244$	−15.5371	−0.994664
$245$	0	0
$246$	0	0
$247$	9.56263	0.608455
$248$	−1.95907	−0.124401
$249$	0	0
$250$	0	0
$251$	9.27137	0.585204	0.292602	−	0.956234i	$-0.405479\pi$
$251$	9.27137	0.585204	0.292602	+	0.956234i	$0.405479\pi$
$252$	0	0
$253$	1.83669	0.115472
$254$	11.4868	0.720747
$255$	0	0
$256$	−17.8457	−1.11536
$257$	14.1129	0.880337	0.440168	−	0.897915i	$-0.354919\pi$
$257$	14.1129	0.880337	0.440168	+	0.897915i	$0.354919\pi$
$258$	0	0
$259$	−15.3085	−0.951227
$260$	0	0
$261$	0	0
$262$	−45.3907	−2.80425
$263$	−6.97962	−0.430382	−0.215191	−	0.976572i	$-0.569037\pi$
$263$	−6.97962	−0.430382	−0.215191	+	0.976572i	$0.569037\pi$
$264$	0	0
$265$	0	0
$266$	15.3085	0.938627
$267$	0	0
$268$	21.3630	1.30495
$269$	7.29257	0.444636	0.222318	−	0.974974i	$-0.428638\pi$
$269$	7.29257	0.444636	0.222318	+	0.974974i	$0.428638\pi$
$270$	0	0
$271$	−23.3617	−1.41912	−0.709561	−	0.704644i	$-0.751107\pi$
$271$	−23.3617	−1.41912	−0.709561	+	0.704644i	$0.751107\pi$
$272$	0.546506	0.0331368
$273$	0	0
$274$	14.4228	0.871316
$275$	0	0
$276$	0	0
$277$	2.91337	0.175047	0.0875236	−	0.996162i	$-0.472105\pi$
$277$	2.91337	0.175047	0.0875236	+	0.996162i	$0.472105\pi$
$278$	50.4425	3.02534
$279$	0	0
$280$	0	0
$281$	−30.1730	−1.79997	−0.899986	−	0.435918i	$-0.856424\pi$
$281$	−30.1730	−1.79997	−0.899986	+	0.435918i	$0.856424\pi$
$282$	0	0
$283$	2.01341	0.119685	0.0598425	−	0.998208i	$-0.480940\pi$
$283$	2.01341	0.119685	0.0598425	+	0.998208i	$0.480940\pi$
$284$	−6.60112	−0.391704
$285$	0	0
$286$	−34.3857	−2.03327
$287$	0	0
$288$	0	0
$289$	−13.5108	−0.794752
$290$	0	0
$291$	0	0
$292$	34.8601	2.04003
$293$	−10.8691	−0.634982	−0.317491	−	0.948261i	$-0.602840\pi$
$293$	−10.8691	−0.634982	−0.317491	+	0.948261i	$0.602840\pi$
$294$	0	0
$295$	0	0
$296$	−11.7154	−0.680945
$297$	0	0
$298$	−9.77877	−0.566469
$299$	−3.89369	−0.225178
$300$	0	0
$301$	41.9256	2.41655
$302$	−26.0356	−1.49818
$303$	0	0
$304$	0.497206	0.0285167
$305$	0	0
$306$	0	0
$307$	9.02429	0.515043	0.257522	−	0.966273i	$-0.417094\pi$
$307$	9.02429	0.515043	0.257522	+	0.966273i	$0.417094\pi$
$308$	−34.2768	−1.95310
$309$	0	0
$310$	0	0
$311$	11.1143	0.630234	0.315117	−	0.949053i	$-0.397956\pi$
$311$	11.1143	0.630234	0.315117	+	0.949053i	$0.397956\pi$
$312$	0	0
$313$	25.7365	1.45471	0.727356	−	0.686260i	$-0.240749\pi$
$313$	25.7365	1.45471	0.727356	+	0.686260i	$0.240749\pi$
$314$	−9.48682	−0.535372
$315$	0	0
$316$	−6.77656	−0.381211
$317$	0.837444	0.0470355	0.0235178	−	0.999723i	$-0.492513\pi$
$317$	0.837444	0.0470355	0.0235178	+	0.999723i	$0.492513\pi$
$318$	0	0
$319$	−2.65427	−0.148611
$320$	0	0
$321$	0	0
$322$	−6.23330	−0.347368
$323$	3.17446	0.176632
$324$	0	0
$325$	0	0
$326$	7.36170	0.407727
$327$	0	0
$328$	0	0
$329$	19.3085	1.06451
$330$	0	0
$331$	22.0691	1.21303	0.606515	−	0.795072i	$-0.292567\pi$
$331$	22.0691	1.21303	0.606515	+	0.795072i	$0.292567\pi$
$332$	6.16526	0.338363
$333$	0	0
$334$	−33.6251	−1.83988
$335$	0	0
$336$	0	0
$337$	−7.74780	−0.422049	−0.211025	−	0.977481i	$-0.567680\pi$
$337$	−7.74780	−0.422049	−0.211025	+	0.977481i	$0.567680\pi$
$338$	42.9660	2.33704
$339$	0	0
$340$	0	0
$341$	1.73662	0.0940433
$342$	0	0
$343$	5.11984	0.276445
$344$	32.0851	1.72991
$345$	0	0
$346$	20.7234	1.11410
$347$	9.33972	0.501383	0.250691	−	0.968067i	$-0.419342\pi$
$347$	9.33972	0.501383	0.250691	+	0.968067i	$0.419342\pi$
$348$	0	0
$349$	−21.6623	−1.15955	−0.579777	−	0.814775i	$-0.696860\pi$
$349$	−21.6623	−1.15955	−0.579777	+	0.814775i	$0.696860\pi$
$350$	0	0
$351$	0	0
$352$	14.1073	0.751924
$353$	−20.5623	−1.09442	−0.547211	−	0.836995i	$-0.684310\pi$
$353$	−20.5623	−1.09442	−0.547211	+	0.836995i	$0.684310\pi$
$354$	0	0
$355$	0	0
$356$	−10.9201	−0.578762
$357$	0	0
$358$	49.0585	2.59282
$359$	−30.5639	−1.61310	−0.806551	−	0.591164i	$-0.798669\pi$
$359$	−30.5639	−1.61310	−0.806551	+	0.591164i	$0.798669\pi$
$360$	0	0
$361$	−16.1119	−0.847995
$362$	−39.0718	−2.05357
$363$	0	0
$364$	72.6650	3.80868
$365$	0	0
$366$	0	0
$367$	−13.7825	−0.719441	−0.359721	−	0.933060i	$-0.617128\pi$
$367$	−13.7825	−0.719441	−0.359721	+	0.933060i	$0.617128\pi$
$368$	−0.202451	−0.0105535
$369$	0	0
$370$	0	0
$371$	30.0160	1.55835
$372$	0	0
$373$	−27.2659	−1.41178	−0.705888	−	0.708324i	$-0.749452\pi$
$373$	−27.2659	−1.41178	−0.705888	+	0.708324i	$0.749452\pi$
$374$	−11.4149	−0.590248
$375$	0	0
$376$	14.7766	0.762043
$377$	5.62692	0.289801
$378$	0	0
$379$	16.4069	0.842764	0.421382	−	0.906883i	$-0.361545\pi$
$379$	16.4069	0.842764	0.421382	+	0.906883i	$0.361545\pi$
$380$	0	0
$381$	0	0
$382$	19.3551	0.990293
$383$	11.7378	0.599776	0.299888	−	0.953974i	$-0.403051\pi$
$383$	11.7378	0.599776	0.299888	+	0.953974i	$0.403051\pi$
$384$	0	0
$385$	0	0
$386$	1.82173	0.0927236
$387$	0	0
$388$	−1.26974	−0.0644610
$389$	12.0000	0.608424	0.304212	−	0.952604i	$-0.401607\pi$
$389$	12.0000	0.608424	0.304212	+	0.952604i	$0.401607\pi$
$390$	0	0
$391$	−1.29257	−0.0653681
$392$	24.8780	1.25653
$393$	0	0
$394$	−56.4204	−2.84242
$395$	0	0
$396$	0	0
$397$	6.03349	0.302812	0.151406	−	0.988472i	$-0.451620\pi$
$397$	6.03349	0.302812	0.151406	+	0.988472i	$0.451620\pi$
$398$	44.4540	2.22828
$399$	0	0
$400$	0	0
$401$	21.4656	1.07194	0.535971	−	0.844237i	$-0.319946\pi$
$401$	21.4656	1.07194	0.535971	+	0.844237i	$0.319946\pi$
$402$	0	0
$403$	−3.68155	−0.183391
$404$	−39.3085	−1.95567
$405$	0	0
$406$	9.00799	0.447059
$407$	10.3851	0.514773
$408$	0	0
$409$	−4.49481	−0.222254	−0.111127	−	0.993806i	$-0.535446\pi$
$409$	−4.49481	−0.222254	−0.111127	+	0.993806i	$0.535446\pi$
$410$	0	0
$411$	0	0
$412$	−49.3713	−2.43235
$413$	−39.1261	−1.92527
$414$	0	0
$415$	0	0
$416$	−29.9068	−1.46630
$417$	0	0
$418$	−10.3851	−0.507954
$419$	−4.19665	−0.205020	−0.102510	−	0.994732i	$-0.532687\pi$
$419$	−4.19665	−0.205020	−0.102510	+	0.994732i	$0.532687\pi$
$420$	0	0
$421$	−18.5108	−0.902160	−0.451080	−	0.892483i	$-0.648961\pi$
$421$	−18.5108	−0.902160	−0.451080	+	0.892483i	$0.648961\pi$
$422$	44.3316	2.15803
$423$	0	0
$424$	22.9708	1.11556
$425$	0	0
$426$	0	0
$427$	−18.4184	−0.891327
$428$	12.9138	0.624213
$429$	0	0
$430$	0	0
$431$	−37.1279	−1.78839	−0.894193	−	0.447681i	$-0.852250\pi$
$431$	−37.1279	−1.78839	−0.894193	+	0.447681i	$0.852250\pi$
$432$	0	0
$433$	−16.8577	−0.810128	−0.405064	−	0.914288i	$-0.632751\pi$
$433$	−16.8577	−0.810128	−0.405064	+	0.914288i	$0.632751\pi$
$434$	−5.89369	−0.282906
$435$	0	0
$436$	−24.9389	−1.19435
$437$	−1.17597	−0.0562543
$438$	0	0
$439$	−9.21821	−0.439961	−0.219981	−	0.975504i	$-0.570599\pi$
$439$	−9.21821	−0.439961	−0.219981	+	0.975504i	$0.570599\pi$
$440$	0	0
$441$	0	0
$442$	24.1989	1.15102
$443$	−18.9023	−0.898078	−0.449039	−	0.893512i	$-0.648234\pi$
$443$	−18.9023	−0.898078	−0.449039	+	0.893512i	$0.648234\pi$
$444$	0	0
$445$	0	0
$446$	25.0240	1.18492
$447$	0	0
$448$	−50.1665	−2.37014
$449$	31.3245	1.47830	0.739148	−	0.673543i	$-0.235228\pi$
$449$	31.3245	1.47830	0.739148	+	0.673543i	$0.235228\pi$
$450$	0	0
$451$	0	0
$452$	61.7019	2.90221
$453$	0	0
$454$	−39.8537	−1.87043
$455$	0	0
$456$	0	0
$457$	−36.6287	−1.71342	−0.856710	−	0.515799i	$-0.827495\pi$
$457$	−36.6287	−1.71342	−0.856710	+	0.515799i	$0.827495\pi$
$458$	43.0700	2.01253
$459$	0	0
$460$	0	0
$461$	−28.8297	−1.34273	−0.671367	−	0.741125i	$-0.734293\pi$
$461$	−28.8297	−1.34273	−0.671367	+	0.741125i	$0.734293\pi$
$462$	0	0
$463$	−29.6409	−1.37753	−0.688766	−	0.724984i	$-0.741847\pi$
$463$	−29.6409	−1.37753	−0.688766	+	0.724984i	$0.741847\pi$
$464$	0.292570	0.0135822
$465$	0	0
$466$	−33.6782	−1.56011
$467$	−26.4746	−1.22510	−0.612550	−	0.790432i	$-0.709856\pi$
$467$	−26.4746	−1.22510	−0.612550	+	0.790432i	$0.709856\pi$
$468$	0	0
$469$	25.3245	1.16938
$470$	0	0
$471$	0	0
$472$	−29.9427	−1.37822
$473$	−28.4419	−1.30776
$474$	0	0
$475$	0	0
$476$	24.1223	1.10564
$477$	0	0
$478$	4.60459	0.210609
$479$	−27.1810	−1.24193	−0.620967	−	0.783837i	$-0.713260\pi$
$479$	−27.1810	−1.24193	−0.620967	+	0.783837i	$0.713260\pi$
$480$	0	0
$481$	−22.0160	−1.00384
$482$	−12.7914	−0.582634
$483$	0	0
$484$	−13.0532	−0.593325
$485$	0	0
$486$	0	0
$487$	32.0922	1.45424	0.727118	−	0.686513i	$-0.240859\pi$
$487$	32.0922	1.45424	0.727118	+	0.686513i	$0.240859\pi$
$488$	−14.0953	−0.638065
$489$	0	0
$490$	0	0
$491$	10.0691	0.454414	0.227207	−	0.973847i	$-0.427041\pi$
$491$	10.0691	0.454414	0.227207	+	0.973847i	$0.427041\pi$
$492$	0	0
$493$	1.86794	0.0841280
$494$	22.0160	0.990546
$495$	0	0
$496$	−0.191421	−0.00859506
$497$	−7.82523	−0.351009
$498$	0	0
$499$	−29.9416	−1.34037	−0.670185	−	0.742194i	$-0.733785\pi$
$499$	−29.9416	−1.34037	−0.670185	+	0.742194i	$0.733785\pi$
$500$	0	0
$501$	0	0
$502$	21.3454	0.952693
$503$	24.0140	1.07073	0.535366	−	0.844620i	$-0.320174\pi$
$503$	24.0140	1.07073	0.535366	+	0.844620i	$0.320174\pi$
$504$	0	0
$505$	0	0
$506$	4.22860	0.187984
$507$	0	0
$508$	16.4674	0.730625
$509$	21.0771	0.934227	0.467113	−	0.884197i	$-0.345294\pi$
$509$	21.0771	0.934227	0.467113	+	0.884197i	$0.345294\pi$
$510$	0	0
$511$	41.3245	1.82809
$512$	−3.30707	−0.146153
$513$	0	0
$514$	32.4920	1.43316
$515$	0	0
$516$	0	0
$517$	−13.0987	−0.576080
$518$	−35.2448	−1.54857
$519$	0	0
$520$	0	0
$521$	−11.6623	−0.510933	−0.255466	−	0.966818i	$-0.582229\pi$
$521$	−11.6623	−0.510933	−0.255466	+	0.966818i	$0.582229\pi$
$522$	0	0
$523$	−36.3895	−1.59120	−0.795601	−	0.605821i	$-0.792845\pi$
$523$	−36.3895	−1.59120	−0.795601	+	0.605821i	$0.792845\pi$
$524$	−65.0719	−2.84268
$525$	0	0
$526$	−16.0691	−0.700647
$527$	−1.22215	−0.0532376
$528$	0	0
$529$	−22.5212	−0.979181
$530$	0	0
$531$	0	0
$532$	21.9463	0.951491
$533$	0	0
$534$	0	0
$535$	0	0
$536$	19.3805	0.837110
$537$	0	0
$538$	16.7896	0.723853
$539$	−22.0532	−0.949897
$540$	0	0
$541$	9.92564	0.426737	0.213368	−	0.976972i	$-0.431557\pi$
$541$	9.92564	0.426737	0.213368	+	0.976972i	$0.431557\pi$
$542$	−53.7855	−2.31029
$543$	0	0
$544$	−9.92804	−0.425661
$545$	0	0
$546$	0	0
$547$	9.03245	0.386200	0.193100	−	0.981179i	$-0.438146\pi$
$547$	9.03245	0.386200	0.193100	+	0.981179i	$0.438146\pi$
$548$	20.6765	0.883258
$549$	0	0
$550$	0	0
$551$	1.69944	0.0723986
$552$	0	0
$553$	−8.03321	−0.341607
$554$	6.70743	0.284971
$555$	0	0
$556$	72.3141	3.06680
$557$	−13.7145	−0.581101	−0.290550	−	0.956860i	$-0.593838\pi$
$557$	−13.7145	−0.581101	−0.290550	+	0.956860i	$0.593838\pi$
$558$	0	0
$559$	60.2953	2.55022
$560$	0	0
$561$	0	0
$562$	−69.4672	−2.93030
$563$	20.2781	0.854621	0.427311	−	0.904105i	$-0.359461\pi$
$563$	20.2781	0.854621	0.427311	+	0.904105i	$0.359461\pi$
$564$	0	0
$565$	0	0
$566$	4.63547	0.194843
$567$	0	0
$568$	−5.98854	−0.251273
$569$	−13.3085	−0.557923	−0.278962	−	0.960302i	$-0.589990\pi$
$569$	−13.3085	−0.557923	−0.278962	+	0.960302i	$0.589990\pi$
$570$	0	0
$571$	−16.4204	−0.687174	−0.343587	−	0.939121i	$-0.611642\pi$
$571$	−16.4204	−0.687174	−0.343587	+	0.939121i	$0.611642\pi$
$572$	−49.2951	−2.06113
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−14.7505	−0.614071	−0.307036	−	0.951698i	$-0.599337\pi$
$577$	−14.7505	−0.614071	−0.307036	+	0.951698i	$0.599337\pi$
$578$	−31.1058	−1.29383
$579$	0	0
$580$	0	0
$581$	7.30855	0.303210
$582$	0	0
$583$	−20.3625	−0.843329
$584$	31.6251	1.30866
$585$	0	0
$586$	−25.0240	−1.03373
$587$	6.36385	0.262664	0.131332	−	0.991338i	$-0.458075\pi$
$587$	6.36385	0.262664	0.131332	+	0.991338i	$0.458075\pi$
$588$	0	0
$589$	−1.11190	−0.0458150
$590$	0	0
$591$	0	0
$592$	−1.14472	−0.0470475
$593$	−30.0706	−1.23485	−0.617426	−	0.786629i	$-0.711824\pi$
$593$	−30.0706	−1.23485	−0.617426	+	0.786629i	$0.711824\pi$
$594$	0	0
$595$	0	0
$596$	−14.0188	−0.574233
$597$	0	0
$598$	−8.96442	−0.366582
$599$	25.0587	1.02387	0.511936	−	0.859023i	$-0.328928\pi$
$599$	25.0587	1.02387	0.511936	+	0.859023i	$0.328928\pi$
$600$	0	0
$601$	40.0320	1.63294	0.816469	−	0.577390i	$-0.195929\pi$
$601$	40.0320	1.63294	0.816469	+	0.577390i	$0.195929\pi$
$602$	96.5252	3.93407
$603$	0	0
$604$	−37.3245	−1.51871
$605$	0	0
$606$	0	0
$607$	−41.3557	−1.67858	−0.839288	−	0.543687i	$-0.817028\pi$
$607$	−41.3557	−1.67858	−0.839288	+	0.543687i	$0.817028\pi$
$608$	−9.03245	−0.366314
$609$	0	0
$610$	0	0
$611$	27.7686	1.12340
$612$	0	0
$613$	40.5183	1.63652	0.818258	−	0.574851i	$-0.194940\pi$
$613$	40.5183	1.63652	0.818258	+	0.574851i	$0.194940\pi$
$614$	20.7766	0.838474
$615$	0	0
$616$	−31.0959	−1.25289
$617$	−1.76865	−0.0712033	−0.0356016	−	0.999366i	$-0.511335\pi$
$617$	−1.76865	−0.0712033	−0.0356016	+	0.999366i	$0.511335\pi$
$618$	0	0
$619$	−0.249804	−0.0100405	−0.00502023	−	0.999987i	$-0.501598\pi$
$619$	−0.249804	−0.0100405	−0.00502023	+	0.999987i	$0.501598\pi$
$620$	0	0
$621$	0	0
$622$	25.5884	1.02600
$623$	−12.9451	−0.518633
$624$	0	0
$625$	0	0
$626$	59.2530	2.36823
$627$	0	0
$628$	−13.6003	−0.542709
$629$	−7.30855	−0.291411
$630$	0	0
$631$	9.48922	0.377760	0.188880	−	0.982000i	$-0.439514\pi$
$631$	9.48922	0.377760	0.188880	+	0.982000i	$0.439514\pi$
$632$	−6.14770	−0.244542
$633$	0	0
$634$	1.92804	0.0765723
$635$	0	0
$636$	0	0
$637$	46.7516	1.85236
$638$	−6.11092	−0.241934
$639$	0	0
$640$	0	0
$641$	49.9416	1.97258	0.986288	−	0.165035i	$-0.0527738\pi$
$641$	49.9416	1.97258	0.986288	+	0.165035i	$0.0527738\pi$
$642$	0	0
$643$	25.8589	1.01977	0.509887	−	0.860241i	$-0.329687\pi$
$643$	25.8589	1.01977	0.509887	+	0.860241i	$0.329687\pi$
$644$	−8.93603	−0.352129
$645$	0	0
$646$	7.30855	0.287551
$647$	32.1384	1.26349	0.631745	−	0.775177i	$-0.282339\pi$
$647$	32.1384	1.26349	0.631745	+	0.775177i	$0.282339\pi$
$648$	0	0
$649$	26.5427	1.04189
$650$	0	0
$651$	0	0
$652$	10.5537	0.413315
$653$	9.79247	0.383209	0.191604	−	0.981472i	$-0.438631\pi$
$653$	9.79247	0.383209	0.191604	+	0.981472i	$0.438631\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	44.4540	1.73300
$659$	7.55835	0.294432	0.147216	−	0.989104i	$-0.452969\pi$
$659$	7.55835	0.294432	0.147216	+	0.989104i	$0.452969\pi$
$660$	0	0
$661$	−10.9041	−0.424119	−0.212060	−	0.977257i	$-0.568017\pi$
$661$	−10.9041	−0.424119	−0.212060	+	0.977257i	$0.568017\pi$
$662$	50.8096	1.97477
$663$	0	0
$664$	5.59313	0.217056
$665$	0	0
$666$	0	0
$667$	−0.691975	−0.0267934
$668$	−48.2048	−1.86510
$669$	0	0
$670$	0	0
$671$	12.4948	0.482357
$672$	0	0
$673$	12.6296	0.486836	0.243418	−	0.969921i	$-0.421731\pi$
$673$	12.6296	0.486836	0.243418	+	0.969921i	$0.421731\pi$
$674$	−17.8377	−0.687083
$675$	0	0
$676$	61.5959	2.36907
$677$	8.87065	0.340927	0.170463	−	0.985364i	$-0.445474\pi$
$677$	8.87065	0.340927	0.170463	+	0.985364i	$0.445474\pi$
$678$	0	0
$679$	−1.50519	−0.0577641
$680$	0	0
$681$	0	0
$682$	3.99821	0.153099
$683$	−24.4749	−0.936507	−0.468254	−	0.883594i	$-0.655117\pi$
$683$	−24.4749	−0.936507	−0.468254	+	0.883594i	$0.655117\pi$
$684$	0	0
$685$	0	0
$686$	11.7874	0.450044
$687$	0	0
$688$	3.13504	0.119522
$689$	43.1675	1.64455
$690$	0	0
$691$	−3.30855	−0.125863	−0.0629315	−	0.998018i	$-0.520045\pi$
$691$	−3.30855	−0.125863	−0.0629315	+	0.998018i	$0.520045\pi$
$692$	29.7090	1.12937
$693$	0	0
$694$	21.5028	0.816235
$695$	0	0
$696$	0	0
$697$	0	0
$698$	−49.8729	−1.88772
$699$	0	0
$700$	0	0
$701$	10.8485	0.409743	0.204871	−	0.978789i	$-0.434322\pi$
$701$	10.8485	0.409743	0.204871	+	0.978789i	$0.434322\pi$
$702$	0	0
$703$	−6.64926	−0.250781
$704$	34.0324	1.28264
$705$	0	0
$706$	−47.3405	−1.78168
$707$	−46.5979	−1.75250
$708$	0	0
$709$	−31.1675	−1.17052	−0.585259	−	0.810846i	$-0.699007\pi$
$709$	−31.1675	−1.17052	−0.585259	+	0.810846i	$0.699007\pi$
$710$	0	0
$711$	0	0
$712$	−9.90668	−0.371269
$713$	0.452741	0.0169553
$714$	0	0
$715$	0	0
$716$	70.3301	2.62836
$717$	0	0
$718$	−70.3672	−2.62608
$719$	13.0056	0.485027	0.242513	−	0.970148i	$-0.422028\pi$
$719$	13.0056	0.485027	0.242513	+	0.970148i	$0.422028\pi$
$720$	0	0
$721$	−58.5268	−2.17965
$722$	−37.0943	−1.38051
$723$	0	0
$724$	−56.0132	−2.08171
$725$	0	0
$726$	0	0
$727$	1.19089	0.0441677	0.0220839	−	0.999756i	$-0.492970\pi$
$727$	1.19089	0.0441677	0.0220839	+	0.999756i	$0.492970\pi$
$728$	65.9218	2.44322
$729$	0	0
$730$	0	0
$731$	20.0160	0.740318
$732$	0	0
$733$	−34.5990	−1.27794	−0.638971	−	0.769231i	$-0.720640\pi$
$733$	−34.5990	−1.27794	−0.638971	+	0.769231i	$0.720640\pi$
$734$	−31.7314	−1.17123
$735$	0	0
$736$	3.67781	0.135566
$737$	−17.1799	−0.632829
$738$	0	0
$739$	−41.7821	−1.53698	−0.768491	−	0.639861i	$-0.778992\pi$
$739$	−41.7821	−1.53698	−0.768491	+	0.639861i	$0.778992\pi$
$740$	0	0
$741$	0	0
$742$	69.1056	2.53695
$743$	−4.02130	−0.147527	−0.0737636	−	0.997276i	$-0.523501\pi$
$743$	−4.02130	−0.147527	−0.0737636	+	0.997276i	$0.523501\pi$
$744$	0	0
$745$	0	0
$746$	−62.7742	−2.29833
$747$	0	0
$748$	−16.3643	−0.598338
$749$	15.3085	0.559362
$750$	0	0
$751$	−0.497206	−0.0181433	−0.00907166	−	0.999959i	$-0.502888\pi$
$751$	−0.497206	−0.0181433	−0.00907166	+	0.999959i	$0.502888\pi$
$752$	1.44382	0.0526507
$753$	0	0
$754$	12.9548	0.471787
$755$	0	0
$756$	0	0
$757$	18.6944	0.679458	0.339729	−	0.940523i	$-0.389665\pi$
$757$	18.6944	0.679458	0.339729	+	0.940523i	$0.389665\pi$
$758$	37.7734	1.37199
$759$	0	0
$760$	0	0
$761$	3.38291	0.122630	0.0613151	−	0.998118i	$-0.480471\pi$
$761$	3.38291	0.122630	0.0613151	+	0.998118i	$0.480471\pi$
$762$	0	0
$763$	−29.5635	−1.07027
$764$	27.7474	1.00386
$765$	0	0
$766$	27.0240	0.976416
$767$	−56.2692	−2.03176
$768$	0	0
$769$	−7.41486	−0.267387	−0.133693	−	0.991023i	$-0.542684\pi$
$769$	−7.41486	−0.267387	−0.133693	+	0.991023i	$0.542684\pi$
$770$	0	0
$771$	0	0
$772$	2.61162	0.0939944
$773$	−33.2775	−1.19691	−0.598455	−	0.801156i	$-0.704218\pi$
$773$	−33.2775	−1.19691	−0.598455	+	0.801156i	$0.704218\pi$
$774$	0	0
$775$	0	0
$776$	−1.15190	−0.0413510
$777$	0	0
$778$	27.6275	0.990495
$779$	0	0
$780$	0	0
$781$	5.30855	0.189955
$782$	−2.97588	−0.106417
$783$	0	0
$784$	2.43084	0.0868156
$785$	0	0
$786$	0	0
$787$	−18.2878	−0.651890	−0.325945	−	0.945389i	$-0.605682\pi$
$787$	−18.2878	−0.651890	−0.325945	+	0.945389i	$0.605682\pi$
$788$	−80.8841	−2.88138
$789$	0	0
$790$	0	0
$791$	73.1439	2.60070
$792$	0	0
$793$	−26.4883	−0.940629
$794$	13.8909	0.492968
$795$	0	0
$796$	63.7290	2.25881
$797$	−44.5845	−1.57926	−0.789632	−	0.613581i	$-0.789729\pi$
$797$	−44.5845	−1.57926	−0.789632	+	0.613581i	$0.789729\pi$
$798$	0	0
$799$	9.21821	0.326117
$800$	0	0
$801$	0	0
$802$	49.4202	1.74509
$803$	−28.0341	−0.989302
$804$	0	0
$805$	0	0
$806$	−8.47600	−0.298554
$807$	0	0
$808$	−35.6607	−1.25454
$809$	−14.1063	−0.495952	−0.247976	−	0.968766i	$-0.579765\pi$
$809$	−14.1063	−0.495952	−0.247976	+	0.968766i	$0.579765\pi$
$810$	0	0
$811$	28.0000	0.983213	0.491606	−	0.870817i	$-0.336410\pi$
$811$	28.0000	0.983213	0.491606	+	0.870817i	$0.336410\pi$
$812$	12.9138	0.453186
$813$	0	0
$814$	23.9097	0.838033
$815$	0	0
$816$	0	0
$817$	18.2104	0.637100
$818$	−10.3484	−0.361822
$819$	0	0
$820$	0	0
$821$	−18.1571	−0.633686	−0.316843	−	0.948478i	$-0.602623\pi$
$821$	−18.1571	−0.633686	−0.316843	+	0.948478i	$0.602623\pi$
$822$	0	0
$823$	−3.25189	−0.113354	−0.0566770	−	0.998393i	$-0.518051\pi$
$823$	−3.25189	−0.113354	−0.0566770	+	0.998393i	$0.518051\pi$
$824$	−44.7897	−1.56032
$825$	0	0
$826$	−90.0799	−3.13428
$827$	10.7155	0.372615	0.186307	−	0.982492i	$-0.440348\pi$
$827$	10.7155	0.372615	0.186307	+	0.982492i	$0.440348\pi$
$828$	0	0
$829$	52.3461	1.81805	0.909027	−	0.416736i	$-0.136826\pi$
$829$	52.3461	1.81805	0.909027	+	0.416736i	$0.136826\pi$
$830$	0	0
$831$	0	0
$832$	−72.1469	−2.50124
$833$	15.5199	0.537733
$834$	0	0
$835$	0	0
$836$	−14.8881	−0.514916
$837$	0	0
$838$	−9.66192	−0.333765
$839$	−24.6703	−0.851712	−0.425856	−	0.904791i	$-0.640027\pi$
$839$	−24.6703	−0.851712	−0.425856	+	0.904791i	$0.640027\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	−42.6173	−1.46869
$843$	0	0
$844$	63.5535	2.18760
$845$	0	0
$846$	0	0
$847$	−15.4737	−0.531684
$848$	2.24448	0.0770758
$849$	0	0
$850$	0	0
$851$	2.70743	0.0928095
$852$	0	0
$853$	28.1115	0.962520	0.481260	−	0.876578i	$-0.340179\pi$
$853$	28.1115	0.962520	0.481260	+	0.876578i	$0.340179\pi$
$854$	−42.4045	−1.45105
$855$	0	0
$856$	11.7154	0.400425
$857$	8.39482	0.286762	0.143381	−	0.989668i	$-0.454203\pi$
$857$	8.39482	0.286762	0.143381	+	0.989668i	$0.454203\pi$
$858$	0	0
$859$	17.8405	0.608711	0.304356	−	0.952559i	$-0.401559\pi$
$859$	17.8405	0.608711	0.304356	+	0.952559i	$0.401559\pi$
$860$	0	0
$861$	0	0
$862$	−85.4793	−2.91144
$863$	−2.48249	−0.0845049	−0.0422524	−	0.999107i	$-0.513453\pi$
$863$	−2.48249	−0.0845049	−0.0422524	+	0.999107i	$0.513453\pi$
$864$	0	0
$865$	0	0
$866$	−38.8113	−1.31886
$867$	0	0
$868$	−8.44916	−0.286783
$869$	5.44964	0.184866
$870$	0	0
$871$	36.4204	1.23406
$872$	−22.6245	−0.766164
$873$	0	0
$874$	−2.70743	−0.0915802
$875$	0	0
$876$	0	0
$877$	25.9813	0.877325	0.438662	−	0.898652i	$-0.355452\pi$
$877$	25.9813	0.877325	0.438662	+	0.898652i	$0.355452\pi$
$878$	−21.2230	−0.716243
$879$	0	0
$880$	0	0
$881$	−25.1279	−0.846580	−0.423290	−	0.905994i	$-0.639125\pi$
$881$	−25.1279	−0.846580	−0.423290	+	0.905994i	$0.639125\pi$
$882$	0	0
$883$	−10.9684	−0.369117	−0.184559	−	0.982822i	$-0.559086\pi$
$883$	−10.9684	−0.369117	−0.184559	+	0.982822i	$0.559086\pi$
$884$	34.6915	1.16680
$885$	0	0
$886$	−43.5188	−1.46204
$887$	2.99897	0.100695	0.0503477	−	0.998732i	$-0.483967\pi$
$887$	2.99897	0.100695	0.0503477	+	0.998732i	$0.483967\pi$
$888$	0	0
$889$	19.5212	0.654719
$890$	0	0
$891$	0	0
$892$	35.8742	1.20116
$893$	8.38665	0.280649
$894$	0	0
$895$	0	0
$896$	−73.9073	−2.46907
$897$	0	0
$898$	72.1183	2.40662
$899$	−0.654273	−0.0218212
$900$	0	0
$901$	14.3301	0.477405
$902$	0	0
$903$	0	0
$904$	55.9760	1.86173
$905$	0	0
$906$	0	0
$907$	−50.5567	−1.67871	−0.839354	−	0.543585i	$-0.817066\pi$
$907$	−50.5567	−1.67871	−0.839354	+	0.543585i	$0.817066\pi$
$908$	−57.1341	−1.89606
$909$	0	0
$910$	0	0
$911$	−35.9044	−1.18957	−0.594784	−	0.803886i	$-0.702762\pi$
$911$	−35.9044	−1.18957	−0.594784	+	0.803886i	$0.702762\pi$
$912$	0	0
$913$	−4.95804	−0.164087
$914$	−84.3301	−2.78939
$915$	0	0
$916$	61.7450	2.04011
$917$	−77.1388	−2.54735
$918$	0	0
$919$	15.2022	0.501475	0.250738	−	0.968055i	$-0.419327\pi$
$919$	15.2022	0.501475	0.250738	+	0.968055i	$0.419327\pi$
$920$	0	0
$921$	0	0
$922$	−66.3745	−2.18593
$923$	−11.2538	−0.370425
$924$	0	0
$925$	0	0
$926$	−68.2422	−2.24258
$927$	0	0
$928$	−5.31495	−0.174472
$929$	6.00000	0.196854	0.0984268	−	0.995144i	$-0.468619\pi$
$929$	6.00000	0.196854	0.0984268	+	0.995144i	$0.468619\pi$
$930$	0	0
$931$	14.1199	0.462761
$932$	−48.2810	−1.58150
$933$	0	0
$934$	−60.9524	−1.99442
$935$	0	0
$936$	0	0
$937$	32.1859	1.05147	0.525734	−	0.850649i	$-0.323791\pi$
$937$	32.1859	1.05147	0.525734	+	0.850649i	$0.323791\pi$
$938$	58.3045	1.90371
$939$	0	0
$940$	0	0
$941$	−22.3697	−0.729231	−0.364616	−	0.931158i	$-0.618800\pi$
$941$	−22.3697	−0.729231	−0.364616	+	0.931158i	$0.618800\pi$
$942$	0	0
$943$	0	0
$944$	−2.92570	−0.0952236
$945$	0	0
$946$	−65.4816	−2.12899
$947$	0.122381	0.00397684	0.00198842	−	0.999998i	$-0.499367\pi$
$947$	0.122381	0.00397684	0.00198842	+	0.999998i	$0.499367\pi$
$948$	0	0
$949$	59.4308	1.92921
$950$	0	0
$951$	0	0
$952$	21.8838	0.709257
$953$	−55.4917	−1.79755	−0.898776	−	0.438409i	$-0.855542\pi$
$953$	−55.4917	−1.79755	−0.898776	+	0.438409i	$0.855542\pi$
$954$	0	0
$955$	0	0
$956$	6.60112	0.213495
$957$	0	0
$958$	−62.5787	−2.02183
$959$	24.5108	0.791494
$960$	0	0
$961$	−30.5719	−0.986191
$962$	−50.6873	−1.63422
$963$	0	0
$964$	−18.3377	−0.590619
$965$	0	0
$966$	0	0
$967$	−49.0722	−1.57806	−0.789028	−	0.614357i	$-0.789416\pi$
$967$	−49.0722	−1.57806	−0.789028	+	0.614357i	$0.789416\pi$
$968$	−11.8418	−0.380611
$969$	0	0
$970$	0	0
$971$	−2.08793	−0.0670050	−0.0335025	−	0.999439i	$-0.510666\pi$
$971$	−2.08793	−0.0670050	−0.0335025	+	0.999439i	$0.510666\pi$
$972$	0	0
$973$	85.7241	2.74819
$974$	73.8856	2.36745
$975$	0	0
$976$	−1.37725	−0.0440849
$977$	−36.5119	−1.16812	−0.584059	−	0.811711i	$-0.698536\pi$
$977$	−36.5119	−1.16812	−0.584059	+	0.811711i	$0.698536\pi$
$978$	0	0
$979$	8.78179	0.280667
$980$	0	0
$981$	0	0
$982$	23.1821	0.739771
$983$	32.8698	1.04838	0.524191	−	0.851601i	$-0.324368\pi$
$983$	32.8698	1.04838	0.524191	+	0.851601i	$0.324368\pi$
$984$	0	0
$985$	0	0
$986$	4.30056	0.136958
$987$	0	0
$988$	31.5620	1.00412
$989$	−7.41486	−0.235779
$990$	0	0
$991$	−36.0160	−1.14409	−0.572043	−	0.820224i	$-0.693849\pi$
$991$	−36.0160	−1.14409	−0.572043	+	0.820224i	$0.693849\pi$
$992$	3.47743	0.110409
$993$	0	0
$994$	−18.0160	−0.571432
$995$	0	0
$996$	0	0
$997$	56.8063	1.79907	0.899537	−	0.436844i	$-0.143904\pi$
$997$	56.8063	1.79907	0.899537	+	0.436844i	$0.143904\pi$
$998$	−68.9344	−2.18208
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6525.2.a.bt.1.5		6
3.2	odd	2		725.2.a.l.1.2		6
5.2	odd	4		1305.2.c.h.784.5		6
5.3	odd	4		1305.2.c.h.784.2		6
5.4	even	2	inner	6525.2.a.bt.1.2		6
15.2	even	4		145.2.b.c.59.2	✓	6
15.8	even	4		145.2.b.c.59.5	yes	6
15.14	odd	2		725.2.a.l.1.5		6
60.23	odd	4		2320.2.d.g.929.6		6
60.47	odd	4		2320.2.d.g.929.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
145.2.b.c.59.2	✓	6	15.2	even	4
145.2.b.c.59.5	yes	6	15.8	even	4
725.2.a.l.1.2		6	3.2	odd	2
725.2.a.l.1.5		6	15.14	odd	2
1305.2.c.h.784.2		6	5.3	odd	4
1305.2.c.h.784.5		6	5.2	odd	4
2320.2.d.g.929.1		6	60.47	odd	4
2320.2.d.g.929.6		6	60.23	odd	4
6525.2.a.bt.1.2		6	5.4	even	2	inner
6525.2.a.bt.1.5		6	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 6525 = 3^{2} \cdot 5^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6525.a (trivial)

\(f(q)\)	\(=\)	\(q+2.30229 q^{2} +3.30056 q^{4} +3.91261 q^{7} +2.99427 q^{8} -2.65427 q^{11} +5.62692 q^{13} +9.00799 q^{14} +0.292570 q^{16} +1.86794 q^{17} +1.69944 q^{19} -6.11092 q^{22} -0.691975 q^{23} +12.9548 q^{26} +12.9138 q^{28} +1.00000 q^{29} -0.654273 q^{31} -5.31495 q^{32} +4.30056 q^{34} -3.91261 q^{37} +3.91261 q^{38} +10.7155 q^{43} -8.76059 q^{44} -1.59313 q^{46} +4.93495 q^{47} +8.30855 q^{49} +18.5720 q^{52} +7.67159 q^{53} +11.7154 q^{56} +2.30229 q^{58} -10.0000 q^{59} -4.70743 q^{61} -1.50633 q^{62} -12.8217 q^{64} +6.47253 q^{67} +6.16526 q^{68} -2.00000 q^{71} +10.5619 q^{73} -9.00799 q^{74} +5.60911 q^{76} -10.3851 q^{77} -2.05316 q^{79} +1.86794 q^{83} +24.6703 q^{86} -7.94761 q^{88} -3.30855 q^{89} +22.0160 q^{91} -2.28390 q^{92} +11.3617 q^{94} -0.384703 q^{97} +19.1287 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 14 q^{4} + 10 q^{11} + 8 q^{14} + 42 q^{16} + 16 q^{19} + 46 q^{26} + 6 q^{29} + 22 q^{31} + 20 q^{34} + 2 q^{44} - 44 q^{46} - 2 q^{49} - 16 q^{56} - 60 q^{59} + 12 q^{61} + 38 q^{64} - 12 q^{71}+ \cdots + 2 q^{94}+O(q^{100}) \) 6 * q + 14 * q^4 + 10 * q^11 + 8 * q^14 + 42 * q^16 + 16 * q^19 + 46 * q^26 + 6 * q^29 + 22 * q^31 + 20 * q^34 + 2 * q^44 - 44 * q^46 - 2 * q^49 - 16 * q^56 - 60 * q^59 + 12 * q^61 + 38 * q^64 - 12 * q^71 - 8 * q^74 - 24 * q^76 + 2 * q^79 + 30 * q^86 + 32 * q^89 + 40 * q^91 + 2 * q^94

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