LMFDB - Embedded newform 9464.2.a.bs.1.15

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [9464,2,Mod(1,9464)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9464, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 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("9464.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 9464 = 2^{3} \cdot 7 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9464.a (trivial)

Newform invariants

comment:select newform

sage:traces = [15,0,-3,0,4,0,15,0,16,0,15,0,0,0,8,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	yes
Analytic conductor:	$75.5704204729$
Analytic rank:	$0$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - 3 x^{14} - 26 x^{13} + 78 x^{12} + 253 x^{11} - 782 x^{10} - 1087 x^{9} + 3776 x^{8} + \cdots - 344 $ x^15 - 3x^14 - 26x^13 + 78x^12 + 253x^11 - 782x^10 - 1087x^9 + 3776x^8 + 1576x^7 - 8849x^6 + 1791x^5 + 8496x^4 - 5353x^3 - 1252x^2 + 1724x - 344
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.15
Root			$-2.83411$ of defining polynomial
Character	$\chi$	$=$	9464.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.83411 q^{3} +1.08830 q^{5} +1.00000 q^{7} +5.03219 q^{9} +0.595045 q^{11} +3.08435 q^{15} +0.179097 q^{17} +0.901358 q^{19} +2.83411 q^{21} -3.08049 q^{23} -3.81561 q^{25} +5.75945 q^{27} +2.37949 q^{29} -0.0668924 q^{31} +1.68642 q^{33} +1.08830 q^{35} +5.61160 q^{37} +4.67510 q^{41} -5.08524 q^{43} +5.47651 q^{45} +4.63523 q^{47} +1.00000 q^{49} +0.507580 q^{51} -3.91804 q^{53} +0.647585 q^{55} +2.55455 q^{57} +8.26679 q^{59} +13.1642 q^{61} +5.03219 q^{63} +10.4840 q^{67} -8.73046 q^{69} -1.74320 q^{71} +4.20557 q^{73} -10.8139 q^{75} +0.595045 q^{77} +10.3440 q^{79} +1.22635 q^{81} +3.57005 q^{83} +0.194910 q^{85} +6.74375 q^{87} +1.87564 q^{89} -0.189580 q^{93} +0.980945 q^{95} +15.2613 q^{97} +2.99438 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q - 3 q^{3} + 4 q^{5} + 15 q^{7} + 16 q^{9} + 15 q^{11} + 8 q^{15} + 2 q^{17} + 13 q^{19} - 3 q^{21} - 10 q^{23} + 23 q^{25} - 9 q^{27} + 25 q^{29} + 19 q^{31} - 24 q^{33} + 4 q^{35} - 2 q^{37} + 30 q^{41}+ \cdots + 70 q^{99}+O(q^{100}) $ 15 * q - 3 * q^3 + 4 * q^5 + 15 * q^7 + 16 * q^9 + 15 * q^11 + 8 * q^15 + 2 * q^17 + 13 * q^19 - 3 * q^21 - 10 * q^23 + 23 * q^25 - 9 * q^27 + 25 * q^29 + 19 * q^31 - 24 * q^33 + 4 * q^35 - 2 * q^37 + 30 * q^41 + 15 * q^43 - 2 * q^45 + 13 * q^47 + 15 * q^49 + q^51 + 62 * q^53 - 3 * q^55 - 2 * q^57 + 22 * q^59 - q^61 + 16 * q^63 + 29 * q^67 + 32 * q^69 + 20 * q^71 + 21 * q^73 - 46 * q^75 + 15 * q^77 + 29 * q^79 + 7 * q^81 + 23 * q^83 + 8 * q^85 - 22 * q^87 + 9 * q^89 + 10 * q^93 + 28 * q^95 + 11 * q^97 + 70 * q^99

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
$3$	2.83411	1.63628	0.818138	−	0.575023i	$-0.195007\pi$
$3$	2.83411	1.63628	0.818138	+	0.575023i	$0.195007\pi$
$4$	0	0
$5$	1.08830	0.486701	0.243350	−	0.969938i	$-0.421753\pi$
$5$	1.08830	0.486701	0.243350	+	0.969938i	$0.421753\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	5.03219	1.67740
$10$	0	0
$11$	0.595045	0.179413	0.0897064	−	0.995968i	$-0.471407\pi$
$11$	0.595045	0.179413	0.0897064	+	0.995968i	$0.471407\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	3.08435	0.796377
$16$	0	0
$17$	0.179097	0.0434374	0.0217187	−	0.999764i	$-0.493086\pi$
$17$	0.179097	0.0434374	0.0217187	+	0.999764i	$0.493086\pi$
$18$	0	0
$19$	0.901358	0.206786	0.103393	−	0.994641i	$-0.467030\pi$
$19$	0.901358	0.206786	0.103393	+	0.994641i	$0.467030\pi$
$20$	0	0
$21$	2.83411	0.618454
$22$	0	0
$23$	−3.08049	−0.642327	−0.321164	−	0.947024i	$-0.604074\pi$
$23$	−3.08049	−0.642327	−0.321164	+	0.947024i	$0.604074\pi$
$24$	0	0
$25$	−3.81561	−0.763122
$26$	0	0
$27$	5.75945	1.10841
$28$	0	0
$29$	2.37949	0.441861	0.220931	−	0.975290i	$-0.429091\pi$
$29$	2.37949	0.441861	0.220931	+	0.975290i	$0.429091\pi$
$30$	0	0
$31$	−0.0668924	−0.0120142	−0.00600711	−	0.999982i	$-0.501912\pi$
$31$	−0.0668924	−0.0120142	−0.00600711	+	0.999982i	$0.501912\pi$
$32$	0	0
$33$	1.68642	0.293569
$34$	0	0
$35$	1.08830	0.183956
$36$	0	0
$37$	5.61160	0.922541	0.461270	−	0.887260i	$-0.347394\pi$
$37$	5.61160	0.922541	0.461270	+	0.887260i	$0.347394\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	4.67510	0.730129	0.365064	−	0.930982i	$-0.381047\pi$
$41$	4.67510	0.730129	0.365064	+	0.930982i	$0.381047\pi$
$42$	0	0
$43$	−5.08524	−0.775493	−0.387746	−	0.921766i	$-0.626746\pi$
$43$	−5.08524	−0.775493	−0.387746	+	0.921766i	$0.626746\pi$
$44$	0	0
$45$	5.47651	0.816390
$46$	0	0
$47$	4.63523	0.676118	0.338059	−	0.941125i	$-0.390230\pi$
$47$	4.63523	0.676118	0.338059	+	0.941125i	$0.390230\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0.507580	0.0710755
$52$	0	0
$53$	−3.91804	−0.538185	−0.269092	−	0.963114i	$-0.586724\pi$
$53$	−3.91804	−0.538185	−0.269092	+	0.963114i	$0.586724\pi$
$54$	0	0
$55$	0.647585	0.0873203
$56$	0	0
$57$	2.55455	0.338358
$58$	0	0
$59$	8.26679	1.07624	0.538122	−	0.842867i	$-0.319134\pi$
$59$	8.26679	1.07624	0.538122	+	0.842867i	$0.319134\pi$
$60$	0	0
$61$	13.1642	1.68551	0.842754	−	0.538299i	$-0.180933\pi$
$61$	13.1642	1.68551	0.842754	+	0.538299i	$0.180933\pi$
$62$	0	0
$63$	5.03219	0.633996
$64$	0	0
$65$	0	0
$66$	0	0
$67$	10.4840	1.28082	0.640410	−	0.768033i	$-0.278764\pi$
$67$	10.4840	1.28082	0.640410	+	0.768033i	$0.278764\pi$
$68$	0	0
$69$	−8.73046	−1.05102
$70$	0	0
$71$	−1.74320	−0.206880	−0.103440	−	0.994636i	$-0.532985\pi$
$71$	−1.74320	−0.206880	−0.103440	+	0.994636i	$0.532985\pi$
$72$	0	0
$73$	4.20557	0.492224	0.246112	−	0.969241i	$-0.420847\pi$
$73$	4.20557	0.492224	0.246112	+	0.969241i	$0.420847\pi$
$74$	0	0
$75$	−10.8139	−1.24868
$76$	0	0
$77$	0.595045	0.0678116
$78$	0	0
$79$	10.3440	1.16379	0.581896	−	0.813264i	$-0.302311\pi$
$79$	10.3440	1.16379	0.581896	+	0.813264i	$0.302311\pi$
$80$	0	0
$81$	1.22635	0.136261
$82$	0	0
$83$	3.57005	0.391864	0.195932	−	0.980617i	$-0.437227\pi$
$83$	3.57005	0.391864	0.195932	+	0.980617i	$0.437227\pi$
$84$	0	0
$85$	0.194910	0.0211410
$86$	0	0
$87$	6.74375	0.723006
$88$	0	0
$89$	1.87564	0.198818	0.0994088	−	0.995047i	$-0.468305\pi$
$89$	1.87564	0.198818	0.0994088	+	0.995047i	$0.468305\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−0.189580	−0.0196586
$94$	0	0
$95$	0.980945	0.100643
$96$	0	0
$97$	15.2613	1.54955	0.774777	−	0.632235i	$-0.217862\pi$
$97$	15.2613	1.54955	0.774777	+	0.632235i	$0.217862\pi$
$98$	0	0
$99$	2.99438	0.300946
$100$	0	0
$101$	11.1795	1.11240	0.556199	−	0.831049i	$-0.312259\pi$
$101$	11.1795	1.11240	0.556199	+	0.831049i	$0.312259\pi$
$102$	0	0
$103$	4.67220	0.460365	0.230183	−	0.973147i	$-0.426068\pi$
$103$	4.67220	0.460365	0.230183	+	0.973147i	$0.426068\pi$
$104$	0	0
$105$	3.08435	0.301002
$106$	0	0
$107$	−7.65362	−0.739904	−0.369952	−	0.929051i	$-0.620626\pi$
$107$	−7.65362	−0.739904	−0.369952	+	0.929051i	$0.620626\pi$
$108$	0	0
$109$	−11.7828	−1.12859	−0.564294	−	0.825574i	$-0.690852\pi$
$109$	−11.7828	−1.12859	−0.564294	+	0.825574i	$0.690852\pi$
$110$	0	0
$111$	15.9039	1.50953
$112$	0	0
$113$	7.93667	0.746619	0.373310	−	0.927707i	$-0.378223\pi$
$113$	7.93667	0.746619	0.373310	+	0.927707i	$0.378223\pi$
$114$	0	0
$115$	−3.35249	−0.312621
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0.179097	0.0164178
$120$	0	0
$121$	−10.6459	−0.967811
$122$	0	0
$123$	13.2498	1.19469
$124$	0	0
$125$	−9.59400	−0.858113
$126$	0	0
$127$	−9.19647	−0.816055	−0.408027	−	0.912970i	$-0.633783\pi$
$127$	−9.19647	−0.816055	−0.408027	+	0.912970i	$0.633783\pi$
$128$	0	0
$129$	−14.4122	−1.26892
$130$	0	0
$131$	15.7858	1.37921	0.689604	−	0.724187i	$-0.257785\pi$
$131$	15.7858	1.37921	0.689604	+	0.724187i	$0.257785\pi$
$132$	0	0
$133$	0.901358	0.0781577
$134$	0	0
$135$	6.26799	0.539462
$136$	0	0
$137$	2.38414	0.203691	0.101845	−	0.994800i	$-0.467525\pi$
$137$	2.38414	0.203691	0.101845	+	0.994800i	$0.467525\pi$
$138$	0	0
$139$	−19.5994	−1.66240	−0.831202	−	0.555971i	$-0.812347\pi$
$139$	−19.5994	−1.66240	−0.831202	+	0.555971i	$0.812347\pi$
$140$	0	0
$141$	13.1368	1.10631
$142$	0	0
$143$	0	0
$144$	0	0
$145$	2.58960	0.215054
$146$	0	0
$147$	2.83411	0.233754
$148$	0	0
$149$	18.8830	1.54695	0.773477	−	0.633825i	$-0.218516\pi$
$149$	18.8830	1.54695	0.773477	+	0.633825i	$0.218516\pi$
$150$	0	0
$151$	17.4886	1.42320	0.711602	−	0.702583i	$-0.247970\pi$
$151$	17.4886	1.42320	0.711602	+	0.702583i	$0.247970\pi$
$152$	0	0
$153$	0.901249	0.0728617
$154$	0	0
$155$	−0.0727987	−0.00584734
$156$	0	0
$157$	17.2851	1.37950	0.689750	−	0.724048i	$-0.257721\pi$
$157$	17.2851	1.37950	0.689750	+	0.724048i	$0.257721\pi$
$158$	0	0
$159$	−11.1042	−0.880618
$160$	0	0
$161$	−3.08049	−0.242777
$162$	0	0
$163$	−0.641674	−0.0502598	−0.0251299	−	0.999684i	$-0.508000\pi$
$163$	−0.641674	−0.0502598	−0.0251299	+	0.999684i	$0.508000\pi$
$164$	0	0
$165$	1.83533	0.142880
$166$	0	0
$167$	−14.6880	−1.13659	−0.568297	−	0.822823i	$-0.692398\pi$
$167$	−14.6880	−1.13659	−0.568297	+	0.822823i	$0.692398\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	4.53580	0.346862
$172$	0	0
$173$	−12.6107	−0.958777	−0.479389	−	0.877603i	$-0.659142\pi$
$173$	−12.6107	−0.958777	−0.479389	+	0.877603i	$0.659142\pi$
$174$	0	0
$175$	−3.81561	−0.288433
$176$	0	0
$177$	23.4290	1.76103
$178$	0	0
$179$	−20.7799	−1.55317	−0.776583	−	0.630015i	$-0.783049\pi$
$179$	−20.7799	−1.55317	−0.776583	+	0.630015i	$0.783049\pi$
$180$	0	0
$181$	−18.4978	−1.37493	−0.687466	−	0.726217i	$-0.741277\pi$
$181$	−18.4978	−1.37493	−0.687466	+	0.726217i	$0.741277\pi$
$182$	0	0
$183$	37.3089	2.75795
$184$	0	0
$185$	6.10708	0.449002
$186$	0	0
$187$	0.106571	0.00779322
$188$	0	0
$189$	5.75945	0.418938
$190$	0	0
$191$	3.47678	0.251571	0.125785	−	0.992057i	$-0.459855\pi$
$191$	3.47678	0.251571	0.125785	+	0.992057i	$0.459855\pi$
$192$	0	0
$193$	−27.4838	−1.97833	−0.989165	−	0.146810i	$-0.953099\pi$
$193$	−27.4838	−1.97833	−0.989165	+	0.146810i	$0.953099\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−18.0630	−1.28694	−0.643468	−	0.765473i	$-0.722505\pi$
$197$	−18.0630	−1.28694	−0.643468	+	0.765473i	$0.722505\pi$
$198$	0	0
$199$	16.5923	1.17620	0.588098	−	0.808790i	$-0.299877\pi$
$199$	16.5923	1.17620	0.588098	+	0.808790i	$0.299877\pi$
$200$	0	0
$201$	29.7127	2.09577
$202$	0	0
$203$	2.37949	0.167008
$204$	0	0
$205$	5.08790	0.355354
$206$	0	0
$207$	−15.5016	−1.07744
$208$	0	0
$209$	0.536348	0.0371000
$210$	0	0
$211$	−19.8478	−1.36638	−0.683189	−	0.730241i	$-0.739408\pi$
$211$	−19.8478	−1.36638	−0.683189	+	0.730241i	$0.739408\pi$
$212$	0	0
$213$	−4.94042	−0.338512
$214$	0	0
$215$	−5.53425	−0.377433
$216$	0	0
$217$	−0.0668924	−0.00454095
$218$	0	0
$219$	11.9190	0.805414
$220$	0	0
$221$	0	0
$222$	0	0
$223$	6.97273	0.466929	0.233464	−	0.972365i	$-0.424994\pi$
$223$	6.97273	0.466929	0.233464	+	0.972365i	$0.424994\pi$
$224$	0	0
$225$	−19.2009	−1.28006
$226$	0	0
$227$	−4.22492	−0.280418	−0.140209	−	0.990122i	$-0.544777\pi$
$227$	−4.22492	−0.280418	−0.140209	+	0.990122i	$0.544777\pi$
$228$	0	0
$229$	3.90217	0.257863	0.128931	−	0.991654i	$-0.458845\pi$
$229$	3.90217	0.257863	0.128931	+	0.991654i	$0.458845\pi$
$230$	0	0
$231$	1.68642	0.110958
$232$	0	0
$233$	2.76932	0.181424	0.0907119	−	0.995877i	$-0.471086\pi$
$233$	2.76932	0.181424	0.0907119	+	0.995877i	$0.471086\pi$
$234$	0	0
$235$	5.04450	0.329067
$236$	0	0
$237$	29.3161	1.90428
$238$	0	0
$239$	−13.1126	−0.848186	−0.424093	−	0.905619i	$-0.639407\pi$
$239$	−13.1126	−0.848186	−0.424093	+	0.905619i	$0.639407\pi$
$240$	0	0
$241$	4.86930	0.313659	0.156830	−	0.987626i	$-0.449873\pi$
$241$	4.86930	0.313659	0.156830	+	0.987626i	$0.449873\pi$
$242$	0	0
$243$	−13.8027	−0.885445
$244$	0	0
$245$	1.08830	0.0695287
$246$	0	0
$247$	0	0
$248$	0	0
$249$	10.1179	0.641198
$250$	0	0
$251$	6.24197	0.393990	0.196995	−	0.980405i	$-0.436882\pi$
$251$	6.24197	0.393990	0.196995	+	0.980405i	$0.436882\pi$
$252$	0	0
$253$	−1.83303	−0.115242
$254$	0	0
$255$	0.552398	0.0345925
$256$	0	0
$257$	−18.5069	−1.15443	−0.577214	−	0.816593i	$-0.695860\pi$
$257$	−18.5069	−1.15443	−0.577214	+	0.816593i	$0.695860\pi$
$258$	0	0
$259$	5.61160	0.348688
$260$	0	0
$261$	11.9741	0.741176
$262$	0	0
$263$	−1.47962	−0.0912371	−0.0456186	−	0.998959i	$-0.514526\pi$
$263$	−1.47962	−0.0912371	−0.0456186	+	0.998959i	$0.514526\pi$
$264$	0	0
$265$	−4.26399	−0.261935
$266$	0	0
$267$	5.31578	0.325320
$268$	0	0
$269$	−24.2542	−1.47880	−0.739402	−	0.673264i	$-0.764892\pi$
$269$	−24.2542	−1.47880	−0.739402	+	0.673264i	$0.764892\pi$
$270$	0	0
$271$	−19.8562	−1.20618	−0.603090	−	0.797673i	$-0.706064\pi$
$271$	−19.8562	−1.20618	−0.603090	+	0.797673i	$0.706064\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−2.27046	−0.136914
$276$	0	0
$277$	2.24632	0.134968	0.0674842	−	0.997720i	$-0.478503\pi$
$277$	2.24632	0.134968	0.0674842	+	0.997720i	$0.478503\pi$
$278$	0	0
$279$	−0.336615	−0.0201526
$280$	0	0
$281$	18.9143	1.12833	0.564167	−	0.825661i	$-0.309197\pi$
$281$	18.9143	1.12833	0.564167	+	0.825661i	$0.309197\pi$
$282$	0	0
$283$	3.12973	0.186043	0.0930217	−	0.995664i	$-0.470347\pi$
$283$	3.12973	0.186043	0.0930217	+	0.995664i	$0.470347\pi$
$284$	0	0
$285$	2.78011	0.164679
$286$	0	0
$287$	4.67510	0.275963
$288$	0	0
$289$	−16.9679	−0.998113
$290$	0	0
$291$	43.2523	2.53550
$292$	0	0
$293$	28.9431	1.69087	0.845436	−	0.534077i	$-0.179341\pi$
$293$	28.9431	1.69087	0.845436	+	0.534077i	$0.179341\pi$
$294$	0	0
$295$	8.99672	0.523809
$296$	0	0
$297$	3.42713	0.198862
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−5.08524	−0.293109
$302$	0	0
$303$	31.6838	1.82019
$304$	0	0
$305$	14.3266	0.820338
$306$	0	0
$307$	−32.1617	−1.83556	−0.917781	−	0.397086i	$-0.870021\pi$
$307$	−32.1617	−1.83556	−0.917781	+	0.397086i	$0.870021\pi$
$308$	0	0
$309$	13.2415	0.753284
$310$	0	0
$311$	−7.82274	−0.443587	−0.221793	−	0.975094i	$-0.571191\pi$
$311$	−7.82274	−0.443587	−0.221793	+	0.975094i	$0.571191\pi$
$312$	0	0
$313$	−12.1517	−0.686855	−0.343428	−	0.939179i	$-0.611588\pi$
$313$	−12.1517	−0.686855	−0.343428	+	0.939179i	$0.611588\pi$
$314$	0	0
$315$	5.47651	0.308567
$316$	0	0
$317$	−23.6556	−1.32863	−0.664316	−	0.747452i	$-0.731277\pi$
$317$	−23.6556	−1.32863	−0.664316	+	0.747452i	$0.731277\pi$
$318$	0	0
$319$	1.41591	0.0792755
$320$	0	0
$321$	−21.6912	−1.21069
$322$	0	0
$323$	0.161430	0.00898223
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−33.3938	−1.84668
$328$	0	0
$329$	4.63523	0.255548
$330$	0	0
$331$	−5.56404	−0.305827	−0.152914	−	0.988240i	$-0.548866\pi$
$331$	−5.56404	−0.305827	−0.152914	+	0.988240i	$0.548866\pi$
$332$	0	0
$333$	28.2386	1.54747
$334$	0	0
$335$	11.4097	0.623376
$336$	0	0
$337$	8.23184	0.448417	0.224208	−	0.974541i	$-0.428020\pi$
$337$	8.23184	0.448417	0.224208	+	0.974541i	$0.428020\pi$
$338$	0	0
$339$	22.4934	1.22167
$340$	0	0
$341$	−0.0398040	−0.00215551
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−9.50133	−0.511534
$346$	0	0
$347$	3.33268	0.178908	0.0894538	−	0.995991i	$-0.471488\pi$
$347$	3.33268	0.178908	0.0894538	+	0.995991i	$0.471488\pi$
$348$	0	0
$349$	13.5626	0.725991	0.362996	−	0.931791i	$-0.381754\pi$
$349$	13.5626	0.725991	0.362996	+	0.931791i	$0.381754\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−10.9916	−0.585025	−0.292512	−	0.956262i	$-0.594491\pi$
$353$	−10.9916	−0.585025	−0.292512	+	0.956262i	$0.594491\pi$
$354$	0	0
$355$	−1.89712	−0.100689
$356$	0	0
$357$	0.507580	0.0268640
$358$	0	0
$359$	26.3444	1.39040	0.695201	−	0.718815i	$-0.255315\pi$
$359$	26.3444	1.39040	0.695201	+	0.718815i	$0.255315\pi$
$360$	0	0
$361$	−18.1876	−0.957240
$362$	0	0
$363$	−30.1717	−1.58361
$364$	0	0
$365$	4.57690	0.239566
$366$	0	0
$367$	8.90682	0.464932	0.232466	−	0.972605i	$-0.425321\pi$
$367$	8.90682	0.464932	0.232466	+	0.972605i	$0.425321\pi$
$368$	0	0
$369$	23.5260	1.22471
$370$	0	0
$371$	−3.91804	−0.203415
$372$	0	0
$373$	0.643889	0.0333393	0.0166697	−	0.999861i	$-0.494694\pi$
$373$	0.643889	0.0333393	0.0166697	+	0.999861i	$0.494694\pi$
$374$	0	0
$375$	−27.1905	−1.40411
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−6.53441	−0.335650	−0.167825	−	0.985817i	$-0.553674\pi$
$379$	−6.53441	−0.335650	−0.167825	+	0.985817i	$0.553674\pi$
$380$	0	0
$381$	−26.0638	−1.33529
$382$	0	0
$383$	7.52143	0.384327	0.192164	−	0.981363i	$-0.438450\pi$
$383$	7.52143	0.384327	0.192164	+	0.981363i	$0.438450\pi$
$384$	0	0
$385$	0.647585	0.0330040
$386$	0	0
$387$	−25.5899	−1.30081
$388$	0	0
$389$	−10.5706	−0.535951	−0.267976	−	0.963426i	$-0.586355\pi$
$389$	−10.5706	−0.535951	−0.267976	+	0.963426i	$0.586355\pi$
$390$	0	0
$391$	−0.551707	−0.0279010
$392$	0	0
$393$	44.7386	2.25676
$394$	0	0
$395$	11.2573	0.566418
$396$	0	0
$397$	−4.38390	−0.220021	−0.110011	−	0.993930i	$-0.535089\pi$
$397$	−4.38390	−0.220021	−0.110011	+	0.993930i	$0.535089\pi$
$398$	0	0
$399$	2.55455	0.127887
$400$	0	0
$401$	−24.1646	−1.20672	−0.603362	−	0.797468i	$-0.706172\pi$
$401$	−24.1646	−1.20672	−0.603362	+	0.797468i	$0.706172\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	1.33464	0.0663186
$406$	0	0
$407$	3.33915	0.165516
$408$	0	0
$409$	16.4283	0.812327	0.406164	−	0.913800i	$-0.366866\pi$
$409$	16.4283	0.812327	0.406164	+	0.913800i	$0.366866\pi$
$410$	0	0
$411$	6.75692	0.333294
$412$	0	0
$413$	8.26679	0.406782
$414$	0	0
$415$	3.88528	0.190721
$416$	0	0
$417$	−55.5470	−2.72015
$418$	0	0
$419$	16.7125	0.816461	0.408230	−	0.912879i	$-0.366146\pi$
$419$	16.7125	0.816461	0.408230	+	0.912879i	$0.366146\pi$
$420$	0	0
$421$	0.000826147 0	4.02639e−5 0	2.01320e−5	−	1.00000i	$-0.499994\pi$
$421$	0.000826147 0	4.02639e−5 0	2.01320e−5		1.00000i	$0.499994\pi$
$422$	0	0
$423$	23.3253	1.13412
$424$	0	0
$425$	−0.683364	−0.0331480
$426$	0	0
$427$	13.1642	0.637062
$428$	0	0
$429$	0	0
$430$	0	0
$431$	9.20094	0.443194	0.221597	−	0.975138i	$-0.428873\pi$
$431$	9.20094	0.443194	0.221597	+	0.975138i	$0.428873\pi$
$432$	0	0
$433$	−1.93157	−0.0928252	−0.0464126	−	0.998922i	$-0.514779\pi$
$433$	−1.93157	−0.0928252	−0.0464126	+	0.998922i	$0.514779\pi$
$434$	0	0
$435$	7.33920	0.351888
$436$	0	0
$437$	−2.77663	−0.132824
$438$	0	0
$439$	−9.16760	−0.437546	−0.218773	−	0.975776i	$-0.570205\pi$
$439$	−9.16760	−0.437546	−0.218773	+	0.975776i	$0.570205\pi$
$440$	0	0
$441$	5.03219	0.239628
$442$	0	0
$443$	−18.9330	−0.899534	−0.449767	−	0.893146i	$-0.648493\pi$
$443$	−18.9330	−0.899534	−0.449767	+	0.893146i	$0.648493\pi$
$444$	0	0
$445$	2.04125	0.0967647
$446$	0	0
$447$	53.5165	2.53124
$448$	0	0
$449$	−31.6338	−1.49289	−0.746446	−	0.665446i	$-0.768241\pi$
$449$	−31.6338	−1.49289	−0.746446	+	0.665446i	$0.768241\pi$
$450$	0	0
$451$	2.78190	0.130994
$452$	0	0
$453$	49.5647	2.32875
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−7.70978	−0.360648	−0.180324	−	0.983607i	$-0.557715\pi$
$457$	−7.70978	−0.360648	−0.180324	+	0.983607i	$0.557715\pi$
$458$	0	0
$459$	1.03150	0.0481462
$460$	0	0
$461$	0.397439	0.0185106	0.00925530	−	0.999957i	$-0.497054\pi$
$461$	0.397439	0.0185106	0.00925530	+	0.999957i	$0.497054\pi$
$462$	0	0
$463$	4.46487	0.207500	0.103750	−	0.994603i	$-0.466916\pi$
$463$	4.46487	0.207500	0.103750	+	0.994603i	$0.466916\pi$
$464$	0	0
$465$	−0.206320	−0.00956785
$466$	0	0
$467$	16.7918	0.777032	0.388516	−	0.921442i	$-0.372988\pi$
$467$	16.7918	0.777032	0.388516	+	0.921442i	$0.372988\pi$
$468$	0	0
$469$	10.4840	0.484105
$470$	0	0
$471$	48.9878	2.25724
$472$	0	0
$473$	−3.02595	−0.139133
$474$	0	0
$475$	−3.43923	−0.157803
$476$	0	0
$477$	−19.7163	−0.902749
$478$	0	0
$479$	8.25106	0.377000	0.188500	−	0.982073i	$-0.439637\pi$
$479$	8.25106	0.377000	0.188500	+	0.982073i	$0.439637\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	−8.73046	−0.397250
$484$	0	0
$485$	16.6088	0.754169
$486$	0	0
$487$	9.58778	0.434464	0.217232	−	0.976120i	$-0.430297\pi$
$487$	9.58778	0.434464	0.217232	+	0.976120i	$0.430297\pi$
$488$	0	0
$489$	−1.81858	−0.0822388
$490$	0	0
$491$	19.9245	0.899179	0.449589	−	0.893235i	$-0.351570\pi$
$491$	19.9245	0.899179	0.449589	+	0.893235i	$0.351570\pi$
$492$	0	0
$493$	0.426160	0.0191933
$494$	0	0
$495$	3.25877	0.146471
$496$	0	0
$497$	−1.74320	−0.0781932
$498$	0	0
$499$	27.5852	1.23488	0.617441	−	0.786617i	$-0.288170\pi$
$499$	27.5852	1.23488	0.617441	+	0.786617i	$0.288170\pi$
$500$	0	0
$501$	−41.6275	−1.85978
$502$	0	0
$503$	−34.4161	−1.53454	−0.767270	−	0.641324i	$-0.778385\pi$
$503$	−34.4161	−1.53454	−0.767270	+	0.641324i	$0.778385\pi$
$504$	0	0
$505$	12.1666	0.541405
$506$	0	0
$507$	0	0
$508$	0	0
$509$	22.7108	1.00664	0.503319	−	0.864100i	$-0.332112\pi$
$509$	22.7108	1.00664	0.503319	+	0.864100i	$0.332112\pi$
$510$	0	0
$511$	4.20557	0.186043
$512$	0	0
$513$	5.19133	0.229203
$514$	0	0
$515$	5.08473	0.224060
$516$	0	0
$517$	2.75817	0.121304
$518$	0	0
$519$	−35.7403	−1.56882
$520$	0	0
$521$	−9.36576	−0.410321	−0.205161	−	0.978728i	$-0.565772\pi$
$521$	−9.36576	−0.410321	−0.205161	+	0.978728i	$0.565772\pi$
$522$	0	0
$523$	−40.6110	−1.77579	−0.887897	−	0.460042i	$-0.847834\pi$
$523$	−40.6110	−1.77579	−0.887897	+	0.460042i	$0.847834\pi$
$524$	0	0
$525$	−10.8139	−0.471956
$526$	0	0
$527$	−0.0119802	−0.000521866 0
$528$	0	0
$529$	−13.5106	−0.587416
$530$	0	0
$531$	41.6001	1.80529
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−8.32941	−0.360112
$536$	0	0
$537$	−58.8927	−2.54141
$538$	0	0
$539$	0.595045	0.0256304
$540$	0	0
$541$	33.7687	1.45183	0.725916	−	0.687784i	$-0.241416\pi$
$541$	33.7687	1.45183	0.725916	+	0.687784i	$0.241416\pi$
$542$	0	0
$543$	−52.4249	−2.24977
$544$	0	0
$545$	−12.8232	−0.549285
$546$	0	0
$547$	44.4102	1.89884	0.949421	−	0.314007i	$-0.101672\pi$
$547$	44.4102	1.89884	0.949421	+	0.314007i	$0.101672\pi$
$548$	0	0
$549$	66.2449	2.82726
$550$	0	0
$551$	2.14478	0.0913706
$552$	0	0
$553$	10.3440	0.439872
$554$	0	0
$555$	17.3082	0.734690
$556$	0	0
$557$	−31.9846	−1.35523	−0.677615	−	0.735417i	$-0.736986\pi$
$557$	−31.9846	−1.35523	−0.677615	+	0.735417i	$0.736986\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0.302033	0.0127518
$562$	0	0
$563$	−9.09013	−0.383103	−0.191552	−	0.981483i	$-0.561352\pi$
$563$	−9.09013	−0.383103	−0.191552	+	0.981483i	$0.561352\pi$
$564$	0	0
$565$	8.63745	0.363380
$566$	0	0
$567$	1.22635	0.0515020
$568$	0	0
$569$	−8.91472	−0.373725	−0.186862	−	0.982386i	$-0.559832\pi$
$569$	−8.91472	−0.373725	−0.186862	+	0.982386i	$0.559832\pi$
$570$	0	0
$571$	−11.4622	−0.479678	−0.239839	−	0.970813i	$-0.577095\pi$
$571$	−11.4622	−0.479678	−0.239839	+	0.970813i	$0.577095\pi$
$572$	0	0
$573$	9.85357	0.411639
$574$	0	0
$575$	11.7540	0.490174
$576$	0	0
$577$	8.75839	0.364616	0.182308	−	0.983241i	$-0.441643\pi$
$577$	8.75839	0.364616	0.182308	+	0.983241i	$0.441643\pi$
$578$	0	0
$579$	−77.8922	−3.23709
$580$	0	0
$581$	3.57005	0.148111
$582$	0	0
$583$	−2.33141	−0.0965572
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−42.6093	−1.75868	−0.879338	−	0.476199i	$-0.842014\pi$
$587$	−42.6093	−1.75868	−0.879338	+	0.476199i	$0.842014\pi$
$588$	0	0
$589$	−0.0602940	−0.00248437
$590$	0	0
$591$	−51.1926	−2.10578
$592$	0	0
$593$	−7.37369	−0.302801	−0.151401	−	0.988472i	$-0.548378\pi$
$593$	−7.37369	−0.302801	−0.151401	+	0.988472i	$0.548378\pi$
$594$	0	0
$595$	0.194910	0.00799055
$596$	0	0
$597$	47.0244	1.92458
$598$	0	0
$599$	−44.8387	−1.83206	−0.916030	−	0.401110i	$-0.868625\pi$
$599$	−44.8387	−1.83206	−0.916030	+	0.401110i	$0.868625\pi$
$600$	0	0
$601$	23.2763	0.949462	0.474731	−	0.880131i	$-0.342545\pi$
$601$	23.2763	0.949462	0.474731	+	0.880131i	$0.342545\pi$
$602$	0	0
$603$	52.7573	2.14844
$604$	0	0
$605$	−11.5859	−0.471035
$606$	0	0
$607$	−25.6497	−1.04109	−0.520545	−	0.853834i	$-0.674271\pi$
$607$	−25.6497	−1.04109	−0.520545	+	0.853834i	$0.674271\pi$
$608$	0	0
$609$	6.74375	0.273271
$610$	0	0
$611$	0	0
$612$	0	0
$613$	3.02437	0.122153	0.0610765	−	0.998133i	$-0.480547\pi$
$613$	3.02437	0.122153	0.0610765	+	0.998133i	$0.480547\pi$
$614$	0	0
$615$	14.4197	0.581457
$616$	0	0
$617$	−18.0611	−0.727112	−0.363556	−	0.931572i	$-0.618437\pi$
$617$	−18.0611	−0.727112	−0.363556	+	0.931572i	$0.618437\pi$
$618$	0	0
$619$	−3.16452	−0.127193	−0.0635963	−	0.997976i	$-0.520257\pi$
$619$	−3.16452	−0.127193	−0.0635963	+	0.997976i	$0.520257\pi$
$620$	0	0
$621$	−17.7419	−0.711960
$622$	0	0
$623$	1.87564	0.0751460
$624$	0	0
$625$	8.63694	0.345478
$626$	0	0
$627$	1.52007	0.0607058
$628$	0	0
$629$	1.00502	0.0400727
$630$	0	0
$631$	17.4422	0.694362	0.347181	−	0.937798i	$-0.387139\pi$
$631$	17.4422	0.694362	0.347181	+	0.937798i	$0.387139\pi$
$632$	0	0
$633$	−56.2509	−2.23577
$634$	0	0
$635$	−10.0085	−0.397175
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−8.77211	−0.347019
$640$	0	0
$641$	14.8473	0.586433	0.293216	−	0.956046i	$-0.405274\pi$
$641$	14.8473	0.586433	0.293216	+	0.956046i	$0.405274\pi$
$642$	0	0
$643$	−11.5130	−0.454027	−0.227014	−	0.973892i	$-0.572896\pi$
$643$	−11.5130	−0.454027	−0.227014	+	0.973892i	$0.572896\pi$
$644$	0	0
$645$	−15.6847	−0.617584
$646$	0	0
$647$	9.07480	0.356767	0.178384	−	0.983961i	$-0.442913\pi$
$647$	9.07480	0.356767	0.178384	+	0.983961i	$0.442913\pi$
$648$	0	0
$649$	4.91911	0.193092
$650$	0	0
$651$	−0.189580	−0.00743024
$652$	0	0
$653$	20.5583	0.804508	0.402254	−	0.915528i	$-0.368227\pi$
$653$	20.5583	0.804508	0.402254	+	0.915528i	$0.368227\pi$
$654$	0	0
$655$	17.1796	0.671262
$656$	0	0
$657$	21.1632	0.825655
$658$	0	0
$659$	−23.9581	−0.933274	−0.466637	−	0.884449i	$-0.654535\pi$
$659$	−23.9581	−0.933274	−0.466637	+	0.884449i	$0.654535\pi$
$660$	0	0
$661$	−12.9464	−0.503555	−0.251778	−	0.967785i	$-0.581015\pi$
$661$	−12.9464	−0.503555	−0.251778	+	0.967785i	$0.581015\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0.980945	0.0380394
$666$	0	0
$667$	−7.33002	−0.283819
$668$	0	0
$669$	19.7615	0.764024
$670$	0	0
$671$	7.83331	0.302402
$672$	0	0
$673$	37.6151	1.44996	0.724978	−	0.688772i	$-0.241850\pi$
$673$	37.6151	1.44996	0.724978	+	0.688772i	$0.241850\pi$
$674$	0	0
$675$	−21.9758	−0.845849
$676$	0	0
$677$	5.60130	0.215275	0.107638	−	0.994190i	$-0.465671\pi$
$677$	5.60130	0.215275	0.107638	+	0.994190i	$0.465671\pi$
$678$	0	0
$679$	15.2613	0.585676
$680$	0	0
$681$	−11.9739	−0.458841
$682$	0	0
$683$	−46.7612	−1.78927	−0.894634	−	0.446800i	$-0.852564\pi$
$683$	−46.7612	−1.78927	−0.894634	+	0.446800i	$0.852564\pi$
$684$	0	0
$685$	2.59465	0.0991366
$686$	0	0
$687$	11.0592	0.421934
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−4.77400	−0.181612	−0.0908058	−	0.995869i	$-0.528944\pi$
$691$	−4.77400	−0.181612	−0.0908058	+	0.995869i	$0.528944\pi$
$692$	0	0
$693$	2.99438	0.113747
$694$	0	0
$695$	−21.3300	−0.809093
$696$	0	0
$697$	0.837296	0.0317149
$698$	0	0
$699$	7.84855	0.296859
$700$	0	0
$701$	41.7541	1.57703	0.788515	−	0.615015i	$-0.210850\pi$
$701$	41.7541	1.57703	0.788515	+	0.615015i	$0.210850\pi$
$702$	0	0
$703$	5.05806	0.190768
$704$	0	0
$705$	14.2967	0.538444
$706$	0	0
$707$	11.1795	0.420447
$708$	0	0
$709$	−3.96643	−0.148962	−0.0744812	−	0.997222i	$-0.523730\pi$
$709$	−3.96643	−0.148962	−0.0744812	+	0.997222i	$0.523730\pi$
$710$	0	0
$711$	52.0530	1.95214
$712$	0	0
$713$	0.206062	0.00771707
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−37.1627	−1.38787
$718$	0	0
$719$	48.6818	1.81553	0.907763	−	0.419483i	$-0.137789\pi$
$719$	48.6818	1.81553	0.907763	+	0.419483i	$0.137789\pi$
$720$	0	0
$721$	4.67220	0.174002
$722$	0	0
$723$	13.8001	0.513233
$724$	0	0
$725$	−9.07922	−0.337194
$726$	0	0
$727$	9.67959	0.358996	0.179498	−	0.983758i	$-0.442553\pi$
$727$	9.67959	0.358996	0.179498	+	0.983758i	$0.442553\pi$
$728$	0	0
$729$	−42.7975	−1.58509
$730$	0	0
$731$	−0.910751	−0.0336854
$732$	0	0
$733$	5.48902	0.202742	0.101371	−	0.994849i	$-0.467677\pi$
$733$	5.48902	0.202742	0.101371	+	0.994849i	$0.467677\pi$
$734$	0	0
$735$	3.08435	0.113768
$736$	0	0
$737$	6.23843	0.229795
$738$	0	0
$739$	−5.82812	−0.214391	−0.107195	−	0.994238i	$-0.534187\pi$
$739$	−5.82812	−0.214391	−0.107195	+	0.994238i	$0.534187\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−22.3078	−0.818395	−0.409197	−	0.912446i	$-0.634191\pi$
$743$	−22.3078	−0.818395	−0.409197	+	0.912446i	$0.634191\pi$
$744$	0	0
$745$	20.5503	0.752904
$746$	0	0
$747$	17.9652	0.657312
$748$	0	0
$749$	−7.65362	−0.279657
$750$	0	0
$751$	−15.8840	−0.579616	−0.289808	−	0.957085i	$-0.593591\pi$
$751$	−15.8840	−0.579616	−0.289808	+	0.957085i	$0.593591\pi$
$752$	0	0
$753$	17.6904	0.644675
$754$	0	0
$755$	19.0328	0.692674
$756$	0	0
$757$	8.32857	0.302707	0.151354	−	0.988480i	$-0.451637\pi$
$757$	8.32857	0.302707	0.151354	+	0.988480i	$0.451637\pi$
$758$	0	0
$759$	−5.19502	−0.188567
$760$	0	0
$761$	−32.6299	−1.18283	−0.591416	−	0.806366i	$-0.701431\pi$
$761$	−32.6299	−1.18283	−0.591416	+	0.806366i	$0.701431\pi$
$762$	0	0
$763$	−11.7828	−0.426566
$764$	0	0
$765$	0.980826	0.0354618
$766$	0	0
$767$	0	0
$768$	0	0
$769$	16.7621	0.604457	0.302228	−	0.953236i	$-0.402270\pi$
$769$	16.7621	0.604457	0.302228	+	0.953236i	$0.402270\pi$
$770$	0	0
$771$	−52.4505	−1.88896
$772$	0	0
$773$	−17.8859	−0.643312	−0.321656	−	0.946857i	$-0.604239\pi$
$773$	−17.8859	−0.643312	−0.321656	+	0.946857i	$0.604239\pi$
$774$	0	0
$775$	0.255235	0.00916832
$776$	0	0
$777$	15.9039	0.570549
$778$	0	0
$779$	4.21394	0.150980
$780$	0	0
$781$	−1.03728	−0.0371168
$782$	0	0
$783$	13.7046	0.489762
$784$	0	0
$785$	18.8113	0.671404
$786$	0	0
$787$	23.9820	0.854866	0.427433	−	0.904047i	$-0.359418\pi$
$787$	23.9820	0.854866	0.427433	+	0.904047i	$0.359418\pi$
$788$	0	0
$789$	−4.19340	−0.149289
$790$	0	0
$791$	7.93667	0.282196
$792$	0	0
$793$	0	0
$794$	0	0
$795$	−12.0846	−0.428598
$796$	0	0
$797$	7.00894	0.248270	0.124135	−	0.992265i	$-0.460384\pi$
$797$	7.00894	0.248270	0.124135	+	0.992265i	$0.460384\pi$
$798$	0	0
$799$	0.830155	0.0293688
$800$	0	0
$801$	9.43858	0.333496
$802$	0	0
$803$	2.50250	0.0883113
$804$	0	0
$805$	−3.35249	−0.118160
$806$	0	0
$807$	−68.7391	−2.41973
$808$	0	0
$809$	−12.8384	−0.451375	−0.225688	−	0.974200i	$-0.572463\pi$
$809$	−12.8384	−0.451375	−0.225688	+	0.974200i	$0.572463\pi$
$810$	0	0
$811$	−25.5350	−0.896655	−0.448328	−	0.893869i	$-0.647980\pi$
$811$	−25.5350	−0.896655	−0.448328	+	0.893869i	$0.647980\pi$
$812$	0	0
$813$	−56.2748	−1.97364
$814$	0	0
$815$	−0.698331	−0.0244615
$816$	0	0
$817$	−4.58363	−0.160361
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−26.6109	−0.928728	−0.464364	−	0.885644i	$-0.653717\pi$
$821$	−26.6109	−0.928728	−0.464364	+	0.885644i	$0.653717\pi$
$822$	0	0
$823$	7.53951	0.262811	0.131405	−	0.991329i	$-0.458051\pi$
$823$	7.53951	0.262811	0.131405	+	0.991329i	$0.458051\pi$
$824$	0	0
$825$	−6.43473	−0.224029
$826$	0	0
$827$	4.45894	0.155053	0.0775263	−	0.996990i	$-0.475298\pi$
$827$	4.45894	0.155053	0.0775263	+	0.996990i	$0.475298\pi$
$828$	0	0
$829$	34.4919	1.19795	0.598976	−	0.800767i	$-0.295574\pi$
$829$	34.4919	1.19795	0.598976	+	0.800767i	$0.295574\pi$
$830$	0	0
$831$	6.36633	0.220846
$832$	0	0
$833$	0.179097	0.00620534
$834$	0	0
$835$	−15.9849	−0.553182
$836$	0	0
$837$	−0.385263	−0.0133166
$838$	0	0
$839$	12.4388	0.429436	0.214718	−	0.976676i	$-0.431117\pi$
$839$	12.4388	0.429436	0.214718	+	0.976676i	$0.431117\pi$
$840$	0	0
$841$	−23.3380	−0.804759
$842$	0	0
$843$	53.6053	1.84627
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−10.6459	−0.365798
$848$	0	0
$849$	8.87001	0.304418
$850$	0	0
$851$	−17.2865	−0.592573
$852$	0	0
$853$	−20.6629	−0.707486	−0.353743	−	0.935343i	$-0.615091\pi$
$853$	−20.6629	−0.707486	−0.353743	+	0.935343i	$0.615091\pi$
$854$	0	0
$855$	4.93630	0.168818
$856$	0	0
$857$	31.4121	1.07302	0.536509	−	0.843894i	$-0.319743\pi$
$857$	31.4121	1.07302	0.536509	+	0.843894i	$0.319743\pi$
$858$	0	0
$859$	−18.7844	−0.640915	−0.320458	−	0.947263i	$-0.603837\pi$
$859$	−18.7844	−0.640915	−0.320458	+	0.947263i	$0.603837\pi$
$860$	0	0
$861$	13.2498	0.451551
$862$	0	0
$863$	−30.3150	−1.03194	−0.515968	−	0.856608i	$-0.672568\pi$
$863$	−30.3150	−1.03194	−0.515968	+	0.856608i	$0.672568\pi$
$864$	0	0
$865$	−13.7242	−0.466638
$866$	0	0
$867$	−48.0890	−1.63319
$868$	0	0
$869$	6.15514	0.208799
$870$	0	0
$871$	0	0
$872$	0	0
$873$	76.7979	2.59921
$874$	0	0
$875$	−9.59400	−0.324336
$876$	0	0
$877$	56.0288	1.89196	0.945979	−	0.324227i	$-0.105104\pi$
$877$	56.0288	1.89196	0.945979	+	0.324227i	$0.105104\pi$
$878$	0	0
$879$	82.0279	2.76673
$880$	0	0
$881$	54.5718	1.83857	0.919285	−	0.393593i	$-0.128768\pi$
$881$	54.5718	1.83857	0.919285	+	0.393593i	$0.128768\pi$
$882$	0	0
$883$	−14.7425	−0.496124	−0.248062	−	0.968744i	$-0.579794\pi$
$883$	−14.7425	−0.496124	−0.248062	+	0.968744i	$0.579794\pi$
$884$	0	0
$885$	25.4977	0.857096
$886$	0	0
$887$	−23.7979	−0.799055	−0.399528	−	0.916721i	$-0.630826\pi$
$887$	−23.7979	−0.799055	−0.399528	+	0.916721i	$0.630826\pi$
$888$	0	0
$889$	−9.19647	−0.308440
$890$	0	0
$891$	0.729735	0.0244470
$892$	0	0
$893$	4.17800	0.139812
$894$	0	0
$895$	−22.6147	−0.755927
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−0.159170	−0.00530862
$900$	0	0
$901$	−0.701709	−0.0233773
$902$	0	0
$903$	−14.4122	−0.479606
$904$	0	0
$905$	−20.1311	−0.669181
$906$	0	0
$907$	−52.0706	−1.72897	−0.864487	−	0.502654i	$-0.832357\pi$
$907$	−52.0706	−1.72897	−0.864487	+	0.502654i	$0.832357\pi$
$908$	0	0
$909$	56.2572	1.86593
$910$	0	0
$911$	41.3085	1.36861	0.684306	−	0.729195i	$-0.260105\pi$
$911$	41.3085	1.36861	0.684306	+	0.729195i	$0.260105\pi$
$912$	0	0
$913$	2.12434	0.0703054
$914$	0	0
$915$	40.6032	1.34230
$916$	0	0
$917$	15.7858	0.521291
$918$	0	0
$919$	−8.32080	−0.274478	−0.137239	−	0.990538i	$-0.543823\pi$
$919$	−8.32080	−0.274478	−0.137239	+	0.990538i	$0.543823\pi$
$920$	0	0
$921$	−91.1497	−3.00349
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−21.4117	−0.704012
$926$	0	0
$927$	23.5114	0.772215
$928$	0	0
$929$	40.1526	1.31737	0.658683	−	0.752421i	$-0.271114\pi$
$929$	40.1526	1.31737	0.658683	+	0.752421i	$0.271114\pi$
$930$	0	0
$931$	0.901358	0.0295408
$932$	0	0
$933$	−22.1705	−0.725830
$934$	0	0
$935$	0.115980	0.00379297
$936$	0	0
$937$	44.1238	1.44146	0.720730	−	0.693215i	$-0.243807\pi$
$937$	44.1238	1.44146	0.720730	+	0.693215i	$0.243807\pi$
$938$	0	0
$939$	−34.4393	−1.12388
$940$	0	0
$941$	−6.77543	−0.220873	−0.110436	−	0.993883i	$-0.535225\pi$
$941$	−6.77543	−0.220873	−0.110436	+	0.993883i	$0.535225\pi$
$942$	0	0
$943$	−14.4016	−0.468982
$944$	0	0
$945$	6.26799	0.203898
$946$	0	0
$947$	−21.6953	−0.705001	−0.352501	−	0.935812i	$-0.614669\pi$
$947$	−21.6953	−0.705001	−0.352501	+	0.935812i	$0.614669\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	−67.0427	−2.17401
$952$	0	0
$953$	26.3993	0.855157	0.427578	−	0.903978i	$-0.359367\pi$
$953$	26.3993	0.855157	0.427578	+	0.903978i	$0.359367\pi$
$954$	0	0
$955$	3.78376	0.122440
$956$	0	0
$957$	4.01283	0.129717
$958$	0	0
$959$	2.38414	0.0769879
$960$	0	0
$961$	−30.9955	−0.999856
$962$	0	0
$963$	−38.5145	−1.24111
$964$	0	0
$965$	−29.9106	−0.962855
$966$	0	0
$967$	−58.9553	−1.89587	−0.947937	−	0.318457i	$-0.896835\pi$
$967$	−58.9553	−1.89587	−0.947937	+	0.318457i	$0.896835\pi$
$968$	0	0
$969$	0.457512	0.0146974
$970$	0	0
$971$	−59.3437	−1.90443	−0.952216	−	0.305427i	$-0.901201\pi$
$971$	−59.3437	−1.90443	−0.952216	+	0.305427i	$0.901201\pi$
$972$	0	0
$973$	−19.5994	−0.628330
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−51.0853	−1.63436	−0.817182	−	0.576380i	$-0.804465\pi$
$977$	−51.0853	−1.63436	−0.817182	+	0.576380i	$0.804465\pi$
$978$	0	0
$979$	1.11609	0.0356704
$980$	0	0
$981$	−59.2933	−1.89309
$982$	0	0
$983$	30.6800	0.978540	0.489270	−	0.872132i	$-0.337263\pi$
$983$	30.6800	0.978540	0.489270	+	0.872132i	$0.337263\pi$
$984$	0	0
$985$	−19.6579	−0.626353
$986$	0	0
$987$	13.1368	0.418148
$988$	0	0
$989$	15.6651	0.498120
$990$	0	0
$991$	−16.8960	−0.536718	−0.268359	−	0.963319i	$-0.586481\pi$
$991$	−16.8960	−0.536718	−0.268359	+	0.963319i	$0.586481\pi$
$992$	0	0
$993$	−15.7691	−0.500417
$994$	0	0
$995$	18.0573	0.572456
$996$	0	0
$997$	10.0971	0.319780	0.159890	−	0.987135i	$-0.448886\pi$
$997$	10.0971	0.319780	0.159890	+	0.987135i	$0.448886\pi$
$998$	0	0
$999$	32.3197	1.02255

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9464.2.a.bs.1.15	yes	15
13.12	even	2		9464.2.a.br.1.15	✓	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9464.2.a.br.1.15	✓	15	13.12	even	2
9464.2.a.bs.1.15	yes	15	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:	\( N \)	\(=\)	\( 9464 = 2^{3} \cdot 7 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9464.a (trivial)

\(f(q)\)	\(=\)	\(q+2.83411 q^{3} +1.08830 q^{5} +1.00000 q^{7} +5.03219 q^{9} +0.595045 q^{11} +3.08435 q^{15} +0.179097 q^{17} +0.901358 q^{19} +2.83411 q^{21} -3.08049 q^{23} -3.81561 q^{25} +5.75945 q^{27} +2.37949 q^{29} -0.0668924 q^{31} +1.68642 q^{33} +1.08830 q^{35} +5.61160 q^{37} +4.67510 q^{41} -5.08524 q^{43} +5.47651 q^{45} +4.63523 q^{47} +1.00000 q^{49} +0.507580 q^{51} -3.91804 q^{53} +0.647585 q^{55} +2.55455 q^{57} +8.26679 q^{59} +13.1642 q^{61} +5.03219 q^{63} +10.4840 q^{67} -8.73046 q^{69} -1.74320 q^{71} +4.20557 q^{73} -10.8139 q^{75} +0.595045 q^{77} +10.3440 q^{79} +1.22635 q^{81} +3.57005 q^{83} +0.194910 q^{85} +6.74375 q^{87} +1.87564 q^{89} -0.189580 q^{93} +0.980945 q^{95} +15.2613 q^{97} +2.99438 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q - 3 q^{3} + 4 q^{5} + 15 q^{7} + 16 q^{9} + 15 q^{11} + 8 q^{15} + 2 q^{17} + 13 q^{19} - 3 q^{21} - 10 q^{23} + 23 q^{25} - 9 q^{27} + 25 q^{29} + 19 q^{31} - 24 q^{33} + 4 q^{35} - 2 q^{37} + 30 q^{41}+ \cdots + 70 q^{99}+O(q^{100}) \) 15 * q - 3 * q^3 + 4 * q^5 + 15 * q^7 + 16 * q^9 + 15 * q^11 + 8 * q^15 + 2 * q^17 + 13 * q^19 - 3 * q^21 - 10 * q^23 + 23 * q^25 - 9 * q^27 + 25 * q^29 + 19 * q^31 - 24 * q^33 + 4 * q^35 - 2 * q^37 + 30 * q^41 + 15 * q^43 - 2 * q^45 + 13 * q^47 + 15 * q^49 + q^51 + 62 * q^53 - 3 * q^55 - 2 * q^57 + 22 * q^59 - q^61 + 16 * q^63 + 29 * q^67 + 32 * q^69 + 20 * q^71 + 21 * q^73 - 46 * q^75 + 15 * q^77 + 29 * q^79 + 7 * q^81 + 23 * q^83 + 8 * q^85 - 22 * q^87 + 9 * q^89 + 10 * q^93 + 28 * q^95 + 11 * q^97 + 70 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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