LMFDB - Embedded newform 9464.2.a.br.1.12

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:	$1$
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			1.12
Root			$-2.05925$ 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)

$f(q)$	$=$	$q+2.05925 q^{3} +1.64516 q^{5} -1.00000 q^{7} +1.24052 q^{9} +1.46249 q^{11} +3.38781 q^{15} +1.88478 q^{17} -6.90801 q^{19} -2.05925 q^{21} -0.533833 q^{23} -2.29344 q^{25} -3.62321 q^{27} -5.48263 q^{29} -10.3800 q^{31} +3.01163 q^{33} -1.64516 q^{35} -1.27868 q^{37} +0.723958 q^{41} -2.04797 q^{43} +2.04086 q^{45} -0.882574 q^{47} +1.00000 q^{49} +3.88124 q^{51} -7.05797 q^{53} +2.40603 q^{55} -14.2253 q^{57} +2.76746 q^{59} +12.3097 q^{61} -1.24052 q^{63} -10.3011 q^{67} -1.09930 q^{69} -7.47426 q^{71} -1.90864 q^{73} -4.72277 q^{75} -1.46249 q^{77} -1.00777 q^{79} -11.1827 q^{81} -10.5593 q^{83} +3.10077 q^{85} -11.2901 q^{87} -1.43994 q^{89} -21.3750 q^{93} -11.3648 q^{95} -2.45245 q^{97} +1.81425 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	2.05925	1.18891	0.594455	−	0.804129i	$-0.297368\pi$
$3$	2.05925	1.18891	0.594455	+	0.804129i	$0.297368\pi$
$4$	0	0
$5$	1.64516	0.735739	0.367870	−	0.929877i	$-0.380087\pi$
$5$	1.64516	0.735739	0.367870	+	0.929877i	$0.380087\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.24052	0.413508
$10$	0	0
$11$	1.46249	0.440956	0.220478	−	0.975392i	$-0.429238\pi$
$11$	1.46249	0.440956	0.220478	+	0.975392i	$0.429238\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	3.38781	0.874728
$16$	0	0
$17$	1.88478	0.457126	0.228563	−	0.973529i	$-0.426597\pi$
$17$	1.88478	0.457126	0.228563	+	0.973529i	$0.426597\pi$
$18$	0	0
$19$	−6.90801	−1.58481	−0.792403	−	0.609998i	$-0.791170\pi$
$19$	−6.90801	−1.58481	−0.792403	+	0.609998i	$0.791170\pi$
$20$	0	0
$21$	−2.05925	−0.449366
$22$	0	0
$23$	−0.533833	−0.111312	−0.0556559	−	0.998450i	$-0.517725\pi$
$23$	−0.533833	−0.111312	−0.0556559	+	0.998450i	$0.517725\pi$
$24$	0	0
$25$	−2.29344	−0.458688
$26$	0	0
$27$	−3.62321	−0.697286
$28$	0	0
$29$	−5.48263	−1.01810	−0.509050	−	0.860737i	$-0.670003\pi$
$29$	−5.48263	−1.01810	−0.509050	+	0.860737i	$0.670003\pi$
$30$	0	0
$31$	−10.3800	−1.86430	−0.932150	−	0.362072i	$-0.882070\pi$
$31$	−10.3800	−1.86430	−0.932150	+	0.362072i	$0.882070\pi$
$32$	0	0
$33$	3.01163	0.524257
$34$	0	0
$35$	−1.64516	−0.278083
$36$	0	0
$37$	−1.27868	−0.210214	−0.105107	−	0.994461i	$-0.533518\pi$
$37$	−1.27868	−0.210214	−0.105107	+	0.994461i	$0.533518\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	0.723958	0.113063	0.0565316	−	0.998401i	$-0.481996\pi$
$41$	0.723958	0.113063	0.0565316	+	0.998401i	$0.481996\pi$
$42$	0	0
$43$	−2.04797	−0.312312	−0.156156	−	0.987732i	$-0.549910\pi$
$43$	−2.04797	−0.312312	−0.156156	+	0.987732i	$0.549910\pi$
$44$	0	0
$45$	2.04086	0.304234
$46$	0	0
$47$	−0.882574	−0.128737	−0.0643684	−	0.997926i	$-0.520503\pi$
$47$	−0.882574	−0.128737	−0.0643684	+	0.997926i	$0.520503\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	3.88124	0.543482
$52$	0	0
$53$	−7.05797	−0.969487	−0.484744	−	0.874656i	$-0.661087\pi$
$53$	−7.05797	−0.969487	−0.484744	+	0.874656i	$0.661087\pi$
$54$	0	0
$55$	2.40603	0.324429
$56$	0	0
$57$	−14.2253	−1.88419
$58$	0	0
$59$	2.76746	0.360292	0.180146	−	0.983640i	$-0.442343\pi$
$59$	2.76746	0.360292	0.180146	+	0.983640i	$0.442343\pi$
$60$	0	0
$61$	12.3097	1.57609	0.788047	−	0.615615i	$-0.211092\pi$
$61$	12.3097	1.57609	0.788047	+	0.615615i	$0.211092\pi$
$62$	0	0
$63$	−1.24052	−0.156291
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−10.3011	−1.25848	−0.629239	−	0.777212i	$-0.716633\pi$
$67$	−10.3011	−1.25848	−0.629239	+	0.777212i	$0.716633\pi$
$68$	0	0
$69$	−1.09930	−0.132340
$70$	0	0
$71$	−7.47426	−0.887032	−0.443516	−	0.896266i	$-0.646269\pi$
$71$	−7.47426	−0.887032	−0.443516	+	0.896266i	$0.646269\pi$
$72$	0	0
$73$	−1.90864	−0.223389	−0.111695	−	0.993743i	$-0.535628\pi$
$73$	−1.90864	−0.223389	−0.111695	+	0.993743i	$0.535628\pi$
$74$	0	0
$75$	−4.72277	−0.545338
$76$	0	0
$77$	−1.46249	−0.166666
$78$	0	0
$79$	−1.00777	−0.113383	−0.0566914	−	0.998392i	$-0.518055\pi$
$79$	−1.00777	−0.113383	−0.0566914	+	0.998392i	$0.518055\pi$
$80$	0	0
$81$	−11.1827	−1.24252
$82$	0	0
$83$	−10.5593	−1.15903	−0.579516	−	0.814961i	$-0.696758\pi$
$83$	−10.5593	−1.15903	−0.579516	+	0.814961i	$0.696758\pi$
$84$	0	0
$85$	3.10077	0.336326
$86$	0	0
$87$	−11.2901	−1.21043
$88$	0	0
$89$	−1.43994	−0.152633	−0.0763166	−	0.997084i	$-0.524316\pi$
$89$	−1.43994	−0.152633	−0.0763166	+	0.997084i	$0.524316\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−21.3750	−2.21649
$94$	0	0
$95$	−11.3648	−1.16600
$96$	0	0
$97$	−2.45245	−0.249008	−0.124504	−	0.992219i	$-0.539734\pi$
$97$	−2.45245	−0.249008	−0.124504	+	0.992219i	$0.539734\pi$
$98$	0	0
$99$	1.81425	0.182339
$100$	0	0
$101$	4.97730	0.495260	0.247630	−	0.968855i	$-0.420348\pi$
$101$	4.97730	0.495260	0.247630	+	0.968855i	$0.420348\pi$
$102$	0	0
$103$	5.78198	0.569716	0.284858	−	0.958570i	$-0.408054\pi$
$103$	5.78198	0.569716	0.284858	+	0.958570i	$0.408054\pi$
$104$	0	0
$105$	−3.38781	−0.330616
$106$	0	0
$107$	5.91831	0.572145	0.286072	−	0.958208i	$-0.407650\pi$
$107$	5.91831	0.572145	0.286072	+	0.958208i	$0.407650\pi$
$108$	0	0
$109$	−5.55734	−0.532296	−0.266148	−	0.963932i	$-0.585751\pi$
$109$	−5.55734	−0.532296	−0.266148	+	0.963932i	$0.585751\pi$
$110$	0	0
$111$	−2.63313	−0.249925
$112$	0	0
$113$	−4.65193	−0.437616	−0.218808	−	0.975768i	$-0.570217\pi$
$113$	−4.65193	−0.437616	−0.218808	+	0.975768i	$0.570217\pi$
$114$	0	0
$115$	−0.878242	−0.0818965
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−1.88478	−0.172778
$120$	0	0
$121$	−8.86114	−0.805558
$122$	0	0
$123$	1.49081	0.134422
$124$	0	0
$125$	−11.9989	−1.07321
$126$	0	0
$127$	17.0060	1.50904	0.754518	−	0.656279i	$-0.227870\pi$
$127$	17.0060	1.50904	0.754518	+	0.656279i	$0.227870\pi$
$128$	0	0
$129$	−4.21728	−0.371311
$130$	0	0
$131$	−16.3314	−1.42688	−0.713442	−	0.700714i	$-0.752865\pi$
$131$	−16.3314	−1.42688	−0.713442	+	0.700714i	$0.752865\pi$
$132$	0	0
$133$	6.90801	0.599000
$134$	0	0
$135$	−5.96077	−0.513021
$136$	0	0
$137$	16.0585	1.37197	0.685983	−	0.727617i	$-0.259372\pi$
$137$	16.0585	1.37197	0.685983	+	0.727617i	$0.259372\pi$
$138$	0	0
$139$	12.3584	1.04823	0.524113	−	0.851649i	$-0.324397\pi$
$139$	12.3584	1.04823	0.524113	+	0.851649i	$0.324397\pi$
$140$	0	0
$141$	−1.81744	−0.153056
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−9.01983	−0.749056
$146$	0	0
$147$	2.05925	0.169844
$148$	0	0
$149$	7.65583	0.627190	0.313595	−	0.949557i	$-0.398467\pi$
$149$	7.65583	0.627190	0.313595	+	0.949557i	$0.398467\pi$
$150$	0	0
$151$	16.4042	1.33495	0.667476	−	0.744632i	$-0.267375\pi$
$151$	16.4042	1.33495	0.667476	+	0.744632i	$0.267375\pi$
$152$	0	0
$153$	2.33812	0.189025
$154$	0	0
$155$	−17.0768	−1.37164
$156$	0	0
$157$	15.6073	1.24560	0.622800	−	0.782381i	$-0.285995\pi$
$157$	15.6073	1.24560	0.622800	+	0.782381i	$0.285995\pi$
$158$	0	0
$159$	−14.5342	−1.15263
$160$	0	0
$161$	0.533833	0.0420719
$162$	0	0
$163$	7.83984	0.614063	0.307032	−	0.951699i	$-0.400664\pi$
$163$	7.83984	0.614063	0.307032	+	0.951699i	$0.400664\pi$
$164$	0	0
$165$	4.95462	0.385717
$166$	0	0
$167$	−24.4428	−1.89144	−0.945721	−	0.324980i	$-0.894643\pi$
$167$	−24.4428	−1.89144	−0.945721	+	0.324980i	$0.894643\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	−8.56955	−0.655330
$172$	0	0
$173$	1.78780	0.135924	0.0679618	−	0.997688i	$-0.478350\pi$
$173$	1.78780	0.135924	0.0679618	+	0.997688i	$0.478350\pi$
$174$	0	0
$175$	2.29344	0.173368
$176$	0	0
$177$	5.69890	0.428355
$178$	0	0
$179$	4.37599	0.327077	0.163538	−	0.986537i	$-0.447709\pi$
$179$	4.37599	0.327077	0.163538	+	0.986537i	$0.447709\pi$
$180$	0	0
$181$	3.08125	0.229028	0.114514	−	0.993422i	$-0.463469\pi$
$181$	3.08125	0.229028	0.114514	+	0.993422i	$0.463469\pi$
$182$	0	0
$183$	25.3488	1.87383
$184$	0	0
$185$	−2.10364	−0.154662
$186$	0	0
$187$	2.75646	0.201573
$188$	0	0
$189$	3.62321	0.263549
$190$	0	0
$191$	7.60189	0.550054	0.275027	−	0.961437i	$-0.411313\pi$
$191$	7.60189	0.550054	0.275027	+	0.961437i	$0.411313\pi$
$192$	0	0
$193$	8.40772	0.605201	0.302600	−	0.953118i	$-0.402145\pi$
$193$	8.40772	0.605201	0.302600	+	0.953118i	$0.402145\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	13.5429	0.964892	0.482446	−	0.875926i	$-0.339749\pi$
$197$	13.5429	0.964892	0.482446	+	0.875926i	$0.339749\pi$
$198$	0	0
$199$	−4.12662	−0.292528	−0.146264	−	0.989246i	$-0.546725\pi$
$199$	−4.12662	−0.292528	−0.146264	+	0.989246i	$0.546725\pi$
$200$	0	0
$201$	−21.2125	−1.49622
$202$	0	0
$203$	5.48263	0.384805
$204$	0	0
$205$	1.19103	0.0831851
$206$	0	0
$207$	−0.662233	−0.0460283
$208$	0	0
$209$	−10.1029	−0.698829
$210$	0	0
$211$	25.7743	1.77438	0.887190	−	0.461405i	$-0.152654\pi$
$211$	25.7743	1.77438	0.887190	+	0.461405i	$0.152654\pi$
$212$	0	0
$213$	−15.3914	−1.05460
$214$	0	0
$215$	−3.36924	−0.229780
$216$	0	0
$217$	10.3800	0.704639
$218$	0	0
$219$	−3.93037	−0.265590
$220$	0	0
$221$	0	0
$222$	0	0
$223$	5.78619	0.387472	0.193736	−	0.981054i	$-0.437940\pi$
$223$	5.78619	0.387472	0.193736	+	0.981054i	$0.437940\pi$
$224$	0	0
$225$	−2.84506	−0.189671
$226$	0	0
$227$	−29.5039	−1.95824	−0.979122	−	0.203274i	$-0.934842\pi$
$227$	−29.5039	−1.95824	−0.979122	+	0.203274i	$0.934842\pi$
$228$	0	0
$229$	−21.5561	−1.42447	−0.712233	−	0.701943i	$-0.752316\pi$
$229$	−21.5561	−1.42447	−0.712233	+	0.701943i	$0.752316\pi$
$230$	0	0
$231$	−3.01163	−0.198151
$232$	0	0
$233$	−6.61589	−0.433422	−0.216711	−	0.976236i	$-0.569533\pi$
$233$	−6.61589	−0.433422	−0.216711	+	0.976236i	$0.569533\pi$
$234$	0	0
$235$	−1.45198	−0.0947167
$236$	0	0
$237$	−2.07525	−0.134802
$238$	0	0
$239$	−13.5710	−0.877834	−0.438917	−	0.898528i	$-0.644638\pi$
$239$	−13.5710	−0.877834	−0.438917	+	0.898528i	$0.644638\pi$
$240$	0	0
$241$	4.16102	0.268035	0.134017	−	0.990979i	$-0.457212\pi$
$241$	4.16102	0.268035	0.134017	+	0.990979i	$0.457212\pi$
$242$	0	0
$243$	−12.1583	−0.779958
$244$	0	0
$245$	1.64516	0.105106
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−21.7442	−1.37799
$250$	0	0
$251$	19.7147	1.24438	0.622192	−	0.782865i	$-0.286242\pi$
$251$	19.7147	1.24438	0.622192	+	0.782865i	$0.286242\pi$
$252$	0	0
$253$	−0.780723	−0.0490836
$254$	0	0
$255$	6.38527	0.399861
$256$	0	0
$257$	9.25138	0.577086	0.288543	−	0.957467i	$-0.406829\pi$
$257$	9.25138	0.577086	0.288543	+	0.957467i	$0.406829\pi$
$258$	0	0
$259$	1.27868	0.0794533
$260$	0	0
$261$	−6.80134	−0.420992
$262$	0	0
$263$	0.0391468	0.00241389	0.00120695	−	0.999999i	$-0.499616\pi$
$263$	0.0391468	0.00241389	0.00120695	+	0.999999i	$0.499616\pi$
$264$	0	0
$265$	−11.6115	−0.713290
$266$	0	0
$267$	−2.96520	−0.181467
$268$	0	0
$269$	21.5482	1.31382	0.656909	−	0.753970i	$-0.271863\pi$
$269$	21.5482	1.31382	0.656909	+	0.753970i	$0.271863\pi$
$270$	0	0
$271$	−23.8900	−1.45121	−0.725607	−	0.688109i	$-0.758441\pi$
$271$	−23.8900	−1.45121	−0.725607	+	0.688109i	$0.758441\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−3.35412	−0.202261
$276$	0	0
$277$	−13.2263	−0.794691	−0.397345	−	0.917669i	$-0.630068\pi$
$277$	−13.2263	−0.794691	−0.397345	+	0.917669i	$0.630068\pi$
$278$	0	0
$279$	−12.8766	−0.770903
$280$	0	0
$281$	−15.4631	−0.922452	−0.461226	−	0.887283i	$-0.652590\pi$
$281$	−15.4631	−0.922452	−0.461226	+	0.887283i	$0.652590\pi$
$282$	0	0
$283$	5.38667	0.320204	0.160102	−	0.987100i	$-0.448818\pi$
$283$	5.38667	0.320204	0.160102	+	0.987100i	$0.448818\pi$
$284$	0	0
$285$	−23.4030	−1.38627
$286$	0	0
$287$	−0.723958	−0.0427339
$288$	0	0
$289$	−13.4476	−0.791035
$290$	0	0
$291$	−5.05021	−0.296049
$292$	0	0
$293$	27.1875	1.58831	0.794156	−	0.607714i	$-0.207913\pi$
$293$	27.1875	1.58831	0.794156	+	0.607714i	$0.207913\pi$
$294$	0	0
$295$	4.55292	0.265081
$296$	0	0
$297$	−5.29889	−0.307473
$298$	0	0
$299$	0	0
$300$	0	0
$301$	2.04797	0.118043
$302$	0	0
$303$	10.2495	0.588820
$304$	0	0
$305$	20.2514	1.15959
$306$	0	0
$307$	15.9548	0.910591	0.455295	−	0.890340i	$-0.349534\pi$
$307$	15.9548	0.910591	0.455295	+	0.890340i	$0.349534\pi$
$308$	0	0
$309$	11.9066	0.677341
$310$	0	0
$311$	−8.01609	−0.454551	−0.227276	−	0.973830i	$-0.572982\pi$
$311$	−8.01609	−0.454551	−0.227276	+	0.973830i	$0.572982\pi$
$312$	0	0
$313$	0.842642	0.0476290	0.0238145	−	0.999716i	$-0.492419\pi$
$313$	0.842642	0.0476290	0.0238145	+	0.999716i	$0.492419\pi$
$314$	0	0
$315$	−2.04086	−0.114990
$316$	0	0
$317$	0.538612	0.0302515	0.0151257	−	0.999886i	$-0.495185\pi$
$317$	0.538612	0.0302515	0.0151257	+	0.999886i	$0.495185\pi$
$318$	0	0
$319$	−8.01827	−0.448937
$320$	0	0
$321$	12.1873	0.680229
$322$	0	0
$323$	−13.0201	−0.724456
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−11.4440	−0.632853
$328$	0	0
$329$	0.882574	0.0486579
$330$	0	0
$331$	−29.1497	−1.60221	−0.801106	−	0.598523i	$-0.795755\pi$
$331$	−29.1497	−1.60221	−0.801106	+	0.598523i	$0.795755\pi$
$332$	0	0
$333$	−1.58623	−0.0869250
$334$	0	0
$335$	−16.9470	−0.925911
$336$	0	0
$337$	−23.6345	−1.28746	−0.643728	−	0.765255i	$-0.722613\pi$
$337$	−23.6345	−1.28746	−0.643728	+	0.765255i	$0.722613\pi$
$338$	0	0
$339$	−9.57950	−0.520287
$340$	0	0
$341$	−15.1806	−0.822074
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−1.80852	−0.0973676
$346$	0	0
$347$	−28.1770	−1.51262	−0.756310	−	0.654213i	$-0.773000\pi$
$347$	−28.1770	−1.51262	−0.756310	+	0.654213i	$0.773000\pi$
$348$	0	0
$349$	10.0817	0.539663	0.269832	−	0.962908i	$-0.413032\pi$
$349$	10.0817	0.539663	0.269832	+	0.962908i	$0.413032\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	10.5259	0.560240	0.280120	−	0.959965i	$-0.409626\pi$
$353$	10.5259	0.560240	0.280120	+	0.959965i	$0.409626\pi$
$354$	0	0
$355$	−12.2964	−0.652624
$356$	0	0
$357$	−3.88124	−0.205417
$358$	0	0
$359$	12.7393	0.672355	0.336178	−	0.941799i	$-0.390866\pi$
$359$	12.7393	0.672355	0.336178	+	0.941799i	$0.390866\pi$
$360$	0	0
$361$	28.7206	1.51161
$362$	0	0
$363$	−18.2473	−0.957736
$364$	0	0
$365$	−3.14002	−0.164356
$366$	0	0
$367$	22.8372	1.19209	0.596046	−	0.802950i	$-0.296738\pi$
$367$	22.8372	1.19209	0.596046	+	0.802950i	$0.296738\pi$
$368$	0	0
$369$	0.898087	0.0467525
$370$	0	0
$371$	7.05797	0.366432
$372$	0	0
$373$	18.2193	0.943358	0.471679	−	0.881770i	$-0.343648\pi$
$373$	18.2193	0.943358	0.471679	+	0.881770i	$0.343648\pi$
$374$	0	0
$375$	−24.7088	−1.27596
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−8.86458	−0.455343	−0.227671	−	0.973738i	$-0.573111\pi$
$379$	−8.86458	−0.455343	−0.227671	+	0.973738i	$0.573111\pi$
$380$	0	0
$381$	35.0196	1.79411
$382$	0	0
$383$	−6.59484	−0.336981	−0.168490	−	0.985703i	$-0.553889\pi$
$383$	−6.59484	−0.336981	−0.168490	+	0.985703i	$0.553889\pi$
$384$	0	0
$385$	−2.40603	−0.122622
$386$	0	0
$387$	−2.54055	−0.129143
$388$	0	0
$389$	−19.3942	−0.983327	−0.491664	−	0.870785i	$-0.663611\pi$
$389$	−19.3942	−0.983327	−0.491664	+	0.870785i	$0.663611\pi$
$390$	0	0
$391$	−1.00616	−0.0508836
$392$	0	0
$393$	−33.6306	−1.69644
$394$	0	0
$395$	−1.65794	−0.0834201
$396$	0	0
$397$	−22.2360	−1.11599	−0.557996	−	0.829843i	$-0.688430\pi$
$397$	−22.2360	−1.11599	−0.557996	+	0.829843i	$0.688430\pi$
$398$	0	0
$399$	14.2253	0.712158
$400$	0	0
$401$	1.52525	0.0761674	0.0380837	−	0.999275i	$-0.487875\pi$
$401$	1.52525	0.0761674	0.0380837	+	0.999275i	$0.487875\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	−18.3973	−0.914170
$406$	0	0
$407$	−1.87005	−0.0926949
$408$	0	0
$409$	16.9520	0.838225	0.419112	−	0.907934i	$-0.362341\pi$
$409$	16.9520	0.838225	0.419112	+	0.907934i	$0.362341\pi$
$410$	0	0
$411$	33.0684	1.63115
$412$	0	0
$413$	−2.76746	−0.136178
$414$	0	0
$415$	−17.3718	−0.852746
$416$	0	0
$417$	25.4491	1.24625
$418$	0	0
$419$	−4.18688	−0.204542	−0.102271	−	0.994757i	$-0.532611\pi$
$419$	−4.18688	−0.204542	−0.102271	+	0.994757i	$0.532611\pi$
$420$	0	0
$421$	17.2196	0.839234	0.419617	−	0.907701i	$-0.362164\pi$
$421$	17.2196	0.839234	0.419617	+	0.907701i	$0.362164\pi$
$422$	0	0
$423$	−1.09485	−0.0532337
$424$	0	0
$425$	−4.32263	−0.209678
$426$	0	0
$427$	−12.3097	−0.595708
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−31.4854	−1.51660	−0.758299	−	0.651907i	$-0.773969\pi$
$431$	−31.4854	−1.51660	−0.758299	+	0.651907i	$0.773969\pi$
$432$	0	0
$433$	5.59350	0.268807	0.134403	−	0.990927i	$-0.457088\pi$
$433$	5.59350	0.268807	0.134403	+	0.990927i	$0.457088\pi$
$434$	0	0
$435$	−18.5741	−0.890560
$436$	0	0
$437$	3.68772	0.176408
$438$	0	0
$439$	24.7585	1.18166	0.590830	−	0.806796i	$-0.298801\pi$
$439$	24.7585	1.18166	0.590830	+	0.806796i	$0.298801\pi$
$440$	0	0
$441$	1.24052	0.0590726
$442$	0	0
$443$	26.9828	1.28199	0.640997	−	0.767544i	$-0.278521\pi$
$443$	26.9828	1.28199	0.640997	+	0.767544i	$0.278521\pi$
$444$	0	0
$445$	−2.36893	−0.112298
$446$	0	0
$447$	15.7653	0.745672
$448$	0	0
$449$	−24.5375	−1.15800	−0.578999	−	0.815328i	$-0.696556\pi$
$449$	−24.5375	−1.15800	−0.578999	+	0.815328i	$0.696556\pi$
$450$	0	0
$451$	1.05878	0.0498559
$452$	0	0
$453$	33.7803	1.58714
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−16.9361	−0.792239	−0.396120	−	0.918199i	$-0.629643\pi$
$457$	−16.9361	−0.792239	−0.396120	+	0.918199i	$0.629643\pi$
$458$	0	0
$459$	−6.82895	−0.318748
$460$	0	0
$461$	−17.2753	−0.804591	−0.402296	−	0.915510i	$-0.631788\pi$
$461$	−17.2753	−0.804591	−0.402296	+	0.915510i	$0.631788\pi$
$462$	0	0
$463$	−6.13080	−0.284922	−0.142461	−	0.989800i	$-0.545502\pi$
$463$	−6.13080	−0.284922	−0.142461	+	0.989800i	$0.545502\pi$
$464$	0	0
$465$	−35.1654	−1.63076
$466$	0	0
$467$	−27.3484	−1.26553	−0.632765	−	0.774344i	$-0.718080\pi$
$467$	−27.3484	−1.26553	−0.632765	+	0.774344i	$0.718080\pi$
$468$	0	0
$469$	10.3011	0.475660
$470$	0	0
$471$	32.1394	1.48091
$472$	0	0
$473$	−2.99512	−0.137716
$474$	0	0
$475$	15.8431	0.726930
$476$	0	0
$477$	−8.75559	−0.400891
$478$	0	0
$479$	−8.24677	−0.376804	−0.188402	−	0.982092i	$-0.560331\pi$
$479$	−8.24677	−0.376804	−0.188402	+	0.982092i	$0.560331\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	1.09930	0.0500197
$484$	0	0
$485$	−4.03468	−0.183205
$486$	0	0
$487$	32.2050	1.45935	0.729674	−	0.683795i	$-0.239672\pi$
$487$	32.2050	1.45935	0.729674	+	0.683795i	$0.239672\pi$
$488$	0	0
$489$	16.1442	0.730066
$490$	0	0
$491$	6.91256	0.311959	0.155980	−	0.987760i	$-0.450147\pi$
$491$	6.91256	0.311959	0.155980	+	0.987760i	$0.450147\pi$
$492$	0	0
$493$	−10.3336	−0.465400
$494$	0	0
$495$	2.98473	0.134154
$496$	0	0
$497$	7.47426	0.335267
$498$	0	0
$499$	23.7281	1.06222	0.531108	−	0.847304i	$-0.321776\pi$
$499$	23.7281	1.06222	0.531108	+	0.847304i	$0.321776\pi$
$500$	0	0
$501$	−50.3340	−2.24876
$502$	0	0
$503$	27.7891	1.23906	0.619528	−	0.784974i	$-0.287324\pi$
$503$	27.7891	1.23906	0.619528	+	0.784974i	$0.287324\pi$
$504$	0	0
$505$	8.18848	0.364382
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−34.7614	−1.54077	−0.770386	−	0.637577i	$-0.779937\pi$
$509$	−34.7614	−1.54077	−0.770386	+	0.637577i	$0.779937\pi$
$510$	0	0
$511$	1.90864	0.0844331
$512$	0	0
$513$	25.0291	1.10506
$514$	0	0
$515$	9.51231	0.419162
$516$	0	0
$517$	−1.29075	−0.0567672
$518$	0	0
$519$	3.68152	0.161601
$520$	0	0
$521$	34.1588	1.49653	0.748263	−	0.663403i	$-0.230888\pi$
$521$	34.1588	1.49653	0.748263	+	0.663403i	$0.230888\pi$
$522$	0	0
$523$	−33.0357	−1.44455	−0.722276	−	0.691605i	$-0.756904\pi$
$523$	−33.0357	−1.44455	−0.722276	+	0.691605i	$0.756904\pi$
$524$	0	0
$525$	4.72277	0.206119
$526$	0	0
$527$	−19.5640	−0.852221
$528$	0	0
$529$	−22.7150	−0.987610
$530$	0	0
$531$	3.43310	0.148984
$532$	0	0
$533$	0	0
$534$	0	0
$535$	9.73659	0.420949
$536$	0	0
$537$	9.01127	0.388865
$538$	0	0
$539$	1.46249	0.0629937
$540$	0	0
$541$	18.1268	0.779330	0.389665	−	0.920957i	$-0.372591\pi$
$541$	18.1268	0.779330	0.389665	+	0.920957i	$0.372591\pi$
$542$	0	0
$543$	6.34508	0.272293
$544$	0	0
$545$	−9.14273	−0.391631
$546$	0	0
$547$	−31.7489	−1.35749	−0.678743	−	0.734375i	$-0.737475\pi$
$547$	−31.7489	−1.35749	−0.678743	+	0.734375i	$0.737475\pi$
$548$	0	0
$549$	15.2705	0.651728
$550$	0	0
$551$	37.8741	1.61349
$552$	0	0
$553$	1.00777	0.0428546
$554$	0	0
$555$	−4.33192	−0.183880
$556$	0	0
$557$	−22.7912	−0.965693	−0.482846	−	0.875705i	$-0.660397\pi$
$557$	−22.7912	−0.965693	−0.482846	+	0.875705i	$0.660397\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	5.67626	0.239652
$562$	0	0
$563$	−3.90099	−0.164407	−0.0822036	−	0.996616i	$-0.526196\pi$
$563$	−3.90099	−0.164407	−0.0822036	+	0.996616i	$0.526196\pi$
$564$	0	0
$565$	−7.65318	−0.321972
$566$	0	0
$567$	11.1827	0.469628
$568$	0	0
$569$	24.7330	1.03686	0.518430	−	0.855120i	$-0.326517\pi$
$569$	24.7330	1.03686	0.518430	+	0.855120i	$0.326517\pi$
$570$	0	0
$571$	0.706622	0.0295712	0.0147856	−	0.999891i	$-0.495293\pi$
$571$	0.706622	0.0295712	0.0147856	+	0.999891i	$0.495293\pi$
$572$	0	0
$573$	15.6542	0.653965
$574$	0	0
$575$	1.22431	0.0510574
$576$	0	0
$577$	−31.9976	−1.33208	−0.666038	−	0.745918i	$-0.732011\pi$
$577$	−31.9976	−1.33208	−0.666038	+	0.745918i	$0.732011\pi$
$578$	0	0
$579$	17.3136	0.719529
$580$	0	0
$581$	10.5593	0.438073
$582$	0	0
$583$	−10.3222	−0.427501
$584$	0	0
$585$	0	0
$586$	0	0
$587$	37.0795	1.53043	0.765217	−	0.643772i	$-0.222632\pi$
$587$	37.0795	1.53043	0.765217	+	0.643772i	$0.222632\pi$
$588$	0	0
$589$	71.7050	2.95455
$590$	0	0
$591$	27.8883	1.14717
$592$	0	0
$593$	41.2754	1.69498	0.847489	−	0.530813i	$-0.178114\pi$
$593$	41.2754	1.69498	0.847489	+	0.530813i	$0.178114\pi$
$594$	0	0
$595$	−3.10077	−0.127119
$596$	0	0
$597$	−8.49775	−0.347790
$598$	0	0
$599$	−10.1108	−0.413116	−0.206558	−	0.978434i	$-0.566226\pi$
$599$	−10.1108	−0.413116	−0.206558	+	0.978434i	$0.566226\pi$
$600$	0	0
$601$	−31.1600	−1.27104	−0.635522	−	0.772083i	$-0.719215\pi$
$601$	−31.1600	−1.27104	−0.635522	+	0.772083i	$0.719215\pi$
$602$	0	0
$603$	−12.7787	−0.520390
$604$	0	0
$605$	−14.5780	−0.592681
$606$	0	0
$607$	−22.3778	−0.908288	−0.454144	−	0.890928i	$-0.650055\pi$
$607$	−22.3778	−0.908288	−0.454144	+	0.890928i	$0.650055\pi$
$608$	0	0
$609$	11.2901	0.457499
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−11.8146	−0.477187	−0.238594	−	0.971119i	$-0.576686\pi$
$613$	−11.8146	−0.477187	−0.238594	+	0.971119i	$0.576686\pi$
$614$	0	0
$615$	2.45263	0.0988996
$616$	0	0
$617$	15.4370	0.621469	0.310734	−	0.950497i	$-0.399425\pi$
$617$	15.4370	0.621469	0.310734	+	0.950497i	$0.399425\pi$
$618$	0	0
$619$	−12.6793	−0.509624	−0.254812	−	0.966991i	$-0.582014\pi$
$619$	−12.6793	−0.509624	−0.254812	+	0.966991i	$0.582014\pi$
$620$	0	0
$621$	1.93419	0.0776162
$622$	0	0
$623$	1.43994	0.0576899
$624$	0	0
$625$	−8.27296	−0.330918
$626$	0	0
$627$	−20.8043	−0.830845
$628$	0	0
$629$	−2.41003	−0.0960942
$630$	0	0
$631$	44.9754	1.79044	0.895222	−	0.445621i	$-0.147017\pi$
$631$	44.9754	1.79044	0.895222	+	0.445621i	$0.147017\pi$
$632$	0	0
$633$	53.0759	2.10958
$634$	0	0
$635$	27.9776	1.11026
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−9.27201	−0.366795
$640$	0	0
$641$	27.9911	1.10558	0.552790	−	0.833320i	$-0.313563\pi$
$641$	27.9911	1.10558	0.552790	+	0.833320i	$0.313563\pi$
$642$	0	0
$643$	−31.0171	−1.22319	−0.611597	−	0.791169i	$-0.709473\pi$
$643$	−31.0171	−1.22319	−0.611597	+	0.791169i	$0.709473\pi$
$644$	0	0
$645$	−6.93811	−0.273188
$646$	0	0
$647$	−26.5959	−1.04559	−0.522796	−	0.852458i	$-0.675111\pi$
$647$	−26.5959	−1.04559	−0.522796	+	0.852458i	$0.675111\pi$
$648$	0	0
$649$	4.04737	0.158873
$650$	0	0
$651$	21.3750	0.837753
$652$	0	0
$653$	−35.1180	−1.37427	−0.687136	−	0.726528i	$-0.741133\pi$
$653$	−35.1180	−1.37427	−0.687136	+	0.726528i	$0.741133\pi$
$654$	0	0
$655$	−26.8679	−1.04981
$656$	0	0
$657$	−2.36771	−0.0923732
$658$	0	0
$659$	−7.64603	−0.297847	−0.148924	−	0.988849i	$-0.547581\pi$
$659$	−7.64603	−0.297847	−0.148924	+	0.988849i	$0.547581\pi$
$660$	0	0
$661$	−28.2973	−1.10064	−0.550318	−	0.834955i	$-0.685493\pi$
$661$	−28.2973	−1.10064	−0.550318	+	0.834955i	$0.685493\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	11.3648	0.440708
$666$	0	0
$667$	2.92681	0.113327
$668$	0	0
$669$	11.9152	0.460669
$670$	0	0
$671$	18.0027	0.694988
$672$	0	0
$673$	−30.2328	−1.16539	−0.582695	−	0.812691i	$-0.698002\pi$
$673$	−30.2328	−1.16539	−0.582695	+	0.812691i	$0.698002\pi$
$674$	0	0
$675$	8.30960	0.319837
$676$	0	0
$677$	−21.6630	−0.832575	−0.416288	−	0.909233i	$-0.636669\pi$
$677$	−21.6630	−0.832575	−0.416288	+	0.909233i	$0.636669\pi$
$678$	0	0
$679$	2.45245	0.0941163
$680$	0	0
$681$	−60.7560	−2.32818
$682$	0	0
$683$	−21.3097	−0.815392	−0.407696	−	0.913118i	$-0.633668\pi$
$683$	−21.3097	−0.815392	−0.407696	+	0.913118i	$0.633668\pi$
$684$	0	0
$685$	26.4188	1.00941
$686$	0	0
$687$	−44.3894	−1.69356
$688$	0	0
$689$	0	0
$690$	0	0
$691$	10.6692	0.405874	0.202937	−	0.979192i	$-0.434951\pi$
$691$	10.6692	0.405874	0.202937	+	0.979192i	$0.434951\pi$
$692$	0	0
$693$	−1.81425	−0.0689176
$694$	0	0
$695$	20.3316	0.771221
$696$	0	0
$697$	1.36450	0.0516842
$698$	0	0
$699$	−13.6238	−0.515299
$700$	0	0
$701$	−9.20768	−0.347769	−0.173885	−	0.984766i	$-0.555632\pi$
$701$	−9.20768	−0.347769	−0.173885	+	0.984766i	$0.555632\pi$
$702$	0	0
$703$	8.83313	0.333148
$704$	0	0
$705$	−2.98999	−0.112610
$706$	0	0
$707$	−4.97730	−0.187191
$708$	0	0
$709$	6.66479	0.250301	0.125151	−	0.992138i	$-0.460059\pi$
$709$	6.66479	0.250301	0.125151	+	0.992138i	$0.460059\pi$
$710$	0	0
$711$	−1.25016	−0.0468847
$712$	0	0
$713$	5.54118	0.207519
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−27.9461	−1.04367
$718$	0	0
$719$	20.5647	0.766935	0.383468	−	0.923554i	$-0.374730\pi$
$719$	20.5647	0.766935	0.383468	+	0.923554i	$0.374730\pi$
$720$	0	0
$721$	−5.78198	−0.215332
$722$	0	0
$723$	8.56858	0.318669
$724$	0	0
$725$	12.5741	0.466989
$726$	0	0
$727$	−22.1146	−0.820184	−0.410092	−	0.912044i	$-0.634503\pi$
$727$	−22.1146	−0.820184	−0.410092	+	0.912044i	$0.634503\pi$
$728$	0	0
$729$	8.51092	0.315219
$730$	0	0
$731$	−3.85996	−0.142766
$732$	0	0
$733$	−15.4786	−0.571714	−0.285857	−	0.958272i	$-0.592278\pi$
$733$	−15.4786	−0.571714	−0.285857	+	0.958272i	$0.592278\pi$
$734$	0	0
$735$	3.38781	0.124961
$736$	0	0
$737$	−15.0652	−0.554933
$738$	0	0
$739$	−36.0964	−1.32783	−0.663914	−	0.747809i	$-0.731106\pi$
$739$	−36.0964	−1.32783	−0.663914	+	0.747809i	$0.731106\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−5.22097	−0.191539	−0.0957693	−	0.995404i	$-0.530531\pi$
$743$	−5.22097	−0.191539	−0.0957693	+	0.995404i	$0.530531\pi$
$744$	0	0
$745$	12.5951	0.461448
$746$	0	0
$747$	−13.0991	−0.479269
$748$	0	0
$749$	−5.91831	−0.216250
$750$	0	0
$751$	−5.06179	−0.184707	−0.0923536	−	0.995726i	$-0.529439\pi$
$751$	−5.06179	−0.184707	−0.0923536	+	0.995726i	$0.529439\pi$
$752$	0	0
$753$	40.5977	1.47946
$754$	0	0
$755$	26.9875	0.982176
$756$	0	0
$757$	51.3460	1.86620	0.933100	−	0.359616i	$-0.117092\pi$
$757$	51.3460	1.86620	0.933100	+	0.359616i	$0.117092\pi$
$758$	0	0
$759$	−1.60771	−0.0583560
$760$	0	0
$761$	39.8180	1.44340	0.721701	−	0.692205i	$-0.243361\pi$
$761$	39.8180	1.44340	0.721701	+	0.692205i	$0.243361\pi$
$762$	0	0
$763$	5.55734	0.201189
$764$	0	0
$765$	3.84658	0.139073
$766$	0	0
$767$	0	0
$768$	0	0
$769$	44.4938	1.60449	0.802244	−	0.596997i	$-0.203639\pi$
$769$	44.4938	1.60449	0.802244	+	0.596997i	$0.203639\pi$
$770$	0	0
$771$	19.0509	0.686103
$772$	0	0
$773$	1.48644	0.0534637	0.0267318	−	0.999643i	$-0.491490\pi$
$773$	1.48644	0.0534637	0.0267318	+	0.999643i	$0.491490\pi$
$774$	0	0
$775$	23.8058	0.855131
$776$	0	0
$777$	2.63313	0.0944629
$778$	0	0
$779$	−5.00110	−0.179183
$780$	0	0
$781$	−10.9310	−0.391142
$782$	0	0
$783$	19.8647	0.709907
$784$	0	0
$785$	25.6766	0.916437
$786$	0	0
$787$	−0.0645794	−0.00230201	−0.00115100	−	0.999999i	$-0.500366\pi$
$787$	−0.0645794	−0.00230201	−0.00115100	+	0.999999i	$0.500366\pi$
$788$	0	0
$789$	0.0806131	0.00286990
$790$	0	0
$791$	4.65193	0.165403
$792$	0	0
$793$	0	0
$794$	0	0
$795$	−23.9111	−0.848038
$796$	0	0
$797$	50.7829	1.79882	0.899411	−	0.437103i	$-0.143995\pi$
$797$	50.7829	1.79882	0.899411	+	0.437103i	$0.143995\pi$
$798$	0	0
$799$	−1.66346	−0.0588489
$800$	0	0
$801$	−1.78628	−0.0631150
$802$	0	0
$803$	−2.79135	−0.0985047
$804$	0	0
$805$	0.878242	0.0309540
$806$	0	0
$807$	44.3733	1.56201
$808$	0	0
$809$	−37.3298	−1.31245	−0.656223	−	0.754567i	$-0.727847\pi$
$809$	−37.3298	−1.31245	−0.656223	+	0.754567i	$0.727847\pi$
$810$	0	0
$811$	11.7849	0.413823	0.206912	−	0.978360i	$-0.433659\pi$
$811$	11.7849	0.413823	0.206912	+	0.978360i	$0.433659\pi$
$812$	0	0
$813$	−49.1956	−1.72536
$814$	0	0
$815$	12.8978	0.451791
$816$	0	0
$817$	14.1474	0.494953
$818$	0	0
$819$	0	0
$820$	0	0
$821$	5.33380	0.186151	0.0930755	−	0.995659i	$-0.470330\pi$
$821$	5.33380	0.186151	0.0930755	+	0.995659i	$0.470330\pi$
$822$	0	0
$823$	−40.1278	−1.39877	−0.699383	−	0.714747i	$-0.746542\pi$
$823$	−40.1278	−1.39877	−0.699383	+	0.714747i	$0.746542\pi$
$824$	0	0
$825$	−6.90698	−0.240470
$826$	0	0
$827$	−43.2429	−1.50370	−0.751851	−	0.659333i	$-0.770839\pi$
$827$	−43.2429	−1.50370	−0.751851	+	0.659333i	$0.770839\pi$
$828$	0	0
$829$	51.3707	1.78418	0.892088	−	0.451861i	$-0.149240\pi$
$829$	51.3707	1.78418	0.892088	+	0.451861i	$0.149240\pi$
$830$	0	0
$831$	−27.2363	−0.944816
$832$	0	0
$833$	1.88478	0.0653038
$834$	0	0
$835$	−40.2124	−1.39161
$836$	0	0
$837$	37.6088	1.29995
$838$	0	0
$839$	10.0033	0.345354	0.172677	−	0.984979i	$-0.444758\pi$
$839$	10.0033	0.345354	0.172677	+	0.984979i	$0.444758\pi$
$840$	0	0
$841$	1.05925	0.0365260
$842$	0	0
$843$	−31.8425	−1.09671
$844$	0	0
$845$	0	0
$846$	0	0
$847$	8.86114	0.304472
$848$	0	0
$849$	11.0925	0.380694
$850$	0	0
$851$	0.682601	0.0233993
$852$	0	0
$853$	−33.1886	−1.13636	−0.568178	−	0.822905i	$-0.692352\pi$
$853$	−33.1886	−1.13636	−0.568178	+	0.822905i	$0.692352\pi$
$854$	0	0
$855$	−14.0983	−0.482152
$856$	0	0
$857$	−36.4808	−1.24616	−0.623080	−	0.782158i	$-0.714119\pi$
$857$	−36.4808	−1.24616	−0.623080	+	0.782158i	$0.714119\pi$
$858$	0	0
$859$	−35.2037	−1.20114	−0.600568	−	0.799574i	$-0.705059\pi$
$859$	−35.2037	−1.20114	−0.600568	+	0.799574i	$0.705059\pi$
$860$	0	0
$861$	−1.49081	−0.0508067
$862$	0	0
$863$	52.3803	1.78305	0.891523	−	0.452976i	$-0.149638\pi$
$863$	52.3803	1.78305	0.891523	+	0.452976i	$0.149638\pi$
$864$	0	0
$865$	2.94122	0.100004
$866$	0	0
$867$	−27.6920	−0.940470
$868$	0	0
$869$	−1.47385	−0.0499968
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−3.04232	−0.102967
$874$	0	0
$875$	11.9989	0.405637
$876$	0	0
$877$	−0.973957	−0.0328882	−0.0164441	−	0.999865i	$-0.505235\pi$
$877$	−0.973957	−0.0328882	−0.0164441	+	0.999865i	$0.505235\pi$
$878$	0	0
$879$	55.9860	1.88836
$880$	0	0
$881$	−5.36952	−0.180904	−0.0904519	−	0.995901i	$-0.528831\pi$
$881$	−5.36952	−0.180904	−0.0904519	+	0.995901i	$0.528831\pi$
$882$	0	0
$883$	−54.4177	−1.83130	−0.915651	−	0.401973i	$-0.868324\pi$
$883$	−54.4177	−1.83130	−0.915651	+	0.401973i	$0.868324\pi$
$884$	0	0
$885$	9.37562	0.315158
$886$	0	0
$887$	−17.6694	−0.593281	−0.296640	−	0.954989i	$-0.595866\pi$
$887$	−17.6694	−0.593281	−0.296640	+	0.954989i	$0.595866\pi$
$888$	0	0
$889$	−17.0060	−0.570362
$890$	0	0
$891$	−16.3545	−0.547896
$892$	0	0
$893$	6.09683	0.204023
$894$	0	0
$895$	7.19921	0.240643
$896$	0	0
$897$	0	0
$898$	0	0
$899$	56.9096	1.89804
$900$	0	0
$901$	−13.3027	−0.443178
$902$	0	0
$903$	4.21728	0.140342
$904$	0	0
$905$	5.06916	0.168505
$906$	0	0
$907$	8.16241	0.271028	0.135514	−	0.990775i	$-0.456731\pi$
$907$	8.16241	0.271028	0.135514	+	0.990775i	$0.456731\pi$
$908$	0	0
$909$	6.17446	0.204794
$910$	0	0
$911$	−23.0876	−0.764928	−0.382464	−	0.923970i	$-0.624924\pi$
$911$	−23.0876	−0.764928	−0.382464	+	0.923970i	$0.624924\pi$
$912$	0	0
$913$	−15.4428	−0.511082
$914$	0	0
$915$	41.7029	1.37865
$916$	0	0
$917$	16.3314	0.539311
$918$	0	0
$919$	23.8236	0.785869	0.392935	−	0.919566i	$-0.371460\pi$
$919$	23.8236	0.785869	0.392935	+	0.919566i	$0.371460\pi$
$920$	0	0
$921$	32.8551	1.08261
$922$	0	0
$923$	0	0
$924$	0	0
$925$	2.93257	0.0964224
$926$	0	0
$927$	7.17269	0.235582
$928$	0	0
$929$	−26.6656	−0.874871	−0.437436	−	0.899250i	$-0.644113\pi$
$929$	−26.6656	−0.874871	−0.437436	+	0.899250i	$0.644113\pi$
$930$	0	0
$931$	−6.90801	−0.226401
$932$	0	0
$933$	−16.5072	−0.540420
$934$	0	0
$935$	4.53483	0.148305
$936$	0	0
$937$	−26.7757	−0.874724	−0.437362	−	0.899286i	$-0.644087\pi$
$937$	−26.7757	−0.874724	−0.437362	+	0.899286i	$0.644087\pi$
$938$	0	0
$939$	1.73521	0.0566266
$940$	0	0
$941$	38.5864	1.25788	0.628940	−	0.777454i	$-0.283489\pi$
$941$	38.5864	1.25788	0.628940	+	0.777454i	$0.283489\pi$
$942$	0	0
$943$	−0.386472	−0.0125853
$944$	0	0
$945$	5.96077	0.193904
$946$	0	0
$947$	−17.3057	−0.562360	−0.281180	−	0.959655i	$-0.590726\pi$
$947$	−17.3057	−0.562360	−0.281180	+	0.959655i	$0.590726\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	1.10914	0.0359663
$952$	0	0
$953$	46.9452	1.52071	0.760353	−	0.649510i	$-0.225026\pi$
$953$	46.9452	1.52071	0.760353	+	0.649510i	$0.225026\pi$
$954$	0	0
$955$	12.5064	0.404696
$956$	0	0
$957$	−16.5116	−0.533746
$958$	0	0
$959$	−16.0585	−0.518555
$960$	0	0
$961$	76.7441	2.47562
$962$	0	0
$963$	7.34181	0.236586
$964$	0	0
$965$	13.8321	0.445270
$966$	0	0
$967$	−35.6294	−1.14576	−0.572882	−	0.819638i	$-0.694175\pi$
$967$	−35.6294	−1.14576	−0.572882	+	0.819638i	$0.694175\pi$
$968$	0	0
$969$	−26.8116	−0.861314
$970$	0	0
$971$	−45.5394	−1.46143	−0.730714	−	0.682683i	$-0.760813\pi$
$971$	−45.5394	−1.46143	−0.730714	+	0.682683i	$0.760813\pi$
$972$	0	0
$973$	−12.3584	−0.396192
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−42.4632	−1.35852	−0.679259	−	0.733898i	$-0.737699\pi$
$977$	−42.4632	−1.35852	−0.679259	+	0.733898i	$0.737699\pi$
$978$	0	0
$979$	−2.10589	−0.0673045
$980$	0	0
$981$	−6.89401	−0.220109
$982$	0	0
$983$	−22.9849	−0.733103	−0.366552	−	0.930398i	$-0.619462\pi$
$983$	−22.9849	−0.733103	−0.366552	+	0.930398i	$0.619462\pi$
$984$	0	0
$985$	22.2803	0.709909
$986$	0	0
$987$	1.81744	0.0578499
$988$	0	0
$989$	1.09327	0.0347640
$990$	0	0
$991$	−8.79024	−0.279231	−0.139616	−	0.990206i	$-0.544587\pi$
$991$	−8.79024	−0.279231	−0.139616	+	0.990206i	$0.544587\pi$
$992$	0	0
$993$	−60.0266	−1.90489
$994$	0	0
$995$	−6.78896	−0.215224
$996$	0	0
$997$	−21.5086	−0.681184	−0.340592	−	0.940211i	$-0.610628\pi$
$997$	−21.5086	−0.681184	−0.340592	+	0.940211i	$0.610628\pi$
$998$	0	0
$999$	4.63292	0.146579

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9464.2.a.br.1.12	✓	15
13.12	even	2		9464.2.a.bs.1.12	yes	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9464.2.a.br.1.12	✓	15	1.1	even	1	trivial
9464.2.a.bs.1.12	yes	15	13.12	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+2.05925 q^{3} +1.64516 q^{5} -1.00000 q^{7} +1.24052 q^{9} +1.46249 q^{11} +3.38781 q^{15} +1.88478 q^{17} -6.90801 q^{19} -2.05925 q^{21} -0.533833 q^{23} -2.29344 q^{25} -3.62321 q^{27} -5.48263 q^{29} -10.3800 q^{31} +3.01163 q^{33} -1.64516 q^{35} -1.27868 q^{37} +0.723958 q^{41} -2.04797 q^{43} +2.04086 q^{45} -0.882574 q^{47} +1.00000 q^{49} +3.88124 q^{51} -7.05797 q^{53} +2.40603 q^{55} -14.2253 q^{57} +2.76746 q^{59} +12.3097 q^{61} -1.24052 q^{63} -10.3011 q^{67} -1.09930 q^{69} -7.47426 q^{71} -1.90864 q^{73} -4.72277 q^{75} -1.46249 q^{77} -1.00777 q^{79} -11.1827 q^{81} -10.5593 q^{83} +3.10077 q^{85} -11.2901 q^{87} -1.43994 q^{89} -21.3750 q^{93} -11.3648 q^{95} -2.45245 q^{97} +1.81425 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q - 3 q^{3} - 4 q^{5} - 15 q^{7} + 16 q^{9} - 15 q^{11} - 8 q^{15} + 2 q^{17} - 13 q^{19} + 3 q^{21} - 10 q^{23} + 23 q^{25} - 9 q^{27} + 25 q^{29} - 19 q^{31} + 24 q^{33} + 4 q^{35} + 2 q^{37} - 30 q^{41}+ \cdots - 70 q^{99}+O(q^{100}) \) 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

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Database

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