LMFDB - Embedded newform 4732.2.a.q.1.3

Error: no document with id 251262196 found in table mf_hecke_traces.

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4732,2,Mod(1,4732)] 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("4732.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4732, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4732 = 2^{2} \cdot 7 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4732.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,6,0,0,0,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$37.7852102365$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.2854789.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 3x^{5} - 6x^{4} + 17x^{3} + 11x^{2} - 20x - 13 $ x^6 - 3x^5 - 6x^4 + 17x^3 + 11x^2 - 20*x - 13
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.875901$ of defining polynomial
Character	$\chi$	$=$	4732.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.76089 q^{3} -2.41317 q^{5} -1.00000 q^{7} +0.100737 q^{9} -6.01705 q^{11} -4.24934 q^{15} +2.95045 q^{17} -0.290293 q^{19} -1.76089 q^{21} +0.329450 q^{23} +0.823409 q^{25} -5.10529 q^{27} +4.40808 q^{29} +2.88630 q^{31} -10.5954 q^{33} +2.41317 q^{35} -4.01600 q^{37} +7.46676 q^{41} +9.64655 q^{43} -0.243097 q^{45} +6.00009 q^{47} +1.00000 q^{49} +5.19542 q^{51} +4.12592 q^{53} +14.5202 q^{55} -0.511175 q^{57} -7.46445 q^{59} -8.05128 q^{61} -0.100737 q^{63} +11.3097 q^{67} +0.580125 q^{69} +1.38708 q^{71} -7.54224 q^{73} +1.44993 q^{75} +6.01705 q^{77} +7.40516 q^{79} -9.29206 q^{81} +16.9577 q^{83} -7.11994 q^{85} +7.76215 q^{87} +11.7438 q^{89} +5.08246 q^{93} +0.700528 q^{95} -5.74851 q^{97} -0.606142 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{3} - 6 q^{7} + 20 q^{9} - 4 q^{11} - 3 q^{15} - 3 q^{17} - 5 q^{19} - 6 q^{21} + 18 q^{23} + 2 q^{25} + 24 q^{27} + 17 q^{29} + 17 q^{31} - 21 q^{33} - q^{37} - 14 q^{41} + 14 q^{43} + 29 q^{45}+ \cdots - 3 q^{99}+O(q^{100}) $ 6 * q + 6 * q^3 - 6 * q^7 + 20 * q^9 - 4 * q^11 - 3 * q^15 - 3 * q^17 - 5 * q^19 - 6 * q^21 + 18 * q^23 + 2 * q^25 + 24 * q^27 + 17 * q^29 + 17 * q^31 - 21 * q^33 - q^37 - 14 * q^41 + 14 * q^43 + 29 * q^45 + 6 * q^47 + 6 * q^49 + 2 * q^51 + 3 * q^53 + 18 * q^55 + 14 * q^59 - 4 * q^61 - 20 * q^63 - 4 * q^67 + 39 * q^69 - 3 * q^71 + 31 * q^73 + 37 * q^75 + 4 * q^77 + 22 * q^79 + 6 * q^81 + 15 * q^83 - 51 * q^85 + 51 * q^87 + 20 * q^89 - 13 * q^93 + 19 * q^95 + 29 * q^97 - 3 * 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
$3$	1.76089	1.01665	0.508325	−	0.861165i	$-0.330265\pi$
$3$	1.76089	1.01665	0.508325	+	0.861165i	$0.330265\pi$
$4$	0	0
$5$	−2.41317	−1.07920	−0.539602	−	0.841920i	$-0.681425\pi$
$5$	−2.41317	−1.07920	−0.539602	+	0.841920i	$0.681425\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	0.100737	0.0335792
$10$	0	0
$11$	−6.01705	−1.81421	−0.907104	−	0.420906i	$-0.861712\pi$
$11$	−6.01705	−1.81421	−0.907104	+	0.420906i	$0.861712\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	−4.24934	−1.09717
$16$	0	0
$17$	2.95045	0.715588	0.357794	−	0.933800i	$-0.383529\pi$
$17$	2.95045	0.715588	0.357794	+	0.933800i	$0.383529\pi$
$18$	0	0
$19$	−0.290293	−0.0665978	−0.0332989	−	0.999445i	$-0.510601\pi$
$19$	−0.290293	−0.0665978	−0.0332989	+	0.999445i	$0.510601\pi$
$20$	0	0
$21$	−1.76089	−0.384258
$22$	0	0
$23$	0.329450	0.0686950	0.0343475	−	0.999410i	$-0.489065\pi$
$23$	0.329450	0.0686950	0.0343475	+	0.999410i	$0.489065\pi$
$24$	0	0
$25$	0.823409	0.164682
$26$	0	0
$27$	−5.10529	−0.982513
$28$	0	0
$29$	4.40808	0.818560	0.409280	−	0.912409i	$-0.365780\pi$
$29$	4.40808	0.818560	0.409280	+	0.912409i	$0.365780\pi$
$30$	0	0
$31$	2.88630	0.518394	0.259197	−	0.965824i	$-0.416542\pi$
$31$	2.88630	0.518394	0.259197	+	0.965824i	$0.416542\pi$
$32$	0	0
$33$	−10.5954	−1.84442
$34$	0	0
$35$	2.41317	0.407901
$36$	0	0
$37$	−4.01600	−0.660226	−0.330113	−	0.943941i	$-0.607087\pi$
$37$	−4.01600	−0.660226	−0.330113	+	0.943941i	$0.607087\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	7.46676	1.16611	0.583056	−	0.812432i	$-0.301857\pi$
$41$	7.46676	1.16611	0.583056	+	0.812432i	$0.301857\pi$
$42$	0	0
$43$	9.64655	1.47109	0.735543	−	0.677478i	$-0.236927\pi$
$43$	9.64655	1.47109	0.735543	+	0.677478i	$0.236927\pi$
$44$	0	0
$45$	−0.243097	−0.0362388
$46$	0	0
$47$	6.00009	0.875203	0.437602	−	0.899169i	$-0.355828\pi$
$47$	6.00009	0.875203	0.437602	+	0.899169i	$0.355828\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	5.19542	0.727504
$52$	0	0
$53$	4.12592	0.566738	0.283369	−	0.959011i	$-0.408548\pi$
$53$	4.12592	0.566738	0.283369	+	0.959011i	$0.408548\pi$
$54$	0	0
$55$	14.5202	1.95790
$56$	0	0
$57$	−0.511175	−0.0677067
$58$	0	0
$59$	−7.46445	−0.971789	−0.485894	−	0.874018i	$-0.661506\pi$
$59$	−7.46445	−0.971789	−0.485894	+	0.874018i	$0.661506\pi$
$60$	0	0
$61$	−8.05128	−1.03086	−0.515430	−	0.856932i	$-0.672368\pi$
$61$	−8.05128	−1.03086	−0.515430	+	0.856932i	$0.672368\pi$
$62$	0	0
$63$	−0.100737	−0.0126917
$64$	0	0
$65$	0	0
$66$	0	0
$67$	11.3097	1.38170	0.690852	−	0.722996i	$-0.257235\pi$
$67$	11.3097	1.38170	0.690852	+	0.722996i	$0.257235\pi$
$68$	0	0
$69$	0.580125	0.0698388
$70$	0	0
$71$	1.38708	0.164616	0.0823078	−	0.996607i	$-0.473771\pi$
$71$	1.38708	0.164616	0.0823078	+	0.996607i	$0.473771\pi$
$72$	0	0
$73$	−7.54224	−0.882753	−0.441376	−	0.897322i	$-0.645510\pi$
$73$	−7.54224	−0.882753	−0.441376	+	0.897322i	$0.645510\pi$
$74$	0	0
$75$	1.44993	0.167424
$76$	0	0
$77$	6.01705	0.685706
$78$	0	0
$79$	7.40516	0.833146	0.416573	−	0.909102i	$-0.363231\pi$
$79$	7.40516	0.833146	0.416573	+	0.909102i	$0.363231\pi$
$80$	0	0
$81$	−9.29206	−1.03245
$82$	0	0
$83$	16.9577	1.86135	0.930676	−	0.365843i	$-0.119219\pi$
$83$	16.9577	1.86135	0.930676	+	0.365843i	$0.119219\pi$
$84$	0	0
$85$	−7.11994	−0.772266
$86$	0	0
$87$	7.76215	0.832190
$88$	0	0
$89$	11.7438	1.24484	0.622418	−	0.782685i	$-0.286150\pi$
$89$	11.7438	1.24484	0.622418	+	0.782685i	$0.286150\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	5.08246	0.527026
$94$	0	0
$95$	0.700528	0.0718727
$96$	0	0
$97$	−5.74851	−0.583672	−0.291836	−	0.956468i	$-0.594266\pi$
$97$	−5.74851	−0.583672	−0.291836	+	0.956468i	$0.594266\pi$
$98$	0	0
$99$	−0.606142	−0.0609196
$100$	0	0
$101$	10.4816	1.04296	0.521478	−	0.853265i	$-0.325381\pi$
$101$	10.4816	1.04296	0.521478	+	0.853265i	$0.325381\pi$
$102$	0	0
$103$	−17.5315	−1.72743	−0.863713	−	0.503983i	$-0.831867\pi$
$103$	−17.5315	−1.72743	−0.863713	+	0.503983i	$0.831867\pi$
$104$	0	0
$105$	4.24934	0.414693
$106$	0	0
$107$	9.33384	0.902336	0.451168	−	0.892439i	$-0.351008\pi$
$107$	9.33384	0.902336	0.451168	+	0.892439i	$0.351008\pi$
$108$	0	0
$109$	1.39103	0.133236	0.0666181	−	0.997779i	$-0.478779\pi$
$109$	1.39103	0.133236	0.0666181	+	0.997779i	$0.478779\pi$
$110$	0	0
$111$	−7.07174	−0.671220
$112$	0	0
$113$	−9.11232	−0.857215	−0.428607	−	0.903491i	$-0.640996\pi$
$113$	−9.11232	−0.857215	−0.428607	+	0.903491i	$0.640996\pi$
$114$	0	0
$115$	−0.795019	−0.0741359
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−2.95045	−0.270467
$120$	0	0
$121$	25.2049	2.29135
$122$	0	0
$123$	13.1481	1.18553
$124$	0	0
$125$	10.0788	0.901479
$126$	0	0
$127$	12.2226	1.08458	0.542289	−	0.840192i	$-0.317558\pi$
$127$	12.2226	1.08458	0.542289	+	0.840192i	$0.317558\pi$
$128$	0	0
$129$	16.9865	1.49558
$130$	0	0
$131$	−10.2889	−0.898942	−0.449471	−	0.893295i	$-0.648387\pi$
$131$	−10.2889	−0.898942	−0.449471	+	0.893295i	$0.648387\pi$
$132$	0	0
$133$	0.290293	0.0251716
$134$	0	0
$135$	12.3199	1.06033
$136$	0	0
$137$	−17.5059	−1.49563	−0.747816	−	0.663906i	$-0.768897\pi$
$137$	−17.5059	−1.49563	−0.747816	+	0.663906i	$0.768897\pi$
$138$	0	0
$139$	5.26079	0.446214	0.223107	−	0.974794i	$-0.428380\pi$
$139$	5.26079	0.446214	0.223107	+	0.974794i	$0.428380\pi$
$140$	0	0
$141$	10.5655	0.889776
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−10.6375	−0.883393
$146$	0	0
$147$	1.76089	0.145236
$148$	0	0
$149$	22.8551	1.87236	0.936180	−	0.351520i	$-0.114335\pi$
$149$	22.8551	1.87236	0.936180	+	0.351520i	$0.114335\pi$
$150$	0	0
$151$	−24.1534	−1.96558	−0.982790	−	0.184729i	$-0.940859\pi$
$151$	−24.1534	−1.96558	−0.982790	+	0.184729i	$0.940859\pi$
$152$	0	0
$153$	0.297221	0.0240289
$154$	0	0
$155$	−6.96514	−0.559453
$156$	0	0
$157$	1.15459	0.0921466	0.0460733	−	0.998938i	$-0.485329\pi$
$157$	1.15459	0.0921466	0.0460733	+	0.998938i	$0.485329\pi$
$158$	0	0
$159$	7.26529	0.576175
$160$	0	0
$161$	−0.329450	−0.0259643
$162$	0	0
$163$	20.6011	1.61360	0.806802	−	0.590821i	$-0.201196\pi$
$163$	20.6011	1.61360	0.806802	+	0.590821i	$0.201196\pi$
$164$	0	0
$165$	25.5685	1.99050
$166$	0	0
$167$	11.8060	0.913573	0.456787	−	0.889576i	$-0.349000\pi$
$167$	11.8060	0.913573	0.456787	+	0.889576i	$0.349000\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	−0.0292434	−0.00223630
$172$	0	0
$173$	20.4935	1.55809	0.779047	−	0.626965i	$-0.215703\pi$
$173$	20.4935	1.55809	0.779047	+	0.626965i	$0.215703\pi$
$174$	0	0
$175$	−0.823409	−0.0622439
$176$	0	0
$177$	−13.1441	−0.987970
$178$	0	0
$179$	−8.65834	−0.647155	−0.323577	−	0.946202i	$-0.604886\pi$
$179$	−8.65834	−0.647155	−0.323577	+	0.946202i	$0.604886\pi$
$180$	0	0
$181$	−6.73893	−0.500901	−0.250450	−	0.968129i	$-0.580579\pi$
$181$	−6.73893	−0.500901	−0.250450	+	0.968129i	$0.580579\pi$
$182$	0	0
$183$	−14.1774	−1.04803
$184$	0	0
$185$	9.69131	0.712519
$186$	0	0
$187$	−17.7530	−1.29823
$188$	0	0
$189$	5.10529	0.371355
$190$	0	0
$191$	19.4098	1.40444	0.702221	−	0.711959i	$-0.252192\pi$
$191$	19.4098	1.40444	0.702221	+	0.711959i	$0.252192\pi$
$192$	0	0
$193$	6.36015	0.457814	0.228907	−	0.973448i	$-0.426485\pi$
$193$	6.36015	0.457814	0.228907	+	0.973448i	$0.426485\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−9.94515	−0.708562	−0.354281	−	0.935139i	$-0.615274\pi$
$197$	−9.94515	−0.708562	−0.354281	+	0.935139i	$0.615274\pi$
$198$	0	0
$199$	−22.3660	−1.58549	−0.792743	−	0.609556i	$-0.791348\pi$
$199$	−22.3660	−1.58549	−0.792743	+	0.609556i	$0.791348\pi$
$200$	0	0
$201$	19.9152	1.40471
$202$	0	0
$203$	−4.40808	−0.309387
$204$	0	0
$205$	−18.0186	−1.25847
$206$	0	0
$207$	0.0331879	0.00230672
$208$	0	0
$209$	1.74671	0.120822
$210$	0	0
$211$	23.6517	1.62825	0.814126	−	0.580689i	$-0.197217\pi$
$211$	23.6517	1.62825	0.814126	+	0.580689i	$0.197217\pi$
$212$	0	0
$213$	2.44249	0.167357
$214$	0	0
$215$	−23.2788	−1.58760
$216$	0	0
$217$	−2.88630	−0.195935
$218$	0	0
$219$	−13.2811	−0.897451
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−8.71399	−0.583532	−0.291766	−	0.956490i	$-0.594243\pi$
$223$	−8.71399	−0.583532	−0.291766	+	0.956490i	$0.594243\pi$
$224$	0	0
$225$	0.0829481	0.00552988
$226$	0	0
$227$	−4.95812	−0.329082	−0.164541	−	0.986370i	$-0.552614\pi$
$227$	−4.95812	−0.329082	−0.164541	+	0.986370i	$0.552614\pi$
$228$	0	0
$229$	−19.5942	−1.29482	−0.647411	−	0.762141i	$-0.724148\pi$
$229$	−19.5942	−1.29482	−0.647411	+	0.762141i	$0.724148\pi$
$230$	0	0
$231$	10.5954	0.697124
$232$	0	0
$233$	−27.4664	−1.79938	−0.899690	−	0.436529i	$-0.856208\pi$
$233$	−27.4664	−1.79938	−0.899690	+	0.436529i	$0.856208\pi$
$234$	0	0
$235$	−14.4793	−0.944523
$236$	0	0
$237$	13.0397	0.847018
$238$	0	0
$239$	13.8340	0.894846	0.447423	−	0.894323i	$-0.352342\pi$
$239$	13.8340	0.894846	0.447423	+	0.894323i	$0.352342\pi$
$240$	0	0
$241$	−1.83821	−0.118410	−0.0592049	−	0.998246i	$-0.518857\pi$
$241$	−1.83821	−0.118410	−0.0592049	+	0.998246i	$0.518857\pi$
$242$	0	0
$243$	−1.04646	−0.0671302
$244$	0	0
$245$	−2.41317	−0.154172
$246$	0	0
$247$	0	0
$248$	0	0
$249$	29.8607	1.89235
$250$	0	0
$251$	−20.1333	−1.27080	−0.635402	−	0.772181i	$-0.719166\pi$
$251$	−20.1333	−1.27080	−0.635402	+	0.772181i	$0.719166\pi$
$252$	0	0
$253$	−1.98231	−0.124627
$254$	0	0
$255$	−12.5374	−0.785125
$256$	0	0
$257$	4.62824	0.288702	0.144351	−	0.989527i	$-0.453891\pi$
$257$	4.62824	0.288702	0.144351	+	0.989527i	$0.453891\pi$
$258$	0	0
$259$	4.01600	0.249542
$260$	0	0
$261$	0.444059	0.0274866
$262$	0	0
$263$	−0.544003	−0.0335447	−0.0167723	−	0.999859i	$-0.505339\pi$
$263$	−0.544003	−0.0335447	−0.0167723	+	0.999859i	$0.505339\pi$
$264$	0	0
$265$	−9.95656	−0.611626
$266$	0	0
$267$	20.6795	1.26556
$268$	0	0
$269$	16.6044	1.01239	0.506194	−	0.862420i	$-0.331052\pi$
$269$	16.6044	1.01239	0.506194	+	0.862420i	$0.331052\pi$
$270$	0	0
$271$	16.6709	1.01269	0.506344	−	0.862332i	$-0.330997\pi$
$271$	16.6709	1.01269	0.506344	+	0.862332i	$0.330997\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−4.95449	−0.298767
$276$	0	0
$277$	19.9098	1.19626	0.598132	−	0.801398i	$-0.295910\pi$
$277$	19.9098	1.19626	0.598132	+	0.801398i	$0.295910\pi$
$278$	0	0
$279$	0.290758	0.0174073
$280$	0	0
$281$	5.58343	0.333080	0.166540	−	0.986035i	$-0.446741\pi$
$281$	5.58343	0.333080	0.166540	+	0.986035i	$0.446741\pi$
$282$	0	0
$283$	6.21921	0.369694	0.184847	−	0.982767i	$-0.440821\pi$
$283$	6.21921	0.369694	0.184847	+	0.982767i	$0.440821\pi$
$284$	0	0
$285$	1.23355	0.0730694
$286$	0	0
$287$	−7.46676	−0.440749
$288$	0	0
$289$	−8.29486	−0.487933
$290$	0	0
$291$	−10.1225	−0.593391
$292$	0	0
$293$	10.8695	0.635000	0.317500	−	0.948258i	$-0.397157\pi$
$293$	10.8695	0.635000	0.317500	+	0.948258i	$0.397157\pi$
$294$	0	0
$295$	18.0130	1.04876
$296$	0	0
$297$	30.7188	1.78248
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−9.64655	−0.556018
$302$	0	0
$303$	18.4569	1.06032
$304$	0	0
$305$	19.4291	1.11251
$306$	0	0
$307$	3.35325	0.191380	0.0956900	−	0.995411i	$-0.469494\pi$
$307$	3.35325	0.191380	0.0956900	+	0.995411i	$0.469494\pi$
$308$	0	0
$309$	−30.8710	−1.75619
$310$	0	0
$311$	22.3596	1.26790	0.633949	−	0.773375i	$-0.281433\pi$
$311$	22.3596	1.26790	0.633949	+	0.773375i	$0.281433\pi$
$312$	0	0
$313$	−19.0923	−1.07916	−0.539579	−	0.841935i	$-0.681417\pi$
$313$	−19.0923	−1.07916	−0.539579	+	0.841935i	$0.681417\pi$
$314$	0	0
$315$	0.243097	0.0136970
$316$	0	0
$317$	30.9317	1.73730	0.868648	−	0.495430i	$-0.164990\pi$
$317$	30.9317	1.73730	0.868648	+	0.495430i	$0.164990\pi$
$318$	0	0
$319$	−26.5236	−1.48504
$320$	0	0
$321$	16.4359	0.917361
$322$	0	0
$323$	−0.856495	−0.0476566
$324$	0	0
$325$	0	0
$326$	0	0
$327$	2.44945	0.135455
$328$	0	0
$329$	−6.00009	−0.330796
$330$	0	0
$331$	−29.9464	−1.64601	−0.823003	−	0.568037i	$-0.807703\pi$
$331$	−29.9464	−1.64601	−0.823003	+	0.568037i	$0.807703\pi$
$332$	0	0
$333$	−0.404562	−0.0221698
$334$	0	0
$335$	−27.2924	−1.49114
$336$	0	0
$337$	−15.4765	−0.843056	−0.421528	−	0.906815i	$-0.638506\pi$
$337$	−15.4765	−0.843056	−0.421528	+	0.906815i	$0.638506\pi$
$338$	0	0
$339$	−16.0458	−0.871488
$340$	0	0
$341$	−17.3670	−0.940476
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−1.39994	−0.0753704
$346$	0	0
$347$	−10.2122	−0.548220	−0.274110	−	0.961698i	$-0.588383\pi$
$347$	−10.2122	−0.548220	−0.274110	+	0.961698i	$0.588383\pi$
$348$	0	0
$349$	28.4820	1.52461	0.762304	−	0.647219i	$-0.224068\pi$
$349$	28.4820	1.52461	0.762304	+	0.647219i	$0.224068\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	4.16002	0.221416	0.110708	−	0.993853i	$-0.464688\pi$
$353$	4.16002	0.221416	0.110708	+	0.993853i	$0.464688\pi$
$354$	0	0
$355$	−3.34726	−0.177654
$356$	0	0
$357$	−5.19542	−0.274971
$358$	0	0
$359$	−6.71704	−0.354512	−0.177256	−	0.984165i	$-0.556722\pi$
$359$	−6.71704	−0.354512	−0.177256	+	0.984165i	$0.556722\pi$
$360$	0	0
$361$	−18.9157	−0.995565
$362$	0	0
$363$	44.3831	2.32951
$364$	0	0
$365$	18.2007	0.952670
$366$	0	0
$367$	6.73106	0.351358	0.175679	−	0.984447i	$-0.443788\pi$
$367$	6.73106	0.351358	0.175679	+	0.984447i	$0.443788\pi$
$368$	0	0
$369$	0.752182	0.0391571
$370$	0	0
$371$	−4.12592	−0.214207
$372$	0	0
$373$	−17.8800	−0.925791	−0.462895	−	0.886413i	$-0.653189\pi$
$373$	−17.8800	−0.925791	−0.462895	+	0.886413i	$0.653189\pi$
$374$	0	0
$375$	17.7477	0.916489
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−5.92742	−0.304471	−0.152236	−	0.988344i	$-0.548647\pi$
$379$	−5.92742	−0.304471	−0.152236	+	0.988344i	$0.548647\pi$
$380$	0	0
$381$	21.5226	1.10264
$382$	0	0
$383$	−21.3131	−1.08905	−0.544523	−	0.838746i	$-0.683289\pi$
$383$	−21.3131	−1.08905	−0.544523	+	0.838746i	$0.683289\pi$
$384$	0	0
$385$	−14.5202	−0.740017
$386$	0	0
$387$	0.971770	0.0493978
$388$	0	0
$389$	0.893641	0.0453094	0.0226547	−	0.999743i	$-0.492788\pi$
$389$	0.893641	0.0453094	0.0226547	+	0.999743i	$0.492788\pi$
$390$	0	0
$391$	0.972024	0.0491573
$392$	0	0
$393$	−18.1176	−0.913910
$394$	0	0
$395$	−17.8699	−0.899134
$396$	0	0
$397$	9.89710	0.496721	0.248361	−	0.968668i	$-0.420108\pi$
$397$	9.89710	0.496721	0.248361	+	0.968668i	$0.420108\pi$
$398$	0	0
$399$	0.511175	0.0255907
$400$	0	0
$401$	−5.00557	−0.249966	−0.124983	−	0.992159i	$-0.539888\pi$
$401$	−5.00557	−0.249966	−0.124983	+	0.992159i	$0.539888\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	22.4234	1.11423
$406$	0	0
$407$	24.1645	1.19779
$408$	0	0
$409$	1.25220	0.0619175	0.0309588	−	0.999521i	$-0.490144\pi$
$409$	1.25220	0.0619175	0.0309588	+	0.999521i	$0.490144\pi$
$410$	0	0
$411$	−30.8260	−1.52053
$412$	0	0
$413$	7.46445	0.367302
$414$	0	0
$415$	−40.9220	−2.00878
$416$	0	0
$417$	9.26367	0.453644
$418$	0	0
$419$	27.0663	1.32228	0.661138	−	0.750264i	$-0.270074\pi$
$419$	27.0663	1.32228	0.661138	+	0.750264i	$0.270074\pi$
$420$	0	0
$421$	18.9252	0.922357	0.461179	−	0.887307i	$-0.347427\pi$
$421$	18.9252	0.922357	0.461179	+	0.887307i	$0.347427\pi$
$422$	0	0
$423$	0.604434	0.0293886
$424$	0	0
$425$	2.42942	0.117844
$426$	0	0
$427$	8.05128	0.389629
$428$	0	0
$429$	0	0
$430$	0	0
$431$	1.91738	0.0923567	0.0461784	−	0.998933i	$-0.485296\pi$
$431$	1.91738	0.0923567	0.0461784	+	0.998933i	$0.485296\pi$
$432$	0	0
$433$	31.8526	1.53074	0.765369	−	0.643592i	$-0.222557\pi$
$433$	31.8526	1.53074	0.765369	+	0.643592i	$0.222557\pi$
$434$	0	0
$435$	−18.7314	−0.898103
$436$	0	0
$437$	−0.0956370	−0.00457494
$438$	0	0
$439$	16.7930	0.801485	0.400743	−	0.916191i	$-0.368752\pi$
$439$	16.7930	0.801485	0.400743	+	0.916191i	$0.368752\pi$
$440$	0	0
$441$	0.100737	0.00479702
$442$	0	0
$443$	1.64455	0.0781351	0.0390675	−	0.999237i	$-0.487561\pi$
$443$	1.64455	0.0781351	0.0390675	+	0.999237i	$0.487561\pi$
$444$	0	0
$445$	−28.3398	−1.34343
$446$	0	0
$447$	40.2453	1.90354
$448$	0	0
$449$	−11.7806	−0.555959	−0.277979	−	0.960587i	$-0.589665\pi$
$449$	−11.7806	−0.555959	−0.277979	+	0.960587i	$0.589665\pi$
$450$	0	0
$451$	−44.9279	−2.11557
$452$	0	0
$453$	−42.5316	−1.99831
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−32.7044	−1.52985	−0.764924	−	0.644120i	$-0.777224\pi$
$457$	−32.7044	−1.52985	−0.764924	+	0.644120i	$0.777224\pi$
$458$	0	0
$459$	−15.0629	−0.703075
$460$	0	0
$461$	−13.4366	−0.625803	−0.312901	−	0.949786i	$-0.601301\pi$
$461$	−13.4366	−0.625803	−0.312901	+	0.949786i	$0.601301\pi$
$462$	0	0
$463$	−37.1196	−1.72509	−0.862547	−	0.505976i	$-0.831132\pi$
$463$	−37.1196	−1.72509	−0.862547	+	0.505976i	$0.831132\pi$
$464$	0	0
$465$	−12.2649	−0.568769
$466$	0	0
$467$	10.6831	0.494357	0.247178	−	0.968970i	$-0.420497\pi$
$467$	10.6831	0.494357	0.247178	+	0.968970i	$0.420497\pi$
$468$	0	0
$469$	−11.3097	−0.522235
$470$	0	0
$471$	2.03311	0.0936809
$472$	0	0
$473$	−58.0438	−2.66886
$474$	0	0
$475$	−0.239030	−0.0109674
$476$	0	0
$477$	0.415634	0.0190306
$478$	0	0
$479$	23.7348	1.08447	0.542235	−	0.840227i	$-0.317578\pi$
$479$	23.7348	1.08447	0.542235	+	0.840227i	$0.317578\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	−0.580125	−0.0263966
$484$	0	0
$485$	13.8721	0.629902
$486$	0	0
$487$	1.87394	0.0849162	0.0424581	−	0.999098i	$-0.486481\pi$
$487$	1.87394	0.0849162	0.0424581	+	0.999098i	$0.486481\pi$
$488$	0	0
$489$	36.2763	1.64047
$490$	0	0
$491$	44.0373	1.98737	0.993687	−	0.112184i	$-0.0357847\pi$
$491$	44.0373	1.98737	0.993687	+	0.112184i	$0.0357847\pi$
$492$	0	0
$493$	13.0058	0.585752
$494$	0	0
$495$	1.46273	0.0657447
$496$	0	0
$497$	−1.38708	−0.0622189
$498$	0	0
$499$	23.6495	1.05870	0.529348	−	0.848405i	$-0.322437\pi$
$499$	23.6495	1.05870	0.529348	+	0.848405i	$0.322437\pi$
$500$	0	0
$501$	20.7890	0.928785
$502$	0	0
$503$	15.5968	0.695428	0.347714	−	0.937601i	$-0.386958\pi$
$503$	15.5968	0.695428	0.347714	+	0.937601i	$0.386958\pi$
$504$	0	0
$505$	−25.2939	−1.12556
$506$	0	0
$507$	0	0
$508$	0	0
$509$	32.4384	1.43781	0.718904	−	0.695110i	$-0.244644\pi$
$509$	32.4384	1.43781	0.718904	+	0.695110i	$0.244644\pi$
$510$	0	0
$511$	7.54224	0.333649
$512$	0	0
$513$	1.48203	0.0654332
$514$	0	0
$515$	42.3065	1.86425
$516$	0	0
$517$	−36.1028	−1.58780
$518$	0	0
$519$	36.0869	1.58404
$520$	0	0
$521$	−14.7595	−0.646626	−0.323313	−	0.946292i	$-0.604797\pi$
$521$	−14.7595	−0.646626	−0.323313	+	0.946292i	$0.604797\pi$
$522$	0	0
$523$	−36.6349	−1.60193	−0.800966	−	0.598710i	$-0.795680\pi$
$523$	−36.6349	−1.60193	−0.800966	+	0.598710i	$0.795680\pi$
$524$	0	0
$525$	−1.44993	−0.0632803
$526$	0	0
$527$	8.51587	0.370957
$528$	0	0
$529$	−22.8915	−0.995281
$530$	0	0
$531$	−0.751950	−0.0326319
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−22.5242	−0.973805
$536$	0	0
$537$	−15.2464	−0.657930
$538$	0	0
$539$	−6.01705	−0.259173
$540$	0	0
$541$	34.6618	1.49023	0.745113	−	0.666938i	$-0.232395\pi$
$541$	34.6618	1.49023	0.745113	+	0.666938i	$0.232395\pi$
$542$	0	0
$543$	−11.8665	−0.509241
$544$	0	0
$545$	−3.35679	−0.143789
$546$	0	0
$547$	−33.5692	−1.43532	−0.717659	−	0.696395i	$-0.754786\pi$
$547$	−33.5692	−1.43532	−0.717659	+	0.696395i	$0.754786\pi$
$548$	0	0
$549$	−0.811065	−0.0346154
$550$	0	0
$551$	−1.27964	−0.0545143
$552$	0	0
$553$	−7.40516	−0.314899
$554$	0	0
$555$	17.0653	0.724383
$556$	0	0
$557$	−9.05561	−0.383699	−0.191849	−	0.981424i	$-0.561448\pi$
$557$	−9.05561	−0.383699	−0.191849	+	0.981424i	$0.561448\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−31.2611	−1.31984
$562$	0	0
$563$	−10.0815	−0.424883	−0.212442	−	0.977174i	$-0.568142\pi$
$563$	−10.0815	−0.424883	−0.212442	+	0.977174i	$0.568142\pi$
$564$	0	0
$565$	21.9896	0.925110
$566$	0	0
$567$	9.29206	0.390230
$568$	0	0
$569$	−15.3185	−0.642185	−0.321093	−	0.947048i	$-0.604050\pi$
$569$	−15.3185	−0.642185	−0.321093	+	0.947048i	$0.604050\pi$
$570$	0	0
$571$	3.10370	0.129886	0.0649428	−	0.997889i	$-0.479313\pi$
$571$	3.10370	0.129886	0.0649428	+	0.997889i	$0.479313\pi$
$572$	0	0
$573$	34.1785	1.42783
$574$	0	0
$575$	0.271272	0.0113128
$576$	0	0
$577$	−28.1391	−1.17145	−0.585723	−	0.810511i	$-0.699189\pi$
$577$	−28.1391	−1.17145	−0.585723	+	0.810511i	$0.699189\pi$
$578$	0	0
$579$	11.1995	0.465437
$580$	0	0
$581$	−16.9577	−0.703525
$582$	0	0
$583$	−24.8258	−1.02818
$584$	0	0
$585$	0	0
$586$	0	0
$587$	14.6993	0.606706	0.303353	−	0.952878i	$-0.401894\pi$
$587$	14.6993	0.606706	0.303353	+	0.952878i	$0.401894\pi$
$588$	0	0
$589$	−0.837873	−0.0345239
$590$	0	0
$591$	−17.5123	−0.720361
$592$	0	0
$593$	−22.2976	−0.915652	−0.457826	−	0.889042i	$-0.651372\pi$
$593$	−22.2976	−0.915652	−0.457826	+	0.889042i	$0.651372\pi$
$594$	0	0
$595$	7.11994	0.291889
$596$	0	0
$597$	−39.3842	−1.61189
$598$	0	0
$599$	−4.57091	−0.186762	−0.0933812	−	0.995630i	$-0.529768\pi$
$599$	−4.57091	−0.186762	−0.0933812	+	0.995630i	$0.529768\pi$
$600$	0	0
$601$	27.7160	1.13056	0.565280	−	0.824899i	$-0.308768\pi$
$601$	27.7160	1.13056	0.565280	+	0.824899i	$0.308768\pi$
$602$	0	0
$603$	1.13932	0.0463965
$604$	0	0
$605$	−60.8238	−2.47284
$606$	0	0
$607$	14.4157	0.585117	0.292558	−	0.956248i	$-0.405493\pi$
$607$	14.4157	0.585117	0.292558	+	0.956248i	$0.405493\pi$
$608$	0	0
$609$	−7.76215	−0.314538
$610$	0	0
$611$	0	0
$612$	0	0
$613$	7.86871	0.317814	0.158907	−	0.987294i	$-0.449203\pi$
$613$	7.86871	0.317814	0.158907	+	0.987294i	$0.449203\pi$
$614$	0	0
$615$	−31.7288	−1.27943
$616$	0	0
$617$	2.73041	0.109922	0.0549611	−	0.998488i	$-0.482497\pi$
$617$	2.73041	0.109922	0.0549611	+	0.998488i	$0.482497\pi$
$618$	0	0
$619$	29.6881	1.19326	0.596632	−	0.802515i	$-0.296505\pi$
$619$	29.6881	1.19326	0.596632	+	0.802515i	$0.296505\pi$
$620$	0	0
$621$	−1.68193	−0.0674937
$622$	0	0
$623$	−11.7438	−0.470504
$624$	0	0
$625$	−28.4390	−1.13756
$626$	0	0
$627$	3.07576	0.122834
$628$	0	0
$629$	−11.8490	−0.472450
$630$	0	0
$631$	27.6160	1.09938	0.549689	−	0.835370i	$-0.314746\pi$
$631$	27.6160	1.09938	0.549689	+	0.835370i	$0.314746\pi$
$632$	0	0
$633$	41.6481	1.65536
$634$	0	0
$635$	−29.4952	−1.17048
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0.139731	0.00552765
$640$	0	0
$641$	−37.2581	−1.47161	−0.735803	−	0.677195i	$-0.763195\pi$
$641$	−37.2581	−1.47161	−0.735803	+	0.677195i	$0.763195\pi$
$642$	0	0
$643$	22.3550	0.881593	0.440797	−	0.897607i	$-0.354696\pi$
$643$	22.3550	0.881593	0.440797	+	0.897607i	$0.354696\pi$
$644$	0	0
$645$	−40.9915	−1.61404
$646$	0	0
$647$	28.2847	1.11199	0.555993	−	0.831187i	$-0.312338\pi$
$647$	28.2847	1.11199	0.555993	+	0.831187i	$0.312338\pi$
$648$	0	0
$649$	44.9140	1.76303
$650$	0	0
$651$	−5.08246	−0.199197
$652$	0	0
$653$	28.3115	1.10791	0.553957	−	0.832545i	$-0.313117\pi$
$653$	28.3115	1.10791	0.553957	+	0.832545i	$0.313117\pi$
$654$	0	0
$655$	24.8288	0.970142
$656$	0	0
$657$	−0.759786	−0.0296421
$658$	0	0
$659$	−19.9190	−0.775936	−0.387968	−	0.921673i	$-0.626823\pi$
$659$	−19.9190	−0.775936	−0.387968	+	0.921673i	$0.626823\pi$
$660$	0	0
$661$	−29.6064	−1.15155	−0.575777	−	0.817607i	$-0.695300\pi$
$661$	−29.6064	−1.15155	−0.575777	+	0.817607i	$0.695300\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−0.700528	−0.0271653
$666$	0	0
$667$	1.45224	0.0562310
$668$	0	0
$669$	−15.3444	−0.593249
$670$	0	0
$671$	48.4449	1.87020
$672$	0	0
$673$	32.8303	1.26552	0.632758	−	0.774350i	$-0.281923\pi$
$673$	32.8303	1.26552	0.632758	+	0.774350i	$0.281923\pi$
$674$	0	0
$675$	−4.20374	−0.161802
$676$	0	0
$677$	25.4406	0.977762	0.488881	−	0.872350i	$-0.337405\pi$
$677$	25.4406	0.977762	0.488881	+	0.872350i	$0.337405\pi$
$678$	0	0
$679$	5.74851	0.220607
$680$	0	0
$681$	−8.73072	−0.334562
$682$	0	0
$683$	−30.3282	−1.16048	−0.580238	−	0.814447i	$-0.697041\pi$
$683$	−30.3282	−1.16048	−0.580238	+	0.814447i	$0.697041\pi$
$684$	0	0
$685$	42.2448	1.61409
$686$	0	0
$687$	−34.5033	−1.31638
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−23.7078	−0.901889	−0.450944	−	0.892552i	$-0.648913\pi$
$691$	−23.7078	−0.901889	−0.450944	+	0.892552i	$0.648913\pi$
$692$	0	0
$693$	0.606142	0.0230254
$694$	0	0
$695$	−12.6952	−0.481556
$696$	0	0
$697$	22.0303	0.834456
$698$	0	0
$699$	−48.3653	−1.82934
$700$	0	0
$701$	−28.7608	−1.08628	−0.543141	−	0.839641i	$-0.682765\pi$
$701$	−28.7608	−1.08628	−0.543141	+	0.839641i	$0.682765\pi$
$702$	0	0
$703$	1.16582	0.0439696
$704$	0	0
$705$	−25.4964	−0.960250
$706$	0	0
$707$	−10.4816	−0.394200
$708$	0	0
$709$	27.2335	1.02278	0.511388	−	0.859350i	$-0.329132\pi$
$709$	27.2335	1.02278	0.511388	+	0.859350i	$0.329132\pi$
$710$	0	0
$711$	0.745977	0.0279763
$712$	0	0
$713$	0.950890	0.0356111
$714$	0	0
$715$	0	0
$716$	0	0
$717$	24.3601	0.909746
$718$	0	0
$719$	1.05310	0.0392740	0.0196370	−	0.999807i	$-0.493749\pi$
$719$	1.05310	0.0392740	0.0196370	+	0.999807i	$0.493749\pi$
$720$	0	0
$721$	17.5315	0.652906
$722$	0	0
$723$	−3.23689	−0.120381
$724$	0	0
$725$	3.62965	0.134802
$726$	0	0
$727$	23.0992	0.856700	0.428350	−	0.903613i	$-0.359095\pi$
$727$	23.0992	0.856700	0.428350	+	0.903613i	$0.359095\pi$
$728$	0	0
$729$	26.0335	0.964204
$730$	0	0
$731$	28.4616	1.05269
$732$	0	0
$733$	−29.0249	−1.07206	−0.536030	−	0.844199i	$-0.680077\pi$
$733$	−29.0249	−1.07206	−0.536030	+	0.844199i	$0.680077\pi$
$734$	0	0
$735$	−4.24934	−0.156739
$736$	0	0
$737$	−68.0513	−2.50670
$738$	0	0
$739$	39.7179	1.46105	0.730523	−	0.682888i	$-0.239276\pi$
$739$	39.7179	1.46105	0.730523	+	0.682888i	$0.239276\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	14.2264	0.521917	0.260958	−	0.965350i	$-0.415961\pi$
$743$	14.2264	0.521917	0.260958	+	0.965350i	$0.415961\pi$
$744$	0	0
$745$	−55.1533	−2.02066
$746$	0	0
$747$	1.70828	0.0625027
$748$	0	0
$749$	−9.33384	−0.341051
$750$	0	0
$751$	21.1280	0.770972	0.385486	−	0.922714i	$-0.374034\pi$
$751$	21.1280	0.770972	0.385486	+	0.922714i	$0.374034\pi$
$752$	0	0
$753$	−35.4526	−1.29196
$754$	0	0
$755$	58.2865	2.12126
$756$	0	0
$757$	−17.8609	−0.649165	−0.324583	−	0.945857i	$-0.605224\pi$
$757$	−17.8609	−0.649165	−0.324583	+	0.945857i	$0.605224\pi$
$758$	0	0
$759$	−3.49064	−0.126702
$760$	0	0
$761$	16.1336	0.584844	0.292422	−	0.956289i	$-0.405539\pi$
$761$	16.1336	0.584844	0.292422	+	0.956289i	$0.405539\pi$
$762$	0	0
$763$	−1.39103	−0.0503586
$764$	0	0
$765$	−0.717245	−0.0259320
$766$	0	0
$767$	0	0
$768$	0	0
$769$	37.2607	1.34366	0.671828	−	0.740707i	$-0.265510\pi$
$769$	37.2607	1.34366	0.671828	+	0.740707i	$0.265510\pi$
$770$	0	0
$771$	8.14983	0.293509
$772$	0	0
$773$	44.2031	1.58987	0.794937	−	0.606691i	$-0.207504\pi$
$773$	44.2031	1.58987	0.794937	+	0.606691i	$0.207504\pi$
$774$	0	0
$775$	2.37660	0.0853701
$776$	0	0
$777$	7.07174	0.253697
$778$	0	0
$779$	−2.16755	−0.0776605
$780$	0	0
$781$	−8.34610	−0.298647
$782$	0	0
$783$	−22.5045	−0.804246
$784$	0	0
$785$	−2.78623	−0.0994450
$786$	0	0
$787$	37.5148	1.33726	0.668630	−	0.743595i	$-0.266881\pi$
$787$	37.5148	1.33726	0.668630	+	0.743595i	$0.266881\pi$
$788$	0	0
$789$	−0.957930	−0.0341032
$790$	0	0
$791$	9.11232	0.323997
$792$	0	0
$793$	0	0
$794$	0	0
$795$	−17.5324	−0.621810
$796$	0	0
$797$	−48.0553	−1.70220	−0.851102	−	0.525000i	$-0.824065\pi$
$797$	−48.0553	−1.70220	−0.851102	+	0.525000i	$0.824065\pi$
$798$	0	0
$799$	17.7029	0.626285
$800$	0	0
$801$	1.18304	0.0418006
$802$	0	0
$803$	45.3820	1.60150
$804$	0	0
$805$	0.795019	0.0280207
$806$	0	0
$807$	29.2385	1.02924
$808$	0	0
$809$	18.9484	0.666192	0.333096	−	0.942893i	$-0.391907\pi$
$809$	18.9484	0.666192	0.333096	+	0.942893i	$0.391907\pi$
$810$	0	0
$811$	14.5958	0.512529	0.256265	−	0.966607i	$-0.417508\pi$
$811$	14.5958	0.512529	0.256265	+	0.966607i	$0.417508\pi$
$812$	0	0
$813$	29.3557	1.02955
$814$	0	0
$815$	−49.7141	−1.74141
$816$	0	0
$817$	−2.80033	−0.0979711
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−28.2334	−0.985354	−0.492677	−	0.870212i	$-0.663982\pi$
$821$	−28.2334	−0.985354	−0.492677	+	0.870212i	$0.663982\pi$
$822$	0	0
$823$	1.24398	0.0433625	0.0216813	−	0.999765i	$-0.493098\pi$
$823$	1.24398	0.0433625	0.0216813	+	0.999765i	$0.493098\pi$
$824$	0	0
$825$	−8.72432	−0.303742
$826$	0	0
$827$	48.2789	1.67882	0.839411	−	0.543497i	$-0.182900\pi$
$827$	48.2789	1.67882	0.839411	+	0.543497i	$0.182900\pi$
$828$	0	0
$829$	10.2769	0.356932	0.178466	−	0.983946i	$-0.442886\pi$
$829$	10.2769	0.356932	0.178466	+	0.983946i	$0.442886\pi$
$830$	0	0
$831$	35.0590	1.21618
$832$	0	0
$833$	2.95045	0.102227
$834$	0	0
$835$	−28.4899	−0.985932
$836$	0	0
$837$	−14.7354	−0.509329
$838$	0	0
$839$	−15.4398	−0.533041	−0.266520	−	0.963829i	$-0.585874\pi$
$839$	−15.4398	−0.533041	−0.266520	+	0.963829i	$0.585874\pi$
$840$	0	0
$841$	−9.56883	−0.329960
$842$	0	0
$843$	9.83182	0.338626
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−25.2049	−0.866050
$848$	0	0
$849$	10.9513	0.375849
$850$	0	0
$851$	−1.32307	−0.0453543
$852$	0	0
$853$	41.1350	1.40844	0.704218	−	0.709983i	$-0.251298\pi$
$853$	41.1350	1.40844	0.704218	+	0.709983i	$0.251298\pi$
$854$	0	0
$855$	0.0705694	0.00241342
$856$	0	0
$857$	−20.3347	−0.694621	−0.347311	−	0.937750i	$-0.612905\pi$
$857$	−20.3347	−0.694621	−0.347311	+	0.937750i	$0.612905\pi$
$858$	0	0
$859$	43.9668	1.50013	0.750064	−	0.661366i	$-0.230023\pi$
$859$	43.9668	1.50013	0.750064	+	0.661366i	$0.230023\pi$
$860$	0	0
$861$	−13.1481	−0.448088
$862$	0	0
$863$	−2.44223	−0.0831344	−0.0415672	−	0.999136i	$-0.513235\pi$
$863$	−2.44223	−0.0831344	−0.0415672	+	0.999136i	$0.513235\pi$
$864$	0	0
$865$	−49.4545	−1.68150
$866$	0	0
$867$	−14.6064	−0.496058
$868$	0	0
$869$	−44.5572	−1.51150
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−0.579090	−0.0195992
$874$	0	0
$875$	−10.0788	−0.340727
$876$	0	0
$877$	29.1380	0.983921	0.491960	−	0.870618i	$-0.336281\pi$
$877$	29.1380	0.983921	0.491960	+	0.870618i	$0.336281\pi$
$878$	0	0
$879$	19.1399	0.645574
$880$	0	0
$881$	30.7832	1.03711	0.518555	−	0.855044i	$-0.326470\pi$
$881$	30.7832	1.03711	0.518555	+	0.855044i	$0.326470\pi$
$882$	0	0
$883$	−43.2493	−1.45546	−0.727728	−	0.685866i	$-0.759423\pi$
$883$	−43.2493	−1.45546	−0.727728	+	0.685866i	$0.759423\pi$
$884$	0	0
$885$	31.7190	1.06622
$886$	0	0
$887$	35.5377	1.19324	0.596620	−	0.802524i	$-0.296510\pi$
$887$	35.5377	1.19324	0.596620	+	0.802524i	$0.296510\pi$
$888$	0	0
$889$	−12.2226	−0.409932
$890$	0	0
$891$	55.9108	1.87308
$892$	0	0
$893$	−1.74179	−0.0582866
$894$	0	0
$895$	20.8941	0.698412
$896$	0	0
$897$	0	0
$898$	0	0
$899$	12.7230	0.424337
$900$	0	0
$901$	12.1733	0.405551
$902$	0	0
$903$	−16.9865	−0.565276
$904$	0	0
$905$	16.2622	0.540574
$906$	0	0
$907$	17.3274	0.575347	0.287674	−	0.957728i	$-0.407118\pi$
$907$	17.3274	0.575347	0.287674	+	0.957728i	$0.407118\pi$
$908$	0	0
$909$	1.05589	0.0350216
$910$	0	0
$911$	47.6645	1.57919	0.789597	−	0.613625i	$-0.210289\pi$
$911$	47.6645	1.57919	0.789597	+	0.613625i	$0.210289\pi$
$912$	0	0
$913$	−102.036	−3.37688
$914$	0	0
$915$	34.2126	1.13103
$916$	0	0
$917$	10.2889	0.339768
$918$	0	0
$919$	−25.1946	−0.831095	−0.415547	−	0.909572i	$-0.636410\pi$
$919$	−25.1946	−0.831095	−0.415547	+	0.909572i	$0.636410\pi$
$920$	0	0
$921$	5.90470	0.194567
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−3.30681	−0.108727
$926$	0	0
$927$	−1.76608	−0.0580055
$928$	0	0
$929$	9.63386	0.316077	0.158038	−	0.987433i	$-0.449483\pi$
$929$	9.63386	0.316077	0.158038	+	0.987433i	$0.449483\pi$
$930$	0	0
$931$	−0.290293	−0.00951398
$932$	0	0
$933$	39.3728	1.28901
$934$	0	0
$935$	42.8410	1.40105
$936$	0	0
$937$	−31.6012	−1.03236	−0.516182	−	0.856479i	$-0.672647\pi$
$937$	−31.6012	−1.03236	−0.516182	+	0.856479i	$0.672647\pi$
$938$	0	0
$939$	−33.6194	−1.09713
$940$	0	0
$941$	25.6805	0.837161	0.418581	−	0.908180i	$-0.362528\pi$
$941$	25.6805	0.837161	0.418581	+	0.908180i	$0.362528\pi$
$942$	0	0
$943$	2.45992	0.0801060
$944$	0	0
$945$	−12.3199	−0.400768
$946$	0	0
$947$	−39.4573	−1.28219	−0.641095	−	0.767461i	$-0.721520\pi$
$947$	−39.4573	−1.28219	−0.641095	+	0.767461i	$0.721520\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	54.4673	1.76622
$952$	0	0
$953$	−10.3605	−0.335611	−0.167806	−	0.985820i	$-0.553668\pi$
$953$	−10.3605	−0.335611	−0.167806	+	0.985820i	$0.553668\pi$
$954$	0	0
$955$	−46.8392	−1.51568
$956$	0	0
$957$	−46.7052	−1.50977
$958$	0	0
$959$	17.5059	0.565295
$960$	0	0
$961$	−22.6693	−0.731267
$962$	0	0
$963$	0.940267	0.0302997
$964$	0	0
$965$	−15.3481	−0.494074
$966$	0	0
$967$	12.9100	0.415159	0.207579	−	0.978218i	$-0.433441\pi$
$967$	12.9100	0.415159	0.207579	+	0.978218i	$0.433441\pi$
$968$	0	0
$969$	−1.50819	−0.0484502
$970$	0	0
$971$	−4.45619	−0.143006	−0.0715030	−	0.997440i	$-0.522780\pi$
$971$	−4.45619	−0.143006	−0.0715030	+	0.997440i	$0.522780\pi$
$972$	0	0
$973$	−5.26079	−0.168653
$974$	0	0
$975$	0	0
$976$	0	0
$977$	28.7785	0.920707	0.460354	−	0.887736i	$-0.347723\pi$
$977$	28.7785	0.920707	0.460354	+	0.887736i	$0.347723\pi$
$978$	0	0
$979$	−70.6628	−2.25839
$980$	0	0
$981$	0.140129	0.00447396
$982$	0	0
$983$	−0.0882026	−0.00281323	−0.00140661	−	0.999999i	$-0.500448\pi$
$983$	−0.0882026	−0.00281323	−0.00140661	+	0.999999i	$0.500448\pi$
$984$	0	0
$985$	23.9994	0.764684
$986$	0	0
$987$	−10.5655	−0.336304
$988$	0	0
$989$	3.17805	0.101056
$990$	0	0
$991$	62.3761	1.98144	0.990721	−	0.135911i	$-0.0433961\pi$
$991$	62.3761	1.98144	0.990721	+	0.135911i	$0.0433961\pi$
$992$	0	0
$993$	−52.7324	−1.67341
$994$	0	0
$995$	53.9731	1.71106
$996$	0	0
$997$	15.1566	0.480014	0.240007	−	0.970771i	$-0.422850\pi$
$997$	15.1566	0.480014	0.240007	+	0.970771i	$0.422850\pi$
$998$	0	0
$999$	20.5028	0.648681

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4732.2.a.q.1.3	✓	6
13.5	odd	4		4732.2.g.j.337.6		12
13.8	odd	4		4732.2.g.j.337.5		12
13.12	even	2		4732.2.a.r.1.3	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4732.2.a.q.1.3	✓	6	1.1	even	1	trivial
4732.2.a.r.1.3	yes	6	13.12	even	2
4732.2.g.j.337.5		12	13.8	odd	4
4732.2.g.j.337.6		12	13.5	odd	4

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:	\( N \)	\(=\)	\( 4732 = 2^{2} \cdot 7 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4732.a (trivial)

\(f(q)\)	\(=\)	\(q+1.76089 q^{3} -2.41317 q^{5} -1.00000 q^{7} +0.100737 q^{9} -6.01705 q^{11} -4.24934 q^{15} +2.95045 q^{17} -0.290293 q^{19} -1.76089 q^{21} +0.329450 q^{23} +0.823409 q^{25} -5.10529 q^{27} +4.40808 q^{29} +2.88630 q^{31} -10.5954 q^{33} +2.41317 q^{35} -4.01600 q^{37} +7.46676 q^{41} +9.64655 q^{43} -0.243097 q^{45} +6.00009 q^{47} +1.00000 q^{49} +5.19542 q^{51} +4.12592 q^{53} +14.5202 q^{55} -0.511175 q^{57} -7.46445 q^{59} -8.05128 q^{61} -0.100737 q^{63} +11.3097 q^{67} +0.580125 q^{69} +1.38708 q^{71} -7.54224 q^{73} +1.44993 q^{75} +6.01705 q^{77} +7.40516 q^{79} -9.29206 q^{81} +16.9577 q^{83} -7.11994 q^{85} +7.76215 q^{87} +11.7438 q^{89} +5.08246 q^{93} +0.700528 q^{95} -5.74851 q^{97} -0.606142 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{3} - 6 q^{7} + 20 q^{9} - 4 q^{11} - 3 q^{15} - 3 q^{17} - 5 q^{19} - 6 q^{21} + 18 q^{23} + 2 q^{25} + 24 q^{27} + 17 q^{29} + 17 q^{31} - 21 q^{33} - q^{37} - 14 q^{41} + 14 q^{43} + 29 q^{45}+ \cdots - 3 q^{99}+O(q^{100}) \) 6 * q + 6 * q^3 - 6 * q^7 + 20 * q^9 - 4 * q^11 - 3 * q^15 - 3 * q^17 - 5 * q^19 - 6 * q^21 + 18 * q^23 + 2 * q^25 + 24 * q^27 + 17 * q^29 + 17 * q^31 - 21 * q^33 - q^37 - 14 * q^41 + 14 * q^43 + 29 * q^45 + 6 * q^47 + 6 * q^49 + 2 * q^51 + 3 * q^53 + 18 * q^55 + 14 * q^59 - 4 * q^61 - 20 * q^63 - 4 * q^67 + 39 * q^69 - 3 * q^71 + 31 * q^73 + 37 * q^75 + 4 * q^77 + 22 * q^79 + 6 * q^81 + 15 * q^83 - 51 * q^85 + 51 * q^87 + 20 * q^89 - 13 * q^93 + 19 * q^95 + 29 * q^97 - 3 * q^99

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