LMFDB - Embedded newform 676.4.a.g.1.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [676,4,Mod(1,676)] 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("676.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(676, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 676 = 2^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	676.a (trivial)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$39.8852911639$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 114x^{6} - 224x^{5} + 3123x^{4} + 10080x^{3} - 7598x^{2} - 46368x - 33663 $ x^8 - 114x^6 - 224x^5 + 3123x^4 + 10080x^3 - 7598x^2 - 46368x - 33663
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6}\cdot 3 $
Twist minimal:	no (minimal twist has level 52)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-6.81371$ of defining polynomial
Character	$\chi$	$=$	676.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.28694 q^{3} +15.8968 q^{5} +1.18417 q^{7} -25.3438 q^{9} +38.9119 q^{11} +20.4582 q^{15} -36.9203 q^{17} +42.9418 q^{19} +1.52395 q^{21} +203.934 q^{23} +127.707 q^{25} -67.3632 q^{27} +58.9916 q^{29} -77.3896 q^{31} +50.0772 q^{33} +18.8244 q^{35} +258.294 q^{37} -169.607 q^{41} +366.335 q^{43} -402.884 q^{45} +249.834 q^{47} -341.598 q^{49} -47.5141 q^{51} +157.459 q^{53} +618.574 q^{55} +55.2634 q^{57} -672.229 q^{59} +580.251 q^{61} -30.0113 q^{63} -180.869 q^{67} +262.451 q^{69} -519.611 q^{71} +982.663 q^{73} +164.351 q^{75} +46.0782 q^{77} -1265.85 q^{79} +597.590 q^{81} -1026.42 q^{83} -586.913 q^{85} +75.9185 q^{87} +1485.05 q^{89} -99.5956 q^{93} +682.635 q^{95} -350.236 q^{97} -986.176 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 140 q^{9} + 176 q^{17} - 40 q^{23} + 84 q^{25} - 432 q^{27} + 968 q^{29} - 80 q^{35} + 1008 q^{43} + 1844 q^{49} + 1808 q^{51} - 1164 q^{53} + 2256 q^{55} + 2448 q^{61} + 3476 q^{69} + 2896 q^{75}+ \cdots + 4800 q^{95}+O(q^{100}) $ 8 * q + 140 * q^9 + 176 * q^17 - 40 * q^23 + 84 * q^25 - 432 * q^27 + 968 * q^29 - 80 * q^35 + 1008 * q^43 + 1844 * q^49 + 1808 * q^51 - 1164 * q^53 + 2256 * q^55 + 2448 * q^61 + 3476 * q^69 + 2896 * q^75 + 3972 * q^77 - 3968 * q^79 + 8264 * q^81 + 3320 * q^87 + 4800 * q^95

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
$3$	1.28694	0.247671	0.123836	−	0.992303i	$-0.460480\pi$
$3$	1.28694	0.247671	0.123836	+	0.992303i	$0.460480\pi$
$4$	0	0
$5$	15.8968	1.42185	0.710925	−	0.703268i	$-0.248276\pi$
$5$	15.8968	1.42185	0.710925	+	0.703268i	$0.248276\pi$
$6$	0	0
$7$	1.18417	0.0639389	0.0319695	−	0.999489i	$-0.489822\pi$
$7$	1.18417	0.0639389	0.0319695	+	0.999489i	$0.489822\pi$
$8$	0	0
$9$	−25.3438	−0.938659
$10$	0	0
$11$	38.9119	1.06658	0.533290	−	0.845932i	$-0.320955\pi$
$11$	38.9119	1.06658	0.533290	+	0.845932i	$0.320955\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	20.4582	0.352151
$16$	0	0
$17$	−36.9203	−0.526734	−0.263367	−	0.964696i	$-0.584833\pi$
$17$	−36.9203	−0.526734	−0.263367	+	0.964696i	$0.584833\pi$
$18$	0	0
$19$	42.9418	0.518501	0.259250	−	0.965810i	$-0.416525\pi$
$19$	42.9418	0.518501	0.259250	+	0.965810i	$0.416525\pi$
$20$	0	0
$21$	1.52395	0.0158358
$22$	0	0
$23$	203.934	1.84884	0.924418	−	0.381380i	$-0.124551\pi$
$23$	203.934	1.84884	0.924418	+	0.381380i	$0.124551\pi$
$24$	0	0
$25$	127.707	1.02166
$26$	0	0
$27$	−67.3632	−0.480150
$28$	0	0
$29$	58.9916	0.377740	0.188870	−	0.982002i	$-0.439518\pi$
$29$	58.9916	0.377740	0.188870	+	0.982002i	$0.439518\pi$
$30$	0	0
$31$	−77.3896	−0.448374	−0.224187	−	0.974546i	$-0.571973\pi$
$31$	−77.3896	−0.448374	−0.224187	+	0.974546i	$0.571973\pi$
$32$	0	0
$33$	50.0772	0.264161
$34$	0	0
$35$	18.8244	0.0909116
$36$	0	0
$37$	258.294	1.14765	0.573827	−	0.818976i	$-0.305458\pi$
$37$	258.294	1.14765	0.573827	+	0.818976i	$0.305458\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−169.607	−0.646052	−0.323026	−	0.946390i	$-0.604700\pi$
$41$	−169.607	−0.646052	−0.323026	+	0.946390i	$0.604700\pi$
$42$	0	0
$43$	366.335	1.29920	0.649600	−	0.760276i	$-0.274937\pi$
$43$	366.335	1.29920	0.649600	+	0.760276i	$0.274937\pi$
$44$	0	0
$45$	−402.884	−1.33463
$46$	0	0
$47$	249.834	0.775362	0.387681	−	0.921794i	$-0.373276\pi$
$47$	249.834	0.775362	0.387681	+	0.921794i	$0.373276\pi$
$48$	0	0
$49$	−341.598	−0.995912
$50$	0	0
$51$	−47.5141	−0.130457
$52$	0	0
$53$	157.459	0.408087	0.204044	−	0.978962i	$-0.434592\pi$
$53$	157.459	0.408087	0.204044	+	0.978962i	$0.434592\pi$
$54$	0	0
$55$	618.574	1.51652
$56$	0	0
$57$	55.2634	0.128418
$58$	0	0
$59$	−672.229	−1.48334	−0.741668	−	0.670768i	$-0.765965\pi$
$59$	−672.229	−1.48334	−0.741668	+	0.670768i	$0.765965\pi$
$60$	0	0
$61$	580.251	1.21793	0.608963	−	0.793199i	$-0.291586\pi$
$61$	580.251	1.21793	0.608963	+	0.793199i	$0.291586\pi$
$62$	0	0
$63$	−30.0113	−0.0600169
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−180.869	−0.329800	−0.164900	−	0.986310i	$-0.552730\pi$
$67$	−180.869	−0.329800	−0.164900	+	0.986310i	$0.552730\pi$
$68$	0	0
$69$	262.451	0.457904
$70$	0	0
$71$	−519.611	−0.868541	−0.434271	−	0.900782i	$-0.642994\pi$
$71$	−519.611	−0.868541	−0.434271	+	0.900782i	$0.642994\pi$
$72$	0	0
$73$	982.663	1.57551	0.787753	−	0.615991i	$-0.211244\pi$
$73$	982.663	1.57551	0.787753	+	0.615991i	$0.211244\pi$
$74$	0	0
$75$	164.351	0.253035
$76$	0	0
$77$	46.0782	0.0681960
$78$	0	0
$79$	−1265.85	−1.80278	−0.901388	−	0.433013i	$-0.857450\pi$
$79$	−1265.85	−1.80278	−0.901388	+	0.433013i	$0.857450\pi$
$80$	0	0
$81$	597.590	0.819740
$82$	0	0
$83$	−1026.42	−1.35740	−0.678702	−	0.734414i	$-0.737457\pi$
$83$	−1026.42	−1.35740	−0.678702	+	0.734414i	$0.737457\pi$
$84$	0	0
$85$	−586.913	−0.748937
$86$	0	0
$87$	75.9185	0.0935553
$88$	0	0
$89$	1485.05	1.76871	0.884355	−	0.466816i	$-0.154599\pi$
$89$	1485.05	1.76871	0.884355	+	0.466816i	$0.154599\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−99.5956	−0.111049
$94$	0	0
$95$	682.635	0.737231
$96$	0	0
$97$	−350.236	−0.366609	−0.183305	−	0.983056i	$-0.558679\pi$
$97$	−350.236	−0.366609	−0.183305	+	0.983056i	$0.558679\pi$
$98$	0	0
$99$	−986.176	−1.00116
$100$	0	0
$101$	1431.71	1.41050	0.705248	−	0.708961i	$-0.250836\pi$
$101$	1431.71	1.41050	0.705248	+	0.708961i	$0.250836\pi$
$102$	0	0
$103$	1492.10	1.42739	0.713695	−	0.700456i	$-0.247020\pi$
$103$	1492.10	1.42739	0.713695	+	0.700456i	$0.247020\pi$
$104$	0	0
$105$	24.2258	0.0225162
$106$	0	0
$107$	1161.82	1.04969	0.524847	−	0.851197i	$-0.324123\pi$
$107$	1161.82	1.04969	0.524847	+	0.851197i	$0.324123\pi$
$108$	0	0
$109$	1599.05	1.40515	0.702574	−	0.711611i	$-0.252034\pi$
$109$	1599.05	1.40515	0.702574	+	0.711611i	$0.252034\pi$
$110$	0	0
$111$	332.408	0.284241
$112$	0	0
$113$	374.714	0.311948	0.155974	−	0.987761i	$-0.450148\pi$
$113$	374.714	0.311948	0.155974	+	0.987761i	$0.450148\pi$
$114$	0	0
$115$	3241.90	2.62877
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−43.7197	−0.0336788
$120$	0	0
$121$	183.138	0.137595
$122$	0	0
$123$	−218.273	−0.160009
$124$	0	0
$125$	43.0377	0.0307952
$126$	0	0
$127$	649.296	0.453667	0.226833	−	0.973934i	$-0.427163\pi$
$127$	649.296	0.453667	0.226833	+	0.973934i	$0.427163\pi$
$128$	0	0
$129$	471.450	0.321774
$130$	0	0
$131$	−612.966	−0.408818	−0.204409	−	0.978886i	$-0.565527\pi$
$131$	−612.966	−0.408818	−0.204409	+	0.978886i	$0.565527\pi$
$132$	0	0
$133$	50.8502	0.0331524
$134$	0	0
$135$	−1070.86	−0.682702
$136$	0	0
$137$	−208.801	−0.130212	−0.0651062	−	0.997878i	$-0.520739\pi$
$137$	−208.801	−0.130212	−0.0651062	+	0.997878i	$0.520739\pi$
$138$	0	0
$139$	−369.589	−0.225526	−0.112763	−	0.993622i	$-0.535970\pi$
$139$	−369.589	−0.225526	−0.112763	+	0.993622i	$0.535970\pi$
$140$	0	0
$141$	321.521	0.192035
$142$	0	0
$143$	0	0
$144$	0	0
$145$	937.775	0.537090
$146$	0	0
$147$	−439.615	−0.246659
$148$	0	0
$149$	−990.203	−0.544433	−0.272217	−	0.962236i	$-0.587757\pi$
$149$	−990.203	−0.544433	−0.272217	+	0.962236i	$0.587757\pi$
$150$	0	0
$151$	2855.12	1.53872	0.769360	−	0.638816i	$-0.220575\pi$
$151$	2855.12	1.53872	0.769360	+	0.638816i	$0.220575\pi$
$152$	0	0
$153$	935.700	0.494424
$154$	0	0
$155$	−1230.24	−0.637520
$156$	0	0
$157$	−947.845	−0.481823	−0.240912	−	0.970547i	$-0.577446\pi$
$157$	−947.845	−0.481823	−0.240912	+	0.970547i	$0.577446\pi$
$158$	0	0
$159$	202.640	0.101071
$160$	0	0
$161$	241.492	0.118213
$162$	0	0
$163$	−2690.70	−1.29296	−0.646479	−	0.762932i	$-0.723759\pi$
$163$	−2690.70	−1.29296	−0.646479	+	0.762932i	$0.723759\pi$
$164$	0	0
$165$	796.066	0.375598
$166$	0	0
$167$	−1805.83	−0.836760	−0.418380	−	0.908272i	$-0.637402\pi$
$167$	−1805.83	−0.836760	−0.418380	+	0.908272i	$0.637402\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	−1088.31	−0.486695
$172$	0	0
$173$	354.043	0.155592	0.0777959	−	0.996969i	$-0.475212\pi$
$173$	354.043	0.155592	0.0777959	+	0.996969i	$0.475212\pi$
$174$	0	0
$175$	151.227	0.0653238
$176$	0	0
$177$	−865.117	−0.367380
$178$	0	0
$179$	−2594.18	−1.08323	−0.541615	−	0.840627i	$-0.682187\pi$
$179$	−2594.18	−1.08323	−0.541615	+	0.840627i	$0.682187\pi$
$180$	0	0
$181$	2338.97	0.960522	0.480261	−	0.877126i	$-0.340542\pi$
$181$	2338.97	0.960522	0.480261	+	0.877126i	$0.340542\pi$
$182$	0	0
$183$	746.747	0.301645
$184$	0	0
$185$	4106.04	1.63179
$186$	0	0
$187$	−1436.64	−0.561805
$188$	0	0
$189$	−79.7692	−0.0307003
$190$	0	0
$191$	30.6008	0.0115927	0.00579633	−	0.999983i	$-0.498155\pi$
$191$	30.6008	0.0115927	0.00579633	+	0.999983i	$0.498155\pi$
$192$	0	0
$193$	−3042.05	−1.13457	−0.567283	−	0.823523i	$-0.692005\pi$
$193$	−3042.05	−1.13457	−0.567283	+	0.823523i	$0.692005\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−4170.80	−1.50841	−0.754205	−	0.656639i	$-0.771978\pi$
$197$	−4170.80	−1.50841	−0.754205	+	0.656639i	$0.771978\pi$
$198$	0	0
$199$	4631.60	1.64988	0.824938	−	0.565223i	$-0.191210\pi$
$199$	4631.60	1.64988	0.824938	+	0.565223i	$0.191210\pi$
$200$	0	0
$201$	−232.767	−0.0816820
$202$	0	0
$203$	69.8558	0.0241523
$204$	0	0
$205$	−2696.20	−0.918590
$206$	0	0
$207$	−5168.47	−1.73543
$208$	0	0
$209$	1670.95	0.553023
$210$	0	0
$211$	−5403.64	−1.76304	−0.881522	−	0.472143i	$-0.843480\pi$
$211$	−5403.64	−1.76304	−0.881522	+	0.472143i	$0.843480\pi$
$212$	0	0
$213$	−668.707	−0.215113
$214$	0	0
$215$	5823.54	1.84727
$216$	0	0
$217$	−91.6421	−0.0286685
$218$	0	0
$219$	1264.63	0.390208
$220$	0	0
$221$	0	0
$222$	0	0
$223$	4988.09	1.49788	0.748941	−	0.662637i	$-0.230563\pi$
$223$	4988.09	1.49788	0.748941	+	0.662637i	$0.230563\pi$
$224$	0	0
$225$	−3236.59	−0.958989
$226$	0	0
$227$	−2536.58	−0.741667	−0.370834	−	0.928699i	$-0.620928\pi$
$227$	−2536.58	−0.741667	−0.370834	+	0.928699i	$0.620928\pi$
$228$	0	0
$229$	−747.558	−0.215721	−0.107860	−	0.994166i	$-0.534400\pi$
$229$	−747.558	−0.215721	−0.107860	+	0.994166i	$0.534400\pi$
$230$	0	0
$231$	59.2997	0.0168902
$232$	0	0
$233$	1661.81	0.467248	0.233624	−	0.972327i	$-0.424942\pi$
$233$	1661.81	0.467248	0.233624	+	0.972327i	$0.424942\pi$
$234$	0	0
$235$	3971.55	1.10245
$236$	0	0
$237$	−1629.07	−0.446496
$238$	0	0
$239$	1995.23	0.540002	0.270001	−	0.962860i	$-0.412976\pi$
$239$	1995.23	0.540002	0.270001	+	0.962860i	$0.412976\pi$
$240$	0	0
$241$	−1304.22	−0.348598	−0.174299	−	0.984693i	$-0.555766\pi$
$241$	−1304.22	−0.348598	−0.174299	+	0.984693i	$0.555766\pi$
$242$	0	0
$243$	2587.87	0.683176
$244$	0	0
$245$	−5430.30	−1.41604
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−1320.94	−0.336190
$250$	0	0
$251$	−5057.63	−1.27185	−0.635926	−	0.771750i	$-0.719381\pi$
$251$	−5057.63	−1.27185	−0.635926	+	0.771750i	$0.719381\pi$
$252$	0	0
$253$	7935.48	1.97193
$254$	0	0
$255$	−755.321	−0.185490
$256$	0	0
$257$	4451.31	1.08041	0.540204	−	0.841534i	$-0.318347\pi$
$257$	4451.31	1.08041	0.540204	+	0.841534i	$0.318347\pi$
$258$	0	0
$259$	305.863	0.0733798
$260$	0	0
$261$	−1495.07	−0.354569
$262$	0	0
$263$	−4575.88	−1.07286	−0.536428	−	0.843946i	$-0.680227\pi$
$263$	−4575.88	−1.07286	−0.536428	+	0.843946i	$0.680227\pi$
$264$	0	0
$265$	2503.09	0.580239
$266$	0	0
$267$	1911.17	0.438058
$268$	0	0
$269$	−5411.32	−1.22652	−0.613260	−	0.789881i	$-0.710142\pi$
$269$	−5411.32	−1.22652	−0.613260	+	0.789881i	$0.710142\pi$
$270$	0	0
$271$	−372.296	−0.0834516	−0.0417258	−	0.999129i	$-0.513286\pi$
$271$	−372.296	−0.0834516	−0.0417258	+	0.999129i	$0.513286\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	4969.34	1.08968
$276$	0	0
$277$	−5907.85	−1.28147	−0.640737	−	0.767761i	$-0.721371\pi$
$277$	−5907.85	−1.28147	−0.640737	+	0.767761i	$0.721371\pi$
$278$	0	0
$279$	1961.35	0.420870
$280$	0	0
$281$	−3598.50	−0.763946	−0.381973	−	0.924174i	$-0.624755\pi$
$281$	−3598.50	−0.763946	−0.381973	+	0.924174i	$0.624755\pi$
$282$	0	0
$283$	−3413.30	−0.716960	−0.358480	−	0.933537i	$-0.616705\pi$
$283$	−3413.30	−0.716960	−0.358480	+	0.933537i	$0.616705\pi$
$284$	0	0
$285$	878.509	0.182591
$286$	0	0
$287$	−200.843	−0.0413079
$288$	0	0
$289$	−3549.89	−0.722551
$290$	0	0
$291$	−450.732	−0.0907985
$292$	0	0
$293$	1127.82	0.224873	0.112436	−	0.993659i	$-0.464135\pi$
$293$	1127.82	0.224873	0.112436	+	0.993659i	$0.464135\pi$
$294$	0	0
$295$	−10686.3	−2.10908
$296$	0	0
$297$	−2621.23	−0.512119
$298$	0	0
$299$	0	0
$300$	0	0
$301$	433.801	0.0830694
$302$	0	0
$303$	1842.52	0.349339
$304$	0	0
$305$	9224.11	1.73171
$306$	0	0
$307$	−7338.57	−1.36428	−0.682141	−	0.731221i	$-0.738951\pi$
$307$	−7338.57	−1.36428	−0.682141	+	0.731221i	$0.738951\pi$
$308$	0	0
$309$	1920.24	0.353524
$310$	0	0
$311$	−7550.86	−1.37675	−0.688376	−	0.725354i	$-0.741676\pi$
$311$	−7550.86	−1.37675	−0.688376	+	0.725354i	$0.741676\pi$
$312$	0	0
$313$	−1262.18	−0.227932	−0.113966	−	0.993485i	$-0.536356\pi$
$313$	−1262.18	−0.227932	−0.113966	+	0.993485i	$0.536356\pi$
$314$	0	0
$315$	−477.082	−0.0853350
$316$	0	0
$317$	4085.51	0.723865	0.361933	−	0.932204i	$-0.382117\pi$
$317$	4085.51	0.723865	0.361933	+	0.932204i	$0.382117\pi$
$318$	0	0
$319$	2295.48	0.402890
$320$	0	0
$321$	1495.19	0.259979
$322$	0	0
$323$	−1585.42	−0.273112
$324$	0	0
$325$	0	0
$326$	0	0
$327$	2057.88	0.348015
$328$	0	0
$329$	295.845	0.0495758
$330$	0	0
$331$	−4515.66	−0.749859	−0.374930	−	0.927053i	$-0.622333\pi$
$331$	−4515.66	−0.749859	−0.374930	+	0.927053i	$0.622333\pi$
$332$	0	0
$333$	−6546.14	−1.07726
$334$	0	0
$335$	−2875.23	−0.468927
$336$	0	0
$337$	123.448	0.0199545	0.00997723	−	0.999950i	$-0.496824\pi$
$337$	123.448	0.0199545	0.00997723	+	0.999950i	$0.496824\pi$
$338$	0	0
$339$	482.233	0.0772605
$340$	0	0
$341$	−3011.38	−0.478227
$342$	0	0
$343$	−810.677	−0.127616
$344$	0	0
$345$	4172.12	0.651070
$346$	0	0
$347$	2452.90	0.379478	0.189739	−	0.981835i	$-0.439236\pi$
$347$	2452.90	0.379478	0.189739	+	0.981835i	$0.439236\pi$
$348$	0	0
$349$	−8778.28	−1.34639	−0.673196	−	0.739464i	$-0.735079\pi$
$349$	−8778.28	−1.34639	−0.673196	+	0.739464i	$0.735079\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−3920.87	−0.591180	−0.295590	−	0.955315i	$-0.595516\pi$
$353$	−3920.87	−0.591180	−0.295590	+	0.955315i	$0.595516\pi$
$354$	0	0
$355$	−8260.13	−1.23494
$356$	0	0
$357$	−56.2646	−0.00834128
$358$	0	0
$359$	9182.02	1.34988	0.674942	−	0.737870i	$-0.264168\pi$
$359$	9182.02	1.34988	0.674942	+	0.737870i	$0.264168\pi$
$360$	0	0
$361$	−5015.00	−0.731157
$362$	0	0
$363$	235.688	0.0340782
$364$	0	0
$365$	15621.2	2.24013
$366$	0	0
$367$	−8415.78	−1.19700	−0.598502	−	0.801121i	$-0.704237\pi$
$367$	−8415.78	−1.19700	−0.598502	+	0.801121i	$0.704237\pi$
$368$	0	0
$369$	4298.48	0.606423
$370$	0	0
$371$	186.457	0.0260927
$372$	0	0
$373$	12424.4	1.72470	0.862350	−	0.506313i	$-0.168992\pi$
$373$	12424.4	1.72470	0.862350	+	0.506313i	$0.168992\pi$
$374$	0	0
$375$	55.3868	0.00762710
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−1197.13	−0.162249	−0.0811246	−	0.996704i	$-0.525851\pi$
$379$	−1197.13	−0.162249	−0.0811246	+	0.996704i	$0.525851\pi$
$380$	0	0
$381$	835.603	0.112360
$382$	0	0
$383$	898.733	0.119904	0.0599519	−	0.998201i	$-0.480905\pi$
$383$	898.733	0.119904	0.0599519	+	0.998201i	$0.480905\pi$
$384$	0	0
$385$	732.494	0.0969646
$386$	0	0
$387$	−9284.32	−1.21951
$388$	0	0
$389$	1927.09	0.251175	0.125588	−	0.992083i	$-0.459918\pi$
$389$	1927.09	0.251175	0.125588	+	0.992083i	$0.459918\pi$
$390$	0	0
$391$	−7529.31	−0.973846
$392$	0	0
$393$	−788.850	−0.101252
$394$	0	0
$395$	−20122.9	−2.56328
$396$	0	0
$397$	2171.32	0.274497	0.137248	−	0.990537i	$-0.456174\pi$
$397$	2171.32	0.274497	0.137248	+	0.990537i	$0.456174\pi$
$398$	0	0
$399$	65.4410	0.00821090
$400$	0	0
$401$	−10451.7	−1.30158	−0.650789	−	0.759259i	$-0.725562\pi$
$401$	−10451.7	−1.30158	−0.650789	+	0.759259i	$0.725562\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	9499.75	1.16555
$406$	0	0
$407$	10050.7	1.22407
$408$	0	0
$409$	−4319.91	−0.522263	−0.261132	−	0.965303i	$-0.584096\pi$
$409$	−4319.91	−0.522263	−0.261132	+	0.965303i	$0.584096\pi$
$410$	0	0
$411$	−268.714	−0.0322499
$412$	0	0
$413$	−796.031	−0.0948429
$414$	0	0
$415$	−16316.8	−1.93002
$416$	0	0
$417$	−475.639	−0.0558564
$418$	0	0
$419$	−2143.72	−0.249946	−0.124973	−	0.992160i	$-0.539884\pi$
$419$	−2143.72	−0.249946	−0.124973	+	0.992160i	$0.539884\pi$
$420$	0	0
$421$	11308.5	1.30912	0.654561	−	0.756009i	$-0.272853\pi$
$421$	11308.5	1.30912	0.654561	+	0.756009i	$0.272853\pi$
$422$	0	0
$423$	−6331.74	−0.727801
$424$	0	0
$425$	−4714.99	−0.538143
$426$	0	0
$427$	687.113	0.0778729
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−8276.70	−0.924999	−0.462499	−	0.886620i	$-0.653047\pi$
$431$	−8276.70	−0.924999	−0.462499	+	0.886620i	$0.653047\pi$
$432$	0	0
$433$	313.808	0.0348283	0.0174142	−	0.999848i	$-0.494457\pi$
$433$	313.808	0.0348283	0.0174142	+	0.999848i	$0.494457\pi$
$434$	0	0
$435$	1206.86	0.133022
$436$	0	0
$437$	8757.30	0.958623
$438$	0	0
$439$	4798.46	0.521681	0.260840	−	0.965382i	$-0.416000\pi$
$439$	4798.46	0.521681	0.260840	+	0.965382i	$0.416000\pi$
$440$	0	0
$441$	8657.38	0.934822
$442$	0	0
$443$	−871.652	−0.0934841	−0.0467420	−	0.998907i	$-0.514884\pi$
$443$	−871.652	−0.0934841	−0.0467420	+	0.998907i	$0.514884\pi$
$444$	0	0
$445$	23607.5	2.51484
$446$	0	0
$447$	−1274.33	−0.134840
$448$	0	0
$449$	−7970.06	−0.837707	−0.418853	−	0.908054i	$-0.637568\pi$
$449$	−7970.06	−0.837707	−0.418853	+	0.908054i	$0.637568\pi$
$450$	0	0
$451$	−6599.73	−0.689067
$452$	0	0
$453$	3674.37	0.381097
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−9041.34	−0.925462	−0.462731	−	0.886499i	$-0.653130\pi$
$457$	−9041.34	−0.925462	−0.462731	+	0.886499i	$0.653130\pi$
$458$	0	0
$459$	2487.07	0.252912
$460$	0	0
$461$	−2296.87	−0.232052	−0.116026	−	0.993246i	$-0.537016\pi$
$461$	−2296.87	−0.232052	−0.116026	+	0.993246i	$0.537016\pi$
$462$	0	0
$463$	−10243.3	−1.02817	−0.514087	−	0.857738i	$-0.671869\pi$
$463$	−10243.3	−1.02817	−0.514087	+	0.857738i	$0.671869\pi$
$464$	0	0
$465$	−1583.25	−0.157895
$466$	0	0
$467$	11561.2	1.14558	0.572792	−	0.819701i	$-0.305860\pi$
$467$	11561.2	1.14558	0.572792	+	0.819701i	$0.305860\pi$
$468$	0	0
$469$	−214.178	−0.0210871
$470$	0	0
$471$	−1219.82	−0.119334
$472$	0	0
$473$	14254.8	1.38570
$474$	0	0
$475$	5483.98	0.529731
$476$	0	0
$477$	−3990.60	−0.383055
$478$	0	0
$479$	4008.98	0.382412	0.191206	−	0.981550i	$-0.438760\pi$
$479$	4008.98	0.382412	0.191206	+	0.981550i	$0.438760\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	310.785	0.0292779
$484$	0	0
$485$	−5567.62	−0.521263
$486$	0	0
$487$	2963.97	0.275791	0.137896	−	0.990447i	$-0.455966\pi$
$487$	2963.97	0.275791	0.137896	+	0.990447i	$0.455966\pi$
$488$	0	0
$489$	−3462.77	−0.320228
$490$	0	0
$491$	17955.2	1.65032	0.825160	−	0.564900i	$-0.191085\pi$
$491$	17955.2	1.65032	0.825160	+	0.564900i	$0.191085\pi$
$492$	0	0
$493$	−2177.99	−0.198969
$494$	0	0
$495$	−15677.0	−1.42349
$496$	0	0
$497$	−615.305	−0.0555336
$498$	0	0
$499$	−15327.6	−1.37507	−0.687533	−	0.726153i	$-0.741306\pi$
$499$	−15327.6	−1.37507	−0.687533	+	0.726153i	$0.741306\pi$
$500$	0	0
$501$	−2323.98	−0.207241
$502$	0	0
$503$	−12826.7	−1.13701	−0.568505	−	0.822680i	$-0.692478\pi$
$503$	−12826.7	−1.13701	−0.568505	+	0.822680i	$0.692478\pi$
$504$	0	0
$505$	22759.5	2.00551
$506$	0	0
$507$	0	0
$508$	0	0
$509$	2405.07	0.209436	0.104718	−	0.994502i	$-0.466606\pi$
$509$	2405.07	0.209436	0.104718	+	0.994502i	$0.466606\pi$
$510$	0	0
$511$	1163.64	0.100736
$512$	0	0
$513$	−2892.69	−0.248958
$514$	0	0
$515$	23719.6	2.02954
$516$	0	0
$517$	9721.52	0.826986
$518$	0	0
$519$	455.631	0.0385356
$520$	0	0
$521$	13258.4	1.11490	0.557450	−	0.830210i	$-0.311780\pi$
$521$	13258.4	1.11490	0.557450	+	0.830210i	$0.311780\pi$
$522$	0	0
$523$	4514.94	0.377484	0.188742	−	0.982027i	$-0.439559\pi$
$523$	4514.94	0.377484	0.188742	+	0.982027i	$0.439559\pi$
$524$	0	0
$525$	194.619	0.0161788
$526$	0	0
$527$	2857.24	0.236174
$528$	0	0
$529$	29422.2	2.41820
$530$	0	0
$531$	17036.8	1.39235
$532$	0	0
$533$	0	0
$534$	0	0
$535$	18469.2	1.49251
$536$	0	0
$537$	−3338.55	−0.268285
$538$	0	0
$539$	−13292.2	−1.06222
$540$	0	0
$541$	−7456.65	−0.592581	−0.296291	−	0.955098i	$-0.595750\pi$
$541$	−7456.65	−0.592581	−0.296291	+	0.955098i	$0.595750\pi$
$542$	0	0
$543$	3010.11	0.237894
$544$	0	0
$545$	25419.7	1.99791
$546$	0	0
$547$	−6132.43	−0.479349	−0.239674	−	0.970853i	$-0.577041\pi$
$547$	−6132.43	−0.479349	−0.239674	+	0.970853i	$0.577041\pi$
$548$	0	0
$549$	−14705.8	−1.14322
$550$	0	0
$551$	2533.20	0.195859
$552$	0	0
$553$	−1498.98	−0.115268
$554$	0	0
$555$	5284.21	0.404148
$556$	0	0
$557$	10428.8	0.793327	0.396664	−	0.917964i	$-0.370168\pi$
$557$	10428.8	0.793327	0.396664	+	0.917964i	$0.370168\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−1848.87	−0.139143
$562$	0	0
$563$	22784.5	1.70560	0.852801	−	0.522236i	$-0.174902\pi$
$563$	22784.5	1.70560	0.852801	+	0.522236i	$0.174902\pi$
$564$	0	0
$565$	5956.74	0.443543
$566$	0	0
$567$	707.646	0.0524133
$568$	0	0
$569$	12879.5	0.948922	0.474461	−	0.880276i	$-0.342643\pi$
$569$	12879.5	0.948922	0.474461	+	0.880276i	$0.342643\pi$
$570$	0	0
$571$	−6071.36	−0.444971	−0.222485	−	0.974936i	$-0.571417\pi$
$571$	−6071.36	−0.444971	−0.222485	+	0.974936i	$0.571417\pi$
$572$	0	0
$573$	39.3813	0.00287117
$574$	0	0
$575$	26043.9	1.88888
$576$	0	0
$577$	−2429.06	−0.175257	−0.0876283	−	0.996153i	$-0.527929\pi$
$577$	−2429.06	−0.175257	−0.0876283	+	0.996153i	$0.527929\pi$
$578$	0	0
$579$	−3914.92	−0.280999
$580$	0	0
$581$	−1215.45	−0.0867909
$582$	0	0
$583$	6127.02	0.435258
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−14624.2	−1.02829	−0.514144	−	0.857704i	$-0.671890\pi$
$587$	−14624.2	−1.02829	−0.514144	+	0.857704i	$0.671890\pi$
$588$	0	0
$589$	−3323.25	−0.232482
$590$	0	0
$591$	−5367.55	−0.373590
$592$	0	0
$593$	15282.4	1.05830	0.529149	−	0.848529i	$-0.322511\pi$
$593$	15282.4	1.05830	0.529149	+	0.848529i	$0.322511\pi$
$594$	0	0
$595$	−695.003	−0.0478863
$596$	0	0
$597$	5960.58	0.408627
$598$	0	0
$599$	−11067.6	−0.754940	−0.377470	−	0.926022i	$-0.623206\pi$
$599$	−11067.6	−0.754940	−0.377470	+	0.926022i	$0.623206\pi$
$600$	0	0
$601$	18287.2	1.24118	0.620590	−	0.784135i	$-0.286893\pi$
$601$	18287.2	1.24118	0.620590	+	0.784135i	$0.286893\pi$
$602$	0	0
$603$	4583.90	0.309570
$604$	0	0
$605$	2911.31	0.195639
$606$	0	0
$607$	2762.58	0.184728	0.0923638	−	0.995725i	$-0.470558\pi$
$607$	2762.58	0.184728	0.0923638	+	0.995725i	$0.470558\pi$
$608$	0	0
$609$	89.9000	0.00598183
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−26080.1	−1.71837	−0.859187	−	0.511661i	$-0.829030\pi$
$613$	−26080.1	−1.71837	−0.859187	+	0.511661i	$0.829030\pi$
$614$	0	0
$615$	−3469.84	−0.227508
$616$	0	0
$617$	−9169.83	−0.598320	−0.299160	−	0.954203i	$-0.596706\pi$
$617$	−9169.83	−0.598320	−0.299160	+	0.954203i	$0.596706\pi$
$618$	0	0
$619$	−17765.0	−1.15353	−0.576767	−	0.816909i	$-0.695686\pi$
$619$	−17765.0	−1.15353	−0.576767	+	0.816909i	$0.695686\pi$
$620$	0	0
$621$	−13737.7	−0.887719
$622$	0	0
$623$	1758.55	0.113089
$624$	0	0
$625$	−15279.3	−0.977872
$626$	0	0
$627$	2150.40	0.136968
$628$	0	0
$629$	−9536.28	−0.604509
$630$	0	0
$631$	−29148.9	−1.83898	−0.919492	−	0.393110i	$-0.871399\pi$
$631$	−29148.9	−1.83898	−0.919492	+	0.393110i	$0.871399\pi$
$632$	0	0
$633$	−6954.15	−0.436655
$634$	0	0
$635$	10321.7	0.645046
$636$	0	0
$637$	0	0
$638$	0	0
$639$	13168.9	0.815264
$640$	0	0
$641$	24613.4	1.51665	0.758323	−	0.651879i	$-0.226019\pi$
$641$	24613.4	1.51665	0.758323	+	0.651879i	$0.226019\pi$
$642$	0	0
$643$	−27133.6	−1.66414	−0.832071	−	0.554668i	$-0.812845\pi$
$643$	−27133.6	−1.66414	−0.832071	+	0.554668i	$0.812845\pi$
$644$	0	0
$645$	7494.54	0.457515
$646$	0	0
$647$	−23734.3	−1.44218	−0.721092	−	0.692839i	$-0.756360\pi$
$647$	−23734.3	−1.44218	−0.721092	+	0.692839i	$0.756360\pi$
$648$	0	0
$649$	−26157.7	−1.58210
$650$	0	0
$651$	−117.938	−0.00710037
$652$	0	0
$653$	19376.4	1.16119	0.580596	−	0.814192i	$-0.302819\pi$
$653$	19376.4	1.16119	0.580596	+	0.814192i	$0.302819\pi$
$654$	0	0
$655$	−9744.19	−0.581278
$656$	0	0
$657$	−24904.4	−1.47886
$658$	0	0
$659$	11271.6	0.666279	0.333139	−	0.942878i	$-0.391892\pi$
$659$	11271.6	0.666279	0.333139	+	0.942878i	$0.391892\pi$
$660$	0	0
$661$	−8384.13	−0.493351	−0.246675	−	0.969098i	$-0.579338\pi$
$661$	−8384.13	−0.493351	−0.246675	+	0.969098i	$0.579338\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	808.353	0.0471377
$666$	0	0
$667$	12030.4	0.698379
$668$	0	0
$669$	6419.37	0.370982
$670$	0	0
$671$	22578.7	1.29902
$672$	0	0
$673$	5407.30	0.309712	0.154856	−	0.987937i	$-0.450509\pi$
$673$	5407.30	0.309712	0.154856	+	0.987937i	$0.450509\pi$
$674$	0	0
$675$	−8602.77	−0.490549
$676$	0	0
$677$	−10090.8	−0.572854	−0.286427	−	0.958102i	$-0.592468\pi$
$677$	−10090.8	−0.572854	−0.286427	+	0.958102i	$0.592468\pi$
$678$	0	0
$679$	−414.737	−0.0234406
$680$	0	0
$681$	−3264.41	−0.183690
$682$	0	0
$683$	13234.6	0.741444	0.370722	−	0.928744i	$-0.379110\pi$
$683$	13234.6	0.741444	0.370722	+	0.928744i	$0.379110\pi$
$684$	0	0
$685$	−3319.27	−0.185143
$686$	0	0
$687$	−962.061	−0.0534278
$688$	0	0
$689$	0	0
$690$	0	0
$691$	2778.75	0.152979	0.0764895	−	0.997070i	$-0.475629\pi$
$691$	2778.75	0.152979	0.0764895	+	0.997070i	$0.475629\pi$
$692$	0	0
$693$	−1167.80	−0.0640128
$694$	0	0
$695$	−5875.28	−0.320665
$696$	0	0
$697$	6261.93	0.340298
$698$	0	0
$699$	2138.65	0.115724
$700$	0	0
$701$	19737.6	1.06345	0.531725	−	0.846917i	$-0.321544\pi$
$701$	19737.6	1.06345	0.531725	+	0.846917i	$0.321544\pi$
$702$	0	0
$703$	11091.6	0.595060
$704$	0	0
$705$	5111.14	0.273045
$706$	0	0
$707$	1695.38	0.0901856
$708$	0	0
$709$	−6053.73	−0.320667	−0.160333	−	0.987063i	$-0.551257\pi$
$709$	−6053.73	−0.320667	−0.160333	+	0.987063i	$0.551257\pi$
$710$	0	0
$711$	32081.4	1.69219
$712$	0	0
$713$	−15782.4	−0.828969
$714$	0	0
$715$	0	0
$716$	0	0
$717$	2567.73	0.133743
$718$	0	0
$719$	−13757.6	−0.713590	−0.356795	−	0.934183i	$-0.616131\pi$
$719$	−13757.6	−0.713590	−0.356795	+	0.934183i	$0.616131\pi$
$720$	0	0
$721$	1766.90	0.0912658
$722$	0	0
$723$	−1678.45	−0.0863377
$724$	0	0
$725$	7533.66	0.385921
$726$	0	0
$727$	18384.8	0.937903	0.468951	−	0.883224i	$-0.344632\pi$
$727$	18384.8	0.937903	0.468951	+	0.883224i	$0.344632\pi$
$728$	0	0
$729$	−12804.5	−0.650537
$730$	0	0
$731$	−13525.2	−0.684333
$732$	0	0
$733$	7848.53	0.395487	0.197744	−	0.980254i	$-0.436639\pi$
$733$	7848.53	0.395487	0.197744	+	0.980254i	$0.436639\pi$
$734$	0	0
$735$	−6988.46	−0.350712
$736$	0	0
$737$	−7037.95	−0.351759
$738$	0	0
$739$	16080.1	0.800430	0.400215	−	0.916421i	$-0.368935\pi$
$739$	16080.1	0.800430	0.400215	+	0.916421i	$0.368935\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−9383.70	−0.463331	−0.231665	−	0.972796i	$-0.574417\pi$
$743$	−9383.70	−0.463331	−0.231665	+	0.972796i	$0.574417\pi$
$744$	0	0
$745$	−15741.0	−0.774103
$746$	0	0
$747$	26013.4	1.27414
$748$	0	0
$749$	1375.79	0.0671163
$750$	0	0
$751$	−8110.58	−0.394087	−0.197043	−	0.980395i	$-0.563134\pi$
$751$	−8110.58	−0.394087	−0.197043	+	0.980395i	$0.563134\pi$
$752$	0	0
$753$	−6508.85	−0.315001
$754$	0	0
$755$	45387.3	2.18783
$756$	0	0
$757$	−25980.7	−1.24740	−0.623701	−	0.781663i	$-0.714372\pi$
$757$	−25980.7	−1.24740	−0.623701	+	0.781663i	$0.714372\pi$
$758$	0	0
$759$	10212.5	0.488391
$760$	0	0
$761$	30759.0	1.46520	0.732598	−	0.680662i	$-0.238308\pi$
$761$	30759.0	1.46520	0.732598	+	0.680662i	$0.238308\pi$
$762$	0	0
$763$	1893.54	0.0898437
$764$	0	0
$765$	14874.6	0.702997
$766$	0	0
$767$	0	0
$768$	0	0
$769$	18655.4	0.874814	0.437407	−	0.899264i	$-0.355897\pi$
$769$	18655.4	0.874814	0.437407	+	0.899264i	$0.355897\pi$
$770$	0	0
$771$	5728.56	0.267586
$772$	0	0
$773$	−16955.2	−0.788920	−0.394460	−	0.918913i	$-0.629068\pi$
$773$	−16955.2	−0.788920	−0.394460	+	0.918913i	$0.629068\pi$
$774$	0	0
$775$	−9883.22	−0.458085
$776$	0	0
$777$	393.626	0.0181741
$778$	0	0
$779$	−7283.22	−0.334979
$780$	0	0
$781$	−20219.1	−0.926370
$782$	0	0
$783$	−3973.86	−0.181372
$784$	0	0
$785$	−15067.7	−0.685081
$786$	0	0
$787$	27338.4	1.23826	0.619129	−	0.785289i	$-0.287486\pi$
$787$	27338.4	1.23826	0.619129	+	0.785289i	$0.287486\pi$
$788$	0	0
$789$	−5888.87	−0.265715
$790$	0	0
$791$	443.723	0.0199456
$792$	0	0
$793$	0	0
$794$	0	0
$795$	3221.31	0.143709
$796$	0	0
$797$	17372.2	0.772090	0.386045	−	0.922480i	$-0.373841\pi$
$797$	17372.2	0.772090	0.386045	+	0.922480i	$0.373841\pi$
$798$	0	0
$799$	−9223.94	−0.408410
$800$	0	0
$801$	−37636.8	−1.66021
$802$	0	0
$803$	38237.3	1.68041
$804$	0	0
$805$	3838.94	0.168081
$806$	0	0
$807$	−6964.03	−0.303774
$808$	0	0
$809$	25492.6	1.10788	0.553939	−	0.832557i	$-0.313124\pi$
$809$	25492.6	1.10788	0.553939	+	0.832557i	$0.313124\pi$
$810$	0	0
$811$	7742.96	0.335255	0.167628	−	0.985850i	$-0.446389\pi$
$811$	7742.96	0.335255	0.167628	+	0.985850i	$0.446389\pi$
$812$	0	0
$813$	−479.122	−0.0206686
$814$	0	0
$815$	−42773.5	−1.83839
$816$	0	0
$817$	15731.1	0.673636
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−13490.6	−0.573476	−0.286738	−	0.958009i	$-0.592571\pi$
$821$	−13490.6	−0.573476	−0.286738	+	0.958009i	$0.592571\pi$
$822$	0	0
$823$	−33100.3	−1.40195	−0.700975	−	0.713186i	$-0.747252\pi$
$823$	−33100.3	−1.40195	−0.700975	+	0.713186i	$0.747252\pi$
$824$	0	0
$825$	6395.23	0.269883
$826$	0	0
$827$	5211.34	0.219125	0.109562	−	0.993980i	$-0.465055\pi$
$827$	5211.34	0.219125	0.109562	+	0.993980i	$0.465055\pi$
$828$	0	0
$829$	14099.9	0.590725	0.295363	−	0.955385i	$-0.404560\pi$
$829$	14099.9	0.590725	0.295363	+	0.955385i	$0.404560\pi$
$830$	0	0
$831$	−7603.03	−0.317384
$832$	0	0
$833$	12611.9	0.524581
$834$	0	0
$835$	−28706.8	−1.18975
$836$	0	0
$837$	5213.21	0.215287
$838$	0	0
$839$	−7315.45	−0.301022	−0.150511	−	0.988608i	$-0.548092\pi$
$839$	−7315.45	−0.301022	−0.150511	+	0.988608i	$0.548092\pi$
$840$	0	0
$841$	−20909.0	−0.857313
$842$	0	0
$843$	−4631.05	−0.189207
$844$	0	0
$845$	0	0
$846$	0	0
$847$	216.866	0.00879765
$848$	0	0
$849$	−4392.70	−0.177570
$850$	0	0
$851$	52674.9	2.12183
$852$	0	0
$853$	−24221.1	−0.972234	−0.486117	−	0.873894i	$-0.661587\pi$
$853$	−24221.1	−0.972234	−0.486117	+	0.873894i	$0.661587\pi$
$854$	0	0
$855$	−17300.6	−0.692008
$856$	0	0
$857$	9191.63	0.366371	0.183186	−	0.983078i	$-0.441359\pi$
$857$	9191.63	0.366371	0.183186	+	0.983078i	$0.441359\pi$
$858$	0	0
$859$	−1899.06	−0.0754309	−0.0377155	−	0.999289i	$-0.512008\pi$
$859$	−1899.06	−0.0754309	−0.0377155	+	0.999289i	$0.512008\pi$
$860$	0	0
$861$	−258.472	−0.0102308
$862$	0	0
$863$	18688.4	0.737149	0.368574	−	0.929598i	$-0.379846\pi$
$863$	18688.4	0.737149	0.368574	+	0.929598i	$0.379846\pi$
$864$	0	0
$865$	5628.14	0.221228
$866$	0	0
$867$	−4568.49	−0.178955
$868$	0	0
$869$	−49256.7	−1.92281
$870$	0	0
$871$	0	0
$872$	0	0
$873$	8876.31	0.344121
$874$	0	0
$875$	50.9637	0.00196902
$876$	0	0
$877$	30018.6	1.15582	0.577910	−	0.816100i	$-0.303868\pi$
$877$	30018.6	1.15582	0.577910	+	0.816100i	$0.303868\pi$
$878$	0	0
$879$	1451.43	0.0556945
$880$	0	0
$881$	11938.3	0.456539	0.228270	−	0.973598i	$-0.426693\pi$
$881$	11938.3	0.456539	0.228270	+	0.973598i	$0.426693\pi$
$882$	0	0
$883$	22169.9	0.844933	0.422467	−	0.906379i	$-0.361164\pi$
$883$	22169.9	0.844933	0.422467	+	0.906379i	$0.361164\pi$
$884$	0	0
$885$	−13752.6	−0.522359
$886$	0	0
$887$	−728.777	−0.0275873	−0.0137936	−	0.999905i	$-0.504391\pi$
$887$	−728.777	−0.0275873	−0.0137936	+	0.999905i	$0.504391\pi$
$888$	0	0
$889$	768.874	0.0290070
$890$	0	0
$891$	23253.4	0.874318
$892$	0	0
$893$	10728.3	0.402026
$894$	0	0
$895$	−41239.1	−1.54019
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−4565.33	−0.169369
$900$	0	0
$901$	−5813.42	−0.214954
$902$	0	0
$903$	558.275	0.0205739
$904$	0	0
$905$	37182.1	1.36572
$906$	0	0
$907$	−39131.3	−1.43256	−0.716280	−	0.697813i	$-0.754157\pi$
$907$	−39131.3	−1.43256	−0.716280	+	0.697813i	$0.754157\pi$
$908$	0	0
$909$	−36284.9	−1.32397
$910$	0	0
$911$	40955.0	1.48946	0.744730	−	0.667366i	$-0.232578\pi$
$911$	40955.0	1.48946	0.744730	+	0.667366i	$0.232578\pi$
$912$	0	0
$913$	−39940.1	−1.44778
$914$	0	0
$915$	11870.9	0.428895
$916$	0	0
$917$	−725.854	−0.0261394
$918$	0	0
$919$	−12567.8	−0.451114	−0.225557	−	0.974230i	$-0.572420\pi$
$919$	−12567.8	−0.451114	−0.225557	+	0.974230i	$0.572420\pi$
$920$	0	0
$921$	−9444.28	−0.337893
$922$	0	0
$923$	0	0
$924$	0	0
$925$	32986.0	1.17251
$926$	0	0
$927$	−37815.5	−1.33983
$928$	0	0
$929$	−9858.84	−0.348179	−0.174089	−	0.984730i	$-0.555698\pi$
$929$	−9858.84	−0.348179	−0.174089	+	0.984730i	$0.555698\pi$
$930$	0	0
$931$	−14668.8	−0.516381
$932$	0	0
$933$	−9717.49	−0.340982
$934$	0	0
$935$	−22837.9	−0.798802
$936$	0	0
$937$	14064.4	0.490355	0.245177	−	0.969478i	$-0.421154\pi$
$937$	14064.4	0.490355	0.245177	+	0.969478i	$0.421154\pi$
$938$	0	0
$939$	−1624.35	−0.0564523
$940$	0	0
$941$	−37759.0	−1.30808	−0.654042	−	0.756458i	$-0.726928\pi$
$941$	−37759.0	−1.30808	−0.654042	+	0.756458i	$0.726928\pi$
$942$	0	0
$943$	−34588.7	−1.19444
$944$	0	0
$945$	−1268.07	−0.0436512
$946$	0	0
$947$	−42748.0	−1.46687	−0.733433	−	0.679761i	$-0.762083\pi$
$947$	−42748.0	−1.46687	−0.733433	+	0.679761i	$0.762083\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	5257.80	0.179281
$952$	0	0
$953$	−29589.2	−1.00576	−0.502879	−	0.864357i	$-0.667726\pi$
$953$	−29589.2	−1.00576	−0.502879	+	0.864357i	$0.667726\pi$
$954$	0	0
$955$	486.454	0.0164830
$956$	0	0
$957$	2954.13	0.0997843
$958$	0	0
$959$	−247.255	−0.00832565
$960$	0	0
$961$	−23801.9	−0.798961
$962$	0	0
$963$	−29444.9	−0.985305
$964$	0	0
$965$	−48358.7	−1.61318
$966$	0	0
$967$	7140.75	0.237468	0.118734	−	0.992926i	$-0.462117\pi$
$967$	7140.75	0.237468	0.118734	+	0.992926i	$0.462117\pi$
$968$	0	0
$969$	−2040.34	−0.0676420
$970$	0	0
$971$	34866.5	1.15234	0.576169	−	0.817331i	$-0.304547\pi$
$971$	34866.5	1.15234	0.576169	+	0.817331i	$0.304547\pi$
$972$	0	0
$973$	−437.655	−0.0144199
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−32240.8	−1.05576	−0.527878	−	0.849320i	$-0.677012\pi$
$977$	−32240.8	−1.05576	−0.527878	+	0.849320i	$0.677012\pi$
$978$	0	0
$979$	57786.2	1.88647
$980$	0	0
$981$	−40526.0	−1.31896
$982$	0	0
$983$	16467.0	0.534298	0.267149	−	0.963655i	$-0.413918\pi$
$983$	16467.0	0.534298	0.267149	+	0.963655i	$0.413918\pi$
$984$	0	0
$985$	−66302.2	−2.14473
$986$	0	0
$987$	380.734	0.0122785
$988$	0	0
$989$	74708.3	2.40201
$990$	0	0
$991$	12348.7	0.395833	0.197916	−	0.980219i	$-0.436583\pi$
$991$	12348.7	0.395833	0.197916	+	0.980219i	$0.436583\pi$
$992$	0	0
$993$	−5811.38	−0.185719
$994$	0	0
$995$	73627.5	2.34588
$996$	0	0
$997$	3884.94	0.123408	0.0617038	−	0.998095i	$-0.480347\pi$
$997$	3884.94	0.123408	0.0617038	+	0.998095i	$0.480347\pi$
$998$	0	0
$999$	−17399.5	−0.551047

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	676.4.a.g.1.6		8
13.2	odd	12		676.4.h.e.485.2		8
13.3	even	3		676.4.e.h.529.4		16
13.4	even	6		676.4.e.h.653.3		16
13.5	odd	4		676.4.d.d.337.5		8
13.6	odd	12		52.4.h.a.49.2	yes	8
13.7	odd	12		676.4.h.e.361.2		8
13.8	odd	4		676.4.d.d.337.6		8
13.9	even	3		676.4.e.h.653.4		16
13.10	even	6		676.4.e.h.529.3		16
13.11	odd	12		52.4.h.a.17.2	✓	8
13.12	even	2	inner	676.4.a.g.1.5		8
39.11	even	12		468.4.t.g.433.1		8
39.32	even	12		468.4.t.g.361.4		8
52.11	even	12		208.4.w.c.17.3		8
52.19	even	12		208.4.w.c.49.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
52.4.h.a.17.2	✓	8	13.11	odd	12
52.4.h.a.49.2	yes	8	13.6	odd	12
208.4.w.c.17.3		8	52.11	even	12
208.4.w.c.49.3		8	52.19	even	12
468.4.t.g.361.4		8	39.32	even	12
468.4.t.g.433.1		8	39.11	even	12
676.4.a.g.1.5		8	13.12	even	2	inner
676.4.a.g.1.6		8	1.1	even	1	trivial
676.4.d.d.337.5		8	13.5	odd	4
676.4.d.d.337.6		8	13.8	odd	4
676.4.e.h.529.3		16	13.10	even	6
676.4.e.h.529.4		16	13.3	even	3
676.4.e.h.653.3		16	13.4	even	6
676.4.e.h.653.4		16	13.9	even	3
676.4.h.e.361.2		8	13.7	odd	12
676.4.h.e.485.2		8	13.2	odd	12

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:	\( N \)	\(=\)	\( 676 = 2^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	676.a (trivial)

\(f(q)\)	\(=\)	\(q+1.28694 q^{3} +15.8968 q^{5} +1.18417 q^{7} -25.3438 q^{9} +38.9119 q^{11} +20.4582 q^{15} -36.9203 q^{17} +42.9418 q^{19} +1.52395 q^{21} +203.934 q^{23} +127.707 q^{25} -67.3632 q^{27} +58.9916 q^{29} -77.3896 q^{31} +50.0772 q^{33} +18.8244 q^{35} +258.294 q^{37} -169.607 q^{41} +366.335 q^{43} -402.884 q^{45} +249.834 q^{47} -341.598 q^{49} -47.5141 q^{51} +157.459 q^{53} +618.574 q^{55} +55.2634 q^{57} -672.229 q^{59} +580.251 q^{61} -30.0113 q^{63} -180.869 q^{67} +262.451 q^{69} -519.611 q^{71} +982.663 q^{73} +164.351 q^{75} +46.0782 q^{77} -1265.85 q^{79} +597.590 q^{81} -1026.42 q^{83} -586.913 q^{85} +75.9185 q^{87} +1485.05 q^{89} -99.5956 q^{93} +682.635 q^{95} -350.236 q^{97} -986.176 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 140 q^{9} + 176 q^{17} - 40 q^{23} + 84 q^{25} - 432 q^{27} + 968 q^{29} - 80 q^{35} + 1008 q^{43} + 1844 q^{49} + 1808 q^{51} - 1164 q^{53} + 2256 q^{55} + 2448 q^{61} + 3476 q^{69} + 2896 q^{75}+ \cdots + 4800 q^{95}+O(q^{100}) \) 8 * q + 140 * q^9 + 176 * q^17 - 40 * q^23 + 84 * q^25 - 432 * q^27 + 968 * q^29 - 80 * q^35 + 1008 * q^43 + 1844 * q^49 + 1808 * q^51 - 1164 * q^53 + 2256 * q^55 + 2448 * q^61 + 3476 * q^69 + 2896 * q^75 + 3972 * q^77 - 3968 * q^79 + 8264 * q^81 + 3320 * q^87 + 4800 * q^95

Introduction

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