LMFDB - Embedded newform 8424.2.a.v.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8424,2,Mod(1,8424)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8424, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8424.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8424 = 2^{3} \cdot 3^{4} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8424.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$67.2659786627$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.20396961.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 3x^{5} - 4x^{4} + 11x^{3} + 7x^{2} - 7x - 2 $ x^6 - 3x^5 - 4x^4 + 11x^3 + 7x^2 - 7*x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 936)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$2.15391$ of defining polynomial
Character	$\chi$	$=$	8424.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.74770 q^{5} -0.351433 q^{7} -3.29530 q^{11} +1.00000 q^{13} +3.88461 q^{17} -4.45847 q^{19} -4.66850 q^{23} +9.04525 q^{25} -4.44539 q^{29} +0.782066 q^{31} -1.31707 q^{35} -11.4322 q^{37} -6.84266 q^{41} -9.54092 q^{43} -2.06612 q^{47} -6.87649 q^{49} -2.97755 q^{53} -12.3498 q^{55} +0.940619 q^{59} +14.4198 q^{61} +3.74770 q^{65} +4.25497 q^{67} -5.25904 q^{71} +7.09427 q^{73} +1.15808 q^{77} -14.2136 q^{79} -12.5543 q^{83} +14.5583 q^{85} -5.56256 q^{89} -0.351433 q^{91} -16.7090 q^{95} -1.70659 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + q^{5} - 2 q^{7} + q^{11} + 6 q^{13} + 6 q^{17} - 7 q^{19} - 19 q^{23} - 5 q^{25} + 2 q^{29} - 8 q^{31} + q^{35} - 17 q^{37} + 2 q^{41} - 15 q^{43} + 9 q^{47} - 12 q^{49} + 16 q^{53} - 5 q^{55}+ \cdots - 12 q^{97}+O(q^{100}) $ 6 * q + q^5 - 2 * q^7 + q^11 + 6 * q^13 + 6 * q^17 - 7 * q^19 - 19 * q^23 - 5 * q^25 + 2 * q^29 - 8 * q^31 + q^35 - 17 * q^37 + 2 * q^41 - 15 * q^43 + 9 * q^47 - 12 * q^49 + 16 * q^53 - 5 * q^55 - 5 * q^59 - 6 * q^61 + q^65 - 8 * q^67 - 21 * q^71 - q^73 - 3 * q^77 - 34 * q^79 - 44 * q^83 - 7 * q^85 + 11 * q^89 - 2 * q^91 - 6 * q^95 - 12 * q^97

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	3.74770	1.67602	0.838011	−	0.545653i	$-0.183718\pi$
$5$	3.74770	1.67602	0.838011	+	0.545653i	$0.183718\pi$
$6$	0	0
$7$	−0.351433	−0.132829	−0.0664147	−	0.997792i	$-0.521156\pi$
$7$	−0.351433	−0.132829	−0.0664147	+	0.997792i	$0.521156\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−3.29530	−0.993572	−0.496786	−	0.867873i	$-0.665487\pi$
$11$	−3.29530	−0.993572	−0.496786	+	0.867873i	$0.665487\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.88461	0.942155	0.471078	−	0.882092i	$-0.343865\pi$
$17$	3.88461	0.942155	0.471078	+	0.882092i	$0.343865\pi$
$18$	0	0
$19$	−4.45847	−1.02284	−0.511421	−	0.859330i	$-0.670881\pi$
$19$	−4.45847	−1.02284	−0.511421	+	0.859330i	$0.670881\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−4.66850	−0.973450	−0.486725	−	0.873555i	$-0.661809\pi$
$23$	−4.66850	−0.973450	−0.486725	+	0.873555i	$0.661809\pi$
$24$	0	0
$25$	9.04525	1.80905
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−4.44539	−0.825488	−0.412744	−	0.910847i	$-0.635430\pi$
$29$	−4.44539	−0.825488	−0.412744	+	0.910847i	$0.635430\pi$
$30$	0	0
$31$	0.782066	0.140463	0.0702316	−	0.997531i	$-0.477626\pi$
$31$	0.782066	0.140463	0.0702316	+	0.997531i	$0.477626\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−1.31707	−0.222625
$36$	0	0
$37$	−11.4322	−1.87944	−0.939721	−	0.341943i	$-0.888915\pi$
$37$	−11.4322	−1.87944	−0.939721	+	0.341943i	$0.888915\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−6.84266	−1.06864	−0.534322	−	0.845281i	$-0.679433\pi$
$41$	−6.84266	−1.06864	−0.534322	+	0.845281i	$0.679433\pi$
$42$	0	0
$43$	−9.54092	−1.45498	−0.727488	−	0.686120i	$-0.759312\pi$
$43$	−9.54092	−1.45498	−0.727488	+	0.686120i	$0.759312\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−2.06612	−0.301375	−0.150687	−	0.988581i	$-0.548149\pi$
$47$	−2.06612	−0.301375	−0.150687	+	0.988581i	$0.548149\pi$
$48$	0	0
$49$	−6.87649	−0.982356
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−2.97755	−0.408998	−0.204499	−	0.978867i	$-0.565557\pi$
$53$	−2.97755	−0.408998	−0.204499	+	0.978867i	$0.565557\pi$
$54$	0	0
$55$	−12.3498	−1.66525
$56$	0	0
$57$	0	0
$58$	0	0
$59$	0.940619	0.122458	0.0612291	−	0.998124i	$-0.480498\pi$
$59$	0.940619	0.122458	0.0612291	+	0.998124i	$0.480498\pi$
$60$	0	0
$61$	14.4198	1.84626	0.923130	−	0.384487i	$-0.125622\pi$
$61$	14.4198	1.84626	0.923130	+	0.384487i	$0.125622\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	3.74770	0.464845
$66$	0	0
$67$	4.25497	0.519827	0.259914	−	0.965632i	$-0.416306\pi$
$67$	4.25497	0.519827	0.259914	+	0.965632i	$0.416306\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−5.25904	−0.624133	−0.312067	−	0.950060i	$-0.601021\pi$
$71$	−5.25904	−0.624133	−0.312067	+	0.950060i	$0.601021\pi$
$72$	0	0
$73$	7.09427	0.830322	0.415161	−	0.909748i	$-0.363725\pi$
$73$	7.09427	0.830322	0.415161	+	0.909748i	$0.363725\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1.15808	0.131976
$78$	0	0
$79$	−14.2136	−1.59915	−0.799577	−	0.600564i	$-0.794943\pi$
$79$	−14.2136	−1.59915	−0.799577	+	0.600564i	$0.794943\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−12.5543	−1.37802	−0.689009	−	0.724753i	$-0.741954\pi$
$83$	−12.5543	−1.37802	−0.689009	+	0.724753i	$0.741954\pi$
$84$	0	0
$85$	14.5583	1.57907
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−5.56256	−0.589631	−0.294815	−	0.955554i	$-0.595258\pi$
$89$	−5.56256	−0.589631	−0.294815	+	0.955554i	$0.595258\pi$
$90$	0	0
$91$	−0.351433	−0.0368402
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−16.7090	−1.71431
$96$	0	0
$97$	−1.70659	−0.173278	−0.0866391	−	0.996240i	$-0.527613\pi$
$97$	−1.70659	−0.173278	−0.0866391	+	0.996240i	$0.527613\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−2.62874	−0.261569	−0.130784	−	0.991411i	$-0.541750\pi$
$101$	−2.62874	−0.261569	−0.130784	+	0.991411i	$0.541750\pi$
$102$	0	0
$103$	13.5643	1.33653	0.668267	−	0.743921i	$-0.267036\pi$
$103$	13.5643	1.33653	0.668267	+	0.743921i	$0.267036\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	3.58218	0.346302	0.173151	−	0.984895i	$-0.444605\pi$
$107$	3.58218	0.346302	0.173151	+	0.984895i	$0.444605\pi$
$108$	0	0
$109$	11.1068	1.06384	0.531918	−	0.846796i	$-0.321472\pi$
$109$	11.1068	1.06384	0.531918	+	0.846796i	$0.321472\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	2.06695	0.194442	0.0972210	−	0.995263i	$-0.469005\pi$
$113$	2.06695	0.194442	0.0972210	+	0.995263i	$0.469005\pi$
$114$	0	0
$115$	−17.4961	−1.63152
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−1.36518	−0.125146
$120$	0	0
$121$	−0.140965	−0.0128150
$122$	0	0
$123$	0	0
$124$	0	0
$125$	15.1604	1.35599
$126$	0	0
$127$	−15.0773	−1.33789	−0.668945	−	0.743312i	$-0.733254\pi$
$127$	−15.0773	−1.33789	−0.668945	+	0.743312i	$0.733254\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	13.6401	1.19174	0.595869	−	0.803082i	$-0.296808\pi$
$131$	13.6401	1.19174	0.595869	+	0.803082i	$0.296808\pi$
$132$	0	0
$133$	1.56685	0.135863
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−6.23525	−0.532713	−0.266356	−	0.963875i	$-0.585820\pi$
$137$	−6.23525	−0.532713	−0.266356	+	0.963875i	$0.585820\pi$
$138$	0	0
$139$	1.67198	0.141815	0.0709076	−	0.997483i	$-0.477410\pi$
$139$	1.67198	0.141815	0.0709076	+	0.997483i	$0.477410\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−3.29530	−0.275567
$144$	0	0
$145$	−16.6600	−1.38354
$146$	0	0
$147$	0	0
$148$	0	0
$149$	16.5896	1.35907	0.679536	−	0.733643i	$-0.262181\pi$
$149$	16.5896	1.35907	0.679536	+	0.733643i	$0.262181\pi$
$150$	0	0
$151$	−15.5999	−1.26950	−0.634751	−	0.772717i	$-0.718897\pi$
$151$	−15.5999	−1.26950	−0.634751	+	0.772717i	$0.718897\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	2.93095	0.235420
$156$	0	0
$157$	−16.3530	−1.30511	−0.652556	−	0.757741i	$-0.726303\pi$
$157$	−16.3530	−1.30511	−0.652556	+	0.757741i	$0.726303\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	1.64067	0.129303
$162$	0	0
$163$	5.60740	0.439206	0.219603	−	0.975589i	$-0.429524\pi$
$163$	5.60740	0.439206	0.219603	+	0.975589i	$0.429524\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−4.72887	−0.365931	−0.182966	−	0.983119i	$-0.558570\pi$
$167$	−4.72887	−0.365931	−0.182966	+	0.983119i	$0.558570\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−16.7949	−1.27689	−0.638447	−	0.769666i	$-0.720423\pi$
$173$	−16.7949	−1.27689	−0.638447	+	0.769666i	$0.720423\pi$
$174$	0	0
$175$	−3.17881	−0.240295
$176$	0	0
$177$	0	0
$178$	0	0
$179$	11.7878	0.881062	0.440531	−	0.897737i	$-0.354790\pi$
$179$	11.7878	0.881062	0.440531	+	0.897737i	$0.354790\pi$
$180$	0	0
$181$	−26.0931	−1.93948	−0.969742	−	0.244132i	$-0.921497\pi$
$181$	−26.0931	−1.93948	−0.969742	+	0.244132i	$0.921497\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−42.8444	−3.14999
$186$	0	0
$187$	−12.8010	−0.936099
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−15.8392	−1.14608	−0.573041	−	0.819527i	$-0.694236\pi$
$191$	−15.8392	−1.14608	−0.573041	+	0.819527i	$0.694236\pi$
$192$	0	0
$193$	11.0744	0.797155	0.398578	−	0.917135i	$-0.369504\pi$
$193$	11.0744	0.797155	0.398578	+	0.917135i	$0.369504\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	8.36629	0.596073	0.298037	−	0.954554i	$-0.403668\pi$
$197$	8.36629	0.596073	0.298037	+	0.954554i	$0.403668\pi$
$198$	0	0
$199$	2.63356	0.186688	0.0933441	−	0.995634i	$-0.470244\pi$
$199$	2.63356	0.186688	0.0933441	+	0.995634i	$0.470244\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	1.56226	0.109649
$204$	0	0
$205$	−25.6442	−1.79107
$206$	0	0
$207$	0	0
$208$	0	0
$209$	14.6920	1.01627
$210$	0	0
$211$	−24.3555	−1.67670	−0.838351	−	0.545131i	$-0.816480\pi$
$211$	−24.3555	−1.67670	−0.838351	+	0.545131i	$0.816480\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−35.7565	−2.43857
$216$	0	0
$217$	−0.274844	−0.0186576
$218$	0	0
$219$	0	0
$220$	0	0
$221$	3.88461	0.261307
$222$	0	0
$223$	25.7203	1.72236	0.861178	−	0.508303i	$-0.169727\pi$
$223$	25.7203	1.72236	0.861178	+	0.508303i	$0.169727\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	22.3523	1.48358	0.741788	−	0.670635i	$-0.233978\pi$
$227$	22.3523	1.48358	0.741788	+	0.670635i	$0.233978\pi$
$228$	0	0
$229$	4.84045	0.319866	0.159933	−	0.987128i	$-0.448872\pi$
$229$	4.84045	0.319866	0.159933	+	0.987128i	$0.448872\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	25.0840	1.64331	0.821655	−	0.569986i	$-0.193051\pi$
$233$	25.0840	1.64331	0.821655	+	0.569986i	$0.193051\pi$
$234$	0	0
$235$	−7.74321	−0.505111
$236$	0	0
$237$	0	0
$238$	0	0
$239$	24.6184	1.59243	0.796217	−	0.605011i	$-0.206831\pi$
$239$	24.6184	1.59243	0.796217	+	0.605011i	$0.206831\pi$
$240$	0	0
$241$	−20.1819	−1.30003	−0.650015	−	0.759922i	$-0.725237\pi$
$241$	−20.1819	−1.30003	−0.650015	+	0.759922i	$0.725237\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−25.7710	−1.64645
$246$	0	0
$247$	−4.45847	−0.283685
$248$	0	0
$249$	0	0
$250$	0	0
$251$	26.3117	1.66078	0.830391	−	0.557181i	$-0.188117\pi$
$251$	26.3117	1.66078	0.830391	+	0.557181i	$0.188117\pi$
$252$	0	0
$253$	15.3841	0.967192
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−9.04219	−0.564036	−0.282018	−	0.959409i	$-0.591004\pi$
$257$	−9.04219	−0.564036	−0.282018	+	0.959409i	$0.591004\pi$
$258$	0	0
$259$	4.01766	0.249645
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−1.90602	−0.117530	−0.0587650	−	0.998272i	$-0.518716\pi$
$263$	−1.90602	−0.117530	−0.0587650	+	0.998272i	$0.518716\pi$
$264$	0	0
$265$	−11.1590	−0.685490
$266$	0	0
$267$	0	0
$268$	0	0
$269$	13.5293	0.824896	0.412448	−	0.910981i	$-0.364674\pi$
$269$	13.5293	0.824896	0.412448	+	0.910981i	$0.364674\pi$
$270$	0	0
$271$	8.66113	0.526126	0.263063	−	0.964779i	$-0.415267\pi$
$271$	8.66113	0.526126	0.263063	+	0.964779i	$0.415267\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−29.8069	−1.79742
$276$	0	0
$277$	25.4902	1.53156	0.765779	−	0.643103i	$-0.222353\pi$
$277$	25.4902	1.53156	0.765779	+	0.643103i	$0.222353\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−22.7193	−1.35532	−0.677659	−	0.735376i	$-0.737006\pi$
$281$	−22.7193	−1.35532	−0.677659	+	0.735376i	$0.737006\pi$
$282$	0	0
$283$	0.811261	0.0482245	0.0241122	−	0.999709i	$-0.492324\pi$
$283$	0.811261	0.0482245	0.0241122	+	0.999709i	$0.492324\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	2.40474	0.141947
$288$	0	0
$289$	−1.90983	−0.112343
$290$	0	0
$291$	0	0
$292$	0	0
$293$	16.4591	0.961549	0.480775	−	0.876844i	$-0.340355\pi$
$293$	16.4591	0.961549	0.480775	+	0.876844i	$0.340355\pi$
$294$	0	0
$295$	3.52516	0.205243
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−4.66850	−0.269986
$300$	0	0
$301$	3.35300	0.193264
$302$	0	0
$303$	0	0
$304$	0	0
$305$	54.0409	3.09437
$306$	0	0
$307$	−1.61680	−0.0922754	−0.0461377	−	0.998935i	$-0.514691\pi$
$307$	−1.61680	−0.0922754	−0.0461377	+	0.998935i	$0.514691\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	29.7730	1.68827	0.844136	−	0.536128i	$-0.180114\pi$
$311$	29.7730	1.68827	0.844136	+	0.536128i	$0.180114\pi$
$312$	0	0
$313$	2.01229	0.113741	0.0568707	−	0.998382i	$-0.481888\pi$
$313$	2.01229	0.113741	0.0568707	+	0.998382i	$0.481888\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	13.0331	0.732012	0.366006	−	0.930612i	$-0.380725\pi$
$317$	13.0331	0.732012	0.366006	+	0.930612i	$0.380725\pi$
$318$	0	0
$319$	14.6489	0.820182
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−17.3194	−0.963676
$324$	0	0
$325$	9.04525	0.501740
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0.726105	0.0400314
$330$	0	0
$331$	2.42161	0.133104	0.0665520	−	0.997783i	$-0.478800\pi$
$331$	2.42161	0.133104	0.0665520	+	0.997783i	$0.478800\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	15.9464	0.871242
$336$	0	0
$337$	−31.1180	−1.69510	−0.847552	−	0.530712i	$-0.821925\pi$
$337$	−31.1180	−1.69510	−0.847552	+	0.530712i	$0.821925\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−2.57715	−0.139560
$342$	0	0
$343$	4.87666	0.263315
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−19.7606	−1.06081	−0.530403	−	0.847745i	$-0.677960\pi$
$347$	−19.7606	−1.06081	−0.530403	+	0.847745i	$0.677960\pi$
$348$	0	0
$349$	4.93903	0.264380	0.132190	−	0.991224i	$-0.457799\pi$
$349$	4.93903	0.264380	0.132190	+	0.991224i	$0.457799\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	8.06760	0.429395	0.214698	−	0.976681i	$-0.431123\pi$
$353$	8.06760	0.429395	0.214698	+	0.976681i	$0.431123\pi$
$354$	0	0
$355$	−19.7093	−1.04606
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−5.70600	−0.301151	−0.150576	−	0.988598i	$-0.548113\pi$
$359$	−5.70600	−0.301151	−0.150576	+	0.988598i	$0.548113\pi$
$360$	0	0
$361$	0.877919	0.0462063
$362$	0	0
$363$	0	0
$364$	0	0
$365$	26.5872	1.39164
$366$	0	0
$367$	−10.4935	−0.547757	−0.273879	−	0.961764i	$-0.588307\pi$
$367$	−10.4935	−0.547757	−0.273879	+	0.961764i	$0.588307\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	1.04641	0.0543270
$372$	0	0
$373$	−28.9262	−1.49774	−0.748870	−	0.662717i	$-0.769403\pi$
$373$	−28.9262	−1.49774	−0.748870	+	0.662717i	$0.769403\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−4.44539	−0.228949
$378$	0	0
$379$	−35.6342	−1.83040	−0.915202	−	0.402995i	$-0.867969\pi$
$379$	−35.6342	−1.83040	−0.915202	+	0.402995i	$0.867969\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−20.7302	−1.05926	−0.529632	−	0.848227i	$-0.677670\pi$
$383$	−20.7302	−1.05926	−0.529632	+	0.848227i	$0.677670\pi$
$384$	0	0
$385$	4.34014	0.221194
$386$	0	0
$387$	0	0
$388$	0	0
$389$	20.1644	1.02238	0.511189	−	0.859468i	$-0.329205\pi$
$389$	20.1644	1.02238	0.511189	+	0.859468i	$0.329205\pi$
$390$	0	0
$391$	−18.1353	−0.917141
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−53.2683	−2.68022
$396$	0	0
$397$	−2.27323	−0.114090	−0.0570450	−	0.998372i	$-0.518168\pi$
$397$	−2.27323	−0.114090	−0.0570450	+	0.998372i	$0.518168\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	26.1927	1.30800	0.653999	−	0.756495i	$-0.273090\pi$
$401$	26.1927	1.30800	0.653999	+	0.756495i	$0.273090\pi$
$402$	0	0
$403$	0.782066	0.0389575
$404$	0	0
$405$	0	0
$406$	0	0
$407$	37.6726	1.86736
$408$	0	0
$409$	8.06517	0.398797	0.199399	−	0.979918i	$-0.436101\pi$
$409$	8.06517	0.398797	0.199399	+	0.979918i	$0.436101\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−0.330565	−0.0162660
$414$	0	0
$415$	−47.0499	−2.30959
$416$	0	0
$417$	0	0
$418$	0	0
$419$	17.4849	0.854191	0.427096	−	0.904207i	$-0.359537\pi$
$419$	17.4849	0.854191	0.427096	+	0.904207i	$0.359537\pi$
$420$	0	0
$421$	−20.0442	−0.976894	−0.488447	−	0.872594i	$-0.662436\pi$
$421$	−20.0442	−0.976894	−0.488447	+	0.872594i	$0.662436\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	35.1373	1.70441
$426$	0	0
$427$	−5.06759	−0.245238
$428$	0	0
$429$	0	0
$430$	0	0
$431$	8.01436	0.386038	0.193019	−	0.981195i	$-0.438172\pi$
$431$	8.01436	0.386038	0.193019	+	0.981195i	$0.438172\pi$
$432$	0	0
$433$	−25.8390	−1.24174	−0.620872	−	0.783912i	$-0.713221\pi$
$433$	−25.8390	−1.24174	−0.620872	+	0.783912i	$0.713221\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	20.8144	0.995685
$438$	0	0
$439$	2.44177	0.116539	0.0582697	−	0.998301i	$-0.481442\pi$
$439$	2.44177	0.116539	0.0582697	+	0.998301i	$0.481442\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−17.4136	−0.827343	−0.413672	−	0.910426i	$-0.635754\pi$
$443$	−17.4136	−0.827343	−0.413672	+	0.910426i	$0.635754\pi$
$444$	0	0
$445$	−20.8468	−0.988234
$446$	0	0
$447$	0	0
$448$	0	0
$449$	18.3826	0.867528	0.433764	−	0.901026i	$-0.357185\pi$
$449$	18.3826	0.867528	0.433764	+	0.901026i	$0.357185\pi$
$450$	0	0
$451$	22.5486	1.06177
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−1.31707	−0.0617451
$456$	0	0
$457$	−15.9285	−0.745104	−0.372552	−	0.928011i	$-0.621517\pi$
$457$	−15.9285	−0.745104	−0.372552	+	0.928011i	$0.621517\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	27.6294	1.28683	0.643414	−	0.765518i	$-0.277517\pi$
$461$	27.6294	1.28683	0.643414	+	0.765518i	$0.277517\pi$
$462$	0	0
$463$	20.1291	0.935478	0.467739	−	0.883867i	$-0.345069\pi$
$463$	20.1291	0.935478	0.467739	+	0.883867i	$0.345069\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−1.34267	−0.0621315	−0.0310657	−	0.999517i	$-0.509890\pi$
$467$	−1.34267	−0.0621315	−0.0310657	+	0.999517i	$0.509890\pi$
$468$	0	0
$469$	−1.49534	−0.0690483
$470$	0	0
$471$	0	0
$472$	0	0
$473$	31.4402	1.44562
$474$	0	0
$475$	−40.3280	−1.85037
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−10.3562	−0.473188	−0.236594	−	0.971609i	$-0.576031\pi$
$479$	−10.3562	−0.473188	−0.236594	+	0.971609i	$0.576031\pi$
$480$	0	0
$481$	−11.4322	−0.521263
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−6.39579	−0.290418
$486$	0	0
$487$	40.8998	1.85335	0.926674	−	0.375867i	$-0.122655\pi$
$487$	40.8998	1.85335	0.926674	+	0.375867i	$0.122655\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	16.0072	0.722393	0.361197	−	0.932490i	$-0.382368\pi$
$491$	16.0072	0.722393	0.361197	+	0.932490i	$0.382368\pi$
$492$	0	0
$493$	−17.2686	−0.777738
$494$	0	0
$495$	0	0
$496$	0	0
$497$	1.84820	0.0829032
$498$	0	0
$499$	−2.36239	−0.105755	−0.0528775	−	0.998601i	$-0.516839\pi$
$499$	−2.36239	−0.105755	−0.0528775	+	0.998601i	$0.516839\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−25.8661	−1.15331	−0.576655	−	0.816988i	$-0.695642\pi$
$503$	−25.8661	−1.15331	−0.576655	+	0.816988i	$0.695642\pi$
$504$	0	0
$505$	−9.85171	−0.438395
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−20.7208	−0.918433	−0.459217	−	0.888324i	$-0.651870\pi$
$509$	−20.7208	−0.918433	−0.459217	+	0.888324i	$0.651870\pi$
$510$	0	0
$511$	−2.49317	−0.110291
$512$	0	0
$513$	0	0
$514$	0	0
$515$	50.8351	2.24006
$516$	0	0
$517$	6.80850	0.299438
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−39.8677	−1.74664	−0.873318	−	0.487150i	$-0.838036\pi$
$521$	−39.8677	−1.74664	−0.873318	+	0.487150i	$0.838036\pi$
$522$	0	0
$523$	−12.3303	−0.539167	−0.269583	−	0.962977i	$-0.586886\pi$
$523$	−12.3303	−0.539167	−0.269583	+	0.962977i	$0.586886\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	3.03802	0.132338
$528$	0	0
$529$	−1.20510	−0.0523957
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−6.84266	−0.296388
$534$	0	0
$535$	13.4249	0.580411
$536$	0	0
$537$	0	0
$538$	0	0
$539$	22.6601	0.976042
$540$	0	0
$541$	22.9129	0.985104	0.492552	−	0.870283i	$-0.336064\pi$
$541$	22.9129	0.985104	0.492552	+	0.870283i	$0.336064\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	41.6248	1.78301
$546$	0	0
$547$	−7.49423	−0.320430	−0.160215	−	0.987082i	$-0.551219\pi$
$547$	−7.49423	−0.320430	−0.160215	+	0.987082i	$0.551219\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	19.8196	0.844344
$552$	0	0
$553$	4.99513	0.212415
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−26.2053	−1.11036	−0.555178	−	0.831732i	$-0.687350\pi$
$557$	−26.2053	−1.11036	−0.555178	+	0.831732i	$0.687350\pi$
$558$	0	0
$559$	−9.54092	−0.403538
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−26.0256	−1.09685	−0.548424	−	0.836200i	$-0.684772\pi$
$563$	−26.0256	−1.09685	−0.548424	+	0.836200i	$0.684772\pi$
$564$	0	0
$565$	7.74630	0.325889
$566$	0	0
$567$	0	0
$568$	0	0
$569$	29.8500	1.25138	0.625689	−	0.780072i	$-0.284818\pi$
$569$	29.8500	1.25138	0.625689	+	0.780072i	$0.284818\pi$
$570$	0	0
$571$	8.97421	0.375559	0.187780	−	0.982211i	$-0.439871\pi$
$571$	8.97421	0.375559	0.187780	+	0.982211i	$0.439871\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−42.2278	−1.76102
$576$	0	0
$577$	−32.4990	−1.35295	−0.676476	−	0.736465i	$-0.736494\pi$
$577$	−32.4990	−1.35295	−0.676476	+	0.736465i	$0.736494\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	4.41202	0.183041
$582$	0	0
$583$	9.81194	0.406369
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−14.2273	−0.587223	−0.293612	−	0.955925i	$-0.594857\pi$
$587$	−14.2273	−0.587223	−0.293612	+	0.955925i	$0.594857\pi$
$588$	0	0
$589$	−3.48682	−0.143672
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−14.7975	−0.607662	−0.303831	−	0.952726i	$-0.598266\pi$
$593$	−14.7975	−0.607662	−0.303831	+	0.952726i	$0.598266\pi$
$594$	0	0
$595$	−5.11629	−0.209747
$596$	0	0
$597$	0	0
$598$	0	0
$599$	1.34362	0.0548989	0.0274495	−	0.999623i	$-0.491261\pi$
$599$	1.34362	0.0548989	0.0274495	+	0.999623i	$0.491261\pi$
$600$	0	0
$601$	28.7195	1.17149	0.585746	−	0.810495i	$-0.300802\pi$
$601$	28.7195	1.17149	0.585746	+	0.810495i	$0.300802\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−0.528295	−0.0214782
$606$	0	0
$607$	−3.12460	−0.126824	−0.0634118	−	0.997987i	$-0.520198\pi$
$607$	−3.12460	−0.126824	−0.0634118	+	0.997987i	$0.520198\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−2.06612	−0.0835864
$612$	0	0
$613$	19.6800	0.794867	0.397433	−	0.917631i	$-0.369901\pi$
$613$	19.6800	0.794867	0.397433	+	0.917631i	$0.369901\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	10.0591	0.404964	0.202482	−	0.979286i	$-0.435099\pi$
$617$	10.0591	0.404964	0.202482	+	0.979286i	$0.435099\pi$
$618$	0	0
$619$	−42.5605	−1.71065	−0.855326	−	0.518090i	$-0.826643\pi$
$619$	−42.5605	−1.71065	−0.855326	+	0.518090i	$0.826643\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	1.95487	0.0783203
$624$	0	0
$625$	11.5904	0.463614
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−44.4096	−1.77073
$630$	0	0
$631$	20.2946	0.807916	0.403958	−	0.914778i	$-0.367634\pi$
$631$	20.2946	0.807916	0.403958	+	0.914778i	$0.367634\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−56.5051	−2.24233
$636$	0	0
$637$	−6.87649	−0.272457
$638$	0	0
$639$	0	0
$640$	0	0
$641$	3.49158	0.137909	0.0689546	−	0.997620i	$-0.478034\pi$
$641$	3.49158	0.137909	0.0689546	+	0.997620i	$0.478034\pi$
$642$	0	0
$643$	−35.0558	−1.38247	−0.691233	−	0.722632i	$-0.742932\pi$
$643$	−35.0558	−1.38247	−0.691233	+	0.722632i	$0.742932\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	44.6001	1.75341	0.876705	−	0.481028i	$-0.159736\pi$
$647$	44.6001	1.75341	0.876705	+	0.481028i	$0.159736\pi$
$648$	0	0
$649$	−3.09963	−0.121671
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−11.7937	−0.461525	−0.230762	−	0.973010i	$-0.574122\pi$
$653$	−11.7937	−0.461525	−0.230762	+	0.973010i	$0.574122\pi$
$654$	0	0
$655$	51.1188	1.99738
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−18.1674	−0.707700	−0.353850	−	0.935302i	$-0.615128\pi$
$659$	−18.1674	−0.707700	−0.353850	+	0.935302i	$0.615128\pi$
$660$	0	0
$661$	4.20156	0.163422	0.0817108	−	0.996656i	$-0.473962\pi$
$661$	4.20156	0.163422	0.0817108	+	0.996656i	$0.473962\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	5.87210	0.227710
$666$	0	0
$667$	20.7533	0.803571
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−47.5175	−1.83439
$672$	0	0
$673$	−44.4630	−1.71392	−0.856961	−	0.515382i	$-0.827650\pi$
$673$	−44.4630	−1.71392	−0.856961	+	0.515382i	$0.827650\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	7.48675	0.287739	0.143870	−	0.989597i	$-0.454045\pi$
$677$	7.48675	0.287739	0.143870	+	0.989597i	$0.454045\pi$
$678$	0	0
$679$	0.599753	0.0230164
$680$	0	0
$681$	0	0
$682$	0	0
$683$	38.6166	1.47762	0.738811	−	0.673912i	$-0.235387\pi$
$683$	38.6166	1.47762	0.738811	+	0.673912i	$0.235387\pi$
$684$	0	0
$685$	−23.3678	−0.892839
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−2.97755	−0.113436
$690$	0	0
$691$	20.4095	0.776413	0.388206	−	0.921572i	$-0.373095\pi$
$691$	20.4095	0.776413	0.388206	+	0.921572i	$0.373095\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	6.26606	0.237685
$696$	0	0
$697$	−26.5810	−1.00683
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−19.9987	−0.755341	−0.377670	−	0.925940i	$-0.623275\pi$
$701$	−19.9987	−0.755341	−0.377670	+	0.925940i	$0.623275\pi$
$702$	0	0
$703$	50.9701	1.92237
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0.923826	0.0347440
$708$	0	0
$709$	−13.2097	−0.496100	−0.248050	−	0.968747i	$-0.579790\pi$
$709$	−13.2097	−0.496100	−0.248050	+	0.968747i	$0.579790\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−3.65108	−0.136734
$714$	0	0
$715$	−12.3498	−0.461857
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−36.4662	−1.35996	−0.679979	−	0.733231i	$-0.738011\pi$
$719$	−36.4662	−1.35996	−0.679979	+	0.733231i	$0.738011\pi$
$720$	0	0
$721$	−4.76696	−0.177531
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−40.2097	−1.49335
$726$	0	0
$727$	41.1655	1.52674	0.763371	−	0.645960i	$-0.223543\pi$
$727$	41.1655	1.52674	0.763371	+	0.645960i	$0.223543\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−37.0627	−1.37081
$732$	0	0
$733$	−29.5029	−1.08971	−0.544857	−	0.838529i	$-0.683416\pi$
$733$	−29.5029	−1.08971	−0.544857	+	0.838529i	$0.683416\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−14.0214	−0.516486
$738$	0	0
$739$	44.3622	1.63189	0.815945	−	0.578129i	$-0.196217\pi$
$739$	44.3622	1.63189	0.815945	+	0.578129i	$0.196217\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−10.1590	−0.372697	−0.186349	−	0.982484i	$-0.559665\pi$
$743$	−10.1590	−0.372697	−0.186349	+	0.982484i	$0.559665\pi$
$744$	0	0
$745$	62.1728	2.27783
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−1.25890	−0.0459991
$750$	0	0
$751$	7.62962	0.278409	0.139204	−	0.990264i	$-0.455545\pi$
$751$	7.62962	0.278409	0.139204	+	0.990264i	$0.455545\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−58.4637	−2.12771
$756$	0	0
$757$	49.6248	1.80364	0.901822	−	0.432107i	$-0.142230\pi$
$757$	49.6248	1.80364	0.901822	+	0.432107i	$0.142230\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−19.7198	−0.714841	−0.357421	−	0.933944i	$-0.616344\pi$
$761$	−19.7198	−0.714841	−0.357421	+	0.933944i	$0.616344\pi$
$762$	0	0
$763$	−3.90329	−0.141309
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0.940619	0.0339638
$768$	0	0
$769$	−51.3471	−1.85162	−0.925812	−	0.377984i	$-0.876618\pi$
$769$	−51.3471	−1.85162	−0.925812	+	0.377984i	$0.876618\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−1.28424	−0.0461910	−0.0230955	−	0.999733i	$-0.507352\pi$
$773$	−1.28424	−0.0461910	−0.0230955	+	0.999733i	$0.507352\pi$
$774$	0	0
$775$	7.07399	0.254105
$776$	0	0
$777$	0	0
$778$	0	0
$779$	30.5078	1.09305
$780$	0	0
$781$	17.3301	0.620121
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−61.2861	−2.18740
$786$	0	0
$787$	−0.0261866	−0.000933451 0	−0.000466726	−	1.00000i	$-0.500149\pi$
$787$	−0.0261866	−0.000933451 0	−0.000466726		1.00000i	$0.500149\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−0.726395	−0.0258276
$792$	0	0
$793$	14.4198	0.512061
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−14.9734	−0.530384	−0.265192	−	0.964196i	$-0.585435\pi$
$797$	−14.9734	−0.530384	−0.265192	+	0.964196i	$0.585435\pi$
$798$	0	0
$799$	−8.02607	−0.283942
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−23.3778	−0.824985
$804$	0	0
$805$	6.14873	0.216714
$806$	0	0
$807$	0	0
$808$	0	0
$809$	41.1952	1.44834	0.724172	−	0.689619i	$-0.242222\pi$
$809$	41.1952	1.44834	0.724172	+	0.689619i	$0.242222\pi$
$810$	0	0
$811$	−49.0285	−1.72162	−0.860811	−	0.508924i	$-0.830043\pi$
$811$	−49.0285	−1.72162	−0.860811	+	0.508924i	$0.830043\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	21.0149	0.736118
$816$	0	0
$817$	42.5379	1.48821
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−9.77995	−0.341323	−0.170661	−	0.985330i	$-0.554590\pi$
$821$	−9.77995	−0.341323	−0.170661	+	0.985330i	$0.554590\pi$
$822$	0	0
$823$	−35.2008	−1.22702	−0.613512	−	0.789686i	$-0.710244\pi$
$823$	−35.2008	−1.22702	−0.613512	+	0.789686i	$0.710244\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−4.45020	−0.154749	−0.0773743	−	0.997002i	$-0.524654\pi$
$827$	−4.45020	−0.154749	−0.0773743	+	0.997002i	$0.524654\pi$
$828$	0	0
$829$	31.8398	1.10584	0.552921	−	0.833234i	$-0.313513\pi$
$829$	31.8398	1.10584	0.552921	+	0.833234i	$0.313513\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−26.7125	−0.925532
$834$	0	0
$835$	−17.7224	−0.613309
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−54.1699	−1.87015	−0.935076	−	0.354448i	$-0.884669\pi$
$839$	−54.1699	−1.87015	−0.935076	+	0.354448i	$0.884669\pi$
$840$	0	0
$841$	−9.23851	−0.318569
$842$	0	0
$843$	0	0
$844$	0	0
$845$	3.74770	0.128925
$846$	0	0
$847$	0.0495398	0.00170221
$848$	0	0
$849$	0	0
$850$	0	0
$851$	53.3712	1.82954
$852$	0	0
$853$	−40.7048	−1.39371	−0.696853	−	0.717214i	$-0.745417\pi$
$853$	−40.7048	−1.39371	−0.696853	+	0.717214i	$0.745417\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	2.60350	0.0889338	0.0444669	−	0.999011i	$-0.485841\pi$
$857$	2.60350	0.0889338	0.0444669	+	0.999011i	$0.485841\pi$
$858$	0	0
$859$	−16.4524	−0.561347	−0.280674	−	0.959803i	$-0.590558\pi$
$859$	−16.4524	−0.561347	−0.280674	+	0.959803i	$0.590558\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	18.2597	0.621566	0.310783	−	0.950481i	$-0.399409\pi$
$863$	18.2597	0.621566	0.310783	+	0.950481i	$0.399409\pi$
$864$	0	0
$865$	−62.9423	−2.14010
$866$	0	0
$867$	0	0
$868$	0	0
$869$	46.8381	1.58887
$870$	0	0
$871$	4.25497	0.144174
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−5.32787	−0.180115
$876$	0	0
$877$	−42.6740	−1.44100	−0.720498	−	0.693457i	$-0.756087\pi$
$877$	−42.6740	−1.44100	−0.720498	+	0.693457i	$0.756087\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	26.5618	0.894890	0.447445	−	0.894311i	$-0.352334\pi$
$881$	26.5618	0.894890	0.447445	+	0.894311i	$0.352334\pi$
$882$	0	0
$883$	47.3009	1.59180	0.795900	−	0.605428i	$-0.206998\pi$
$883$	47.3009	1.59180	0.795900	+	0.605428i	$0.206998\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−42.6762	−1.43293	−0.716464	−	0.697624i	$-0.754241\pi$
$887$	−42.6762	−1.43293	−0.716464	+	0.697624i	$0.754241\pi$
$888$	0	0
$889$	5.29865	0.177711
$890$	0	0
$891$	0	0
$892$	0	0
$893$	9.21174	0.308259
$894$	0	0
$895$	44.1772	1.47668
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−3.47659	−0.115951
$900$	0	0
$901$	−11.5666	−0.385340
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−97.7891	−3.25062
$906$	0	0
$907$	−40.3848	−1.34095	−0.670477	−	0.741930i	$-0.733910\pi$
$907$	−40.3848	−1.34095	−0.670477	+	0.741930i	$0.733910\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−53.2907	−1.76560	−0.882799	−	0.469750i	$-0.844344\pi$
$911$	−53.2907	−1.76560	−0.882799	+	0.469750i	$0.844344\pi$
$912$	0	0
$913$	41.3704	1.36916
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−4.79357	−0.158298
$918$	0	0
$919$	−5.72441	−0.188831	−0.0944155	−	0.995533i	$-0.530098\pi$
$919$	−5.72441	−0.188831	−0.0944155	+	0.995533i	$0.530098\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−5.25904	−0.173103
$924$	0	0
$925$	−103.407	−3.40001
$926$	0	0
$927$	0	0
$928$	0	0
$929$	4.72533	0.155033	0.0775165	−	0.996991i	$-0.475301\pi$
$929$	4.72533	0.155033	0.0775165	+	0.996991i	$0.475301\pi$
$930$	0	0
$931$	30.6586	1.00480
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−47.9742	−1.56892
$936$	0	0
$937$	−1.19269	−0.0389635	−0.0194818	−	0.999810i	$-0.506202\pi$
$937$	−1.19269	−0.0389635	−0.0194818	+	0.999810i	$0.506202\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	42.5472	1.38700	0.693500	−	0.720457i	$-0.256068\pi$
$941$	42.5472	1.38700	0.693500	+	0.720457i	$0.256068\pi$
$942$	0	0
$943$	31.9450	1.04027
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−19.1494	−0.622272	−0.311136	−	0.950365i	$-0.600709\pi$
$947$	−19.1494	−0.622272	−0.311136	+	0.950365i	$0.600709\pi$
$948$	0	0
$949$	7.09427	0.230290
$950$	0	0
$951$	0	0
$952$	0	0
$953$	42.6090	1.38024	0.690120	−	0.723695i	$-0.257558\pi$
$953$	42.6090	1.38024	0.690120	+	0.723695i	$0.257558\pi$
$954$	0	0
$955$	−59.3604	−1.92086
$956$	0	0
$957$	0	0
$958$	0	0
$959$	2.19127	0.0707599
$960$	0	0
$961$	−30.3884	−0.980270
$962$	0	0
$963$	0	0
$964$	0	0
$965$	41.5037	1.33605
$966$	0	0
$967$	−22.4484	−0.721893	−0.360947	−	0.932587i	$-0.617546\pi$
$967$	−22.4484	−0.721893	−0.360947	+	0.932587i	$0.617546\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	36.1977	1.16164	0.580820	−	0.814032i	$-0.302732\pi$
$971$	36.1977	1.16164	0.580820	+	0.814032i	$0.302732\pi$
$972$	0	0
$973$	−0.587588	−0.0188372
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−5.12812	−0.164063	−0.0820316	−	0.996630i	$-0.526141\pi$
$977$	−5.12812	−0.164063	−0.0820316	+	0.996630i	$0.526141\pi$
$978$	0	0
$979$	18.3303	0.585840
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−25.5444	−0.814738	−0.407369	−	0.913264i	$-0.633554\pi$
$983$	−25.5444	−0.814738	−0.407369	+	0.913264i	$0.633554\pi$
$984$	0	0
$985$	31.3543	0.999032
$986$	0	0
$987$	0	0
$988$	0	0
$989$	44.5418	1.41635
$990$	0	0
$991$	−6.62586	−0.210477	−0.105239	−	0.994447i	$-0.533561\pi$
$991$	−6.62586	−0.210477	−0.105239	+	0.994447i	$0.533561\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	9.86979	0.312894
$996$	0	0
$997$	−18.9208	−0.599228	−0.299614	−	0.954060i	$-0.596858\pi$
$997$	−18.9208	−0.599228	−0.299614	+	0.954060i	$0.596858\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8424.2.a.v.1.6		6
3.2	odd	2		8424.2.a.u.1.1		6
9.2	odd	6		2808.2.q.d.1873.6		12
9.4	even	3		936.2.q.d.313.5	✓	12
9.5	odd	6		2808.2.q.d.937.6		12
9.7	even	3		936.2.q.d.625.5	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
936.2.q.d.313.5	✓	12	9.4	even	3
936.2.q.d.625.5	yes	12	9.7	even	3
2808.2.q.d.937.6		12	9.5	odd	6
2808.2.q.d.1873.6		12	9.2	odd	6
8424.2.a.u.1.1		6	3.2	odd	2
8424.2.a.v.1.6		6	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8424 = 2^{3} \cdot 3^{4} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8424.a (trivial)

\(f(q)\)	\(=\)	\(q+3.74770 q^{5} -0.351433 q^{7} -3.29530 q^{11} +1.00000 q^{13} +3.88461 q^{17} -4.45847 q^{19} -4.66850 q^{23} +9.04525 q^{25} -4.44539 q^{29} +0.782066 q^{31} -1.31707 q^{35} -11.4322 q^{37} -6.84266 q^{41} -9.54092 q^{43} -2.06612 q^{47} -6.87649 q^{49} -2.97755 q^{53} -12.3498 q^{55} +0.940619 q^{59} +14.4198 q^{61} +3.74770 q^{65} +4.25497 q^{67} -5.25904 q^{71} +7.09427 q^{73} +1.15808 q^{77} -14.2136 q^{79} -12.5543 q^{83} +14.5583 q^{85} -5.56256 q^{89} -0.351433 q^{91} -16.7090 q^{95} -1.70659 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + q^{5} - 2 q^{7} + q^{11} + 6 q^{13} + 6 q^{17} - 7 q^{19} - 19 q^{23} - 5 q^{25} + 2 q^{29} - 8 q^{31} + q^{35} - 17 q^{37} + 2 q^{41} - 15 q^{43} + 9 q^{47} - 12 q^{49} + 16 q^{53} - 5 q^{55}+ \cdots - 12 q^{97}+O(q^{100}) \) 6 * q + q^5 - 2 * q^7 + q^11 + 6 * q^13 + 6 * q^17 - 7 * q^19 - 19 * q^23 - 5 * q^25 + 2 * q^29 - 8 * q^31 + q^35 - 17 * q^37 + 2 * q^41 - 15 * q^43 + 9 * q^47 - 12 * q^49 + 16 * q^53 - 5 * q^55 - 5 * q^59 - 6 * q^61 + q^65 - 8 * q^67 - 21 * q^71 - q^73 - 3 * q^77 - 34 * q^79 - 44 * q^83 - 7 * q^85 + 11 * q^89 - 2 * q^91 - 6 * q^95 - 12 * q^97

Introduction

L-functions

Modular forms

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