LMFDB - Embedded newform 6975.2.a.cb.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6975,2,Mod(1,6975)] 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("6975.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6975, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 6975 = 3^{2} \cdot 5^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6975.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,0,12,0,0,-8,0,0,0,0,0,0,0,0,8,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	yes
Analytic conductor:	$55.6956554098$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.361944768.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 12x^{4} + 40x^{2} - 27 $ x^6 - 12x^4 + 40x^2 - 27
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 279)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$0.948456$ of defining polynomial
Character	$\chi$	$=$	6975.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+0.948456 q^{2} -1.10043 q^{4} -2.51225 q^{7} -2.94062 q^{8} -5.50933 q^{11} -6.20086 q^{13} -2.38275 q^{14} -0.588185 q^{16} -7.40624 q^{17} +1.48775 q^{19} -5.22536 q^{22} -7.77816 q^{23} -5.88124 q^{26} +2.76456 q^{28} +1.89691 q^{29} +1.00000 q^{31} +5.32338 q^{32} -7.02449 q^{34} -5.17637 q^{37} +1.41107 q^{38} -1.13441 q^{41} +8.20086 q^{43} +6.06265 q^{44} -7.37723 q^{46} -9.67507 q^{47} -0.688617 q^{49} +6.82363 q^{52} +7.40624 q^{53} +7.38757 q^{56} +1.79914 q^{58} -0.762500 q^{59} +3.02449 q^{61} +0.948456 q^{62} +6.22536 q^{64} +4.00000 q^{67} +8.15007 q^{68} -3.03132 q^{71} +3.22536 q^{73} -4.90956 q^{74} -1.63717 q^{76} +13.8408 q^{77} -13.4262 q^{79} -1.07594 q^{82} +1.52500 q^{83} +7.77816 q^{86} +16.2009 q^{88} +13.4689 q^{89} +15.5781 q^{91} +8.55933 q^{92} -9.17637 q^{94} -13.7376 q^{97} -0.653123 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 12 q^{4} - 8 q^{7} + 8 q^{16} + 16 q^{19} + 20 q^{22} - 18 q^{28} + 6 q^{31} - 28 q^{34} - 8 q^{37} + 12 q^{43} + 16 q^{46} + 26 q^{49} + 64 q^{52} + 48 q^{58} + 4 q^{61} - 14 q^{64} + 24 q^{67} - 32 q^{73}+ \cdots - 24 q^{97}+O(q^{100}) $ 6 * q + 12 * q^4 - 8 * q^7 + 8 * q^16 + 16 * q^19 + 20 * q^22 - 18 * q^28 + 6 * q^31 - 28 * q^34 - 8 * q^37 + 12 * q^43 + 16 * q^46 + 26 * q^49 + 64 * q^52 + 48 * q^58 + 4 * q^61 - 14 * q^64 + 24 * q^67 - 32 * q^73 + 30 * q^76 + 8 * q^79 - 2 * q^82 + 60 * q^88 - 4 * q^91 - 32 * q^94 - 24 * 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 $

$2$	0.948456	0.670659	0.335330	−	0.942101i	$-0.391152\pi$
$2$	0.948456	0.670659	0.335330	+	0.942101i	$0.391152\pi$
$3$	0	0
$3$	0	0
$4$	−1.10043	−0.550216
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−2.51225	−0.949540	−0.474770	−	0.880110i	$-0.657469\pi$
$7$	−2.51225	−0.949540	−0.474770	+	0.880110i	$0.657469\pi$
$8$	−2.94062	−1.03967
$9$	0	0
$10$	0	0
$11$	−5.50933	−1.66113	−0.830563	−	0.556925i	$-0.811981\pi$
$11$	−5.50933	−1.66113	−0.830563	+	0.556925i	$0.811981\pi$
$12$	0	0
$13$	−6.20086	−1.71981	−0.859905	−	0.510454i	$-0.829477\pi$
$13$	−6.20086	−1.71981	−0.859905	+	0.510454i	$0.829477\pi$
$14$	−2.38275	−0.636818
$15$	0	0
$16$	−0.588185	−0.147046
$17$	−7.40624	−1.79628	−0.898139	−	0.439712i	$-0.855081\pi$
$17$	−7.40624	−1.79628	−0.898139	+	0.439712i	$0.855081\pi$
$18$	0	0
$19$	1.48775	0.341314	0.170657	−	0.985330i	$-0.445411\pi$
$19$	1.48775	0.341314	0.170657	+	0.985330i	$0.445411\pi$
$20$	0	0
$21$	0	0
$22$	−5.22536	−1.11405
$23$	−7.77816	−1.62186	−0.810929	−	0.585145i	$-0.801038\pi$
$23$	−7.77816	−1.62186	−0.810929	+	0.585145i	$0.801038\pi$
$24$	0	0
$25$	0	0
$26$	−5.88124	−1.15341
$27$	0	0
$28$	2.76456	0.522452
$29$	1.89691	0.352248	0.176124	−	0.984368i	$-0.443644\pi$
$29$	1.89691	0.352248	0.176124	+	0.984368i	$0.443644\pi$
$30$	0	0
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	5.32338	0.941049
$33$	0	0
$34$	−7.02449	−1.20469
$35$	0	0
$36$	0	0
$37$	−5.17637	−0.850990	−0.425495	−	0.904961i	$-0.639900\pi$
$37$	−5.17637	−0.850990	−0.425495	+	0.904961i	$0.639900\pi$
$38$	1.41107	0.228905
$39$	0	0
$40$	0	0
$41$	−1.13441	−0.177165	−0.0885827	−	0.996069i	$-0.528234\pi$
$41$	−1.13441	−0.177165	−0.0885827	+	0.996069i	$0.528234\pi$
$42$	0	0
$43$	8.20086	1.25062	0.625310	−	0.780376i	$-0.284973\pi$
$43$	8.20086	1.25062	0.625310	+	0.780376i	$0.284973\pi$
$44$	6.06265	0.913978
$45$	0	0
$46$	−7.37723	−1.08771
$47$	−9.67507	−1.41125	−0.705627	−	0.708584i	$-0.749334\pi$
$47$	−9.67507	−1.41125	−0.705627	+	0.708584i	$0.749334\pi$
$48$	0	0
$49$	−0.688617	−0.0983739
$50$	0	0
$51$	0	0
$52$	6.82363	0.946267
$53$	7.40624	1.01733	0.508663	−	0.860966i	$-0.330140\pi$
$53$	7.40624	1.01733	0.508663	+	0.860966i	$0.330140\pi$
$54$	0	0
$55$	0	0
$56$	7.38757	0.987205
$57$	0	0
$58$	1.79914	0.236238
$59$	−0.762500	−0.0992690	−0.0496345	−	0.998767i	$-0.515806\pi$
$59$	−0.762500	−0.0992690	−0.0496345	+	0.998767i	$0.515806\pi$
$60$	0	0
$61$	3.02449	0.387247	0.193623	−	0.981076i	$-0.437976\pi$
$61$	3.02449	0.387247	0.193623	+	0.981076i	$0.437976\pi$
$62$	0.948456	0.120454
$63$	0	0
$64$	6.22536	0.778170
$65$	0	0
$66$	0	0
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	8.15007	0.988341
$69$	0	0
$70$	0	0
$71$	−3.03132	−0.359752	−0.179876	−	0.983689i	$-0.557570\pi$
$71$	−3.03132	−0.359752	−0.179876	+	0.983689i	$0.557570\pi$
$72$	0	0
$73$	3.22536	0.377499	0.188750	−	0.982025i	$-0.439557\pi$
$73$	3.22536	0.377499	0.188750	+	0.982025i	$0.439557\pi$
$74$	−4.90956	−0.570725
$75$	0	0
$76$	−1.63717	−0.187796
$77$	13.8408	1.57731
$78$	0	0
$79$	−13.4262	−1.51057	−0.755284	−	0.655398i	$-0.772501\pi$
$79$	−13.4262	−1.51057	−0.755284	+	0.655398i	$0.772501\pi$
$80$	0	0
$81$	0	0
$82$	−1.07594	−0.118818
$83$	1.52500	0.167390	0.0836952	−	0.996491i	$-0.473328\pi$
$83$	1.52500	0.167390	0.0836952	+	0.996491i	$0.473328\pi$
$84$	0	0
$85$	0	0
$86$	7.77816	0.838740
$87$	0	0
$88$	16.2009	1.72702
$89$	13.4689	1.42770	0.713850	−	0.700299i	$-0.246950\pi$
$89$	13.4689	1.42770	0.713850	+	0.700299i	$0.246950\pi$
$90$	0	0
$91$	15.5781	1.63303
$92$	8.55933	0.892372
$93$	0	0
$94$	−9.17637	−0.946470
$95$	0	0
$96$	0	0
$97$	−13.7376	−1.39484	−0.697421	−	0.716662i	$-0.745669\pi$
$97$	−13.7376	−1.39484	−0.697421	+	0.716662i	$0.745669\pi$
$98$	−0.653123	−0.0659754
$99$	0	0
$100$	0	0
$101$	−3.40323	−0.338634	−0.169317	−	0.985562i	$-0.554156\pi$
$101$	−3.40323	−0.338634	−0.169317	+	0.985562i	$0.554156\pi$
$102$	0	0
$103$	−2.51225	−0.247539	−0.123770	−	0.992311i	$-0.539498\pi$
$103$	−2.51225	−0.247539	−0.123770	+	0.992311i	$0.539498\pi$
$104$	18.2344	1.78803
$105$	0	0
$106$	7.02449	0.682279
$107$	−11.7812	−1.13893	−0.569464	−	0.822016i	$-0.692849\pi$
$107$	−11.7812	−1.13893	−0.569464	+	0.822016i	$0.692849\pi$
$108$	0	0
$109$	4.31138	0.412956	0.206478	−	0.978451i	$-0.433800\pi$
$109$	4.31138	0.412956	0.206478	+	0.978451i	$0.433800\pi$
$110$	0	0
$111$	0	0
$112$	1.47767	0.139626
$113$	13.6781	1.28673	0.643363	−	0.765562i	$-0.277539\pi$
$113$	13.6781	1.28673	0.643363	+	0.765562i	$0.277539\pi$
$114$	0	0
$115$	0	0
$116$	−2.08742	−0.193812
$117$	0	0
$118$	−0.723197	−0.0665757
$119$	18.6063	1.70564
$120$	0	0
$121$	19.3527	1.75934
$122$	2.86860	0.259711
$123$	0	0
$124$	−1.10043	−0.0988217
$125$	0	0
$126$	0	0
$127$	−0.200864	−0.0178238	−0.00891190	−	0.999960i	$-0.502837\pi$
$127$	−0.200864	−0.0178238	−0.00891190	+	0.999960i	$0.502837\pi$
$128$	−4.74228	−0.419162
$129$	0	0
$130$	0	0
$131$	−8.93124	−0.780326	−0.390163	−	0.920746i	$-0.627581\pi$
$131$	−8.93124	−0.780326	−0.390163	+	0.920746i	$0.627581\pi$
$132$	0	0
$133$	−3.73760	−0.324091
$134$	3.79382	0.327736
$135$	0	0
$136$	21.7790	1.86753
$137$	−7.95956	−0.680031	−0.340015	−	0.940420i	$-0.610432\pi$
$137$	−7.95956	−0.680031	−0.340015	+	0.940420i	$0.610432\pi$
$138$	0	0
$139$	−10.2499	−0.869381	−0.434690	−	0.900580i	$-0.643142\pi$
$139$	−10.2499	−0.869381	−0.434690	+	0.900580i	$0.643142\pi$
$140$	0	0
$141$	0	0
$142$	−2.87507	−0.241271
$143$	34.1626	2.85682
$144$	0	0
$145$	0	0
$146$	3.05911	0.253174
$147$	0	0
$148$	5.69624	0.468228
$149$	−11.0187	−0.902684	−0.451342	−	0.892351i	$-0.649055\pi$
$149$	−11.0187	−0.902684	−0.451342	+	0.892351i	$0.649055\pi$
$150$	0	0
$151$	−1.84812	−0.150398	−0.0751990	−	0.997169i	$-0.523959\pi$
$151$	−1.84812	−0.150398	−0.0751990	+	0.997169i	$0.523959\pi$
$152$	−4.37492	−0.354853
$153$	0	0
$154$	13.1274	1.05783
$155$	0	0
$156$	0	0
$157$	−20.0903	−1.60338	−0.801692	−	0.597737i	$-0.796067\pi$
$157$	−20.0903	−1.60338	−0.801692	+	0.597737i	$0.796067\pi$
$158$	−12.7342	−1.01308
$159$	0	0
$160$	0	0
$161$	19.5406	1.54002
$162$	0	0
$163$	17.8895	1.40121	0.700606	−	0.713548i	$-0.252913\pi$
$163$	17.8895	1.40121	0.700606	+	0.713548i	$0.252913\pi$
$164$	1.24834	0.0974792
$165$	0	0
$166$	1.44639	0.112262
$167$	−0.553313	−0.0428167	−0.0214083	−	0.999771i	$-0.506815\pi$
$167$	−0.553313	−0.0428167	−0.0214083	+	0.999771i	$0.506815\pi$
$168$	0	0
$169$	25.4507	1.95775
$170$	0	0
$171$	0	0
$172$	−9.02449	−0.688111
$173$	−14.8125	−1.12617	−0.563086	−	0.826398i	$-0.690386\pi$
$173$	−14.8125	−1.12617	−0.563086	+	0.826398i	$0.690386\pi$
$174$	0	0
$175$	0	0
$176$	3.24051	0.244263
$177$	0	0
$178$	12.7746	0.957500
$179$	−11.0187	−0.823574	−0.411787	−	0.911280i	$-0.635095\pi$
$179$	−11.0187	−0.823574	−0.411787	+	0.911280i	$0.635095\pi$
$180$	0	0
$181$	−11.0245	−0.819444	−0.409722	−	0.912210i	$-0.634374\pi$
$181$	−11.0245	−0.819444	−0.409722	+	0.912210i	$0.634374\pi$
$182$	14.7751	1.09521
$183$	0	0
$184$	22.8726	1.68619
$185$	0	0
$186$	0	0
$187$	40.8035	2.98384
$188$	10.6468	0.776494
$189$	0	0
$190$	0	0
$191$	−18.5876	−1.34495	−0.672477	−	0.740118i	$-0.734770\pi$
$191$	−18.5876	−1.34495	−0.672477	+	0.740118i	$0.734770\pi$
$192$	0	0
$193$	−10.6641	−0.767620	−0.383810	−	0.923412i	$-0.625388\pi$
$193$	−10.6641	−0.767620	−0.383810	+	0.923412i	$0.625388\pi$
$194$	−13.0295	−0.935464
$195$	0	0
$196$	0.757777	0.0541269
$197$	3.42191	0.243801	0.121901	−	0.992542i	$-0.461101\pi$
$197$	3.42191	0.243801	0.121901	+	0.992542i	$0.461101\pi$
$198$	0	0
$199$	0.200864	0.0142389	0.00711943	−	0.999975i	$-0.497734\pi$
$199$	0.200864	0.0142389	0.00711943	+	0.999975i	$0.497734\pi$
$200$	0	0
$201$	0	0
$202$	−3.22782	−0.227108
$203$	−4.76551	−0.334473
$204$	0	0
$205$	0	0
$206$	−2.38275	−0.166014
$207$	0	0
$208$	3.64726	0.252892
$209$	−8.19653	−0.566966
$210$	0	0
$211$	1.48775	0.102421	0.0512106	−	0.998688i	$-0.483692\pi$
$211$	1.48775	0.102421	0.0512106	+	0.998688i	$0.483692\pi$
$212$	−8.15007	−0.559749
$213$	0	0
$214$	−11.1739	−0.763833
$215$	0	0
$216$	0	0
$217$	−2.51225	−0.170542
$218$	4.08915	0.276953
$219$	0	0
$220$	0	0
$221$	45.9251	3.08926
$222$	0	0
$223$	0.823629	0.0551543	0.0275771	−	0.999620i	$-0.491221\pi$
$223$	0.823629	0.0551543	0.0275771	+	0.999620i	$0.491221\pi$
$224$	−13.3736	−0.893563
$225$	0	0
$226$	12.9730	0.862954
$227$	−1.34360	−0.0891777	−0.0445889	−	0.999005i	$-0.514198\pi$
$227$	−1.34360	−0.0891777	−0.0445889	+	0.999005i	$0.514198\pi$
$228$	0	0
$229$	−13.1764	−0.870719	−0.435360	−	0.900257i	$-0.643379\pi$
$229$	−13.1764	−0.870719	−0.435360	+	0.900257i	$0.643379\pi$
$230$	0	0
$231$	0	0
$232$	−5.57810	−0.366220
$233$	8.35925	0.547633	0.273816	−	0.961782i	$-0.411714\pi$
$233$	8.35925	0.547633	0.273816	+	0.961782i	$0.411714\pi$
$234$	0	0
$235$	0	0
$236$	0.839079	0.0546194
$237$	0	0
$238$	17.6473	1.14390
$239$	2.26882	0.146758	0.0733790	−	0.997304i	$-0.476622\pi$
$239$	2.26882	0.146758	0.0733790	+	0.997304i	$0.476622\pi$
$240$	0	0
$241$	−11.6271	−0.748966	−0.374483	−	0.927234i	$-0.622180\pi$
$241$	−11.6271	−0.748966	−0.374483	+	0.927234i	$0.622180\pi$
$242$	18.3552	1.17992
$243$	0	0
$244$	−3.32825	−0.213069
$245$	0	0
$246$	0	0
$247$	−9.22536	−0.586995
$248$	−2.94062	−0.186730
$249$	0	0
$250$	0	0
$251$	−13.8408	−0.873624	−0.436812	−	0.899553i	$-0.643892\pi$
$251$	−13.8408	−0.873624	−0.436812	+	0.899553i	$0.643892\pi$
$252$	0	0
$253$	42.8524	2.69411
$254$	−0.190511	−0.0119537
$255$	0	0
$256$	−16.9486	−1.05928
$257$	−2.65941	−0.165889	−0.0829447	−	0.996554i	$-0.526433\pi$
$257$	−2.65941	−0.165889	−0.0829447	+	0.996554i	$0.526433\pi$
$258$	0	0
$259$	13.0043	0.808049
$260$	0	0
$261$	0	0
$262$	−8.47089	−0.523333
$263$	−11.7625	−0.725306	−0.362653	−	0.931924i	$-0.618129\pi$
$263$	−11.7625	−0.725306	−0.362653	+	0.931924i	$0.618129\pi$
$264$	0	0
$265$	0	0
$266$	−3.54495	−0.217355
$267$	0	0
$268$	−4.40173	−0.268878
$269$	−15.1844	−0.925809	−0.462905	−	0.886408i	$-0.653193\pi$
$269$	−15.1844	−0.925809	−0.462905	+	0.886408i	$0.653193\pi$
$270$	0	0
$271$	−23.3772	−1.42007	−0.710033	−	0.704168i	$-0.751320\pi$
$271$	−23.3772	−1.42007	−0.710033	+	0.704168i	$0.751320\pi$
$272$	4.35624	0.264136
$273$	0	0
$274$	−7.54929	−0.456069
$275$	0	0
$276$	0	0
$277$	0.0489861	0.00294329	0.00147164	−	0.999999i	$-0.499532\pi$
$277$	0.0489861	0.00294329	0.00147164	+	0.999999i	$0.499532\pi$
$278$	−9.72153	−0.583058
$279$	0	0
$280$	0	0
$281$	−23.5345	−1.40395	−0.701977	−	0.712200i	$-0.747699\pi$
$281$	−23.5345	−1.40395	−0.701977	+	0.712200i	$0.747699\pi$
$282$	0	0
$283$	1.95101	0.115976	0.0579879	−	0.998317i	$-0.481532\pi$
$283$	1.95101	0.115976	0.0579879	+	0.998317i	$0.481532\pi$
$284$	3.33576	0.197941
$285$	0	0
$286$	32.4017	1.91595
$287$	2.84992	0.168226
$288$	0	0
$289$	37.8524	2.22661
$290$	0	0
$291$	0	0
$292$	−3.54929	−0.207706
$293$	22.7812	1.33089	0.665445	−	0.746447i	$-0.268242\pi$
$293$	22.7812	1.33089	0.665445	+	0.746447i	$0.268242\pi$
$294$	0	0
$295$	0	0
$296$	15.2218	0.884746
$297$	0	0
$298$	−10.4507	−0.605394
$299$	48.2313	2.78929
$300$	0	0
$301$	−20.6026	−1.18751
$302$	−1.75286	−0.100866
$303$	0	0
$304$	−0.875075	−0.0501890
$305$	0	0
$306$	0	0
$307$	−3.53674	−0.201852	−0.100926	−	0.994894i	$-0.532181\pi$
$307$	−3.53674	−0.201852	−0.100926	+	0.994894i	$0.532181\pi$
$308$	−15.2309	−0.867859
$309$	0	0
$310$	0	0
$311$	−14.0500	−0.796702	−0.398351	−	0.917233i	$-0.630417\pi$
$311$	−14.0500	−0.796702	−0.398351	+	0.917233i	$0.630417\pi$
$312$	0	0
$313$	27.8279	1.57293	0.786464	−	0.617636i	$-0.211909\pi$
$313$	27.8279	1.57293	0.786464	+	0.617636i	$0.211909\pi$
$314$	−19.0548	−1.07532
$315$	0	0
$316$	14.7746	0.831139
$317$	−8.72206	−0.489879	−0.244940	−	0.969538i	$-0.578768\pi$
$317$	−8.72206	−0.489879	−0.244940	+	0.969538i	$0.578768\pi$
$318$	0	0
$319$	−10.4507	−0.585128
$320$	0	0
$321$	0	0
$322$	18.5334	1.03283
$323$	−11.0187	−0.613095
$324$	0	0
$325$	0	0
$326$	16.9674	0.939736
$327$	0	0
$328$	3.33588	0.184193
$329$	24.3062	1.34004
$330$	0	0
$331$	11.0735	0.608653	0.304327	−	0.952568i	$-0.401569\pi$
$331$	11.0735	0.608653	0.304327	+	0.952568i	$0.401569\pi$
$332$	−1.67816	−0.0921009
$333$	0	0
$334$	−0.524793	−0.0287154
$335$	0	0
$336$	0	0
$337$	−16.0490	−0.874244	−0.437122	−	0.899402i	$-0.644002\pi$
$337$	−16.0490	−0.874244	−0.437122	+	0.899402i	$0.644002\pi$
$338$	24.1389	1.31298
$339$	0	0
$340$	0	0
$341$	−5.50933	−0.298347
$342$	0	0
$343$	19.3157	1.04295
$344$	−24.1156	−1.30023
$345$	0	0
$346$	−14.0490	−0.755278
$347$	22.5906	1.21273	0.606365	−	0.795187i	$-0.292627\pi$
$347$	22.5906	1.21273	0.606365	+	0.795187i	$0.292627\pi$
$348$	0	0
$349$	4.04899	0.216737	0.108369	−	0.994111i	$-0.465437\pi$
$349$	4.04899	0.216737	0.108369	+	0.994111i	$0.465437\pi$
$350$	0	0
$351$	0	0
$352$	−29.3282	−1.56320
$353$	−18.9782	−1.01011	−0.505054	−	0.863088i	$-0.668528\pi$
$353$	−18.9782	−1.01011	−0.505054	+	0.863088i	$0.668528\pi$
$354$	0	0
$355$	0	0
$356$	−14.8216	−0.785543
$357$	0	0
$358$	−10.4507	−0.552337
$359$	−3.03132	−0.159987	−0.0799935	−	0.996795i	$-0.525490\pi$
$359$	−3.03132	−0.159987	−0.0799935	+	0.996795i	$0.525490\pi$
$360$	0	0
$361$	−16.7866	−0.883505
$362$	−10.4562	−0.549568
$363$	0	0
$364$	−17.1426	−0.898518
$365$	0	0
$366$	0	0
$367$	31.7790	1.65885	0.829424	−	0.558619i	$-0.188669\pi$
$367$	31.7790	1.65885	0.829424	+	0.558619i	$0.188669\pi$
$368$	4.57500	0.238488
$369$	0	0
$370$	0	0
$371$	−18.6063	−0.965991
$372$	0	0
$373$	11.2869	0.584413	0.292206	−	0.956355i	$-0.405611\pi$
$373$	11.2869	0.584413	0.292206	+	0.956355i	$0.405611\pi$
$374$	38.7003	2.00114
$375$	0	0
$376$	28.4507	1.46723
$377$	−11.7625	−0.605799
$378$	0	0
$379$	30.4507	1.56415	0.782074	−	0.623186i	$-0.214162\pi$
$379$	30.4507	1.56415	0.782074	+	0.623186i	$0.214162\pi$
$380$	0	0
$381$	0	0
$382$	−17.6295	−0.902006
$383$	−8.00602	−0.409088	−0.204544	−	0.978857i	$-0.565571\pi$
$383$	−8.00602	−0.409088	−0.204544	+	0.978857i	$0.565571\pi$
$384$	0	0
$385$	0	0
$386$	−10.1144	−0.514812
$387$	0	0
$388$	15.1173	0.767465
$389$	−4.16573	−0.211211	−0.105606	−	0.994408i	$-0.533678\pi$
$389$	−4.16573	−0.211211	−0.105606	+	0.994408i	$0.533678\pi$
$390$	0	0
$391$	57.6069	2.91331
$392$	2.02496	0.102276
$393$	0	0
$394$	3.24553	0.163507
$395$	0	0
$396$	0	0
$397$	−10.1393	−0.508878	−0.254439	−	0.967089i	$-0.581891\pi$
$397$	−10.1393	−0.508878	−0.254439	+	0.967089i	$0.581891\pi$
$398$	0.190511	0.00954943
$399$	0	0
$400$	0	0
$401$	−29.2157	−1.45896	−0.729481	−	0.684001i	$-0.760239\pi$
$401$	−29.2157	−1.45896	−0.729481	+	0.684001i	$0.760239\pi$
$402$	0	0
$403$	−6.20086	−0.308887
$404$	3.74503	0.186322
$405$	0	0
$406$	−4.51987	−0.224318
$407$	28.5183	1.41360
$408$	0	0
$409$	24.5536	1.21410	0.607049	−	0.794665i	$-0.292353\pi$
$409$	24.5536	1.21410	0.607049	+	0.794665i	$0.292353\pi$
$410$	0	0
$411$	0	0
$412$	2.76456	0.136200
$413$	1.91559	0.0942599
$414$	0	0
$415$	0	0
$416$	−33.0095	−1.61843
$417$	0	0
$418$	−7.77404	−0.380241
$419$	13.2688	0.648224	0.324112	−	0.946019i	$-0.394935\pi$
$419$	13.2688	0.648224	0.324112	+	0.946019i	$0.394935\pi$
$420$	0	0
$421$	33.1148	1.61392	0.806959	−	0.590607i	$-0.201112\pi$
$421$	33.1148	1.61392	0.806959	+	0.590607i	$0.201112\pi$
$422$	1.41107	0.0686897
$423$	0	0
$424$	−21.7790	−1.05768
$425$	0	0
$426$	0	0
$427$	−7.59827	−0.367706
$428$	12.9644	0.626656
$429$	0	0
$430$	0	0
$431$	2.45022	0.118023	0.0590116	−	0.998257i	$-0.481205\pi$
$431$	2.45022	0.118023	0.0590116	+	0.998257i	$0.481205\pi$
$432$	0	0
$433$	−18.8035	−0.903636	−0.451818	−	0.892110i	$-0.649224\pi$
$433$	−18.8035	−0.903636	−0.451818	+	0.892110i	$0.649224\pi$
$434$	−2.38275	−0.114376
$435$	0	0
$436$	−4.74438	−0.227215
$437$	−11.5720	−0.553563
$438$	0	0
$439$	2.51225	0.119903	0.0599515	−	0.998201i	$-0.480905\pi$
$439$	2.51225	0.119903	0.0599515	+	0.998201i	$0.480905\pi$
$440$	0	0
$441$	0	0
$442$	43.5579	2.07184
$443$	22.3815	1.06338	0.531688	−	0.846941i	$-0.321558\pi$
$443$	22.3815	1.06338	0.531688	+	0.846941i	$0.321558\pi$
$444$	0	0
$445$	0	0
$446$	0.781176	0.0369897
$447$	0	0
$448$	−15.6396	−0.738903
$449$	−28.8347	−1.36079	−0.680397	−	0.732844i	$-0.738192\pi$
$449$	−28.8347	−1.36079	−0.680397	+	0.732844i	$0.738192\pi$
$450$	0	0
$451$	6.24985	0.294294
$452$	−15.0518	−0.707977
$453$	0	0
$454$	−1.27434	−0.0598079
$455$	0	0
$456$	0	0
$457$	−16.1519	−0.755553	−0.377776	−	0.925897i	$-0.623311\pi$
$457$	−16.1519	−0.755553	−0.377776	+	0.925897i	$0.623311\pi$
$458$	−12.4972	−0.583956
$459$	0	0
$460$	0	0
$461$	5.88124	0.273917	0.136958	−	0.990577i	$-0.456267\pi$
$461$	5.88124	0.273917	0.136958	+	0.990577i	$0.456267\pi$
$462$	0	0
$463$	−23.0735	−1.07232	−0.536158	−	0.844118i	$-0.680125\pi$
$463$	−23.0735	−1.07232	−0.536158	+	0.844118i	$0.680125\pi$
$464$	−1.11574	−0.0517967
$465$	0	0
$466$	7.92838	0.367275
$467$	−13.2688	−0.614007	−0.307004	−	0.951708i	$-0.599326\pi$
$467$	−13.2688	−0.614007	−0.307004	+	0.951708i	$0.599326\pi$
$468$	0	0
$469$	−10.0490	−0.464019
$470$	0	0
$471$	0	0
$472$	2.24222	0.103207
$473$	−45.1813	−2.07744
$474$	0	0
$475$	0	0
$476$	−20.4750	−0.938469
$477$	0	0
$478$	2.15188	0.0984246
$479$	15.5750	0.711639	0.355820	−	0.934555i	$-0.384202\pi$
$479$	15.5750	0.711639	0.355820	+	0.934555i	$0.384202\pi$
$480$	0	0
$481$	32.0980	1.46354
$482$	−11.0278	−0.502301
$483$	0	0
$484$	−21.2964	−0.968017
$485$	0	0
$486$	0	0
$487$	21.2254	0.961813	0.480906	−	0.876772i	$-0.340308\pi$
$487$	21.2254	0.961813	0.480906	+	0.876772i	$0.340308\pi$
$488$	−8.89389	−0.402608
$489$	0	0
$490$	0	0
$491$	−6.29051	−0.283887	−0.141943	−	0.989875i	$-0.545335\pi$
$491$	−6.29051	−0.283887	−0.141943	+	0.989875i	$0.545335\pi$
$492$	0	0
$493$	−14.0490	−0.632735
$494$	−8.74984	−0.393674
$495$	0	0
$496$	−0.588185	−0.0264103
$497$	7.61543	0.341599
$498$	0	0
$499$	−13.4262	−0.601040	−0.300520	−	0.953775i	$-0.597160\pi$
$499$	−13.4262	−0.601040	−0.300520	+	0.953775i	$0.597160\pi$
$500$	0	0
$501$	0	0
$502$	−13.1274	−0.585904
$503$	−41.7316	−1.86072	−0.930360	−	0.366648i	$-0.880505\pi$
$503$	−41.7316	−1.86072	−0.930360	+	0.366648i	$0.880505\pi$
$504$	0	0
$505$	0	0
$506$	40.6436	1.80683
$507$	0	0
$508$	0.221037	0.00980694
$509$	−15.9656	−0.707662	−0.353831	−	0.935309i	$-0.615121\pi$
$509$	−15.9656	−0.707662	−0.353831	+	0.935309i	$0.615121\pi$
$510$	0	0
$511$	−8.10289	−0.358451
$512$	−6.59039	−0.291257
$513$	0	0
$514$	−2.52233	−0.111255
$515$	0	0
$516$	0	0
$517$	53.3032	2.34427
$518$	12.3340	0.541926
$519$	0	0
$520$	0	0
$521$	37.5936	1.64701	0.823504	−	0.567311i	$-0.192016\pi$
$521$	37.5936	1.64701	0.823504	+	0.567311i	$0.192016\pi$
$522$	0	0
$523$	26.0288	1.13816	0.569080	−	0.822282i	$-0.307299\pi$
$523$	26.0288	1.13816	0.569080	+	0.822282i	$0.307299\pi$
$524$	9.82823	0.429348
$525$	0	0
$526$	−11.1562	−0.486433
$527$	−7.40624	−0.322621
$528$	0	0
$529$	37.4997	1.63042
$530$	0	0
$531$	0	0
$532$	4.11298	0.178320
$533$	7.03433	0.304691
$534$	0	0
$535$	0	0
$536$	−11.7625	−0.508062
$537$	0	0
$538$	−14.4017	−0.620903
$539$	3.79382	0.163411
$540$	0	0
$541$	20.0903	0.863751	0.431876	−	0.901933i	$-0.357852\pi$
$541$	20.0903	0.863751	0.431876	+	0.901933i	$0.357852\pi$
$542$	−22.1723	−0.952381
$543$	0	0
$544$	−39.4262	−1.69039
$545$	0	0
$546$	0	0
$547$	−28.7419	−1.22892	−0.614458	−	0.788950i	$-0.710625\pi$
$547$	−28.7419	−1.22892	−0.614458	+	0.788950i	$0.710625\pi$
$548$	8.75895	0.374164
$549$	0	0
$550$	0	0
$551$	2.82214	0.120227
$552$	0	0
$553$	33.7300	1.43434
$554$	0.0464611	0.00197394
$555$	0	0
$556$	11.2793	0.478347
$557$	34.1717	1.44790	0.723951	−	0.689851i	$-0.242324\pi$
$557$	34.1717	1.44790	0.723951	+	0.689851i	$0.242324\pi$
$558$	0	0
$559$	−50.8524	−2.15083
$560$	0	0
$561$	0	0
$562$	−22.3215	−0.941574
$563$	22.3815	0.943266	0.471633	−	0.881795i	$-0.343665\pi$
$563$	22.3815	0.943266	0.471633	+	0.881795i	$0.343665\pi$
$564$	0	0
$565$	0	0
$566$	1.85045	0.0777802
$567$	0	0
$568$	8.91397	0.374022
$569$	−5.46287	−0.229015	−0.114508	−	0.993422i	$-0.536529\pi$
$569$	−5.46287	−0.229015	−0.114508	+	0.993422i	$0.536529\pi$
$570$	0	0
$571$	27.2743	1.14140	0.570698	−	0.821160i	$-0.306673\pi$
$571$	27.2743	1.14140	0.570698	+	0.821160i	$0.306673\pi$
$572$	−37.5936	−1.57187
$573$	0	0
$574$	2.70302	0.112822
$575$	0	0
$576$	0	0
$577$	16.3527	0.680774	0.340387	−	0.940286i	$-0.389442\pi$
$577$	16.3527	0.680774	0.340387	+	0.940286i	$0.389442\pi$
$578$	35.9014	1.49330
$579$	0	0
$580$	0	0
$581$	−3.83117	−0.158944
$582$	0	0
$583$	−40.8035	−1.68991
$584$	−9.48456	−0.392474
$585$	0	0
$586$	21.6069	0.892573
$587$	−3.79382	−0.156588	−0.0782939	−	0.996930i	$-0.524947\pi$
$587$	−3.79382	−0.156588	−0.0782939	+	0.996930i	$0.524947\pi$
$588$	0	0
$589$	1.48775	0.0613018
$590$	0	0
$591$	0	0
$592$	3.04467	0.125135
$593$	11.3719	0.466988	0.233494	−	0.972358i	$-0.424984\pi$
$593$	11.3719	0.466988	0.233494	+	0.972358i	$0.424984\pi$
$594$	0	0
$595$	0	0
$596$	12.1253	0.496671
$597$	0	0
$598$	45.7452	1.87066
$599$	−18.5876	−0.759470	−0.379735	−	0.925095i	$-0.623985\pi$
$599$	−18.5876	−0.759470	−0.379735	+	0.925095i	$0.623985\pi$
$600$	0	0
$601$	10.8236	0.441505	0.220753	−	0.975330i	$-0.429149\pi$
$601$	10.8236	0.441505	0.220753	+	0.975330i	$0.429149\pi$
$602$	−19.5406	−0.796417
$603$	0	0
$604$	2.03373	0.0827514
$605$	0	0
$606$	0	0
$607$	18.0490	0.732586	0.366293	−	0.930500i	$-0.380627\pi$
$607$	18.0490	0.732586	0.366293	+	0.930500i	$0.380627\pi$
$608$	7.91987	0.321193
$609$	0	0
$610$	0	0
$611$	59.9938	2.42709
$612$	0	0
$613$	14.0980	0.569412	0.284706	−	0.958615i	$-0.408104\pi$
$613$	14.0980	0.569412	0.284706	+	0.958615i	$0.408104\pi$
$614$	−3.35444	−0.135374
$615$	0	0
$616$	−40.7006	−1.63987
$617$	−31.5683	−1.27089	−0.635447	−	0.772145i	$-0.719184\pi$
$617$	−31.5683	−1.27089	−0.635447	+	0.772145i	$0.719184\pi$
$618$	0	0
$619$	−9.82795	−0.395019	−0.197509	−	0.980301i	$-0.563285\pi$
$619$	−9.82795	−0.395019	−0.197509	+	0.980301i	$0.563285\pi$
$620$	0	0
$621$	0	0
$622$	−13.3258	−0.534315
$623$	−33.8372	−1.35566
$624$	0	0
$625$	0	0
$626$	26.3936	1.05490
$627$	0	0
$628$	22.1081	0.882208
$629$	38.3375	1.52861
$630$	0	0
$631$	−29.5242	−1.17534	−0.587670	−	0.809101i	$-0.699955\pi$
$631$	−29.5242	−1.17534	−0.587670	+	0.809101i	$0.699955\pi$
$632$	39.4814	1.57049
$633$	0	0
$634$	−8.27248	−0.328542
$635$	0	0
$636$	0	0
$637$	4.27002	0.169184
$638$	−9.91204	−0.392421
$639$	0	0
$640$	0	0
$641$	37.4122	1.47769	0.738847	−	0.673873i	$-0.235370\pi$
$641$	37.4122	1.47769	0.738847	+	0.673873i	$0.235370\pi$
$642$	0	0
$643$	−37.8279	−1.49179	−0.745894	−	0.666064i	$-0.767978\pi$
$643$	−37.8279	−1.49179	−0.745894	+	0.666064i	$0.767978\pi$
$644$	−21.5031	−0.847343
$645$	0	0
$646$	−10.4507	−0.411178
$647$	15.7468	0.619071	0.309536	−	0.950888i	$-0.399826\pi$
$647$	15.7468	0.619071	0.309536	+	0.950888i	$0.399826\pi$
$648$	0	0
$649$	4.20086	0.164898
$650$	0	0
$651$	0	0
$652$	−19.6862	−0.770969
$653$	25.0873	0.981743	0.490871	−	0.871232i	$-0.336678\pi$
$653$	25.0873	0.981743	0.490871	+	0.871232i	$0.336678\pi$
$654$	0	0
$655$	0	0
$656$	0.667244	0.0260515
$657$	0	0
$658$	23.0533	0.898711
$659$	−38.3561	−1.49414	−0.747072	−	0.664744i	$-0.768541\pi$
$659$	−38.3561	−1.49414	−0.747072	+	0.664744i	$0.768541\pi$
$660$	0	0
$661$	−28.0903	−1.09259	−0.546294	−	0.837594i	$-0.683962\pi$
$661$	−28.0903	−1.09259	−0.546294	+	0.837594i	$0.683962\pi$
$662$	10.5027	0.408199
$663$	0	0
$664$	−4.48445	−0.174030
$665$	0	0
$666$	0	0
$667$	−14.7545	−0.571295
$668$	0.608884	0.0235584
$669$	0	0
$670$	0	0
$671$	−16.6629	−0.643266
$672$	0	0
$673$	29.9798	1.15564	0.577819	−	0.816165i	$-0.303904\pi$
$673$	29.9798	1.15564	0.577819	+	0.816165i	$0.303904\pi$
$674$	−15.2218	−0.586320
$675$	0	0
$676$	−28.0068	−1.07718
$677$	8.37793	0.321990	0.160995	−	0.986955i	$-0.448530\pi$
$677$	8.37793	0.321990	0.160995	+	0.986955i	$0.448530\pi$
$678$	0	0
$679$	34.5122	1.32446
$680$	0	0
$681$	0	0
$682$	−5.22536	−0.200089
$683$	25.8125	0.987687	0.493843	−	0.869551i	$-0.335592\pi$
$683$	25.8125	0.987687	0.493843	+	0.869551i	$0.335592\pi$
$684$	0	0
$685$	0	0
$686$	18.3201	0.699464
$687$	0	0
$688$	−4.82363	−0.183899
$689$	−45.9251	−1.74961
$690$	0	0
$691$	7.84050	0.298267	0.149133	−	0.988817i	$-0.452352\pi$
$691$	7.84050	0.298267	0.149133	+	0.988817i	$0.452352\pi$
$692$	16.3001	0.619638
$693$	0	0
$694$	21.4262	0.813328
$695$	0	0
$696$	0	0
$697$	8.40173	0.318238
$698$	3.84028	0.145357
$699$	0	0
$700$	0	0
$701$	36.8220	1.39075	0.695374	−	0.718648i	$-0.255239\pi$
$701$	36.8220	1.39075	0.695374	+	0.718648i	$0.255239\pi$
$702$	0	0
$703$	−7.70116	−0.290455
$704$	−34.2976	−1.29264
$705$	0	0
$706$	−18.0000	−0.677439
$707$	8.54976	0.321547
$708$	0	0
$709$	6.62277	0.248723	0.124362	−	0.992237i	$-0.460312\pi$
$709$	6.62277	0.248723	0.124362	+	0.992237i	$0.460312\pi$
$710$	0	0
$711$	0	0
$712$	−39.6069	−1.48433
$713$	−7.77816	−0.291294
$714$	0	0
$715$	0	0
$716$	12.1253	0.453143
$717$	0	0
$718$	−2.87507	−0.107297
$719$	−37.1753	−1.38640	−0.693202	−	0.720744i	$-0.743801\pi$
$719$	−37.1753	−1.38640	−0.693202	+	0.720744i	$0.743801\pi$
$720$	0	0
$721$	6.31138	0.235048
$722$	−15.9213	−0.592531
$723$	0	0
$724$	12.1317	0.450871
$725$	0	0
$726$	0	0
$727$	11.7174	0.434575	0.217288	−	0.976108i	$-0.430279\pi$
$727$	11.7174	0.434575	0.217288	+	0.976108i	$0.430279\pi$
$728$	−45.8093	−1.69781
$729$	0	0
$730$	0	0
$731$	−60.7376	−2.24646
$732$	0	0
$733$	25.3359	0.935802	0.467901	−	0.883781i	$-0.345010\pi$
$733$	25.3359	0.935802	0.467901	+	0.883781i	$0.345010\pi$
$734$	30.1409	1.11252
$735$	0	0
$736$	−41.4060	−1.52625
$737$	−22.0373	−0.811755
$738$	0	0
$739$	−18.1519	−0.667728	−0.333864	−	0.942621i	$-0.608353\pi$
$739$	−18.1519	−0.667728	−0.333864	+	0.942621i	$0.608353\pi$
$740$	0	0
$741$	0	0
$742$	−17.6473	−0.647851
$743$	−53.2849	−1.95483	−0.977417	−	0.211322i	$-0.932223\pi$
$743$	−53.2849	−1.95483	−0.977417	+	0.211322i	$0.932223\pi$
$744$	0	0
$745$	0	0
$746$	10.7051	0.391942
$747$	0	0
$748$	−44.9014	−1.64176
$749$	29.5972	1.08146
$750$	0	0
$751$	30.8160	1.12449	0.562246	−	0.826970i	$-0.309937\pi$
$751$	30.8160	1.12449	0.562246	+	0.826970i	$0.309937\pi$
$752$	5.69073	0.207520
$753$	0	0
$754$	−11.1562	−0.406285
$755$	0	0
$756$	0	0
$757$	11.1274	0.404432	0.202216	−	0.979341i	$-0.435186\pi$
$757$	11.1274	0.404432	0.202216	+	0.979341i	$0.435186\pi$
$758$	28.8812	1.04901
$759$	0	0
$760$	0	0
$761$	−47.8220	−1.73355	−0.866773	−	0.498702i	$-0.833810\pi$
$761$	−47.8220	−1.73355	−0.866773	+	0.498702i	$0.833810\pi$
$762$	0	0
$763$	−10.8313	−0.392118
$764$	20.4544	0.740015
$765$	0	0
$766$	−7.59335	−0.274359
$767$	4.72816	0.170724
$768$	0	0
$769$	−14.5411	−0.524364	−0.262182	−	0.965018i	$-0.584442\pi$
$769$	−14.5411	−0.524364	−0.262182	+	0.965018i	$0.584442\pi$
$770$	0	0
$771$	0	0
$772$	11.7351	0.422357
$773$	−52.3597	−1.88325	−0.941623	−	0.336669i	$-0.890700\pi$
$773$	−52.3597	−1.88325	−0.941623	+	0.336669i	$0.890700\pi$
$774$	0	0
$775$	0	0
$776$	40.3971	1.45017
$777$	0	0
$778$	−3.95101	−0.141651
$779$	−1.68772	−0.0604690
$780$	0	0
$781$	16.7006	0.597593
$782$	54.6376	1.95384
$783$	0	0
$784$	0.405035	0.0144655
$785$	0	0
$786$	0	0
$787$	27.4752	0.979385	0.489693	−	0.871895i	$-0.337109\pi$
$787$	27.4752	0.979385	0.489693	+	0.871895i	$0.337109\pi$
$788$	−3.76558	−0.134143
$789$	0	0
$790$	0	0
$791$	−34.3627	−1.22180
$792$	0	0
$793$	−18.7545	−0.665991
$794$	−9.61671	−0.341284
$795$	0	0
$796$	−0.221037	−0.00783445
$797$	−27.1192	−0.960611	−0.480305	−	0.877101i	$-0.659474\pi$
$797$	−27.1192	−0.960611	−0.480305	+	0.877101i	$0.659474\pi$
$798$	0	0
$799$	71.6559	2.53500
$800$	0	0
$801$	0	0
$802$	−27.7098	−0.978467
$803$	−17.7696	−0.627074
$804$	0	0
$805$	0	0
$806$	−5.88124	−0.207158
$807$	0	0
$808$	10.0076	0.352067
$809$	1.47854	0.0519826	0.0259913	−	0.999662i	$-0.491726\pi$
$809$	1.47854	0.0519826	0.0259913	+	0.999662i	$0.491726\pi$
$810$	0	0
$811$	26.1470	0.918144	0.459072	−	0.888399i	$-0.348182\pi$
$811$	26.1470	0.918144	0.459072	+	0.888399i	$0.348182\pi$
$812$	5.24412	0.184032
$813$	0	0
$814$	27.0484	0.948045
$815$	0	0
$816$	0	0
$817$	12.2009	0.426854
$818$	23.2880	0.814246
$819$	0	0
$820$	0	0
$821$	15.4123	0.537892	0.268946	−	0.963155i	$-0.413325\pi$
$821$	15.4123	0.537892	0.268946	+	0.963155i	$0.413325\pi$
$822$	0	0
$823$	4.70548	0.164023	0.0820114	−	0.996631i	$-0.473866\pi$
$823$	4.70548	0.164023	0.0820114	+	0.996631i	$0.473866\pi$
$824$	7.38757	0.257358
$825$	0	0
$826$	1.81685	0.0632163
$827$	−12.7715	−0.444109	−0.222055	−	0.975034i	$-0.571276\pi$
$827$	−12.7715	−0.444109	−0.222055	+	0.975034i	$0.571276\pi$
$828$	0	0
$829$	2.00000	0.0694629	0.0347314	−	0.999397i	$-0.488942\pi$
$829$	2.00000	0.0694629	0.0347314	+	0.999397i	$0.488942\pi$
$830$	0	0
$831$	0	0
$832$	−38.6026	−1.33830
$833$	5.10007	0.176707
$834$	0	0
$835$	0	0
$836$	9.01972	0.311954
$837$	0	0
$838$	12.5849	0.434737
$839$	−21.0565	−0.726952	−0.363476	−	0.931604i	$-0.618410\pi$
$839$	−21.0565	−0.726952	−0.363476	+	0.931604i	$0.618410\pi$
$840$	0	0
$841$	−25.4017	−0.875922
$842$	31.4080	1.08239
$843$	0	0
$844$	−1.63717	−0.0563538
$845$	0	0
$846$	0	0
$847$	−48.6189	−1.67056
$848$	−4.35624	−0.149594
$849$	0	0
$850$	0	0
$851$	40.2626	1.38018
$852$	0	0
$853$	−32.3527	−1.10774	−0.553868	−	0.832604i	$-0.686849\pi$
$853$	−32.3527	−1.10774	−0.553868	+	0.832604i	$0.686849\pi$
$854$	−7.20662	−0.246606
$855$	0	0
$856$	34.6440	1.18411
$857$	−20.4568	−0.698790	−0.349395	−	0.936976i	$-0.613613\pi$
$857$	−20.4568	−0.698790	−0.349395	+	0.936976i	$0.613613\pi$
$858$	0	0
$859$	37.9308	1.29418	0.647092	−	0.762412i	$-0.275985\pi$
$859$	37.9308	1.29418	0.647092	+	0.762412i	$0.275985\pi$
$860$	0	0
$861$	0	0
$862$	2.32393	0.0791533
$863$	−29.2995	−0.997367	−0.498684	−	0.866784i	$-0.666183\pi$
$863$	−29.2995	−0.997367	−0.498684	+	0.866784i	$0.666183\pi$
$864$	0	0
$865$	0	0
$866$	−17.8342	−0.606032
$867$	0	0
$868$	2.76456	0.0938352
$869$	73.9695	2.50924
$870$	0	0
$871$	−24.8035	−0.840433
$872$	−12.6781	−0.429336
$873$	0	0
$874$	−10.9755	−0.371252
$875$	0	0
$876$	0	0
$877$	−2.76210	−0.0932694	−0.0466347	−	0.998912i	$-0.514850\pi$
$877$	−2.76210	−0.0932694	−0.0466347	+	0.998912i	$0.514850\pi$
$878$	2.38275	0.0804141
$879$	0	0
$880$	0	0
$881$	−32.4936	−1.09474	−0.547368	−	0.836892i	$-0.684370\pi$
$881$	−32.4936	−1.09474	−0.547368	+	0.836892i	$0.684370\pi$
$882$	0	0
$883$	−30.4507	−1.02475	−0.512374	−	0.858762i	$-0.671234\pi$
$883$	−30.4507	−1.02475	−0.512374	+	0.858762i	$0.671234\pi$
$884$	−50.5375	−1.69976
$885$	0	0
$886$	21.2278	0.713162
$887$	−12.5623	−0.421802	−0.210901	−	0.977507i	$-0.567640\pi$
$887$	−12.5623	−0.421802	−0.210901	+	0.977507i	$0.567640\pi$
$888$	0	0
$889$	0.504620	0.0169244
$890$	0	0
$891$	0	0
$892$	−0.906348	−0.0303468
$893$	−14.3941	−0.481681
$894$	0	0
$895$	0	0
$896$	11.9138	0.398011
$897$	0	0
$898$	−27.3484	−0.912629
$899$	1.89691	0.0632655
$900$	0	0
$901$	−54.8524	−1.82740
$902$	5.92771	0.197371
$903$	0	0
$904$	−40.2221	−1.33777
$905$	0	0
$906$	0	0
$907$	−5.26672	−0.174878	−0.0874392	−	0.996170i	$-0.527868\pi$
$907$	−5.26672	−0.174878	−0.0874392	+	0.996170i	$0.527868\pi$
$908$	1.47854	0.0490670
$909$	0	0
$910$	0	0
$911$	5.89035	0.195156	0.0975781	−	0.995228i	$-0.468890\pi$
$911$	5.89035	0.195156	0.0975781	+	0.995228i	$0.468890\pi$
$912$	0	0
$913$	−8.40173	−0.278057
$914$	−15.3193	−0.506719
$915$	0	0
$916$	14.4997	0.479084
$917$	22.4375	0.740951
$918$	0	0
$919$	−8.80346	−0.290399	−0.145200	−	0.989402i	$-0.546382\pi$
$919$	−8.80346	−0.290399	−0.145200	+	0.989402i	$0.546382\pi$
$920$	0	0
$921$	0	0
$922$	5.57810	0.183705
$923$	18.7968	0.618705
$924$	0	0
$925$	0	0
$926$	−21.8842	−0.719158
$927$	0	0
$928$	10.0980	0.331482
$929$	47.4966	1.55831	0.779156	−	0.626830i	$-0.215648\pi$
$929$	47.4966	1.55831	0.779156	+	0.626830i	$0.215648\pi$
$930$	0	0
$931$	−1.02449	−0.0335764
$932$	−9.19879	−0.301316
$933$	0	0
$934$	−12.5849	−0.411790
$935$	0	0
$936$	0	0
$937$	12.0490	0.393623	0.196812	−	0.980441i	$-0.436941\pi$
$937$	12.0490	0.393623	0.196812	+	0.980441i	$0.436941\pi$
$938$	−9.53102	−0.311199
$939$	0	0
$940$	0	0
$941$	−19.9499	−0.650348	−0.325174	−	0.945654i	$-0.605423\pi$
$941$	−19.9499	−0.650348	−0.325174	+	0.945654i	$0.605423\pi$
$942$	0	0
$943$	8.82363	0.287337
$944$	0.448491	0.0145971
$945$	0	0
$946$	−42.8524	−1.39325
$947$	−55.9721	−1.81885	−0.909424	−	0.415870i	$-0.863477\pi$
$947$	−55.9721	−1.81885	−0.909424	+	0.415870i	$0.863477\pi$
$948$	0	0
$949$	−20.0000	−0.649227
$950$	0	0
$951$	0	0
$952$	−54.7141	−1.77329
$953$	35.8317	1.16070	0.580351	−	0.814367i	$-0.302915\pi$
$953$	35.8317	1.16070	0.580351	+	0.814367i	$0.302915\pi$
$954$	0	0
$955$	0	0
$956$	−2.49669	−0.0807486
$957$	0	0
$958$	14.7722	0.477268
$959$	19.9964	0.645716
$960$	0	0
$961$	1.00000	0.0322581
$962$	30.4435	0.981538
$963$	0	0
$964$	12.7948	0.412093
$965$	0	0
$966$	0	0
$967$	−18.8726	−0.606902	−0.303451	−	0.952847i	$-0.598139\pi$
$967$	−18.8726	−0.606902	−0.303451	+	0.952847i	$0.598139\pi$
$968$	−56.9091	−1.82913
$969$	0	0
$970$	0	0
$971$	−18.0065	−0.577857	−0.288929	−	0.957351i	$-0.593299\pi$
$971$	−18.0065	−0.577857	−0.288929	+	0.957351i	$0.593299\pi$
$972$	0	0
$973$	25.7501	0.825512
$974$	20.1313	0.645049
$975$	0	0
$976$	−1.77896	−0.0569432
$977$	12.1531	0.388811	0.194406	−	0.980921i	$-0.437722\pi$
$977$	12.1531	0.388811	0.194406	+	0.980921i	$0.437722\pi$
$978$	0	0
$979$	−74.2046	−2.37159
$980$	0	0
$981$	0	0
$982$	−5.96627	−0.190391
$983$	15.0403	0.479713	0.239856	−	0.970808i	$-0.422900\pi$
$983$	15.0403	0.479713	0.239856	+	0.970808i	$0.422900\pi$
$984$	0	0
$985$	0	0
$986$	−13.3248	−0.424349
$987$	0	0
$988$	10.1519	0.322974
$989$	−63.7876	−2.02833
$990$	0	0
$991$	−28.0000	−0.889449	−0.444725	−	0.895667i	$-0.646698\pi$
$991$	−28.0000	−0.889449	−0.444725	+	0.895667i	$0.646698\pi$
$992$	5.32338	0.169017
$993$	0	0
$994$	7.22290	0.229096
$995$	0	0
$996$	0	0
$997$	37.3359	1.18244	0.591220	−	0.806511i	$-0.298647\pi$
$997$	37.3359	1.18244	0.591220	+	0.806511i	$0.298647\pi$
$998$	−12.7342	−0.403093
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6975.2.a.cb.1.4		6
3.2	odd	2	inner	6975.2.a.cb.1.3		6
5.4	even	2		279.2.a.d.1.3	✓	6
15.14	odd	2		279.2.a.d.1.4	yes	6
20.19	odd	2		4464.2.a.bt.1.3		6
60.59	even	2		4464.2.a.bt.1.4		6
155.154	odd	2		8649.2.a.bb.1.3		6
465.464	even	2		8649.2.a.bb.1.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
279.2.a.d.1.3	✓	6	5.4	even	2
279.2.a.d.1.4	yes	6	15.14	odd	2
4464.2.a.bt.1.3		6	20.19	odd	2
4464.2.a.bt.1.4		6	60.59	even	2
6975.2.a.cb.1.3		6	3.2	odd	2	inner
6975.2.a.cb.1.4		6	1.1	even	1	trivial
8649.2.a.bb.1.3		6	155.154	odd	2
8649.2.a.bb.1.4		6	465.464	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6975 = 3^{2} \cdot 5^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6975.a (trivial)

\(f(q)\)	\(=\)	\(q+0.948456 q^{2} -1.10043 q^{4} -2.51225 q^{7} -2.94062 q^{8} -5.50933 q^{11} -6.20086 q^{13} -2.38275 q^{14} -0.588185 q^{16} -7.40624 q^{17} +1.48775 q^{19} -5.22536 q^{22} -7.77816 q^{23} -5.88124 q^{26} +2.76456 q^{28} +1.89691 q^{29} +1.00000 q^{31} +5.32338 q^{32} -7.02449 q^{34} -5.17637 q^{37} +1.41107 q^{38} -1.13441 q^{41} +8.20086 q^{43} +6.06265 q^{44} -7.37723 q^{46} -9.67507 q^{47} -0.688617 q^{49} +6.82363 q^{52} +7.40624 q^{53} +7.38757 q^{56} +1.79914 q^{58} -0.762500 q^{59} +3.02449 q^{61} +0.948456 q^{62} +6.22536 q^{64} +4.00000 q^{67} +8.15007 q^{68} -3.03132 q^{71} +3.22536 q^{73} -4.90956 q^{74} -1.63717 q^{76} +13.8408 q^{77} -13.4262 q^{79} -1.07594 q^{82} +1.52500 q^{83} +7.77816 q^{86} +16.2009 q^{88} +13.4689 q^{89} +15.5781 q^{91} +8.55933 q^{92} -9.17637 q^{94} -13.7376 q^{97} -0.653123 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 12 q^{4} - 8 q^{7} + 8 q^{16} + 16 q^{19} + 20 q^{22} - 18 q^{28} + 6 q^{31} - 28 q^{34} - 8 q^{37} + 12 q^{43} + 16 q^{46} + 26 q^{49} + 64 q^{52} + 48 q^{58} + 4 q^{61} - 14 q^{64} + 24 q^{67} - 32 q^{73}+ \cdots - 24 q^{97}+O(q^{100}) \) 6 * q + 12 * q^4 - 8 * q^7 + 8 * q^16 + 16 * q^19 + 20 * q^22 - 18 * q^28 + 6 * q^31 - 28 * q^34 - 8 * q^37 + 12 * q^43 + 16 * q^46 + 26 * q^49 + 64 * q^52 + 48 * q^58 + 4 * q^61 - 14 * q^64 + 24 * q^67 - 32 * q^73 + 30 * q^76 + 8 * q^79 - 2 * q^82 + 60 * q^88 - 4 * q^91 - 32 * q^94 - 24 * q^97

Introduction

L-functions

Modular forms

Varieties

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Database

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