LMFDB - Embedded newform 2012.2.a.b.1.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2012,2,Mod(1,2012)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2012, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2012.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2012 = 2^{2} \cdot 503 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2012.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:	$16.0659008867$
Analytic rank:	$0$
Dimension:	$21$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.13
Character	$\chi$	$=$	2012.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.32334 q^{3} +2.33991 q^{5} +2.12211 q^{7} -1.24878 q^{9} +O(q^{10})$	$q+1.32334 q^{3} +2.33991 q^{5} +2.12211 q^{7} -1.24878 q^{9} +1.69672 q^{11} -4.85037 q^{13} +3.09649 q^{15} +7.73650 q^{17} -0.742375 q^{19} +2.80827 q^{21} +2.11455 q^{23} +0.475191 q^{25} -5.62257 q^{27} +4.46715 q^{29} +3.40159 q^{31} +2.24533 q^{33} +4.96556 q^{35} +0.982569 q^{37} -6.41867 q^{39} +7.22768 q^{41} +4.18992 q^{43} -2.92205 q^{45} -4.64253 q^{47} -2.49664 q^{49} +10.2380 q^{51} +10.5040 q^{53} +3.97017 q^{55} -0.982411 q^{57} +0.980586 q^{59} +2.61825 q^{61} -2.65006 q^{63} -11.3495 q^{65} -9.59050 q^{67} +2.79826 q^{69} +9.87610 q^{71} +4.75443 q^{73} +0.628836 q^{75} +3.60063 q^{77} -15.6527 q^{79} -3.69418 q^{81} +8.39562 q^{83} +18.1027 q^{85} +5.91154 q^{87} -13.6200 q^{89} -10.2930 q^{91} +4.50145 q^{93} -1.73709 q^{95} -8.72503 q^{97} -2.11884 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 21 q + 10 q^{3} + 3 q^{5} + 13 q^{7} + 21 q^{9}+O(q^{10}) $ 21 * q + 10 * q^3 + 3 * q^5 + 13 * q^7 + 21 * q^9	$ 21 q + 10 q^{3} + 3 q^{5} + 13 q^{7} + 21 q^{9} + 7 q^{11} + 12 q^{13} + 14 q^{15} + q^{17} + 14 q^{19} + 14 q^{21} + 26 q^{23} + 18 q^{25} + 37 q^{27} + 9 q^{29} + 28 q^{31} + 3 q^{33} + 20 q^{35} + 31 q^{37} + 29 q^{39} + 4 q^{41} + 38 q^{43} + 24 q^{45} + 9 q^{47} + 16 q^{49} + 15 q^{51} + 22 q^{53} + 35 q^{55} - q^{57} + 10 q^{59} + 22 q^{61} + 35 q^{63} - 14 q^{65} + 58 q^{67} + 15 q^{69} + 27 q^{71} - 6 q^{73} + 48 q^{75} + 16 q^{77} + 47 q^{79} + 29 q^{81} + 22 q^{83} + 14 q^{85} + 29 q^{87} + q^{89} + 51 q^{91} + 34 q^{93} + 23 q^{95} - 2 q^{97} + 22 q^{99}+O(q^{100}) $ 21 * q + 10 * q^3 + 3 * q^5 + 13 * q^7 + 21 * q^9 + 7 * q^11 + 12 * q^13 + 14 * q^15 + q^17 + 14 * q^19 + 14 * q^21 + 26 * q^23 + 18 * q^25 + 37 * q^27 + 9 * q^29 + 28 * q^31 + 3 * q^33 + 20 * q^35 + 31 * q^37 + 29 * q^39 + 4 * q^41 + 38 * q^43 + 24 * q^45 + 9 * q^47 + 16 * q^49 + 15 * q^51 + 22 * q^53 + 35 * q^55 - q^57 + 10 * q^59 + 22 * q^61 + 35 * q^63 - 14 * q^65 + 58 * q^67 + 15 * q^69 + 27 * q^71 - 6 * q^73 + 48 * q^75 + 16 * q^77 + 47 * q^79 + 29 * q^81 + 22 * q^83 + 14 * q^85 + 29 * q^87 + q^89 + 51 * q^91 + 34 * q^93 + 23 * q^95 - 2 * q^97 + 22 * q^99

Display 100 coefficients 10 coefficients

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
$3$	1.32334	0.764028	0.382014	−	0.924157i	$-0.375231\pi$
$3$	1.32334	0.764028	0.382014	+	0.924157i	$0.375231\pi$
$4$	0	0
$5$	2.33991	1.04644	0.523220	−	0.852197i	$-0.324730\pi$
$5$	2.33991	1.04644	0.523220	+	0.852197i	$0.324730\pi$
$6$	0	0
$7$	2.12211	0.802083	0.401042	−	0.916060i	$-0.368648\pi$
$7$	2.12211	0.802083	0.401042	+	0.916060i	$0.368648\pi$
$8$	0	0
$9$	−1.24878	−0.416261
$10$	0	0
$11$	1.69672	0.511580	0.255790	−	0.966732i	$-0.417664\pi$
$11$	1.69672	0.511580	0.255790	+	0.966732i	$0.417664\pi$
$12$	0	0
$13$	−4.85037	−1.34525	−0.672626	−	0.739983i	$-0.734834\pi$
$13$	−4.85037	−1.34525	−0.672626	+	0.739983i	$0.734834\pi$
$14$	0	0
$15$	3.09649	0.799510
$16$	0	0
$17$	7.73650	1.87638	0.938189	−	0.346124i	$-0.112502\pi$
$17$	7.73650	1.87638	0.938189	+	0.346124i	$0.112502\pi$
$18$	0	0
$19$	−0.742375	−0.170312	−0.0851562	−	0.996368i	$-0.527139\pi$
$19$	−0.742375	−0.170312	−0.0851562	+	0.996368i	$0.527139\pi$
$20$	0	0
$21$	2.80827	0.612814
$22$	0	0
$23$	2.11455	0.440915	0.220458	−	0.975397i	$-0.429245\pi$
$23$	2.11455	0.440915	0.220458	+	0.975397i	$0.429245\pi$
$24$	0	0
$25$	0.475191	0.0950381
$26$	0	0
$27$	−5.62257	−1.08206
$28$	0	0
$29$	4.46715	0.829529	0.414765	−	0.909929i	$-0.363864\pi$
$29$	4.46715	0.829529	0.414765	+	0.909929i	$0.363864\pi$
$30$	0	0
$31$	3.40159	0.610944	0.305472	−	0.952201i	$-0.401186\pi$
$31$	3.40159	0.610944	0.305472	+	0.952201i	$0.401186\pi$
$32$	0	0
$33$	2.24533	0.390861
$34$	0	0
$35$	4.96556	0.839332
$36$	0	0
$37$	0.982569	0.161533	0.0807667	−	0.996733i	$-0.474263\pi$
$37$	0.982569	0.161533	0.0807667	+	0.996733i	$0.474263\pi$
$38$	0	0
$39$	−6.41867	−1.02781
$40$	0	0
$41$	7.22768	1.12877	0.564387	−	0.825510i	$-0.309113\pi$
$41$	7.22768	1.12877	0.564387	+	0.825510i	$0.309113\pi$
$42$	0	0
$43$	4.18992	0.638956	0.319478	−	0.947594i	$-0.396492\pi$
$43$	4.18992	0.638956	0.319478	+	0.947594i	$0.396492\pi$
$44$	0	0
$45$	−2.92205	−0.435593
$46$	0	0
$47$	−4.64253	−0.677183	−0.338592	−	0.940933i	$-0.609951\pi$
$47$	−4.64253	−0.677183	−0.338592	+	0.940933i	$0.609951\pi$
$48$	0	0
$49$	−2.49664	−0.356663
$50$	0	0
$51$	10.2380	1.43360
$52$	0	0
$53$	10.5040	1.44283	0.721414	−	0.692504i	$-0.243492\pi$
$53$	10.5040	1.44283	0.721414	+	0.692504i	$0.243492\pi$
$54$	0	0
$55$	3.97017	0.535338
$56$	0	0
$57$	−0.982411	−0.130123
$58$	0	0
$59$	0.980586	0.127661	0.0638307	−	0.997961i	$-0.479668\pi$
$59$	0.980586	0.127661	0.0638307	+	0.997961i	$0.479668\pi$
$60$	0	0
$61$	2.61825	0.335233	0.167616	−	0.985852i	$-0.446393\pi$
$61$	2.61825	0.335233	0.167616	+	0.985852i	$0.446393\pi$
$62$	0	0
$63$	−2.65006	−0.333876
$64$	0	0
$65$	−11.3495	−1.40773
$66$	0	0
$67$	−9.59050	−1.17167	−0.585833	−	0.810432i	$-0.699232\pi$
$67$	−9.59050	−1.17167	−0.585833	+	0.810432i	$0.699232\pi$
$68$	0	0
$69$	2.79826	0.336871
$70$	0	0
$71$	9.87610	1.17208	0.586038	−	0.810283i	$-0.300687\pi$
$71$	9.87610	1.17208	0.586038	+	0.810283i	$0.300687\pi$
$72$	0	0
$73$	4.75443	0.556464	0.278232	−	0.960514i	$-0.410252\pi$
$73$	4.75443	0.556464	0.278232	+	0.960514i	$0.410252\pi$
$74$	0	0
$75$	0.628836	0.0726118
$76$	0	0
$77$	3.60063	0.410330
$78$	0	0
$79$	−15.6527	−1.76107	−0.880535	−	0.473982i	$-0.842816\pi$
$79$	−15.6527	−1.76107	−0.880535	+	0.473982i	$0.842816\pi$
$80$	0	0
$81$	−3.69418	−0.410465
$82$	0	0
$83$	8.39562	0.921539	0.460770	−	0.887520i	$-0.347573\pi$
$83$	8.39562	0.921539	0.460770	+	0.887520i	$0.347573\pi$
$84$	0	0
$85$	18.1027	1.96352
$86$	0	0
$87$	5.91154	0.633783
$88$	0	0
$89$	−13.6200	−1.44372	−0.721860	−	0.692039i	$-0.756712\pi$
$89$	−13.6200	−1.44372	−0.721860	+	0.692039i	$0.756712\pi$
$90$	0	0
$91$	−10.2930	−1.07900
$92$	0	0
$93$	4.50145	0.466778
$94$	0	0
$95$	−1.73709	−0.178222
$96$	0	0
$97$	−8.72503	−0.885893	−0.442946	−	0.896548i	$-0.646067\pi$
$97$	−8.72503	−0.885893	−0.442946	+	0.896548i	$0.646067\pi$
$98$	0	0
$99$	−2.11884	−0.212951
$100$	0	0
$101$	−19.0477	−1.89531	−0.947657	−	0.319292i	$-0.896555\pi$
$101$	−19.0477	−1.89531	−0.947657	+	0.319292i	$0.896555\pi$
$102$	0	0
$103$	−10.1478	−0.999894	−0.499947	−	0.866056i	$-0.666647\pi$
$103$	−10.1478	−0.999894	−0.499947	+	0.866056i	$0.666647\pi$
$104$	0	0
$105$	6.57110	0.641273
$106$	0	0
$107$	−4.74329	−0.458551	−0.229276	−	0.973362i	$-0.573636\pi$
$107$	−4.74329	−0.458551	−0.229276	+	0.973362i	$0.573636\pi$
$108$	0	0
$109$	4.41670	0.423043	0.211522	−	0.977373i	$-0.432158\pi$
$109$	4.41670	0.423043	0.211522	+	0.977373i	$0.432158\pi$
$110$	0	0
$111$	1.30027	0.123416
$112$	0	0
$113$	−1.21371	−0.114176	−0.0570881	−	0.998369i	$-0.518182\pi$
$113$	−1.21371	−0.114176	−0.0570881	+	0.998369i	$0.518182\pi$
$114$	0	0
$115$	4.94787	0.461391
$116$	0	0
$117$	6.05707	0.559976
$118$	0	0
$119$	16.4177	1.50501
$120$	0	0
$121$	−8.12114	−0.738286
$122$	0	0
$123$	9.56464	0.862414
$124$	0	0
$125$	−10.5877	−0.946989
$126$	0	0
$127$	15.8203	1.40383	0.701913	−	0.712262i	$-0.252329\pi$
$127$	15.8203	1.40383	0.701913	+	0.712262i	$0.252329\pi$
$128$	0	0
$129$	5.54466	0.488180
$130$	0	0
$131$	18.5655	1.62207	0.811037	−	0.584995i	$-0.198903\pi$
$131$	18.5655	1.62207	0.811037	+	0.584995i	$0.198903\pi$
$132$	0	0
$133$	−1.57540	−0.136605
$134$	0	0
$135$	−13.1563	−1.13231
$136$	0	0
$137$	−17.4193	−1.48823	−0.744114	−	0.668053i	$-0.767128\pi$
$137$	−17.4193	−1.48823	−0.744114	+	0.668053i	$0.767128\pi$
$138$	0	0
$139$	−2.49522	−0.211642	−0.105821	−	0.994385i	$-0.533747\pi$
$139$	−2.49522	−0.211642	−0.105821	+	0.994385i	$0.533747\pi$
$140$	0	0
$141$	−6.14363	−0.517387
$142$	0	0
$143$	−8.22972	−0.688204
$144$	0	0
$145$	10.4527	0.868053
$146$	0	0
$147$	−3.30389	−0.272500
$148$	0	0
$149$	−5.96388	−0.488580	−0.244290	−	0.969702i	$-0.578555\pi$
$149$	−5.96388	−0.488580	−0.244290	+	0.969702i	$0.578555\pi$
$150$	0	0
$151$	−0.336567	−0.0273894	−0.0136947	−	0.999906i	$-0.504359\pi$
$151$	−0.336567	−0.0273894	−0.0136947	+	0.999906i	$0.504359\pi$
$152$	0	0
$153$	−9.66122	−0.781064
$154$	0	0
$155$	7.95943	0.639317
$156$	0	0
$157$	−15.0673	−1.20250	−0.601249	−	0.799062i	$-0.705330\pi$
$157$	−15.0673	−1.20250	−0.601249	+	0.799062i	$0.705330\pi$
$158$	0	0
$159$	13.9002	1.10236
$160$	0	0
$161$	4.48732	0.353651
$162$	0	0
$163$	21.9972	1.72295	0.861475	−	0.507799i	$-0.169541\pi$
$163$	21.9972	1.72295	0.861475	+	0.507799i	$0.169541\pi$
$164$	0	0
$165$	5.25387	0.409013
$166$	0	0
$167$	9.88866	0.765207	0.382604	−	0.923913i	$-0.375027\pi$
$167$	9.88866	0.765207	0.382604	+	0.923913i	$0.375027\pi$
$168$	0	0
$169$	10.5261	0.809703
$170$	0	0
$171$	0.927066	0.0708945
$172$	0	0
$173$	21.5847	1.64105	0.820526	−	0.571609i	$-0.193681\pi$
$173$	21.5847	1.64105	0.820526	+	0.571609i	$0.193681\pi$
$174$	0	0
$175$	1.00841	0.0762285
$176$	0	0
$177$	1.29764	0.0975369
$178$	0	0
$179$	11.6643	0.871832	0.435916	−	0.899987i	$-0.356425\pi$
$179$	11.6643	0.871832	0.435916	+	0.899987i	$0.356425\pi$
$180$	0	0
$181$	16.6896	1.24053	0.620266	−	0.784392i	$-0.287025\pi$
$181$	16.6896	1.24053	0.620266	+	0.784392i	$0.287025\pi$
$182$	0	0
$183$	3.46482	0.256127
$184$	0	0
$185$	2.29913	0.169035
$186$	0	0
$187$	13.1267	0.959918
$188$	0	0
$189$	−11.9317	−0.867905
$190$	0	0
$191$	0.214614	0.0155289	0.00776447	−	0.999970i	$-0.497528\pi$
$191$	0.214614	0.0155289	0.00776447	+	0.999970i	$0.497528\pi$
$192$	0	0
$193$	5.25840	0.378508	0.189254	−	0.981928i	$-0.439393\pi$
$193$	5.25840	0.378508	0.189254	+	0.981928i	$0.439393\pi$
$194$	0	0
$195$	−15.0191	−1.07554
$196$	0	0
$197$	−25.8901	−1.84460	−0.922298	−	0.386479i	$-0.873691\pi$
$197$	−25.8901	−1.84460	−0.922298	+	0.386479i	$0.873691\pi$
$198$	0	0
$199$	−8.89667	−0.630669	−0.315334	−	0.948981i	$-0.602117\pi$
$199$	−8.89667	−0.630669	−0.315334	+	0.948981i	$0.602117\pi$
$200$	0	0
$201$	−12.6914	−0.895185
$202$	0	0
$203$	9.47980	0.665351
$204$	0	0
$205$	16.9121	1.18119
$206$	0	0
$207$	−2.64062	−0.183536
$208$	0	0
$209$	−1.25960	−0.0871285
$210$	0	0
$211$	13.0811	0.900539	0.450269	−	0.892893i	$-0.351328\pi$
$211$	13.0811	0.900539	0.450269	+	0.892893i	$0.351328\pi$
$212$	0	0
$213$	13.0694	0.895499
$214$	0	0
$215$	9.80404	0.668630
$216$	0	0
$217$	7.21856	0.490028
$218$	0	0
$219$	6.29170	0.425154
$220$	0	0
$221$	−37.5249	−2.52420
$222$	0	0
$223$	−24.7787	−1.65930	−0.829651	−	0.558282i	$-0.811461\pi$
$223$	−24.7787	−1.65930	−0.829651	+	0.558282i	$0.811461\pi$
$224$	0	0
$225$	−0.593411	−0.0395607
$226$	0	0
$227$	−26.4873	−1.75803	−0.879013	−	0.476797i	$-0.841798\pi$
$227$	−26.4873	−1.75803	−0.879013	+	0.476797i	$0.841798\pi$
$228$	0	0
$229$	1.97029	0.130201	0.0651004	−	0.997879i	$-0.479263\pi$
$229$	1.97029	0.130201	0.0651004	+	0.997879i	$0.479263\pi$
$230$	0	0
$231$	4.76484	0.313503
$232$	0	0
$233$	−4.48678	−0.293938	−0.146969	−	0.989141i	$-0.546952\pi$
$233$	−4.48678	−0.293938	−0.146969	+	0.989141i	$0.546952\pi$
$234$	0	0
$235$	−10.8631	−0.708632
$236$	0	0
$237$	−20.7138	−1.34551
$238$	0	0
$239$	5.31245	0.343634	0.171817	−	0.985129i	$-0.445036\pi$
$239$	5.31245	0.343634	0.171817	+	0.985129i	$0.445036\pi$
$240$	0	0
$241$	−10.1072	−0.651061	−0.325531	−	0.945532i	$-0.605543\pi$
$241$	−10.1072	−0.651061	−0.325531	+	0.945532i	$0.605543\pi$
$242$	0	0
$243$	11.9791	0.768457
$244$	0	0
$245$	−5.84192	−0.373226
$246$	0	0
$247$	3.60080	0.229113
$248$	0	0
$249$	11.1102	0.704082
$250$	0	0
$251$	−28.1164	−1.77469	−0.887345	−	0.461106i	$-0.847453\pi$
$251$	−28.1164	−1.77469	−0.887345	+	0.461106i	$0.847453\pi$
$252$	0	0
$253$	3.58781	0.225563
$254$	0	0
$255$	23.9560	1.50018
$256$	0	0
$257$	−7.90727	−0.493242	−0.246621	−	0.969112i	$-0.579320\pi$
$257$	−7.90727	−0.493242	−0.246621	+	0.969112i	$0.579320\pi$
$258$	0	0
$259$	2.08512	0.129563
$260$	0	0
$261$	−5.57851	−0.345301
$262$	0	0
$263$	8.62142	0.531620	0.265810	−	0.964025i	$-0.414361\pi$
$263$	8.62142	0.531620	0.265810	+	0.964025i	$0.414361\pi$
$264$	0	0
$265$	24.5783	1.50983
$266$	0	0
$267$	−18.0239	−1.10304
$268$	0	0
$269$	3.62340	0.220923	0.110461	−	0.993880i	$-0.464767\pi$
$269$	3.62340	0.220923	0.110461	+	0.993880i	$0.464767\pi$
$270$	0	0
$271$	2.71700	0.165046	0.0825229	−	0.996589i	$-0.473702\pi$
$271$	2.71700	0.165046	0.0825229	+	0.996589i	$0.473702\pi$
$272$	0	0
$273$	−13.6211	−0.824389
$274$	0	0
$275$	0.806265	0.0486196
$276$	0	0
$277$	−28.2909	−1.69984	−0.849919	−	0.526914i	$-0.823349\pi$
$277$	−28.2909	−1.69984	−0.849919	+	0.526914i	$0.823349\pi$
$278$	0	0
$279$	−4.24785	−0.254312
$280$	0	0
$281$	1.31533	0.0784659	0.0392329	−	0.999230i	$-0.487509\pi$
$281$	1.31533	0.0784659	0.0392329	+	0.999230i	$0.487509\pi$
$282$	0	0
$283$	3.89661	0.231629	0.115815	−	0.993271i	$-0.463052\pi$
$283$	3.89661	0.231629	0.115815	+	0.993271i	$0.463052\pi$
$284$	0	0
$285$	−2.29875	−0.136166
$286$	0	0
$287$	15.3379	0.905370
$288$	0	0
$289$	42.8535	2.52079
$290$	0	0
$291$	−11.5461	−0.676847
$292$	0	0
$293$	−15.9626	−0.932545	−0.466273	−	0.884641i	$-0.654403\pi$
$293$	−15.9626	−0.932545	−0.466273	+	0.884641i	$0.654403\pi$
$294$	0	0
$295$	2.29449	0.133590
$296$	0	0
$297$	−9.53992	−0.553562
$298$	0	0
$299$	−10.2564	−0.593142
$300$	0	0
$301$	8.89147	0.512496
$302$	0	0
$303$	−25.2064	−1.44807
$304$	0	0
$305$	6.12648	0.350801
$306$	0	0
$307$	9.95786	0.568325	0.284163	−	0.958776i	$-0.408284\pi$
$307$	9.95786	0.568325	0.284163	+	0.958776i	$0.408284\pi$
$308$	0	0
$309$	−13.4290	−0.763947
$310$	0	0
$311$	31.6261	1.79335	0.896676	−	0.442687i	$-0.145975\pi$
$311$	31.6261	1.79335	0.896676	+	0.442687i	$0.145975\pi$
$312$	0	0
$313$	−0.518837	−0.0293264	−0.0146632	−	0.999892i	$-0.504668\pi$
$313$	−0.518837	−0.0293264	−0.0146632	+	0.999892i	$0.504668\pi$
$314$	0	0
$315$	−6.20091	−0.349382
$316$	0	0
$317$	27.1749	1.52630	0.763148	−	0.646224i	$-0.223653\pi$
$317$	27.1749	1.52630	0.763148	+	0.646224i	$0.223653\pi$
$318$	0	0
$319$	7.57950	0.424371
$320$	0	0
$321$	−6.27696	−0.350346
$322$	0	0
$323$	−5.74339	−0.319571
$324$	0	0
$325$	−2.30485	−0.127850
$326$	0	0
$327$	5.84477	0.323217
$328$	0	0
$329$	−9.85198	−0.543157
$330$	0	0
$331$	−13.1334	−0.721878	−0.360939	−	0.932590i	$-0.617544\pi$
$331$	−13.1334	−0.721878	−0.360939	+	0.932590i	$0.617544\pi$
$332$	0	0
$333$	−1.22702	−0.0672401
$334$	0	0
$335$	−22.4409	−1.22608
$336$	0	0
$337$	−26.6321	−1.45074	−0.725372	−	0.688357i	$-0.758332\pi$
$337$	−26.6321	−1.45074	−0.725372	+	0.688357i	$0.758332\pi$
$338$	0	0
$339$	−1.60614	−0.0872337
$340$	0	0
$341$	5.77155	0.312547
$342$	0	0
$343$	−20.1529	−1.08816
$344$	0	0
$345$	6.54769	0.352516
$346$	0	0
$347$	−25.3518	−1.36096	−0.680478	−	0.732769i	$-0.738228\pi$
$347$	−25.3518	−1.36096	−0.680478	+	0.732769i	$0.738228\pi$
$348$	0	0
$349$	6.18147	0.330886	0.165443	−	0.986219i	$-0.447095\pi$
$349$	6.18147	0.330886	0.165443	+	0.986219i	$0.447095\pi$
$350$	0	0
$351$	27.2715	1.45565
$352$	0	0
$353$	−5.23561	−0.278663	−0.139332	−	0.990246i	$-0.544495\pi$
$353$	−5.23561	−0.278663	−0.139332	+	0.990246i	$0.544495\pi$
$354$	0	0
$355$	23.1092	1.22651
$356$	0	0
$357$	21.7262	1.14987
$358$	0	0
$359$	−22.8101	−1.20387	−0.601936	−	0.798544i	$-0.705604\pi$
$359$	−22.8101	−1.20387	−0.601936	+	0.798544i	$0.705604\pi$
$360$	0	0
$361$	−18.4489	−0.970994
$362$	0	0
$363$	−10.7470	−0.564071
$364$	0	0
$365$	11.1250	0.582307
$366$	0	0
$367$	13.4411	0.701619	0.350810	−	0.936447i	$-0.385906\pi$
$367$	13.4411	0.701619	0.350810	+	0.936447i	$0.385906\pi$
$368$	0	0
$369$	−9.02581	−0.469865
$370$	0	0
$371$	22.2906	1.15727
$372$	0	0
$373$	5.18995	0.268725	0.134363	−	0.990932i	$-0.457101\pi$
$373$	5.18995	0.268725	0.134363	+	0.990932i	$0.457101\pi$
$374$	0	0
$375$	−14.0110	−0.723526
$376$	0	0
$377$	−21.6674	−1.11593
$378$	0	0
$379$	3.50797	0.180192	0.0900962	−	0.995933i	$-0.471283\pi$
$379$	3.50797	0.180192	0.0900962	+	0.995933i	$0.471283\pi$
$380$	0	0
$381$	20.9356	1.07256
$382$	0	0
$383$	7.47537	0.381974	0.190987	−	0.981593i	$-0.438831\pi$
$383$	7.47537	0.381974	0.190987	+	0.981593i	$0.438831\pi$
$384$	0	0
$385$	8.42516	0.429386
$386$	0	0
$387$	−5.23230	−0.265973
$388$	0	0
$389$	−17.1858	−0.871356	−0.435678	−	0.900103i	$-0.643491\pi$
$389$	−17.1858	−0.871356	−0.435678	+	0.900103i	$0.643491\pi$
$390$	0	0
$391$	16.3593	0.827323
$392$	0	0
$393$	24.5684	1.23931
$394$	0	0
$395$	−36.6260	−1.84285
$396$	0	0
$397$	−22.1202	−1.11018	−0.555090	−	0.831791i	$-0.687316\pi$
$397$	−22.1202	−1.11018	−0.555090	+	0.831791i	$0.687316\pi$
$398$	0	0
$399$	−2.08479	−0.104370
$400$	0	0
$401$	23.4967	1.17337	0.586684	−	0.809816i	$-0.300433\pi$
$401$	23.4967	1.17337	0.586684	+	0.809816i	$0.300433\pi$
$402$	0	0
$403$	−16.4990	−0.821873
$404$	0	0
$405$	−8.64407	−0.429527
$406$	0	0
$407$	1.66714	0.0826372
$408$	0	0
$409$	−28.9637	−1.43216	−0.716081	−	0.698017i	$-0.754066\pi$
$409$	−28.9637	−1.43216	−0.716081	+	0.698017i	$0.754066\pi$
$410$	0	0
$411$	−23.0515	−1.13705
$412$	0	0
$413$	2.08091	0.102395
$414$	0	0
$415$	19.6450	0.964336
$416$	0	0
$417$	−3.30202	−0.161700
$418$	0	0
$419$	2.55648	0.124892	0.0624461	−	0.998048i	$-0.480110\pi$
$419$	2.55648	0.124892	0.0624461	+	0.998048i	$0.480110\pi$
$420$	0	0
$421$	19.1067	0.931203	0.465602	−	0.884994i	$-0.345838\pi$
$421$	19.1067	0.931203	0.465602	+	0.884994i	$0.345838\pi$
$422$	0	0
$423$	5.79752	0.281885
$424$	0	0
$425$	3.67631	0.178327
$426$	0	0
$427$	5.55623	0.268885
$428$	0	0
$429$	−10.8907	−0.525807
$430$	0	0
$431$	6.88486	0.331632	0.165816	−	0.986157i	$-0.446974\pi$
$431$	6.88486	0.331632	0.165816	+	0.986157i	$0.446974\pi$
$432$	0	0
$433$	−5.65446	−0.271736	−0.135868	−	0.990727i	$-0.543382\pi$
$433$	−5.65446	−0.271736	−0.135868	+	0.990727i	$0.543382\pi$
$434$	0	0
$435$	13.8325	0.663217
$436$	0	0
$437$	−1.56979	−0.0750933
$438$	0	0
$439$	−28.4032	−1.35561	−0.677806	−	0.735241i	$-0.737069\pi$
$439$	−28.4032	−1.35561	−0.677806	+	0.735241i	$0.737069\pi$
$440$	0	0
$441$	3.11776	0.148465
$442$	0	0
$443$	−31.8962	−1.51543	−0.757716	−	0.652584i	$-0.773685\pi$
$443$	−31.8962	−1.51543	−0.757716	+	0.652584i	$0.773685\pi$
$444$	0	0
$445$	−31.8697	−1.51077
$446$	0	0
$447$	−7.89221	−0.373289
$448$	0	0
$449$	1.97690	0.0932958	0.0466479	−	0.998911i	$-0.485146\pi$
$449$	1.97690	0.0932958	0.0466479	+	0.998911i	$0.485146\pi$
$450$	0	0
$451$	12.2633	0.577458
$452$	0	0
$453$	−0.445390	−0.0209263
$454$	0	0
$455$	−24.0848	−1.12911
$456$	0	0
$457$	−8.24045	−0.385472	−0.192736	−	0.981251i	$-0.561736\pi$
$457$	−8.24045	−0.385472	−0.192736	+	0.981251i	$0.561736\pi$
$458$	0	0
$459$	−43.4990	−2.03036
$460$	0	0
$461$	−23.2432	−1.08255	−0.541273	−	0.840847i	$-0.682057\pi$
$461$	−23.2432	−1.08255	−0.541273	+	0.840847i	$0.682057\pi$
$462$	0	0
$463$	24.4110	1.13447	0.567237	−	0.823555i	$-0.308012\pi$
$463$	24.4110	1.13447	0.567237	+	0.823555i	$0.308012\pi$
$464$	0	0
$465$	10.5330	0.488456
$466$	0	0
$467$	24.9478	1.15445	0.577223	−	0.816587i	$-0.304137\pi$
$467$	24.9478	1.15445	0.577223	+	0.816587i	$0.304137\pi$
$468$	0	0
$469$	−20.3521	−0.939773
$470$	0	0
$471$	−19.9390	−0.918742
$472$	0	0
$473$	7.10911	0.326877
$474$	0	0
$475$	−0.352770	−0.0161862
$476$	0	0
$477$	−13.1172	−0.600594
$478$	0	0
$479$	−13.4445	−0.614293	−0.307146	−	0.951662i	$-0.599374\pi$
$479$	−13.4445	−0.614293	−0.307146	+	0.951662i	$0.599374\pi$
$480$	0	0
$481$	−4.76583	−0.217303
$482$	0	0
$483$	5.93823	0.270199
$484$	0	0
$485$	−20.4158	−0.927034
$486$	0	0
$487$	31.7361	1.43810	0.719049	−	0.694959i	$-0.244578\pi$
$487$	31.7361	1.43810	0.719049	+	0.694959i	$0.244578\pi$
$488$	0	0
$489$	29.1096	1.31638
$490$	0	0
$491$	8.07750	0.364532	0.182266	−	0.983249i	$-0.441657\pi$
$491$	8.07750	0.364532	0.182266	+	0.983249i	$0.441657\pi$
$492$	0	0
$493$	34.5601	1.55651
$494$	0	0
$495$	−4.95789	−0.222841
$496$	0	0
$497$	20.9582	0.940103
$498$	0	0
$499$	5.23448	0.234328	0.117164	−	0.993113i	$-0.462620\pi$
$499$	5.23448	0.234328	0.117164	+	0.993113i	$0.462620\pi$
$500$	0	0
$501$	13.0860	0.584640
$502$	0	0
$503$	−1.00000	−0.0445878
$503$	−1.00000	−0.0445878
$504$	0	0
$505$	−44.5699	−1.98333
$506$	0	0
$507$	13.9296	0.618635
$508$	0	0
$509$	22.2258	0.985143	0.492572	−	0.870272i	$-0.336057\pi$
$509$	22.2258	0.985143	0.492572	+	0.870272i	$0.336057\pi$
$510$	0	0
$511$	10.0894	0.446330
$512$	0	0
$513$	4.17405	0.184289
$514$	0	0
$515$	−23.7450	−1.04633
$516$	0	0
$517$	−7.87708	−0.346434
$518$	0	0
$519$	28.5638	1.25381
$520$	0	0
$521$	−29.9794	−1.31342	−0.656711	−	0.754142i	$-0.728053\pi$
$521$	−29.9794	−1.31342	−0.656711	+	0.754142i	$0.728053\pi$
$522$	0	0
$523$	−14.2912	−0.624911	−0.312455	−	0.949932i	$-0.601151\pi$
$523$	−14.2912	−0.624911	−0.312455	+	0.949932i	$0.601151\pi$
$524$	0	0
$525$	1.33446	0.0582407
$526$	0	0
$527$	26.3164	1.14636
$528$	0	0
$529$	−18.5287	−0.805594
$530$	0	0
$531$	−1.22454	−0.0531405
$532$	0	0
$533$	−35.0569	−1.51848
$534$	0	0
$535$	−11.0989	−0.479847
$536$	0	0
$537$	15.4358	0.666104
$538$	0	0
$539$	−4.23610	−0.182462
$540$	0	0
$541$	16.6005	0.713711	0.356856	−	0.934160i	$-0.383849\pi$
$541$	16.6005	0.713711	0.356856	+	0.934160i	$0.383849\pi$
$542$	0	0
$543$	22.0860	0.947801
$544$	0	0
$545$	10.3347	0.442689
$546$	0	0
$547$	−2.73063	−0.116754	−0.0583768	−	0.998295i	$-0.518592\pi$
$547$	−2.73063	−0.116754	−0.0583768	+	0.998295i	$0.518592\pi$
$548$	0	0
$549$	−3.26963	−0.139544
$550$	0	0
$551$	−3.31630	−0.141279
$552$	0	0
$553$	−33.2168	−1.41252
$554$	0	0
$555$	3.04251	0.129147
$556$	0	0
$557$	−35.8228	−1.51786	−0.758931	−	0.651172i	$-0.774278\pi$
$557$	−35.8228	−1.51786	−0.758931	+	0.651172i	$0.774278\pi$
$558$	0	0
$559$	−20.3227	−0.859557
$560$	0	0
$561$	17.3710	0.733404
$562$	0	0
$563$	13.0475	0.549888	0.274944	−	0.961460i	$-0.411341\pi$
$563$	13.0475	0.549888	0.274944	+	0.961460i	$0.411341\pi$
$564$	0	0
$565$	−2.83997	−0.119479
$566$	0	0
$567$	−7.83948	−0.329227
$568$	0	0
$569$	−43.6881	−1.83150	−0.915749	−	0.401750i	$-0.868402\pi$
$569$	−43.6881	−1.83150	−0.915749	+	0.401750i	$0.868402\pi$
$570$	0	0
$571$	−18.6778	−0.781641	−0.390821	−	0.920467i	$-0.627809\pi$
$571$	−18.6778	−0.781641	−0.390821	+	0.920467i	$0.627809\pi$
$572$	0	0
$573$	0.284006	0.0118645
$574$	0	0
$575$	1.00482	0.0419037
$576$	0	0
$577$	−15.8306	−0.659037	−0.329518	−	0.944149i	$-0.606886\pi$
$577$	−15.8306	−0.659037	−0.329518	+	0.944149i	$0.606886\pi$
$578$	0	0
$579$	6.95862	0.289191
$580$	0	0
$581$	17.8165	0.739151
$582$	0	0
$583$	17.8223	0.738122
$584$	0	0
$585$	14.1730	0.585982
$586$	0	0
$587$	29.4131	1.21401	0.607003	−	0.794699i	$-0.292371\pi$
$587$	29.4131	1.21401	0.607003	+	0.794699i	$0.292371\pi$
$588$	0	0
$589$	−2.52526	−0.104051
$590$	0	0
$591$	−34.2613	−1.40932
$592$	0	0
$593$	−22.0741	−0.906476	−0.453238	−	0.891390i	$-0.649731\pi$
$593$	−22.0741	−0.906476	−0.453238	+	0.891390i	$0.649731\pi$
$594$	0	0
$595$	38.4161	1.57490
$596$	0	0
$597$	−11.7733	−0.481848
$598$	0	0
$599$	−23.0487	−0.941745	−0.470873	−	0.882201i	$-0.656061\pi$
$599$	−23.0487	−0.941745	−0.470873	+	0.882201i	$0.656061\pi$
$600$	0	0
$601$	2.63605	0.107527	0.0537634	−	0.998554i	$-0.482878\pi$
$601$	2.63605	0.107527	0.0537634	+	0.998554i	$0.482878\pi$
$602$	0	0
$603$	11.9765	0.487719
$604$	0	0
$605$	−19.0028	−0.772572
$606$	0	0
$607$	37.6682	1.52890	0.764452	−	0.644680i	$-0.223010\pi$
$607$	37.6682	1.52890	0.764452	+	0.644680i	$0.223010\pi$
$608$	0	0
$609$	12.5449	0.508347
$610$	0	0
$611$	22.5180	0.910982
$612$	0	0
$613$	−20.4202	−0.824766	−0.412383	−	0.911011i	$-0.635303\pi$
$613$	−20.4202	−0.824766	−0.412383	+	0.911011i	$0.635303\pi$
$614$	0	0
$615$	22.3804	0.902466
$616$	0	0
$617$	30.2485	1.21776	0.608879	−	0.793263i	$-0.291620\pi$
$617$	30.2485	1.21776	0.608879	+	0.793263i	$0.291620\pi$
$618$	0	0
$619$	−4.50342	−0.181008	−0.0905038	−	0.995896i	$-0.528848\pi$
$619$	−4.50342	−0.181008	−0.0905038	+	0.995896i	$0.528848\pi$
$620$	0	0
$621$	−11.8892	−0.477098
$622$	0	0
$623$	−28.9032	−1.15798
$624$	0	0
$625$	−27.1501	−1.08601
$626$	0	0
$627$	−1.66688	−0.0665686
$628$	0	0
$629$	7.60165	0.303098
$630$	0	0
$631$	−25.5741	−1.01809	−0.509045	−	0.860740i	$-0.670001\pi$
$631$	−25.5741	−1.01809	−0.509045	+	0.860740i	$0.670001\pi$
$632$	0	0
$633$	17.3106	0.688037
$634$	0	0
$635$	37.0182	1.46902
$636$	0	0
$637$	12.1096	0.479801
$638$	0	0
$639$	−12.3331	−0.487890
$640$	0	0
$641$	14.5945	0.576449	0.288225	−	0.957563i	$-0.406935\pi$
$641$	14.5945	0.576449	0.288225	+	0.957563i	$0.406935\pi$
$642$	0	0
$643$	14.3349	0.565314	0.282657	−	0.959221i	$-0.408784\pi$
$643$	14.3349	0.565314	0.282657	+	0.959221i	$0.408784\pi$
$644$	0	0
$645$	12.9740	0.510852
$646$	0	0
$647$	29.3688	1.15461	0.577304	−	0.816529i	$-0.304105\pi$
$647$	29.3688	1.15461	0.577304	+	0.816529i	$0.304105\pi$
$648$	0	0
$649$	1.66378	0.0653091
$650$	0	0
$651$	9.55257	0.374395
$652$	0	0
$653$	−2.96264	−0.115937	−0.0579685	−	0.998318i	$-0.518462\pi$
$653$	−2.96264	−0.115937	−0.0579685	+	0.998318i	$0.518462\pi$
$654$	0	0
$655$	43.4416	1.69740
$656$	0	0
$657$	−5.93726	−0.231635
$658$	0	0
$659$	42.6914	1.66302	0.831510	−	0.555510i	$-0.187477\pi$
$659$	42.6914	1.66302	0.831510	+	0.555510i	$0.187477\pi$
$660$	0	0
$661$	49.0072	1.90616	0.953079	−	0.302721i	$-0.0978950\pi$
$661$	49.0072	1.90616	0.953079	+	0.302721i	$0.0978950\pi$
$662$	0	0
$663$	−49.6581	−1.92856
$664$	0	0
$665$	−3.68630	−0.142949
$666$	0	0
$667$	9.44604	0.365752
$668$	0	0
$669$	−32.7905	−1.26775
$670$	0	0
$671$	4.44244	0.171498
$672$	0	0
$673$	27.4475	1.05802	0.529012	−	0.848614i	$-0.322563\pi$
$673$	27.4475	1.05802	0.529012	+	0.848614i	$0.322563\pi$
$674$	0	0
$675$	−2.67179	−0.102837
$676$	0	0
$677$	−25.9569	−0.997603	−0.498802	−	0.866716i	$-0.666226\pi$
$677$	−25.9569	−0.997603	−0.498802	+	0.866716i	$0.666226\pi$
$678$	0	0
$679$	−18.5155	−0.710560
$680$	0	0
$681$	−35.0516	−1.34318
$682$	0	0
$683$	−37.5603	−1.43721	−0.718603	−	0.695420i	$-0.755218\pi$
$683$	−37.5603	−1.43721	−0.718603	+	0.695420i	$0.755218\pi$
$684$	0	0
$685$	−40.7595	−1.55734
$686$	0	0
$687$	2.60736	0.0994770
$688$	0	0
$689$	−50.9481	−1.94097
$690$	0	0
$691$	−19.7656	−0.751918	−0.375959	−	0.926636i	$-0.622687\pi$
$691$	−19.7656	−0.751918	−0.375959	+	0.926636i	$0.622687\pi$
$692$	0	0
$693$	−4.49641	−0.170804
$694$	0	0
$695$	−5.83861	−0.221471
$696$	0	0
$697$	55.9170	2.11801
$698$	0	0
$699$	−5.93751	−0.224577
$700$	0	0
$701$	12.4520	0.470306	0.235153	−	0.971958i	$-0.424441\pi$
$701$	12.4520	0.470306	0.235153	+	0.971958i	$0.424441\pi$
$702$	0	0
$703$	−0.729434	−0.0275111
$704$	0	0
$705$	−14.3756	−0.541415
$706$	0	0
$707$	−40.4213	−1.52020
$708$	0	0
$709$	32.7697	1.23069	0.615346	−	0.788257i	$-0.289016\pi$
$709$	32.7697	1.23069	0.615346	+	0.788257i	$0.289016\pi$
$710$	0	0
$711$	19.5469	0.733065
$712$	0	0
$713$	7.19285	0.269374
$714$	0	0
$715$	−19.2568	−0.720165
$716$	0	0
$717$	7.03015	0.262546
$718$	0	0
$719$	42.2138	1.57431	0.787154	−	0.616757i	$-0.211554\pi$
$719$	42.2138	1.57431	0.787154	+	0.616757i	$0.211554\pi$
$720$	0	0
$721$	−21.5348	−0.801998
$722$	0	0
$723$	−13.3752	−0.497429
$724$	0	0
$725$	2.12275	0.0788369
$726$	0	0
$727$	−27.4396	−1.01768	−0.508839	−	0.860862i	$-0.669925\pi$
$727$	−27.4396	−1.01768	−0.508839	+	0.860862i	$0.669925\pi$
$728$	0	0
$729$	26.9349	0.997587
$730$	0	0
$731$	32.4153	1.19892
$732$	0	0
$733$	−31.9552	−1.18029	−0.590146	−	0.807297i	$-0.700930\pi$
$733$	−31.9552	−1.18029	−0.590146	+	0.807297i	$0.700930\pi$
$734$	0	0
$735$	−7.73081	−0.285155
$736$	0	0
$737$	−16.2724	−0.599401
$738$	0	0
$739$	−9.85286	−0.362443	−0.181222	−	0.983442i	$-0.558005\pi$
$739$	−9.85286	−0.362443	−0.181222	+	0.983442i	$0.558005\pi$
$740$	0	0
$741$	4.76506	0.175049
$742$	0	0
$743$	30.3227	1.11243	0.556217	−	0.831037i	$-0.312252\pi$
$743$	30.3227	1.11243	0.556217	+	0.831037i	$0.312252\pi$
$744$	0	0
$745$	−13.9550	−0.511270
$746$	0	0
$747$	−10.4843	−0.383601
$748$	0	0
$749$	−10.0658	−0.367796
$750$	0	0
$751$	−9.76022	−0.356155	−0.178078	−	0.984016i	$-0.556988\pi$
$751$	−9.76022	−0.356155	−0.178078	+	0.984016i	$0.556988\pi$
$752$	0	0
$753$	−37.2074	−1.35591
$754$	0	0
$755$	−0.787536	−0.0286614
$756$	0	0
$757$	−19.1999	−0.697833	−0.348916	−	0.937154i	$-0.613450\pi$
$757$	−19.1999	−0.697833	−0.348916	+	0.937154i	$0.613450\pi$
$758$	0	0
$759$	4.74787	0.172337
$760$	0	0
$761$	−30.4899	−1.10526	−0.552629	−	0.833427i	$-0.686375\pi$
$761$	−30.4899	−1.10526	−0.552629	+	0.833427i	$0.686375\pi$
$762$	0	0
$763$	9.37273	0.339316
$764$	0	0
$765$	−22.6064	−0.817337
$766$	0	0
$767$	−4.75621	−0.171737
$768$	0	0
$769$	−27.7156	−0.999451	−0.499725	−	0.866184i	$-0.666566\pi$
$769$	−27.7156	−0.999451	−0.499725	+	0.866184i	$0.666566\pi$
$770$	0	0
$771$	−10.4640	−0.376850
$772$	0	0
$773$	1.37663	0.0495139	0.0247569	−	0.999693i	$-0.492119\pi$
$773$	1.37663	0.0495139	0.0247569	+	0.999693i	$0.492119\pi$
$774$	0	0
$775$	1.61640	0.0580630
$776$	0	0
$777$	2.75931	0.0989899
$778$	0	0
$779$	−5.36565	−0.192244
$780$	0	0
$781$	16.7570	0.599611
$782$	0	0
$783$	−25.1169	−0.897603
$784$	0	0
$785$	−35.2561	−1.25834
$786$	0	0
$787$	33.4878	1.19371	0.596855	−	0.802349i	$-0.296417\pi$
$787$	33.4878	1.19371	0.596855	+	0.802349i	$0.296417\pi$
$788$	0	0
$789$	11.4090	0.406172
$790$	0	0
$791$	−2.57563	−0.0915787
$792$	0	0
$793$	−12.6995	−0.450973
$794$	0	0
$795$	32.5254	1.15356
$796$	0	0
$797$	10.2714	0.363833	0.181917	−	0.983314i	$-0.441770\pi$
$797$	10.2714	0.363833	0.181917	+	0.983314i	$0.441770\pi$
$798$	0	0
$799$	−35.9170	−1.27065
$800$	0	0
$801$	17.0085	0.600965
$802$	0	0
$803$	8.06693	0.284676
$804$	0	0
$805$	10.4999	0.370074
$806$	0	0
$807$	4.79498	0.168791
$808$	0	0
$809$	8.41096	0.295713	0.147857	−	0.989009i	$-0.452763\pi$
$809$	8.41096	0.295713	0.147857	+	0.989009i	$0.452763\pi$
$810$	0	0
$811$	48.5840	1.70602	0.853008	−	0.521897i	$-0.174776\pi$
$811$	48.5840	1.70602	0.853008	+	0.521897i	$0.174776\pi$
$812$	0	0
$813$	3.59550	0.126100
$814$	0	0
$815$	51.4714	1.80297
$816$	0	0
$817$	−3.11049	−0.108822
$818$	0	0
$819$	12.8538	0.449148
$820$	0	0
$821$	34.0409	1.18804	0.594018	−	0.804451i	$-0.297541\pi$
$821$	34.0409	1.18804	0.594018	+	0.804451i	$0.297541\pi$
$822$	0	0
$823$	−1.27092	−0.0443014	−0.0221507	−	0.999755i	$-0.507051\pi$
$823$	−1.27092	−0.0443014	−0.0221507	+	0.999755i	$0.507051\pi$
$824$	0	0
$825$	1.06696	0.0371467
$826$	0	0
$827$	−32.0117	−1.11316	−0.556578	−	0.830795i	$-0.687886\pi$
$827$	−32.0117	−1.11316	−0.556578	+	0.830795i	$0.687886\pi$
$828$	0	0
$829$	12.7070	0.441333	0.220667	−	0.975349i	$-0.429177\pi$
$829$	12.7070	0.441333	0.220667	+	0.975349i	$0.429177\pi$
$830$	0	0
$831$	−37.4384	−1.29872
$832$	0	0
$833$	−19.3153	−0.669234
$834$	0	0
$835$	23.1386	0.800744
$836$	0	0
$837$	−19.1257	−0.661080
$838$	0	0
$839$	10.9875	0.379329	0.189665	−	0.981849i	$-0.439260\pi$
$839$	10.9875	0.379329	0.189665	+	0.981849i	$0.439260\pi$
$840$	0	0
$841$	−9.04455	−0.311881
$842$	0	0
$843$	1.74062	0.0599501
$844$	0	0
$845$	24.6302	0.847306
$846$	0	0
$847$	−17.2340	−0.592167
$848$	0	0
$849$	5.15652	0.176971
$850$	0	0
$851$	2.07770	0.0712225
$852$	0	0
$853$	25.2555	0.864733	0.432367	−	0.901698i	$-0.357679\pi$
$853$	25.2555	0.864733	0.432367	+	0.901698i	$0.357679\pi$
$854$	0	0
$855$	2.16925	0.0741869
$856$	0	0
$857$	−35.7576	−1.22146	−0.610729	−	0.791840i	$-0.709123\pi$
$857$	−35.7576	−1.22146	−0.610729	+	0.791840i	$0.709123\pi$
$858$	0	0
$859$	−17.5952	−0.600342	−0.300171	−	0.953885i	$-0.597044\pi$
$859$	−17.5952	−0.600342	−0.300171	+	0.953885i	$0.597044\pi$
$860$	0	0
$861$	20.2972	0.691728
$862$	0	0
$863$	−27.3819	−0.932090	−0.466045	−	0.884761i	$-0.654321\pi$
$863$	−27.3819	−0.932090	−0.466045	+	0.884761i	$0.654321\pi$
$864$	0	0
$865$	50.5062	1.71726
$866$	0	0
$867$	56.7095	1.92596
$868$	0	0
$869$	−26.5583	−0.900928
$870$	0	0
$871$	46.5175	1.57619
$872$	0	0
$873$	10.8957	0.368763
$874$	0	0
$875$	−22.4682	−0.759564
$876$	0	0
$877$	35.5226	1.19951	0.599757	−	0.800182i	$-0.295264\pi$
$877$	35.5226	1.19951	0.599757	+	0.800182i	$0.295264\pi$
$878$	0	0
$879$	−21.1239	−0.712491
$880$	0	0
$881$	3.58437	0.120760	0.0603802	−	0.998175i	$-0.480769\pi$
$881$	3.58437	0.120760	0.0603802	+	0.998175i	$0.480769\pi$
$882$	0	0
$883$	−28.6457	−0.964004	−0.482002	−	0.876170i	$-0.660090\pi$
$883$	−28.6457	−0.964004	−0.482002	+	0.876170i	$0.660090\pi$
$884$	0	0
$885$	3.03637	0.102067
$886$	0	0
$887$	34.8142	1.16895	0.584474	−	0.811413i	$-0.301301\pi$
$887$	34.8142	1.16895	0.584474	+	0.811413i	$0.301301\pi$
$888$	0	0
$889$	33.5725	1.12599
$890$	0	0
$891$	−6.26799	−0.209986
$892$	0	0
$893$	3.44650	0.115333
$894$	0	0
$895$	27.2935	0.912320
$896$	0	0
$897$	−13.5726	−0.453177
$898$	0	0
$899$	15.1954	0.506796
$900$	0	0
$901$	81.2639	2.70729
$902$	0	0
$903$	11.7664	0.391561
$904$	0	0
$905$	39.0523	1.29814
$906$	0	0
$907$	24.5129	0.813937	0.406968	−	0.913442i	$-0.366586\pi$
$907$	24.5129	0.813937	0.406968	+	0.913442i	$0.366586\pi$
$908$	0	0
$909$	23.7864	0.788946
$910$	0	0
$911$	−25.4920	−0.844589	−0.422294	−	0.906459i	$-0.638775\pi$
$911$	−25.4920	−0.844589	−0.422294	+	0.906459i	$0.638775\pi$
$912$	0	0
$913$	14.2450	0.471441
$914$	0	0
$915$	8.10739	0.268022
$916$	0	0
$917$	39.3980	1.30104
$918$	0	0
$919$	21.9929	0.725480	0.362740	−	0.931890i	$-0.381841\pi$
$919$	21.9929	0.725480	0.362740	+	0.931890i	$0.381841\pi$
$920$	0	0
$921$	13.1776	0.434216
$922$	0	0
$923$	−47.9028	−1.57674
$924$	0	0
$925$	0.466908	0.0153518
$926$	0	0
$927$	12.6724	0.416217
$928$	0	0
$929$	−26.0594	−0.854982	−0.427491	−	0.904020i	$-0.640602\pi$
$929$	−26.0594	−0.854982	−0.427491	+	0.904020i	$0.640602\pi$
$930$	0	0
$931$	1.85344	0.0607441
$932$	0	0
$933$	41.8519	1.37017
$934$	0	0
$935$	30.7153	1.00450
$936$	0	0
$937$	45.7606	1.49493	0.747467	−	0.664298i	$-0.231270\pi$
$937$	45.7606	1.49493	0.747467	+	0.664298i	$0.231270\pi$
$938$	0	0
$939$	−0.686595	−0.0224062
$940$	0	0
$941$	−1.09811	−0.0357972	−0.0178986	−	0.999840i	$-0.505698\pi$
$941$	−1.09811	−0.0357972	−0.0178986	+	0.999840i	$0.505698\pi$
$942$	0	0
$943$	15.2833	0.497693
$944$	0	0
$945$	−27.9192	−0.908211
$946$	0	0
$947$	8.40050	0.272979	0.136490	−	0.990641i	$-0.456418\pi$
$947$	8.40050	0.272979	0.136490	+	0.990641i	$0.456418\pi$
$948$	0	0
$949$	−23.0608	−0.748584
$950$	0	0
$951$	35.9615	1.16613
$952$	0	0
$953$	17.6158	0.570633	0.285316	−	0.958433i	$-0.407901\pi$
$953$	17.6158	0.570633	0.285316	+	0.958433i	$0.407901\pi$
$954$	0	0
$955$	0.502178	0.0162501
$956$	0	0
$957$	10.0302	0.324231
$958$	0	0
$959$	−36.9656	−1.19368
$960$	0	0
$961$	−19.4292	−0.626748
$962$	0	0
$963$	5.92335	0.190877
$964$	0	0
$965$	12.3042	0.396086
$966$	0	0
$967$	21.9285	0.705174	0.352587	−	0.935779i	$-0.385302\pi$
$967$	21.9285	0.705174	0.352587	+	0.935779i	$0.385302\pi$
$968$	0	0
$969$	−7.60042	−0.244161
$970$	0	0
$971$	26.0655	0.836482	0.418241	−	0.908336i	$-0.362647\pi$
$971$	26.0655	0.836482	0.418241	+	0.908336i	$0.362647\pi$
$972$	0	0
$973$	−5.29515	−0.169755
$974$	0	0
$975$	−3.05009	−0.0976811
$976$	0	0
$977$	−29.9618	−0.958563	−0.479281	−	0.877661i	$-0.659103\pi$
$977$	−29.9618	−0.958563	−0.479281	+	0.877661i	$0.659103\pi$
$978$	0	0
$979$	−23.1094	−0.738578
$980$	0	0
$981$	−5.51550	−0.176097
$982$	0	0
$983$	−2.45674	−0.0783578	−0.0391789	−	0.999232i	$-0.512474\pi$
$983$	−2.45674	−0.0783578	−0.0391789	+	0.999232i	$0.512474\pi$
$984$	0	0
$985$	−60.5807	−1.93026
$986$	0	0
$987$	−13.0375	−0.414987
$988$	0	0
$989$	8.85980	0.281725
$990$	0	0
$991$	−27.5478	−0.875086	−0.437543	−	0.899197i	$-0.644151\pi$
$991$	−27.5478	−0.875086	−0.437543	+	0.899197i	$0.644151\pi$
$992$	0	0
$993$	−17.3799	−0.551535
$994$	0	0
$995$	−20.8174	−0.659957
$996$	0	0
$997$	49.0816	1.55443	0.777215	−	0.629235i	$-0.216632\pi$
$997$	49.0816	1.55443	0.777215	+	0.629235i	$0.216632\pi$
$998$	0	0
$999$	−5.52456	−0.174789

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2012.2.a.b.1.13	✓	21
4.3	odd	2		8048.2.a.s.1.9		21

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2012.2.a.b.1.13	✓	21	1.1	even	1	trivial
8048.2.a.s.1.9		21	4.3	odd	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 2012 = 2^{2} \cdot 503 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2012.a (trivial)

\(f(q)\)	\(=\)	\(q+1.32334 q^{3} +2.33991 q^{5} +2.12211 q^{7} -1.24878 q^{9} +O(q^{10})\)	\(q+1.32334 q^{3} +2.33991 q^{5} +2.12211 q^{7} -1.24878 q^{9} +1.69672 q^{11} -4.85037 q^{13} +3.09649 q^{15} +7.73650 q^{17} -0.742375 q^{19} +2.80827 q^{21} +2.11455 q^{23} +0.475191 q^{25} -5.62257 q^{27} +4.46715 q^{29} +3.40159 q^{31} +2.24533 q^{33} +4.96556 q^{35} +0.982569 q^{37} -6.41867 q^{39} +7.22768 q^{41} +4.18992 q^{43} -2.92205 q^{45} -4.64253 q^{47} -2.49664 q^{49} +10.2380 q^{51} +10.5040 q^{53} +3.97017 q^{55} -0.982411 q^{57} +0.980586 q^{59} +2.61825 q^{61} -2.65006 q^{63} -11.3495 q^{65} -9.59050 q^{67} +2.79826 q^{69} +9.87610 q^{71} +4.75443 q^{73} +0.628836 q^{75} +3.60063 q^{77} -15.6527 q^{79} -3.69418 q^{81} +8.39562 q^{83} +18.1027 q^{85} +5.91154 q^{87} -13.6200 q^{89} -10.2930 q^{91} +4.50145 q^{93} -1.73709 q^{95} -8.72503 q^{97} -2.11884 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 21 q + 10 q^{3} + 3 q^{5} + 13 q^{7} + 21 q^{9}+O(q^{10}) \) 21 * q + 10 * q^3 + 3 * q^5 + 13 * q^7 + 21 * q^9	\( 21 q + 10 q^{3} + 3 q^{5} + 13 q^{7} + 21 q^{9} + 7 q^{11} + 12 q^{13} + 14 q^{15} + q^{17} + 14 q^{19} + 14 q^{21} + 26 q^{23} + 18 q^{25} + 37 q^{27} + 9 q^{29} + 28 q^{31} + 3 q^{33} + 20 q^{35} + 31 q^{37} + 29 q^{39} + 4 q^{41} + 38 q^{43} + 24 q^{45} + 9 q^{47} + 16 q^{49} + 15 q^{51} + 22 q^{53} + 35 q^{55} - q^{57} + 10 q^{59} + 22 q^{61} + 35 q^{63} - 14 q^{65} + 58 q^{67} + 15 q^{69} + 27 q^{71} - 6 q^{73} + 48 q^{75} + 16 q^{77} + 47 q^{79} + 29 q^{81} + 22 q^{83} + 14 q^{85} + 29 q^{87} + q^{89} + 51 q^{91} + 34 q^{93} + 23 q^{95} - 2 q^{97} + 22 q^{99}+O(q^{100}) \) 21 * q + 10 * q^3 + 3 * q^5 + 13 * q^7 + 21 * q^9 + 7 * q^11 + 12 * q^13 + 14 * q^15 + q^17 + 14 * q^19 + 14 * q^21 + 26 * q^23 + 18 * q^25 + 37 * q^27 + 9 * q^29 + 28 * q^31 + 3 * q^33 + 20 * q^35 + 31 * q^37 + 29 * q^39 + 4 * q^41 + 38 * q^43 + 24 * q^45 + 9 * q^47 + 16 * q^49 + 15 * q^51 + 22 * q^53 + 35 * q^55 - q^57 + 10 * q^59 + 22 * q^61 + 35 * q^63 - 14 * q^65 + 58 * q^67 + 15 * q^69 + 27 * q^71 - 6 * q^73 + 48 * q^75 + 16 * q^77 + 47 * q^79 + 29 * q^81 + 22 * q^83 + 14 * q^85 + 29 * q^87 + q^89 + 51 * q^91 + 34 * q^93 + 23 * q^95 - 2 * q^97 + 22 * q^99

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