LMFDB - Embedded newform 8020.2.a.f.1.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8020,2,Mod(1,8020)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8020, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("8020.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8020 = 2^{2} \cdot 5 \cdot 401 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8020.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:	$64.0400224211$
Analytic rank:	$0$
Dimension:	$37$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.15
Character	$\chi$	$=$	8020.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.446086 q^{3} +1.00000 q^{5} +3.25733 q^{7} -2.80101 q^{9} +O(q^{10})$	$q-0.446086 q^{3} +1.00000 q^{5} +3.25733 q^{7} -2.80101 q^{9} -1.99296 q^{11} -3.05673 q^{13} -0.446086 q^{15} +8.07028 q^{17} +2.16727 q^{19} -1.45305 q^{21} -4.14546 q^{23} +1.00000 q^{25} +2.58775 q^{27} +4.82170 q^{29} -3.36075 q^{31} +0.889032 q^{33} +3.25733 q^{35} -1.82227 q^{37} +1.36356 q^{39} +7.04183 q^{41} +6.10534 q^{43} -2.80101 q^{45} -3.36467 q^{47} +3.61018 q^{49} -3.60004 q^{51} -1.14924 q^{53} -1.99296 q^{55} -0.966791 q^{57} -8.30566 q^{59} -3.17178 q^{61} -9.12379 q^{63} -3.05673 q^{65} -3.60913 q^{67} +1.84923 q^{69} +3.46600 q^{71} +2.96942 q^{73} -0.446086 q^{75} -6.49172 q^{77} +13.1268 q^{79} +7.24866 q^{81} +1.13221 q^{83} +8.07028 q^{85} -2.15089 q^{87} +12.5148 q^{89} -9.95676 q^{91} +1.49918 q^{93} +2.16727 q^{95} -6.31243 q^{97} +5.58229 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 37 q + 3 q^{3} + 37 q^{5} + 4 q^{7} + 50 q^{9}+O(q^{10}) $ 37 * q + 3 * q^3 + 37 * q^5 + 4 * q^7 + 50 * q^9	$ 37 q + 3 q^{3} + 37 q^{5} + 4 q^{7} + 50 q^{9} + 2 q^{11} + 27 q^{13} + 3 q^{15} + 36 q^{17} - 6 q^{19} + 20 q^{21} + 17 q^{23} + 37 q^{25} + 9 q^{27} + 29 q^{29} + 5 q^{31} + 36 q^{33} + 4 q^{35} + 35 q^{37} + 21 q^{39} + 24 q^{41} + 11 q^{43} + 50 q^{45} + 19 q^{47} + 57 q^{49} + 8 q^{51} + 65 q^{53} + 2 q^{55} + 62 q^{57} - 9 q^{59} + 13 q^{61} + 26 q^{63} + 27 q^{65} + 13 q^{67} + 20 q^{69} + 33 q^{71} + 67 q^{73} + 3 q^{75} + 62 q^{77} + 23 q^{79} + 97 q^{81} + 2 q^{83} + 36 q^{85} + 32 q^{87} + 34 q^{89} + q^{91} + 41 q^{93} - 6 q^{95} + 66 q^{97} - 7 q^{99}+O(q^{100}) $ 37 * q + 3 * q^3 + 37 * q^5 + 4 * q^7 + 50 * q^9 + 2 * q^11 + 27 * q^13 + 3 * q^15 + 36 * q^17 - 6 * q^19 + 20 * q^21 + 17 * q^23 + 37 * q^25 + 9 * q^27 + 29 * q^29 + 5 * q^31 + 36 * q^33 + 4 * q^35 + 35 * q^37 + 21 * q^39 + 24 * q^41 + 11 * q^43 + 50 * q^45 + 19 * q^47 + 57 * q^49 + 8 * q^51 + 65 * q^53 + 2 * q^55 + 62 * q^57 - 9 * q^59 + 13 * q^61 + 26 * q^63 + 27 * q^65 + 13 * q^67 + 20 * q^69 + 33 * q^71 + 67 * q^73 + 3 * q^75 + 62 * q^77 + 23 * q^79 + 97 * q^81 + 2 * q^83 + 36 * q^85 + 32 * q^87 + 34 * q^89 + q^91 + 41 * q^93 - 6 * q^95 + 66 * q^97 - 7 * 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
$3$	−0.446086	−0.257548	−0.128774	−	0.991674i	$-0.541104\pi$
$3$	−0.446086	−0.257548	−0.128774	+	0.991674i	$0.541104\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	3.25733	1.23115	0.615577	−	0.788077i	$-0.288923\pi$
$7$	3.25733	1.23115	0.615577	+	0.788077i	$0.288923\pi$
$8$	0	0
$9$	−2.80101	−0.933669
$10$	0	0
$11$	−1.99296	−0.600900	−0.300450	−	0.953798i	$-0.597137\pi$
$11$	−1.99296	−0.600900	−0.300450	+	0.953798i	$0.597137\pi$
$12$	0	0
$13$	−3.05673	−0.847783	−0.423892	−	0.905713i	$-0.639336\pi$
$13$	−3.05673	−0.847783	−0.423892	+	0.905713i	$0.639336\pi$
$14$	0	0
$15$	−0.446086	−0.115179
$16$	0	0
$17$	8.07028	1.95733	0.978666	−	0.205459i	$-0.0658687\pi$
$17$	8.07028	1.95733	0.978666	+	0.205459i	$0.0658687\pi$
$18$	0	0
$19$	2.16727	0.497207	0.248603	−	0.968605i	$-0.420028\pi$
$19$	2.16727	0.497207	0.248603	+	0.968605i	$0.420028\pi$
$20$	0	0
$21$	−1.45305	−0.317081
$22$	0	0
$23$	−4.14546	−0.864388	−0.432194	−	0.901781i	$-0.642260\pi$
$23$	−4.14546	−0.864388	−0.432194	+	0.901781i	$0.642260\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	2.58775	0.498013
$28$	0	0
$29$	4.82170	0.895367	0.447683	−	0.894192i	$-0.352249\pi$
$29$	4.82170	0.895367	0.447683	+	0.894192i	$0.352249\pi$
$30$	0	0
$31$	−3.36075	−0.603608	−0.301804	−	0.953370i	$-0.597589\pi$
$31$	−3.36075	−0.603608	−0.301804	+	0.953370i	$0.597589\pi$
$32$	0	0
$33$	0.889032	0.154761
$34$	0	0
$35$	3.25733	0.550589
$36$	0	0
$37$	−1.82227	−0.299579	−0.149789	−	0.988718i	$-0.547860\pi$
$37$	−1.82227	−0.299579	−0.149789	+	0.988718i	$0.547860\pi$
$38$	0	0
$39$	1.36356	0.218345
$40$	0	0
$41$	7.04183	1.09975	0.549874	−	0.835247i	$-0.314676\pi$
$41$	7.04183	1.09975	0.549874	+	0.835247i	$0.314676\pi$
$42$	0	0
$43$	6.10534	0.931056	0.465528	−	0.885033i	$-0.345864\pi$
$43$	6.10534	0.931056	0.465528	+	0.885033i	$0.345864\pi$
$44$	0	0
$45$	−2.80101	−0.417549
$46$	0	0
$47$	−3.36467	−0.490788	−0.245394	−	0.969423i	$-0.578917\pi$
$47$	−3.36467	−0.490788	−0.245394	+	0.969423i	$0.578917\pi$
$48$	0	0
$49$	3.61018	0.515740
$50$	0	0
$51$	−3.60004	−0.504107
$52$	0	0
$53$	−1.14924	−0.157861	−0.0789303	−	0.996880i	$-0.525150\pi$
$53$	−1.14924	−0.157861	−0.0789303	+	0.996880i	$0.525150\pi$
$54$	0	0
$55$	−1.99296	−0.268730
$56$	0	0
$57$	−0.966791	−0.128055
$58$	0	0
$59$	−8.30566	−1.08130	−0.540652	−	0.841246i	$-0.681823\pi$
$59$	−8.30566	−1.08130	−0.540652	+	0.841246i	$0.681823\pi$
$60$	0	0
$61$	−3.17178	−0.406105	−0.203052	−	0.979168i	$-0.565086\pi$
$61$	−3.17178	−0.406105	−0.203052	+	0.979168i	$0.565086\pi$
$62$	0	0
$63$	−9.12379	−1.14949
$64$	0	0
$65$	−3.05673	−0.379140
$66$	0	0
$67$	−3.60913	−0.440925	−0.220463	−	0.975395i	$-0.570757\pi$
$67$	−3.60913	−0.440925	−0.220463	+	0.975395i	$0.570757\pi$
$68$	0	0
$69$	1.84923	0.222621
$70$	0	0
$71$	3.46600	0.411339	0.205669	−	0.978622i	$-0.434063\pi$
$71$	3.46600	0.411339	0.205669	+	0.978622i	$0.434063\pi$
$72$	0	0
$73$	2.96942	0.347545	0.173772	−	0.984786i	$-0.444404\pi$
$73$	2.96942	0.347545	0.173772	+	0.984786i	$0.444404\pi$
$74$	0	0
$75$	−0.446086	−0.0515096
$76$	0	0
$77$	−6.49172	−0.739800
$78$	0	0
$79$	13.1268	1.47688	0.738439	−	0.674320i	$-0.235563\pi$
$79$	13.1268	1.47688	0.738439	+	0.674320i	$0.235563\pi$
$80$	0	0
$81$	7.24866	0.805407
$82$	0	0
$83$	1.13221	0.124276	0.0621378	−	0.998068i	$-0.480208\pi$
$83$	1.13221	0.124276	0.0621378	+	0.998068i	$0.480208\pi$
$84$	0	0
$85$	8.07028	0.875345
$86$	0	0
$87$	−2.15089	−0.230600
$88$	0	0
$89$	12.5148	1.32657	0.663283	−	0.748369i	$-0.269163\pi$
$89$	12.5148	1.32657	0.663283	+	0.748369i	$0.269163\pi$
$90$	0	0
$91$	−9.95676	−1.04375
$92$	0	0
$93$	1.49918	0.155458
$94$	0	0
$95$	2.16727	0.222358
$96$	0	0
$97$	−6.31243	−0.640930	−0.320465	−	0.947260i	$-0.603839\pi$
$97$	−6.31243	−0.640930	−0.320465	+	0.947260i	$0.603839\pi$
$98$	0	0
$99$	5.58229	0.561041
$100$	0	0
$101$	−12.7093	−1.26462	−0.632311	−	0.774715i	$-0.717893\pi$
$101$	−12.7093	−1.26462	−0.632311	+	0.774715i	$0.717893\pi$
$102$	0	0
$103$	5.23712	0.516029	0.258015	−	0.966141i	$-0.416932\pi$
$103$	5.23712	0.516029	0.258015	+	0.966141i	$0.416932\pi$
$104$	0	0
$105$	−1.45305	−0.141803
$106$	0	0
$107$	7.69180	0.743595	0.371797	−	0.928314i	$-0.378742\pi$
$107$	7.69180	0.743595	0.371797	+	0.928314i	$0.378742\pi$
$108$	0	0
$109$	12.1383	1.16264	0.581319	−	0.813676i	$-0.302537\pi$
$109$	12.1383	1.16264	0.581319	+	0.813676i	$0.302537\pi$
$110$	0	0
$111$	0.812888	0.0771559
$112$	0	0
$113$	−14.4736	−1.36156	−0.680780	−	0.732488i	$-0.738359\pi$
$113$	−14.4736	−1.36156	−0.680780	+	0.732488i	$0.738359\pi$
$114$	0	0
$115$	−4.14546	−0.386566
$116$	0	0
$117$	8.56191	0.791549
$118$	0	0
$119$	26.2876	2.40978
$120$	0	0
$121$	−7.02812	−0.638920
$122$	0	0
$123$	−3.14126	−0.283238
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	15.9216	1.41282	0.706408	−	0.707805i	$-0.250314\pi$
$127$	15.9216	1.41282	0.706408	+	0.707805i	$0.250314\pi$
$128$	0	0
$129$	−2.72351	−0.239792
$130$	0	0
$131$	2.72220	0.237840	0.118920	−	0.992904i	$-0.462057\pi$
$131$	2.72220	0.237840	0.118920	+	0.992904i	$0.462057\pi$
$132$	0	0
$133$	7.05952	0.612138
$134$	0	0
$135$	2.58775	0.222718
$136$	0	0
$137$	9.14775	0.781545	0.390772	−	0.920487i	$-0.372208\pi$
$137$	9.14775	0.781545	0.390772	+	0.920487i	$0.372208\pi$
$138$	0	0
$139$	11.4970	0.975161	0.487581	−	0.873078i	$-0.337880\pi$
$139$	11.4970	0.975161	0.487581	+	0.873078i	$0.337880\pi$
$140$	0	0
$141$	1.50094	0.126402
$142$	0	0
$143$	6.09193	0.509433
$144$	0	0
$145$	4.82170	0.400420
$146$	0	0
$147$	−1.61045	−0.132828
$148$	0	0
$149$	4.20659	0.344617	0.172309	−	0.985043i	$-0.444877\pi$
$149$	4.20659	0.344617	0.172309	+	0.985043i	$0.444877\pi$
$150$	0	0
$151$	12.0190	0.978092	0.489046	−	0.872258i	$-0.337345\pi$
$151$	12.0190	0.978092	0.489046	+	0.872258i	$0.337345\pi$
$152$	0	0
$153$	−22.6049	−1.82750
$154$	0	0
$155$	−3.36075	−0.269942
$156$	0	0
$157$	20.7730	1.65787	0.828934	−	0.559346i	$-0.188948\pi$
$157$	20.7730	1.65787	0.828934	+	0.559346i	$0.188948\pi$
$158$	0	0
$159$	0.512661	0.0406567
$160$	0	0
$161$	−13.5031	−1.06419
$162$	0	0
$163$	−21.9751	−1.72122	−0.860611	−	0.509264i	$-0.829918\pi$
$163$	−21.9751	−1.72122	−0.860611	+	0.509264i	$0.829918\pi$
$164$	0	0
$165$	0.889032	0.0692110
$166$	0	0
$167$	−16.0150	−1.23928	−0.619639	−	0.784887i	$-0.712721\pi$
$167$	−16.0150	−1.23928	−0.619639	+	0.784887i	$0.712721\pi$
$168$	0	0
$169$	−3.65642	−0.281263
$170$	0	0
$171$	−6.07055	−0.464226
$172$	0	0
$173$	7.92101	0.602223	0.301112	−	0.953589i	$-0.402642\pi$
$173$	7.92101	0.602223	0.301112	+	0.953589i	$0.402642\pi$
$174$	0	0
$175$	3.25733	0.246231
$176$	0	0
$177$	3.70504	0.278488
$178$	0	0
$179$	11.5243	0.861369	0.430684	−	0.902503i	$-0.358272\pi$
$179$	11.5243	0.861369	0.430684	+	0.902503i	$0.358272\pi$
$180$	0	0
$181$	2.23143	0.165861	0.0829306	−	0.996555i	$-0.473572\pi$
$181$	2.23143	0.165861	0.0829306	+	0.996555i	$0.473572\pi$
$182$	0	0
$183$	1.41489	0.104592
$184$	0	0
$185$	−1.82227	−0.133976
$186$	0	0
$187$	−16.0837	−1.17616
$188$	0	0
$189$	8.42915	0.613130
$190$	0	0
$191$	12.1879	0.881886	0.440943	−	0.897535i	$-0.354644\pi$
$191$	12.1879	0.881886	0.440943	+	0.897535i	$0.354644\pi$
$192$	0	0
$193$	−4.49527	−0.323577	−0.161788	−	0.986825i	$-0.551726\pi$
$193$	−4.49527	−0.323577	−0.161788	+	0.986825i	$0.551726\pi$
$194$	0	0
$195$	1.36356	0.0976469
$196$	0	0
$197$	14.5813	1.03888	0.519439	−	0.854508i	$-0.326141\pi$
$197$	14.5813	1.03888	0.519439	+	0.854508i	$0.326141\pi$
$198$	0	0
$199$	17.8535	1.26560	0.632800	−	0.774315i	$-0.281905\pi$
$199$	17.8535	1.26560	0.632800	+	0.774315i	$0.281905\pi$
$200$	0	0
$201$	1.60998	0.113559
$202$	0	0
$203$	15.7058	1.10233
$204$	0	0
$205$	7.04183	0.491823
$206$	0	0
$207$	11.6115	0.807052
$208$	0	0
$209$	−4.31929	−0.298771
$210$	0	0
$211$	−14.3157	−0.985532	−0.492766	−	0.870162i	$-0.664014\pi$
$211$	−14.3157	−0.985532	−0.492766	+	0.870162i	$0.664014\pi$
$212$	0	0
$213$	−1.54614	−0.105940
$214$	0	0
$215$	6.10534	0.416381
$216$	0	0
$217$	−10.9470	−0.743134
$218$	0	0
$219$	−1.32462	−0.0895094
$220$	0	0
$221$	−24.6687	−1.65939
$222$	0	0
$223$	−4.19433	−0.280873	−0.140437	−	0.990090i	$-0.544851\pi$
$223$	−4.19433	−0.280873	−0.140437	+	0.990090i	$0.544851\pi$
$224$	0	0
$225$	−2.80101	−0.186734
$226$	0	0
$227$	−11.1282	−0.738605	−0.369302	−	0.929309i	$-0.620403\pi$
$227$	−11.1282	−0.738605	−0.369302	+	0.929309i	$0.620403\pi$
$228$	0	0
$229$	0.350369	0.0231530	0.0115765	−	0.999933i	$-0.496315\pi$
$229$	0.350369	0.0231530	0.0115765	+	0.999933i	$0.496315\pi$
$230$	0	0
$231$	2.89587	0.190534
$232$	0	0
$233$	6.92655	0.453773	0.226887	−	0.973921i	$-0.427145\pi$
$233$	6.92655	0.453773	0.226887	+	0.973921i	$0.427145\pi$
$234$	0	0
$235$	−3.36467	−0.219487
$236$	0	0
$237$	−5.85568	−0.380367
$238$	0	0
$239$	−10.4210	−0.674079	−0.337039	−	0.941491i	$-0.609426\pi$
$239$	−10.4210	−0.674079	−0.337039	+	0.941491i	$0.609426\pi$
$240$	0	0
$241$	4.99157	0.321535	0.160768	−	0.986992i	$-0.448603\pi$
$241$	4.99157	0.321535	0.160768	+	0.986992i	$0.448603\pi$
$242$	0	0
$243$	−10.9968	−0.705444
$244$	0	0
$245$	3.61018	0.230646
$246$	0	0
$247$	−6.62476	−0.421524
$248$	0	0
$249$	−0.505061	−0.0320070
$250$	0	0
$251$	−2.34122	−0.147777	−0.0738883	−	0.997267i	$-0.523541\pi$
$251$	−2.34122	−0.147777	−0.0738883	+	0.997267i	$0.523541\pi$
$252$	0	0
$253$	8.26172	0.519410
$254$	0	0
$255$	−3.60004	−0.225444
$256$	0	0
$257$	−2.81393	−0.175528	−0.0877641	−	0.996141i	$-0.527972\pi$
$257$	−2.81393	−0.175528	−0.0877641	+	0.996141i	$0.527972\pi$
$258$	0	0
$259$	−5.93572	−0.368827
$260$	0	0
$261$	−13.5056	−0.835976
$262$	0	0
$263$	−7.45102	−0.459450	−0.229725	−	0.973256i	$-0.573783\pi$
$263$	−7.45102	−0.459450	−0.229725	+	0.973256i	$0.573783\pi$
$264$	0	0
$265$	−1.14924	−0.0705974
$266$	0	0
$267$	−5.58268	−0.341655
$268$	0	0
$269$	−19.3132	−1.17755	−0.588774	−	0.808298i	$-0.700389\pi$
$269$	−19.3132	−1.17755	−0.588774	+	0.808298i	$0.700389\pi$
$270$	0	0
$271$	24.1071	1.46440	0.732202	−	0.681087i	$-0.238493\pi$
$271$	24.1071	1.46440	0.732202	+	0.681087i	$0.238493\pi$
$272$	0	0
$273$	4.44157	0.268816
$274$	0	0
$275$	−1.99296	−0.120180
$276$	0	0
$277$	28.0206	1.68360	0.841799	−	0.539792i	$-0.181497\pi$
$277$	28.0206	1.68360	0.841799	+	0.539792i	$0.181497\pi$
$278$	0	0
$279$	9.41347	0.563570
$280$	0	0
$281$	28.2341	1.68430	0.842152	−	0.539241i	$-0.181289\pi$
$281$	28.2341	1.68430	0.842152	+	0.539241i	$0.181289\pi$
$282$	0	0
$283$	27.9370	1.66068	0.830341	−	0.557256i	$-0.188146\pi$
$283$	27.9370	1.66068	0.830341	+	0.557256i	$0.188146\pi$
$284$	0	0
$285$	−0.966791	−0.0572678
$286$	0	0
$287$	22.9375	1.35396
$288$	0	0
$289$	48.1295	2.83115
$290$	0	0
$291$	2.81589	0.165070
$292$	0	0
$293$	10.2699	0.599973	0.299987	−	0.953943i	$-0.403018\pi$
$293$	10.2699	0.599973	0.299987	+	0.953943i	$0.403018\pi$
$294$	0	0
$295$	−8.30566	−0.483574
$296$	0	0
$297$	−5.15728	−0.299256
$298$	0	0
$299$	12.6715	0.732814
$300$	0	0
$301$	19.8871	1.14627
$302$	0	0
$303$	5.66944	0.325701
$304$	0	0
$305$	−3.17178	−0.181616
$306$	0	0
$307$	−13.7756	−0.786217	−0.393108	−	0.919492i	$-0.628600\pi$
$307$	−13.7756	−0.786217	−0.393108	+	0.919492i	$0.628600\pi$
$308$	0	0
$309$	−2.33621	−0.132902
$310$	0	0
$311$	−0.410205	−0.0232606	−0.0116303	−	0.999932i	$-0.503702\pi$
$311$	−0.410205	−0.0232606	−0.0116303	+	0.999932i	$0.503702\pi$
$312$	0	0
$313$	17.4578	0.986775	0.493387	−	0.869810i	$-0.335758\pi$
$313$	17.4578	0.986775	0.493387	+	0.869810i	$0.335758\pi$
$314$	0	0
$315$	−9.12379	−0.514068
$316$	0	0
$317$	−5.97441	−0.335556	−0.167778	−	0.985825i	$-0.553659\pi$
$317$	−5.97441	−0.335556	−0.167778	+	0.985825i	$0.553659\pi$
$318$	0	0
$319$	−9.60944	−0.538025
$320$	0	0
$321$	−3.43121	−0.191511
$322$	0	0
$323$	17.4905	0.973198
$324$	0	0
$325$	−3.05673	−0.169557
$326$	0	0
$327$	−5.41473	−0.299435
$328$	0	0
$329$	−10.9598	−0.604236
$330$	0	0
$331$	−23.1436	−1.27209	−0.636043	−	0.771653i	$-0.719430\pi$
$331$	−23.1436	−1.27209	−0.636043	+	0.771653i	$0.719430\pi$
$332$	0	0
$333$	5.10418	0.279707
$334$	0	0
$335$	−3.60913	−0.197188
$336$	0	0
$337$	−24.0824	−1.31185	−0.655925	−	0.754826i	$-0.727721\pi$
$337$	−24.0824	−1.31185	−0.655925	+	0.754826i	$0.727721\pi$
$338$	0	0
$339$	6.45647	0.350667
$340$	0	0
$341$	6.69783	0.362708
$342$	0	0
$343$	−11.0418	−0.596199
$344$	0	0
$345$	1.84923	0.0995593
$346$	0	0
$347$	3.74519	0.201052	0.100526	−	0.994934i	$-0.467947\pi$
$347$	3.74519	0.201052	0.100526	+	0.994934i	$0.467947\pi$
$348$	0	0
$349$	−14.8245	−0.793540	−0.396770	−	0.917918i	$-0.629869\pi$
$349$	−14.8245	−0.793540	−0.396770	+	0.917918i	$0.629869\pi$
$350$	0	0
$351$	−7.91005	−0.422207
$352$	0	0
$353$	11.4489	0.609362	0.304681	−	0.952455i	$-0.401450\pi$
$353$	11.4489	0.609362	0.304681	+	0.952455i	$0.401450\pi$
$354$	0	0
$355$	3.46600	0.183956
$356$	0	0
$357$	−11.7265	−0.620633
$358$	0	0
$359$	−11.8391	−0.624846	−0.312423	−	0.949943i	$-0.601141\pi$
$359$	−11.8391	−0.624846	−0.312423	+	0.949943i	$0.601141\pi$
$360$	0	0
$361$	−14.3029	−0.752786
$362$	0	0
$363$	3.13515	0.164553
$364$	0	0
$365$	2.96942	0.155427
$366$	0	0
$367$	−18.9777	−0.990626	−0.495313	−	0.868714i	$-0.664947\pi$
$367$	−18.9777	−0.990626	−0.495313	+	0.868714i	$0.664947\pi$
$368$	0	0
$369$	−19.7242	−1.02680
$370$	0	0
$371$	−3.74346	−0.194351
$372$	0	0
$373$	−19.0254	−0.985097	−0.492548	−	0.870285i	$-0.663935\pi$
$373$	−19.0254	−0.985097	−0.492548	+	0.870285i	$0.663935\pi$
$374$	0	0
$375$	−0.446086	−0.0230358
$376$	0	0
$377$	−14.7386	−0.759077
$378$	0	0
$379$	8.42843	0.432939	0.216470	−	0.976289i	$-0.430546\pi$
$379$	8.42843	0.432939	0.216470	+	0.976289i	$0.430546\pi$
$380$	0	0
$381$	−7.10242	−0.363868
$382$	0	0
$383$	36.9367	1.88737	0.943687	−	0.330839i	$-0.107332\pi$
$383$	36.9367	1.88737	0.943687	+	0.330839i	$0.107332\pi$
$384$	0	0
$385$	−6.49172	−0.330848
$386$	0	0
$387$	−17.1011	−0.869298
$388$	0	0
$389$	2.21739	0.112426	0.0562130	−	0.998419i	$-0.482097\pi$
$389$	2.21739	0.112426	0.0562130	+	0.998419i	$0.482097\pi$
$390$	0	0
$391$	−33.4550	−1.69189
$392$	0	0
$393$	−1.21434	−0.0612553
$394$	0	0
$395$	13.1268	0.660480
$396$	0	0
$397$	−5.94147	−0.298194	−0.149097	−	0.988823i	$-0.547637\pi$
$397$	−5.94147	−0.298194	−0.149097	+	0.988823i	$0.547637\pi$
$398$	0	0
$399$	−3.14915	−0.157655
$400$	0	0
$401$	1.00000	0.0499376
$401$	1.00000	0.0499376
$402$	0	0
$403$	10.2729	0.511729
$404$	0	0
$405$	7.24866	0.360189
$406$	0	0
$407$	3.63170	0.180017
$408$	0	0
$409$	26.3635	1.30359	0.651795	−	0.758396i	$-0.274016\pi$
$409$	26.3635	1.30359	0.651795	+	0.758396i	$0.274016\pi$
$410$	0	0
$411$	−4.08068	−0.201285
$412$	0	0
$413$	−27.0542	−1.33125
$414$	0	0
$415$	1.13221	0.0555778
$416$	0	0
$417$	−5.12865	−0.251151
$418$	0	0
$419$	9.01129	0.440230	0.220115	−	0.975474i	$-0.429357\pi$
$419$	9.01129	0.440230	0.220115	+	0.975474i	$0.429357\pi$
$420$	0	0
$421$	−18.6124	−0.907113	−0.453556	−	0.891228i	$-0.649845\pi$
$421$	−18.6124	−0.907113	−0.453556	+	0.891228i	$0.649845\pi$
$422$	0	0
$423$	9.42447	0.458234
$424$	0	0
$425$	8.07028	0.391466
$426$	0	0
$427$	−10.3315	−0.499977
$428$	0	0
$429$	−2.71753	−0.131203
$430$	0	0
$431$	12.8101	0.617041	0.308521	−	0.951218i	$-0.400166\pi$
$431$	12.8101	0.617041	0.308521	+	0.951218i	$0.400166\pi$
$432$	0	0
$433$	12.7128	0.610937	0.305469	−	0.952202i	$-0.401187\pi$
$433$	12.7128	0.610937	0.305469	+	0.952202i	$0.401187\pi$
$434$	0	0
$435$	−2.15089	−0.103127
$436$	0	0
$437$	−8.98434	−0.429779
$438$	0	0
$439$	−19.0665	−0.909995	−0.454998	−	0.890493i	$-0.650360\pi$
$439$	−19.0665	−0.909995	−0.454998	+	0.890493i	$0.650360\pi$
$440$	0	0
$441$	−10.1121	−0.481530
$442$	0	0
$443$	26.0806	1.23912	0.619562	−	0.784947i	$-0.287310\pi$
$443$	26.0806	1.23912	0.619562	+	0.784947i	$0.287310\pi$
$444$	0	0
$445$	12.5148	0.593258
$446$	0	0
$447$	−1.87650	−0.0887556
$448$	0	0
$449$	4.10115	0.193545	0.0967727	−	0.995307i	$-0.469148\pi$
$449$	4.10115	0.193545	0.0967727	+	0.995307i	$0.469148\pi$
$450$	0	0
$451$	−14.0341	−0.660838
$452$	0	0
$453$	−5.36151	−0.251906
$454$	0	0
$455$	−9.95676	−0.466780
$456$	0	0
$457$	−7.41868	−0.347031	−0.173516	−	0.984831i	$-0.555513\pi$
$457$	−7.41868	−0.347031	−0.173516	+	0.984831i	$0.555513\pi$
$458$	0	0
$459$	20.8839	0.974776
$460$	0	0
$461$	26.2918	1.22453	0.612265	−	0.790653i	$-0.290259\pi$
$461$	26.2918	1.22453	0.612265	+	0.790653i	$0.290259\pi$
$462$	0	0
$463$	−13.6062	−0.632332	−0.316166	−	0.948704i	$-0.602396\pi$
$463$	−13.6062	−0.632332	−0.316166	+	0.948704i	$0.602396\pi$
$464$	0	0
$465$	1.49918	0.0695230
$466$	0	0
$467$	36.5923	1.69329	0.846645	−	0.532159i	$-0.178619\pi$
$467$	36.5923	1.69329	0.846645	+	0.532159i	$0.178619\pi$
$468$	0	0
$469$	−11.7561	−0.542847
$470$	0	0
$471$	−9.26657	−0.426981
$472$	0	0
$473$	−12.1677	−0.559471
$474$	0	0
$475$	2.16727	0.0994413
$476$	0	0
$477$	3.21903	0.147389
$478$	0	0
$479$	−23.6398	−1.08013	−0.540065	−	0.841623i	$-0.681600\pi$
$479$	−23.6398	−1.08013	−0.540065	+	0.841623i	$0.681600\pi$
$480$	0	0
$481$	5.57017	0.253978
$482$	0	0
$483$	6.02355	0.274081
$484$	0	0
$485$	−6.31243	−0.286633
$486$	0	0
$487$	−7.74008	−0.350736	−0.175368	−	0.984503i	$-0.556112\pi$
$487$	−7.74008	−0.350736	−0.175368	+	0.984503i	$0.556112\pi$
$488$	0	0
$489$	9.80278	0.443297
$490$	0	0
$491$	−9.20142	−0.415254	−0.207627	−	0.978208i	$-0.566574\pi$
$491$	−9.20142	−0.415254	−0.207627	+	0.978208i	$0.566574\pi$
$492$	0	0
$493$	38.9125	1.75253
$494$	0	0
$495$	5.58229	0.250905
$496$	0	0
$497$	11.2899	0.506421
$498$	0	0
$499$	28.1502	1.26018	0.630088	−	0.776524i	$-0.283019\pi$
$499$	28.1502	1.26018	0.630088	+	0.776524i	$0.283019\pi$
$500$	0	0
$501$	7.14407	0.319174
$502$	0	0
$503$	−2.89025	−0.128870	−0.0644350	−	0.997922i	$-0.520525\pi$
$503$	−2.89025	−0.128870	−0.0644350	+	0.997922i	$0.520525\pi$
$504$	0	0
$505$	−12.7093	−0.565556
$506$	0	0
$507$	1.63108	0.0724388
$508$	0	0
$509$	−1.94915	−0.0863945	−0.0431973	−	0.999067i	$-0.513754\pi$
$509$	−1.94915	−0.0863945	−0.0431973	+	0.999067i	$0.513754\pi$
$510$	0	0
$511$	9.67238	0.427881
$512$	0	0
$513$	5.60836	0.247615
$514$	0	0
$515$	5.23712	0.230775
$516$	0	0
$517$	6.70565	0.294914
$518$	0	0
$519$	−3.53346	−0.155101
$520$	0	0
$521$	33.5804	1.47119	0.735593	−	0.677424i	$-0.236904\pi$
$521$	33.5804	1.47119	0.735593	+	0.677424i	$0.236904\pi$
$522$	0	0
$523$	−7.82774	−0.342283	−0.171142	−	0.985246i	$-0.554746\pi$
$523$	−7.82774	−0.342283	−0.171142	+	0.985246i	$0.554746\pi$
$524$	0	0
$525$	−1.45305	−0.0634163
$526$	0	0
$527$	−27.1222	−1.18146
$528$	0	0
$529$	−5.81518	−0.252834
$530$	0	0
$531$	23.2642	1.00958
$532$	0	0
$533$	−21.5249	−0.932349
$534$	0	0
$535$	7.69180	0.332546
$536$	0	0
$537$	−5.14085	−0.221844
$538$	0	0
$539$	−7.19493	−0.309908
$540$	0	0
$541$	33.9125	1.45801	0.729006	−	0.684507i	$-0.239983\pi$
$541$	33.9125	1.45801	0.729006	+	0.684507i	$0.239983\pi$
$542$	0	0
$543$	−0.995412	−0.0427172
$544$	0	0
$545$	12.1383	0.519947
$546$	0	0
$547$	−22.2298	−0.950480	−0.475240	−	0.879856i	$-0.657639\pi$
$547$	−22.2298	−0.950480	−0.475240	+	0.879856i	$0.657639\pi$
$548$	0	0
$549$	8.88418	0.379167
$550$	0	0
$551$	10.4499	0.445182
$552$	0	0
$553$	42.7582	1.81826
$554$	0	0
$555$	0.812888	0.0345052
$556$	0	0
$557$	21.8721	0.926752	0.463376	−	0.886162i	$-0.346638\pi$
$557$	21.8721	0.926752	0.463376	+	0.886162i	$0.346638\pi$
$558$	0	0
$559$	−18.6624	−0.789334
$560$	0	0
$561$	7.17474	0.302918
$562$	0	0
$563$	−4.64970	−0.195961	−0.0979806	−	0.995188i	$-0.531238\pi$
$563$	−4.64970	−0.195961	−0.0979806	+	0.995188i	$0.531238\pi$
$564$	0	0
$565$	−14.4736	−0.608908
$566$	0	0
$567$	23.6113	0.991580
$568$	0	0
$569$	−8.36789	−0.350800	−0.175400	−	0.984497i	$-0.556122\pi$
$569$	−8.36789	−0.350800	−0.175400	+	0.984497i	$0.556122\pi$
$570$	0	0
$571$	−43.3064	−1.81232	−0.906158	−	0.422939i	$-0.860999\pi$
$571$	−43.3064	−1.81232	−0.906158	+	0.422939i	$0.860999\pi$
$572$	0	0
$573$	−5.43686	−0.227128
$574$	0	0
$575$	−4.14546	−0.172878
$576$	0	0
$577$	39.9622	1.66365	0.831824	−	0.555040i	$-0.187297\pi$
$577$	39.9622	1.66365	0.831824	+	0.555040i	$0.187297\pi$
$578$	0	0
$579$	2.00528	0.0833366
$580$	0	0
$581$	3.68796	0.153002
$582$	0	0
$583$	2.29039	0.0948583
$584$	0	0
$585$	8.56191	0.353992
$586$	0	0
$587$	−9.29856	−0.383793	−0.191896	−	0.981415i	$-0.561464\pi$
$587$	−9.29856	−0.383793	−0.191896	+	0.981415i	$0.561464\pi$
$588$	0	0
$589$	−7.28366	−0.300118
$590$	0	0
$591$	−6.50454	−0.267561
$592$	0	0
$593$	26.6803	1.09563	0.547815	−	0.836600i	$-0.315460\pi$
$593$	26.6803	1.09563	0.547815	+	0.836600i	$0.315460\pi$
$594$	0	0
$595$	26.2876	1.07768
$596$	0	0
$597$	−7.96420	−0.325953
$598$	0	0
$599$	−10.1887	−0.416299	−0.208149	−	0.978097i	$-0.566744\pi$
$599$	−10.1887	−0.416299	−0.208149	+	0.978097i	$0.566744\pi$
$600$	0	0
$601$	43.1041	1.75825	0.879126	−	0.476589i	$-0.158127\pi$
$601$	43.1041	1.75825	0.879126	+	0.476589i	$0.158127\pi$
$602$	0	0
$603$	10.1092	0.411678
$604$	0	0
$605$	−7.02812	−0.285734
$606$	0	0
$607$	36.5420	1.48320	0.741598	−	0.670845i	$-0.234068\pi$
$607$	36.5420	1.48320	0.741598	+	0.670845i	$0.234068\pi$
$608$	0	0
$609$	−7.00616	−0.283904
$610$	0	0
$611$	10.2849	0.416082
$612$	0	0
$613$	31.6675	1.27904	0.639518	−	0.768776i	$-0.279134\pi$
$613$	31.6675	1.27904	0.639518	+	0.768776i	$0.279134\pi$
$614$	0	0
$615$	−3.14126	−0.126668
$616$	0	0
$617$	24.0255	0.967231	0.483615	−	0.875281i	$-0.339323\pi$
$617$	24.0255	0.967231	0.483615	+	0.875281i	$0.339323\pi$
$618$	0	0
$619$	−36.7400	−1.47671	−0.738353	−	0.674415i	$-0.764396\pi$
$619$	−36.7400	−1.47671	−0.738353	+	0.674415i	$0.764396\pi$
$620$	0	0
$621$	−10.7274	−0.430476
$622$	0	0
$623$	40.7648	1.63321
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	1.92677	0.0769480
$628$	0	0
$629$	−14.7062	−0.586375
$630$	0	0
$631$	−24.1601	−0.961797	−0.480899	−	0.876776i	$-0.659689\pi$
$631$	−24.1601	−0.961797	−0.480899	+	0.876776i	$0.659689\pi$
$632$	0	0
$633$	6.38603	0.253822
$634$	0	0
$635$	15.9216	0.631831
$636$	0	0
$637$	−11.0353	−0.437235
$638$	0	0
$639$	−9.70830	−0.384054
$640$	0	0
$641$	44.2383	1.74731	0.873654	−	0.486548i	$-0.161744\pi$
$641$	44.2383	1.74731	0.873654	+	0.486548i	$0.161744\pi$
$642$	0	0
$643$	−40.0885	−1.58093	−0.790467	−	0.612504i	$-0.790162\pi$
$643$	−40.0885	−1.58093	−0.790467	+	0.612504i	$0.790162\pi$
$644$	0	0
$645$	−2.72351	−0.107238
$646$	0	0
$647$	10.6963	0.420514	0.210257	−	0.977646i	$-0.432570\pi$
$647$	10.6963	0.420514	0.210257	+	0.977646i	$0.432570\pi$
$648$	0	0
$649$	16.5528	0.649755
$650$	0	0
$651$	4.88333	0.191393
$652$	0	0
$653$	−6.02145	−0.235637	−0.117819	−	0.993035i	$-0.537590\pi$
$653$	−6.02145	−0.235637	−0.117819	+	0.993035i	$0.537590\pi$
$654$	0	0
$655$	2.72220	0.106365
$656$	0	0
$657$	−8.31737	−0.324492
$658$	0	0
$659$	10.6261	0.413933	0.206966	−	0.978348i	$-0.433641\pi$
$659$	10.6261	0.413933	0.206966	+	0.978348i	$0.433641\pi$
$660$	0	0
$661$	−12.5446	−0.487927	−0.243964	−	0.969784i	$-0.578448\pi$
$661$	−12.5446	−0.487927	−0.243964	+	0.969784i	$0.578448\pi$
$662$	0	0
$663$	11.0044	0.427374
$664$	0	0
$665$	7.05952	0.273756
$666$	0	0
$667$	−19.9881	−0.773944
$668$	0	0
$669$	1.87104	0.0723384
$670$	0	0
$671$	6.32123	0.244028
$672$	0	0
$673$	−18.5774	−0.716105	−0.358052	−	0.933702i	$-0.616559\pi$
$673$	−18.5774	−0.716105	−0.358052	+	0.933702i	$0.616559\pi$
$674$	0	0
$675$	2.58775	0.0996026
$676$	0	0
$677$	20.1732	0.775319	0.387660	−	0.921803i	$-0.373284\pi$
$677$	20.1732	0.775319	0.387660	+	0.921803i	$0.373284\pi$
$678$	0	0
$679$	−20.5616	−0.789083
$680$	0	0
$681$	4.96414	0.190226
$682$	0	0
$683$	14.6509	0.560600	0.280300	−	0.959912i	$-0.409566\pi$
$683$	14.6509	0.560600	0.280300	+	0.959912i	$0.409566\pi$
$684$	0	0
$685$	9.14775	0.349517
$686$	0	0
$687$	−0.156295	−0.00596302
$688$	0	0
$689$	3.51292	0.133832
$690$	0	0
$691$	−24.5336	−0.933300	−0.466650	−	0.884442i	$-0.654539\pi$
$691$	−24.5336	−0.933300	−0.466650	+	0.884442i	$0.654539\pi$
$692$	0	0
$693$	18.1833	0.690728
$694$	0	0
$695$	11.4970	0.436105
$696$	0	0
$697$	56.8296	2.15257
$698$	0	0
$699$	−3.08984	−0.116868
$700$	0	0
$701$	−25.5173	−0.963775	−0.481887	−	0.876233i	$-0.660049\pi$
$701$	−25.5173	−0.963775	−0.481887	+	0.876233i	$0.660049\pi$
$702$	0	0
$703$	−3.94935	−0.148953
$704$	0	0
$705$	1.50094	0.0565285
$706$	0	0
$707$	−41.3983	−1.55694
$708$	0	0
$709$	−9.13068	−0.342910	−0.171455	−	0.985192i	$-0.554847\pi$
$709$	−9.13068	−0.342910	−0.171455	+	0.985192i	$0.554847\pi$
$710$	0	0
$711$	−36.7682	−1.37891
$712$	0	0
$713$	13.9318	0.521751
$714$	0	0
$715$	6.09193	0.227825
$716$	0	0
$717$	4.64867	0.173608
$718$	0	0
$719$	36.2454	1.35172	0.675862	−	0.737028i	$-0.263772\pi$
$719$	36.2454	1.35172	0.675862	+	0.737028i	$0.263772\pi$
$720$	0	0
$721$	17.0590	0.635311
$722$	0	0
$723$	−2.22667	−0.0828108
$724$	0	0
$725$	4.82170	0.179073
$726$	0	0
$727$	13.8891	0.515119	0.257559	−	0.966262i	$-0.417082\pi$
$727$	13.8891	0.515119	0.257559	+	0.966262i	$0.417082\pi$
$728$	0	0
$729$	−16.8405	−0.623721
$730$	0	0
$731$	49.2718	1.82238
$732$	0	0
$733$	28.9266	1.06843	0.534215	−	0.845349i	$-0.320607\pi$
$733$	28.9266	1.06843	0.534215	+	0.845349i	$0.320607\pi$
$734$	0	0
$735$	−1.61045	−0.0594024
$736$	0	0
$737$	7.19284	0.264952
$738$	0	0
$739$	−12.9659	−0.476959	−0.238480	−	0.971147i	$-0.576649\pi$
$739$	−12.9659	−0.476959	−0.238480	+	0.971147i	$0.576649\pi$
$740$	0	0
$741$	2.95522	0.108563
$742$	0	0
$743$	52.7962	1.93690	0.968452	−	0.249200i	$-0.0801676\pi$
$743$	52.7962	1.93690	0.968452	+	0.249200i	$0.0801676\pi$
$744$	0	0
$745$	4.20659	0.154118
$746$	0	0
$747$	−3.17131	−0.116032
$748$	0	0
$749$	25.0547	0.915479
$750$	0	0
$751$	28.8417	1.05245	0.526225	−	0.850345i	$-0.323607\pi$
$751$	28.8417	1.05245	0.526225	+	0.850345i	$0.323607\pi$
$752$	0	0
$753$	1.04439	0.0380596
$754$	0	0
$755$	12.0190	0.437416
$756$	0	0
$757$	27.7946	1.01021	0.505106	−	0.863058i	$-0.331454\pi$
$757$	27.7946	1.01021	0.505106	+	0.863058i	$0.331454\pi$
$758$	0	0
$759$	−3.68544	−0.133773
$760$	0	0
$761$	−0.867190	−0.0314356	−0.0157178	−	0.999876i	$-0.505003\pi$
$761$	−0.867190	−0.0314356	−0.0157178	+	0.999876i	$0.505003\pi$
$762$	0	0
$763$	39.5384	1.43139
$764$	0	0
$765$	−22.6049	−0.817283
$766$	0	0
$767$	25.3881	0.916712
$768$	0	0
$769$	−20.4897	−0.738879	−0.369439	−	0.929255i	$-0.620450\pi$
$769$	−20.4897	−0.738879	−0.369439	+	0.929255i	$0.620450\pi$
$770$	0	0
$771$	1.25526	0.0452070
$772$	0	0
$773$	18.5745	0.668078	0.334039	−	0.942559i	$-0.391588\pi$
$773$	18.5745	0.668078	0.334039	+	0.942559i	$0.391588\pi$
$774$	0	0
$775$	−3.36075	−0.120722
$776$	0	0
$777$	2.64784	0.0949908
$778$	0	0
$779$	15.2616	0.546802
$780$	0	0
$781$	−6.90760	−0.247173
$782$	0	0
$783$	12.4773	0.445904
$784$	0	0
$785$	20.7730	0.741421
$786$	0	0
$787$	−33.1043	−1.18004	−0.590021	−	0.807388i	$-0.700880\pi$
$787$	−33.1043	−1.18004	−0.590021	+	0.807388i	$0.700880\pi$
$788$	0	0
$789$	3.32380	0.118330
$790$	0	0
$791$	−47.1452	−1.67629
$792$	0	0
$793$	9.69526	0.344289
$794$	0	0
$795$	0.512661	0.0181822
$796$	0	0
$797$	47.2972	1.67535	0.837676	−	0.546167i	$-0.183914\pi$
$797$	47.2972	1.67535	0.837676	+	0.546167i	$0.183914\pi$
$798$	0	0
$799$	−27.1539	−0.960635
$800$	0	0
$801$	−35.0540	−1.23857
$802$	0	0
$803$	−5.91793	−0.208839
$804$	0	0
$805$	−13.5031	−0.475922
$806$	0	0
$807$	8.61536	0.303275
$808$	0	0
$809$	49.5457	1.74194	0.870968	−	0.491340i	$-0.163493\pi$
$809$	49.5457	1.74194	0.870968	+	0.491340i	$0.163493\pi$
$810$	0	0
$811$	−9.36120	−0.328716	−0.164358	−	0.986401i	$-0.552555\pi$
$811$	−9.36120	−0.328716	−0.164358	+	0.986401i	$0.552555\pi$
$812$	0	0
$813$	−10.7539	−0.377155
$814$	0	0
$815$	−21.9751	−0.769753
$816$	0	0
$817$	13.2319	0.462927
$818$	0	0
$819$	27.8889	0.974519
$820$	0	0
$821$	−11.2880	−0.393954	−0.196977	−	0.980408i	$-0.563112\pi$
$821$	−11.2880	−0.393954	−0.196977	+	0.980408i	$0.563112\pi$
$822$	0	0
$823$	3.36919	0.117442	0.0587212	−	0.998274i	$-0.481298\pi$
$823$	3.36919	0.117442	0.0587212	+	0.998274i	$0.481298\pi$
$824$	0	0
$825$	0.889032	0.0309521
$826$	0	0
$827$	−16.1823	−0.562712	−0.281356	−	0.959603i	$-0.590784\pi$
$827$	−16.1823	−0.562712	−0.281356	+	0.959603i	$0.590784\pi$
$828$	0	0
$829$	3.21441	0.111641	0.0558206	−	0.998441i	$-0.482223\pi$
$829$	3.21441	0.111641	0.0558206	+	0.998441i	$0.482223\pi$
$830$	0	0
$831$	−12.4996	−0.433607
$832$	0	0
$833$	29.1352	1.00947
$834$	0	0
$835$	−16.0150	−0.554222
$836$	0	0
$837$	−8.69677	−0.300604
$838$	0	0
$839$	1.65776	0.0572323	0.0286161	−	0.999590i	$-0.490890\pi$
$839$	1.65776	0.0572323	0.0286161	+	0.999590i	$0.490890\pi$
$840$	0	0
$841$	−5.75125	−0.198319
$842$	0	0
$843$	−12.5948	−0.433789
$844$	0	0
$845$	−3.65642	−0.125785
$846$	0	0
$847$	−22.8929	−0.786608
$848$	0	0
$849$	−12.4623	−0.427705
$850$	0	0
$851$	7.55413	0.258952
$852$	0	0
$853$	−32.1302	−1.10012	−0.550058	−	0.835126i	$-0.685395\pi$
$853$	−32.1302	−1.10012	−0.550058	+	0.835126i	$0.685395\pi$
$854$	0	0
$855$	−6.07055	−0.207608
$856$	0	0
$857$	1.96537	0.0671358	0.0335679	−	0.999436i	$-0.489313\pi$
$857$	1.96537	0.0671358	0.0335679	+	0.999436i	$0.489313\pi$
$858$	0	0
$859$	−49.4350	−1.68670	−0.843351	−	0.537363i	$-0.819420\pi$
$859$	−49.4350	−1.68670	−0.843351	+	0.537363i	$0.819420\pi$
$860$	0	0
$861$	−10.2321	−0.348710
$862$	0	0
$863$	−9.91021	−0.337348	−0.168674	−	0.985672i	$-0.553948\pi$
$863$	−9.91021	−0.337348	−0.168674	+	0.985672i	$0.553948\pi$
$864$	0	0
$865$	7.92101	0.269322
$866$	0	0
$867$	−21.4699	−0.729156
$868$	0	0
$869$	−26.1611	−0.887455
$870$	0	0
$871$	11.0321	0.373809
$872$	0	0
$873$	17.6811	0.598416
$874$	0	0
$875$	3.25733	0.110118
$876$	0	0
$877$	−19.9386	−0.673279	−0.336639	−	0.941634i	$-0.609290\pi$
$877$	−19.9386	−0.673279	−0.336639	+	0.941634i	$0.609290\pi$
$878$	0	0
$879$	−4.58126	−0.154522
$880$	0	0
$881$	−3.38461	−0.114030	−0.0570152	−	0.998373i	$-0.518158\pi$
$881$	−3.38461	−0.114030	−0.0570152	+	0.998373i	$0.518158\pi$
$882$	0	0
$883$	29.2134	0.983109	0.491555	−	0.870847i	$-0.336429\pi$
$883$	29.2134	0.983109	0.491555	+	0.870847i	$0.336429\pi$
$884$	0	0
$885$	3.70504	0.124544
$886$	0	0
$887$	−4.31238	−0.144795	−0.0723977	−	0.997376i	$-0.523065\pi$
$887$	−4.31238	−0.144795	−0.0723977	+	0.997376i	$0.523065\pi$
$888$	0	0
$889$	51.8620	1.73939
$890$	0	0
$891$	−14.4463	−0.483969
$892$	0	0
$893$	−7.29217	−0.244023
$894$	0	0
$895$	11.5243	0.385216
$896$	0	0
$897$	−5.65260	−0.188735
$898$	0	0
$899$	−16.2045	−0.540450
$900$	0	0
$901$	−9.27471	−0.308985
$902$	0	0
$903$	−8.87136	−0.295220
$904$	0	0
$905$	2.23143	0.0741754
$906$	0	0
$907$	25.3003	0.840083	0.420042	−	0.907505i	$-0.362015\pi$
$907$	25.3003	0.840083	0.420042	+	0.907505i	$0.362015\pi$
$908$	0	0
$909$	35.5988	1.18074
$910$	0	0
$911$	9.88016	0.327344	0.163672	−	0.986515i	$-0.447666\pi$
$911$	9.88016	0.327344	0.163672	+	0.986515i	$0.447666\pi$
$912$	0	0
$913$	−2.25644	−0.0746772
$914$	0	0
$915$	1.41489	0.0467748
$916$	0	0
$917$	8.86711	0.292818
$918$	0	0
$919$	−15.1849	−0.500904	−0.250452	−	0.968129i	$-0.580579\pi$
$919$	−15.1849	−0.500904	−0.250452	+	0.968129i	$0.580579\pi$
$920$	0	0
$921$	6.14512	0.202489
$922$	0	0
$923$	−10.5946	−0.348726
$924$	0	0
$925$	−1.82227	−0.0599157
$926$	0	0
$927$	−14.6692	−0.481801
$928$	0	0
$929$	32.9570	1.08128	0.540642	−	0.841253i	$-0.318181\pi$
$929$	32.9570	1.08128	0.540642	+	0.841253i	$0.318181\pi$
$930$	0	0
$931$	7.82424	0.256429
$932$	0	0
$933$	0.182987	0.00599072
$934$	0	0
$935$	−16.0837	−0.525995
$936$	0	0
$937$	28.9160	0.944645	0.472322	−	0.881426i	$-0.343416\pi$
$937$	28.9160	0.944645	0.472322	+	0.881426i	$0.343416\pi$
$938$	0	0
$939$	−7.78770	−0.254142
$940$	0	0
$941$	−25.2404	−0.822815	−0.411407	−	0.911452i	$-0.634963\pi$
$941$	−25.2404	−0.822815	−0.411407	+	0.911452i	$0.634963\pi$
$942$	0	0
$943$	−29.1916	−0.950609
$944$	0	0
$945$	8.42915	0.274200
$946$	0	0
$947$	−22.6582	−0.736293	−0.368146	−	0.929768i	$-0.620007\pi$
$947$	−22.6582	−0.736293	−0.368146	+	0.929768i	$0.620007\pi$
$948$	0	0
$949$	−9.07671	−0.294642
$950$	0	0
$951$	2.66510	0.0864218
$952$	0	0
$953$	−50.8027	−1.64566	−0.822831	−	0.568287i	$-0.807606\pi$
$953$	−50.8027	−1.64566	−0.822831	+	0.568287i	$0.807606\pi$
$954$	0	0
$955$	12.1879	0.394391
$956$	0	0
$957$	4.28664	0.138567
$958$	0	0
$959$	29.7972	0.962202
$960$	0	0
$961$	−19.7054	−0.635658
$962$	0	0
$963$	−21.5448	−0.694271
$964$	0	0
$965$	−4.49527	−0.144708
$966$	0	0
$967$	−6.47882	−0.208345	−0.104172	−	0.994559i	$-0.533219\pi$
$967$	−6.47882	−0.208345	−0.104172	+	0.994559i	$0.533219\pi$
$968$	0	0
$969$	−7.80228	−0.250645
$970$	0	0
$971$	−56.0318	−1.79815	−0.899073	−	0.437798i	$-0.855758\pi$
$971$	−56.0318	−1.79815	−0.899073	+	0.437798i	$0.855758\pi$
$972$	0	0
$973$	37.4494	1.20057
$974$	0	0
$975$	1.36356	0.0436690
$976$	0	0
$977$	−58.0840	−1.85827	−0.929136	−	0.369737i	$-0.879448\pi$
$977$	−58.0840	−1.85827	−0.929136	+	0.369737i	$0.879448\pi$
$978$	0	0
$979$	−24.9415	−0.797133
$980$	0	0
$981$	−33.9994	−1.08552
$982$	0	0
$983$	−6.79356	−0.216681	−0.108340	−	0.994114i	$-0.534554\pi$
$983$	−6.79356	−0.216681	−0.108340	+	0.994114i	$0.534554\pi$
$984$	0	0
$985$	14.5813	0.464600
$986$	0	0
$987$	4.88904	0.155620
$988$	0	0
$989$	−25.3094	−0.804793
$990$	0	0
$991$	−31.6897	−1.00666	−0.503328	−	0.864095i	$-0.667891\pi$
$991$	−31.6897	−1.00666	−0.503328	+	0.864095i	$0.667891\pi$
$992$	0	0
$993$	10.3240	0.327624
$994$	0	0
$995$	17.8535	0.565994
$996$	0	0
$997$	34.0968	1.07986	0.539928	−	0.841711i	$-0.318451\pi$
$997$	34.0968	1.07986	0.539928	+	0.841711i	$0.318451\pi$
$998$	0	0
$999$	−4.71557	−0.149194

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8020.2.a.f.1.15	✓	37

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8020.2.a.f.1.15	✓	37	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:	\( N \)	\(=\)	\( 8020 = 2^{2} \cdot 5 \cdot 401 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8020.a (trivial)

\(f(q)\)	\(=\)	\(q-0.446086 q^{3} +1.00000 q^{5} +3.25733 q^{7} -2.80101 q^{9} +O(q^{10})\)	\(q-0.446086 q^{3} +1.00000 q^{5} +3.25733 q^{7} -2.80101 q^{9} -1.99296 q^{11} -3.05673 q^{13} -0.446086 q^{15} +8.07028 q^{17} +2.16727 q^{19} -1.45305 q^{21} -4.14546 q^{23} +1.00000 q^{25} +2.58775 q^{27} +4.82170 q^{29} -3.36075 q^{31} +0.889032 q^{33} +3.25733 q^{35} -1.82227 q^{37} +1.36356 q^{39} +7.04183 q^{41} +6.10534 q^{43} -2.80101 q^{45} -3.36467 q^{47} +3.61018 q^{49} -3.60004 q^{51} -1.14924 q^{53} -1.99296 q^{55} -0.966791 q^{57} -8.30566 q^{59} -3.17178 q^{61} -9.12379 q^{63} -3.05673 q^{65} -3.60913 q^{67} +1.84923 q^{69} +3.46600 q^{71} +2.96942 q^{73} -0.446086 q^{75} -6.49172 q^{77} +13.1268 q^{79} +7.24866 q^{81} +1.13221 q^{83} +8.07028 q^{85} -2.15089 q^{87} +12.5148 q^{89} -9.95676 q^{91} +1.49918 q^{93} +2.16727 q^{95} -6.31243 q^{97} +5.58229 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 37 q + 3 q^{3} + 37 q^{5} + 4 q^{7} + 50 q^{9}+O(q^{10}) \) 37 * q + 3 * q^3 + 37 * q^5 + 4 * q^7 + 50 * q^9	\( 37 q + 3 q^{3} + 37 q^{5} + 4 q^{7} + 50 q^{9} + 2 q^{11} + 27 q^{13} + 3 q^{15} + 36 q^{17} - 6 q^{19} + 20 q^{21} + 17 q^{23} + 37 q^{25} + 9 q^{27} + 29 q^{29} + 5 q^{31} + 36 q^{33} + 4 q^{35} + 35 q^{37} + 21 q^{39} + 24 q^{41} + 11 q^{43} + 50 q^{45} + 19 q^{47} + 57 q^{49} + 8 q^{51} + 65 q^{53} + 2 q^{55} + 62 q^{57} - 9 q^{59} + 13 q^{61} + 26 q^{63} + 27 q^{65} + 13 q^{67} + 20 q^{69} + 33 q^{71} + 67 q^{73} + 3 q^{75} + 62 q^{77} + 23 q^{79} + 97 q^{81} + 2 q^{83} + 36 q^{85} + 32 q^{87} + 34 q^{89} + q^{91} + 41 q^{93} - 6 q^{95} + 66 q^{97} - 7 q^{99}+O(q^{100}) \) 37 * q + 3 * q^3 + 37 * q^5 + 4 * q^7 + 50 * q^9 + 2 * q^11 + 27 * q^13 + 3 * q^15 + 36 * q^17 - 6 * q^19 + 20 * q^21 + 17 * q^23 + 37 * q^25 + 9 * q^27 + 29 * q^29 + 5 * q^31 + 36 * q^33 + 4 * q^35 + 35 * q^37 + 21 * q^39 + 24 * q^41 + 11 * q^43 + 50 * q^45 + 19 * q^47 + 57 * q^49 + 8 * q^51 + 65 * q^53 + 2 * q^55 + 62 * q^57 - 9 * q^59 + 13 * q^61 + 26 * q^63 + 27 * q^65 + 13 * q^67 + 20 * q^69 + 33 * q^71 + 67 * q^73 + 3 * q^75 + 62 * q^77 + 23 * q^79 + 97 * q^81 + 2 * q^83 + 36 * q^85 + 32 * q^87 + 34 * q^89 + q^91 + 41 * q^93 - 6 * q^95 + 66 * q^97 - 7 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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