LMFDB - Embedded newform 8020.2.a.c.1.20

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:	$1$
Dimension:	$28$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.20
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)

$f(q)$	$=$	$q+1.67955 q^{3} -1.00000 q^{5} -2.58133 q^{7} -0.179108 q^{9} +O(q^{10})$	$q+1.67955 q^{3} -1.00000 q^{5} -2.58133 q^{7} -0.179108 q^{9} +4.59416 q^{11} +6.44343 q^{13} -1.67955 q^{15} +3.22423 q^{17} -7.28131 q^{19} -4.33547 q^{21} -7.91521 q^{23} +1.00000 q^{25} -5.33947 q^{27} -9.25560 q^{29} -3.08945 q^{31} +7.71613 q^{33} +2.58133 q^{35} +2.24163 q^{37} +10.8221 q^{39} +2.13339 q^{41} +7.12005 q^{43} +0.179108 q^{45} -1.12364 q^{47} -0.336745 q^{49} +5.41526 q^{51} -13.2332 q^{53} -4.59416 q^{55} -12.2293 q^{57} +3.97345 q^{59} +8.25721 q^{61} +0.462336 q^{63} -6.44343 q^{65} +0.637996 q^{67} -13.2940 q^{69} -3.08049 q^{71} -5.63615 q^{73} +1.67955 q^{75} -11.8590 q^{77} +8.87088 q^{79} -8.43060 q^{81} +12.7834 q^{83} -3.22423 q^{85} -15.5453 q^{87} +5.95597 q^{89} -16.6326 q^{91} -5.18889 q^{93} +7.28131 q^{95} -15.2444 q^{97} -0.822850 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q + 3 q^{3} - 28 q^{5} - 4 q^{7} + 17 q^{9}+O(q^{10}) $ 28 * q + 3 * q^3 - 28 * q^5 - 4 * q^7 + 17 * q^9	$ 28 q + 3 q^{3} - 28 q^{5} - 4 q^{7} + 17 q^{9} + 2 q^{11} + 3 q^{13} - 3 q^{15} - 10 q^{17} - 2 q^{19} - 12 q^{21} - 23 q^{23} + 28 q^{25} + 9 q^{27} - 37 q^{29} - 11 q^{31} + 2 q^{33} + 4 q^{35} - 3 q^{37} - 19 q^{39} - 30 q^{41} + 13 q^{43} - 17 q^{45} - 15 q^{47} + 12 q^{49} - 8 q^{51} - 35 q^{53} - 2 q^{55} - 22 q^{57} - q^{59} - 33 q^{61} - 20 q^{63} - 3 q^{65} + 19 q^{67} - 8 q^{69} - 31 q^{71} + 31 q^{73} + 3 q^{75} - 42 q^{77} - 29 q^{79} - 36 q^{81} + 14 q^{83} + 10 q^{85} - 32 q^{87} - 32 q^{89} - 7 q^{91} - 11 q^{93} + 2 q^{95} + 2 q^{97} - 39 q^{99}+O(q^{100}) $ 28 * q + 3 * q^3 - 28 * q^5 - 4 * q^7 + 17 * q^9 + 2 * q^11 + 3 * q^13 - 3 * q^15 - 10 * q^17 - 2 * q^19 - 12 * q^21 - 23 * q^23 + 28 * q^25 + 9 * q^27 - 37 * q^29 - 11 * q^31 + 2 * q^33 + 4 * q^35 - 3 * q^37 - 19 * q^39 - 30 * q^41 + 13 * q^43 - 17 * q^45 - 15 * q^47 + 12 * q^49 - 8 * q^51 - 35 * q^53 - 2 * q^55 - 22 * q^57 - q^59 - 33 * q^61 - 20 * q^63 - 3 * q^65 + 19 * q^67 - 8 * q^69 - 31 * q^71 + 31 * q^73 + 3 * q^75 - 42 * q^77 - 29 * q^79 - 36 * q^81 + 14 * q^83 + 10 * q^85 - 32 * q^87 - 32 * q^89 - 7 * q^91 - 11 * q^93 + 2 * q^95 + 2 * q^97 - 39 * 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
$3$	1.67955	0.969689	0.484845	−	0.874600i	$-0.338876\pi$
$3$	1.67955	0.969689	0.484845	+	0.874600i	$0.338876\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−2.58133	−0.975650	−0.487825	−	0.872941i	$-0.662210\pi$
$7$	−2.58133	−0.975650	−0.487825	+	0.872941i	$0.662210\pi$
$8$	0	0
$9$	−0.179108	−0.0597026
$10$	0	0
$11$	4.59416	1.38519	0.692596	−	0.721326i	$-0.256467\pi$
$11$	4.59416	1.38519	0.692596	+	0.721326i	$0.256467\pi$
$12$	0	0
$13$	6.44343	1.78709	0.893543	−	0.448978i	$-0.148212\pi$
$13$	6.44343	1.78709	0.893543	+	0.448978i	$0.148212\pi$
$14$	0	0
$15$	−1.67955	−0.433658
$16$	0	0
$17$	3.22423	0.781990	0.390995	−	0.920393i	$-0.372131\pi$
$17$	3.22423	0.781990	0.390995	+	0.920393i	$0.372131\pi$
$18$	0	0
$19$	−7.28131	−1.67045	−0.835224	−	0.549910i	$-0.814662\pi$
$19$	−7.28131	−1.67045	−0.835224	+	0.549910i	$0.814662\pi$
$20$	0	0
$21$	−4.33547	−0.946078
$22$	0	0
$23$	−7.91521	−1.65044	−0.825218	−	0.564815i	$-0.808948\pi$
$23$	−7.91521	−1.65044	−0.825218	+	0.564815i	$0.808948\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−5.33947	−1.02758
$28$	0	0
$29$	−9.25560	−1.71872	−0.859361	−	0.511369i	$-0.829139\pi$
$29$	−9.25560	−1.71872	−0.859361	+	0.511369i	$0.829139\pi$
$30$	0	0
$31$	−3.08945	−0.554882	−0.277441	−	0.960743i	$-0.589486\pi$
$31$	−3.08945	−0.554882	−0.277441	+	0.960743i	$0.589486\pi$
$32$	0	0
$33$	7.71613	1.34321
$34$	0	0
$35$	2.58133	0.436324
$36$	0	0
$37$	2.24163	0.368521	0.184260	−	0.982877i	$-0.441011\pi$
$37$	2.24163	0.368521	0.184260	+	0.982877i	$0.441011\pi$
$38$	0	0
$39$	10.8221	1.73292
$40$	0	0
$41$	2.13339	0.333179	0.166589	−	0.986026i	$-0.446725\pi$
$41$	2.13339	0.333179	0.166589	+	0.986026i	$0.446725\pi$
$42$	0	0
$43$	7.12005	1.08580	0.542898	−	0.839798i	$-0.317327\pi$
$43$	7.12005	1.08580	0.542898	+	0.839798i	$0.317327\pi$
$44$	0	0
$45$	0.179108	0.0266998
$46$	0	0
$47$	−1.12364	−0.163899	−0.0819497	−	0.996636i	$-0.526115\pi$
$47$	−1.12364	−0.163899	−0.0819497	+	0.996636i	$0.526115\pi$
$48$	0	0
$49$	−0.336745	−0.0481064
$50$	0	0
$51$	5.41526	0.758288
$52$	0	0
$53$	−13.2332	−1.81772	−0.908858	−	0.417106i	$-0.863044\pi$
$53$	−13.2332	−1.81772	−0.908858	+	0.417106i	$0.863044\pi$
$54$	0	0
$55$	−4.59416	−0.619476
$56$	0	0
$57$	−12.2293	−1.61982
$58$	0	0
$59$	3.97345	0.517299	0.258650	−	0.965971i	$-0.416723\pi$
$59$	3.97345	0.517299	0.258650	+	0.965971i	$0.416723\pi$
$60$	0	0
$61$	8.25721	1.05723	0.528614	−	0.848862i	$-0.322712\pi$
$61$	8.25721	1.05723	0.528614	+	0.848862i	$0.322712\pi$
$62$	0	0
$63$	0.462336	0.0582489
$64$	0	0
$65$	−6.44343	−0.799209
$66$	0	0
$67$	0.637996	0.0779436	0.0389718	−	0.999240i	$-0.487592\pi$
$67$	0.637996	0.0779436	0.0389718	+	0.999240i	$0.487592\pi$
$68$	0	0
$69$	−13.2940	−1.60041
$70$	0	0
$71$	−3.08049	−0.365587	−0.182794	−	0.983151i	$-0.558514\pi$
$71$	−3.08049	−0.365587	−0.182794	+	0.983151i	$0.558514\pi$
$72$	0	0
$73$	−5.63615	−0.659661	−0.329831	−	0.944040i	$-0.606992\pi$
$73$	−5.63615	−0.659661	−0.329831	+	0.944040i	$0.606992\pi$
$74$	0	0
$75$	1.67955	0.193938
$76$	0	0
$77$	−11.8590	−1.35146
$78$	0	0
$79$	8.87088	0.998052	0.499026	−	0.866587i	$-0.333691\pi$
$79$	8.87088	0.998052	0.499026	+	0.866587i	$0.333691\pi$
$80$	0	0
$81$	−8.43060	−0.936733
$82$	0	0
$83$	12.7834	1.40316	0.701578	−	0.712593i	$-0.252479\pi$
$83$	12.7834	1.40316	0.701578	+	0.712593i	$0.252479\pi$
$84$	0	0
$85$	−3.22423	−0.349717
$86$	0	0
$87$	−15.5453	−1.66663
$88$	0	0
$89$	5.95597	0.631332	0.315666	−	0.948870i	$-0.397772\pi$
$89$	5.95597	0.631332	0.315666	+	0.948870i	$0.397772\pi$
$90$	0	0
$91$	−16.6326	−1.74357
$92$	0	0
$93$	−5.18889	−0.538063
$94$	0	0
$95$	7.28131	0.747047
$96$	0	0
$97$	−15.2444	−1.54784	−0.773918	−	0.633286i	$-0.781706\pi$
$97$	−15.2444	−1.54784	−0.773918	+	0.633286i	$0.781706\pi$
$98$	0	0
$99$	−0.822850	−0.0826996
$100$	0	0
$101$	−9.63749	−0.958966	−0.479483	−	0.877551i	$-0.659176\pi$
$101$	−9.63749	−0.958966	−0.479483	+	0.877551i	$0.659176\pi$
$102$	0	0
$103$	−7.20489	−0.709919	−0.354960	−	0.934882i	$-0.615505\pi$
$103$	−7.20489	−0.709919	−0.354960	+	0.934882i	$0.615505\pi$
$104$	0	0
$105$	4.33547	0.423099
$106$	0	0
$107$	7.91334	0.765011	0.382506	−	0.923953i	$-0.375061\pi$
$107$	7.91334	0.765011	0.382506	+	0.923953i	$0.375061\pi$
$108$	0	0
$109$	−5.89407	−0.564550	−0.282275	−	0.959334i	$-0.591089\pi$
$109$	−5.89407	−0.564550	−0.282275	+	0.959334i	$0.591089\pi$
$110$	0	0
$111$	3.76492	0.357351
$112$	0	0
$113$	−16.3182	−1.53508	−0.767542	−	0.640999i	$-0.778520\pi$
$113$	−16.3182	−1.53508	−0.767542	+	0.640999i	$0.778520\pi$
$114$	0	0
$115$	7.91521	0.738097
$116$	0	0
$117$	−1.15407	−0.106694
$118$	0	0
$119$	−8.32279	−0.762949
$120$	0	0
$121$	10.1063	0.918755
$122$	0	0
$123$	3.58313	0.323080
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−4.59767	−0.407977	−0.203988	−	0.978973i	$-0.565390\pi$
$127$	−4.59767	−0.407977	−0.203988	+	0.978973i	$0.565390\pi$
$128$	0	0
$129$	11.9585	1.05289
$130$	0	0
$131$	3.86247	0.337466	0.168733	−	0.985662i	$-0.446032\pi$
$131$	3.86247	0.337466	0.168733	+	0.985662i	$0.446032\pi$
$132$	0	0
$133$	18.7955	1.62977
$134$	0	0
$135$	5.33947	0.459549
$136$	0	0
$137$	−20.5234	−1.75343	−0.876714	−	0.481012i	$-0.840269\pi$
$137$	−20.5234	−1.75343	−0.876714	+	0.481012i	$0.840269\pi$
$138$	0	0
$139$	−0.583586	−0.0494991	−0.0247496	−	0.999694i	$-0.507879\pi$
$139$	−0.583586	−0.0494991	−0.0247496	+	0.999694i	$0.507879\pi$
$140$	0	0
$141$	−1.88721	−0.158931
$142$	0	0
$143$	29.6021	2.47546
$144$	0	0
$145$	9.25560	0.768636
$146$	0	0
$147$	−0.565580	−0.0466483
$148$	0	0
$149$	−3.08589	−0.252806	−0.126403	−	0.991979i	$-0.540343\pi$
$149$	−3.08589	−0.252806	−0.126403	+	0.991979i	$0.540343\pi$
$150$	0	0
$151$	15.0643	1.22592	0.612958	−	0.790116i	$-0.289980\pi$
$151$	15.0643	1.22592	0.612958	+	0.790116i	$0.289980\pi$
$152$	0	0
$153$	−0.577485	−0.0466869
$154$	0	0
$155$	3.08945	0.248151
$156$	0	0
$157$	−13.7025	−1.09358	−0.546788	−	0.837271i	$-0.684150\pi$
$157$	−13.7025	−1.09358	−0.546788	+	0.837271i	$0.684150\pi$
$158$	0	0
$159$	−22.2258	−1.76262
$160$	0	0
$161$	20.4318	1.61025
$162$	0	0
$163$	−12.0417	−0.943181	−0.471591	−	0.881818i	$-0.656320\pi$
$163$	−12.0417	−0.943181	−0.471591	+	0.881818i	$0.656320\pi$
$164$	0	0
$165$	−7.71613	−0.600700
$166$	0	0
$167$	−7.41046	−0.573439	−0.286719	−	0.958015i	$-0.592565\pi$
$167$	−7.41046	−0.573439	−0.286719	+	0.958015i	$0.592565\pi$
$168$	0	0
$169$	28.5178	2.19368
$170$	0	0
$171$	1.30414	0.0997302
$172$	0	0
$173$	−4.53388	−0.344704	−0.172352	−	0.985035i	$-0.555137\pi$
$173$	−4.53388	−0.344704	−0.172352	+	0.985035i	$0.555137\pi$
$174$	0	0
$175$	−2.58133	−0.195130
$176$	0	0
$177$	6.67361	0.501619
$178$	0	0
$179$	−3.99188	−0.298367	−0.149184	−	0.988810i	$-0.547665\pi$
$179$	−3.99188	−0.298367	−0.149184	+	0.988810i	$0.547665\pi$
$180$	0	0
$181$	−2.55190	−0.189681	−0.0948407	−	0.995492i	$-0.530234\pi$
$181$	−2.55190	−0.189681	−0.0948407	+	0.995492i	$0.530234\pi$
$182$	0	0
$183$	13.8684	1.02518
$184$	0	0
$185$	−2.24163	−0.164808
$186$	0	0
$187$	14.8126	1.08321
$188$	0	0
$189$	13.7829	1.00256
$190$	0	0
$191$	6.34859	0.459368	0.229684	−	0.973265i	$-0.426231\pi$
$191$	6.34859	0.459368	0.229684	+	0.973265i	$0.426231\pi$
$192$	0	0
$193$	1.62059	0.116653	0.0583263	−	0.998298i	$-0.481424\pi$
$193$	1.62059	0.116653	0.0583263	+	0.998298i	$0.481424\pi$
$194$	0	0
$195$	−10.8221	−0.774985
$196$	0	0
$197$	7.14222	0.508862	0.254431	−	0.967091i	$-0.418112\pi$
$197$	7.14222	0.508862	0.254431	+	0.967091i	$0.418112\pi$
$198$	0	0
$199$	2.55292	0.180972	0.0904858	−	0.995898i	$-0.471158\pi$
$199$	2.55292	0.180972	0.0904858	+	0.995898i	$0.471158\pi$
$200$	0	0
$201$	1.07155	0.0755811
$202$	0	0
$203$	23.8918	1.67687
$204$	0	0
$205$	−2.13339	−0.149002
$206$	0	0
$207$	1.41768	0.0985354
$208$	0	0
$209$	−33.4515	−2.31389
$210$	0	0
$211$	−8.33625	−0.573891	−0.286946	−	0.957947i	$-0.592640\pi$
$211$	−8.33625	−0.573891	−0.286946	+	0.957947i	$0.592640\pi$
$212$	0	0
$213$	−5.17385	−0.354506
$214$	0	0
$215$	−7.12005	−0.485583
$216$	0	0
$217$	7.97489	0.541371
$218$	0	0
$219$	−9.46620	−0.639667
$220$	0	0
$221$	20.7751	1.39748
$222$	0	0
$223$	−7.57485	−0.507249	−0.253625	−	0.967303i	$-0.581623\pi$
$223$	−7.57485	−0.507249	−0.253625	+	0.967303i	$0.581623\pi$
$224$	0	0
$225$	−0.179108	−0.0119405
$226$	0	0
$227$	−10.2642	−0.681261	−0.340631	−	0.940197i	$-0.610641\pi$
$227$	−10.2642	−0.681261	−0.340631	+	0.940197i	$0.610641\pi$
$228$	0	0
$229$	−6.42972	−0.424888	−0.212444	−	0.977173i	$-0.568142\pi$
$229$	−6.42972	−0.424888	−0.212444	+	0.977173i	$0.568142\pi$
$230$	0	0
$231$	−19.9179	−1.31050
$232$	0	0
$233$	16.4905	1.08033	0.540165	−	0.841559i	$-0.318362\pi$
$233$	16.4905	1.08033	0.540165	+	0.841559i	$0.318362\pi$
$234$	0	0
$235$	1.12364	0.0732980
$236$	0	0
$237$	14.8991	0.967800
$238$	0	0
$239$	1.59350	0.103075	0.0515374	−	0.998671i	$-0.483588\pi$
$239$	1.59350	0.103075	0.0515374	+	0.998671i	$0.483588\pi$
$240$	0	0
$241$	−2.70370	−0.174161	−0.0870804	−	0.996201i	$-0.527754\pi$
$241$	−2.70370	−0.174161	−0.0870804	+	0.996201i	$0.527754\pi$
$242$	0	0
$243$	1.85881	0.119242
$244$	0	0
$245$	0.336745	0.0215138
$246$	0	0
$247$	−46.9166	−2.98523
$248$	0	0
$249$	21.4703	1.36063
$250$	0	0
$251$	−0.787430	−0.0497021	−0.0248511	−	0.999691i	$-0.507911\pi$
$251$	−0.787430	−0.0497021	−0.0248511	+	0.999691i	$0.507911\pi$
$252$	0	0
$253$	−36.3637	−2.28617
$254$	0	0
$255$	−5.41526	−0.339116
$256$	0	0
$257$	8.52813	0.531970	0.265985	−	0.963977i	$-0.414303\pi$
$257$	8.52813	0.531970	0.265985	+	0.963977i	$0.414303\pi$
$258$	0	0
$259$	−5.78637	−0.359548
$260$	0	0
$261$	1.65775	0.102612
$262$	0	0
$263$	−13.0631	−0.805506	−0.402753	−	0.915309i	$-0.631947\pi$
$263$	−13.0631	−0.805506	−0.402753	+	0.915309i	$0.631947\pi$
$264$	0	0
$265$	13.2332	0.812907
$266$	0	0
$267$	10.0034	0.612196
$268$	0	0
$269$	−28.0367	−1.70943	−0.854715	−	0.519097i	$-0.826268\pi$
$269$	−28.0367	−1.70943	−0.854715	+	0.519097i	$0.826268\pi$
$270$	0	0
$271$	−14.1917	−0.862087	−0.431043	−	0.902331i	$-0.641854\pi$
$271$	−14.1917	−0.862087	−0.431043	+	0.902331i	$0.641854\pi$
$272$	0	0
$273$	−27.9353	−1.69072
$274$	0	0
$275$	4.59416	0.277038
$276$	0	0
$277$	−8.46869	−0.508834	−0.254417	−	0.967095i	$-0.581884\pi$
$277$	−8.46869	−0.508834	−0.254417	+	0.967095i	$0.581884\pi$
$278$	0	0
$279$	0.553346	0.0331279
$280$	0	0
$281$	−10.3898	−0.619804	−0.309902	−	0.950768i	$-0.600296\pi$
$281$	−10.3898	−0.619804	−0.309902	+	0.950768i	$0.600296\pi$
$282$	0	0
$283$	4.49892	0.267433	0.133717	−	0.991020i	$-0.457309\pi$
$283$	4.49892	0.267433	0.133717	+	0.991020i	$0.457309\pi$
$284$	0	0
$285$	12.2293	0.724403
$286$	0	0
$287$	−5.50697	−0.325066
$288$	0	0
$289$	−6.60435	−0.388491
$290$	0	0
$291$	−25.6038	−1.50092
$292$	0	0
$293$	12.2358	0.714822	0.357411	−	0.933947i	$-0.383659\pi$
$293$	12.2358	0.714822	0.357411	+	0.933947i	$0.383659\pi$
$294$	0	0
$295$	−3.97345	−0.231343
$296$	0	0
$297$	−24.5304	−1.42340
$298$	0	0
$299$	−51.0011	−2.94947
$300$	0	0
$301$	−18.3792	−1.05936
$302$	0	0
$303$	−16.1867	−0.929899
$304$	0	0
$305$	−8.25721	−0.472807
$306$	0	0
$307$	−6.04890	−0.345229	−0.172614	−	0.984989i	$-0.555221\pi$
$307$	−6.04890	−0.345229	−0.172614	+	0.984989i	$0.555221\pi$
$308$	0	0
$309$	−12.1010	−0.688401
$310$	0	0
$311$	−24.9972	−1.41746	−0.708730	−	0.705479i	$-0.750732\pi$
$311$	−24.9972	−1.41746	−0.708730	+	0.705479i	$0.750732\pi$
$312$	0	0
$313$	4.23082	0.239140	0.119570	−	0.992826i	$-0.461848\pi$
$313$	4.23082	0.239140	0.119570	+	0.992826i	$0.461848\pi$
$314$	0	0
$315$	−0.462336	−0.0260497
$316$	0	0
$317$	−32.3166	−1.81508	−0.907539	−	0.419967i	$-0.862042\pi$
$317$	−32.3166	−1.81508	−0.907539	+	0.419967i	$0.862042\pi$
$318$	0	0
$319$	−42.5217	−2.38076
$320$	0	0
$321$	13.2909	0.741823
$322$	0	0
$323$	−23.4766	−1.30627
$324$	0	0
$325$	6.44343	0.357417
$326$	0	0
$327$	−9.89940	−0.547438
$328$	0	0
$329$	2.90048	0.159908
$330$	0	0
$331$	−9.99656	−0.549461	−0.274730	−	0.961521i	$-0.588589\pi$
$331$	−9.99656	−0.549461	−0.274730	+	0.961521i	$0.588589\pi$
$332$	0	0
$333$	−0.401493	−0.0220017
$334$	0	0
$335$	−0.637996	−0.0348574
$336$	0	0
$337$	−10.8754	−0.592421	−0.296211	−	0.955123i	$-0.595723\pi$
$337$	−10.8754	−0.592421	−0.296211	+	0.955123i	$0.595723\pi$
$338$	0	0
$339$	−27.4072	−1.48855
$340$	0	0
$341$	−14.1934	−0.768618
$342$	0	0
$343$	18.9385	1.02259
$344$	0	0
$345$	13.2940	0.715725
$346$	0	0
$347$	33.7763	1.81321	0.906603	−	0.421986i	$-0.138667\pi$
$347$	33.7763	1.81321	0.906603	+	0.421986i	$0.138667\pi$
$348$	0	0
$349$	−22.7292	−1.21667	−0.608333	−	0.793682i	$-0.708162\pi$
$349$	−22.7292	−1.21667	−0.608333	+	0.793682i	$0.708162\pi$
$350$	0	0
$351$	−34.4045	−1.83638
$352$	0	0
$353$	11.3547	0.604352	0.302176	−	0.953252i	$-0.402287\pi$
$353$	11.3547	0.604352	0.302176	+	0.953252i	$0.402287\pi$
$354$	0	0
$355$	3.08049	0.163496
$356$	0	0
$357$	−13.9786	−0.739823
$358$	0	0
$359$	−14.4162	−0.760858	−0.380429	−	0.924810i	$-0.624224\pi$
$359$	−14.4162	−0.760858	−0.380429	+	0.924810i	$0.624224\pi$
$360$	0	0
$361$	34.0175	1.79040
$362$	0	0
$363$	16.9741	0.890907
$364$	0	0
$365$	5.63615	0.295010
$366$	0	0
$367$	20.7212	1.08164	0.540819	−	0.841139i	$-0.318114\pi$
$367$	20.7212	1.08164	0.540819	+	0.841139i	$0.318114\pi$
$368$	0	0
$369$	−0.382106	−0.0198917
$370$	0	0
$371$	34.1592	1.77346
$372$	0	0
$373$	31.5493	1.63356	0.816780	−	0.576949i	$-0.195757\pi$
$373$	31.5493	1.63356	0.816780	+	0.576949i	$0.195757\pi$
$374$	0	0
$375$	−1.67955	−0.0867316
$376$	0	0
$377$	−59.6378	−3.07151
$378$	0	0
$379$	−21.3142	−1.09484	−0.547419	−	0.836859i	$-0.684389\pi$
$379$	−21.3142	−1.09484	−0.547419	+	0.836859i	$0.684389\pi$
$380$	0	0
$381$	−7.72201	−0.395611
$382$	0	0
$383$	25.3013	1.29283	0.646417	−	0.762984i	$-0.276267\pi$
$383$	25.3013	1.29283	0.646417	+	0.762984i	$0.276267\pi$
$384$	0	0
$385$	11.8590	0.604392
$386$	0	0
$387$	−1.27526	−0.0648250
$388$	0	0
$389$	16.4409	0.833586	0.416793	−	0.909001i	$-0.363154\pi$
$389$	16.4409	0.833586	0.416793	+	0.909001i	$0.363154\pi$
$390$	0	0
$391$	−25.5204	−1.29062
$392$	0	0
$393$	6.48722	0.327237
$394$	0	0
$395$	−8.87088	−0.446342
$396$	0	0
$397$	−22.6377	−1.13615	−0.568076	−	0.822976i	$-0.692312\pi$
$397$	−22.6377	−1.13615	−0.568076	+	0.822976i	$0.692312\pi$
$398$	0	0
$399$	31.5679	1.58037
$400$	0	0
$401$	1.00000	0.0499376
$401$	1.00000	0.0499376
$402$	0	0
$403$	−19.9067	−0.991622
$404$	0	0
$405$	8.43060	0.418920
$406$	0	0
$407$	10.2984	0.510472
$408$	0	0
$409$	−10.5807	−0.523180	−0.261590	−	0.965179i	$-0.584247\pi$
$409$	−10.5807	−0.523180	−0.261590	+	0.965179i	$0.584247\pi$
$410$	0	0
$411$	−34.4700	−1.70028
$412$	0	0
$413$	−10.2568	−0.504703
$414$	0	0
$415$	−12.7834	−0.627510
$416$	0	0
$417$	−0.980163	−0.0479988
$418$	0	0
$419$	−1.55883	−0.0761537	−0.0380769	−	0.999275i	$-0.512123\pi$
$419$	−1.55883	−0.0761537	−0.0380769	+	0.999275i	$0.512123\pi$
$420$	0	0
$421$	1.07986	0.0526291	0.0263146	−	0.999654i	$-0.491623\pi$
$421$	1.07986	0.0526291	0.0263146	+	0.999654i	$0.491623\pi$
$422$	0	0
$423$	0.201252	0.00978522
$424$	0	0
$425$	3.22423	0.156398
$426$	0	0
$427$	−21.3146	−1.03148
$428$	0	0
$429$	49.7183	2.40042
$430$	0	0
$431$	31.1168	1.49884	0.749421	−	0.662094i	$-0.230332\pi$
$431$	31.1168	1.49884	0.749421	+	0.662094i	$0.230332\pi$
$432$	0	0
$433$	−21.3006	−1.02364	−0.511820	−	0.859092i	$-0.671029\pi$
$433$	−21.3006	−1.02364	−0.511820	+	0.859092i	$0.671029\pi$
$434$	0	0
$435$	15.5453	0.745338
$436$	0	0
$437$	57.6331	2.75697
$438$	0	0
$439$	8.57119	0.409080	0.204540	−	0.978858i	$-0.434430\pi$
$439$	8.57119	0.409080	0.204540	+	0.978858i	$0.434430\pi$
$440$	0	0
$441$	0.0603137	0.00287208
$442$	0	0
$443$	33.2640	1.58042	0.790210	−	0.612836i	$-0.209971\pi$
$443$	33.2640	1.58042	0.790210	+	0.612836i	$0.209971\pi$
$444$	0	0
$445$	−5.95597	−0.282340
$446$	0	0
$447$	−5.18291	−0.245143
$448$	0	0
$449$	−22.4485	−1.05941	−0.529704	−	0.848182i	$-0.677697\pi$
$449$	−22.4485	−1.05941	−0.529704	+	0.848182i	$0.677697\pi$
$450$	0	0
$451$	9.80111	0.461516
$452$	0	0
$453$	25.3013	1.18876
$454$	0	0
$455$	16.6326	0.779749
$456$	0	0
$457$	16.0366	0.750162	0.375081	−	0.926992i	$-0.377615\pi$
$457$	16.0366	0.750162	0.375081	+	0.926992i	$0.377615\pi$
$458$	0	0
$459$	−17.2157	−0.803559
$460$	0	0
$461$	2.43704	0.113504	0.0567521	−	0.998388i	$-0.481926\pi$
$461$	2.43704	0.113504	0.0567521	+	0.998388i	$0.481926\pi$
$462$	0	0
$463$	−37.5880	−1.74686	−0.873432	−	0.486946i	$-0.838111\pi$
$463$	−37.5880	−1.74686	−0.873432	+	0.486946i	$0.838111\pi$
$464$	0	0
$465$	5.18889	0.240629
$466$	0	0
$467$	10.5098	0.486335	0.243168	−	0.969984i	$-0.421813\pi$
$467$	10.5098	0.486335	0.243168	+	0.969984i	$0.421813\pi$
$468$	0	0
$469$	−1.64688	−0.0760457
$470$	0	0
$471$	−23.0140	−1.06043
$472$	0	0
$473$	32.7106	1.50404
$474$	0	0
$475$	−7.28131	−0.334090
$476$	0	0
$477$	2.37017	0.108522
$478$	0	0
$479$	−33.2100	−1.51740	−0.758701	−	0.651439i	$-0.774166\pi$
$479$	−33.2100	−1.51740	−0.758701	+	0.651439i	$0.774166\pi$
$480$	0	0
$481$	14.4438	0.658579
$482$	0	0
$483$	34.3162	1.56144
$484$	0	0
$485$	15.2444	0.692213
$486$	0	0
$487$	30.1639	1.36686	0.683428	−	0.730018i	$-0.260488\pi$
$487$	30.1639	1.36686	0.683428	+	0.730018i	$0.260488\pi$
$488$	0	0
$489$	−20.2247	−0.914593
$490$	0	0
$491$	−0.422756	−0.0190787	−0.00953936	−	0.999954i	$-0.503037\pi$
$491$	−0.422756	−0.0190787	−0.00953936	+	0.999954i	$0.503037\pi$
$492$	0	0
$493$	−29.8422	−1.34402
$494$	0	0
$495$	0.822850	0.0369844
$496$	0	0
$497$	7.95177	0.356685
$498$	0	0
$499$	−6.46635	−0.289474	−0.144737	−	0.989470i	$-0.546234\pi$
$499$	−6.46635	−0.289474	−0.144737	+	0.989470i	$0.546234\pi$
$500$	0	0
$501$	−12.4463	−0.556058
$502$	0	0
$503$	38.2733	1.70652	0.853261	−	0.521485i	$-0.174622\pi$
$503$	38.2733	1.70652	0.853261	+	0.521485i	$0.174622\pi$
$504$	0	0
$505$	9.63749	0.428863
$506$	0	0
$507$	47.8971	2.12718
$508$	0	0
$509$	−31.3957	−1.39159	−0.695794	−	0.718241i	$-0.744947\pi$
$509$	−31.3957	−1.39159	−0.695794	+	0.718241i	$0.744947\pi$
$510$	0	0
$511$	14.5488	0.643599
$512$	0	0
$513$	38.8784	1.71652
$514$	0	0
$515$	7.20489	0.317485
$516$	0	0
$517$	−5.16217	−0.227032
$518$	0	0
$519$	−7.61488	−0.334256
$520$	0	0
$521$	−17.5577	−0.769219	−0.384609	−	0.923079i	$-0.625664\pi$
$521$	−17.5577	−0.769219	−0.384609	+	0.923079i	$0.625664\pi$
$522$	0	0
$523$	1.58125	0.0691432	0.0345716	−	0.999402i	$-0.488993\pi$
$523$	1.58125	0.0691432	0.0345716	+	0.999402i	$0.488993\pi$
$524$	0	0
$525$	−4.33547	−0.189216
$526$	0	0
$527$	−9.96110	−0.433912
$528$	0	0
$529$	39.6506	1.72394
$530$	0	0
$531$	−0.711676	−0.0308841
$532$	0	0
$533$	13.7463	0.595419
$534$	0	0
$535$	−7.91334	−0.342123
$536$	0	0
$537$	−6.70457	−0.289323
$538$	0	0
$539$	−1.54706	−0.0666366
$540$	0	0
$541$	6.50947	0.279864	0.139932	−	0.990161i	$-0.455312\pi$
$541$	6.50947	0.279864	0.139932	+	0.990161i	$0.455312\pi$
$542$	0	0
$543$	−4.28605	−0.183932
$544$	0	0
$545$	5.89407	0.252474
$546$	0	0
$547$	34.4477	1.47288	0.736439	−	0.676504i	$-0.236506\pi$
$547$	34.4477	1.47288	0.736439	+	0.676504i	$0.236506\pi$
$548$	0	0
$549$	−1.47893	−0.0631193
$550$	0	0
$551$	67.3930	2.87104
$552$	0	0
$553$	−22.8987	−0.973750
$554$	0	0
$555$	−3.76492	−0.159812
$556$	0	0
$557$	−43.3036	−1.83483	−0.917416	−	0.397929i	$-0.869729\pi$
$557$	−43.3036	−1.83483	−0.917416	+	0.397929i	$0.869729\pi$
$558$	0	0
$559$	45.8775	1.94041
$560$	0	0
$561$	24.8786	1.05037
$562$	0	0
$563$	−42.4344	−1.78840	−0.894198	−	0.447671i	$-0.852254\pi$
$563$	−42.4344	−1.78840	−0.894198	+	0.447671i	$0.852254\pi$
$564$	0	0
$565$	16.3182	0.686510
$566$	0	0
$567$	21.7621	0.913924
$568$	0	0
$569$	20.3579	0.853448	0.426724	−	0.904382i	$-0.359668\pi$
$569$	20.3579	0.853448	0.426724	+	0.904382i	$0.359668\pi$
$570$	0	0
$571$	24.1569	1.01094	0.505468	−	0.862845i	$-0.331320\pi$
$571$	24.1569	1.01094	0.505468	+	0.862845i	$0.331320\pi$
$572$	0	0
$573$	10.6628	0.445444
$574$	0	0
$575$	−7.91521	−0.330087
$576$	0	0
$577$	−38.9382	−1.62102	−0.810510	−	0.585725i	$-0.800810\pi$
$577$	−38.9382	−1.62102	−0.810510	+	0.585725i	$0.800810\pi$
$578$	0	0
$579$	2.72186	0.113117
$580$	0	0
$581$	−32.9980	−1.36899
$582$	0	0
$583$	−60.7953	−2.51788
$584$	0	0
$585$	1.15407	0.0477149
$586$	0	0
$587$	−16.3066	−0.673047	−0.336524	−	0.941675i	$-0.609251\pi$
$587$	−16.3066	−0.673047	−0.336524	+	0.941675i	$0.609251\pi$
$588$	0	0
$589$	22.4953	0.926902
$590$	0	0
$591$	11.9957	0.493438
$592$	0	0
$593$	30.5859	1.25601	0.628006	−	0.778209i	$-0.283871\pi$
$593$	30.5859	1.25601	0.628006	+	0.778209i	$0.283871\pi$
$594$	0	0
$595$	8.32279	0.341201
$596$	0	0
$597$	4.28776	0.175486
$598$	0	0
$599$	−15.5538	−0.635511	−0.317755	−	0.948173i	$-0.602929\pi$
$599$	−15.5538	−0.635511	−0.317755	+	0.948173i	$0.602929\pi$
$600$	0	0
$601$	−28.9614	−1.18136	−0.590680	−	0.806906i	$-0.701140\pi$
$601$	−28.9614	−1.18136	−0.590680	+	0.806906i	$0.701140\pi$
$602$	0	0
$603$	−0.114270	−0.00465344
$604$	0	0
$605$	−10.1063	−0.410880
$606$	0	0
$607$	6.12304	0.248527	0.124263	−	0.992249i	$-0.460343\pi$
$607$	6.12304	0.248527	0.124263	+	0.992249i	$0.460343\pi$
$608$	0	0
$609$	40.1274	1.62605
$610$	0	0
$611$	−7.24008	−0.292902
$612$	0	0
$613$	−33.8842	−1.36857	−0.684286	−	0.729214i	$-0.739886\pi$
$613$	−33.8842	−1.36857	−0.684286	+	0.729214i	$0.739886\pi$
$614$	0	0
$615$	−3.58313	−0.144486
$616$	0	0
$617$	−5.52236	−0.222322	−0.111161	−	0.993802i	$-0.535457\pi$
$617$	−5.52236	−0.222322	−0.111161	+	0.993802i	$0.535457\pi$
$618$	0	0
$619$	6.63594	0.266721	0.133360	−	0.991068i	$-0.457423\pi$
$619$	6.63594	0.266721	0.133360	+	0.991068i	$0.457423\pi$
$620$	0	0
$621$	42.2631	1.69596
$622$	0	0
$623$	−15.3743	−0.615959
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−56.1835	−2.24375
$628$	0	0
$629$	7.22751	0.288180
$630$	0	0
$631$	−7.23719	−0.288108	−0.144054	−	0.989570i	$-0.546014\pi$
$631$	−7.23719	−0.288108	−0.144054	+	0.989570i	$0.546014\pi$
$632$	0	0
$633$	−14.0012	−0.556496
$634$	0	0
$635$	4.59767	0.182453
$636$	0	0
$637$	−2.16979	−0.0859703
$638$	0	0
$639$	0.551741	0.0218265
$640$	0	0
$641$	24.8354	0.980940	0.490470	−	0.871458i	$-0.336825\pi$
$641$	24.8354	0.980940	0.490470	+	0.871458i	$0.336825\pi$
$642$	0	0
$643$	28.7654	1.13440	0.567198	−	0.823582i	$-0.308028\pi$
$643$	28.7654	1.13440	0.567198	+	0.823582i	$0.308028\pi$
$644$	0	0
$645$	−11.9585	−0.470865
$646$	0	0
$647$	12.7061	0.499530	0.249765	−	0.968307i	$-0.419647\pi$
$647$	12.7061	0.499530	0.249765	+	0.968307i	$0.419647\pi$
$648$	0	0
$649$	18.2547	0.716558
$650$	0	0
$651$	13.3942	0.524962
$652$	0	0
$653$	−12.4665	−0.487851	−0.243926	−	0.969794i	$-0.578435\pi$
$653$	−12.4665	−0.487851	−0.243926	+	0.969794i	$0.578435\pi$
$654$	0	0
$655$	−3.86247	−0.150919
$656$	0	0
$657$	1.00948	0.0393835
$658$	0	0
$659$	−33.1425	−1.29105	−0.645525	−	0.763739i	$-0.723361\pi$
$659$	−33.1425	−1.29105	−0.645525	+	0.763739i	$0.723361\pi$
$660$	0	0
$661$	−4.30159	−0.167313	−0.0836563	−	0.996495i	$-0.526660\pi$
$661$	−4.30159	−0.167313	−0.0836563	+	0.996495i	$0.526660\pi$
$662$	0	0
$663$	34.8928	1.35513
$664$	0	0
$665$	−18.7955	−0.728857
$666$	0	0
$667$	73.2601	2.83664
$668$	0	0
$669$	−12.7223	−0.491874
$670$	0	0
$671$	37.9349	1.46446
$672$	0	0
$673$	40.4514	1.55929	0.779643	−	0.626224i	$-0.215400\pi$
$673$	40.4514	1.55929	0.779643	+	0.626224i	$0.215400\pi$
$674$	0	0
$675$	−5.33947	−0.205516
$676$	0	0
$677$	47.6791	1.83246	0.916229	−	0.400655i	$-0.131217\pi$
$677$	47.6791	1.83246	0.916229	+	0.400655i	$0.131217\pi$
$678$	0	0
$679$	39.3508	1.51015
$680$	0	0
$681$	−17.2393	−0.660612
$682$	0	0
$683$	−47.4403	−1.81525	−0.907626	−	0.419780i	$-0.862107\pi$
$683$	−47.4403	−1.81525	−0.907626	+	0.419780i	$0.862107\pi$
$684$	0	0
$685$	20.5234	0.784157
$686$	0	0
$687$	−10.7991	−0.412009
$688$	0	0
$689$	−85.2670	−3.24841
$690$	0	0
$691$	31.8319	1.21094	0.605470	−	0.795868i	$-0.292985\pi$
$691$	31.8319	1.21094	0.605470	+	0.795868i	$0.292985\pi$
$692$	0	0
$693$	2.12405	0.0806859
$694$	0	0
$695$	0.583586	0.0221367
$696$	0	0
$697$	6.87852	0.260543
$698$	0	0
$699$	27.6967	1.04758
$700$	0	0
$701$	−39.7995	−1.50321	−0.751603	−	0.659616i	$-0.770719\pi$
$701$	−39.7995	−1.50321	−0.751603	+	0.659616i	$0.770719\pi$
$702$	0	0
$703$	−16.3220	−0.615595
$704$	0	0
$705$	1.88721	0.0710763
$706$	0	0
$707$	24.8775	0.935615
$708$	0	0
$709$	9.17219	0.344469	0.172234	−	0.985056i	$-0.444901\pi$
$709$	9.17219	0.344469	0.172234	+	0.985056i	$0.444901\pi$
$710$	0	0
$711$	−1.58884	−0.0595863
$712$	0	0
$713$	24.4537	0.915797
$714$	0	0
$715$	−29.6021	−1.10706
$716$	0	0
$717$	2.67636	0.0999506
$718$	0	0
$719$	−30.8573	−1.15078	−0.575392	−	0.817878i	$-0.695151\pi$
$719$	−30.8573	−1.15078	−0.575392	+	0.817878i	$0.695151\pi$
$720$	0	0
$721$	18.5982	0.692633
$722$	0	0
$723$	−4.54101	−0.168882
$724$	0	0
$725$	−9.25560	−0.343745
$726$	0	0
$727$	31.8367	1.18076	0.590378	−	0.807127i	$-0.298979\pi$
$727$	31.8367	1.18076	0.590378	+	0.807127i	$0.298979\pi$
$728$	0	0
$729$	28.4137	1.05236
$730$	0	0
$731$	22.9567	0.849083
$732$	0	0
$733$	−24.1615	−0.892424	−0.446212	−	0.894927i	$-0.647227\pi$
$733$	−24.1615	−0.892424	−0.446212	+	0.894927i	$0.647227\pi$
$734$	0	0
$735$	0.565580	0.0208617
$736$	0	0
$737$	2.93105	0.107967
$738$	0	0
$739$	39.2435	1.44360	0.721798	−	0.692103i	$-0.243316\pi$
$739$	39.2435	1.44360	0.721798	+	0.692103i	$0.243316\pi$
$740$	0	0
$741$	−78.7989	−2.89475
$742$	0	0
$743$	13.5190	0.495963	0.247982	−	0.968765i	$-0.420233\pi$
$743$	13.5190	0.495963	0.247982	+	0.968765i	$0.420233\pi$
$744$	0	0
$745$	3.08589	0.113058
$746$	0	0
$747$	−2.28960	−0.0837721
$748$	0	0
$749$	−20.4269	−0.746383
$750$	0	0
$751$	−2.36282	−0.0862204	−0.0431102	−	0.999070i	$-0.513727\pi$
$751$	−2.36282	−0.0862204	−0.0431102	+	0.999070i	$0.513727\pi$
$752$	0	0
$753$	−1.32253	−0.0481956
$754$	0	0
$755$	−15.0643	−0.548246
$756$	0	0
$757$	19.4097	0.705459	0.352729	−	0.935725i	$-0.385254\pi$
$757$	19.4097	0.705459	0.352729	+	0.935725i	$0.385254\pi$
$758$	0	0
$759$	−61.0748	−2.21687
$760$	0	0
$761$	27.4865	0.996383	0.498192	−	0.867067i	$-0.333998\pi$
$761$	27.4865	0.996383	0.498192	+	0.867067i	$0.333998\pi$
$762$	0	0
$763$	15.2145	0.550803
$764$	0	0
$765$	0.577485	0.0208790
$766$	0	0
$767$	25.6026	0.924458
$768$	0	0
$769$	29.6475	1.06912	0.534558	−	0.845132i	$-0.320478\pi$
$769$	29.6475	1.06912	0.534558	+	0.845132i	$0.320478\pi$
$770$	0	0
$771$	14.3234	0.515846
$772$	0	0
$773$	−33.7025	−1.21219	−0.606096	−	0.795391i	$-0.707265\pi$
$773$	−33.7025	−1.21219	−0.606096	+	0.795391i	$0.707265\pi$
$774$	0	0
$775$	−3.08945	−0.110976
$776$	0	0
$777$	−9.71851	−0.348649
$778$	0	0
$779$	−15.5338	−0.556558
$780$	0	0
$781$	−14.1523	−0.506408
$782$	0	0
$783$	49.4201	1.76613
$784$	0	0
$785$	13.7025	0.489062
$786$	0	0
$787$	−41.8199	−1.49072	−0.745358	−	0.666664i	$-0.767722\pi$
$787$	−41.8199	−1.49072	−0.745358	+	0.666664i	$0.767722\pi$
$788$	0	0
$789$	−21.9402	−0.781090
$790$	0	0
$791$	42.1225	1.49770
$792$	0	0
$793$	53.2048	1.88936
$794$	0	0
$795$	22.2258	0.788267
$796$	0	0
$797$	8.16193	0.289110	0.144555	−	0.989497i	$-0.453825\pi$
$797$	8.16193	0.289110	0.144555	+	0.989497i	$0.453825\pi$
$798$	0	0
$799$	−3.62286	−0.128168
$800$	0	0
$801$	−1.06676	−0.0376922
$802$	0	0
$803$	−25.8934	−0.913757
$804$	0	0
$805$	−20.4318	−0.720125
$806$	0	0
$807$	−47.0891	−1.65762
$808$	0	0
$809$	26.9762	0.948433	0.474216	−	0.880408i	$-0.342731\pi$
$809$	26.9762	0.948433	0.474216	+	0.880408i	$0.342731\pi$
$810$	0	0
$811$	5.65551	0.198592	0.0992960	−	0.995058i	$-0.468341\pi$
$811$	5.65551	0.198592	0.0992960	+	0.995058i	$0.468341\pi$
$812$	0	0
$813$	−23.8357	−0.835956
$814$	0	0
$815$	12.0417	0.421803
$816$	0	0
$817$	−51.8433	−1.81377
$818$	0	0
$819$	2.97903	0.104096
$820$	0	0
$821$	21.5188	0.751010	0.375505	−	0.926820i	$-0.377469\pi$
$821$	21.5188	0.751010	0.375505	+	0.926820i	$0.377469\pi$
$822$	0	0
$823$	21.0631	0.734214	0.367107	−	0.930179i	$-0.380348\pi$
$823$	21.0631	0.734214	0.367107	+	0.930179i	$0.380348\pi$
$824$	0	0
$825$	7.71613	0.268641
$826$	0	0
$827$	−7.17097	−0.249359	−0.124679	−	0.992197i	$-0.539790\pi$
$827$	−7.17097	−0.249359	−0.124679	+	0.992197i	$0.539790\pi$
$828$	0	0
$829$	−11.9526	−0.415132	−0.207566	−	0.978221i	$-0.566554\pi$
$829$	−11.9526	−0.415132	−0.207566	+	0.978221i	$0.566554\pi$
$830$	0	0
$831$	−14.2236	−0.493411
$832$	0	0
$833$	−1.08574	−0.0376187
$834$	0	0
$835$	7.41046	0.256450
$836$	0	0
$837$	16.4961	0.570187
$838$	0	0
$839$	37.0510	1.27914	0.639572	−	0.768731i	$-0.279112\pi$
$839$	37.0510	1.27914	0.639572	+	0.768731i	$0.279112\pi$
$840$	0	0
$841$	56.6662	1.95401
$842$	0	0
$843$	−17.4502	−0.601018
$844$	0	0
$845$	−28.5178	−0.981042
$846$	0	0
$847$	−26.0877	−0.896383
$848$	0	0
$849$	7.55617	0.259327
$850$	0	0
$851$	−17.7429	−0.608220
$852$	0	0
$853$	32.1374	1.10037	0.550183	−	0.835044i	$-0.314558\pi$
$853$	32.1374	1.10037	0.550183	+	0.835044i	$0.314558\pi$
$854$	0	0
$855$	−1.30414	−0.0446007
$856$	0	0
$857$	−32.0607	−1.09517	−0.547587	−	0.836749i	$-0.684454\pi$
$857$	−32.0607	−1.09517	−0.547587	+	0.836749i	$0.684454\pi$
$858$	0	0
$859$	−15.5109	−0.529224	−0.264612	−	0.964355i	$-0.585244\pi$
$859$	−15.5109	−0.529224	−0.264612	+	0.964355i	$0.585244\pi$
$860$	0	0
$861$	−9.24923	−0.315213
$862$	0	0
$863$	51.8139	1.76376	0.881882	−	0.471470i	$-0.156276\pi$
$863$	51.8139	1.76376	0.881882	+	0.471470i	$0.156276\pi$
$864$	0	0
$865$	4.53388	0.154156
$866$	0	0
$867$	−11.0923	−0.376716
$868$	0	0
$869$	40.7542	1.38249
$870$	0	0
$871$	4.11088	0.139292
$872$	0	0
$873$	2.73039	0.0924099
$874$	0	0
$875$	2.58133	0.0872648
$876$	0	0
$877$	31.2657	1.05577	0.527884	−	0.849317i	$-0.322986\pi$
$877$	31.2657	1.05577	0.527884	+	0.849317i	$0.322986\pi$
$878$	0	0
$879$	20.5506	0.693156
$880$	0	0
$881$	18.7450	0.631536	0.315768	−	0.948836i	$-0.397738\pi$
$881$	18.7450	0.631536	0.315768	+	0.948836i	$0.397738\pi$
$882$	0	0
$883$	8.62047	0.290102	0.145051	−	0.989424i	$-0.453665\pi$
$883$	8.62047	0.290102	0.145051	+	0.989424i	$0.453665\pi$
$884$	0	0
$885$	−6.67361	−0.224331
$886$	0	0
$887$	7.43060	0.249495	0.124748	−	0.992189i	$-0.460188\pi$
$887$	7.43060	0.249495	0.124748	+	0.992189i	$0.460188\pi$
$888$	0	0
$889$	11.8681	0.398043
$890$	0	0
$891$	−38.7315	−1.29755
$892$	0	0
$893$	8.18155	0.273785
$894$	0	0
$895$	3.99188	0.133434
$896$	0	0
$897$	−85.6590	−2.86007
$898$	0	0
$899$	28.5948	0.953688
$900$	0	0
$901$	−42.6668	−1.42144
$902$	0	0
$903$	−30.8688	−1.02725
$904$	0	0
$905$	2.55190	0.0848281
$906$	0	0
$907$	−39.5834	−1.31435	−0.657173	−	0.753740i	$-0.728248\pi$
$907$	−39.5834	−1.31435	−0.657173	+	0.753740i	$0.728248\pi$
$908$	0	0
$909$	1.72615	0.0572528
$910$	0	0
$911$	24.8966	0.824862	0.412431	−	0.910989i	$-0.364680\pi$
$911$	24.8966	0.824862	0.412431	+	0.910989i	$0.364680\pi$
$912$	0	0
$913$	58.7288	1.94364
$914$	0	0
$915$	−13.8684	−0.458475
$916$	0	0
$917$	−9.97031	−0.329249
$918$	0	0
$919$	34.0156	1.12207	0.561035	−	0.827792i	$-0.310403\pi$
$919$	34.0156	1.12207	0.561035	+	0.827792i	$0.310403\pi$
$920$	0	0
$921$	−10.1594	−0.334765
$922$	0	0
$923$	−19.8490	−0.653336
$924$	0	0
$925$	2.24163	0.0737042
$926$	0	0
$927$	1.29045	0.0423840
$928$	0	0
$929$	7.65125	0.251029	0.125515	−	0.992092i	$-0.459942\pi$
$929$	7.65125	0.251029	0.125515	+	0.992092i	$0.459942\pi$
$930$	0	0
$931$	2.45194	0.0803592
$932$	0	0
$933$	−41.9841	−1.37450
$934$	0	0
$935$	−14.8126	−0.484424
$936$	0	0
$937$	−1.89496	−0.0619056	−0.0309528	−	0.999521i	$-0.509854\pi$
$937$	−1.89496	−0.0619056	−0.0309528	+	0.999521i	$0.509854\pi$
$938$	0	0
$939$	7.10588	0.231892
$940$	0	0
$941$	−13.5652	−0.442213	−0.221107	−	0.975250i	$-0.570967\pi$
$941$	−13.5652	−0.442213	−0.221107	+	0.975250i	$0.570967\pi$
$942$	0	0
$943$	−16.8862	−0.549890
$944$	0	0
$945$	−13.7829	−0.448359
$946$	0	0
$947$	−0.317586	−0.0103202	−0.00516008	−	0.999987i	$-0.501643\pi$
$947$	−0.317586	−0.0103202	−0.00516008	+	0.999987i	$0.501643\pi$
$948$	0	0
$949$	−36.3161	−1.17887
$950$	0	0
$951$	−54.2773	−1.76006
$952$	0	0
$953$	−28.1040	−0.910377	−0.455188	−	0.890395i	$-0.650428\pi$
$953$	−28.1040	−0.910377	−0.455188	+	0.890395i	$0.650428\pi$
$954$	0	0
$955$	−6.34859	−0.205435
$956$	0	0
$957$	−71.4174	−2.30860
$958$	0	0
$959$	52.9775	1.71073
$960$	0	0
$961$	−21.4553	−0.692106
$962$	0	0
$963$	−1.41734	−0.0456732
$964$	0	0
$965$	−1.62059	−0.0521686
$966$	0	0
$967$	17.0331	0.547747	0.273873	−	0.961766i	$-0.411695\pi$
$967$	17.0331	0.547747	0.273873	+	0.961766i	$0.411695\pi$
$968$	0	0
$969$	−39.4302	−1.26668
$970$	0	0
$971$	−7.21690	−0.231601	−0.115801	−	0.993272i	$-0.536943\pi$
$971$	−7.21690	−0.231601	−0.115801	+	0.993272i	$0.536943\pi$
$972$	0	0
$973$	1.50643	0.0482938
$974$	0	0
$975$	10.8221	0.346584
$976$	0	0
$977$	−32.0798	−1.02633	−0.513163	−	0.858291i	$-0.671526\pi$
$977$	−32.0798	−1.02633	−0.513163	+	0.858291i	$0.671526\pi$
$978$	0	0
$979$	27.3627	0.874515
$980$	0	0
$981$	1.05568	0.0337051
$982$	0	0
$983$	20.3622	0.649453	0.324726	−	0.945808i	$-0.394728\pi$
$983$	20.3622	0.649453	0.324726	+	0.945808i	$0.394728\pi$
$984$	0	0
$985$	−7.14222	−0.227570
$986$	0	0
$987$	4.87150	0.155061
$988$	0	0
$989$	−56.3567	−1.79204
$990$	0	0
$991$	−46.1409	−1.46571	−0.732857	−	0.680383i	$-0.761813\pi$
$991$	−46.1409	−1.46571	−0.732857	+	0.680383i	$0.761813\pi$
$992$	0	0
$993$	−16.7897	−0.532806
$994$	0	0
$995$	−2.55292	−0.0809330
$996$	0	0
$997$	37.3831	1.18393	0.591967	−	0.805962i	$-0.298351\pi$
$997$	37.3831	1.18393	0.591967	+	0.805962i	$0.298351\pi$
$998$	0	0
$999$	−11.9691	−0.378686

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8020.2.a.c.1.20	✓	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8020.2.a.c.1.20	✓	28	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)

\(f(q)\)	\(=\)	\(q+1.67955 q^{3} -1.00000 q^{5} -2.58133 q^{7} -0.179108 q^{9} +O(q^{10})\)	\(q+1.67955 q^{3} -1.00000 q^{5} -2.58133 q^{7} -0.179108 q^{9} +4.59416 q^{11} +6.44343 q^{13} -1.67955 q^{15} +3.22423 q^{17} -7.28131 q^{19} -4.33547 q^{21} -7.91521 q^{23} +1.00000 q^{25} -5.33947 q^{27} -9.25560 q^{29} -3.08945 q^{31} +7.71613 q^{33} +2.58133 q^{35} +2.24163 q^{37} +10.8221 q^{39} +2.13339 q^{41} +7.12005 q^{43} +0.179108 q^{45} -1.12364 q^{47} -0.336745 q^{49} +5.41526 q^{51} -13.2332 q^{53} -4.59416 q^{55} -12.2293 q^{57} +3.97345 q^{59} +8.25721 q^{61} +0.462336 q^{63} -6.44343 q^{65} +0.637996 q^{67} -13.2940 q^{69} -3.08049 q^{71} -5.63615 q^{73} +1.67955 q^{75} -11.8590 q^{77} +8.87088 q^{79} -8.43060 q^{81} +12.7834 q^{83} -3.22423 q^{85} -15.5453 q^{87} +5.95597 q^{89} -16.6326 q^{91} -5.18889 q^{93} +7.28131 q^{95} -15.2444 q^{97} -0.822850 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 3 q^{3} - 28 q^{5} - 4 q^{7} + 17 q^{9}+O(q^{10}) \) 28 * q + 3 * q^3 - 28 * q^5 - 4 * q^7 + 17 * q^9	\( 28 q + 3 q^{3} - 28 q^{5} - 4 q^{7} + 17 q^{9} + 2 q^{11} + 3 q^{13} - 3 q^{15} - 10 q^{17} - 2 q^{19} - 12 q^{21} - 23 q^{23} + 28 q^{25} + 9 q^{27} - 37 q^{29} - 11 q^{31} + 2 q^{33} + 4 q^{35} - 3 q^{37} - 19 q^{39} - 30 q^{41} + 13 q^{43} - 17 q^{45} - 15 q^{47} + 12 q^{49} - 8 q^{51} - 35 q^{53} - 2 q^{55} - 22 q^{57} - q^{59} - 33 q^{61} - 20 q^{63} - 3 q^{65} + 19 q^{67} - 8 q^{69} - 31 q^{71} + 31 q^{73} + 3 q^{75} - 42 q^{77} - 29 q^{79} - 36 q^{81} + 14 q^{83} + 10 q^{85} - 32 q^{87} - 32 q^{89} - 7 q^{91} - 11 q^{93} + 2 q^{95} + 2 q^{97} - 39 q^{99}+O(q^{100}) \) 28 * q + 3 * q^3 - 28 * q^5 - 4 * q^7 + 17 * q^9 + 2 * q^11 + 3 * q^13 - 3 * q^15 - 10 * q^17 - 2 * q^19 - 12 * q^21 - 23 * q^23 + 28 * q^25 + 9 * q^27 - 37 * q^29 - 11 * q^31 + 2 * q^33 + 4 * q^35 - 3 * q^37 - 19 * q^39 - 30 * q^41 + 13 * q^43 - 17 * q^45 - 15 * q^47 + 12 * q^49 - 8 * q^51 - 35 * q^53 - 2 * q^55 - 22 * q^57 - q^59 - 33 * q^61 - 20 * q^63 - 3 * q^65 + 19 * q^67 - 8 * q^69 - 31 * q^71 + 31 * q^73 + 3 * q^75 - 42 * q^77 - 29 * q^79 - 36 * q^81 + 14 * q^83 + 10 * q^85 - 32 * q^87 - 32 * q^89 - 7 * q^91 - 11 * q^93 + 2 * q^95 + 2 * q^97 - 39 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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