LMFDB - Embedded newform 8020.2.a.c.1.11

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			1.11
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)

$f(q)$	$=$	$q-0.750059 q^{3} -1.00000 q^{5} -4.31732 q^{7} -2.43741 q^{9} +O(q^{10})$	$q-0.750059 q^{3} -1.00000 q^{5} -4.31732 q^{7} -2.43741 q^{9} +1.26278 q^{11} +0.840199 q^{13} +0.750059 q^{15} +1.74261 q^{17} +4.53194 q^{19} +3.23824 q^{21} -7.00253 q^{23} +1.00000 q^{25} +4.07838 q^{27} -5.86477 q^{29} +5.85507 q^{31} -0.947155 q^{33} +4.31732 q^{35} +9.40647 q^{37} -0.630198 q^{39} -6.00340 q^{41} +6.28753 q^{43} +2.43741 q^{45} +1.12131 q^{47} +11.6393 q^{49} -1.30706 q^{51} +0.740586 q^{53} -1.26278 q^{55} -3.39922 q^{57} -1.87130 q^{59} -4.43841 q^{61} +10.5231 q^{63} -0.840199 q^{65} -2.32847 q^{67} +5.25231 q^{69} -11.5920 q^{71} +14.5436 q^{73} -0.750059 q^{75} -5.45181 q^{77} +16.2442 q^{79} +4.25321 q^{81} -6.59783 q^{83} -1.74261 q^{85} +4.39892 q^{87} -9.88080 q^{89} -3.62741 q^{91} -4.39165 q^{93} -4.53194 q^{95} +4.74520 q^{97} -3.07790 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−0.750059	−0.433047	−0.216523	−	0.976277i	$-0.569472\pi$
$3$	−0.750059	−0.433047	−0.216523	+	0.976277i	$0.569472\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−4.31732	−1.63179	−0.815897	−	0.578197i	$-0.803756\pi$
$7$	−4.31732	−1.63179	−0.815897	+	0.578197i	$0.803756\pi$
$8$	0	0
$9$	−2.43741	−0.812471
$10$	0	0
$11$	1.26278	0.380741	0.190371	−	0.981712i	$-0.439031\pi$
$11$	1.26278	0.380741	0.190371	+	0.981712i	$0.439031\pi$
$12$	0	0
$13$	0.840199	0.233029	0.116515	−	0.993189i	$-0.462828\pi$
$13$	0.840199	0.233029	0.116515	+	0.993189i	$0.462828\pi$
$14$	0	0
$15$	0.750059	0.193664
$16$	0	0
$17$	1.74261	0.422646	0.211323	−	0.977416i	$-0.432223\pi$
$17$	1.74261	0.422646	0.211323	+	0.977416i	$0.432223\pi$
$18$	0	0
$19$	4.53194	1.03970	0.519849	−	0.854258i	$-0.325988\pi$
$19$	4.53194	1.03970	0.519849	+	0.854258i	$0.325988\pi$
$20$	0	0
$21$	3.23824	0.706643
$22$	0	0
$23$	−7.00253	−1.46013	−0.730065	−	0.683378i	$-0.760510\pi$
$23$	−7.00253	−1.46013	−0.730065	+	0.683378i	$0.760510\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	4.07838	0.784884
$28$	0	0
$29$	−5.86477	−1.08906	−0.544530	−	0.838741i	$-0.683292\pi$
$29$	−5.86477	−1.08906	−0.544530	+	0.838741i	$0.683292\pi$
$30$	0	0
$31$	5.85507	1.05160	0.525801	−	0.850608i	$-0.323766\pi$
$31$	5.85507	1.05160	0.525801	+	0.850608i	$0.323766\pi$
$32$	0	0
$33$	−0.947155	−0.164879
$34$	0	0
$35$	4.31732	0.729760
$36$	0	0
$37$	9.40647	1.54641	0.773207	−	0.634154i	$-0.218651\pi$
$37$	9.40647	1.54641	0.773207	+	0.634154i	$0.218651\pi$
$38$	0	0
$39$	−0.630198	−0.100913
$40$	0	0
$41$	−6.00340	−0.937574	−0.468787	−	0.883311i	$-0.655309\pi$
$41$	−6.00340	−0.937574	−0.468787	+	0.883311i	$0.655309\pi$
$42$	0	0
$43$	6.28753	0.958840	0.479420	−	0.877586i	$-0.340847\pi$
$43$	6.28753	0.958840	0.479420	+	0.877586i	$0.340847\pi$
$44$	0	0
$45$	2.43741	0.363348
$46$	0	0
$47$	1.12131	0.163559	0.0817797	−	0.996650i	$-0.473940\pi$
$47$	1.12131	0.163559	0.0817797	+	0.996650i	$0.473940\pi$
$48$	0	0
$49$	11.6393	1.66275
$50$	0	0
$51$	−1.30706	−0.183025
$52$	0	0
$53$	0.740586	0.101727	0.0508637	−	0.998706i	$-0.483803\pi$
$53$	0.740586	0.101727	0.0508637	+	0.998706i	$0.483803\pi$
$54$	0	0
$55$	−1.26278	−0.170273
$56$	0	0
$57$	−3.39922	−0.450237
$58$	0	0
$59$	−1.87130	−0.243622	−0.121811	−	0.992553i	$-0.538870\pi$
$59$	−1.87130	−0.243622	−0.121811	+	0.992553i	$0.538870\pi$
$60$	0	0
$61$	−4.43841	−0.568281	−0.284140	−	0.958783i	$-0.591708\pi$
$61$	−4.43841	−0.568281	−0.284140	+	0.958783i	$0.591708\pi$
$62$	0	0
$63$	10.5231	1.32578
$64$	0	0
$65$	−0.840199	−0.104214
$66$	0	0
$67$	−2.32847	−0.284468	−0.142234	−	0.989833i	$-0.545428\pi$
$67$	−2.32847	−0.284468	−0.142234	+	0.989833i	$0.545428\pi$
$68$	0	0
$69$	5.25231	0.632304
$70$	0	0
$71$	−11.5920	−1.37571	−0.687857	−	0.725846i	$-0.741448\pi$
$71$	−11.5920	−1.37571	−0.687857	+	0.725846i	$0.741448\pi$
$72$	0	0
$73$	14.5436	1.70220	0.851099	−	0.525005i	$-0.175936\pi$
$73$	14.5436	1.70220	0.851099	+	0.525005i	$0.175936\pi$
$74$	0	0
$75$	−0.750059	−0.0866093
$76$	0	0
$77$	−5.45181	−0.621291
$78$	0	0
$79$	16.2442	1.82761	0.913805	−	0.406154i	$-0.133130\pi$
$79$	16.2442	1.82761	0.913805	+	0.406154i	$0.133130\pi$
$80$	0	0
$81$	4.25321	0.472579
$82$	0	0
$83$	−6.59783	−0.724205	−0.362103	−	0.932138i	$-0.617941\pi$
$83$	−6.59783	−0.724205	−0.362103	+	0.932138i	$0.617941\pi$
$84$	0	0
$85$	−1.74261	−0.189013
$86$	0	0
$87$	4.39892	0.471614
$88$	0	0
$89$	−9.88080	−1.04736	−0.523681	−	0.851914i	$-0.675442\pi$
$89$	−9.88080	−1.04736	−0.523681	+	0.851914i	$0.675442\pi$
$90$	0	0
$91$	−3.62741	−0.380256
$92$	0	0
$93$	−4.39165	−0.455392
$94$	0	0
$95$	−4.53194	−0.464967
$96$	0	0
$97$	4.74520	0.481802	0.240901	−	0.970550i	$-0.422557\pi$
$97$	4.74520	0.481802	0.240901	+	0.970550i	$0.422557\pi$
$98$	0	0
$99$	−3.07790	−0.309341
$100$	0	0
$101$	−7.87274	−0.783367	−0.391683	−	0.920100i	$-0.628107\pi$
$101$	−7.87274	−0.783367	−0.391683	+	0.920100i	$0.628107\pi$
$102$	0	0
$103$	17.8281	1.75665	0.878325	−	0.478063i	$-0.158661\pi$
$103$	17.8281	1.75665	0.878325	+	0.478063i	$0.158661\pi$
$104$	0	0
$105$	−3.23824	−0.316020
$106$	0	0
$107$	−0.215578	−0.0208407	−0.0104203	−	0.999946i	$-0.503317\pi$
$107$	−0.215578	−0.0208407	−0.0104203	+	0.999946i	$0.503317\pi$
$108$	0	0
$109$	5.40467	0.517674	0.258837	−	0.965921i	$-0.416661\pi$
$109$	5.40467	0.517674	0.258837	+	0.965921i	$0.416661\pi$
$110$	0	0
$111$	−7.05540	−0.669669
$112$	0	0
$113$	15.9257	1.49817	0.749083	−	0.662476i	$-0.230495\pi$
$113$	15.9257	1.49817	0.749083	+	0.662476i	$0.230495\pi$
$114$	0	0
$115$	7.00253	0.652990
$116$	0	0
$117$	−2.04791	−0.189329
$118$	0	0
$119$	−7.52343	−0.689671
$120$	0	0
$121$	−9.40540	−0.855036
$122$	0	0
$123$	4.50290	0.406013
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	1.24363	0.110355	0.0551773	−	0.998477i	$-0.482428\pi$
$127$	1.24363	0.110355	0.0551773	+	0.998477i	$0.482428\pi$
$128$	0	0
$129$	−4.71602	−0.415222
$130$	0	0
$131$	0.986253	0.0861693	0.0430846	−	0.999071i	$-0.486281\pi$
$131$	0.986253	0.0861693	0.0430846	+	0.999071i	$0.486281\pi$
$132$	0	0
$133$	−19.5658	−1.69657
$134$	0	0
$135$	−4.07838	−0.351011
$136$	0	0
$137$	−4.44041	−0.379369	−0.189685	−	0.981845i	$-0.560747\pi$
$137$	−4.44041	−0.379369	−0.189685	+	0.981845i	$0.560747\pi$
$138$	0	0
$139$	−4.08344	−0.346352	−0.173176	−	0.984891i	$-0.555403\pi$
$139$	−4.08344	−0.346352	−0.173176	+	0.984891i	$0.555403\pi$
$140$	0	0
$141$	−0.841045	−0.0708288
$142$	0	0
$143$	1.06098	0.0887238
$144$	0	0
$145$	5.86477	0.487043
$146$	0	0
$147$	−8.73013	−0.720049
$148$	0	0
$149$	12.1336	0.994020	0.497010	−	0.867745i	$-0.334431\pi$
$149$	12.1336	0.994020	0.497010	+	0.867745i	$0.334431\pi$
$150$	0	0
$151$	−7.99445	−0.650579	−0.325290	−	0.945614i	$-0.605462\pi$
$151$	−7.99445	−0.650579	−0.325290	+	0.945614i	$0.605462\pi$
$152$	0	0
$153$	−4.24747	−0.343388
$154$	0	0
$155$	−5.85507	−0.470291
$156$	0	0
$157$	−15.1332	−1.20776	−0.603882	−	0.797074i	$-0.706380\pi$
$157$	−15.1332	−1.20776	−0.603882	+	0.797074i	$0.706380\pi$
$158$	0	0
$159$	−0.555483	−0.0440527
$160$	0	0
$161$	30.2322	2.38263
$162$	0	0
$163$	21.4567	1.68062	0.840308	−	0.542109i	$-0.182374\pi$
$163$	21.4567	1.68062	0.840308	+	0.542109i	$0.182374\pi$
$164$	0	0
$165$	0.947155	0.0737360
$166$	0	0
$167$	9.37763	0.725663	0.362831	−	0.931855i	$-0.381810\pi$
$167$	9.37763	0.725663	0.362831	+	0.931855i	$0.381810\pi$
$168$	0	0
$169$	−12.2941	−0.945697
$170$	0	0
$171$	−11.0462	−0.844724
$172$	0	0
$173$	4.51979	0.343633	0.171817	−	0.985129i	$-0.445036\pi$
$173$	4.51979	0.343633	0.171817	+	0.985129i	$0.445036\pi$
$174$	0	0
$175$	−4.31732	−0.326359
$176$	0	0
$177$	1.40358	0.105500
$178$	0	0
$179$	4.72827	0.353408	0.176704	−	0.984264i	$-0.443456\pi$
$179$	4.72827	0.353408	0.176704	+	0.984264i	$0.443456\pi$
$180$	0	0
$181$	−24.3759	−1.81185	−0.905925	−	0.423438i	$-0.860823\pi$
$181$	−24.3759	−1.81185	−0.905925	+	0.423438i	$0.860823\pi$
$182$	0	0
$183$	3.32907	0.246092
$184$	0	0
$185$	−9.40647	−0.691577
$186$	0	0
$187$	2.20053	0.160919
$188$	0	0
$189$	−17.6077	−1.28077
$190$	0	0
$191$	−17.4655	−1.26376	−0.631878	−	0.775068i	$-0.717716\pi$
$191$	−17.4655	−1.26376	−0.631878	+	0.775068i	$0.717716\pi$
$192$	0	0
$193$	−13.6808	−0.984767	−0.492383	−	0.870378i	$-0.663874\pi$
$193$	−13.6808	−0.984767	−0.492383	+	0.870378i	$0.663874\pi$
$194$	0	0
$195$	0.630198	0.0451294
$196$	0	0
$197$	−5.60348	−0.399231	−0.199616	−	0.979874i	$-0.563969\pi$
$197$	−5.60348	−0.399231	−0.199616	+	0.979874i	$0.563969\pi$
$198$	0	0
$199$	−12.7383	−0.902995	−0.451497	−	0.892273i	$-0.649110\pi$
$199$	−12.7383	−0.902995	−0.451497	+	0.892273i	$0.649110\pi$
$200$	0	0
$201$	1.74649	0.123188
$202$	0	0
$203$	25.3201	1.77712
$204$	0	0
$205$	6.00340	0.419296
$206$	0	0
$207$	17.0681	1.18631
$208$	0	0
$209$	5.72282	0.395856
$210$	0	0
$211$	−8.31509	−0.572435	−0.286217	−	0.958165i	$-0.592398\pi$
$211$	−8.31509	−0.572435	−0.286217	+	0.958165i	$0.592398\pi$
$212$	0	0
$213$	8.69466	0.595748
$214$	0	0
$215$	−6.28753	−0.428806
$216$	0	0
$217$	−25.2782	−1.71600
$218$	0	0
$219$	−10.9085	−0.737131
$220$	0	0
$221$	1.46414	0.0984889
$222$	0	0
$223$	−0.527641	−0.0353334	−0.0176667	−	0.999844i	$-0.505624\pi$
$223$	−0.527641	−0.0353334	−0.0176667	+	0.999844i	$0.505624\pi$
$224$	0	0
$225$	−2.43741	−0.162494
$226$	0	0
$227$	−15.7014	−1.04214	−0.521070	−	0.853514i	$-0.674467\pi$
$227$	−15.7014	−1.04214	−0.521070	+	0.853514i	$0.674467\pi$
$228$	0	0
$229$	0.772461	0.0510457	0.0255228	−	0.999674i	$-0.491875\pi$
$229$	0.772461	0.0510457	0.0255228	+	0.999674i	$0.491875\pi$
$230$	0	0
$231$	4.08917	0.269048
$232$	0	0
$233$	−10.3255	−0.676445	−0.338223	−	0.941066i	$-0.609826\pi$
$233$	−10.3255	−0.676445	−0.338223	+	0.941066i	$0.609826\pi$
$234$	0	0
$235$	−1.12131	−0.0731459
$236$	0	0
$237$	−12.1841	−0.791440
$238$	0	0
$239$	−3.56332	−0.230492	−0.115246	−	0.993337i	$-0.536766\pi$
$239$	−3.56332	−0.230492	−0.115246	+	0.993337i	$0.536766\pi$
$240$	0	0
$241$	2.28273	0.147043	0.0735217	−	0.997294i	$-0.476576\pi$
$241$	2.28273	0.147043	0.0735217	+	0.997294i	$0.476576\pi$
$242$	0	0
$243$	−15.4253	−0.989533
$244$	0	0
$245$	−11.6393	−0.743605
$246$	0	0
$247$	3.80773	0.242280
$248$	0	0
$249$	4.94876	0.313615
$250$	0	0
$251$	12.5000	0.788995	0.394497	−	0.918897i	$-0.370919\pi$
$251$	12.5000	0.788995	0.394497	+	0.918897i	$0.370919\pi$
$252$	0	0
$253$	−8.84263	−0.555931
$254$	0	0
$255$	1.30706	0.0818515
$256$	0	0
$257$	−23.4959	−1.46563	−0.732817	−	0.680426i	$-0.761795\pi$
$257$	−23.4959	−1.46563	−0.732817	+	0.680426i	$0.761795\pi$
$258$	0	0
$259$	−40.6107	−2.52343
$260$	0	0
$261$	14.2949	0.884830
$262$	0	0
$263$	−2.77922	−0.171374	−0.0856871	−	0.996322i	$-0.527309\pi$
$263$	−2.77922	−0.171374	−0.0856871	+	0.996322i	$0.527309\pi$
$264$	0	0
$265$	−0.740586	−0.0454938
$266$	0	0
$267$	7.41118	0.453557
$268$	0	0
$269$	−23.9484	−1.46016	−0.730079	−	0.683363i	$-0.760517\pi$
$269$	−23.9484	−1.46016	−0.730079	+	0.683363i	$0.760517\pi$
$270$	0	0
$271$	−9.32023	−0.566164	−0.283082	−	0.959096i	$-0.591357\pi$
$271$	−9.32023	−0.566164	−0.283082	+	0.959096i	$0.591357\pi$
$272$	0	0
$273$	2.72077	0.164668
$274$	0	0
$275$	1.26278	0.0761482
$276$	0	0
$277$	−13.2903	−0.798536	−0.399268	−	0.916834i	$-0.630736\pi$
$277$	−13.2903	−0.798536	−0.399268	+	0.916834i	$0.630736\pi$
$278$	0	0
$279$	−14.2712	−0.854396
$280$	0	0
$281$	2.56864	0.153232	0.0766162	−	0.997061i	$-0.475588\pi$
$281$	2.56864	0.153232	0.0766162	+	0.997061i	$0.475588\pi$
$282$	0	0
$283$	8.92647	0.530623	0.265312	−	0.964163i	$-0.414525\pi$
$283$	8.92647	0.530623	0.265312	+	0.964163i	$0.414525\pi$
$284$	0	0
$285$	3.39922	0.201352
$286$	0	0
$287$	25.9186	1.52993
$288$	0	0
$289$	−13.9633	−0.821370
$290$	0	0
$291$	−3.55918	−0.208643
$292$	0	0
$293$	−23.2424	−1.35783	−0.678917	−	0.734215i	$-0.737550\pi$
$293$	−23.2424	−1.35783	−0.678917	+	0.734215i	$0.737550\pi$
$294$	0	0
$295$	1.87130	0.108951
$296$	0	0
$297$	5.15007	0.298838
$298$	0	0
$299$	−5.88352	−0.340253
$300$	0	0
$301$	−27.1453	−1.56463
$302$	0	0
$303$	5.90502	0.339234
$304$	0	0
$305$	4.43841	0.254143
$306$	0	0
$307$	−24.7518	−1.41266	−0.706329	−	0.707883i	$-0.749650\pi$
$307$	−24.7518	−1.41266	−0.706329	+	0.707883i	$0.749650\pi$
$308$	0	0
$309$	−13.3721	−0.760712
$310$	0	0
$311$	31.4484	1.78328	0.891639	−	0.452747i	$-0.149556\pi$
$311$	31.4484	1.78328	0.891639	+	0.452747i	$0.149556\pi$
$312$	0	0
$313$	29.0008	1.63922	0.819612	−	0.572919i	$-0.194189\pi$
$313$	29.0008	1.63922	0.819612	+	0.572919i	$0.194189\pi$
$314$	0	0
$315$	−10.5231	−0.592909
$316$	0	0
$317$	−14.8857	−0.836063	−0.418032	−	0.908432i	$-0.637280\pi$
$317$	−14.8857	−0.836063	−0.418032	+	0.908432i	$0.637280\pi$
$318$	0	0
$319$	−7.40589	−0.414650
$320$	0	0
$321$	0.161696	0.00902498
$322$	0	0
$323$	7.89742	0.439424
$324$	0	0
$325$	0.840199	0.0466059
$326$	0	0
$327$	−4.05382	−0.224177
$328$	0	0
$329$	−4.84104	−0.266895
$330$	0	0
$331$	−8.99982	−0.494675	−0.247338	−	0.968929i	$-0.579556\pi$
$331$	−8.99982	−0.494675	−0.247338	+	0.968929i	$0.579556\pi$
$332$	0	0
$333$	−22.9274	−1.25642
$334$	0	0
$335$	2.32847	0.127218
$336$	0	0
$337$	26.7997	1.45987	0.729935	−	0.683516i	$-0.239550\pi$
$337$	26.7997	1.45987	0.729935	+	0.683516i	$0.239550\pi$
$338$	0	0
$339$	−11.9452	−0.648775
$340$	0	0
$341$	7.39364	0.400388
$342$	0	0
$343$	−20.0292	−1.08147
$344$	0	0
$345$	−5.25231	−0.282775
$346$	0	0
$347$	−0.0900549	−0.00483440	−0.00241720	−	0.999997i	$-0.500769\pi$
$347$	−0.0900549	−0.00483440	−0.00241720	+	0.999997i	$0.500769\pi$
$348$	0	0
$349$	−34.9049	−1.86842	−0.934209	−	0.356726i	$-0.883893\pi$
$349$	−34.9049	−1.86842	−0.934209	+	0.356726i	$0.883893\pi$
$350$	0	0
$351$	3.42665	0.182901
$352$	0	0
$353$	−19.4246	−1.03386	−0.516932	−	0.856026i	$-0.672926\pi$
$353$	−19.4246	−1.03386	−0.516932	+	0.856026i	$0.672926\pi$
$354$	0	0
$355$	11.5920	0.615238
$356$	0	0
$357$	5.64301	0.298660
$358$	0	0
$359$	0.828543	0.0437288	0.0218644	−	0.999761i	$-0.493040\pi$
$359$	0.828543	0.0437288	0.0218644	+	0.999761i	$0.493040\pi$
$360$	0	0
$361$	1.53845	0.0809712
$362$	0	0
$363$	7.05460	0.370270
$364$	0	0
$365$	−14.5436	−0.761246
$366$	0	0
$367$	−21.9404	−1.14528	−0.572640	−	0.819807i	$-0.694081\pi$
$367$	−21.9404	−1.14528	−0.572640	+	0.819807i	$0.694081\pi$
$368$	0	0
$369$	14.6328	0.761751
$370$	0	0
$371$	−3.19735	−0.165998
$372$	0	0
$373$	−10.4994	−0.543637	−0.271819	−	0.962349i	$-0.587625\pi$
$373$	−10.4994	−0.543637	−0.271819	+	0.962349i	$0.587625\pi$
$374$	0	0
$375$	0.750059	0.0387329
$376$	0	0
$377$	−4.92757	−0.253783
$378$	0	0
$379$	1.36333	0.0700295	0.0350147	−	0.999387i	$-0.488852\pi$
$379$	1.36333	0.0700295	0.0350147	+	0.999387i	$0.488852\pi$
$380$	0	0
$381$	−0.932798	−0.0477887
$382$	0	0
$383$	15.7867	0.806661	0.403330	−	0.915054i	$-0.367853\pi$
$383$	15.7867	0.806661	0.403330	+	0.915054i	$0.367853\pi$
$384$	0	0
$385$	5.45181	0.277850
$386$	0	0
$387$	−15.3253	−0.779029
$388$	0	0
$389$	−23.7530	−1.20433	−0.602164	−	0.798373i	$-0.705694\pi$
$389$	−23.7530	−1.20433	−0.602164	+	0.798373i	$0.705694\pi$
$390$	0	0
$391$	−12.2027	−0.617118
$392$	0	0
$393$	−0.739747	−0.0373153
$394$	0	0
$395$	−16.2442	−0.817332
$396$	0	0
$397$	9.80619	0.492158	0.246079	−	0.969250i	$-0.420858\pi$
$397$	9.80619	0.492158	0.246079	+	0.969250i	$0.420858\pi$
$398$	0	0
$399$	14.6755	0.734695
$400$	0	0
$401$	1.00000	0.0499376
$401$	1.00000	0.0499376
$402$	0	0
$403$	4.91942	0.245054
$404$	0	0
$405$	−4.25321	−0.211344
$406$	0	0
$407$	11.8783	0.588783
$408$	0	0
$409$	35.4444	1.75261	0.876306	−	0.481754i	$-0.160000\pi$
$409$	35.4444	1.75261	0.876306	+	0.481754i	$0.160000\pi$
$410$	0	0
$411$	3.33056	0.164285
$412$	0	0
$413$	8.07899	0.397541
$414$	0	0
$415$	6.59783	0.323875
$416$	0	0
$417$	3.06282	0.149987
$418$	0	0
$419$	12.9094	0.630667	0.315334	−	0.948981i	$-0.397884\pi$
$419$	12.9094	0.630667	0.315334	+	0.948981i	$0.397884\pi$
$420$	0	0
$421$	19.2983	0.940544	0.470272	−	0.882522i	$-0.344156\pi$
$421$	19.2983	0.940544	0.470272	+	0.882522i	$0.344156\pi$
$422$	0	0
$423$	−2.73309	−0.132887
$424$	0	0
$425$	1.74261	0.0845292
$426$	0	0
$427$	19.1621	0.927317
$428$	0	0
$429$	−0.795799	−0.0384215
$430$	0	0
$431$	12.7813	0.615653	0.307826	−	0.951443i	$-0.400398\pi$
$431$	12.7813	0.615653	0.307826	+	0.951443i	$0.400398\pi$
$432$	0	0
$433$	35.9291	1.72664	0.863321	−	0.504655i	$-0.168380\pi$
$433$	35.9291	1.72664	0.863321	+	0.504655i	$0.168380\pi$
$434$	0	0
$435$	−4.39892	−0.210912
$436$	0	0
$437$	−31.7350	−1.51809
$438$	0	0
$439$	−13.2146	−0.630696	−0.315348	−	0.948976i	$-0.602121\pi$
$439$	−13.2146	−0.630696	−0.315348	+	0.948976i	$0.602121\pi$
$440$	0	0
$441$	−28.3697	−1.35094
$442$	0	0
$443$	33.6761	1.60000	0.800001	−	0.599999i	$-0.204832\pi$
$443$	33.6761	1.60000	0.800001	+	0.599999i	$0.204832\pi$
$444$	0	0
$445$	9.88080	0.468395
$446$	0	0
$447$	−9.10088	−0.430457
$448$	0	0
$449$	−9.39505	−0.443380	−0.221690	−	0.975117i	$-0.571157\pi$
$449$	−9.39505	−0.443380	−0.221690	+	0.975117i	$0.571157\pi$
$450$	0	0
$451$	−7.58095	−0.356973
$452$	0	0
$453$	5.99631	0.281731
$454$	0	0
$455$	3.62741	0.170056
$456$	0	0
$457$	−5.30725	−0.248263	−0.124131	−	0.992266i	$-0.539614\pi$
$457$	−5.30725	−0.248263	−0.124131	+	0.992266i	$0.539614\pi$
$458$	0	0
$459$	7.10704	0.331728
$460$	0	0
$461$	30.7981	1.43441	0.717206	−	0.696861i	$-0.245420\pi$
$461$	30.7981	1.43441	0.717206	+	0.696861i	$0.245420\pi$
$462$	0	0
$463$	37.7678	1.75522	0.877609	−	0.479377i	$-0.159137\pi$
$463$	37.7678	1.75522	0.877609	+	0.479377i	$0.159137\pi$
$464$	0	0
$465$	4.39165	0.203658
$466$	0	0
$467$	−0.924859	−0.0427974	−0.0213987	−	0.999771i	$-0.506812\pi$
$467$	−0.924859	−0.0427974	−0.0213987	+	0.999771i	$0.506812\pi$
$468$	0	0
$469$	10.0527	0.464192
$470$	0	0
$471$	11.3508	0.523018
$472$	0	0
$473$	7.93974	0.365070
$474$	0	0
$475$	4.53194	0.207940
$476$	0	0
$477$	−1.80511	−0.0826505
$478$	0	0
$479$	−0.230945	−0.0105522	−0.00527608	−	0.999986i	$-0.501679\pi$
$479$	−0.230945	−0.0105522	−0.00527608	+	0.999986i	$0.501679\pi$
$480$	0	0
$481$	7.90330	0.360360
$482$	0	0
$483$	−22.6759	−1.03179
$484$	0	0
$485$	−4.74520	−0.215468
$486$	0	0
$487$	−18.2318	−0.826162	−0.413081	−	0.910694i	$-0.635547\pi$
$487$	−18.2318	−0.826162	−0.413081	+	0.910694i	$0.635547\pi$
$488$	0	0
$489$	−16.0938	−0.727785
$490$	0	0
$491$	11.6486	0.525694	0.262847	−	0.964838i	$-0.415339\pi$
$491$	11.6486	0.525694	0.262847	+	0.964838i	$0.415339\pi$
$492$	0	0
$493$	−10.2200	−0.460287
$494$	0	0
$495$	3.07790	0.138341
$496$	0	0
$497$	50.0462	2.24488
$498$	0	0
$499$	1.40138	0.0627344	0.0313672	−	0.999508i	$-0.490014\pi$
$499$	1.40138	0.0627344	0.0313672	+	0.999508i	$0.490014\pi$
$500$	0	0
$501$	−7.03377	−0.314246
$502$	0	0
$503$	−39.1078	−1.74373	−0.871865	−	0.489746i	$-0.837089\pi$
$503$	−39.1078	−1.74373	−0.871865	+	0.489746i	$0.837089\pi$
$504$	0	0
$505$	7.87274	0.350332
$506$	0	0
$507$	9.22127	0.409531
$508$	0	0
$509$	−40.3500	−1.78848	−0.894242	−	0.447584i	$-0.852285\pi$
$509$	−40.3500	−1.78848	−0.894242	+	0.447584i	$0.852285\pi$
$510$	0	0
$511$	−62.7893	−2.77764
$512$	0	0
$513$	18.4830	0.816042
$514$	0	0
$515$	−17.8281	−0.785598
$516$	0	0
$517$	1.41596	0.0622737
$518$	0	0
$519$	−3.39011	−0.148809
$520$	0	0
$521$	−30.1444	−1.32065	−0.660324	−	0.750981i	$-0.729581\pi$
$521$	−30.1444	−1.32065	−0.660324	+	0.750981i	$0.729581\pi$
$522$	0	0
$523$	2.56645	0.112223	0.0561115	−	0.998425i	$-0.482130\pi$
$523$	2.56645	0.112223	0.0561115	+	0.998425i	$0.482130\pi$
$524$	0	0
$525$	3.23824	0.141329
$526$	0	0
$527$	10.2031	0.444455
$528$	0	0
$529$	26.0355	1.13198
$530$	0	0
$531$	4.56112	0.197936
$532$	0	0
$533$	−5.04405	−0.218482
$534$	0	0
$535$	0.215578	0.00932023
$536$	0	0
$537$	−3.54648	−0.153042
$538$	0	0
$539$	14.6978	0.633078
$540$	0	0
$541$	5.30796	0.228207	0.114104	−	0.993469i	$-0.463600\pi$
$541$	5.30796	0.228207	0.114104	+	0.993469i	$0.463600\pi$
$542$	0	0
$543$	18.2834	0.784615
$544$	0	0
$545$	−5.40467	−0.231511
$546$	0	0
$547$	−19.6184	−0.838821	−0.419411	−	0.907797i	$-0.637763\pi$
$547$	−19.6184	−0.838821	−0.419411	+	0.907797i	$0.637763\pi$
$548$	0	0
$549$	10.8182	0.461711
$550$	0	0
$551$	−26.5788	−1.13229
$552$	0	0
$553$	−70.1312	−2.98228
$554$	0	0
$555$	7.05540	0.299485
$556$	0	0
$557$	−27.9723	−1.18522	−0.592612	−	0.805488i	$-0.701903\pi$
$557$	−27.9723	−1.18522	−0.592612	+	0.805488i	$0.701903\pi$
$558$	0	0
$559$	5.28278	0.223438
$560$	0	0
$561$	−1.65053	−0.0696853
$562$	0	0
$563$	30.2689	1.27568	0.637842	−	0.770168i	$-0.279827\pi$
$563$	30.2689	1.27568	0.637842	+	0.770168i	$0.279827\pi$
$564$	0	0
$565$	−15.9257	−0.670000
$566$	0	0
$567$	−18.3625	−0.771152
$568$	0	0
$569$	32.2680	1.35275	0.676373	−	0.736560i	$-0.263551\pi$
$569$	32.2680	1.35275	0.676373	+	0.736560i	$0.263551\pi$
$570$	0	0
$571$	−21.3512	−0.893518	−0.446759	−	0.894654i	$-0.647422\pi$
$571$	−21.3512	−0.893518	−0.446759	+	0.894654i	$0.647422\pi$
$572$	0	0
$573$	13.1001	0.547265
$574$	0	0
$575$	−7.00253	−0.292026
$576$	0	0
$577$	6.18332	0.257415	0.128708	−	0.991683i	$-0.458917\pi$
$577$	6.18332	0.257415	0.128708	+	0.991683i	$0.458917\pi$
$578$	0	0
$579$	10.2614	0.426450
$580$	0	0
$581$	28.4849	1.18175
$582$	0	0
$583$	0.935194	0.0387318
$584$	0	0
$585$	2.04791	0.0846707
$586$	0	0
$587$	17.3892	0.717731	0.358866	−	0.933389i	$-0.383164\pi$
$587$	17.3892	0.717731	0.358866	+	0.933389i	$0.383164\pi$
$588$	0	0
$589$	26.5348	1.09335
$590$	0	0
$591$	4.20294	0.172886
$592$	0	0
$593$	−14.7534	−0.605849	−0.302924	−	0.953015i	$-0.597963\pi$
$593$	−14.7534	−0.605849	−0.302924	+	0.953015i	$0.597963\pi$
$594$	0	0
$595$	7.52343	0.308430
$596$	0	0
$597$	9.55447	0.391039
$598$	0	0
$599$	32.6529	1.33416	0.667081	−	0.744985i	$-0.267543\pi$
$599$	32.6529	1.33416	0.667081	+	0.744985i	$0.267543\pi$
$600$	0	0
$601$	44.1895	1.80253	0.901263	−	0.433273i	$-0.142641\pi$
$601$	44.1895	1.80253	0.901263	+	0.433273i	$0.142641\pi$
$602$	0	0
$603$	5.67543	0.231122
$604$	0	0
$605$	9.40540	0.382384
$606$	0	0
$607$	19.3881	0.786940	0.393470	−	0.919338i	$-0.371275\pi$
$607$	19.3881	0.786940	0.393470	+	0.919338i	$0.371275\pi$
$608$	0	0
$609$	−18.9916	−0.769577
$610$	0	0
$611$	0.942120	0.0381141
$612$	0	0
$613$	−30.4845	−1.23126	−0.615629	−	0.788036i	$-0.711098\pi$
$613$	−30.4845	−1.23126	−0.615629	+	0.788036i	$0.711098\pi$
$614$	0	0
$615$	−4.50290	−0.181575
$616$	0	0
$617$	−39.5693	−1.59300	−0.796499	−	0.604640i	$-0.793317\pi$
$617$	−39.5693	−1.59300	−0.796499	+	0.604640i	$0.793317\pi$
$618$	0	0
$619$	9.87274	0.396819	0.198410	−	0.980119i	$-0.436422\pi$
$619$	9.87274	0.396819	0.198410	+	0.980119i	$0.436422\pi$
$620$	0	0
$621$	−28.5590	−1.14603
$622$	0	0
$623$	42.6586	1.70908
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−4.29245	−0.171424
$628$	0	0
$629$	16.3918	0.653586
$630$	0	0
$631$	−18.1507	−0.722568	−0.361284	−	0.932456i	$-0.617661\pi$
$631$	−18.1507	−0.722568	−0.361284	+	0.932456i	$0.617661\pi$
$632$	0	0
$633$	6.23681	0.247891
$634$	0	0
$635$	−1.24363	−0.0493521
$636$	0	0
$637$	9.77929	0.387470
$638$	0	0
$639$	28.2544	1.11773
$640$	0	0
$641$	22.5290	0.889841	0.444921	−	0.895570i	$-0.353232\pi$
$641$	22.5290	0.889841	0.444921	+	0.895570i	$0.353232\pi$
$642$	0	0
$643$	−10.7285	−0.423091	−0.211546	−	0.977368i	$-0.567850\pi$
$643$	−10.7285	−0.423091	−0.211546	+	0.977368i	$0.567850\pi$
$644$	0	0
$645$	4.71602	0.185693
$646$	0	0
$647$	−39.3358	−1.54645	−0.773225	−	0.634132i	$-0.781358\pi$
$647$	−39.3358	−1.54645	−0.773225	+	0.634132i	$0.781358\pi$
$648$	0	0
$649$	−2.36303	−0.0927570
$650$	0	0
$651$	18.9601	0.743107
$652$	0	0
$653$	22.7548	0.890464	0.445232	−	0.895415i	$-0.353121\pi$
$653$	22.7548	0.890464	0.445232	+	0.895415i	$0.353121\pi$
$654$	0	0
$655$	−0.986253	−0.0385361
$656$	0	0
$657$	−35.4487	−1.38299
$658$	0	0
$659$	18.7924	0.732050	0.366025	−	0.930605i	$-0.380719\pi$
$659$	18.7924	0.732050	0.366025	+	0.930605i	$0.380719\pi$
$660$	0	0
$661$	−39.4244	−1.53343	−0.766716	−	0.641987i	$-0.778110\pi$
$661$	−39.4244	−1.53343	−0.766716	+	0.641987i	$0.778110\pi$
$662$	0	0
$663$	−1.09819	−0.0426503
$664$	0	0
$665$	19.5658	0.758730
$666$	0	0
$667$	41.0683	1.59017
$668$	0	0
$669$	0.395762	0.0153010
$670$	0	0
$671$	−5.60472	−0.216368
$672$	0	0
$673$	−43.2657	−1.66777	−0.833885	−	0.551938i	$-0.813889\pi$
$673$	−43.2657	−1.66777	−0.833885	+	0.551938i	$0.813889\pi$
$674$	0	0
$675$	4.07838	0.156977
$676$	0	0
$677$	−12.5311	−0.481608	−0.240804	−	0.970574i	$-0.577411\pi$
$677$	−12.5311	−0.481608	−0.240804	+	0.970574i	$0.577411\pi$
$678$	0	0
$679$	−20.4866	−0.786202
$680$	0	0
$681$	11.7770	0.451295
$682$	0	0
$683$	−5.48199	−0.209763	−0.104881	−	0.994485i	$-0.533446\pi$
$683$	−5.48199	−0.209763	−0.104881	+	0.994485i	$0.533446\pi$
$684$	0	0
$685$	4.44041	0.169659
$686$	0	0
$687$	−0.579391	−0.0221052
$688$	0	0
$689$	0.622240	0.0237054
$690$	0	0
$691$	−42.2665	−1.60790	−0.803948	−	0.594700i	$-0.797271\pi$
$691$	−42.2665	−1.60790	−0.803948	+	0.594700i	$0.797271\pi$
$692$	0	0
$693$	13.2883	0.504781
$694$	0	0
$695$	4.08344	0.154894
$696$	0	0
$697$	−10.4616	−0.396262
$698$	0	0
$699$	7.74472	0.292932
$700$	0	0
$701$	19.7589	0.746284	0.373142	−	0.927774i	$-0.378280\pi$
$701$	19.7589	0.746284	0.373142	+	0.927774i	$0.378280\pi$
$702$	0	0
$703$	42.6295	1.60780
$704$	0	0
$705$	0.841045	0.0316756
$706$	0	0
$707$	33.9891	1.27829
$708$	0	0
$709$	−31.3628	−1.17786	−0.588928	−	0.808186i	$-0.700450\pi$
$709$	−31.3628	−1.17786	−0.588928	+	0.808186i	$0.700450\pi$
$710$	0	0
$711$	−39.5937	−1.48488
$712$	0	0
$713$	−41.0003	−1.53547
$714$	0	0
$715$	−1.06098	−0.0396785
$716$	0	0
$717$	2.67270	0.0998138
$718$	0	0
$719$	−15.8563	−0.591341	−0.295670	−	0.955290i	$-0.595543\pi$
$719$	−15.8563	−0.591341	−0.295670	+	0.955290i	$0.595543\pi$
$720$	0	0
$721$	−76.9694	−2.86649
$722$	0	0
$723$	−1.71218	−0.0636766
$724$	0	0
$725$	−5.86477	−0.217812
$726$	0	0
$727$	−17.3941	−0.645110	−0.322555	−	0.946551i	$-0.604542\pi$
$727$	−17.3941	−0.645110	−0.322555	+	0.946551i	$0.604542\pi$
$728$	0	0
$729$	−1.18977	−0.0440656
$730$	0	0
$731$	10.9567	0.405250
$732$	0	0
$733$	39.1108	1.44459	0.722295	−	0.691585i	$-0.243087\pi$
$733$	39.1108	1.44459	0.722295	+	0.691585i	$0.243087\pi$
$734$	0	0
$735$	8.73013	0.322016
$736$	0	0
$737$	−2.94033	−0.108308
$738$	0	0
$739$	12.7603	0.469395	0.234697	−	0.972069i	$-0.424590\pi$
$739$	12.7603	0.469395	0.234697	+	0.972069i	$0.424590\pi$
$740$	0	0
$741$	−2.85602	−0.104919
$742$	0	0
$743$	28.8539	1.05855	0.529274	−	0.848451i	$-0.322464\pi$
$743$	28.8539	1.05855	0.529274	+	0.848451i	$0.322464\pi$
$744$	0	0
$745$	−12.1336	−0.444539
$746$	0	0
$747$	16.0816	0.588396
$748$	0	0
$749$	0.930718	0.0340077
$750$	0	0
$751$	−4.43265	−0.161750	−0.0808749	−	0.996724i	$-0.525771\pi$
$751$	−4.43265	−0.161750	−0.0808749	+	0.996724i	$0.525771\pi$
$752$	0	0
$753$	−9.37576	−0.341672
$754$	0	0
$755$	7.99445	0.290948
$756$	0	0
$757$	−24.7646	−0.900084	−0.450042	−	0.893007i	$-0.648591\pi$
$757$	−24.7646	−0.900084	−0.450042	+	0.893007i	$0.648591\pi$
$758$	0	0
$759$	6.63249	0.240744
$760$	0	0
$761$	−35.9830	−1.30438	−0.652191	−	0.758055i	$-0.726150\pi$
$761$	−35.9830	−1.30438	−0.652191	+	0.758055i	$0.726150\pi$
$762$	0	0
$763$	−23.3337	−0.844737
$764$	0	0
$765$	4.24747	0.153568
$766$	0	0
$767$	−1.57226	−0.0567711
$768$	0	0
$769$	38.3042	1.38128	0.690642	−	0.723197i	$-0.257328\pi$
$769$	38.3042	1.38128	0.690642	+	0.723197i	$0.257328\pi$
$770$	0	0
$771$	17.6233	0.634688
$772$	0	0
$773$	26.9499	0.969320	0.484660	−	0.874703i	$-0.338943\pi$
$773$	26.9499	0.969320	0.484660	+	0.874703i	$0.338943\pi$
$774$	0	0
$775$	5.85507	0.210320
$776$	0	0
$777$	30.4604	1.09276
$778$	0	0
$779$	−27.2070	−0.974793
$780$	0	0
$781$	−14.6381	−0.523791
$782$	0	0
$783$	−23.9188	−0.854786
$784$	0	0
$785$	15.1332	0.540129
$786$	0	0
$787$	−33.5166	−1.19474	−0.597369	−	0.801966i	$-0.703787\pi$
$787$	−33.5166	−1.19474	−0.597369	+	0.801966i	$0.703787\pi$
$788$	0	0
$789$	2.08458	0.0742130
$790$	0	0
$791$	−68.7564	−2.44470
$792$	0	0
$793$	−3.72915	−0.132426
$794$	0	0
$795$	0.555483	0.0197009
$796$	0	0
$797$	40.5353	1.43584	0.717918	−	0.696128i	$-0.245095\pi$
$797$	40.5353	1.43584	0.717918	+	0.696128i	$0.245095\pi$
$798$	0	0
$799$	1.95400	0.0691277
$800$	0	0
$801$	24.0836	0.850951
$802$	0	0
$803$	18.3653	0.648097
$804$	0	0
$805$	−30.2322	−1.06554
$806$	0	0
$807$	17.9627	0.632316
$808$	0	0
$809$	−29.3086	−1.03044	−0.515218	−	0.857059i	$-0.672289\pi$
$809$	−29.3086	−1.03044	−0.515218	+	0.857059i	$0.672289\pi$
$810$	0	0
$811$	−9.48650	−0.333116	−0.166558	−	0.986032i	$-0.553265\pi$
$811$	−9.48650	−0.333116	−0.166558	+	0.986032i	$0.553265\pi$
$812$	0	0
$813$	6.99072	0.245175
$814$	0	0
$815$	−21.4567	−0.751594
$816$	0	0
$817$	28.4947	0.996904
$818$	0	0
$819$	8.84149	0.308947
$820$	0	0
$821$	−36.9014	−1.28787	−0.643934	−	0.765081i	$-0.722699\pi$
$821$	−36.9014	−1.28787	−0.643934	+	0.765081i	$0.722699\pi$
$822$	0	0
$823$	37.9089	1.32142	0.660711	−	0.750641i	$-0.270255\pi$
$823$	37.9089	1.32142	0.660711	+	0.750641i	$0.270255\pi$
$824$	0	0
$825$	−0.947155	−0.0329757
$826$	0	0
$827$	9.35095	0.325165	0.162582	−	0.986695i	$-0.448018\pi$
$827$	9.35095	0.325165	0.162582	+	0.986695i	$0.448018\pi$
$828$	0	0
$829$	−25.9832	−0.902433	−0.451216	−	0.892415i	$-0.649010\pi$
$829$	−25.9832	−0.902433	−0.451216	+	0.892415i	$0.649010\pi$
$830$	0	0
$831$	9.96850	0.345803
$832$	0	0
$833$	20.2827	0.702755
$834$	0	0
$835$	−9.37763	−0.324526
$836$	0	0
$837$	23.8792	0.825385
$838$	0	0
$839$	22.1025	0.763063	0.381531	−	0.924356i	$-0.375397\pi$
$839$	22.1025	0.763063	0.381531	+	0.924356i	$0.375397\pi$
$840$	0	0
$841$	5.39554	0.186053
$842$	0	0
$843$	−1.92663	−0.0663568
$844$	0	0
$845$	12.2941	0.422929
$846$	0	0
$847$	40.6061	1.39524
$848$	0	0
$849$	−6.69537	−0.229785
$850$	0	0
$851$	−65.8691	−2.25796
$852$	0	0
$853$	−12.4183	−0.425193	−0.212596	−	0.977140i	$-0.568192\pi$
$853$	−12.4183	−0.425193	−0.212596	+	0.977140i	$0.568192\pi$
$854$	0	0
$855$	11.0462	0.377772
$856$	0	0
$857$	−51.5011	−1.75924	−0.879622	−	0.475673i	$-0.842205\pi$
$857$	−51.5011	−1.75924	−0.879622	+	0.475673i	$0.842205\pi$
$858$	0	0
$859$	−21.0721	−0.718969	−0.359485	−	0.933151i	$-0.617047\pi$
$859$	−21.0721	−0.718969	−0.359485	+	0.933151i	$0.617047\pi$
$860$	0	0
$861$	−19.4405	−0.662530
$862$	0	0
$863$	14.4552	0.492061	0.246030	−	0.969262i	$-0.420874\pi$
$863$	14.4552	0.492061	0.246030	+	0.969262i	$0.420874\pi$
$864$	0	0
$865$	−4.51979	−0.153677
$866$	0	0
$867$	10.4733	0.355692
$868$	0	0
$869$	20.5127	0.695846
$870$	0	0
$871$	−1.95638	−0.0662893
$872$	0	0
$873$	−11.5660	−0.391450
$874$	0	0
$875$	4.31732	0.145952
$876$	0	0
$877$	−15.1108	−0.510255	−0.255127	−	0.966907i	$-0.582117\pi$
$877$	−15.1108	−0.510255	−0.255127	+	0.966907i	$0.582117\pi$
$878$	0	0
$879$	17.4331	0.588005
$880$	0	0
$881$	−41.3918	−1.39452	−0.697262	−	0.716816i	$-0.745599\pi$
$881$	−41.3918	−1.39452	−0.697262	+	0.716816i	$0.745599\pi$
$882$	0	0
$883$	48.7179	1.63949	0.819744	−	0.572729i	$-0.194115\pi$
$883$	48.7179	1.63949	0.819744	+	0.572729i	$0.194115\pi$
$884$	0	0
$885$	−1.40358	−0.0471809
$886$	0	0
$887$	2.80750	0.0942667	0.0471334	−	0.998889i	$-0.484991\pi$
$887$	2.80750	0.0942667	0.0471334	+	0.998889i	$0.484991\pi$
$888$	0	0
$889$	−5.36917	−0.180076
$890$	0	0
$891$	5.37085	0.179930
$892$	0	0
$893$	5.08169	0.170052
$894$	0	0
$895$	−4.72827	−0.158049
$896$	0	0
$897$	4.41299	0.147345
$898$	0	0
$899$	−34.3386	−1.14526
$900$	0	0
$901$	1.29056	0.0429947
$902$	0	0
$903$	20.3606	0.677557
$904$	0	0
$905$	24.3759	0.810284
$906$	0	0
$907$	2.03530	0.0675811	0.0337905	−	0.999429i	$-0.489242\pi$
$907$	2.03530	0.0675811	0.0337905	+	0.999429i	$0.489242\pi$
$908$	0	0
$909$	19.1891	0.636463
$910$	0	0
$911$	−47.0929	−1.56026	−0.780129	−	0.625618i	$-0.784847\pi$
$911$	−47.0929	−1.56026	−0.780129	+	0.625618i	$0.784847\pi$
$912$	0	0
$913$	−8.33157	−0.275735
$914$	0	0
$915$	−3.32907	−0.110056
$916$	0	0
$917$	−4.25797	−0.140611
$918$	0	0
$919$	19.2868	0.636214	0.318107	−	0.948055i	$-0.396953\pi$
$919$	19.2868	0.636214	0.318107	+	0.948055i	$0.396953\pi$
$920$	0	0
$921$	18.5653	0.611747
$922$	0	0
$923$	−9.73956	−0.320582
$924$	0	0
$925$	9.40647	0.309283
$926$	0	0
$927$	−43.4543	−1.42723
$928$	0	0
$929$	−37.9926	−1.24650	−0.623249	−	0.782024i	$-0.714188\pi$
$929$	−37.9926	−1.24650	−0.623249	+	0.782024i	$0.714188\pi$
$930$	0	0
$931$	52.7484	1.72876
$932$	0	0
$933$	−23.5882	−0.772242
$934$	0	0
$935$	−2.20053	−0.0719651
$936$	0	0
$937$	46.3448	1.51402	0.757010	−	0.653404i	$-0.226660\pi$
$937$	46.3448	1.51402	0.757010	+	0.653404i	$0.226660\pi$
$938$	0	0
$939$	−21.7523	−0.709860
$940$	0	0
$941$	40.5059	1.32045	0.660227	−	0.751066i	$-0.270460\pi$
$941$	40.5059	1.32045	0.660227	+	0.751066i	$0.270460\pi$
$942$	0	0
$943$	42.0390	1.36898
$944$	0	0
$945$	17.6077	0.572777
$946$	0	0
$947$	59.7355	1.94114	0.970571	−	0.240815i	$-0.0774147\pi$
$947$	59.7355	1.94114	0.970571	+	0.240815i	$0.0774147\pi$
$948$	0	0
$949$	12.2195	0.396662
$950$	0	0
$951$	11.1651	0.362054
$952$	0	0
$953$	−23.6121	−0.764872	−0.382436	−	0.923982i	$-0.624915\pi$
$953$	−23.6121	−0.764872	−0.382436	+	0.923982i	$0.624915\pi$
$954$	0	0
$955$	17.4655	0.565169
$956$	0	0
$957$	5.55485	0.179563
$958$	0	0
$959$	19.1707	0.619053
$960$	0	0
$961$	3.28184	0.105866
$962$	0	0
$963$	0.525452	0.0169324
$964$	0	0
$965$	13.6808	0.440401
$966$	0	0
$967$	−41.5091	−1.33484	−0.667421	−	0.744681i	$-0.732602\pi$
$967$	−41.5091	−1.33484	−0.667421	+	0.744681i	$0.732602\pi$
$968$	0	0
$969$	−5.92353	−0.190291
$970$	0	0
$971$	50.1683	1.60998	0.804988	−	0.593291i	$-0.202172\pi$
$971$	50.1683	1.60998	0.804988	+	0.593291i	$0.202172\pi$
$972$	0	0
$973$	17.6295	0.565176
$974$	0	0
$975$	−0.630198	−0.0201825
$976$	0	0
$977$	−2.47811	−0.0792818	−0.0396409	−	0.999214i	$-0.512621\pi$
$977$	−2.47811	−0.0792818	−0.0396409	+	0.999214i	$0.512621\pi$
$978$	0	0
$979$	−12.4772	−0.398774
$980$	0	0
$981$	−13.1734	−0.420595
$982$	0	0
$983$	2.36744	0.0755096	0.0377548	−	0.999287i	$-0.487979\pi$
$983$	2.36744	0.0755096	0.0377548	+	0.999287i	$0.487979\pi$
$984$	0	0
$985$	5.60348	0.178542
$986$	0	0
$987$	3.63106	0.115578
$988$	0	0
$989$	−44.0287	−1.40003
$990$	0	0
$991$	42.5633	1.35207	0.676034	−	0.736870i	$-0.263697\pi$
$991$	42.5633	1.35207	0.676034	+	0.736870i	$0.263697\pi$
$992$	0	0
$993$	6.75040	0.214217
$994$	0	0
$995$	12.7383	0.403831
$996$	0	0
$997$	24.2874	0.769189	0.384594	−	0.923086i	$-0.374341\pi$
$997$	24.2874	0.769189	0.384594	+	0.923086i	$0.374341\pi$
$998$	0	0
$999$	38.3631	1.21376

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8020.2.a.c.1.11	✓	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8020.2.a.c.1.11	✓	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)

\(f(q)\)	\(=\)	\(q-0.750059 q^{3} -1.00000 q^{5} -4.31732 q^{7} -2.43741 q^{9} +O(q^{10})\)	\(q-0.750059 q^{3} -1.00000 q^{5} -4.31732 q^{7} -2.43741 q^{9} +1.26278 q^{11} +0.840199 q^{13} +0.750059 q^{15} +1.74261 q^{17} +4.53194 q^{19} +3.23824 q^{21} -7.00253 q^{23} +1.00000 q^{25} +4.07838 q^{27} -5.86477 q^{29} +5.85507 q^{31} -0.947155 q^{33} +4.31732 q^{35} +9.40647 q^{37} -0.630198 q^{39} -6.00340 q^{41} +6.28753 q^{43} +2.43741 q^{45} +1.12131 q^{47} +11.6393 q^{49} -1.30706 q^{51} +0.740586 q^{53} -1.26278 q^{55} -3.39922 q^{57} -1.87130 q^{59} -4.43841 q^{61} +10.5231 q^{63} -0.840199 q^{65} -2.32847 q^{67} +5.25231 q^{69} -11.5920 q^{71} +14.5436 q^{73} -0.750059 q^{75} -5.45181 q^{77} +16.2442 q^{79} +4.25321 q^{81} -6.59783 q^{83} -1.74261 q^{85} +4.39892 q^{87} -9.88080 q^{89} -3.62741 q^{91} -4.39165 q^{93} -4.53194 q^{95} +4.74520 q^{97} -3.07790 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 3 q^{3} - 28 q^{5} - 4 q^{7} + 17 q^{9}+O(q^{10}) \) 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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