LMFDB - Embedded newform 4024.2.a.f.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4024,2,Mod(1,4024)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4024, 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("4024.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4024 = 2^{3} \cdot 503 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4024.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:	$32.1318017734$
Analytic rank:	$0$
Dimension:	$33$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Character	$\chi$	$=$	4024.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.05859 q^{3} +3.84987 q^{5} -4.00015 q^{7} +1.23779 q^{9} +O(q^{10})$	$q-2.05859 q^{3} +3.84987 q^{5} -4.00015 q^{7} +1.23779 q^{9} +3.28247 q^{11} -3.96224 q^{13} -7.92531 q^{15} +0.871284 q^{17} -3.06253 q^{19} +8.23467 q^{21} -1.95121 q^{23} +9.82152 q^{25} +3.62767 q^{27} +0.855098 q^{29} -6.85755 q^{31} -6.75725 q^{33} -15.4001 q^{35} -1.69962 q^{37} +8.15663 q^{39} +6.10417 q^{41} +2.26468 q^{43} +4.76534 q^{45} +3.98291 q^{47} +9.00121 q^{49} -1.79362 q^{51} -0.873984 q^{53} +12.6371 q^{55} +6.30449 q^{57} +6.66917 q^{59} +12.1660 q^{61} -4.95135 q^{63} -15.2541 q^{65} -15.5579 q^{67} +4.01675 q^{69} +6.93395 q^{71} +3.14252 q^{73} -20.2185 q^{75} -13.1304 q^{77} -0.0135297 q^{79} -11.1812 q^{81} -2.36812 q^{83} +3.35433 q^{85} -1.76030 q^{87} +12.8225 q^{89} +15.8496 q^{91} +14.1169 q^{93} -11.7904 q^{95} -7.16848 q^{97} +4.06301 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 33 q - 2 q^{3} + 12 q^{5} + 4 q^{7} + 43 q^{9}+O(q^{10}) $ 33 * q - 2 * q^3 + 12 * q^5 + 4 * q^7 + 43 * q^9	$ 33 q - 2 q^{3} + 12 q^{5} + 4 q^{7} + 43 q^{9} + 22 q^{11} + 25 q^{13} - 4 q^{15} + 17 q^{17} + 6 q^{19} + 18 q^{21} + 16 q^{23} + 47 q^{25} - 20 q^{27} + 47 q^{29} - 7 q^{31} - 6 q^{33} + 19 q^{35} + 75 q^{37} + 21 q^{39} + 22 q^{41} - 5 q^{43} + 33 q^{45} + 10 q^{47} + 31 q^{49} + 9 q^{51} + 64 q^{53} - 3 q^{55} + 5 q^{57} + 28 q^{59} + 49 q^{61} - 10 q^{63} + 46 q^{65} - 14 q^{67} + 30 q^{69} + 35 q^{71} + 19 q^{73} - 33 q^{75} + 32 q^{77} - 12 q^{79} + 57 q^{81} + 82 q^{85} - 5 q^{87} + 42 q^{89} - 15 q^{91} + 55 q^{93} + 33 q^{95} + 4 q^{97} + 22 q^{99}+O(q^{100}) $ 33 * q - 2 * q^3 + 12 * q^5 + 4 * q^7 + 43 * q^9 + 22 * q^11 + 25 * q^13 - 4 * q^15 + 17 * q^17 + 6 * q^19 + 18 * q^21 + 16 * q^23 + 47 * q^25 - 20 * q^27 + 47 * q^29 - 7 * q^31 - 6 * q^33 + 19 * q^35 + 75 * q^37 + 21 * q^39 + 22 * q^41 - 5 * q^43 + 33 * q^45 + 10 * q^47 + 31 * q^49 + 9 * q^51 + 64 * q^53 - 3 * q^55 + 5 * q^57 + 28 * q^59 + 49 * q^61 - 10 * q^63 + 46 * q^65 - 14 * q^67 + 30 * q^69 + 35 * q^71 + 19 * q^73 - 33 * q^75 + 32 * q^77 - 12 * q^79 + 57 * q^81 + 82 * q^85 - 5 * q^87 + 42 * q^89 - 15 * q^91 + 55 * q^93 + 33 * q^95 + 4 * q^97 + 22 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−2.05859	−1.18853	−0.594264	−	0.804270i	$-0.702556\pi$
$3$	−2.05859	−1.18853	−0.594264	+	0.804270i	$0.702556\pi$
$4$	0	0
$5$	3.84987	1.72172	0.860858	−	0.508846i	$-0.169928\pi$
$5$	3.84987	1.72172	0.860858	+	0.508846i	$0.169928\pi$
$6$	0	0
$7$	−4.00015	−1.51192	−0.755958	−	0.654621i	$-0.772828\pi$
$7$	−4.00015	−1.51192	−0.755958	+	0.654621i	$0.772828\pi$
$8$	0	0
$9$	1.23779	0.412597
$10$	0	0
$11$	3.28247	0.989701	0.494851	−	0.868978i	$-0.335223\pi$
$11$	3.28247	0.989701	0.494851	+	0.868978i	$0.335223\pi$
$12$	0	0
$13$	−3.96224	−1.09893	−0.549464	−	0.835517i	$-0.685168\pi$
$13$	−3.96224	−1.09893	−0.549464	+	0.835517i	$0.685168\pi$
$14$	0	0
$15$	−7.92531	−2.04631
$16$	0	0
$17$	0.871284	0.211317	0.105659	−	0.994402i	$-0.466305\pi$
$17$	0.871284	0.211317	0.105659	+	0.994402i	$0.466305\pi$
$18$	0	0
$19$	−3.06253	−0.702593	−0.351296	−	0.936264i	$-0.614259\pi$
$19$	−3.06253	−0.702593	−0.351296	+	0.936264i	$0.614259\pi$
$20$	0	0
$21$	8.23467	1.79695
$22$	0	0
$23$	−1.95121	−0.406856	−0.203428	−	0.979090i	$-0.565208\pi$
$23$	−1.95121	−0.406856	−0.203428	+	0.979090i	$0.565208\pi$
$24$	0	0
$25$	9.82152	1.96430
$26$	0	0
$27$	3.62767	0.698144
$28$	0	0
$29$	0.855098	0.158788	0.0793938	−	0.996843i	$-0.474702\pi$
$29$	0.855098	0.158788	0.0793938	+	0.996843i	$0.474702\pi$
$30$	0	0
$31$	−6.85755	−1.23165	−0.615826	−	0.787882i	$-0.711178\pi$
$31$	−6.85755	−1.23165	−0.615826	+	0.787882i	$0.711178\pi$
$32$	0	0
$33$	−6.75725	−1.17629
$34$	0	0
$35$	−15.4001	−2.60309
$36$	0	0
$37$	−1.69962	−0.279416	−0.139708	−	0.990193i	$-0.544616\pi$
$37$	−1.69962	−0.279416	−0.139708	+	0.990193i	$0.544616\pi$
$38$	0	0
$39$	8.15663	1.30611
$40$	0	0
$41$	6.10417	0.953311	0.476655	−	0.879090i	$-0.341849\pi$
$41$	6.10417	0.953311	0.476655	+	0.879090i	$0.341849\pi$
$42$	0	0
$43$	2.26468	0.345361	0.172680	−	0.984978i	$-0.444757\pi$
$43$	2.26468	0.345361	0.172680	+	0.984978i	$0.444757\pi$
$44$	0	0
$45$	4.76534	0.710375
$46$	0	0
$47$	3.98291	0.580966	0.290483	−	0.956880i	$-0.406184\pi$
$47$	3.98291	0.580966	0.290483	+	0.956880i	$0.406184\pi$
$48$	0	0
$49$	9.00121	1.28589
$50$	0	0
$51$	−1.79362	−0.251157
$52$	0	0
$53$	−0.873984	−0.120051	−0.0600254	−	0.998197i	$-0.519118\pi$
$53$	−0.873984	−0.120051	−0.0600254	+	0.998197i	$0.519118\pi$
$54$	0	0
$55$	12.6371	1.70398
$56$	0	0
$57$	6.30449	0.835050
$58$	0	0
$59$	6.66917	0.868252	0.434126	−	0.900852i	$-0.357057\pi$
$59$	6.66917	0.868252	0.434126	+	0.900852i	$0.357057\pi$
$60$	0	0
$61$	12.1660	1.55769	0.778846	−	0.627215i	$-0.215805\pi$
$61$	12.1660	1.55769	0.778846	+	0.627215i	$0.215805\pi$
$62$	0	0
$63$	−4.95135	−0.623812
$64$	0	0
$65$	−15.2541	−1.89204
$66$	0	0
$67$	−15.5579	−1.90070	−0.950352	−	0.311178i	$-0.899276\pi$
$67$	−15.5579	−1.90070	−0.950352	+	0.311178i	$0.899276\pi$
$68$	0	0
$69$	4.01675	0.483559
$70$	0	0
$71$	6.93395	0.822908	0.411454	−	0.911430i	$-0.365021\pi$
$71$	6.93395	0.822908	0.411454	+	0.911430i	$0.365021\pi$
$72$	0	0
$73$	3.14252	0.367805	0.183902	−	0.982945i	$-0.441127\pi$
$73$	3.14252	0.367805	0.183902	+	0.982945i	$0.441127\pi$
$74$	0	0
$75$	−20.2185	−2.33463
$76$	0	0
$77$	−13.1304	−1.49634
$78$	0	0
$79$	−0.0135297	−0.00152221	−0.000761104	−	1.00000i	$-0.500242\pi$
$79$	−0.0135297	−0.00152221	−0.000761104		1.00000i	$0.500242\pi$
$80$	0	0
$81$	−11.1812	−1.24236
$82$	0	0
$83$	−2.36812	−0.259935	−0.129968	−	0.991518i	$-0.541487\pi$
$83$	−2.36812	−0.259935	−0.129968	+	0.991518i	$0.541487\pi$
$84$	0	0
$85$	3.35433	0.363828
$86$	0	0
$87$	−1.76030	−0.188723
$88$	0	0
$89$	12.8225	1.35919	0.679594	−	0.733589i	$-0.262156\pi$
$89$	12.8225	1.35919	0.679594	+	0.733589i	$0.262156\pi$
$90$	0	0
$91$	15.8496	1.66149
$92$	0	0
$93$	14.1169	1.46385
$94$	0	0
$95$	−11.7904	−1.20966
$96$	0	0
$97$	−7.16848	−0.727849	−0.363924	−	0.931428i	$-0.618563\pi$
$97$	−7.16848	−0.727849	−0.363924	+	0.931428i	$0.618563\pi$
$98$	0	0
$99$	4.06301	0.408348
$100$	0	0
$101$	−1.37929	−0.137244	−0.0686221	−	0.997643i	$-0.521860\pi$
$101$	−1.37929	−0.137244	−0.0686221	+	0.997643i	$0.521860\pi$
$102$	0	0
$103$	17.4096	1.71542	0.857709	−	0.514135i	$-0.171887\pi$
$103$	17.4096	1.71542	0.857709	+	0.514135i	$0.171887\pi$
$104$	0	0
$105$	31.7024	3.09384
$106$	0	0
$107$	3.12887	0.302480	0.151240	−	0.988497i	$-0.451673\pi$
$107$	3.12887	0.302480	0.151240	+	0.988497i	$0.451673\pi$
$108$	0	0
$109$	14.6408	1.40233	0.701166	−	0.712998i	$-0.252663\pi$
$109$	14.6408	1.40233	0.701166	+	0.712998i	$0.252663\pi$
$110$	0	0
$111$	3.49882	0.332093
$112$	0	0
$113$	−4.64883	−0.437325	−0.218662	−	0.975801i	$-0.570169\pi$
$113$	−4.64883	−0.437325	−0.218662	+	0.975801i	$0.570169\pi$
$114$	0	0
$115$	−7.51192	−0.700490
$116$	0	0
$117$	−4.90443	−0.453414
$118$	0	0
$119$	−3.48527	−0.319494
$120$	0	0
$121$	−0.225410	−0.0204919
$122$	0	0
$123$	−12.5660	−1.13304
$124$	0	0
$125$	18.5622	1.66026
$126$	0	0
$127$	13.3906	1.18822	0.594111	−	0.804383i	$-0.297504\pi$
$127$	13.3906	1.18822	0.594111	+	0.804383i	$0.297504\pi$
$128$	0	0
$129$	−4.66205	−0.410470
$130$	0	0
$131$	−6.66227	−0.582085	−0.291043	−	0.956710i	$-0.594002\pi$
$131$	−6.66227	−0.582085	−0.291043	+	0.956710i	$0.594002\pi$
$132$	0	0
$133$	12.2506	1.06226
$134$	0	0
$135$	13.9660	1.20201
$136$	0	0
$137$	4.57864	0.391179	0.195590	−	0.980686i	$-0.437338\pi$
$137$	4.57864	0.391179	0.195590	+	0.980686i	$0.437338\pi$
$138$	0	0
$139$	−11.1450	−0.945307	−0.472654	−	0.881248i	$-0.656704\pi$
$139$	−11.1450	−0.945307	−0.472654	+	0.881248i	$0.656704\pi$
$140$	0	0
$141$	−8.19917	−0.690494
$142$	0	0
$143$	−13.0059	−1.08761
$144$	0	0
$145$	3.29202	0.273387
$146$	0	0
$147$	−18.5298	−1.52831
$148$	0	0
$149$	−20.1221	−1.64846	−0.824232	−	0.566252i	$-0.808393\pi$
$149$	−20.1221	−1.64846	−0.824232	+	0.566252i	$0.808393\pi$
$150$	0	0
$151$	9.85909	0.802321	0.401161	−	0.916008i	$-0.368607\pi$
$151$	9.85909	0.802321	0.401161	+	0.916008i	$0.368607\pi$
$152$	0	0
$153$	1.07847	0.0871889
$154$	0	0
$155$	−26.4007	−2.12055
$156$	0	0
$157$	11.0889	0.884989	0.442494	−	0.896771i	$-0.354094\pi$
$157$	11.0889	0.884989	0.442494	+	0.896771i	$0.354094\pi$
$158$	0	0
$159$	1.79917	0.142684
$160$	0	0
$161$	7.80515	0.615132
$162$	0	0
$163$	−6.18166	−0.484185	−0.242092	−	0.970253i	$-0.577834\pi$
$163$	−6.18166	−0.484185	−0.242092	+	0.970253i	$0.577834\pi$
$164$	0	0
$165$	−26.0146	−2.02523
$166$	0	0
$167$	−14.4992	−1.12198	−0.560992	−	0.827821i	$-0.689580\pi$
$167$	−14.4992	−1.12198	−0.560992	+	0.827821i	$0.689580\pi$
$168$	0	0
$169$	2.69935	0.207643
$170$	0	0
$171$	−3.79077	−0.289888
$172$	0	0
$173$	5.65634	0.430044	0.215022	−	0.976609i	$-0.431018\pi$
$173$	5.65634	0.430044	0.215022	+	0.976609i	$0.431018\pi$
$174$	0	0
$175$	−39.2876	−2.96986
$176$	0	0
$177$	−13.7291	−1.03194
$178$	0	0
$179$	−5.23294	−0.391128	−0.195564	−	0.980691i	$-0.562654\pi$
$179$	−5.23294	−0.391128	−0.195564	+	0.980691i	$0.562654\pi$
$180$	0	0
$181$	8.96553	0.666403	0.333201	−	0.942856i	$-0.391871\pi$
$181$	8.96553	0.666403	0.333201	+	0.942856i	$0.391871\pi$
$182$	0	0
$183$	−25.0447	−1.85136
$184$	0	0
$185$	−6.54332	−0.481075
$186$	0	0
$187$	2.85996	0.209141
$188$	0	0
$189$	−14.5112	−1.05554
$190$	0	0
$191$	26.5550	1.92145	0.960725	−	0.277501i	$-0.0895062\pi$
$191$	26.5550	1.92145	0.960725	+	0.277501i	$0.0895062\pi$
$192$	0	0
$193$	13.8223	0.994951	0.497475	−	0.867478i	$-0.334260\pi$
$193$	13.8223	0.994951	0.497475	+	0.867478i	$0.334260\pi$
$194$	0	0
$195$	31.4020	2.24874
$196$	0	0
$197$	7.26629	0.517702	0.258851	−	0.965917i	$-0.416656\pi$
$197$	7.26629	0.517702	0.258851	+	0.965917i	$0.416656\pi$
$198$	0	0
$199$	15.6741	1.11111	0.555553	−	0.831481i	$-0.312506\pi$
$199$	15.6741	1.11111	0.555553	+	0.831481i	$0.312506\pi$
$200$	0	0
$201$	32.0274	2.25904
$202$	0	0
$203$	−3.42052	−0.240074
$204$	0	0
$205$	23.5003	1.64133
$206$	0	0
$207$	−2.41519	−0.167868
$208$	0	0
$209$	−10.0527	−0.695357
$210$	0	0
$211$	16.7601	1.15381	0.576906	−	0.816811i	$-0.304260\pi$
$211$	16.7601	1.15381	0.576906	+	0.816811i	$0.304260\pi$
$212$	0	0
$213$	−14.2742	−0.978049
$214$	0	0
$215$	8.71873	0.594613
$216$	0	0
$217$	27.4312	1.86215
$218$	0	0
$219$	−6.46917	−0.437146
$220$	0	0
$221$	−3.45224	−0.232223
$222$	0	0
$223$	−18.2167	−1.21988	−0.609942	−	0.792446i	$-0.708807\pi$
$223$	−18.2167	−1.21988	−0.609942	+	0.792446i	$0.708807\pi$
$224$	0	0
$225$	12.1570	0.810466
$226$	0	0
$227$	−6.55769	−0.435249	−0.217625	−	0.976033i	$-0.569831\pi$
$227$	−6.55769	−0.435249	−0.217625	+	0.976033i	$0.569831\pi$
$228$	0	0
$229$	15.2286	1.00633	0.503166	−	0.864190i	$-0.332168\pi$
$229$	15.2286	1.00633	0.503166	+	0.864190i	$0.332168\pi$
$230$	0	0
$231$	27.0300	1.77845
$232$	0	0
$233$	−8.67074	−0.568039	−0.284019	−	0.958819i	$-0.591668\pi$
$233$	−8.67074	−0.568039	−0.284019	+	0.958819i	$0.591668\pi$
$234$	0	0
$235$	15.3337	1.00026
$236$	0	0
$237$	0.0278520	0.00180918
$238$	0	0
$239$	17.8842	1.15683	0.578416	−	0.815742i	$-0.303671\pi$
$239$	17.8842	1.15683	0.578416	+	0.815742i	$0.303671\pi$
$240$	0	0
$241$	16.4136	1.05729	0.528646	−	0.848842i	$-0.322700\pi$
$241$	16.4136	1.05729	0.528646	+	0.848842i	$0.322700\pi$
$242$	0	0
$243$	12.1346	0.778435
$244$	0	0
$245$	34.6535	2.21393
$246$	0	0
$247$	12.1345	0.772099
$248$	0	0
$249$	4.87500	0.308940
$250$	0	0
$251$	−17.4168	−1.09934	−0.549671	−	0.835381i	$-0.685247\pi$
$251$	−17.4168	−1.09934	−0.549671	+	0.835381i	$0.685247\pi$
$252$	0	0
$253$	−6.40479	−0.402666
$254$	0	0
$255$	−6.90519	−0.432420
$256$	0	0
$257$	−8.07759	−0.503866	−0.251933	−	0.967745i	$-0.581066\pi$
$257$	−8.07759	−0.503866	−0.251933	+	0.967745i	$0.581066\pi$
$258$	0	0
$259$	6.79874	0.422453
$260$	0	0
$261$	1.05843	0.0655153
$262$	0	0
$263$	24.4984	1.51064	0.755318	−	0.655358i	$-0.227482\pi$
$263$	24.4984	1.51064	0.755318	+	0.655358i	$0.227482\pi$
$264$	0	0
$265$	−3.36473	−0.206693
$266$	0	0
$267$	−26.3964	−1.61543
$268$	0	0
$269$	−14.5843	−0.889222	−0.444611	−	0.895724i	$-0.646658\pi$
$269$	−14.5843	−0.889222	−0.444611	+	0.895724i	$0.646658\pi$
$270$	0	0
$271$	25.1327	1.52670	0.763351	−	0.645984i	$-0.223553\pi$
$271$	25.1327	1.52670	0.763351	+	0.645984i	$0.223553\pi$
$272$	0	0
$273$	−32.6277	−1.97472
$274$	0	0
$275$	32.2388	1.94407
$276$	0	0
$277$	17.9674	1.07956	0.539780	−	0.841806i	$-0.318508\pi$
$277$	17.9674	1.07956	0.539780	+	0.841806i	$0.318508\pi$
$278$	0	0
$279$	−8.48821	−0.508176
$280$	0	0
$281$	16.7721	1.00054	0.500270	−	0.865869i	$-0.333234\pi$
$281$	16.7721	1.00054	0.500270	+	0.865869i	$0.333234\pi$
$282$	0	0
$283$	11.8216	0.702721	0.351360	−	0.936240i	$-0.385719\pi$
$283$	11.8216	0.702721	0.351360	+	0.936240i	$0.385719\pi$
$284$	0	0
$285$	24.2715	1.43772
$286$	0	0
$287$	−24.4176	−1.44133
$288$	0	0
$289$	−16.2409	−0.955345
$290$	0	0
$291$	14.7570	0.865068
$292$	0	0
$293$	7.45609	0.435590	0.217795	−	0.975995i	$-0.430114\pi$
$293$	7.45609	0.435590	0.217795	+	0.975995i	$0.430114\pi$
$294$	0	0
$295$	25.6755	1.49488
$296$	0	0
$297$	11.9077	0.690954
$298$	0	0
$299$	7.73118	0.447105
$300$	0	0
$301$	−9.05907	−0.522156
$302$	0	0
$303$	2.83938	0.163118
$304$	0	0
$305$	46.8374	2.68190
$306$	0	0
$307$	32.7488	1.86907	0.934535	−	0.355871i	$-0.115816\pi$
$307$	32.7488	1.86907	0.934535	+	0.355871i	$0.115816\pi$
$308$	0	0
$309$	−35.8392	−2.03882
$310$	0	0
$311$	18.1499	1.02918	0.514592	−	0.857435i	$-0.327943\pi$
$311$	18.1499	1.02918	0.514592	+	0.857435i	$0.327943\pi$
$312$	0	0
$313$	−31.1499	−1.76070	−0.880349	−	0.474327i	$-0.842692\pi$
$313$	−31.1499	−1.76070	−0.880349	+	0.474327i	$0.842692\pi$
$314$	0	0
$315$	−19.0621	−1.07403
$316$	0	0
$317$	1.05563	0.0592900	0.0296450	−	0.999560i	$-0.490562\pi$
$317$	1.05563	0.0592900	0.0296450	+	0.999560i	$0.490562\pi$
$318$	0	0
$319$	2.80683	0.157152
$320$	0	0
$321$	−6.44107	−0.359505
$322$	0	0
$323$	−2.66833	−0.148470
$324$	0	0
$325$	−38.9152	−2.15863
$326$	0	0
$327$	−30.1393	−1.66671
$328$	0	0
$329$	−15.9322	−0.878372
$330$	0	0
$331$	21.3147	1.17156	0.585781	−	0.810469i	$-0.300788\pi$
$331$	21.3147	1.17156	0.585781	+	0.810469i	$0.300788\pi$
$332$	0	0
$333$	−2.10377	−0.115286
$334$	0	0
$335$	−59.8960	−3.27247
$336$	0	0
$337$	14.1978	0.773403	0.386702	−	0.922205i	$-0.373614\pi$
$337$	14.1978	0.773403	0.386702	+	0.922205i	$0.373614\pi$
$338$	0	0
$339$	9.57002	0.519772
$340$	0	0
$341$	−22.5097	−1.21897
$342$	0	0
$343$	−8.00515	−0.432237
$344$	0	0
$345$	15.4640	0.832552
$346$	0	0
$347$	34.9785	1.87774	0.938871	−	0.344270i	$-0.111873\pi$
$347$	34.9785	1.87774	0.938871	+	0.344270i	$0.111873\pi$
$348$	0	0
$349$	18.2119	0.974858	0.487429	−	0.873163i	$-0.337935\pi$
$349$	18.2119	0.974858	0.487429	+	0.873163i	$0.337935\pi$
$350$	0	0
$351$	−14.3737	−0.767210
$352$	0	0
$353$	−29.0603	−1.54672	−0.773362	−	0.633965i	$-0.781426\pi$
$353$	−29.0603	−1.54672	−0.773362	+	0.633965i	$0.781426\pi$
$354$	0	0
$355$	26.6948	1.41681
$356$	0	0
$357$	7.17474	0.379727
$358$	0	0
$359$	−24.3824	−1.28685	−0.643427	−	0.765508i	$-0.722488\pi$
$359$	−24.3824	−1.28685	−0.643427	+	0.765508i	$0.722488\pi$
$360$	0	0
$361$	−9.62091	−0.506364
$362$	0	0
$363$	0.464027	0.0243551
$364$	0	0
$365$	12.0983	0.633255
$366$	0	0
$367$	3.22785	0.168492	0.0842462	−	0.996445i	$-0.473152\pi$
$367$	3.22785	0.168492	0.0842462	+	0.996445i	$0.473152\pi$
$368$	0	0
$369$	7.55568	0.393333
$370$	0	0
$371$	3.49607	0.181507
$372$	0	0
$373$	−6.20919	−0.321500	−0.160750	−	0.986995i	$-0.551391\pi$
$373$	−6.20919	−0.321500	−0.160750	+	0.986995i	$0.551391\pi$
$374$	0	0
$375$	−38.2120	−1.97326
$376$	0	0
$377$	−3.38810	−0.174496
$378$	0	0
$379$	19.6114	1.00737	0.503686	−	0.863887i	$-0.331977\pi$
$379$	19.6114	1.00737	0.503686	+	0.863887i	$0.331977\pi$
$380$	0	0
$381$	−27.5657	−1.41223
$382$	0	0
$383$	15.4157	0.787704	0.393852	−	0.919174i	$-0.371142\pi$
$383$	15.4157	0.787704	0.393852	+	0.919174i	$0.371142\pi$
$384$	0	0
$385$	−50.5502	−2.57628
$386$	0	0
$387$	2.80320	0.142495
$388$	0	0
$389$	16.1791	0.820313	0.410157	−	0.912015i	$-0.365474\pi$
$389$	16.1791	0.820313	0.410157	+	0.912015i	$0.365474\pi$
$390$	0	0
$391$	−1.70006	−0.0859758
$392$	0	0
$393$	13.7149	0.691824
$394$	0	0
$395$	−0.0520875	−0.00262081
$396$	0	0
$397$	7.70760	0.386834	0.193417	−	0.981117i	$-0.438043\pi$
$397$	7.70760	0.386834	0.193417	+	0.981117i	$0.438043\pi$
$398$	0	0
$399$	−25.2189	−1.26253
$400$	0	0
$401$	7.27557	0.363324	0.181662	−	0.983361i	$-0.441852\pi$
$401$	7.27557	0.363324	0.181662	+	0.983361i	$0.441852\pi$
$402$	0	0
$403$	27.1713	1.35350
$404$	0	0
$405$	−43.0464	−2.13899
$406$	0	0
$407$	−5.57895	−0.276538
$408$	0	0
$409$	−36.9234	−1.82575	−0.912873	−	0.408243i	$-0.866142\pi$
$409$	−36.9234	−1.82575	−0.912873	+	0.408243i	$0.866142\pi$
$410$	0	0
$411$	−9.42553	−0.464927
$412$	0	0
$413$	−26.6777	−1.31272
$414$	0	0
$415$	−9.11698	−0.447535
$416$	0	0
$417$	22.9430	1.12352
$418$	0	0
$419$	19.3937	0.947442	0.473721	−	0.880675i	$-0.342911\pi$
$419$	19.3937	0.947442	0.473721	+	0.880675i	$0.342911\pi$
$420$	0	0
$421$	−9.26954	−0.451770	−0.225885	−	0.974154i	$-0.572527\pi$
$421$	−9.26954	−0.451770	−0.225885	+	0.974154i	$0.572527\pi$
$422$	0	0
$423$	4.93000	0.239705
$424$	0	0
$425$	8.55733	0.415092
$426$	0	0
$427$	−48.6657	−2.35510
$428$	0	0
$429$	26.7739	1.29265
$430$	0	0
$431$	16.2808	0.784220	0.392110	−	0.919918i	$-0.371745\pi$
$431$	16.2808	0.784220	0.392110	+	0.919918i	$0.371745\pi$
$432$	0	0
$433$	27.1840	1.30638	0.653190	−	0.757194i	$-0.273430\pi$
$433$	27.1840	1.30638	0.653190	+	0.757194i	$0.273430\pi$
$434$	0	0
$435$	−6.77691	−0.324928
$436$	0	0
$437$	5.97565	0.285854
$438$	0	0
$439$	16.5485	0.789815	0.394907	−	0.918721i	$-0.370777\pi$
$439$	16.5485	0.789815	0.394907	+	0.918721i	$0.370777\pi$
$440$	0	0
$441$	11.1416	0.530553
$442$	0	0
$443$	30.7067	1.45892	0.729459	−	0.684025i	$-0.239772\pi$
$443$	30.7067	1.45892	0.729459	+	0.684025i	$0.239772\pi$
$444$	0	0
$445$	49.3652	2.34013
$446$	0	0
$447$	41.4231	1.95924
$448$	0	0
$449$	−28.5979	−1.34962	−0.674808	−	0.737993i	$-0.735774\pi$
$449$	−28.5979	−1.34962	−0.674808	+	0.737993i	$0.735774\pi$
$450$	0	0
$451$	20.0367	0.943493
$452$	0	0
$453$	−20.2958	−0.953581
$454$	0	0
$455$	61.0188	2.86061
$456$	0	0
$457$	−25.9708	−1.21486	−0.607431	−	0.794373i	$-0.707800\pi$
$457$	−25.9708	−1.21486	−0.607431	+	0.794373i	$0.707800\pi$
$458$	0	0
$459$	3.16073	0.147530
$460$	0	0
$461$	−36.0499	−1.67901	−0.839507	−	0.543349i	$-0.817156\pi$
$461$	−36.0499	−1.67901	−0.839507	+	0.543349i	$0.817156\pi$
$462$	0	0
$463$	6.42476	0.298584	0.149292	−	0.988793i	$-0.452301\pi$
$463$	6.42476	0.298584	0.149292	+	0.988793i	$0.452301\pi$
$464$	0	0
$465$	54.3482	2.52034
$466$	0	0
$467$	17.8439	0.825718	0.412859	−	0.910795i	$-0.364530\pi$
$467$	17.8439	0.825718	0.412859	+	0.910795i	$0.364530\pi$
$468$	0	0
$469$	62.2341	2.87370
$470$	0	0
$471$	−22.8274	−1.05183
$472$	0	0
$473$	7.43374	0.341804
$474$	0	0
$475$	−30.0787	−1.38011
$476$	0	0
$477$	−1.08181	−0.0495326
$478$	0	0
$479$	2.74048	0.125216	0.0626078	−	0.998038i	$-0.480058\pi$
$479$	2.74048	0.125216	0.0626078	+	0.998038i	$0.480058\pi$
$480$	0	0
$481$	6.73430	0.307058
$482$	0	0
$483$	−16.0676	−0.731101
$484$	0	0
$485$	−27.5977	−1.25315
$486$	0	0
$487$	−20.1162	−0.911551	−0.455775	−	0.890095i	$-0.650638\pi$
$487$	−20.1162	−0.911551	−0.455775	+	0.890095i	$0.650638\pi$
$488$	0	0
$489$	12.7255	0.575467
$490$	0	0
$491$	5.68626	0.256617	0.128309	−	0.991734i	$-0.459045\pi$
$491$	5.68626	0.256617	0.128309	+	0.991734i	$0.459045\pi$
$492$	0	0
$493$	0.745033	0.0335546
$494$	0	0
$495$	15.6421	0.703058
$496$	0	0
$497$	−27.7368	−1.24417
$498$	0	0
$499$	−35.3326	−1.58171	−0.790853	−	0.612007i	$-0.790363\pi$
$499$	−35.3326	−1.58171	−0.790853	+	0.612007i	$0.790363\pi$
$500$	0	0
$501$	29.8480	1.33351
$502$	0	0
$503$	−1.00000	−0.0445878
$503$	−1.00000	−0.0445878
$504$	0	0
$505$	−5.31008	−0.236295
$506$	0	0
$507$	−5.55686	−0.246789
$508$	0	0
$509$	12.3290	0.546473	0.273236	−	0.961947i	$-0.411906\pi$
$509$	12.3290	0.546473	0.273236	+	0.961947i	$0.411906\pi$
$510$	0	0
$511$	−12.5706	−0.556090
$512$	0	0
$513$	−11.1098	−0.490511
$514$	0	0
$515$	67.0247	2.95346
$516$	0	0
$517$	13.0738	0.574983
$518$	0	0
$519$	−11.6441	−0.511119
$520$	0	0
$521$	18.0200	0.789473	0.394736	−	0.918794i	$-0.370836\pi$
$521$	18.0200	0.789473	0.394736	+	0.918794i	$0.370836\pi$
$522$	0	0
$523$	−27.3857	−1.19749	−0.598746	−	0.800939i	$-0.704334\pi$
$523$	−27.3857	−1.19749	−0.598746	+	0.800939i	$0.704334\pi$
$524$	0	0
$525$	80.8770	3.52976
$526$	0	0
$527$	−5.97487	−0.260270
$528$	0	0
$529$	−19.1928	−0.834468
$530$	0	0
$531$	8.25504	0.358238
$532$	0	0
$533$	−24.1862	−1.04762
$534$	0	0
$535$	12.0458	0.520784
$536$	0	0
$537$	10.7725	0.464866
$538$	0	0
$539$	29.5462	1.27264
$540$	0	0
$541$	1.65120	0.0709907	0.0354954	−	0.999370i	$-0.488699\pi$
$541$	1.65120	0.0709907	0.0354954	+	0.999370i	$0.488699\pi$
$542$	0	0
$543$	−18.4563	−0.792038
$544$	0	0
$545$	56.3651	2.41442
$546$	0	0
$547$	−15.5289	−0.663967	−0.331983	−	0.943285i	$-0.607718\pi$
$547$	−15.5289	−0.663967	−0.331983	+	0.943285i	$0.607718\pi$
$548$	0	0
$549$	15.0589	0.642699
$550$	0	0
$551$	−2.61876	−0.111563
$552$	0	0
$553$	0.0541207	0.00230145
$554$	0	0
$555$	13.4700	0.571770
$556$	0	0
$557$	13.2317	0.560644	0.280322	−	0.959906i	$-0.409559\pi$
$557$	13.2317	0.560644	0.280322	+	0.959906i	$0.409559\pi$
$558$	0	0
$559$	−8.97321	−0.379526
$560$	0	0
$561$	−5.88749	−0.248570
$562$	0	0
$563$	−19.0552	−0.803080	−0.401540	−	0.915841i	$-0.631525\pi$
$563$	−19.0552	−0.803080	−0.401540	+	0.915841i	$0.631525\pi$
$564$	0	0
$565$	−17.8974	−0.752949
$566$	0	0
$567$	44.7267	1.87834
$568$	0	0
$569$	−30.0664	−1.26045	−0.630224	−	0.776413i	$-0.717037\pi$
$569$	−30.0664	−1.26045	−0.630224	+	0.776413i	$0.717037\pi$
$570$	0	0
$571$	32.8673	1.37545	0.687726	−	0.725970i	$-0.258609\pi$
$571$	32.8673	1.37545	0.687726	+	0.725970i	$0.258609\pi$
$572$	0	0
$573$	−54.6658	−2.28370
$574$	0	0
$575$	−19.1639	−0.799189
$576$	0	0
$577$	7.75032	0.322650	0.161325	−	0.986901i	$-0.448423\pi$
$577$	7.75032	0.322650	0.161325	+	0.986901i	$0.448423\pi$
$578$	0	0
$579$	−28.4544	−1.18253
$580$	0	0
$581$	9.47286	0.393000
$582$	0	0
$583$	−2.86882	−0.118814
$584$	0	0
$585$	−18.8814	−0.780650
$586$	0	0
$587$	−36.3549	−1.50053	−0.750263	−	0.661140i	$-0.770073\pi$
$587$	−36.3549	−1.50053	−0.750263	+	0.661140i	$0.770073\pi$
$588$	0	0
$589$	21.0015	0.865350
$590$	0	0
$591$	−14.9583	−0.615303
$592$	0	0
$593$	1.55824	0.0639892	0.0319946	−	0.999488i	$-0.489814\pi$
$593$	1.55824	0.0639892	0.0319946	+	0.999488i	$0.489814\pi$
$594$	0	0
$595$	−13.4178	−0.550078
$596$	0	0
$597$	−32.2665	−1.32058
$598$	0	0
$599$	−39.2861	−1.60519	−0.802593	−	0.596527i	$-0.796547\pi$
$599$	−39.2861	−1.60519	−0.802593	+	0.596527i	$0.796547\pi$
$600$	0	0
$601$	−16.2158	−0.661457	−0.330729	−	0.943726i	$-0.607294\pi$
$601$	−16.2158	−0.661457	−0.330729	+	0.943726i	$0.607294\pi$
$602$	0	0
$603$	−19.2575	−0.784224
$604$	0	0
$605$	−0.867801	−0.0352811
$606$	0	0
$607$	−35.2063	−1.42898	−0.714489	−	0.699646i	$-0.753341\pi$
$607$	−35.2063	−1.42898	−0.714489	+	0.699646i	$0.753341\pi$
$608$	0	0
$609$	7.04145	0.285334
$610$	0	0
$611$	−15.7812	−0.638440
$612$	0	0
$613$	2.94564	0.118973	0.0594866	−	0.998229i	$-0.481054\pi$
$613$	2.94564	0.118973	0.0594866	+	0.998229i	$0.481054\pi$
$614$	0	0
$615$	−48.3774	−1.95077
$616$	0	0
$617$	36.2495	1.45935	0.729675	−	0.683795i	$-0.239672\pi$
$617$	36.2495	1.45935	0.729675	+	0.683795i	$0.239672\pi$
$618$	0	0
$619$	21.4223	0.861036	0.430518	−	0.902582i	$-0.358331\pi$
$619$	21.4223	0.861036	0.430518	+	0.902582i	$0.358331\pi$
$620$	0	0
$621$	−7.07835	−0.284044
$622$	0	0
$623$	−51.2921	−2.05498
$624$	0	0
$625$	22.3547	0.894187
$626$	0	0
$627$	20.6943	0.826450
$628$	0	0
$629$	−1.48085	−0.0590454
$630$	0	0
$631$	−30.6737	−1.22110	−0.610550	−	0.791978i	$-0.709052\pi$
$631$	−30.6737	−1.22110	−0.610550	+	0.791978i	$0.709052\pi$
$632$	0	0
$633$	−34.5021	−1.37134
$634$	0	0
$635$	51.5520	2.04578
$636$	0	0
$637$	−35.6650	−1.41310
$638$	0	0
$639$	8.58278	0.339529
$640$	0	0
$641$	3.28605	0.129791	0.0648956	−	0.997892i	$-0.479329\pi$
$641$	3.28605	0.129791	0.0648956	+	0.997892i	$0.479329\pi$
$642$	0	0
$643$	−46.0517	−1.81610	−0.908050	−	0.418861i	$-0.862429\pi$
$643$	−46.0517	−1.81610	−0.908050	+	0.418861i	$0.862429\pi$
$644$	0	0
$645$	−17.9483	−0.706713
$646$	0	0
$647$	−25.6924	−1.01007	−0.505036	−	0.863098i	$-0.668521\pi$
$647$	−25.6924	−1.01007	−0.505036	+	0.863098i	$0.668521\pi$
$648$	0	0
$649$	21.8913	0.859310
$650$	0	0
$651$	−56.4697	−2.21322
$652$	0	0
$653$	6.17888	0.241798	0.120899	−	0.992665i	$-0.461422\pi$
$653$	6.17888	0.241798	0.120899	+	0.992665i	$0.461422\pi$
$654$	0	0
$655$	−25.6489	−1.00219
$656$	0	0
$657$	3.88979	0.151755
$658$	0	0
$659$	41.1285	1.60214	0.801070	−	0.598571i	$-0.204265\pi$
$659$	41.1285	1.60214	0.801070	+	0.598571i	$0.204265\pi$
$660$	0	0
$661$	−5.85140	−0.227593	−0.113796	−	0.993504i	$-0.536301\pi$
$661$	−5.85140	−0.227593	−0.113796	+	0.993504i	$0.536301\pi$
$662$	0	0
$663$	7.10674	0.276003
$664$	0	0
$665$	47.1632	1.82891
$666$	0	0
$667$	−1.66848	−0.0646037
$668$	0	0
$669$	37.5008	1.44986
$670$	0	0
$671$	39.9344	1.54165
$672$	0	0
$673$	−15.8918	−0.612582	−0.306291	−	0.951938i	$-0.599088\pi$
$673$	−15.8918	−0.612582	−0.306291	+	0.951938i	$0.599088\pi$
$674$	0	0
$675$	35.6292	1.37137
$676$	0	0
$677$	−15.2243	−0.585118	−0.292559	−	0.956247i	$-0.594507\pi$
$677$	−15.2243	−0.585118	−0.292559	+	0.956247i	$0.594507\pi$
$678$	0	0
$679$	28.6750	1.10045
$680$	0	0
$681$	13.4996	0.517306
$682$	0	0
$683$	−9.28193	−0.355163	−0.177582	−	0.984106i	$-0.556827\pi$
$683$	−9.28193	−0.355163	−0.177582	+	0.984106i	$0.556827\pi$
$684$	0	0
$685$	17.6272	0.673500
$686$	0	0
$687$	−31.3494	−1.19605
$688$	0	0
$689$	3.46293	0.131927
$690$	0	0
$691$	19.1787	0.729590	0.364795	−	0.931088i	$-0.381139\pi$
$691$	19.1787	0.729590	0.364795	+	0.931088i	$0.381139\pi$
$692$	0	0
$693$	−16.2526	−0.617387
$694$	0	0
$695$	−42.9069	−1.62755
$696$	0	0
$697$	5.31846	0.201451
$698$	0	0
$699$	17.8495	0.675130
$700$	0	0
$701$	−1.25754	−0.0474967	−0.0237483	−	0.999718i	$-0.507560\pi$
$701$	−1.25754	−0.0474967	−0.0237483	+	0.999718i	$0.507560\pi$
$702$	0	0
$703$	5.20514	0.196316
$704$	0	0
$705$	−31.5657	−1.18883
$706$	0	0
$707$	5.51736	0.207502
$708$	0	0
$709$	35.2024	1.32205	0.661026	−	0.750363i	$-0.270121\pi$
$709$	35.2024	1.32205	0.661026	+	0.750363i	$0.270121\pi$
$710$	0	0
$711$	−0.0167469	−0.000628058 0
$712$	0	0
$713$	13.3805	0.501105
$714$	0	0
$715$	−50.0712	−1.87256
$716$	0	0
$717$	−36.8162	−1.37493
$718$	0	0
$719$	−22.9839	−0.857156	−0.428578	−	0.903505i	$-0.640985\pi$
$719$	−22.9839	−0.857156	−0.428578	+	0.903505i	$0.640985\pi$
$720$	0	0
$721$	−69.6410	−2.59357
$722$	0	0
$723$	−33.7888	−1.25662
$724$	0	0
$725$	8.39836	0.311907
$726$	0	0
$727$	5.11695	0.189777	0.0948886	−	0.995488i	$-0.469751\pi$
$727$	5.11695	0.189777	0.0948886	+	0.995488i	$0.469751\pi$
$728$	0	0
$729$	8.56358	0.317169
$730$	0	0
$731$	1.97318	0.0729807
$732$	0	0
$733$	19.4202	0.717300	0.358650	−	0.933472i	$-0.383237\pi$
$733$	19.4202	0.717300	0.358650	+	0.933472i	$0.383237\pi$
$734$	0	0
$735$	−71.3374	−2.63132
$736$	0	0
$737$	−51.0684	−1.88113
$738$	0	0
$739$	34.4538	1.26740	0.633702	−	0.773577i	$-0.281535\pi$
$739$	34.4538	1.26740	0.633702	+	0.773577i	$0.281535\pi$
$740$	0	0
$741$	−24.9799	−0.917660
$742$	0	0
$743$	34.2956	1.25818	0.629092	−	0.777331i	$-0.283427\pi$
$743$	34.2956	1.25818	0.629092	+	0.777331i	$0.283427\pi$
$744$	0	0
$745$	−77.4674	−2.83819
$746$	0	0
$747$	−2.93124	−0.107249
$748$	0	0
$749$	−12.5160	−0.457323
$750$	0	0
$751$	−45.8704	−1.67383	−0.836917	−	0.547330i	$-0.815644\pi$
$751$	−45.8704	−1.67383	−0.836917	+	0.547330i	$0.815644\pi$
$752$	0	0
$753$	35.8541	1.30660
$754$	0	0
$755$	37.9562	1.38137
$756$	0	0
$757$	18.1237	0.658719	0.329359	−	0.944205i	$-0.393167\pi$
$757$	18.1237	0.658719	0.329359	+	0.944205i	$0.393167\pi$
$758$	0	0
$759$	13.1848	0.478579
$760$	0	0
$761$	28.4866	1.03264	0.516320	−	0.856396i	$-0.327301\pi$
$761$	28.4866	1.03264	0.516320	+	0.856396i	$0.327301\pi$
$762$	0	0
$763$	−58.5653	−2.12021
$764$	0	0
$765$	4.15196	0.150115
$766$	0	0
$767$	−26.4249	−0.954146
$768$	0	0
$769$	−3.25802	−0.117487	−0.0587435	−	0.998273i	$-0.518709\pi$
$769$	−3.25802	−0.117487	−0.0587435	+	0.998273i	$0.518709\pi$
$770$	0	0
$771$	16.6284	0.598858
$772$	0	0
$773$	−3.80437	−0.136834	−0.0684168	−	0.997657i	$-0.521795\pi$
$773$	−3.80437	−0.136834	−0.0684168	+	0.997657i	$0.521795\pi$
$774$	0	0
$775$	−67.3516	−2.41934
$776$	0	0
$777$	−13.9958	−0.502097
$778$	0	0
$779$	−18.6942	−0.669789
$780$	0	0
$781$	22.7605	0.814433
$782$	0	0
$783$	3.10201	0.110857
$784$	0	0
$785$	42.6908	1.52370
$786$	0	0
$787$	−11.3921	−0.406086	−0.203043	−	0.979170i	$-0.565083\pi$
$787$	−11.3921	−0.406086	−0.203043	+	0.979170i	$0.565083\pi$
$788$	0	0
$789$	−50.4322	−1.79543
$790$	0	0
$791$	18.5960	0.661198
$792$	0	0
$793$	−48.2045	−1.71179
$794$	0	0
$795$	6.92659	0.245661
$796$	0	0
$797$	−40.6380	−1.43947	−0.719736	−	0.694248i	$-0.755737\pi$
$797$	−40.6380	−1.43947	−0.719736	+	0.694248i	$0.755737\pi$
$798$	0	0
$799$	3.47024	0.122768
$800$	0	0
$801$	15.8716	0.560797
$802$	0	0
$803$	10.3152	0.364017
$804$	0	0
$805$	30.0488	1.05908
$806$	0	0
$807$	30.0231	1.05686
$808$	0	0
$809$	25.5847	0.899510	0.449755	−	0.893152i	$-0.351511\pi$
$809$	25.5847	0.899510	0.449755	+	0.893152i	$0.351511\pi$
$810$	0	0
$811$	−11.7808	−0.413679	−0.206839	−	0.978375i	$-0.566318\pi$
$811$	−11.7808	−0.413679	−0.206839	+	0.978375i	$0.566318\pi$
$812$	0	0
$813$	−51.7379	−1.81453
$814$	0	0
$815$	−23.7986	−0.833628
$816$	0	0
$817$	−6.93565	−0.242648
$818$	0	0
$819$	19.6184	0.685524
$820$	0	0
$821$	51.1237	1.78423	0.892114	−	0.451809i	$-0.149221\pi$
$821$	51.1237	1.78423	0.892114	+	0.451809i	$0.149221\pi$
$822$	0	0
$823$	8.42687	0.293742	0.146871	−	0.989156i	$-0.453080\pi$
$823$	8.42687	0.293742	0.146871	+	0.989156i	$0.453080\pi$
$824$	0	0
$825$	−66.3665	−2.31058
$826$	0	0
$827$	20.9554	0.728691	0.364346	−	0.931264i	$-0.381293\pi$
$827$	20.9554	0.728691	0.364346	+	0.931264i	$0.381293\pi$
$828$	0	0
$829$	−38.5517	−1.33896	−0.669478	−	0.742832i	$-0.733482\pi$
$829$	−38.5517	−1.33896	−0.669478	+	0.742832i	$0.733482\pi$
$830$	0	0
$831$	−36.9876	−1.28309
$832$	0	0
$833$	7.84261	0.271730
$834$	0	0
$835$	−55.8202	−1.93174
$836$	0	0
$837$	−24.8769	−0.859871
$838$	0	0
$839$	−57.1462	−1.97291	−0.986454	−	0.164041i	$-0.947547\pi$
$839$	−57.1462	−1.97291	−0.986454	+	0.164041i	$0.947547\pi$
$840$	0	0
$841$	−28.2688	−0.974786
$842$	0	0
$843$	−34.5269	−1.18917
$844$	0	0
$845$	10.3922	0.357501
$846$	0	0
$847$	0.901676	0.0309819
$848$	0	0
$849$	−24.3358	−0.835203
$850$	0	0
$851$	3.31632	0.113682
$852$	0	0
$853$	−17.1642	−0.587690	−0.293845	−	0.955853i	$-0.594935\pi$
$853$	−17.1642	−0.587690	−0.293845	+	0.955853i	$0.594935\pi$
$854$	0	0
$855$	−14.5940	−0.499104
$856$	0	0
$857$	−27.0786	−0.924988	−0.462494	−	0.886622i	$-0.653045\pi$
$857$	−27.0786	−0.924988	−0.462494	+	0.886622i	$0.653045\pi$
$858$	0	0
$859$	−39.1498	−1.33577	−0.667886	−	0.744263i	$-0.732801\pi$
$859$	−39.1498	−1.33577	−0.667886	+	0.744263i	$0.732801\pi$
$860$	0	0
$861$	50.2658	1.71305
$862$	0	0
$863$	−3.39480	−0.115560	−0.0577801	−	0.998329i	$-0.518402\pi$
$863$	−3.39480	−0.115560	−0.0577801	+	0.998329i	$0.518402\pi$
$864$	0	0
$865$	21.7762	0.740413
$866$	0	0
$867$	33.4333	1.13545
$868$	0	0
$869$	−0.0444107	−0.00150653
$870$	0	0
$871$	61.6442	2.08874
$872$	0	0
$873$	−8.87308	−0.300308
$874$	0	0
$875$	−74.2518	−2.51017
$876$	0	0
$877$	−47.5485	−1.60560	−0.802799	−	0.596250i	$-0.796657\pi$
$877$	−47.5485	−1.60560	−0.802799	+	0.596250i	$0.796657\pi$
$878$	0	0
$879$	−15.3490	−0.517710
$880$	0	0
$881$	44.2695	1.49148	0.745739	−	0.666238i	$-0.232097\pi$
$881$	44.2695	1.49148	0.745739	+	0.666238i	$0.232097\pi$
$882$	0	0
$883$	−13.8810	−0.467133	−0.233566	−	0.972341i	$-0.575040\pi$
$883$	−13.8810	−0.467133	−0.233566	+	0.972341i	$0.575040\pi$
$884$	0	0
$885$	−52.8552	−1.77671
$886$	0	0
$887$	−7.88915	−0.264892	−0.132446	−	0.991190i	$-0.542283\pi$
$887$	−7.88915	−0.264892	−0.132446	+	0.991190i	$0.542283\pi$
$888$	0	0
$889$	−53.5643	−1.79649
$890$	0	0
$891$	−36.7021	−1.22957
$892$	0	0
$893$	−12.1978	−0.408183
$894$	0	0
$895$	−20.1461	−0.673411
$896$	0	0
$897$	−15.9153	−0.531397
$898$	0	0
$899$	−5.86388	−0.195571
$900$	0	0
$901$	−0.761488	−0.0253688
$902$	0	0
$903$	18.6489	0.620596
$904$	0	0
$905$	34.5161	1.14736
$906$	0	0
$907$	−16.8663	−0.560037	−0.280019	−	0.959995i	$-0.590341\pi$
$907$	−16.8663	−0.560037	−0.280019	+	0.959995i	$0.590341\pi$
$908$	0	0
$909$	−1.70727	−0.0566265
$910$	0	0
$911$	6.46407	0.214164	0.107082	−	0.994250i	$-0.465849\pi$
$911$	6.46407	0.214164	0.107082	+	0.994250i	$0.465849\pi$
$912$	0	0
$913$	−7.77329	−0.257258
$914$	0	0
$915$	−96.4190	−3.18752
$916$	0	0
$917$	26.6501	0.880063
$918$	0	0
$919$	18.4896	0.609916	0.304958	−	0.952366i	$-0.401358\pi$
$919$	18.4896	0.609916	0.304958	+	0.952366i	$0.401358\pi$
$920$	0	0
$921$	−67.4163	−2.22144
$922$	0	0
$923$	−27.4740	−0.904317
$924$	0	0
$925$	−16.6929	−0.548858
$926$	0	0
$927$	21.5494	0.707776
$928$	0	0
$929$	−37.7932	−1.23995	−0.619977	−	0.784620i	$-0.712858\pi$
$929$	−37.7932	−1.23995	−0.619977	+	0.784620i	$0.712858\pi$
$930$	0	0
$931$	−27.5665	−0.903455
$932$	0	0
$933$	−37.3631	−1.22321
$934$	0	0
$935$	11.0105	0.360081
$936$	0	0
$937$	−0.141195	−0.00461263	−0.00230631	−	0.999997i	$-0.500734\pi$
$937$	−0.141195	−0.00461263	−0.00230631	+	0.999997i	$0.500734\pi$
$938$	0	0
$939$	64.1249	2.09264
$940$	0	0
$941$	2.59540	0.0846076	0.0423038	−	0.999105i	$-0.486530\pi$
$941$	2.59540	0.0846076	0.0423038	+	0.999105i	$0.486530\pi$
$942$	0	0
$943$	−11.9105	−0.387860
$944$	0	0
$945$	−55.8663	−1.81733
$946$	0	0
$947$	−36.3251	−1.18041	−0.590203	−	0.807255i	$-0.700952\pi$
$947$	−36.3251	−1.18041	−0.590203	+	0.807255i	$0.700952\pi$
$948$	0	0
$949$	−12.4514	−0.404191
$950$	0	0
$951$	−2.17311	−0.0704678
$952$	0	0
$953$	−46.6809	−1.51214	−0.756071	−	0.654490i	$-0.772883\pi$
$953$	−46.6809	−1.51214	−0.756071	+	0.654490i	$0.772883\pi$
$954$	0	0
$955$	102.233	3.30819
$956$	0	0
$957$	−5.77811	−0.186780
$958$	0	0
$959$	−18.3152	−0.591430
$960$	0	0
$961$	16.0260	0.516967
$962$	0	0
$963$	3.87289	0.124802
$964$	0	0
$965$	53.2141	1.71302
$966$	0	0
$967$	24.3623	0.783439	0.391719	−	0.920085i	$-0.371880\pi$
$967$	24.3623	0.783439	0.391719	+	0.920085i	$0.371880\pi$
$968$	0	0
$969$	5.49300	0.176461
$970$	0	0
$971$	−17.5521	−0.563272	−0.281636	−	0.959521i	$-0.590877\pi$
$971$	−17.5521	−0.563272	−0.281636	+	0.959521i	$0.590877\pi$
$972$	0	0
$973$	44.5817	1.42922
$974$	0	0
$975$	80.1105	2.56559
$976$	0	0
$977$	17.4651	0.558758	0.279379	−	0.960181i	$-0.409871\pi$
$977$	17.4651	0.558758	0.279379	+	0.960181i	$0.409871\pi$
$978$	0	0
$979$	42.0896	1.34519
$980$	0	0
$981$	18.1222	0.578598
$982$	0	0
$983$	−28.6031	−0.912296	−0.456148	−	0.889904i	$-0.650771\pi$
$983$	−28.6031	−0.912296	−0.456148	+	0.889904i	$0.650771\pi$
$984$	0	0
$985$	27.9743	0.891335
$986$	0	0
$987$	32.7979	1.04397
$988$	0	0
$989$	−4.41887	−0.140512
$990$	0	0
$991$	−31.6846	−1.00649	−0.503247	−	0.864142i	$-0.667861\pi$
$991$	−31.6846	−1.00649	−0.503247	+	0.864142i	$0.667861\pi$
$992$	0	0
$993$	−43.8782	−1.39243
$994$	0	0
$995$	60.3432	1.91301
$996$	0	0
$997$	−4.75656	−0.150642	−0.0753209	−	0.997159i	$-0.523998\pi$
$997$	−4.75656	−0.150642	−0.0753209	+	0.997159i	$0.523998\pi$
$998$	0	0
$999$	−6.16565	−0.195073

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4024.2.a.f.1.9	✓	33
4.3	odd	2		8048.2.a.y.1.25		33

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4024.2.a.f.1.9	✓	33	1.1	even	1	trivial
8048.2.a.y.1.25		33	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4024 = 2^{3} \cdot 503 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4024.a (trivial)

\(f(q)\)	\(=\)	\(q-2.05859 q^{3} +3.84987 q^{5} -4.00015 q^{7} +1.23779 q^{9} +O(q^{10})\)	\(q-2.05859 q^{3} +3.84987 q^{5} -4.00015 q^{7} +1.23779 q^{9} +3.28247 q^{11} -3.96224 q^{13} -7.92531 q^{15} +0.871284 q^{17} -3.06253 q^{19} +8.23467 q^{21} -1.95121 q^{23} +9.82152 q^{25} +3.62767 q^{27} +0.855098 q^{29} -6.85755 q^{31} -6.75725 q^{33} -15.4001 q^{35} -1.69962 q^{37} +8.15663 q^{39} +6.10417 q^{41} +2.26468 q^{43} +4.76534 q^{45} +3.98291 q^{47} +9.00121 q^{49} -1.79362 q^{51} -0.873984 q^{53} +12.6371 q^{55} +6.30449 q^{57} +6.66917 q^{59} +12.1660 q^{61} -4.95135 q^{63} -15.2541 q^{65} -15.5579 q^{67} +4.01675 q^{69} +6.93395 q^{71} +3.14252 q^{73} -20.2185 q^{75} -13.1304 q^{77} -0.0135297 q^{79} -11.1812 q^{81} -2.36812 q^{83} +3.35433 q^{85} -1.76030 q^{87} +12.8225 q^{89} +15.8496 q^{91} +14.1169 q^{93} -11.7904 q^{95} -7.16848 q^{97} +4.06301 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 33 q - 2 q^{3} + 12 q^{5} + 4 q^{7} + 43 q^{9}+O(q^{10}) \) 33 * q - 2 * q^3 + 12 * q^5 + 4 * q^7 + 43 * q^9	\( 33 q - 2 q^{3} + 12 q^{5} + 4 q^{7} + 43 q^{9} + 22 q^{11} + 25 q^{13} - 4 q^{15} + 17 q^{17} + 6 q^{19} + 18 q^{21} + 16 q^{23} + 47 q^{25} - 20 q^{27} + 47 q^{29} - 7 q^{31} - 6 q^{33} + 19 q^{35} + 75 q^{37} + 21 q^{39} + 22 q^{41} - 5 q^{43} + 33 q^{45} + 10 q^{47} + 31 q^{49} + 9 q^{51} + 64 q^{53} - 3 q^{55} + 5 q^{57} + 28 q^{59} + 49 q^{61} - 10 q^{63} + 46 q^{65} - 14 q^{67} + 30 q^{69} + 35 q^{71} + 19 q^{73} - 33 q^{75} + 32 q^{77} - 12 q^{79} + 57 q^{81} + 82 q^{85} - 5 q^{87} + 42 q^{89} - 15 q^{91} + 55 q^{93} + 33 q^{95} + 4 q^{97} + 22 q^{99}+O(q^{100}) \) 33 * q - 2 * q^3 + 12 * q^5 + 4 * q^7 + 43 * q^9 + 22 * q^11 + 25 * q^13 - 4 * q^15 + 17 * q^17 + 6 * q^19 + 18 * q^21 + 16 * q^23 + 47 * q^25 - 20 * q^27 + 47 * q^29 - 7 * q^31 - 6 * q^33 + 19 * q^35 + 75 * q^37 + 21 * q^39 + 22 * q^41 - 5 * q^43 + 33 * q^45 + 10 * q^47 + 31 * q^49 + 9 * q^51 + 64 * q^53 - 3 * q^55 + 5 * q^57 + 28 * q^59 + 49 * q^61 - 10 * q^63 + 46 * q^65 - 14 * q^67 + 30 * q^69 + 35 * q^71 + 19 * q^73 - 33 * q^75 + 32 * q^77 - 12 * q^79 + 57 * q^81 + 82 * q^85 - 5 * q^87 + 42 * q^89 - 15 * q^91 + 55 * q^93 + 33 * q^95 + 4 * q^97 + 22 * q^99

Introduction

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