LMFDB - Embedded newform 416.2.b.b.209.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [416,2,Mod(209,416)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(416, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("416.209");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 416 = 2^{5} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	416.b (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$3.32177672409$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 104)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			209.4
Root			$-0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	416.209
Dual form			416.2.b.b.209.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000i q^{3} +3.46410i q^{5} +4.73205 q^{7} -1.00000 q^{9} +O(q^{10})$	$q+2.00000i q^{3} +3.46410i q^{5} +4.73205 q^{7} -1.00000 q^{9} -1.26795i q^{11} -1.00000i q^{13} -6.92820 q^{15} -1.46410 q^{17} -2.73205i q^{19} +9.46410i q^{21} -4.00000 q^{23} -7.00000 q^{25} +4.00000i q^{27} -2.00000i q^{29} +3.26795 q^{31} +2.53590 q^{33} +16.3923i q^{35} -4.92820i q^{37} +2.00000 q^{39} -4.92820 q^{41} -7.46410i q^{43} -3.46410i q^{45} -3.26795 q^{47} +15.3923 q^{49} -2.92820i q^{51} -10.9282i q^{53} +4.39230 q^{55} +5.46410 q^{57} +0.196152i q^{59} +10.9282i q^{61} -4.73205 q^{63} +3.46410 q^{65} -2.73205i q^{67} -8.00000i q^{69} -2.19615 q^{71} -0.535898 q^{73} -14.0000i q^{75} -6.00000i q^{77} +1.46410 q^{79} -11.0000 q^{81} +6.73205i q^{83} -5.07180i q^{85} +4.00000 q^{87} +17.3205 q^{89} -4.73205i q^{91} +6.53590i q^{93} +9.46410 q^{95} -14.3923 q^{97} +1.26795i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 12 q^{7} - 4 q^{9}+O(q^{10}) $ 4 * q + 12 * q^7 - 4 * q^9	$ 4 q + 12 q^{7} - 4 q^{9} + 8 q^{17} - 16 q^{23} - 28 q^{25} + 20 q^{31} + 24 q^{33} + 8 q^{39} + 8 q^{41} - 20 q^{47} + 20 q^{49} - 24 q^{55} + 8 q^{57} - 12 q^{63} + 12 q^{71} - 16 q^{73} - 8 q^{79} - 44 q^{81} + 16 q^{87} + 24 q^{95} - 16 q^{97}+O(q^{100}) $ 4 * q + 12 * q^7 - 4 * q^9 + 8 * q^17 - 16 * q^23 - 28 * q^25 + 20 * q^31 + 24 * q^33 + 8 * q^39 + 8 * q^41 - 20 * q^47 + 20 * q^49 - 24 * q^55 + 8 * q^57 - 12 * q^63 + 12 * q^71 - 16 * q^73 - 8 * q^79 - 44 * q^81 + 16 * q^87 + 24 * q^95 - 16 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/416\mathbb{Z}\right)^\times$.

$n$	$261$	$287$	$353$
$\chi(n)$	$-1$	$1$	$1$

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
$3$	2.00000i	1.15470i	0.816497	+	0.577350i	$0.195913\pi$
$3$	2.00000i	1.15470i	−0.816497	+	0.577350i	$0.804087\pi$
$4$	0	0
$5$	3.46410i	1.54919i	0.632456	+	0.774597i	$0.282047\pi$
$5$	3.46410i	1.54919i	−0.632456	+	0.774597i	$0.717953\pi$
$6$	0	0
$7$	4.73205	1.78855	0.894274	−	0.447521i	$-0.147693\pi$
$7$	4.73205	1.78855	0.894274	+	0.447521i	$0.147693\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	− 1.26795i	− 0.382301i	−0.981561	−	0.191151i	$-0.938778\pi$
$11$	− 1.26795i	− 0.382301i	0.981561	−	0.191151i	$-0.0612219\pi$
$12$	0	0
$13$	− 1.00000i	− 0.277350i
$13$	− 1.00000i	− 0.277350i
$14$	0	0
$15$	−6.92820	−1.78885
$16$	0	0
$17$	−1.46410	−0.355097	−0.177548	−	0.984112i	$-0.556817\pi$
$17$	−1.46410	−0.355097	−0.177548	+	0.984112i	$0.556817\pi$
$18$	0	0
$19$	− 2.73205i	− 0.626775i	−0.949625	−	0.313388i	$-0.898536\pi$
$19$	− 2.73205i	− 0.626775i	0.949625	−	0.313388i	$-0.101464\pi$
$20$	0	0
$21$	9.46410i	2.06524i
$22$	0	0
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	0	0
$25$	−7.00000	−1.40000
$26$	0	0
$27$	4.00000i	0.769800i
$28$	0	0
$29$	− 2.00000i	− 0.371391i	−0.982607	−	0.185695i	$-0.940546\pi$
$29$	− 2.00000i	− 0.371391i	0.982607	−	0.185695i	$-0.0594537\pi$
$30$	0	0
$31$	3.26795	0.586941	0.293471	−	0.955968i	$-0.405190\pi$
$31$	3.26795	0.586941	0.293471	+	0.955968i	$0.405190\pi$
$32$	0	0
$33$	2.53590	0.441443
$34$	0	0
$35$	16.3923i	2.77081i
$36$	0	0
$37$	− 4.92820i	− 0.810192i	−0.914274	−	0.405096i	$-0.867238\pi$
$37$	− 4.92820i	− 0.810192i	0.914274	−	0.405096i	$-0.132762\pi$
$38$	0	0
$39$	2.00000	0.320256
$40$	0	0
$41$	−4.92820	−0.769656	−0.384828	−	0.922988i	$-0.625739\pi$
$41$	−4.92820	−0.769656	−0.384828	+	0.922988i	$0.625739\pi$
$42$	0	0
$43$	− 7.46410i	− 1.13826i	−0.822246	−	0.569132i	$-0.807279\pi$
$43$	− 7.46410i	− 1.13826i	0.822246	−	0.569132i	$-0.192721\pi$
$44$	0	0
$45$	− 3.46410i	− 0.516398i
$46$	0	0
$47$	−3.26795	−0.476679	−0.238340	−	0.971182i	$-0.576603\pi$
$47$	−3.26795	−0.476679	−0.238340	+	0.971182i	$0.576603\pi$
$48$	0	0
$49$	15.3923	2.19890
$50$	0	0
$51$	− 2.92820i	− 0.410030i
$52$	0	0
$53$	− 10.9282i	− 1.50110i	−0.660811	−	0.750552i	$-0.729788\pi$
$53$	− 10.9282i	− 1.50110i	0.660811	−	0.750552i	$-0.270212\pi$
$54$	0	0
$55$	4.39230	0.592258
$56$	0	0
$57$	5.46410	0.723738
$58$	0	0
$59$	0.196152i	0.0255369i	0.999918	+	0.0127684i	$0.00406443\pi$
$59$	0.196152i	0.0255369i	−0.999918	+	0.0127684i	$0.995936\pi$
$60$	0	0
$61$	10.9282i	1.39921i	0.714528	+	0.699607i	$0.246641\pi$
$61$	10.9282i	1.39921i	−0.714528	+	0.699607i	$0.753359\pi$
$62$	0	0
$63$	−4.73205	−0.596182
$64$	0	0
$65$	3.46410	0.429669
$66$	0	0
$67$	− 2.73205i	− 0.333773i	−0.985976	−	0.166887i	$-0.946629\pi$
$67$	− 2.73205i	− 0.333773i	0.985976	−	0.166887i	$-0.0533714\pi$
$68$	0	0
$69$	− 8.00000i	− 0.963087i
$70$	0	0
$71$	−2.19615	−0.260635	−0.130318	−	0.991472i	$-0.541600\pi$
$71$	−2.19615	−0.260635	−0.130318	+	0.991472i	$0.541600\pi$
$72$	0	0
$73$	−0.535898	−0.0627222	−0.0313611	−	0.999508i	$-0.509984\pi$
$73$	−0.535898	−0.0627222	−0.0313611	+	0.999508i	$0.509984\pi$
$74$	0	0
$75$	− 14.0000i	− 1.61658i
$76$	0	0
$77$	− 6.00000i	− 0.683763i
$78$	0	0
$79$	1.46410	0.164724	0.0823622	−	0.996602i	$-0.473754\pi$
$79$	1.46410	0.164724	0.0823622	+	0.996602i	$0.473754\pi$
$80$	0	0
$81$	−11.0000	−1.22222
$82$	0	0
$83$	6.73205i	0.738939i	0.929243	+	0.369469i	$0.120461\pi$
$83$	6.73205i	0.738939i	−0.929243	+	0.369469i	$0.879539\pi$
$84$	0	0
$85$	− 5.07180i	− 0.550114i
$86$	0	0
$87$	4.00000	0.428845
$88$	0	0
$89$	17.3205	1.83597	0.917985	−	0.396615i	$-0.129815\pi$
$89$	17.3205	1.83597	0.917985	+	0.396615i	$0.129815\pi$
$90$	0	0
$91$	− 4.73205i	− 0.496054i
$92$	0	0
$93$	6.53590i	0.677741i
$94$	0	0
$95$	9.46410	0.970996
$96$	0	0
$97$	−14.3923	−1.46132	−0.730659	−	0.682743i	$-0.760787\pi$
$97$	−14.3923	−1.46132	−0.730659	+	0.682743i	$0.760787\pi$
$98$	0	0
$99$	1.26795i	0.127434i
$100$	0	0
$101$	12.0000i	1.19404i	0.802225	+	0.597022i	$0.203650\pi$
$101$	12.0000i	1.19404i	−0.802225	+	0.597022i	$0.796350\pi$
$102$	0	0
$103$	6.92820	0.682656	0.341328	−	0.939944i	$-0.389123\pi$
$103$	6.92820	0.682656	0.341328	+	0.939944i	$0.389123\pi$
$104$	0	0
$105$	−32.7846	−3.19945
$106$	0	0
$107$	− 8.92820i	− 0.863122i	−0.902084	−	0.431561i	$-0.857963\pi$
$107$	− 8.92820i	− 0.863122i	0.902084	−	0.431561i	$-0.142037\pi$
$108$	0	0
$109$	2.00000i	0.191565i	0.995402	+	0.0957826i	$0.0305354\pi$
$109$	2.00000i	0.191565i	−0.995402	+	0.0957826i	$0.969465\pi$
$110$	0	0
$111$	9.85641	0.935529
$112$	0	0
$113$	9.46410	0.890308	0.445154	−	0.895454i	$-0.353149\pi$
$113$	9.46410	0.890308	0.445154	+	0.895454i	$0.353149\pi$
$114$	0	0
$115$	− 13.8564i	− 1.29212i
$116$	0	0
$117$	1.00000i	0.0924500i
$118$	0	0
$119$	−6.92820	−0.635107
$120$	0	0
$121$	9.39230	0.853846
$122$	0	0
$123$	− 9.85641i	− 0.888722i
$124$	0	0
$125$	− 6.92820i	− 0.619677i
$126$	0	0
$127$	4.00000	0.354943	0.177471	−	0.984126i	$-0.443208\pi$
$127$	4.00000	0.354943	0.177471	+	0.984126i	$0.443208\pi$
$128$	0	0
$129$	14.9282	1.31436
$130$	0	0
$131$	− 7.85641i	− 0.686417i	−0.939259	−	0.343209i	$-0.888486\pi$
$131$	− 7.85641i	− 0.686417i	0.939259	−	0.343209i	$-0.111514\pi$
$132$	0	0
$133$	− 12.9282i	− 1.12102i
$134$	0	0
$135$	−13.8564	−1.19257
$136$	0	0
$137$	−0.928203	−0.0793018	−0.0396509	−	0.999214i	$-0.512625\pi$
$137$	−0.928203	−0.0793018	−0.0396509	+	0.999214i	$0.512625\pi$
$138$	0	0
$139$	− 10.0000i	− 0.848189i	−0.905618	−	0.424094i	$-0.860592\pi$
$139$	− 10.0000i	− 0.848189i	0.905618	−	0.424094i	$-0.139408\pi$
$140$	0	0
$141$	− 6.53590i	− 0.550422i
$142$	0	0
$143$	−1.26795	−0.106031
$144$	0	0
$145$	6.92820	0.575356
$146$	0	0
$147$	30.7846i	2.53907i
$148$	0	0
$149$	− 0.928203i	− 0.0760414i	−0.999277	−	0.0380207i	$-0.987895\pi$
$149$	− 0.928203i	− 0.0760414i	0.999277	−	0.0380207i	$-0.0121053\pi$
$150$	0	0
$151$	−17.1244	−1.39356	−0.696780	−	0.717285i	$-0.745385\pi$
$151$	−17.1244	−1.39356	−0.696780	+	0.717285i	$0.745385\pi$
$152$	0	0
$153$	1.46410	0.118366
$154$	0	0
$155$	11.3205i	0.909285i
$156$	0	0
$157$	3.07180i	0.245156i	0.992459	+	0.122578i	$0.0391162\pi$
$157$	3.07180i	0.245156i	−0.992459	+	0.122578i	$0.960884\pi$
$158$	0	0
$159$	21.8564	1.73333
$160$	0	0
$161$	−18.9282	−1.49175
$162$	0	0
$163$	13.2679i	1.03923i	0.854402	+	0.519613i	$0.173924\pi$
$163$	13.2679i	1.03923i	−0.854402	+	0.519613i	$0.826076\pi$
$164$	0	0
$165$	8.78461i	0.683881i
$166$	0	0
$167$	−11.6603	−0.902298	−0.451149	−	0.892449i	$-0.648986\pi$
$167$	−11.6603	−0.902298	−0.451149	+	0.892449i	$0.648986\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	2.73205i	0.208925i
$172$	0	0
$173$	6.92820i	0.526742i	0.964695	+	0.263371i	$0.0848343\pi$
$173$	6.92820i	0.526742i	−0.964695	+	0.263371i	$0.915166\pi$
$174$	0	0
$175$	−33.1244	−2.50397
$176$	0	0
$177$	−0.392305	−0.0294874
$178$	0	0
$179$	10.3923i	0.776757i	0.921500	+	0.388379i	$0.126965\pi$
$179$	10.3923i	0.776757i	−0.921500	+	0.388379i	$0.873035\pi$
$180$	0	0
$181$	− 4.92820i	− 0.366310i	−0.983084	−	0.183155i	$-0.941369\pi$
$181$	− 4.92820i	− 0.366310i	0.983084	−	0.183155i	$-0.0586311\pi$
$182$	0	0
$183$	−21.8564	−1.61567
$184$	0	0
$185$	17.0718	1.25514
$186$	0	0
$187$	1.85641i	0.135754i
$188$	0	0
$189$	18.9282i	1.37682i
$190$	0	0
$191$	21.4641	1.55309	0.776544	−	0.630063i	$-0.216971\pi$
$191$	21.4641	1.55309	0.776544	+	0.630063i	$0.216971\pi$
$192$	0	0
$193$	−22.3923	−1.61183	−0.805917	−	0.592029i	$-0.798327\pi$
$193$	−22.3923	−1.61183	−0.805917	+	0.592029i	$0.798327\pi$
$194$	0	0
$195$	6.92820i	0.496139i
$196$	0	0
$197$	16.9282i	1.20608i	0.797709	+	0.603042i	$0.206045\pi$
$197$	16.9282i	1.20608i	−0.797709	+	0.603042i	$0.793955\pi$
$198$	0	0
$199$	24.7846	1.75693	0.878467	−	0.477803i	$-0.158567\pi$
$199$	24.7846	1.75693	0.878467	+	0.477803i	$0.158567\pi$
$200$	0	0
$201$	5.46410	0.385408
$202$	0	0
$203$	− 9.46410i	− 0.664250i
$204$	0	0
$205$	− 17.0718i	− 1.19235i
$206$	0	0
$207$	4.00000	0.278019
$208$	0	0
$209$	−3.46410	−0.239617
$210$	0	0
$211$	19.8564i	1.36697i	0.729964	+	0.683486i	$0.239537\pi$
$211$	19.8564i	1.36697i	−0.729964	+	0.683486i	$0.760463\pi$
$212$	0	0
$213$	− 4.39230i	− 0.300956i
$214$	0	0
$215$	25.8564	1.76339
$216$	0	0
$217$	15.4641	1.04977
$218$	0	0
$219$	− 1.07180i	− 0.0724253i
$220$	0	0
$221$	1.46410i	0.0984861i
$222$	0	0
$223$	−10.1962	−0.682785	−0.341392	−	0.939921i	$-0.610899\pi$
$223$	−10.1962	−0.682785	−0.341392	+	0.939921i	$0.610899\pi$
$224$	0	0
$225$	7.00000	0.466667
$226$	0	0
$227$	27.1244i	1.80031i	0.435573	+	0.900153i	$0.356546\pi$
$227$	27.1244i	1.80031i	−0.435573	+	0.900153i	$0.643454\pi$
$228$	0	0
$229$	− 29.3205i	− 1.93755i	−0.247934	−	0.968777i	$-0.579752\pi$
$229$	− 29.3205i	− 1.93755i	0.247934	−	0.968777i	$-0.420248\pi$
$230$	0	0
$231$	12.0000	0.789542
$232$	0	0
$233$	3.07180	0.201240	0.100620	−	0.994925i	$-0.467917\pi$
$233$	3.07180	0.201240	0.100620	+	0.994925i	$0.467917\pi$
$234$	0	0
$235$	− 11.3205i	− 0.738469i
$236$	0	0
$237$	2.92820i	0.190207i
$238$	0	0
$239$	−15.2679	−0.987602	−0.493801	−	0.869575i	$-0.664393\pi$
$239$	−15.2679	−0.987602	−0.493801	+	0.869575i	$0.664393\pi$
$240$	0	0
$241$	−9.60770	−0.618886	−0.309443	−	0.950918i	$-0.600143\pi$
$241$	−9.60770	−0.618886	−0.309443	+	0.950918i	$0.600143\pi$
$242$	0	0
$243$	− 10.0000i	− 0.641500i
$244$	0	0
$245$	53.3205i	3.40652i
$246$	0	0
$247$	−2.73205	−0.173836
$248$	0	0
$249$	−13.4641	−0.853253
$250$	0	0
$251$	− 14.3923i	− 0.908434i	−0.890891	−	0.454217i	$-0.849919\pi$
$251$	− 14.3923i	− 0.908434i	0.890891	−	0.454217i	$-0.150081\pi$
$252$	0	0
$253$	5.07180i	0.318861i
$254$	0	0
$255$	10.1436	0.635216
$256$	0	0
$257$	3.85641	0.240556	0.120278	−	0.992740i	$-0.461621\pi$
$257$	3.85641	0.240556	0.120278	+	0.992740i	$0.461621\pi$
$258$	0	0
$259$	− 23.3205i	− 1.44907i
$260$	0	0
$261$	2.00000i	0.123797i
$262$	0	0
$263$	−7.32051	−0.451402	−0.225701	−	0.974197i	$-0.572467\pi$
$263$	−7.32051	−0.451402	−0.225701	+	0.974197i	$0.572467\pi$
$264$	0	0
$265$	37.8564	2.32550
$266$	0	0
$267$	34.6410i	2.12000i
$268$	0	0
$269$	− 19.8564i	− 1.21067i	−0.795972	−	0.605333i	$-0.793040\pi$
$269$	− 19.8564i	− 1.21067i	0.795972	−	0.605333i	$-0.206960\pi$
$270$	0	0
$271$	9.80385	0.595541	0.297771	−	0.954637i	$-0.403757\pi$
$271$	9.80385	0.595541	0.297771	+	0.954637i	$0.403757\pi$
$272$	0	0
$273$	9.46410	0.572793
$274$	0	0
$275$	8.87564i	0.535221i
$276$	0	0
$277$	25.8564i	1.55356i	0.629771	+	0.776780i	$0.283149\pi$
$277$	25.8564i	1.55356i	−0.629771	+	0.776780i	$0.716851\pi$
$278$	0	0
$279$	−3.26795	−0.195647
$280$	0	0
$281$	−25.3205	−1.51049	−0.755247	−	0.655440i	$-0.772483\pi$
$281$	−25.3205	−1.51049	−0.755247	+	0.655440i	$0.772483\pi$
$282$	0	0
$283$	12.5359i	0.745182i	0.927996	+	0.372591i	$0.121531\pi$
$283$	12.5359i	0.745182i	−0.927996	+	0.372591i	$0.878469\pi$
$284$	0	0
$285$	18.9282i	1.12121i
$286$	0	0
$287$	−23.3205	−1.37657
$288$	0	0
$289$	−14.8564	−0.873906
$290$	0	0
$291$	− 28.7846i	− 1.68738i
$292$	0	0
$293$	− 19.0718i	− 1.11419i	−0.830450	−	0.557093i	$-0.811917\pi$
$293$	− 19.0718i	− 1.11419i	0.830450	−	0.557093i	$-0.188083\pi$
$294$	0	0
$295$	−0.679492	−0.0395615
$296$	0	0
$297$	5.07180	0.294295
$298$	0	0
$299$	4.00000i	0.231326i
$300$	0	0
$301$	− 35.3205i	− 2.03584i
$302$	0	0
$303$	−24.0000	−1.37876
$304$	0	0
$305$	−37.8564	−2.16765
$306$	0	0
$307$	− 2.73205i	− 0.155926i	−0.996956	−	0.0779632i	$-0.975158\pi$
$307$	− 2.73205i	− 0.155926i	0.996956	−	0.0779632i	$-0.0248417\pi$
$308$	0	0
$309$	13.8564i	0.788263i
$310$	0	0
$311$	−14.9282	−0.846501	−0.423250	−	0.906013i	$-0.639111\pi$
$311$	−14.9282	−0.846501	−0.423250	+	0.906013i	$0.639111\pi$
$312$	0	0
$313$	−20.3923	−1.15264	−0.576321	−	0.817224i	$-0.695512\pi$
$313$	−20.3923	−1.15264	−0.576321	+	0.817224i	$0.695512\pi$
$314$	0	0
$315$	− 16.3923i	− 0.923602i
$316$	0	0
$317$	3.46410i	0.194563i	0.995257	+	0.0972817i	$0.0310148\pi$
$317$	3.46410i	0.194563i	−0.995257	+	0.0972817i	$0.968985\pi$
$318$	0	0
$319$	−2.53590	−0.141983
$320$	0	0
$321$	17.8564	0.996647
$322$	0	0
$323$	4.00000i	0.222566i
$324$	0	0
$325$	7.00000i	0.388290i
$326$	0	0
$327$	−4.00000	−0.221201
$328$	0	0
$329$	−15.4641	−0.852564
$330$	0	0
$331$	− 27.5167i	− 1.51245i	−0.654310	−	0.756226i	$-0.727041\pi$
$331$	− 27.5167i	− 1.51245i	0.654310	−	0.756226i	$-0.272959\pi$
$332$	0	0
$333$	4.92820i	0.270064i
$334$	0	0
$335$	9.46410	0.517079
$336$	0	0
$337$	−5.46410	−0.297649	−0.148824	−	0.988864i	$-0.547549\pi$
$337$	−5.46410	−0.297649	−0.148824	+	0.988864i	$0.547549\pi$
$338$	0	0
$339$	18.9282i	1.02804i
$340$	0	0
$341$	− 4.14359i	− 0.224388i
$342$	0	0
$343$	39.7128	2.14429
$344$	0	0
$345$	27.7128	1.49201
$346$	0	0
$347$	− 20.9282i	− 1.12348i	−0.827312	−	0.561742i	$-0.810131\pi$
$347$	− 20.9282i	− 1.12348i	0.827312	−	0.561742i	$-0.189869\pi$
$348$	0	0
$349$	− 30.3923i	− 1.62686i	−0.581661	−	0.813431i	$-0.697597\pi$
$349$	− 30.3923i	− 1.62686i	0.581661	−	0.813431i	$-0.302403\pi$
$350$	0	0
$351$	4.00000	0.213504
$352$	0	0
$353$	−24.9282	−1.32679	−0.663397	−	0.748267i	$-0.730886\pi$
$353$	−24.9282	−1.32679	−0.663397	+	0.748267i	$0.730886\pi$
$354$	0	0
$355$	− 7.60770i	− 0.403775i
$356$	0	0
$357$	− 13.8564i	− 0.733359i
$358$	0	0
$359$	13.5167	0.713382	0.356691	−	0.934222i	$-0.383905\pi$
$359$	13.5167	0.713382	0.356691	+	0.934222i	$0.383905\pi$
$360$	0	0
$361$	11.5359	0.607153
$362$	0	0
$363$	18.7846i	0.985936i
$364$	0	0
$365$	− 1.85641i	− 0.0971688i
$366$	0	0
$367$	−26.2487	−1.37017	−0.685086	−	0.728462i	$-0.740235\pi$
$367$	−26.2487	−1.37017	−0.685086	+	0.728462i	$0.740235\pi$
$368$	0	0
$369$	4.92820	0.256552
$370$	0	0
$371$	− 51.7128i	− 2.68480i
$372$	0	0
$373$	− 26.7846i	− 1.38685i	−0.720527	−	0.693427i	$-0.756100\pi$
$373$	− 26.7846i	− 1.38685i	0.720527	−	0.693427i	$-0.243900\pi$
$374$	0	0
$375$	13.8564	0.715542
$376$	0	0
$377$	−2.00000	−0.103005
$378$	0	0
$379$	− 16.5885i	− 0.852092i	−0.904702	−	0.426046i	$-0.859906\pi$
$379$	− 16.5885i	− 0.852092i	0.904702	−	0.426046i	$-0.140094\pi$
$380$	0	0
$381$	8.00000i	0.409852i
$382$	0	0
$383$	27.6603	1.41337	0.706686	−	0.707527i	$-0.250189\pi$
$383$	27.6603	1.41337	0.706686	+	0.707527i	$0.250189\pi$
$384$	0	0
$385$	20.7846	1.05928
$386$	0	0
$387$	7.46410i	0.379422i
$388$	0	0
$389$	− 2.00000i	− 0.101404i	−0.998714	−	0.0507020i	$-0.983854\pi$
$389$	− 2.00000i	− 0.101404i	0.998714	−	0.0507020i	$-0.0161459\pi$
$390$	0	0
$391$	5.85641	0.296171
$392$	0	0
$393$	15.7128	0.792607
$394$	0	0
$395$	5.07180i	0.255190i
$396$	0	0
$397$	11.4641i	0.575367i	0.957726	+	0.287683i	$0.0928851\pi$
$397$	11.4641i	0.575367i	−0.957726	+	0.287683i	$0.907115\pi$
$398$	0	0
$399$	25.8564	1.29444
$400$	0	0
$401$	11.4641	0.572490	0.286245	−	0.958156i	$-0.407593\pi$
$401$	11.4641	0.572490	0.286245	+	0.958156i	$0.407593\pi$
$402$	0	0
$403$	− 3.26795i	− 0.162788i
$404$	0	0
$405$	− 38.1051i	− 1.89346i
$406$	0	0
$407$	−6.24871	−0.309737
$408$	0	0
$409$	−0.928203	−0.0458967	−0.0229483	−	0.999737i	$-0.507305\pi$
$409$	−0.928203	−0.0458967	−0.0229483	+	0.999737i	$0.507305\pi$
$410$	0	0
$411$	− 1.85641i	− 0.0915698i
$412$	0	0
$413$	0.928203i	0.0456739i
$414$	0	0
$415$	−23.3205	−1.14476
$416$	0	0
$417$	20.0000	0.979404
$418$	0	0
$419$	10.7846i	0.526863i	0.964678	+	0.263431i	$0.0848542\pi$
$419$	10.7846i	0.526863i	−0.964678	+	0.263431i	$0.915146\pi$
$420$	0	0
$421$	24.2487i	1.18181i	0.806741	+	0.590905i	$0.201229\pi$
$421$	24.2487i	1.18181i	−0.806741	+	0.590905i	$0.798771\pi$
$422$	0	0
$423$	3.26795	0.158893
$424$	0	0
$425$	10.2487	0.497136
$426$	0	0
$427$	51.7128i	2.50256i
$428$	0	0
$429$	− 2.53590i	− 0.122434i
$430$	0	0
$431$	−14.8756	−0.716535	−0.358267	−	0.933619i	$-0.616632\pi$
$431$	−14.8756	−0.716535	−0.358267	+	0.933619i	$0.616632\pi$
$432$	0	0
$433$	34.7846	1.67164	0.835821	−	0.549002i	$-0.184992\pi$
$433$	34.7846	1.67164	0.835821	+	0.549002i	$0.184992\pi$
$434$	0	0
$435$	13.8564i	0.664364i
$436$	0	0
$437$	10.9282i	0.522767i
$438$	0	0
$439$	10.9282	0.521575	0.260787	−	0.965396i	$-0.416018\pi$
$439$	10.9282	0.521575	0.260787	+	0.965396i	$0.416018\pi$
$440$	0	0
$441$	−15.3923	−0.732967
$442$	0	0
$443$	− 30.0000i	− 1.42534i	−0.701498	−	0.712672i	$-0.747485\pi$
$443$	− 30.0000i	− 1.42534i	0.701498	−	0.712672i	$-0.252515\pi$
$444$	0	0
$445$	60.0000i	2.84427i
$446$	0	0
$447$	1.85641	0.0878050
$448$	0	0
$449$	38.3923	1.81184	0.905922	−	0.423444i	$-0.139179\pi$
$449$	38.3923	1.81184	0.905922	+	0.423444i	$0.139179\pi$
$450$	0	0
$451$	6.24871i	0.294240i
$452$	0	0
$453$	− 34.2487i	− 1.60914i
$454$	0	0
$455$	16.3923	0.768483
$456$	0	0
$457$	−14.0000	−0.654892	−0.327446	−	0.944870i	$-0.606188\pi$
$457$	−14.0000	−0.654892	−0.327446	+	0.944870i	$0.606188\pi$
$458$	0	0
$459$	− 5.85641i	− 0.273354i
$460$	0	0
$461$	− 7.07180i	− 0.329366i	−0.986347	−	0.164683i	$-0.947340\pi$
$461$	− 7.07180i	− 0.329366i	0.986347	−	0.164683i	$-0.0526602\pi$
$462$	0	0
$463$	15.6603	0.727794	0.363897	−	0.931439i	$-0.381446\pi$
$463$	15.6603	0.727794	0.363897	+	0.931439i	$0.381446\pi$
$464$	0	0
$465$	−22.6410	−1.04995
$466$	0	0
$467$	− 12.2487i	− 0.566803i	−0.959001	−	0.283401i	$-0.908537\pi$
$467$	− 12.2487i	− 0.566803i	0.959001	−	0.283401i	$-0.0914629\pi$
$468$	0	0
$469$	− 12.9282i	− 0.596969i
$470$	0	0
$471$	−6.14359	−0.283082
$472$	0	0
$473$	−9.46410	−0.435160
$474$	0	0
$475$	19.1244i	0.877486i
$476$	0	0
$477$	10.9282i	0.500368i
$478$	0	0
$479$	−24.7321	−1.13004	−0.565018	−	0.825078i	$-0.691131\pi$
$479$	−24.7321	−1.13004	−0.565018	+	0.825078i	$0.691131\pi$
$480$	0	0
$481$	−4.92820	−0.224707
$482$	0	0
$483$	− 37.8564i	− 1.72253i
$484$	0	0
$485$	− 49.8564i	− 2.26386i
$486$	0	0
$487$	−16.4449	−0.745188	−0.372594	−	0.927994i	$-0.621532\pi$
$487$	−16.4449	−0.745188	−0.372594	+	0.927994i	$0.621532\pi$
$488$	0	0
$489$	−26.5359	−1.19999
$490$	0	0
$491$	24.2487i	1.09433i	0.837025	+	0.547165i	$0.184293\pi$
$491$	24.2487i	1.09433i	−0.837025	+	0.547165i	$0.815707\pi$
$492$	0	0
$493$	2.92820i	0.131880i
$494$	0	0
$495$	−4.39230	−0.197419
$496$	0	0
$497$	−10.3923	−0.466159
$498$	0	0
$499$	− 10.7321i	− 0.480433i	−0.970719	−	0.240216i	$-0.922782\pi$
$499$	− 10.7321i	− 0.480433i	0.970719	−	0.240216i	$-0.0772184\pi$
$500$	0	0
$501$	− 23.3205i	− 1.04188i
$502$	0	0
$503$	−37.4641	−1.67044	−0.835221	−	0.549915i	$-0.814660\pi$
$503$	−37.4641	−1.67044	−0.835221	+	0.549915i	$0.814660\pi$
$504$	0	0
$505$	−41.5692	−1.84981
$506$	0	0
$507$	− 2.00000i	− 0.0888231i
$508$	0	0
$509$	6.39230i	0.283334i	0.989914	+	0.141667i	$0.0452462\pi$
$509$	6.39230i	0.283334i	−0.989914	+	0.141667i	$0.954754\pi$
$510$	0	0
$511$	−2.53590	−0.112182
$512$	0	0
$513$	10.9282	0.482492
$514$	0	0
$515$	24.0000i	1.05757i
$516$	0	0
$517$	4.14359i	0.182235i
$518$	0	0
$519$	−13.8564	−0.608229
$520$	0	0
$521$	37.1769	1.62875	0.814375	−	0.580339i	$-0.197080\pi$
$521$	37.1769	1.62875	0.814375	+	0.580339i	$0.197080\pi$
$522$	0	0
$523$	14.0000i	0.612177i	0.952003	+	0.306089i	$0.0990204\pi$
$523$	14.0000i	0.612177i	−0.952003	+	0.306089i	$0.900980\pi$
$524$	0	0
$525$	− 66.2487i	− 2.89133i
$526$	0	0
$527$	−4.78461	−0.208421
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	− 0.196152i	− 0.00851229i
$532$	0	0
$533$	4.92820i	0.213464i
$534$	0	0
$535$	30.9282	1.33714
$536$	0	0
$537$	−20.7846	−0.896922
$538$	0	0
$539$	− 19.5167i	− 0.840642i
$540$	0	0
$541$	16.9282i	0.727800i	0.931438	+	0.363900i	$0.118555\pi$
$541$	16.9282i	0.727800i	−0.931438	+	0.363900i	$0.881445\pi$
$542$	0	0
$543$	9.85641	0.422979
$544$	0	0
$545$	−6.92820	−0.296772
$546$	0	0
$547$	15.8564i	0.677971i	0.940792	+	0.338985i	$0.110084\pi$
$547$	15.8564i	0.677971i	−0.940792	+	0.338985i	$0.889916\pi$
$548$	0	0
$549$	− 10.9282i	− 0.466404i
$550$	0	0
$551$	−5.46410	−0.232779
$552$	0	0
$553$	6.92820	0.294617
$554$	0	0
$555$	34.1436i	1.44931i
$556$	0	0
$557$	6.78461i	0.287473i	0.989616	+	0.143737i	$0.0459118\pi$
$557$	6.78461i	0.287473i	−0.989616	+	0.143737i	$0.954088\pi$
$558$	0	0
$559$	−7.46410	−0.315698
$560$	0	0
$561$	−3.71281	−0.156755
$562$	0	0
$563$	− 0.535898i	− 0.0225854i	−0.999936	−	0.0112927i	$-0.996405\pi$
$563$	− 0.535898i	− 0.0225854i	0.999936	−	0.0112927i	$-0.00359466\pi$
$564$	0	0
$565$	32.7846i	1.37926i
$566$	0	0
$567$	−52.0526	−2.18600
$568$	0	0
$569$	−14.0000	−0.586911	−0.293455	−	0.955973i	$-0.594805\pi$
$569$	−14.0000	−0.586911	−0.293455	+	0.955973i	$0.594805\pi$
$570$	0	0
$571$	37.3205i	1.56181i	0.624647	+	0.780907i	$0.285243\pi$
$571$	37.3205i	1.56181i	−0.624647	+	0.780907i	$0.714757\pi$
$572$	0	0
$573$	42.9282i	1.79335i
$574$	0	0
$575$	28.0000	1.16768
$576$	0	0
$577$	−20.9282	−0.871253	−0.435626	−	0.900128i	$-0.643473\pi$
$577$	−20.9282	−0.871253	−0.435626	+	0.900128i	$0.643473\pi$
$578$	0	0
$579$	− 44.7846i	− 1.86118i
$580$	0	0
$581$	31.8564i	1.32163i
$582$	0	0
$583$	−13.8564	−0.573874
$584$	0	0
$585$	−3.46410	−0.143223
$586$	0	0
$587$	− 21.6603i	− 0.894014i	−0.894530	−	0.447007i	$-0.852490\pi$
$587$	− 21.6603i	− 0.894014i	0.894530	−	0.447007i	$-0.147510\pi$
$588$	0	0
$589$	− 8.92820i	− 0.367880i
$590$	0	0
$591$	−33.8564	−1.39267
$592$	0	0
$593$	19.8564	0.815405	0.407702	−	0.913115i	$-0.366330\pi$
$593$	19.8564	0.815405	0.407702	+	0.913115i	$0.366330\pi$
$594$	0	0
$595$	− 24.0000i	− 0.983904i
$596$	0	0
$597$	49.5692i	2.02873i
$598$	0	0
$599$	−17.1769	−0.701830	−0.350915	−	0.936407i	$-0.614129\pi$
$599$	−17.1769	−0.701830	−0.350915	+	0.936407i	$0.614129\pi$
$600$	0	0
$601$	16.3923	0.668656	0.334328	−	0.942457i	$-0.391491\pi$
$601$	16.3923	0.668656	0.334328	+	0.942457i	$0.391491\pi$
$602$	0	0
$603$	2.73205i	0.111258i
$604$	0	0
$605$	32.5359i	1.32277i
$606$	0	0
$607$	27.3205	1.10891	0.554453	−	0.832215i	$-0.312928\pi$
$607$	27.3205	1.10891	0.554453	+	0.832215i	$0.312928\pi$
$608$	0	0
$609$	18.9282	0.767010
$610$	0	0
$611$	3.26795i	0.132207i
$612$	0	0
$613$	44.6410i	1.80303i	0.432744	+	0.901517i	$0.357545\pi$
$613$	44.6410i	1.80303i	−0.432744	+	0.901517i	$0.642455\pi$
$614$	0	0
$615$	34.1436	1.37680
$616$	0	0
$617$	6.67949	0.268906	0.134453	−	0.990920i	$-0.457072\pi$
$617$	6.67949	0.268906	0.134453	+	0.990920i	$0.457072\pi$
$618$	0	0
$619$	12.5885i	0.505973i	0.967470	+	0.252986i	$0.0814128\pi$
$619$	12.5885i	0.505973i	−0.967470	+	0.252986i	$0.918587\pi$
$620$	0	0
$621$	− 16.0000i	− 0.642058i
$622$	0	0
$623$	81.9615	3.28372
$624$	0	0
$625$	−11.0000	−0.440000
$626$	0	0
$627$	− 6.92820i	− 0.276686i
$628$	0	0
$629$	7.21539i	0.287696i
$630$	0	0
$631$	3.94744	0.157145	0.0785726	−	0.996908i	$-0.474964\pi$
$631$	3.94744	0.157145	0.0785726	+	0.996908i	$0.474964\pi$
$632$	0	0
$633$	−39.7128	−1.57844
$634$	0	0
$635$	13.8564i	0.549875i
$636$	0	0
$637$	− 15.3923i	− 0.609865i
$638$	0	0
$639$	2.19615	0.0868784
$640$	0	0
$641$	26.2487	1.03676	0.518381	−	0.855150i	$-0.326535\pi$
$641$	26.2487	1.03676	0.518381	+	0.855150i	$0.326535\pi$
$642$	0	0
$643$	− 9.26795i	− 0.365492i	−0.983160	−	0.182746i	$-0.941501\pi$
$643$	− 9.26795i	− 0.365492i	0.983160	−	0.182746i	$-0.0584986\pi$
$644$	0	0
$645$	51.7128i	2.03619i
$646$	0	0
$647$	−10.1436	−0.398786	−0.199393	−	0.979920i	$-0.563897\pi$
$647$	−10.1436	−0.398786	−0.199393	+	0.979920i	$0.563897\pi$
$648$	0	0
$649$	0.248711	0.00976277
$650$	0	0
$651$	30.9282i	1.21217i
$652$	0	0
$653$	16.9282i	0.662452i	0.943551	+	0.331226i	$0.107462\pi$
$653$	16.9282i	0.662452i	−0.943551	+	0.331226i	$0.892538\pi$
$654$	0	0
$655$	27.2154	1.06339
$656$	0	0
$657$	0.535898	0.0209074
$658$	0	0
$659$	14.0000i	0.545363i	0.962104	+	0.272681i	$0.0879105\pi$
$659$	14.0000i	0.545363i	−0.962104	+	0.272681i	$0.912090\pi$
$660$	0	0
$661$	17.3205i	0.673690i	0.941560	+	0.336845i	$0.109360\pi$
$661$	17.3205i	0.673690i	−0.941560	+	0.336845i	$0.890640\pi$
$662$	0	0
$663$	−2.92820	−0.113722
$664$	0	0
$665$	44.7846	1.73667
$666$	0	0
$667$	8.00000i	0.309761i
$668$	0	0
$669$	− 20.3923i	− 0.788412i
$670$	0	0
$671$	13.8564	0.534921
$672$	0	0
$673$	−29.1769	−1.12469	−0.562344	−	0.826904i	$-0.690100\pi$
$673$	−29.1769	−1.12469	−0.562344	+	0.826904i	$0.690100\pi$
$674$	0	0
$675$	− 28.0000i	− 1.07772i
$676$	0	0
$677$	− 34.9282i	− 1.34240i	−0.741276	−	0.671200i	$-0.765779\pi$
$677$	− 34.9282i	− 1.34240i	0.741276	−	0.671200i	$-0.234221\pi$
$678$	0	0
$679$	−68.1051	−2.61363
$680$	0	0
$681$	−54.2487	−2.07882
$682$	0	0
$683$	− 28.1962i	− 1.07890i	−0.842019	−	0.539448i	$-0.818633\pi$
$683$	− 28.1962i	− 1.07890i	0.842019	−	0.539448i	$-0.181367\pi$
$684$	0	0
$685$	− 3.21539i	− 0.122854i
$686$	0	0
$687$	58.6410	2.23729
$688$	0	0
$689$	−10.9282	−0.416331
$690$	0	0
$691$	46.8372i	1.78177i	0.454229	+	0.890885i	$0.349915\pi$
$691$	46.8372i	1.78177i	−0.454229	+	0.890885i	$0.650085\pi$
$692$	0	0
$693$	6.00000i	0.227921i
$694$	0	0
$695$	34.6410	1.31401
$696$	0	0
$697$	7.21539	0.273302
$698$	0	0
$699$	6.14359i	0.232372i
$700$	0	0
$701$	− 28.6410i	− 1.08176i	−0.841101	−	0.540878i	$-0.818092\pi$
$701$	− 28.6410i	− 1.08176i	0.841101	−	0.540878i	$-0.181908\pi$
$702$	0	0
$703$	−13.4641	−0.507808
$704$	0	0
$705$	22.6410	0.852710
$706$	0	0
$707$	56.7846i	2.13561i
$708$	0	0
$709$	5.32051i	0.199816i	0.994997	+	0.0999079i	$0.0318548\pi$
$709$	5.32051i	0.199816i	−0.994997	+	0.0999079i	$0.968145\pi$
$710$	0	0
$711$	−1.46410	−0.0549081
$712$	0	0
$713$	−13.0718	−0.489543
$714$	0	0
$715$	− 4.39230i	− 0.164263i
$716$	0	0
$717$	− 30.5359i	− 1.14038i
$718$	0	0
$719$	−17.0718	−0.636671	−0.318335	−	0.947978i	$-0.603124\pi$
$719$	−17.0718	−0.636671	−0.318335	+	0.947978i	$0.603124\pi$
$720$	0	0
$721$	32.7846	1.22096
$722$	0	0
$723$	− 19.2154i	− 0.714628i
$724$	0	0
$725$	14.0000i	0.519947i
$726$	0	0
$727$	9.07180	0.336454	0.168227	−	0.985748i	$-0.446196\pi$
$727$	9.07180	0.336454	0.168227	+	0.985748i	$0.446196\pi$
$728$	0	0
$729$	−13.0000	−0.481481
$730$	0	0
$731$	10.9282i	0.404194i
$732$	0	0
$733$	− 2.67949i	− 0.0989693i	−0.998775	−	0.0494846i	$-0.984242\pi$
$733$	− 2.67949i	− 0.0989693i	0.998775	−	0.0494846i	$-0.0157579\pi$
$734$	0	0
$735$	−106.641	−3.93351
$736$	0	0
$737$	−3.46410	−0.127602
$738$	0	0
$739$	− 53.7654i	− 1.97779i	−0.148612	−	0.988896i	$-0.547481\pi$
$739$	− 53.7654i	− 1.97779i	0.148612	−	0.988896i	$-0.452519\pi$
$740$	0	0
$741$	− 5.46410i	− 0.200729i
$742$	0	0
$743$	21.8038	0.799906	0.399953	−	0.916536i	$-0.369027\pi$
$743$	21.8038	0.799906	0.399953	+	0.916536i	$0.369027\pi$
$744$	0	0
$745$	3.21539	0.117803
$746$	0	0
$747$	− 6.73205i	− 0.246313i
$748$	0	0
$749$	− 42.2487i	− 1.54373i
$750$	0	0
$751$	−14.9282	−0.544738	−0.272369	−	0.962193i	$-0.587807\pi$
$751$	−14.9282	−0.544738	−0.272369	+	0.962193i	$0.587807\pi$
$752$	0	0
$753$	28.7846	1.04897
$754$	0	0
$755$	− 59.3205i	− 2.15889i
$756$	0	0
$757$	0.784610i	0.0285171i	0.999898	+	0.0142586i	$0.00453880\pi$
$757$	0.784610i	0.0285171i	−0.999898	+	0.0142586i	$0.995461\pi$
$758$	0	0
$759$	−10.1436	−0.368189
$760$	0	0
$761$	−22.7846	−0.825941	−0.412971	−	0.910744i	$-0.635509\pi$
$761$	−22.7846	−0.825941	−0.412971	+	0.910744i	$0.635509\pi$
$762$	0	0
$763$	9.46410i	0.342623i
$764$	0	0
$765$	5.07180i	0.183371i
$766$	0	0
$767$	0.196152	0.00708265
$768$	0	0
$769$	24.5359	0.884787	0.442394	−	0.896821i	$-0.354129\pi$
$769$	24.5359	0.884787	0.442394	+	0.896821i	$0.354129\pi$
$770$	0	0
$771$	7.71281i	0.277770i
$772$	0	0
$773$	− 20.5359i	− 0.738625i	−0.929305	−	0.369312i	$-0.879593\pi$
$773$	− 20.5359i	− 0.738625i	0.929305	−	0.369312i	$-0.120407\pi$
$774$	0	0
$775$	−22.8756	−0.821717
$776$	0	0
$777$	46.6410	1.67324
$778$	0	0
$779$	13.4641i	0.482402i
$780$	0	0
$781$	2.78461i	0.0996412i
$782$	0	0
$783$	8.00000	0.285897
$784$	0	0
$785$	−10.6410	−0.379794
$786$	0	0
$787$	− 4.87564i	− 0.173798i	−0.996217	−	0.0868990i	$-0.972304\pi$
$787$	− 4.87564i	− 0.173798i	0.996217	−	0.0868990i	$-0.0276957\pi$
$788$	0	0
$789$	− 14.6410i	− 0.521234i
$790$	0	0
$791$	44.7846	1.59236
$792$	0	0
$793$	10.9282	0.388072
$794$	0	0
$795$	75.7128i	2.68526i
$796$	0	0
$797$	30.0000i	1.06265i	0.847167	+	0.531327i	$0.178307\pi$
$797$	30.0000i	1.06265i	−0.847167	+	0.531327i	$0.821693\pi$
$798$	0	0
$799$	4.78461	0.169267
$800$	0	0
$801$	−17.3205	−0.611990
$802$	0	0
$803$	0.679492i	0.0239787i
$804$	0	0
$805$	− 65.5692i	− 2.31101i
$806$	0	0
$807$	39.7128	1.39796
$808$	0	0
$809$	1.46410	0.0514751	0.0257375	−	0.999669i	$-0.491807\pi$
$809$	1.46410	0.0514751	0.0257375	+	0.999669i	$0.491807\pi$
$810$	0	0
$811$	52.5885i	1.84663i	0.384043	+	0.923315i	$0.374531\pi$
$811$	52.5885i	1.84663i	−0.384043	+	0.923315i	$0.625469\pi$
$812$	0	0
$813$	19.6077i	0.687672i
$814$	0	0
$815$	−45.9615	−1.60996
$816$	0	0
$817$	−20.3923	−0.713436
$818$	0	0
$819$	4.73205i	0.165351i
$820$	0	0
$821$	− 0.248711i	− 0.00868008i	−0.999991	−	0.00434004i	$-0.998619\pi$
$821$	− 0.248711i	− 0.00868008i	0.999991	−	0.00434004i	$-0.00138148\pi$
$822$	0	0
$823$	−20.0000	−0.697156	−0.348578	−	0.937280i	$-0.613335\pi$
$823$	−20.0000	−0.697156	−0.348578	+	0.937280i	$0.613335\pi$
$824$	0	0
$825$	−17.7513	−0.618021
$826$	0	0
$827$	5.26795i	0.183185i	0.995797	+	0.0915923i	$0.0291956\pi$
$827$	5.26795i	0.183185i	−0.995797	+	0.0915923i	$0.970804\pi$
$828$	0	0
$829$	− 12.7846i	− 0.444028i	−0.975043	−	0.222014i	$-0.928737\pi$
$829$	− 12.7846i	− 0.444028i	0.975043	−	0.222014i	$-0.0712630\pi$
$830$	0	0
$831$	−51.7128	−1.79390
$832$	0	0
$833$	−22.5359	−0.780823
$834$	0	0
$835$	− 40.3923i	− 1.39783i
$836$	0	0
$837$	13.0718i	0.451827i
$838$	0	0
$839$	−30.9808	−1.06957	−0.534787	−	0.844987i	$-0.679608\pi$
$839$	−30.9808	−1.06957	−0.534787	+	0.844987i	$0.679608\pi$
$840$	0	0
$841$	25.0000	0.862069
$842$	0	0
$843$	− 50.6410i	− 1.74417i
$844$	0	0
$845$	− 3.46410i	− 0.119169i
$846$	0	0
$847$	44.4449	1.52714
$848$	0	0
$849$	−25.0718	−0.860462
$850$	0	0
$851$	19.7128i	0.675747i
$852$	0	0
$853$	− 7.17691i	− 0.245733i	−0.992423	−	0.122866i	$-0.960791\pi$
$853$	− 7.17691i	− 0.245733i	0.992423	−	0.122866i	$-0.0392087\pi$
$854$	0	0
$855$	−9.46410	−0.323665
$856$	0	0
$857$	19.8564	0.678282	0.339141	−	0.940736i	$-0.389864\pi$
$857$	19.8564	0.678282	0.339141	+	0.940736i	$0.389864\pi$
$858$	0	0
$859$	18.0000i	0.614152i	0.951685	+	0.307076i	$0.0993506\pi$
$859$	18.0000i	0.614152i	−0.951685	+	0.307076i	$0.900649\pi$
$860$	0	0
$861$	− 46.6410i	− 1.58952i
$862$	0	0
$863$	−4.73205	−0.161081	−0.0805404	−	0.996751i	$-0.525665\pi$
$863$	−4.73205	−0.161081	−0.0805404	+	0.996751i	$0.525665\pi$
$864$	0	0
$865$	−24.0000	−0.816024
$866$	0	0
$867$	− 29.7128i	− 1.00910i
$868$	0	0
$869$	− 1.85641i	− 0.0629743i
$870$	0	0
$871$	−2.73205	−0.0925720
$872$	0	0
$873$	14.3923	0.487106
$874$	0	0
$875$	− 32.7846i	− 1.10832i
$876$	0	0
$877$	16.5359i	0.558378i	0.960236	+	0.279189i	$0.0900655\pi$
$877$	16.5359i	0.558378i	−0.960236	+	0.279189i	$0.909934\pi$
$878$	0	0
$879$	38.1436	1.28655
$880$	0	0
$881$	21.7128	0.731523	0.365762	−	0.930709i	$-0.380809\pi$
$881$	21.7128	0.731523	0.365762	+	0.930709i	$0.380809\pi$
$882$	0	0
$883$	24.5359i	0.825699i	0.910799	+	0.412849i	$0.135466\pi$
$883$	24.5359i	0.825699i	−0.910799	+	0.412849i	$0.864534\pi$
$884$	0	0
$885$	− 1.35898i	− 0.0456817i
$886$	0	0
$887$	18.2487	0.612732	0.306366	−	0.951914i	$-0.400887\pi$
$887$	18.2487	0.612732	0.306366	+	0.951914i	$0.400887\pi$
$888$	0	0
$889$	18.9282	0.634832
$890$	0	0
$891$	13.9474i	0.467257i
$892$	0	0
$893$	8.92820i	0.298771i
$894$	0	0
$895$	−36.0000	−1.20335
$896$	0	0
$897$	−8.00000	−0.267112
$898$	0	0
$899$	− 6.53590i	− 0.217984i
$900$	0	0
$901$	16.0000i	0.533037i
$902$	0	0
$903$	70.6410	2.35079
$904$	0	0
$905$	17.0718	0.567486
$906$	0	0
$907$	− 54.1051i	− 1.79653i	−0.439453	−	0.898265i	$-0.644828\pi$
$907$	− 54.1051i	− 1.79653i	0.439453	−	0.898265i	$-0.355172\pi$
$908$	0	0
$909$	− 12.0000i	− 0.398015i
$910$	0	0
$911$	31.3205	1.03769	0.518847	−	0.854867i	$-0.326361\pi$
$911$	31.3205	1.03769	0.518847	+	0.854867i	$0.326361\pi$
$912$	0	0
$913$	8.53590	0.282497
$914$	0	0
$915$	− 75.7128i	− 2.50299i
$916$	0	0
$917$	− 37.1769i	− 1.22769i
$918$	0	0
$919$	0.679492	0.0224144	0.0112072	−	0.999937i	$-0.496433\pi$
$919$	0.679492	0.0224144	0.0112072	+	0.999937i	$0.496433\pi$
$920$	0	0
$921$	5.46410	0.180048
$922$	0	0
$923$	2.19615i	0.0722872i
$924$	0	0
$925$	34.4974i	1.13427i
$926$	0	0
$927$	−6.92820	−0.227552
$928$	0	0
$929$	27.4641	0.901068	0.450534	−	0.892759i	$-0.351234\pi$
$929$	27.4641	0.901068	0.450534	+	0.892759i	$0.351234\pi$
$930$	0	0
$931$	− 42.0526i	− 1.37822i
$932$	0	0
$933$	− 29.8564i	− 0.977455i
$934$	0	0
$935$	−6.43078	−0.210309
$936$	0	0
$937$	41.7128	1.36270	0.681349	−	0.731959i	$-0.261394\pi$
$937$	41.7128	1.36270	0.681349	+	0.731959i	$0.261394\pi$
$938$	0	0
$939$	− 40.7846i	− 1.33096i
$940$	0	0
$941$	16.2487i	0.529693i	0.964291	+	0.264846i	$0.0853213\pi$
$941$	16.2487i	0.529693i	−0.964291	+	0.264846i	$0.914679\pi$
$942$	0	0
$943$	19.7128	0.641938
$944$	0	0
$945$	−65.5692	−2.13297
$946$	0	0
$947$	10.4449i	0.339412i	0.985495	+	0.169706i	$0.0542819\pi$
$947$	10.4449i	0.339412i	−0.985495	+	0.169706i	$0.945718\pi$
$948$	0	0
$949$	0.535898i	0.0173960i
$950$	0	0
$951$	−6.92820	−0.224662
$952$	0	0
$953$	−4.14359	−0.134224	−0.0671121	−	0.997745i	$-0.521379\pi$
$953$	−4.14359	−0.134224	−0.0671121	+	0.997745i	$0.521379\pi$
$954$	0	0
$955$	74.3538i	2.40603i
$956$	0	0
$957$	− 5.07180i	− 0.163948i
$958$	0	0
$959$	−4.39230	−0.141835
$960$	0	0
$961$	−20.3205	−0.655500
$962$	0	0
$963$	8.92820i	0.287707i
$964$	0	0
$965$	− 77.5692i	− 2.49704i
$966$	0	0
$967$	−42.9808	−1.38217	−0.691084	−	0.722774i	$-0.742867\pi$
$967$	−42.9808	−1.38217	−0.691084	+	0.722774i	$0.742867\pi$
$968$	0	0
$969$	−8.00000	−0.256997
$970$	0	0
$971$	− 43.8564i	− 1.40742i	−0.710488	−	0.703710i	$-0.751526\pi$
$971$	− 43.8564i	− 1.40742i	0.710488	−	0.703710i	$-0.248474\pi$
$972$	0	0
$973$	− 47.3205i	− 1.51703i
$974$	0	0
$975$	−14.0000	−0.448359
$976$	0	0
$977$	−37.6077	−1.20318	−0.601588	−	0.798806i	$-0.705465\pi$
$977$	−37.6077	−1.20318	−0.601588	+	0.798806i	$0.705465\pi$
$978$	0	0
$979$	− 21.9615i	− 0.701893i
$980$	0	0
$981$	− 2.00000i	− 0.0638551i
$982$	0	0
$983$	−5.80385	−0.185114	−0.0925570	−	0.995707i	$-0.529504\pi$
$983$	−5.80385	−0.185114	−0.0925570	+	0.995707i	$0.529504\pi$
$984$	0	0
$985$	−58.6410	−1.86846
$986$	0	0
$987$	− 30.9282i	− 0.984456i
$988$	0	0
$989$	29.8564i	0.949378i
$990$	0	0
$991$	43.3205	1.37612	0.688061	−	0.725653i	$-0.258462\pi$
$991$	43.3205	1.37612	0.688061	+	0.725653i	$0.258462\pi$
$992$	0	0
$993$	55.0333	1.74643
$994$	0	0
$995$	85.8564i	2.72183i
$996$	0	0
$997$	49.5692i	1.56987i	0.619576	+	0.784936i	$0.287304\pi$
$997$	49.5692i	1.56987i	−0.619576	+	0.784936i	$0.712696\pi$
$998$	0	0
$999$	19.7128	0.623686

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	416.2.b.b.209.4		4
3.2	odd	2		3744.2.g.b.1873.2		4
4.3	odd	2		104.2.b.b.53.3	✓	4
8.3	odd	2		104.2.b.b.53.4	yes	4
8.5	even	2	inner	416.2.b.b.209.1		4
12.11	even	2		936.2.g.b.469.2		4
16.3	odd	4		3328.2.a.bd.1.2		2
16.5	even	4		3328.2.a.bc.1.1		2
16.11	odd	4		3328.2.a.n.1.1		2
16.13	even	4		3328.2.a.m.1.2		2
24.5	odd	2		3744.2.g.b.1873.4		4
24.11	even	2		936.2.g.b.469.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
104.2.b.b.53.3	✓	4	4.3	odd	2
104.2.b.b.53.4	yes	4	8.3	odd	2
416.2.b.b.209.1		4	8.5	even	2	inner
416.2.b.b.209.4		4	1.1	even	1	trivial
936.2.g.b.469.1		4	24.11	even	2
936.2.g.b.469.2		4	12.11	even	2
3328.2.a.m.1.2		2	16.13	even	4
3328.2.a.n.1.1		2	16.11	odd	4
3328.2.a.bc.1.1		2	16.5	even	4
3328.2.a.bd.1.2		2	16.3	odd	4
3744.2.g.b.1873.2		4	3.2	odd	2
3744.2.g.b.1873.4		4	24.5	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:	\( N \)	\(=\)	\( 416 = 2^{5} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	416.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.00000i q^{3} +3.46410i q^{5} +4.73205 q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q+2.00000i q^{3} +3.46410i q^{5} +4.73205 q^{7} -1.00000 q^{9} -1.26795i q^{11} -1.00000i q^{13} -6.92820 q^{15} -1.46410 q^{17} -2.73205i q^{19} +9.46410i q^{21} -4.00000 q^{23} -7.00000 q^{25} +4.00000i q^{27} -2.00000i q^{29} +3.26795 q^{31} +2.53590 q^{33} +16.3923i q^{35} -4.92820i q^{37} +2.00000 q^{39} -4.92820 q^{41} -7.46410i q^{43} -3.46410i q^{45} -3.26795 q^{47} +15.3923 q^{49} -2.92820i q^{51} -10.9282i q^{53} +4.39230 q^{55} +5.46410 q^{57} +0.196152i q^{59} +10.9282i q^{61} -4.73205 q^{63} +3.46410 q^{65} -2.73205i q^{67} -8.00000i q^{69} -2.19615 q^{71} -0.535898 q^{73} -14.0000i q^{75} -6.00000i q^{77} +1.46410 q^{79} -11.0000 q^{81} +6.73205i q^{83} -5.07180i q^{85} +4.00000 q^{87} +17.3205 q^{89} -4.73205i q^{91} +6.53590i q^{93} +9.46410 q^{95} -14.3923 q^{97} +1.26795i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 12 q^{7} - 4 q^{9}+O(q^{10}) \) 4 * q + 12 * q^7 - 4 * q^9	\( 4 q + 12 q^{7} - 4 q^{9} + 8 q^{17} - 16 q^{23} - 28 q^{25} + 20 q^{31} + 24 q^{33} + 8 q^{39} + 8 q^{41} - 20 q^{47} + 20 q^{49} - 24 q^{55} + 8 q^{57} - 12 q^{63} + 12 q^{71} - 16 q^{73} - 8 q^{79} - 44 q^{81} + 16 q^{87} + 24 q^{95} - 16 q^{97}+O(q^{100}) \) 4 * q + 12 * q^7 - 4 * q^9 + 8 * q^17 - 16 * q^23 - 28 * q^25 + 20 * q^31 + 24 * q^33 + 8 * q^39 + 8 * q^41 - 20 * q^47 + 20 * q^49 - 24 * q^55 + 8 * q^57 - 12 * q^63 + 12 * q^71 - 16 * q^73 - 8 * q^79 - 44 * q^81 + 16 * q^87 + 24 * q^95 - 16 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more