LMFDB - Embedded newform 3328.2.b.j.1665.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3328,2,Mod(1665,3328)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3328, 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("3328.1665");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3328 = 2^{8} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3328.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:	$26.5742137927$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 26)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1665.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	3328.1665
Dual form			3328.2.b.j.1665.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} -3.00000i q^{5} -1.00000 q^{7} +2.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{3} -3.00000i q^{5} -1.00000 q^{7} +2.00000 q^{9} -6.00000i q^{11} -1.00000i q^{13} +3.00000 q^{15} -3.00000 q^{17} +2.00000i q^{19} -1.00000i q^{21} -4.00000 q^{25} +5.00000i q^{27} -6.00000i q^{29} +4.00000 q^{31} +6.00000 q^{33} +3.00000i q^{35} -7.00000i q^{37} +1.00000 q^{39} +1.00000i q^{43} -6.00000i q^{45} -3.00000 q^{47} -6.00000 q^{49} -3.00000i q^{51} -18.0000 q^{55} -2.00000 q^{57} +6.00000i q^{59} -8.00000i q^{61} -2.00000 q^{63} -3.00000 q^{65} +14.0000i q^{67} -3.00000 q^{71} -2.00000 q^{73} -4.00000i q^{75} +6.00000i q^{77} -8.00000 q^{79} +1.00000 q^{81} +12.0000i q^{83} +9.00000i q^{85} +6.00000 q^{87} +6.00000 q^{89} +1.00000i q^{91} +4.00000i q^{93} +6.00000 q^{95} -10.0000 q^{97} -12.0000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{7} + 4 q^{9}+O(q^{10}) $ 2 * q - 2 * q^7 + 4 * q^9	$ 2 q - 2 q^{7} + 4 q^{9} + 6 q^{15} - 6 q^{17} - 8 q^{25} + 8 q^{31} + 12 q^{33} + 2 q^{39} - 6 q^{47} - 12 q^{49} - 36 q^{55} - 4 q^{57} - 4 q^{63} - 6 q^{65} - 6 q^{71} - 4 q^{73} - 16 q^{79} + 2 q^{81} + 12 q^{87} + 12 q^{89} + 12 q^{95} - 20 q^{97}+O(q^{100}) $ 2 * q - 2 * q^7 + 4 * q^9 + 6 * q^15 - 6 * q^17 - 8 * q^25 + 8 * q^31 + 12 * q^33 + 2 * q^39 - 6 * q^47 - 12 * q^49 - 36 * q^55 - 4 * q^57 - 4 * q^63 - 6 * q^65 - 6 * q^71 - 4 * q^73 - 16 * q^79 + 2 * q^81 + 12 * q^87 + 12 * q^89 + 12 * q^95 - 20 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3328\mathbb{Z}\right)^\times$.

$n$	$261$	$769$	$1535$
$\chi(n)$	$-1$	$1$	$1$

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
$3$	1.00000i	0.577350i	0.957427	+	0.288675i	$0.0932147\pi$
$3$	1.00000i	0.577350i	−0.957427	+	0.288675i	$0.906785\pi$
$4$	0	0
$5$	− 3.00000i	− 1.34164i	−0.741620	−	0.670820i	$-0.765942\pi$
$5$	− 3.00000i	− 1.34164i	0.741620	−	0.670820i	$-0.234058\pi$
$6$	0	0
$7$	−1.00000	−0.377964	−0.188982	−	0.981981i	$-0.560519\pi$
$7$	−1.00000	−0.377964	−0.188982	+	0.981981i	$0.560519\pi$
$8$	0	0
$9$	2.00000	0.666667
$10$	0	0
$11$	− 6.00000i	− 1.80907i	−0.426401	−	0.904534i	$-0.640219\pi$
$11$	− 6.00000i	− 1.80907i	0.426401	−	0.904534i	$-0.359781\pi$
$12$	0	0
$13$	− 1.00000i	− 0.277350i
$13$	− 1.00000i	− 0.277350i
$14$	0	0
$15$	3.00000	0.774597
$16$	0	0
$17$	−3.00000	−0.727607	−0.363803	−	0.931476i	$-0.618522\pi$
$17$	−3.00000	−0.727607	−0.363803	+	0.931476i	$0.618522\pi$
$18$	0	0
$19$	2.00000i	0.458831i	0.973329	+	0.229416i	$0.0736815\pi$
$19$	2.00000i	0.458831i	−0.973329	+	0.229416i	$0.926318\pi$
$20$	0	0
$21$	− 1.00000i	− 0.218218i
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	−4.00000	−0.800000
$26$	0	0
$27$	5.00000i	0.962250i
$28$	0	0
$29$	− 6.00000i	− 1.11417i	−0.830455	−	0.557086i	$-0.811919\pi$
$29$	− 6.00000i	− 1.11417i	0.830455	−	0.557086i	$-0.188081\pi$
$30$	0	0
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	0	0
$33$	6.00000	1.04447
$34$	0	0
$35$	3.00000i	0.507093i
$36$	0	0
$37$	− 7.00000i	− 1.15079i	−0.817875	−	0.575396i	$-0.804848\pi$
$37$	− 7.00000i	− 1.15079i	0.817875	−	0.575396i	$-0.195152\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	1.00000i	0.152499i	0.997089	+	0.0762493i	$0.0242945\pi$
$43$	1.00000i	0.152499i	−0.997089	+	0.0762493i	$0.975706\pi$
$44$	0	0
$45$	− 6.00000i	− 0.894427i
$46$	0	0
$47$	−3.00000	−0.437595	−0.218797	−	0.975770i	$-0.570213\pi$
$47$	−3.00000	−0.437595	−0.218797	+	0.975770i	$0.570213\pi$
$48$	0	0
$49$	−6.00000	−0.857143
$50$	0	0
$51$	− 3.00000i	− 0.420084i
$52$	0	0
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	0	0
$55$	−18.0000	−2.42712
$56$	0	0
$57$	−2.00000	−0.264906
$58$	0	0
$59$	6.00000i	0.781133i	0.920575	+	0.390567i	$0.127721\pi$
$59$	6.00000i	0.781133i	−0.920575	+	0.390567i	$0.872279\pi$
$60$	0	0
$61$	− 8.00000i	− 1.02430i	−0.858898	−	0.512148i	$-0.828850\pi$
$61$	− 8.00000i	− 1.02430i	0.858898	−	0.512148i	$-0.171150\pi$
$62$	0	0
$63$	−2.00000	−0.251976
$64$	0	0
$65$	−3.00000	−0.372104
$66$	0	0
$67$	14.0000i	1.71037i	0.518321	+	0.855186i	$0.326557\pi$
$67$	14.0000i	1.71037i	−0.518321	+	0.855186i	$0.673443\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−3.00000	−0.356034	−0.178017	−	0.984027i	$-0.556968\pi$
$71$	−3.00000	−0.356034	−0.178017	+	0.984027i	$0.556968\pi$
$72$	0	0
$73$	−2.00000	−0.234082	−0.117041	−	0.993127i	$-0.537341\pi$
$73$	−2.00000	−0.234082	−0.117041	+	0.993127i	$0.537341\pi$
$74$	0	0
$75$	− 4.00000i	− 0.461880i
$76$	0	0
$77$	6.00000i	0.683763i
$78$	0	0
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	12.0000i	1.31717i	0.752506	+	0.658586i	$0.228845\pi$
$83$	12.0000i	1.31717i	−0.752506	+	0.658586i	$0.771155\pi$
$84$	0	0
$85$	9.00000i	0.976187i
$86$	0	0
$87$	6.00000	0.643268
$88$	0	0
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	1.00000i	0.104828i
$92$	0	0
$93$	4.00000i	0.414781i
$94$	0	0
$95$	6.00000	0.615587
$96$	0	0
$97$	−10.0000	−1.01535	−0.507673	−	0.861550i	$-0.669494\pi$
$97$	−10.0000	−1.01535	−0.507673	+	0.861550i	$0.669494\pi$
$98$	0	0
$99$	− 12.0000i	− 1.20605i
$100$	0	0
$101$	− 12.0000i	− 1.19404i	−0.802225	−	0.597022i	$-0.796350\pi$
$101$	− 12.0000i	− 1.19404i	0.802225	−	0.597022i	$-0.203650\pi$
$102$	0	0
$103$	−4.00000	−0.394132	−0.197066	−	0.980390i	$-0.563141\pi$
$103$	−4.00000	−0.394132	−0.197066	+	0.980390i	$0.563141\pi$
$104$	0	0
$105$	−3.00000	−0.292770
$106$	0	0
$107$	− 12.0000i	− 1.16008i	−0.814587	−	0.580042i	$-0.803036\pi$
$107$	− 12.0000i	− 1.16008i	0.814587	−	0.580042i	$-0.196964\pi$
$108$	0	0
$109$	7.00000i	0.670478i	0.942133	+	0.335239i	$0.108817\pi$
$109$	7.00000i	0.670478i	−0.942133	+	0.335239i	$0.891183\pi$
$110$	0	0
$111$	7.00000	0.664411
$112$	0	0
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 2.00000i	− 0.184900i
$118$	0	0
$119$	3.00000	0.275010
$120$	0	0
$121$	−25.0000	−2.27273
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 3.00000i	− 0.268328i
$126$	0	0
$127$	−20.0000	−1.77471	−0.887357	−	0.461084i	$-0.847461\pi$
$127$	−20.0000	−1.77471	−0.887357	+	0.461084i	$0.847461\pi$
$128$	0	0
$129$	−1.00000	−0.0880451
$130$	0	0
$131$	− 21.0000i	− 1.83478i	−0.397991	−	0.917389i	$-0.630293\pi$
$131$	− 21.0000i	− 1.83478i	0.397991	−	0.917389i	$-0.369707\pi$
$132$	0	0
$133$	− 2.00000i	− 0.173422i
$134$	0	0
$135$	15.0000	1.29099
$136$	0	0
$137$	0	0		−	1.00000i	$-0.5\pi$
$137$	0	0			1.00000i	$0.5\pi$
$138$	0	0
$139$	13.0000i	1.10265i	0.834292	+	0.551323i	$0.185877\pi$
$139$	13.0000i	1.10265i	−0.834292	+	0.551323i	$0.814123\pi$
$140$	0	0
$141$	− 3.00000i	− 0.252646i
$142$	0	0
$143$	−6.00000	−0.501745
$144$	0	0
$145$	−18.0000	−1.49482
$146$	0	0
$147$	− 6.00000i	− 0.494872i
$148$	0	0
$149$	− 6.00000i	− 0.491539i	−0.969328	−	0.245770i	$-0.920959\pi$
$149$	− 6.00000i	− 0.491539i	0.969328	−	0.245770i	$-0.0790407\pi$
$150$	0	0
$151$	17.0000	1.38344	0.691720	−	0.722166i	$-0.256853\pi$
$151$	17.0000	1.38344	0.691720	+	0.722166i	$0.256853\pi$
$152$	0	0
$153$	−6.00000	−0.485071
$154$	0	0
$155$	− 12.0000i	− 0.963863i
$156$	0	0
$157$	− 14.0000i	− 1.11732i	−0.829396	−	0.558661i	$-0.811315\pi$
$157$	− 14.0000i	− 1.11732i	0.829396	−	0.558661i	$-0.188685\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	− 16.0000i	− 1.25322i	−0.779334	−	0.626608i	$-0.784443\pi$
$163$	− 16.0000i	− 1.25322i	0.779334	−	0.626608i	$-0.215557\pi$
$164$	0	0
$165$	− 18.0000i	− 1.40130i
$166$	0	0
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	4.00000i	0.305888i
$172$	0	0
$173$	0	0	1.00000			$0$
$173$	0	0	−1.00000			$\pi$
$174$	0	0
$175$	4.00000	0.302372
$176$	0	0
$177$	−6.00000	−0.450988
$178$	0	0
$179$	3.00000i	0.224231i	0.993695	+	0.112115i	$0.0357626\pi$
$179$	3.00000i	0.224231i	−0.993695	+	0.112115i	$0.964237\pi$
$180$	0	0
$181$	20.0000i	1.48659i	0.668965	+	0.743294i	$0.266738\pi$
$181$	20.0000i	1.48659i	−0.668965	+	0.743294i	$0.733262\pi$
$182$	0	0
$183$	8.00000	0.591377
$184$	0	0
$185$	−21.0000	−1.54395
$186$	0	0
$187$	18.0000i	1.31629i
$188$	0	0
$189$	− 5.00000i	− 0.363696i
$190$	0	0
$191$	18.0000	1.30243	0.651217	−	0.758891i	$-0.274259\pi$
$191$	18.0000	1.30243	0.651217	+	0.758891i	$0.274259\pi$
$192$	0	0
$193$	−4.00000	−0.287926	−0.143963	−	0.989583i	$-0.545985\pi$
$193$	−4.00000	−0.287926	−0.143963	+	0.989583i	$0.545985\pi$
$194$	0	0
$195$	− 3.00000i	− 0.214834i
$196$	0	0
$197$	3.00000i	0.213741i	0.994273	+	0.106871i	$0.0340831\pi$
$197$	3.00000i	0.213741i	−0.994273	+	0.106871i	$0.965917\pi$
$198$	0	0
$199$	2.00000	0.141776	0.0708881	−	0.997484i	$-0.477417\pi$
$199$	2.00000	0.141776	0.0708881	+	0.997484i	$0.477417\pi$
$200$	0	0
$201$	−14.0000	−0.987484
$202$	0	0
$203$	6.00000i	0.421117i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	12.0000	0.830057
$210$	0	0
$211$	− 13.0000i	− 0.894957i	−0.894295	−	0.447478i	$-0.852322\pi$
$211$	− 13.0000i	− 0.894957i	0.894295	−	0.447478i	$-0.147678\pi$
$212$	0	0
$213$	− 3.00000i	− 0.205557i
$214$	0	0
$215$	3.00000	0.204598
$216$	0	0
$217$	−4.00000	−0.271538
$218$	0	0
$219$	− 2.00000i	− 0.135147i
$220$	0	0
$221$	3.00000i	0.201802i
$222$	0	0
$223$	19.0000	1.27233	0.636167	−	0.771551i	$-0.280519\pi$
$223$	19.0000	1.27233	0.636167	+	0.771551i	$0.280519\pi$
$224$	0	0
$225$	−8.00000	−0.533333
$226$	0	0
$227$	0	0	1.00000			$0$
$227$	0	0	−1.00000			$\pi$
$228$	0	0
$229$	− 13.0000i	− 0.859064i	−0.903052	−	0.429532i	$-0.858679\pi$
$229$	− 13.0000i	− 0.859064i	0.903052	−	0.429532i	$-0.141321\pi$
$230$	0	0
$231$	−6.00000	−0.394771
$232$	0	0
$233$	27.0000	1.76883	0.884414	−	0.466702i	$-0.154558\pi$
$233$	27.0000	1.76883	0.884414	+	0.466702i	$0.154558\pi$
$234$	0	0
$235$	9.00000i	0.587095i
$236$	0	0
$237$	− 8.00000i	− 0.519656i
$238$	0	0
$239$	−15.0000	−0.970269	−0.485135	−	0.874439i	$-0.661229\pi$
$239$	−15.0000	−0.970269	−0.485135	+	0.874439i	$0.661229\pi$
$240$	0	0
$241$	−10.0000	−0.644157	−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000	−0.644157	−0.322078	+	0.946713i	$0.604381\pi$
$242$	0	0
$243$	16.0000i	1.02640i
$244$	0	0
$245$	18.0000i	1.14998i
$246$	0	0
$247$	2.00000	0.127257
$248$	0	0
$249$	−12.0000	−0.760469
$250$	0	0
$251$	− 24.0000i	− 1.51487i	−0.652913	−	0.757433i	$-0.726453\pi$
$251$	− 24.0000i	− 1.51487i	0.652913	−	0.757433i	$-0.273547\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−9.00000	−0.563602
$256$	0	0
$257$	9.00000	0.561405	0.280702	−	0.959795i	$-0.409433\pi$
$257$	9.00000	0.561405	0.280702	+	0.959795i	$0.409433\pi$
$258$	0	0
$259$	7.00000i	0.434959i
$260$	0	0
$261$	− 12.0000i	− 0.742781i
$262$	0	0
$263$	−12.0000	−0.739952	−0.369976	−	0.929041i	$-0.620634\pi$
$263$	−12.0000	−0.739952	−0.369976	+	0.929041i	$0.620634\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	6.00000i	0.367194i
$268$	0	0
$269$	− 24.0000i	− 1.46331i	−0.681677	−	0.731653i	$-0.738749\pi$
$269$	− 24.0000i	− 1.46331i	0.681677	−	0.731653i	$-0.261251\pi$
$270$	0	0
$271$	−11.0000	−0.668202	−0.334101	−	0.942537i	$-0.608433\pi$
$271$	−11.0000	−0.668202	−0.334101	+	0.942537i	$0.608433\pi$
$272$	0	0
$273$	−1.00000	−0.0605228
$274$	0	0
$275$	24.0000i	1.44725i
$276$	0	0
$277$	− 28.0000i	− 1.68236i	−0.540758	−	0.841178i	$-0.681862\pi$
$277$	− 28.0000i	− 1.68236i	0.540758	−	0.841178i	$-0.318138\pi$
$278$	0	0
$279$	8.00000	0.478947
$280$	0	0
$281$	6.00000	0.357930	0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000	0.357930	0.178965	+	0.983855i	$0.442725\pi$
$282$	0	0
$283$	4.00000i	0.237775i	0.992908	+	0.118888i	$0.0379328\pi$
$283$	4.00000i	0.237775i	−0.992908	+	0.118888i	$0.962067\pi$
$284$	0	0
$285$	6.00000i	0.355409i
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−8.00000	−0.470588
$290$	0	0
$291$	− 10.0000i	− 0.586210i
$292$	0	0
$293$	21.0000i	1.22683i	0.789760	+	0.613417i	$0.210205\pi$
$293$	21.0000i	1.22683i	−0.789760	+	0.613417i	$0.789795\pi$
$294$	0	0
$295$	18.0000	1.04800
$296$	0	0
$297$	30.0000	1.74078
$298$	0	0
$299$	0	0
$300$	0	0
$301$	− 1.00000i	− 0.0576390i
$302$	0	0
$303$	12.0000	0.689382
$304$	0	0
$305$	−24.0000	−1.37424
$306$	0	0
$307$	2.00000i	0.114146i	0.998370	+	0.0570730i	$0.0181768\pi$
$307$	2.00000i	0.114146i	−0.998370	+	0.0570730i	$0.981823\pi$
$308$	0	0
$309$	− 4.00000i	− 0.227552i
$310$	0	0
$311$	−30.0000	−1.70114	−0.850572	−	0.525859i	$-0.823744\pi$
$311$	−30.0000	−1.70114	−0.850572	+	0.525859i	$0.823744\pi$
$312$	0	0
$313$	1.00000	0.0565233	0.0282617	−	0.999601i	$-0.491003\pi$
$313$	1.00000	0.0565233	0.0282617	+	0.999601i	$0.491003\pi$
$314$	0	0
$315$	6.00000i	0.338062i
$316$	0	0
$317$	6.00000i	0.336994i	0.985702	+	0.168497i	$0.0538913\pi$
$317$	6.00000i	0.336994i	−0.985702	+	0.168497i	$0.946109\pi$
$318$	0	0
$319$	−36.0000	−2.01561
$320$	0	0
$321$	12.0000	0.669775
$322$	0	0
$323$	− 6.00000i	− 0.333849i
$324$	0	0
$325$	4.00000i	0.221880i
$326$	0	0
$327$	−7.00000	−0.387101
$328$	0	0
$329$	3.00000	0.165395
$330$	0	0
$331$	− 8.00000i	− 0.439720i	−0.975531	−	0.219860i	$-0.929440\pi$
$331$	− 8.00000i	− 0.439720i	0.975531	−	0.219860i	$-0.0705600\pi$
$332$	0	0
$333$	− 14.0000i	− 0.767195i
$334$	0	0
$335$	42.0000	2.29471
$336$	0	0
$337$	23.0000	1.25289	0.626445	−	0.779466i	$-0.284509\pi$
$337$	23.0000	1.25289	0.626445	+	0.779466i	$0.284509\pi$
$338$	0	0
$339$	− 6.00000i	− 0.325875i
$340$	0	0
$341$	− 24.0000i	− 1.29967i
$342$	0	0
$343$	13.0000	0.701934
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 3.00000i	− 0.161048i	−0.996753	−	0.0805242i	$-0.974341\pi$
$347$	− 3.00000i	− 0.161048i	0.996753	−	0.0805242i	$-0.0256594\pi$
$348$	0	0
$349$	19.0000i	1.01705i	0.861048	+	0.508523i	$0.169808\pi$
$349$	19.0000i	1.01705i	−0.861048	+	0.508523i	$0.830192\pi$
$350$	0	0
$351$	5.00000	0.266880
$352$	0	0
$353$	24.0000	1.27739	0.638696	−	0.769460i	$-0.279474\pi$
$353$	24.0000	1.27739	0.638696	+	0.769460i	$0.279474\pi$
$354$	0	0
$355$	9.00000i	0.477670i
$356$	0	0
$357$	3.00000i	0.158777i
$358$	0	0
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	15.0000	0.789474
$362$	0	0
$363$	− 25.0000i	− 1.31216i
$364$	0	0
$365$	6.00000i	0.314054i
$366$	0	0
$367$	−26.0000	−1.35719	−0.678594	−	0.734513i	$-0.737411\pi$
$367$	−26.0000	−1.35719	−0.678594	+	0.734513i	$0.737411\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	− 4.00000i	− 0.207112i	−0.994624	−	0.103556i	$-0.966978\pi$
$373$	− 4.00000i	− 0.207112i	0.994624	−	0.103556i	$-0.0330221\pi$
$374$	0	0
$375$	3.00000	0.154919
$376$	0	0
$377$	−6.00000	−0.309016
$378$	0	0
$379$	− 20.0000i	− 1.02733i	−0.857991	−	0.513665i	$-0.828287\pi$
$379$	− 20.0000i	− 1.02733i	0.857991	−	0.513665i	$-0.171713\pi$
$380$	0	0
$381$	− 20.0000i	− 1.02463i
$382$	0	0
$383$	−21.0000	−1.07305	−0.536525	−	0.843884i	$-0.680263\pi$
$383$	−21.0000	−1.07305	−0.536525	+	0.843884i	$0.680263\pi$
$384$	0	0
$385$	18.0000	0.917365
$386$	0	0
$387$	2.00000i	0.101666i
$388$	0	0
$389$	− 6.00000i	− 0.304212i	−0.988364	−	0.152106i	$-0.951394\pi$
$389$	− 6.00000i	− 0.304212i	0.988364	−	0.152106i	$-0.0486055\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	21.0000	1.05931
$394$	0	0
$395$	24.0000i	1.20757i
$396$	0	0
$397$	34.0000i	1.70641i	0.521575	+	0.853206i	$0.325345\pi$
$397$	34.0000i	1.70641i	−0.521575	+	0.853206i	$0.674655\pi$
$398$	0	0
$399$	2.00000	0.100125
$400$	0	0
$401$	36.0000	1.79775	0.898877	−	0.438201i	$-0.144384\pi$
$401$	36.0000	1.79775	0.898877	+	0.438201i	$0.144384\pi$
$402$	0	0
$403$	− 4.00000i	− 0.199254i
$404$	0	0
$405$	− 3.00000i	− 0.149071i
$406$	0	0
$407$	−42.0000	−2.08186
$408$	0	0
$409$	−32.0000	−1.58230	−0.791149	−	0.611623i	$-0.790517\pi$
$409$	−32.0000	−1.58230	−0.791149	+	0.611623i	$0.790517\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 6.00000i	− 0.295241i
$414$	0	0
$415$	36.0000	1.76717
$416$	0	0
$417$	−13.0000	−0.636613
$418$	0	0
$419$	9.00000i	0.439679i	0.975536	+	0.219839i	$0.0705533\pi$
$419$	9.00000i	0.439679i	−0.975536	+	0.219839i	$0.929447\pi$
$420$	0	0
$421$	17.0000i	0.828529i	0.910156	+	0.414265i	$0.135961\pi$
$421$	17.0000i	0.828529i	−0.910156	+	0.414265i	$0.864039\pi$
$422$	0	0
$423$	−6.00000	−0.291730
$424$	0	0
$425$	12.0000	0.582086
$426$	0	0
$427$	8.00000i	0.387147i
$428$	0	0
$429$	− 6.00000i	− 0.289683i
$430$	0	0
$431$	33.0000	1.58955	0.794777	−	0.606902i	$-0.207588\pi$
$431$	33.0000	1.58955	0.794777	+	0.606902i	$0.207588\pi$
$432$	0	0
$433$	−25.0000	−1.20142	−0.600712	−	0.799466i	$-0.705116\pi$
$433$	−25.0000	−1.20142	−0.600712	+	0.799466i	$0.705116\pi$
$434$	0	0
$435$	− 18.0000i	− 0.863034i
$436$	0	0
$437$	0	0
$438$	0	0
$439$	26.0000	1.24091	0.620456	−	0.784241i	$-0.286947\pi$
$439$	26.0000	1.24091	0.620456	+	0.784241i	$0.286947\pi$
$440$	0	0
$441$	−12.0000	−0.571429
$442$	0	0
$443$	− 21.0000i	− 0.997740i	−0.866677	−	0.498870i	$-0.833748\pi$
$443$	− 21.0000i	− 0.997740i	0.866677	−	0.498870i	$-0.166252\pi$
$444$	0	0
$445$	− 18.0000i	− 0.853282i
$446$	0	0
$447$	6.00000	0.283790
$448$	0	0
$449$	6.00000	0.283158	0.141579	−	0.989927i	$-0.454782\pi$
$449$	6.00000	0.283158	0.141579	+	0.989927i	$0.454782\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	17.0000i	0.798730i
$454$	0	0
$455$	3.00000	0.140642
$456$	0	0
$457$	10.0000	0.467780	0.233890	−	0.972263i	$-0.424854\pi$
$457$	10.0000	0.467780	0.233890	+	0.972263i	$0.424854\pi$
$458$	0	0
$459$	− 15.0000i	− 0.700140i
$460$	0	0
$461$	− 9.00000i	− 0.419172i	−0.977790	−	0.209586i	$-0.932788\pi$
$461$	− 9.00000i	− 0.419172i	0.977790	−	0.209586i	$-0.0672116\pi$
$462$	0	0
$463$	40.0000	1.85896	0.929479	−	0.368875i	$-0.120257\pi$
$463$	40.0000	1.85896	0.929479	+	0.368875i	$0.120257\pi$
$464$	0	0
$465$	12.0000	0.556487
$466$	0	0
$467$	36.0000i	1.66588i	0.553362	+	0.832941i	$0.313345\pi$
$467$	36.0000i	1.66588i	−0.553362	+	0.832941i	$0.686655\pi$
$468$	0	0
$469$	− 14.0000i	− 0.646460i
$470$	0	0
$471$	14.0000	0.645086
$472$	0	0
$473$	6.00000	0.275880
$474$	0	0
$475$	− 8.00000i	− 0.367065i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	21.0000	0.959514	0.479757	−	0.877401i	$-0.340725\pi$
$479$	21.0000	0.959514	0.479757	+	0.877401i	$0.340725\pi$
$480$	0	0
$481$	−7.00000	−0.319173
$482$	0	0
$483$	0	0
$484$	0	0
$485$	30.0000i	1.36223i
$486$	0	0
$487$	−16.0000	−0.725029	−0.362515	−	0.931978i	$-0.618082\pi$
$487$	−16.0000	−0.725029	−0.362515	+	0.931978i	$0.618082\pi$
$488$	0	0
$489$	16.0000	0.723545
$490$	0	0
$491$	9.00000i	0.406164i	0.979162	+	0.203082i	$0.0650959\pi$
$491$	9.00000i	0.406164i	−0.979162	+	0.203082i	$0.934904\pi$
$492$	0	0
$493$	18.0000i	0.810679i
$494$	0	0
$495$	−36.0000	−1.61808
$496$	0	0
$497$	3.00000	0.134568
$498$	0	0
$499$	− 40.0000i	− 1.79065i	−0.445418	−	0.895323i	$-0.646945\pi$
$499$	− 40.0000i	− 1.79065i	0.445418	−	0.895323i	$-0.353055\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−30.0000	−1.33763	−0.668817	−	0.743427i	$-0.733199\pi$
$503$	−30.0000	−1.33763	−0.668817	+	0.743427i	$0.733199\pi$
$504$	0	0
$505$	−36.0000	−1.60198
$506$	0	0
$507$	− 1.00000i	− 0.0444116i
$508$	0	0
$509$	18.0000i	0.797836i	0.916987	+	0.398918i	$0.130614\pi$
$509$	18.0000i	0.797836i	−0.916987	+	0.398918i	$0.869386\pi$
$510$	0	0
$511$	2.00000	0.0884748
$512$	0	0
$513$	−10.0000	−0.441511
$514$	0	0
$515$	12.0000i	0.528783i
$516$	0	0
$517$	18.0000i	0.791639i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	9.00000	0.394297	0.197149	−	0.980374i	$-0.436832\pi$
$521$	9.00000	0.394297	0.197149	+	0.980374i	$0.436832\pi$
$522$	0	0
$523$	− 20.0000i	− 0.874539i	−0.899331	−	0.437269i	$-0.855946\pi$
$523$	− 20.0000i	− 0.874539i	0.899331	−	0.437269i	$-0.144054\pi$
$524$	0	0
$525$	4.00000i	0.174574i
$526$	0	0
$527$	−12.0000	−0.522728
$528$	0	0
$529$	−23.0000	−1.00000
$530$	0	0
$531$	12.0000i	0.520756i
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−36.0000	−1.55642
$536$	0	0
$537$	−3.00000	−0.129460
$538$	0	0
$539$	36.0000i	1.55063i
$540$	0	0
$541$	− 11.0000i	− 0.472927i	−0.971640	−	0.236463i	$-0.924012\pi$
$541$	− 11.0000i	− 0.472927i	0.971640	−	0.236463i	$-0.0759884\pi$
$542$	0	0
$543$	−20.0000	−0.858282
$544$	0	0
$545$	21.0000	0.899541
$546$	0	0
$547$	17.0000i	0.726868i	0.931620	+	0.363434i	$0.118396\pi$
$547$	17.0000i	0.726868i	−0.931620	+	0.363434i	$0.881604\pi$
$548$	0	0
$549$	− 16.0000i	− 0.682863i
$550$	0	0
$551$	12.0000	0.511217
$552$	0	0
$553$	8.00000	0.340195
$554$	0	0
$555$	− 21.0000i	− 0.891400i
$556$	0	0
$557$	− 3.00000i	− 0.127114i	−0.997978	−	0.0635570i	$-0.979756\pi$
$557$	− 3.00000i	− 0.127114i	0.997978	−	0.0635570i	$-0.0202445\pi$
$558$	0	0
$559$	1.00000	0.0422955
$560$	0	0
$561$	−18.0000	−0.759961
$562$	0	0
$563$	39.0000i	1.64365i	0.569737	+	0.821827i	$0.307045\pi$
$563$	39.0000i	1.64365i	−0.569737	+	0.821827i	$0.692955\pi$
$564$	0	0
$565$	18.0000i	0.757266i
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	−15.0000	−0.628833	−0.314416	−	0.949285i	$-0.601809\pi$
$569$	−15.0000	−0.628833	−0.314416	+	0.949285i	$0.601809\pi$
$570$	0	0
$571$	− 5.00000i	− 0.209243i	−0.994512	−	0.104622i	$-0.966637\pi$
$571$	− 5.00000i	− 0.209243i	0.994512	−	0.104622i	$-0.0333632\pi$
$572$	0	0
$573$	18.0000i	0.751961i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	38.0000	1.58196	0.790980	−	0.611842i	$-0.209571\pi$
$577$	38.0000	1.58196	0.790980	+	0.611842i	$0.209571\pi$
$578$	0	0
$579$	− 4.00000i	− 0.166234i
$580$	0	0
$581$	− 12.0000i	− 0.497844i
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−6.00000	−0.248069
$586$	0	0
$587$	− 24.0000i	− 0.990586i	−0.868726	−	0.495293i	$-0.835061\pi$
$587$	− 24.0000i	− 0.990586i	0.868726	−	0.495293i	$-0.164939\pi$
$588$	0	0
$589$	8.00000i	0.329634i
$590$	0	0
$591$	−3.00000	−0.123404
$592$	0	0
$593$	18.0000	0.739171	0.369586	−	0.929197i	$-0.379500\pi$
$593$	18.0000	0.739171	0.369586	+	0.929197i	$0.379500\pi$
$594$	0	0
$595$	− 9.00000i	− 0.368964i
$596$	0	0
$597$	2.00000i	0.0818546i
$598$	0	0
$599$	6.00000	0.245153	0.122577	−	0.992459i	$-0.460884\pi$
$599$	6.00000	0.245153	0.122577	+	0.992459i	$0.460884\pi$
$600$	0	0
$601$	19.0000	0.775026	0.387513	−	0.921864i	$-0.373334\pi$
$601$	19.0000	0.775026	0.387513	+	0.921864i	$0.373334\pi$
$602$	0	0
$603$	28.0000i	1.14025i
$604$	0	0
$605$	75.0000i	3.04918i
$606$	0	0
$607$	−14.0000	−0.568242	−0.284121	−	0.958788i	$-0.591702\pi$
$607$	−14.0000	−0.568242	−0.284121	+	0.958788i	$0.591702\pi$
$608$	0	0
$609$	−6.00000	−0.243132
$610$	0	0
$611$	3.00000i	0.121367i
$612$	0	0
$613$	38.0000i	1.53481i	0.641165	+	0.767403i	$0.278451\pi$
$613$	38.0000i	1.53481i	−0.641165	+	0.767403i	$0.721549\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	24.0000	0.966204	0.483102	−	0.875564i	$-0.339510\pi$
$617$	24.0000	0.966204	0.483102	+	0.875564i	$0.339510\pi$
$618$	0	0
$619$	28.0000i	1.12542i	0.826656	+	0.562708i	$0.190240\pi$
$619$	28.0000i	1.12542i	−0.826656	+	0.562708i	$0.809760\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−6.00000	−0.240385
$624$	0	0
$625$	−29.0000	−1.16000
$626$	0	0
$627$	12.0000i	0.479234i
$628$	0	0
$629$	21.0000i	0.837325i
$630$	0	0
$631$	29.0000	1.15447	0.577236	−	0.816577i	$-0.304131\pi$
$631$	29.0000	1.15447	0.577236	+	0.816577i	$0.304131\pi$
$632$	0	0
$633$	13.0000	0.516704
$634$	0	0
$635$	60.0000i	2.38103i
$636$	0	0
$637$	6.00000i	0.237729i
$638$	0	0
$639$	−6.00000	−0.237356
$640$	0	0
$641$	−18.0000	−0.710957	−0.355479	−	0.934684i	$-0.615682\pi$
$641$	−18.0000	−0.710957	−0.355479	+	0.934684i	$0.615682\pi$
$642$	0	0
$643$	14.0000i	0.552106i	0.961142	+	0.276053i	$0.0890266\pi$
$643$	14.0000i	0.552106i	−0.961142	+	0.276053i	$0.910973\pi$
$644$	0	0
$645$	3.00000i	0.118125i
$646$	0	0
$647$	−6.00000	−0.235884	−0.117942	−	0.993020i	$-0.537630\pi$
$647$	−6.00000	−0.235884	−0.117942	+	0.993020i	$0.537630\pi$
$648$	0	0
$649$	36.0000	1.41312
$650$	0	0
$651$	− 4.00000i	− 0.156772i
$652$	0	0
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	0	0
$655$	−63.0000	−2.46161
$656$	0	0
$657$	−4.00000	−0.156055
$658$	0	0
$659$	− 36.0000i	− 1.40236i	−0.712984	−	0.701180i	$-0.752657\pi$
$659$	− 36.0000i	− 1.40236i	0.712984	−	0.701180i	$-0.247343\pi$
$660$	0	0
$661$	− 22.0000i	− 0.855701i	−0.903850	−	0.427850i	$-0.859271\pi$
$661$	− 22.0000i	− 0.855701i	0.903850	−	0.427850i	$-0.140729\pi$
$662$	0	0
$663$	−3.00000	−0.116510
$664$	0	0
$665$	−6.00000	−0.232670
$666$	0	0
$667$	0	0
$668$	0	0
$669$	19.0000i	0.734582i
$670$	0	0
$671$	−48.0000	−1.85302
$672$	0	0
$673$	−19.0000	−0.732396	−0.366198	−	0.930537i	$-0.619341\pi$
$673$	−19.0000	−0.732396	−0.366198	+	0.930537i	$0.619341\pi$
$674$	0	0
$675$	− 20.0000i	− 0.769800i
$676$	0	0
$677$	48.0000i	1.84479i	0.386248	+	0.922395i	$0.373771\pi$
$677$	48.0000i	1.84479i	−0.386248	+	0.922395i	$0.626229\pi$
$678$	0	0
$679$	10.0000	0.383765
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 24.0000i	− 0.918334i	−0.888350	−	0.459167i	$-0.848148\pi$
$683$	− 24.0000i	− 0.918334i	0.888350	−	0.459167i	$-0.151852\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	13.0000	0.495981
$688$	0	0
$689$	0	0
$690$	0	0
$691$	8.00000i	0.304334i	0.988355	+	0.152167i	$0.0486252\pi$
$691$	8.00000i	0.304334i	−0.988355	+	0.152167i	$0.951375\pi$
$692$	0	0
$693$	12.0000i	0.455842i
$694$	0	0
$695$	39.0000	1.47935
$696$	0	0
$697$	0	0
$698$	0	0
$699$	27.0000i	1.02123i
$700$	0	0
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	0	0
$703$	14.0000	0.528020
$704$	0	0
$705$	−9.00000	−0.338960
$706$	0	0
$707$	12.0000i	0.451306i
$708$	0	0
$709$	26.0000i	0.976450i	0.872718	+	0.488225i	$0.162356\pi$
$709$	26.0000i	0.976450i	−0.872718	+	0.488225i	$0.837644\pi$
$710$	0	0
$711$	−16.0000	−0.600047
$712$	0	0
$713$	0	0
$714$	0	0
$715$	18.0000i	0.673162i
$716$	0	0
$717$	− 15.0000i	− 0.560185i
$718$	0	0
$719$	−6.00000	−0.223762	−0.111881	−	0.993722i	$-0.535688\pi$
$719$	−6.00000	−0.223762	−0.111881	+	0.993722i	$0.535688\pi$
$720$	0	0
$721$	4.00000	0.148968
$722$	0	0
$723$	− 10.0000i	− 0.371904i
$724$	0	0
$725$	24.0000i	0.891338i
$726$	0	0
$727$	−10.0000	−0.370879	−0.185440	−	0.982656i	$-0.559371\pi$
$727$	−10.0000	−0.370879	−0.185440	+	0.982656i	$0.559371\pi$
$728$	0	0
$729$	−13.0000	−0.481481
$730$	0	0
$731$	− 3.00000i	− 0.110959i
$732$	0	0
$733$	− 23.0000i	− 0.849524i	−0.905305	−	0.424762i	$-0.860358\pi$
$733$	− 23.0000i	− 0.849524i	0.905305	−	0.424762i	$-0.139642\pi$
$734$	0	0
$735$	−18.0000	−0.663940
$736$	0	0
$737$	84.0000	3.09418
$738$	0	0
$739$	20.0000i	0.735712i	0.929883	+	0.367856i	$0.119908\pi$
$739$	20.0000i	0.735712i	−0.929883	+	0.367856i	$0.880092\pi$
$740$	0	0
$741$	2.00000i	0.0734718i
$742$	0	0
$743$	−9.00000	−0.330178	−0.165089	−	0.986279i	$-0.552791\pi$
$743$	−9.00000	−0.330178	−0.165089	+	0.986279i	$0.552791\pi$
$744$	0	0
$745$	−18.0000	−0.659469
$746$	0	0
$747$	24.0000i	0.878114i
$748$	0	0
$749$	12.0000i	0.438470i
$750$	0	0
$751$	40.0000	1.45962	0.729810	−	0.683650i	$-0.239608\pi$
$751$	40.0000	1.45962	0.729810	+	0.683650i	$0.239608\pi$
$752$	0	0
$753$	24.0000	0.874609
$754$	0	0
$755$	− 51.0000i	− 1.85608i
$756$	0	0
$757$	− 16.0000i	− 0.581530i	−0.956795	−	0.290765i	$-0.906090\pi$
$757$	− 16.0000i	− 0.581530i	0.956795	−	0.290765i	$-0.0939098\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	6.00000	0.217500	0.108750	−	0.994069i	$-0.465315\pi$
$761$	6.00000	0.217500	0.108750	+	0.994069i	$0.465315\pi$
$762$	0	0
$763$	− 7.00000i	− 0.253417i
$764$	0	0
$765$	18.0000i	0.650791i
$766$	0	0
$767$	6.00000	0.216647
$768$	0	0
$769$	32.0000	1.15395	0.576975	−	0.816762i	$-0.304233\pi$
$769$	32.0000	1.15395	0.576975	+	0.816762i	$0.304233\pi$
$770$	0	0
$771$	9.00000i	0.324127i
$772$	0	0
$773$	− 39.0000i	− 1.40273i	−0.712801	−	0.701366i	$-0.752574\pi$
$773$	− 39.0000i	− 1.40273i	0.712801	−	0.701366i	$-0.247426\pi$
$774$	0	0
$775$	−16.0000	−0.574737
$776$	0	0
$777$	−7.00000	−0.251124
$778$	0	0
$779$	0	0
$780$	0	0
$781$	18.0000i	0.644091i
$782$	0	0
$783$	30.0000	1.07211
$784$	0	0
$785$	−42.0000	−1.49904
$786$	0	0
$787$	− 40.0000i	− 1.42585i	−0.701242	−	0.712923i	$-0.747371\pi$
$787$	− 40.0000i	− 1.42585i	0.701242	−	0.712923i	$-0.252629\pi$
$788$	0	0
$789$	− 12.0000i	− 0.427211i
$790$	0	0
$791$	6.00000	0.213335
$792$	0	0
$793$	−8.00000	−0.284088
$794$	0	0
$795$	0	0
$796$	0	0
$797$	42.0000i	1.48772i	0.668338	+	0.743858i	$0.267006\pi$
$797$	42.0000i	1.48772i	−0.668338	+	0.743858i	$0.732994\pi$
$798$	0	0
$799$	9.00000	0.318397
$800$	0	0
$801$	12.0000	0.423999
$802$	0	0
$803$	12.0000i	0.423471i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	24.0000	0.844840
$808$	0	0
$809$	33.0000	1.16022	0.580109	−	0.814539i	$-0.303010\pi$
$809$	33.0000	1.16022	0.580109	+	0.814539i	$0.303010\pi$
$810$	0	0
$811$	− 20.0000i	− 0.702295i	−0.936320	−	0.351147i	$-0.885792\pi$
$811$	− 20.0000i	− 0.702295i	0.936320	−	0.351147i	$-0.114208\pi$
$812$	0	0
$813$	− 11.0000i	− 0.385787i
$814$	0	0
$815$	−48.0000	−1.68137
$816$	0	0
$817$	−2.00000	−0.0699711
$818$	0	0
$819$	2.00000i	0.0698857i
$820$	0	0
$821$	− 3.00000i	− 0.104701i	−0.998629	−	0.0523504i	$-0.983329\pi$
$821$	− 3.00000i	− 0.104701i	0.998629	−	0.0523504i	$-0.0166713\pi$
$822$	0	0
$823$	14.0000	0.488009	0.244005	−	0.969774i	$-0.421539\pi$
$823$	14.0000	0.488009	0.244005	+	0.969774i	$0.421539\pi$
$824$	0	0
$825$	−24.0000	−0.835573
$826$	0	0
$827$	− 18.0000i	− 0.625921i	−0.949766	−	0.312961i	$-0.898679\pi$
$827$	− 18.0000i	− 0.625921i	0.949766	−	0.312961i	$-0.101321\pi$
$828$	0	0
$829$	− 38.0000i	− 1.31979i	−0.751356	−	0.659897i	$-0.770600\pi$
$829$	− 38.0000i	− 1.31979i	0.751356	−	0.659897i	$-0.229400\pi$
$830$	0	0
$831$	28.0000	0.971309
$832$	0	0
$833$	18.0000	0.623663
$834$	0	0
$835$	0	0
$836$	0	0
$837$	20.0000i	0.691301i
$838$	0	0
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	−7.00000	−0.241379
$842$	0	0
$843$	6.00000i	0.206651i
$844$	0	0
$845$	3.00000i	0.103203i
$846$	0	0
$847$	25.0000	0.859010
$848$	0	0
$849$	−4.00000	−0.137280
$850$	0	0
$851$	0	0
$852$	0	0
$853$	− 37.0000i	− 1.26686i	−0.773802	−	0.633428i	$-0.781647\pi$
$853$	− 37.0000i	− 1.26686i	0.773802	−	0.633428i	$-0.218353\pi$
$854$	0	0
$855$	12.0000	0.410391
$856$	0	0
$857$	42.0000	1.43469	0.717346	−	0.696717i	$-0.245357\pi$
$857$	42.0000	1.43469	0.717346	+	0.696717i	$0.245357\pi$
$858$	0	0
$859$	4.00000i	0.136478i	0.997669	+	0.0682391i	$0.0217381\pi$
$859$	4.00000i	0.136478i	−0.997669	+	0.0682391i	$0.978262\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	45.0000	1.53182	0.765909	−	0.642949i	$-0.222289\pi$
$863$	45.0000	1.53182	0.765909	+	0.642949i	$0.222289\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 8.00000i	− 0.271694i
$868$	0	0
$869$	48.0000i	1.62829i
$870$	0	0
$871$	14.0000	0.474372
$872$	0	0
$873$	−20.0000	−0.676897
$874$	0	0
$875$	3.00000i	0.101419i
$876$	0	0
$877$	13.0000i	0.438979i	0.975615	+	0.219489i	$0.0704391\pi$
$877$	13.0000i	0.438979i	−0.975615	+	0.219489i	$0.929561\pi$
$878$	0	0
$879$	−21.0000	−0.708312
$880$	0	0
$881$	21.0000	0.707508	0.353754	−	0.935339i	$-0.384905\pi$
$881$	21.0000	0.707508	0.353754	+	0.935339i	$0.384905\pi$
$882$	0	0
$883$	29.0000i	0.975928i	0.872864	+	0.487964i	$0.162260\pi$
$883$	29.0000i	0.975928i	−0.872864	+	0.487964i	$0.837740\pi$
$884$	0	0
$885$	18.0000i	0.605063i
$886$	0	0
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	0	0
$889$	20.0000	0.670778
$890$	0	0
$891$	− 6.00000i	− 0.201008i
$892$	0	0
$893$	− 6.00000i	− 0.200782i
$894$	0	0
$895$	9.00000	0.300837
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 24.0000i	− 0.800445i
$900$	0	0
$901$	0	0
$902$	0	0
$903$	1.00000	0.0332779
$904$	0	0
$905$	60.0000	1.99447
$906$	0	0
$907$	37.0000i	1.22856i	0.789086	+	0.614282i	$0.210554\pi$
$907$	37.0000i	1.22856i	−0.789086	+	0.614282i	$0.789446\pi$
$908$	0	0
$909$	− 24.0000i	− 0.796030i
$910$	0	0
$911$	−30.0000	−0.993944	−0.496972	−	0.867766i	$-0.665555\pi$
$911$	−30.0000	−0.993944	−0.496972	+	0.867766i	$0.665555\pi$
$912$	0	0
$913$	72.0000	2.38285
$914$	0	0
$915$	− 24.0000i	− 0.793416i
$916$	0	0
$917$	21.0000i	0.693481i
$918$	0	0
$919$	−16.0000	−0.527791	−0.263896	−	0.964551i	$-0.585007\pi$
$919$	−16.0000	−0.527791	−0.263896	+	0.964551i	$0.585007\pi$
$920$	0	0
$921$	−2.00000	−0.0659022
$922$	0	0
$923$	3.00000i	0.0987462i
$924$	0	0
$925$	28.0000i	0.920634i
$926$	0	0
$927$	−8.00000	−0.262754
$928$	0	0
$929$	36.0000	1.18112	0.590561	−	0.806993i	$-0.298907\pi$
$929$	36.0000	1.18112	0.590561	+	0.806993i	$0.298907\pi$
$930$	0	0
$931$	− 12.0000i	− 0.393284i
$932$	0	0
$933$	− 30.0000i	− 0.982156i
$934$	0	0
$935$	54.0000	1.76599
$936$	0	0
$937$	34.0000	1.11073	0.555366	−	0.831606i	$-0.312578\pi$
$937$	34.0000	1.11073	0.555366	+	0.831606i	$0.312578\pi$
$938$	0	0
$939$	1.00000i	0.0326338i
$940$	0	0
$941$	21.0000i	0.684580i	0.939594	+	0.342290i	$0.111203\pi$
$941$	21.0000i	0.684580i	−0.939594	+	0.342290i	$0.888797\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	−15.0000	−0.487950
$946$	0	0
$947$	6.00000i	0.194974i	0.995237	+	0.0974869i	$0.0310804\pi$
$947$	6.00000i	0.194974i	−0.995237	+	0.0974869i	$0.968920\pi$
$948$	0	0
$949$	2.00000i	0.0649227i
$950$	0	0
$951$	−6.00000	−0.194563
$952$	0	0
$953$	−15.0000	−0.485898	−0.242949	−	0.970039i	$-0.578115\pi$
$953$	−15.0000	−0.485898	−0.242949	+	0.970039i	$0.578115\pi$
$954$	0	0
$955$	− 54.0000i	− 1.74740i
$956$	0	0
$957$	− 36.0000i	− 1.16371i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	− 24.0000i	− 0.773389i
$964$	0	0
$965$	12.0000i	0.386294i
$966$	0	0
$967$	−31.0000	−0.996893	−0.498446	−	0.866921i	$-0.666096\pi$
$967$	−31.0000	−0.996893	−0.498446	+	0.866921i	$0.666096\pi$
$968$	0	0
$969$	6.00000	0.192748
$970$	0	0
$971$	3.00000i	0.0962746i	0.998841	+	0.0481373i	$0.0153285\pi$
$971$	3.00000i	0.0962746i	−0.998841	+	0.0481373i	$0.984672\pi$
$972$	0	0
$973$	− 13.0000i	− 0.416761i
$974$	0	0
$975$	−4.00000	−0.128103
$976$	0	0
$977$	−54.0000	−1.72761	−0.863807	−	0.503824i	$-0.831926\pi$
$977$	−54.0000	−1.72761	−0.863807	+	0.503824i	$0.831926\pi$
$978$	0	0
$979$	− 36.0000i	− 1.15056i
$980$	0	0
$981$	14.0000i	0.446986i
$982$	0	0
$983$	39.0000	1.24391	0.621953	−	0.783054i	$-0.286339\pi$
$983$	39.0000	1.24391	0.621953	+	0.783054i	$0.286339\pi$
$984$	0	0
$985$	9.00000	0.286764
$986$	0	0
$987$	3.00000i	0.0954911i
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−2.00000	−0.0635321	−0.0317660	−	0.999495i	$-0.510113\pi$
$991$	−2.00000	−0.0635321	−0.0317660	+	0.999495i	$0.510113\pi$
$992$	0	0
$993$	8.00000	0.253872
$994$	0	0
$995$	− 6.00000i	− 0.190213i
$996$	0	0
$997$	− 46.0000i	− 1.45683i	−0.685134	−	0.728417i	$-0.740256\pi$
$997$	− 46.0000i	− 1.45683i	0.685134	−	0.728417i	$-0.259744\pi$
$998$	0	0
$999$	35.0000	1.10735

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3328.2.b.j.1665.2		2
4.3	odd	2		3328.2.b.m.1665.1		2
8.3	odd	2		3328.2.b.m.1665.2		2
8.5	even	2	inner	3328.2.b.j.1665.1		2
16.3	odd	4		26.2.a.a.1.1	✓	1
16.5	even	4		832.2.a.i.1.1		1
16.11	odd	4		832.2.a.d.1.1		1
16.13	even	4		208.2.a.a.1.1		1
48.5	odd	4		7488.2.a.h.1.1		1
48.11	even	4		7488.2.a.g.1.1		1
48.29	odd	4		1872.2.a.q.1.1		1
48.35	even	4		234.2.a.e.1.1		1
80.3	even	4		650.2.b.d.599.2		2
80.19	odd	4		650.2.a.j.1.1		1
80.29	even	4		5200.2.a.x.1.1		1
80.67	even	4		650.2.b.d.599.1		2
112.3	even	12		1274.2.f.r.79.1		2
112.19	even	12		1274.2.f.r.1145.1		2
112.51	odd	12		1274.2.f.p.1145.1		2
112.67	odd	12		1274.2.f.p.79.1		2
112.83	even	4		1274.2.a.d.1.1		1
144.67	odd	12		2106.2.e.ba.703.1		2
144.83	even	12		2106.2.e.b.1405.1		2
144.115	odd	12		2106.2.e.ba.1405.1		2
144.131	even	12		2106.2.e.b.703.1		2
176.131	even	4		3146.2.a.n.1.1		1
208.3	odd	12		338.2.c.d.191.1		2
208.19	even	12		338.2.e.a.23.2		4
208.35	odd	12		338.2.c.d.315.1		2
208.51	odd	4		338.2.a.f.1.1		1
208.67	even	12		338.2.e.a.147.1		4
208.77	even	4		2704.2.a.f.1.1		1
208.83	even	4		338.2.b.c.337.2		2
208.99	even	4		338.2.b.c.337.1		2
208.109	odd	4		2704.2.f.d.337.2		2
208.115	even	12		338.2.e.a.147.2		4
208.125	odd	4		2704.2.f.d.337.1		2
208.147	odd	12		338.2.c.a.315.1		2
208.163	even	12		338.2.e.a.23.1		4
208.179	odd	12		338.2.c.a.191.1		2
240.83	odd	4		5850.2.e.a.5149.1		2
240.179	even	4		5850.2.a.p.1.1		1
240.227	odd	4		5850.2.e.a.5149.2		2
272.67	odd	4		7514.2.a.c.1.1		1
304.227	even	4		9386.2.a.j.1.1		1
624.83	odd	4		3042.2.b.a.1351.1		2
624.467	even	4		3042.2.a.a.1.1		1
624.515	odd	4		3042.2.b.a.1351.2		2
1040.259	odd	4		8450.2.a.c.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
26.2.a.a.1.1	✓	1	16.3	odd	4
208.2.a.a.1.1		1	16.13	even	4
234.2.a.e.1.1		1	48.35	even	4
338.2.a.f.1.1		1	208.51	odd	4
338.2.b.c.337.1		2	208.99	even	4
338.2.b.c.337.2		2	208.83	even	4
338.2.c.a.191.1		2	208.179	odd	12
338.2.c.a.315.1		2	208.147	odd	12
338.2.c.d.191.1		2	208.3	odd	12
338.2.c.d.315.1		2	208.35	odd	12
338.2.e.a.23.1		4	208.163	even	12
338.2.e.a.23.2		4	208.19	even	12
338.2.e.a.147.1		4	208.67	even	12
338.2.e.a.147.2		4	208.115	even	12
650.2.a.j.1.1		1	80.19	odd	4
650.2.b.d.599.1		2	80.67	even	4
650.2.b.d.599.2		2	80.3	even	4
832.2.a.d.1.1		1	16.11	odd	4
832.2.a.i.1.1		1	16.5	even	4
1274.2.a.d.1.1		1	112.83	even	4
1274.2.f.p.79.1		2	112.67	odd	12
1274.2.f.p.1145.1		2	112.51	odd	12
1274.2.f.r.79.1		2	112.3	even	12
1274.2.f.r.1145.1		2	112.19	even	12
1872.2.a.q.1.1		1	48.29	odd	4
2106.2.e.b.703.1		2	144.131	even	12
2106.2.e.b.1405.1		2	144.83	even	12
2106.2.e.ba.703.1		2	144.67	odd	12
2106.2.e.ba.1405.1		2	144.115	odd	12
2704.2.a.f.1.1		1	208.77	even	4
2704.2.f.d.337.1		2	208.125	odd	4
2704.2.f.d.337.2		2	208.109	odd	4
3042.2.a.a.1.1		1	624.467	even	4
3042.2.b.a.1351.1		2	624.83	odd	4
3042.2.b.a.1351.2		2	624.515	odd	4
3146.2.a.n.1.1		1	176.131	even	4
3328.2.b.j.1665.1		2	8.5	even	2	inner
3328.2.b.j.1665.2		2	1.1	even	1	trivial
3328.2.b.m.1665.1		2	4.3	odd	2
3328.2.b.m.1665.2		2	8.3	odd	2
5200.2.a.x.1.1		1	80.29	even	4
5850.2.a.p.1.1		1	240.179	even	4
5850.2.e.a.5149.1		2	240.83	odd	4
5850.2.e.a.5149.2		2	240.227	odd	4
7488.2.a.g.1.1		1	48.11	even	4
7488.2.a.h.1.1		1	48.5	odd	4
7514.2.a.c.1.1		1	272.67	odd	4
8450.2.a.c.1.1		1	1040.259	odd	4
9386.2.a.j.1.1		1	304.227	even	4

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:	\( N \)	\(=\)	\( 3328 = 2^{8} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3328.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} -3.00000i q^{5} -1.00000 q^{7} +2.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{3} -3.00000i q^{5} -1.00000 q^{7} +2.00000 q^{9} -6.00000i q^{11} -1.00000i q^{13} +3.00000 q^{15} -3.00000 q^{17} +2.00000i q^{19} -1.00000i q^{21} -4.00000 q^{25} +5.00000i q^{27} -6.00000i q^{29} +4.00000 q^{31} +6.00000 q^{33} +3.00000i q^{35} -7.00000i q^{37} +1.00000 q^{39} +1.00000i q^{43} -6.00000i q^{45} -3.00000 q^{47} -6.00000 q^{49} -3.00000i q^{51} -18.0000 q^{55} -2.00000 q^{57} +6.00000i q^{59} -8.00000i q^{61} -2.00000 q^{63} -3.00000 q^{65} +14.0000i q^{67} -3.00000 q^{71} -2.00000 q^{73} -4.00000i q^{75} +6.00000i q^{77} -8.00000 q^{79} +1.00000 q^{81} +12.0000i q^{83} +9.00000i q^{85} +6.00000 q^{87} +6.00000 q^{89} +1.00000i q^{91} +4.00000i q^{93} +6.00000 q^{95} -10.0000 q^{97} -12.0000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{7} + 4 q^{9}+O(q^{10}) \) 2 * q - 2 * q^7 + 4 * q^9	\( 2 q - 2 q^{7} + 4 q^{9} + 6 q^{15} - 6 q^{17} - 8 q^{25} + 8 q^{31} + 12 q^{33} + 2 q^{39} - 6 q^{47} - 12 q^{49} - 36 q^{55} - 4 q^{57} - 4 q^{63} - 6 q^{65} - 6 q^{71} - 4 q^{73} - 16 q^{79} + 2 q^{81} + 12 q^{87} + 12 q^{89} + 12 q^{95} - 20 q^{97}+O(q^{100}) \) 2 * q - 2 * q^7 + 4 * q^9 + 6 * q^15 - 6 * q^17 - 8 * q^25 + 8 * q^31 + 12 * q^33 + 2 * q^39 - 6 * q^47 - 12 * q^49 - 36 * q^55 - 4 * q^57 - 4 * q^63 - 6 * q^65 - 6 * q^71 - 4 * q^73 - 16 * q^79 + 2 * q^81 + 12 * q^87 + 12 * q^89 + 12 * q^95 - 20 * q^97

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