LMFDB - Embedded newform 2704.2.f.d.337.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2704,2,Mod(337,2704)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2704, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2704.337");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2704 = 2^{4} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2704.f (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:	$21.5915487066$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 26)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.2
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	2704.337
Dual form			2704.2.f.d.337.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

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

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2704\mathbb{Z}\right)^\times$.

$n$	$677$	$1185$	$2367$
$\chi(n)$	$1$	$-1$	$1$

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
$3$	−1.00000	−0.577350	−0.288675	−	0.957427i	$-0.593215\pi$
$3$	−1.00000	−0.577350	−0.288675	+	0.957427i	$0.593215\pi$
$4$	0	0
$5$	3.00000i	1.34164i	0.741620	+	0.670820i	$0.234058\pi$
$5$	3.00000i	1.34164i	−0.741620	+	0.670820i	$0.765942\pi$
$6$	0	0
$7$	1.00000i	0.377964i	0.981981	+	0.188982i	$0.0605189\pi$
$7$	1.00000i	0.377964i	−0.981981	+	0.188982i	$0.939481\pi$
$8$	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
$13$	0	0
$13$	0	0
$14$	0	0
$15$	− 3.00000i	− 0.774597i
$16$	0	0
$17$	3.00000	0.727607	0.363803	−	0.931476i	$-0.381478\pi$
$17$	3.00000	0.727607	0.363803	+	0.931476i	$0.381478\pi$
$18$	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
$27$	5.00000	0.962250
$28$	0	0
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	0	0
$31$	− 4.00000i	− 0.718421i	−0.933257	−	0.359211i	$-0.883046\pi$
$31$	− 4.00000i	− 0.718421i	0.933257	−	0.359211i	$-0.116954\pi$
$32$	0	0
$33$	6.00000i	1.04447i
$34$	0	0
$35$	−3.00000	−0.507093
$36$	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
$39$	0	0
$40$	0	0
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	0	0
$43$	−1.00000	−0.152499	−0.0762493	−	0.997089i	$-0.524294\pi$
$43$	−1.00000	−0.152499	−0.0762493	+	0.997089i	$0.524294\pi$
$44$	0	0
$45$	− 6.00000i	− 0.894427i
$46$	0	0
$47$	− 3.00000i	− 0.437595i	−0.975770	−	0.218797i	$-0.929787\pi$
$47$	− 3.00000i	− 0.437595i	0.975770	−	0.218797i	$-0.0702134\pi$
$48$	0	0
$49$	6.00000	0.857143
$50$	0	0
$51$	−3.00000	−0.420084
$52$	0	0
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	0	0
$55$	18.0000	2.42712
$56$	0	0
$57$	− 2.00000i	− 0.264906i
$58$	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
$61$	8.00000	1.02430	0.512148	−	0.858898i	$-0.328850\pi$
$61$	8.00000	1.02430	0.512148	+	0.858898i	$0.328850\pi$
$62$	0	0
$63$	− 2.00000i	− 0.251976i
$64$	0	0
$65$	0	0
$66$	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
$71$	− 3.00000i	− 0.356034i	−0.984027	−	0.178017i	$-0.943032\pi$
$71$	− 3.00000i	− 0.356034i	0.984027	−	0.178017i	$-0.0569683\pi$
$72$	0	0
$73$	2.00000i	0.234082i	0.993127	+	0.117041i	$0.0373409\pi$
$73$	2.00000i	0.234082i	−0.993127	+	0.117041i	$0.962659\pi$
$74$	0	0
$75$	4.00000	0.461880
$76$	0	0
$77$	6.00000	0.683763
$78$	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
$89$	− 6.00000i	− 0.635999i	−0.948091	−	0.317999i	$-0.896989\pi$
$89$	− 6.00000i	− 0.635999i	0.948091	−	0.317999i	$-0.103011\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	4.00000i	0.414781i
$94$	0	0
$95$	−6.00000	−0.615587
$96$	0	0
$97$	10.0000i	1.01535i	0.861550	+	0.507673i	$0.169494\pi$
$97$	10.0000i	1.01535i	−0.861550	+	0.507673i	$0.830506\pi$
$98$	0	0
$99$	12.0000i	1.20605i
$100$	0	0
$101$	12.0000	1.19404	0.597022	−	0.802225i	$-0.296350\pi$
$101$	12.0000	1.19404	0.597022	+	0.802225i	$0.296350\pi$
$102$	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
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	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
$111$	7.00000i	0.664411i
$112$	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
$117$	0	0
$118$	0	0
$119$	3.00000i	0.275010i
$120$	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	$-0.152539\pi$
$127$	20.0000	1.77471	0.887357	+	0.461084i	$0.152539\pi$
$128$	0	0
$129$	1.00000	0.0880451
$130$	0	0
$131$	21.0000	1.83478	0.917389	−	0.397991i	$-0.130293\pi$
$131$	21.0000	1.83478	0.917389	+	0.397991i	$0.130293\pi$
$132$	0	0
$133$	−2.00000	−0.173422
$134$	0	0
$135$	15.0000i	1.29099i
$136$	0	0
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	0	0
$139$	13.0000	1.10265	0.551323	−	0.834292i	$-0.314123\pi$
$139$	13.0000	1.10265	0.551323	+	0.834292i	$0.314123\pi$
$140$	0	0
$141$	3.00000i	0.252646i
$142$	0	0
$143$	0	0
$144$	0	0
$145$	18.0000i	1.49482i
$146$	0	0
$147$	−6.00000	−0.494872
$148$	0	0
$149$	6.00000i	0.491539i	0.969328	+	0.245770i	$0.0790407\pi$
$149$	6.00000i	0.491539i	−0.969328	+	0.245770i	$0.920959\pi$
$150$	0	0
$151$	− 17.0000i	− 1.38344i	−0.722166	−	0.691720i	$-0.756853\pi$
$151$	− 17.0000i	− 1.38344i	0.722166	−	0.691720i	$-0.243147\pi$
$152$	0	0
$153$	−6.00000	−0.485071
$154$	0	0
$155$	12.0000	0.963863
$156$	0	0
$157$	14.0000	1.11732	0.558661	−	0.829396i	$-0.311315\pi$
$157$	14.0000	1.11732	0.558661	+	0.829396i	$0.311315\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	16.0000i	1.25322i	0.779334	+	0.626608i	$0.215557\pi$
$163$	16.0000i	1.25322i	−0.779334	+	0.626608i	$0.784443\pi$
$164$	0	0
$165$	−18.0000	−1.40130
$166$	0	0
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	− 4.00000i	− 0.305888i
$172$	0	0
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	0	0
$175$	− 4.00000i	− 0.302372i
$176$	0	0
$177$	− 6.00000i	− 0.450988i
$178$	0	0
$179$	3.00000	0.224231	0.112115	−	0.993695i	$-0.464237\pi$
$179$	3.00000	0.224231	0.112115	+	0.993695i	$0.464237\pi$
$180$	0	0
$181$	−20.0000	−1.48659	−0.743294	−	0.668965i	$-0.766738\pi$
$181$	−20.0000	−1.48659	−0.743294	+	0.668965i	$0.766738\pi$
$182$	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
$193$	− 4.00000i	− 0.287926i	−0.989583	−	0.143963i	$-0.954015\pi$
$193$	− 4.00000i	− 0.287926i	0.989583	−	0.143963i	$-0.0459847\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 3.00000i	− 0.213741i	−0.994273	−	0.106871i	$-0.965917\pi$
$197$	− 3.00000i	− 0.213741i	0.994273	−	0.106871i	$-0.0340831\pi$
$198$	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
$201$	− 14.0000i	− 0.987484i
$202$	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
$211$	13.0000	0.894957	0.447478	−	0.894295i	$-0.352322\pi$
$211$	13.0000	0.894957	0.447478	+	0.894295i	$0.352322\pi$
$212$	0	0
$213$	3.00000i	0.205557i
$214$	0	0
$215$	− 3.00000i	− 0.204598i
$216$	0	0
$217$	4.00000	0.271538
$218$	0	0
$219$	− 2.00000i	− 0.135147i
$220$	0	0
$221$	0	0
$222$	0	0
$223$	− 19.0000i	− 1.27233i	−0.771551	−	0.636167i	$-0.780519\pi$
$223$	− 19.0000i	− 1.27233i	0.771551	−	0.636167i	$-0.219481\pi$
$224$	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
$235$	9.00000	0.587095
$236$	0	0
$237$	8.00000	0.519656
$238$	0	0
$239$	15.0000i	0.970269i	0.874439	+	0.485135i	$0.161229\pi$
$239$	15.0000i	0.970269i	−0.874439	+	0.485135i	$0.838771\pi$
$240$	0	0
$241$	− 10.0000i	− 0.644157i	−0.946713	−	0.322078i	$-0.895619\pi$
$241$	− 10.0000i	− 0.644157i	0.946713	−	0.322078i	$-0.104381\pi$
$242$	0	0
$243$	−16.0000	−1.02640
$244$	0	0
$245$	18.0000i	1.14998i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	− 12.0000i	− 0.760469i
$250$	0	0
$251$	24.0000	1.51487	0.757433	−	0.652913i	$-0.226453\pi$
$251$	24.0000	1.51487	0.757433	+	0.652913i	$0.226453\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	− 9.00000i	− 0.563602i
$256$	0	0
$257$	−9.00000	−0.561405	−0.280702	−	0.959795i	$-0.590567\pi$
$257$	−9.00000	−0.561405	−0.280702	+	0.959795i	$0.590567\pi$
$258$	0	0
$259$	7.00000	0.434959
$260$	0	0
$261$	−12.0000	−0.742781
$262$	0	0
$263$	12.0000	0.739952	0.369976	−	0.929041i	$-0.379366\pi$
$263$	12.0000	0.739952	0.369976	+	0.929041i	$0.379366\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	6.00000i	0.367194i
$268$	0	0
$269$	24.0000	1.46331	0.731653	−	0.681677i	$-0.238749\pi$
$269$	24.0000	1.46331	0.731653	+	0.681677i	$0.238749\pi$
$270$	0	0
$271$	− 11.0000i	− 0.668202i	−0.942537	−	0.334101i	$-0.891567\pi$
$271$	− 11.0000i	− 0.668202i	0.942537	−	0.334101i	$-0.108433\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	24.0000i	1.44725i
$276$	0	0
$277$	28.0000	1.68236	0.841178	−	0.540758i	$-0.181862\pi$
$277$	28.0000	1.68236	0.841178	+	0.540758i	$0.181862\pi$
$278$	0	0
$279$	8.00000i	0.478947i
$280$	0	0
$281$	− 6.00000i	− 0.357930i	−0.983855	−	0.178965i	$-0.942725\pi$
$281$	− 6.00000i	− 0.357930i	0.983855	−	0.178965i	$-0.0572749\pi$
$282$	0	0
$283$	−4.00000	−0.237775	−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000	−0.237775	−0.118888	+	0.992908i	$0.537933\pi$
$284$	0	0
$285$	6.00000	0.355409
$286$	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
$297$	− 30.0000i	− 1.74078i
$298$	0	0
$299$	0	0
$300$	0	0
$301$	− 1.00000i	− 0.0576390i
$302$	0	0
$303$	−12.0000	−0.689382
$304$	0	0
$305$	24.0000i	1.37424i
$306$	0	0
$307$	− 2.00000i	− 0.114146i	−0.998370	−	0.0570730i	$-0.981823\pi$
$307$	− 2.00000i	− 0.114146i	0.998370	−	0.0570730i	$-0.0181768\pi$
$308$	0	0
$309$	4.00000	0.227552
$310$	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	$-0.508997\pi$
$313$	−1.00000	−0.0565233	−0.0282617	+	0.999601i	$0.508997\pi$
$314$	0	0
$315$	6.00000	0.338062
$316$	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
$319$	− 36.0000i	− 2.01561i
$320$	0	0
$321$	12.0000	0.669775
$322$	0	0
$323$	6.00000i	0.333849i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 7.00000i	− 0.387101i
$328$	0	0
$329$	3.00000	0.165395
$330$	0	0
$331$	8.00000i	0.439720i	0.975531	+	0.219860i	$0.0705600\pi$
$331$	8.00000i	0.439720i	−0.975531	+	0.219860i	$0.929440\pi$
$332$	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	$-0.715491\pi$
$337$	−23.0000	−1.25289	−0.626445	+	0.779466i	$0.715491\pi$
$338$	0	0
$339$	6.00000	0.325875
$340$	0	0
$341$	−24.0000	−1.29967
$342$	0	0
$343$	13.0000i	0.701934i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−3.00000	−0.161048	−0.0805242	−	0.996753i	$-0.525659\pi$
$347$	−3.00000	−0.161048	−0.0805242	+	0.996753i	$0.525659\pi$
$348$	0	0
$349$	− 19.0000i	− 1.01705i	−0.861048	−	0.508523i	$-0.830192\pi$
$349$	− 19.0000i	− 1.01705i	0.861048	−	0.508523i	$-0.169808\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 24.0000i	− 1.27739i	−0.769460	−	0.638696i	$-0.779474\pi$
$353$	− 24.0000i	− 1.27739i	0.769460	−	0.638696i	$-0.220526\pi$
$354$	0	0
$355$	9.00000	0.477670
$356$	0	0
$357$	− 3.00000i	− 0.158777i
$358$	0	0
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	0	0
$361$	15.0000	0.789474
$362$	0	0
$363$	25.0000	1.31216
$364$	0	0
$365$	−6.00000	−0.314054
$366$	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
$373$	−4.00000	−0.207112	−0.103556	−	0.994624i	$-0.533022\pi$
$373$	−4.00000	−0.207112	−0.103556	+	0.994624i	$0.533022\pi$
$374$	0	0
$375$	− 3.00000i	− 0.154919i
$376$	0	0
$377$	0	0
$378$	0	0
$379$	20.0000i	1.02733i	0.857991	+	0.513665i	$0.171713\pi$
$379$	20.0000i	1.02733i	−0.857991	+	0.513665i	$0.828287\pi$
$380$	0	0
$381$	−20.0000	−1.02463
$382$	0	0
$383$	21.0000i	1.07305i	0.843884	+	0.536525i	$0.180263\pi$
$383$	21.0000i	1.07305i	−0.843884	+	0.536525i	$0.819737\pi$
$384$	0	0
$385$	18.0000i	0.917365i
$386$	0	0
$387$	2.00000	0.101666
$388$	0	0
$389$	6.00000	0.304212	0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000	0.304212	0.152106	+	0.988364i	$0.451394\pi$
$390$	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	$-0.674655\pi$
$397$	− 34.0000i	− 1.70641i	0.521575	−	0.853206i	$-0.325345\pi$
$398$	0	0
$399$	2.00000	0.100125
$400$	0	0
$401$	36.0000i	1.79775i	0.438201	+	0.898877i	$0.355616\pi$
$401$	36.0000i	1.79775i	−0.438201	+	0.898877i	$0.644384\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	3.00000i	0.149071i
$406$	0	0
$407$	−42.0000	−2.08186
$408$	0	0
$409$	− 32.0000i	− 1.58230i	−0.611623	−	0.791149i	$-0.709483\pi$
$409$	− 32.0000i	− 1.58230i	0.611623	−	0.791149i	$-0.290517\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−6.00000	−0.295241
$414$	0	0
$415$	−36.0000	−1.76717
$416$	0	0
$417$	−13.0000	−0.636613
$418$	0	0
$419$	−9.00000	−0.439679	−0.219839	−	0.975536i	$-0.570553\pi$
$419$	−9.00000	−0.439679	−0.219839	+	0.975536i	$0.570553\pi$
$420$	0	0
$421$	− 17.0000i	− 0.828529i	−0.910156	−	0.414265i	$-0.864039\pi$
$421$	− 17.0000i	− 0.828529i	0.910156	−	0.414265i	$-0.135961\pi$
$422$	0	0
$423$	6.00000i	0.291730i
$424$	0	0
$425$	−12.0000	−0.582086
$426$	0	0
$427$	8.00000i	0.387147i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 33.0000i	− 1.58955i	−0.606902	−	0.794777i	$-0.707588\pi$
$431$	− 33.0000i	− 1.58955i	0.606902	−	0.794777i	$-0.292412\pi$
$432$	0	0
$433$	25.0000	1.20142	0.600712	−	0.799466i	$-0.294884\pi$
$433$	25.0000	1.20142	0.600712	+	0.799466i	$0.294884\pi$
$434$	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
$443$	−21.0000	−0.997740	−0.498870	−	0.866677i	$-0.666252\pi$
$443$	−21.0000	−0.997740	−0.498870	+	0.866677i	$0.666252\pi$
$444$	0	0
$445$	18.0000	0.853282
$446$	0	0
$447$	− 6.00000i	− 0.283790i
$448$	0	0
$449$	6.00000i	0.283158i	0.989927	+	0.141579i	$0.0452178\pi$
$449$	6.00000i	0.283158i	−0.989927	+	0.141579i	$0.954782\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	17.0000i	0.798730i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	10.0000i	0.467780i	0.972263	+	0.233890i	$0.0751456\pi$
$457$	10.0000i	0.467780i	−0.972263	+	0.233890i	$0.924854\pi$
$458$	0	0
$459$	15.0000	0.700140
$460$	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
$463$	40.0000i	1.85896i	0.368875	+	0.929479i	$0.379743\pi$
$463$	40.0000i	1.85896i	−0.368875	+	0.929479i	$0.620257\pi$
$464$	0	0
$465$	−12.0000	−0.556487
$466$	0	0
$467$	36.0000	1.66588	0.832941	−	0.553362i	$-0.186655\pi$
$467$	36.0000	1.66588	0.832941	+	0.553362i	$0.186655\pi$
$468$	0	0
$469$	−14.0000	−0.646460
$470$	0	0
$471$	−14.0000	−0.645086
$472$	0	0
$473$	6.00000i	0.275880i
$474$	0	0
$475$	− 8.00000i	− 0.367065i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	21.0000i	0.959514i	0.877401	+	0.479757i	$0.159275\pi$
$479$	21.0000i	0.959514i	−0.877401	+	0.479757i	$0.840725\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−30.0000	−1.36223
$486$	0	0
$487$	− 16.0000i	− 0.725029i	−0.931978	−	0.362515i	$-0.881918\pi$
$487$	− 16.0000i	− 0.725029i	0.931978	−	0.362515i	$-0.118082\pi$
$488$	0	0
$489$	− 16.0000i	− 0.723545i
$490$	0	0
$491$	−9.00000	−0.406164	−0.203082	−	0.979162i	$-0.565096\pi$
$491$	−9.00000	−0.406164	−0.203082	+	0.979162i	$0.565096\pi$
$492$	0	0
$493$	18.0000	0.810679
$494$	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	$-0.266801\pi$
$503$	30.0000	1.33763	0.668817	+	0.743427i	$0.266801\pi$
$504$	0	0
$505$	36.0000i	1.60198i
$506$	0	0
$507$	0	0
$508$	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
$513$	10.0000i	0.441511i
$514$	0	0
$515$	− 12.0000i	− 0.528783i
$516$	0	0
$517$	−18.0000	−0.791639
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−9.00000	−0.394297	−0.197149	−	0.980374i	$-0.563168\pi$
$521$	−9.00000	−0.394297	−0.197149	+	0.980374i	$0.563168\pi$
$522$	0	0
$523$	−20.0000	−0.874539	−0.437269	−	0.899331i	$-0.644054\pi$
$523$	−20.0000	−0.874539	−0.437269	+	0.899331i	$0.644054\pi$
$524$	0	0
$525$	4.00000i	0.174574i
$526$	0	0
$527$	− 12.0000i	− 0.522728i
$528$	0	0
$529$	−23.0000	−1.00000
$530$	0	0
$531$	− 12.0000i	− 0.520756i
$532$	0	0
$533$	0	0
$534$	0	0
$535$	− 36.0000i	− 1.55642i
$536$	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	$0.0759884\pi$
$541$	11.0000i	0.472927i	−0.971640	+	0.236463i	$0.924012\pi$
$542$	0	0
$543$	20.0000	0.858282
$544$	0	0
$545$	−21.0000	−0.899541
$546$	0	0
$547$	−17.0000	−0.726868	−0.363434	−	0.931620i	$-0.618396\pi$
$547$	−17.0000	−0.726868	−0.363434	+	0.931620i	$0.618396\pi$
$548$	0	0
$549$	−16.0000	−0.682863
$550$	0	0
$551$	12.0000i	0.511217i
$552$	0	0
$553$	− 8.00000i	− 0.340195i
$554$	0	0
$555$	−21.0000	−0.891400
$556$	0	0
$557$	3.00000i	0.127114i	0.997978	+	0.0635570i	$0.0202445\pi$
$557$	3.00000i	0.127114i	−0.997978	+	0.0635570i	$0.979756\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	18.0000i	0.759961i
$562$	0	0
$563$	39.0000	1.64365	0.821827	−	0.569737i	$-0.192955\pi$
$563$	39.0000	1.64365	0.821827	+	0.569737i	$0.192955\pi$
$564$	0	0
$565$	− 18.0000i	− 0.757266i
$566$	0	0
$567$	1.00000i	0.0419961i
$568$	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
$571$	5.00000	0.209243	0.104622	−	0.994512i	$-0.466637\pi$
$571$	5.00000	0.209243	0.104622	+	0.994512i	$0.466637\pi$
$572$	0	0
$573$	−18.0000	−0.751961
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 38.0000i	− 1.58196i	−0.611842	−	0.790980i	$-0.709571\pi$
$577$	− 38.0000i	− 1.58196i	0.611842	−	0.790980i	$-0.290429\pi$
$578$	0	0
$579$	4.00000i	0.166234i
$580$	0	0
$581$	−12.0000	−0.497844
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	24.0000i	0.990586i	0.868726	+	0.495293i	$0.164939\pi$
$587$	24.0000i	0.990586i	−0.868726	+	0.495293i	$0.835061\pi$
$588$	0	0
$589$	8.00000	0.329634
$590$	0	0
$591$	3.00000i	0.123404i
$592$	0	0
$593$	18.0000i	0.739171i	0.929197	+	0.369586i	$0.120500\pi$
$593$	18.0000i	0.739171i	−0.929197	+	0.369586i	$0.879500\pi$
$594$	0	0
$595$	−9.00000	−0.368964
$596$	0	0
$597$	−2.00000	−0.0818546
$598$	0	0
$599$	−6.00000	−0.245153	−0.122577	−	0.992459i	$-0.539116\pi$
$599$	−6.00000	−0.245153	−0.122577	+	0.992459i	$0.539116\pi$
$600$	0	0
$601$	−19.0000	−0.775026	−0.387513	−	0.921864i	$-0.626666\pi$
$601$	−19.0000	−0.775026	−0.387513	+	0.921864i	$0.626666\pi$
$602$	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
$609$	− 6.00000i	− 0.243132i
$610$	0	0
$611$	0	0
$612$	0	0
$613$	− 38.0000i	− 1.53481i	−0.641165	−	0.767403i	$-0.721549\pi$
$613$	− 38.0000i	− 1.53481i	0.641165	−	0.767403i	$-0.278451\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	24.0000i	0.966204i	0.875564	+	0.483102i	$0.160490\pi$
$617$	24.0000i	0.966204i	−0.875564	+	0.483102i	$0.839510\pi$
$618$	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
$627$	−12.0000	−0.479234
$628$	0	0
$629$	− 21.0000i	− 0.837325i
$630$	0	0
$631$	− 29.0000i	− 1.15447i	−0.816577	−	0.577236i	$-0.804131\pi$
$631$	− 29.0000i	− 1.15447i	0.816577	−	0.577236i	$-0.195869\pi$
$632$	0	0
$633$	−13.0000	−0.516704
$634$	0	0
$635$	60.0000i	2.38103i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	6.00000i	0.237356i
$640$	0	0
$641$	18.0000	0.710957	0.355479	−	0.934684i	$-0.384318\pi$
$641$	18.0000	0.710957	0.355479	+	0.934684i	$0.384318\pi$
$642$	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
$651$	−4.00000	−0.156772
$652$	0	0
$653$	0	0		−	1.00000i	$-0.5\pi$
$653$	0	0			1.00000i	$0.5\pi$
$654$	0	0
$655$	63.0000i	2.46161i
$656$	0	0
$657$	− 4.00000i	− 0.156055i
$658$	0	0
$659$	36.0000	1.40236	0.701180	−	0.712984i	$-0.252657\pi$
$659$	36.0000	1.40236	0.701180	+	0.712984i	$0.252657\pi$
$660$	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
$663$	0	0
$664$	0	0
$665$	− 6.00000i	− 0.232670i
$666$	0	0
$667$	0	0
$668$	0	0
$669$	19.0000i	0.734582i
$670$	0	0
$671$	− 48.0000i	− 1.85302i
$672$	0	0
$673$	19.0000	0.732396	0.366198	−	0.930537i	$-0.380659\pi$
$673$	19.0000	0.732396	0.366198	+	0.930537i	$0.380659\pi$
$674$	0	0
$675$	−20.0000	−0.769800
$676$	0	0
$677$	48.0000	1.84479	0.922395	−	0.386248i	$-0.126229\pi$
$677$	48.0000	1.84479	0.922395	+	0.386248i	$0.126229\pi$
$678$	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
$687$	13.0000i	0.495981i
$688$	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
$693$	−12.0000	−0.455842
$694$	0	0
$695$	39.0000i	1.47935i
$696$	0	0
$697$	0	0
$698$	0	0
$699$	−27.0000	−1.02123
$700$	0	0
$701$	0	0		−	1.00000i	$-0.5\pi$
$701$	0	0			1.00000i	$0.5\pi$
$702$	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
$715$	0	0
$716$	0	0
$717$	− 15.0000i	− 0.560185i
$718$	0	0
$719$	6.00000	0.223762	0.111881	−	0.993722i	$-0.464312\pi$
$719$	6.00000	0.223762	0.111881	+	0.993722i	$0.464312\pi$
$720$	0	0
$721$	− 4.00000i	− 0.148968i
$722$	0	0
$723$	10.0000i	0.371904i
$724$	0	0
$725$	−24.0000	−0.891338
$726$	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
$731$	−3.00000	−0.110959
$732$	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
$735$	− 18.0000i	− 0.663940i
$736$	0	0
$737$	84.0000	3.09418
$738$	0	0
$739$	− 20.0000i	− 0.735712i	−0.929883	−	0.367856i	$-0.880092\pi$
$739$	− 20.0000i	− 0.735712i	0.929883	−	0.367856i	$-0.119908\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 9.00000i	− 0.330178i	−0.986279	−	0.165089i	$-0.947209\pi$
$743$	− 9.00000i	− 0.330178i	0.986279	−	0.165089i	$-0.0527911\pi$
$744$	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	$-0.760392\pi$
$751$	−40.0000	−1.45962	−0.729810	+	0.683650i	$0.760392\pi$
$752$	0	0
$753$	−24.0000	−0.874609
$754$	0	0
$755$	51.0000	1.85608
$756$	0	0
$757$	−16.0000	−0.581530	−0.290765	−	0.956795i	$-0.593910\pi$
$757$	−16.0000	−0.581530	−0.290765	+	0.956795i	$0.593910\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 6.00000i	− 0.217500i	−0.994069	−	0.108750i	$-0.965315\pi$
$761$	− 6.00000i	− 0.217500i	0.994069	−	0.108750i	$-0.0346848\pi$
$762$	0	0
$763$	−7.00000	−0.253417
$764$	0	0
$765$	− 18.0000i	− 0.650791i
$766$	0	0
$767$	0	0
$768$	0	0
$769$	− 32.0000i	− 1.15395i	−0.816762	−	0.576975i	$-0.804233\pi$
$769$	− 32.0000i	− 1.15395i	0.816762	−	0.576975i	$-0.195767\pi$
$770$	0	0
$771$	9.00000	0.324127
$772$	0	0
$773$	39.0000i	1.40273i	0.712801	+	0.701366i	$0.247426\pi$
$773$	39.0000i	1.40273i	−0.712801	+	0.701366i	$0.752574\pi$
$774$	0	0
$775$	16.0000i	0.574737i
$776$	0	0
$777$	−7.00000	−0.251124
$778$	0	0
$779$	0	0
$780$	0	0
$781$	−18.0000	−0.644091
$782$	0	0
$783$	30.0000	1.07211
$784$	0	0
$785$	42.0000i	1.49904i
$786$	0	0
$787$	40.0000i	1.42585i	0.701242	+	0.712923i	$0.252629\pi$
$787$	40.0000i	1.42585i	−0.701242	+	0.712923i	$0.747371\pi$
$788$	0	0
$789$	−12.0000	−0.427211
$790$	0	0
$791$	− 6.00000i	− 0.213335i
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	42.0000	1.48772	0.743858	−	0.668338i	$-0.232994\pi$
$797$	42.0000	1.48772	0.743858	+	0.668338i	$0.232994\pi$
$798$	0	0
$799$	− 9.00000i	− 0.318397i
$800$	0	0
$801$	12.0000i	0.423999i
$802$	0	0
$803$	12.0000	0.423471
$804$	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	$-0.696990\pi$
$809$	−33.0000	−1.16022	−0.580109	+	0.814539i	$0.696990\pi$
$810$	0	0
$811$	20.0000i	0.702295i	0.936320	+	0.351147i	$0.114208\pi$
$811$	20.0000i	0.702295i	−0.936320	+	0.351147i	$0.885792\pi$
$812$	0	0
$813$	11.0000i	0.385787i
$814$	0	0
$815$	−48.0000	−1.68137
$816$	0	0
$817$	− 2.00000i	− 0.0699711i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	3.00000i	0.104701i	0.998629	+	0.0523504i	$0.0166713\pi$
$821$	3.00000i	0.104701i	−0.998629	+	0.0523504i	$0.983329\pi$
$822$	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
$825$	− 24.0000i	− 0.835573i
$826$	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
$829$	−38.0000	−1.31979	−0.659897	−	0.751356i	$-0.729400\pi$
$829$	−38.0000	−1.31979	−0.659897	+	0.751356i	$0.729400\pi$
$830$	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	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	0	0
$843$	6.00000i	0.206651i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 25.0000i	− 0.859010i
$848$	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
$859$	4.00000	0.136478	0.0682391	−	0.997669i	$-0.478262\pi$
$859$	4.00000	0.136478	0.0682391	+	0.997669i	$0.478262\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 45.0000i	− 1.53182i	−0.642949	−	0.765909i	$-0.722289\pi$
$863$	− 45.0000i	− 1.53182i	0.642949	−	0.765909i	$-0.277711\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	8.00000	0.271694
$868$	0	0
$869$	48.0000i	1.62829i
$870$	0	0
$871$	0	0
$872$	0	0
$873$	− 20.0000i	− 0.676897i
$874$	0	0
$875$	−3.00000	−0.101419
$876$	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
$879$	− 21.0000i	− 0.708312i
$880$	0	0
$881$	−21.0000	−0.707508	−0.353754	−	0.935339i	$-0.615095\pi$
$881$	−21.0000	−0.707508	−0.353754	+	0.935339i	$0.615095\pi$
$882$	0	0
$883$	29.0000	0.975928	0.487964	−	0.872864i	$-0.337740\pi$
$883$	29.0000	0.975928	0.487964	+	0.872864i	$0.337740\pi$
$884$	0	0
$885$	18.0000	0.605063
$886$	0	0
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	0	0
$889$	20.0000i	0.670778i
$890$	0	0
$891$	− 6.00000i	− 0.201008i
$892$	0	0
$893$	6.00000	0.200782
$894$	0	0
$895$	9.00000i	0.300837i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 24.0000i	− 0.800445i
$900$	0	0
$901$	0	0
$902$	0	0
$903$	1.00000i	0.0332779i
$904$	0	0
$905$	− 60.0000i	− 1.99447i
$906$	0	0
$907$	−37.0000	−1.22856	−0.614282	−	0.789086i	$-0.710554\pi$
$907$	−37.0000	−1.22856	−0.614282	+	0.789086i	$0.710554\pi$
$908$	0	0
$909$	−24.0000	−0.796030
$910$	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	$-0.414993\pi$
$919$	16.0000	0.527791	0.263896	+	0.964551i	$0.414993\pi$
$920$	0	0
$921$	2.00000i	0.0659022i
$922$	0	0
$923$	0	0
$924$	0	0
$925$	28.0000i	0.920634i
$926$	0	0
$927$	8.00000	0.262754
$928$	0	0
$929$	− 36.0000i	− 1.18112i	−0.806993	−	0.590561i	$-0.798907\pi$
$929$	− 36.0000i	− 1.18112i	0.806993	−	0.590561i	$-0.201093\pi$
$930$	0	0
$931$	12.0000i	0.393284i
$932$	0	0
$933$	30.0000	0.982156
$934$	0	0
$935$	54.0000	1.76599
$936$	0	0
$937$	−34.0000	−1.11073	−0.555366	−	0.831606i	$-0.687422\pi$
$937$	−34.0000	−1.11073	−0.555366	+	0.831606i	$0.687422\pi$
$938$	0	0
$939$	1.00000	0.0326338
$940$	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	$-0.968920\pi$
$947$	− 6.00000i	− 0.194974i	0.995237	−	0.0974869i	$-0.0310804\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	− 6.00000i	− 0.194563i
$952$	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
$963$	24.0000	0.773389
$964$	0	0
$965$	12.0000	0.386294
$966$	0	0
$967$	− 31.0000i	− 0.996893i	−0.866921	−	0.498446i	$-0.833904\pi$
$967$	− 31.0000i	− 0.996893i	0.866921	−	0.498446i	$-0.166096\pi$
$968$	0	0
$969$	− 6.00000i	− 0.192748i
$970$	0	0
$971$	3.00000	0.0962746	0.0481373	−	0.998841i	$-0.484672\pi$
$971$	3.00000	0.0962746	0.0481373	+	0.998841i	$0.484672\pi$
$972$	0	0
$973$	13.0000i	0.416761i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	54.0000i	1.72761i	0.503824	+	0.863807i	$0.331926\pi$
$977$	54.0000i	1.72761i	−0.503824	+	0.863807i	$0.668074\pi$
$978$	0	0
$979$	−36.0000	−1.15056
$980$	0	0
$981$	− 14.0000i	− 0.446986i
$982$	0	0
$983$	− 39.0000i	− 1.24391i	−0.783054	−	0.621953i	$-0.786339\pi$
$983$	− 39.0000i	− 1.24391i	0.783054	−	0.621953i	$-0.213661\pi$
$984$	0	0
$985$	9.00000	0.286764
$986$	0	0
$987$	−3.00000	−0.0954911
$988$	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
$993$	− 8.00000i	− 0.253872i
$994$	0	0
$995$	6.00000i	0.190213i
$996$	0	0
$997$	−46.0000	−1.45683	−0.728417	−	0.685134i	$-0.759744\pi$
$997$	−46.0000	−1.45683	−0.728417	+	0.685134i	$0.759744\pi$
$998$	0	0
$999$	− 35.0000i	− 1.10735i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2704.2.f.d.337.2		2
4.3	odd	2		338.2.b.c.337.2		2
12.11	even	2		3042.2.b.a.1351.1		2
13.5	odd	4		2704.2.a.f.1.1		1
13.8	odd	4		208.2.a.a.1.1		1
13.12	even	2	inner	2704.2.f.d.337.1		2
39.8	even	4		1872.2.a.q.1.1		1
52.3	odd	6		338.2.e.a.147.1		4
52.7	even	12		338.2.c.d.315.1		2
52.11	even	12		338.2.c.d.191.1		2
52.15	even	12		338.2.c.a.191.1		2
52.19	even	12		338.2.c.a.315.1		2
52.23	odd	6		338.2.e.a.147.2		4
52.31	even	4		338.2.a.f.1.1		1
52.35	odd	6		338.2.e.a.23.2		4
52.43	odd	6		338.2.e.a.23.1		4
52.47	even	4		26.2.a.a.1.1	✓	1
52.51	odd	2		338.2.b.c.337.1		2
65.34	odd	4		5200.2.a.x.1.1		1
104.21	odd	4		832.2.a.i.1.1		1
104.99	even	4		832.2.a.d.1.1		1
156.47	odd	4		234.2.a.e.1.1		1
156.83	odd	4		3042.2.a.a.1.1		1
156.155	even	2		3042.2.b.a.1351.2		2
208.21	odd	4		3328.2.b.j.1665.2		2
208.99	even	4		3328.2.b.m.1665.2		2
208.125	odd	4		3328.2.b.j.1665.1		2
208.203	even	4		3328.2.b.m.1665.1		2
260.47	odd	4		650.2.b.d.599.1		2
260.99	even	4		650.2.a.j.1.1		1
260.203	odd	4		650.2.b.d.599.2		2
260.239	even	4		8450.2.a.c.1.1		1
312.125	even	4		7488.2.a.h.1.1		1
312.203	odd	4		7488.2.a.g.1.1		1
364.47	odd	12		1274.2.f.r.1145.1		2
364.151	even	12		1274.2.f.p.79.1		2
364.255	odd	12		1274.2.f.r.79.1		2
364.307	odd	4		1274.2.a.d.1.1		1
364.359	even	12		1274.2.f.p.1145.1		2
468.47	odd	12		2106.2.e.b.1405.1		2
468.151	even	12		2106.2.e.ba.1405.1		2
468.203	odd	12		2106.2.e.b.703.1		2
468.463	even	12		2106.2.e.ba.703.1		2
572.307	odd	4		3146.2.a.n.1.1		1
780.47	even	4		5850.2.e.a.5149.2		2
780.203	even	4		5850.2.e.a.5149.1		2
780.359	odd	4		5850.2.a.p.1.1		1
884.203	even	4		7514.2.a.c.1.1		1
988.151	odd	4		9386.2.a.j.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
26.2.a.a.1.1	✓	1	52.47	even	4
208.2.a.a.1.1		1	13.8	odd	4
234.2.a.e.1.1		1	156.47	odd	4
338.2.a.f.1.1		1	52.31	even	4
338.2.b.c.337.1		2	52.51	odd	2
338.2.b.c.337.2		2	4.3	odd	2
338.2.c.a.191.1		2	52.15	even	12
338.2.c.a.315.1		2	52.19	even	12
338.2.c.d.191.1		2	52.11	even	12
338.2.c.d.315.1		2	52.7	even	12
338.2.e.a.23.1		4	52.43	odd	6
338.2.e.a.23.2		4	52.35	odd	6
338.2.e.a.147.1		4	52.3	odd	6
338.2.e.a.147.2		4	52.23	odd	6
650.2.a.j.1.1		1	260.99	even	4
650.2.b.d.599.1		2	260.47	odd	4
650.2.b.d.599.2		2	260.203	odd	4
832.2.a.d.1.1		1	104.99	even	4
832.2.a.i.1.1		1	104.21	odd	4
1274.2.a.d.1.1		1	364.307	odd	4
1274.2.f.p.79.1		2	364.151	even	12
1274.2.f.p.1145.1		2	364.359	even	12
1274.2.f.r.79.1		2	364.255	odd	12
1274.2.f.r.1145.1		2	364.47	odd	12
1872.2.a.q.1.1		1	39.8	even	4
2106.2.e.b.703.1		2	468.203	odd	12
2106.2.e.b.1405.1		2	468.47	odd	12
2106.2.e.ba.703.1		2	468.463	even	12
2106.2.e.ba.1405.1		2	468.151	even	12
2704.2.a.f.1.1		1	13.5	odd	4
2704.2.f.d.337.1		2	13.12	even	2	inner
2704.2.f.d.337.2		2	1.1	even	1	trivial
3042.2.a.a.1.1		1	156.83	odd	4
3042.2.b.a.1351.1		2	12.11	even	2
3042.2.b.a.1351.2		2	156.155	even	2
3146.2.a.n.1.1		1	572.307	odd	4
3328.2.b.j.1665.1		2	208.125	odd	4
3328.2.b.j.1665.2		2	208.21	odd	4
3328.2.b.m.1665.1		2	208.203	even	4
3328.2.b.m.1665.2		2	208.99	even	4
5200.2.a.x.1.1		1	65.34	odd	4
5850.2.a.p.1.1		1	780.359	odd	4
5850.2.e.a.5149.1		2	780.203	even	4
5850.2.e.a.5149.2		2	780.47	even	4
7488.2.a.g.1.1		1	312.203	odd	4
7488.2.a.h.1.1		1	312.125	even	4
7514.2.a.c.1.1		1	884.203	even	4
8450.2.a.c.1.1		1	260.239	even	4
9386.2.a.j.1.1		1	988.151	odd	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:	\( N \)	\(=\)	\( 2704 = 2^{4} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2704.f (of order \(2\), degree \(1\), not minimal)

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

Introduction

L-functions

Modular forms

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