LMFDB - Embedded newform 2704.2.f.f.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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 52)
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.f.337.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000i q^{5} -2.00000i q^{7} -3.00000 q^{9} +O(q^{10})$	$q+2.00000i q^{5} -2.00000i q^{7} -3.00000 q^{9} -2.00000i q^{11} -6.00000 q^{17} +6.00000i q^{19} +8.00000 q^{23} +1.00000 q^{25} +2.00000 q^{29} -10.0000i q^{31} +4.00000 q^{35} +6.00000i q^{37} -6.00000i q^{41} +4.00000 q^{43} -6.00000i q^{45} -2.00000i q^{47} +3.00000 q^{49} +6.00000 q^{53} +4.00000 q^{55} -10.0000i q^{59} -2.00000 q^{61} +6.00000i q^{63} -10.0000i q^{67} -10.0000i q^{71} -2.00000i q^{73} -4.00000 q^{77} +4.00000 q^{79} +9.00000 q^{81} +6.00000i q^{83} -12.0000i q^{85} +6.00000i q^{89} -12.0000 q^{95} +2.00000i q^{97} +6.00000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 6 q^{9}+O(q^{10}) $ 2 * q - 6 * q^9	$ 2 q - 6 q^{9} - 12 q^{17} + 16 q^{23} + 2 q^{25} + 4 q^{29} + 8 q^{35} + 8 q^{43} + 6 q^{49} + 12 q^{53} + 8 q^{55} - 4 q^{61} - 8 q^{77} + 8 q^{79} + 18 q^{81} - 24 q^{95}+O(q^{100}) $ 2 * q - 6 * q^9 - 12 * q^17 + 16 * q^23 + 2 * q^25 + 4 * q^29 + 8 * q^35 + 8 * q^43 + 6 * q^49 + 12 * q^53 + 8 * q^55 - 4 * q^61 - 8 * q^77 + 8 * q^79 + 18 * q^81 - 24 * 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
$3$	0	0		−	1.00000i	$-0.5\pi$
$3$	0	0			1.00000i	$0.5\pi$
$4$	0	0
$5$	2.00000i	0.894427i	0.894427	+	0.447214i	$0.147584\pi$
$5$	2.00000i	0.894427i	−0.894427	+	0.447214i	$0.852416\pi$
$6$	0	0
$7$	− 2.00000i	− 0.755929i	−0.925820	−	0.377964i	$-0.876624\pi$
$7$	− 2.00000i	− 0.755929i	0.925820	−	0.377964i	$-0.123376\pi$
$8$	0	0
$9$	−3.00000	−1.00000
$10$	0	0
$11$	− 2.00000i	− 0.603023i	−0.953463	−	0.301511i	$-0.902509\pi$
$11$	− 2.00000i	− 0.603023i	0.953463	−	0.301511i	$-0.0974911\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−6.00000	−1.45521	−0.727607	−	0.685994i	$-0.759367\pi$
$17$	−6.00000	−1.45521	−0.727607	+	0.685994i	$0.759367\pi$
$18$	0	0
$19$	6.00000i	1.37649i	0.725476	+	0.688247i	$0.241620\pi$
$19$	6.00000i	1.37649i	−0.725476	+	0.688247i	$0.758380\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	8.00000	1.66812	0.834058	−	0.551677i	$-0.186012\pi$
$23$	8.00000	1.66812	0.834058	+	0.551677i	$0.186012\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	0	0
$31$	− 10.0000i	− 1.79605i	−0.439941	−	0.898027i	$-0.645001\pi$
$31$	− 10.0000i	− 1.79605i	0.439941	−	0.898027i	$-0.354999\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	4.00000	0.676123
$36$	0	0
$37$	6.00000i	0.986394i	0.869918	+	0.493197i	$0.164172\pi$
$37$	6.00000i	0.986394i	−0.869918	+	0.493197i	$0.835828\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 6.00000i	− 0.937043i	−0.883452	−	0.468521i	$-0.844787\pi$
$41$	− 6.00000i	− 0.937043i	0.883452	−	0.468521i	$-0.155213\pi$
$42$	0	0
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	0	0
$45$	− 6.00000i	− 0.894427i
$46$	0	0
$47$	− 2.00000i	− 0.291730i	−0.989305	−	0.145865i	$-0.953403\pi$
$47$	− 2.00000i	− 0.291730i	0.989305	−	0.145865i	$-0.0465965\pi$
$48$	0	0
$49$	3.00000	0.428571
$50$	0	0
$51$	0	0
$52$	0	0
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	0	0
$55$	4.00000	0.539360
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 10.0000i	− 1.30189i	−0.759125	−	0.650945i	$-0.774373\pi$
$59$	− 10.0000i	− 1.30189i	0.759125	−	0.650945i	$-0.225627\pi$
$60$	0	0
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	0	0
$63$	6.00000i	0.755929i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 10.0000i	− 1.22169i	−0.791748	−	0.610847i	$-0.790829\pi$
$67$	− 10.0000i	− 1.22169i	0.791748	−	0.610847i	$-0.209171\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 10.0000i	− 1.18678i	−0.804914	−	0.593391i	$-0.797789\pi$
$71$	− 10.0000i	− 1.18678i	0.804914	−	0.593391i	$-0.202211\pi$
$72$	0	0
$73$	− 2.00000i	− 0.234082i	−0.993127	−	0.117041i	$-0.962659\pi$
$73$	− 2.00000i	− 0.234082i	0.993127	−	0.117041i	$-0.0373409\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−4.00000	−0.455842
$78$	0	0
$79$	4.00000	0.450035	0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000	0.450035	0.225018	+	0.974355i	$0.427756\pi$
$80$	0	0
$81$	9.00000	1.00000
$82$	0	0
$83$	6.00000i	0.658586i	0.944228	+	0.329293i	$0.106810\pi$
$83$	6.00000i	0.658586i	−0.944228	+	0.329293i	$0.893190\pi$
$84$	0	0
$85$	− 12.0000i	− 1.30158i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	6.00000i	0.635999i	0.948091	+	0.317999i	$0.103011\pi$
$89$	6.00000i	0.635999i	−0.948091	+	0.317999i	$0.896989\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−12.0000	−1.23117
$96$	0	0
$97$	2.00000i	0.203069i	0.994832	+	0.101535i	$0.0323753\pi$
$97$	2.00000i	0.203069i	−0.994832	+	0.101535i	$0.967625\pi$
$98$	0	0
$99$	6.00000i	0.603023i
$100$	0	0
$101$	2.00000	0.199007	0.0995037	−	0.995037i	$-0.468274\pi$
$101$	2.00000	0.199007	0.0995037	+	0.995037i	$0.468274\pi$
$102$	0	0
$103$	−8.00000	−0.788263	−0.394132	−	0.919054i	$-0.628955\pi$
$103$	−8.00000	−0.788263	−0.394132	+	0.919054i	$0.628955\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	16.0000	1.54678	0.773389	−	0.633932i	$-0.218560\pi$
$107$	16.0000	1.54678	0.773389	+	0.633932i	$0.218560\pi$
$108$	0	0
$109$	− 14.0000i	− 1.34096i	−0.741929	−	0.670478i	$-0.766089\pi$
$109$	− 14.0000i	− 1.34096i	0.741929	−	0.670478i	$-0.233911\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	14.0000	1.31701	0.658505	−	0.752577i	$-0.271189\pi$
$113$	14.0000	1.31701	0.658505	+	0.752577i	$0.271189\pi$
$114$	0	0
$115$	16.0000i	1.49201i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	12.0000i	1.10004i
$120$	0	0
$121$	7.00000	0.636364
$122$	0	0
$123$	0	0
$124$	0	0
$125$	12.0000i	1.07331i
$126$	0	0
$127$	−8.00000	−0.709885	−0.354943	−	0.934888i	$-0.615500\pi$
$127$	−8.00000	−0.709885	−0.354943	+	0.934888i	$0.615500\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	16.0000	1.39793	0.698963	−	0.715158i	$-0.253645\pi$
$131$	16.0000	1.39793	0.698963	+	0.715158i	$0.253645\pi$
$132$	0	0
$133$	12.0000	1.04053
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 18.0000i	− 1.53784i	−0.639343	−	0.768922i	$-0.720793\pi$
$137$	− 18.0000i	− 1.53784i	0.639343	−	0.768922i	$-0.279207\pi$
$138$	0	0
$139$	−16.0000	−1.35710	−0.678551	−	0.734553i	$-0.737392\pi$
$139$	−16.0000	−1.35710	−0.678551	+	0.734553i	$0.737392\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	4.00000i	0.332182i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	18.0000i	1.47462i	0.675556	+	0.737309i	$0.263904\pi$
$149$	18.0000i	1.47462i	−0.675556	+	0.737309i	$0.736096\pi$
$150$	0	0
$151$	6.00000i	0.488273i	0.969741	+	0.244137i	$0.0785045\pi$
$151$	6.00000i	0.488273i	−0.969741	+	0.244137i	$0.921495\pi$
$152$	0	0
$153$	18.0000	1.45521
$154$	0	0
$155$	20.0000	1.60644
$156$	0	0
$157$	2.00000	0.159617	0.0798087	−	0.996810i	$-0.474569\pi$
$157$	2.00000	0.159617	0.0798087	+	0.996810i	$0.474569\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 16.0000i	− 1.26098i
$162$	0	0
$163$	− 10.0000i	− 0.783260i	−0.920123	−	0.391630i	$-0.871911\pi$
$163$	− 10.0000i	− 0.783260i	0.920123	−	0.391630i	$-0.128089\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	6.00000i	0.464294i	0.972681	+	0.232147i	$0.0745750\pi$
$167$	6.00000i	0.464294i	−0.972681	+	0.232147i	$0.925425\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	− 18.0000i	− 1.37649i
$172$	0	0
$173$	10.0000	0.760286	0.380143	−	0.924928i	$-0.375875\pi$
$173$	10.0000	0.760286	0.380143	+	0.924928i	$0.375875\pi$
$174$	0	0
$175$	− 2.00000i	− 0.151186i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	0	0
$181$	6.00000	0.445976	0.222988	−	0.974821i	$-0.428419\pi$
$181$	6.00000	0.445976	0.222988	+	0.974821i	$0.428419\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−12.0000	−0.882258
$186$	0	0
$187$	12.0000i	0.877527i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−4.00000	−0.289430	−0.144715	−	0.989473i	$-0.546227\pi$
$191$	−4.00000	−0.289430	−0.144715	+	0.989473i	$0.546227\pi$
$192$	0	0
$193$	− 2.00000i	− 0.143963i	−0.997406	−	0.0719816i	$-0.977068\pi$
$193$	− 2.00000i	− 0.143963i	0.997406	−	0.0719816i	$-0.0229323\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 6.00000i	− 0.427482i	−0.976890	−	0.213741i	$-0.931435\pi$
$197$	− 6.00000i	− 0.427482i	0.976890	−	0.213741i	$-0.0685649\pi$
$198$	0	0
$199$	−16.0000	−1.13421	−0.567105	−	0.823646i	$-0.691937\pi$
$199$	−16.0000	−1.13421	−0.567105	+	0.823646i	$0.691937\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 4.00000i	− 0.280745i
$204$	0	0
$205$	12.0000	0.838116
$206$	0	0
$207$	−24.0000	−1.66812
$208$	0	0
$209$	12.0000	0.830057
$210$	0	0
$211$	−8.00000	−0.550743	−0.275371	−	0.961338i	$-0.588801\pi$
$211$	−8.00000	−0.550743	−0.275371	+	0.961338i	$0.588801\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	8.00000i	0.545595i
$216$	0	0
$217$	−20.0000	−1.35769
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	6.00000i	0.401790i	0.979613	+	0.200895i	$0.0643850\pi$
$223$	6.00000i	0.401790i	−0.979613	+	0.200895i	$0.935615\pi$
$224$	0	0
$225$	−3.00000	−0.200000
$226$	0	0
$227$	− 18.0000i	− 1.19470i	−0.801980	−	0.597351i	$-0.796220\pi$
$227$	− 18.0000i	− 1.19470i	0.801980	−	0.597351i	$-0.203780\pi$
$228$	0	0
$229$	− 18.0000i	− 1.18947i	−0.803921	−	0.594737i	$-0.797256\pi$
$229$	− 18.0000i	− 1.18947i	0.803921	−	0.594737i	$-0.202744\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−10.0000	−0.655122	−0.327561	−	0.944830i	$-0.606227\pi$
$233$	−10.0000	−0.655122	−0.327561	+	0.944830i	$0.606227\pi$
$234$	0	0
$235$	4.00000	0.260931
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 18.0000i	− 1.16432i	−0.813073	−	0.582162i	$-0.802207\pi$
$239$	− 18.0000i	− 1.16432i	0.813073	−	0.582162i	$-0.197793\pi$
$240$	0	0
$241$	6.00000i	0.386494i	0.981150	+	0.193247i	$0.0619019\pi$
$241$	6.00000i	0.386494i	−0.981150	+	0.193247i	$0.938098\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	6.00000i	0.383326i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	12.0000	0.757433	0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000	0.757433	0.378717	+	0.925513i	$0.376365\pi$
$252$	0	0
$253$	− 16.0000i	− 1.00591i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−18.0000	−1.12281	−0.561405	−	0.827541i	$-0.689739\pi$
$257$	−18.0000	−1.12281	−0.561405	+	0.827541i	$0.689739\pi$
$258$	0	0
$259$	12.0000	0.745644
$260$	0	0
$261$	−6.00000	−0.371391
$262$	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
$265$	12.0000i	0.737154i
$266$	0	0
$267$	0	0
$268$	0	0
$269$	18.0000	1.09748	0.548740	−	0.835993i	$-0.315108\pi$
$269$	18.0000	1.09748	0.548740	+	0.835993i	$0.315108\pi$
$270$	0	0
$271$	− 2.00000i	− 0.121491i	−0.998153	−	0.0607457i	$-0.980652\pi$
$271$	− 2.00000i	− 0.121491i	0.998153	−	0.0607457i	$-0.0193479\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 2.00000i	− 0.120605i
$276$	0	0
$277$	−30.0000	−1.80253	−0.901263	−	0.433273i	$-0.857359\pi$
$277$	−30.0000	−1.80253	−0.901263	+	0.433273i	$0.857359\pi$
$278$	0	0
$279$	30.0000i	1.79605i
$280$	0	0
$281$	14.0000i	0.835170i	0.908638	+	0.417585i	$0.137123\pi$
$281$	14.0000i	0.835170i	−0.908638	+	0.417585i	$0.862877\pi$
$282$	0	0
$283$	4.00000	0.237775	0.118888	−	0.992908i	$-0.462067\pi$
$283$	4.00000	0.237775	0.118888	+	0.992908i	$0.462067\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−12.0000	−0.708338
$288$	0	0
$289$	19.0000	1.11765
$290$	0	0
$291$	0	0
$292$	0	0
$293$	14.0000i	0.817889i	0.912559	+	0.408944i	$0.134103\pi$
$293$	14.0000i	0.817889i	−0.912559	+	0.408944i	$0.865897\pi$
$294$	0	0
$295$	20.0000	1.16445
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	− 8.00000i	− 0.461112i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 4.00000i	− 0.229039i
$306$	0	0
$307$	− 34.0000i	− 1.94048i	−0.242140	−	0.970241i	$-0.577849\pi$
$307$	− 34.0000i	− 1.94048i	0.242140	−	0.970241i	$-0.422151\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	−10.0000	−0.565233	−0.282617	−	0.959233i	$-0.591202\pi$
$313$	−10.0000	−0.565233	−0.282617	+	0.959233i	$0.591202\pi$
$314$	0	0
$315$	−12.0000	−0.676123
$316$	0	0
$317$	18.0000i	1.01098i	0.862832	+	0.505490i	$0.168688\pi$
$317$	18.0000i	1.01098i	−0.862832	+	0.505490i	$0.831312\pi$
$318$	0	0
$319$	− 4.00000i	− 0.223957i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 36.0000i	− 2.00309i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−4.00000	−0.220527
$330$	0	0
$331$	22.0000i	1.20923i	0.796518	+	0.604615i	$0.206673\pi$
$331$	22.0000i	1.20923i	−0.796518	+	0.604615i	$0.793327\pi$
$332$	0	0
$333$	− 18.0000i	− 0.986394i
$334$	0	0
$335$	20.0000	1.09272
$336$	0	0
$337$	10.0000	0.544735	0.272367	−	0.962193i	$-0.412193\pi$
$337$	10.0000	0.544735	0.272367	+	0.962193i	$0.412193\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−20.0000	−1.08306
$342$	0	0
$343$	− 20.0000i	− 1.07990i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	24.0000	1.28839	0.644194	−	0.764862i	$-0.277193\pi$
$347$	24.0000	1.28839	0.644194	+	0.764862i	$0.277193\pi$
$348$	0	0
$349$	− 10.0000i	− 0.535288i	−0.963518	−	0.267644i	$-0.913755\pi$
$349$	− 10.0000i	− 0.535288i	0.963518	−	0.267644i	$-0.0862451\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 14.0000i	− 0.745145i	−0.928003	−	0.372572i	$-0.878476\pi$
$353$	− 14.0000i	− 0.745145i	0.928003	−	0.372572i	$-0.121524\pi$
$354$	0	0
$355$	20.0000	1.06149
$356$	0	0
$357$	0	0
$358$	0	0
$359$	30.0000i	1.58334i	0.610949	+	0.791670i	$0.290788\pi$
$359$	30.0000i	1.58334i	−0.610949	+	0.791670i	$0.709212\pi$
$360$	0	0
$361$	−17.0000	−0.894737
$362$	0	0
$363$	0	0
$364$	0	0
$365$	4.00000	0.209370
$366$	0	0
$367$	4.00000	0.208798	0.104399	−	0.994535i	$-0.466708\pi$
$367$	4.00000	0.208798	0.104399	+	0.994535i	$0.466708\pi$
$368$	0	0
$369$	18.0000i	0.937043i
$370$	0	0
$371$	− 12.0000i	− 0.623009i
$372$	0	0
$373$	−22.0000	−1.13912	−0.569558	−	0.821951i	$-0.692886\pi$
$373$	−22.0000	−1.13912	−0.569558	+	0.821951i	$0.692886\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	− 10.0000i	− 0.513665i	−0.966456	−	0.256833i	$-0.917321\pi$
$379$	− 10.0000i	− 0.513665i	0.966456	−	0.256833i	$-0.0826790\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 18.0000i	− 0.919757i	−0.887982	−	0.459879i	$-0.847893\pi$
$383$	− 18.0000i	− 0.919757i	0.887982	−	0.459879i	$-0.152107\pi$
$384$	0	0
$385$	− 8.00000i	− 0.407718i
$386$	0	0
$387$	−12.0000	−0.609994
$388$	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
$391$	−48.0000	−2.42746
$392$	0	0
$393$	0	0
$394$	0	0
$395$	8.00000i	0.402524i
$396$	0	0
$397$	− 2.00000i	− 0.100377i	−0.998740	−	0.0501886i	$-0.984018\pi$
$397$	− 2.00000i	− 0.100377i	0.998740	−	0.0501886i	$-0.0159822\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	6.00000i	0.299626i	0.988714	+	0.149813i	$0.0478671\pi$
$401$	6.00000i	0.299626i	−0.988714	+	0.149813i	$0.952133\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	18.0000i	0.894427i
$406$	0	0
$407$	12.0000	0.594818
$408$	0	0
$409$	− 14.0000i	− 0.692255i	−0.938187	−	0.346128i	$-0.887496\pi$
$409$	− 14.0000i	− 0.692255i	0.938187	−	0.346128i	$-0.112504\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−20.0000	−0.984136
$414$	0	0
$415$	−12.0000	−0.589057
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−24.0000	−1.17248	−0.586238	−	0.810139i	$-0.699392\pi$
$419$	−24.0000	−1.17248	−0.586238	+	0.810139i	$0.699392\pi$
$420$	0	0
$421$	10.0000i	0.487370i	0.969854	+	0.243685i	$0.0783563\pi$
$421$	10.0000i	0.487370i	−0.969854	+	0.243685i	$0.921644\pi$
$422$	0	0
$423$	6.00000i	0.291730i
$424$	0	0
$425$	−6.00000	−0.291043
$426$	0	0
$427$	4.00000i	0.193574i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	30.0000i	1.44505i	0.691345	+	0.722525i	$0.257018\pi$
$431$	30.0000i	1.44505i	−0.691345	+	0.722525i	$0.742982\pi$
$432$	0	0
$433$	14.0000	0.672797	0.336399	−	0.941720i	$-0.390791\pi$
$433$	14.0000	0.672797	0.336399	+	0.941720i	$0.390791\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	48.0000i	2.29615i
$438$	0	0
$439$	−16.0000	−0.763638	−0.381819	−	0.924237i	$-0.624702\pi$
$439$	−16.0000	−0.763638	−0.381819	+	0.924237i	$0.624702\pi$
$440$	0	0
$441$	−9.00000	−0.428571
$442$	0	0
$443$	24.0000	1.14027	0.570137	−	0.821549i	$-0.306890\pi$
$443$	24.0000	1.14027	0.570137	+	0.821549i	$0.306890\pi$
$444$	0	0
$445$	−12.0000	−0.568855
$446$	0	0
$447$	0	0
$448$	0	0
$449$	38.0000i	1.79333i	0.442709	+	0.896665i	$0.354018\pi$
$449$	38.0000i	1.79333i	−0.442709	+	0.896665i	$0.645982\pi$
$450$	0	0
$451$	−12.0000	−0.565058
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	34.0000i	1.59045i	0.606313	+	0.795226i	$0.292648\pi$
$457$	34.0000i	1.59045i	−0.606313	+	0.795226i	$0.707352\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 6.00000i	− 0.279448i	−0.990190	−	0.139724i	$-0.955378\pi$
$461$	− 6.00000i	− 0.279448i	0.990190	−	0.139724i	$-0.0446215\pi$
$462$	0	0
$463$	22.0000i	1.02243i	0.859454	+	0.511213i	$0.170804\pi$
$463$	22.0000i	1.02243i	−0.859454	+	0.511213i	$0.829196\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	20.0000	0.925490	0.462745	−	0.886492i	$-0.346865\pi$
$467$	20.0000	0.925490	0.462745	+	0.886492i	$0.346865\pi$
$468$	0	0
$469$	−20.0000	−0.923514
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 8.00000i	− 0.367840i
$474$	0	0
$475$	6.00000i	0.275299i
$476$	0	0
$477$	−18.0000	−0.824163
$478$	0	0
$479$	− 18.0000i	− 0.822441i	−0.911536	−	0.411220i	$-0.865103\pi$
$479$	− 18.0000i	− 0.822441i	0.911536	−	0.411220i	$-0.134897\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−4.00000	−0.181631
$486$	0	0
$487$	− 2.00000i	− 0.0906287i	−0.998973	−	0.0453143i	$-0.985571\pi$
$487$	− 2.00000i	− 0.0906287i	0.998973	−	0.0453143i	$-0.0144289\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−20.0000	−0.902587	−0.451294	−	0.892375i	$-0.649037\pi$
$491$	−20.0000	−0.902587	−0.451294	+	0.892375i	$0.649037\pi$
$492$	0	0
$493$	−12.0000	−0.540453
$494$	0	0
$495$	−12.0000	−0.539360
$496$	0	0
$497$	−20.0000	−0.897123
$498$	0	0
$499$	14.0000i	0.626726i	0.949633	+	0.313363i	$0.101456\pi$
$499$	14.0000i	0.626726i	−0.949633	+	0.313363i	$0.898544\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−28.0000	−1.24846	−0.624229	−	0.781241i	$-0.714587\pi$
$503$	−28.0000	−1.24846	−0.624229	+	0.781241i	$0.714587\pi$
$504$	0	0
$505$	4.00000i	0.177998i
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 14.0000i	− 0.620539i	−0.950649	−	0.310270i	$-0.899581\pi$
$509$	− 14.0000i	− 0.620539i	0.950649	−	0.310270i	$-0.100419\pi$
$510$	0	0
$511$	−4.00000	−0.176950
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 16.0000i	− 0.705044i
$516$	0	0
$517$	−4.00000	−0.175920
$518$	0	0
$519$	0	0
$520$	0	0
$521$	6.00000	0.262865	0.131432	−	0.991325i	$-0.458042\pi$
$521$	6.00000	0.262865	0.131432	+	0.991325i	$0.458042\pi$
$522$	0	0
$523$	−24.0000	−1.04945	−0.524723	−	0.851273i	$-0.675831\pi$
$523$	−24.0000	−1.04945	−0.524723	+	0.851273i	$0.675831\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	60.0000i	2.61364i
$528$	0	0
$529$	41.0000	1.78261
$530$	0	0
$531$	30.0000i	1.30189i
$532$	0	0
$533$	0	0
$534$	0	0
$535$	32.0000i	1.38348i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 6.00000i	− 0.258438i
$540$	0	0
$541$	− 26.0000i	− 1.11783i	−0.829226	−	0.558914i	$-0.811218\pi$
$541$	− 26.0000i	− 1.11783i	0.829226	−	0.558914i	$-0.188782\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	28.0000	1.19939
$546$	0	0
$547$	32.0000	1.36822	0.684111	−	0.729378i	$-0.260191\pi$
$547$	32.0000	1.36822	0.684111	+	0.729378i	$0.260191\pi$
$548$	0	0
$549$	6.00000	0.256074
$550$	0	0
$551$	12.0000i	0.511217i
$552$	0	0
$553$	− 8.00000i	− 0.340195i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 42.0000i	− 1.77960i	−0.456354	−	0.889799i	$-0.650845\pi$
$557$	− 42.0000i	− 1.77960i	0.456354	−	0.889799i	$-0.349155\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−4.00000	−0.168580	−0.0842900	−	0.996441i	$-0.526862\pi$
$563$	−4.00000	−0.168580	−0.0842900	+	0.996441i	$0.526862\pi$
$564$	0	0
$565$	28.0000i	1.17797i
$566$	0	0
$567$	− 18.0000i	− 0.755929i
$568$	0	0
$569$	−10.0000	−0.419222	−0.209611	−	0.977785i	$-0.567220\pi$
$569$	−10.0000	−0.419222	−0.209611	+	0.977785i	$0.567220\pi$
$570$	0	0
$571$	−12.0000	−0.502184	−0.251092	−	0.967963i	$-0.580790\pi$
$571$	−12.0000	−0.502184	−0.251092	+	0.967963i	$0.580790\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	8.00000	0.333623
$576$	0	0
$577$	− 6.00000i	− 0.249783i	−0.992170	−	0.124892i	$-0.960142\pi$
$577$	− 6.00000i	− 0.249783i	0.992170	−	0.124892i	$-0.0398583\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	12.0000	0.497844
$582$	0	0
$583$	− 12.0000i	− 0.496989i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 26.0000i	− 1.07313i	−0.843857	−	0.536567i	$-0.819721\pi$
$587$	− 26.0000i	− 1.07313i	0.843857	−	0.536567i	$-0.180279\pi$
$588$	0	0
$589$	60.0000	2.47226
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 26.0000i	− 1.06769i	−0.845582	−	0.533846i	$-0.820746\pi$
$593$	− 26.0000i	− 1.06769i	0.845582	−	0.533846i	$-0.179254\pi$
$594$	0	0
$595$	−24.0000	−0.983904
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−4.00000	−0.163436	−0.0817178	−	0.996656i	$-0.526041\pi$
$599$	−4.00000	−0.163436	−0.0817178	+	0.996656i	$0.526041\pi$
$600$	0	0
$601$	−26.0000	−1.06056	−0.530281	−	0.847822i	$-0.677914\pi$
$601$	−26.0000	−1.06056	−0.530281	+	0.847822i	$0.677914\pi$
$602$	0	0
$603$	30.0000i	1.22169i
$604$	0	0
$605$	14.0000i	0.569181i
$606$	0	0
$607$	12.0000	0.487065	0.243532	−	0.969893i	$-0.421694\pi$
$607$	12.0000	0.487065	0.243532	+	0.969893i	$0.421694\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	26.0000i	1.05013i	0.851062	+	0.525065i	$0.175959\pi$
$613$	26.0000i	1.05013i	−0.851062	+	0.525065i	$0.824041\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	34.0000i	1.36879i	0.729112	+	0.684394i	$0.239933\pi$
$617$	34.0000i	1.36879i	−0.729112	+	0.684394i	$0.760067\pi$
$618$	0	0
$619$	− 2.00000i	− 0.0803868i	−0.999192	−	0.0401934i	$-0.987203\pi$
$619$	− 2.00000i	− 0.0803868i	0.999192	−	0.0401934i	$-0.0127974\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	12.0000	0.480770
$624$	0	0
$625$	−19.0000	−0.760000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 36.0000i	− 1.43541i
$630$	0	0
$631$	22.0000i	0.875806i	0.899022	+	0.437903i	$0.144279\pi$
$631$	22.0000i	0.875806i	−0.899022	+	0.437903i	$0.855721\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 16.0000i	− 0.634941i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	30.0000i	1.18678i
$640$	0	0
$641$	26.0000	1.02694	0.513469	−	0.858108i	$-0.328360\pi$
$641$	26.0000	1.02694	0.513469	+	0.858108i	$0.328360\pi$
$642$	0	0
$643$	− 2.00000i	− 0.0788723i	−0.999222	−	0.0394362i	$-0.987444\pi$
$643$	− 2.00000i	− 0.0788723i	0.999222	−	0.0394362i	$-0.0125562\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−32.0000	−1.25805	−0.629025	−	0.777385i	$-0.716546\pi$
$647$	−32.0000	−1.25805	−0.629025	+	0.777385i	$0.716546\pi$
$648$	0	0
$649$	−20.0000	−0.785069
$650$	0	0
$651$	0	0
$652$	0	0
$653$	18.0000	0.704394	0.352197	−	0.935926i	$-0.385435\pi$
$653$	18.0000	0.704394	0.352197	+	0.935926i	$0.385435\pi$
$654$	0	0
$655$	32.0000i	1.25034i
$656$	0	0
$657$	6.00000i	0.234082i
$658$	0	0
$659$	0	0		−	1.00000i	$-0.5\pi$
$659$	0	0			1.00000i	$0.5\pi$
$660$	0	0
$661$	− 2.00000i	− 0.0777910i	−0.999243	−	0.0388955i	$-0.987616\pi$
$661$	− 2.00000i	− 0.0777910i	0.999243	−	0.0388955i	$-0.0123839\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	24.0000i	0.930680i
$666$	0	0
$667$	16.0000	0.619522
$668$	0	0
$669$	0	0
$670$	0	0
$671$	4.00000i	0.154418i
$672$	0	0
$673$	26.0000	1.00223	0.501113	−	0.865382i	$-0.332924\pi$
$673$	26.0000	1.00223	0.501113	+	0.865382i	$0.332924\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	6.00000	0.230599	0.115299	−	0.993331i	$-0.463217\pi$
$677$	6.00000	0.230599	0.115299	+	0.993331i	$0.463217\pi$
$678$	0	0
$679$	4.00000	0.153506
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 2.00000i	− 0.0765279i	−0.999268	−	0.0382639i	$-0.987817\pi$
$683$	− 2.00000i	− 0.0765279i	0.999268	−	0.0382639i	$-0.0121828\pi$
$684$	0	0
$685$	36.0000	1.37549
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	− 34.0000i	− 1.29342i	−0.762736	−	0.646710i	$-0.776144\pi$
$691$	− 34.0000i	− 1.29342i	0.762736	−	0.646710i	$-0.223856\pi$
$692$	0	0
$693$	12.0000	0.455842
$694$	0	0
$695$	− 32.0000i	− 1.21383i
$696$	0	0
$697$	36.0000i	1.36360i
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−18.0000	−0.679851	−0.339925	−	0.940452i	$-0.610402\pi$
$701$	−18.0000	−0.679851	−0.339925	+	0.940452i	$0.610402\pi$
$702$	0	0
$703$	−36.0000	−1.35777
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 4.00000i	− 0.150435i
$708$	0	0
$709$	22.0000i	0.826227i	0.910679	+	0.413114i	$0.135559\pi$
$709$	22.0000i	0.826227i	−0.910679	+	0.413114i	$0.864441\pi$
$710$	0	0
$711$	−12.0000	−0.450035
$712$	0	0
$713$	− 80.0000i	− 2.99602i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−40.0000	−1.49175	−0.745874	−	0.666087i	$-0.767968\pi$
$719$	−40.0000	−1.49175	−0.745874	+	0.666087i	$0.767968\pi$
$720$	0	0
$721$	16.0000i	0.595871i
$722$	0	0
$723$	0	0
$724$	0	0
$725$	2.00000	0.0742781
$726$	0	0
$727$	−32.0000	−1.18681	−0.593407	−	0.804902i	$-0.702218\pi$
$727$	−32.0000	−1.18681	−0.593407	+	0.804902i	$0.702218\pi$
$728$	0	0
$729$	−27.0000	−1.00000
$730$	0	0
$731$	−24.0000	−0.887672
$732$	0	0
$733$	26.0000i	0.960332i	0.877178	+	0.480166i	$0.159424\pi$
$733$	26.0000i	0.960332i	−0.877178	+	0.480166i	$0.840576\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−20.0000	−0.736709
$738$	0	0
$739$	6.00000i	0.220714i	0.993892	+	0.110357i	$0.0351994\pi$
$739$	6.00000i	0.220714i	−0.993892	+	0.110357i	$0.964801\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	14.0000i	0.513610i	0.966463	+	0.256805i	$0.0826698\pi$
$743$	14.0000i	0.513610i	−0.966463	+	0.256805i	$0.917330\pi$
$744$	0	0
$745$	−36.0000	−1.31894
$746$	0	0
$747$	− 18.0000i	− 0.658586i
$748$	0	0
$749$	− 32.0000i	− 1.16925i
$750$	0	0
$751$	0	0		−	1.00000i	$-0.5\pi$
$751$	0	0			1.00000i	$0.5\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−12.0000	−0.436725
$756$	0	0
$757$	−42.0000	−1.52652	−0.763258	−	0.646094i	$-0.776401\pi$
$757$	−42.0000	−1.52652	−0.763258	+	0.646094i	$0.776401\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 18.0000i	− 0.652499i	−0.945284	−	0.326250i	$-0.894215\pi$
$761$	− 18.0000i	− 0.652499i	0.945284	−	0.326250i	$-0.105785\pi$
$762$	0	0
$763$	−28.0000	−1.01367
$764$	0	0
$765$	36.0000i	1.30158i
$766$	0	0
$767$	0	0
$768$	0	0
$769$	− 54.0000i	− 1.94729i	−0.228069	−	0.973645i	$-0.573241\pi$
$769$	− 54.0000i	− 1.94729i	0.228069	−	0.973645i	$-0.426759\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 46.0000i	− 1.65451i	−0.561830	−	0.827253i	$-0.689903\pi$
$773$	− 46.0000i	− 1.65451i	0.561830	−	0.827253i	$-0.310097\pi$
$774$	0	0
$775$	− 10.0000i	− 0.359211i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	36.0000	1.28983
$780$	0	0
$781$	−20.0000	−0.715656
$782$	0	0
$783$	0	0
$784$	0	0
$785$	4.00000i	0.142766i
$786$	0	0
$787$	38.0000i	1.35455i	0.735728	+	0.677277i	$0.236840\pi$
$787$	38.0000i	1.35455i	−0.735728	+	0.677277i	$0.763160\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 28.0000i	− 0.995565i
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−18.0000	−0.637593	−0.318796	−	0.947823i	$-0.603279\pi$
$797$	−18.0000	−0.637593	−0.318796	+	0.947823i	$0.603279\pi$
$798$	0	0
$799$	12.0000i	0.424529i
$800$	0	0
$801$	− 18.0000i	− 0.635999i
$802$	0	0
$803$	−4.00000	−0.141157
$804$	0	0
$805$	32.0000	1.12785
$806$	0	0
$807$	0	0
$808$	0	0
$809$	6.00000	0.210949	0.105474	−	0.994422i	$-0.466364\pi$
$809$	6.00000	0.210949	0.105474	+	0.994422i	$0.466364\pi$
$810$	0	0
$811$	22.0000i	0.772524i	0.922389	+	0.386262i	$0.126234\pi$
$811$	22.0000i	0.772524i	−0.922389	+	0.386262i	$0.873766\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	20.0000	0.700569
$816$	0	0
$817$	24.0000i	0.839654i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	2.00000i	0.0698005i	0.999391	+	0.0349002i	$0.0111113\pi$
$821$	2.00000i	0.0698005i	−0.999391	+	0.0349002i	$0.988889\pi$
$822$	0	0
$823$	−24.0000	−0.836587	−0.418294	−	0.908312i	$-0.637372\pi$
$823$	−24.0000	−0.836587	−0.418294	+	0.908312i	$0.637372\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	6.00000i	0.208640i	0.994544	+	0.104320i	$0.0332667\pi$
$827$	6.00000i	0.208640i	−0.994544	+	0.104320i	$0.966733\pi$
$828$	0	0
$829$	−22.0000	−0.764092	−0.382046	−	0.924143i	$-0.624780\pi$
$829$	−22.0000	−0.764092	−0.382046	+	0.924143i	$0.624780\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−18.0000	−0.623663
$834$	0	0
$835$	−12.0000	−0.415277
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 10.0000i	− 0.345238i	−0.984989	−	0.172619i	$-0.944777\pi$
$839$	− 10.0000i	− 0.345238i	0.984989	−	0.172619i	$-0.0552230\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 14.0000i	− 0.481046i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	48.0000i	1.64542i
$852$	0	0
$853$	38.0000i	1.30110i	0.759465	+	0.650548i	$0.225461\pi$
$853$	38.0000i	1.30110i	−0.759465	+	0.650548i	$0.774539\pi$
$854$	0	0
$855$	36.0000	1.23117
$856$	0	0
$857$	38.0000	1.29806	0.649028	−	0.760765i	$-0.275176\pi$
$857$	38.0000	1.29806	0.649028	+	0.760765i	$0.275176\pi$
$858$	0	0
$859$	−8.00000	−0.272956	−0.136478	−	0.990643i	$-0.543578\pi$
$859$	−8.00000	−0.272956	−0.136478	+	0.990643i	$0.543578\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 34.0000i	− 1.15737i	−0.815550	−	0.578687i	$-0.803565\pi$
$863$	− 34.0000i	− 1.15737i	0.815550	−	0.578687i	$-0.196435\pi$
$864$	0	0
$865$	20.0000i	0.680020i
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 8.00000i	− 0.271381i
$870$	0	0
$871$	0	0
$872$	0	0
$873$	− 6.00000i	− 0.203069i
$874$	0	0
$875$	24.0000	0.811348
$876$	0	0
$877$	18.0000i	0.607817i	0.952701	+	0.303908i	$0.0982917\pi$
$877$	18.0000i	0.607817i	−0.952701	+	0.303908i	$0.901708\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−18.0000	−0.606435	−0.303218	−	0.952921i	$-0.598061\pi$
$881$	−18.0000	−0.606435	−0.303218	+	0.952921i	$0.598061\pi$
$882$	0	0
$883$	28.0000	0.942275	0.471138	−	0.882060i	$-0.343844\pi$
$883$	28.0000	0.942275	0.471138	+	0.882060i	$0.343844\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−4.00000	−0.134307	−0.0671534	−	0.997743i	$-0.521392\pi$
$887$	−4.00000	−0.134307	−0.0671534	+	0.997743i	$0.521392\pi$
$888$	0	0
$889$	16.0000i	0.536623i
$890$	0	0
$891$	− 18.0000i	− 0.603023i
$892$	0	0
$893$	12.0000	0.401565
$894$	0	0
$895$	24.0000i	0.802232i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 20.0000i	− 0.667037i
$900$	0	0
$901$	−36.0000	−1.19933
$902$	0	0
$903$	0	0
$904$	0	0
$905$	12.0000i	0.398893i
$906$	0	0
$907$	36.0000	1.19536	0.597680	−	0.801735i	$-0.296089\pi$
$907$	36.0000	1.19536	0.597680	+	0.801735i	$0.296089\pi$
$908$	0	0
$909$	−6.00000	−0.199007
$910$	0	0
$911$	36.0000	1.19273	0.596367	−	0.802712i	$-0.296610\pi$
$911$	36.0000	1.19273	0.596367	+	0.802712i	$0.296610\pi$
$912$	0	0
$913$	12.0000	0.397142
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 32.0000i	− 1.05673i
$918$	0	0
$919$	12.0000	0.395843	0.197922	−	0.980218i	$-0.436581\pi$
$919$	12.0000	0.395843	0.197922	+	0.980218i	$0.436581\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	6.00000i	0.197279i
$926$	0	0
$927$	24.0000	0.788263
$928$	0	0
$929$	− 54.0000i	− 1.77168i	−0.463988	−	0.885841i	$-0.653582\pi$
$929$	− 54.0000i	− 1.77168i	0.463988	−	0.885841i	$-0.346418\pi$
$930$	0	0
$931$	18.0000i	0.589926i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−24.0000	−0.784884
$936$	0	0
$937$	42.0000	1.37208	0.686040	−	0.727564i	$-0.259347\pi$
$937$	42.0000	1.37208	0.686040	+	0.727564i	$0.259347\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	10.0000i	0.325991i	0.986627	+	0.162995i	$0.0521156\pi$
$941$	10.0000i	0.325991i	−0.986627	+	0.162995i	$0.947884\pi$
$942$	0	0
$943$	− 48.0000i	− 1.56310i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 42.0000i	− 1.36482i	−0.730971	−	0.682408i	$-0.760933\pi$
$947$	− 42.0000i	− 1.36482i	0.730971	−	0.682408i	$-0.239067\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−26.0000	−0.842223	−0.421111	−	0.907009i	$-0.638360\pi$
$953$	−26.0000	−0.842223	−0.421111	+	0.907009i	$0.638360\pi$
$954$	0	0
$955$	− 8.00000i	− 0.258874i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−36.0000	−1.16250
$960$	0	0
$961$	−69.0000	−2.22581
$962$	0	0
$963$	−48.0000	−1.54678
$964$	0	0
$965$	4.00000	0.128765
$966$	0	0
$967$	− 50.0000i	− 1.60789i	−0.594703	−	0.803946i	$-0.702730\pi$
$967$	− 50.0000i	− 1.60789i	0.594703	−	0.803946i	$-0.297270\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	48.0000	1.54039	0.770197	−	0.637806i	$-0.220158\pi$
$971$	48.0000	1.54039	0.770197	+	0.637806i	$0.220158\pi$
$972$	0	0
$973$	32.0000i	1.02587i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	18.0000i	0.575871i	0.957650	+	0.287936i	$0.0929689\pi$
$977$	18.0000i	0.575871i	−0.957650	+	0.287936i	$0.907031\pi$
$978$	0	0
$979$	12.0000	0.383522
$980$	0	0
$981$	42.0000i	1.34096i
$982$	0	0
$983$	14.0000i	0.446531i	0.974758	+	0.223265i	$0.0716716\pi$
$983$	14.0000i	0.446531i	−0.974758	+	0.223265i	$0.928328\pi$
$984$	0	0
$985$	12.0000	0.382352
$986$	0	0
$987$	0	0
$988$	0	0
$989$	32.0000	1.01754
$990$	0	0
$991$	−12.0000	−0.381193	−0.190596	−	0.981669i	$-0.561042\pi$
$991$	−12.0000	−0.381193	−0.190596	+	0.981669i	$0.561042\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 32.0000i	− 1.01447i
$996$	0	0
$997$	−58.0000	−1.83688	−0.918439	−	0.395562i	$-0.870550\pi$
$997$	−58.0000	−1.83688	−0.918439	+	0.395562i	$0.870550\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2704.2.f.f.337.2		2
4.3	odd	2		676.2.d.c.337.2		2
12.11	even	2		6084.2.b.m.4393.1		2
13.5	odd	4		208.2.a.c.1.1		1
13.8	odd	4		2704.2.a.g.1.1		1
13.12	even	2	inner	2704.2.f.f.337.1		2
39.5	even	4		1872.2.a.f.1.1		1
52.3	odd	6		676.2.h.c.485.2		4
52.7	even	12		676.2.e.b.653.1		2
52.11	even	12		676.2.e.b.529.1		2
52.15	even	12		676.2.e.c.529.1		2
52.19	even	12		676.2.e.c.653.1		2
52.23	odd	6		676.2.h.c.485.1		4
52.31	even	4		52.2.a.a.1.1	✓	1
52.35	odd	6		676.2.h.c.361.2		4
52.43	odd	6		676.2.h.c.361.1		4
52.47	even	4		676.2.a.c.1.1		1
52.51	odd	2		676.2.d.c.337.1		2
65.44	odd	4		5200.2.a.q.1.1		1
104.5	odd	4		832.2.a.f.1.1		1
104.83	even	4		832.2.a.e.1.1		1
156.47	odd	4		6084.2.a.m.1.1		1
156.83	odd	4		468.2.a.b.1.1		1
156.155	even	2		6084.2.b.m.4393.2		2
208.5	odd	4		3328.2.b.e.1665.2		2
208.83	even	4		3328.2.b.q.1665.1		2
208.109	odd	4		3328.2.b.e.1665.1		2
208.187	even	4		3328.2.b.q.1665.2		2
260.83	odd	4		1300.2.c.c.1249.2		2
260.187	odd	4		1300.2.c.c.1249.1		2
260.239	even	4		1300.2.a.d.1.1		1
312.5	even	4		7488.2.a.bw.1.1		1
312.83	odd	4		7488.2.a.bn.1.1		1
364.31	odd	12		2548.2.j.f.1353.1		2
364.83	odd	4		2548.2.a.e.1.1		1
364.135	even	12		2548.2.j.e.1145.1		2
364.187	odd	12		2548.2.j.f.1145.1		2
364.291	even	12		2548.2.j.e.1353.1		2
468.31	even	12		4212.2.i.d.2809.1		2
468.83	odd	12		4212.2.i.i.1405.1		2
468.187	even	12		4212.2.i.d.1405.1		2
468.239	odd	12		4212.2.i.i.2809.1		2
572.395	odd	4		6292.2.a.g.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
52.2.a.a.1.1	✓	1	52.31	even	4
208.2.a.c.1.1		1	13.5	odd	4
468.2.a.b.1.1		1	156.83	odd	4
676.2.a.c.1.1		1	52.47	even	4
676.2.d.c.337.1		2	52.51	odd	2
676.2.d.c.337.2		2	4.3	odd	2
676.2.e.b.529.1		2	52.11	even	12
676.2.e.b.653.1		2	52.7	even	12
676.2.e.c.529.1		2	52.15	even	12
676.2.e.c.653.1		2	52.19	even	12
676.2.h.c.361.1		4	52.43	odd	6
676.2.h.c.361.2		4	52.35	odd	6
676.2.h.c.485.1		4	52.23	odd	6
676.2.h.c.485.2		4	52.3	odd	6
832.2.a.e.1.1		1	104.83	even	4
832.2.a.f.1.1		1	104.5	odd	4
1300.2.a.d.1.1		1	260.239	even	4
1300.2.c.c.1249.1		2	260.187	odd	4
1300.2.c.c.1249.2		2	260.83	odd	4
1872.2.a.f.1.1		1	39.5	even	4
2548.2.a.e.1.1		1	364.83	odd	4
2548.2.j.e.1145.1		2	364.135	even	12
2548.2.j.e.1353.1		2	364.291	even	12
2548.2.j.f.1145.1		2	364.187	odd	12
2548.2.j.f.1353.1		2	364.31	odd	12
2704.2.a.g.1.1		1	13.8	odd	4
2704.2.f.f.337.1		2	13.12	even	2	inner
2704.2.f.f.337.2		2	1.1	even	1	trivial
3328.2.b.e.1665.1		2	208.109	odd	4
3328.2.b.e.1665.2		2	208.5	odd	4
3328.2.b.q.1665.1		2	208.83	even	4
3328.2.b.q.1665.2		2	208.187	even	4
4212.2.i.d.1405.1		2	468.187	even	12
4212.2.i.d.2809.1		2	468.31	even	12
4212.2.i.i.1405.1		2	468.83	odd	12
4212.2.i.i.2809.1		2	468.239	odd	12
5200.2.a.q.1.1		1	65.44	odd	4
6084.2.a.m.1.1		1	156.47	odd	4
6084.2.b.m.4393.1		2	12.11	even	2
6084.2.b.m.4393.2		2	156.155	even	2
6292.2.a.g.1.1		1	572.395	odd	4
7488.2.a.bn.1.1		1	312.83	odd	4
7488.2.a.bw.1.1		1	312.5	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:	\( N \)	\(=\)	\( 2704 = 2^{4} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2704.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.00000i q^{5} -2.00000i q^{7} -3.00000 q^{9} +O(q^{10})\)	\(q+2.00000i q^{5} -2.00000i q^{7} -3.00000 q^{9} -2.00000i q^{11} -6.00000 q^{17} +6.00000i q^{19} +8.00000 q^{23} +1.00000 q^{25} +2.00000 q^{29} -10.0000i q^{31} +4.00000 q^{35} +6.00000i q^{37} -6.00000i q^{41} +4.00000 q^{43} -6.00000i q^{45} -2.00000i q^{47} +3.00000 q^{49} +6.00000 q^{53} +4.00000 q^{55} -10.0000i q^{59} -2.00000 q^{61} +6.00000i q^{63} -10.0000i q^{67} -10.0000i q^{71} -2.00000i q^{73} -4.00000 q^{77} +4.00000 q^{79} +9.00000 q^{81} +6.00000i q^{83} -12.0000i q^{85} +6.00000i q^{89} -12.0000 q^{95} +2.00000i q^{97} +6.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{9}+O(q^{10}) \) 2 * q - 6 * q^9	\( 2 q - 6 q^{9} - 12 q^{17} + 16 q^{23} + 2 q^{25} + 4 q^{29} + 8 q^{35} + 8 q^{43} + 6 q^{49} + 12 q^{53} + 8 q^{55} - 4 q^{61} - 8 q^{77} + 8 q^{79} + 18 q^{81} - 24 q^{95}+O(q^{100}) \) 2 * q - 6 * q^9 - 12 * q^17 + 16 * q^23 + 2 * q^25 + 4 * q^29 + 8 * q^35 + 8 * q^43 + 6 * q^49 + 12 * q^53 + 8 * q^55 - 4 * q^61 - 8 * q^77 + 8 * q^79 + 18 * q^81 - 24 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more