LMFDB - Embedded newform 7056.2.b.l.1567.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7056,2,Mod(1567,7056)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7056, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 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("7056.1567");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7056 = 2^{4} \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7056.b (of order $2$, degree $1$, 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:	$56.3424436662$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 336)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1567.2
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	7056.1567
Dual form			7056.2.b.l.1567.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.46410i q^{5} +O(q^{10})$	$q+3.46410i q^{5} -3.46410i q^{11} +5.19615i q^{13} -6.92820i q^{17} +7.00000 q^{19} -7.00000 q^{25} +5.00000 q^{31} +1.00000 q^{37} -10.3923i q^{41} +1.73205i q^{43} +6.00000 q^{47} +12.0000 q^{55} -18.0000 q^{65} -1.73205i q^{67} +3.46410i q^{71} -8.66025i q^{73} -15.5885i q^{79} +6.00000 q^{83} +24.0000 q^{85} -6.92820i q^{89} +24.2487i q^{95} -6.92820i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q+O(q^{10}) $ 2 * q	$ 2 q + 14 q^{19} - 14 q^{25} + 10 q^{31} + 2 q^{37} + 12 q^{47} + 24 q^{55} - 36 q^{65} + 12 q^{83} + 48 q^{85}+O(q^{100}) $ 2 * q + 14 * q^19 - 14 * q^25 + 10 * q^31 + 2 * q^37 + 12 * q^47 + 24 * q^55 - 36 * q^65 + 12 * q^83 + 48 * q^85

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/7056\mathbb{Z}\right)^\times$.

$n$	$785$	$1765$	$4609$	$6175$
$\chi(n)$	$1$	$1$	$-1$	$-1$

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
$3$	0	0
$4$	0	0
$5$	3.46410i	1.54919i	0.632456	+	0.774597i	$0.282047\pi$
$5$	3.46410i	1.54919i	−0.632456	+	0.774597i	$0.717953\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 3.46410i	− 1.04447i	−0.852803	−	0.522233i	$-0.825099\pi$
$11$	− 3.46410i	− 1.04447i	0.852803	−	0.522233i	$-0.174901\pi$
$12$	0	0
$13$	5.19615i	1.44115i	0.693375	+	0.720577i	$0.256123\pi$
$13$	5.19615i	1.44115i	−0.693375	+	0.720577i	$0.743877\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 6.92820i	− 1.68034i	−0.542326	−	0.840168i	$-0.682456\pi$
$17$	− 6.92820i	− 1.68034i	0.542326	−	0.840168i	$-0.317544\pi$
$18$	0	0
$19$	7.00000	1.60591	0.802955	−	0.596040i	$-0.203260\pi$
$19$	7.00000	1.60591	0.802955	+	0.596040i	$0.203260\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	0	0
$25$	−7.00000	−1.40000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	5.00000	0.898027	0.449013	−	0.893525i	$-0.351776\pi$
$31$	5.00000	0.898027	0.449013	+	0.893525i	$0.351776\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	1.00000	0.164399	0.0821995	−	0.996616i	$-0.473806\pi$
$37$	1.00000	0.164399	0.0821995	+	0.996616i	$0.473806\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 10.3923i	− 1.62301i	−0.584349	−	0.811503i	$-0.698650\pi$
$41$	− 10.3923i	− 1.62301i	0.584349	−	0.811503i	$-0.301350\pi$
$42$	0	0
$43$	1.73205i	0.264135i	0.991241	+	0.132068i	$0.0421616\pi$
$43$	1.73205i	0.264135i	−0.991241	+	0.132068i	$0.957838\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	6.00000	0.875190	0.437595	−	0.899172i	$-0.355830\pi$
$47$	6.00000	0.875190	0.437595	+	0.899172i	$0.355830\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	0	0
$52$	0	0
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	0	0
$55$	12.0000	1.61808
$56$	0	0
$57$	0	0
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−18.0000	−2.23263
$66$	0	0
$67$	− 1.73205i	− 0.211604i	−0.994387	−	0.105802i	$-0.966259\pi$
$67$	− 1.73205i	− 0.211604i	0.994387	−	0.105802i	$-0.0337409\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	3.46410i	0.411113i	0.978645	+	0.205557i	$0.0659005\pi$
$71$	3.46410i	0.411113i	−0.978645	+	0.205557i	$0.934100\pi$
$72$	0	0
$73$	− 8.66025i	− 1.01361i	−0.862062	−	0.506803i	$-0.830827\pi$
$73$	− 8.66025i	− 1.01361i	0.862062	−	0.506803i	$-0.169173\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	− 15.5885i	− 1.75384i	−0.480638	−	0.876919i	$-0.659595\pi$
$79$	− 15.5885i	− 1.75384i	0.480638	−	0.876919i	$-0.340405\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	6.00000	0.658586	0.329293	−	0.944228i	$-0.393190\pi$
$83$	6.00000	0.658586	0.329293	+	0.944228i	$0.393190\pi$
$84$	0	0
$85$	24.0000	2.60317
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 6.92820i	− 0.734388i	−0.930144	−	0.367194i	$-0.880318\pi$
$89$	− 6.92820i	− 0.734388i	0.930144	−	0.367194i	$-0.119682\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	24.2487i	2.48787i
$96$	0	0
$97$	− 6.92820i	− 0.703452i	−0.936103	−	0.351726i	$-0.885595\pi$
$97$	− 6.92820i	− 0.703452i	0.936103	−	0.351726i	$-0.114405\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 3.46410i	− 0.344691i	−0.985037	−	0.172345i	$-0.944865\pi$
$101$	− 3.46410i	− 0.344691i	0.985037	−	0.172345i	$-0.0551346\pi$
$102$	0	0
$103$	5.00000	0.492665	0.246332	−	0.969185i	$-0.420775\pi$
$103$	5.00000	0.492665	0.246332	+	0.969185i	$0.420775\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	6.92820i	0.669775i	0.942258	+	0.334887i	$0.108698\pi$
$107$	6.92820i	0.669775i	−0.942258	+	0.334887i	$0.891302\pi$
$108$	0	0
$109$	5.00000	0.478913	0.239457	−	0.970907i	$-0.423031\pi$
$109$	5.00000	0.478913	0.239457	+	0.970907i	$0.423031\pi$
$110$	0	0
$111$	0	0
$112$	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
$119$	0	0
$120$	0	0
$121$	−1.00000	−0.0909091
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 6.92820i	− 0.619677i
$126$	0	0
$127$	− 15.5885i	− 1.38325i	−0.722256	−	0.691626i	$-0.756895\pi$
$127$	− 15.5885i	− 1.38325i	0.722256	−	0.691626i	$-0.243105\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−18.0000	−1.57267	−0.786334	−	0.617802i	$-0.788023\pi$
$131$	−18.0000	−1.57267	−0.786334	+	0.617802i	$0.788023\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	12.0000	1.02523	0.512615	−	0.858619i	$-0.328677\pi$
$137$	12.0000	1.02523	0.512615	+	0.858619i	$0.328677\pi$
$138$	0	0
$139$	5.00000	0.424094	0.212047	−	0.977259i	$-0.431987\pi$
$139$	5.00000	0.424094	0.212047	+	0.977259i	$0.431987\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	18.0000	1.50524
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	12.0000	0.983078	0.491539	−	0.870855i	$-0.336434\pi$
$149$	12.0000	0.983078	0.491539	+	0.870855i	$0.336434\pi$
$150$	0	0
$151$	10.3923i	0.845714i	0.906196	+	0.422857i	$0.138973\pi$
$151$	10.3923i	0.845714i	−0.906196	+	0.422857i	$0.861027\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	17.3205i	1.39122i
$156$	0	0
$157$	13.8564i	1.10586i	0.833227	+	0.552931i	$0.186491\pi$
$157$	13.8564i	1.10586i	−0.833227	+	0.552931i	$0.813509\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	− 3.46410i	− 0.271329i	−0.990755	−	0.135665i	$-0.956683\pi$
$163$	− 3.46410i	− 0.271329i	0.990755	−	0.135665i	$-0.0433170\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−6.00000	−0.464294	−0.232147	−	0.972681i	$-0.574575\pi$
$167$	−6.00000	−0.464294	−0.232147	+	0.972681i	$0.574575\pi$
$168$	0	0
$169$	−14.0000	−1.07692
$170$	0	0
$171$	0	0
$172$	0	0
$173$	0	0	1.00000			$0$
$173$	0	0	−1.00000			$\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 17.3205i	− 1.29460i	−0.762237	−	0.647298i	$-0.775899\pi$
$179$	− 17.3205i	− 1.29460i	0.762237	−	0.647298i	$-0.224101\pi$
$180$	0	0
$181$	15.5885i	1.15868i	0.815086	+	0.579340i	$0.196690\pi$
$181$	15.5885i	1.15868i	−0.815086	+	0.579340i	$0.803310\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	3.46410i	0.254686i
$186$	0	0
$187$	−24.0000	−1.75505
$188$	0	0
$189$	0	0
$190$	0	0
$191$	3.46410i	0.250654i	0.992116	+	0.125327i	$0.0399979\pi$
$191$	3.46410i	0.250654i	−0.992116	+	0.125327i	$0.960002\pi$
$192$	0	0
$193$	13.0000	0.935760	0.467880	−	0.883792i	$-0.345018\pi$
$193$	13.0000	0.935760	0.467880	+	0.883792i	$0.345018\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−12.0000	−0.854965	−0.427482	−	0.904024i	$-0.640599\pi$
$197$	−12.0000	−0.854965	−0.427482	+	0.904024i	$0.640599\pi$
$198$	0	0
$199$	16.0000	1.13421	0.567105	−	0.823646i	$-0.308063\pi$
$199$	16.0000	1.13421	0.567105	+	0.823646i	$0.308063\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	36.0000	2.51435
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 24.2487i	− 1.67732i
$210$	0	0
$211$	17.3205i	1.19239i	0.802839	+	0.596196i	$0.203322\pi$
$211$	17.3205i	1.19239i	−0.802839	+	0.596196i	$0.796678\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−6.00000	−0.409197
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	36.0000	2.42162
$222$	0	0
$223$	−8.00000	−0.535720	−0.267860	−	0.963458i	$-0.586316\pi$
$223$	−8.00000	−0.535720	−0.267860	+	0.963458i	$0.586316\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	30.0000	1.99117	0.995585	−	0.0938647i	$-0.0299221\pi$
$227$	30.0000	1.99117	0.995585	+	0.0938647i	$0.0299221\pi$
$228$	0	0
$229$	1.73205i	0.114457i	0.998361	+	0.0572286i	$0.0182264\pi$
$229$	1.73205i	0.114457i	−0.998361	+	0.0572286i	$0.981774\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	18.0000	1.17922	0.589610	−	0.807688i	$-0.299282\pi$
$233$	18.0000	1.17922	0.589610	+	0.807688i	$0.299282\pi$
$234$	0	0
$235$	20.7846i	1.35584i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	17.3205i	1.12037i	0.828367	+	0.560185i	$0.189270\pi$
$239$	17.3205i	1.12037i	−0.828367	+	0.560185i	$0.810730\pi$
$240$	0	0
$241$	13.8564i	0.892570i	0.894891	+	0.446285i	$0.147253\pi$
$241$	13.8564i	0.892570i	−0.894891	+	0.446285i	$0.852747\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	36.3731i	2.31436i
$248$	0	0
$249$	0	0
$250$	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
$255$	0	0
$256$	0	0
$257$	10.3923i	0.648254i	0.946014	+	0.324127i	$0.105071\pi$
$257$	10.3923i	0.648254i	−0.946014	+	0.324127i	$0.894929\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	27.7128i	1.70885i	0.519579	+	0.854423i	$0.326089\pi$
$263$	27.7128i	1.70885i	−0.519579	+	0.854423i	$0.673911\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 31.1769i	− 1.90089i	−0.310893	−	0.950445i	$-0.600628\pi$
$269$	− 31.1769i	− 1.90089i	0.310893	−	0.950445i	$-0.399372\pi$
$270$	0	0
$271$	−16.0000	−0.971931	−0.485965	−	0.873978i	$-0.661532\pi$
$271$	−16.0000	−0.971931	−0.485965	+	0.873978i	$0.661532\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	24.2487i	1.46225i
$276$	0	0
$277$	−17.0000	−1.02143	−0.510716	−	0.859750i	$-0.670619\pi$
$277$	−17.0000	−1.02143	−0.510716	+	0.859750i	$0.670619\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−12.0000	−0.715860	−0.357930	−	0.933748i	$-0.616517\pi$
$281$	−12.0000	−0.715860	−0.357930	+	0.933748i	$0.616517\pi$
$282$	0	0
$283$	−5.00000	−0.297219	−0.148610	−	0.988896i	$-0.547480\pi$
$283$	−5.00000	−0.297219	−0.148610	+	0.988896i	$0.547480\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−31.0000	−1.82353
$290$	0	0
$291$	0	0
$292$	0	0
$293$	20.7846i	1.21425i	0.794606	+	0.607125i	$0.207677\pi$
$293$	20.7846i	1.21425i	−0.794606	+	0.607125i	$0.792323\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	25.0000	1.42683	0.713413	−	0.700744i	$-0.247149\pi$
$307$	25.0000	1.42683	0.713413	+	0.700744i	$0.247149\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−18.0000	−1.02069	−0.510343	−	0.859971i	$-0.670482\pi$
$311$	−18.0000	−1.02069	−0.510343	+	0.859971i	$0.670482\pi$
$312$	0	0
$313$	15.5885i	0.881112i	0.897725	+	0.440556i	$0.145219\pi$
$313$	15.5885i	0.881112i	−0.897725	+	0.440556i	$0.854781\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	12.0000	0.673987	0.336994	−	0.941507i	$-0.390590\pi$
$317$	12.0000	0.673987	0.336994	+	0.941507i	$0.390590\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 48.4974i	− 2.69847i
$324$	0	0
$325$	− 36.3731i	− 2.01761i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	8.66025i	0.476011i	0.971264	+	0.238005i	$0.0764936\pi$
$331$	8.66025i	0.476011i	−0.971264	+	0.238005i	$0.923506\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	6.00000	0.327815
$336$	0	0
$337$	−19.0000	−1.03500	−0.517498	−	0.855684i	$-0.673136\pi$
$337$	−19.0000	−1.03500	−0.517498	+	0.855684i	$0.673136\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 17.3205i	− 0.937958i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	34.6410i	1.85963i	0.368031	+	0.929814i	$0.380032\pi$
$347$	34.6410i	1.85963i	−0.368031	+	0.929814i	$0.619968\pi$
$348$	0	0
$349$	− 13.8564i	− 0.741716i	−0.928689	−	0.370858i	$-0.879064\pi$
$349$	− 13.8564i	− 0.741716i	0.928689	−	0.370858i	$-0.120936\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	17.3205i	0.921878i	0.887432	+	0.460939i	$0.152487\pi$
$353$	17.3205i	0.921878i	−0.887432	+	0.460939i	$0.847513\pi$
$354$	0	0
$355$	−12.0000	−0.636894
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 6.92820i	− 0.365657i	−0.983145	−	0.182828i	$-0.941475\pi$
$359$	− 6.92820i	− 0.365657i	0.983145	−	0.182828i	$-0.0585252\pi$
$360$	0	0
$361$	30.0000	1.57895
$362$	0	0
$363$	0	0
$364$	0	0
$365$	30.0000	1.57027
$366$	0	0
$367$	37.0000	1.93138	0.965692	−	0.259690i	$-0.0836203\pi$
$367$	37.0000	1.93138	0.965692	+	0.259690i	$0.0836203\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	11.0000	0.569558	0.284779	−	0.958593i	$-0.408080\pi$
$373$	11.0000	0.569558	0.284779	+	0.958593i	$0.408080\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	− 12.1244i	− 0.622786i	−0.950281	−	0.311393i	$-0.899204\pi$
$379$	− 12.1244i	− 0.622786i	0.950281	−	0.311393i	$-0.100796\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−36.0000	−1.83951	−0.919757	−	0.392488i	$-0.871614\pi$
$383$	−36.0000	−1.83951	−0.919757	+	0.392488i	$0.871614\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−30.0000	−1.52106	−0.760530	−	0.649303i	$-0.775061\pi$
$389$	−30.0000	−1.52106	−0.760530	+	0.649303i	$0.775061\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	54.0000	2.71703
$396$	0	0
$397$	5.19615i	0.260787i	0.991462	+	0.130394i	$0.0416241\pi$
$397$	5.19615i	0.260787i	−0.991462	+	0.130394i	$0.958376\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−24.0000	−1.19850	−0.599251	−	0.800561i	$-0.704535\pi$
$401$	−24.0000	−1.19850	−0.599251	+	0.800561i	$0.704535\pi$
$402$	0	0
$403$	25.9808i	1.29419i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 3.46410i	− 0.171709i
$408$	0	0
$409$	− 19.0526i	− 0.942088i	−0.882110	−	0.471044i	$-0.843877\pi$
$409$	− 19.0526i	− 0.942088i	0.882110	−	0.471044i	$-0.156123\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	20.7846i	1.02028i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	30.0000	1.46560	0.732798	−	0.680446i	$-0.238214\pi$
$419$	30.0000	1.46560	0.732798	+	0.680446i	$0.238214\pi$
$420$	0	0
$421$	−29.0000	−1.41337	−0.706687	−	0.707527i	$-0.749811\pi$
$421$	−29.0000	−1.41337	−0.706687	+	0.707527i	$0.749811\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	48.4974i	2.35247i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	17.3205i	0.834300i	0.908838	+	0.417150i	$0.136971\pi$
$431$	17.3205i	0.834300i	−0.908838	+	0.417150i	$0.863029\pi$
$432$	0	0
$433$	− 29.4449i	− 1.41503i	−0.706698	−	0.707515i	$-0.749816\pi$
$433$	− 29.4449i	− 1.41503i	0.706698	−	0.707515i	$-0.250184\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	16.0000	0.763638	0.381819	−	0.924237i	$-0.375298\pi$
$439$	16.0000	0.763638	0.381819	+	0.924237i	$0.375298\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 6.92820i	− 0.329169i	−0.986363	−	0.164584i	$-0.947372\pi$
$443$	− 6.92820i	− 0.329169i	0.986363	−	0.164584i	$-0.0526283\pi$
$444$	0	0
$445$	24.0000	1.13771
$446$	0	0
$447$	0	0
$448$	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
$451$	−36.0000	−1.69517
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	5.00000	0.233890	0.116945	−	0.993138i	$-0.462690\pi$
$457$	5.00000	0.233890	0.116945	+	0.993138i	$0.462690\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	13.8564i	0.645357i	0.946509	+	0.322679i	$0.104583\pi$
$461$	13.8564i	0.645357i	−0.946509	+	0.322679i	$0.895417\pi$
$462$	0	0
$463$	12.1244i	0.563467i	0.959493	+	0.281733i	$0.0909093\pi$
$463$	12.1244i	0.563467i	−0.959493	+	0.281733i	$0.909091\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	6.00000	0.277647	0.138823	−	0.990317i	$-0.455668\pi$
$467$	6.00000	0.277647	0.138823	+	0.990317i	$0.455668\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	6.00000	0.275880
$474$	0	0
$475$	−49.0000	−2.24827
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−24.0000	−1.09659	−0.548294	−	0.836286i	$-0.684723\pi$
$479$	−24.0000	−1.09659	−0.548294	+	0.836286i	$0.684723\pi$
$480$	0	0
$481$	5.19615i	0.236924i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	24.0000	1.08978
$486$	0	0
$487$	− 12.1244i	− 0.549407i	−0.961529	−	0.274703i	$-0.911420\pi$
$487$	− 12.1244i	− 0.549407i	0.961529	−	0.274703i	$-0.0885797\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	20.7846i	0.937996i	0.883199	+	0.468998i	$0.155385\pi$
$491$	20.7846i	0.937996i	−0.883199	+	0.468998i	$0.844615\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	− 12.1244i	− 0.542761i	−0.962472	−	0.271380i	$-0.912520\pi$
$499$	− 12.1244i	− 0.542761i	0.962472	−	0.271380i	$-0.0874801\pi$
$500$	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
$505$	12.0000	0.533993
$506$	0	0
$507$	0	0
$508$	0	0
$509$	3.46410i	0.153544i	0.997049	+	0.0767718i	$0.0244613\pi$
$509$	3.46410i	0.153544i	−0.997049	+	0.0767718i	$0.975539\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	17.3205i	0.763233i
$516$	0	0
$517$	− 20.7846i	− 0.914106i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 13.8564i	− 0.607060i	−0.952822	−	0.303530i	$-0.901835\pi$
$521$	− 13.8564i	− 0.607060i	0.952822	−	0.303530i	$-0.0981653\pi$
$522$	0	0
$523$	−1.00000	−0.0437269	−0.0218635	−	0.999761i	$-0.506960\pi$
$523$	−1.00000	−0.0437269	−0.0218635	+	0.999761i	$0.506960\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 34.6410i	− 1.50899i
$528$	0	0
$529$	23.0000	1.00000
$530$	0	0
$531$	0	0
$532$	0	0
$533$	54.0000	2.33900
$534$	0	0
$535$	−24.0000	−1.03761
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−41.0000	−1.76273	−0.881364	−	0.472438i	$-0.843374\pi$
$541$	−41.0000	−1.76273	−0.881364	+	0.472438i	$0.843374\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	17.3205i	0.741929i
$546$	0	0
$547$	10.3923i	0.444343i	0.975008	+	0.222171i	$0.0713145\pi$
$547$	10.3923i	0.444343i	−0.975008	+	0.222171i	$0.928686\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−18.0000	−0.762684	−0.381342	−	0.924434i	$-0.624538\pi$
$557$	−18.0000	−0.762684	−0.381342	+	0.924434i	$0.624538\pi$
$558$	0	0
$559$	−9.00000	−0.380659
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−18.0000	−0.758610	−0.379305	−	0.925272i	$-0.623837\pi$
$563$	−18.0000	−0.758610	−0.379305	+	0.925272i	$0.623837\pi$
$564$	0	0
$565$	− 20.7846i	− 0.874415i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	6.00000	0.251533	0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000	0.251533	0.125767	+	0.992060i	$0.459861\pi$
$570$	0	0
$571$	25.9808i	1.08726i	0.839325	+	0.543631i	$0.182951\pi$
$571$	25.9808i	1.08726i	−0.839325	+	0.543631i	$0.817049\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	1.73205i	0.0721062i	0.999350	+	0.0360531i	$0.0114785\pi$
$577$	1.73205i	0.0721062i	−0.999350	+	0.0360531i	$0.988521\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−12.0000	−0.495293	−0.247647	−	0.968850i	$-0.579657\pi$
$587$	−12.0000	−0.495293	−0.247647	+	0.968850i	$0.579657\pi$
$588$	0	0
$589$	35.0000	1.44215
$590$	0	0
$591$	0	0
$592$	0	0
$593$	38.1051i	1.56479i	0.622783	+	0.782395i	$0.286002\pi$
$593$	38.1051i	1.56479i	−0.622783	+	0.782395i	$0.713998\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	6.92820i	0.283079i	0.989933	+	0.141539i	$0.0452052\pi$
$599$	6.92820i	0.283079i	−0.989933	+	0.141539i	$0.954795\pi$
$600$	0	0
$601$	− 25.9808i	− 1.05978i	−0.848067	−	0.529889i	$-0.822234\pi$
$601$	− 25.9808i	− 1.05978i	0.848067	−	0.529889i	$-0.177766\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 3.46410i	− 0.140836i
$606$	0	0
$607$	37.0000	1.50178	0.750892	−	0.660425i	$-0.229624\pi$
$607$	37.0000	1.50178	0.750892	+	0.660425i	$0.229624\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	31.1769i	1.26128i
$612$	0	0
$613$	−22.0000	−0.888572	−0.444286	−	0.895885i	$-0.646543\pi$
$613$	−22.0000	−0.888572	−0.444286	+	0.895885i	$0.646543\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	42.0000	1.69086	0.845428	−	0.534089i	$-0.179345\pi$
$617$	42.0000	1.69086	0.845428	+	0.534089i	$0.179345\pi$
$618$	0	0
$619$	35.0000	1.40677	0.703384	−	0.710810i	$-0.251671\pi$
$619$	35.0000	1.40677	0.703384	+	0.710810i	$0.251671\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−11.0000	−0.440000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 6.92820i	− 0.276246i
$630$	0	0
$631$	3.46410i	0.137904i	0.997620	+	0.0689519i	$0.0219655\pi$
$631$	3.46410i	0.137904i	−0.997620	+	0.0689519i	$0.978035\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	54.0000	2.14292
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	36.0000	1.42191	0.710957	−	0.703235i	$-0.248262\pi$
$641$	36.0000	1.42191	0.710957	+	0.703235i	$0.248262\pi$
$642$	0	0
$643$	43.0000	1.69575	0.847877	−	0.530193i	$-0.177880\pi$
$643$	43.0000	1.69575	0.847877	+	0.530193i	$0.177880\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	18.0000	0.707653	0.353827	−	0.935311i	$-0.384880\pi$
$647$	18.0000	0.707653	0.353827	+	0.935311i	$0.384880\pi$
$648$	0	0
$649$	0	0
$650$	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
$655$	− 62.3538i	− 2.43637i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 20.7846i	− 0.809653i	−0.914393	−	0.404827i	$-0.867332\pi$
$659$	− 20.7846i	− 0.809653i	0.914393	−	0.404827i	$-0.132668\pi$
$660$	0	0
$661$	− 19.0526i	− 0.741059i	−0.928821	−	0.370529i	$-0.879176\pi$
$661$	− 19.0526i	− 0.741059i	0.928821	−	0.370529i	$-0.120824\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	1.00000	0.0385472	0.0192736	−	0.999814i	$-0.493865\pi$
$673$	1.00000	0.0385472	0.0192736	+	0.999814i	$0.493865\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 34.6410i	− 1.33136i	−0.746236	−	0.665681i	$-0.768141\pi$
$677$	− 34.6410i	− 1.33136i	0.746236	−	0.665681i	$-0.231859\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 41.5692i	− 1.59060i	−0.606215	−	0.795301i	$-0.707313\pi$
$683$	− 41.5692i	− 1.59060i	0.606215	−	0.795301i	$-0.292687\pi$
$684$	0	0
$685$	41.5692i	1.58828i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−13.0000	−0.494543	−0.247272	−	0.968946i	$-0.579534\pi$
$691$	−13.0000	−0.494543	−0.247272	+	0.968946i	$0.579534\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	17.3205i	0.657004i
$696$	0	0
$697$	−72.0000	−2.72719
$698$	0	0
$699$	0	0
$700$	0	0
$701$	0	0		−	1.00000i	$-0.5\pi$
$701$	0	0			1.00000i	$0.5\pi$
$702$	0	0
$703$	7.00000	0.264010
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−10.0000	−0.375558	−0.187779	−	0.982211i	$-0.560129\pi$
$709$	−10.0000	−0.375558	−0.187779	+	0.982211i	$0.560129\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	0	0
$714$	0	0
$715$	62.3538i	2.33190i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	18.0000	0.671287	0.335643	−	0.941989i	$-0.391046\pi$
$719$	18.0000	0.671287	0.335643	+	0.941989i	$0.391046\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−1.00000	−0.0370879	−0.0185440	−	0.999828i	$-0.505903\pi$
$727$	−1.00000	−0.0370879	−0.0185440	+	0.999828i	$0.505903\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	12.0000	0.443836
$732$	0	0
$733$	− 36.3731i	− 1.34347i	−0.740792	−	0.671735i	$-0.765549\pi$
$733$	− 36.3731i	− 1.34347i	0.740792	−	0.671735i	$-0.234451\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−6.00000	−0.221013
$738$	0	0
$739$	− 36.3731i	− 1.33800i	−0.743260	−	0.669002i	$-0.766722\pi$
$739$	− 36.3731i	− 1.33800i	0.743260	−	0.669002i	$-0.233278\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 3.46410i	− 0.127086i	−0.997979	−	0.0635428i	$-0.979760\pi$
$743$	− 3.46410i	− 0.127086i	0.997979	−	0.0635428i	$-0.0202399\pi$
$744$	0	0
$745$	41.5692i	1.52298i
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	− 46.7654i	− 1.70649i	−0.521508	−	0.853246i	$-0.674630\pi$
$751$	− 46.7654i	− 1.70649i	0.521508	−	0.853246i	$-0.325370\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−36.0000	−1.31017
$756$	0	0
$757$	−22.0000	−0.799604	−0.399802	−	0.916602i	$-0.630921\pi$
$757$	−22.0000	−0.799604	−0.399802	+	0.916602i	$0.630921\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 34.6410i	− 1.25574i	−0.778320	−	0.627868i	$-0.783928\pi$
$761$	− 34.6410i	− 1.25574i	0.778320	−	0.627868i	$-0.216072\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	− 29.4449i	− 1.06181i	−0.847432	−	0.530904i	$-0.821852\pi$
$769$	− 29.4449i	− 1.06181i	0.847432	−	0.530904i	$-0.178148\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 31.1769i	− 1.12136i	−0.828034	−	0.560678i	$-0.810541\pi$
$773$	− 31.1769i	− 1.12136i	0.828034	−	0.560678i	$-0.189459\pi$
$774$	0	0
$775$	−35.0000	−1.25724
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 72.7461i	− 2.60640i
$780$	0	0
$781$	12.0000	0.429394
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−48.0000	−1.71319
$786$	0	0
$787$	8.00000	0.285169	0.142585	−	0.989783i	$-0.454459\pi$
$787$	8.00000	0.285169	0.142585	+	0.989783i	$0.454459\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 41.5692i	− 1.47246i	−0.676733	−	0.736229i	$-0.736605\pi$
$797$	− 41.5692i	− 1.47246i	0.676733	−	0.736229i	$-0.263395\pi$
$798$	0	0
$799$	− 41.5692i	− 1.47061i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−30.0000	−1.05868
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−54.0000	−1.89854	−0.949269	−	0.314464i	$-0.898175\pi$
$809$	−54.0000	−1.89854	−0.949269	+	0.314464i	$0.898175\pi$
$810$	0	0
$811$	−16.0000	−0.561836	−0.280918	−	0.959732i	$-0.590639\pi$
$811$	−16.0000	−0.561836	−0.280918	+	0.959732i	$0.590639\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	12.0000	0.420342
$816$	0	0
$817$	12.1244i	0.424178i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	6.00000	0.209401	0.104701	−	0.994504i	$-0.466612\pi$
$821$	6.00000	0.209401	0.104701	+	0.994504i	$0.466612\pi$
$822$	0	0
$823$	17.3205i	0.603755i	0.953347	+	0.301877i	$0.0976134\pi$
$823$	17.3205i	0.603755i	−0.953347	+	0.301877i	$0.902387\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 17.3205i	− 0.602293i	−0.953578	−	0.301147i	$-0.902631\pi$
$827$	− 17.3205i	− 0.602293i	0.953578	−	0.301147i	$-0.0973693\pi$
$828$	0	0
$829$	22.5167i	0.782036i	0.920383	+	0.391018i	$0.127877\pi$
$829$	22.5167i	0.782036i	−0.920383	+	0.391018i	$0.872123\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	− 20.7846i	− 0.719281i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	24.0000	0.828572	0.414286	−	0.910147i	$-0.364031\pi$
$839$	24.0000	0.828572	0.414286	+	0.910147i	$0.364031\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 48.4974i	− 1.66836i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	1.73205i	0.0593043i	0.999560	+	0.0296521i	$0.00943995\pi$
$853$	1.73205i	0.0593043i	−0.999560	+	0.0296521i	$0.990560\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	0	0	1.00000			$0$
$857$	0	0	−1.00000			$\pi$
$858$	0	0
$859$	40.0000	1.36478	0.682391	−	0.730987i	$-0.260940\pi$
$859$	40.0000	1.36478	0.682391	+	0.730987i	$0.260940\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	31.1769i	1.06127i	0.847599	+	0.530637i	$0.178047\pi$
$863$	31.1769i	1.06127i	−0.847599	+	0.530637i	$0.821953\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−54.0000	−1.83182
$870$	0	0
$871$	9.00000	0.304953
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−22.0000	−0.742887	−0.371444	−	0.928456i	$-0.621137\pi$
$877$	−22.0000	−0.742887	−0.371444	+	0.928456i	$0.621137\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	27.7128i	0.933668i	0.884345	+	0.466834i	$0.154606\pi$
$881$	27.7128i	0.933668i	−0.884345	+	0.466834i	$0.845394\pi$
$882$	0	0
$883$	− 12.1244i	− 0.408017i	−0.978969	−	0.204009i	$-0.934603\pi$
$883$	− 12.1244i	− 0.408017i	0.978969	−	0.204009i	$-0.0653970\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	30.0000	1.00730	0.503651	−	0.863907i	$-0.331990\pi$
$887$	30.0000	1.00730	0.503651	+	0.863907i	$0.331990\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	42.0000	1.40548
$894$	0	0
$895$	60.0000	2.00558
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−54.0000	−1.79502
$906$	0	0
$907$	15.5885i	0.517606i	0.965930	+	0.258803i	$0.0833281\pi$
$907$	15.5885i	0.517606i	−0.965930	+	0.258803i	$0.916672\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	20.7846i	0.688625i	0.938855	+	0.344312i	$0.111888\pi$
$911$	20.7846i	0.688625i	−0.938855	+	0.344312i	$0.888112\pi$
$912$	0	0
$913$	− 20.7846i	− 0.687870i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	1.73205i	0.0571351i	0.999592	+	0.0285675i	$0.00909457\pi$
$919$	1.73205i	0.0571351i	−0.999592	+	0.0285675i	$0.990905\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−18.0000	−0.592477
$924$	0	0
$925$	−7.00000	−0.230159
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 3.46410i	− 0.113653i	−0.998384	−	0.0568267i	$-0.981902\pi$
$929$	− 3.46410i	− 0.113653i	0.998384	−	0.0568267i	$-0.0180983\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 83.1384i	− 2.71892i
$936$	0	0
$937$	29.4449i	0.961922i	0.876742	+	0.480961i	$0.159712\pi$
$937$	29.4449i	0.961922i	−0.876742	+	0.480961i	$0.840288\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	48.4974i	1.58097i	0.612481	+	0.790485i	$0.290172\pi$
$941$	48.4974i	1.58097i	−0.612481	+	0.790485i	$0.709828\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	17.3205i	0.562841i	0.959585	+	0.281420i	$0.0908056\pi$
$947$	17.3205i	0.562841i	−0.959585	+	0.281420i	$0.909194\pi$
$948$	0	0
$949$	45.0000	1.46076
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−36.0000	−1.16615	−0.583077	−	0.812417i	$-0.698151\pi$
$953$	−36.0000	−1.16615	−0.583077	+	0.812417i	$0.698151\pi$
$954$	0	0
$955$	−12.0000	−0.388311
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−6.00000	−0.193548
$962$	0	0
$963$	0	0
$964$	0	0
$965$	45.0333i	1.44967i
$966$	0	0
$967$	− 22.5167i	− 0.724087i	−0.932161	−	0.362043i	$-0.882079\pi$
$967$	− 22.5167i	− 0.724087i	0.932161	−	0.362043i	$-0.117921\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−12.0000	−0.385098	−0.192549	−	0.981287i	$-0.561675\pi$
$971$	−12.0000	−0.385098	−0.192549	+	0.981287i	$0.561675\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	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
$979$	−24.0000	−0.767043
$980$	0	0
$981$	0	0
$982$	0	0
$983$	0	0		−	1.00000i	$-0.5\pi$
$983$	0	0			1.00000i	$0.5\pi$
$984$	0	0
$985$	− 41.5692i	− 1.32451i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	− 12.1244i	− 0.385143i	−0.981283	−	0.192571i	$-0.938317\pi$
$991$	− 12.1244i	− 0.385143i	0.981283	−	0.192571i	$-0.0616827\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	55.4256i	1.75711i
$996$	0	0
$997$	− 43.3013i	− 1.37136i	−0.727901	−	0.685682i	$-0.759504\pi$
$997$	− 43.3013i	− 1.37136i	0.727901	−	0.685682i	$-0.240496\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7056.2.b.l.1567.2		2
3.2	odd	2		2352.2.b.h.1567.1		2
4.3	odd	2		7056.2.b.a.1567.2		2
7.2	even	3		1008.2.cs.n.703.1		2
7.3	odd	6		1008.2.cs.m.271.1		2
7.6	odd	2		7056.2.b.a.1567.1		2
12.11	even	2		2352.2.b.a.1567.1		2
21.2	odd	6		336.2.bl.a.31.1	✓	2
21.5	even	6		2352.2.bl.l.31.1		2
21.11	odd	6		2352.2.bl.f.607.1		2
21.17	even	6		336.2.bl.e.271.1	yes	2
21.20	even	2		2352.2.b.a.1567.2		2
28.3	even	6		1008.2.cs.n.271.1		2
28.23	odd	6		1008.2.cs.m.703.1		2
28.27	even	2	inner	7056.2.b.l.1567.1		2
84.11	even	6		2352.2.bl.l.607.1		2
84.23	even	6		336.2.bl.e.31.1	yes	2
84.47	odd	6		2352.2.bl.f.31.1		2
84.59	odd	6		336.2.bl.a.271.1	yes	2
84.83	odd	2		2352.2.b.h.1567.2		2
168.59	odd	6		1344.2.bl.h.1279.1		2
168.101	even	6		1344.2.bl.d.1279.1		2
168.107	even	6		1344.2.bl.d.703.1		2
168.149	odd	6		1344.2.bl.h.703.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
336.2.bl.a.31.1	✓	2	21.2	odd	6
336.2.bl.a.271.1	yes	2	84.59	odd	6
336.2.bl.e.31.1	yes	2	84.23	even	6
336.2.bl.e.271.1	yes	2	21.17	even	6
1008.2.cs.m.271.1		2	7.3	odd	6
1008.2.cs.m.703.1		2	28.23	odd	6
1008.2.cs.n.271.1		2	28.3	even	6
1008.2.cs.n.703.1		2	7.2	even	3
1344.2.bl.d.703.1		2	168.107	even	6
1344.2.bl.d.1279.1		2	168.101	even	6
1344.2.bl.h.703.1		2	168.149	odd	6
1344.2.bl.h.1279.1		2	168.59	odd	6
2352.2.b.a.1567.1		2	12.11	even	2
2352.2.b.a.1567.2		2	21.20	even	2
2352.2.b.h.1567.1		2	3.2	odd	2
2352.2.b.h.1567.2		2	84.83	odd	2
2352.2.bl.f.31.1		2	84.47	odd	6
2352.2.bl.f.607.1		2	21.11	odd	6
2352.2.bl.l.31.1		2	21.5	even	6
2352.2.bl.l.607.1		2	84.11	even	6
7056.2.b.a.1567.1		2	7.6	odd	2
7056.2.b.a.1567.2		2	4.3	odd	2
7056.2.b.l.1567.1		2	28.27	even	2	inner
7056.2.b.l.1567.2		2	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 7056 = 2^{4} \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7056.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+3.46410i q^{5} +O(q^{10})\)	\(q+3.46410i q^{5} -3.46410i q^{11} +5.19615i q^{13} -6.92820i q^{17} +7.00000 q^{19} -7.00000 q^{25} +5.00000 q^{31} +1.00000 q^{37} -10.3923i q^{41} +1.73205i q^{43} +6.00000 q^{47} +12.0000 q^{55} -18.0000 q^{65} -1.73205i q^{67} +3.46410i q^{71} -8.66025i q^{73} -15.5885i q^{79} +6.00000 q^{83} +24.0000 q^{85} -6.92820i q^{89} +24.2487i q^{95} -6.92820i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q+O(q^{10}) \) 2 * q	\( 2 q + 14 q^{19} - 14 q^{25} + 10 q^{31} + 2 q^{37} + 12 q^{47} + 24 q^{55} - 36 q^{65} + 12 q^{83} + 48 q^{85}+O(q^{100}) \) 2 * q + 14 * q^19 - 14 * q^25 + 10 * q^31 + 2 * q^37 + 12 * q^47 + 24 * q^55 - 36 * q^65 + 12 * q^83 + 48 * q^85

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