LMFDB - Embedded newform 7056.2.b.o.1567.1

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:	$4$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} + 2x^{2} + x + 1 $ x^4 - x^3 + 2*x^2 + x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{29}]$
Coefficient ring index:	$ 2^{4}\cdot 3 $
Twist minimal:	no (minimal twist has level 1008)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1567.1
Root			$-0.309017 + 0.535233i$ of defining polynomial
Character	$\chi$	$=$	7056.1567
Dual form			7056.2.b.o.1567.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.87298i q^{5} +O(q^{10})$	$q-3.87298i q^{5} +3.87298i q^{11} -3.46410i q^{13} -4.00000 q^{19} +7.74597i q^{23} -10.0000 q^{25} +6.70820 q^{29} +1.00000 q^{31} +4.00000 q^{37} -7.74597i q^{41} +6.92820i q^{43} -13.4164 q^{47} -6.70820 q^{53} +15.0000 q^{55} -6.70820 q^{59} +10.3923i q^{61} -13.4164 q^{65} +6.92820i q^{67} -7.74597i q^{71} +6.92820i q^{73} +12.1244i q^{79} -6.70820 q^{83} -7.74597i q^{89} +15.4919i q^{95} +5.19615i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q - 16 q^{19} - 40 q^{25} + 4 q^{31} + 16 q^{37} + 60 q^{55}+O(q^{100}) $ 4 * q - 16 * q^19 - 40 * q^25 + 4 * q^31 + 16 * q^37 + 60 * q^55

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
$5$	− 3.87298i	− 1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$5$	− 3.87298i	− 1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	0	0
$10$	0	0
$11$	3.87298i	1.16775i	0.811844	+	0.583874i	$0.198464\pi$
$11$	3.87298i	1.16775i	−0.811844	+	0.583874i	$0.801536\pi$
$12$	0	0
$13$	− 3.46410i	− 0.960769i	−0.877058	−	0.480384i	$-0.840497\pi$
$13$	− 3.46410i	− 0.960769i	0.877058	−	0.480384i	$-0.159503\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	0	0
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	7.74597i	1.61515i	0.589768	+	0.807573i	$0.299219\pi$
$23$	7.74597i	1.61515i	−0.589768	+	0.807573i	$0.700781\pi$
$24$	0	0
$25$	−10.0000	−2.00000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	6.70820	1.24568	0.622841	−	0.782348i	$-0.285978\pi$
$29$	6.70820	1.24568	0.622841	+	0.782348i	$0.285978\pi$
$30$	0	0
$31$	1.00000	0.179605	0.0898027	−	0.995960i	$-0.471376\pi$
$31$	1.00000	0.179605	0.0898027	+	0.995960i	$0.471376\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	4.00000	0.657596	0.328798	−	0.944400i	$-0.393356\pi$
$37$	4.00000	0.657596	0.328798	+	0.944400i	$0.393356\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 7.74597i	− 1.20972i	−0.796333	−	0.604858i	$-0.793230\pi$
$41$	− 7.74597i	− 1.20972i	0.796333	−	0.604858i	$-0.206770\pi$
$42$	0	0
$43$	6.92820i	1.05654i	0.849076	+	0.528271i	$0.177159\pi$
$43$	6.92820i	1.05654i	−0.849076	+	0.528271i	$0.822841\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−13.4164	−1.95698	−0.978492	−	0.206284i	$-0.933863\pi$
$47$	−13.4164	−1.95698	−0.978492	+	0.206284i	$0.933863\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−6.70820	−0.921443	−0.460721	−	0.887545i	$-0.652409\pi$
$53$	−6.70820	−0.921443	−0.460721	+	0.887545i	$0.652409\pi$
$54$	0	0
$55$	15.0000	2.02260
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−6.70820	−0.873334	−0.436667	−	0.899623i	$-0.643841\pi$
$59$	−6.70820	−0.873334	−0.436667	+	0.899623i	$0.643841\pi$
$60$	0	0
$61$	10.3923i	1.33060i	0.746577	+	0.665299i	$0.231696\pi$
$61$	10.3923i	1.33060i	−0.746577	+	0.665299i	$0.768304\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−13.4164	−1.66410
$66$	0	0
$67$	6.92820i	0.846415i	0.906033	+	0.423207i	$0.139096\pi$
$67$	6.92820i	0.846415i	−0.906033	+	0.423207i	$0.860904\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 7.74597i	− 0.919277i	−0.888106	−	0.459639i	$-0.847979\pi$
$71$	− 7.74597i	− 0.919277i	0.888106	−	0.459639i	$-0.152021\pi$
$72$	0	0
$73$	6.92820i	0.810885i	0.914121	+	0.405442i	$0.132883\pi$
$73$	6.92820i	0.810885i	−0.914121	+	0.405442i	$0.867117\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	12.1244i	1.36410i	0.731307	+	0.682048i	$0.238911\pi$
$79$	12.1244i	1.36410i	−0.731307	+	0.682048i	$0.761089\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−6.70820	−0.736321	−0.368161	−	0.929762i	$-0.620012\pi$
$83$	−6.70820	−0.736321	−0.368161	+	0.929762i	$0.620012\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 7.74597i	− 0.821071i	−0.911845	−	0.410535i	$-0.865342\pi$
$89$	− 7.74597i	− 0.821071i	0.911845	−	0.410535i	$-0.134658\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	15.4919i	1.58944i
$96$	0	0
$97$	5.19615i	0.527589i	0.964579	+	0.263795i	$0.0849741\pi$
$97$	5.19615i	0.527589i	−0.964579	+	0.263795i	$0.915026\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	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
$107$	11.6190i	1.12325i	0.827393	+	0.561623i	$0.189823\pi$
$107$	11.6190i	1.12325i	−0.827393	+	0.561623i	$0.810177\pi$
$108$	0	0
$109$	−4.00000	−0.383131	−0.191565	−	0.981480i	$-0.561356\pi$
$109$	−4.00000	−0.383131	−0.191565	+	0.981480i	$0.561356\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−13.4164	−1.26211	−0.631055	−	0.775738i	$-0.717378\pi$
$113$	−13.4164	−1.26211	−0.631055	+	0.775738i	$0.717378\pi$
$114$	0	0
$115$	30.0000	2.79751
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−4.00000	−0.363636
$122$	0	0
$123$	0	0
$124$	0	0
$125$	19.3649i	1.73205i
$126$	0	0
$127$	− 12.1244i	− 1.07586i	−0.842989	−	0.537931i	$-0.819206\pi$
$127$	− 12.1244i	− 1.07586i	0.842989	−	0.537931i	$-0.180794\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−6.70820	−0.586098	−0.293049	−	0.956097i	$-0.594670\pi$
$131$	−6.70820	−0.586098	−0.293049	+	0.956097i	$0.594670\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	0	0		−	1.00000i	$-0.5\pi$
$137$	0	0			1.00000i	$0.5\pi$
$138$	0	0
$139$	−2.00000	−0.169638	−0.0848189	−	0.996396i	$-0.527031\pi$
$139$	−2.00000	−0.169638	−0.0848189	+	0.996396i	$0.527031\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	13.4164	1.12194
$144$	0	0
$145$	− 25.9808i	− 2.15758i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	13.4164	1.09911	0.549557	−	0.835456i	$-0.314796\pi$
$149$	13.4164	1.09911	0.549557	+	0.835456i	$0.314796\pi$
$150$	0	0
$151$	15.5885i	1.26857i	0.773099	+	0.634285i	$0.218706\pi$
$151$	15.5885i	1.26857i	−0.773099	+	0.634285i	$0.781294\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 3.87298i	− 0.311086i
$156$	0	0
$157$	20.7846i	1.65879i	0.558661	+	0.829396i	$0.311315\pi$
$157$	20.7846i	1.65879i	−0.558661	+	0.829396i	$0.688685\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	− 10.3923i	− 0.813988i	−0.913431	−	0.406994i	$-0.866577\pi$
$163$	− 10.3923i	− 0.813988i	0.913431	−	0.406994i	$-0.133423\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0	0
$172$	0	0
$173$	15.4919i	1.17783i	0.808195	+	0.588915i	$0.200445\pi$
$173$	15.4919i	1.17783i	−0.808195	+	0.588915i	$0.799555\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 23.2379i	− 1.73688i	−0.495792	−	0.868441i	$-0.665122\pi$
$179$	− 23.2379i	− 1.73688i	0.495792	−	0.868441i	$-0.334878\pi$
$180$	0	0
$181$	24.2487i	1.80239i	0.433411	+	0.901196i	$0.357310\pi$
$181$	24.2487i	1.80239i	−0.433411	+	0.901196i	$0.642690\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 15.4919i	− 1.13899i
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 15.4919i	− 1.12096i	−0.828169	−	0.560478i	$-0.810617\pi$
$191$	− 15.4919i	− 1.12096i	0.828169	−	0.560478i	$-0.189383\pi$
$192$	0	0
$193$	7.00000	0.503871	0.251936	−	0.967744i	$-0.418933\pi$
$193$	7.00000	0.503871	0.251936	+	0.967744i	$0.418933\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	13.4164	0.955879	0.477940	−	0.878393i	$-0.341384\pi$
$197$	13.4164	0.955879	0.477940	+	0.878393i	$0.341384\pi$
$198$	0	0
$199$	8.00000	0.567105	0.283552	−	0.958957i	$-0.408487\pi$
$199$	8.00000	0.567105	0.283552	+	0.958957i	$0.408487\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−30.0000	−2.09529
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 15.4919i	− 1.07160i
$210$	0	0
$211$	10.3923i	0.715436i	0.933830	+	0.357718i	$0.116445\pi$
$211$	10.3923i	0.715436i	−0.933830	+	0.357718i	$0.883555\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	26.8328	1.82998
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	23.0000	1.54019	0.770097	−	0.637927i	$-0.220208\pi$
$223$	23.0000	1.54019	0.770097	+	0.637927i	$0.220208\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−6.70820	−0.445239	−0.222620	−	0.974905i	$-0.571461\pi$
$227$	−6.70820	−0.445239	−0.222620	+	0.974905i	$0.571461\pi$
$228$	0	0
$229$	− 17.3205i	− 1.14457i	−0.820054	−	0.572286i	$-0.806057\pi$
$229$	− 17.3205i	− 1.14457i	0.820054	−	0.572286i	$-0.193943\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	0	0		−	1.00000i	$-0.5\pi$
$233$	0	0			1.00000i	$0.5\pi$
$234$	0	0
$235$	51.9615i	3.38960i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	23.2379i	1.50313i	0.659656	+	0.751567i	$0.270702\pi$
$239$	23.2379i	1.50313i	−0.659656	+	0.751567i	$0.729298\pi$
$240$	0	0
$241$	− 15.5885i	− 1.00414i	−0.864827	−	0.502070i	$-0.832572\pi$
$241$	− 15.5885i	− 1.00414i	0.864827	−	0.502070i	$-0.167428\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	13.8564i	0.881662i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	6.70820	0.423418	0.211709	−	0.977333i	$-0.432097\pi$
$251$	6.70820	0.423418	0.211709	+	0.977333i	$0.432097\pi$
$252$	0	0
$253$	−30.0000	−1.88608
$254$	0	0
$255$	0	0
$256$	0	0
$257$	7.74597i	0.483180i	0.970378	+	0.241590i	$0.0776689\pi$
$257$	7.74597i	0.483180i	−0.970378	+	0.241590i	$0.922331\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 15.4919i	− 0.955274i	−0.878557	−	0.477637i	$-0.841494\pi$
$263$	− 15.4919i	− 0.955274i	0.878557	−	0.477637i	$-0.158506\pi$
$264$	0	0
$265$	25.9808i	1.59599i
$266$	0	0
$267$	0	0
$268$	0	0
$269$	27.1109i	1.65298i	0.562952	+	0.826490i	$0.309666\pi$
$269$	27.1109i	1.65298i	−0.562952	+	0.826490i	$0.690334\pi$
$270$	0	0
$271$	7.00000	0.425220	0.212610	−	0.977137i	$-0.431804\pi$
$271$	7.00000	0.425220	0.212610	+	0.977137i	$0.431804\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 38.7298i	− 2.33550i
$276$	0	0
$277$	−20.0000	−1.20168	−0.600842	−	0.799368i	$-0.705168\pi$
$277$	−20.0000	−1.20168	−0.600842	+	0.799368i	$0.705168\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−26.8328	−1.60071	−0.800356	−	0.599525i	$-0.795356\pi$
$281$	−26.8328	−1.60071	−0.800356	+	0.599525i	$0.795356\pi$
$282$	0	0
$283$	20.0000	1.18888	0.594438	−	0.804141i	$-0.297374\pi$
$283$	20.0000	1.18888	0.594438	+	0.804141i	$0.297374\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	17.0000	1.00000
$290$	0	0
$291$	0	0
$292$	0	0
$293$	11.6190i	0.678786i	0.940645	+	0.339393i	$0.110222\pi$
$293$	11.6190i	0.678786i	−0.940645	+	0.339393i	$0.889778\pi$
$294$	0	0
$295$	25.9808i	1.51266i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	26.8328	1.55178
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	40.2492	2.30466
$306$	0	0
$307$	14.0000	0.799022	0.399511	−	0.916728i	$-0.369180\pi$
$307$	14.0000	0.799022	0.399511	+	0.916728i	$0.369180\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−13.4164	−0.760775	−0.380387	−	0.924827i	$-0.624209\pi$
$311$	−13.4164	−0.760775	−0.380387	+	0.924827i	$0.624209\pi$
$312$	0	0
$313$	1.73205i	0.0979013i	0.998801	+	0.0489506i	$0.0155877\pi$
$313$	1.73205i	0.0979013i	−0.998801	+	0.0489506i	$0.984412\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	20.1246	1.13031	0.565155	−	0.824984i	$-0.308816\pi$
$317$	20.1246	1.13031	0.565155	+	0.824984i	$0.308816\pi$
$318$	0	0
$319$	25.9808i	1.45464i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	34.6410i	1.92154i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	− 6.92820i	− 0.380808i	−0.981706	−	0.190404i	$-0.939020\pi$
$331$	− 6.92820i	− 0.380808i	0.981706	−	0.190404i	$-0.0609799\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	26.8328	1.46603
$336$	0	0
$337$	−1.00000	−0.0544735	−0.0272367	−	0.999629i	$-0.508671\pi$
$337$	−1.00000	−0.0544735	−0.0272367	+	0.999629i	$0.508671\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	3.87298i	0.209734i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	7.74597i	0.415825i	0.978147	+	0.207913i	$0.0666670\pi$
$347$	7.74597i	0.415825i	−0.978147	+	0.207913i	$0.933333\pi$
$348$	0	0
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	15.4919i	0.824552i	0.911059	+	0.412276i	$0.135266\pi$
$353$	15.4919i	0.824552i	−0.911059	+	0.412276i	$0.864734\pi$
$354$	0	0
$355$	−30.0000	−1.59223
$356$	0	0
$357$	0	0
$358$	0	0
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	0	0
$363$	0	0
$364$	0	0
$365$	26.8328	1.40449
$366$	0	0
$367$	11.0000	0.574195	0.287098	−	0.957901i	$-0.407310\pi$
$367$	11.0000	0.574195	0.287098	+	0.957901i	$0.407310\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−10.0000	−0.517780	−0.258890	−	0.965907i	$-0.583357\pi$
$373$	−10.0000	−0.517780	−0.258890	+	0.965907i	$0.583357\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 23.2379i	− 1.19681i
$378$	0	0
$379$	− 17.3205i	− 0.889695i	−0.895606	−	0.444847i	$-0.853258\pi$
$379$	− 17.3205i	− 0.889695i	0.895606	−	0.444847i	$-0.146742\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	13.4164	0.685546	0.342773	−	0.939418i	$-0.388634\pi$
$383$	13.4164	0.685546	0.342773	+	0.939418i	$0.388634\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−13.4164	−0.680239	−0.340119	−	0.940382i	$-0.610468\pi$
$389$	−13.4164	−0.680239	−0.340119	+	0.940382i	$0.610468\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	46.9574	2.36268
$396$	0	0
$397$	13.8564i	0.695433i	0.937600	+	0.347717i	$0.113043\pi$
$397$	13.8564i	0.695433i	−0.937600	+	0.347717i	$0.886957\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	0	0		−	1.00000i	$-0.5\pi$
$401$	0	0			1.00000i	$0.5\pi$
$402$	0	0
$403$	− 3.46410i	− 0.172559i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	15.4919i	0.767907i
$408$	0	0
$409$	8.66025i	0.428222i	0.976809	+	0.214111i	$0.0686854\pi$
$409$	8.66025i	0.428222i	−0.976809	+	0.214111i	$0.931315\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	25.9808i	1.27535i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	0	0		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	0	0
$421$	34.0000	1.65706	0.828529	−	0.559946i	$-0.189178\pi$
$421$	34.0000	1.65706	0.828529	+	0.559946i	$0.189178\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 38.7298i	− 1.86555i	−0.360459	−	0.932775i	$-0.617380\pi$
$431$	− 38.7298i	− 1.86555i	0.360459	−	0.932775i	$-0.382620\pi$
$432$	0	0
$433$	− 6.92820i	− 0.332948i	−0.986046	−	0.166474i	$-0.946762\pi$
$433$	− 6.92820i	− 0.332948i	0.986046	−	0.166474i	$-0.0532382\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 30.9839i	− 1.48216i
$438$	0	0
$439$	−37.0000	−1.76591	−0.882957	−	0.469454i	$-0.844451\pi$
$439$	−37.0000	−1.76591	−0.882957	+	0.469454i	$0.844451\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	3.87298i	0.184011i	0.995758	+	0.0920055i	$0.0293277\pi$
$443$	3.87298i	0.184011i	−0.995758	+	0.0920055i	$0.970672\pi$
$444$	0	0
$445$	−30.0000	−1.42214
$446$	0	0
$447$	0	0
$448$	0	0
$449$	13.4164	0.633159	0.316580	−	0.948566i	$-0.397466\pi$
$449$	13.4164	0.633159	0.316580	+	0.948566i	$0.397466\pi$
$450$	0	0
$451$	30.0000	1.41264
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	29.0000	1.35656	0.678281	−	0.734802i	$-0.262725\pi$
$457$	29.0000	1.35656	0.678281	+	0.734802i	$0.262725\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	15.4919i	0.721531i	0.932657	+	0.360766i	$0.117485\pi$
$461$	15.4919i	0.721531i	−0.932657	+	0.360766i	$0.882515\pi$
$462$	0	0
$463$	− 24.2487i	− 1.12693i	−0.826139	−	0.563467i	$-0.809467\pi$
$463$	− 24.2487i	− 1.12693i	0.826139	−	0.563467i	$-0.190533\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	0	0		−	1.00000i	$-0.5\pi$
$467$	0	0			1.00000i	$0.5\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−26.8328	−1.23377
$474$	0	0
$475$	40.0000	1.83533
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−26.8328	−1.22602	−0.613011	−	0.790074i	$-0.710042\pi$
$479$	−26.8328	−1.22602	−0.613011	+	0.790074i	$0.710042\pi$
$480$	0	0
$481$	− 13.8564i	− 0.631798i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	20.1246	0.913812
$486$	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
$491$	19.3649i	0.873926i	0.899479	+	0.436963i	$0.143946\pi$
$491$	19.3649i	0.873926i	−0.899479	+	0.436963i	$0.856054\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	17.3205i	0.775372i	0.921791	+	0.387686i	$0.126726\pi$
$499$	17.3205i	0.775372i	−0.921791	+	0.387686i	$0.873274\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−26.8328	−1.19642	−0.598208	−	0.801341i	$-0.704120\pi$
$503$	−26.8328	−1.19642	−0.598208	+	0.801341i	$0.704120\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 11.6190i	− 0.515001i	−0.966278	−	0.257500i	$-0.917101\pi$
$509$	− 11.6190i	− 0.515001i	0.966278	−	0.257500i	$-0.0828989\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	30.9839i	1.36531i
$516$	0	0
$517$	− 51.9615i	− 2.28527i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 15.4919i	− 0.678714i	−0.940658	−	0.339357i	$-0.889791\pi$
$521$	− 15.4919i	− 0.678714i	0.940658	−	0.339357i	$-0.110209\pi$
$522$	0	0
$523$	22.0000	0.961993	0.480996	−	0.876723i	$-0.340275\pi$
$523$	22.0000	0.961993	0.480996	+	0.876723i	$0.340275\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−37.0000	−1.60870
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−26.8328	−1.16226
$534$	0	0
$535$	45.0000	1.94552
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−32.0000	−1.37579	−0.687894	−	0.725811i	$-0.741464\pi$
$541$	−32.0000	−1.37579	−0.687894	+	0.725811i	$0.741464\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	15.4919i	0.663602i
$546$	0	0
$547$	41.5692i	1.77737i	0.458517	+	0.888686i	$0.348381\pi$
$547$	41.5692i	1.77737i	−0.458517	+	0.888686i	$0.651619\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−26.8328	−1.14312
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	33.5410	1.42118	0.710589	−	0.703607i	$-0.248428\pi$
$557$	33.5410	1.42118	0.710589	+	0.703607i	$0.248428\pi$
$558$	0	0
$559$	24.0000	1.01509
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−33.5410	−1.41359	−0.706793	−	0.707421i	$-0.749859\pi$
$563$	−33.5410	−1.41359	−0.706793	+	0.707421i	$0.749859\pi$
$564$	0	0
$565$	51.9615i	2.18604i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−26.8328	−1.12489	−0.562445	−	0.826835i	$-0.690139\pi$
$569$	−26.8328	−1.12489	−0.562445	+	0.826835i	$0.690139\pi$
$570$	0	0
$571$	− 17.3205i	− 0.724841i	−0.932015	−	0.362420i	$-0.881950\pi$
$571$	− 17.3205i	− 0.724841i	0.932015	−	0.362420i	$-0.118050\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 77.4597i	− 3.23029i
$576$	0	0
$577$	22.5167i	0.937381i	0.883362	+	0.468690i	$0.155274\pi$
$577$	22.5167i	0.937381i	−0.883362	+	0.468690i	$0.844726\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	− 25.9808i	− 1.07601i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−6.70820	−0.276877	−0.138439	−	0.990371i	$-0.544208\pi$
$587$	−6.70820	−0.276877	−0.138439	+	0.990371i	$0.544208\pi$
$588$	0	0
$589$	−4.00000	−0.164817
$590$	0	0
$591$	0	0
$592$	0	0
$593$	7.74597i	0.318089i	0.987271	+	0.159044i	$0.0508413\pi$
$593$	7.74597i	0.318089i	−0.987271	+	0.159044i	$0.949159\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	30.9839i	1.26597i	0.774165	+	0.632983i	$0.218170\pi$
$599$	30.9839i	1.26597i	−0.774165	+	0.632983i	$0.781830\pi$
$600$	0	0
$601$	43.3013i	1.76630i	0.469095	+	0.883148i	$0.344580\pi$
$601$	43.3013i	1.76630i	−0.469095	+	0.883148i	$0.655420\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	15.4919i	0.629837i
$606$	0	0
$607$	11.0000	0.446476	0.223238	−	0.974764i	$-0.428337\pi$
$607$	11.0000	0.446476	0.223238	+	0.974764i	$0.428337\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	46.4758i	1.88021i
$612$	0	0
$613$	−10.0000	−0.403896	−0.201948	−	0.979396i	$-0.564727\pi$
$613$	−10.0000	−0.403896	−0.201948	+	0.979396i	$0.564727\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−40.2492	−1.62037	−0.810186	−	0.586172i	$-0.800634\pi$
$617$	−40.2492	−1.62037	−0.810186	+	0.586172i	$0.800634\pi$
$618$	0	0
$619$	−38.0000	−1.52735	−0.763674	−	0.645601i	$-0.776607\pi$
$619$	−38.0000	−1.52735	−0.763674	+	0.645601i	$0.776607\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	25.0000	1.00000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	36.3731i	1.44799i	0.689806	+	0.723994i	$0.257696\pi$
$631$	36.3731i	1.44799i	−0.689806	+	0.723994i	$0.742304\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−46.9574	−1.86345
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−13.4164	−0.529916	−0.264958	−	0.964260i	$-0.585358\pi$
$641$	−13.4164	−0.529916	−0.264958	+	0.964260i	$0.585358\pi$
$642$	0	0
$643$	8.00000	0.315489	0.157745	−	0.987480i	$-0.449578\pi$
$643$	8.00000	0.315489	0.157745	+	0.987480i	$0.449578\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	40.2492	1.58236	0.791180	−	0.611583i	$-0.209467\pi$
$647$	40.2492	1.58236	0.791180	+	0.611583i	$0.209467\pi$
$648$	0	0
$649$	− 25.9808i	− 1.01983i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−46.9574	−1.83759	−0.918793	−	0.394740i	$-0.870835\pi$
$653$	−46.9574	−1.83759	−0.918793	+	0.394740i	$0.870835\pi$
$654$	0	0
$655$	25.9808i	1.01515i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 7.74597i	− 0.301740i	−0.988554	−	0.150870i	$-0.951793\pi$
$659$	− 7.74597i	− 0.301740i	0.988554	−	0.150870i	$-0.0482075\pi$
$660$	0	0
$661$	6.92820i	0.269476i	0.990881	+	0.134738i	$0.0430193\pi$
$661$	6.92820i	0.269476i	−0.990881	+	0.134738i	$0.956981\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	51.9615i	2.01196i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−40.2492	−1.55380
$672$	0	0
$673$	13.0000	0.501113	0.250557	−	0.968102i	$-0.419386\pi$
$673$	13.0000	0.501113	0.250557	+	0.968102i	$0.419386\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 3.87298i	− 0.148851i	−0.997227	−	0.0744254i	$-0.976288\pi$
$677$	− 3.87298i	− 0.148851i	0.997227	−	0.0744254i	$-0.0237123\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	3.87298i	0.148196i	0.997251	+	0.0740978i	$0.0236077\pi$
$683$	3.87298i	0.148196i	−0.997251	+	0.0740978i	$0.976392\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	23.2379i	0.885293i
$690$	0	0
$691$	−14.0000	−0.532585	−0.266293	−	0.963892i	$-0.585799\pi$
$691$	−14.0000	−0.532585	−0.266293	+	0.963892i	$0.585799\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	7.74597i	0.293821i
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	20.1246	0.760096	0.380048	−	0.924967i	$-0.375907\pi$
$701$	20.1246	0.760096	0.380048	+	0.924967i	$0.375907\pi$
$702$	0	0
$703$	−16.0000	−0.603451
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−22.0000	−0.826227	−0.413114	−	0.910679i	$-0.635559\pi$
$709$	−22.0000	−0.826227	−0.413114	+	0.910679i	$0.635559\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	7.74597i	0.290089i
$714$	0	0
$715$	− 51.9615i	− 1.94325i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−40.2492	−1.50104	−0.750521	−	0.660846i	$-0.770198\pi$
$719$	−40.2492	−1.50104	−0.750521	+	0.660846i	$0.770198\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−67.0820	−2.49136
$726$	0	0
$727$	−5.00000	−0.185440	−0.0927199	−	0.995692i	$-0.529556\pi$
$727$	−5.00000	−0.185440	−0.0927199	+	0.995692i	$0.529556\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	6.92820i	0.255899i	0.991781	+	0.127950i	$0.0408395\pi$
$733$	6.92820i	0.255899i	−0.991781	+	0.127950i	$0.959160\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−26.8328	−0.988399
$738$	0	0
$739$	3.46410i	0.127429i	0.997968	+	0.0637145i	$0.0202947\pi$
$739$	3.46410i	0.127429i	−0.997968	+	0.0637145i	$0.979705\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	15.4919i	0.568344i	0.958773	+	0.284172i	$0.0917187\pi$
$743$	15.4919i	0.568344i	−0.958773	+	0.284172i	$0.908281\pi$
$744$	0	0
$745$	− 51.9615i	− 1.90372i
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	− 50.2295i	− 1.83290i	−0.400150	−	0.916450i	$-0.631042\pi$
$751$	− 50.2295i	− 1.83290i	0.400150	−	0.916450i	$-0.368958\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	60.3738	2.19723
$756$	0	0
$757$	−34.0000	−1.23575	−0.617876	−	0.786276i	$-0.712006\pi$
$757$	−34.0000	−1.23575	−0.617876	+	0.786276i	$0.712006\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 23.2379i	− 0.842373i	−0.906974	−	0.421187i	$-0.861614\pi$
$761$	− 23.2379i	− 0.842373i	0.906974	−	0.421187i	$-0.138386\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	23.2379i	0.839072i
$768$	0	0
$769$	− 12.1244i	− 0.437215i	−0.975813	−	0.218608i	$-0.929848\pi$
$769$	− 12.1244i	− 0.437215i	0.975813	−	0.218608i	$-0.0701515\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 15.4919i	− 0.557206i	−0.960406	−	0.278603i	$-0.910129\pi$
$773$	− 15.4919i	− 0.557206i	0.960406	−	0.278603i	$-0.0898714\pi$
$774$	0	0
$775$	−10.0000	−0.359211
$776$	0	0
$777$	0	0
$778$	0	0
$779$	30.9839i	1.11011i
$780$	0	0
$781$	30.0000	1.07348
$782$	0	0
$783$	0	0
$784$	0	0
$785$	80.4984	2.87311
$786$	0	0
$787$	−14.0000	−0.499046	−0.249523	−	0.968369i	$-0.580274\pi$
$787$	−14.0000	−0.499046	−0.249523	+	0.968369i	$0.580274\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	36.0000	1.27840
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 50.3488i	− 1.78345i	−0.452582	−	0.891723i	$-0.649497\pi$
$797$	− 50.3488i	− 1.78345i	0.452582	−	0.891723i	$-0.350503\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−26.8328	−0.946910
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−26.8328	−0.943392	−0.471696	−	0.881761i	$-0.656358\pi$
$809$	−26.8328	−0.943392	−0.471696	+	0.881761i	$0.656358\pi$
$810$	0	0
$811$	−44.0000	−1.54505	−0.772524	−	0.634985i	$-0.781006\pi$
$811$	−44.0000	−1.54505	−0.772524	+	0.634985i	$0.781006\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−40.2492	−1.40987
$816$	0	0
$817$	− 27.7128i	− 0.969549i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−6.70820	−0.234118	−0.117059	−	0.993125i	$-0.537347\pi$
$821$	−6.70820	−0.234118	−0.117059	+	0.993125i	$0.537347\pi$
$822$	0	0
$823$	10.3923i	0.362253i	0.983460	+	0.181126i	$0.0579743\pi$
$823$	10.3923i	0.362253i	−0.983460	+	0.181126i	$0.942026\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 3.87298i	− 0.134677i	−0.997730	−	0.0673384i	$-0.978549\pi$
$827$	− 3.87298i	− 0.134677i	0.997730	−	0.0673384i	$-0.0214507\pi$
$828$	0	0
$829$	13.8564i	0.481253i	0.970618	+	0.240626i	$0.0773529\pi$
$829$	13.8564i	0.481253i	−0.970618	+	0.240626i	$0.922647\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−26.8328	−0.926372	−0.463186	−	0.886261i	$-0.653294\pi$
$839$	−26.8328	−0.926372	−0.463186	+	0.886261i	$0.653294\pi$
$840$	0	0
$841$	16.0000	0.551724
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 3.87298i	− 0.133235i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	30.9839i	1.06211i
$852$	0	0
$853$	6.92820i	0.237217i	0.992941	+	0.118609i	$0.0378434\pi$
$853$	6.92820i	0.237217i	−0.992941	+	0.118609i	$0.962157\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 54.2218i	− 1.85218i	−0.377302	−	0.926090i	$-0.623148\pi$
$857$	− 54.2218i	− 1.85218i	0.377302	−	0.926090i	$-0.376852\pi$
$858$	0	0
$859$	−16.0000	−0.545913	−0.272956	−	0.962026i	$-0.588002\pi$
$859$	−16.0000	−0.545913	−0.272956	+	0.962026i	$0.588002\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	23.2379i	0.791027i	0.918460	+	0.395514i	$0.129433\pi$
$863$	23.2379i	0.791027i	−0.918460	+	0.395514i	$0.870567\pi$
$864$	0	0
$865$	60.0000	2.04006
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−46.9574	−1.59292
$870$	0	0
$871$	24.0000	0.813209
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−4.00000	−0.135070	−0.0675352	−	0.997717i	$-0.521513\pi$
$877$	−4.00000	−0.135070	−0.0675352	+	0.997717i	$0.521513\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	15.4919i	0.521936i	0.965347	+	0.260968i	$0.0840418\pi$
$881$	15.4919i	0.521936i	−0.965347	+	0.260968i	$0.915958\pi$
$882$	0	0
$883$	3.46410i	0.116576i	0.998300	+	0.0582882i	$0.0185642\pi$
$883$	3.46410i	0.116576i	−0.998300	+	0.0582882i	$0.981436\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	13.4164	0.450479	0.225239	−	0.974303i	$-0.427684\pi$
$887$	13.4164	0.450479	0.225239	+	0.974303i	$0.427684\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	53.6656	1.79585
$894$	0	0
$895$	−90.0000	−3.00837
$896$	0	0
$897$	0	0
$898$	0	0
$899$	6.70820	0.223731
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	93.9149	3.12184
$906$	0	0
$907$	3.46410i	0.115024i	0.998345	+	0.0575118i	$0.0183167\pi$
$907$	3.46410i	0.115024i	−0.998345	+	0.0575118i	$0.981683\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	23.2379i	0.769906i	0.922936	+	0.384953i	$0.125782\pi$
$911$	23.2379i	0.769906i	−0.922936	+	0.384953i	$0.874218\pi$
$912$	0	0
$913$	− 25.9808i	− 0.859838i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	− 38.1051i	− 1.25697i	−0.777821	−	0.628486i	$-0.783675\pi$
$919$	− 38.1051i	− 1.25697i	0.777821	−	0.628486i	$-0.216325\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−26.8328	−0.883213
$924$	0	0
$925$	−40.0000	−1.31519
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 46.4758i	− 1.52482i	−0.647093	−	0.762411i	$-0.724016\pi$
$929$	− 46.4758i	− 1.52482i	0.647093	−	0.762411i	$-0.275984\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	12.1244i	0.396085i	0.980193	+	0.198043i	$0.0634585\pi$
$937$	12.1244i	0.396085i	−0.980193	+	0.198043i	$0.936542\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	19.3649i	0.631278i	0.948879	+	0.315639i	$0.102219\pi$
$941$	19.3649i	0.631278i	−0.948879	+	0.315639i	$0.897781\pi$
$942$	0	0
$943$	60.0000	1.95387
$944$	0	0
$945$	0	0
$946$	0	0
$947$	7.74597i	0.251710i	0.992049	+	0.125855i	$0.0401674\pi$
$947$	7.74597i	0.251710i	−0.992049	+	0.125855i	$0.959833\pi$
$948$	0	0
$949$	24.0000	0.779073
$950$	0	0
$951$	0	0
$952$	0	0
$953$	26.8328	0.869200	0.434600	−	0.900624i	$-0.356890\pi$
$953$	26.8328	0.869200	0.434600	+	0.900624i	$0.356890\pi$
$954$	0	0
$955$	−60.0000	−1.94155
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−30.0000	−0.967742
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 27.1109i	− 0.872730i
$966$	0	0
$967$	− 29.4449i	− 0.946883i	−0.880825	−	0.473441i	$-0.843012\pi$
$967$	− 29.4449i	− 0.946883i	0.880825	−	0.473441i	$-0.156988\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	46.9574	1.50694	0.753468	−	0.657485i	$-0.228380\pi$
$971$	46.9574	1.50694	0.753468	+	0.657485i	$0.228380\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	53.6656	1.71692	0.858458	−	0.512884i	$-0.171423\pi$
$977$	53.6656	1.71692	0.858458	+	0.512884i	$0.171423\pi$
$978$	0	0
$979$	30.0000	0.958804
$980$	0	0
$981$	0	0
$982$	0	0
$983$	13.4164	0.427917	0.213958	−	0.976843i	$-0.431364\pi$
$983$	13.4164	0.427917	0.213958	+	0.976843i	$0.431364\pi$
$984$	0	0
$985$	− 51.9615i	− 1.65563i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−53.6656	−1.70647
$990$	0	0
$991$	1.73205i	0.0550204i	0.999622	+	0.0275102i	$0.00875787\pi$
$991$	1.73205i	0.0550204i	−0.999622	+	0.0275102i	$0.991242\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 30.9839i	− 0.982255i
$996$	0	0
$997$	55.4256i	1.75535i	0.479259	+	0.877674i	$0.340906\pi$
$997$	55.4256i	1.75535i	−0.479259	+	0.877674i	$0.659094\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7056.2.b.o.1567.1		4
3.2	odd	2	inner	7056.2.b.o.1567.3		4
4.3	odd	2		7056.2.b.v.1567.1		4
7.2	even	3		1008.2.cs.p.703.1	yes	4
7.3	odd	6		1008.2.cs.o.271.1	✓	4
7.6	odd	2		7056.2.b.v.1567.4		4
12.11	even	2		7056.2.b.v.1567.3		4
21.2	odd	6		1008.2.cs.p.703.2	yes	4
21.17	even	6		1008.2.cs.o.271.2	yes	4
21.20	even	2		7056.2.b.v.1567.2		4
28.3	even	6		1008.2.cs.p.271.1	yes	4
28.23	odd	6		1008.2.cs.o.703.1	yes	4
28.27	even	2	inner	7056.2.b.o.1567.4		4
84.23	even	6		1008.2.cs.o.703.2	yes	4
84.59	odd	6		1008.2.cs.p.271.2	yes	4
84.83	odd	2	inner	7056.2.b.o.1567.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1008.2.cs.o.271.1	✓	4	7.3	odd	6
1008.2.cs.o.271.2	yes	4	21.17	even	6
1008.2.cs.o.703.1	yes	4	28.23	odd	6
1008.2.cs.o.703.2	yes	4	84.23	even	6
1008.2.cs.p.271.1	yes	4	28.3	even	6
1008.2.cs.p.271.2	yes	4	84.59	odd	6
1008.2.cs.p.703.1	yes	4	7.2	even	3
1008.2.cs.p.703.2	yes	4	21.2	odd	6
7056.2.b.o.1567.1		4	1.1	even	1	trivial
7056.2.b.o.1567.2		4	84.83	odd	2	inner
7056.2.b.o.1567.3		4	3.2	odd	2	inner
7056.2.b.o.1567.4		4	28.27	even	2	inner
7056.2.b.v.1567.1		4	4.3	odd	2
7056.2.b.v.1567.2		4	21.20	even	2
7056.2.b.v.1567.3		4	12.11	even	2
7056.2.b.v.1567.4		4	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-3.87298i q^{5} +O(q^{10})\)	\(q-3.87298i q^{5} +3.87298i q^{11} -3.46410i q^{13} -4.00000 q^{19} +7.74597i q^{23} -10.0000 q^{25} +6.70820 q^{29} +1.00000 q^{31} +4.00000 q^{37} -7.74597i q^{41} +6.92820i q^{43} -13.4164 q^{47} -6.70820 q^{53} +15.0000 q^{55} -6.70820 q^{59} +10.3923i q^{61} -13.4164 q^{65} +6.92820i q^{67} -7.74597i q^{71} +6.92820i q^{73} +12.1244i q^{79} -6.70820 q^{83} -7.74597i q^{89} +15.4919i q^{95} +5.19615i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q - 16 q^{19} - 40 q^{25} + 4 q^{31} + 16 q^{37} + 60 q^{55}+O(q^{100}) \) 4 * q - 16 * q^19 - 40 * q^25 + 4 * q^31 + 16 * q^37 + 60 * q^55

Introduction

L-functions

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