LMFDB - Embedded newform 7056.2.k.d.881.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7056,2,Mod(881,7056)]

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([0, 0, 1, 1]))

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

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

chi := DirichletCharacter("7056.881");

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.k (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:	$56.3424436662$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{16})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 1 $ x^8 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 882)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			881.6
Root			$-0.382683 - 0.923880i$ of defining polynomial
Character	$\chi$	$=$	7056.881
Dual form			7056.2.k.d.881.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.765367 q^{5} +O(q^{10})$	$q+0.765367 q^{5} +4.00000i q^{11} -4.90923i q^{13} -5.54328 q^{17} +7.39104i q^{19} -5.65685i q^{23} -4.41421 q^{25} +1.65685i q^{29} +4.32957i q^{31} +8.24264 q^{37} +1.84776 q^{41} +1.65685 q^{43} -7.39104 q^{47} -8.24264i q^{53} +3.06147i q^{55} +4.32957 q^{59} -14.0167i q^{61} -3.75736i q^{65} -4.00000 q^{67} -13.6569i q^{71} -3.82683i q^{73} -5.65685 q^{79} -13.5140 q^{83} -4.24264 q^{85} -18.6089 q^{89} +5.65685i q^{95} +3.82683i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q - 24 q^{25} + 32 q^{37} - 32 q^{43} - 32 q^{67}+O(q^{100}) $ 8 * q - 24 * q^25 + 32 * q^37 - 32 * q^43 - 32 * q^67

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
$5$	0.765367	0.342282	0.171141	−	0.985247i	$-0.445255\pi$
$5$	0.765367	0.342282	0.171141	+	0.985247i	$0.445255\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	0	0
$10$	0	0
$11$	4.00000i	1.20605i	0.797724	+	0.603023i	$0.206037\pi$
$11$	4.00000i	1.20605i	−0.797724	+	0.603023i	$0.793963\pi$
$12$	0	0
$13$	− 4.90923i	− 1.36157i	−0.732481	−	0.680787i	$-0.761638\pi$
$13$	− 4.90923i	− 1.36157i	0.732481	−	0.680787i	$-0.238362\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−5.54328	−1.34444	−0.672221	−	0.740350i	$-0.734660\pi$
$17$	−5.54328	−1.34444	−0.672221	+	0.740350i	$0.734660\pi$
$18$	0	0
$19$	7.39104i	1.69562i	0.530300	+	0.847810i	$0.322079\pi$
$19$	7.39104i	1.69562i	−0.530300	+	0.847810i	$0.677921\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 5.65685i	− 1.17954i	−0.807573	−	0.589768i	$-0.799219\pi$
$23$	− 5.65685i	− 1.17954i	0.807573	−	0.589768i	$-0.200781\pi$
$24$	0	0
$25$	−4.41421	−0.882843
$26$	0	0
$27$	0	0
$28$	0	0
$29$	1.65685i	0.307670i	0.988097	+	0.153835i	$0.0491624\pi$
$29$	1.65685i	0.307670i	−0.988097	+	0.153835i	$0.950838\pi$
$30$	0	0
$31$	4.32957i	0.777614i	0.921319	+	0.388807i	$0.127113\pi$
$31$	4.32957i	0.777614i	−0.921319	+	0.388807i	$0.872887\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	8.24264	1.35508	0.677541	−	0.735485i	$-0.263046\pi$
$37$	8.24264	1.35508	0.677541	+	0.735485i	$0.263046\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	1.84776	0.288571	0.144286	−	0.989536i	$-0.453912\pi$
$41$	1.84776	0.288571	0.144286	+	0.989536i	$0.453912\pi$
$42$	0	0
$43$	1.65685	0.252668	0.126334	−	0.991988i	$-0.459679\pi$
$43$	1.65685	0.252668	0.126334	+	0.991988i	$0.459679\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−7.39104	−1.07809	−0.539047	−	0.842276i	$-0.681215\pi$
$47$	−7.39104	−1.07809	−0.539047	+	0.842276i	$0.681215\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 8.24264i	− 1.13221i	−0.824332	−	0.566107i	$-0.808449\pi$
$53$	− 8.24264i	− 1.13221i	0.824332	−	0.566107i	$-0.191551\pi$
$54$	0	0
$55$	3.06147i	0.412808i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	4.32957	0.563662	0.281831	−	0.959464i	$-0.409058\pi$
$59$	4.32957	0.563662	0.281831	+	0.959464i	$0.409058\pi$
$60$	0	0
$61$	− 14.0167i	− 1.79466i	−0.441365	−	0.897328i	$-0.645506\pi$
$61$	− 14.0167i	− 1.79466i	0.441365	−	0.897328i	$-0.354494\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 3.75736i	− 0.466043i
$66$	0	0
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 13.6569i	− 1.62077i	−0.585897	−	0.810385i	$-0.699258\pi$
$71$	− 13.6569i	− 1.62077i	0.585897	−	0.810385i	$-0.300742\pi$
$72$	0	0
$73$	− 3.82683i	− 0.447897i	−0.974601	−	0.223949i	$-0.928105\pi$
$73$	− 3.82683i	− 0.447897i	0.974601	−	0.223949i	$-0.0718948\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−5.65685	−0.636446	−0.318223	−	0.948016i	$-0.603086\pi$
$79$	−5.65685	−0.636446	−0.318223	+	0.948016i	$0.603086\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−13.5140	−1.48335	−0.741676	−	0.670759i	$-0.765969\pi$
$83$	−13.5140	−1.48335	−0.741676	+	0.670759i	$0.765969\pi$
$84$	0	0
$85$	−4.24264	−0.460179
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−18.6089	−1.97254	−0.986270	−	0.165140i	$-0.947192\pi$
$89$	−18.6089	−1.97254	−0.986270	+	0.165140i	$0.947192\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	5.65685i	0.580381i
$96$	0	0
$97$	3.82683i	0.388556i	0.980946	+	0.194278i	$0.0622364\pi$
$97$	3.82683i	0.388556i	−0.980946	+	0.194278i	$0.937764\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	5.35757	0.533098	0.266549	−	0.963821i	$-0.414117\pi$
$101$	5.35757	0.533098	0.266549	+	0.963821i	$0.414117\pi$
$102$	0	0
$103$	− 13.5140i	− 1.33157i	−0.746143	−	0.665786i	$-0.768097\pi$
$103$	− 13.5140i	− 1.33157i	0.746143	−	0.665786i	$-0.231903\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	7.31371i	0.707043i	0.935426	+	0.353521i	$0.115016\pi$
$107$	7.31371i	0.707043i	−0.935426	+	0.353521i	$0.884984\pi$
$108$	0	0
$109$	−2.58579	−0.247673	−0.123837	−	0.992303i	$-0.539520\pi$
$109$	−2.58579	−0.247673	−0.123837	+	0.992303i	$0.539520\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	1.41421i	0.133038i	0.997785	+	0.0665190i	$0.0211893\pi$
$113$	1.41421i	0.133038i	−0.997785	+	0.0665190i	$0.978811\pi$
$114$	0	0
$115$	− 4.32957i	− 0.403734i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−5.00000	−0.454545
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−7.20533	−0.644464
$126$	0	0
$127$	2.34315	0.207921	0.103960	−	0.994581i	$-0.466849\pi$
$127$	2.34315	0.207921	0.103960	+	0.994581i	$0.466849\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−1.26810	−0.110795	−0.0553973	−	0.998464i	$-0.517643\pi$
$131$	−1.26810	−0.110795	−0.0553973	+	0.998464i	$0.517643\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	19.3137i	1.65008i	0.565073	+	0.825041i	$0.308848\pi$
$137$	19.3137i	1.65008i	−0.565073	+	0.825041i	$0.691152\pi$
$138$	0	0
$139$	− 22.1731i	− 1.88070i	−0.340211	−	0.940349i	$-0.610498\pi$
$139$	− 22.1731i	− 1.88070i	0.340211	−	0.940349i	$-0.389502\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	19.6369	1.64212
$144$	0	0
$145$	1.26810i	0.105310i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 10.5858i	− 0.867221i	−0.901100	−	0.433611i	$-0.857239\pi$
$149$	− 10.5858i	− 0.867221i	0.901100	−	0.433611i	$-0.142761\pi$
$150$	0	0
$151$	−13.6569	−1.11138	−0.555690	−	0.831390i	$-0.687546\pi$
$151$	−13.6569	−1.11138	−0.555690	+	0.831390i	$0.687546\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	3.31371i	0.266163i
$156$	0	0
$157$	− 4.90923i	− 0.391799i	−0.980624	−	0.195899i	$-0.937237\pi$
$157$	− 4.90923i	− 0.391799i	0.980624	−	0.195899i	$-0.0627626\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	17.6569	1.38299	0.691496	−	0.722380i	$-0.256952\pi$
$163$	17.6569	1.38299	0.691496	+	0.722380i	$0.256952\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	8.65914	0.670064	0.335032	−	0.942207i	$-0.391253\pi$
$167$	8.65914	0.670064	0.335032	+	0.942207i	$0.391253\pi$
$168$	0	0
$169$	−11.1005	−0.853885
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−12.3003	−0.935172	−0.467586	−	0.883948i	$-0.654876\pi$
$173$	−12.3003	−0.935172	−0.467586	+	0.883948i	$0.654876\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	15.3137i	1.14460i	0.820044	+	0.572300i	$0.193949\pi$
$179$	15.3137i	1.14460i	−0.820044	+	0.572300i	$0.806051\pi$
$180$	0	0
$181$	− 9.42450i	− 0.700518i	−0.936653	−	0.350259i	$-0.886094\pi$
$181$	− 9.42450i	− 0.700518i	0.936653	−	0.350259i	$-0.113906\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	6.30864	0.463821
$186$	0	0
$187$	− 22.1731i	− 1.62146i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 11.3137i	− 0.818631i	−0.912393	−	0.409316i	$-0.865768\pi$
$191$	− 11.3137i	− 0.818631i	0.912393	−	0.409316i	$-0.134232\pi$
$192$	0	0
$193$	19.3137	1.39023	0.695116	−	0.718898i	$-0.255353\pi$
$193$	19.3137	1.39023	0.695116	+	0.718898i	$0.255353\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 5.41421i	− 0.385747i	−0.981224	−	0.192873i	$-0.938219\pi$
$197$	− 5.41421i	− 0.385747i	0.981224	−	0.192873i	$-0.0617807\pi$
$198$	0	0
$199$	7.39104i	0.523937i	0.965076	+	0.261968i	$0.0843716\pi$
$199$	7.39104i	0.523937i	−0.965076	+	0.261968i	$0.915628\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	1.41421	0.0987730
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−29.5641	−2.04499
$210$	0	0
$211$	1.65685	0.114063	0.0570313	−	0.998372i	$-0.481837\pi$
$211$	1.65685	0.114063	0.0570313	+	0.998372i	$0.481837\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	1.26810	0.0864838
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	27.2132i	1.83056i
$222$	0	0
$223$	− 3.06147i	− 0.205011i	−0.994732	−	0.102506i	$-0.967314\pi$
$223$	− 3.06147i	− 0.205011i	0.994732	−	0.102506i	$-0.0326859\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−26.5027	−1.75904	−0.879522	−	0.475858i	$-0.842138\pi$
$227$	−26.5027	−1.75904	−0.879522	+	0.475858i	$0.842138\pi$
$228$	0	0
$229$	7.89377i	0.521635i	0.965388	+	0.260818i	$0.0839921\pi$
$229$	7.89377i	0.521635i	−0.965388	+	0.260818i	$0.916008\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 16.6274i	− 1.08930i	−0.838664	−	0.544649i	$-0.816663\pi$
$233$	− 16.6274i	− 1.08930i	0.838664	−	0.544649i	$-0.183337\pi$
$234$	0	0
$235$	−5.65685	−0.369012
$236$	0	0
$237$	0	0
$238$	0	0
$239$	22.6274i	1.46365i	0.681495	+	0.731823i	$0.261330\pi$
$239$	22.6274i	1.46365i	−0.681495	+	0.731823i	$0.738670\pi$
$240$	0	0
$241$	3.11586i	0.200710i	0.994952	+	0.100355i	$0.0319979\pi$
$241$	3.11586i	0.200710i	−0.994952	+	0.100355i	$0.968002\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	36.2843	2.30871
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−11.7206	−0.739798	−0.369899	−	0.929072i	$-0.620608\pi$
$251$	−11.7206	−0.739798	−0.369899	+	0.929072i	$0.620608\pi$
$252$	0	0
$253$	22.6274	1.42257
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−6.81138	−0.424882	−0.212441	−	0.977174i	$-0.568141\pi$
$257$	−6.81138	−0.424882	−0.212441	+	0.977174i	$0.568141\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 3.31371i	− 0.204332i	−0.994767	−	0.102166i	$-0.967423\pi$
$263$	− 3.31371i	− 0.204332i	0.994767	−	0.102166i	$-0.0325773\pi$
$264$	0	0
$265$	− 6.30864i	− 0.387537i
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−27.0823	−1.65124	−0.825620	−	0.564227i	$-0.809174\pi$
$269$	−27.0823	−1.65124	−0.825620	+	0.564227i	$0.809174\pi$
$270$	0	0
$271$	− 7.39104i	− 0.448973i	−0.974477	−	0.224487i	$-0.927929\pi$
$271$	− 7.39104i	− 0.448973i	0.974477	−	0.224487i	$-0.0720705\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 17.6569i	− 1.06475i
$276$	0	0
$277$	−20.9706	−1.26000	−0.630000	−	0.776596i	$-0.716945\pi$
$277$	−20.9706	−1.26000	−0.630000	+	0.776596i	$0.716945\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 22.0000i	− 1.31241i	−0.754583	−	0.656205i	$-0.772161\pi$
$281$	− 22.0000i	− 1.31241i	0.754583	−	0.656205i	$-0.227839\pi$
$282$	0	0
$283$	4.32957i	0.257366i	0.991686	+	0.128683i	$0.0410750\pi$
$283$	4.32957i	0.257366i	−0.991686	+	0.128683i	$0.958925\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	13.7279	0.807525
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−0.765367	−0.0447132	−0.0223566	−	0.999750i	$-0.507117\pi$
$293$	−0.765367	−0.0447132	−0.0223566	+	0.999750i	$0.507117\pi$
$294$	0	0
$295$	3.31371	0.192932
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−27.7708	−1.60603
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 10.7279i	− 0.614279i
$306$	0	0
$307$	− 3.06147i	− 0.174727i	−0.996176	−	0.0873636i	$-0.972156\pi$
$307$	− 3.06147i	− 0.174727i	0.996176	−	0.0873636i	$-0.0278442\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−11.7206	−0.664615	−0.332307	−	0.943171i	$-0.607827\pi$
$311$	−11.7206	−0.664615	−0.332307	+	0.943171i	$0.607827\pi$
$312$	0	0
$313$	− 10.5069i	− 0.593885i	−0.954895	−	0.296942i	$-0.904033\pi$
$313$	− 10.5069i	− 0.593885i	0.954895	−	0.296942i	$-0.0959670\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 0.242641i	− 0.0136281i	−0.999977	−	0.00681403i	$-0.997831\pi$
$317$	− 0.242641i	− 0.0136281i	0.999977	−	0.00681403i	$-0.00216899\pi$
$318$	0	0
$319$	−6.62742	−0.371064
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 40.9706i	− 2.27966i
$324$	0	0
$325$	21.6704i	1.20206i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−25.6569	−1.41023	−0.705114	−	0.709094i	$-0.749104\pi$
$331$	−25.6569	−1.41023	−0.705114	+	0.709094i	$0.749104\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−3.06147	−0.167266
$336$	0	0
$337$	25.8995	1.41084	0.705418	−	0.708792i	$-0.250759\pi$
$337$	25.8995	1.41084	0.705418	+	0.708792i	$0.250759\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−17.3183	−0.937837
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 8.68629i	− 0.466305i	−0.972440	−	0.233152i	$-0.925096\pi$
$347$	− 8.68629i	− 0.466305i	0.972440	−	0.233152i	$-0.0749041\pi$
$348$	0	0
$349$	19.6913i	1.05405i	0.849849	+	0.527026i	$0.176693\pi$
$349$	19.6913i	1.05405i	−0.849849	+	0.527026i	$0.823307\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	27.7933	1.47929	0.739644	−	0.672998i	$-0.234994\pi$
$353$	27.7933	1.47929	0.739644	+	0.672998i	$0.234994\pi$
$354$	0	0
$355$	− 10.4525i	− 0.554761i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 24.0000i	− 1.26667i	−0.773877	−	0.633336i	$-0.781685\pi$
$359$	− 24.0000i	− 1.26667i	0.773877	−	0.633336i	$-0.218315\pi$
$360$	0	0
$361$	−35.6274	−1.87513
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 2.92893i	− 0.153307i
$366$	0	0
$367$	− 23.9665i	− 1.25104i	−0.780208	−	0.625520i	$-0.784887\pi$
$367$	− 23.9665i	− 1.25104i	0.780208	−	0.625520i	$-0.215113\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	1.65685	0.0857887	0.0428943	−	0.999080i	$-0.486342\pi$
$373$	1.65685	0.0857887	0.0428943	+	0.999080i	$0.486342\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	8.13387	0.418916
$378$	0	0
$379$	24.2843	1.24740	0.623700	−	0.781664i	$-0.285629\pi$
$379$	24.2843	1.24740	0.623700	+	0.781664i	$0.285629\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−38.7485	−1.97996	−0.989979	−	0.141214i	$-0.954900\pi$
$383$	−38.7485	−1.97996	−0.989979	+	0.141214i	$0.954900\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	14.3431i	0.727226i	0.931550	+	0.363613i	$0.118457\pi$
$389$	14.3431i	0.727226i	−0.931550	+	0.363613i	$0.881543\pi$
$390$	0	0
$391$	31.3575i	1.58582i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−4.32957	−0.217844
$396$	0	0
$397$	− 18.6089i	− 0.933954i	−0.884269	−	0.466977i	$-0.845343\pi$
$397$	− 18.6089i	− 0.933954i	0.884269	−	0.466977i	$-0.154657\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 14.0000i	− 0.699127i	−0.936913	−	0.349563i	$-0.886330\pi$
$401$	− 14.0000i	− 0.699127i	0.936913	−	0.349563i	$-0.113670\pi$
$402$	0	0
$403$	21.2548	1.05878
$404$	0	0
$405$	0	0
$406$	0	0
$407$	32.9706i	1.63429i
$408$	0	0
$409$	13.0112i	0.643364i	0.946848	+	0.321682i	$0.104248\pi$
$409$	13.0112i	0.643364i	−0.946848	+	0.321682i	$0.895752\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−10.3431	−0.507725
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−29.5641	−1.44430	−0.722151	−	0.691735i	$-0.756847\pi$
$419$	−29.5641	−1.44430	−0.722151	+	0.691735i	$0.756847\pi$
$420$	0	0
$421$	−1.65685	−0.0807501	−0.0403751	−	0.999185i	$-0.512855\pi$
$421$	−1.65685	−0.0807501	−0.0403751	+	0.999185i	$0.512855\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	24.4692	1.18693
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	27.3137i	1.31566i	0.753169	+	0.657828i	$0.228524\pi$
$431$	27.3137i	1.31566i	−0.753169	+	0.657828i	$0.771476\pi$
$432$	0	0
$433$	− 31.8602i	− 1.53111i	−0.643373	−	0.765553i	$-0.722466\pi$
$433$	− 31.8602i	− 1.53111i	0.643373	−	0.765553i	$-0.277534\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	41.8100	2.00004
$438$	0	0
$439$	14.7821i	0.705510i	0.935716	+	0.352755i	$0.114755\pi$
$439$	14.7821i	0.705510i	−0.935716	+	0.352755i	$0.885245\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	6.34315i	0.301372i	0.988582	+	0.150686i	$0.0481482\pi$
$443$	6.34315i	0.301372i	−0.988582	+	0.150686i	$0.951852\pi$
$444$	0	0
$445$	−14.2426	−0.675166
$446$	0	0
$447$	0	0
$448$	0	0
$449$	23.5563i	1.11169i	0.831285	+	0.555846i	$0.187606\pi$
$449$	23.5563i	1.11169i	−0.831285	+	0.555846i	$0.812394\pi$
$450$	0	0
$451$	7.39104i	0.348030i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	26.0000	1.21623	0.608114	−	0.793849i	$-0.291926\pi$
$457$	26.0000	1.21623	0.608114	+	0.793849i	$0.291926\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	2.48181	0.115589	0.0577947	−	0.998328i	$-0.481593\pi$
$461$	2.48181	0.115589	0.0577947	+	0.998328i	$0.481593\pi$
$462$	0	0
$463$	−29.6569	−1.37827	−0.689135	−	0.724633i	$-0.742010\pi$
$463$	−29.6569	−1.37827	−0.689135	+	0.724633i	$0.742010\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−13.5140	−0.625352	−0.312676	−	0.949860i	$-0.601225\pi$
$467$	−13.5140	−0.625352	−0.312676	+	0.949860i	$0.601225\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	6.62742i	0.304729i
$474$	0	0
$475$	− 32.6256i	− 1.49697i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	16.0502	0.733351	0.366676	−	0.930349i	$-0.380496\pi$
$479$	16.0502	0.733351	0.366676	+	0.930349i	$0.380496\pi$
$480$	0	0
$481$	− 40.4650i	− 1.84504i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	2.92893i	0.132996i
$486$	0	0
$487$	−11.3137	−0.512673	−0.256337	−	0.966588i	$-0.582516\pi$
$487$	−11.3137	−0.512673	−0.256337	+	0.966588i	$0.582516\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 12.9706i	− 0.585353i	−0.956211	−	0.292677i	$-0.905454\pi$
$491$	− 12.9706i	− 0.585353i	0.956211	−	0.292677i	$-0.0945460\pi$
$492$	0	0
$493$	− 9.18440i	− 0.413645i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−17.6569	−0.790429	−0.395215	−	0.918589i	$-0.629330\pi$
$499$	−17.6569	−0.790429	−0.395215	+	0.918589i	$0.629330\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−23.9665	−1.06861	−0.534306	−	0.845291i	$-0.679427\pi$
$503$	−23.9665	−1.06861	−0.534306	+	0.845291i	$0.679427\pi$
$504$	0	0
$505$	4.10051	0.182470
$506$	0	0
$507$	0	0
$508$	0	0
$509$	0.0543929	0.00241093	0.00120546	−	0.999999i	$-0.499616\pi$
$509$	0.0543929	0.00241093	0.00120546	+	0.999999i	$0.499616\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 10.3431i	− 0.455773i
$516$	0	0
$517$	− 29.5641i	− 1.30023i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−15.5474	−0.681146	−0.340573	−	0.940218i	$-0.610621\pi$
$521$	−15.5474	−0.681146	−0.340573	+	0.940218i	$0.610621\pi$
$522$	0	0
$523$	2.53620i	0.110900i	0.998461	+	0.0554502i	$0.0176594\pi$
$523$	2.53620i	0.110900i	−0.998461	+	0.0554502i	$0.982341\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 24.0000i	− 1.04546i
$528$	0	0
$529$	−9.00000	−0.391304
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 9.07107i	− 0.392912i
$534$	0	0
$535$	5.59767i	0.242008i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−20.6274	−0.886842	−0.443421	−	0.896313i	$-0.646235\pi$
$541$	−20.6274	−0.886842	−0.443421	+	0.896313i	$0.646235\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−1.97908	−0.0847743
$546$	0	0
$547$	−4.00000	−0.171028	−0.0855138	−	0.996337i	$-0.527253\pi$
$547$	−4.00000	−0.171028	−0.0855138	+	0.996337i	$0.527253\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−12.2459	−0.521692
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	8.72792i	0.369814i	0.982756	+	0.184907i	$0.0591984\pi$
$557$	8.72792i	0.369814i	−0.982756	+	0.184907i	$0.940802\pi$
$558$	0	0
$559$	− 8.13387i	− 0.344026i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−14.7821	−0.622990	−0.311495	−	0.950248i	$-0.600830\pi$
$563$	−14.7821	−0.622990	−0.311495	+	0.950248i	$0.600830\pi$
$564$	0	0
$565$	1.08239i	0.0455366i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	41.9411i	1.75826i	0.476579	+	0.879132i	$0.341877\pi$
$569$	41.9411i	1.75826i	−0.476579	+	0.879132i	$0.658123\pi$
$570$	0	0
$571$	26.6274	1.11432	0.557161	−	0.830404i	$-0.311890\pi$
$571$	26.6274	1.11432	0.557161	+	0.830404i	$0.311890\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	24.9706i	1.04134i
$576$	0	0
$577$	6.88830i	0.286764i	0.989667	+	0.143382i	$0.0457977\pi$
$577$	6.88830i	0.286764i	−0.989667	+	0.143382i	$0.954202\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	32.9706	1.36550
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−1.26810	−0.0523401	−0.0261701	−	0.999658i	$-0.508331\pi$
$587$	−1.26810	−0.0523401	−0.0261701	+	0.999658i	$0.508331\pi$
$588$	0	0
$589$	−32.0000	−1.31854
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−10.5069	−0.431467	−0.215733	−	0.976452i	$-0.569214\pi$
$593$	−10.5069	−0.431467	−0.215733	+	0.976452i	$0.569214\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	37.6569i	1.53862i	0.638877	+	0.769309i	$0.279399\pi$
$599$	37.6569i	1.53862i	−0.638877	+	0.769309i	$0.720601\pi$
$600$	0	0
$601$	− 5.54328i	− 0.226115i	−0.993588	−	0.113057i	$-0.963936\pi$
$601$	− 5.54328i	− 0.226115i	0.993588	−	0.113057i	$-0.0360644\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−3.82683	−0.155583
$606$	0	0
$607$	− 9.92724i	− 0.402934i	−0.979495	−	0.201467i	$-0.935429\pi$
$607$	− 9.92724i	− 0.402934i	0.979495	−	0.201467i	$-0.0645709\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	36.2843i	1.46790i
$612$	0	0
$613$	33.2132	1.34147	0.670734	−	0.741698i	$-0.265979\pi$
$613$	33.2132	1.34147	0.670734	+	0.741698i	$0.265979\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 16.0000i	− 0.644136i	−0.946717	−	0.322068i	$-0.895622\pi$
$617$	− 16.0000i	− 0.644136i	0.946717	−	0.322068i	$-0.104378\pi$
$618$	0	0
$619$	− 46.1396i	− 1.85451i	−0.374435	−	0.927253i	$-0.622163\pi$
$619$	− 46.1396i	− 1.85451i	0.374435	−	0.927253i	$-0.377837\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	16.5563	0.662254
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−45.6912	−1.82183
$630$	0	0
$631$	−41.9411	−1.66965	−0.834825	−	0.550516i	$-0.814431\pi$
$631$	−41.9411	−1.66965	−0.834825	+	0.550516i	$0.814431\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	1.79337	0.0711676
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	2.00000i	0.0789953i	0.999220	+	0.0394976i	$0.0125758\pi$
$641$	2.00000i	0.0789953i	−0.999220	+	0.0394976i	$0.987424\pi$
$642$	0	0
$643$	19.1116i	0.753690i	0.926276	+	0.376845i	$0.122991\pi$
$643$	19.1116i	0.753690i	−0.926276	+	0.376845i	$0.877009\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	33.1509	1.30330	0.651648	−	0.758522i	$-0.274078\pi$
$647$	33.1509	1.30330	0.651648	+	0.758522i	$0.274078\pi$
$648$	0	0
$649$	17.3183i	0.679802i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 14.3431i	− 0.561291i	−0.959812	−	0.280645i	$-0.909452\pi$
$653$	− 14.3431i	− 0.561291i	0.959812	−	0.280645i	$-0.0905485\pi$
$654$	0	0
$655$	−0.970563	−0.0379230
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 20.0000i	− 0.779089i	−0.921008	−	0.389545i	$-0.872632\pi$
$659$	− 20.0000i	− 0.779089i	0.921008	−	0.389545i	$-0.127368\pi$
$660$	0	0
$661$	− 28.2417i	− 1.09847i	−0.835667	−	0.549236i	$-0.814919\pi$
$661$	− 28.2417i	− 1.09847i	0.835667	−	0.549236i	$-0.185081\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	9.37258	0.362908
$668$	0	0
$669$	0	0
$670$	0	0
$671$	56.0668	2.16444
$672$	0	0
$673$	−18.8701	−0.727387	−0.363694	−	0.931519i	$-0.618484\pi$
$673$	−18.8701	−0.727387	−0.363694	+	0.931519i	$0.618484\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−9.87285	−0.379444	−0.189722	−	0.981838i	$-0.560759\pi$
$677$	−9.87285	−0.379444	−0.189722	+	0.981838i	$0.560759\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	4.97056i	0.190193i	0.995468	+	0.0950966i	$0.0303160\pi$
$683$	4.97056i	0.190193i	−0.995468	+	0.0950966i	$0.969684\pi$
$684$	0	0
$685$	14.7821i	0.564794i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−40.4650	−1.54159
$690$	0	0
$691$	− 15.3073i	− 0.582319i	−0.956675	−	0.291159i	$-0.905959\pi$
$691$	− 15.3073i	− 0.582319i	0.956675	−	0.291159i	$-0.0940410\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 16.9706i	− 0.643730i
$696$	0	0
$697$	−10.2426	−0.387968
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 2.00000i	− 0.0755390i	−0.999286	−	0.0377695i	$-0.987975\pi$
$701$	− 2.00000i	− 0.0755390i	0.999286	−	0.0377695i	$-0.0120253\pi$
$702$	0	0
$703$	60.9217i	2.29770i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	28.5269	1.07135	0.535675	−	0.844424i	$-0.320057\pi$
$709$	28.5269	1.07135	0.535675	+	0.844424i	$0.320057\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	24.4917	0.917223
$714$	0	0
$715$	15.0294	0.562069
$716$	0	0
$717$	0	0
$718$	0	0
$719$	43.0781	1.60654	0.803271	−	0.595613i	$-0.203091\pi$
$719$	43.0781	1.60654	0.803271	+	0.595613i	$0.203091\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 7.31371i	− 0.271624i
$726$	0	0
$727$	− 0.525265i	− 0.0194810i	−0.999953	−	0.00974050i	$-0.996899\pi$
$727$	− 0.525265i	− 0.0194810i	0.999953	−	0.00974050i	$-0.00310055\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−9.18440	−0.339697
$732$	0	0
$733$	20.1396i	0.743875i	0.928258	+	0.371937i	$0.121306\pi$
$733$	20.1396i	0.743875i	−0.928258	+	0.371937i	$0.878694\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 16.0000i	− 0.589368i
$738$	0	0
$739$	−37.9411	−1.39569	−0.697843	−	0.716250i	$-0.745857\pi$
$739$	−37.9411	−1.39569	−0.697843	+	0.716250i	$0.745857\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 27.3137i	− 1.00204i	−0.865435	−	0.501021i	$-0.832958\pi$
$743$	− 27.3137i	− 1.00204i	0.865435	−	0.501021i	$-0.167042\pi$
$744$	0	0
$745$	− 8.10201i	− 0.296835i
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	26.3431	0.961275	0.480638	−	0.876919i	$-0.340405\pi$
$751$	26.3431	0.961275	0.480638	+	0.876919i	$0.340405\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−10.4525	−0.380406
$756$	0	0
$757$	19.5563	0.710788	0.355394	−	0.934717i	$-0.384347\pi$
$757$	19.5563	0.710788	0.355394	+	0.934717i	$0.384347\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−20.1396	−0.730061	−0.365031	−	0.930995i	$-0.618942\pi$
$761$	−20.1396	−0.730061	−0.365031	+	0.930995i	$0.618942\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 21.2548i	− 0.767468i
$768$	0	0
$769$	27.2680i	0.983311i	0.870790	+	0.491655i	$0.163608\pi$
$769$	27.2680i	0.983311i	−0.870790	+	0.491655i	$0.836392\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	8.71353	0.313404	0.156702	−	0.987646i	$-0.449914\pi$
$773$	8.71353	0.313404	0.156702	+	0.987646i	$0.449914\pi$
$774$	0	0
$775$	− 19.1116i	− 0.686510i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	13.6569i	0.489308i
$780$	0	0
$781$	54.6274	1.95472
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 3.75736i	− 0.134106i
$786$	0	0
$787$	− 9.92724i	− 0.353868i	−0.984223	−	0.176934i	$-0.943382\pi$
$787$	− 9.92724i	− 0.353868i	0.984223	−	0.176934i	$-0.0566179\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−68.8112	−2.44356
$794$	0	0
$795$	0	0
$796$	0	0
$797$	6.88830	0.243996	0.121998	−	0.992530i	$-0.461070\pi$
$797$	6.88830	0.243996	0.121998	+	0.992530i	$0.461070\pi$
$798$	0	0
$799$	40.9706	1.44943
$800$	0	0
$801$	0	0
$802$	0	0
$803$	15.3073	0.540184
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	11.2721i	0.396305i	0.980171	+	0.198153i	$0.0634942\pi$
$809$	11.2721i	0.396305i	−0.980171	+	0.198153i	$0.936506\pi$
$810$	0	0
$811$	− 22.6984i	− 0.797048i	−0.917158	−	0.398524i	$-0.869523\pi$
$811$	− 22.6984i	− 0.797048i	0.917158	−	0.398524i	$-0.130477\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	13.5140	0.473374
$816$	0	0
$817$	12.2459i	0.428429i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 39.3553i	− 1.37351i	−0.726889	−	0.686755i	$-0.759034\pi$
$821$	− 39.3553i	− 1.37351i	0.726889	−	0.686755i	$-0.240966\pi$
$822$	0	0
$823$	−29.6569	−1.03377	−0.516886	−	0.856054i	$-0.672909\pi$
$823$	−29.6569	−1.03377	−0.516886	+	0.856054i	$0.672909\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 11.0294i	− 0.383531i	−0.981441	−	0.191766i	$-0.938579\pi$
$827$	− 11.0294i	− 0.383531i	0.981441	−	0.191766i	$-0.0614213\pi$
$828$	0	0
$829$	8.60474i	0.298855i	0.988773	+	0.149428i	$0.0477431\pi$
$829$	8.60474i	0.298855i	−0.988773	+	0.149428i	$0.952257\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	6.62742	0.229351
$836$	0	0
$837$	0	0
$838$	0	0
$839$	50.4692	1.74239	0.871194	−	0.490938i	$-0.163346\pi$
$839$	50.4692	1.74239	0.871194	+	0.490938i	$0.163346\pi$
$840$	0	0
$841$	26.2548	0.905339
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−8.49596	−0.292270
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 46.6274i	− 1.59837i
$852$	0	0
$853$	29.5098i	1.01039i	0.863004	+	0.505197i	$0.168580\pi$
$853$	29.5098i	1.01039i	−0.863004	+	0.505197i	$0.831420\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	15.5474	0.531090	0.265545	−	0.964098i	$-0.414448\pi$
$857$	15.5474	0.531090	0.265545	+	0.964098i	$0.414448\pi$
$858$	0	0
$859$	40.5419i	1.38327i	0.722246	+	0.691636i	$0.243110\pi$
$859$	40.5419i	1.38327i	−0.722246	+	0.691636i	$0.756890\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 28.2843i	− 0.962808i	−0.876499	−	0.481404i	$-0.840127\pi$
$863$	− 28.2843i	− 0.962808i	0.876499	−	0.481404i	$-0.159873\pi$
$864$	0	0
$865$	−9.41421	−0.320093
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 22.6274i	− 0.767583i
$870$	0	0
$871$	19.6369i	0.665371i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−2.58579	−0.0873158	−0.0436579	−	0.999047i	$-0.513901\pi$
$877$	−2.58579	−0.0873158	−0.0436579	+	0.999047i	$0.513901\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	20.2166	0.681113	0.340557	−	0.940224i	$-0.389384\pi$
$881$	20.2166	0.681113	0.340557	+	0.940224i	$0.389384\pi$
$882$	0	0
$883$	57.2548	1.92678	0.963389	−	0.268106i	$-0.0863979\pi$
$883$	57.2548	1.92678	0.963389	+	0.268106i	$0.0863979\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	25.2346	0.847294	0.423647	−	0.905827i	$-0.360750\pi$
$887$	25.2346	0.847294	0.423647	+	0.905827i	$0.360750\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 54.6274i	− 1.82804i
$894$	0	0
$895$	11.7206i	0.391777i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−7.17346	−0.239248
$900$	0	0
$901$	45.6912i	1.52220i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 7.21320i	− 0.239775i
$906$	0	0
$907$	−12.0000	−0.398453	−0.199227	−	0.979953i	$-0.563843\pi$
$907$	−12.0000	−0.398453	−0.199227	+	0.979953i	$0.563843\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 14.6274i	− 0.484628i	−0.970198	−	0.242314i	$-0.922094\pi$
$911$	− 14.6274i	− 0.484628i	0.970198	−	0.242314i	$-0.0779064\pi$
$912$	0	0
$913$	− 54.0559i	− 1.78899i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−50.9117	−1.67942	−0.839711	−	0.543034i	$-0.817276\pi$
$919$	−50.9117	−1.67942	−0.839711	+	0.543034i	$0.817276\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−67.0446	−2.20680
$924$	0	0
$925$	−36.3848	−1.19632
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−29.3240	−0.962091	−0.481045	−	0.876696i	$-0.659743\pi$
$929$	−29.3240	−0.962091	−0.481045	+	0.876696i	$0.659743\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 16.9706i	− 0.554997i
$936$	0	0
$937$	20.3253i	0.664000i	0.943279	+	0.332000i	$0.107723\pi$
$937$	20.3253i	0.664000i	−0.943279	+	0.332000i	$0.892277\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−28.3504	−0.924198	−0.462099	−	0.886828i	$-0.652903\pi$
$941$	−28.3504	−0.924198	−0.462099	+	0.886828i	$0.652903\pi$
$942$	0	0
$943$	− 10.4525i	− 0.340380i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	58.6274i	1.90514i	0.304327	+	0.952568i	$0.401568\pi$
$947$	58.6274i	1.90514i	−0.304327	+	0.952568i	$0.598432\pi$
$948$	0	0
$949$	−18.7868	−0.609845
$950$	0	0
$951$	0	0
$952$	0	0
$953$	30.1005i	0.975051i	0.873109	+	0.487526i	$0.162100\pi$
$953$	30.1005i	0.975051i	−0.873109	+	0.487526i	$0.837900\pi$
$954$	0	0
$955$	− 8.65914i	− 0.280203i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	12.2548	0.395317
$962$	0	0
$963$	0	0
$964$	0	0
$965$	14.7821	0.475852
$966$	0	0
$967$	−16.9706	−0.545737	−0.272868	−	0.962051i	$-0.587972\pi$
$967$	−16.9706	−0.545737	−0.272868	+	0.962051i	$0.587972\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	37.4804	1.20280	0.601402	−	0.798946i	$-0.294609\pi$
$971$	37.4804	1.20280	0.601402	+	0.798946i	$0.294609\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 25.9411i	− 0.829930i	−0.909837	−	0.414965i	$-0.863794\pi$
$977$	− 25.9411i	− 0.829930i	0.909837	−	0.414965i	$-0.136206\pi$
$978$	0	0
$979$	− 74.4356i	− 2.37897i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	14.0392	0.447782	0.223891	−	0.974614i	$-0.428124\pi$
$983$	14.0392	0.447782	0.223891	+	0.974614i	$0.428124\pi$
$984$	0	0
$985$	− 4.14386i	− 0.132034i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 9.37258i	− 0.298031i
$990$	0	0
$991$	−12.6863	−0.402993	−0.201497	−	0.979489i	$-0.564581\pi$
$991$	−12.6863	−0.402993	−0.201497	+	0.979489i	$0.564581\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	5.65685i	0.179334i
$996$	0	0
$997$	30.7779i	0.974744i	0.873194	+	0.487372i	$0.162044\pi$
$997$	30.7779i	0.974744i	−0.873194	+	0.487372i	$0.837956\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7056.2.k.d.881.6		8
3.2	odd	2	inner	7056.2.k.d.881.3		8
4.3	odd	2		882.2.d.b.881.7	yes	8
7.6	odd	2	inner	7056.2.k.d.881.4		8
12.11	even	2		882.2.d.b.881.2	✓	8
21.20	even	2	inner	7056.2.k.d.881.5		8
28.3	even	6		882.2.k.b.215.3		16
28.11	odd	6		882.2.k.b.215.2		16
28.19	even	6		882.2.k.b.521.7		16
28.23	odd	6		882.2.k.b.521.6		16
28.27	even	2		882.2.d.b.881.6	yes	8
84.11	even	6		882.2.k.b.215.7		16
84.23	even	6		882.2.k.b.521.3		16
84.47	odd	6		882.2.k.b.521.2		16
84.59	odd	6		882.2.k.b.215.6		16
84.83	odd	2		882.2.d.b.881.3	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
882.2.d.b.881.2	✓	8	12.11	even	2
882.2.d.b.881.3	yes	8	84.83	odd	2
882.2.d.b.881.6	yes	8	28.27	even	2
882.2.d.b.881.7	yes	8	4.3	odd	2
882.2.k.b.215.2		16	28.11	odd	6
882.2.k.b.215.3		16	28.3	even	6
882.2.k.b.215.6		16	84.59	odd	6
882.2.k.b.215.7		16	84.11	even	6
882.2.k.b.521.2		16	84.47	odd	6
882.2.k.b.521.3		16	84.23	even	6
882.2.k.b.521.6		16	28.23	odd	6
882.2.k.b.521.7		16	28.19	even	6
7056.2.k.d.881.3		8	3.2	odd	2	inner
7056.2.k.d.881.4		8	7.6	odd	2	inner
7056.2.k.d.881.5		8	21.20	even	2	inner
7056.2.k.d.881.6		8	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.k (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+0.765367 q^{5} +O(q^{10})\)	\(q+0.765367 q^{5} +4.00000i q^{11} -4.90923i q^{13} -5.54328 q^{17} +7.39104i q^{19} -5.65685i q^{23} -4.41421 q^{25} +1.65685i q^{29} +4.32957i q^{31} +8.24264 q^{37} +1.84776 q^{41} +1.65685 q^{43} -7.39104 q^{47} -8.24264i q^{53} +3.06147i q^{55} +4.32957 q^{59} -14.0167i q^{61} -3.75736i q^{65} -4.00000 q^{67} -13.6569i q^{71} -3.82683i q^{73} -5.65685 q^{79} -13.5140 q^{83} -4.24264 q^{85} -18.6089 q^{89} +5.65685i q^{95} +3.82683i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q - 24 q^{25} + 32 q^{37} - 32 q^{43} - 32 q^{67}+O(q^{100}) \) 8 * q - 24 * q^25 + 32 * q^37 - 32 * q^43 - 32 * q^67

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