LMFDB - Embedded newform 7056.2.k.e.881.8

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 1764)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			881.8
Root			$-0.923880 - 0.382683i$ of defining polynomial
Character	$\chi$	$=$	7056.881
Dual form			7056.2.k.e.881.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.84776 q^{5} +O(q^{10})$	$q+1.84776 q^{5} +2.00000i q^{11} -4.46088i q^{13} -2.29610 q^{17} -1.53073i q^{19} +8.82843i q^{23} -1.58579 q^{25} +1.17157i q^{29} +5.86030i q^{31} +8.24264 q^{37} -11.8519 q^{41} -1.17157 q^{43} -8.02509 q^{47} -3.75736i q^{53} +3.69552i q^{55} +9.81845 q^{59} +12.3003i q^{61} -8.24264i q^{65} -12.4853 q^{67} +13.3137i q^{71} +2.74444i q^{73} -11.3137 q^{79} +10.4525 q^{83} -4.24264 q^{85} +14.4650 q^{89} -2.82843i q^{95} -2.74444i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	1.84776	0.826343	0.413171	−	0.910653i	$-0.364421\pi$
$5$	1.84776	0.826343	0.413171	+	0.910653i	$0.364421\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	0	0
$10$	0	0
$11$	2.00000i	0.603023i	0.953463	+	0.301511i	$0.0974911\pi$
$11$	2.00000i	0.603023i	−0.953463	+	0.301511i	$0.902509\pi$
$12$	0	0
$13$	− 4.46088i	− 1.23723i	−0.785695	−	0.618613i	$-0.787695\pi$
$13$	− 4.46088i	− 1.23723i	0.785695	−	0.618613i	$-0.212305\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−2.29610	−0.556886	−0.278443	−	0.960453i	$-0.589818\pi$
$17$	−2.29610	−0.556886	−0.278443	+	0.960453i	$0.589818\pi$
$18$	0	0
$19$	− 1.53073i	− 0.351174i	−0.984464	−	0.175587i	$-0.943818\pi$
$19$	− 1.53073i	− 0.351174i	0.984464	−	0.175587i	$-0.0561824\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	8.82843i	1.84085i	0.390914	+	0.920427i	$0.372159\pi$
$23$	8.82843i	1.84085i	−0.390914	+	0.920427i	$0.627841\pi$
$24$	0	0
$25$	−1.58579	−0.317157
$26$	0	0
$27$	0	0
$28$	0	0
$29$	1.17157i	0.217556i	0.994066	+	0.108778i	$0.0346937\pi$
$29$	1.17157i	0.217556i	−0.994066	+	0.108778i	$0.965306\pi$
$30$	0	0
$31$	5.86030i	1.05254i	0.850317	+	0.526271i	$0.176410\pi$
$31$	5.86030i	1.05254i	−0.850317	+	0.526271i	$0.823590\pi$
$32$	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
$41$	−11.8519	−1.85096	−0.925480	−	0.378798i	$-0.876338\pi$
$41$	−11.8519	−1.85096	−0.925480	+	0.378798i	$0.876338\pi$
$42$	0	0
$43$	−1.17157	−0.178663	−0.0893316	−	0.996002i	$-0.528473\pi$
$43$	−1.17157	−0.178663	−0.0893316	+	0.996002i	$0.528473\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−8.02509	−1.17058	−0.585290	−	0.810824i	$-0.699019\pi$
$47$	−8.02509	−1.17058	−0.585290	+	0.810824i	$0.699019\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 3.75736i	− 0.516113i	−0.966130	−	0.258056i	$-0.916918\pi$
$53$	− 3.75736i	− 0.516113i	0.966130	−	0.258056i	$-0.0830821\pi$
$54$	0	0
$55$	3.69552i	0.498304i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	9.81845	1.27825	0.639127	−	0.769101i	$-0.279296\pi$
$59$	9.81845	1.27825	0.639127	+	0.769101i	$0.279296\pi$
$60$	0	0
$61$	12.3003i	1.57489i	0.616387	+	0.787444i	$0.288596\pi$
$61$	12.3003i	1.57489i	−0.616387	+	0.787444i	$0.711404\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 8.24264i	− 1.02237i
$66$	0	0
$67$	−12.4853	−1.52532	−0.762660	−	0.646800i	$-0.776107\pi$
$67$	−12.4853	−1.52532	−0.762660	+	0.646800i	$0.776107\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	13.3137i	1.58005i	0.613077	+	0.790023i	$0.289932\pi$
$71$	13.3137i	1.58005i	−0.613077	+	0.790023i	$0.710068\pi$
$72$	0	0
$73$	2.74444i	0.321213i	0.987019	+	0.160606i	$0.0513450\pi$
$73$	2.74444i	0.321213i	−0.987019	+	0.160606i	$0.948655\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−11.3137	−1.27289	−0.636446	−	0.771321i	$-0.719596\pi$
$79$	−11.3137	−1.27289	−0.636446	+	0.771321i	$0.719596\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	10.4525	1.14731	0.573656	−	0.819097i	$-0.305525\pi$
$83$	10.4525	1.14731	0.573656	+	0.819097i	$0.305525\pi$
$84$	0	0
$85$	−4.24264	−0.460179
$86$	0	0
$87$	0	0
$88$	0	0
$89$	14.4650	1.53329	0.766646	−	0.642070i	$-0.221924\pi$
$89$	14.4650	1.53329	0.766646	+	0.642070i	$0.221924\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 2.82843i	− 0.290191i
$96$	0	0
$97$	− 2.74444i	− 0.278656i	−0.990246	−	0.139328i	$-0.955506\pi$
$97$	− 2.74444i	− 0.278656i	0.990246	−	0.139328i	$-0.0444942\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−2.74444	−0.273082	−0.136541	−	0.990634i	$-0.543599\pi$
$101$	−2.74444	−0.273082	−0.136541	+	0.990634i	$0.543599\pi$
$102$	0	0
$103$	0.634051i	0.0624749i	0.999512	+	0.0312374i	$0.00994480\pi$
$103$	0.634051i	0.0624749i	−0.999512	+	0.0312374i	$0.990055\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	12.8284i	1.24017i	0.784534	+	0.620085i	$0.212902\pi$
$107$	12.8284i	1.24017i	−0.784534	+	0.620085i	$0.787098\pi$
$108$	0	0
$109$	3.07107	0.294155	0.147077	−	0.989125i	$-0.453013\pi$
$109$	3.07107	0.294155	0.147077	+	0.989125i	$0.453013\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	13.4142i	1.26190i	0.775822	+	0.630952i	$0.217335\pi$
$113$	13.4142i	1.26190i	−0.775822	+	0.630952i	$0.782665\pi$
$114$	0	0
$115$	16.3128i	1.52118i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	7.00000	0.636364
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−12.1689	−1.08842
$126$	0	0
$127$	−6.82843	−0.605925	−0.302962	−	0.953002i	$-0.597976\pi$
$127$	−6.82843	−0.605925	−0.302962	+	0.953002i	$0.597976\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	11.3492	0.991583	0.495792	−	0.868442i	$-0.334878\pi$
$131$	11.3492	0.991583	0.495792	+	0.868442i	$0.334878\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 5.17157i	− 0.441837i	−0.975292	−	0.220919i	$-0.929094\pi$
$137$	− 5.17157i	− 0.441837i	0.975292	−	0.220919i	$-0.0709055\pi$
$138$	0	0
$139$	6.49435i	0.550844i	0.961323	+	0.275422i	$0.0888176\pi$
$139$	6.49435i	0.550844i	−0.961323	+	0.275422i	$0.911182\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	8.92177	0.746076
$144$	0	0
$145$	2.16478i	0.179776i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 7.07107i	− 0.579284i	−0.957135	−	0.289642i	$-0.906464\pi$
$149$	− 7.07107i	− 0.579284i	0.957135	−	0.289642i	$-0.0935363\pi$
$150$	0	0
$151$	18.1421	1.47639	0.738193	−	0.674590i	$-0.235679\pi$
$151$	18.1421	1.47639	0.738193	+	0.674590i	$0.235679\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	10.8284i	0.869760i
$156$	0	0
$157$	11.2179i	0.895284i	0.894213	+	0.447642i	$0.147736\pi$
$157$	11.2179i	0.895284i	−0.894213	+	0.447642i	$0.852264\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	2.82843	0.221540	0.110770	−	0.993846i	$-0.464668\pi$
$163$	2.82843	0.221540	0.110770	+	0.993846i	$0.464668\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−16.3128	−1.26232	−0.631161	−	0.775652i	$-0.717421\pi$
$167$	−16.3128	−1.26232	−0.631161	+	0.775652i	$0.717421\pi$
$168$	0	0
$169$	−6.89949	−0.530730
$170$	0	0
$171$	0	0
$172$	0	0
$173$	12.1146	0.921052	0.460526	−	0.887646i	$-0.347661\pi$
$173$	12.1146	0.921052	0.460526	+	0.887646i	$0.347661\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	13.3137i	0.995113i	0.867431	+	0.497557i	$0.165769\pi$
$179$	13.3137i	0.995113i	−0.867431	+	0.497557i	$0.834231\pi$
$180$	0	0
$181$	− 14.4650i	− 1.07518i	−0.843207	−	0.537589i	$-0.819335\pi$
$181$	− 14.4650i	− 1.07518i	0.843207	−	0.537589i	$-0.180665\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	15.2304	1.11976
$186$	0	0
$187$	− 4.59220i	− 0.335815i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 5.31371i	− 0.384486i	−0.981347	−	0.192243i	$-0.938424\pi$
$191$	− 5.31371i	− 0.384486i	0.981347	−	0.192243i	$-0.0615763\pi$
$192$	0	0
$193$	1.65685	0.119263	0.0596315	−	0.998220i	$-0.481007\pi$
$193$	1.65685	0.119263	0.0596315	+	0.998220i	$0.481007\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	11.0711i	0.788781i	0.918943	+	0.394390i	$0.129044\pi$
$197$	11.0711i	0.788781i	−0.918943	+	0.394390i	$0.870956\pi$
$198$	0	0
$199$	3.06147i	0.217022i	0.994095	+	0.108511i	$0.0346082\pi$
$199$	3.06147i	0.217022i	−0.994095	+	0.108511i	$0.965392\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−21.8995	−1.52953
$206$	0	0
$207$	0	0
$208$	0	0
$209$	3.06147	0.211766
$210$	0	0
$211$	−26.6274	−1.83311	−0.916553	−	0.399912i	$-0.869041\pi$
$211$	−26.6274	−1.83311	−0.916553	+	0.399912i	$0.869041\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−2.16478	−0.147637
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	10.2426i	0.688995i
$222$	0	0
$223$	− 21.2764i	− 1.42477i	−0.701786	−	0.712387i	$-0.747614\pi$
$223$	− 21.2764i	− 1.42477i	0.701786	−	0.712387i	$-0.252386\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	5.86030	0.388962	0.194481	−	0.980906i	$-0.437698\pi$
$227$	5.86030	0.388962	0.194481	+	0.980906i	$0.437698\pi$
$228$	0	0
$229$	− 13.1969i	− 0.872079i	−0.899928	−	0.436039i	$-0.856381\pi$
$229$	− 13.1969i	− 0.872079i	0.899928	−	0.436039i	$-0.143619\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	20.8284i	1.36452i	0.731112	+	0.682258i	$0.239002\pi$
$233$	20.8284i	1.36452i	−0.731112	+	0.682258i	$0.760998\pi$
$234$	0	0
$235$	−14.8284	−0.967300
$236$	0	0
$237$	0	0
$238$	0	0
$239$	3.17157i	0.205152i	0.994725	+	0.102576i	$0.0327085\pi$
$239$	3.17157i	0.205152i	−0.994725	+	0.102576i	$0.967292\pi$
$240$	0	0
$241$	− 6.25425i	− 0.402872i	−0.979502	−	0.201436i	$-0.935439\pi$
$241$	− 6.25425i	− 0.402872i	0.979502	−	0.201436i	$-0.0645608\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−6.82843	−0.434482
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−19.7457	−1.24634	−0.623169	−	0.782088i	$-0.714155\pi$
$251$	−19.7457	−1.24634	−0.623169	+	0.782088i	$0.714155\pi$
$252$	0	0
$253$	−17.6569	−1.11008
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−13.1200	−0.818405	−0.409202	−	0.912444i	$-0.634193\pi$
$257$	−13.1200	−0.818405	−0.409202	+	0.912444i	$0.634193\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 8.34315i	− 0.514460i	−0.966350	−	0.257230i	$-0.917190\pi$
$263$	− 8.34315i	− 0.514460i	0.966350	−	0.257230i	$-0.0828099\pi$
$264$	0	0
$265$	− 6.94269i	− 0.426486i
$266$	0	0
$267$	0	0
$268$	0	0
$269$	24.7318	1.50793	0.753964	−	0.656916i	$-0.228140\pi$
$269$	24.7318	1.50793	0.753964	+	0.656916i	$0.228140\pi$
$270$	0	0
$271$	23.7038i	1.43991i	0.694023	+	0.719953i	$0.255837\pi$
$271$	23.7038i	1.43991i	−0.694023	+	0.719953i	$0.744163\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 3.17157i	− 0.191253i
$276$	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
$281$	6.48528i	0.386879i	0.981112	+	0.193440i	$0.0619644\pi$
$281$	6.48528i	0.386879i	−0.981112	+	0.193440i	$0.938036\pi$
$282$	0	0
$283$	− 31.9916i	− 1.90170i	−0.309650	−	0.950850i	$-0.600212\pi$
$283$	− 31.9916i	− 1.90170i	0.309650	−	0.950850i	$-0.399788\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−11.7279	−0.689878
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−8.34211	−0.487351	−0.243676	−	0.969857i	$-0.578353\pi$
$293$	−8.34211	−0.487351	−0.243676	+	0.969857i	$0.578353\pi$
$294$	0	0
$295$	18.1421	1.05628
$296$	0	0
$297$	0	0
$298$	0	0
$299$	39.3826	2.27755
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	22.7279i	1.30140i
$306$	0	0
$307$	− 28.5587i	− 1.62993i	−0.579510	−	0.814965i	$-0.696756\pi$
$307$	− 28.5587i	− 1.62993i	0.579510	−	0.814965i	$-0.303244\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	21.9105	1.24243	0.621215	−	0.783641i	$-0.286640\pi$
$311$	21.9105	1.24243	0.621215	+	0.783641i	$0.286640\pi$
$312$	0	0
$313$	22.6758i	1.28171i	0.767660	+	0.640857i	$0.221421\pi$
$313$	22.6758i	1.28171i	−0.767660	+	0.640857i	$0.778579\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	24.2426i	1.36160i	0.732468	+	0.680801i	$0.238368\pi$
$317$	24.2426i	1.36160i	−0.732468	+	0.680801i	$0.761632\pi$
$318$	0	0
$319$	−2.34315	−0.131191
$320$	0	0
$321$	0	0
$322$	0	0
$323$	3.51472i	0.195564i
$324$	0	0
$325$	7.07401i	0.392396i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	4.68629	0.257582	0.128791	−	0.991672i	$-0.458890\pi$
$331$	4.68629	0.257582	0.128791	+	0.991672i	$0.458890\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−23.0698	−1.26044
$336$	0	0
$337$	−12.9289	−0.704284	−0.352142	−	0.935947i	$-0.614547\pi$
$337$	−12.9289	−0.704284	−0.352142	+	0.935947i	$0.614547\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−11.7206	−0.634706
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	9.79899i	0.526037i	0.964791	+	0.263019i	$0.0847181\pi$
$347$	9.79899i	0.526037i	−0.964791	+	0.263019i	$0.915282\pi$
$348$	0	0
$349$	30.0669i	1.60944i	0.593652	+	0.804722i	$0.297685\pi$
$349$	30.0669i	1.60944i	−0.593652	+	0.804722i	$0.702315\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	32.5712	1.73359	0.866796	−	0.498664i	$-0.166176\pi$
$353$	32.5712	1.73359	0.866796	+	0.498664i	$0.166176\pi$
$354$	0	0
$355$	24.6005i	1.30566i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 31.4558i	− 1.66018i	−0.557632	−	0.830088i	$-0.688290\pi$
$359$	− 31.4558i	− 1.66018i	0.557632	−	0.830088i	$-0.311710\pi$
$360$	0	0
$361$	16.6569	0.876677
$362$	0	0
$363$	0	0
$364$	0	0
$365$	5.07107i	0.265432i
$366$	0	0
$367$	27.9246i	1.45765i	0.684698	+	0.728827i	$0.259934\pi$
$367$	27.9246i	1.45765i	−0.684698	+	0.728827i	$0.740066\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−14.6274	−0.757379	−0.378689	−	0.925524i	$-0.623625\pi$
$373$	−14.6274	−0.757379	−0.378689	+	0.925524i	$0.623625\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	5.22625	0.269166
$378$	0	0
$379$	−20.2843	−1.04193	−0.520967	−	0.853577i	$-0.674428\pi$
$379$	−20.2843	−1.04193	−0.520967	+	0.853577i	$0.674428\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−6.12293	−0.312867	−0.156434	−	0.987688i	$-0.550000\pi$
$383$	−6.12293	−0.312867	−0.156434	+	0.987688i	$0.550000\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	6.82843i	0.346215i	0.984903	+	0.173107i	$0.0553808\pi$
$389$	6.82843i	0.346215i	−0.984903	+	0.173107i	$0.944619\pi$
$390$	0	0
$391$	− 20.2710i	− 1.02515i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−20.9050	−1.05185
$396$	0	0
$397$	− 7.97069i	− 0.400038i	−0.979792	−	0.200019i	$-0.935900\pi$
$397$	− 7.97069i	− 0.400038i	0.979792	−	0.200019i	$-0.0641003\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	22.4853i	1.12286i	0.827524	+	0.561431i	$0.189749\pi$
$401$	22.4853i	1.12286i	−0.827524	+	0.561431i	$0.810251\pi$
$402$	0	0
$403$	26.1421	1.30223
$404$	0	0
$405$	0	0
$406$	0	0
$407$	16.4853i	0.817145i
$408$	0	0
$409$	28.6131i	1.41483i	0.706801	+	0.707413i	$0.250138\pi$
$409$	28.6131i	1.41483i	−0.706801	+	0.707413i	$0.749862\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	19.3137	0.948073
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−14.5194	−0.709321	−0.354661	−	0.934995i	$-0.615404\pi$
$419$	−14.5194	−0.709321	−0.354661	+	0.934995i	$0.615404\pi$
$420$	0	0
$421$	−7.31371	−0.356448	−0.178224	−	0.983990i	$-0.557035\pi$
$421$	−7.31371	−0.356448	−0.178224	+	0.983990i	$0.557035\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	3.64113	0.176621
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 10.6863i	− 0.514741i	−0.966313	−	0.257370i	$-0.917144\pi$
$431$	− 10.6863i	− 0.514741i	0.966313	−	0.257370i	$-0.0828560\pi$
$432$	0	0
$433$	− 19.6913i	− 0.946303i	−0.880981	−	0.473152i	$-0.843116\pi$
$433$	− 19.6913i	− 0.946303i	0.880981	−	0.473152i	$-0.156884\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	13.5140	0.646461
$438$	0	0
$439$	− 18.7402i	− 0.894422i	−0.894428	−	0.447211i	$-0.852417\pi$
$439$	− 18.7402i	− 0.894422i	0.894428	−	0.447211i	$-0.147583\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 6.68629i	− 0.317675i	−0.987305	−	0.158838i	$-0.949225\pi$
$443$	− 6.68629i	− 0.317675i	0.987305	−	0.158838i	$-0.0507746\pi$
$444$	0	0
$445$	26.7279	1.26703
$446$	0	0
$447$	0	0
$448$	0	0
$449$	19.5563i	0.922921i	0.887161	+	0.461461i	$0.152674\pi$
$449$	19.5563i	0.922921i	−0.887161	+	0.461461i	$0.847326\pi$
$450$	0	0
$451$	− 23.7038i	− 1.11617i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−22.0000	−1.02912	−0.514558	−	0.857455i	$-0.672044\pi$
$457$	−22.0000	−1.02912	−0.514558	+	0.857455i	$0.672044\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	28.1647	1.31176	0.655881	−	0.754864i	$-0.272297\pi$
$461$	28.1647	1.31176	0.655881	+	0.754864i	$0.272297\pi$
$462$	0	0
$463$	−2.82843	−0.131448	−0.0657241	−	0.997838i	$-0.520936\pi$
$463$	−2.82843	−0.131448	−0.0657241	+	0.997838i	$0.520936\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	37.2178	1.72224	0.861118	−	0.508406i	$-0.169765\pi$
$467$	37.2178	1.72224	0.861118	+	0.508406i	$0.169765\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 2.34315i	− 0.107738i
$474$	0	0
$475$	2.42742i	0.111378i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	24.9719	1.14100	0.570499	−	0.821299i	$-0.306750\pi$
$479$	24.9719	1.14100	0.570499	+	0.821299i	$0.306750\pi$
$480$	0	0
$481$	− 36.7695i	− 1.67654i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 5.07107i	− 0.230265i
$486$	0	0
$487$	19.7990	0.897178	0.448589	−	0.893738i	$-0.351927\pi$
$487$	19.7990	0.897178	0.448589	+	0.893738i	$0.351927\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	17.5147i	0.790428i	0.918589	+	0.395214i	$0.129330\pi$
$491$	17.5147i	0.790428i	−0.918589	+	0.395214i	$0.870670\pi$
$492$	0	0
$493$	− 2.69005i	− 0.121154i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	2.14214	0.0958952	0.0479476	−	0.998850i	$-0.484732\pi$
$499$	2.14214	0.0958952	0.0479476	+	0.998850i	$0.484732\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−12.2459	−0.546016	−0.273008	−	0.962012i	$-0.588019\pi$
$503$	−12.2459	−0.546016	−0.273008	+	0.962012i	$0.588019\pi$
$504$	0	0
$505$	−5.07107	−0.225660
$506$	0	0
$507$	0	0
$508$	0	0
$509$	5.35757	0.237470	0.118735	−	0.992926i	$-0.462116\pi$
$509$	5.35757	0.237470	0.118735	+	0.992926i	$0.462116\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	1.17157i	0.0516257i
$516$	0	0
$517$	− 16.0502i	− 0.705886i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−15.9958	−0.700788	−0.350394	−	0.936602i	$-0.613952\pi$
$521$	−15.9958	−0.700788	−0.350394	+	0.936602i	$0.613952\pi$
$522$	0	0
$523$	29.1927i	1.27651i	0.769826	+	0.638254i	$0.220343\pi$
$523$	29.1927i	1.27651i	−0.769826	+	0.638254i	$0.779657\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 13.4558i	− 0.586146i
$528$	0	0
$529$	−54.9411	−2.38874
$530$	0	0
$531$	0	0
$532$	0	0
$533$	52.8701i	2.29006i
$534$	0	0
$535$	23.7038i	1.02481i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	19.6569	0.845114	0.422557	−	0.906336i	$-0.361133\pi$
$541$	19.6569	0.845114	0.422557	+	0.906336i	$0.361133\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	5.67459	0.243073
$546$	0	0
$547$	−16.0000	−0.684111	−0.342055	−	0.939680i	$-0.611123\pi$
$547$	−16.0000	−0.684111	−0.342055	+	0.939680i	$0.611123\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	1.79337	0.0764000
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 8.72792i	− 0.369814i	−0.982756	−	0.184907i	$-0.940802\pi$
$557$	− 8.72792i	− 0.369814i	0.982756	−	0.184907i	$-0.0591984\pi$
$558$	0	0
$559$	5.22625i	0.221047i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−42.8155	−1.80446	−0.902229	−	0.431258i	$-0.858070\pi$
$563$	−42.8155	−1.80446	−0.902229	+	0.431258i	$0.858070\pi$
$564$	0	0
$565$	24.7862i	1.04276i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 16.4853i	− 0.691099i	−0.938401	−	0.345549i	$-0.887693\pi$
$569$	− 16.4853i	− 0.691099i	0.938401	−	0.345549i	$-0.112307\pi$
$570$	0	0
$571$	−42.6274	−1.78390	−0.891951	−	0.452132i	$-0.850664\pi$
$571$	−42.6274	−1.78390	−0.891951	+	0.452132i	$0.850664\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 14.0000i	− 0.583840i
$576$	0	0
$577$	14.7277i	0.613121i	0.951851	+	0.306561i	$0.0991782\pi$
$577$	14.7277i	0.613121i	−0.951851	+	0.306561i	$0.900822\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	7.51472	0.311228
$584$	0	0
$585$	0	0
$586$	0	0
$587$	15.9414	0.657971	0.328986	−	0.944335i	$-0.393293\pi$
$587$	15.9414	0.657971	0.328986	+	0.944335i	$0.393293\pi$
$588$	0	0
$589$	8.97056	0.369626
$590$	0	0
$591$	0	0
$592$	0	0
$593$	4.08947	0.167934	0.0839671	−	0.996469i	$-0.473241\pi$
$593$	4.08947	0.167934	0.0839671	+	0.996469i	$0.473241\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 13.3137i	− 0.543983i	−0.962300	−	0.271992i	$-0.912318\pi$
$599$	− 13.3137i	− 0.543983i	0.962300	−	0.271992i	$-0.0876823\pi$
$600$	0	0
$601$	42.0501i	1.71526i	0.514267	+	0.857630i	$0.328064\pi$
$601$	42.0501i	1.71526i	−0.514267	+	0.857630i	$0.671936\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	12.9343	0.525855
$606$	0	0
$607$	− 0.896683i	− 0.0363952i	−0.999834	−	0.0181976i	$-0.994207\pi$
$607$	− 0.896683i	− 0.0363952i	0.999834	−	0.0181976i	$-0.00579280\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	35.7990i	1.44827i
$612$	0	0
$613$	−2.78680	−0.112558	−0.0562788	−	0.998415i	$-0.517924\pi$
$613$	−2.78680	−0.112558	−0.0562788	+	0.998415i	$0.517924\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 33.4558i	− 1.34688i	−0.739241	−	0.673441i	$-0.764816\pi$
$617$	− 33.4558i	− 1.34688i	0.739241	−	0.673441i	$-0.235184\pi$
$618$	0	0
$619$	25.2346i	1.01426i	0.861869	+	0.507132i	$0.169294\pi$
$619$	25.2346i	1.01426i	−0.861869	+	0.507132i	$0.830706\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−14.5563	−0.582254
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−18.9259	−0.754626
$630$	0	0
$631$	−4.48528	−0.178556	−0.0892781	−	0.996007i	$-0.528456\pi$
$631$	−4.48528	−0.178556	−0.0892781	+	0.996007i	$0.528456\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−12.6173	−0.500702
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 27.4558i	− 1.08444i	−0.840236	−	0.542220i	$-0.817584\pi$
$641$	− 27.4558i	− 1.08444i	0.840236	−	0.542220i	$-0.182416\pi$
$642$	0	0
$643$	47.1451i	1.85922i	0.368546	+	0.929610i	$0.379856\pi$
$643$	47.1451i	1.85922i	−0.368546	+	0.929610i	$0.620144\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	36.3211	1.42793	0.713965	−	0.700181i	$-0.246897\pi$
$647$	36.3211	1.42793	0.713965	+	0.700181i	$0.246897\pi$
$648$	0	0
$649$	19.6369i	0.770816i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	34.1421i	1.33609i	0.744123	+	0.668043i	$0.232868\pi$
$653$	34.1421i	1.33609i	−0.744123	+	0.668043i	$0.767132\pi$
$654$	0	0
$655$	20.9706	0.819388
$656$	0	0
$657$	0	0
$658$	0	0
$659$	30.9706i	1.20644i	0.797574	+	0.603221i	$0.206116\pi$
$659$	30.9706i	1.20644i	−0.797574	+	0.603221i	$0.793884\pi$
$660$	0	0
$661$	1.02800i	0.0399845i	0.999800	+	0.0199923i	$0.00636416\pi$
$661$	1.02800i	0.0399845i	−0.999800	+	0.0199923i	$0.993636\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−10.3431	−0.400488
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−24.6005	−0.949693
$672$	0	0
$673$	−28.0416	−1.08093	−0.540463	−	0.841368i	$-0.681751\pi$
$673$	−28.0416	−1.08093	−0.540463	+	0.841368i	$0.681751\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	10.0586	0.386582	0.193291	−	0.981142i	$-0.438084\pi$
$677$	10.0586	0.386582	0.193291	+	0.981142i	$0.438084\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	13.0294i	0.498558i	0.968432	+	0.249279i	$0.0801935\pi$
$683$	13.0294i	0.498558i	−0.968432	+	0.249279i	$0.919806\pi$
$684$	0	0
$685$	− 9.55582i	− 0.365109i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−16.7611	−0.638549
$690$	0	0
$691$	− 32.2542i	− 1.22701i	−0.789692	−	0.613504i	$-0.789760\pi$
$691$	− 32.2542i	− 1.22701i	0.789692	−	0.613504i	$-0.210240\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	12.0000i	0.455186i
$696$	0	0
$697$	27.2132	1.03077
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 35.4558i	− 1.33915i	−0.742745	−	0.669574i	$-0.766477\pi$
$701$	− 35.4558i	− 1.33915i	0.742745	−	0.669574i	$-0.233523\pi$
$702$	0	0
$703$	− 12.6173i	− 0.475870i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	22.8701	0.858903	0.429452	−	0.903090i	$-0.358707\pi$
$709$	22.8701	0.858903	0.429452	+	0.903090i	$0.358707\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−51.7373	−1.93758
$714$	0	0
$715$	16.4853	0.616515
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−26.5027	−0.988383	−0.494192	−	0.869353i	$-0.664536\pi$
$719$	−26.5027	−0.988383	−0.494192	+	0.869353i	$0.664536\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 1.85786i	− 0.0689994i
$726$	0	0
$727$	− 25.3434i	− 0.939933i	−0.882684	−	0.469967i	$-0.844266\pi$
$727$	− 25.3434i	− 0.939933i	0.882684	−	0.469967i	$-0.155734\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	2.69005	0.0994951
$732$	0	0
$733$	1.58513i	0.0585480i	0.999571	+	0.0292740i	$0.00931953\pi$
$733$	1.58513i	0.0585480i	−0.999571	+	0.0292740i	$0.990680\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 24.9706i	− 0.919803i
$738$	0	0
$739$	−53.4558	−1.96641	−0.983203	−	0.182518i	$-0.941575\pi$
$739$	−53.4558	−1.96641	−0.983203	+	0.182518i	$0.941575\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 32.3431i	− 1.18655i	−0.804998	−	0.593277i	$-0.797834\pi$
$743$	− 32.3431i	− 1.18655i	0.804998	−	0.593277i	$-0.202166\pi$
$744$	0	0
$745$	− 13.0656i	− 0.478688i
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−20.2843	−0.740184	−0.370092	−	0.928995i	$-0.620674\pi$
$751$	−20.2843	−0.740184	−0.370092	+	0.928995i	$0.620674\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	33.5223	1.22000
$756$	0	0
$757$	37.8995	1.37748	0.688740	−	0.725008i	$-0.258164\pi$
$757$	37.8995	1.37748	0.688740	+	0.725008i	$0.258164\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	14.5738	0.528301	0.264151	−	0.964481i	$-0.414908\pi$
$761$	14.5738	0.528301	0.264151	+	0.964481i	$0.414908\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 43.7990i	− 1.58149i
$768$	0	0
$769$	− 13.5684i	− 0.489288i	−0.969613	−	0.244644i	$-0.921329\pi$
$769$	− 13.5684i	− 0.489288i	0.969613	−	0.244644i	$-0.0786710\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−18.2375	−0.655957	−0.327978	−	0.944685i	$-0.606367\pi$
$773$	−18.2375	−0.655957	−0.327978	+	0.944685i	$0.606367\pi$
$774$	0	0
$775$	− 9.29319i	− 0.333821i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	18.1421i	0.650009i
$780$	0	0
$781$	−26.6274	−0.952804
$782$	0	0
$783$	0	0
$784$	0	0
$785$	20.7279i	0.739811i
$786$	0	0
$787$	− 3.58673i	− 0.127853i	−0.997955	−	0.0639266i	$-0.979638\pi$
$787$	− 3.58673i	− 0.127853i	0.997955	−	0.0639266i	$-0.0203624\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	54.8701	1.94849
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−30.4064	−1.07705	−0.538526	−	0.842609i	$-0.681018\pi$
$797$	−30.4064	−1.07705	−0.538526	+	0.842609i	$0.681018\pi$
$798$	0	0
$799$	18.4264	0.651879
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−5.48888	−0.193699
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	21.2132i	0.745817i	0.927868	+	0.372908i	$0.121639\pi$
$809$	21.2132i	0.745817i	−0.927868	+	0.372908i	$0.878361\pi$
$810$	0	0
$811$	− 13.1426i	− 0.461497i	−0.973013	−	0.230749i	$-0.925882\pi$
$811$	− 13.1426i	− 0.461497i	0.973013	−	0.230749i	$-0.0741175\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	5.22625	0.183068
$816$	0	0
$817$	1.79337i	0.0627419i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 36.9289i	− 1.28883i	−0.764677	−	0.644414i	$-0.777101\pi$
$821$	− 36.9289i	− 1.28883i	0.764677	−	0.644414i	$-0.222899\pi$
$822$	0	0
$823$	−16.2843	−0.567634	−0.283817	−	0.958878i	$-0.591601\pi$
$823$	−16.2843	−0.567634	−0.283817	+	0.958878i	$0.591601\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1.51472i	0.0526719i	0.999653	+	0.0263360i	$0.00838397\pi$
$827$	1.51472i	0.0526719i	−0.999653	+	0.0263360i	$0.991616\pi$
$828$	0	0
$829$	− 6.99709i	− 0.243019i	−0.992590	−	0.121509i	$-0.961227\pi$
$829$	− 6.99709i	− 0.243019i	0.992590	−	0.121509i	$-0.0387735\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−30.1421	−1.04311
$836$	0	0
$837$	0	0
$838$	0	0
$839$	16.6842	0.576003	0.288002	−	0.957630i	$-0.407009\pi$
$839$	16.6842	0.576003	0.288002	+	0.957630i	$0.407009\pi$
$840$	0	0
$841$	27.6274	0.952670
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−12.7486	−0.438565
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	72.7696i	2.49451i
$852$	0	0
$853$	− 23.7264i	− 0.812376i	−0.913790	−	0.406188i	$-0.866858\pi$
$853$	− 23.7264i	− 0.812376i	0.913790	−	0.406188i	$-0.133142\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−40.2249	−1.37406	−0.687028	−	0.726631i	$-0.741085\pi$
$857$	−40.2249	−1.37406	−0.687028	+	0.726631i	$0.741085\pi$
$858$	0	0
$859$	33.2597i	1.13481i	0.823441	+	0.567403i	$0.192052\pi$
$859$	33.2597i	1.13481i	−0.823441	+	0.567403i	$0.807948\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	37.1127i	1.26333i	0.775241	+	0.631665i	$0.217628\pi$
$863$	37.1127i	1.26333i	−0.775241	+	0.631665i	$0.782372\pi$
$864$	0	0
$865$	22.3848	0.761105
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 22.6274i	− 0.767583i
$870$	0	0
$871$	55.6954i	1.88717i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	15.0711	0.508914	0.254457	−	0.967084i	$-0.418103\pi$
$877$	15.0711	0.508914	0.254457	+	0.967084i	$0.418103\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	16.8155	0.566530	0.283265	−	0.959042i	$-0.408582\pi$
$881$	16.8155	0.566530	0.283265	+	0.959042i	$0.408582\pi$
$882$	0	0
$883$	50.1421	1.68742	0.843709	−	0.536801i	$-0.180368\pi$
$883$	50.1421	1.68742	0.843709	+	0.536801i	$0.180368\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−48.0417	−1.61308	−0.806542	−	0.591177i	$-0.798663\pi$
$887$	−48.0417	−1.61308	−0.806542	+	0.591177i	$0.798663\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	12.2843i	0.411077i
$894$	0	0
$895$	24.6005i	0.822305i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−6.86577	−0.228986
$900$	0	0
$901$	8.62727i	0.287416i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 26.7279i	− 0.888466i
$906$	0	0
$907$	15.5147	0.515158	0.257579	−	0.966257i	$-0.417075\pi$
$907$	15.5147	0.515158	0.257579	+	0.966257i	$0.417075\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	14.2843i	0.473259i	0.971600	+	0.236630i	$0.0760428\pi$
$911$	14.2843i	0.473259i	−0.971600	+	0.236630i	$0.923957\pi$
$912$	0	0
$913$	20.9050i	0.691855i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	32.4853	1.07159	0.535795	−	0.844348i	$-0.320012\pi$
$919$	32.4853	1.07159	0.535795	+	0.844348i	$0.320012\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	59.3909	1.95488
$924$	0	0
$925$	−13.0711	−0.429774
$926$	0	0
$927$	0	0
$928$	0	0
$929$	1.58513	0.0520063	0.0260032	−	0.999662i	$-0.491722\pi$
$929$	1.58513	0.0520063	0.0260032	+	0.999662i	$0.491722\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 8.48528i	− 0.277498i
$936$	0	0
$937$	− 45.1116i	− 1.47373i	−0.676039	−	0.736866i	$-0.736305\pi$
$937$	− 45.1116i	− 1.47373i	0.676039	−	0.736866i	$-0.263695\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−42.3127	−1.37936	−0.689678	−	0.724116i	$-0.742248\pi$
$941$	−42.3127	−1.37936	−0.689678	+	0.724116i	$0.742248\pi$
$942$	0	0
$943$	− 104.634i	− 3.40735i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	3.17157i	0.103062i	0.998671	+	0.0515311i	$0.0164101\pi$
$947$	3.17157i	0.103062i	−0.998671	+	0.0515311i	$0.983590\pi$
$948$	0	0
$949$	12.2426	0.397413
$950$	0	0
$951$	0	0
$952$	0	0
$953$	5.12994i	0.166175i	0.996542	+	0.0830876i	$0.0264781\pi$
$953$	5.12994i	0.166175i	−0.996542	+	0.0830876i	$0.973522\pi$
$954$	0	0
$955$	− 9.81845i	− 0.317718i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−3.34315	−0.107843
$962$	0	0
$963$	0	0
$964$	0	0
$965$	3.06147	0.0985521
$966$	0	0
$967$	−32.4853	−1.04466	−0.522328	−	0.852745i	$-0.674936\pi$
$967$	−32.4853	−1.04466	−0.522328	+	0.852745i	$0.674936\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	1.79337	0.0575519	0.0287759	−	0.999586i	$-0.490839\pi$
$971$	1.79337	0.0575519	0.0287759	+	0.999586i	$0.490839\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 16.4853i	− 0.527411i	−0.964603	−	0.263705i	$-0.915055\pi$
$977$	− 16.4853i	− 0.527411i	0.964603	−	0.263705i	$-0.0849447\pi$
$978$	0	0
$979$	28.9301i	0.924610i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−27.3994	−0.873904	−0.436952	−	0.899485i	$-0.643942\pi$
$983$	−27.3994	−0.873904	−0.436952	+	0.899485i	$0.643942\pi$
$984$	0	0
$985$	20.4567i	0.651804i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 10.3431i	− 0.328893i
$990$	0	0
$991$	38.1421	1.21162	0.605812	−	0.795608i	$-0.292848\pi$
$991$	38.1421	1.21162	0.605812	+	0.795608i	$0.292848\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	5.65685i	0.179334i
$996$	0	0
$997$	− 41.0446i	− 1.29990i	−0.759978	−	0.649949i	$-0.774790\pi$
$997$	− 41.0446i	− 1.29990i	0.759978	−	0.649949i	$-0.225210\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7056.2.k.e.881.8		8
3.2	odd	2	inner	7056.2.k.e.881.1		8
4.3	odd	2		1764.2.f.b.881.7	yes	8
7.6	odd	2	inner	7056.2.k.e.881.2		8
12.11	even	2		1764.2.f.b.881.2	yes	8
21.20	even	2	inner	7056.2.k.e.881.7		8
28.3	even	6		1764.2.t.c.1097.7		16
28.11	odd	6		1764.2.t.c.1097.1		16
28.19	even	6		1764.2.t.c.521.8		16
28.23	odd	6		1764.2.t.c.521.2		16
28.27	even	2		1764.2.f.b.881.1	✓	8
84.11	even	6		1764.2.t.c.1097.8		16
84.23	even	6		1764.2.t.c.521.7		16
84.47	odd	6		1764.2.t.c.521.1		16
84.59	odd	6		1764.2.t.c.1097.2		16
84.83	odd	2		1764.2.f.b.881.8	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1764.2.f.b.881.1	✓	8	28.27	even	2
1764.2.f.b.881.2	yes	8	12.11	even	2
1764.2.f.b.881.7	yes	8	4.3	odd	2
1764.2.f.b.881.8	yes	8	84.83	odd	2
1764.2.t.c.521.1		16	84.47	odd	6
1764.2.t.c.521.2		16	28.23	odd	6
1764.2.t.c.521.7		16	84.23	even	6
1764.2.t.c.521.8		16	28.19	even	6
1764.2.t.c.1097.1		16	28.11	odd	6
1764.2.t.c.1097.2		16	84.59	odd	6
1764.2.t.c.1097.7		16	28.3	even	6
1764.2.t.c.1097.8		16	84.11	even	6
7056.2.k.e.881.1		8	3.2	odd	2	inner
7056.2.k.e.881.2		8	7.6	odd	2	inner
7056.2.k.e.881.7		8	21.20	even	2	inner
7056.2.k.e.881.8		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)

\(f(q)\)	\(=\)	\(q+1.84776 q^{5} +O(q^{10})\)	\(q+1.84776 q^{5} +2.00000i q^{11} -4.46088i q^{13} -2.29610 q^{17} -1.53073i q^{19} +8.82843i q^{23} -1.58579 q^{25} +1.17157i q^{29} +5.86030i q^{31} +8.24264 q^{37} -11.8519 q^{41} -1.17157 q^{43} -8.02509 q^{47} -3.75736i q^{53} +3.69552i q^{55} +9.81845 q^{59} +12.3003i q^{61} -8.24264i q^{65} -12.4853 q^{67} +13.3137i q^{71} +2.74444i q^{73} -11.3137 q^{79} +10.4525 q^{83} -4.24264 q^{85} +14.4650 q^{89} -2.82843i q^{95} -2.74444i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more