LMFDB - Embedded newform 7056.2.h.n.4607.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7056,2,Mod(4607,7056)]

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, 1, 0]))

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

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

chi := DirichletCharacter("7056.4607");

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.h (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:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 28x^{10} + 276x^{8} + 1178x^{6} + 2292x^{4} + 1888x^{2} + 529 $ x^12 + 28x^10 + 276x^8 + 1178x^6 + 2292x^4 + 1888*x^2 + 529
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{10}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 1008)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			4607.1
Root			$-3.39892i$ of defining polynomial
Character	$\chi$	$=$	7056.4607
Dual form			7056.2.h.n.4607.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.19116i q^{5} -1.57338 q^{11} +0.670500 q^{13} +5.91633i q^{17} +3.28885i q^{19} +3.95386 q^{23} -5.18348 q^{25} -0.585502i q^{29} -9.54875i q^{31} +7.69645 q^{37} -3.77666i q^{41} -12.3446i q^{43} -7.34847 q^{47} +13.3501i q^{53} +5.02090i q^{55} -4.96798 q^{59} +6.51298 q^{61} -2.13967i q^{65} -5.51394i q^{67} +6.85278 q^{71} +5.84248 q^{73} -6.08465i q^{79} -9.48111 q^{83} +18.8799 q^{85} -14.8676i q^{89} +10.4952 q^{95} -0.183476 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 4 q^{13} - 4 q^{25} - 12 q^{37} + 32 q^{61} + 36 q^{73} + 64 q^{85} + 56 q^{97}+O(q^{100}) $ 12 * q - 4 * q^13 - 4 * q^25 - 12 * q^37 + 32 * q^61 + 36 * q^73 + 64 * q^85 + 56 * q^97

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
$5$	− 3.19116i	− 1.42713i	−0.700590	−	0.713564i	$-0.747080\pi$
$5$	− 3.19116i	− 1.42713i	0.700590	−	0.713564i	$-0.252920\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−1.57338	−0.474392	−0.237196	−	0.971462i	$-0.576228\pi$
$11$	−1.57338	−0.474392	−0.237196	+	0.971462i	$0.576228\pi$
$12$	0	0
$13$	0.670500	0.185963	0.0929817	−	0.995668i	$-0.470360\pi$
$13$	0.670500	0.185963	0.0929817	+	0.995668i	$0.470360\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	5.91633i	1.43492i	0.696599	+	0.717460i	$0.254696\pi$
$17$	5.91633i	1.43492i	−0.696599	+	0.717460i	$0.745304\pi$
$18$	0	0
$19$	3.28885i	0.754514i	0.926109	+	0.377257i	$0.123133\pi$
$19$	3.28885i	0.754514i	−0.926109	+	0.377257i	$0.876867\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	3.95386	0.824438	0.412219	−	0.911085i	$-0.364754\pi$
$23$	3.95386	0.824438	0.412219	+	0.911085i	$0.364754\pi$
$24$	0	0
$25$	−5.18348	−1.03670
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 0.585502i	− 0.108725i	−0.998521	−	0.0543625i	$-0.982687\pi$
$29$	− 0.585502i	− 0.108725i	0.998521	−	0.0543625i	$-0.0173127\pi$
$30$	0	0
$31$	− 9.54875i	− 1.71501i	−0.514478	−	0.857503i	$-0.672014\pi$
$31$	− 9.54875i	− 1.71501i	0.514478	−	0.857503i	$-0.327986\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	7.69645	1.26529	0.632644	−	0.774442i	$-0.281969\pi$
$37$	7.69645	1.26529	0.632644	+	0.774442i	$0.281969\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 3.77666i	− 0.589815i	−0.955526	−	0.294907i	$-0.904711\pi$
$41$	− 3.77666i	− 0.589815i	0.955526	−	0.294907i	$-0.0952888\pi$
$42$	0	0
$43$	− 12.3446i	− 1.88253i	−0.337672	−	0.941264i	$-0.609639\pi$
$43$	− 12.3446i	− 1.88253i	0.337672	−	0.941264i	$-0.390361\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−7.34847	−1.07188	−0.535942	−	0.844255i	$-0.680044\pi$
$47$	−7.34847	−1.07188	−0.535942	+	0.844255i	$0.680044\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	0	0
$52$	0	0
$53$	13.3501i	1.83378i	0.399139	+	0.916890i	$0.369309\pi$
$53$	13.3501i	1.83378i	−0.399139	+	0.916890i	$0.630691\pi$
$54$	0	0
$55$	5.02090i	0.677018i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−4.96798	−0.646776	−0.323388	−	0.946266i	$-0.604822\pi$
$59$	−4.96798	−0.646776	−0.323388	+	0.946266i	$0.604822\pi$
$60$	0	0
$61$	6.51298	0.833901	0.416951	−	0.908929i	$-0.363099\pi$
$61$	6.51298	0.833901	0.416951	+	0.908929i	$0.363099\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 2.13967i	− 0.265394i
$66$	0	0
$67$	− 5.51394i	− 0.673635i	−0.941570	−	0.336818i	$-0.890649\pi$
$67$	− 5.51394i	− 0.673635i	0.941570	−	0.336818i	$-0.109351\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	6.85278	0.813275	0.406638	−	0.913590i	$-0.366701\pi$
$71$	6.85278	0.813275	0.406638	+	0.913590i	$0.366701\pi$
$72$	0	0
$73$	5.84248	0.683810	0.341905	−	0.939735i	$-0.388928\pi$
$73$	5.84248	0.683810	0.341905	+	0.939735i	$0.388928\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	− 6.08465i	− 0.684577i	−0.939595	−	0.342288i	$-0.888798\pi$
$79$	− 6.08465i	− 0.684577i	0.939595	−	0.342288i	$-0.111202\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−9.48111	−1.04069	−0.520343	−	0.853957i	$-0.674196\pi$
$83$	−9.48111	−1.04069	−0.520343	+	0.853957i	$0.674196\pi$
$84$	0	0
$85$	18.8799	2.04782
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 14.8676i	− 1.57596i	−0.615700	−	0.787981i	$-0.711127\pi$
$89$	− 14.8676i	− 1.57596i	0.615700	−	0.787981i	$-0.288873\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	10.4952	1.07679
$96$	0	0
$97$	−0.183476	−0.0186292	−0.00931460	−	0.999957i	$-0.502965\pi$
$97$	−0.183476	−0.0186292	−0.00931460	+	0.999957i	$0.502965\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 3.77666i	− 0.375792i	−0.982189	−	0.187896i	$-0.939833\pi$
$101$	− 3.77666i	− 0.375792i	0.982189	−	0.187896i	$-0.0601667\pi$
$102$	0	0
$103$	− 13.7787i	− 1.35766i	−0.734296	−	0.678830i	$-0.762487\pi$
$103$	− 13.7787i	− 1.35766i	0.734296	−	0.678830i	$-0.237513\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	19.1692	1.85316	0.926580	−	0.376098i	$-0.122735\pi$
$107$	19.1692	1.85316	0.926580	+	0.376098i	$0.122735\pi$
$108$	0	0
$109$	8.20943	0.786321	0.393160	−	0.919470i	$-0.371382\pi$
$109$	8.20943	0.786321	0.393160	+	0.919470i	$0.371382\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 4.70862i	− 0.442950i	−0.975166	−	0.221475i	$-0.928913\pi$
$113$	− 4.70862i	− 0.442950i	0.975166	−	0.221475i	$-0.0710871\pi$
$114$	0	0
$115$	− 12.6174i	− 1.17658i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−8.52448	−0.774952
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0.585502i	0.0523689i
$126$	0	0
$127$	− 14.3495i	− 1.27331i	−0.771150	−	0.636654i	$-0.780318\pi$
$127$	− 14.3495i	− 1.27331i	0.771150	−	0.636654i	$-0.219682\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−13.1236	−1.14661	−0.573305	−	0.819342i	$-0.694339\pi$
$131$	−13.1236	−1.14661	−0.573305	+	0.819342i	$0.694339\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 12.7646i	− 1.09056i	−0.838255	−	0.545278i	$-0.816424\pi$
$137$	− 12.7646i	− 1.09056i	0.838255	−	0.545278i	$-0.183576\pi$
$138$	0	0
$139$	− 4.52786i	− 0.384048i	−0.981390	−	0.192024i	$-0.938495\pi$
$139$	− 4.52786i	− 0.384048i	0.981390	−	0.192024i	$-0.0615051\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−1.05495	−0.0882195
$144$	0	0
$145$	−1.86843	−0.155164
$146$	0	0
$147$	0	0
$148$	0	0
$149$	12.7646i	1.04572i	0.852419	+	0.522859i	$0.175135\pi$
$149$	12.7646i	1.04572i	−0.852419	+	0.522859i	$0.824865\pi$
$150$	0	0
$151$	14.2718i	1.16142i	0.814110	+	0.580711i	$0.197225\pi$
$151$	14.2718i	1.16142i	−0.814110	+	0.580711i	$0.802775\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−30.4716	−2.44754
$156$	0	0
$157$	−1.02595	−0.0818799	−0.0409399	−	0.999162i	$-0.513035\pi$
$157$	−1.02595	−0.0818799	−0.0409399	+	0.999162i	$0.513035\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	12.6174i	0.988271i	0.869385	+	0.494135i	$0.164515\pi$
$163$	12.6174i	0.988271i	−0.869385	+	0.494135i	$0.835485\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	10.8066	0.836243	0.418122	−	0.908391i	$-0.362689\pi$
$167$	10.8066	0.836243	0.418122	+	0.908391i	$0.362689\pi$
$168$	0	0
$169$	−12.5504	−0.965418
$170$	0	0
$171$	0	0
$172$	0	0
$173$	10.6617i	0.810591i	0.914186	+	0.405295i	$0.132831\pi$
$173$	10.6617i	0.810591i	−0.914186	+	0.405295i	$0.867169\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	11.3023	0.844776	0.422388	−	0.906415i	$-0.361192\pi$
$179$	11.3023	0.844776	0.422388	+	0.906415i	$0.361192\pi$
$180$	0	0
$181$	−14.5245	−1.07960	−0.539798	−	0.841795i	$-0.681499\pi$
$181$	−14.5245	−1.07960	−0.539798	+	0.841795i	$0.681499\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 24.5606i	− 1.80573i
$186$	0	0
$187$	− 9.30863i	− 0.680714i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−25.0079	−1.80951	−0.904754	−	0.425935i	$-0.859945\pi$
$191$	−25.0079	−1.80951	−0.904754	+	0.425935i	$0.859945\pi$
$192$	0	0
$193$	−13.7109	−0.986932	−0.493466	−	0.869765i	$-0.664270\pi$
$193$	−13.7109	−0.986932	−0.493466	+	0.869765i	$0.664270\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 3.03494i	− 0.216230i	−0.994138	−	0.108115i	$-0.965518\pi$
$197$	− 3.03494i	− 0.216230i	0.994138	−	0.108115i	$-0.0344815\pi$
$198$	0	0
$199$	10.0418i	0.711844i	0.934516	+	0.355922i	$0.115833\pi$
$199$	10.0418i	0.711844i	−0.934516	+	0.355922i	$0.884167\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−12.0519	−0.841742
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 5.17461i	− 0.357935i
$210$	0	0
$211$	7.46620i	0.513994i	0.966412	+	0.256997i	$0.0827331\pi$
$211$	7.46620i	0.513994i	−0.966412	+	0.256997i	$0.917267\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−39.3934	−2.68661
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	3.96690i	0.266843i
$222$	0	0
$223$	− 14.2718i	− 0.955709i	−0.878439	−	0.477855i	$-0.841415\pi$
$223$	− 14.2718i	− 0.955709i	0.878439	−	0.477855i	$-0.158585\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−16.2703	−1.07990	−0.539949	−	0.841697i	$-0.681557\pi$
$227$	−16.2703	−1.07990	−0.539949	+	0.841697i	$0.681557\pi$
$228$	0	0
$229$	6.50147	0.429630	0.214815	−	0.976655i	$-0.431085\pi$
$229$	6.50147	0.429630	0.214815	+	0.976655i	$0.431085\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	17.0073i	1.11418i	0.830451	+	0.557091i	$0.188083\pi$
$233$	17.0073i	1.11418i	−0.830451	+	0.557091i	$0.811917\pi$
$234$	0	0
$235$	23.4501i	1.52972i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−28.3389	−1.83309	−0.916546	−	0.399928i	$-0.869035\pi$
$239$	−28.3389	−1.83309	−0.916546	+	0.399928i	$0.869035\pi$
$240$	0	0
$241$	22.5504	1.45260	0.726300	−	0.687377i	$-0.241238\pi$
$241$	22.5504	1.45260	0.726300	+	0.687377i	$0.241238\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	2.20517i	0.140312i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−11.2615	−0.710820	−0.355410	−	0.934711i	$-0.615659\pi$
$251$	−11.2615	−0.710820	−0.355410	+	0.934711i	$0.615659\pi$
$252$	0	0
$253$	−6.22093	−0.391106
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 20.7472i	− 1.29418i	−0.762415	−	0.647088i	$-0.775987\pi$
$257$	− 20.7472i	− 1.29418i	0.762415	−	0.647088i	$-0.224013\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−0.311415	−0.0192027	−0.00960133	−	0.999954i	$-0.503056\pi$
$263$	−0.311415	−0.0192027	−0.00960133	+	0.999954i	$0.503056\pi$
$264$	0	0
$265$	42.6023	2.61704
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 24.4411i	− 1.49020i	−0.666954	−	0.745099i	$-0.732402\pi$
$269$	− 24.4411i	− 1.49020i	0.666954	−	0.745099i	$-0.267598\pi$
$270$	0	0
$271$	− 31.0466i	− 1.88595i	−0.332867	−	0.942974i	$-0.608016\pi$
$271$	− 31.0466i	− 1.88595i	0.332867	−	0.942974i	$-0.391984\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	8.15558	0.491800
$276$	0	0
$277$	11.8944	0.714664	0.357332	−	0.933977i	$-0.383686\pi$
$277$	11.8944	0.714664	0.357332	+	0.933977i	$0.383686\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 25.5292i	− 1.52295i	−0.648196	−	0.761474i	$-0.724476\pi$
$281$	− 25.5292i	− 1.52295i	0.648196	−	0.761474i	$-0.275524\pi$
$282$	0	0
$283$	2.93834i	0.174666i	0.996179	+	0.0873331i	$0.0278345\pi$
$283$	2.93834i	0.174666i	−0.996179	+	0.0873331i	$0.972166\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−18.0029	−1.05900
$290$	0	0
$291$	0	0
$292$	0	0
$293$	16.3851i	0.957225i	0.878026	+	0.478613i	$0.158860\pi$
$293$	16.3851i	0.957225i	−0.878026	+	0.478613i	$0.841140\pi$
$294$	0	0
$295$	15.8536i	0.923033i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	2.65107	0.153315
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 20.7839i	− 1.19008i
$306$	0	0
$307$	− 13.3306i	− 0.760820i	−0.924818	−	0.380410i	$-0.875783\pi$
$307$	− 13.3306i	− 0.760820i	0.924818	−	0.380410i	$-0.124217\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−30.2646	−1.71615	−0.858073	−	0.513528i	$-0.828338\pi$
$311$	−30.2646	−1.71615	−0.858073	+	0.513528i	$0.828338\pi$
$312$	0	0
$313$	7.97405	0.450720	0.225360	−	0.974276i	$-0.427644\pi$
$313$	7.97405	0.450720	0.225360	+	0.974276i	$0.427644\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 26.1147i	− 1.46675i	−0.679824	−	0.733375i	$-0.737944\pi$
$317$	− 26.1147i	− 1.46675i	0.679824	−	0.733375i	$-0.262056\pi$
$318$	0	0
$319$	0.921217i	0.0515782i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−19.4579	−1.08267
$324$	0	0
$325$	−3.47552	−0.192787
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	19.0198i	1.04543i	0.852509	+	0.522713i	$0.175080\pi$
$331$	19.0198i	1.04543i	−0.852509	+	0.522713i	$0.824920\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−17.5959	−0.961364
$336$	0	0
$337$	5.02300	0.273620	0.136810	−	0.990597i	$-0.456315\pi$
$337$	5.02300	0.273620	0.136810	+	0.990597i	$0.456315\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	15.0238i	0.813585i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	22.1090	1.18687	0.593436	−	0.804881i	$-0.297771\pi$
$347$	22.1090	1.18687	0.593436	+	0.804881i	$0.297771\pi$
$348$	0	0
$349$	−22.0519	−1.18041	−0.590206	−	0.807253i	$-0.700953\pi$
$349$	−22.0519	−1.18041	−0.590206	+	0.807253i	$0.700953\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	1.67369i	0.0890814i	0.999008	+	0.0445407i	$0.0141824\pi$
$353$	1.67369i	0.0890814i	−0.999008	+	0.0445407i	$0.985818\pi$
$354$	0	0
$355$	− 21.8683i	− 1.16065i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	7.90773	0.417354	0.208677	−	0.977985i	$-0.433084\pi$
$359$	7.90773	0.417354	0.208677	+	0.977985i	$0.433084\pi$
$360$	0	0
$361$	8.18348	0.430709
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 18.6443i	− 0.975885i
$366$	0	0
$367$	8.84774i	0.461848i	0.972972	+	0.230924i	$0.0741749\pi$
$367$	8.84774i	0.461848i	−0.972972	+	0.230924i	$0.925825\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	7.52743	0.389755	0.194878	−	0.980828i	$-0.437569\pi$
$373$	7.52743	0.389755	0.194878	+	0.980828i	$0.437569\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 0.392579i	− 0.0202189i
$378$	0	0
$379$	− 25.1571i	− 1.29224i	−0.763238	−	0.646118i	$-0.776392\pi$
$379$	− 25.1571i	− 1.29224i	0.763238	−	0.646118i	$-0.223608\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−11.5502	−0.590187	−0.295093	−	0.955468i	$-0.595351\pi$
$383$	−11.5502	−0.590187	−0.295093	+	0.955468i	$0.595351\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	32.5799i	1.65187i	0.563769	+	0.825933i	$0.309351\pi$
$389$	32.5799i	1.65187i	−0.563769	+	0.825933i	$0.690649\pi$
$390$	0	0
$391$	23.3924i	1.18300i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−19.4171	−0.976979
$396$	0	0
$397$	34.2843	1.72068	0.860341	−	0.509719i	$-0.170251\pi$
$397$	34.2843	1.72068	0.860341	+	0.509719i	$0.170251\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	16.0386i	0.800929i	0.916312	+	0.400465i	$0.131151\pi$
$401$	16.0386i	0.800929i	−0.916312	+	0.400465i	$0.868849\pi$
$402$	0	0
$403$	− 6.40244i	− 0.318928i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−12.1094	−0.600243
$408$	0	0
$409$	−22.4448	−1.10982	−0.554912	−	0.831909i	$-0.687248\pi$
$409$	−22.4448	−1.10982	−0.554912	+	0.831909i	$0.687248\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	30.2557i	1.48519i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	23.8894	1.16707	0.583536	−	0.812087i	$-0.301669\pi$
$419$	23.8894	1.16707	0.583536	+	0.812087i	$0.301669\pi$
$420$	0	0
$421$	16.3555	0.797116	0.398558	−	0.917143i	$-0.369511\pi$
$421$	16.3555	0.797116	0.398558	+	0.917143i	$0.369511\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 30.6672i	− 1.48758i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	14.4491	0.695988	0.347994	−	0.937497i	$-0.386863\pi$
$431$	14.4491	0.695988	0.347994	+	0.937497i	$0.386863\pi$
$432$	0	0
$433$	−8.20943	−0.394520	−0.197260	−	0.980351i	$-0.563204\pi$
$433$	−8.20943	−0.394520	−0.197260	+	0.980351i	$0.563204\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	13.0037i	0.622050i
$438$	0	0
$439$	− 10.9630i	− 0.523236i	−0.965171	−	0.261618i	$-0.915744\pi$
$439$	− 10.9630i	− 0.523236i	0.965171	−	0.261618i	$-0.0842561\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−10.8475	−0.515379	−0.257690	−	0.966228i	$-0.582961\pi$
$443$	−10.8475	−0.515379	−0.257690	+	0.966228i	$0.582961\pi$
$444$	0	0
$445$	−47.4448	−2.24910
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 4.48168i	− 0.211503i	−0.994393	−	0.105752i	$-0.966275\pi$
$449$	− 4.48168i	− 0.211503i	0.994393	−	0.105752i	$-0.0337249\pi$
$450$	0	0
$451$	5.94212i	0.279803i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	27.4189	1.28260	0.641300	−	0.767290i	$-0.278395\pi$
$457$	27.4189	1.28260	0.641300	+	0.767290i	$0.278395\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 22.4943i	− 1.04766i	−0.851821	−	0.523832i	$-0.824502\pi$
$461$	− 22.4943i	− 1.04766i	0.851821	−	0.523832i	$-0.175498\pi$
$462$	0	0
$463$	− 2.20517i	− 0.102483i	−0.998686	−	0.0512416i	$-0.983682\pi$
$463$	− 2.20517i	− 0.102483i	0.998686	−	0.0512416i	$-0.0163178\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−7.03705	−0.325636	−0.162818	−	0.986656i	$-0.552058\pi$
$467$	−7.03705	−0.325636	−0.162818	+	0.986656i	$0.552058\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	19.4227i	0.893056i
$474$	0	0
$475$	− 17.0477i	− 0.782201i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	1.92563	0.0879841	0.0439921	−	0.999032i	$-0.485992\pi$
$479$	1.92563	0.0879841	0.0439921	+	0.999032i	$0.485992\pi$
$480$	0	0
$481$	5.16047	0.235297
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0.585502i	0.0265863i
$486$	0	0
$487$	− 20.9271i	− 0.948300i	−0.880444	−	0.474150i	$-0.842756\pi$
$487$	− 20.9271i	− 0.948300i	0.880444	−	0.474150i	$-0.157244\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	1.32553	0.0598205	0.0299103	−	0.999553i	$-0.490478\pi$
$491$	1.32553	0.0598205	0.0299103	+	0.999553i	$0.490478\pi$
$492$	0	0
$493$	3.46402	0.156012
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	− 13.7787i	− 0.616821i	−0.951253	−	0.308411i	$-0.900203\pi$
$499$	− 13.7787i	− 0.616821i	0.951253	−	0.308411i	$-0.0997971\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	16.7887	0.748573	0.374287	−	0.927313i	$-0.377888\pi$
$503$	16.7887	0.748573	0.374287	+	0.927313i	$0.377888\pi$
$504$	0	0
$505$	−12.0519	−0.536303
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 18.3712i	− 0.814289i	−0.913364	−	0.407144i	$-0.866525\pi$
$509$	− 18.3712i	− 0.814289i	0.913364	−	0.407144i	$-0.133475\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−43.9701	−1.93755
$516$	0	0
$517$	11.5619	0.508493
$518$	0	0
$519$	0	0
$520$	0	0
$521$	2.64236i	0.115764i	0.998323	+	0.0578819i	$0.0184347\pi$
$521$	2.64236i	0.115764i	−0.998323	+	0.0578819i	$0.981565\pi$
$522$	0	0
$523$	6.75295i	0.295286i	0.989041	+	0.147643i	$0.0471686\pi$
$523$	6.75295i	0.295286i	−0.989041	+	0.147643i	$0.952831\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	56.4936	2.46090
$528$	0	0
$529$	−7.36695	−0.320302
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 2.53225i	− 0.109684i
$534$	0	0
$535$	− 61.1720i	− 2.64470i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−15.4044	−0.662287	−0.331144	−	0.943580i	$-0.607434\pi$
$541$	−15.4044	−0.662287	−0.331144	+	0.943580i	$0.607434\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 26.1976i	− 1.12218i
$546$	0	0
$547$	− 4.15743i	− 0.177759i	−0.996042	−	0.0888794i	$-0.971671\pi$
$547$	− 4.15743i	− 0.177759i	0.996042	−	0.0888794i	$-0.0283286\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	1.92563	0.0820345
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	39.5477i	1.67569i	0.545909	+	0.837845i	$0.316185\pi$
$557$	39.5477i	1.67569i	−0.545909	+	0.837845i	$0.683815\pi$
$558$	0	0
$559$	− 8.27703i	− 0.350081i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−17.3888	−0.732852	−0.366426	−	0.930447i	$-0.619419\pi$
$563$	−17.3888	−0.732852	−0.366426	+	0.930447i	$0.619419\pi$
$564$	0	0
$565$	−15.0260	−0.632147
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 20.3546i	− 0.853311i	−0.904414	−	0.426655i	$-0.859692\pi$
$569$	− 20.3546i	− 0.853311i	0.904414	−	0.426655i	$-0.140308\pi$
$570$	0	0
$571$	11.3983i	0.477005i	0.971142	+	0.238502i	$0.0766564\pi$
$571$	11.3983i	0.477005i	−0.971142	+	0.238502i	$0.923344\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−20.4948	−0.854691
$576$	0	0
$577$	0.369903	0.0153993	0.00769963	−	0.999970i	$-0.497549\pi$
$577$	0.369903	0.0153993	0.00769963	+	0.999970i	$0.497549\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	− 21.0048i	− 0.869930i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−42.8924	−1.77036	−0.885180	−	0.465249i	$-0.845965\pi$
$587$	−42.8924	−1.77036	−0.885180	+	0.465249i	$0.845965\pi$
$588$	0	0
$589$	31.4044	1.29400
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 30.9429i	− 1.27067i	−0.772236	−	0.635336i	$-0.780862\pi$
$593$	− 30.9429i	− 1.27067i	0.772236	−	0.635336i	$-0.219138\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	1.61421	0.0659549	0.0329775	−	0.999456i	$-0.489501\pi$
$599$	1.61421	0.0659549	0.0329775	+	0.999456i	$0.489501\pi$
$600$	0	0
$601$	32.6850	1.33325	0.666623	−	0.745395i	$-0.267739\pi$
$601$	32.6850	1.33325	0.666623	+	0.745395i	$0.267739\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	27.2029i	1.10596i
$606$	0	0
$607$	33.7320i	1.36914i	0.728947	+	0.684571i	$0.240010\pi$
$607$	33.7320i	1.36914i	−0.728947	+	0.684571i	$0.759990\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−4.92715	−0.199331
$612$	0	0
$613$	4.85693	0.196169	0.0980847	−	0.995178i	$-0.468728\pi$
$613$	4.85693	0.196169	0.0980847	+	0.995178i	$0.468728\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	36.1542i	1.45551i	0.685835	+	0.727757i	$0.259437\pi$
$617$	36.1542i	1.45551i	−0.685835	+	0.727757i	$0.740563\pi$
$618$	0	0
$619$	− 43.0156i	− 1.72894i	−0.502681	−	0.864472i	$-0.667653\pi$
$619$	− 43.0156i	− 1.72894i	0.502681	−	0.864472i	$-0.332347\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−24.0490	−0.961958
$626$	0	0
$627$	0	0
$628$	0	0
$629$	45.5347i	1.81559i
$630$	0	0
$631$	− 27.7378i	− 1.10423i	−0.833769	−	0.552113i	$-0.813822\pi$
$631$	− 27.7378i	− 1.10423i	0.833769	−	0.552113i	$-0.186178\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−45.7913	−1.81717
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 36.8225i	− 1.45440i	−0.686425	−	0.727201i	$-0.740821\pi$
$641$	− 36.8225i	− 1.45440i	0.686425	−	0.727201i	$-0.259179\pi$
$642$	0	0
$643$	32.3306i	1.27499i	0.770453	+	0.637497i	$0.220030\pi$
$643$	32.3306i	1.27499i	−0.770453	+	0.637497i	$0.779970\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−19.9355	−0.783746	−0.391873	−	0.920019i	$-0.628173\pi$
$647$	−19.9355	−0.783746	−0.391873	+	0.920019i	$0.628173\pi$
$648$	0	0
$649$	7.81652	0.306825
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 13.7794i	− 0.539230i	−0.962968	−	0.269615i	$-0.913104\pi$
$653$	− 13.7794i	− 0.539230i	0.962968	−	0.269615i	$-0.0868964\pi$
$654$	0	0
$655$	41.8793i	1.63636i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	36.3102	1.41445	0.707223	−	0.706991i	$-0.249948\pi$
$659$	36.3102	1.41445	0.707223	+	0.706991i	$0.249948\pi$
$660$	0	0
$661$	−12.7224	−0.494844	−0.247422	−	0.968908i	$-0.579583\pi$
$661$	−12.7224	−0.494844	−0.247422	+	0.968908i	$0.579583\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 2.31500i	− 0.0896370i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−10.2474	−0.395596
$672$	0	0
$673$	2.02595	0.0780947	0.0390474	−	0.999237i	$-0.487568\pi$
$673$	2.02595	0.0780947	0.0390474	+	0.999237i	$0.487568\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 9.14419i	− 0.351440i	−0.984440	−	0.175720i	$-0.943775\pi$
$677$	− 9.14419i	− 0.351440i	0.984440	−	0.175720i	$-0.0562253\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	20.7834	0.795256	0.397628	−	0.917547i	$-0.369833\pi$
$683$	20.7834	0.795256	0.397628	+	0.917547i	$0.369833\pi$
$684$	0	0
$685$	−40.7339	−1.55636
$686$	0	0
$687$	0	0
$688$	0	0
$689$	8.95126i	0.341016i
$690$	0	0
$691$	− 13.6234i	− 0.518259i	−0.965843	−	0.259129i	$-0.916564\pi$
$691$	− 13.6234i	− 0.518259i	0.965843	−	0.259129i	$-0.0834356\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−14.4491	−0.548085
$696$	0	0
$697$	22.3440	0.846337
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 24.0118i	− 0.906912i	−0.891279	−	0.453456i	$-0.850191\pi$
$701$	− 24.0118i	− 0.906912i	0.891279	−	0.453456i	$-0.149809\pi$
$702$	0	0
$703$	25.3125i	0.954678i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	50.1268	1.88255	0.941276	−	0.337638i	$-0.109628\pi$
$709$	50.1268	1.88255	0.941276	+	0.337638i	$0.109628\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 37.7545i	− 1.41392i
$714$	0	0
$715$	3.36651i	0.125901i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	25.9176	0.966564	0.483282	−	0.875465i	$-0.339445\pi$
$719$	25.9176	0.966564	0.483282	+	0.875465i	$0.339445\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	3.03494i	0.112715i
$726$	0	0
$727$	− 21.2777i	− 0.789144i	−0.918865	−	0.394572i	$-0.870893\pi$
$727$	− 21.2777i	− 0.789144i	0.918865	−	0.394572i	$-0.129107\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	73.0345	2.70128
$732$	0	0
$733$	−27.4044	−1.01220	−0.506102	−	0.862473i	$-0.668914\pi$
$733$	−27.4044	−1.01220	−0.506102	+	0.862473i	$0.668914\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	8.67552i	0.319567i
$738$	0	0
$739$	− 20.1613i	− 0.741644i	−0.928704	−	0.370822i	$-0.879076\pi$
$739$	− 20.1613i	− 0.741644i	0.928704	−	0.370822i	$-0.120924\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	31.7971	1.16652	0.583261	−	0.812285i	$-0.301776\pi$
$743$	31.7971	1.16652	0.583261	+	0.812285i	$0.301776\pi$
$744$	0	0
$745$	40.7339	1.49237
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	− 15.4909i	− 0.565270i	−0.959228	−	0.282635i	$-0.908792\pi$
$751$	− 15.4909i	− 0.565270i	0.959228	−	0.282635i	$-0.0912085\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	45.5435	1.65750
$756$	0	0
$757$	9.17198	0.333361	0.166681	−	0.986011i	$-0.446695\pi$
$757$	9.17198	0.333361	0.166681	+	0.986011i	$0.446695\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	27.4787i	0.996101i	0.867148	+	0.498051i	$0.165951\pi$
$761$	27.4787i	0.996101i	−0.867148	+	0.498051i	$0.834049\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−3.33103	−0.120277
$768$	0	0
$769$	−16.0779	−0.579782	−0.289891	−	0.957060i	$-0.593619\pi$
$769$	−16.0779	−0.579782	−0.289891	+	0.957060i	$0.593619\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 9.30041i	− 0.334512i	−0.985913	−	0.167256i	$-0.946509\pi$
$773$	− 9.30041i	− 0.334512i	0.985913	−	0.167256i	$-0.0534907\pi$
$774$	0	0
$775$	49.4957i	1.77794i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	12.4209	0.445023
$780$	0	0
$781$	−10.7820	−0.385811
$782$	0	0
$783$	0	0
$784$	0	0
$785$	3.27397i	0.116853i
$786$	0	0
$787$	− 10.0995i	− 0.360010i	−0.983666	−	0.180005i	$-0.942389\pi$
$787$	− 10.0995i	− 0.360010i	0.983666	−	0.180005i	$-0.0576114\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	4.36695	0.155075
$794$	0	0
$795$	0	0
$796$	0	0
$797$	39.1184i	1.38565i	0.721108	+	0.692823i	$0.243633\pi$
$797$	39.1184i	1.38565i	−0.721108	+	0.692823i	$0.756367\pi$
$798$	0	0
$799$	− 43.4760i	− 1.53807i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−9.19243	−0.324394
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 4.94766i	− 0.173951i	−0.996210	−	0.0869753i	$-0.972280\pi$
$809$	− 4.94766i	− 0.173951i	0.996210	−	0.0869753i	$-0.0277201\pi$
$810$	0	0
$811$	42.8005i	1.50293i	0.659773	+	0.751465i	$0.270653\pi$
$811$	42.8005i	1.50293i	−0.659773	+	0.751465i	$0.729347\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	40.2641	1.41039
$816$	0	0
$817$	40.5994	1.42039
$818$	0	0
$819$	0	0
$820$	0	0
$821$	46.0034i	1.60553i	0.596296	+	0.802765i	$0.296639\pi$
$821$	46.0034i	1.60553i	−0.596296	+	0.802765i	$0.703361\pi$
$822$	0	0
$823$	− 30.1254i	− 1.05011i	−0.851070	−	0.525053i	$-0.824046\pi$
$823$	− 30.1254i	− 1.05011i	0.851070	−	0.525053i	$-0.175954\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	7.94856	0.276399	0.138199	−	0.990404i	$-0.455869\pi$
$827$	7.94856	0.276399	0.138199	+	0.990404i	$0.455869\pi$
$828$	0	0
$829$	21.1413	0.734266	0.367133	−	0.930168i	$-0.380339\pi$
$829$	21.1413	0.734266	0.367133	+	0.930168i	$0.380339\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	− 34.4857i	− 1.19343i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−35.0074	−1.20859	−0.604295	−	0.796760i	$-0.706545\pi$
$839$	−35.0074	−1.20859	−0.604295	+	0.796760i	$0.706545\pi$
$840$	0	0
$841$	28.6572	0.988179
$842$	0	0
$843$	0	0
$844$	0	0
$845$	40.0504i	1.37777i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	30.4307	1.04315
$852$	0	0
$853$	6.26133	0.214384	0.107192	−	0.994238i	$-0.465814\pi$
$853$	6.26133	0.214384	0.107192	+	0.994238i	$0.465814\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 35.4125i	− 1.20967i	−0.796352	−	0.604834i	$-0.793240\pi$
$857$	− 35.4125i	− 1.20967i	0.796352	−	0.604834i	$-0.206760\pi$
$858$	0	0
$859$	− 28.3407i	− 0.966972i	−0.875352	−	0.483486i	$-0.839370\pi$
$859$	− 28.3407i	− 0.966972i	0.875352	−	0.483486i	$-0.160630\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	20.7426	0.706087	0.353043	−	0.935607i	$-0.385147\pi$
$863$	20.7426	0.706087	0.353043	+	0.935607i	$0.385147\pi$
$864$	0	0
$865$	34.0230	1.15682
$866$	0	0
$867$	0	0
$868$	0	0
$869$	9.57347i	0.324758i
$870$	0	0
$871$	− 3.69710i	− 0.125271i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	17.3640	0.586341	0.293170	−	0.956060i	$-0.405290\pi$
$877$	17.3640	0.586341	0.293170	+	0.956060i	$0.405290\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 51.9905i	− 1.75160i	−0.482671	−	0.875802i	$-0.660333\pi$
$881$	− 51.9905i	− 1.75160i	0.482671	−	0.875802i	$-0.339667\pi$
$882$	0	0
$883$	− 15.9062i	− 0.535288i	−0.963518	−	0.267644i	$-0.913755\pi$
$883$	− 15.9062i	− 0.535288i	0.963518	−	0.267644i	$-0.0862451\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	21.5497	0.723569	0.361784	−	0.932262i	$-0.382168\pi$
$887$	21.5497	0.723569	0.361784	+	0.932262i	$0.382168\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 24.1680i	− 0.808751i
$894$	0	0
$895$	− 36.0675i	− 1.20560i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−5.59081	−0.186464
$900$	0	0
$901$	−78.9837	−2.63133
$902$	0	0
$903$	0	0
$904$	0	0
$905$	46.3499i	1.54072i
$906$	0	0
$907$	− 27.0317i	− 0.897574i	−0.893639	−	0.448787i	$-0.851856\pi$
$907$	− 27.0317i	− 0.897574i	0.893639	−	0.448787i	$-0.148144\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	3.95386	0.130997	0.0654987	−	0.997853i	$-0.479136\pi$
$911$	3.95386	0.130997	0.0654987	+	0.997853i	$0.479136\pi$
$912$	0	0
$913$	14.9174	0.493693
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	25.3125i	0.834981i	0.908681	+	0.417491i	$0.137090\pi$
$919$	25.3125i	0.834981i	−0.908681	+	0.417491i	$0.862910\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	4.59479	0.151239
$924$	0	0
$925$	−39.8944	−1.31172
$926$	0	0
$927$	0	0
$928$	0	0
$929$	34.9832i	1.14776i	0.818939	+	0.573881i	$0.194563\pi$
$929$	34.9832i	1.14776i	−0.818939	+	0.573881i	$0.805437\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−29.7053	−0.971467
$936$	0	0
$937$	−10.0779	−0.329229	−0.164615	−	0.986358i	$-0.552638\pi$
$937$	−10.0779	−0.329229	−0.164615	+	0.986358i	$0.552638\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 15.6921i	− 0.511549i	−0.966736	−	0.255774i	$-0.917670\pi$
$941$	− 15.6921i	− 0.511549i	0.966736	−	0.255774i	$-0.0823304\pi$
$942$	0	0
$943$	− 14.9324i	− 0.486266i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−39.7048	−1.29023	−0.645117	−	0.764084i	$-0.723191\pi$
$947$	−39.7048	−1.29023	−0.645117	+	0.764084i	$0.723191\pi$
$948$	0	0
$949$	3.91738	0.127164
$950$	0	0
$951$	0	0
$952$	0	0
$953$	51.6901i	1.67441i	0.546891	+	0.837203i	$0.315811\pi$
$953$	51.6901i	1.67441i	−0.546891	+	0.837203i	$0.684189\pi$
$954$	0	0
$955$	79.8041i	2.58240i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−60.1787	−1.94125
$962$	0	0
$963$	0	0
$964$	0	0
$965$	43.7536i	1.40848i
$966$	0	0
$967$	13.9090i	0.447285i	0.974671	+	0.223642i	$0.0717948\pi$
$967$	13.9090i	0.447285i	−0.974671	+	0.223642i	$0.928205\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	2.31692	0.0743534	0.0371767	−	0.999309i	$-0.488164\pi$
$971$	2.31692	0.0743534	0.0371767	+	0.999309i	$0.488164\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 30.6305i	− 0.979955i	−0.871735	−	0.489977i	$-0.837005\pi$
$977$	− 30.6305i	− 0.979955i	0.871735	−	0.489977i	$-0.162995\pi$
$978$	0	0
$979$	23.3924i	0.747623i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	11.5502	0.368394	0.184197	−	0.982889i	$-0.441032\pi$
$983$	11.5502	0.368394	0.184197	+	0.982889i	$0.441032\pi$
$984$	0	0
$985$	−9.68495	−0.308588
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 48.8087i	− 1.55203i
$990$	0	0
$991$	− 20.2916i	− 0.644583i	−0.946641	−	0.322291i	$-0.895547\pi$
$991$	− 20.2916i	− 0.644583i	0.946641	−	0.322291i	$-0.104453\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	32.0449	1.01589
$996$	0	0
$997$	−12.5015	−0.395926	−0.197963	−	0.980210i	$-0.563433\pi$
$997$	−12.5015	−0.395926	−0.197963	+	0.980210i	$0.563433\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7056.2.h.n.4607.1		12
3.2	odd	2	inner	7056.2.h.n.4607.12		12
4.3	odd	2	inner	7056.2.h.n.4607.2		12
7.2	even	3		1008.2.cq.c.431.1	yes	12
7.4	even	3		1008.2.cq.b.863.6	yes	12
7.6	odd	2		7056.2.h.o.4607.11		12
12.11	even	2	inner	7056.2.h.n.4607.11		12
21.2	odd	6		1008.2.cq.c.431.6	yes	12
21.11	odd	6		1008.2.cq.b.863.1	yes	12
21.20	even	2		7056.2.h.o.4607.2		12
28.11	odd	6		1008.2.cq.c.863.6	yes	12
28.23	odd	6		1008.2.cq.b.431.1	✓	12
28.27	even	2		7056.2.h.o.4607.12		12
84.11	even	6		1008.2.cq.c.863.1	yes	12
84.23	even	6		1008.2.cq.b.431.6	yes	12
84.83	odd	2		7056.2.h.o.4607.1		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1008.2.cq.b.431.1	✓	12	28.23	odd	6
1008.2.cq.b.431.6	yes	12	84.23	even	6
1008.2.cq.b.863.1	yes	12	21.11	odd	6
1008.2.cq.b.863.6	yes	12	7.4	even	3
1008.2.cq.c.431.1	yes	12	7.2	even	3
1008.2.cq.c.431.6	yes	12	21.2	odd	6
1008.2.cq.c.863.1	yes	12	84.11	even	6
1008.2.cq.c.863.6	yes	12	28.11	odd	6
7056.2.h.n.4607.1		12	1.1	even	1	trivial
7056.2.h.n.4607.2		12	4.3	odd	2	inner
7056.2.h.n.4607.11		12	12.11	even	2	inner
7056.2.h.n.4607.12		12	3.2	odd	2	inner
7056.2.h.o.4607.1		12	84.83	odd	2
7056.2.h.o.4607.2		12	21.20	even	2
7056.2.h.o.4607.11		12	7.6	odd	2
7056.2.h.o.4607.12		12	28.27	even	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.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.19116i q^{5} -1.57338 q^{11} +0.670500 q^{13} +5.91633i q^{17} +3.28885i q^{19} +3.95386 q^{23} -5.18348 q^{25} -0.585502i q^{29} -9.54875i q^{31} +7.69645 q^{37} -3.77666i q^{41} -12.3446i q^{43} -7.34847 q^{47} +13.3501i q^{53} +5.02090i q^{55} -4.96798 q^{59} +6.51298 q^{61} -2.13967i q^{65} -5.51394i q^{67} +6.85278 q^{71} +5.84248 q^{73} -6.08465i q^{79} -9.48111 q^{83} +18.8799 q^{85} -14.8676i q^{89} +10.4952 q^{95} -0.183476 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 4 q^{13} - 4 q^{25} - 12 q^{37} + 32 q^{61} + 36 q^{73} + 64 q^{85} + 56 q^{97}+O(q^{100}) \) 12 * q - 4 * q^13 - 4 * q^25 - 12 * q^37 + 32 * q^61 + 36 * q^73 + 64 * q^85 + 56 * q^97

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