LMFDB - Embedded newform 7056.2.b.w.1567.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

chi = DirichletCharacter(H, H._module([1, 0, 0, 1]))

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

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

chi := DirichletCharacter("7056.1567");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$56.3424436662$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.339738624.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{6} + 14x^{4} - 8x^{2} + 4 $ x^8 - 4x^6 + 14x^4 - 8*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 2352)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1567.3
Root			$1.60021 - 0.923880i$ of defining polynomial
Character	$\chi$	$=$	7056.1567
Dual form			7056.2.b.w.1567.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.68412i q^{5} +O(q^{10})$	$q-1.68412i q^{5} -3.53188i q^{11} +2.93015i q^{13} +2.01140i q^{17} +1.69764 q^{19} +1.59851i q^{23} +2.16373 q^{25} -7.94028 q^{29} -4.95367 q^{31} -10.4663 q^{37} -2.86237i q^{41} +11.7518i q^{43} +6.70319 q^{47} -2.92109 q^{53} -5.94812 q^{55} +7.75532 q^{59} +12.5925i q^{61} +4.93473 q^{65} +1.70195i q^{67} +6.13052i q^{71} +2.43166i q^{73} +0.865467i q^{79} -14.5319 q^{83} +3.38744 q^{85} -15.9387i q^{89} -2.85903i q^{95} -9.82270i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q - 24 q^{25} - 16 q^{29} - 32 q^{31} + 16 q^{47} - 16 q^{53} - 64 q^{55} - 48 q^{59} + 16 q^{65} - 64 q^{85}+O(q^{100}) $ 8 * q - 24 * q^25 - 16 * q^29 - 32 * q^31 + 16 * q^47 - 16 * q^53 - 64 * q^55 - 48 * q^59 + 16 * q^65 - 64 * q^85

Display 100 coefficients 10 coefficients

Character values

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

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

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	− 1.68412i	− 0.753163i	−0.926384	−	0.376581i	$-0.877100\pi$
$5$	− 1.68412i	− 0.753163i	0.926384	−	0.376581i	$-0.122900\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 3.53188i	− 1.06490i	−0.846461	−	0.532451i	$-0.821271\pi$
$11$	− 3.53188i	− 1.06490i	0.846461	−	0.532451i	$-0.178729\pi$
$12$	0	0
$13$	2.93015i	0.812678i	0.913722	+	0.406339i	$0.133195\pi$
$13$	2.93015i	0.812678i	−0.913722	+	0.406339i	$0.866805\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2.01140i	0.487835i	0.969796	+	0.243918i	$0.0784326\pi$
$17$	2.01140i	0.487835i	−0.969796	+	0.243918i	$0.921567\pi$
$18$	0	0
$19$	1.69764	0.389465	0.194733	−	0.980856i	$-0.437616\pi$
$19$	1.69764	0.389465	0.194733	+	0.980856i	$0.437616\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	1.59851i	0.333313i	0.986015	+	0.166657i	$0.0532971\pi$
$23$	1.59851i	0.333313i	−0.986015	+	0.166657i	$0.946703\pi$
$24$	0	0
$25$	2.16373	0.432746
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−7.94028	−1.47447	−0.737237	−	0.675635i	$-0.763870\pi$
$29$	−7.94028	−1.47447	−0.737237	+	0.675635i	$0.763870\pi$
$30$	0	0
$31$	−4.95367	−0.889705	−0.444853	−	0.895604i	$-0.646744\pi$
$31$	−4.95367	−0.889705	−0.444853	+	0.895604i	$0.646744\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−10.4663	−1.72066	−0.860328	−	0.509740i	$-0.829742\pi$
$37$	−10.4663	−1.72066	−0.860328	+	0.509740i	$0.829742\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 2.86237i	− 0.447027i	−0.974701	−	0.223514i	$-0.928247\pi$
$41$	− 2.86237i	− 0.447027i	0.974701	−	0.223514i	$-0.0717527\pi$
$42$	0	0
$43$	11.7518i	1.79214i	0.443917	+	0.896068i	$0.353589\pi$
$43$	11.7518i	1.79214i	−0.443917	+	0.896068i	$0.646411\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	6.70319	0.977760	0.488880	−	0.872351i	$-0.337406\pi$
$47$	6.70319	0.977760	0.488880	+	0.872351i	$0.337406\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−2.92109	−0.401242	−0.200621	−	0.979669i	$-0.564296\pi$
$53$	−2.92109	−0.401242	−0.200621	+	0.979669i	$0.564296\pi$
$54$	0	0
$55$	−5.94812	−0.802045
$56$	0	0
$57$	0	0
$58$	0	0
$59$	7.75532	1.00966	0.504828	−	0.863220i	$-0.331556\pi$
$59$	7.75532	1.00966	0.504828	+	0.863220i	$0.331556\pi$
$60$	0	0
$61$	12.5925i	1.61231i	0.591704	+	0.806155i	$0.298455\pi$
$61$	12.5925i	1.61231i	−0.591704	+	0.806155i	$0.701545\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	4.93473	0.612079
$66$	0	0
$67$	1.70195i	0.207926i	0.994581	+	0.103963i	$0.0331524\pi$
$67$	1.70195i	0.207926i	−0.994581	+	0.103963i	$0.966848\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	6.13052i	0.727558i	0.931485	+	0.363779i	$0.118514\pi$
$71$	6.13052i	0.727558i	−0.931485	+	0.363779i	$0.881486\pi$
$72$	0	0
$73$	2.43166i	0.284604i	0.989823	+	0.142302i	$0.0454505\pi$
$73$	2.43166i	0.284604i	−0.989823	+	0.142302i	$0.954550\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	0.865467i	0.0973727i	0.998814	+	0.0486863i	$0.0155035\pi$
$79$	0.865467i	0.0973727i	−0.998814	+	0.0486863i	$0.984497\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−14.5319	−1.59508	−0.797540	−	0.603266i	$-0.793866\pi$
$83$	−14.5319	−1.59508	−0.797540	+	0.603266i	$0.793866\pi$
$84$	0	0
$85$	3.38744	0.367419
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 15.9387i	− 1.68950i	−0.535160	−	0.844751i	$-0.679749\pi$
$89$	− 15.9387i	− 1.68950i	0.535160	−	0.844751i	$-0.320251\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 2.85903i	− 0.293331i
$96$	0	0
$97$	− 9.82270i	− 0.997344i	−0.866791	−	0.498672i	$-0.833821\pi$
$97$	− 9.82270i	− 0.997344i	0.866791	−	0.498672i	$-0.166179\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 9.42764i	− 0.938086i	−0.883175	−	0.469043i	$-0.844599\pi$
$101$	− 9.42764i	− 0.938086i	0.883175	−	0.469043i	$-0.155401\pi$
$102$	0	0
$103$	−15.6627	−1.54329	−0.771644	−	0.636055i	$-0.780565\pi$
$103$	−15.6627	−1.54329	−0.771644	+	0.636055i	$0.780565\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	18.9792i	1.83479i	0.397977	+	0.917395i	$0.369712\pi$
$107$	18.9792i	1.83479i	−0.397977	+	0.917395i	$0.630288\pi$
$108$	0	0
$109$	10.5590	1.01137	0.505685	−	0.862718i	$-0.331240\pi$
$109$	10.5590	1.01137	0.505685	+	0.862718i	$0.331240\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−15.4530	−1.45369	−0.726846	−	0.686800i	$-0.759015\pi$
$113$	−15.4530	−1.45369	−0.726846	+	0.686800i	$0.759015\pi$
$114$	0	0
$115$	2.69209	0.251039
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−1.47419	−0.134017
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 12.0646i	− 1.07909i
$126$	0	0
$127$	2.16478i	0.192094i	0.995377	+	0.0960468i	$0.0306198\pi$
$127$	2.16478i	0.192094i	−0.995377	+	0.0960468i	$0.969380\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	15.3219	1.33868	0.669341	−	0.742955i	$-0.266577\pi$
$131$	15.3219	1.33868	0.669341	+	0.742955i	$0.266577\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−12.8614	−1.09882	−0.549410	−	0.835553i	$-0.685148\pi$
$137$	−12.8614	−1.09882	−0.549410	+	0.835553i	$0.685148\pi$
$138$	0	0
$139$	−0.743971	−0.0631028	−0.0315514	−	0.999502i	$-0.510045\pi$
$139$	−0.743971	−0.0631028	−0.0315514	+	0.999502i	$0.510045\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	10.3489	0.865423
$144$	0	0
$145$	13.3724i	1.11052i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	7.30262	0.598254	0.299127	−	0.954213i	$-0.403305\pi$
$149$	7.30262	0.598254	0.299127	+	0.954213i	$0.403305\pi$
$150$	0	0
$151$	6.28343i	0.511338i	0.966764	+	0.255669i	$0.0822958\pi$
$151$	6.28343i	0.511338i	−0.966764	+	0.255669i	$0.917704\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	8.34259i	0.670093i
$156$	0	0
$157$	8.91152i	0.711217i	0.934635	+	0.355608i	$0.115726\pi$
$157$	8.91152i	0.711217i	−0.934635	+	0.355608i	$0.884274\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	0.242138i	0.0189657i	0.999955	+	0.00948287i	$0.00301854\pi$
$163$	0.242138i	0.0189657i	−0.999955	+	0.00948287i	$0.996981\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	5.82817	0.450997	0.225499	−	0.974243i	$-0.427599\pi$
$167$	5.82817	0.450997	0.225499	+	0.974243i	$0.427599\pi$
$168$	0	0
$169$	4.41421	0.339555
$170$	0	0
$171$	0	0
$172$	0	0
$173$	16.7935i	1.27678i	0.769712	+	0.638392i	$0.220400\pi$
$173$	16.7935i	1.27678i	−0.769712	+	0.638392i	$0.779600\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	21.2399i	1.58754i	0.608217	+	0.793771i	$0.291885\pi$
$179$	21.2399i	1.58754i	−0.608217	+	0.793771i	$0.708115\pi$
$180$	0	0
$181$	− 2.74444i	− 0.203993i	−0.994785	−	0.101996i	$-0.967477\pi$
$181$	− 2.74444i	− 0.203993i	0.994785	−	0.101996i	$-0.0325230\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	17.6266i	1.29593i
$186$	0	0
$187$	7.10401	0.519497
$188$	0	0
$189$	0	0
$190$	0	0
$191$	1.60924i	0.116440i	0.998304	+	0.0582201i	$0.0185425\pi$
$191$	1.60924i	0.116440i	−0.998304	+	0.0582201i	$0.981457\pi$
$192$	0	0
$193$	4.01109	0.288725	0.144362	−	0.989525i	$-0.453887\pi$
$193$	4.01109	0.288725	0.144362	+	0.989525i	$0.453887\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−11.7413	−0.836534	−0.418267	−	0.908324i	$-0.637362\pi$
$197$	−11.7413	−0.836534	−0.418267	+	0.908324i	$0.637362\pi$
$198$	0	0
$199$	17.3603	1.23064	0.615319	−	0.788278i	$-0.289027\pi$
$199$	17.3603	1.23064	0.615319	+	0.788278i	$0.289027\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−4.82058	−0.336684
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 5.99586i	− 0.414743i
$210$	0	0
$211$	24.1172i	1.66030i	0.557543	+	0.830148i	$0.311744\pi$
$211$	24.1172i	1.66030i	−0.557543	+	0.830148i	$0.688256\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	19.7915	1.34977
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−5.89369	−0.396453
$222$	0	0
$223$	−20.1154	−1.34702	−0.673512	−	0.739176i	$-0.735215\pi$
$223$	−20.1154	−1.34702	−0.673512	+	0.739176i	$0.735215\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	17.5893	1.16744	0.583721	−	0.811954i	$-0.301596\pi$
$227$	17.5893	1.16744	0.583721	+	0.811954i	$0.301596\pi$
$228$	0	0
$229$	3.99205i	0.263802i	0.991263	+	0.131901i	$0.0421081\pi$
$229$	3.99205i	0.263802i	−0.991263	+	0.131901i	$0.957892\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−3.26967	−0.214203	−0.107102	−	0.994248i	$-0.534157\pi$
$233$	−3.26967	−0.214203	−0.107102	+	0.994248i	$0.534157\pi$
$234$	0	0
$235$	− 11.2890i	− 0.736412i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	28.9127i	1.87021i	0.354371	+	0.935105i	$0.384695\pi$
$239$	28.9127i	1.87021i	−0.354371	+	0.935105i	$0.615305\pi$
$240$	0	0
$241$	6.11269i	0.393753i	0.980428	+	0.196876i	$0.0630798\pi$
$241$	6.11269i	0.393753i	−0.980428	+	0.196876i	$0.936920\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	4.97434i	0.316510i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−27.7471	−1.75138	−0.875691	−	0.482872i	$-0.839594\pi$
$251$	−27.7471	−1.75138	−0.875691	+	0.482872i	$0.839594\pi$
$252$	0	0
$253$	5.64576	0.354946
$254$	0	0
$255$	0	0
$256$	0	0
$257$	2.86237i	0.178550i	0.996007	+	0.0892749i	$0.0284550\pi$
$257$	2.86237i	0.178550i	−0.996007	+	0.0892749i	$0.971545\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 19.4528i	− 1.19951i	−0.800184	−	0.599755i	$-0.795265\pi$
$263$	− 19.4528i	− 1.19951i	0.800184	−	0.599755i	$-0.204735\pi$
$264$	0	0
$265$	4.91947i	0.302201i
$266$	0	0
$267$	0	0
$268$	0	0
$269$	16.6831i	1.01719i	0.861007	+	0.508594i	$0.169835\pi$
$269$	16.6831i	1.01719i	−0.861007	+	0.508594i	$0.830165\pi$
$270$	0	0
$271$	19.9057	1.20918	0.604591	−	0.796536i	$-0.293336\pi$
$271$	19.9057	1.20918	0.604591	+	0.796536i	$0.293336\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 7.64204i	− 0.460832i
$276$	0	0
$277$	−14.3275	−0.860854	−0.430427	−	0.902625i	$-0.641637\pi$
$277$	−14.3275	−0.860854	−0.430427	+	0.902625i	$0.641637\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−16.5450	−0.986992	−0.493496	−	0.869748i	$-0.664281\pi$
$281$	−16.5450	−0.986992	−0.493496	+	0.869748i	$0.664281\pi$
$282$	0	0
$283$	29.8591	1.77494	0.887469	−	0.460868i	$-0.152462\pi$
$283$	29.8591	1.77494	0.887469	+	0.460868i	$0.152462\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	12.9543	0.762017
$290$	0	0
$291$	0	0
$292$	0	0
$293$	6.40183i	0.373999i	0.982360	+	0.187000i	$0.0598763\pi$
$293$	6.40183i	0.373999i	−0.982360	+	0.187000i	$0.940124\pi$
$294$	0	0
$295$	− 13.0609i	− 0.760436i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−4.68389	−0.270876
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	21.2074	1.21433
$306$	0	0
$307$	11.6511	0.664961	0.332480	−	0.943110i	$-0.392114\pi$
$307$	11.6511	0.664961	0.332480	+	0.943110i	$0.392114\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	23.0772	1.30859	0.654295	−	0.756239i	$-0.272966\pi$
$311$	23.0772	1.30859	0.654295	+	0.756239i	$0.272966\pi$
$312$	0	0
$313$	− 25.9505i	− 1.46681i	−0.679794	−	0.733404i	$-0.737931\pi$
$313$	− 25.9505i	− 1.46681i	0.679794	−	0.733404i	$-0.262069\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−20.5430	−1.15381	−0.576904	−	0.816812i	$-0.695739\pi$
$317$	−20.5430	−1.15381	−0.576904	+	0.816812i	$0.695739\pi$
$318$	0	0
$319$	28.0441i	1.57017i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	3.41462i	0.189995i
$324$	0	0
$325$	6.34006i	0.351683i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	− 32.7763i	− 1.80155i	−0.434286	−	0.900775i	$-0.642999\pi$
$331$	− 32.7763i	− 1.80155i	0.434286	−	0.900775i	$-0.357001\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	2.86630	0.156602
$336$	0	0
$337$	−33.2791	−1.81283	−0.906414	−	0.422391i	$-0.861191\pi$
$337$	−33.2791	−1.81283	−0.906414	+	0.422391i	$0.861191\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	17.4958i	0.947449i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	9.65926i	0.518536i	0.965805	+	0.259268i	$0.0834813\pi$
$347$	9.65926i	0.518536i	−0.965805	+	0.259268i	$0.916519\pi$
$348$	0	0
$349$	11.9525i	0.639802i	0.947451	+	0.319901i	$0.103650\pi$
$349$	11.9525i	0.639802i	−0.947451	+	0.319901i	$0.896350\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	12.9230i	0.687820i	0.939003	+	0.343910i	$0.111751\pi$
$353$	12.9230i	0.687820i	−0.939003	+	0.343910i	$0.888249\pi$
$354$	0	0
$355$	10.3245	0.547970
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 11.0629i	− 0.583879i	−0.956437	−	0.291939i	$-0.905699\pi$
$359$	− 11.0629i	− 0.583879i	0.956437	−	0.291939i	$-0.0943005\pi$
$360$	0	0
$361$	−16.1180	−0.848317
$362$	0	0
$363$	0	0
$364$	0	0
$365$	4.09522	0.214353
$366$	0	0
$367$	−11.8888	−0.620589	−0.310294	−	0.950641i	$-0.600428\pi$
$367$	−11.8888	−0.620589	−0.310294	+	0.950641i	$0.600428\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−9.52581	−0.493228	−0.246614	−	0.969114i	$-0.579318\pi$
$373$	−9.52581	−0.493228	−0.246614	+	0.969114i	$0.579318\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 23.2662i	− 1.19827i
$378$	0	0
$379$	− 4.52128i	− 0.232243i	−0.993235	−	0.116121i	$-0.962954\pi$
$379$	− 4.52128i	− 0.232243i	0.993235	−	0.116121i	$-0.0370461\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	2.34315	0.119729	0.0598646	−	0.998207i	$-0.480933\pi$
$383$	2.34315	0.119729	0.0598646	+	0.998207i	$0.480933\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−18.0981	−0.917610	−0.458805	−	0.888537i	$-0.651722\pi$
$389$	−18.0981	−0.917610	−0.458805	+	0.888537i	$0.651722\pi$
$390$	0	0
$391$	−3.21524	−0.162602
$392$	0	0
$393$	0	0
$394$	0	0
$395$	1.45755	0.0733375
$396$	0	0
$397$	2.81508i	0.141285i	0.997502	+	0.0706425i	$0.0225050\pi$
$397$	2.81508i	0.141285i	−0.997502	+	0.0706425i	$0.977495\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−6.08242	−0.303741	−0.151871	−	0.988400i	$-0.548530\pi$
$401$	−6.08242	−0.303741	−0.151871	+	0.988400i	$0.548530\pi$
$402$	0	0
$403$	− 14.5150i	− 0.723044i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	36.9659i	1.83233i
$408$	0	0
$409$	8.41275i	0.415984i	0.978131	+	0.207992i	$0.0666928\pi$
$409$	8.41275i	0.415984i	−0.978131	+	0.207992i	$0.933307\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	24.4735i	1.20135i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−18.6496	−0.911094	−0.455547	−	0.890212i	$-0.650556\pi$
$419$	−18.6496	−0.911094	−0.455547	+	0.890212i	$0.650556\pi$
$420$	0	0
$421$	−22.6274	−1.10279	−0.551396	−	0.834243i	$-0.685905\pi$
$421$	−22.6274	−1.10279	−0.551396	+	0.834243i	$0.685905\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	4.35212i	0.211109i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	15.9070i	0.766215i	0.923704	+	0.383107i	$0.125146\pi$
$431$	15.9070i	0.766215i	−0.923704	+	0.383107i	$0.874854\pi$
$432$	0	0
$433$	− 14.4650i	− 0.695146i	−0.937653	−	0.347573i	$-0.887006\pi$
$433$	− 14.4650i	− 0.695146i	0.937653	−	0.347573i	$-0.112994\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	2.71370i	0.129814i
$438$	0	0
$439$	21.7961	1.04027	0.520136	−	0.854084i	$-0.325881\pi$
$439$	21.7961	1.04027	0.520136	+	0.854084i	$0.325881\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	11.8637i	0.563664i	0.959464	+	0.281832i	$0.0909420\pi$
$443$	11.8637i	0.563664i	−0.959464	+	0.281832i	$0.909058\pi$
$444$	0	0
$445$	−26.8428	−1.27247
$446$	0	0
$447$	0	0
$448$	0	0
$449$	38.6472	1.82388	0.911938	−	0.410329i	$-0.134586\pi$
$449$	38.6472	1.82388	0.911938	+	0.410329i	$0.134586\pi$
$450$	0	0
$451$	−10.1096	−0.476040
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	18.9438	0.886153	0.443076	−	0.896484i	$-0.353887\pi$
$457$	18.9438	0.886153	0.443076	+	0.896484i	$0.353887\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	24.5269i	1.14233i	0.820834	+	0.571167i	$0.193509\pi$
$461$	24.5269i	1.14233i	−0.820834	+	0.571167i	$0.806491\pi$
$462$	0	0
$463$	7.59791i	0.353105i	0.984291	+	0.176552i	$0.0564945\pi$
$463$	7.59791i	0.353105i	−0.984291	+	0.176552i	$0.943505\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−11.0188	−0.509891	−0.254945	−	0.966955i	$-0.582057\pi$
$467$	−11.0188	−0.509891	−0.254945	+	0.966955i	$0.582057\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	41.5060	1.90845
$474$	0	0
$475$	3.67323	0.168540
$476$	0	0
$477$	0	0
$478$	0	0
$479$	36.2027	1.65415	0.827073	−	0.562095i	$-0.190005\pi$
$479$	36.2027	1.65415	0.827073	+	0.562095i	$0.190005\pi$
$480$	0	0
$481$	− 30.6680i	− 1.39834i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−16.5426	−0.751162
$486$	0	0
$487$	17.4119i	0.789010i	0.918894	+	0.394505i	$0.129084\pi$
$487$	17.4119i	0.789010i	−0.918894	+	0.394505i	$0.870916\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	21.5154i	0.970977i	0.874243	+	0.485489i	$0.161358\pi$
$491$	21.5154i	0.970977i	−0.874243	+	0.485489i	$0.838642\pi$
$492$	0	0
$493$	− 15.9710i	− 0.719300i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	36.4384i	1.63121i	0.578610	+	0.815604i	$0.303595\pi$
$499$	36.4384i	1.63121i	−0.578610	+	0.815604i	$0.696405\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−39.5840	−1.76496	−0.882482	−	0.470347i	$-0.844129\pi$
$503$	−39.5840	−1.76496	−0.882482	+	0.470347i	$0.844129\pi$
$504$	0	0
$505$	−15.8773	−0.706531
$506$	0	0
$507$	0	0
$508$	0	0
$509$	35.1759i	1.55914i	0.626313	+	0.779572i	$0.284563\pi$
$509$	35.1759i	1.55914i	−0.626313	+	0.779572i	$0.715437\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	26.3778i	1.16235i
$516$	0	0
$517$	− 23.6749i	− 1.04122i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	19.5734i	0.857525i	0.903417	+	0.428762i	$0.141050\pi$
$521$	19.5734i	0.857525i	−0.903417	+	0.428762i	$0.858950\pi$
$522$	0	0
$523$	8.66240	0.378780	0.189390	−	0.981902i	$-0.439349\pi$
$523$	8.66240	0.378780	0.189390	+	0.981902i	$0.439349\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 9.96379i	− 0.434029i
$528$	0	0
$529$	20.4448	0.888902
$530$	0	0
$531$	0	0
$532$	0	0
$533$	8.38718	0.363289
$534$	0	0
$535$	31.9633	1.38190
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	25.0905	1.07873	0.539363	−	0.842074i	$-0.318665\pi$
$541$	25.0905	1.07873	0.539363	+	0.842074i	$0.318665\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 17.7827i	− 0.761726i
$546$	0	0
$547$	0.523032i	0.0223632i	0.999937	+	0.0111816i	$0.00355929\pi$
$547$	0.523032i	0.0223632i	−0.999937	+	0.0111816i	$0.996441\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−13.4797	−0.574256
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−22.1339	−0.937845	−0.468922	−	0.883239i	$-0.655358\pi$
$557$	−22.1339	−0.937845	−0.468922	+	0.883239i	$0.655358\pi$
$558$	0	0
$559$	−34.4346	−1.45643
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−10.6076	−0.447059	−0.223529	−	0.974697i	$-0.571758\pi$
$563$	−10.6076	−0.447059	−0.223529	+	0.974697i	$0.571758\pi$
$564$	0	0
$565$	26.0247i	1.09487i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−10.3988	−0.435940	−0.217970	−	0.975955i	$-0.569943\pi$
$569$	−10.3988	−0.435940	−0.217970	+	0.975955i	$0.569943\pi$
$570$	0	0
$571$	− 21.3518i	− 0.893544i	−0.894648	−	0.446772i	$-0.852573\pi$
$571$	− 21.3518i	− 0.893544i	0.894648	−	0.446772i	$-0.147427\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	3.45875i	0.144240i
$576$	0	0
$577$	− 0.922093i	− 0.0383872i	−0.999816	−	0.0191936i	$-0.993890\pi$
$577$	− 0.922093i	− 0.0383872i	0.999816	−	0.0191936i	$-0.00610989\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	10.3169i	0.427284i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−9.11149	−0.376071	−0.188036	−	0.982162i	$-0.560212\pi$
$587$	−9.11149	−0.376071	−0.188036	+	0.982162i	$0.560212\pi$
$588$	0	0
$589$	−8.40955	−0.346509
$590$	0	0
$591$	0	0
$592$	0	0
$593$	12.2075i	0.501304i	0.968077	+	0.250652i	$0.0806450\pi$
$593$	12.2075i	0.501304i	−0.968077	+	0.250652i	$0.919355\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 35.2167i	− 1.43891i	−0.694537	−	0.719457i	$-0.744391\pi$
$599$	− 35.2167i	− 1.43891i	0.694537	−	0.719457i	$-0.255609\pi$
$600$	0	0
$601$	− 30.6430i	− 1.24995i	−0.780644	−	0.624976i	$-0.785109\pi$
$601$	− 30.6430i	− 1.24995i	0.780644	−	0.624976i	$-0.214891\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	2.48272i	0.100937i
$606$	0	0
$607$	−9.96502	−0.404468	−0.202234	−	0.979337i	$-0.564820\pi$
$607$	−9.96502	−0.404468	−0.202234	+	0.979337i	$0.564820\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	19.6413i	0.794604i
$612$	0	0
$613$	−15.1259	−0.610928	−0.305464	−	0.952204i	$-0.598812\pi$
$613$	−15.1259	−0.610928	−0.305464	+	0.952204i	$0.598812\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−11.7055	−0.471245	−0.235622	−	0.971845i	$-0.575713\pi$
$617$	−11.7055	−0.471245	−0.235622	+	0.971845i	$0.575713\pi$
$618$	0	0
$619$	7.26763	0.292111	0.146055	−	0.989276i	$-0.453342\pi$
$619$	7.26763	0.292111	0.146055	+	0.989276i	$0.453342\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−9.49962	−0.379985
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 21.0520i	− 0.839397i
$630$	0	0
$631$	− 1.40366i	− 0.0558789i	−0.999610	−	0.0279395i	$-0.991105\pi$
$631$	− 1.40366i	− 0.0558789i	0.999610	−	0.0279395i	$-0.00889456\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	3.64576	0.144678
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	33.2272	1.31240	0.656198	−	0.754589i	$-0.272164\pi$
$641$	33.2272	1.31240	0.656198	+	0.754589i	$0.272164\pi$
$642$	0	0
$643$	46.2604	1.82433	0.912166	−	0.409820i	$-0.134409\pi$
$643$	46.2604	1.82433	0.912166	+	0.409820i	$0.134409\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−34.6332	−1.36157	−0.680786	−	0.732482i	$-0.738362\pi$
$647$	−34.6332	−1.36157	−0.680786	+	0.732482i	$0.738362\pi$
$648$	0	0
$649$	− 27.3909i	− 1.07519i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−8.22863	−0.322011	−0.161006	−	0.986953i	$-0.551474\pi$
$653$	−8.22863	−0.322011	−0.161006	+	0.986953i	$0.551474\pi$
$654$	0	0
$655$	− 25.8040i	− 1.00825i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 10.9381i	− 0.426087i	−0.977043	−	0.213044i	$-0.931662\pi$
$659$	− 10.9381i	− 0.426087i	0.977043	−	0.213044i	$-0.0683376\pi$
$660$	0	0
$661$	− 25.1953i	− 0.979985i	−0.871727	−	0.489993i	$-0.836999\pi$
$661$	− 25.1953i	− 0.979985i	0.871727	−	0.489993i	$-0.163001\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 12.6926i	− 0.491461i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	44.4754	1.71695
$672$	0	0
$673$	11.5831	0.446497	0.223248	−	0.974762i	$-0.428334\pi$
$673$	11.5831	0.446497	0.223248	+	0.974762i	$0.428334\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	34.7641i	1.33609i	0.744120	+	0.668046i	$0.232869\pi$
$677$	34.7641i	1.33609i	−0.744120	+	0.668046i	$0.767131\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	11.7692i	0.450335i	0.974320	+	0.225167i	$0.0722929\pi$
$683$	11.7692i	0.450335i	−0.974320	+	0.225167i	$0.927707\pi$
$684$	0	0
$685$	21.6601i	0.827591i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 8.55923i	− 0.326081i
$690$	0	0
$691$	11.3031	0.429991	0.214996	−	0.976615i	$-0.431026\pi$
$691$	11.3031	0.429991	0.214996	+	0.976615i	$0.431026\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	1.25294i	0.0475267i
$696$	0	0
$697$	5.75736	0.218076
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−24.9265	−0.941462	−0.470731	−	0.882277i	$-0.656010\pi$
$701$	−24.9265	−0.941462	−0.470731	+	0.882277i	$0.656010\pi$
$702$	0	0
$703$	−17.7681	−0.670136
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	44.2380	1.66139	0.830697	−	0.556724i	$-0.187942\pi$
$709$	44.2380	1.66139	0.830697	+	0.556724i	$0.187942\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 7.91851i	− 0.296551i
$714$	0	0
$715$	− 17.4289i	− 0.651804i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	5.40637	0.201624	0.100812	−	0.994906i	$-0.467856\pi$
$719$	5.40637	0.201624	0.100812	+	0.994906i	$0.467856\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−17.1806	−0.638072
$726$	0	0
$727$	−36.5459	−1.35541	−0.677706	−	0.735333i	$-0.737026\pi$
$727$	−36.5459	−1.35541	−0.677706	+	0.735333i	$0.737026\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−23.6376	−0.874267
$732$	0	0
$733$	− 27.3245i	− 1.00925i	−0.863338	−	0.504626i	$-0.831630\pi$
$733$	− 27.3245i	− 1.00925i	0.863338	−	0.504626i	$-0.168370\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	6.01109	0.221421
$738$	0	0
$739$	− 17.7874i	− 0.654320i	−0.944969	−	0.327160i	$-0.893908\pi$
$739$	− 17.7874i	− 0.654320i	0.944969	−	0.327160i	$-0.106092\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	30.1701i	1.10683i	0.832904	+	0.553417i	$0.186676\pi$
$743$	30.1701i	1.10683i	−0.832904	+	0.553417i	$0.813324\pi$
$744$	0	0
$745$	− 12.2985i	− 0.450582i
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	42.9930i	1.56884i	0.620232	+	0.784418i	$0.287038\pi$
$751$	42.9930i	1.56884i	−0.620232	+	0.784418i	$0.712962\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	10.5821	0.385121
$756$	0	0
$757$	−13.2021	−0.479839	−0.239919	−	0.970793i	$-0.577121\pi$
$757$	−13.2021	−0.479839	−0.239919	+	0.970793i	$0.577121\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 18.1325i	− 0.657302i	−0.944451	−	0.328651i	$-0.893406\pi$
$761$	− 18.1325i	− 0.657302i	0.944451	−	0.328651i	$-0.106594\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	22.7243i	0.820525i
$768$	0	0
$769$	45.0980i	1.62628i	0.582070	+	0.813139i	$0.302243\pi$
$769$	45.0980i	1.62628i	−0.582070	+	0.813139i	$0.697757\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 36.0470i	− 1.29652i	−0.761420	−	0.648259i	$-0.775497\pi$
$773$	− 36.0470i	− 1.29652i	0.761420	−	0.648259i	$-0.224503\pi$
$774$	0	0
$775$	−10.7184	−0.385016
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 4.85927i	− 0.174102i
$780$	0	0
$781$	21.6523	0.774779
$782$	0	0
$783$	0	0
$784$	0	0
$785$	15.0081	0.535662
$786$	0	0
$787$	−26.3844	−0.940503	−0.470251	−	0.882533i	$-0.655837\pi$
$787$	−26.3844	−0.940503	−0.470251	+	0.882533i	$0.655837\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−36.8981	−1.31029
$794$	0	0
$795$	0	0
$796$	0	0
$797$	52.8839i	1.87324i	0.350344	+	0.936621i	$0.386065\pi$
$797$	52.8839i	1.87324i	−0.350344	+	0.936621i	$0.613935\pi$
$798$	0	0
$799$	13.4828i	0.476986i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	8.58834	0.303076
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−16.1702	−0.568512	−0.284256	−	0.958748i	$-0.591747\pi$
$809$	−16.1702	−0.568512	−0.284256	+	0.958748i	$0.591747\pi$
$810$	0	0
$811$	−4.18532	−0.146967	−0.0734833	−	0.997296i	$-0.523412\pi$
$811$	−4.18532	−0.146967	−0.0734833	+	0.997296i	$0.523412\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0.407791	0.0142843
$816$	0	0
$817$	19.9504i	0.697975i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	4.89720	0.170913	0.0854567	−	0.996342i	$-0.472765\pi$
$821$	4.89720	0.170913	0.0854567	+	0.996342i	$0.472765\pi$
$822$	0	0
$823$	− 6.49031i	− 0.226238i	−0.993581	−	0.113119i	$-0.963916\pi$
$823$	− 6.49031i	− 0.226238i	0.993581	−	0.113119i	$-0.0360841\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 18.8437i	− 0.655258i	−0.944806	−	0.327629i	$-0.893750\pi$
$827$	− 18.8437i	− 0.655258i	0.944806	−	0.327629i	$-0.106250\pi$
$828$	0	0
$829$	− 15.8101i	− 0.549106i	−0.961572	−	0.274553i	$-0.911470\pi$
$829$	− 15.8101i	− 0.549106i	0.961572	−	0.274553i	$-0.0885300\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	− 9.81535i	− 0.339674i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	25.1313	0.867629	0.433814	−	0.901002i	$-0.357167\pi$
$839$	25.1313	0.867629	0.433814	+	0.901002i	$0.357167\pi$
$840$	0	0
$841$	34.0481	1.17407
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 7.43408i	− 0.255740i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 16.7306i	− 0.573518i
$852$	0	0
$853$	− 31.5399i	− 1.07991i	−0.841695	−	0.539953i	$-0.818442\pi$
$853$	− 31.5399i	− 1.07991i	0.841695	−	0.539953i	$-0.181558\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	28.3495i	0.968401i	0.874957	+	0.484201i	$0.160890\pi$
$857$	28.3495i	0.968401i	−0.874957	+	0.484201i	$0.839110\pi$
$858$	0	0
$859$	−34.6798	−1.18326	−0.591630	−	0.806210i	$-0.701515\pi$
$859$	−34.6798	−1.18326	−0.591630	+	0.806210i	$0.701515\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 18.4754i	− 0.628910i	−0.949272	−	0.314455i	$-0.898178\pi$
$863$	− 18.4754i	− 0.628910i	0.949272	−	0.314455i	$-0.101822\pi$
$864$	0	0
$865$	28.2823	0.961626
$866$	0	0
$867$	0	0
$868$	0	0
$869$	3.05673	0.103692
$870$	0	0
$871$	−4.98697	−0.168977
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−42.9784	−1.45128	−0.725639	−	0.688076i	$-0.758456\pi$
$877$	−42.9784	−1.45128	−0.725639	+	0.688076i	$0.758456\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	4.50973i	0.151937i	0.997110	+	0.0759684i	$0.0242048\pi$
$881$	4.50973i	0.151937i	−0.997110	+	0.0759684i	$0.975795\pi$
$882$	0	0
$883$	42.8824i	1.44311i	0.692359	+	0.721553i	$0.256571\pi$
$883$	42.8824i	1.44311i	−0.692359	+	0.721553i	$0.743429\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	4.75725	0.159733	0.0798665	−	0.996806i	$-0.474551\pi$
$887$	4.75725	0.159733	0.0798665	+	0.996806i	$0.474551\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	11.3796	0.380804
$894$	0	0
$895$	35.7705	1.19568
$896$	0	0
$897$	0	0
$898$	0	0
$899$	39.3335	1.31185
$900$	0	0
$901$	− 5.87547i	− 0.195740i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−4.62198	−0.153640
$906$	0	0
$907$	15.8103i	0.524971i	0.964936	+	0.262486i	$0.0845422\pi$
$907$	15.8103i	0.524971i	−0.964936	+	0.262486i	$0.915458\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 15.5236i	− 0.514320i	−0.966369	−	0.257160i	$-0.917213\pi$
$911$	− 15.5236i	− 0.514320i	0.966369	−	0.257160i	$-0.0827868\pi$
$912$	0	0
$913$	51.3248i	1.69860i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	− 58.5264i	− 1.93061i	−0.261129	−	0.965304i	$-0.584095\pi$
$919$	− 58.5264i	− 1.93061i	0.261129	−	0.965304i	$-0.415905\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−17.9633	−0.591271
$924$	0	0
$925$	−22.6464	−0.744607
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 53.1650i	− 1.74429i	−0.489249	−	0.872144i	$-0.662729\pi$
$929$	− 53.1650i	− 1.74429i	0.489249	−	0.872144i	$-0.337271\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 11.9640i	− 0.391266i
$936$	0	0
$937$	− 17.0577i	− 0.557250i	−0.960400	−	0.278625i	$-0.910121\pi$
$937$	− 17.0577i	− 0.557250i	0.960400	−	0.278625i	$-0.0898787\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 16.4321i	− 0.535671i	−0.963465	−	0.267836i	$-0.913692\pi$
$941$	− 16.4321i	− 0.535671i	0.963465	−	0.267836i	$-0.0863084\pi$
$942$	0	0
$943$	4.57554	0.149000
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 25.8513i	− 0.840054i	−0.907512	−	0.420027i	$-0.862021\pi$
$947$	− 25.8513i	− 0.840054i	0.907512	−	0.420027i	$-0.137979\pi$
$948$	0	0
$949$	−7.12514	−0.231292
$950$	0	0
$951$	0	0
$952$	0	0
$953$	3.79848	0.123045	0.0615224	−	0.998106i	$-0.480404\pi$
$953$	3.79848	0.123045	0.0615224	+	0.998106i	$0.480404\pi$
$954$	0	0
$955$	2.71015	0.0876984
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−6.46117	−0.208425
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 6.75517i	− 0.217457i
$966$	0	0
$967$	− 20.7285i	− 0.666582i	−0.942824	−	0.333291i	$-0.891841\pi$
$967$	− 20.7285i	− 0.666582i	0.942824	−	0.333291i	$-0.108159\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−39.5188	−1.26822	−0.634110	−	0.773243i	$-0.718633\pi$
$971$	−39.5188	−1.26822	−0.634110	+	0.773243i	$0.718633\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	21.1235	0.675799	0.337900	−	0.941182i	$-0.390284\pi$
$977$	21.1235	0.675799	0.337900	+	0.941182i	$0.390284\pi$
$978$	0	0
$979$	−56.2937	−1.79915
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−45.9575	−1.46582	−0.732909	−	0.680327i	$-0.761838\pi$
$983$	−45.9575	−1.46582	−0.732909	+	0.680327i	$0.761838\pi$
$984$	0	0
$985$	19.7738i	0.630046i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−18.7855	−0.597343
$990$	0	0
$991$	− 3.93322i	− 0.124943i	−0.998047	−	0.0624714i	$-0.980102\pi$
$991$	− 3.93322i	− 0.124943i	0.998047	−	0.0624714i	$-0.0198982\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 29.2369i	− 0.926871i
$996$	0	0
$997$	10.7049i	0.339027i	0.985528	+	0.169514i	$0.0542197\pi$
$997$	10.7049i	0.339027i	−0.985528	+	0.169514i	$0.945780\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7056.2.b.w.1567.3		8
3.2	odd	2		2352.2.b.l.1567.6	yes	8
4.3	odd	2		7056.2.b.x.1567.3		8
7.6	odd	2		7056.2.b.x.1567.6		8
12.11	even	2		2352.2.b.k.1567.6	yes	8
21.2	odd	6		2352.2.bl.q.31.3		8
21.5	even	6		2352.2.bl.t.31.2		8
21.11	odd	6		2352.2.bl.o.607.2		8
21.17	even	6		2352.2.bl.r.607.3		8
21.20	even	2		2352.2.b.k.1567.3	✓	8
28.27	even	2	inner	7056.2.b.w.1567.6		8
84.11	even	6		2352.2.bl.t.607.2		8
84.23	even	6		2352.2.bl.r.31.3		8
84.47	odd	6		2352.2.bl.o.31.2		8
84.59	odd	6		2352.2.bl.q.607.3		8
84.83	odd	2		2352.2.b.l.1567.3	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2352.2.b.k.1567.3	✓	8	21.20	even	2
2352.2.b.k.1567.6	yes	8	12.11	even	2
2352.2.b.l.1567.3	yes	8	84.83	odd	2
2352.2.b.l.1567.6	yes	8	3.2	odd	2
2352.2.bl.o.31.2		8	84.47	odd	6
2352.2.bl.o.607.2		8	21.11	odd	6
2352.2.bl.q.31.3		8	21.2	odd	6
2352.2.bl.q.607.3		8	84.59	odd	6
2352.2.bl.r.31.3		8	84.23	even	6
2352.2.bl.r.607.3		8	21.17	even	6
2352.2.bl.t.31.2		8	21.5	even	6
2352.2.bl.t.607.2		8	84.11	even	6
7056.2.b.w.1567.3		8	1.1	even	1	trivial
7056.2.b.w.1567.6		8	28.27	even	2	inner
7056.2.b.x.1567.3		8	4.3	odd	2
7056.2.b.x.1567.6		8	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-1.68412i q^{5} +O(q^{10})\)	\(q-1.68412i q^{5} -3.53188i q^{11} +2.93015i q^{13} +2.01140i q^{17} +1.69764 q^{19} +1.59851i q^{23} +2.16373 q^{25} -7.94028 q^{29} -4.95367 q^{31} -10.4663 q^{37} -2.86237i q^{41} +11.7518i q^{43} +6.70319 q^{47} -2.92109 q^{53} -5.94812 q^{55} +7.75532 q^{59} +12.5925i q^{61} +4.93473 q^{65} +1.70195i q^{67} +6.13052i q^{71} +2.43166i q^{73} +0.865467i q^{79} -14.5319 q^{83} +3.38744 q^{85} -15.9387i q^{89} -2.85903i q^{95} -9.82270i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q - 24 q^{25} - 16 q^{29} - 32 q^{31} + 16 q^{47} - 16 q^{53} - 64 q^{55} - 48 q^{59} + 16 q^{65} - 64 q^{85}+O(q^{100}) \) 8 * q - 24 * q^25 - 16 * q^29 - 32 * q^31 + 16 * q^47 - 16 * q^53 - 64 * q^55 - 48 * q^59 + 16 * q^65 - 64 * q^85

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