LMFDB - Embedded newform 5292.2.f.e.2645.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5292,2,Mod(2645,5292)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5292, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("5292.2645"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 5292 = 2^{2} \cdot 3^{3} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5292.f (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,48] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(37)] == traces)

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

Self dual:	no
Analytic conductor:	$42.2568327497$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	12.0.17213603549184.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 5x^{10} + 19x^{8} - 28x^{6} + 31x^{4} - 6x^{2} + 1 $ x^12 - 5x^10 + 19x^8 - 28x^6 + 31x^4 - 6*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{9} $
Twist minimal:	no (minimal twist has level 756)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2645.4
Root			$-1.07992 - 0.623490i$ of defining polynomial
Character	$\chi$	$=$	5292.2645
Dual form			5292.2.f.e.2645.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5292\mathbb{Z}\right)^\times$.

$n$	$785$	$1081$	$2647$
$\chi(n)$	$-1$	$-1$	$1$

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
$5$	−2.15983	−0.965906	−0.482953	−	0.875646i	$-0.660436\pi$
$5$	−2.15983	−0.965906	−0.482953	+	0.875646i	$0.660436\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	0	0
$10$	0	0
$11$	1.33513i	0.402556i	0.979534	+	0.201278i	$0.0645094\pi$
$11$	1.33513i	0.402556i	−0.979534	+	0.201278i	$0.935491\pi$
$12$	0	0
$13$	2.35021i	0.651832i	0.945399	+	0.325916i	$0.105673\pi$
$13$	2.35021i	0.651832i	−0.945399	+	0.325916i	$0.894327\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	7.20331	1.74706	0.873530	−	0.486771i	$-0.161825\pi$
$17$	7.20331	1.74706	0.873530	+	0.486771i	$0.161825\pi$
$18$	0	0
$19$	− 1.19862i	− 0.274981i	−0.990503	−	0.137491i	$-0.956096\pi$
$19$	− 1.19862i	− 0.274981i	0.990503	−	0.137491i	$-0.0439037\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	2.40581i	0.501647i	0.968033	+	0.250823i	$0.0807013\pi$
$23$	2.40581i	0.501647i	−0.968033	+	0.250823i	$0.919299\pi$
$24$	0	0
$25$	−0.335126	−0.0670251
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 6.14675i	− 1.14142i	−0.821151	−	0.570712i	$-0.806667\pi$
$29$	− 6.14675i	− 1.14142i	0.821151	−	0.570712i	$-0.193333\pi$
$30$	0	0
$31$	− 8.21155i	− 1.47484i	−0.675436	−	0.737419i	$-0.736045\pi$
$31$	− 8.21155i	− 1.47484i	0.675436	−	0.737419i	$-0.263955\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−5.14675	−0.846121	−0.423060	−	0.906101i	$-0.639044\pi$
$37$	−5.14675	−0.846121	−0.423060	+	0.906101i	$0.639044\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	4.47234	0.698462	0.349231	−	0.937037i	$-0.386443\pi$
$41$	4.47234	0.698462	0.349231	+	0.937037i	$0.386443\pi$
$42$	0	0
$43$	−0.664874	−0.101392	−0.0506962	−	0.998714i	$-0.516144\pi$
$43$	−0.664874	−0.101392	−0.0506962	+	0.998714i	$0.516144\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−2.15983	−0.315044	−0.157522	−	0.987515i	$-0.550350\pi$
$47$	−2.15983	−0.315044	−0.157522	+	0.987515i	$0.550350\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	0	0
$52$	0	0
$53$	12.7409i	1.75010i	0.484030	+	0.875051i	$0.339172\pi$
$53$	12.7409i	1.75010i	−0.484030	+	0.875051i	$0.660828\pi$
$54$	0	0
$55$	− 2.88365i	− 0.388831i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	9.66849	1.25873	0.629365	−	0.777110i	$-0.283315\pi$
$59$	9.66849	1.25873	0.629365	+	0.777110i	$0.283315\pi$
$60$	0	0
$61$	− 12.2938i	− 1.57406i	−0.616914	−	0.787031i	$-0.711617\pi$
$61$	− 12.2938i	− 1.57406i	0.616914	−	0.787031i	$-0.288383\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 5.07606i	− 0.629608i
$66$	0	0
$67$	−3.73556	−0.456372	−0.228186	−	0.973618i	$-0.573279\pi$
$67$	−3.73556	−0.456372	−0.228186	+	0.973618i	$0.573279\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	11.4058i	1.35362i	0.736157	+	0.676810i	$0.236638\pi$
$71$	11.4058i	1.35362i	−0.736157	+	0.676810i	$0.763362\pi$
$72$	0	0
$73$	10.4770i	1.22624i	0.789988	+	0.613122i	$0.210087\pi$
$73$	10.4770i	1.22624i	−0.789988	+	0.613122i	$0.789913\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−9.07606	−1.02114	−0.510569	−	0.859837i	$-0.670565\pi$
$79$	−9.07606	−1.02114	−0.510569	+	0.859837i	$0.670565\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−10.7992	−1.18536	−0.592681	−	0.805437i	$-0.701930\pi$
$83$	−10.7992	−1.18536	−0.592681	+	0.805437i	$0.701930\pi$
$84$	0	0
$85$	−15.5579	−1.68750
$86$	0	0
$87$	0	0
$88$	0	0
$89$	5.19615	0.550791	0.275396	−	0.961331i	$-0.411191\pi$
$89$	5.19615	0.550791	0.275396	+	0.961331i	$0.411191\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	2.58881i	0.265606i
$96$	0	0
$97$	− 1.23632i	− 0.125530i	−0.998028	−	0.0627648i	$-0.980008\pi$
$97$	− 1.23632i	− 0.125530i	0.998028	−	0.0627648i	$-0.0199918\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	3.91281	0.389339	0.194670	−	0.980869i	$-0.437637\pi$
$101$	3.91281	0.389339	0.194670	+	0.980869i	$0.437637\pi$
$102$	0	0
$103$	4.12928i	0.406870i	0.979088	+	0.203435i	$0.0652106\pi$
$103$	4.12928i	0.406870i	−0.979088	+	0.203435i	$0.934789\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 3.92394i	− 0.379341i	−0.981848	−	0.189671i	$-0.939258\pi$
$107$	− 3.92394i	− 0.379341i	0.981848	−	0.189671i	$-0.0607420\pi$
$108$	0	0
$109$	1.32975	0.127367	0.0636834	−	0.997970i	$-0.479715\pi$
$109$	1.32975	0.127367	0.0636834	+	0.997970i	$0.479715\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	16.4819i	1.55048i	0.631664	+	0.775242i	$0.282372\pi$
$113$	16.4819i	1.55048i	−0.631664	+	0.775242i	$0.717628\pi$
$114$	0	0
$115$	− 5.19615i	− 0.484544i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	9.21744	0.837949
$122$	0	0
$123$	0	0
$124$	0	0
$125$	11.5230	1.03065
$126$	0	0
$127$	14.8877	1.32107	0.660534	−	0.750796i	$-0.270330\pi$
$127$	14.8877	1.32107	0.660534	+	0.750796i	$0.270330\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	15.8426	1.38418	0.692089	−	0.721812i	$-0.256691\pi$
$131$	15.8426	1.38418	0.692089	+	0.721812i	$0.256691\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	22.8116i	1.94893i	0.224542	+	0.974464i	$0.427911\pi$
$137$	22.8116i	1.94893i	−0.224542	+	0.974464i	$0.572089\pi$
$138$	0	0
$139$	16.3761i	1.38900i	0.719492	+	0.694500i	$0.244374\pi$
$139$	16.3761i	1.38900i	−0.719492	+	0.694500i	$0.755626\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−3.13783	−0.262398
$144$	0	0
$145$	13.2760i	1.10251i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 8.73556i	− 0.715645i	−0.933790	−	0.357823i	$-0.883519\pi$
$149$	− 8.73556i	− 0.715645i	0.933790	−	0.357823i	$-0.116481\pi$
$150$	0	0
$151$	7.14675	0.581594	0.290797	−	0.956785i	$-0.406079\pi$
$151$	7.14675	0.581594	0.290797	+	0.956785i	$0.406079\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	17.7356i	1.42455i
$156$	0	0
$157$	− 12.2468i	− 0.977400i	−0.872452	−	0.488700i	$-0.837471\pi$
$157$	− 12.2468i	− 0.977400i	0.872452	−	0.488700i	$-0.162529\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	16.0761	1.25917	0.629587	−	0.776930i	$-0.283224\pi$
$163$	16.0761	1.25917	0.629587	+	0.776930i	$0.283224\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	11.3703	0.879860	0.439930	−	0.898032i	$-0.355003\pi$
$167$	11.3703	0.879860	0.439930	+	0.898032i	$0.355003\pi$
$168$	0	0
$169$	7.47650	0.575115
$170$	0	0
$171$	0	0
$172$	0	0
$173$	12.7048	0.965929	0.482964	−	0.875640i	$-0.339560\pi$
$173$	12.7048	0.965929	0.482964	+	0.875640i	$0.339560\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 16.0344i	− 1.19847i	−0.800573	−	0.599235i	$-0.795471\pi$
$179$	− 16.0344i	− 1.19847i	0.800573	−	0.599235i	$-0.204529\pi$
$180$	0	0
$181$	− 14.0729i	− 1.04603i	−0.852324	−	0.523015i	$-0.824807\pi$
$181$	− 14.0729i	− 1.04603i	0.852324	−	0.523015i	$-0.175193\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	11.1161	0.817273
$186$	0	0
$187$	9.61733i	0.703288i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 26.2881i	− 1.90214i	−0.308972	−	0.951071i	$-0.599985\pi$
$191$	− 26.2881i	− 1.90214i	0.308972	−	0.951071i	$-0.400015\pi$
$192$	0	0
$193$	3.07606	0.221420	0.110710	−	0.993853i	$-0.464688\pi$
$193$	3.07606	0.221420	0.110710	+	0.993853i	$0.464688\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	9.08144i	0.647026i	0.946224	+	0.323513i	$0.104864\pi$
$197$	9.08144i	0.647026i	−0.946224	+	0.323513i	$0.895136\pi$
$198$	0	0
$199$	22.6861i	1.60818i	0.594510	+	0.804088i	$0.297346\pi$
$199$	22.6861i	1.60818i	−0.594510	+	0.804088i	$0.702654\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−9.65950	−0.674649
$206$	0	0
$207$	0	0
$208$	0	0
$209$	1.60030	0.110695
$210$	0	0
$211$	9.81700	0.675830	0.337915	−	0.941177i	$-0.390278\pi$
$211$	9.81700	0.675830	0.337915	+	0.941177i	$0.390278\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	1.43602	0.0979355
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	16.9293i	1.13879i
$222$	0	0
$223$	7.05064i	0.472146i	0.971735	+	0.236073i	$0.0758604\pi$
$223$	7.05064i	0.472146i	−0.971735	+	0.236073i	$0.924140\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	19.7554	1.31122	0.655608	−	0.755102i	$-0.272413\pi$
$227$	19.7554	1.31122	0.655608	+	0.755102i	$0.272413\pi$
$228$	0	0
$229$	− 3.33235i	− 0.220208i	−0.993920	−	0.110104i	$-0.964882\pi$
$229$	− 3.33235i	− 0.220208i	0.993920	−	0.110104i	$-0.0351184\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	6.32975i	0.414676i	0.978269	+	0.207338i	$0.0664799\pi$
$233$	6.32975i	0.414676i	−0.978269	+	0.207338i	$0.933520\pi$
$234$	0	0
$235$	4.66487	0.304303
$236$	0	0
$237$	0	0
$238$	0	0
$239$	27.8877i	1.80390i	0.431836	+	0.901952i	$0.357866\pi$
$239$	27.8877i	1.80390i	−0.431836	+	0.901952i	$0.642134\pi$
$240$	0	0
$241$	8.99916i	0.579687i	0.957074	+	0.289844i	$0.0936033\pi$
$241$	8.99916i	0.579687i	−0.957074	+	0.289844i	$0.906397\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	2.81700	0.179242
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−11.5230	−0.727324	−0.363662	−	0.931531i	$-0.618474\pi$
$251$	−11.5230	−0.727324	−0.363662	+	0.931531i	$0.618474\pi$
$252$	0	0
$253$	−3.21206	−0.201941
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−3.13783	−0.195732	−0.0978662	−	0.995200i	$-0.531202\pi$
$257$	−3.13783	−0.195732	−0.0978662	+	0.995200i	$0.531202\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	11.4058i	0.703313i	0.936129	+	0.351656i	$0.114381\pi$
$263$	11.4058i	0.703313i	−0.936129	+	0.351656i	$0.885619\pi$
$264$	0	0
$265$	− 27.5183i	− 1.69043i
$266$	0	0
$267$	0	0
$268$	0	0
$269$	21.3441	1.30138	0.650688	−	0.759346i	$-0.274481\pi$
$269$	21.3441	1.30138	0.650688	+	0.759346i	$0.274481\pi$
$270$	0	0
$271$	17.9480i	1.09026i	0.838351	+	0.545131i	$0.183520\pi$
$271$	17.9480i	1.09026i	−0.838351	+	0.545131i	$0.816480\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 0.447435i	− 0.0269813i
$276$	0	0
$277$	−31.7700	−1.90887	−0.954437	−	0.298412i	$-0.903543\pi$
$277$	−31.7700	−1.90887	−0.954437	+	0.298412i	$0.903543\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 3.55794i	− 0.212249i	−0.994353	−	0.106125i	$-0.966156\pi$
$281$	− 3.55794i	− 0.212249i	0.994353	−	0.106125i	$-0.0338442\pi$
$282$	0	0
$283$	7.75353i	0.460899i	0.973084	+	0.230450i	$0.0740197\pi$
$283$	7.75353i	0.460899i	−0.973084	+	0.230450i	$0.925980\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	34.8877	2.05222
$290$	0	0
$291$	0	0
$292$	0	0
$293$	5.04348	0.294643	0.147322	−	0.989089i	$-0.452935\pi$
$293$	5.04348	0.294643	0.147322	+	0.989089i	$0.452935\pi$
$294$	0	0
$295$	−20.8823	−1.21582
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−5.65417	−0.326989
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	26.5526i	1.52040i
$306$	0	0
$307$	− 15.0080i	− 0.856552i	−0.903648	−	0.428276i	$-0.859121\pi$
$307$	− 15.0080i	− 0.856552i	0.903648	−	0.428276i	$-0.140879\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	10.0870	0.571979	0.285989	−	0.958233i	$-0.407678\pi$
$311$	10.0870	0.571979	0.285989	+	0.958233i	$0.407678\pi$
$312$	0	0
$313$	− 17.9857i	− 1.01661i	−0.861177	−	0.508306i	$-0.830272\pi$
$313$	− 17.9857i	− 1.01661i	0.861177	−	0.508306i	$-0.169728\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	4.73019i	0.265674i	0.991138	+	0.132837i	$0.0424086\pi$
$317$	4.73019i	0.265674i	−0.991138	+	0.132837i	$0.957591\pi$
$318$	0	0
$319$	8.20669	0.459486
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 8.63401i	− 0.480409i
$324$	0	0
$325$	− 0.787616i	− 0.0436891i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−2.77718	−0.152648	−0.0763239	−	0.997083i	$-0.524318\pi$
$331$	−2.77718	−0.152648	−0.0763239	+	0.997083i	$0.524318\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	8.06819	0.440812
$336$	0	0
$337$	−6.99462	−0.381021	−0.190511	−	0.981685i	$-0.561014\pi$
$337$	−6.99462	−0.381021	−0.190511	+	0.981685i	$0.561014\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	10.9634	0.593704
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	20.3043i	1.08999i	0.838439	+	0.544995i	$0.183468\pi$
$347$	20.3043i	1.08999i	−0.838439	+	0.544995i	$0.816532\pi$
$348$	0	0
$349$	9.77414i	0.523198i	0.965177	+	0.261599i	$0.0842498\pi$
$349$	9.77414i	0.523198i	−0.965177	+	0.261599i	$0.915750\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	12.8575	0.684335	0.342167	−	0.939639i	$-0.388839\pi$
$353$	12.8575	0.684335	0.342167	+	0.939639i	$0.388839\pi$
$354$	0	0
$355$	− 24.6346i	− 1.30747i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 19.3351i	− 1.02047i	−0.860035	−	0.510234i	$-0.829559\pi$
$359$	− 19.3351i	− 1.02047i	0.860035	−	0.510234i	$-0.170441\pi$
$360$	0	0
$361$	17.5633	0.924385
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 22.6286i	− 1.18444i
$366$	0	0
$367$	− 26.8908i	− 1.40369i	−0.712330	−	0.701845i	$-0.752360\pi$
$367$	− 26.8908i	− 1.40369i	0.712330	−	0.701845i	$-0.247640\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−38.4456	−1.99064	−0.995320	−	0.0966369i	$-0.969191\pi$
$373$	−38.4456	−1.99064	−0.995320	+	0.0966369i	$0.969191\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	14.4462	0.744016
$378$	0	0
$379$	3.05993	0.157178	0.0785891	−	0.996907i	$-0.474958\pi$
$379$	3.05993	0.157178	0.0785891	+	0.996907i	$0.474958\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−13.6828	−0.699159	−0.349579	−	0.936907i	$-0.613676\pi$
$383$	−13.6828	−0.699159	−0.349579	+	0.936907i	$0.613676\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 0.887691i	− 0.0450077i	−0.999747	−	0.0225039i	$-0.992836\pi$
$389$	− 0.887691i	− 0.0450077i	0.999747	−	0.0225039i	$-0.00716380\pi$
$390$	0	0
$391$	17.3298i	0.876407i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	19.6028	0.986323
$396$	0	0
$397$	34.1923i	1.71606i	0.513597	+	0.858031i	$0.328313\pi$
$397$	34.1923i	1.71606i	−0.513597	+	0.858031i	$0.671687\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	27.8877i	1.39264i	0.717729	+	0.696322i	$0.245182\pi$
$401$	27.8877i	1.39264i	−0.717729	+	0.696322i	$0.754818\pi$
$402$	0	0
$403$	19.2989	0.961346
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 6.87156i	− 0.340611i
$408$	0	0
$409$	13.4924i	0.667158i	0.942722	+	0.333579i	$0.108256\pi$
$409$	13.4924i	0.667158i	−0.942722	+	0.333579i	$0.891744\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	23.3244	1.14495
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−27.9368	−1.36480	−0.682400	−	0.730979i	$-0.739064\pi$
$419$	−27.9368	−1.36480	−0.682400	+	0.730979i	$0.739064\pi$
$420$	0	0
$421$	12.5472	0.611513	0.305756	−	0.952110i	$-0.401091\pi$
$421$	12.5472	0.611513	0.305756	+	0.952110i	$0.401091\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−2.41401	−0.117097
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	17.2881i	0.832740i	0.909195	+	0.416370i	$0.136698\pi$
$431$	17.2881i	0.832740i	−0.909195	+	0.416370i	$0.863302\pi$
$432$	0	0
$433$	27.1920i	1.30676i	0.757028	+	0.653382i	$0.226651\pi$
$433$	27.1920i	1.30676i	−0.757028	+	0.653382i	$0.773349\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	2.88365	0.137944
$438$	0	0
$439$	− 10.9634i	− 0.523257i	−0.965169	−	0.261628i	$-0.915741\pi$
$439$	− 10.9634i	− 0.523257i	0.965169	−	0.261628i	$-0.0842595\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	18.4403i	0.876123i	0.898945	+	0.438062i	$0.144335\pi$
$443$	18.4403i	0.876123i	−0.898945	+	0.438062i	$0.855665\pi$
$444$	0	0
$445$	−11.2228	−0.532013
$446$	0	0
$447$	0	0
$448$	0	0
$449$	23.6179i	1.11460i	0.830312	+	0.557298i	$0.188162\pi$
$449$	23.6179i	1.11460i	−0.830312	+	0.557298i	$0.811838\pi$
$450$	0	0
$451$	5.97113i	0.281170i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−29.2282	−1.36724	−0.683619	−	0.729839i	$-0.739595\pi$
$457$	−29.2282	−1.36724	−0.683619	+	0.729839i	$0.739595\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−5.91997	−0.275720	−0.137860	−	0.990452i	$-0.544022\pi$
$461$	−5.91997	−0.275720	−0.137860	+	0.990452i	$0.544022\pi$
$462$	0	0
$463$	34.4349	1.60032	0.800162	−	0.599784i	$-0.204747\pi$
$463$	34.4349	1.60032	0.800162	+	0.599784i	$0.204747\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−38.6228	−1.78725	−0.893625	−	0.448815i	$-0.851846\pi$
$467$	−38.6228	−1.78725	−0.893625	+	0.448815i	$0.851846\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 0.887691i	− 0.0408161i
$474$	0	0
$475$	0.401687i	0.0184307i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−3.03632	−0.138733	−0.0693665	−	0.997591i	$-0.522098\pi$
$479$	−3.03632	−0.138733	−0.0693665	+	0.997591i	$0.522098\pi$
$480$	0	0
$481$	− 12.0960i	− 0.551528i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	2.67025i	0.121250i
$486$	0	0
$487$	−26.7700	−1.21306	−0.606532	−	0.795059i	$-0.707440\pi$
$487$	−26.7700	−1.21306	−0.606532	+	0.795059i	$0.707440\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 10.9584i	− 0.494545i	−0.968946	−	0.247272i	$-0.920466\pi$
$491$	− 10.9584i	− 0.494545i	0.968946	−	0.247272i	$-0.0795342\pi$
$492$	0	0
$493$	− 44.2770i	− 1.99413i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−30.0998	−1.34745	−0.673725	−	0.738982i	$-0.735307\pi$
$499$	−30.0998	−1.34745	−0.673725	+	0.738982i	$0.735307\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	12.0941	0.539250	0.269625	−	0.962965i	$-0.413100\pi$
$503$	12.0941	0.539250	0.269625	+	0.962965i	$0.413100\pi$
$504$	0	0
$505$	−8.45101	−0.376065
$506$	0	0
$507$	0	0
$508$	0	0
$509$	25.0531	1.11046	0.555230	−	0.831697i	$-0.312630\pi$
$509$	25.0531	1.11046	0.555230	+	0.831697i	$0.312630\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 8.91856i	− 0.392999i
$516$	0	0
$517$	− 2.88365i	− 0.126823i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	16.7191	0.732478	0.366239	−	0.930521i	$-0.380645\pi$
$521$	16.7191	0.732478	0.366239	+	0.930521i	$0.380645\pi$
$522$	0	0
$523$	26.0185i	1.13771i	0.822438	+	0.568854i	$0.192613\pi$
$523$	26.0185i	1.13771i	−0.822438	+	0.568854i	$0.807387\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 59.1503i	− 2.57663i
$528$	0	0
$529$	17.2121	0.748351
$530$	0	0
$531$	0	0
$532$	0	0
$533$	10.5109i	0.455280i
$534$	0	0
$535$	8.47504i	0.366408i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	30.9946	1.33256	0.666281	−	0.745700i	$-0.267885\pi$
$541$	30.9946	1.33256	0.666281	+	0.745700i	$0.267885\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−2.87203	−0.123024
$546$	0	0
$547$	−36.3696	−1.55505	−0.777525	−	0.628852i	$-0.783525\pi$
$547$	−36.3696	−1.55505	−0.777525	+	0.628852i	$0.783525\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−7.36760	−0.313870
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	21.5579i	0.913439i	0.889611	+	0.456720i	$0.150976\pi$
$557$	21.5579i	0.913439i	−0.889611	+	0.456720i	$0.849024\pi$
$558$	0	0
$559$	− 1.56260i	− 0.0660908i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	0.712202	0.0300157	0.0150079	−	0.999887i	$-0.495223\pi$
$563$	0.712202	0.0300157	0.0150079	+	0.999887i	$0.495223\pi$
$564$	0	0
$565$	− 35.5981i	− 1.49762i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	34.3803i	1.44130i	0.693300	+	0.720649i	$0.256156\pi$
$569$	34.3803i	1.44130i	−0.693300	+	0.720649i	$0.743844\pi$
$570$	0	0
$571$	−25.4403	−1.06464	−0.532321	−	0.846543i	$-0.678680\pi$
$571$	−25.4403	−1.06464	−0.532321	+	0.846543i	$0.678680\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 0.806250i	− 0.0336229i
$576$	0	0
$577$	8.61323i	0.358574i	0.983797	+	0.179287i	$0.0573790\pi$
$577$	8.61323i	0.358574i	−0.983797	+	0.179287i	$0.942621\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−17.0108	−0.704513
$584$	0	0
$585$	0	0
$586$	0	0
$587$	31.6853	1.30779	0.653896	−	0.756585i	$-0.273134\pi$
$587$	31.6853	1.30779	0.653896	+	0.756585i	$0.273134\pi$
$588$	0	0
$589$	−9.84249	−0.405553
$590$	0	0
$591$	0	0
$592$	0	0
$593$	48.2913	1.98309	0.991543	−	0.129781i	$-0.0414274\pi$
$593$	48.2913	1.98309	0.991543	+	0.129781i	$0.0414274\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 29.6703i	− 1.21229i	−0.795353	−	0.606147i	$-0.792714\pi$
$599$	− 29.6703i	− 1.21229i	0.795353	−	0.606147i	$-0.207286\pi$
$600$	0	0
$601$	41.0577i	1.67478i	0.546606	+	0.837390i	$0.315920\pi$
$601$	41.0577i	1.67478i	−0.546606	+	0.837390i	$0.684080\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−19.9081	−0.809380
$606$	0	0
$607$	− 22.8649i	− 0.928057i	−0.885820	−	0.464028i	$-0.846404\pi$
$607$	− 22.8649i	− 0.928057i	0.885820	−	0.464028i	$-0.153596\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 5.07606i	− 0.205356i
$612$	0	0
$613$	25.4058	1.02613	0.513066	−	0.858349i	$-0.328510\pi$
$613$	25.4058	1.02613	0.513066	+	0.858349i	$0.328510\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 14.9638i	− 0.602418i	−0.953558	−	0.301209i	$-0.902610\pi$
$617$	− 14.9638i	− 0.602418i	0.953558	−	0.301209i	$-0.0973902\pi$
$618$	0	0
$619$	− 16.9472i	− 0.681166i	−0.940214	−	0.340583i	$-0.889376\pi$
$619$	− 16.9472i	− 0.681166i	0.940214	−	0.340583i	$-0.110624\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−23.2121	−0.928483
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−37.0737	−1.47822
$630$	0	0
$631$	−21.3534	−0.850067	−0.425033	−	0.905178i	$-0.639738\pi$
$631$	−21.3534	−0.850067	−0.425033	+	0.905178i	$0.639738\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−32.1549	−1.27603
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 44.8098i	− 1.76988i	−0.465703	−	0.884941i	$-0.654199\pi$
$641$	− 44.8098i	− 1.76988i	0.465703	−	0.884941i	$-0.345801\pi$
$642$	0	0
$643$	− 18.8487i	− 0.743321i	−0.928369	−	0.371660i	$-0.878789\pi$
$643$	− 18.8487i	− 0.743321i	0.928369	−	0.371660i	$-0.121211\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−39.0808	−1.53643	−0.768213	−	0.640195i	$-0.778854\pi$
$647$	−39.0808	−1.53643	−0.768213	+	0.640195i	$0.778854\pi$
$648$	0	0
$649$	12.9086i	0.506709i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	11.8532i	0.463853i	0.972733	+	0.231927i	$0.0745030\pi$
$653$	11.8532i	0.463853i	−0.972733	+	0.231927i	$0.925497\pi$
$654$	0	0
$655$	−34.2174	−1.33699
$656$	0	0
$657$	0	0
$658$	0	0
$659$	33.3226i	1.29806i	0.760762	+	0.649032i	$0.224826\pi$
$659$	33.3226i	1.29806i	−0.760762	+	0.649032i	$0.775174\pi$
$660$	0	0
$661$	− 3.10943i	− 0.120943i	−0.998170	−	0.0604715i	$-0.980740\pi$
$661$	− 3.10943i	− 0.120943i	0.998170	−	0.0604715i	$-0.0192604\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	14.7879	0.572591
$668$	0	0
$669$	0	0
$670$	0	0
$671$	16.4138	0.633647
$672$	0	0
$673$	2.68100	0.103345	0.0516726	−	0.998664i	$-0.483545\pi$
$673$	2.68100	0.103345	0.0516726	+	0.998664i	$0.483545\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	28.5474	1.09717	0.548584	−	0.836096i	$-0.315167\pi$
$677$	28.5474	1.09717	0.548584	+	0.836096i	$0.315167\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 43.9221i	− 1.68063i	−0.542096	−	0.840317i	$-0.682369\pi$
$683$	− 43.9221i	− 1.68063i	0.542096	−	0.840317i	$-0.317631\pi$
$684$	0	0
$685$	− 49.2693i	− 1.88248i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−29.9439	−1.14077
$690$	0	0
$691$	− 31.2993i	− 1.19068i	−0.803473	−	0.595342i	$-0.797017\pi$
$691$	− 31.2993i	− 1.19068i	0.803473	−	0.595342i	$-0.202983\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 35.3696i	− 1.34164i
$696$	0	0
$697$	32.2156	1.22025
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 23.2591i	− 0.878483i	−0.898369	−	0.439241i	$-0.855247\pi$
$701$	− 23.2591i	− 0.878483i	0.898369	−	0.439241i	$-0.144753\pi$
$702$	0	0
$703$	6.16898i	0.232668i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	20.5924	0.773363	0.386682	−	0.922213i	$-0.373621\pi$
$709$	20.5924	0.773363	0.386682	+	0.922213i	$0.373621\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	19.7554	0.739847
$714$	0	0
$715$	6.77718	0.253452
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−7.52027	−0.280459	−0.140229	−	0.990119i	$-0.544784\pi$
$719$	−7.52027	−0.280459	−0.140229	+	0.990119i	$0.544784\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	2.05993i	0.0765040i
$726$	0	0
$727$	− 33.2949i	− 1.23484i	−0.786634	−	0.617420i	$-0.788178\pi$
$727$	− 33.2949i	− 1.23484i	0.786634	−	0.617420i	$-0.211822\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−4.78930	−0.177139
$732$	0	0
$733$	− 15.4567i	− 0.570907i	−0.958393	−	0.285453i	$-0.907856\pi$
$733$	− 15.4567i	− 0.570907i	0.958393	−	0.285453i	$-0.0921441\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 4.98744i	− 0.183715i
$738$	0	0
$739$	6.21744	0.228712	0.114356	−	0.993440i	$-0.463520\pi$
$739$	6.21744	0.228712	0.114356	+	0.993440i	$0.463520\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	12.0291i	0.441304i	0.975353	+	0.220652i	$0.0708184\pi$
$743$	12.0291i	0.441304i	−0.975353	+	0.220652i	$0.929182\pi$
$744$	0	0
$745$	18.8673i	0.691246i
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−33.7754	−1.23248	−0.616241	−	0.787558i	$-0.711345\pi$
$751$	−33.7754	−1.23248	−0.616241	+	0.787558i	$0.711345\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−15.4358	−0.561766
$756$	0	0
$757$	49.5870	1.80227	0.901135	−	0.433538i	$-0.142735\pi$
$757$	49.5870	1.80227	0.901135	+	0.433538i	$0.142735\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	39.9294	1.44744	0.723719	−	0.690095i	$-0.242431\pi$
$761$	39.9294	1.44744	0.723719	+	0.690095i	$0.242431\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	22.7230i	0.820480i
$768$	0	0
$769$	15.4693i	0.557839i	0.960315	+	0.278919i	$0.0899762\pi$
$769$	15.4693i	0.557839i	−0.960315	+	0.278919i	$0.910024\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−34.7332	−1.24927	−0.624633	−	0.780918i	$-0.714752\pi$
$773$	−34.7332	−1.24927	−0.624633	+	0.780918i	$0.714752\pi$
$774$	0	0
$775$	2.75190i	0.0988512i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 5.36062i	− 0.192064i
$780$	0	0
$781$	−15.2282	−0.544908
$782$	0	0
$783$	0	0
$784$	0	0
$785$	26.4510i	0.944077i
$786$	0	0
$787$	− 26.0748i	− 0.929466i	−0.885451	−	0.464733i	$-0.846150\pi$
$787$	− 26.0748i	− 0.929466i	0.885451	−	0.464733i	$-0.153850\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	28.8931	1.02602
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−14.1013	−0.499493	−0.249746	−	0.968311i	$-0.580347\pi$
$797$	−14.1013	−0.499493	−0.249746	+	0.968311i	$0.580347\pi$
$798$	0	0
$799$	−15.5579	−0.550400
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−13.9882	−0.493631
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 31.7302i	− 1.11557i	−0.829984	−	0.557787i	$-0.811651\pi$
$809$	− 31.7302i	− 1.11557i	0.829984	−	0.557787i	$-0.188349\pi$
$810$	0	0
$811$	− 18.3590i	− 0.644671i	−0.946625	−	0.322336i	$-0.895532\pi$
$811$	− 18.3590i	− 0.644671i	0.946625	−	0.322336i	$-0.104468\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−34.7216	−1.21624
$816$	0	0
$817$	0.796929i	0.0278810i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	19.4980i	0.680485i	0.940338	+	0.340243i	$0.110509\pi$
$821$	19.4980i	0.680485i	−0.940338	+	0.340243i	$0.889491\pi$
$822$	0	0
$823$	0.635808	0.0221629	0.0110814	−	0.999939i	$-0.496473\pi$
$823$	0.635808	0.0221629	0.0110814	+	0.999939i	$0.496473\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 28.6939i	− 0.997786i	−0.866664	−	0.498893i	$-0.833740\pi$
$827$	− 28.6939i	− 0.997786i	0.866664	−	0.498893i	$-0.166260\pi$
$828$	0	0
$829$	34.1169i	1.18493i	0.805597	+	0.592464i	$0.201845\pi$
$829$	34.1169i	1.18493i	−0.805597	+	0.592464i	$0.798155\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−24.5579	−0.849863
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−14.6105	−0.504409	−0.252205	−	0.967674i	$-0.581156\pi$
$839$	−14.6105	−0.504409	−0.252205	+	0.967674i	$0.581156\pi$
$840$	0	0
$841$	−8.78256	−0.302847
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−16.1480	−0.555508
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 12.3821i	− 0.424454i
$852$	0	0
$853$	27.1794i	0.930604i	0.885152	+	0.465302i	$0.154054\pi$
$853$	27.1794i	0.930604i	−0.885152	+	0.465302i	$0.845946\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−2.61785	−0.0894241	−0.0447121	−	0.999000i	$-0.514237\pi$
$857$	−2.61785	−0.0894241	−0.0447121	+	0.999000i	$0.514237\pi$
$858$	0	0
$859$	45.0083i	1.53566i	0.640653	+	0.767831i	$0.278664\pi$
$859$	45.0083i	1.53566i	−0.640653	+	0.767831i	$0.721336\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	29.8461i	1.01597i	0.861365	+	0.507986i	$0.169610\pi$
$863$	29.8461i	1.01597i	−0.861365	+	0.507986i	$0.830390\pi$
$864$	0	0
$865$	−27.4403	−0.932997
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 12.1177i	− 0.411064i
$870$	0	0
$871$	− 8.77936i	− 0.297477i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	40.7284	1.37530	0.687650	−	0.726042i	$-0.258642\pi$
$877$	40.7284	1.37530	0.687650	+	0.726042i	$0.258642\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	0.774980	0.0261097	0.0130549	−	0.999915i	$-0.495844\pi$
$881$	0.774980	0.0261097	0.0130549	+	0.999915i	$0.495844\pi$
$882$	0	0
$883$	7.22819	0.243248	0.121624	−	0.992576i	$-0.461190\pi$
$883$	7.22819	0.243248	0.121624	+	0.992576i	$0.461190\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−53.9175	−1.81037	−0.905187	−	0.425015i	$-0.860269\pi$
$887$	−53.9175	−1.81037	−0.905187	+	0.425015i	$0.860269\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	2.58881i	0.0866312i
$894$	0	0
$895$	34.6317i	1.15761i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−50.4743	−1.68341
$900$	0	0
$901$	91.7769i	3.05753i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	30.3951i	1.01037i
$906$	0	0
$907$	34.2935	1.13870	0.569349	−	0.822096i	$-0.307195\pi$
$907$	34.2935	1.13870	0.569349	+	0.822096i	$0.307195\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	3.55794i	0.117880i	0.998262	+	0.0589399i	$0.0187720\pi$
$911$	3.55794i	0.117880i	−0.998262	+	0.0589399i	$0.981228\pi$
$912$	0	0
$913$	− 14.4182i	− 0.477174i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	40.7861	1.34541	0.672705	−	0.739911i	$-0.265132\pi$
$919$	40.7861	1.34541	0.672705	+	0.739911i	$0.265132\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−26.8061	−0.882333
$924$	0	0
$925$	1.72481	0.0567114
$926$	0	0
$927$	0	0
$928$	0	0
$929$	23.8713	0.783191	0.391596	−	0.920137i	$-0.371923\pi$
$929$	23.8713	0.783191	0.391596	+	0.920137i	$0.371923\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 20.7718i	− 0.679311i
$936$	0	0
$937$	48.4063i	1.58136i	0.612228	+	0.790682i	$0.290274\pi$
$937$	48.4063i	1.58136i	−0.612228	+	0.790682i	$0.709726\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	8.89351	0.289920	0.144960	−	0.989438i	$-0.453695\pi$
$941$	8.89351	0.289920	0.144960	+	0.989438i	$0.453695\pi$
$942$	0	0
$943$	10.7596i	0.350381i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	19.3481i	0.628728i	0.949303	+	0.314364i	$0.101791\pi$
$947$	19.3481i	0.628728i	−0.949303	+	0.314364i	$0.898209\pi$
$948$	0	0
$949$	−24.6233	−0.799305
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 21.7409i	− 0.704258i	−0.935951	−	0.352129i	$-0.885458\pi$
$953$	− 21.7409i	− 0.704258i	0.935951	−	0.352129i	$-0.114542\pi$
$954$	0	0
$955$	56.7779i	1.83729i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−36.4295	−1.17515
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−6.64378	−0.213871
$966$	0	0
$967$	−12.1866	−0.391894	−0.195947	−	0.980615i	$-0.562778\pi$
$967$	−12.1866	−0.391894	−0.195947	+	0.980615i	$0.562778\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−5.34883	−0.171652	−0.0858260	−	0.996310i	$-0.527353\pi$
$971$	−5.34883	−0.171652	−0.0858260	+	0.996310i	$0.527353\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	4.62863i	0.148083i	0.997255	+	0.0740415i	$0.0235897\pi$
$977$	4.62863i	0.148083i	−0.997255	+	0.0740415i	$0.976410\pi$
$978$	0	0
$979$	6.93752i	0.221724i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−4.06548	−0.129669	−0.0648344	−	0.997896i	$-0.520652\pi$
$983$	−4.06548	−0.129669	−0.0648344	+	0.997896i	$0.520652\pi$
$984$	0	0
$985$	− 19.6144i	− 0.624966i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 1.59956i	− 0.0508632i
$990$	0	0
$991$	51.5163	1.63647	0.818235	−	0.574884i	$-0.194953\pi$
$991$	51.5163	1.63647	0.818235	+	0.574884i	$0.194953\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 48.9982i	− 1.55335i
$996$	0	0
$997$	− 6.34775i	− 0.201035i	−0.994935	−	0.100518i	$-0.967950\pi$
$997$	− 6.34775i	− 0.201035i	0.994935	−	0.100518i	$-0.0320499\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5292.2.f.e.2645.4		12
3.2	odd	2	inner	5292.2.f.e.2645.9		12
7.4	even	3		756.2.t.e.593.5	yes	12
7.5	odd	6		756.2.t.e.269.2	✓	12
7.6	odd	2	inner	5292.2.f.e.2645.10		12
21.5	even	6		756.2.t.e.269.5	yes	12
21.11	odd	6		756.2.t.e.593.2	yes	12
21.20	even	2	inner	5292.2.f.e.2645.3		12
63.4	even	3		2268.2.bm.i.593.2		12
63.5	even	6		2268.2.w.i.269.5		12
63.11	odd	6		2268.2.w.i.1349.2		12
63.25	even	3		2268.2.w.i.1349.5		12
63.32	odd	6		2268.2.bm.i.593.5		12
63.40	odd	6		2268.2.w.i.269.2		12
63.47	even	6		2268.2.bm.i.1025.2		12
63.61	odd	6		2268.2.bm.i.1025.5		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
756.2.t.e.269.2	✓	12	7.5	odd	6
756.2.t.e.269.5	yes	12	21.5	even	6
756.2.t.e.593.2	yes	12	21.11	odd	6
756.2.t.e.593.5	yes	12	7.4	even	3
2268.2.w.i.269.2		12	63.40	odd	6
2268.2.w.i.269.5		12	63.5	even	6
2268.2.w.i.1349.2		12	63.11	odd	6
2268.2.w.i.1349.5		12	63.25	even	3
2268.2.bm.i.593.2		12	63.4	even	3
2268.2.bm.i.593.5		12	63.32	odd	6
2268.2.bm.i.1025.2		12	63.47	even	6
2268.2.bm.i.1025.5		12	63.61	odd	6
5292.2.f.e.2645.3		12	21.20	even	2	inner
5292.2.f.e.2645.4		12	1.1	even	1	trivial
5292.2.f.e.2645.9		12	3.2	odd	2	inner
5292.2.f.e.2645.10		12	7.6	odd	2	inner

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 5292 = 2^{2} \cdot 3^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5292.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.15983 q^{5} +1.33513i q^{11} +2.35021i q^{13} +7.20331 q^{17} -1.19862i q^{19} +2.40581i q^{23} -0.335126 q^{25} -6.14675i q^{29} -8.21155i q^{31} -5.14675 q^{37} +4.47234 q^{41} -0.664874 q^{43} -2.15983 q^{47} +12.7409i q^{53} -2.88365i q^{55} +9.66849 q^{59} -12.2938i q^{61} -5.07606i q^{65} -3.73556 q^{67} +11.4058i q^{71} +10.4770i q^{73} -9.07606 q^{79} -10.7992 q^{83} -15.5579 q^{85} +5.19615 q^{89} +2.58881i q^{95} -1.23632i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 48 q^{37} - 12 q^{43} - 48 q^{79} - 12 q^{85}+O(q^{100}) \) 12 * q + 48 * q^37 - 12 * q^43 - 48 * q^79 - 12 * q^85

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