LMFDB - Embedded newform 5292.2.f.e.2645.5

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			2645.5
Root			$-0.385418 - 0.222521i$ of defining polynomial
Character	$\chi$	$=$	5292.2645
Dual form			5292.2.f.e.2645.6

$q$-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
$5$	−0.770835	−0.344728	−0.172364	−	0.985033i	$-0.555141\pi$
$5$	−0.770835	−0.344728	−0.172364	+	0.985033i	$0.555141\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 5.40581i	− 1.62991i	−0.579521	−	0.814957i	$-0.696760\pi$
$11$	− 5.40581i	− 1.62991i	0.579521	−	0.814957i	$-0.303240\pi$
$12$	0	0
$13$	− 5.28088i	− 1.46465i	−0.680954	−	0.732326i	$-0.738435\pi$
$13$	− 5.28088i	− 1.46465i	0.680954	−	0.732326i	$-0.261565\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	5.70866	1.38455	0.692277	−	0.721632i	$-0.256608\pi$
$17$	5.70866	1.38455	0.692277	+	0.721632i	$0.256608\pi$
$18$	0	0
$19$	− 0.618162i	− 0.141816i	−0.997483	−	0.0709080i	$-0.977410\pi$
$19$	− 0.618162i	− 0.141816i	0.997483	−	0.0709080i	$-0.0225897\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	6.74094i	1.40558i	0.711396	+	0.702791i	$0.248063\pi$
$23$	6.74094i	1.40558i	−0.711396	+	0.702791i	$0.751937\pi$
$24$	0	0
$25$	−4.40581	−0.881163
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 8.07606i	− 1.49969i	−0.661615	−	0.749844i	$-0.730129\pi$
$29$	− 8.07606i	− 1.49969i	0.661615	−	0.749844i	$-0.269871\pi$
$30$	0	0
$31$	0.580455i	0.104253i	0.998640	+	0.0521264i	$0.0165999\pi$
$31$	0.580455i	0.104253i	−0.998640	+	0.0521264i	$0.983400\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	9.07606	1.49210	0.746048	−	0.665892i	$-0.231949\pi$
$37$	9.07606	1.49210	0.746048	+	0.665892i	$0.231949\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−8.59231	−1.34189	−0.670947	−	0.741506i	$-0.734112\pi$
$41$	−8.59231	−1.34189	−0.670947	+	0.741506i	$0.734112\pi$
$42$	0	0
$43$	3.40581	0.519382	0.259691	−	0.965692i	$-0.416379\pi$
$43$	3.40581	0.519382	0.259691	+	0.965692i	$0.416379\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−0.770835	−0.112438	−0.0562189	−	0.998418i	$-0.517904\pi$
$47$	−0.770835	−0.112438	−0.0562189	+	0.998418i	$0.517904\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 7.66487i	− 1.05285i	−0.850221	−	0.526426i	$-0.823532\pi$
$53$	− 7.66487i	− 1.05285i	0.850221	−	0.526426i	$-0.176468\pi$
$54$	0	0
$55$	4.16699i	0.561877i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−13.7885	−1.79510	−0.897552	−	0.440908i	$-0.854657\pi$
$59$	−13.7885	−1.79510	−0.897552	+	0.440908i	$0.854657\pi$
$60$	0	0
$61$	4.12928i	0.528701i	0.964427	+	0.264350i	$0.0851575\pi$
$61$	4.12928i	0.528701i	−0.964427	+	0.264350i	$0.914842\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	4.07069i	0.504907i
$66$	0	0
$67$	13.5526	1.65571	0.827855	−	0.560943i	$-0.189561\pi$
$67$	13.5526	1.65571	0.827855	+	0.560943i	$0.189561\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 2.25906i	− 0.268101i	−0.990974	−	0.134051i	$-0.957202\pi$
$71$	− 2.25906i	− 0.268101i	0.990974	−	0.134051i	$-0.0427985\pi$
$72$	0	0
$73$	2.26549i	0.265155i	0.991173	+	0.132577i	$0.0423253\pi$
$73$	2.26549i	0.265155i	−0.991173	+	0.132577i	$0.957675\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−8.07069	−0.908023	−0.454012	−	0.890996i	$-0.650008\pi$
$79$	−8.07069	−0.908023	−0.454012	+	0.890996i	$0.650008\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−3.85418	−0.423051	−0.211525	−	0.977373i	$-0.567843\pi$
$83$	−3.85418	−0.423051	−0.211525	+	0.977373i	$0.567843\pi$
$84$	0	0
$85$	−4.40044	−0.477294
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−5.19615	−0.550791	−0.275396	−	0.961331i	$-0.588809\pi$
$89$	−5.19615	−0.550791	−0.275396	+	0.961331i	$0.588809\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0.476501i	0.0488880i
$96$	0	0
$97$	14.0259i	1.42411i	0.702123	+	0.712055i	$0.252235\pi$
$97$	14.0259i	1.42411i	−0.702123	+	0.712055i	$0.747765\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−12.7048	−1.26418	−0.632088	−	0.774897i	$-0.717802\pi$
$101$	−12.7048	−1.26418	−0.632088	+	0.774897i	$0.717802\pi$
$102$	0	0
$103$	2.96837i	0.292483i	0.989249	+	0.146241i	$0.0467176\pi$
$103$	2.96837i	0.292483i	−0.989249	+	0.146241i	$0.953282\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	4.92931i	0.476535i	0.971200	+	0.238267i	$0.0765794\pi$
$107$	4.92931i	0.476535i	−0.971200	+	0.238267i	$0.923421\pi$
$108$	0	0
$109$	−6.81163	−0.652435	−0.326218	−	0.945295i	$-0.605774\pi$
$109$	−6.81163	−0.652435	−0.326218	+	0.945295i	$0.605774\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 6.32975i	− 0.595453i	−0.954651	−	0.297726i	$-0.903772\pi$
$113$	− 6.32975i	− 0.595453i	0.954651	−	0.297726i	$-0.0962283\pi$
$114$	0	0
$115$	− 5.19615i	− 0.484544i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−18.2228	−1.65662
$122$	0	0
$123$	0	0
$124$	0	0
$125$	7.25033	0.648489
$126$	0	0
$127$	−4.41119	−0.391430	−0.195715	−	0.980661i	$-0.562703\pi$
$127$	−4.41119	−0.391430	−0.195715	+	0.980661i	$0.562703\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	8.79200	0.768161	0.384080	−	0.923300i	$-0.374519\pi$
$131$	8.79200	0.768161	0.384080	+	0.923300i	$0.374519\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 4.51812i	− 0.386009i	−0.981198	−	0.193005i	$-0.938177\pi$
$137$	− 4.51812i	− 0.386009i	0.981198	−	0.193005i	$-0.0618232\pi$
$138$	0	0
$139$	− 7.67811i	− 0.651249i	−0.945499	−	0.325624i	$-0.894425\pi$
$139$	− 7.67811i	− 0.651249i	0.945499	−	0.325624i	$-0.105575\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−28.5474	−2.38726
$144$	0	0
$145$	6.22531i	0.516984i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 8.55257i	− 0.700653i	−0.936628	−	0.350327i	$-0.886071\pi$
$149$	− 8.55257i	− 0.700653i	0.936628	−	0.350327i	$-0.113929\pi$
$150$	0	0
$151$	−7.07606	−0.575842	−0.287921	−	0.957654i	$-0.592964\pi$
$151$	−7.07606	−0.575842	−0.287921	+	0.957654i	$0.592964\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 0.447435i	− 0.0359388i
$156$	0	0
$157$	10.6465i	0.849682i	0.905268	+	0.424841i	$0.139670\pi$
$157$	10.6465i	0.849682i	−0.905268	+	0.424841i	$0.860330\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	15.0707	1.18043	0.590214	−	0.807247i	$-0.299043\pi$
$163$	15.0707	1.18043	0.590214	+	0.807247i	$0.299043\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	17.3843	1.34524	0.672619	−	0.739989i	$-0.265169\pi$
$167$	17.3843	1.34524	0.672619	+	0.739989i	$0.265169\pi$
$168$	0	0
$169$	−14.8877	−1.14521
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−19.7554	−1.50198	−0.750989	−	0.660314i	$-0.770423\pi$
$173$	−19.7554	−1.50198	−0.750989	+	0.660314i	$0.770423\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 17.4873i	− 1.30706i	−0.756901	−	0.653529i	$-0.773288\pi$
$179$	− 17.4873i	− 1.30706i	0.756901	−	0.653529i	$-0.226712\pi$
$180$	0	0
$181$	− 4.11997i	− 0.306235i	−0.988208	−	0.153118i	$-0.951069\pi$
$181$	− 4.11997i	− 0.306235i	0.988208	−	0.153118i	$-0.0489313\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−6.99615	−0.514367
$186$	0	0
$187$	− 30.8600i	− 2.25670i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 14.3696i	− 1.03975i	−0.854244	−	0.519873i	$-0.825979\pi$
$191$	− 14.3696i	− 1.03975i	0.854244	−	0.519873i	$-0.174021\pi$
$192$	0	0
$193$	2.07069	0.149051	0.0745257	−	0.997219i	$-0.476256\pi$
$193$	2.07069	0.149051	0.0745257	+	0.997219i	$0.476256\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 20.2881i	− 1.44547i	−0.691126	−	0.722735i	$-0.742885\pi$
$197$	− 20.2881i	− 1.44547i	0.691126	−	0.722735i	$-0.257115\pi$
$198$	0	0
$199$	6.26302i	0.443974i	0.975050	+	0.221987i	$0.0712542\pi$
$199$	6.26302i	0.443974i	−0.975050	+	0.221987i	$0.928746\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	6.62325	0.462588
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−3.34167	−0.231148
$210$	0	0
$211$	3.73556	0.257167	0.128583	−	0.991699i	$-0.458957\pi$
$211$	3.73556	0.257167	0.128583	+	0.991699i	$0.458957\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−2.62532	−0.179045
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 30.1468i	− 2.02789i
$222$	0	0
$223$	− 15.8426i	− 1.06090i	−0.847716	−	0.530451i	$-0.822023\pi$
$223$	− 15.8426i	− 1.06090i	0.847716	−	0.530451i	$-0.177977\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−3.91281	−0.259702	−0.129851	−	0.991534i	$-0.541450\pi$
$227$	−3.91281	−0.259702	−0.129851	+	0.991534i	$0.541450\pi$
$228$	0	0
$229$	− 5.07372i	− 0.335281i	−0.985848	−	0.167640i	$-0.946385\pi$
$229$	− 5.07372i	− 0.335281i	0.985848	−	0.167640i	$-0.0536147\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	1.81163i	0.118684i	0.998238	+	0.0593418i	$0.0189002\pi$
$233$	1.81163i	0.118684i	−0.998238	+	0.0593418i	$0.981100\pi$
$234$	0	0
$235$	0.594187	0.0387605
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 8.58881i	− 0.555564i	−0.960644	−	0.277782i	$-0.910401\pi$
$239$	− 8.58881i	− 0.555564i	0.960644	−	0.277782i	$-0.0895993\pi$
$240$	0	0
$241$	− 23.8470i	− 1.53612i	−0.640377	−	0.768061i	$-0.721222\pi$
$241$	− 23.8470i	− 1.53612i	0.640377	−	0.768061i	$-0.278778\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−3.26444	−0.207711
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−7.25033	−0.457637	−0.228818	−	0.973469i	$-0.573486\pi$
$251$	−7.25033	−0.457637	−0.228818	+	0.973469i	$0.573486\pi$
$252$	0	0
$253$	36.4403	2.29098
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−28.5474	−1.78074	−0.890370	−	0.455237i	$-0.849555\pi$
$257$	−28.5474	−1.78074	−0.890370	+	0.455237i	$0.849555\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 2.25906i	− 0.139300i	−0.997572	−	0.0696498i	$-0.977812\pi$
$263$	− 2.25906i	− 0.139300i	0.997572	−	0.0696498i	$-0.0221882\pi$
$264$	0	0
$265$	5.90835i	0.362947i
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−16.6721	−1.01652	−0.508258	−	0.861205i	$-0.669710\pi$
$269$	−16.6721	−1.01652	−0.508258	+	0.861205i	$0.669710\pi$
$270$	0	0
$271$	31.4688i	1.91159i	0.294028	+	0.955797i	$0.405004\pi$
$271$	31.4688i	1.91159i	−0.294028	+	0.955797i	$0.594996\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	23.8170i	1.43622i
$276$	0	0
$277$	19.0398	1.14399	0.571996	−	0.820256i	$-0.306169\pi$
$277$	19.0398	1.14399	0.571996	+	0.820256i	$0.306169\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 7.59956i	− 0.453352i	−0.973970	−	0.226676i	$-0.927214\pi$
$281$	− 7.59956i	− 0.453352i	0.973970	−	0.226676i	$-0.0727858\pi$
$282$	0	0
$283$	− 30.9824i	− 1.84171i	−0.389903	−	0.920856i	$-0.627491\pi$
$283$	− 30.9824i	− 1.84171i	0.389903	−	0.920856i	$-0.372509\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	15.5888	0.916989
$290$	0	0
$291$	0	0
$292$	0	0
$293$	4.93783	0.288471	0.144235	−	0.989543i	$-0.453928\pi$
$293$	4.93783	0.288471	0.144235	+	0.989543i	$0.453928\pi$
$294$	0	0
$295$	10.6286	0.618823
$296$	0	0
$297$	0	0
$298$	0	0
$299$	35.5981	2.05869
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 3.18300i	− 0.182258i
$306$	0	0
$307$	− 7.95736i	− 0.454151i	−0.973877	−	0.227075i	$-0.927084\pi$
$307$	− 7.95736i	− 0.454151i	0.973877	−	0.227075i	$-0.0729164\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	9.87565	0.559997	0.279998	−	0.960000i	$-0.409666\pi$
$311$	9.87565	0.559997	0.279998	+	0.960000i	$0.409666\pi$
$312$	0	0
$313$	− 16.8248i	− 0.950993i	−0.879718	−	0.475496i	$-0.842268\pi$
$313$	− 16.8248i	− 0.950993i	0.879718	−	0.475496i	$-0.157732\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	24.7700i	1.39122i	0.718419	+	0.695611i	$0.244866\pi$
$317$	24.7700i	1.39122i	−0.718419	+	0.695611i	$0.755134\pi$
$318$	0	0
$319$	−43.6577	−2.44436
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 3.52888i	− 0.196352i
$324$	0	0
$325$	23.2666i	1.29060i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−18.0054	−0.989665	−0.494833	−	0.868988i	$-0.664771\pi$
$331$	−18.0054	−0.989665	−0.494833	+	0.868988i	$0.664771\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−10.4468	−0.570769
$336$	0	0
$337$	5.21744	0.284212	0.142106	−	0.989851i	$-0.454613\pi$
$337$	5.21744	0.284212	0.142106	+	0.989851i	$0.454613\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	3.13783	0.169923
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 16.2828i	− 0.874104i	−0.899436	−	0.437052i	$-0.856023\pi$
$347$	− 16.2828i	− 0.874104i	0.899436	−	0.437052i	$-0.143977\pi$
$348$	0	0
$349$	17.4052i	0.931681i	0.884869	+	0.465840i	$0.154248\pi$
$349$	17.4052i	0.931681i	−0.884869	+	0.465840i	$0.845752\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−29.8894	−1.59085	−0.795427	−	0.606050i	$-0.792753\pi$
$353$	−29.8894	−1.59085	−0.795427	+	0.606050i	$0.792753\pi$
$354$	0	0
$355$	1.74136i	0.0924220i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	23.4058i	1.23531i	0.786449	+	0.617656i	$0.211917\pi$
$359$	23.4058i	1.23531i	−0.786449	+	0.617656i	$0.788083\pi$
$360$	0	0
$361$	18.6179	0.979888
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 1.74632i	− 0.0914063i
$366$	0	0
$367$	20.0567i	1.04695i	0.852041	+	0.523474i	$0.175364\pi$
$367$	20.0567i	1.04695i	−0.852041	+	0.523474i	$0.824636\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−7.98925	−0.413667	−0.206834	−	0.978376i	$-0.566316\pi$
$373$	−7.98925	−0.413667	−0.206834	+	0.978376i	$0.566316\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−42.6487	−2.19652
$378$	0	0
$379$	−34.5816	−1.77634	−0.888170	−	0.459516i	$-0.848023\pi$
$379$	−34.5816	−1.77634	−0.888170	+	0.459516i	$0.848023\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−8.02117	−0.409862	−0.204931	−	0.978776i	$-0.565697\pi$
$383$	−8.02117	−0.409862	−0.204931	+	0.978776i	$0.565697\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 18.4112i	− 0.933484i	−0.884393	−	0.466742i	$-0.845428\pi$
$389$	− 18.4112i	− 0.933484i	0.884393	−	0.466742i	$-0.154572\pi$
$390$	0	0
$391$	38.4817i	1.94610i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	6.22117	0.313021
$396$	0	0
$397$	25.4003i	1.27480i	0.770531	+	0.637402i	$0.219991\pi$
$397$	25.4003i	1.27480i	−0.770531	+	0.637402i	$0.780009\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 8.58881i	− 0.428905i	−0.976734	−	0.214452i	$-0.931203\pi$
$401$	− 8.58881i	− 0.428905i	0.976734	−	0.214452i	$-0.0687967\pi$
$402$	0	0
$403$	3.06531	0.152694
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 49.0635i	− 2.43199i
$408$	0	0
$409$	− 3.51112i	− 0.173614i	−0.996225	−	0.0868069i	$-0.972334\pi$
$409$	− 3.51112i	− 0.173614i	0.996225	−	0.0868069i	$-0.0276663\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	2.97093	0.145837
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−29.5725	−1.44471	−0.722355	−	0.691523i	$-0.756940\pi$
$419$	−29.5725	−1.44471	−0.722355	+	0.691523i	$0.756940\pi$
$420$	0	0
$421$	−23.0344	−1.12263	−0.561315	−	0.827602i	$-0.689704\pi$
$421$	−23.0344	−1.12263	−0.561315	+	0.827602i	$0.689704\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−25.1513	−1.22002
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	23.3696i	1.12567i	0.826568	+	0.562836i	$0.190290\pi$
$431$	23.3696i	1.12567i	−0.826568	+	0.562836i	$0.809710\pi$
$432$	0	0
$433$	− 37.9199i	− 1.82231i	−0.412059	−	0.911157i	$-0.635190\pi$
$433$	− 37.9199i	− 1.82231i	0.412059	−	0.911157i	$-0.364810\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	4.16699	0.199334
$438$	0	0
$439$	3.13783i	0.149760i	0.997193	+	0.0748802i	$0.0238574\pi$
$439$	3.13783i	0.149760i	−0.997193	+	0.0748802i	$0.976143\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	24.2282i	1.15112i	0.817761	+	0.575558i	$0.195215\pi$
$443$	24.2282i	1.15112i	−0.817761	+	0.575558i	$0.804785\pi$
$444$	0	0
$445$	4.00538	0.189873
$446$	0	0
$447$	0	0
$448$	0	0
$449$	25.1812i	1.18837i	0.804327	+	0.594187i	$0.202526\pi$
$449$	25.1812i	1.18837i	−0.804327	+	0.594187i	$0.797474\pi$
$450$	0	0
$451$	46.4484i	2.18717i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−26.2121	−1.22615	−0.613074	−	0.790025i	$-0.710067\pi$
$457$	−26.2121	−1.22615	−0.613074	+	0.790025i	$0.710067\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	1.80000	0.0838342	0.0419171	−	0.999121i	$-0.486653\pi$
$461$	1.80000	0.0838342	0.0419171	+	0.999121i	$0.486653\pi$
$462$	0	0
$463$	−20.4456	−0.950189	−0.475095	−	0.879935i	$-0.657586\pi$
$463$	−20.4456	−0.950189	−0.475095	+	0.879935i	$0.657586\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	10.5054	0.486133	0.243067	−	0.970010i	$-0.421847\pi$
$467$	10.5054	0.486133	0.243067	+	0.970010i	$0.421847\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 18.4112i	− 0.846547i
$474$	0	0
$475$	2.72351i	0.124963i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	5.96699	0.272639	0.136319	−	0.990665i	$-0.456473\pi$
$479$	5.96699	0.272639	0.136319	+	0.990665i	$0.456473\pi$
$480$	0	0
$481$	− 47.9296i	− 2.18540i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 10.8116i	− 0.490931i
$486$	0	0
$487$	24.0398	1.08935	0.544674	−	0.838648i	$-0.316653\pi$
$487$	24.0398	1.08935	0.544674	+	0.838648i	$0.316653\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 21.5579i	− 0.972896i	−0.873709	−	0.486448i	$-0.838292\pi$
$491$	− 21.5579i	− 0.972896i	0.873709	−	0.486448i	$-0.161708\pi$
$492$	0	0
$493$	− 46.1035i	− 2.07640i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	28.8514	1.29157	0.645784	−	0.763520i	$-0.276531\pi$
$499$	28.8514	1.29157	0.645784	+	0.763520i	$0.276531\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	20.7805	0.926555	0.463278	−	0.886213i	$-0.346673\pi$
$503$	20.7805	0.926555	0.463278	+	0.886213i	$0.346673\pi$
$504$	0	0
$505$	9.79331	0.435797
$506$	0	0
$507$	0	0
$508$	0	0
$509$	25.4055	1.12608	0.563039	−	0.826430i	$-0.309632\pi$
$509$	25.4055	1.12608	0.563039	+	0.826430i	$0.309632\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 2.28813i	− 0.100827i
$516$	0	0
$517$	4.16699i	0.183264i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	2.05418	0.0899952	0.0449976	−	0.998987i	$-0.485672\pi$
$521$	2.05418	0.0899952	0.0449976	+	0.998987i	$0.485672\pi$
$522$	0	0
$523$	11.3367i	0.495721i	0.968796	+	0.247860i	$0.0797275\pi$
$523$	11.3367i	0.495721i	−0.968796	+	0.247860i	$0.920273\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	3.31362i	0.144343i
$528$	0	0
$529$	−22.4403	−0.975663
$530$	0	0
$531$	0	0
$532$	0	0
$533$	45.3749i	1.96541i
$534$	0	0
$535$	− 3.79969i	− 0.164275i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	18.7826	0.807525	0.403763	−	0.914864i	$-0.367702\pi$
$541$	18.7826	0.807525	0.403763	+	0.914864i	$0.367702\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	5.25064	0.224913
$546$	0	0
$547$	−6.91856	−0.295816	−0.147908	−	0.989001i	$-0.547254\pi$
$547$	−6.91856	−0.295816	−0.147908	+	0.989001i	$0.547254\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−4.99231	−0.212680
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 10.4004i	− 0.440681i	−0.975423	−	0.220340i	$-0.929283\pi$
$557$	− 10.4004i	− 0.440681i	0.975423	−	0.220340i	$-0.0707168\pi$
$558$	0	0
$559$	− 17.9857i	− 0.760714i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−6.02148	−0.253775	−0.126887	−	0.991917i	$-0.540499\pi$
$563$	−6.02148	−0.253775	−0.126887	+	0.991917i	$0.540499\pi$
$564$	0	0
$565$	4.87919i	0.205269i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 29.3534i	− 1.23056i	−0.788309	−	0.615280i	$-0.789043\pi$
$569$	− 29.3534i	− 1.23056i	0.788309	−	0.615280i	$-0.210957\pi$
$570$	0	0
$571$	17.2282	0.720977	0.360489	−	0.932764i	$-0.382610\pi$
$571$	17.2282	0.720977	0.360489	+	0.932764i	$0.382610\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 29.6993i	− 1.23855i
$576$	0	0
$577$	2.14305i	0.0892164i	0.999005	+	0.0446082i	$0.0142039\pi$
$577$	2.14305i	0.0892164i	−0.999005	+	0.0446082i	$0.985796\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−41.4349	−1.71606
$584$	0	0
$585$	0	0
$586$	0	0
$587$	17.5840	0.725769	0.362885	−	0.931834i	$-0.381792\pi$
$587$	17.5840	0.725769	0.362885	+	0.931834i	$0.381792\pi$
$588$	0	0
$589$	0.358815	0.0147847
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−24.2939	−0.997630	−0.498815	−	0.866708i	$-0.666231\pi$
$593$	−24.2939	−0.997630	−0.498815	+	0.866708i	$0.666231\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	37.8116i	1.54494i	0.635051	+	0.772471i	$0.280979\pi$
$599$	37.8116i	1.54494i	−0.635051	+	0.772471i	$0.719021\pi$
$600$	0	0
$601$	− 2.90227i	− 0.118386i	−0.998247	−	0.0591931i	$-0.981147\pi$
$601$	− 2.90227i	− 0.118386i	0.998247	−	0.0591931i	$-0.0188528\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	14.0468	0.571083
$606$	0	0
$607$	− 11.1706i	− 0.453401i	−0.973965	−	0.226701i	$-0.927206\pi$
$607$	− 11.1706i	− 0.453401i	0.973965	−	0.226701i	$-0.0727939\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	4.07069i	0.164682i
$612$	0	0
$613$	16.2591	0.656697	0.328349	−	0.944557i	$-0.393508\pi$
$613$	16.2591	0.656697	0.328349	+	0.944557i	$0.393508\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 5.34050i	− 0.215001i	−0.994205	−	0.107500i	$-0.965715\pi$
$617$	− 5.34050i	− 0.215001i	0.994205	−	0.107500i	$-0.0342846\pi$
$618$	0	0
$619$	21.2082i	0.852431i	0.904622	+	0.426216i	$0.140154\pi$
$619$	21.2082i	0.852431i	−0.904622	+	0.426216i	$0.859846\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	16.4403	0.657610
$626$	0	0
$627$	0	0
$628$	0	0
$629$	51.8122	2.06589
$630$	0	0
$631$	44.7338	1.78082	0.890411	−	0.455157i	$-0.150417\pi$
$631$	44.7338	1.78082	0.890411	+	0.455157i	$0.150417\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	3.40030	0.134937
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 27.3096i	− 1.07867i	−0.842093	−	0.539333i	$-0.818676\pi$
$641$	− 27.3096i	− 1.07867i	0.842093	−	0.539333i	$-0.181324\pi$
$642$	0	0
$643$	35.7298i	1.40905i	0.709681	+	0.704524i	$0.248839\pi$
$643$	35.7298i	1.40905i	−0.709681	+	0.704524i	$0.751161\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	40.9074	1.60823	0.804117	−	0.594471i	$-0.202638\pi$
$647$	40.9074	1.60823	0.804117	+	0.594471i	$0.202638\pi$
$648$	0	0
$649$	74.5378i	2.92587i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 26.0761i	− 1.02044i	−0.860045	−	0.510218i	$-0.829565\pi$
$653$	− 26.0761i	− 1.02044i	0.860045	−	0.510218i	$-0.170435\pi$
$654$	0	0
$655$	−6.77718	−0.264806
$656$	0	0
$657$	0	0
$658$	0	0
$659$	40.8568i	1.59156i	0.605589	+	0.795778i	$0.292938\pi$
$659$	40.8568i	1.59156i	−0.605589	+	0.795778i	$0.707062\pi$
$660$	0	0
$661$	− 7.25780i	− 0.282296i	−0.989989	−	0.141148i	$-0.954921\pi$
$661$	− 7.25780i	− 0.282296i	0.989989	−	0.141148i	$-0.0450793\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	54.4403	2.10794
$668$	0	0
$669$	0	0
$670$	0	0
$671$	22.3221	0.861736
$672$	0	0
$673$	35.2465	1.35865	0.679326	−	0.733836i	$-0.262272\pi$
$673$	35.2465	1.35865	0.679326	+	0.733836i	$0.262272\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−10.9634	−0.421359	−0.210680	−	0.977555i	$-0.567568\pi$
$677$	−10.9634	−0.421359	−0.210680	+	0.977555i	$0.567568\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 8.89844i	− 0.340490i	−0.985402	−	0.170245i	$-0.945544\pi$
$683$	− 8.89844i	− 0.340490i	0.985402	−	0.170245i	$-0.0544558\pi$
$684$	0	0
$685$	3.48273i	0.133068i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−40.4773	−1.54206
$690$	0	0
$691$	− 8.40607i	− 0.319782i	−0.987135	−	0.159891i	$-0.948886\pi$
$691$	− 8.40607i	− 0.319782i	0.987135	−	0.159891i	$-0.0511143\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	5.91856i	0.224504i
$696$	0	0
$697$	−49.0506	−1.85792
$698$	0	0
$699$	0	0
$700$	0	0
$701$	28.3351i	1.07020i	0.844788	+	0.535101i	$0.179727\pi$
$701$	28.3351i	1.07020i	−0.844788	+	0.535101i	$0.820273\pi$
$702$	0	0
$703$	− 5.61048i	− 0.211603i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−24.0868	−0.904599	−0.452300	−	0.891866i	$-0.649396\pi$
$709$	−24.0868	−0.904599	−0.452300	+	0.891866i	$0.649396\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−3.91281	−0.146536
$714$	0	0
$715$	22.0054	0.822954
$716$	0	0
$717$	0	0
$718$	0	0
$719$	5.14166	0.191752	0.0958759	−	0.995393i	$-0.469435\pi$
$719$	5.14166	0.191752	0.0958759	+	0.995393i	$0.469435\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	35.5816i	1.32147i
$726$	0	0
$727$	− 6.91889i	− 0.256607i	−0.991735	−	0.128304i	$-0.959047\pi$
$727$	− 6.91889i	− 0.256607i	0.991735	−	0.128304i	$-0.0409532\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	19.4426	0.719112
$732$	0	0
$733$	− 17.1981i	− 0.635225i	−0.948221	−	0.317613i	$-0.897119\pi$
$733$	− 17.1981i	− 0.635225i	0.948221	−	0.317613i	$-0.102881\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 73.2626i	− 2.69866i
$738$	0	0
$739$	−21.2228	−0.780694	−0.390347	−	0.920668i	$-0.627645\pi$
$739$	−21.2228	−0.780694	−0.390347	+	0.920668i	$0.627645\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	33.7047i	1.23651i	0.785979	+	0.618253i	$0.212159\pi$
$743$	33.7047i	1.23651i	−0.785979	+	0.618253i	$0.787841\pi$
$744$	0	0
$745$	6.59262i	0.241535i
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	4.82238	0.175971	0.0879856	−	0.996122i	$-0.471957\pi$
$751$	4.82238	0.175971	0.0879856	+	0.996122i	$0.471957\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	5.45448	0.198509
$756$	0	0
$757$	−7.30426	−0.265478	−0.132739	−	0.991151i	$-0.542377\pi$
$757$	−7.30426	−0.265478	−0.132739	+	0.991151i	$0.542377\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	15.8385	0.574145	0.287072	−	0.957909i	$-0.407318\pi$
$761$	15.8385	0.574145	0.287072	+	0.957909i	$0.407318\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	72.8152i	2.62920i
$768$	0	0
$769$	− 47.3208i	− 1.70643i	−0.521559	−	0.853215i	$-0.674649\pi$
$769$	− 47.3208i	− 1.70643i	0.521559	−	0.853215i	$-0.325351\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−21.0346	−0.756564	−0.378282	−	0.925690i	$-0.623485\pi$
$773$	−21.0346	−0.756564	−0.378282	+	0.925690i	$0.623485\pi$
$774$	0	0
$775$	− 2.55737i	− 0.0918636i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	5.31144i	0.190302i
$780$	0	0
$781$	−12.2121	−0.436982
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 8.20669i	− 0.292909i
$786$	0	0
$787$	− 39.0152i	− 1.39074i	−0.718652	−	0.695370i	$-0.755240\pi$
$787$	− 39.0152i	− 1.39074i	0.718652	−	0.695370i	$-0.244760\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	21.8062	0.774363
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−31.6853	−1.12235	−0.561175	−	0.827697i	$-0.689651\pi$
$797$	−31.6853	−1.12235	−0.561175	+	0.827697i	$0.689651\pi$
$798$	0	0
$799$	−4.40044	−0.155676
$800$	0	0
$801$	0	0
$802$	0	0
$803$	12.2468	0.432180
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	2.23000i	0.0784025i	0.999231	+	0.0392012i	$0.0124813\pi$
$809$	2.23000i	0.0784025i	−0.999231	+	0.0392012i	$0.987519\pi$
$810$	0	0
$811$	− 55.3535i	− 1.94373i	−0.235548	−	0.971863i	$-0.575688\pi$
$811$	− 55.3535i	− 1.94373i	0.235548	−	0.971863i	$-0.424312\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−11.6170	−0.406926
$816$	0	0
$817$	− 2.10534i	− 0.0736566i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 45.9821i	− 1.60479i	−0.596797	−	0.802393i	$-0.703560\pi$
$821$	− 45.9821i	− 1.60479i	0.596797	−	0.802393i	$-0.296440\pi$
$822$	0	0
$823$	42.2989	1.47445	0.737223	−	0.675649i	$-0.236137\pi$
$823$	42.2989	1.47445	0.737223	+	0.675649i	$0.236137\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 21.1105i	− 0.734084i	−0.930204	−	0.367042i	$-0.880370\pi$
$827$	− 21.1105i	− 0.734084i	0.930204	−	0.367042i	$-0.119630\pi$
$828$	0	0
$829$	54.6884i	1.89941i	0.313154	+	0.949703i	$0.398615\pi$
$829$	54.6884i	1.89941i	−0.313154	+	0.949703i	$0.601385\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−13.4004	−0.463741
$836$	0	0
$837$	0	0
$838$	0	0
$839$	43.3651	1.49713	0.748564	−	0.663062i	$-0.230744\pi$
$839$	43.3651	1.49713	0.748564	+	0.663062i	$0.230744\pi$
$840$	0	0
$841$	−36.2228	−1.24906
$842$	0	0
$843$	0	0
$844$	0	0
$845$	11.4760	0.394785
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	61.1812i	2.09726i
$852$	0	0
$853$	26.5989i	0.910730i	0.890305	+	0.455365i	$0.150491\pi$
$853$	26.5989i	0.910730i	−0.890305	+	0.455365i	$0.849509\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	29.6311	1.01218	0.506090	−	0.862481i	$-0.331091\pi$
$857$	29.6311	1.01218	0.506090	+	0.862481i	$0.331091\pi$
$858$	0	0
$859$	− 4.84149i	− 0.165189i	−0.996583	−	0.0825947i	$-0.973679\pi$
$859$	− 4.84149i	− 0.165189i	0.996583	−	0.0825947i	$-0.0263207\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	21.9691i	0.747838i	0.927461	+	0.373919i	$0.121986\pi$
$863$	21.9691i	0.747838i	−0.927461	+	0.373919i	$0.878014\pi$
$864$	0	0
$865$	15.2282	0.517774
$866$	0	0
$867$	0	0
$868$	0	0
$869$	43.6286i	1.48000i
$870$	0	0
$871$	− 71.5695i	− 2.42504i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−42.5978	−1.43842	−0.719212	−	0.694791i	$-0.755497\pi$
$877$	−42.5978	−1.43842	−0.719212	+	0.694791i	$0.755497\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−41.2523	−1.38982	−0.694912	−	0.719095i	$-0.744557\pi$
$881$	−41.2523	−1.38982	−0.694912	+	0.719095i	$0.744557\pi$
$882$	0	0
$883$	4.21206	0.141747	0.0708736	−	0.997485i	$-0.477421\pi$
$883$	4.21206	0.141747	0.0708736	+	0.997485i	$0.477421\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−3.59170	−0.120598	−0.0602988	−	0.998180i	$-0.519205\pi$
$887$	−3.59170	−0.120598	−0.0602988	+	0.998180i	$0.519205\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	0.476501i	0.0159455i
$894$	0	0
$895$	13.4798i	0.450580i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	4.68779	0.156347
$900$	0	0
$901$	− 43.7562i	− 1.45773i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	3.17582i	0.105568i
$906$	0	0
$907$	5.84787	0.194175	0.0970877	−	0.995276i	$-0.469047\pi$
$907$	5.84787	0.194175	0.0970877	+	0.995276i	$0.469047\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	7.59956i	0.251785i	0.992044	+	0.125892i	$0.0401794\pi$
$911$	7.59956i	0.251785i	−0.992044	+	0.125892i	$0.959821\pi$
$912$	0	0
$913$	20.8350i	0.689536i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	26.6125	0.877865	0.438933	−	0.898520i	$-0.355357\pi$
$919$	26.6125	0.877865	0.438933	+	0.898520i	$0.355357\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−11.9298	−0.392675
$924$	0	0
$925$	−39.9874	−1.31478
$926$	0	0
$927$	0	0
$928$	0	0
$929$	52.4113	1.71956	0.859779	−	0.510667i	$-0.170601\pi$
$929$	52.4113	1.71956	0.859779	+	0.510667i	$0.170601\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	23.7879i	0.777949i
$936$	0	0
$937$	49.0719i	1.60311i	0.597922	+	0.801554i	$0.295993\pi$
$937$	49.0719i	1.60311i	−0.597922	+	0.801554i	$0.704007\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	27.4638	0.895294	0.447647	−	0.894210i	$-0.352262\pi$
$941$	27.4638	0.895294	0.447647	+	0.894210i	$0.352262\pi$
$942$	0	0
$943$	− 57.9202i	− 1.88614i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	58.9512i	1.91566i	0.287341	+	0.957828i	$0.407229\pi$
$947$	58.9512i	1.91566i	−0.287341	+	0.957828i	$0.592771\pi$
$948$	0	0
$949$	11.9638	0.388360
$950$	0	0
$951$	0	0
$952$	0	0
$953$	16.6649i	0.539828i	0.962884	+	0.269914i	$0.0869953\pi$
$953$	16.6649i	0.539828i	−0.962884	+	0.269914i	$0.913005\pi$
$954$	0	0
$955$	11.0766i	0.358429i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	30.6631	0.989131
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−1.59616	−0.0513822
$966$	0	0
$967$	23.3459	0.750753	0.375376	−	0.926872i	$-0.377513\pi$
$967$	23.3459	0.750753	0.375376	+	0.926872i	$0.377513\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	15.3301	0.491967	0.245984	−	0.969274i	$-0.420889\pi$
$971$	15.3301	0.491967	0.245984	+	0.969274i	$0.420889\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	19.7463i	0.631741i	0.948802	+	0.315870i	$0.102296\pi$
$977$	19.7463i	0.631741i	−0.948802	+	0.315870i	$0.897704\pi$
$978$	0	0
$979$	28.0894i	0.897742i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	22.8388	0.728444	0.364222	−	0.931312i	$-0.381335\pi$
$983$	22.8388	0.728444	0.364222	+	0.931312i	$0.381335\pi$
$984$	0	0
$985$	15.6388i	0.498294i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	22.9584i	0.730034i
$990$	0	0
$991$	7.84249	0.249125	0.124563	−	0.992212i	$-0.460247\pi$
$991$	7.84249	0.249125	0.124563	+	0.992212i	$0.460247\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 4.82776i	− 0.153050i
$996$	0	0
$997$	0.702889i	0.0222607i	0.999938	+	0.0111304i	$0.00354298\pi$
$997$	0.702889i	0.0222607i	−0.999938	+	0.0111304i	$0.996457\pi$
$998$	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.2645.5		12
3.2	odd	2	inner	5292.2.f.e.2645.8		12
7.4	even	3		756.2.t.e.593.4	yes	12
7.5	odd	6		756.2.t.e.269.3	✓	12
7.6	odd	2	inner	5292.2.f.e.2645.7		12
21.5	even	6		756.2.t.e.269.4	yes	12
21.11	odd	6		756.2.t.e.593.3	yes	12
21.20	even	2	inner	5292.2.f.e.2645.6		12
63.4	even	3		2268.2.bm.i.593.3		12
63.5	even	6		2268.2.w.i.269.4		12
63.11	odd	6		2268.2.w.i.1349.3		12
63.25	even	3		2268.2.w.i.1349.4		12
63.32	odd	6		2268.2.bm.i.593.4		12
63.40	odd	6		2268.2.w.i.269.3		12
63.47	even	6		2268.2.bm.i.1025.3		12
63.61	odd	6		2268.2.bm.i.1025.4		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
756.2.t.e.269.3	✓	12	7.5	odd	6
756.2.t.e.269.4	yes	12	21.5	even	6
756.2.t.e.593.3	yes	12	21.11	odd	6
756.2.t.e.593.4	yes	12	7.4	even	3
2268.2.w.i.269.3		12	63.40	odd	6
2268.2.w.i.269.4		12	63.5	even	6
2268.2.w.i.1349.3		12	63.11	odd	6
2268.2.w.i.1349.4		12	63.25	even	3
2268.2.bm.i.593.3		12	63.4	even	3
2268.2.bm.i.593.4		12	63.32	odd	6
2268.2.bm.i.1025.3		12	63.47	even	6
2268.2.bm.i.1025.4		12	63.61	odd	6
5292.2.f.e.2645.5		12	1.1	even	1	trivial
5292.2.f.e.2645.6		12	21.20	even	2	inner
5292.2.f.e.2645.7		12	7.6	odd	2	inner
5292.2.f.e.2645.8		12	3.2	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)

\(f(q)\)	\(=\)	\(q-0.770835 q^{5} -5.40581i q^{11} -5.28088i q^{13} +5.70866 q^{17} -0.618162i q^{19} +6.74094i q^{23} -4.40581 q^{25} -8.07606i q^{29} +0.580455i q^{31} +9.07606 q^{37} -8.59231 q^{41} +3.40581 q^{43} -0.770835 q^{47} -7.66487i q^{53} +4.16699i q^{55} -13.7885 q^{59} +4.12928i q^{61} +4.07069i q^{65} +13.5526 q^{67} -2.25906i q^{71} +2.26549i q^{73} -8.07069 q^{79} -3.85418 q^{83} -4.40044 q^{85} -5.19615 q^{89} +0.476501i q^{95} +14.0259i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 48 q^{37} - 12 q^{43} - 48 q^{79} - 12 q^{85}+O(q^{100}) \) 12 * q + 48 * q^37 - 12 * q^43 - 48 * q^79 - 12 * q^85

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