LMFDB - Embedded newform 7200.2.b.i.4751.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7200,2,Mod(4751,7200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7200, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("7200.4751");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7200 = 2^{5} \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7200.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$57.4922894553$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 6x^{14} + 28x^{12} + 16x^{10} - 40x^{8} + 610x^{6} + 1625x^{4} - 524x^{2} + 1444 $ x^16 - 6x^14 + 28x^12 + 16x^10 - 40x^8 + 610x^6 + 1625x^4 - 524*x^2 + 1444
Coefficient ring:	$\Z[a_1, \ldots, a_{53}]$
Coefficient ring index:	$ 2^{24} $
Twist minimal:	no (minimal twist has level 360)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			4751.7
Root			$-0.328458 - 1.49331i$ of defining polynomial
Character	$\chi$	$=$	7200.4751
Dual form			7200.2.b.i.4751.10

$q$-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 $\left(\mathbb{Z}/7200\mathbb{Z}\right)^\times$.

$n$	$577$	$901$	$6401$	$6751$
$\chi(n)$	$1$	$-1$	$-1$	$-1$

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
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 0.936426i	− 0.353936i	−0.984217	−	0.176968i	$-0.943371\pi$
$7$	− 0.936426i	− 0.353936i	0.984217	−	0.176968i	$-0.0566289\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	2.20837i	0.665848i	0.942954	+	0.332924i	$0.108035\pi$
$11$	2.20837i	0.665848i	−0.942954	+	0.332924i	$0.891965\pi$
$12$	0	0
$13$	− 3.33513i	− 0.924999i	−0.886619	−	0.462500i	$-0.846953\pi$
$13$	− 3.33513i	− 0.924999i	0.886619	−	0.462500i	$-0.153047\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 1.54417i	− 0.374517i	−0.982311	−	0.187259i	$-0.940040\pi$
$17$	− 1.54417i	− 0.374517i	0.982311	−	0.187259i	$-0.0599602\pi$
$18$	0	0
$19$	−3.12311	−0.716490	−0.358245	−	0.933628i	$-0.616625\pi$
$19$	−3.12311	−0.716490	−0.358245	+	0.933628i	$0.616625\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−3.39228	−0.707340	−0.353670	−	0.935370i	$-0.615066\pi$
$23$	−3.39228	−0.707340	−0.353670	+	0.935370i	$0.615066\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−8.44804	−1.56876	−0.784380	−	0.620280i	$-0.787019\pi$
$29$	−8.44804	−1.56876	−0.784380	+	0.620280i	$0.787019\pi$
$30$	0	0
$31$	8.30571i	1.49175i	0.666086	+	0.745875i	$0.267968\pi$
$31$	8.30571i	1.49175i	−0.666086	+	0.745875i	$0.732032\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	7.60669i	1.25053i	0.780412	+	0.625266i	$0.215010\pi$
$37$	7.60669i	1.25053i	−0.780412	+	0.625266i	$0.784990\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 5.83095i	− 0.910642i	−0.890327	−	0.455321i	$-0.849525\pi$
$41$	− 5.83095i	− 0.910642i	0.890327	−	0.455321i	$-0.150475\pi$
$42$	0	0
$43$	7.77769	1.18609	0.593043	−	0.805171i	$-0.297926\pi$
$43$	7.77769	1.18609	0.593043	+	0.805171i	$0.297926\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	10.7575	1.56914	0.784570	−	0.620040i	$-0.212884\pi$
$47$	10.7575	1.56914	0.784570	+	0.620040i	$0.212884\pi$
$48$	0	0
$49$	6.12311	0.874729
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−5.08842	−0.698949	−0.349474	−	0.936946i	$-0.613640\pi$
$53$	−5.08842	−0.698949	−0.349474	+	0.936946i	$0.613640\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	10.6937i	1.39219i	0.717947	+	0.696097i	$0.245082\pi$
$59$	10.6937i	1.39219i	−0.717947	+	0.696097i	$0.754918\pi$
$60$	0	0
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	12.1453	1.48378	0.741890	−	0.670521i	$-0.233929\pi$
$67$	12.1453	1.48378	0.741890	+	0.670521i	$0.233929\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	11.7460	1.39400	0.697000	−	0.717071i	$-0.254518\pi$
$71$	11.7460	1.39400	0.697000	+	0.717071i	$0.254518\pi$
$72$	0	0
$73$	5.59390	0.654716	0.327358	−	0.944900i	$-0.393842\pi$
$73$	5.59390	0.654716	0.327358	+	0.944900i	$0.393842\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	2.06798	0.235668
$78$	0	0
$79$	− 1.02248i	− 0.115038i	−0.998344	−	0.0575190i	$-0.981681\pi$
$79$	− 1.02248i	− 0.115038i	0.998344	−	0.0575190i	$-0.0183190\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	14.0877i	1.54632i	0.634210	+	0.773161i	$0.281325\pi$
$83$	14.0877i	1.54632i	−0.634210	+	0.773161i	$0.718675\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 13.0761i	− 1.38607i	−0.720906	−	0.693033i	$-0.756274\pi$
$89$	− 13.0761i	− 1.38607i	0.720906	−	0.693033i	$-0.243726\pi$
$90$	0	0
$91$	−3.12311	−0.327390
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−2.18379	−0.221730	−0.110865	−	0.993835i	$-0.535362\pi$
$97$	−2.18379	−0.221730	−0.110865	+	0.993835i	$0.535362\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	3.29801	0.328165	0.164082	−	0.986447i	$-0.447534\pi$
$101$	3.29801	0.328165	0.164082	+	0.986447i	$0.447534\pi$
$102$	0	0
$103$	− 18.0227i	− 1.77583i	−0.460012	−	0.887913i	$-0.652155\pi$
$103$	− 18.0227i	− 1.77583i	0.460012	−	0.887913i	$-0.347845\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	10.9993i	1.06334i	0.846950	+	0.531672i	$0.178436\pi$
$107$	10.9993i	1.06334i	−0.846950	+	0.531672i	$0.821564\pi$
$108$	0	0
$109$	− 11.9473i	− 1.14435i	−0.820133	−	0.572173i	$-0.806100\pi$
$109$	− 11.9473i	− 1.14435i	0.820133	−	0.572173i	$-0.193900\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	12.5435i	1.17999i	0.807406	+	0.589996i	$0.200871\pi$
$113$	12.5435i	1.17999i	−0.807406	+	0.589996i	$0.799129\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−1.44600	−0.132555
$120$	0	0
$121$	6.12311	0.556646
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 4.68213i	− 0.415472i	−0.978185	−	0.207736i	$-0.933391\pi$
$127$	− 4.68213i	− 0.415472i	0.978185	−	0.207736i	$-0.0666095\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	0.620058i	0.0541747i	0.999633	+	0.0270874i	$0.00862323\pi$
$131$	0.620058i	0.0541747i	−0.999633	+	0.0270874i	$0.991377\pi$
$132$	0	0
$133$	2.92456i	0.253591i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 1.54417i	− 0.131928i	−0.997822	−	0.0659638i	$-0.978988\pi$
$137$	− 1.54417i	− 0.131928i	0.997822	−	0.0659638i	$-0.0210122\pi$
$138$	0	0
$139$	−12.0000	−1.01783	−0.508913	−	0.860818i	$-0.669953\pi$
$139$	−12.0000	−1.01783	−0.508913	+	0.860818i	$0.669953\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	7.36520	0.615909
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−1.85201	−0.151722	−0.0758612	−	0.997118i	$-0.524171\pi$
$149$	−1.85201	−0.151722	−0.0758612	+	0.997118i	$0.524171\pi$
$150$	0	0
$151$	− 17.6339i	− 1.43503i	−0.696545	−	0.717513i	$-0.745280\pi$
$151$	− 17.6339i	− 1.43503i	0.696545	−	0.717513i	$-0.254720\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 2.80928i	− 0.224205i	−0.993697	−	0.112102i	$-0.964242\pi$
$157$	− 2.80928i	− 0.224205i	0.993697	−	0.112102i	$-0.0357585\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	3.17662i	0.250353i
$162$	0	0
$163$	15.5554	1.21839	0.609195	−	0.793020i	$-0.291492\pi$
$163$	15.5554	1.21839	0.609195	+	0.793020i	$0.291492\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−9.43318	−0.729961	−0.364981	−	0.931015i	$-0.618924\pi$
$167$	−9.43318	−0.729961	−0.364981	+	0.931015i	$0.618924\pi$
$168$	0	0
$169$	1.87689	0.144376
$170$	0	0
$171$	0	0
$172$	0	0
$173$	1.69614	0.128955	0.0644776	−	0.997919i	$-0.479462\pi$
$173$	1.69614	0.128955	0.0644776	+	0.997919i	$0.479462\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	8.21342i	0.613900i	0.951726	+	0.306950i	$0.0993084\pi$
$179$	8.21342i	0.613900i	−0.951726	+	0.306950i	$0.900692\pi$
$180$	0	0
$181$	4.66410i	0.346680i	0.984862	+	0.173340i	$0.0554559\pi$
$181$	4.66410i	0.346680i	−0.984862	+	0.173340i	$0.944544\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	3.41011	0.249372
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−13.1921	−0.954544	−0.477272	−	0.878756i	$-0.658374\pi$
$191$	−13.1921	−0.954544	−0.477272	+	0.878756i	$0.658374\pi$
$192$	0	0
$193$	23.3331	1.67955	0.839775	−	0.542934i	$-0.182687\pi$
$193$	23.3331	1.67955	0.839775	+	0.542934i	$0.182687\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	4.92539	0.350920	0.175460	−	0.984487i	$-0.443859\pi$
$197$	4.92539	0.350920	0.175460	+	0.984487i	$0.443859\pi$
$198$	0	0
$199$	17.6339i	1.25003i	0.780611	+	0.625017i	$0.214908\pi$
$199$	17.6339i	1.25003i	−0.780611	+	0.625017i	$0.785092\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	7.91096i	0.555241i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 6.89697i	− 0.477073i
$210$	0	0
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	7.77769	0.527984
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−5.15002	−0.346428
$222$	0	0
$223$	21.9989i	1.47316i	0.676352	+	0.736578i	$0.263560\pi$
$223$	21.9989i	1.47316i	−0.676352	+	0.736578i	$0.736440\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	3.08835i	0.204981i	0.994734	+	0.102490i	$0.0326811\pi$
$227$	3.08835i	0.204981i	−0.994734	+	0.102490i	$0.967319\pi$
$228$	0	0
$229$	− 2.61914i	− 0.173077i	−0.996248	−	0.0865387i	$-0.972419\pi$
$229$	− 2.61914i	− 0.173077i	0.996248	−	0.0865387i	$-0.0275806\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	20.4544i	1.34002i	0.742354	+	0.670008i	$0.233709\pi$
$233$	20.4544i	1.34002i	−0.742354	+	0.670008i	$0.766291\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	23.4921	1.51958	0.759789	−	0.650170i	$-0.225302\pi$
$239$	23.4921	1.51958	0.759789	+	0.650170i	$0.225302\pi$
$240$	0	0
$241$	−4.00000	−0.257663	−0.128831	−	0.991667i	$-0.541123\pi$
$241$	−4.00000	−0.257663	−0.128831	+	0.991667i	$0.541123\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	10.4160i	0.662752i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	3.44849i	0.217666i	0.994060	+	0.108833i	$0.0347114\pi$
$251$	3.44849i	0.217666i	−0.994060	+	0.108833i	$0.965289\pi$
$252$	0	0
$253$	− 7.49141i	− 0.470981i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	15.6318i	0.975087i	0.873099	+	0.487543i	$0.162107\pi$
$257$	15.6318i	0.975087i	−0.873099	+	0.487543i	$0.837893\pi$
$258$	0	0
$259$	7.12311	0.442608
$260$	0	0
$261$	0	0
$262$	0	0
$263$	14.7304	0.908316	0.454158	−	0.890921i	$-0.349940\pi$
$263$	14.7304	0.908316	0.454158	+	0.890921i	$0.349940\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	15.0441	0.917253	0.458626	−	0.888629i	$-0.348342\pi$
$269$	15.0441	0.917253	0.458626	+	0.888629i	$0.348342\pi$
$270$	0	0
$271$	10.3507i	0.628758i	0.949297	+	0.314379i	$0.101796\pi$
$271$	10.3507i	0.628758i	−0.949297	+	0.314379i	$0.898204\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 23.3459i	− 1.40272i	−0.712807	−	0.701360i	$-0.752576\pi$
$277$	− 23.3459i	− 1.40272i	0.712807	−	0.701360i	$-0.247424\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 3.00252i	− 0.179116i	−0.995982	−	0.0895578i	$-0.971455\pi$
$281$	− 3.00252i	− 0.179116i	0.995982	−	0.0895578i	$-0.0285454\pi$
$282$	0	0
$283$	23.3331	1.38701	0.693503	−	0.720453i	$-0.256066\pi$
$283$	23.3331	1.38701	0.693503	+	0.720453i	$0.256066\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−5.46026	−0.322309
$288$	0	0
$289$	14.6155	0.859737
$290$	0	0
$291$	0	0
$292$	0	0
$293$	29.8326	1.74284	0.871421	−	0.490536i	$-0.163199\pi$
$293$	29.8326	1.74284	0.871421	+	0.490536i	$0.163199\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	11.3137i	0.654289i
$300$	0	0
$301$	− 7.28323i	− 0.419799i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−4.36758	−0.249271	−0.124636	−	0.992203i	$-0.539776\pi$
$307$	−4.36758	−0.249271	−0.124636	+	0.992203i	$0.539776\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−10.3000	−0.584062	−0.292031	−	0.956409i	$-0.594331\pi$
$311$	−10.3000	−0.584062	−0.292031	+	0.956409i	$0.594331\pi$
$312$	0	0
$313$	32.0682	1.81260	0.906302	−	0.422631i	$-0.138893\pi$
$313$	32.0682	1.81260	0.906302	+	0.422631i	$0.138893\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−24.5354	−1.37805	−0.689023	−	0.724739i	$-0.741960\pi$
$317$	−24.5354	−1.37805	−0.689023	+	0.724739i	$0.741960\pi$
$318$	0	0
$319$	− 18.6564i	− 1.04456i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	4.82262i	0.268338i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 10.0736i	− 0.555375i
$330$	0	0
$331$	1.36932	0.0752645	0.0376322	−	0.999292i	$-0.488018\pi$
$331$	1.36932	0.0752645	0.0376322	+	0.999292i	$0.488018\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	29.8844	1.62791	0.813954	−	0.580929i	$-0.197310\pi$
$337$	29.8844	1.62791	0.813954	+	0.580929i	$0.197310\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−18.3421	−0.993279
$342$	0	0
$343$	− 12.2888i	− 0.663534i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 14.0877i	− 0.756265i	−0.925751	−	0.378133i	$-0.876566\pi$
$347$	− 14.0877i	− 0.756265i	0.925751	−	0.378133i	$-0.123434\pi$
$348$	0	0
$349$	33.2228i	1.77838i	0.457540	+	0.889189i	$0.348731\pi$
$349$	33.2228i	1.77838i	−0.457540	+	0.889189i	$0.651269\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 7.72087i	− 0.410940i	−0.978663	−	0.205470i	$-0.934128\pi$
$353$	− 7.72087i	− 0.410940i	0.978663	−	0.205470i	$-0.0658724\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−1.44600	−0.0763172	−0.0381586	−	0.999272i	$-0.512149\pi$
$359$	−1.44600	−0.0763172	−0.0381586	+	0.999272i	$0.512149\pi$
$360$	0	0
$361$	−9.24621	−0.486643
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 32.4149i	− 1.69204i	−0.533148	−	0.846022i	$-0.678991\pi$
$367$	− 32.4149i	− 1.69204i	0.533148	−	0.846022i	$-0.321009\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	4.76493i	0.247383i
$372$	0	0
$373$	− 9.47954i	− 0.490832i	−0.969418	−	0.245416i	$-0.921075\pi$
$373$	− 9.47954i	− 0.490832i	0.969418	−	0.245416i	$-0.0789246\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	28.1753i	1.45110i
$378$	0	0
$379$	−16.4924	−0.847159	−0.423579	−	0.905859i	$-0.639227\pi$
$379$	−16.4924	−0.847159	−0.423579	+	0.905859i	$0.639227\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−10.0138	−0.511682	−0.255841	−	0.966719i	$-0.582352\pi$
$383$	−10.0138	−0.511682	−0.255841	+	0.966719i	$0.582352\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−9.89404	−0.501648	−0.250824	−	0.968033i	$-0.580702\pi$
$389$	−9.89404	−0.501648	−0.250824	+	0.968033i	$0.580702\pi$
$390$	0	0
$391$	5.23827i	0.264911i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 10.3007i	− 0.516977i	−0.966014	−	0.258488i	$-0.916776\pi$
$397$	− 10.3007i	− 0.516977i	0.966014	−	0.258488i	$-0.0832244\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 0.522293i	− 0.0260821i	−0.999915	−	0.0130410i	$-0.995849\pi$
$401$	− 0.522293i	− 0.0260821i	0.999915	−	0.0130410i	$-0.00415121\pi$
$402$	0	0
$403$	27.7006	1.37987
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−16.7984	−0.832665
$408$	0	0
$409$	−3.12311	−0.154428	−0.0772138	−	0.997015i	$-0.524602\pi$
$409$	−3.12311	−0.154428	−0.0772138	+	0.997015i	$0.524602\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	10.0138	0.492748
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 32.4291i	− 1.58427i	−0.610348	−	0.792134i	$-0.708970\pi$
$419$	− 32.4291i	− 1.58427i	0.610348	−	0.792134i	$-0.291030\pi$
$420$	0	0
$421$	− 30.6037i	− 1.49153i	−0.666207	−	0.745767i	$-0.732083\pi$
$421$	− 30.6037i	− 1.49153i	0.666207	−	0.745767i	$-0.267917\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−1.44600	−0.0696516	−0.0348258	−	0.999393i	$-0.511088\pi$
$431$	−1.44600	−0.0696516	−0.0348258	+	0.999393i	$0.511088\pi$
$432$	0	0
$433$	15.5554	0.747544	0.373772	−	0.927521i	$-0.378064\pi$
$433$	15.5554	0.747544	0.373772	+	0.927521i	$0.378064\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	10.5945	0.506802
$438$	0	0
$439$	26.9621i	1.28683i	0.765517	+	0.643415i	$0.222483\pi$
$439$	26.9621i	1.28683i	−0.765517	+	0.643415i	$0.777517\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	25.0870i	1.19192i	0.803015	+	0.595959i	$0.203228\pi$
$443$	25.0870i	1.19192i	−0.803015	+	0.595959i	$0.796772\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	9.89949i	0.467186i	0.972334	+	0.233593i	$0.0750483\pi$
$449$	9.89949i	0.467186i	−0.972334	+	0.233593i	$0.924952\pi$
$450$	0	0
$451$	12.8769	0.606349
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	13.3716	0.625496	0.312748	−	0.949836i	$-0.398750\pi$
$457$	13.3716	0.625496	0.312748	+	0.949836i	$0.398750\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	1.85201	0.0862567	0.0431283	−	0.999070i	$-0.486268\pi$
$461$	1.85201	0.0862567	0.0431283	+	0.999070i	$0.486268\pi$
$462$	0	0
$463$	13.2252i	0.614629i	0.951608	+	0.307315i	$0.0994304\pi$
$463$	13.2252i	0.614629i	−0.951608	+	0.307315i	$0.900570\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	18.9103i	0.875063i	0.899203	+	0.437532i	$0.144147\pi$
$467$	18.9103i	0.875063i	−0.899203	+	0.437532i	$0.855853\pi$
$468$	0	0
$469$	− 11.3732i	− 0.525163i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	17.1760i	0.789754i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	14.6381	0.668830	0.334415	−	0.942426i	$-0.391461\pi$
$479$	14.6381	0.668830	0.334415	+	0.942426i	$0.391461\pi$
$480$	0	0
$481$	25.3693	1.15674
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	29.4903i	1.33633i	0.744011	+	0.668167i	$0.232921\pi$
$487$	29.4903i	1.33633i	−0.744011	+	0.668167i	$0.767079\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	16.3505i	0.737888i	0.929452	+	0.368944i	$0.120281\pi$
$491$	16.3505i	0.737888i	−0.929452	+	0.368944i	$0.879719\pi$
$492$	0	0
$493$	13.0452i	0.587528i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 10.9993i	− 0.493387i
$498$	0	0
$499$	16.8769	0.755514	0.377757	−	0.925905i	$-0.376696\pi$
$499$	16.8769	0.755514	0.377757	+	0.925905i	$0.376696\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	16.0547	0.715844	0.357922	−	0.933751i	$-0.383485\pi$
$503$	16.0547	0.715844	0.357922	+	0.933751i	$0.383485\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−3.29801	−0.146182	−0.0730909	−	0.997325i	$-0.523286\pi$
$509$	−3.29801	−0.146182	−0.0730909	+	0.997325i	$0.523286\pi$
$510$	0	0
$511$	− 5.23827i	− 0.231728i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	23.7565i	1.04481i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 2.65433i	− 0.116288i	−0.998308	−	0.0581441i	$-0.981482\pi$
$521$	− 2.65433i	− 0.116288i	0.998308	−	0.0581441i	$-0.0185183\pi$
$522$	0	0
$523$	−4.36758	−0.190981	−0.0954905	−	0.995430i	$-0.530442\pi$
$523$	−4.36758	−0.190981	−0.0954905	+	0.995430i	$0.530442\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	12.8255	0.558686
$528$	0	0
$529$	−11.4924	−0.499671
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−19.4470	−0.842343
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	13.5221i	0.582437i
$540$	0	0
$541$	9.90237i	0.425736i	0.977081	+	0.212868i	$0.0682804\pi$
$541$	9.90237i	0.425736i	−0.977081	+	0.212868i	$0.931720\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−16.5129	−0.706039	−0.353019	−	0.935616i	$-0.614845\pi$
$547$	−16.5129	−0.706039	−0.353019	+	0.935616i	$0.614845\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	26.3841	1.12400
$552$	0	0
$553$	−0.957477	−0.0407161
$554$	0	0
$555$	0	0
$556$	0	0
$557$	32.6443	1.38318	0.691591	−	0.722289i	$-0.256910\pi$
$557$	32.6443	1.38318	0.691591	+	0.722289i	$0.256910\pi$
$558$	0	0
$559$	− 25.9396i	− 1.09713i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 14.0877i	− 0.593724i	−0.954920	−	0.296862i	$-0.904060\pi$
$563$	− 14.0877i	− 0.593724i	0.954920	−	0.296862i	$-0.0959401\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	30.0467i	1.25962i	0.776748	+	0.629811i	$0.216868\pi$
$569$	30.0467i	1.25962i	−0.776748	+	0.629811i	$0.783132\pi$
$570$	0	0
$571$	−33.8617	−1.41707	−0.708535	−	0.705676i	$-0.750644\pi$
$571$	−33.8617	−1.41707	−0.708535	+	0.705676i	$0.750644\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−7.77769	−0.323789	−0.161895	−	0.986808i	$-0.551760\pi$
$577$	−7.77769	−0.323789	−0.161895	+	0.986808i	$0.551760\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	13.1921	0.547299
$582$	0	0
$583$	− 11.2371i	− 0.465394i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 10.9993i	− 0.453990i	−0.973896	−	0.226995i	$-0.927110\pi$
$587$	− 10.9993i	− 0.453990i	0.973896	−	0.226995i	$-0.0728901\pi$
$588$	0	0
$589$	− 25.9396i	− 1.06882i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	43.8071i	1.79894i	0.436978	+	0.899472i	$0.356049\pi$
$593$	43.8071i	1.79894i	−0.436978	+	0.899472i	$0.643951\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−45.5382	−1.86064	−0.930320	−	0.366749i	$-0.880471\pi$
$599$	−45.5382	−1.86064	−0.930320	+	0.366749i	$0.880471\pi$
$600$	0	0
$601$	−23.3693	−0.953254	−0.476627	−	0.879106i	$-0.658141\pi$
$601$	−23.3693	−0.953254	−0.476627	+	0.879106i	$0.658141\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	9.71010i	0.394121i	0.980391	+	0.197060i	$0.0631395\pi$
$607$	9.71010i	0.394121i	−0.980391	+	0.197060i	$0.936860\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 35.8776i	− 1.45145i
$612$	0	0
$613$	− 23.8718i	− 0.964172i	−0.876124	−	0.482086i	$-0.839879\pi$
$613$	− 23.8718i	− 0.964172i	0.876124	−	0.482086i	$-0.160121\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 15.6318i	− 0.629314i	−0.949205	−	0.314657i	$-0.898111\pi$
$617$	− 15.6318i	− 0.629314i	0.949205	−	0.314657i	$-0.101889\pi$
$618$	0	0
$619$	18.7386	0.753169	0.376585	−	0.926382i	$-0.377098\pi$
$619$	18.7386	0.753169	0.376585	+	0.926382i	$0.377098\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−12.2448	−0.490578
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	11.7460	0.468346
$630$	0	0
$631$	− 17.6339i	− 0.701995i	−0.936376	−	0.350997i	$-0.885843\pi$
$631$	− 17.6339i	− 0.701995i	0.936376	−	0.350997i	$-0.114157\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 20.4214i	− 0.809124i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 7.07107i	− 0.279290i	−0.990202	−	0.139645i	$-0.955404\pi$
$641$	− 7.07107i	− 0.279290i	0.990202	−	0.139645i	$-0.0445962\pi$
$642$	0	0
$643$	−8.73516	−0.344481	−0.172241	−	0.985055i	$-0.555101\pi$
$643$	−8.73516	−0.344481	−0.172241	+	0.985055i	$0.555101\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−1.32431	−0.0520639	−0.0260319	−	0.999661i	$-0.508287\pi$
$647$	−1.32431	−0.0520639	−0.0260319	+	0.999661i	$0.508287\pi$
$648$	0	0
$649$	−23.6155	−0.926991
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−15.8459	−0.620098	−0.310049	−	0.950721i	$-0.600346\pi$
$653$	−15.8459	−0.620098	−0.310049	+	0.950721i	$0.600346\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 33.3211i	− 1.29800i	−0.760786	−	0.649002i	$-0.775187\pi$
$659$	− 33.3211i	− 1.29800i	0.760786	−	0.649002i	$-0.224813\pi$
$660$	0	0
$661$	44.5960i	1.73458i	0.497800	+	0.867292i	$0.334141\pi$
$661$	44.5960i	1.73458i	−0.497800	+	0.867292i	$0.665859\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	28.6581	1.10965
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	0	0		−	1.00000i	$-0.5\pi$
$673$	0	0			1.00000i	$0.5\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	5.25145	0.201830	0.100915	−	0.994895i	$-0.467823\pi$
$677$	5.25145	0.201830	0.100915	+	0.994895i	$0.467823\pi$
$678$	0	0
$679$	2.04496i	0.0784783i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	20.2644i	0.775394i	0.921787	+	0.387697i	$0.126729\pi$
$683$	20.2644i	0.775394i	−0.921787	+	0.387697i	$0.873271\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	16.9706i	0.646527i
$690$	0	0
$691$	23.1231	0.879644	0.439822	−	0.898085i	$-0.355041\pi$
$691$	23.1231	0.879644	0.439822	+	0.898085i	$0.355041\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−9.00400	−0.341051
$698$	0	0
$699$	0	0
$700$	0	0
$701$	20.1941	0.762720	0.381360	−	0.924427i	$-0.375456\pi$
$701$	20.1941	0.762720	0.381360	+	0.924427i	$0.375456\pi$
$702$	0	0
$703$	− 23.7565i	− 0.895993i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 3.08835i	− 0.116149i
$708$	0	0
$709$	30.6037i	1.14935i	0.818383	+	0.574673i	$0.194871\pi$
$709$	30.6037i	1.14935i	−0.818383	+	0.574673i	$0.805129\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 28.1753i	− 1.05517i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−32.3461	−1.20631	−0.603154	−	0.797625i	$-0.706089\pi$
$719$	−32.3461	−1.20631	−0.603154	+	0.797625i	$0.706089\pi$
$720$	0	0
$721$	−16.8769	−0.628528
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	9.47954i	0.351577i	0.984428	+	0.175788i	$0.0562475\pi$
$727$	9.47954i	0.351577i	−0.984428	+	0.175788i	$0.943753\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 12.0101i	− 0.444210i
$732$	0	0
$733$	− 19.6002i	− 0.723951i	−0.932188	−	0.361975i	$-0.882102\pi$
$733$	− 19.6002i	− 0.723951i	0.932188	−	0.361975i	$-0.117898\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	26.8212i	0.987973i
$738$	0	0
$739$	35.6155	1.31014	0.655069	−	0.755569i	$-0.272639\pi$
$739$	35.6155	1.31014	0.655069	+	0.755569i	$0.272639\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−10.0138	−0.367371	−0.183686	−	0.982985i	$-0.558803\pi$
$743$	−10.0138	−0.367371	−0.183686	+	0.982985i	$0.558803\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	10.3000	0.376355
$750$	0	0
$751$	41.5286i	1.51540i	0.652604	+	0.757699i	$0.273677\pi$
$751$	41.5286i	1.51540i	−0.652604	+	0.757699i	$0.726323\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	3.33513i	0.121217i	0.998162	+	0.0606087i	$0.0193042\pi$
$757$	3.33513i	0.121217i	−0.998162	+	0.0606087i	$0.980696\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	49.3019i	1.78719i	0.448870	+	0.893597i	$0.351827\pi$
$761$	49.3019i	1.78719i	−0.448870	+	0.893597i	$0.648173\pi$
$762$	0	0
$763$	−11.1878	−0.405025
$764$	0	0
$765$	0	0
$766$	0	0
$767$	35.6647	1.28778
$768$	0	0
$769$	−25.6155	−0.923720	−0.461860	−	0.886953i	$-0.652818\pi$
$769$	−25.6155	−0.923720	−0.461860	+	0.886953i	$0.652818\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	2.27678	0.0818901	0.0409450	−	0.999161i	$-0.486963\pi$
$773$	2.27678	0.0818901	0.0409450	+	0.999161i	$0.486963\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	18.2107i	0.652465i
$780$	0	0
$781$	25.9396i	0.928192i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	19.9230	0.710177	0.355088	−	0.934833i	$-0.384451\pi$
$787$	19.9230	0.710177	0.355088	+	0.934833i	$0.384451\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	11.7460	0.417641
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−15.6829	−0.555516	−0.277758	−	0.960651i	$-0.589591\pi$
$797$	−15.6829	−0.555516	−0.277758	+	0.960651i	$0.589591\pi$
$798$	0	0
$799$	− 16.6114i	− 0.587670i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	12.3534i	0.435942i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	20.6695i	0.726700i	0.931653	+	0.363350i	$0.118367\pi$
$809$	20.6695i	0.726700i	−0.931653	+	0.363350i	$0.881633\pi$
$810$	0	0
$811$	49.4773	1.73738	0.868691	−	0.495354i	$-0.164962\pi$
$811$	49.4773	1.73738	0.868691	+	0.495354i	$0.164962\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−24.2905	−0.849818
$818$	0	0
$819$	0	0
$820$	0	0
$821$	12.1521	0.424110	0.212055	−	0.977258i	$-0.431984\pi$
$821$	12.1521	0.424110	0.212055	+	0.977258i	$0.431984\pi$
$822$	0	0
$823$	− 28.6692i	− 0.999345i	−0.866214	−	0.499673i	$-0.833454\pi$
$823$	− 28.6692i	− 0.999345i	0.866214	−	0.499673i	$-0.166546\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	39.1746i	1.36224i	0.732174	+	0.681118i	$0.238506\pi$
$827$	39.1746i	1.36224i	−0.732174	+	0.681118i	$0.761494\pi$
$828$	0	0
$829$	− 4.66410i	− 0.161991i	−0.996714	−	0.0809954i	$-0.974190\pi$
$829$	− 4.66410i	− 0.161991i	0.996714	−	0.0809954i	$-0.0258099\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 9.45514i	− 0.327601i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−24.9381	−0.860959	−0.430479	−	0.902600i	$-0.641656\pi$
$839$	−24.9381	−0.860959	−0.430479	+	0.902600i	$0.641656\pi$
$840$	0	0
$841$	42.3693	1.46101
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 5.73384i	− 0.197017i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 25.8040i	− 0.884551i
$852$	0	0
$853$	45.5249i	1.55874i	0.626563	+	0.779371i	$0.284461\pi$
$853$	45.5249i	1.55874i	−0.626563	+	0.779371i	$0.715539\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 40.7188i	− 1.39093i	−0.718561	−	0.695464i	$-0.755199\pi$
$857$	− 40.7188i	− 1.39093i	0.718561	−	0.695464i	$-0.244801\pi$
$858$	0	0
$859$	−33.7538	−1.15166	−0.575832	−	0.817568i	$-0.695322\pi$
$859$	−33.7538	−1.15166	−0.575832	+	0.817568i	$0.695322\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−45.6786	−1.55492	−0.777458	−	0.628935i	$-0.783491\pi$
$863$	−45.6786	−1.55492	−0.777458	+	0.628935i	$0.783491\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	2.25801	0.0765978
$870$	0	0
$871$	− 40.5061i	− 1.37250i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	46.5766i	1.57278i	0.617731	+	0.786389i	$0.288052\pi$
$877$	46.5766i	1.57278i	−0.617731	+	0.786389i	$0.711948\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	5.83095i	0.196450i	0.995164	+	0.0982249i	$0.0313164\pi$
$881$	5.83095i	0.196450i	−0.995164	+	0.0982249i	$0.968684\pi$
$882$	0	0
$883$	0	0		−	1.00000i	$-0.5\pi$
$883$	0	0			1.00000i	$0.5\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	16.2177	0.544538	0.272269	−	0.962221i	$-0.412226\pi$
$887$	16.2177	0.544538	0.272269	+	0.962221i	$0.412226\pi$
$888$	0	0
$889$	−4.38447	−0.147050
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−33.5968	−1.12427
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 70.1670i	− 2.34020i
$900$	0	0
$901$	7.85741i	0.261768i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	16.5129	0.548300	0.274150	−	0.961687i	$-0.411603\pi$
$907$	16.5129	0.548300	0.274150	+	0.961687i	$0.411603\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−36.6842	−1.21540	−0.607700	−	0.794167i	$-0.707908\pi$
$911$	−36.6842	−1.21540	−0.607700	+	0.794167i	$0.707908\pi$
$912$	0	0
$913$	−31.1107	−1.02962
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0.580639	0.0191744
$918$	0	0
$919$	− 30.1554i	− 0.994735i	−0.867540	−	0.497368i	$-0.834300\pi$
$919$	− 30.1554i	− 0.994735i	0.867540	−	0.497368i	$-0.165700\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 39.1746i	− 1.28945i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 3.00252i	− 0.0985096i	−0.998786	−	0.0492548i	$-0.984315\pi$
$929$	− 3.00252i	− 0.0985096i	0.998786	−	0.0492548i	$-0.0156846\pi$
$930$	0	0
$931$	−19.1231	−0.626734
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	0	0		−	1.00000i	$-0.5\pi$
$937$	0	0			1.00000i	$0.5\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−3.29801	−0.107512	−0.0537561	−	0.998554i	$-0.517119\pi$
$941$	−3.29801	−0.107512	−0.0537561	+	0.998554i	$0.517119\pi$
$942$	0	0
$943$	19.7802i	0.644133i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 32.9979i	− 1.07229i	−0.844126	−	0.536144i	$-0.819880\pi$
$947$	− 32.9979i	− 1.07229i	0.844126	−	0.536144i	$-0.180120\pi$
$948$	0	0
$949$	− 18.6564i	− 0.605612i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 12.5435i	− 0.406323i	−0.979145	−	0.203162i	$-0.934878\pi$
$953$	− 12.5435i	− 0.406323i	0.979145	−	0.203162i	$-0.0651217\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−1.44600	−0.0466939
$960$	0	0
$961$	−37.9848	−1.22532
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 15.0981i	− 0.485522i	−0.970086	−	0.242761i	$-0.921947\pi$
$967$	− 15.0981i	− 0.485522i	0.970086	−	0.242761i	$-0.0780531\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	19.8753i	0.637829i	0.947783	+	0.318915i	$0.103318\pi$
$971$	19.8753i	0.637829i	−0.947783	+	0.318915i	$0.896682\pi$
$972$	0	0
$973$	11.2371i	0.360245i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 37.6305i	− 1.20390i	−0.798532	−	0.601952i	$-0.794390\pi$
$977$	− 37.6305i	− 1.20390i	0.798532	−	0.601952i	$-0.205610\pi$
$978$	0	0
$979$	28.8769	0.922910
$980$	0	0
$981$	0	0
$982$	0	0
$983$	53.0438	1.69183	0.845917	−	0.533315i	$-0.179054\pi$
$983$	53.0438	1.69183	0.845917	+	0.533315i	$0.179054\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−26.3841	−0.838966
$990$	0	0
$991$	17.6339i	0.560159i	0.959977	+	0.280080i	$0.0903609\pi$
$991$	17.6339i	0.560159i	−0.959977	+	0.280080i	$0.909639\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	25.5141i	0.808039i	0.914750	+	0.404019i	$0.132387\pi$
$997$	25.5141i	0.808039i	−0.914750	+	0.404019i	$0.867613\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7200.2.b.i.4751.7		16
3.2	odd	2	inner	7200.2.b.i.4751.6		16
4.3	odd	2		1800.2.b.g.251.10		16
5.2	odd	4		1440.2.m.c.719.12		16
5.3	odd	4		1440.2.m.c.719.9		16
5.4	even	2	inner	7200.2.b.i.4751.12		16
8.3	odd	2	inner	7200.2.b.i.4751.11		16
8.5	even	2		1800.2.b.g.251.5		16
12.11	even	2		1800.2.b.g.251.8		16
15.2	even	4		1440.2.m.c.719.6		16
15.8	even	4		1440.2.m.c.719.7		16
15.14	odd	2	inner	7200.2.b.i.4751.9		16
20.3	even	4		360.2.m.c.179.2	yes	16
20.7	even	4		360.2.m.c.179.16	yes	16
20.19	odd	2		1800.2.b.g.251.7		16
24.5	odd	2		1800.2.b.g.251.11		16
24.11	even	2	inner	7200.2.b.i.4751.10		16
40.3	even	4		1440.2.m.c.719.8		16
40.13	odd	4		360.2.m.c.179.3	yes	16
40.19	odd	2	inner	7200.2.b.i.4751.8		16
40.27	even	4		1440.2.m.c.719.5		16
40.29	even	2		1800.2.b.g.251.12		16
40.37	odd	4		360.2.m.c.179.13	yes	16
60.23	odd	4		360.2.m.c.179.15	yes	16
60.47	odd	4		360.2.m.c.179.1	✓	16
60.59	even	2		1800.2.b.g.251.9		16
120.29	odd	2		1800.2.b.g.251.6		16
120.53	even	4		360.2.m.c.179.14	yes	16
120.59	even	2	inner	7200.2.b.i.4751.5		16
120.77	even	4		360.2.m.c.179.4	yes	16
120.83	odd	4		1440.2.m.c.719.10		16
120.107	odd	4		1440.2.m.c.719.11		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
360.2.m.c.179.1	✓	16	60.47	odd	4
360.2.m.c.179.2	yes	16	20.3	even	4
360.2.m.c.179.3	yes	16	40.13	odd	4
360.2.m.c.179.4	yes	16	120.77	even	4
360.2.m.c.179.13	yes	16	40.37	odd	4
360.2.m.c.179.14	yes	16	120.53	even	4
360.2.m.c.179.15	yes	16	60.23	odd	4
360.2.m.c.179.16	yes	16	20.7	even	4
1440.2.m.c.719.5		16	40.27	even	4
1440.2.m.c.719.6		16	15.2	even	4
1440.2.m.c.719.7		16	15.8	even	4
1440.2.m.c.719.8		16	40.3	even	4
1440.2.m.c.719.9		16	5.3	odd	4
1440.2.m.c.719.10		16	120.83	odd	4
1440.2.m.c.719.11		16	120.107	odd	4
1440.2.m.c.719.12		16	5.2	odd	4
1800.2.b.g.251.5		16	8.5	even	2
1800.2.b.g.251.6		16	120.29	odd	2
1800.2.b.g.251.7		16	20.19	odd	2
1800.2.b.g.251.8		16	12.11	even	2
1800.2.b.g.251.9		16	60.59	even	2
1800.2.b.g.251.10		16	4.3	odd	2
1800.2.b.g.251.11		16	24.5	odd	2
1800.2.b.g.251.12		16	40.29	even	2
7200.2.b.i.4751.5		16	120.59	even	2	inner
7200.2.b.i.4751.6		16	3.2	odd	2	inner
7200.2.b.i.4751.7		16	1.1	even	1	trivial
7200.2.b.i.4751.8		16	40.19	odd	2	inner
7200.2.b.i.4751.9		16	15.14	odd	2	inner
7200.2.b.i.4751.10		16	24.11	even	2	inner
7200.2.b.i.4751.11		16	8.3	odd	2	inner
7200.2.b.i.4751.12		16	5.4	even	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 7200 = 2^{5} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7200.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-0.936426i q^{7} +2.20837i q^{11} -3.33513i q^{13} -1.54417i q^{17} -3.12311 q^{19} -3.39228 q^{23} -8.44804 q^{29} +8.30571i q^{31} +7.60669i q^{37} -5.83095i q^{41} +7.77769 q^{43} +10.7575 q^{47} +6.12311 q^{49} -5.08842 q^{53} +10.6937i q^{59} +12.1453 q^{67} +11.7460 q^{71} +5.59390 q^{73} +2.06798 q^{77} -1.02248i q^{79} +14.0877i q^{83} -13.0761i q^{89} -3.12311 q^{91} -2.18379 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 16 q^{19} + 32 q^{49} + 16 q^{91}+O(q^{100}) \) 16 * q + 16 * q^19 + 32 * q^49 + 16 * q^91

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