LMFDB - Embedded newform 1152.3.h.e.449.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1152,3,Mod(449,1152)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1152, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1152.449");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1152 = 2^{7} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1152.h (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$31.3897264543$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.40960000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 7x^{4} + 1 $ x^8 + 7*x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{18} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.7
Root			$-1.14412 + 1.14412i$ of defining polynomial
Character	$\chi$	$=$	1152.449
Dual form			1152.3.h.e.449.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.24264 q^{5} +12.6491 q^{7} +O(q^{10})$	$q+4.24264 q^{5} +12.6491 q^{7} -17.8885 q^{11} -10.0000i q^{13} -24.0416i q^{17} -25.2982i q^{19} -17.8885i q^{23} -7.00000 q^{25} -15.5563 q^{29} -12.6491 q^{31} +53.6656 q^{35} -64.0000i q^{37} -12.7279i q^{41} +50.5964i q^{43} +17.8885i q^{47} +111.000 q^{49} -18.3848 q^{53} -75.8947 q^{55} -42.4264i q^{65} +75.8947i q^{67} +125.220i q^{71} +96.0000 q^{73} -226.274 q^{77} +63.2456 q^{79} -125.220 q^{83} -102.000i q^{85} -55.1543i q^{89} -126.491i q^{91} -107.331i q^{95} +64.0000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q - 56 q^{25} + 888 q^{49} + 768 q^{73} + 512 q^{97}+O(q^{100}) $ 8 * q - 56 * q^25 + 888 * q^49 + 768 * q^73 + 512 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1152\mathbb{Z}\right)^\times$.

$n$	$127$	$641$	$901$
$\chi(n)$	$1$	$-1$	$-1$

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
$5$	4.24264	0.848528	0.424264	−	0.905539i	$-0.360533\pi$
$5$	4.24264	0.848528	0.424264	+	0.905539i	$0.360533\pi$
$6$	0	0
$7$	12.6491	1.80702	0.903508	−	0.428571i	$-0.140983\pi$
$7$	12.6491	1.80702	0.903508	+	0.428571i	$0.140983\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−17.8885	−1.62623	−0.813116	−	0.582102i	$-0.802230\pi$
$11$	−17.8885	−1.62623	−0.813116	+	0.582102i	$0.802230\pi$
$12$	0	0
$13$	− 10.0000i	− 0.769231i	−0.923077	−	0.384615i	$-0.874334\pi$
$13$	− 10.0000i	− 0.769231i	0.923077	−	0.384615i	$-0.125666\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 24.0416i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$17$	− 24.0416i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$18$	0	0
$19$	− 25.2982i	− 1.33149i	−0.746181	−	0.665743i	$-0.768115\pi$
$19$	− 25.2982i	− 1.33149i	0.746181	−	0.665743i	$-0.231885\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 17.8885i	− 0.777763i	−0.921288	−	0.388881i	$-0.872862\pi$
$23$	− 17.8885i	− 0.777763i	0.921288	−	0.388881i	$-0.127138\pi$
$24$	0	0
$25$	−7.00000	−0.280000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−15.5563	−0.536426	−0.268213	−	0.963360i	$-0.586433\pi$
$29$	−15.5563	−0.536426	−0.268213	+	0.963360i	$0.586433\pi$
$30$	0	0
$31$	−12.6491	−0.408036	−0.204018	−	0.978967i	$-0.565400\pi$
$31$	−12.6491	−0.408036	−0.204018	+	0.978967i	$0.565400\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	53.6656	1.53330
$36$	0	0
$37$	− 64.0000i	− 1.72973i	−0.502005	−	0.864865i	$-0.667404\pi$
$37$	− 64.0000i	− 1.72973i	0.502005	−	0.864865i	$-0.332596\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 12.7279i	− 0.310437i	−0.987880	−	0.155219i	$-0.950392\pi$
$41$	− 12.7279i	− 0.310437i	0.987880	−	0.155219i	$-0.0496082\pi$
$42$	0	0
$43$	50.5964i	1.17666i	0.808620	+	0.588331i	$0.200215\pi$
$43$	50.5964i	1.17666i	−0.808620	+	0.588331i	$0.799785\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	17.8885i	0.380607i	0.981725	+	0.190304i	$0.0609473\pi$
$47$	17.8885i	0.380607i	−0.981725	+	0.190304i	$0.939053\pi$
$48$	0	0
$49$	111.000	2.26531
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−18.3848	−0.346883	−0.173441	−	0.984844i	$-0.555489\pi$
$53$	−18.3848	−0.346883	−0.173441	+	0.984844i	$0.555489\pi$
$54$	0	0
$55$	−75.8947	−1.37990
$56$	0	0
$57$	0	0
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 42.4264i	− 0.652714i
$66$	0	0
$67$	75.8947i	1.13276i	0.824146	+	0.566378i	$0.191656\pi$
$67$	75.8947i	1.13276i	−0.824146	+	0.566378i	$0.808344\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	125.220i	1.76366i	0.471568	+	0.881830i	$0.343688\pi$
$71$	125.220i	1.76366i	−0.471568	+	0.881830i	$0.656312\pi$
$72$	0	0
$73$	96.0000	1.31507	0.657534	−	0.753425i	$-0.271599\pi$
$73$	96.0000	1.31507	0.657534	+	0.753425i	$0.271599\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−226.274	−2.93863
$78$	0	0
$79$	63.2456	0.800577	0.400288	−	0.916389i	$-0.368910\pi$
$79$	63.2456	0.800577	0.400288	+	0.916389i	$0.368910\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−125.220	−1.50867	−0.754336	−	0.656488i	$-0.772041\pi$
$83$	−125.220	−1.50867	−0.754336	+	0.656488i	$0.772041\pi$
$84$	0	0
$85$	− 102.000i	− 1.20000i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 55.1543i	− 0.619712i	−0.950784	−	0.309856i	$-0.899719\pi$
$89$	− 55.1543i	− 0.619712i	0.950784	−	0.309856i	$-0.100281\pi$
$90$	0	0
$91$	− 126.491i	− 1.39001i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 107.331i	− 1.12980i
$96$	0	0
$97$	64.0000	0.659794	0.329897	−	0.944017i	$-0.392986\pi$
$97$	64.0000	0.659794	0.329897	+	0.944017i	$0.392986\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	41.0122	0.406061	0.203031	−	0.979172i	$-0.434921\pi$
$101$	41.0122	0.406061	0.203031	+	0.979172i	$0.434921\pi$
$102$	0	0
$103$	139.140	1.35088	0.675438	−	0.737417i	$-0.263955\pi$
$103$	139.140	1.35088	0.675438	+	0.737417i	$0.263955\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	143.108	1.33746	0.668731	−	0.743505i	$-0.266838\pi$
$107$	143.108	1.33746	0.668731	+	0.743505i	$0.266838\pi$
$108$	0	0
$109$	− 118.000i	− 1.08257i	−0.840840	−	0.541284i	$-0.817938\pi$
$109$	− 118.000i	− 1.08257i	0.840840	−	0.541284i	$-0.182062\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 24.0416i	− 0.212758i	−0.994326	−	0.106379i	$-0.966074\pi$
$113$	− 24.0416i	− 0.212758i	0.994326	−	0.106379i	$-0.0339256\pi$
$114$	0	0
$115$	− 75.8947i	− 0.659954i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 304.105i	− 2.55551i
$120$	0	0
$121$	199.000	1.64463
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−135.765	−1.08612
$126$	0	0
$127$	−63.2456	−0.497996	−0.248998	−	0.968504i	$-0.580101\pi$
$127$	−63.2456	−0.497996	−0.248998	+	0.968504i	$0.580101\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−107.331	−0.819323	−0.409661	−	0.912238i	$-0.634353\pi$
$131$	−107.331	−0.819323	−0.409661	+	0.912238i	$0.634353\pi$
$132$	0	0
$133$	− 320.000i	− 2.40602i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 123.037i	− 0.898077i	−0.893512	−	0.449039i	$-0.851767\pi$
$137$	− 123.037i	− 0.898077i	0.893512	−	0.449039i	$-0.148233\pi$
$138$	0	0
$139$	− 126.491i	− 0.910008i	−0.890489	−	0.455004i	$-0.849638\pi$
$139$	− 126.491i	− 0.910008i	0.890489	−	0.455004i	$-0.150362\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	178.885i	1.25095i
$144$	0	0
$145$	−66.0000	−0.455172
$146$	0	0
$147$	0	0
$148$	0	0
$149$	199.404	1.33828	0.669141	−	0.743135i	$-0.266662\pi$
$149$	199.404	1.33828	0.669141	+	0.743135i	$0.266662\pi$
$150$	0	0
$151$	−37.9473	−0.251307	−0.125653	−	0.992074i	$-0.540103\pi$
$151$	−37.9473	−0.251307	−0.125653	+	0.992074i	$0.540103\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−53.6656	−0.346230
$156$	0	0
$157$	256.000i	1.63057i	0.579058	+	0.815287i	$0.303421\pi$
$157$	256.000i	1.63057i	−0.579058	+	0.815287i	$0.696579\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 226.274i	− 1.40543i
$162$	0	0
$163$	101.193i	0.620815i	0.950604	+	0.310408i	$0.100466\pi$
$163$	101.193i	0.620815i	−0.950604	+	0.310408i	$0.899534\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	214.663i	1.28540i	0.766116	+	0.642702i	$0.222187\pi$
$167$	214.663i	1.28540i	−0.766116	+	0.642702i	$0.777813\pi$
$168$	0	0
$169$	69.0000	0.408284
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−173.948	−1.00548	−0.502741	−	0.864437i	$-0.667675\pi$
$173$	−173.948	−1.00548	−0.502741	+	0.864437i	$0.667675\pi$
$174$	0	0
$175$	−88.5438	−0.505964
$176$	0	0
$177$	0	0
$178$	0	0
$179$	250.440	1.39910	0.699552	−	0.714582i	$-0.253383\pi$
$179$	250.440	1.39910	0.699552	+	0.714582i	$0.253383\pi$
$180$	0	0
$181$	− 218.000i	− 1.20442i	−0.798338	−	0.602210i	$-0.794287\pi$
$181$	− 218.000i	− 1.20442i	0.798338	−	0.602210i	$-0.205713\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 271.529i	− 1.46772i
$186$	0	0
$187$	430.070i	2.29984i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	71.5542i	0.374629i	0.982300	+	0.187315i	$0.0599784\pi$
$191$	71.5542i	0.374629i	−0.982300	+	0.187315i	$0.940022\pi$
$192$	0	0
$193$	−30.0000	−0.155440	−0.0777202	−	0.996975i	$-0.524764\pi$
$193$	−30.0000	−0.155440	−0.0777202	+	0.996975i	$0.524764\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	131.522	0.667624	0.333812	−	0.942640i	$-0.391665\pi$
$197$	131.522	0.667624	0.333812	+	0.942640i	$0.391665\pi$
$198$	0	0
$199$	−12.6491	−0.0635634	−0.0317817	−	0.999495i	$-0.510118\pi$
$199$	−12.6491	−0.0635634	−0.0317817	+	0.999495i	$0.510118\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−196.774	−0.969330
$204$	0	0
$205$	− 54.0000i	− 0.263415i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	452.548i	2.16530i
$210$	0	0
$211$	126.491i	0.599484i	0.954020	+	0.299742i	$0.0969006\pi$
$211$	126.491i	0.599484i	−0.954020	+	0.299742i	$0.903099\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	214.663i	0.998430i
$216$	0	0
$217$	−160.000	−0.737327
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−240.416	−1.08786
$222$	0	0
$223$	−12.6491	−0.0567225	−0.0283612	−	0.999598i	$-0.509029\pi$
$223$	−12.6491	−0.0567225	−0.0283612	+	0.999598i	$0.509029\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	196.774	0.866846	0.433423	−	0.901191i	$-0.357306\pi$
$227$	196.774	0.866846	0.433423	+	0.901191i	$0.357306\pi$
$228$	0	0
$229$	58.0000i	0.253275i	0.991949	+	0.126638i	$0.0404185\pi$
$229$	58.0000i	0.253275i	−0.991949	+	0.126638i	$0.959581\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	349.311i	1.49919i	0.661898	+	0.749594i	$0.269751\pi$
$233$	349.311i	1.49919i	−0.661898	+	0.749594i	$0.730249\pi$
$234$	0	0
$235$	75.8947i	0.322956i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 321.994i	− 1.34725i	−0.739071	−	0.673627i	$-0.764735\pi$
$239$	− 321.994i	− 1.34725i	0.739071	−	0.673627i	$-0.235265\pi$
$240$	0	0
$241$	160.000	0.663900	0.331950	−	0.943297i	$-0.392293\pi$
$241$	160.000	0.663900	0.331950	+	0.943297i	$0.392293\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	470.933	1.92218
$246$	0	0
$247$	−252.982	−1.02422
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−53.6656	−0.213807	−0.106904	−	0.994269i	$-0.534094\pi$
$251$	−53.6656	−0.213807	−0.106904	+	0.994269i	$0.534094\pi$
$252$	0	0
$253$	320.000i	1.26482i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 179.605i	− 0.698853i	−0.936964	−	0.349426i	$-0.886377\pi$
$257$	− 179.605i	− 0.698853i	0.936964	−	0.349426i	$-0.113623\pi$
$258$	0	0
$259$	− 809.543i	− 3.12565i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 250.440i	− 0.952242i	−0.879380	−	0.476121i	$-0.842042\pi$
$263$	− 250.440i	− 0.952242i	0.879380	−	0.476121i	$-0.157958\pi$
$264$	0	0
$265$	−78.0000	−0.294340
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−369.110	−1.37216	−0.686078	−	0.727528i	$-0.740669\pi$
$269$	−369.110	−1.37216	−0.686078	+	0.727528i	$0.740669\pi$
$270$	0	0
$271$	341.526	1.26024	0.630122	−	0.776496i	$-0.283005\pi$
$271$	341.526	1.26024	0.630122	+	0.776496i	$0.283005\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	125.220	0.455345
$276$	0	0
$277$	230.000i	0.830325i	0.909747	+	0.415162i	$0.136275\pi$
$277$	230.000i	0.830325i	−0.909747	+	0.415162i	$0.863725\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	352.139i	1.25316i	0.779355	+	0.626582i	$0.215547\pi$
$281$	352.139i	1.25316i	−0.779355	+	0.626582i	$0.784453\pi$
$282$	0	0
$283$	− 404.772i	− 1.43029i	−0.698977	−	0.715144i	$-0.746361\pi$
$283$	− 404.772i	− 1.43029i	0.698977	−	0.715144i	$-0.253639\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 160.997i	− 0.560965i
$288$	0	0
$289$	−289.000	−1.00000
$290$	0	0
$291$	0	0
$292$	0	0
$293$	4.24264	0.0144800	0.00724000	−	0.999974i	$-0.497695\pi$
$293$	4.24264	0.0144800	0.00724000	+	0.999974i	$0.497695\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−178.885	−0.598279
$300$	0	0
$301$	640.000i	2.12625i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	379.473i	1.23607i	0.786151	+	0.618035i	$0.212071\pi$
$307$	379.473i	1.23607i	−0.786151	+	0.618035i	$0.787929\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 35.7771i	− 0.115039i	−0.998344	−	0.0575194i	$-0.981681\pi$
$311$	− 35.7771i	− 0.115039i	0.998344	−	0.0575194i	$-0.0183191\pi$
$312$	0	0
$313$	−430.000	−1.37380	−0.686901	−	0.726751i	$-0.741029\pi$
$313$	−430.000	−1.37380	−0.686901	+	0.726751i	$0.741029\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	287.085	0.905632	0.452816	−	0.891604i	$-0.350419\pi$
$317$	287.085	0.905632	0.452816	+	0.891604i	$0.350419\pi$
$318$	0	0
$319$	278.280	0.872352
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−608.210	−1.88300
$324$	0	0
$325$	70.0000i	0.215385i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	226.274i	0.687763i
$330$	0	0
$331$	− 25.2982i	− 0.0764297i	−0.999270	−	0.0382148i	$-0.987833\pi$
$331$	− 25.2982i	− 0.0764297i	0.999270	−	0.0382148i	$-0.0121671\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	321.994i	0.961175i
$336$	0	0
$337$	224.000	0.664688	0.332344	−	0.943158i	$-0.392160\pi$
$337$	224.000	0.664688	0.332344	+	0.943158i	$0.392160\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	226.274	0.663561
$342$	0	0
$343$	784.245	2.28643
$344$	0	0
$345$	0	0
$346$	0	0
$347$	375.659	1.08259	0.541296	−	0.840832i	$-0.317934\pi$
$347$	375.659	1.08259	0.541296	+	0.840832i	$0.317934\pi$
$348$	0	0
$349$	320.000i	0.916905i	0.888719	+	0.458453i	$0.151596\pi$
$349$	320.000i	0.916905i	−0.888719	+	0.458453i	$0.848404\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	179.605i	0.508796i	0.967100	+	0.254398i	$0.0818774\pi$
$353$	179.605i	0.508796i	−0.967100	+	0.254398i	$0.918123\pi$
$354$	0	0
$355$	531.263i	1.49651i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 482.991i	− 1.34538i	−0.739925	−	0.672689i	$-0.765139\pi$
$359$	− 482.991i	− 1.34538i	0.739925	−	0.672689i	$-0.234861\pi$
$360$	0	0
$361$	−279.000	−0.772853
$362$	0	0
$363$	0	0
$364$	0	0
$365$	407.294	1.11587
$366$	0	0
$367$	−720.999	−1.96458	−0.982288	−	0.187378i	$-0.940001\pi$
$367$	−720.999	−1.96458	−0.982288	+	0.187378i	$0.940001\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−232.551	−0.626822
$372$	0	0
$373$	− 256.000i	− 0.686327i	−0.939276	−	0.343164i	$-0.888501\pi$
$373$	− 256.000i	− 0.686327i	0.939276	−	0.343164i	$-0.111499\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	155.563i	0.412635i
$378$	0	0
$379$	− 25.2982i	− 0.0667499i	−0.999443	−	0.0333750i	$-0.989374\pi$
$379$	− 25.2982i	− 0.0667499i	0.999443	−	0.0333750i	$-0.0106256\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	679.765i	1.77484i	0.460959	+	0.887421i	$0.347505\pi$
$383$	679.765i	1.77484i	−0.460959	+	0.887421i	$0.652495\pi$
$384$	0	0
$385$	−960.000	−2.49351
$386$	0	0
$387$	0	0
$388$	0	0
$389$	765.090	1.96681	0.983406	−	0.181421i	$-0.0580696\pi$
$389$	765.090	1.96681	0.983406	+	0.181421i	$0.0580696\pi$
$390$	0	0
$391$	−430.070	−1.09992
$392$	0	0
$393$	0	0
$394$	0	0
$395$	268.328	0.679312
$396$	0	0
$397$	− 64.0000i	− 0.161209i	−0.996746	−	0.0806045i	$-0.974315\pi$
$397$	− 64.0000i	− 0.161209i	0.996746	−	0.0806045i	$-0.0256851\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	337.997i	0.842885i	0.906855	+	0.421443i	$0.138476\pi$
$401$	337.997i	0.842885i	−0.906855	+	0.421443i	$0.861524\pi$
$402$	0	0
$403$	126.491i	0.313874i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1144.87i	2.81294i
$408$	0	0
$409$	−640.000	−1.56479	−0.782396	−	0.622781i	$-0.786003\pi$
$409$	−640.000	−1.56479	−0.782396	+	0.622781i	$0.786003\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−531.263	−1.28015
$416$	0	0
$417$	0	0
$418$	0	0
$419$	232.551	0.555014	0.277507	−	0.960724i	$-0.410492\pi$
$419$	232.551	0.555014	0.277507	+	0.960724i	$0.410492\pi$
$420$	0	0
$421$	518.000i	1.23040i	0.788370	+	0.615202i	$0.210926\pi$
$421$	518.000i	1.23040i	−0.788370	+	0.615202i	$0.789074\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	168.291i	0.395980i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 697.653i	− 1.61868i	−0.587337	−	0.809342i	$-0.699824\pi$
$431$	− 697.653i	− 1.61868i	0.587337	−	0.809342i	$-0.300176\pi$
$432$	0	0
$433$	290.000	0.669746	0.334873	−	0.942263i	$-0.391307\pi$
$433$	290.000	0.669746	0.334873	+	0.942263i	$0.391307\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−452.548	−1.03558
$438$	0	0
$439$	−37.9473	−0.0864404	−0.0432202	−	0.999066i	$-0.513762\pi$
$439$	−37.9473	−0.0864404	−0.0432202	+	0.999066i	$0.513762\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	697.653	1.57484	0.787419	−	0.616418i	$-0.211417\pi$
$443$	697.653	1.57484	0.787419	+	0.616418i	$0.211417\pi$
$444$	0	0
$445$	− 234.000i	− 0.525843i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 722.663i	− 1.60949i	−0.593617	−	0.804747i	$-0.702301\pi$
$449$	− 722.663i	− 1.60949i	0.593617	−	0.804747i	$-0.297699\pi$
$450$	0	0
$451$	227.684i	0.504843i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	− 536.656i	− 1.17946i
$456$	0	0
$457$	64.0000	0.140044	0.0700219	−	0.997545i	$-0.477693\pi$
$457$	64.0000	0.140044	0.0700219	+	0.997545i	$0.477693\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	581.242	1.26083	0.630414	−	0.776259i	$-0.282885\pi$
$461$	581.242	1.26083	0.630414	+	0.776259i	$0.282885\pi$
$462$	0	0
$463$	63.2456	0.136599	0.0682997	−	0.997665i	$-0.478243\pi$
$463$	63.2456	0.136599	0.0682997	+	0.997665i	$0.478243\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−160.997	−0.344747	−0.172374	−	0.985032i	$-0.555144\pi$
$467$	−160.997	−0.344747	−0.172374	+	0.985032i	$0.555144\pi$
$468$	0	0
$469$	960.000i	2.04691i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 905.097i	− 1.91352i
$474$	0	0
$475$	177.088i	0.372816i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 304.105i	− 0.634875i	−0.948279	−	0.317438i	$-0.897178\pi$
$479$	− 304.105i	− 0.634875i	0.948279	−	0.317438i	$-0.102822\pi$
$480$	0	0
$481$	−640.000	−1.33056
$482$	0	0
$483$	0	0
$484$	0	0
$485$	271.529	0.559854
$486$	0	0
$487$	−771.596	−1.58439	−0.792193	−	0.610271i	$-0.791061\pi$
$487$	−771.596	−1.58439	−0.792193	+	0.610271i	$0.791061\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	107.331	0.218597	0.109299	−	0.994009i	$-0.465140\pi$
$491$	107.331	0.218597	0.109299	+	0.994009i	$0.465140\pi$
$492$	0	0
$493$	374.000i	0.758621i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	1583.92i	3.18696i
$498$	0	0
$499$	− 708.350i	− 1.41954i	−0.704434	−	0.709770i	$-0.748799\pi$
$499$	− 708.350i	− 1.41954i	0.704434	−	0.709770i	$-0.251201\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 482.991i	− 0.960220i	−0.877208	−	0.480110i	$-0.840597\pi$
$503$	− 482.991i	− 0.960220i	0.877208	−	0.480110i	$-0.159403\pi$
$504$	0	0
$505$	174.000	0.344554
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−241.831	−0.475109	−0.237555	−	0.971374i	$-0.576346\pi$
$509$	−241.831	−0.475109	−0.237555	+	0.971374i	$0.576346\pi$
$510$	0	0
$511$	1214.31	2.37635
$512$	0	0
$513$	0	0
$514$	0	0
$515$	590.322	1.14626
$516$	0	0
$517$	− 320.000i	− 0.618956i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	284.257i	0.545599i	0.962071	+	0.272799i	$0.0879495\pi$
$521$	284.257i	0.545599i	−0.962071	+	0.272799i	$0.912050\pi$
$522$	0	0
$523$	− 278.280i	− 0.532085i	−0.963961	−	0.266042i	$-0.914284\pi$
$523$	− 278.280i	− 0.532085i	0.963961	−	0.266042i	$-0.0857161\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	304.105i	0.577050i
$528$	0	0
$529$	209.000	0.395085
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−127.279	−0.238798
$534$	0	0
$535$	607.157	1.13487
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−1985.63	−3.68391
$540$	0	0
$541$	− 662.000i	− 1.22366i	−0.790989	−	0.611830i	$-0.790434\pi$
$541$	− 662.000i	− 1.22366i	0.790989	−	0.611830i	$-0.209566\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 500.632i	− 0.918590i
$546$	0	0
$547$	151.789i	0.277494i	0.990328	+	0.138747i	$0.0443075\pi$
$547$	151.789i	0.277494i	−0.990328	+	0.138747i	$0.955692\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	393.548i	0.714243i
$552$	0	0
$553$	800.000	1.44665
$554$	0	0
$555$	0	0
$556$	0	0
$557$	400.222	0.718532	0.359266	−	0.933235i	$-0.383027\pi$
$557$	400.222	0.718532	0.359266	+	0.933235i	$0.383027\pi$
$558$	0	0
$559$	505.964	0.905124
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−268.328	−0.476604	−0.238302	−	0.971191i	$-0.576591\pi$
$563$	−268.328	−0.476604	−0.238302	+	0.971191i	$0.576591\pi$
$564$	0	0
$565$	− 102.000i	− 0.180531i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	578.413i	1.01654i	0.861197	+	0.508272i	$0.169715\pi$
$569$	578.413i	1.01654i	−0.861197	+	0.508272i	$0.830285\pi$
$570$	0	0
$571$	50.5964i	0.0886102i	0.999018	+	0.0443051i	$0.0141074\pi$
$571$	50.5964i	0.0886102i	−0.999018	+	0.0443051i	$0.985893\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	125.220i	0.217774i
$576$	0	0
$577$	−30.0000	−0.0519931	−0.0259965	−	0.999662i	$-0.508276\pi$
$577$	−30.0000	−0.0519931	−0.0259965	+	0.999662i	$0.508276\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−1583.92	−2.72619
$582$	0	0
$583$	328.877	0.564111
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−286.217	−0.487592	−0.243796	−	0.969826i	$-0.578393\pi$
$587$	−286.217	−0.487592	−0.243796	+	0.969826i	$0.578393\pi$
$588$	0	0
$589$	320.000i	0.543294i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	250.316i	0.422118i	0.977473	+	0.211059i	$0.0676912\pi$
$593$	250.316i	0.422118i	−0.977473	+	0.211059i	$0.932309\pi$
$594$	0	0
$595$	− 1290.21i	− 2.16842i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	482.991i	0.806328i	0.915128	+	0.403164i	$0.132090\pi$
$599$	482.991i	0.806328i	−0.915128	+	0.403164i	$0.867910\pi$
$600$	0	0
$601$	658.000	1.09484	0.547421	−	0.836857i	$-0.315610\pi$
$601$	658.000	1.09484	0.547421	+	0.836857i	$0.315610\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	844.285	1.39551
$606$	0	0
$607$	−164.438	−0.270904	−0.135452	−	0.990784i	$-0.543249\pi$
$607$	−164.438	−0.270904	−0.135452	+	0.990784i	$0.543249\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	178.885	0.292775
$612$	0	0
$613$	576.000i	0.939641i	0.882762	+	0.469821i	$0.155681\pi$
$613$	576.000i	0.939641i	−0.882762	+	0.469821i	$0.844319\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	193.747i	0.314015i	0.987597	+	0.157008i	$0.0501847\pi$
$617$	193.747i	0.314015i	−0.987597	+	0.157008i	$0.949815\pi$
$618$	0	0
$619$	556.561i	0.899129i	0.893248	+	0.449565i	$0.148421\pi$
$619$	556.561i	0.899129i	−0.893248	+	0.449565i	$0.851579\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 697.653i	− 1.11983i
$624$	0	0
$625$	−401.000	−0.641600
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−1538.66	−2.44621
$630$	0	0
$631$	−518.614	−0.821891	−0.410946	−	0.911660i	$-0.634801\pi$
$631$	−518.614	−0.821891	−0.410946	+	0.911660i	$0.634801\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−268.328	−0.422564
$636$	0	0
$637$	− 1110.00i	− 1.74254i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 861.256i	− 1.34361i	−0.740727	−	0.671807i	$-0.765519\pi$
$641$	− 861.256i	− 1.34361i	0.740727	−	0.671807i	$-0.234481\pi$
$642$	0	0
$643$	303.579i	0.472129i	0.971737	+	0.236064i	$0.0758576\pi$
$643$	303.579i	0.472129i	−0.971737	+	0.236064i	$0.924142\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 268.328i	− 0.414727i	−0.978264	−	0.207363i	$-0.933512\pi$
$647$	− 268.328i	− 0.414727i	0.978264	−	0.207363i	$-0.0664882\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−490.732	−0.751504	−0.375752	−	0.926720i	$-0.622616\pi$
$653$	−490.732	−0.751504	−0.375752	+	0.926720i	$0.622616\pi$
$654$	0	0
$655$	−455.368	−0.695218
$656$	0	0
$657$	0	0
$658$	0	0
$659$	643.988	0.977219	0.488610	−	0.872502i	$-0.337504\pi$
$659$	643.988	0.977219	0.488610	+	0.872502i	$0.337504\pi$
$660$	0	0
$661$	− 640.000i	− 0.968230i	−0.875004	−	0.484115i	$-0.839142\pi$
$661$	− 640.000i	− 0.968230i	0.875004	−	0.484115i	$-0.160858\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 1357.65i	− 2.04157i
$666$	0	0
$667$	278.280i	0.417212i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	830.000	1.23328	0.616642	−	0.787244i	$-0.288493\pi$
$673$	830.000	1.23328	0.616642	+	0.787244i	$0.288493\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	230.517	0.340498	0.170249	−	0.985401i	$-0.445543\pi$
$677$	230.517	0.340498	0.170249	+	0.985401i	$0.445543\pi$
$678$	0	0
$679$	809.543	1.19226
$680$	0	0
$681$	0	0
$682$	0	0
$683$	1055.42	1.54528	0.772638	−	0.634846i	$-0.218937\pi$
$683$	1055.42	1.54528	0.772638	+	0.634846i	$0.218937\pi$
$684$	0	0
$685$	− 522.000i	− 0.762044i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	183.848i	0.266833i
$690$	0	0
$691$	809.543i	1.17155i	0.810473	+	0.585776i	$0.199210\pi$
$691$	809.543i	1.17155i	−0.810473	+	0.585776i	$0.800790\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 536.656i	− 0.772167i
$696$	0	0
$697$	−306.000	−0.439024
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−196.576	−0.280422	−0.140211	−	0.990122i	$-0.544778\pi$
$701$	−196.576	−0.280422	−0.140211	+	0.990122i	$0.544778\pi$
$702$	0	0
$703$	−1619.09	−2.30311
$704$	0	0
$705$	0	0
$706$	0	0
$707$	518.768	0.733759
$708$	0	0
$709$	698.000i	0.984485i	0.870458	+	0.492243i	$0.163823\pi$
$709$	698.000i	0.984485i	−0.870458	+	0.492243i	$0.836177\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	226.274i	0.317355i
$714$	0	0
$715$	758.947i	1.06146i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	1234.31i	1.71670i	0.513062	+	0.858352i	$0.328511\pi$
$719$	1234.31i	1.71670i	−0.513062	+	0.858352i	$0.671489\pi$
$720$	0	0
$721$	1760.00	2.44105
$722$	0	0
$723$	0	0
$724$	0	0
$725$	108.894	0.150199
$726$	0	0
$727$	−468.017	−0.643765	−0.321882	−	0.946780i	$-0.604316\pi$
$727$	−468.017	−0.643765	−0.321882	+	0.946780i	$0.604316\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	1216.42	1.66405
$732$	0	0
$733$	− 1130.00i	− 1.54161i	−0.637071	−	0.770805i	$-0.719854\pi$
$733$	− 1130.00i	− 1.54161i	0.637071	−	0.770805i	$-0.280146\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 1357.65i	− 1.84212i
$738$	0	0
$739$	− 50.5964i	− 0.0684661i	−0.999414	−	0.0342330i	$-0.989101\pi$
$739$	− 50.5964i	− 0.0684661i	0.999414	−	0.0342330i	$-0.0108988\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	143.108i	0.192609i	0.995352	+	0.0963044i	$0.0307022\pi$
$743$	143.108i	0.192609i	−0.995352	+	0.0963044i	$0.969298\pi$
$744$	0	0
$745$	846.000	1.13557
$746$	0	0
$747$	0	0
$748$	0	0
$749$	1810.19	2.41681
$750$	0	0
$751$	−771.596	−1.02742	−0.513712	−	0.857963i	$-0.671730\pi$
$751$	−771.596	−1.02742	−0.513712	+	0.857963i	$0.671730\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−160.997	−0.213241
$756$	0	0
$757$	90.0000i	0.118890i	0.998232	+	0.0594452i	$0.0189331\pi$
$757$	90.0000i	0.118890i	−0.998232	+	0.0594452i	$0.981067\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 1076.22i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$761$	− 1076.22i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$762$	0	0
$763$	− 1492.60i	− 1.95622i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−482.000	−0.626788	−0.313394	−	0.949623i	$-0.601466\pi$
$769$	−482.000	−0.626788	−0.313394	+	0.949623i	$0.601466\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−1262.89	−1.63376	−0.816878	−	0.576811i	$-0.804297\pi$
$773$	−1262.89	−1.63376	−0.816878	+	0.576811i	$0.804297\pi$
$774$	0	0
$775$	88.5438	0.114250
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−321.994	−0.413342
$780$	0	0
$781$	− 2240.00i	− 2.86812i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	1086.12i	1.38359i
$786$	0	0
$787$	531.263i	0.675048i	0.941317	+	0.337524i	$0.109589\pi$
$787$	531.263i	0.675048i	−0.941317	+	0.337524i	$0.890411\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 304.105i	− 0.384457i
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−603.869	−0.757678	−0.378839	−	0.925463i	$-0.623677\pi$
$797$	−603.869	−0.757678	−0.378839	+	0.925463i	$0.623677\pi$
$798$	0	0
$799$	430.070	0.538260
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−1717.30	−2.13861
$804$	0	0
$805$	− 960.000i	− 1.19255i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	168.291i	0.208024i	0.994576	+	0.104012i	$0.0331680\pi$
$809$	168.291i	0.208024i	−0.994576	+	0.104012i	$0.966832\pi$
$810$	0	0
$811$	− 632.456i	− 0.779847i	−0.920847	−	0.389923i	$-0.872502\pi$
$811$	− 632.456i	− 0.779847i	0.920847	−	0.389923i	$-0.127498\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	429.325i	0.526779i
$816$	0	0
$817$	1280.00	1.56671
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−479.418	−0.583944	−0.291972	−	0.956427i	$-0.594311\pi$
$821$	−479.418	−0.583944	−0.291972	+	0.956427i	$0.594311\pi$
$822$	0	0
$823$	−746.298	−0.906801	−0.453401	−	0.891307i	$-0.649789\pi$
$823$	−746.298	−0.906801	−0.453401	+	0.891307i	$0.649789\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	679.765	0.821965	0.410982	−	0.911643i	$-0.365186\pi$
$827$	679.765	0.821965	0.410982	+	0.911643i	$0.365186\pi$
$828$	0	0
$829$	− 598.000i	− 0.721351i	−0.932691	−	0.360676i	$-0.882546\pi$
$829$	− 598.000i	− 0.721351i	0.932691	−	0.360676i	$-0.117454\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 2668.62i	− 3.20363i
$834$	0	0
$835$	910.736i	1.09070i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	1234.31i	1.47117i	0.677434	+	0.735584i	$0.263092\pi$
$839$	1234.31i	1.47117i	−0.677434	+	0.735584i	$0.736908\pi$
$840$	0	0
$841$	−599.000	−0.712247
$842$	0	0
$843$	0	0
$844$	0	0
$845$	292.742	0.346440
$846$	0	0
$847$	2517.17	2.97187
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−1144.87	−1.34532
$852$	0	0
$853$	− 256.000i	− 0.300117i	−0.988677	−	0.150059i	$-0.952054\pi$
$853$	− 256.000i	− 0.300117i	0.988677	−	0.150059i	$-0.0479462\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	371.938i	0.434000i	0.976172	+	0.217000i	$0.0696272\pi$
$857$	371.938i	0.434000i	−0.976172	+	0.217000i	$0.930373\pi$
$858$	0	0
$859$	860.140i	1.00133i	0.865642	+	0.500663i	$0.166911\pi$
$859$	860.140i	1.00133i	−0.865642	+	0.500663i	$0.833089\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 572.433i	− 0.663306i	−0.943401	−	0.331653i	$-0.892394\pi$
$863$	− 572.433i	− 0.663306i	0.943401	−	0.331653i	$-0.107606\pi$
$864$	0	0
$865$	−738.000	−0.853179
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−1131.37	−1.30192
$870$	0	0
$871$	758.947	0.871351
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−1717.30	−1.96263
$876$	0	0
$877$	704.000i	0.802737i	0.915917	+	0.401368i	$0.131465\pi$
$877$	704.000i	0.802737i	−0.915917	+	0.401368i	$0.868535\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	69.2965i	0.0786566i	0.999226	+	0.0393283i	$0.0125218\pi$
$881$	69.2965i	0.0786566i	−0.999226	+	0.0393283i	$0.987478\pi$
$882$	0	0
$883$	556.561i	0.630307i	0.949041	+	0.315153i	$0.102056\pi$
$883$	556.561i	0.630307i	−0.949041	+	0.315153i	$0.897944\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 1609.97i	− 1.81507i	−0.419974	−	0.907536i	$-0.637961\pi$
$887$	− 1609.97i	− 1.81507i	0.419974	−	0.907536i	$-0.362039\pi$
$888$	0	0
$889$	−800.000	−0.899888
$890$	0	0
$891$	0	0
$892$	0	0
$893$	452.548	0.506773
$894$	0	0
$895$	1062.53	1.18718
$896$	0	0
$897$	0	0
$898$	0	0
$899$	196.774	0.218881
$900$	0	0
$901$	442.000i	0.490566i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 924.896i	− 1.02198i
$906$	0	0
$907$	− 1770.88i	− 1.95245i	−0.216751	−	0.976227i	$-0.569546\pi$
$907$	− 1770.88i	− 1.95245i	0.216751	−	0.976227i	$-0.430454\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 858.650i	− 0.942536i	−0.881990	−	0.471268i	$-0.843797\pi$
$911$	− 858.650i	− 0.942536i	0.881990	−	0.471268i	$-0.156203\pi$
$912$	0	0
$913$	2240.00	2.45345
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−1357.65	−1.48053
$918$	0	0
$919$	−37.9473	−0.0412920	−0.0206460	−	0.999787i	$-0.506572\pi$
$919$	−37.9473	−0.0412920	−0.0206460	+	0.999787i	$0.506572\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	1252.20	1.35666
$924$	0	0
$925$	448.000i	0.484324i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	948.937i	1.02146i	0.859741	+	0.510731i	$0.170625\pi$
$929$	948.937i	1.02146i	−0.859741	+	0.510731i	$0.829375\pi$
$930$	0	0
$931$	− 2808.10i	− 3.01622i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	1824.63i	1.95148i
$936$	0	0
$937$	1810.00	1.93170	0.965848	−	0.259108i	$-0.0834284\pi$
$937$	1810.00	1.93170	0.965848	+	0.259108i	$0.0834284\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−550.129	−0.584622	−0.292311	−	0.956323i	$-0.594424\pi$
$941$	−550.129	−0.584622	−0.292311	+	0.956323i	$0.594424\pi$
$942$	0	0
$943$	−227.684	−0.241446
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−214.663	−0.226676	−0.113338	−	0.993556i	$-0.536154\pi$
$947$	−214.663	−0.226676	−0.113338	+	0.993556i	$0.536154\pi$
$948$	0	0
$949$	− 960.000i	− 1.01159i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	80.6102i	0.0845857i	0.999105	+	0.0422929i	$0.0134662\pi$
$953$	80.6102i	0.0845857i	−0.999105	+	0.0422929i	$0.986534\pi$
$954$	0	0
$955$	303.579i	0.317883i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 1556.30i	− 1.62284i
$960$	0	0
$961$	−801.000	−0.833507
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−127.279	−0.131896
$966$	0	0
$967$	341.526	0.353181	0.176590	−	0.984284i	$-0.443493\pi$
$967$	341.526	0.353181	0.176590	+	0.984284i	$0.443493\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	1198.53	1.23433	0.617164	−	0.786835i	$-0.288282\pi$
$971$	1198.53	1.23433	0.617164	+	0.786835i	$0.288282\pi$
$972$	0	0
$973$	− 1600.00i	− 1.64440i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 428.507i	− 0.438594i	−0.975658	−	0.219297i	$-0.929624\pi$
$977$	− 428.507i	− 0.438594i	0.975658	−	0.219297i	$-0.0703764\pi$
$978$	0	0
$979$	986.631i	1.00779i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 1466.86i	− 1.49223i	−0.665818	−	0.746114i	$-0.731917\pi$
$983$	− 1466.86i	− 1.49223i	0.665818	−	0.746114i	$-0.268083\pi$
$984$	0	0
$985$	558.000	0.566497
$986$	0	0
$987$	0	0
$988$	0	0
$989$	905.097	0.915163
$990$	0	0
$991$	−973.982	−0.982827	−0.491413	−	0.870926i	$-0.663520\pi$
$991$	−973.982	−0.982827	−0.491413	+	0.870926i	$0.663520\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−53.6656	−0.0539353
$996$	0	0
$997$	896.000i	0.898696i	0.893357	+	0.449348i	$0.148344\pi$
$997$	896.000i	0.898696i	−0.893357	+	0.449348i	$0.851656\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1152.3.h.e.449.7	yes	8
3.2	odd	2	inner	1152.3.h.e.449.3	yes	8
4.3	odd	2	inner	1152.3.h.e.449.5	yes	8
8.3	odd	2	inner	1152.3.h.e.449.2	yes	8
8.5	even	2	inner	1152.3.h.e.449.4	yes	8
12.11	even	2	inner	1152.3.h.e.449.1	✓	8
16.3	odd	4		2304.3.e.k.1025.2		4
16.5	even	4		2304.3.e.f.1025.3		4
16.11	odd	4		2304.3.e.f.1025.4		4
16.13	even	4		2304.3.e.k.1025.1		4
24.5	odd	2	inner	1152.3.h.e.449.8	yes	8
24.11	even	2	inner	1152.3.h.e.449.6	yes	8
48.5	odd	4		2304.3.e.f.1025.1		4
48.11	even	4		2304.3.e.f.1025.2		4
48.29	odd	4		2304.3.e.k.1025.3		4
48.35	even	4		2304.3.e.k.1025.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1152.3.h.e.449.1	✓	8	12.11	even	2	inner
1152.3.h.e.449.2	yes	8	8.3	odd	2	inner
1152.3.h.e.449.3	yes	8	3.2	odd	2	inner
1152.3.h.e.449.4	yes	8	8.5	even	2	inner
1152.3.h.e.449.5	yes	8	4.3	odd	2	inner
1152.3.h.e.449.6	yes	8	24.11	even	2	inner
1152.3.h.e.449.7	yes	8	1.1	even	1	trivial
1152.3.h.e.449.8	yes	8	24.5	odd	2	inner
2304.3.e.f.1025.1		4	48.5	odd	4
2304.3.e.f.1025.2		4	48.11	even	4
2304.3.e.f.1025.3		4	16.5	even	4
2304.3.e.f.1025.4		4	16.11	odd	4
2304.3.e.k.1025.1		4	16.13	even	4
2304.3.e.k.1025.2		4	16.3	odd	4
2304.3.e.k.1025.3		4	48.29	odd	4
2304.3.e.k.1025.4		4	48.35	even	4

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:	\( N \)	\(=\)	\( 1152 = 2^{7} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1152.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+4.24264 q^{5} +12.6491 q^{7} +O(q^{10})\)	\(q+4.24264 q^{5} +12.6491 q^{7} -17.8885 q^{11} -10.0000i q^{13} -24.0416i q^{17} -25.2982i q^{19} -17.8885i q^{23} -7.00000 q^{25} -15.5563 q^{29} -12.6491 q^{31} +53.6656 q^{35} -64.0000i q^{37} -12.7279i q^{41} +50.5964i q^{43} +17.8885i q^{47} +111.000 q^{49} -18.3848 q^{53} -75.8947 q^{55} -42.4264i q^{65} +75.8947i q^{67} +125.220i q^{71} +96.0000 q^{73} -226.274 q^{77} +63.2456 q^{79} -125.220 q^{83} -102.000i q^{85} -55.1543i q^{89} -126.491i q^{91} -107.331i q^{95} +64.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q - 56 q^{25} + 888 q^{49} + 768 q^{73} + 512 q^{97}+O(q^{100}) \) 8 * q - 56 * q^25 + 888 * q^49 + 768 * q^73 + 512 * q^97

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