LMFDB - Embedded newform 1296.3.g.d.1135.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1296,3,Mod(1135,1296)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1296.1135"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1296, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 0])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 1296 = 2^{4} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1296.g (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,-6,0,0,0,0,0,0,0,10,0,0,0,-30] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	no
Analytic conductor:	$35.3134422611$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-7}, \sqrt{-15})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 11x^{2} + 4 $ x^4 + 11*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 144)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1135.1
Root			$-3.25937i$ of defining polynomial
Character	$\chi$	$=$	1296.1135
Dual form			1296.3.g.d.1135.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-6.62348 q^{5} -9.77810i q^{7} -11.6190i q^{11} +17.8704 q^{13} -2.37652 q^{17} +14.0337i q^{19} -41.5271i q^{23} +18.8704 q^{25} -11.3765 q^{29} +20.8232i q^{31} +64.7650i q^{35} +35.7409 q^{37} -5.25915 q^{41} -62.9242i q^{43} -4.94868i q^{47} -46.6113 q^{49} -75.7409 q^{53} +76.9578i q^{55} +58.0948i q^{59} -11.8704 q^{61} -118.364 q^{65} +23.8118i q^{67} +46.4758i q^{71} -104.352 q^{73} -113.611 q^{77} -20.8232i q^{79} +28.1866i q^{83} +15.7409 q^{85} -73.0122 q^{89} -174.739i q^{91} -92.9516i q^{95} -142.223 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 6 q^{5} + 10 q^{13} - 30 q^{17} + 14 q^{25} - 66 q^{29} + 20 q^{37} - 144 q^{41} - 2 q^{49} - 180 q^{53} + 14 q^{61} - 330 q^{65} - 110 q^{73} - 270 q^{77} - 60 q^{85} - 456 q^{89} - 200 q^{97}+O(q^{100}) $ 4 * q - 6 * q^5 + 10 * q^13 - 30 * q^17 + 14 * q^25 - 66 * q^29 + 20 * q^37 - 144 * q^41 - 2 * q^49 - 180 * q^53 + 14 * q^61 - 330 * q^65 - 110 * q^73 - 270 * q^77 - 60 * q^85 - 456 * q^89 - 200 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1296\mathbb{Z}\right)^\times$.

$n$	$325$	$1135$	$1217$
$\chi(n)$	$1$	$-1$	$1$

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
$5$	−6.62348	−1.32470	−0.662348	−	0.749197i	$-0.730440\pi$
$5$	−6.62348	−1.32470	−0.662348	+	0.749197i	$0.730440\pi$
$6$	0	0
$7$	− 9.77810i	− 1.39687i	−0.715673	−	0.698436i	$-0.753880\pi$
$7$	− 9.77810i	− 1.39687i	0.715673	−	0.698436i	$-0.246120\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 11.6190i	− 1.05627i	−0.849161	−	0.528134i	$-0.822892\pi$
$11$	− 11.6190i	− 1.05627i	0.849161	−	0.528134i	$-0.177108\pi$
$12$	0	0
$13$	17.8704	1.37465	0.687324	−	0.726351i	$-0.258785\pi$
$13$	17.8704	1.37465	0.687324	+	0.726351i	$0.258785\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−2.37652	−0.139796	−0.0698978	−	0.997554i	$-0.522267\pi$
$17$	−2.37652	−0.139796	−0.0698978	+	0.997554i	$0.522267\pi$
$18$	0	0
$19$	14.0337i	0.738614i	0.929308	+	0.369307i	$0.120405\pi$
$19$	14.0337i	0.738614i	−0.929308	+	0.369307i	$0.879595\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 41.5271i	− 1.80553i	−0.430137	−	0.902763i	$-0.641535\pi$
$23$	− 41.5271i	− 1.80553i	0.430137	−	0.902763i	$-0.358465\pi$
$24$	0	0
$25$	18.8704	0.754817
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−11.3765	−0.392294	−0.196147	−	0.980575i	$-0.562843\pi$
$29$	−11.3765	−0.392294	−0.196147	+	0.980575i	$0.562843\pi$
$30$	0	0
$31$	20.8232i	0.671716i	0.941913	+	0.335858i	$0.109026\pi$
$31$	20.8232i	0.671716i	−0.941913	+	0.335858i	$0.890974\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	64.7650i	1.85043i
$36$	0	0
$37$	35.7409	0.965969	0.482984	−	0.875629i	$-0.339553\pi$
$37$	35.7409	0.965969	0.482984	+	0.875629i	$0.339553\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−5.25915	−0.128272	−0.0641359	−	0.997941i	$-0.520429\pi$
$41$	−5.25915	−0.128272	−0.0641359	+	0.997941i	$0.520429\pi$
$42$	0	0
$43$	− 62.9242i	− 1.46335i	−0.681652	−	0.731676i	$-0.738738\pi$
$43$	− 62.9242i	− 1.46335i	0.681652	−	0.731676i	$-0.261262\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 4.94868i	− 0.105291i	−0.998613	−	0.0526456i	$-0.983235\pi$
$47$	− 4.94868i	− 0.105291i	0.998613	−	0.0526456i	$-0.0167654\pi$
$48$	0	0
$49$	−46.6113	−0.951251
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−75.7409	−1.42907	−0.714536	−	0.699598i	$-0.753362\pi$
$53$	−75.7409	−1.42907	−0.714536	+	0.699598i	$0.753362\pi$
$54$	0	0
$55$	76.9578i	1.39923i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	58.0948i	0.984657i	0.870410	+	0.492328i	$0.163854\pi$
$59$	58.0948i	0.984657i	−0.870410	+	0.492328i	$0.836146\pi$
$60$	0	0
$61$	−11.8704	−0.194597	−0.0972986	−	0.995255i	$-0.531020\pi$
$61$	−11.8704	−0.194597	−0.0972986	+	0.995255i	$0.531020\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−118.364	−1.82099
$66$	0	0
$67$	23.8118i	0.355399i	0.984085	+	0.177700i	$0.0568656\pi$
$67$	23.8118i	0.355399i	−0.984085	+	0.177700i	$0.943134\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	46.4758i	0.654589i	0.944922	+	0.327294i	$0.106137\pi$
$71$	46.4758i	0.654589i	−0.944922	+	0.327294i	$0.893863\pi$
$72$	0	0
$73$	−104.352	−1.42948	−0.714741	−	0.699390i	$-0.753455\pi$
$73$	−104.352	−1.42948	−0.714741	+	0.699390i	$0.753455\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−113.611	−1.47547
$78$	0	0
$79$	− 20.8232i	− 0.263585i	−0.991277	−	0.131792i	$-0.957927\pi$
$79$	− 20.8232i	− 0.263585i	0.991277	−	0.131792i	$-0.0420732\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	28.1866i	0.339597i	0.985479	+	0.169799i	$0.0543117\pi$
$83$	28.1866i	0.339597i	−0.985479	+	0.169799i	$0.945688\pi$
$84$	0	0
$85$	15.7409	0.185186
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−73.0122	−0.820362	−0.410181	−	0.912004i	$-0.634534\pi$
$89$	−73.0122	−0.820362	−0.410181	+	0.912004i	$0.634534\pi$
$90$	0	0
$91$	− 174.739i	− 1.92021i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 92.9516i	− 0.978438i
$96$	0	0
$97$	−142.223	−1.46621	−0.733106	−	0.680114i	$-0.761930\pi$
$97$	−142.223	−1.46621	−0.733106	+	0.680114i	$0.761930\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	27.3521	0.270813	0.135407	−	0.990790i	$-0.456766\pi$
$101$	27.3521	0.270813	0.135407	+	0.990790i	$0.456766\pi$
$102$	0	0
$103$	− 146.672i	− 1.42400i	−0.702182	−	0.711998i	$-0.747791\pi$
$103$	− 146.672i	− 1.42400i	0.702182	−	0.711998i	$-0.252209\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 192.573i	− 1.79975i	−0.436146	−	0.899876i	$-0.643657\pi$
$107$	− 192.573i	− 1.79975i	0.436146	−	0.899876i	$-0.356343\pi$
$108$	0	0
$109$	0.259148	0.00237750	0.00118875	−	0.999999i	$-0.499622\pi$
$109$	0.259148	0.00237750	0.00118875	+	0.999999i	$0.499622\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−198.858	−1.75981	−0.879904	−	0.475152i	$-0.842393\pi$
$113$	−198.858	−1.75981	−0.879904	+	0.475152i	$0.842393\pi$
$114$	0	0
$115$	275.054i	2.39177i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	23.2379i	0.195276i
$120$	0	0
$121$	−14.0000	−0.115702
$122$	0	0
$123$	0	0
$124$	0	0
$125$	40.5991	0.324793
$126$	0	0
$127$	72.2477i	0.568879i	0.958694	+	0.284440i	$0.0918075\pi$
$127$	72.2477i	0.568879i	−0.958694	+	0.284440i	$0.908192\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	184.398i	1.40762i	0.710390	+	0.703808i	$0.248518\pi$
$131$	184.398i	1.40762i	−0.710390	+	0.703808i	$0.751482\pi$
$132$	0	0
$133$	137.223	1.03175
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−139.753	−1.02010	−0.510048	−	0.860146i	$-0.670372\pi$
$137$	−139.753	−1.02010	−0.510048	+	0.860146i	$0.670372\pi$
$138$	0	0
$139$	175.193i	1.26038i	0.776439	+	0.630192i	$0.217024\pi$
$139$	175.193i	1.26038i	−0.776439	+	0.630192i	$0.782976\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 207.636i	− 1.45200i
$144$	0	0
$145$	75.3521	0.519670
$146$	0	0
$147$	0	0
$148$	0	0
$149$	191.069	1.28234	0.641170	−	0.767399i	$-0.278449\pi$
$149$	191.069	1.28234	0.641170	+	0.767399i	$0.278449\pi$
$150$	0	0
$151$	7.24413i	0.0479744i	0.999712	+	0.0239872i	$0.00763609\pi$
$151$	7.24413i	0.0479744i	−0.999712	+	0.0239872i	$0.992364\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 137.922i	− 0.889819i
$156$	0	0
$157$	72.1296	0.459424	0.229712	−	0.973259i	$-0.426222\pi$
$157$	72.1296	0.459424	0.229712	+	0.973259i	$0.426222\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−406.056	−2.52209
$162$	0	0
$163$	− 112.269i	− 0.688769i	−0.938829	−	0.344384i	$-0.888088\pi$
$163$	− 112.269i	− 0.688769i	0.938829	−	0.344384i	$-0.111912\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	18.2892i	0.109516i	0.998500	+	0.0547581i	$0.0174388\pi$
$167$	18.2892i	0.109516i	−0.998500	+	0.0547581i	$0.982561\pi$
$168$	0	0
$169$	150.352	0.889658
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−293.303	−1.69540	−0.847698	−	0.530480i	$-0.822012\pi$
$173$	−293.303	−1.69540	−0.847698	+	0.530480i	$0.822012\pi$
$174$	0	0
$175$	− 184.517i	− 1.05438i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 92.9516i	− 0.519283i	−0.965705	−	0.259641i	$-0.916396\pi$
$179$	− 92.9516i	− 0.519283i	0.965705	−	0.259641i	$-0.0836043\pi$
$180$	0	0
$181$	128.445	0.709642	0.354821	−	0.934934i	$-0.384542\pi$
$181$	128.445	0.709642	0.354821	+	0.934934i	$0.384542\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−236.729	−1.27961
$186$	0	0
$187$	27.6127i	0.147662i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	300.587i	1.57375i	0.617109	+	0.786877i	$0.288304\pi$
$191$	300.587i	1.57375i	−0.617109	+	0.786877i	$0.711696\pi$
$192$	0	0
$193$	−39.2591	−0.203415	−0.101708	−	0.994814i	$-0.532431\pi$
$193$	−39.2591	−0.203415	−0.101708	+	0.994814i	$0.532431\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	94.7530	0.480980	0.240490	−	0.970652i	$-0.422692\pi$
$197$	94.7530	0.480980	0.240490	+	0.970652i	$0.422692\pi$
$198$	0	0
$199$	− 111.360i	− 0.559598i	−0.960059	−	0.279799i	$-0.909732\pi$
$199$	− 111.360i	− 0.559598i	0.960059	−	0.279799i	$-0.0902679\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	111.241i	0.547984i
$204$	0	0
$205$	34.8338	0.169921
$206$	0	0
$207$	0	0
$208$	0	0
$209$	163.056	0.780174
$210$	0	0
$211$	− 118.604i	− 0.562105i	−0.959692	−	0.281053i	$-0.909316\pi$
$211$	− 118.604i	− 0.562105i	0.959692	−	0.281053i	$-0.0906836\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	416.777i	1.93850i
$216$	0	0
$217$	203.611	0.938301
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−42.4695	−0.192170
$222$	0	0
$223$	24.2664i	0.108818i	0.998519	+	0.0544089i	$0.0173274\pi$
$223$	24.2664i	0.108818i	−0.998519	+	0.0544089i	$0.982673\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 11.6190i	− 0.0511848i	−0.999672	−	0.0255924i	$-0.991853\pi$
$227$	− 11.6190i	− 0.0511848i	0.999672	−	0.0255924i	$-0.00814720\pi$
$228$	0	0
$229$	−18.1296	−0.0791684	−0.0395842	−	0.999216i	$-0.512603\pi$
$229$	−18.1296	−0.0791684	−0.0395842	+	0.999216i	$0.512603\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	219.785	0.943283	0.471642	−	0.881790i	$-0.343662\pi$
$233$	219.785	0.943283	0.471642	+	0.881790i	$0.343662\pi$
$234$	0	0
$235$	32.7775i	0.139479i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 161.160i	− 0.674309i	−0.941449	−	0.337154i	$-0.890536\pi$
$239$	− 161.160i	− 0.674309i	0.941449	−	0.337154i	$-0.109464\pi$
$240$	0	0
$241$	−169.186	−0.702016	−0.351008	−	0.936372i	$-0.614161\pi$
$241$	−169.186	−0.702016	−0.351008	+	0.936372i	$0.614161\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	308.729	1.26012
$246$	0	0
$247$	250.787i	1.01533i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	219.255i	0.873524i	0.899577	+	0.436762i	$0.143875\pi$
$251$	219.255i	0.873524i	−0.899577	+	0.436762i	$0.856125\pi$
$252$	0	0
$253$	−482.502	−1.90712
$254$	0	0
$255$	0	0
$256$	0	0
$257$	177.470	0.690543	0.345271	−	0.938503i	$-0.387787\pi$
$257$	177.470	0.690543	0.345271	+	0.938503i	$0.387787\pi$
$258$	0	0
$259$	− 349.478i	− 1.34933i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	68.2082i	0.259347i	0.991557	+	0.129673i	$0.0413929\pi$
$263$	68.2082i	0.259347i	−0.991557	+	0.129673i	$0.958607\pi$
$264$	0	0
$265$	501.668	1.89309
$266$	0	0
$267$	0	0
$268$	0	0
$269$	77.7652	0.289090	0.144545	−	0.989498i	$-0.453828\pi$
$269$	77.7652	0.289090	0.144545	+	0.989498i	$0.453828\pi$
$270$	0	0
$271$	447.259i	1.65040i	0.564840	+	0.825201i	$0.308938\pi$
$271$	447.259i	1.65040i	−0.564840	+	0.825201i	$0.691062\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 219.255i	− 0.797289i
$276$	0	0
$277$	−168.056	−0.606702	−0.303351	−	0.952879i	$-0.598105\pi$
$277$	−168.056	−0.606702	−0.303351	+	0.952879i	$0.598105\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−206.340	−0.734306	−0.367153	−	0.930161i	$-0.619667\pi$
$281$	−206.340	−0.734306	−0.367153	+	0.930161i	$0.619667\pi$
$282$	0	0
$283$	199.363i	0.704463i	0.935913	+	0.352231i	$0.114577\pi$
$283$	199.363i	0.704463i	−0.935913	+	0.352231i	$0.885423\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	51.4245i	0.179179i
$288$	0	0
$289$	−283.352	−0.980457
$290$	0	0
$291$	0	0
$292$	0	0
$293$	57.1906	0.195190	0.0975948	−	0.995226i	$-0.468885\pi$
$293$	57.1906	0.195190	0.0975948	+	0.995226i	$0.468885\pi$
$294$	0	0
$295$	− 384.789i	− 1.30437i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 742.107i	− 2.48196i
$300$	0	0
$301$	−615.279	−2.04412
$302$	0	0
$303$	0	0
$304$	0	0
$305$	78.6235	0.257782
$306$	0	0
$307$	276.775i	0.901549i	0.892638	+	0.450774i	$0.148852\pi$
$307$	276.775i	0.901549i	−0.892638	+	0.450774i	$0.851148\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	24.7434i	0.0795608i	0.999208	+	0.0397804i	$0.0126658\pi$
$311$	24.7434i	0.0795608i	−0.999208	+	0.0397804i	$0.987334\pi$
$312$	0	0
$313$	−64.8140	−0.207074	−0.103537	−	0.994626i	$-0.533016\pi$
$313$	−64.8140	−0.207074	−0.103537	+	0.994626i	$0.533016\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	454.599	1.43407	0.717033	−	0.697039i	$-0.245500\pi$
$317$	454.599	1.43407	0.717033	+	0.697039i	$0.245500\pi$
$318$	0	0
$319$	132.183i	0.414368i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 33.3513i	− 0.103255i
$324$	0	0
$325$	337.223	1.03761
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−48.3887	−0.147078
$330$	0	0
$331$	− 272.520i	− 0.823323i	−0.911337	−	0.411661i	$-0.864949\pi$
$331$	− 272.520i	− 0.823323i	0.911337	−	0.411661i	$-0.135051\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 157.717i	− 0.470796i
$336$	0	0
$337$	−493.890	−1.46555	−0.732775	−	0.680471i	$-0.761775\pi$
$337$	−493.890	−1.46555	−0.732775	+	0.680471i	$0.761775\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	241.944	0.709512
$342$	0	0
$343$	− 23.3572i	− 0.0680967i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 519.842i	− 1.49810i	−0.662512	−	0.749051i	$-0.730510\pi$
$347$	− 519.842i	− 1.49810i	0.662512	−	0.749051i	$-0.269490\pi$
$348$	0	0
$349$	−241.279	−0.691344	−0.345672	−	0.938355i	$-0.612349\pi$
$349$	−241.279	−0.691344	−0.345672	+	0.938355i	$0.612349\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−200.927	−0.569198	−0.284599	−	0.958647i	$-0.591860\pi$
$353$	−200.927	−0.569198	−0.284599	+	0.958647i	$0.591860\pi$
$354$	0	0
$355$	− 307.831i	− 0.867130i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	159.654i	0.444719i	0.974965	+	0.222360i	$0.0713759\pi$
$359$	159.654i	0.444719i	−0.974965	+	0.222360i	$0.928624\pi$
$360$	0	0
$361$	164.056	0.454450
$362$	0	0
$363$	0	0
$364$	0	0
$365$	691.174	1.89363
$366$	0	0
$367$	370.301i	1.00899i	0.863413	+	0.504497i	$0.168322\pi$
$367$	370.301i	1.00899i	−0.863413	+	0.504497i	$0.831678\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	740.602i	1.99623i
$372$	0	0
$373$	326.389	0.875037	0.437518	−	0.899209i	$-0.355857\pi$
$373$	326.389	0.875037	0.437518	+	0.899209i	$0.355857\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−203.303	−0.539266
$378$	0	0
$379$	− 348.114i	− 0.918506i	−0.888305	−	0.459253i	$-0.848117\pi$
$379$	− 348.114i	− 0.918506i	0.888305	−	0.459253i	$-0.151883\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 233.885i	− 0.610665i	−0.952246	−	0.305332i	$-0.901232\pi$
$383$	− 233.885i	− 0.610665i	0.952246	−	0.305332i	$-0.0987675\pi$
$384$	0	0
$385$	752.502	1.95455
$386$	0	0
$387$	0	0
$388$	0	0
$389$	437.611	1.12496	0.562482	−	0.826809i	$-0.309846\pi$
$389$	437.611	1.12496	0.562482	+	0.826809i	$0.309846\pi$
$390$	0	0
$391$	98.6902i	0.252405i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	137.922i	0.349169i
$396$	0	0
$397$	−324.259	−0.816774	−0.408387	−	0.912809i	$-0.633909\pi$
$397$	−324.259	−0.816774	−0.408387	+	0.912809i	$0.633909\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	394.174	0.982977	0.491488	−	0.870884i	$-0.336453\pi$
$401$	394.174	0.982977	0.491488	+	0.870884i	$0.336453\pi$
$402$	0	0
$403$	372.119i	0.923373i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 415.271i	− 1.02032i
$408$	0	0
$409$	−28.2957	−0.0691827	−0.0345914	−	0.999402i	$-0.511013\pi$
$409$	−28.2957	−0.0691827	−0.0345914	+	0.999402i	$0.511013\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	568.056	1.37544
$414$	0	0
$415$	− 186.693i	− 0.449863i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	323.825i	0.772852i	0.922320	+	0.386426i	$0.126291\pi$
$419$	323.825i	0.772852i	−0.922320	+	0.386426i	$0.873709\pi$
$420$	0	0
$421$	−431.352	−1.02459	−0.512295	−	0.858810i	$-0.671204\pi$
$421$	−431.352	−1.02459	−0.512295	+	0.858810i	$0.671204\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−44.8460	−0.105520
$426$	0	0
$427$	116.070i	0.271827i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 345.557i	− 0.801757i	−0.916131	−	0.400879i	$-0.868705\pi$
$431$	− 345.557i	− 0.801757i	0.916131	−	0.400879i	$-0.131295\pi$
$432$	0	0
$433$	369.907	0.854289	0.427144	−	0.904183i	$-0.359520\pi$
$433$	369.907	0.854289	0.427144	+	0.904183i	$0.359520\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	582.777	1.33359
$438$	0	0
$439$	− 413.766i	− 0.942519i	−0.881995	−	0.471259i	$-0.843800\pi$
$439$	− 413.766i	− 0.942519i	0.881995	−	0.471259i	$-0.156200\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	539.636i	1.21814i	0.793116	+	0.609070i	$0.208457\pi$
$443$	539.636i	1.21814i	−0.793116	+	0.609070i	$0.791543\pi$
$444$	0	0
$445$	483.594	1.08673
$446$	0	0
$447$	0	0
$448$	0	0
$449$	325.056	0.723956	0.361978	−	0.932187i	$-0.382101\pi$
$449$	325.056	0.723956	0.361978	+	0.932187i	$0.382101\pi$
$450$	0	0
$451$	61.1058i	0.135490i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	1157.38i	2.54369i
$456$	0	0
$457$	618.149	1.35262	0.676312	−	0.736615i	$-0.263577\pi$
$457$	618.149	1.35262	0.676312	+	0.736615i	$0.263577\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	849.983	1.84378	0.921891	−	0.387450i	$-0.126644\pi$
$461$	849.983	1.84378	0.921891	+	0.387450i	$0.126644\pi$
$462$	0	0
$463$	− 815.741i	− 1.76186i	−0.473247	−	0.880930i	$-0.656918\pi$
$463$	− 815.741i	− 1.76186i	0.473247	−	0.880930i	$-0.343082\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 378.909i	− 0.811368i	−0.914013	−	0.405684i	$-0.867033\pi$
$467$	− 378.909i	− 0.811368i	0.914013	−	0.405684i	$-0.132967\pi$
$468$	0	0
$469$	232.834	0.496447
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−731.113	−1.54569
$474$	0	0
$475$	264.821i	0.557518i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 344.052i	− 0.718271i	−0.933285	−	0.359136i	$-0.883072\pi$
$479$	− 344.052i	− 0.718271i	0.933285	−	0.359136i	$-0.116928\pi$
$480$	0	0
$481$	638.704	1.32787
$482$	0	0
$483$	0	0
$484$	0	0
$485$	942.008	1.94228
$486$	0	0
$487$	− 587.595i	− 1.20656i	−0.797529	−	0.603281i	$-0.793860\pi$
$487$	− 587.595i	− 1.20656i	0.797529	−	0.603281i	$-0.206140\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	496.604i	1.01141i	0.862706	+	0.505707i	$0.168768\pi$
$491$	496.604i	1.01141i	−0.862706	+	0.505707i	$0.831232\pi$
$492$	0	0
$493$	27.0366	0.0548410
$494$	0	0
$495$	0	0
$496$	0	0
$497$	454.445	0.914376
$498$	0	0
$499$	34.8569i	0.0698534i	0.999390	+	0.0349267i	$0.0111198\pi$
$499$	34.8569i	0.0698534i	−0.999390	+	0.0349267i	$0.988880\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 630.866i	− 1.25421i	−0.778936	−	0.627104i	$-0.784240\pi$
$503$	− 630.866i	− 1.25421i	0.778936	−	0.627104i	$-0.215760\pi$
$504$	0	0
$505$	−181.166	−0.358745
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−82.0564	−0.161211	−0.0806055	−	0.996746i	$-0.525685\pi$
$509$	−82.0564	−0.161211	−0.0806055	+	0.996746i	$0.525685\pi$
$510$	0	0
$511$	1020.37i	1.99680i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	971.475i	1.88636i
$516$	0	0
$517$	−57.4985	−0.111216
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−562.797	−1.08023	−0.540113	−	0.841593i	$-0.681618\pi$
$521$	−562.797	−1.08023	−0.540113	+	0.841593i	$0.681618\pi$
$522$	0	0
$523$	909.721i	1.73943i	0.493555	+	0.869714i	$0.335697\pi$
$523$	909.721i	1.73943i	−0.493555	+	0.869714i	$0.664303\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 49.4868i	− 0.0939029i
$528$	0	0
$529$	−1195.50	−2.25993
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−93.9832	−0.176329
$534$	0	0
$535$	1275.51i	2.38412i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	541.574i	1.00478i
$540$	0	0
$541$	468.259	0.865544	0.432772	−	0.901503i	$-0.357536\pi$
$541$	468.259	0.865544	0.432772	+	0.901503i	$0.357536\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−1.71646	−0.00314947
$546$	0	0
$547$	312.996i	0.572205i	0.958199	+	0.286102i	$0.0923598\pi$
$547$	312.996i	0.572205i	−0.958199	+	0.286102i	$0.907640\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 159.654i	− 0.289754i
$552$	0	0
$553$	−203.611	−0.368194
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−382.841	−0.687328	−0.343664	−	0.939093i	$-0.611668\pi$
$557$	−382.841	−0.687328	−0.343664	+	0.939093i	$0.611668\pi$
$558$	0	0
$559$	− 1124.48i	− 2.01160i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 599.021i	− 1.06398i	−0.846751	−	0.531990i	$-0.821444\pi$
$563$	− 599.021i	− 1.06398i	0.846751	−	0.531990i	$-0.178556\pi$
$564$	0	0
$565$	1317.13	2.33121
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−844.174	−1.48361	−0.741805	−	0.670616i	$-0.766030\pi$
$569$	−844.174	−1.48361	−0.741805	+	0.670616i	$0.766030\pi$
$570$	0	0
$571$	− 776.368i	− 1.35966i	−0.733368	−	0.679832i	$-0.762053\pi$
$571$	− 776.368i	− 1.35966i	0.733368	−	0.679832i	$-0.237947\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 783.634i	− 1.36284i
$576$	0	0
$577$	−487.316	−0.844568	−0.422284	−	0.906464i	$-0.638771\pi$
$577$	−487.316	−0.844568	−0.422284	+	0.906464i	$0.638771\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	275.611	0.474374
$582$	0	0
$583$	880.029i	1.50948i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 562.874i	− 0.958900i	−0.877569	−	0.479450i	$-0.840836\pi$
$587$	− 562.874i	− 0.958900i	0.877569	−	0.479450i	$-0.159164\pi$
$588$	0	0
$589$	−292.226	−0.496138
$590$	0	0
$591$	0	0
$592$	0	0
$593$	316.915	0.534426	0.267213	−	0.963637i	$-0.413897\pi$
$593$	316.915	0.534426	0.267213	+	0.963637i	$0.413897\pi$
$594$	0	0
$595$	− 153.916i	− 0.258682i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 88.4351i	− 0.147638i	−0.997272	−	0.0738189i	$-0.976481\pi$
$599$	− 88.4351i	− 0.147638i	0.997272	−	0.0738189i	$-0.0235187\pi$
$600$	0	0
$601$	−111.817	−0.186052	−0.0930258	−	0.995664i	$-0.529654\pi$
$601$	−111.817	−0.186052	−0.0930258	+	0.995664i	$0.529654\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	92.7287	0.153271
$606$	0	0
$607$	− 688.268i	− 1.13388i	−0.823757	−	0.566942i	$-0.808126\pi$
$607$	− 688.268i	− 1.13388i	0.823757	−	0.566942i	$-0.191874\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 88.4351i	− 0.144738i
$612$	0	0
$613$	−1021.67	−1.66667	−0.833334	−	0.552770i	$-0.813571\pi$
$613$	−1021.67	−1.66667	−0.833334	+	0.552770i	$0.813571\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	763.768	1.23787	0.618937	−	0.785441i	$-0.287564\pi$
$617$	763.768	1.23787	0.618937	+	0.785441i	$0.287564\pi$
$618$	0	0
$619$	− 763.698i	− 1.23376i	−0.787057	−	0.616880i	$-0.788396\pi$
$619$	− 763.698i	− 1.23376i	0.787057	−	0.616880i	$-0.211604\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	713.921i	1.14594i
$624$	0	0
$625$	−740.668	−1.18507
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−84.9390	−0.135038
$630$	0	0
$631$	− 166.586i	− 0.264002i	−0.991250	−	0.132001i	$-0.957860\pi$
$631$	− 166.586i	− 0.264002i	0.991250	−	0.132001i	$-0.0421403\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 478.531i	− 0.753592i
$636$	0	0
$637$	−832.963	−1.30763
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−204.271	−0.318676	−0.159338	−	0.987224i	$-0.550936\pi$
$641$	−204.271	−0.318676	−0.159338	+	0.987224i	$0.550936\pi$
$642$	0	0
$643$	− 954.908i	− 1.48508i	−0.669801	−	0.742541i	$-0.733620\pi$
$643$	− 954.908i	− 1.48508i	0.669801	−	0.742541i	$-0.266380\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 424.736i	− 0.656470i	−0.944596	−	0.328235i	$-0.893546\pi$
$647$	− 424.736i	− 0.656470i	0.944596	−	0.328235i	$-0.106454\pi$
$648$	0	0
$649$	675.000	1.04006
$650$	0	0
$651$	0	0
$652$	0	0
$653$	614.355	0.940820	0.470410	−	0.882448i	$-0.344106\pi$
$653$	614.355	0.940820	0.470410	+	0.882448i	$0.344106\pi$
$654$	0	0
$655$	− 1221.35i	− 1.86466i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 974.486i	− 1.47873i	−0.673302	−	0.739367i	$-0.735125\pi$
$659$	− 974.486i	− 1.47873i	0.673302	−	0.739367i	$-0.264875\pi$
$660$	0	0
$661$	−548.761	−0.830198	−0.415099	−	0.909776i	$-0.636253\pi$
$661$	−548.761	−0.830198	−0.415099	+	0.909776i	$0.636253\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−908.890	−1.36675
$666$	0	0
$667$	472.434i	0.708297i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	137.922i	0.205547i
$672$	0	0
$673$	−373.983	−0.555696	−0.277848	−	0.960625i	$-0.589621\pi$
$673$	−373.983	−0.555696	−0.277848	+	0.960625i	$0.589621\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	241.944	0.357376	0.178688	−	0.983906i	$-0.442815\pi$
$677$	241.944	0.357376	0.178688	+	0.983906i	$0.442815\pi$
$678$	0	0
$679$	1390.67i	2.04811i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	524.358i	0.767728i	0.923390	+	0.383864i	$0.125407\pi$
$683$	524.358i	0.767728i	−0.923390	+	0.383864i	$0.874593\pi$
$684$	0	0
$685$	925.651	1.35132
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−1353.52	−1.96447
$690$	0	0
$691$	− 353.994i	− 0.512293i	−0.966638	−	0.256146i	$-0.917547\pi$
$691$	− 353.994i	− 0.512293i	0.966638	−	0.256146i	$-0.0824529\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 1160.39i	− 1.66963i
$696$	0	0
$697$	12.4985	0.0179318
$698$	0	0
$699$	0	0
$700$	0	0
$701$	621.287	0.886286	0.443143	−	0.896451i	$-0.353863\pi$
$701$	621.287	0.886286	0.443143	+	0.896451i	$0.353863\pi$
$702$	0	0
$703$	501.575i	0.713478i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 267.452i	− 0.378291i
$708$	0	0
$709$	787.983	1.11140	0.555700	−	0.831383i	$-0.312450\pi$
$709$	787.983	1.11140	0.555700	+	0.831383i	$0.312450\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	864.727	1.21280
$714$	0	0
$715$	1375.27i	1.92345i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	1248.82i	1.73689i	0.495785	+	0.868445i	$0.334881\pi$
$719$	1248.82i	1.73689i	−0.495785	+	0.868445i	$0.665119\pi$
$720$	0	0
$721$	−1434.17	−1.98914
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−214.680	−0.296110
$726$	0	0
$727$	− 732.642i	− 1.00776i	−0.863773	−	0.503880i	$-0.831905\pi$
$727$	− 732.642i	− 1.00776i	0.863773	−	0.503880i	$-0.168095\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	149.541i	0.204570i
$732$	0	0
$733$	−364.721	−0.497573	−0.248787	−	0.968558i	$-0.580032\pi$
$733$	−364.721	−0.497573	−0.248787	+	0.968558i	$0.580032\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	276.668	0.375397
$738$	0	0
$739$	1162.07i	1.57248i	0.617918	+	0.786242i	$0.287976\pi$
$739$	1162.07i	1.57248i	−0.617918	+	0.786242i	$0.712024\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 180.955i	− 0.243546i	−0.992558	−	0.121773i	$-0.961142\pi$
$743$	− 180.955i	− 0.243546i	0.992558	−	0.121773i	$-0.0388580\pi$
$744$	0	0
$745$	−1265.54	−1.69871
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−1883.00	−2.51402
$750$	0	0
$751$	− 176.557i	− 0.235096i	−0.993067	−	0.117548i	$-0.962497\pi$
$751$	− 176.557i	− 0.235096i	0.993067	−	0.117548i	$-0.0375034\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 47.9813i	− 0.0635514i
$756$	0	0
$757$	146.665	0.193745	0.0968723	−	0.995297i	$-0.469116\pi$
$757$	146.665	0.193745	0.0968723	+	0.995297i	$0.469116\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−806.809	−1.06020	−0.530098	−	0.847936i	$-0.677845\pi$
$761$	−806.809	−1.06020	−0.530098	+	0.847936i	$0.677845\pi$
$762$	0	0
$763$	− 2.53397i	− 0.00332107i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1038.18i	1.35356i
$768$	0	0
$769$	551.133	0.716687	0.358344	−	0.933590i	$-0.383342\pi$
$769$	551.133	0.716687	0.358344	+	0.933590i	$0.383342\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−856.915	−1.10856	−0.554279	−	0.832331i	$-0.687006\pi$
$773$	−856.915	−1.10856	−0.554279	+	0.832331i	$0.687006\pi$
$774$	0	0
$775$	392.942i	0.507023i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 73.8051i	− 0.0947434i
$780$	0	0
$781$	540.000	0.691421
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−477.748	−0.608597
$786$	0	0
$787$	− 538.705i	− 0.684504i	−0.939608	−	0.342252i	$-0.888810\pi$
$787$	− 538.705i	− 0.684504i	0.939608	−	0.342252i	$-0.111190\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	1944.46i	2.45823i
$792$	0	0
$793$	−212.130	−0.267503
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−1348.31	−1.69173	−0.845863	−	0.533399i	$-0.820914\pi$
$797$	−1348.31	−1.69173	−0.845863	+	0.533399i	$0.820914\pi$
$798$	0	0
$799$	11.7607i	0.0147192i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	1212.46i	1.50992i
$804$	0	0
$805$	2689.50	3.34100
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−1163.88	−1.43867	−0.719334	−	0.694664i	$-0.755553\pi$
$809$	−1163.88	−1.43867	−0.719334	+	0.694664i	$0.755553\pi$
$810$	0	0
$811$	− 627.878i	− 0.774202i	−0.922037	−	0.387101i	$-0.873476\pi$
$811$	− 627.878i	− 0.774202i	0.922037	−	0.387101i	$-0.126524\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	743.613i	0.912408i
$816$	0	0
$817$	883.056	1.08085
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−64.3643	−0.0783975	−0.0391987	−	0.999231i	$-0.512481\pi$
$821$	−64.3643	−0.0783975	−0.0391987	+	0.999231i	$0.512481\pi$
$822$	0	0
$823$	− 943.602i	− 1.14654i	−0.819367	−	0.573269i	$-0.805675\pi$
$823$	− 943.602i	− 1.14654i	0.819367	−	0.573269i	$-0.194325\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	571.482i	0.691031i	0.938413	+	0.345515i	$0.112296\pi$
$827$	571.482i	0.691031i	−0.938413	+	0.345515i	$0.887704\pi$
$828$	0	0
$829$	−519.555	−0.626725	−0.313362	−	0.949634i	$-0.601455\pi$
$829$	−519.555	−0.626725	−0.313362	+	0.949634i	$0.601455\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	110.773	0.132981
$834$	0	0
$835$	− 121.138i	− 0.145076i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 1200.84i	− 1.43128i	−0.698470	−	0.715640i	$-0.746135\pi$
$839$	− 1200.84i	− 1.43128i	0.698470	−	0.715640i	$-0.253865\pi$
$840$	0	0
$841$	−711.575	−0.846105
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−995.854	−1.17853
$846$	0	0
$847$	136.893i	0.161622i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 1484.21i	− 1.74408i
$852$	0	0
$853$	−599.166	−0.702422	−0.351211	−	0.936296i	$-0.614230\pi$
$853$	−599.166	−0.702422	−0.351211	+	0.936296i	$0.614230\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−691.328	−0.806683	−0.403342	−	0.915049i	$-0.632151\pi$
$857$	−691.328	−0.806683	−0.403342	+	0.915049i	$0.632151\pi$
$858$	0	0
$859$	1403.79i	1.63422i	0.576485	+	0.817108i	$0.304424\pi$
$859$	1403.79i	1.63422i	−0.576485	+	0.817108i	$0.695576\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 253.038i	− 0.293207i	−0.989195	−	0.146604i	$-0.953166\pi$
$863$	− 253.038i	− 0.293207i	0.989195	−	0.146604i	$-0.0468342\pi$
$864$	0	0
$865$	1942.69	2.24588
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−241.944	−0.278416
$870$	0	0
$871$	425.526i	0.488549i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 396.982i	− 0.453694i
$876$	0	0
$877$	−107.870	−0.122999	−0.0614997	−	0.998107i	$-0.519588\pi$
$877$	−107.870	−0.122999	−0.0614997	+	0.998107i	$0.519588\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	119.927	0.136126	0.0680629	−	0.997681i	$-0.478318\pi$
$881$	119.927	0.136126	0.0680629	+	0.997681i	$0.478318\pi$
$882$	0	0
$883$	− 306.661i	− 0.347295i	−0.984808	−	0.173647i	$-0.944445\pi$
$883$	− 306.661i	− 0.347295i	0.984808	−	0.173647i	$-0.0555553\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 785.572i	− 0.885651i	−0.896608	−	0.442825i	$-0.853976\pi$
$887$	− 785.572i	− 0.885651i	0.896608	−	0.442825i	$-0.146024\pi$
$888$	0	0
$889$	706.445	0.794651
$890$	0	0
$891$	0	0
$892$	0	0
$893$	69.4481	0.0777695
$894$	0	0
$895$	615.663i	0.687891i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 236.896i	− 0.263510i
$900$	0	0
$901$	180.000	0.199778
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−850.753	−0.940059
$906$	0	0
$907$	− 1218.36i	− 1.34329i	−0.740873	−	0.671646i	$-0.765588\pi$
$907$	− 1218.36i	− 1.34329i	0.740873	−	0.671646i	$-0.234412\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	878.524i	0.964351i	0.876075	+	0.482175i	$0.160153\pi$
$911$	878.524i	0.964351i	−0.876075	+	0.482175i	$0.839847\pi$
$912$	0	0
$913$	327.498	0.358706
$914$	0	0
$915$	0	0
$916$	0	0
$917$	1803.06	1.96626
$918$	0	0
$919$	1036.67i	1.12804i	0.825760	+	0.564022i	$0.190747\pi$
$919$	1036.67i	1.12804i	−0.825760	+	0.564022i	$0.809253\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	830.542i	0.899829i
$924$	0	0
$925$	674.445	0.729130
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−249.822	−0.268915	−0.134457	−	0.990919i	$-0.542929\pi$
$929$	−249.822	−0.268915	−0.134457	+	0.990919i	$0.542929\pi$
$930$	0	0
$931$	− 654.127i	− 0.702607i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 182.892i	− 0.195607i
$936$	0	0
$937$	190.744	0.203569	0.101784	−	0.994806i	$-0.467545\pi$
$937$	190.744	0.203569	0.101784	+	0.994806i	$0.467545\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−282.300	−0.300000	−0.150000	−	0.988686i	$-0.547927\pi$
$941$	−282.300	−0.300000	−0.150000	+	0.988686i	$0.547927\pi$
$942$	0	0
$943$	218.397i	0.231598i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 662.712i	− 0.699802i	−0.936787	−	0.349901i	$-0.886215\pi$
$947$	− 662.712i	− 0.699802i	0.936787	−	0.349901i	$-0.113785\pi$
$948$	0	0
$949$	−1864.82	−1.96503
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−1166.70	−1.22424	−0.612119	−	0.790765i	$-0.709683\pi$
$953$	−1166.70	−1.22424	−0.612119	+	0.790765i	$0.709683\pi$
$954$	0	0
$955$	− 1990.93i	− 2.08475i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	1366.52i	1.42494i
$960$	0	0
$961$	527.395	0.548798
$962$	0	0
$963$	0	0
$964$	0	0
$965$	260.032	0.269463
$966$	0	0
$967$	− 1166.84i	− 1.20666i	−0.797490	−	0.603332i	$-0.793839\pi$
$967$	− 1166.84i	− 1.20666i	0.797490	−	0.603332i	$-0.206161\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 1481.20i	− 1.52544i	−0.646728	−	0.762721i	$-0.723863\pi$
$971$	− 1481.20i	− 1.52544i	0.646728	−	0.762721i	$-0.276137\pi$
$972$	0	0
$973$	1713.06	1.76060
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1185.56	1.21347	0.606734	−	0.794905i	$-0.292479\pi$
$977$	1185.56	1.21347	0.606734	+	0.794905i	$0.292479\pi$
$978$	0	0
$979$	848.325i	0.866522i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 64.7650i	− 0.0658851i	−0.999457	−	0.0329425i	$-0.989512\pi$
$983$	− 64.7650i	− 0.0658851i	0.999457	−	0.0329425i	$-0.0104878\pi$
$984$	0	0
$985$	−627.594	−0.637152
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−2613.06	−2.64212
$990$	0	0
$991$	1394.27i	1.40694i	0.710727	+	0.703468i	$0.248366\pi$
$991$	1394.27i	1.40694i	−0.710727	+	0.703468i	$0.751634\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	737.591i	0.741297i
$996$	0	0
$997$	−974.541	−0.977474	−0.488737	−	0.872431i	$-0.662542\pi$
$997$	−974.541	−0.977474	−0.488737	+	0.872431i	$0.662542\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1296.3.g.d.1135.1		4
3.2	odd	2		1296.3.g.h.1135.3		4
4.3	odd	2	inner	1296.3.g.d.1135.2		4
9.2	odd	6		144.3.o.b.31.2	✓	8
9.4	even	3		432.3.o.c.127.4		8
9.5	odd	6		144.3.o.b.79.3	yes	8
9.7	even	3		432.3.o.c.415.3		8
12.11	even	2		1296.3.g.h.1135.4		4
36.7	odd	6		432.3.o.c.415.4		8
36.11	even	6		144.3.o.b.31.3	yes	8
36.23	even	6		144.3.o.b.79.2	yes	8
36.31	odd	6		432.3.o.c.127.3		8
72.5	odd	6		576.3.o.e.511.2		8
72.11	even	6		576.3.o.e.319.2		8
72.13	even	6		1728.3.o.d.127.2		8
72.29	odd	6		576.3.o.e.319.3		8
72.43	odd	6		1728.3.o.d.1279.2		8
72.59	even	6		576.3.o.e.511.3		8
72.61	even	6		1728.3.o.d.1279.1		8
72.67	odd	6		1728.3.o.d.127.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
144.3.o.b.31.2	✓	8	9.2	odd	6
144.3.o.b.31.3	yes	8	36.11	even	6
144.3.o.b.79.2	yes	8	36.23	even	6
144.3.o.b.79.3	yes	8	9.5	odd	6
432.3.o.c.127.3		8	36.31	odd	6
432.3.o.c.127.4		8	9.4	even	3
432.3.o.c.415.3		8	9.7	even	3
432.3.o.c.415.4		8	36.7	odd	6
576.3.o.e.319.2		8	72.11	even	6
576.3.o.e.319.3		8	72.29	odd	6
576.3.o.e.511.2		8	72.5	odd	6
576.3.o.e.511.3		8	72.59	even	6
1296.3.g.d.1135.1		4	1.1	even	1	trivial
1296.3.g.d.1135.2		4	4.3	odd	2	inner
1296.3.g.h.1135.3		4	3.2	odd	2
1296.3.g.h.1135.4		4	12.11	even	2
1728.3.o.d.127.1		8	72.67	odd	6
1728.3.o.d.127.2		8	72.13	even	6
1728.3.o.d.1279.1		8	72.61	even	6
1728.3.o.d.1279.2		8	72.43	odd	6

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:	\( N \)	\(=\)	\( 1296 = 2^{4} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1296.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-6.62348 q^{5} -9.77810i q^{7} -11.6190i q^{11} +17.8704 q^{13} -2.37652 q^{17} +14.0337i q^{19} -41.5271i q^{23} +18.8704 q^{25} -11.3765 q^{29} +20.8232i q^{31} +64.7650i q^{35} +35.7409 q^{37} -5.25915 q^{41} -62.9242i q^{43} -4.94868i q^{47} -46.6113 q^{49} -75.7409 q^{53} +76.9578i q^{55} +58.0948i q^{59} -11.8704 q^{61} -118.364 q^{65} +23.8118i q^{67} +46.4758i q^{71} -104.352 q^{73} -113.611 q^{77} -20.8232i q^{79} +28.1866i q^{83} +15.7409 q^{85} -73.0122 q^{89} -174.739i q^{91} -92.9516i q^{95} -142.223 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 6 q^{5} + 10 q^{13} - 30 q^{17} + 14 q^{25} - 66 q^{29} + 20 q^{37} - 144 q^{41} - 2 q^{49} - 180 q^{53} + 14 q^{61} - 330 q^{65} - 110 q^{73} - 270 q^{77} - 60 q^{85} - 456 q^{89} - 200 q^{97}+O(q^{100}) \) 4 * q - 6 * q^5 + 10 * q^13 - 30 * q^17 + 14 * q^25 - 66 * q^29 + 20 * q^37 - 144 * q^41 - 2 * q^49 - 180 * q^53 + 14 * q^61 - 330 * q^65 - 110 * q^73 - 270 * q^77 - 60 * q^85 - 456 * q^89 - 200 * q^97

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