LMFDB - Embedded newform 324.4.i.a.37.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [324,4,Mod(37,324)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(324, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([0, 14]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("324.37");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 324 = 2^{2} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	324.i (of order $9$, degree $6$, 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:	$19.1166188419$
Analytic rank:	$0$
Dimension:	$54$
Relative dimension:	$9$ over $\Q(\zeta_{9})$
Twist minimal:	no (minimal twist has level 108)
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label			37.2
Character	$\chi$	$=$	324.37
Dual form			324.4.i.a.289.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-13.2490 + 4.82225i) q^{5} +(3.17294 + 17.9946i) q^{7} +O(q^{10})$	$q+(-13.2490 + 4.82225i) q^{5} +(3.17294 + 17.9946i) q^{7} +(-33.5576 - 12.2140i) q^{11} +(-6.15848 - 5.16757i) q^{13} +(58.3572 - 101.078i) q^{17} +(39.9112 + 69.1281i) q^{19} +(15.4931 - 87.8657i) q^{23} +(56.5267 - 47.4316i) q^{25} +(-85.5671 + 71.7993i) q^{29} +(47.1052 - 267.147i) q^{31} +(-128.813 - 223.110i) q^{35} +(180.562 - 312.742i) q^{37} +(10.5230 + 8.82981i) q^{41} +(-383.698 - 139.655i) q^{43} +(-14.3445 - 81.3518i) q^{47} +(8.57561 - 3.12127i) q^{49} +28.8354 q^{53} +503.503 q^{55} +(-629.134 + 228.986i) q^{59} +(-16.7704 - 95.1097i) q^{61} +(106.513 + 38.7676i) q^{65} +(514.271 + 431.525i) q^{67} +(47.7773 - 82.7527i) q^{71} +(-502.908 - 871.062i) q^{73} +(113.309 - 642.610i) q^{77} +(780.366 - 654.805i) q^{79} +(547.955 - 459.789i) q^{83} +(-285.754 + 1620.59i) q^{85} +(-314.627 - 544.950i) q^{89} +(73.4481 - 127.216i) q^{91} +(-862.136 - 723.418i) q^{95} +(-1544.58 - 562.179i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 54 q - 12 q^{5}+O(q^{10}) $ 54 * q - 12 * q^5	$ 54 q - 12 q^{5} + 87 q^{11} - 204 q^{17} - 96 q^{23} - 216 q^{25} - 318 q^{29} - 54 q^{31} - 6 q^{35} - 867 q^{41} - 513 q^{43} + 1548 q^{47} + 594 q^{49} + 1068 q^{53} + 1218 q^{59} - 54 q^{61} - 96 q^{65} - 2997 q^{67} + 120 q^{71} - 216 q^{73} - 3480 q^{77} + 2808 q^{79} - 4464 q^{83} + 2160 q^{85} - 4029 q^{89} + 270 q^{91} + 1650 q^{95} - 3483 q^{97}+O(q^{100}) $ 54 * q - 12 * q^5 + 87 * q^11 - 204 * q^17 - 96 * q^23 - 216 * q^25 - 318 * q^29 - 54 * q^31 - 6 * q^35 - 867 * q^41 - 513 * q^43 + 1548 * q^47 + 594 * q^49 + 1068 * q^53 + 1218 * q^59 - 54 * q^61 - 96 * q^65 - 2997 * q^67 + 120 * q^71 - 216 * q^73 - 3480 * q^77 + 2808 * q^79 - 4464 * q^83 + 2160 * q^85 - 4029 * q^89 + 270 * q^91 + 1650 * q^95 - 3483 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/324\mathbb{Z}\right)^\times$.

$n$	$163$	$245$
$\chi(n)$	$1$	$e\left(\frac{7}{9}\right)$

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
$5$	−13.2490	+	4.82225i	−1.18503	+	0.431315i	−0.857975	−	0.513691i	$-0.828278\pi$
$5$	−13.2490	+	4.82225i	−1.18503	+	0.431315i	−0.327053	+	0.945006i	$0.606055\pi$
$6$		0			0
$7$	3.17294	+	17.9946i	0.171323	+	0.971618i	0.942303	+	0.334760i	$0.108655\pi$
$7$	3.17294	+	17.9946i	0.171323	+	0.971618i	−0.770981	+	0.636858i	$0.780234\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−33.5576	−	12.2140i	−0.919817	−	0.334786i	−0.161651	−	0.986848i	$-0.551682\pi$
$11$	−33.5576	−	12.2140i	−0.919817	−	0.334786i	−0.758166	+	0.652062i	$0.773904\pi$
$12$		0			0
$13$	−6.15848	−	5.16757i	−0.131389	−	0.110248i	0.574725	−	0.818346i	$-0.305109\pi$
$13$	−6.15848	−	5.16757i	−0.131389	−	0.110248i	−0.706114	+	0.708098i	$0.749553\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	58.3572	−	101.078i	0.832570	−	1.44205i	−0.0634231	−	0.997987i	$-0.520202\pi$
$17$	58.3572	−	101.078i	0.832570	−	1.44205i	0.895993	−	0.444067i	$-0.146465\pi$
$18$		0			0
$19$	39.9112	+	69.1281i	0.481908	+	0.834689i	0.999784	−	0.0207667i	$-0.00661072\pi$
$19$	39.9112	+	69.1281i	0.481908	+	0.834689i	−0.517877	+	0.855455i	$0.673277\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	15.4931	−	87.8657i	0.140458	−	0.796576i	−0.830445	−	0.557101i	$-0.811914\pi$
$23$	15.4931	−	87.8657i	0.140458	−	0.796576i	0.970903	−	0.239475i	$-0.0769753\pi$
$24$		0			0
$25$	56.5267	−	47.4316i	0.452214	−	0.379453i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−85.5671	+	71.7993i	−0.547911	+	0.459752i	−0.874233	−	0.485507i	$-0.838635\pi$
$29$	−85.5671	+	71.7993i	−0.547911	+	0.459752i	0.326322	+	0.945259i	$0.394191\pi$
$30$		0			0
$31$	47.1052	−	267.147i	0.272914	−	1.54777i	−0.472596	−	0.881279i	$-0.656683\pi$
$31$	47.1052	−	267.147i	0.272914	−	1.54777i	0.745511	−	0.666494i	$-0.232206\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−128.813	−	223.110i	−0.622095	−	1.07750i
$36$		0			0
$37$	180.562	−	312.742i	0.802275	−	1.38958i	−0.115841	−	0.993268i	$-0.536956\pi$
$37$	180.562	−	312.742i	0.802275	−	1.38958i	0.918115	−	0.396313i	$-0.129710\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	10.5230	+	8.82981i	0.0400832	+	0.0336338i	0.662609	−	0.748965i	$-0.269449\pi$
$41$	10.5230	+	8.82981i	0.0400832	+	0.0336338i	−0.622526	+	0.782599i	$0.713894\pi$
$42$		0			0
$43$	−383.698	−	139.655i	−1.36078	−	0.495282i	−0.444482	−	0.895788i	$-0.646612\pi$
$43$	−383.698	−	139.655i	−1.36078	−	0.495282i	−0.916294	+	0.400506i	$0.868834\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−14.3445	−	81.3518i	−0.0445184	−	0.252476i	0.954424	−	0.298454i	$-0.0964709\pi$
$47$	−14.3445	−	81.3518i	−0.0445184	−	0.252476i	−0.998942	+	0.0459775i	$0.985360\pi$
$48$		0			0
$49$	8.57561	−	3.12127i	0.0250018	−	0.00909990i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	28.8354			0.0747329			0.0373665	−	0.999302i	$-0.488103\pi$
$53$	28.8354			0.0747329			0.0373665	+	0.999302i	$0.488103\pi$
$54$		0			0
$55$	503.503			1.23441
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−629.134	+	228.986i	−1.38824	+	0.505279i	−0.924666	−	0.380779i	$-0.875656\pi$
$59$	−629.134	+	228.986i	−1.38824	+	0.505279i	−0.463575	+	0.886058i	$0.653434\pi$
$60$		0			0
$61$	−16.7704	−	95.1097i	−0.0352005	−	0.199632i	0.962136	−	0.272570i	$-0.0878736\pi$
$61$	−16.7704	−	95.1097i	−0.0352005	−	0.199632i	−0.997336	+	0.0729379i	$0.976763\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	106.513	+	38.7676i	0.203251	+	0.0739773i
$66$		0			0
$67$	514.271	+	431.525i	0.937735	+	0.786853i	0.977190	−	0.212368i	$-0.0681176\pi$
$67$	514.271	+	431.525i	0.937735	+	0.786853i	−0.0394546	+	0.999221i	$0.512562\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	47.7773	−	82.7527i	0.0798609	−	0.138323i	−0.823329	−	0.567565i	$-0.807886\pi$
$71$	47.7773	−	82.7527i	0.0798609	−	0.138323i	0.903190	+	0.429241i	$0.141219\pi$
$72$		0			0
$73$	−502.908	−	871.062i	−0.806314	−	1.39658i	−0.915400	−	0.402545i	$-0.868126\pi$
$73$	−502.908	−	871.062i	−0.806314	−	1.39658i	0.109086	−	0.994032i	$-0.465207\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	113.309	−	642.610i	0.167699	−	0.951068i
$78$		0			0
$79$	780.366	−	654.805i	1.11137	−	0.932548i	0.113231	−	0.993569i	$-0.463880\pi$
$79$	780.366	−	654.805i	1.11137	−	0.932548i	0.998137	+	0.0610203i	$0.0194355\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	547.955	−	459.789i	0.724650	−	0.608053i	−0.204018	−	0.978967i	$-0.565400\pi$
$83$	547.955	−	459.789i	0.724650	−	0.608053i	0.928667	+	0.370914i	$0.120956\pi$
$84$		0			0
$85$	−285.754	+	1620.59i	−0.364640	+	2.06797i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−314.627	−	544.950i	−0.374724	−	0.649041i	0.615562	−	0.788089i	$-0.288929\pi$
$89$	−314.627	−	544.950i	−0.374724	−	0.649041i	−0.990286	+	0.139048i	$0.955596\pi$
$90$		0			0
$91$	73.4481	−	127.216i	0.0846094	−	0.146548i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−862.136	−	723.418i	−0.931088	−	0.781275i
$96$		0			0
$97$	−1544.58	−	562.179i	−1.61678	−	0.588461i	−0.634017	−	0.773319i	$-0.718595\pi$
$97$	−1544.58	−	562.179i	−1.61678	−	0.588461i	−0.982765	+	0.184858i	$0.940817\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	221.812	+	1257.96i	0.218526	+	1.23932i	0.874683	+	0.484696i	$0.161070\pi$
$101$	221.812	+	1257.96i	0.218526	+	1.23932i	−0.656157	+	0.754624i	$0.727819\pi$
$102$		0			0
$103$	−47.4842	+	17.2828i	−0.0454248	+	0.0165333i	−0.364633	−	0.931151i	$-0.618805\pi$
$103$	−47.4842	+	17.2828i	−0.0454248	+	0.0165333i	0.319208	+	0.947685i	$0.396583\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	1585.17			1.43219			0.716095	−	0.698003i	$-0.245928\pi$
$107$	1585.17			1.43219			0.716095	+	0.698003i	$0.245928\pi$
$108$		0			0
$109$	201.612			0.177164			0.0885821	−	0.996069i	$-0.471766\pi$
$109$	201.612			0.177164			0.0885821	+	0.996069i	$0.471766\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−1453.68	+	529.095i	−1.21018	+	0.440470i	−0.866766	−	0.498715i	$-0.833806\pi$
$113$	−1453.68	+	529.095i	−1.21018	+	0.440470i	−0.343414	+	0.939184i	$0.611583\pi$
$114$		0			0
$115$	218.442	+	1238.84i	0.177129	+	1.00455i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	2004.02	+	729.403i	1.54376	+	0.561884i
$120$		0			0
$121$	−42.6753	−	35.8088i	−0.0320626	−	0.0269037i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	361.010	−	625.287i	0.258318	−	0.447419i
$126$		0			0
$127$	1112.61	+	1927.10i	0.777387	+	1.34647i	0.933443	+	0.358725i	$0.116788\pi$
$127$	1112.61	+	1927.10i	0.777387	+	1.34647i	−0.156056	+	0.987748i	$0.549878\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−215.838	+	1224.08i	−0.143953	+	0.816397i	0.824249	+	0.566227i	$0.191598\pi$
$131$	−215.838	+	1224.08i	−0.143953	+	0.816397i	−0.968202	+	0.250170i	$0.919514\pi$
$132$		0			0
$133$	−1117.30	+	937.525i	−0.728437	+	0.611231i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−2055.53	+	1724.79i	−1.28187	+	1.07561i	−0.288883	+	0.957364i	$0.593284\pi$
$137$	−2055.53	+	1724.79i	−1.28187	+	1.07561i	−0.992984	+	0.118250i	$0.962272\pi$
$138$		0			0
$139$	−261.289	+	1481.85i	−0.159441	+	0.904234i	0.795172	+	0.606384i	$0.207381\pi$
$139$	−261.289	+	1481.85i	−0.159441	+	0.904234i	−0.954613	+	0.297850i	$0.903731\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	143.547	+	248.631i	0.0839440	+	0.145395i
$144$		0			0
$145$	787.445	−	1363.90i	0.450992	−	0.781140i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	172.536	+	144.775i	0.0948639	+	0.0796003i	0.688986	−	0.724774i	$-0.258056\pi$
$149$	172.536	+	144.775i	0.0948639	+	0.0796003i	−0.594122	+	0.804375i	$0.702500\pi$
$150$		0			0
$151$	−2151.99	−	783.262i	−1.15978	−	0.422126i	−0.310761	−	0.950488i	$-0.600584\pi$
$151$	−2151.99	−	783.262i	−1.15978	−	0.422126i	−0.849019	+	0.528362i	$0.822806\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	664.150	+	3766.58i	0.344167	+	1.95187i
$156$		0			0
$157$	−1002.97	+	365.052i	−0.509846	+	0.185569i	−0.584117	−	0.811669i	$-0.698559\pi$
$157$	−1002.97	+	365.052i	−0.509846	+	0.185569i	0.0742710	+	0.997238i	$0.476337\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	1630.27			0.798032
$162$		0			0
$163$	565.717			0.271843			0.135921	−	0.990720i	$-0.456601\pi$
$163$	565.717			0.271843			0.135921	+	0.990720i	$0.456601\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	1411.33	−	513.682i	0.653964	−	0.238023i	0.00633579	−	0.999980i	$-0.497983\pi$
$167$	1411.33	−	513.682i	0.653964	−	0.238023i	0.647628	+	0.761956i	$0.275761\pi$
$168$		0			0
$169$	−370.282	−	2099.97i	−0.168540	−	0.955837i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	388.119	+	141.264i	0.170567	+	0.0620814i	0.425892	−	0.904774i	$-0.359960\pi$
$173$	388.119	+	141.264i	0.170567	+	0.0620814i	−0.255325	+	0.966855i	$0.582182\pi$
$174$		0			0
$175$	1032.87	+	866.680i	0.446158	+	0.374371i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	699.370	−	1211.34i	0.292030	−	0.505811i	−0.682260	−	0.731110i	$-0.739003\pi$
$179$	699.370	−	1211.34i	0.292030	−	0.505811i	0.974290	+	0.225299i	$0.0723360\pi$
$180$		0			0
$181$	−807.914	−	1399.35i	−0.331778	−	0.574656i	0.651083	−	0.759007i	$-0.274315\pi$
$181$	−807.914	−	1399.35i	−0.331778	−	0.574656i	−0.982860	+	0.184351i	$0.940982\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−884.145	+	5014.24i	−0.351371	+	1.99272i
$186$		0			0
$187$	−3192.88	+	2679.15i	−1.24859	+	1.04769i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	1340.95	−	1125.19i	0.507998	−	0.426261i	−0.352426	−	0.935840i	$-0.614643\pi$
$191$	1340.95	−	1125.19i	0.507998	−	0.426261i	0.860424	+	0.509579i	$0.170199\pi$
$192$		0			0
$193$	−212.711	+	1206.35i	−0.0793331	+	0.449920i	0.919103	+	0.394017i	$0.128915\pi$
$193$	−212.711	+	1206.35i	−0.0793331	+	0.449920i	−0.998436	+	0.0559032i	$0.982196\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−2719.89	−	4710.98i	−0.983675	−	1.70377i	−0.647683	−	0.761910i	$-0.724262\pi$
$197$	−2719.89	−	4710.98i	−0.983675	−	1.70377i	−0.335992	−	0.941865i	$-0.609071\pi$
$198$		0			0
$199$	1103.11	−	1910.64i	0.392951	−	0.680611i	−0.599886	−	0.800085i	$-0.704788\pi$
$199$	1103.11	−	1910.64i	0.392951	−	0.680611i	0.992837	+	0.119474i	$0.0381209\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−1563.50	−	1311.93i	−0.540572	−	0.453594i
$204$		0			0
$205$	−181.998	−	66.2420i	−0.0620064	−	0.0225685i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−494.993	−	2807.25i	−0.163825	−	0.929097i
$210$		0			0
$211$	−445.805	+	162.260i	−0.145453	+	0.0529404i	−0.413721	−	0.910404i	$-0.635771\pi$
$211$	−445.805	+	162.260i	−0.145453	+	0.0529404i	0.268268	+	0.963344i	$0.413549\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	5757.06			1.82618
$216$		0			0
$217$	4956.67			1.55060
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−881.717	+	320.919i	−0.268374	+	0.0976803i
$222$		0			0
$223$	−257.186	−	1458.57i	−0.0772306	−	0.437996i	−0.998764	−	0.0496981i	$-0.984174\pi$
$223$	−257.186	−	1458.57i	−0.0772306	−	0.437996i	0.921534	−	0.388298i	$-0.126937\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−1055.78	−	384.271i	−0.308697	−	0.112357i	0.183026	−	0.983108i	$-0.441411\pi$
$227$	−1055.78	−	384.271i	−0.308697	−	0.112357i	−0.491723	+	0.870751i	$0.663633\pi$
$228$		0			0
$229$	2643.38	+	2218.06i	0.762793	+	0.640059i	0.938852	−	0.344321i	$-0.111891\pi$
$229$	2643.38	+	2218.06i	0.762793	+	0.640059i	−0.176060	+	0.984380i	$0.556335\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−3359.46	+	5818.75i	−0.944572	+	1.63605i	−0.187967	+	0.982175i	$0.560190\pi$
$233$	−3359.46	+	5818.75i	−0.944572	+	1.63605i	−0.756605	+	0.653872i	$0.773143\pi$
$234$		0			0
$235$	582.350	+	1008.66i	0.161652	+	0.279990i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	1084.54	−	6150.74i	0.293528	−	1.66468i	−0.379598	−	0.925152i	$-0.623938\pi$
$239$	1084.54	−	6150.74i	0.293528	−	1.66468i	0.673126	−	0.739528i	$-0.264951\pi$
$240$		0			0
$241$	−644.213	+	540.559i	−0.172189	+	0.144483i	−0.724809	−	0.688949i	$-0.758072\pi$
$241$	−644.213	+	540.559i	−0.172189	+	0.144483i	0.552621	+	0.833433i	$0.313628\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−98.5668	+	82.7074i	−0.0257029	+	0.0215673i
$246$		0			0
$247$	111.433	−	631.968i	0.0287057	−	0.162798i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−2582.47	−	4472.97i	−0.649419	−	1.12483i	−0.983262	−	0.182198i	$-0.941679\pi$
$251$	−2582.47	−	4472.97i	−0.649419	−	1.12483i	0.333843	−	0.942629i	$-0.391654\pi$
$252$		0			0
$253$	−1593.10	+	2759.33i	−0.395878	+	0.685681i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−3334.32	−	2797.83i	−0.809297	−	0.679081i	0.141143	−	0.989989i	$-0.454922\pi$
$257$	−3334.32	−	2797.83i	−0.809297	−	0.679081i	−0.950440	+	0.310909i	$0.899367\pi$
$258$		0			0
$259$	6200.59	+	2256.83i	1.48759	+	0.541438i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	267.218	+	1515.47i	0.0626517	+	0.355315i	0.999976	+	0.00685846i	$0.00218313\pi$
$263$	267.218	+	1515.47i	0.0626517	+	0.355315i	−0.937325	+	0.348457i	$0.886706\pi$
$264$		0			0
$265$	−382.041	+	139.051i	−0.0885606	+	0.0322334i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−3281.40			−0.743757			−0.371878	−	0.928281i	$-0.621286\pi$
$269$	−3281.40			−0.743757			−0.371878	+	0.928281i	$0.621286\pi$
$270$		0			0
$271$	−4649.81			−1.04227			−0.521136	−	0.853474i	$-0.674492\pi$
$271$	−4649.81			−1.04227			−0.521136	+	0.853474i	$0.674492\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−2476.23	+	901.273i	−0.542990	+	0.197632i
$276$		0			0
$277$	−1143.21	−	6483.47i	−0.247974	−	1.40633i	−0.813482	−	0.581590i	$-0.802431\pi$
$277$	−1143.21	−	6483.47i	−0.247974	−	1.40633i	0.565508	−	0.824743i	$-0.308680\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	4066.26	+	1480.00i	0.863249	+	0.314197i	0.735430	−	0.677601i	$-0.236980\pi$
$281$	4066.26	+	1480.00i	0.863249	+	0.314197i	0.127819	+	0.991798i	$0.459202\pi$
$282$		0			0
$283$	−954.503	−	800.923i	−0.200492	−	0.168233i	0.537014	−	0.843573i	$-0.319552\pi$
$283$	−954.503	−	800.923i	−0.200492	−	0.168233i	−0.737506	+	0.675340i	$0.763997\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−125.500	+	217.373i	−0.0258120	+	0.0447078i
$288$		0			0
$289$	−4354.62	−	7542.42i	−0.886347	−	1.53520i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	584.021	−	3312.15i	0.116447	−	0.660402i	−0.869577	−	0.493797i	$-0.835608\pi$
$293$	584.021	−	3312.15i	0.116447	−	0.660402i	0.986024	−	0.166605i	$-0.0532804\pi$
$294$		0			0
$295$	7231.18	−	6067.68i	1.42717	−	1.19754i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−549.466	+	461.057i	−0.106276	+	0.0891759i
$300$		0			0
$301$	1295.58	−	7347.61i	0.248093	−	1.40701i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	680.834	+	1179.24i	0.127818	+	0.221387i
$306$		0			0
$307$	1343.07	−	2326.27i	0.249685	−	0.432467i	−0.713753	−	0.700397i	$-0.753006\pi$
$307$	1343.07	−	2326.27i	0.249685	−	0.432467i	0.963438	+	0.267930i	$0.0863396\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−607.838	−	510.037i	−0.110827	−	0.0929953i	0.585690	−	0.810535i	$-0.300824\pi$
$311$	−607.838	−	510.037i	−0.110827	−	0.0929953i	−0.696517	+	0.717540i	$0.745268\pi$
$312$		0			0
$313$	1948.47	+	709.186i	0.351866	+	0.128069i	0.511905	−	0.859042i	$-0.328940\pi$
$313$	1948.47	+	709.186i	0.351866	+	0.128069i	−0.160039	+	0.987111i	$0.551162\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−1177.45	−	6677.64i	−0.208619	−	1.18314i	−0.891643	−	0.452739i	$-0.850447\pi$
$317$	−1177.45	−	6677.64i	−0.208619	−	1.18314i	0.683024	−	0.730396i	$-0.260664\pi$
$318$		0			0
$319$	3748.38	−	1364.30i	0.657896	−	0.239455i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	9316.41			1.60489
$324$		0			0
$325$	−593.225			−0.101250
$326$		0			0
$327$		0			0
$328$		0			0
$329$	1418.38	−	516.249i	0.237684	−	0.0865098i
$330$		0			0
$331$	−300.134	−	1702.14i	−0.0498394	−	0.282653i	0.949695	−	0.313178i	$-0.101393\pi$
$331$	−300.134	−	1702.14i	−0.0498394	−	0.282653i	−0.999534	+	0.0305241i	$0.990282\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−8894.51	−	3237.34i	−1.45062	−	0.527984i
$336$		0			0
$337$	−4592.96	−	3853.95i	−0.742417	−	0.622962i	0.191069	−	0.981577i	$-0.438805\pi$
$337$	−4592.96	−	3853.95i	−0.742417	−	0.622962i	−0.933486	+	0.358615i	$0.883249\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−4843.65	+	8389.45i	−0.769204	+	1.33230i
$342$		0			0
$343$	3217.06	+	5572.11i	0.506429	+	0.877160i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−550.059	+	3119.54i	−0.0850971	+	0.482610i	0.912238	+	0.409660i	$0.134352\pi$
$347$	−550.059	+	3119.54i	−0.0850971	+	0.482610i	−0.997336	+	0.0729501i	$0.976759\pi$
$348$		0			0
$349$	−5745.12	+	4820.73i	−0.881173	+	0.739392i	−0.966420	−	0.256968i	$-0.917276\pi$
$349$	−5745.12	+	4820.73i	−0.881173	+	0.739392i	0.0852471	+	0.996360i	$0.472832\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	6735.79	−	5652.00i	1.01561	−	0.852197i	0.0265392	−	0.999648i	$-0.491551\pi$
$353$	6735.79	−	5652.00i	1.01561	−	0.852197i	0.989069	+	0.147451i	$0.0471069\pi$
$354$		0			0
$355$	−233.948	+	1326.79i	−0.0349766	+	0.198362i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−3196.24	−	5536.04i	−0.469891	−	0.813875i	0.529516	−	0.848300i	$-0.322374\pi$
$359$	−3196.24	−	5536.04i	−0.469891	−	0.813875i	−0.999407	+	0.0344245i	$0.989040\pi$
$360$		0			0
$361$	243.700	−	422.101i	0.0355300	−	0.0615397i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	10863.5	+	9115.57i	1.55787	+	1.30721i
$366$		0			0
$367$	10794.5	+	3928.89i	1.53534	+	0.558819i	0.964922	−	0.262535i	$-0.0845585\pi$
$367$	10794.5	+	3928.89i	1.53534	+	0.558819i	0.570419	+	0.821354i	$0.306781\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	91.4929	+	518.882i	0.0128034	+	0.0726119i
$372$		0			0
$373$	2603.65	−	947.652i	0.361426	−	0.131548i	−0.154923	−	0.987927i	$-0.549513\pi$
$373$	2603.65	−	947.652i	0.361426	−	0.131548i	0.516349	+	0.856378i	$0.327291\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	897.991			0.122676
$378$		0			0
$379$	−8765.55			−1.18801			−0.594005	−	0.804461i	$-0.702454\pi$
$379$	−8765.55			−1.18801			−0.594005	+	0.804461i	$0.702454\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	3592.97	−	1307.73i	0.479353	−	0.174470i	−0.0910317	−	0.995848i	$-0.529016\pi$
$383$	3592.97	−	1307.73i	0.479353	−	0.174470i	0.570384	+	0.821378i	$0.306794\pi$
$384$		0			0
$385$	1597.59	+	9060.36i	0.211482	+	1.19937i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−8013.33	−	2916.61i	−1.04445	−	0.380149i	−0.237886	−	0.971293i	$-0.576454\pi$
$389$	−8013.33	−	2916.61i	−1.04445	−	0.380149i	−0.806566	+	0.591144i	$0.798677\pi$
$390$		0			0
$391$	−7977.12	−	6693.60i	−1.03176	−	0.865754i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−7181.45	+	12438.6i	−0.914780	+	1.58445i
$396$		0			0
$397$	6081.30	+	10533.1i	0.768796	+	1.33159i	0.938216	+	0.346050i	$0.112477\pi$
$397$	6081.30	+	10533.1i	0.768796	+	1.33159i	−0.169420	+	0.985544i	$0.554190\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	1077.38	−	6110.10i	0.134168	−	0.760907i	−0.841267	−	0.540620i	$-0.818190\pi$
$401$	1077.38	−	6110.10i	0.134168	−	0.760907i	0.975435	−	0.220287i	$-0.0706993\pi$
$402$		0			0
$403$	−1670.60	+	1401.80i	−0.206497	+	0.173272i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−9879.03	+	8289.49i	−1.20316	+	1.00957i
$408$		0			0
$409$	−773.536	+	4386.94i	−0.0935181	+	0.530367i	0.901673	+	0.432418i	$0.142339\pi$
$409$	−773.536	+	4386.94i	−0.0935181	+	0.530367i	−0.995191	+	0.0979494i	$0.968772\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−6116.72	−	10594.5i	−0.728775	−	1.26228i
$414$		0			0
$415$	−5042.65	+	8734.13i	−0.596467	+	1.03311i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	9991.52	+	8383.88i	1.16496	+	0.977517i	0.999961	−	0.00877723i	$-0.00279391\pi$
$419$	9991.52	+	8383.88i	1.16496	+	0.977517i	0.164998	+	0.986294i	$0.447238\pi$
$420$		0			0
$421$	−15259.8	−	5554.12i	−1.76655	−	0.642972i	−1.00000		0.000788484i	$-0.999749\pi$
$421$	−15259.8	−	5554.12i	−1.76655	−	0.642972i	−0.766551	−	0.642183i	$-0.778029\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−1495.53	−	8481.56i	−0.170691	−	0.968038i
$426$		0			0
$427$	1658.25	−	603.555i	0.187936	−	0.0684029i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	506.152			0.0565672			0.0282836	−	0.999600i	$-0.490996\pi$
$431$	506.152			0.0565672			0.0282836	+	0.999600i	$0.490996\pi$
$432$		0			0
$433$	3810.53			0.422916			0.211458	−	0.977387i	$-0.432179\pi$
$433$	3810.53			0.422916			0.211458	+	0.977387i	$0.432179\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	6692.34	−	2435.81i	0.732581	−	0.266638i
$438$		0			0
$439$	2784.60	+	15792.2i	0.302737	+	1.71691i	0.633969	+	0.773358i	$0.281425\pi$
$439$	2784.60	+	15792.2i	0.302737	+	1.71691i	−0.331232	+	0.943549i	$0.607464\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	7142.68	+	2599.72i	0.766048	+	0.278819i	0.695342	−	0.718679i	$-0.255253\pi$
$443$	7142.68	+	2599.72i	0.766048	+	0.278819i	0.0707054	+	0.997497i	$0.477475\pi$
$444$		0			0
$445$	6796.38	+	5702.84i	0.723999	+	0.607507i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−7613.88	+	13187.6i	−0.800270	+	1.38611i	0.119168	+	0.992874i	$0.461977\pi$
$449$	−7613.88	+	13187.6i	−0.800270	+	1.38611i	−0.919438	+	0.393234i	$0.871356\pi$
$450$		0			0
$451$	−245.278	−	424.834i	−0.0256091	−	0.0443562i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−359.649	+	2039.67i	−0.0370562	+	0.210156i
$456$		0			0
$457$	2349.87	−	1971.78i	0.240531	−	0.201829i	−0.514551	−	0.857460i	$-0.672042\pi$
$457$	2349.87	−	1971.78i	0.240531	−	0.201829i	0.755082	+	0.655630i	$0.227597\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	4161.68	−	3492.06i	0.420453	−	0.352802i	−0.407883	−	0.913034i	$-0.633733\pi$
$461$	4161.68	−	3492.06i	0.420453	−	0.352802i	0.828335	+	0.560233i	$0.189288\pi$
$462$		0			0
$463$	3055.02	−	17325.9i	0.306650	−	1.73910i	−0.308986	−	0.951067i	$-0.599989\pi$
$463$	3055.02	−	17325.9i	0.306650	−	1.73910i	0.615635	−	0.788031i	$-0.288899\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−761.856	−	1319.57i	−0.0754914	−	0.130755i	0.825808	−	0.563951i	$-0.190719\pi$
$467$	−761.856	−	1319.57i	−0.0754914	−	0.130755i	−0.901300	+	0.433196i	$0.857386\pi$
$468$		0			0
$469$	−6133.38	+	10623.3i	−0.603866	+	1.04593i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	11170.2	+	9372.93i	1.08585	+	0.911137i
$474$		0			0
$475$	5534.90	+	2014.54i	0.534650	+	0.194597i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−17.9267	−	101.668i	−0.00171001	−	0.00969793i	0.983941	−	0.178495i	$-0.0571228\pi$
$479$	−17.9267	−	101.668i	−0.00171001	−	0.00969793i	−0.985651	+	0.168797i	$0.946012\pi$
$480$		0			0
$481$	−2728.10	+	992.948i	−0.258609	+	0.0941259i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	23175.1			2.16974
$486$		0			0
$487$	−6424.90			−0.597823			−0.298912	−	0.954281i	$-0.596624\pi$
$487$	−6424.90			−0.597823			−0.298912	+	0.954281i	$0.596624\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	9033.50	−	3287.92i	0.830297	−	0.302203i	0.108316	−	0.994116i	$-0.465454\pi$
$491$	9033.50	−	3287.92i	0.830297	−	0.302203i	0.721981	+	0.691913i	$0.243232\pi$
$492$		0			0
$493$	2263.85	+	12838.9i	0.206813	+	1.17289i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	1640.70	+	597.166i	0.148079	+	0.0538964i
$498$		0			0
$499$	−15561.9	−	13058.0i	−1.39608	−	1.17145i	−0.962806	−	0.270192i	$-0.912913\pi$
$499$	−15561.9	−	13058.0i	−1.39608	−	1.17145i	−0.433277	−	0.901261i	$-0.642643\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−4198.74	+	7272.42i	−0.372192	+	0.644655i	−0.989902	−	0.141751i	$-0.954727\pi$
$503$	−4198.74	+	7272.42i	−0.372192	+	0.644655i	0.617711	+	0.786405i	$0.288060\pi$
$504$		0			0
$505$	−9004.96	−	15597.0i	−0.793496	−	1.37438i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−1856.65	+	10529.6i	−0.161679	+	0.916926i	0.790744	+	0.612147i	$0.209694\pi$
$509$	−1856.65	+	10529.6i	−0.161679	+	0.916926i	−0.952423	+	0.304779i	$0.901417\pi$
$510$		0			0
$511$	14078.7	−	11813.5i	1.21880	−	1.02269i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	545.777	−	457.961i	0.0466986	−	0.0391848i
$516$		0			0
$517$	−512.261	+	2905.17i	−0.0435768	+	0.247136i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	8883.34	+	15386.4i	0.746998	+	1.29384i	0.949255	+	0.314506i	$0.101839\pi$
$521$	8883.34	+	15386.4i	0.746998	+	1.29384i	−0.202257	+	0.979332i	$0.564828\pi$
$522$		0			0
$523$	−9911.83	+	17167.8i	−0.828708	+	1.43536i	0.0703449	+	0.997523i	$0.477590\pi$
$523$	−9911.83	+	17167.8i	−0.828708	+	1.43536i	−0.899052	+	0.437841i	$0.855743\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−24253.6	−	20351.2i	−2.00475	−	1.68219i
$528$		0			0
$529$	3952.90	+	1438.74i	0.324887	+	0.118249i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−19.1767	−	108.756i	−0.00155841	−	0.00883820i
$534$		0			0
$535$	−21002.0	+	7644.09i	−1.69719	+	0.617725i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−325.899			−0.0260436
$540$		0			0
$541$	879.526			0.0698960			0.0349480	−	0.999389i	$-0.488873\pi$
$541$	879.526			0.0698960			0.0349480	+	0.999389i	$0.488873\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−2671.16	+	972.221i	−0.209944	+	0.0764135i
$546$		0			0
$547$	−1578.98	−	8954.82i	−0.123423	−	0.699964i	−0.982232	−	0.187670i	$-0.939907\pi$
$547$	−1578.98	−	8954.82i	−0.123423	−	0.699964i	0.858810	−	0.512295i	$-0.171205\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−8378.43	−	3049.50i	−0.647792	−	0.235777i
$552$		0			0
$553$	14259.0	+	11964.7i	1.09648	+	0.920059i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−11427.3	+	19792.7i	−0.869283	+	1.50564i	−0.00655273	+	0.999979i	$0.502086\pi$
$557$	−11427.3	+	19792.7i	−0.869283	+	1.50564i	−0.862730	+	0.505664i	$0.831248\pi$
$558$		0			0
$559$	1641.32	+	2842.85i	0.124187	+	0.215098i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−999.815	+	5670.23i	−0.0748440	+	0.424462i	0.924245	+	0.381799i	$0.124695\pi$
$563$	−999.815	+	5670.23i	−0.0748440	+	0.424462i	−0.999090	+	0.0426629i	$0.986416\pi$
$564$		0			0
$565$	16708.4	−	14020.0i	1.24412	−	1.04394i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	912.392	−	765.588i	0.0672223	−	0.0564062i	−0.608557	−	0.793510i	$-0.708252\pi$
$569$	912.392	−	765.588i	0.0672223	−	0.0564062i	0.675780	+	0.737104i	$0.263807\pi$
$570$		0			0
$571$	1268.54	−	7194.23i	0.0929713	−	0.527267i	−0.902379	−	0.430944i	$-0.858181\pi$
$571$	1268.54	−	7194.23i	0.0929713	−	0.527267i	0.995350	−	0.0963229i	$-0.0307081\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−3291.83	−	5701.62i	−0.238746	−	0.413520i
$576$		0			0
$577$	5516.75	−	9555.29i	0.398033	−	0.689414i	−0.595450	−	0.803393i	$-0.703026\pi$
$577$	5516.75	−	9555.29i	0.398033	−	0.689414i	0.993483	+	0.113978i	$0.0363594\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	10012.4	+	8401.37i	0.714944	+	0.599910i
$582$		0			0
$583$	−967.646	−	352.194i	−0.0687406	−	0.0250195i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	1937.43	+	10987.7i	0.136229	+	0.772593i	0.973996	+	0.226565i	$0.0727496\pi$
$587$	1937.43	+	10987.7i	0.136229	+	0.772593i	−0.837767	+	0.546028i	$0.816139\pi$
$588$		0			0
$589$	20347.4	−	7405.84i	1.42343	−	0.518085i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−9356.50			−0.647935			−0.323967	−	0.946068i	$-0.605017\pi$
$593$	−9356.50			−0.647935			−0.323967	+	0.946068i	$0.605017\pi$
$594$		0			0
$595$	−30068.6			−2.07175
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−2260.98	+	822.930i	−0.154226	+	0.0561335i	−0.417979	−	0.908457i	$-0.637262\pi$
$599$	−2260.98	+	822.930i	−0.154226	+	0.0561335i	0.263754	+	0.964590i	$0.415039\pi$
$600$		0			0
$601$	1862.04	+	10560.2i	0.126380	+	0.716736i	0.980479	+	0.196625i	$0.0629983\pi$
$601$	1862.04	+	10560.2i	0.126380	+	0.716736i	−0.854099	+	0.520111i	$0.825891\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	738.084	+	268.641i	0.0495990	+	0.0180526i
$606$		0			0
$607$	6535.29	+	5483.76i	0.437000	+	0.366687i	0.834585	−	0.550879i	$-0.185707\pi$
$607$	6535.29	+	5483.76i	0.437000	+	0.366687i	−0.397585	+	0.917565i	$0.630152\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−332.051	+	575.130i	−0.0219859	+	0.0380806i
$612$		0			0
$613$	2670.34	+	4625.17i	0.175945	+	0.304745i	0.940488	−	0.339827i	$-0.110369\pi$
$613$	2670.34	+	4625.17i	0.175945	+	0.304745i	−0.764543	+	0.644573i	$0.777035\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	3028.80	−	17177.2i	0.197625	−	1.12079i	−0.711005	−	0.703187i	$-0.751760\pi$
$617$	3028.80	−	17177.2i	0.197625	−	1.12079i	0.908630	−	0.417601i	$-0.137129\pi$
$618$		0			0
$619$	−6065.97	+	5089.96i	−0.393881	+	0.330505i	−0.818123	−	0.575044i	$-0.804985\pi$
$619$	−6065.97	+	5089.96i	−0.393881	+	0.330505i	0.424242	+	0.905549i	$0.360541\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	8807.88	−	7390.69i	0.566421	−	0.475284i
$624$		0			0
$625$	−3369.43	+	19109.0i	−0.215644	+	1.22298i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−21074.1	−	36501.5i	−1.33590	−	2.31385i
$630$		0			0
$631$	8251.31	−	14291.7i	0.520570	−	0.901653i	−0.479144	−	0.877736i	$-0.659053\pi$
$631$	8251.31	−	14291.7i	0.520570	−	0.901653i	0.999714	−	0.0239171i	$-0.00761378\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−24033.9	−	20166.8i	−1.50198	−	1.26031i
$636$		0			0
$637$	−68.9420	−	25.0928i	−0.00428820	−	0.00156078i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−3265.74	−	18520.9i	−0.201231	−	1.14124i	−0.903262	−	0.429089i	$-0.858835\pi$
$641$	−3265.74	−	18520.9i	−0.201231	−	1.14124i	0.702031	−	0.712146i	$-0.252277\pi$
$642$		0			0
$643$	−955.782	+	347.876i	−0.0586195	+	0.0213358i	−0.371163	−	0.928568i	$-0.621041\pi$
$643$	−955.782	+	347.876i	−0.0586195	+	0.0213358i	0.312544	+	0.949903i	$0.398819\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	20407.3			1.24002			0.620011	−	0.784593i	$-0.287128\pi$
$647$	20407.3			1.24002			0.620011	+	0.784593i	$0.287128\pi$
$648$		0			0
$649$	23909.0			1.44609
$650$		0			0
$651$		0			0
$652$		0			0
$653$	24273.6	−	8834.88i	1.45467	−	0.529457i	0.510780	−	0.859711i	$-0.329357\pi$
$653$	24273.6	−	8834.88i	1.45467	−	0.529457i	0.943892	+	0.330254i	$0.107134\pi$
$654$		0			0
$655$	−3043.16	−	17258.6i	−0.181536	−	1.02954i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	11124.7	+	4049.07i	0.657599	+	0.239347i	0.649199	−	0.760618i	$-0.275104\pi$
$659$	11124.7	+	4049.07i	0.657599	+	0.239347i	0.00839966	+	0.999965i	$0.497326\pi$
$660$		0			0
$661$	−5401.80	−	4532.65i	−0.317860	−	0.266717i	0.469871	−	0.882735i	$-0.344300\pi$
$661$	−5401.80	−	4532.65i	−0.317860	−	0.266717i	−0.787732	+	0.616018i	$0.788745\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	10282.1	−	17809.2i	0.599585	−	1.03851i
$666$		0			0
$667$	4982.99	+	8630.80i	0.289269	+	0.501028i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−598.892	+	3396.48i	−0.0344560	+	0.195410i
$672$		0			0
$673$	−17454.9	+	14646.4i	−0.999756	+	0.838895i	−0.986951	−	0.161023i	$-0.948521\pi$
$673$	−17454.9	+	14646.4i	−0.999756	+	0.838895i	−0.0128057	+	0.999918i	$0.504076\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	8712.25	−	7310.45i	0.494592	−	0.415012i	−0.361076	−	0.932536i	$-0.617591\pi$
$677$	8712.25	−	7310.45i	0.494592	−	0.415012i	0.855669	+	0.517524i	$0.173146\pi$
$678$		0			0
$679$	5215.37	−	29577.8i	0.294768	−	1.67171i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	5758.82	+	9974.56i	0.322628	+	0.558808i	0.981029	−	0.193859i	$-0.0621003\pi$
$683$	5758.82	+	9974.56i	0.322628	+	0.558808i	−0.658401	+	0.752667i	$0.728767\pi$
$684$		0			0
$685$	18916.4	−	32764.1i	1.05512	−	1.82752i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−177.582	−	149.009i	−0.00981907	−	0.00823918i
$690$		0			0
$691$	−13056.2	−	4752.05i	−0.718784	−	0.261616i	−0.0433746	−	0.999059i	$-0.513811\pi$
$691$	−13056.2	−	4752.05i	−0.718784	−	0.261616i	−0.675410	+	0.737443i	$0.736033\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−3684.00	−	20893.0i	−0.201068	−	1.14031i
$696$		0			0
$697$	1506.59	−	548.352i	0.0818738	−	0.0297996i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−25381.5			−1.36754			−0.683771	−	0.729696i	$-0.739661\pi$
$701$	−25381.5			−1.36754			−0.683771	+	0.729696i	$0.739661\pi$
$702$		0			0
$703$	28825.7			1.54649
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−21932.7	+	7982.83i	−1.16671	+	0.424647i
$708$		0			0
$709$	−3197.43	−	18133.5i	−0.169368	−	0.960535i	−0.944445	−	0.328668i	$-0.893400\pi$
$709$	−3197.43	−	18133.5i	−0.169368	−	0.960535i	0.775077	−	0.631867i	$-0.217711\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−22743.2	−	8277.85i	−1.19459	−	0.434794i
$714$		0			0
$715$	−3100.81	−	2601.89i	−0.162187	−	0.136091i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	4178.46	−	7237.30i	0.216732	−	0.375391i	−0.737075	−	0.675811i	$-0.763794\pi$
$719$	4178.46	−	7237.30i	0.216732	−	0.375391i	0.953807	+	0.300420i	$0.0971269\pi$
$720$		0			0
$721$	−461.662	−	799.623i	−0.0238463	−	0.0413031i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−1431.27	+	8117.16i	−0.0733189	+	0.415812i
$726$		0			0
$727$	−2910.27	+	2442.01i	−0.148468	+	0.124579i	−0.713996	−	0.700149i	$-0.753117\pi$
$727$	−2910.27	+	2442.01i	−0.148468	+	0.124579i	0.565528	+	0.824729i	$0.308672\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−36507.5	+	30633.4i	−1.84716	+	1.54996i
$732$		0			0
$733$	−3398.64	+	19274.7i	−0.171257	+	0.971249i	0.771118	+	0.636692i	$0.219698\pi$
$733$	−3398.64	+	19274.7i	−0.171257	+	0.971249i	−0.942375	+	0.334557i	$0.891413\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−11987.1	−	20762.2i	−0.599117	−	1.03770i
$738$		0			0
$739$	−3828.18	+	6630.60i	−0.190557	+	0.330055i	−0.945435	−	0.325811i	$-0.894363\pi$
$739$	−3828.18	+	6630.60i	−0.190557	+	0.330055i	0.754878	+	0.655866i	$0.227696\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	8817.80	+	7399.01i	0.435388	+	0.365334i	0.833980	−	0.551794i	$-0.186057\pi$
$743$	8817.80	+	7399.01i	0.435388	+	0.365334i	−0.398592	+	0.917128i	$0.630501\pi$
$744$		0			0
$745$	−2984.08	−	1086.12i	−0.146749	−	0.0534123i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	5029.65	+	28524.6i	0.245366	+	1.39154i
$750$		0			0
$751$	21813.3	−	7939.39i	1.05989	−	0.385769i	0.247504	−	0.968887i	$-0.420390\pi$
$751$	21813.3	−	7939.39i	1.05989	−	0.385769i	0.812387	+	0.583118i	$0.198168\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	32288.9			1.55644
$756$		0			0
$757$	−2092.13			−0.100449			−0.0502244	−	0.998738i	$-0.515994\pi$
$757$	−2092.13			−0.100449			−0.0502244	+	0.998738i	$0.515994\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−22912.8	+	8339.57i	−1.09144	+	0.397253i	−0.824155	−	0.566364i	$-0.808350\pi$
$761$	−22912.8	+	8339.57i	−1.09144	+	0.397253i	−0.267288	+	0.963617i	$0.586127\pi$
$762$		0			0
$763$	639.701	+	3627.93i	0.0303522	+	0.172136i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	5057.81	+	1840.89i	0.238105	+	0.0866633i
$768$		0			0
$769$	3200.87	+	2685.85i	0.150099	+	0.125948i	0.714745	−	0.699385i	$-0.246543\pi$
$769$	3200.87	+	2685.85i	0.150099	+	0.125948i	−0.564646	+	0.825333i	$0.690987\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	7133.99	−	12356.4i	0.331943	−	0.574942i	−0.650950	−	0.759121i	$-0.725629\pi$
$773$	7133.99	−	12356.4i	0.331943	−	0.574942i	0.982893	+	0.184179i	$0.0589625\pi$
$774$		0			0
$775$	−10008.5	−	17335.2i	−0.463891	−	0.803482i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−190.405	+	1079.84i	−0.00875734	+	0.0496653i
$780$		0			0
$781$	−2614.03	+	2193.43i	−0.119766	+	0.100496i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	11528.0	−	9673.16i	0.524143	−	0.439809i
$786$		0			0
$787$	−3425.45	+	19426.7i	−0.155151	+	0.879908i	0.803496	+	0.595310i	$0.202971\pi$
$787$	−3425.45	+	19426.7i	−0.155151	+	0.879908i	−0.958647	+	0.284597i	$0.908140\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−14133.3	−	24479.6i	−0.635300	−	1.10037i
$792$		0			0
$793$	−388.206	+	672.393i	−0.0173841	+	0.0301102i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	7710.37	+	6469.77i	0.342679	+	0.287542i	0.797843	−	0.602866i	$-0.205975\pi$
$797$	7710.37	+	6469.77i	0.342679	+	0.287542i	−0.455163	+	0.890408i	$0.650419\pi$
$798$		0			0
$799$	−9059.96	−	3297.55i	−0.401149	−	0.146006i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	6237.25	+	35373.2i	0.274107	+	1.55454i
$804$		0			0
$805$	−21599.4	+	7861.55i	−0.945690	+	0.344203i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	22978.1			0.998601			0.499300	−	0.866429i	$-0.333590\pi$
$809$	22978.1			0.998601			0.499300	+	0.866429i	$0.333590\pi$
$810$		0			0
$811$	26002.0			1.12584			0.562919	−	0.826512i	$-0.309678\pi$
$811$	26002.0			1.12584			0.562919	+	0.826512i	$0.309678\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−7495.19	+	2728.03i	−0.322141	+	0.117250i
$816$		0			0
$817$	−5659.76	−	32098.1i	−0.242362	−	1.37450i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	26141.6	+	9514.78i	1.11127	+	0.404468i	0.831458	−	0.555588i	$-0.187507\pi$
$821$	26141.6	+	9514.78i	1.11127	+	0.404468i	0.279809	+	0.960056i	$0.409729\pi$
$822$		0			0
$823$	1726.87	+	1449.01i	0.0731407	+	0.0613723i	0.678625	−	0.734485i	$-0.262576\pi$
$823$	1726.87	+	1449.01i	0.0731407	+	0.0613723i	−0.605484	+	0.795857i	$0.707021\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−821.250	+	1422.45i	−0.0345316	+	0.0598105i	−0.882775	−	0.469797i	$-0.844327\pi$
$827$	−821.250	+	1422.45i	−0.0345316	+	0.0598105i	0.848243	+	0.529607i	$0.177661\pi$
$828$		0			0
$829$	−5372.26	−	9305.02i	−0.225074	−	0.389839i	0.731268	−	0.682090i	$-0.238929\pi$
$829$	−5372.26	−	9305.02i	−0.225074	−	0.389839i	−0.956342	+	0.292251i	$0.905596\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	184.958	−	1048.95i	0.00769318	−	0.0436302i
$834$		0			0
$835$	−16221.6	+	13611.6i	−0.672302	+	0.564129i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−6016.97	+	5048.83i	−0.247591	+	0.207753i	−0.758134	−	0.652099i	$-0.773889\pi$
$839$	−6016.97	+	5048.83i	−0.247591	+	0.207753i	0.510543	+	0.859852i	$0.329444\pi$
$840$		0			0
$841$	−2068.52	+	11731.2i	−0.0848137	+	0.481002i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	15032.5	+	26037.0i	0.611991	+	1.06000i
$846$		0			0
$847$	508.960	−	881.545i	0.0206471	−	0.0357618i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−24681.8	−	20710.5i	−0.994221	−	0.834251i
$852$		0			0
$853$	21669.0	+	7886.87i	0.869792	+	0.316578i	0.738083	−	0.674710i	$-0.235731\pi$
$853$	21669.0	+	7886.87i	0.869792	+	0.316578i	0.131709	+	0.991288i	$0.457954\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	3543.22	+	20094.6i	0.141230	+	0.800955i	0.970317	+	0.241836i	$0.0777495\pi$
$857$	3543.22	+	20094.6i	0.141230	+	0.800955i	−0.829087	+	0.559119i	$0.811139\pi$
$858$		0			0
$859$	9749.16	−	3548.41i	0.387238	−	0.140943i	−0.141062	−	0.990001i	$-0.545052\pi$
$859$	9749.16	−	3548.41i	0.387238	−	0.140943i	0.528300	+	0.849058i	$0.322830\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	4579.50			0.180635			0.0903175	−	0.995913i	$-0.471212\pi$
$863$	4579.50			0.180635			0.0903175	+	0.995913i	$0.471212\pi$
$864$		0			0
$865$	−5823.40			−0.228904
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−34185.0	+	12442.3i	−1.33446	+	0.485703i
$870$		0			0
$871$	−937.191	−	5315.07i	−0.0364587	−	0.206767i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	12397.3	+	4512.24i	0.478976	+	0.174333i
$876$		0			0
$877$	2605.11	+	2185.94i	0.100306	+	0.0841665i	0.691561	−	0.722318i	$-0.256923\pi$
$877$	2605.11	+	2185.94i	0.100306	+	0.0841665i	−0.591256	+	0.806484i	$0.701368\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	12076.5	−	20917.0i	0.461823	−	0.799902i	−0.537228	−	0.843437i	$-0.680529\pi$
$881$	12076.5	−	20917.0i	0.461823	−	0.799902i	0.999052	+	0.0435352i	$0.0138620\pi$
$882$		0			0
$883$	9087.36	+	15739.8i	0.346335	+	0.599870i	0.985595	−	0.169120i	$-0.0540926\pi$
$883$	9087.36	+	15739.8i	0.346335	+	0.599870i	−0.639260	+	0.768991i	$0.720759\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	4251.25	−	24110.0i	0.160928	−	0.912667i	−0.792237	−	0.610214i	$-0.791083\pi$
$887$	4251.25	−	24110.0i	0.160928	−	0.912667i	0.953164	−	0.302453i	$-0.0978055\pi$
$888$		0			0
$889$	−31147.1	+	26135.5i	−1.17507	+	0.986004i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	5051.20	−	4238.46i	0.189285	−	0.158829i
$894$		0			0
$895$	−3424.56	+	19421.6i	−0.127900	+	0.725356i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	15150.3	+	26241.1i	0.562058	+	0.973514i
$900$		0			0
$901$	1682.75	−	2914.61i	0.0622204	−	0.107769i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	17452.1	+	14644.0i	0.641023	+	0.537882i
$906$		0			0
$907$	6108.70	+	2223.38i	0.223634	+	0.0813961i	0.451407	−	0.892318i	$-0.350922\pi$
$907$	6108.70	+	2223.38i	0.223634	+	0.0813961i	−0.227773	+	0.973714i	$0.573144\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	2777.96	+	15754.6i	0.101029	+	0.572966i	0.992732	+	0.120345i	$0.0384000\pi$
$911$	2777.96	+	15754.6i	0.101029	+	0.572966i	−0.891703	+	0.452621i	$0.850489\pi$
$912$		0			0
$913$	−24003.9	+	8736.70i	−0.870113	+	0.316695i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−22711.6			−0.817889
$918$		0			0
$919$	11885.2			0.426611			0.213305	−	0.976986i	$-0.431577\pi$
$919$	11885.2			0.426611			0.213305	+	0.976986i	$0.431577\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−721.866	+	262.738i	−0.0257427	+	0.00936958i
$924$		0			0
$925$	−4627.28	−	26242.6i	−0.164480	−	0.932813i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	16750.5	+	6096.69i	0.591568	+	0.215313i	0.620419	−	0.784271i	$-0.286963\pi$
$929$	16750.5	+	6096.69i	0.591568	+	0.215313i	−0.0288511	+	0.999584i	$0.509185\pi$
$930$		0			0
$931$	558.030	+	468.242i	0.0196441	+	0.0164834i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	29383.0	−	50892.9i	1.02773	−	1.78008i
$936$		0			0
$937$	3408.22	+	5903.21i	0.118828	+	0.205816i	0.919303	−	0.393550i	$-0.128753\pi$
$937$	3408.22	+	5903.21i	0.118828	+	0.205816i	−0.800476	+	0.599365i	$0.795420\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	512.291	−	2905.35i	0.0177473	−	0.100650i	−0.974647	−	0.223746i	$-0.928171\pi$
$941$	512.291	−	2905.35i	0.0177473	−	0.100650i	0.992395	+	0.123096i	$0.0392824\pi$
$942$		0			0
$943$	938.870	−	787.805i	0.0324219	−	0.0272052i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	5209.50	−	4371.29i	0.178760	−	0.149998i	−0.549017	−	0.835811i	$-0.684998\pi$
$947$	5209.50	−	4371.29i	0.178760	−	0.149998i	0.727777	+	0.685813i	$0.240553\pi$
$948$		0			0
$949$	−1404.13	+	7963.23i	−0.0480296	+	0.272389i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	21077.2	+	36506.7i	0.716429	+	1.24089i	0.962406	+	0.271615i	$0.0875579\pi$
$953$	21077.2	+	36506.7i	0.716429	+	1.24089i	−0.245977	+	0.969276i	$0.579109\pi$
$954$		0			0
$955$	−12340.3	+	21374.0i	−0.418139	+	0.724238i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−37559.1	−	31515.8i	−1.26470	−	1.06121i
$960$		0			0
$961$	−41154.1	−	14978.9i	−1.38143	−	0.502798i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−2999.08	−	17008.6i	−0.100045	−	0.567386i
$966$		0			0
$967$	−15811.2	+	5754.81i	−0.525806	+	0.191378i	−0.591265	−	0.806478i	$-0.701371\pi$
$967$	−15811.2	+	5754.81i	−0.525806	+	0.191378i	0.0654587	+	0.997855i	$0.479149\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−44445.5			−1.46892			−0.734461	−	0.678651i	$-0.762565\pi$
$971$	−44445.5			−1.46892			−0.734461	+	0.678651i	$0.762565\pi$
$972$		0			0
$973$	−27494.3			−0.905886
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−14594.3	+	5311.89i	−0.477905	+	0.173943i	−0.569730	−	0.821832i	$-0.692952\pi$
$977$	−14594.3	+	5311.89i	−0.477905	+	0.173943i	0.0918251	+	0.995775i	$0.470730\pi$
$978$		0			0
$979$	3902.12	+	22130.0i	0.127388	+	0.722451i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	50650.6	+	18435.3i	1.64344	+	0.598164i	0.987636	−	0.156765i	$-0.0501066\pi$
$983$	50650.6	+	18435.3i	1.64344	+	0.598164i	0.655806	+	0.754929i	$0.272329\pi$
$984$		0			0
$985$	58753.3	+	49299.9i	1.90055	+	1.59475i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−18215.5	+	31550.2i	−0.585661	+	1.01440i
$990$		0			0
$991$	−14415.8	−	24969.0i	−0.462093	−	0.800369i	0.536972	−	0.843600i	$-0.319568\pi$
$991$	−14415.8	−	24969.0i	−0.462093	−	0.800369i	−0.999065	+	0.0432313i	$0.986235\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−5401.52	+	30633.5i	−0.172100	+	0.976028i
$996$		0			0
$997$	−24165.5	+	20277.2i	−0.767631	+	0.644119i	−0.940101	−	0.340896i	$-0.889270\pi$
$997$	−24165.5	+	20277.2i	−0.767631	+	0.644119i	0.172470	+	0.985015i	$0.444825\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	324.4.i.a.37.2		54
3.2	odd	2		108.4.i.a.49.4	✓	54
27.11	odd	18		108.4.i.a.97.4	yes	54
27.16	even	9	inner	324.4.i.a.289.2		54

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
108.4.i.a.49.4	✓	54	3.2	odd	2
108.4.i.a.97.4	yes	54	27.11	odd	18
324.4.i.a.37.2		54	1.1	even	1	trivial
324.4.i.a.289.2		54	27.16	even	9	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:	\( N \)	\(=\)	\( 324 = 2^{2} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	324.i (of order \(9\), degree \(6\), not minimal)

\(f(q)\)	\(=\)	\(q+(-13.2490 + 4.82225i) q^{5} +(3.17294 + 17.9946i) q^{7} +O(q^{10})\)	\(q+(-13.2490 + 4.82225i) q^{5} +(3.17294 + 17.9946i) q^{7} +(-33.5576 - 12.2140i) q^{11} +(-6.15848 - 5.16757i) q^{13} +(58.3572 - 101.078i) q^{17} +(39.9112 + 69.1281i) q^{19} +(15.4931 - 87.8657i) q^{23} +(56.5267 - 47.4316i) q^{25} +(-85.5671 + 71.7993i) q^{29} +(47.1052 - 267.147i) q^{31} +(-128.813 - 223.110i) q^{35} +(180.562 - 312.742i) q^{37} +(10.5230 + 8.82981i) q^{41} +(-383.698 - 139.655i) q^{43} +(-14.3445 - 81.3518i) q^{47} +(8.57561 - 3.12127i) q^{49} +28.8354 q^{53} +503.503 q^{55} +(-629.134 + 228.986i) q^{59} +(-16.7704 - 95.1097i) q^{61} +(106.513 + 38.7676i) q^{65} +(514.271 + 431.525i) q^{67} +(47.7773 - 82.7527i) q^{71} +(-502.908 - 871.062i) q^{73} +(113.309 - 642.610i) q^{77} +(780.366 - 654.805i) q^{79} +(547.955 - 459.789i) q^{83} +(-285.754 + 1620.59i) q^{85} +(-314.627 - 544.950i) q^{89} +(73.4481 - 127.216i) q^{91} +(-862.136 - 723.418i) q^{95} +(-1544.58 - 562.179i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 54 q - 12 q^{5}+O(q^{10}) \) 54 * q - 12 * q^5	\( 54 q - 12 q^{5} + 87 q^{11} - 204 q^{17} - 96 q^{23} - 216 q^{25} - 318 q^{29} - 54 q^{31} - 6 q^{35} - 867 q^{41} - 513 q^{43} + 1548 q^{47} + 594 q^{49} + 1068 q^{53} + 1218 q^{59} - 54 q^{61} - 96 q^{65} - 2997 q^{67} + 120 q^{71} - 216 q^{73} - 3480 q^{77} + 2808 q^{79} - 4464 q^{83} + 2160 q^{85} - 4029 q^{89} + 270 q^{91} + 1650 q^{95} - 3483 q^{97}+O(q^{100}) \) 54 * q - 12 * q^5 + 87 * q^11 - 204 * q^17 - 96 * q^23 - 216 * q^25 - 318 * q^29 - 54 * q^31 - 6 * q^35 - 867 * q^41 - 513 * q^43 + 1548 * q^47 + 594 * q^49 + 1068 * q^53 + 1218 * q^59 - 54 * q^61 - 96 * q^65 - 2997 * q^67 + 120 * q^71 - 216 * q^73 - 3480 * q^77 + 2808 * q^79 - 4464 * q^83 + 2160 * q^85 - 4029 * q^89 + 270 * q^91 + 1650 * q^95 - 3483 * q^97

Introduction

L-functions

Modular forms

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