LMFDB - Embedded newform 324.4.i.a.37.9

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			37.9
Character	$\chi$	$=$	324.37
Dual form			324.4.i.a.289.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(13.0875 - 4.76347i) q^{5} +(-0.689494 - 3.91032i) q^{7} +O(q^{10})$	$q+(13.0875 - 4.76347i) q^{5} +(-0.689494 - 3.91032i) q^{7} +(41.0823 + 14.9527i) q^{11} +(4.04425 + 3.39352i) q^{13} +(3.06285 - 5.30501i) q^{17} +(-7.42787 - 12.8654i) q^{19} +(21.3706 - 121.199i) q^{23} +(52.8371 - 44.3356i) q^{25} +(108.223 - 90.8101i) q^{29} +(-46.8412 + 265.650i) q^{31} +(-27.6504 - 47.8920i) q^{35} +(100.261 - 173.657i) q^{37} +(-239.406 - 200.886i) q^{41} +(462.032 + 168.166i) q^{43} +(10.6113 + 60.1798i) q^{47} +(307.499 - 111.921i) q^{49} +380.159 q^{53} +608.892 q^{55} +(154.748 - 56.3238i) q^{59} +(-149.877 - 849.997i) q^{61} +(69.0941 + 25.1482i) q^{65} +(159.359 + 133.718i) q^{67} +(451.411 - 781.866i) q^{71} +(353.597 + 612.447i) q^{73} +(30.1439 - 170.954i) q^{77} +(-847.858 + 711.438i) q^{79} +(-652.401 + 547.430i) q^{83} +(14.8148 - 84.0192i) q^{85} +(-214.477 - 371.485i) q^{89} +(10.4813 - 18.1541i) q^{91} +(-158.497 - 132.994i) q^{95} +(-1166.45 - 424.552i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	13.0875	−	4.76347i	1.17058	−	0.426058i	0.317717	−	0.948186i	$-0.397084\pi$
$5$	13.0875	−	4.76347i	1.17058	−	0.426058i	0.852867	+	0.522128i	$0.174862\pi$
$6$		0			0
$7$	−0.689494	−	3.91032i	−0.0372292	−	0.211137i	0.960518	−	0.278216i	$-0.0897432\pi$
$7$	−0.689494	−	3.91032i	−0.0372292	−	0.211137i	−0.997748	+	0.0670790i	$0.978632\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	41.0823	+	14.9527i	1.12607	+	0.409856i	0.836864	−	0.547411i	$-0.184387\pi$
$11$	41.0823	+	14.9527i	1.12607	+	0.409856i	0.289206	+	0.957267i	$0.406609\pi$
$12$		0			0
$13$	4.04425	+	3.39352i	0.0862825	+	0.0723996i	0.684909	−	0.728629i	$-0.259842\pi$
$13$	4.04425	+	3.39352i	0.0862825	+	0.0723996i	−0.598626	+	0.801028i	$0.704287\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	3.06285	−	5.30501i	0.0436970	−	0.0756855i	−0.843350	−	0.537365i	$-0.819420\pi$
$17$	3.06285	−	5.30501i	0.0436970	−	0.0756855i	0.887047	+	0.461680i	$0.152753\pi$
$18$		0			0
$19$	−7.42787	−	12.8654i	−0.0896879	−	0.155344i	0.817691	−	0.575657i	$-0.195254\pi$
$19$	−7.42787	−	12.8654i	−0.0896879	−	0.155344i	−0.907379	+	0.420313i	$0.861920\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	21.3706	−	121.199i	0.193743	−	1.09877i	−0.720455	−	0.693502i	$-0.756067\pi$
$23$	21.3706	−	121.199i	0.193743	−	1.09877i	0.914198	−	0.405268i	$-0.132822\pi$
$24$		0			0
$25$	52.8371	−	44.3356i	0.422697	−	0.354685i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	108.223	−	90.8101i	0.692985	−	0.581483i	−0.226784	−	0.973945i	$-0.572821\pi$
$29$	108.223	−	90.8101i	0.692985	−	0.581483i	0.919768	+	0.392462i	$0.128377\pi$
$30$		0			0
$31$	−46.8412	+	265.650i	−0.271385	+	1.53910i	0.478831	+	0.877907i	$0.341061\pi$
$31$	−46.8412	+	265.650i	−0.271385	+	1.53910i	−0.750216	+	0.661193i	$0.770050\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−27.6504	−	47.8920i	−0.133537	−	0.231292i
$36$		0			0
$37$	100.261	−	173.657i	0.445481	−	0.771595i	−0.552605	−	0.833443i	$-0.686366\pi$
$37$	100.261	−	173.657i	0.445481	−	0.771595i	0.998086	+	0.0618483i	$0.0196995\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−239.406	−	200.886i	−0.911927	−	0.765197i	0.0605578	−	0.998165i	$-0.480712\pi$
$41$	−239.406	−	200.886i	−0.911927	−	0.765197i	−0.972485	+	0.232967i	$0.925156\pi$
$42$		0			0
$43$	462.032	+	168.166i	1.63859	+	0.596397i	0.986791	−	0.161998i	$-0.0517939\pi$
$43$	462.032	+	168.166i	1.63859	+	0.596397i	0.651795	+	0.758395i	$0.274016\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	10.6113	+	60.1798i	0.0329323	+	0.186769i	0.996836	−	0.0794815i	$-0.0253264\pi$
$47$	10.6113	+	60.1798i	0.0329323	+	0.186769i	−0.963904	+	0.266250i	$0.914215\pi$
$48$		0			0
$49$	307.499	−	111.921i	0.896500	−	0.326299i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	380.159			0.985262			0.492631	−	0.870238i	$-0.336035\pi$
$53$	380.159			0.985262			0.492631	+	0.870238i	$0.336035\pi$
$54$		0			0
$55$	608.892			1.49278
$56$		0			0
$57$		0			0
$58$		0			0
$59$	154.748	−	56.3238i	0.341467	−	0.124284i	−0.165594	−	0.986194i	$-0.552954\pi$
$59$	154.748	−	56.3238i	0.341467	−	0.124284i	0.507061	+	0.861910i	$0.330732\pi$
$60$		0			0
$61$	−149.877	−	849.997i	−0.314588	−	1.78411i	−0.574523	−	0.818488i	$-0.694812\pi$
$61$	−149.877	−	849.997i	−0.314588	−	1.78411i	0.259935	−	0.965626i	$-0.416299\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	69.0941	+	25.1482i	0.131847	+	0.0479885i
$66$		0			0
$67$	159.359	+	133.718i	0.290579	+	0.243825i	0.776410	−	0.630228i	$-0.217039\pi$
$67$	159.359	+	133.718i	0.290579	+	0.243825i	−0.485831	+	0.874053i	$0.661483\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	451.411	−	781.866i	0.754543	−	1.30691i	−0.191058	−	0.981579i	$-0.561192\pi$
$71$	451.411	−	781.866i	0.754543	−	1.30691i	0.945601	−	0.325329i	$-0.105475\pi$
$72$		0			0
$73$	353.597	+	612.447i	0.566923	+	0.981939i	0.996868	+	0.0790840i	$0.0251995\pi$
$73$	353.597	+	612.447i	0.566923	+	0.981939i	−0.429945	+	0.902855i	$0.641467\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	30.1439	−	170.954i	0.0446132	−	0.253014i
$78$		0			0
$79$	−847.858	+	711.438i	−1.20749	+	1.01320i	−0.208104	+	0.978107i	$0.566729\pi$
$79$	−847.858	+	711.438i	−1.20749	+	1.01320i	−0.999384	+	0.0350955i	$0.988826\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−652.401	+	547.430i	−0.862775	+	0.723954i	−0.962564	−	0.271054i	$-0.912628\pi$
$83$	−652.401	+	547.430i	−0.862775	+	0.723954i	0.0997890	+	0.995009i	$0.468183\pi$
$84$		0			0
$85$	14.8148	−	84.0192i	0.0189047	−	0.107214i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−214.477	−	371.485i	−0.255444	−	0.442442i	0.709572	−	0.704633i	$-0.248888\pi$
$89$	−214.477	−	371.485i	−0.255444	−	0.442442i	−0.965016	+	0.262191i	$0.915555\pi$
$90$		0			0
$91$	10.4813	−	18.1541i	0.0120740	−	0.0209128i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−158.497	−	132.994i	−0.171173	−	0.143631i
$96$		0			0
$97$	−1166.45	−	424.552i	−1.22098	−	0.444400i	−0.350480	−	0.936570i	$-0.613981\pi$
$97$	−1166.45	−	424.552i	−1.22098	−	0.444400i	−0.870499	+	0.492171i	$0.836204\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	142.775	+	809.715i	0.140659	+	0.797719i	0.970750	+	0.240091i	$0.0771773\pi$
$101$	142.775	+	809.715i	0.140659	+	0.797719i	−0.830091	+	0.557628i	$0.811712\pi$
$102$		0			0
$103$	−1713.66	+	623.722i	−1.63934	+	0.596672i	−0.986924	−	0.161187i	$-0.948468\pi$
$103$	−1713.66	+	623.722i	−1.63934	+	0.596672i	−0.652418	+	0.757859i	$0.726245\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	2093.70			1.89164			0.945819	−	0.324695i	$-0.105262\pi$
$107$	2093.70			1.89164			0.945819	+	0.324695i	$0.105262\pi$
$108$		0			0
$109$	785.096			0.689895			0.344947	−	0.938622i	$-0.387897\pi$
$109$	785.096			0.689895			0.344947	+	0.938622i	$0.387897\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−1484.56	+	540.337i	−1.23589	+	0.449828i	−0.875612	−	0.483015i	$-0.839542\pi$
$113$	−1484.56	+	540.337i	−1.23589	+	0.449828i	−0.360281	+	0.932844i	$0.617319\pi$
$114$		0			0
$115$	−297.638	−	1687.99i	−0.241347	−	1.36875i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−22.8561	−	8.31893i	−0.0176068	−	0.00640836i
$120$		0			0
$121$	444.562	+	373.032i	0.334006	+	0.280265i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−390.150	+	675.760i	−0.279169	+	0.483534i
$126$		0			0
$127$	−197.197	−	341.555i	−0.137783	−	0.238646i	0.788874	−	0.614554i	$-0.210664\pi$
$127$	−197.197	−	341.555i	−0.137783	−	0.238646i	−0.926657	+	0.375908i	$0.877331\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−104.311	+	591.579i	−0.0695704	+	0.394553i	0.930061	+	0.367405i	$0.119754\pi$
$131$	−104.311	+	591.579i	−0.0695704	+	0.394553i	−0.999631	+	0.0271482i	$0.991357\pi$
$132$		0			0
$133$	−45.1865	+	37.9160i	−0.0294599	+	0.0247198i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−875.032	+	734.239i	−0.545686	+	0.457885i	−0.873477	−	0.486865i	$-0.838140\pi$
$137$	−875.032	+	734.239i	−0.545686	+	0.457885i	0.327791	+	0.944750i	$0.393696\pi$
$138$		0			0
$139$	−359.669	+	2039.78i	−0.219473	+	1.24469i	0.653501	+	0.756926i	$0.273300\pi$
$139$	−359.669	+	2039.78i	−0.219473	+	1.24469i	−0.872974	+	0.487767i	$0.837812\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	115.404	+	199.886i	0.0674867	+	0.116890i
$144$		0			0
$145$	983.804	−	1704.00i	0.563451	−	0.975926i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−2383.51	−	2000.00i	−1.31050	−	1.09964i	−0.988227	−	0.152997i	$-0.951108\pi$
$149$	−2383.51	−	2000.00i	−1.31050	−	1.09964i	−0.322276	−	0.946646i	$-0.604448\pi$
$150$		0			0
$151$	−35.5500	−	12.9391i	−0.0191590	−	0.00697332i	0.332423	−	0.943130i	$-0.392134\pi$
$151$	−35.5500	−	12.9391i	−0.0191590	−	0.00697332i	−0.351582	+	0.936157i	$0.614356\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	652.379	+	3699.82i	0.338067	+	1.91727i
$156$		0			0
$157$	−2170.17	+	789.878i	−1.10318	+	0.401523i	−0.828486	−	0.560010i	$-0.810797\pi$
$157$	−2170.17	+	789.878i	−1.10318	+	0.401523i	−0.274689	+	0.961533i	$0.588575\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−488.661			−0.239204
$162$		0			0
$163$	232.306			0.111630			0.0558148	−	0.998441i	$-0.482224\pi$
$163$	232.306			0.111630			0.0558148	+	0.998441i	$0.482224\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−1530.58	+	557.084i	−0.709218	+	0.258134i	−0.671342	−	0.741148i	$-0.734282\pi$
$167$	−1530.58	+	557.084i	−0.709218	+	0.258134i	−0.0378768	+	0.999282i	$0.512059\pi$
$168$		0			0
$169$	−376.665	−	2136.17i	−0.171445	−	0.972314i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	1144.54	+	416.577i	0.502992	+	0.183074i	0.581039	−	0.813876i	$-0.302646\pi$
$173$	1144.54	+	416.577i	0.502992	+	0.183074i	−0.0780474	+	0.996950i	$0.524869\pi$
$174$		0			0
$175$	−209.797	−	176.041i	−0.0906238	−	0.0760424i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−1158.73	+	2006.98i	−0.483842	+	0.838039i	−0.999828	−	0.0185580i	$-0.994092\pi$
$179$	−1158.73	+	2006.98i	−0.483842	+	0.838039i	0.515986	+	0.856597i	$0.327426\pi$
$180$		0			0
$181$	−77.3428	−	133.962i	−0.0317616	−	0.0550127i	0.849708	−	0.527254i	$-0.176778\pi$
$181$	−77.3428	−	133.962i	−0.0317616	−	0.0550127i	−0.881469	+	0.472241i	$0.843445\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	484.957	−	2750.33i	0.192728	−	1.09302i
$186$		0			0
$187$	205.153	−	172.144i	0.0802260	−	0.0673176i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−2149.29	+	1803.47i	−0.814227	+	0.683217i	−0.951613	−	0.307300i	$-0.900574\pi$
$191$	−2149.29	+	1803.47i	−0.814227	+	0.683217i	0.137386	+	0.990518i	$0.456130\pi$
$192$		0			0
$193$	−475.369	+	2695.95i	−0.177294	+	1.00549i	0.758168	+	0.652060i	$0.226095\pi$
$193$	−475.369	+	2695.95i	−0.177294	+	1.00549i	−0.935462	+	0.353427i	$0.885016\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−1571.62	−	2722.12i	−0.568391	−	0.984482i	−0.996725	−	0.0808620i	$-0.974233\pi$
$197$	−1571.62	−	2722.12i	−0.568391	−	0.984482i	0.428334	−	0.903620i	$-0.359101\pi$
$198$		0			0
$199$	−935.924	+	1621.07i	−0.333396	+	0.577459i	−0.983175	−	0.182664i	$-0.941528\pi$
$199$	−935.924	+	1621.07i	−0.333396	+	0.577459i	0.649779	+	0.760123i	$0.274861\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−429.716	−	360.574i	−0.148572	−	0.124667i
$204$		0			0
$205$	−4090.15	−	1488.69i	−1.39350	−	0.507194i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−112.780	−	639.608i	−0.0373262	−	0.211687i
$210$		0			0
$211$	238.052	−	86.6438i	0.0776691	−	0.0282692i	−0.302893	−	0.953024i	$-0.597953\pi$
$211$	238.052	−	86.6438i	0.0776691	−	0.0282692i	0.380562	+	0.924755i	$0.375730\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	6847.91			2.17220
$216$		0			0
$217$	1071.07			0.335065
$218$		0			0
$219$		0			0
$220$		0			0
$221$	30.3896	−	11.0609i	0.00924988	−	0.00336668i
$222$		0			0
$223$	600.687	+	3406.66i	0.180381	+	1.02299i	0.931748	+	0.363106i	$0.118284\pi$
$223$	600.687	+	3406.66i	0.180381	+	1.02299i	−0.751367	+	0.659885i	$0.770605\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	1925.67	+	700.886i	0.563044	+	0.204931i	0.607833	−	0.794065i	$-0.292039\pi$
$227$	1925.67	+	700.886i	0.563044	+	0.204931i	−0.0447883	+	0.998997i	$0.514261\pi$
$228$		0			0
$229$	−693.662	−	582.052i	−0.200168	−	0.167961i	0.537194	−	0.843459i	$-0.319484\pi$
$229$	−693.662	−	582.052i	−0.200168	−	0.167961i	−0.737362	+	0.675498i	$0.763929\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	696.488	−	1206.35i	0.195830	−	0.339188i	−0.751342	−	0.659913i	$-0.770593\pi$
$233$	696.488	−	1206.35i	0.195830	−	0.339188i	0.947172	+	0.320725i	$0.103927\pi$
$234$		0			0
$235$	425.540	+	737.057i	0.118124	+	0.204597i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	43.4332	−	246.322i	0.0117551	−	0.0666663i	−0.978366	−	0.206882i	$-0.933668\pi$
$239$	43.4332	−	246.322i	0.0117551	−	0.0666663i	0.990121	+	0.140216i	$0.0447795\pi$
$240$		0			0
$241$	3775.11	−	3167.70i	1.00903	−	0.846677i	0.0208208	−	0.999783i	$-0.493372\pi$
$241$	3775.11	−	3167.70i	1.00903	−	0.846677i	0.988210	+	0.153106i	$0.0489276\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	3491.28	−	2929.53i	0.910406	−	0.763921i
$246$		0			0
$247$	13.6191	−	77.2377i	0.00350835	−	0.0198968i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−557.386	−	965.420i	−0.140167	−	0.242776i	0.787392	−	0.616452i	$-0.211431\pi$
$251$	−557.386	−	965.420i	−0.140167	−	0.242776i	−0.927559	+	0.373676i	$0.878097\pi$
$252$		0			0
$253$	2690.21	−	4659.57i	0.668505	−	1.15788i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	2365.04	+	1984.51i	0.574037	+	0.481674i	0.882983	−	0.469406i	$-0.155532\pi$
$257$	2365.04	+	1984.51i	0.574037	+	0.481674i	−0.308946	+	0.951080i	$0.599976\pi$
$258$		0			0
$259$	−748.183	−	272.316i	−0.179497	−	0.0653317i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−187.667	−	1064.32i	−0.0440003	−	0.249538i	0.954872	−	0.297018i	$-0.0959921\pi$
$263$	−187.667	−	1064.32i	−0.0440003	−	0.249538i	−0.998872	+	0.0474799i	$0.984881\pi$
$264$		0			0
$265$	4975.34	−	1810.88i	1.15333	−	0.419778i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	5507.29			1.24827			0.624137	−	0.781315i	$-0.285451\pi$
$269$	5507.29			1.24827			0.624137	+	0.781315i	$0.285451\pi$
$270$		0			0
$271$	4075.64			0.913570			0.456785	−	0.889577i	$-0.349001\pi$
$271$	4075.64			0.913570			0.456785	+	0.889577i	$0.349001\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	2833.61	−	1031.35i	0.621356	−	0.226155i
$276$		0			0
$277$	615.311	+	3489.60i	0.133467	+	0.756931i	0.975915	+	0.218152i	$0.0700029\pi$
$277$	615.311	+	3489.60i	0.133467	+	0.756931i	−0.842447	+	0.538779i	$0.818886\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−6900.33	−	2511.51i	−1.46491	−	0.533183i	−0.518195	−	0.855262i	$-0.673396\pi$
$281$	−6900.33	−	2511.51i	−1.46491	−	0.533183i	−0.946713	+	0.322080i	$0.895618\pi$
$282$		0			0
$283$	−102.360	−	85.8899i	−0.0215005	−	0.0180411i	0.631974	−	0.774990i	$-0.282245\pi$
$283$	−102.360	−	85.8899i	−0.0215005	−	0.0180411i	−0.653475	+	0.756948i	$0.726689\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−620.458	+	1074.66i	−0.127611	+	0.221029i
$288$		0			0
$289$	2437.74	+	4222.29i	0.496181	+	0.859411i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−710.916	+	4031.81i	−0.141748	+	0.803893i	0.828173	+	0.560473i	$0.189381\pi$
$293$	−710.916	+	4031.81i	−0.141748	+	0.803893i	−0.969921	+	0.243420i	$0.921731\pi$
$294$		0			0
$295$	1756.98	−	1474.28i	0.346763	−	0.290969i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	497.719	−	417.636i	0.0962671	−	0.0807777i
$300$		0			0
$301$	339.013	−	1922.64i	0.0649183	−	0.368170i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−6010.46	−	10410.4i	−1.12839	−	1.95442i
$306$		0			0
$307$	−4191.33	+	7259.59i	−0.779191	+	1.34960i	0.153217	+	0.988192i	$0.451037\pi$
$307$	−4191.33	+	7259.59i	−0.779191	+	1.34960i	−0.932409	+	0.361406i	$0.882297\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−1493.12	−	1252.88i	−0.272242	−	0.228438i	0.496437	−	0.868073i	$-0.334641\pi$
$311$	−1493.12	−	1252.88i	−0.272242	−	0.228438i	−0.768679	+	0.639635i	$0.779086\pi$
$312$		0			0
$313$	−1191.53	−	433.681i	−0.215173	−	0.0783165i	0.232185	−	0.972672i	$-0.425413\pi$
$313$	−1191.53	−	433.681i	−0.215173	−	0.0783165i	−0.447358	+	0.894355i	$0.647635\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−1155.38	−	6552.49i	−0.204709	−	1.16096i	−0.897898	−	0.440204i	$-0.854906\pi$
$317$	−1155.38	−	6552.49i	−0.204709	−	1.16096i	0.693189	−	0.720756i	$-0.256205\pi$
$318$		0			0
$319$	5803.91	−	2112.45i	1.01867	−	0.370767i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−91.0017			−0.0156764
$324$		0			0
$325$	364.140			0.0621504
$326$		0			0
$327$		0			0
$328$		0			0
$329$	228.005	−	82.9872i	0.0382077	−	0.0139065i
$330$		0			0
$331$	988.638	+	5606.85i	0.164171	+	0.931058i	0.949915	+	0.312507i	$0.101169\pi$
$331$	988.638	+	5606.85i	0.164171	+	0.931058i	−0.785745	+	0.618551i	$0.787720\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	2722.58	+	990.936i	0.444030	+	0.161614i
$336$		0			0
$337$	4026.87	+	3378.95i	0.650913	+	0.546181i	0.907348	−	0.420381i	$-0.138103\pi$
$337$	4026.87	+	3378.95i	0.650913	+	0.546181i	−0.256435	+	0.966561i	$0.582548\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−5896.53	+	10213.1i	−0.936407	+	1.62190i
$342$		0			0
$343$	−1330.63	−	2304.72i	−0.209467	−	0.362808i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	1204.18	−	6829.25i	0.186293	−	1.05652i	−0.737989	−	0.674813i	$-0.764224\pi$
$347$	1204.18	−	6829.25i	0.186293	−	1.05652i	0.924282	−	0.381710i	$-0.124665\pi$
$348$		0			0
$349$	−362.713	+	304.352i	−0.0556320	+	0.0466808i	−0.670180	−	0.742199i	$-0.733783\pi$
$349$	−362.713	+	304.352i	−0.0556320	+	0.0466808i	0.614548	+	0.788880i	$0.289339\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−1133.07	+	950.758i	−0.170842	+	0.143353i	−0.724200	−	0.689590i	$-0.757791\pi$
$353$	−1133.07	+	950.758i	−0.170842	+	0.143353i	0.553358	+	0.832944i	$0.313346\pi$
$354$		0			0
$355$	2183.45	−	12383.0i	0.326438	−	1.85132i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−1718.27	−	2976.14i	−0.252610	−	0.437534i	0.711634	−	0.702551i	$-0.247956\pi$
$359$	−1718.27	−	2976.14i	−0.252610	−	0.437534i	−0.964244	+	0.265017i	$0.914622\pi$
$360$		0			0
$361$	3319.15	−	5748.94i	0.483912	−	0.838160i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	7545.08	+	6331.07i	1.08199	+	0.907900i
$366$		0			0
$367$	8040.63	+	2926.55i	1.14364	+	0.416253i	0.843228	−	0.537556i	$-0.180652\pi$
$367$	8040.63	+	2926.55i	1.14364	+	0.416253i	0.300416	+	0.953808i	$0.402874\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−262.117	−	1486.54i	−0.0366805	−	0.208025i
$372$		0			0
$373$	−8918.15	+	3245.94i	−1.23797	+	0.450586i	−0.876321	−	0.481727i	$-0.840010\pi$
$373$	−8918.15	+	3245.94i	−1.23797	+	0.450586i	−0.361653	+	0.932313i	$0.617787\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	745.848			0.101892
$378$		0			0
$379$	−11784.3			−1.59714			−0.798572	−	0.601899i	$-0.794411\pi$
$379$	−11784.3			−1.59714			−0.798572	+	0.601899i	$0.794411\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	8550.92	−	3112.28i	1.14081	−	0.415222i	0.298606	−	0.954376i	$-0.403478\pi$
$383$	8550.92	−	3112.28i	1.14081	−	0.415222i	0.842207	+	0.539155i	$0.181256\pi$
$384$		0			0
$385$	−419.827	−	2380.96i	−0.0555750	−	0.315182i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	8567.80	+	3118.43i	1.11672	+	0.406454i	0.833455	−	0.552587i	$-0.186359\pi$
$389$	8567.80	+	3118.43i	1.11672	+	0.406454i	0.283267	+	0.959041i	$0.408582\pi$
$390$		0			0
$391$	−577.506	−	484.585i	−0.0746949	−	0.0626765i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−7707.46	+	13349.7i	−0.981783	+	1.70050i
$396$		0			0
$397$	−4029.27	−	6978.90i	−0.509378	−	0.882269i	−0.999941	−	0.0108631i	$-0.996542\pi$
$397$	−4029.27	−	6978.90i	−0.509378	−	0.882269i	0.490563	−	0.871406i	$-0.336791\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−2031.54	+	11521.4i	−0.252993	+	1.43479i	0.548180	+	0.836361i	$0.315321\pi$
$401$	−2031.54	+	11521.4i	−0.252993	+	1.43479i	−0.801173	+	0.598433i	$0.795790\pi$
$402$		0			0
$403$	−1090.93	+	915.396i	−0.134846	+	0.113149i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	6715.58	−	5635.04i	0.817885	−	0.686287i
$408$		0			0
$409$	−489.847	+	2778.06i	−0.0592210	+	0.335859i	−0.999995	−	0.00317545i	$-0.998989\pi$
$409$	−489.847	+	2778.06i	−0.0592210	+	0.335859i	0.940774	+	0.339034i	$0.110100\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−326.942	−	566.281i	−0.0389535	−	0.0674694i
$414$		0			0
$415$	−5930.65	+	10272.2i	−0.701504	+	1.21504i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−7349.28	−	6166.78i	−0.856887	−	0.719014i	0.104408	−	0.994535i	$-0.466705\pi$
$419$	−7349.28	−	6166.78i	−0.856887	−	0.719014i	−0.961295	+	0.275521i	$0.911150\pi$
$420$		0			0
$421$	11950.5	+	4349.63i	1.38345	+	0.503535i	0.923222	−	0.384266i	$-0.125545\pi$
$421$	11950.5	+	4349.63i	1.38345	+	0.503535i	0.460228	+	0.887801i	$0.347768\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−73.3687	−	416.094i	−0.00837389	−	0.0474907i
$426$		0			0
$427$	−3220.42	+	1172.14i	−0.364981	+	0.132842i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	7867.05			0.879216			0.439608	−	0.898190i	$-0.355117\pi$
$431$	7867.05			0.879216			0.439608	+	0.898190i	$0.355117\pi$
$432$		0			0
$433$	−17294.1			−1.91940			−0.959702	−	0.281018i	$-0.909328\pi$
$433$	−17294.1			−1.91940			−0.959702	+	0.281018i	$0.909328\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−1718.02	+	625.306i	−0.188064	+	0.0684496i
$438$		0			0
$439$	578.944	+	3283.35i	0.0629418	+	0.356961i	0.999970	+	0.00770790i	$0.00245352\pi$
$439$	578.944	+	3283.35i	0.0629418	+	0.356961i	−0.937028	+	0.349253i	$0.886435\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−12292.3	−	4474.02i	−1.31834	−	0.479836i	−0.415413	−	0.909633i	$-0.636363\pi$
$443$	−12292.3	−	4474.02i	−1.31834	−	0.479836i	−0.902925	+	0.429797i	$0.858585\pi$
$444$		0			0
$445$	−4576.53	−	3840.17i	−0.487525	−	0.409082i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−9483.66	+	16426.2i	−0.996796	+	1.72650i	−0.429132	+	0.903242i	$0.641180\pi$
$449$	−9483.66	+	16426.2i	−0.996796	+	1.72650i	−0.567665	+	0.823260i	$0.692153\pi$
$450$		0			0
$451$	−6831.56	−	11832.6i	−0.713272	−	1.23542i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	50.6974	−	287.519i	0.00522359	−	0.0296244i
$456$		0			0
$457$	−10486.3	+	8799.07i	−1.07337	+	0.900664i	−0.995353	−	0.0962885i	$-0.969303\pi$
$457$	−10486.3	+	8799.07i	−1.07337	+	0.900664i	−0.0780156	+	0.996952i	$0.524858\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	7991.70	−	6705.83i	0.807398	−	0.677488i	−0.142587	−	0.989782i	$-0.545542\pi$
$461$	7991.70	−	6705.83i	0.807398	−	0.677488i	0.949985	+	0.312295i	$0.101098\pi$
$462$		0			0
$463$	848.557	−	4812.41i	0.0851745	−	0.483048i	−0.912144	−	0.409869i	$-0.865574\pi$
$463$	848.557	−	4812.41i	0.0851745	−	0.483048i	0.997319	−	0.0731793i	$-0.0233145\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−3003.39	−	5202.03i	−0.297602	−	0.515463i	0.677984	−	0.735076i	$-0.262854\pi$
$467$	−3003.39	−	5202.03i	−0.297602	−	0.515463i	−0.975587	+	0.219614i	$0.929520\pi$
$468$		0			0
$469$	413.003	−	715.341i	0.0406624	−	0.0704294i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	16466.8	+	13817.3i	1.60073	+	1.34317i
$474$		0			0
$475$	−962.864	−	350.454i	−0.0930089	−	0.0338525i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−529.337	−	3002.02i	−0.0504928	−	0.286359i	0.949097	−	0.314982i	$-0.101999\pi$
$479$	−529.337	−	3002.02i	−0.0504928	−	0.286359i	−0.999590	+	0.0286237i	$0.990888\pi$
$480$		0			0
$481$	994.788	−	362.073i	0.0943003	−	0.0343225i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−17288.3			−1.61860
$486$		0			0
$487$	−1795.38			−0.167056			−0.0835281	−	0.996505i	$-0.526619\pi$
$487$	−1795.38			−0.167056			−0.0835281	+	0.996505i	$0.526619\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−11759.8	+	4280.20i	−1.08088	+	0.393407i	−0.820234	−	0.572028i	$-0.806157\pi$
$491$	−11759.8	+	4280.20i	−1.08088	+	0.393407i	−0.260643	+	0.965435i	$0.583935\pi$
$492$		0			0
$493$	−150.277	−	852.262i	−0.0137285	−	0.0778580i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−3368.59	−	1226.07i	−0.304028	−	0.110657i
$498$		0			0
$499$	−10407.7	−	8733.06i	−0.933688	−	0.783458i	0.0427875	−	0.999084i	$-0.486376\pi$
$499$	−10407.7	−	8733.06i	−0.933688	−	0.783458i	−0.976476	+	0.215627i	$0.930821\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−6402.30	+	11089.1i	−0.567523	+	0.982979i	0.429287	+	0.903168i	$0.358765\pi$
$503$	−6402.30	+	11089.1i	−0.567523	+	0.982979i	−0.996810	+	0.0798111i	$0.974568\pi$
$504$		0			0
$505$	5725.62	+	9917.06i	0.504528	+	0.873868i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−14.7863	+	83.8573i	−0.00128761	+	0.00730237i	−0.985445	−	0.169996i	$-0.945625\pi$
$509$	−14.7863	+	83.8573i	−0.00128761	+	0.00730237i	0.984157	+	0.177298i	$0.0567357\pi$
$510$		0			0
$511$	2151.06	−	1804.95i	0.186218	−	0.156255i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−19456.5	+	16326.0i	−1.66477	+	1.39691i
$516$		0			0
$517$	−463.914	+	2630.99i	−0.0394641	+	0.223812i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−8164.42	−	14141.2i	−0.686545	−	1.18913i	−0.972949	−	0.231021i	$-0.925793\pi$
$521$	−8164.42	−	14141.2i	−0.686545	−	1.18913i	0.286404	−	0.958109i	$-0.407540\pi$
$522$		0			0
$523$	−5508.00	+	9540.13i	−0.460512	+	0.797630i	−0.998986	−	0.0450116i	$-0.985668\pi$
$523$	−5508.00	+	9540.13i	−0.460512	+	0.797630i	0.538474	+	0.842642i	$0.319001\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	1265.81	+	1062.14i	0.104629	+	0.0877940i
$528$		0			0
$529$	−2799.22	−	1018.83i	−0.230067	−	0.0837374i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−286.507	−	1624.86i	−0.0232833	−	0.132046i
$534$		0			0
$535$	27401.3	−	9973.26i	2.21432	−	0.805947i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	14306.3			1.14326
$540$		0			0
$541$	8618.57			0.684919			0.342460	−	0.939533i	$-0.388740\pi$
$541$	8618.57			0.684919			0.342460	+	0.939533i	$0.388740\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	10275.0	−	3739.78i	0.807580	−	0.293935i
$546$		0			0
$547$	−118.470	−	671.876i	−0.00926034	−	0.0525180i	0.979827	−	0.199845i	$-0.0640440\pi$
$547$	−118.470	−	671.876i	−0.00926034	−	0.0525180i	−0.989088	+	0.147327i	$0.952933\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−1972.18	−	717.815i	−0.152482	−	0.0554990i
$552$		0			0
$553$	3366.54	+	2824.86i	0.258878	+	0.217225i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	706.450	−	1223.61i	0.0537401	−	0.0930806i	−0.837904	−	0.545818i	$-0.816219\pi$
$557$	706.450	−	1223.61i	0.0537401	−	0.0930806i	0.891644	+	0.452737i	$0.149552\pi$
$558$		0			0
$559$	1297.90	+	2248.02i	0.0982024	+	0.170092i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−3353.19	+	19016.9i	−0.251013	+	1.42356i	0.555092	+	0.831789i	$0.312683\pi$
$563$	−3353.19	+	19016.9i	−0.251013	+	1.42356i	−0.806104	+	0.591774i	$0.798428\pi$
$564$		0			0
$565$	−16855.4	+	14143.3i	−1.25506	+	1.05312i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	10023.8	−	8410.99i	0.738525	−	0.619696i	−0.193916	−	0.981018i	$-0.562119\pi$
$569$	10023.8	−	8410.99i	0.738525	−	0.619696i	0.932441	+	0.361322i	$0.117675\pi$
$570$		0			0
$571$	3569.22	−	20242.1i	0.261589	−	1.48354i	−0.516987	−	0.855993i	$-0.672946\pi$
$571$	3569.22	−	20242.1i	0.261589	−	1.48354i	0.778576	−	0.627551i	$-0.215943\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−4244.26	−	7351.28i	−0.307823	−	0.533164i
$576$		0			0
$577$	1699.26	−	2943.21i	0.122602	−	0.212352i	−0.798191	−	0.602404i	$-0.794210\pi$
$577$	1699.26	−	2943.21i	0.122602	−	0.212352i	0.920793	+	0.390052i	$0.127543\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	2590.45	+	2173.65i	0.184974	+	0.155212i
$582$		0			0
$583$	15617.8	+	5684.41i	1.10947	+	0.403815i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−2010.20	−	11400.4i	−0.141346	−	0.801612i	−0.970229	−	0.242190i	$-0.922134\pi$
$587$	−2010.20	−	11400.4i	−0.141346	−	0.801612i	0.828883	−	0.559422i	$-0.188977\pi$
$588$		0			0
$589$	3765.63	−	1370.58i	0.263430	−	0.0958806i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	20424.8			1.41441			0.707205	−	0.707008i	$-0.249956\pi$
$593$	20424.8			1.41441			0.707205	+	0.707008i	$0.249956\pi$
$594$		0			0
$595$	−338.756			−0.0233406
$596$		0			0
$597$		0			0
$598$		0			0
$599$	15491.8	−	5638.56i	1.05672	−	0.384616i	0.245528	−	0.969389i	$-0.421039\pi$
$599$	15491.8	−	5638.56i	1.05672	−	0.384616i	0.811197	+	0.584773i	$0.198816\pi$
$600$		0			0
$601$	−3430.04	−	19452.7i	−0.232803	−	1.32029i	−0.847192	−	0.531286i	$-0.821709\pi$
$601$	−3430.04	−	19452.7i	−0.232803	−	1.32029i	0.614390	−	0.789003i	$-0.289402\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	7595.15	+	2764.41i	0.510391	+	0.185767i
$606$		0			0
$607$	−3232.28	−	2712.21i	−0.216136	−	0.181359i	0.528291	−	0.849063i	$-0.322833\pi$
$607$	−3232.28	−	2712.21i	−0.216136	−	0.181359i	−0.744427	+	0.667704i	$0.767277\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−161.307	+	279.391i	−0.0106805	+	0.0184991i
$612$		0			0
$613$	5986.71	+	10369.3i	0.394455	+	0.683216i	0.993031	−	0.117850i	$-0.0376001\pi$
$613$	5986.71	+	10369.3i	0.394455	+	0.683216i	−0.598577	+	0.801066i	$0.704267\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	766.653	−	4347.91i	0.0500232	−	0.283695i	−0.949527	−	0.313685i	$-0.898436\pi$
$617$	766.653	−	4347.91i	0.0500232	−	0.283695i	0.999550	+	0.0299897i	$0.00954746\pi$
$618$		0			0
$619$	4277.74	−	3589.45i	0.277766	−	0.233073i	−0.493252	−	0.869886i	$-0.664192\pi$
$619$	4277.74	−	3589.45i	0.277766	−	0.233073i	0.771018	+	0.636813i	$0.219748\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−1304.74	+	1094.81i	−0.0839061	+	0.0704055i
$624$		0			0
$625$	−3384.29	+	19193.3i	−0.216594	+	1.22837i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−614.167	−	1063.77i	−0.0389324	−	0.0674328i
$630$		0			0
$631$	6743.30	−	11679.7i	0.425431	−	0.736867i	−0.571030	−	0.820929i	$-0.693456\pi$
$631$	6743.30	−	11679.7i	0.425431	−	0.736867i	0.996461	+	0.0840619i	$0.0267893\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−4207.80	−	3530.77i	−0.262963	−	0.220652i
$636$		0			0
$637$	1623.41	+	590.872i	0.100976	+	0.0367523i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−1301.91	−	7383.47i	−0.0802218	−	0.454960i	−0.998286	−	0.0585281i	$-0.981359\pi$
$641$	−1301.91	−	7383.47i	−0.0802218	−	0.454960i	0.918064	−	0.396432i	$-0.129752\pi$
$642$		0			0
$643$	−18909.9	+	6882.64i	−1.15977	+	0.422122i	−0.849016	−	0.528367i	$-0.822804\pi$
$643$	−18909.9	+	6882.64i	−1.15977	+	0.422122i	−0.310756	+	0.950490i	$0.600582\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	13603.1			0.826575			0.413288	−	0.910601i	$-0.364380\pi$
$647$	13603.1			0.826575			0.413288	+	0.910601i	$0.364380\pi$
$648$		0			0
$649$	7199.61			0.435454
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−21556.3	+	7845.87i	−1.29183	+	0.470188i	−0.894327	−	0.447413i	$-0.852345\pi$
$653$	−21556.3	+	7845.87i	−1.29183	+	0.470188i	−0.397503	+	0.917601i	$0.630123\pi$
$654$		0			0
$655$	1452.79	+	8239.19i	0.0866645	+	0.491499i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−15226.4	−	5541.95i	−0.900054	−	0.327593i	−0.149779	−	0.988719i	$-0.547856\pi$
$659$	−15226.4	−	5541.95i	−0.900054	−	0.327593i	−0.750274	+	0.661127i	$0.770079\pi$
$660$		0			0
$661$	8534.64	+	7161.41i	0.502207	+	0.421402i	0.858377	−	0.513019i	$-0.171473\pi$
$661$	8534.64	+	7161.41i	0.502207	+	0.421402i	−0.356170	+	0.934421i	$0.615918\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−410.768	+	711.470i	−0.0239532	+	0.0414882i
$666$		0			0
$667$	−8693.28	−	15057.2i	−0.504655	−	0.874089i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	6552.46	−	37160.9i	0.376982	−	2.13797i
$672$		0			0
$673$	972.493	−	816.019i	0.0557011	−	0.0467388i	−0.614512	−	0.788907i	$-0.710647\pi$
$673$	972.493	−	816.019i	0.0557011	−	0.0467388i	0.670213	+	0.742169i	$0.266203\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	15101.6	−	12671.8i	0.857315	−	0.719372i	−0.104073	−	0.994570i	$-0.533188\pi$
$677$	15101.6	−	12671.8i	0.857315	−	0.719372i	0.961388	+	0.275197i	$0.0887431\pi$
$678$		0			0
$679$	−855.875	+	4853.91i	−0.0483733	+	0.274339i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−4391.39	−	7606.11i	−0.246020	−	0.426120i	0.716398	−	0.697692i	$-0.245790\pi$
$683$	−4391.39	−	7606.11i	−0.246020	−	0.426120i	−0.962418	+	0.271572i	$0.912456\pi$
$684$		0			0
$685$	−7954.48	+	13777.6i	−0.443686	+	0.768487i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	1537.46	+	1290.08i	0.0850108	+	0.0713325i
$690$		0			0
$691$	9374.91	+	3412.19i	0.516119	+	0.187852i	0.586930	−	0.809638i	$-0.300336\pi$
$691$	9374.91	+	3412.19i	0.516119	+	0.187852i	−0.0708107	+	0.997490i	$0.522559\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	5009.27	+	28409.0i	0.273399	+	1.55053i
$696$		0			0
$697$	−1798.97	+	654.770i	−0.0977628	+	0.0355828i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	4822.63			0.259841			0.129920	−	0.991524i	$-0.458528\pi$
$701$	4822.63			0.259841			0.129920	+	0.991524i	$0.458528\pi$
$702$		0			0
$703$	−2978.90			−0.159817
$704$		0			0
$705$		0			0
$706$		0			0
$707$	3067.80	−	1116.59i	0.163192	−	0.0593969i
$708$		0			0
$709$	2404.44	+	13636.3i	0.127363	+	0.722314i	0.979876	+	0.199607i	$0.0639666\pi$
$709$	2404.44	+	13636.3i	0.127363	+	0.722314i	−0.852513	+	0.522707i	$0.824922\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	31195.4	+	11354.2i	1.63854	+	0.596379i
$714$		0			0
$715$	2462.51	+	2066.29i	0.128801	+	0.108077i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	17247.7	−	29874.0i	0.894621	−	1.54953i	0.0603471	−	0.998177i	$-0.480779\pi$
$719$	17247.7	−	29874.0i	0.894621	−	1.54953i	0.834273	−	0.551351i	$-0.185887\pi$
$720$		0			0
$721$	3620.51	+	6270.91i	0.187011	+	0.323913i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	1692.08	−	9596.29i	0.0866792	−	0.491582i
$726$		0			0
$727$	28116.6	−	23592.6i	1.43437	−	1.20358i	0.491295	−	0.870993i	$-0.336524\pi$
$727$	28116.6	−	23592.6i	1.43437	−	1.20358i	0.943073	−	0.332585i	$-0.107921\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	2307.25	−	1936.02i	0.116740	−	0.0979564i
$732$		0			0
$733$	−2455.96	+	13928.4i	−0.123756	+	0.701854i	0.858283	+	0.513176i	$0.171531\pi$
$733$	−2455.96	+	13928.4i	−0.123756	+	0.701854i	−0.982039	+	0.188678i	$0.939580\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	4547.38	+	7876.28i	0.227279	+	0.393659i
$738$		0			0
$739$	17248.4	−	29875.1i	0.858583	−	1.48711i	−0.0146980	−	0.999892i	$-0.504679\pi$
$739$	17248.4	−	29875.1i	0.858583	−	1.48711i	0.873281	−	0.487217i	$-0.161988\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−29161.7	−	24469.5i	−1.43989	−	1.20821i	−0.939564	−	0.342373i	$-0.888769\pi$
$743$	−29161.7	−	24469.5i	−1.43989	−	1.20821i	−0.500326	−	0.865837i	$-0.666786\pi$
$744$		0			0
$745$	−40721.2	−	14821.3i	−2.00256	−	0.728874i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−1443.59	−	8187.01i	−0.0704241	−	0.399395i
$750$		0			0
$751$	−4734.32	+	1723.15i	−0.230037	+	0.0837266i	−0.454467	−	0.890764i	$-0.650170\pi$
$751$	−4734.32	+	1723.15i	−0.230037	+	0.0837266i	0.224430	+	0.974490i	$0.427948\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−526.896			−0.0253983
$756$		0			0
$757$	39695.9			1.90591			0.952954	−	0.303115i	$-0.0980266\pi$
$757$	39695.9			1.90591			0.952954	+	0.303115i	$0.0980266\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	6617.71	−	2408.65i	0.315233	−	0.114735i	−0.179558	−	0.983747i	$-0.557467\pi$
$761$	6617.71	−	2408.65i	0.315233	−	0.114735i	0.494791	+	0.869012i	$0.335245\pi$
$762$		0			0
$763$	−541.319	−	3069.97i	−0.0256842	−	0.145662i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	816.977	+	297.355i	0.0384607	+	0.0139985i
$768$		0			0
$769$	30249.5	+	25382.4i	1.41850	+	1.19026i	0.952138	+	0.305669i	$0.0988802\pi$
$769$	30249.5	+	25382.4i	1.41850	+	1.19026i	0.466362	+	0.884594i	$0.345564\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−15979.4	+	27677.1i	−0.743516	+	1.28781i	0.207369	+	0.978263i	$0.433510\pi$
$773$	−15979.4	+	27677.1i	−0.743516	+	1.28781i	−0.950885	+	0.309545i	$0.899823\pi$
$774$		0			0
$775$	9302.79	+	16112.9i	0.431182	+	0.746829i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−806.206	+	4572.22i	−0.0370800	+	0.210291i
$780$		0			0
$781$	30236.0	−	25371.0i	1.38531	−	1.16241i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−24639.6	+	20675.1i	−1.12029	+	0.940033i
$786$		0			0
$787$	787.039	−	4463.52i	0.0356479	−	0.202169i	−0.961782	−	0.273816i	$-0.911714\pi$
$787$	787.039	−	4463.52i	0.0356479	−	0.202169i	0.997430	+	0.0716463i	$0.0228253\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	3136.48	+	5432.55i	0.140987	+	0.244196i
$792$		0			0
$793$	2278.35	−	3946.21i	0.102026	−	0.176714i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−3143.57	−	2637.77i	−0.139713	−	0.117233i	0.570253	−	0.821469i	$-0.306845\pi$
$797$	−3143.57	−	2637.77i	−0.139713	−	0.117233i	−0.709966	+	0.704236i	$0.751290\pi$
$798$		0			0
$799$	351.755	+	128.028i	0.0155747	+	0.00566873i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	5368.79	+	30447.9i	0.235941	+	1.33809i
$804$		0			0
$805$	−6395.36	+	2327.72i	−0.280008	+	0.101915i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−3418.34			−0.148557			−0.0742783	−	0.997238i	$-0.523665\pi$
$809$	−3418.34			−0.148557			−0.0742783	+	0.997238i	$0.523665\pi$
$810$		0			0
$811$	−3687.72			−0.159671			−0.0798357	−	0.996808i	$-0.525440\pi$
$811$	−3687.72			−0.159671			−0.0798357	+	0.996808i	$0.525440\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	3040.31	−	1106.58i	0.130672	−	0.0475606i
$816$		0			0
$817$	−1268.38	−	7193.36i	−0.0543147	−	0.308034i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	3319.42	+	1208.17i	0.141107	+	0.0513586i	0.411608	−	0.911361i	$-0.364967\pi$
$821$	3319.42	+	1208.17i	0.141107	+	0.0513586i	−0.270501	+	0.962720i	$0.587189\pi$
$822$		0			0
$823$	−3449.91	−	2894.82i	−0.146119	−	0.122609i	0.566798	−	0.823857i	$-0.308182\pi$
$823$	−3449.91	−	2894.82i	−0.146119	−	0.122609i	−0.712917	+	0.701248i	$0.752626\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	7788.45	−	13490.0i	0.327486	−	0.567223i	−0.654526	−	0.756039i	$-0.727132\pi$
$827$	7788.45	−	13490.0i	0.327486	−	0.567223i	0.982012	+	0.188817i	$0.0604652\pi$
$828$		0			0
$829$	−9335.06	−	16168.8i	−0.391098	−	0.677401i	0.601497	−	0.798875i	$-0.294571\pi$
$829$	−9335.06	−	16168.8i	−0.391098	−	0.677401i	−0.992595	+	0.121474i	$0.961238\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	348.084	−	1974.08i	0.0144783	−	0.0821103i
$834$		0			0
$835$	−17377.8	+	14581.7i	−0.720219	+	0.604336i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−13863.9	+	11633.2i	−0.570482	+	0.478691i	−0.881806	−	0.471613i	$-0.843672\pi$
$839$	−13863.9	+	11633.2i	−0.570482	+	0.478691i	0.311324	+	0.950304i	$0.399228\pi$
$840$		0			0
$841$	−769.305	+	4362.95i	−0.0315431	+	0.178890i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−15105.2	−	26163.0i	−0.614953	−	1.06513i
$846$		0			0
$847$	1152.15	−	1995.58i	0.0467395	−	0.0809552i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−18904.4	−	15862.7i	−0.761497	−	0.638972i
$852$		0			0
$853$	−35220.9	−	12819.4i	−1.41377	−	0.514569i	−0.481533	−	0.876428i	$-0.659920\pi$
$853$	−35220.9	−	12819.4i	−1.41377	−	0.514569i	−0.932233	+	0.361860i	$0.882142\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	3991.51	+	22637.0i	0.159098	+	0.902291i	0.954943	+	0.296790i	$0.0959162\pi$
$857$	3991.51	+	22637.0i	0.159098	+	0.902291i	−0.795844	+	0.605501i	$0.792973\pi$
$858$		0			0
$859$	−29016.9	+	10561.3i	−1.15256	+	0.419496i	−0.846432	−	0.532496i	$-0.821254\pi$
$859$	−29016.9	+	10561.3i	−1.15256	+	0.419496i	−0.306123	+	0.951992i	$0.599032\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	19121.6			0.754236			0.377118	−	0.926165i	$-0.376915\pi$
$863$	19121.6			0.754236			0.377118	+	0.926165i	$0.376915\pi$
$864$		0			0
$865$	16963.5			0.666794
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−45469.9	+	16549.7i	−1.77498	+	0.646040i
$870$		0			0
$871$	190.711	+	1081.58i	0.00741906	+	0.0420756i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	2911.44	+	1059.68i	0.112485	+	0.0409413i
$876$		0			0
$877$	−12696.7	−	10653.8i	−0.488867	−	0.410208i	0.364753	−	0.931104i	$-0.381153\pi$
$877$	−12696.7	−	10653.8i	−0.488867	−	0.410208i	−0.853620	+	0.520896i	$0.825598\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	15481.4	−	26814.5i	0.592032	−	1.02543i	−0.401926	−	0.915672i	$-0.631659\pi$
$881$	15481.4	−	26814.5i	0.592032	−	1.02543i	0.993958	−	0.109758i	$-0.0350075\pi$
$882$		0			0
$883$	12180.7	+	21097.6i	0.464229	+	0.804068i	0.999166	−	0.0408239i	$-0.0129983\pi$
$883$	12180.7	+	21097.6i	0.464229	+	0.804068i	−0.534938	+	0.844891i	$0.679665\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−4420.12	+	25067.7i	−0.167320	+	0.948920i	0.779320	+	0.626627i	$0.215565\pi$
$887$	−4420.12	+	25067.7i	−0.167320	+	0.948920i	−0.946640	+	0.322293i	$0.895546\pi$
$888$		0			0
$889$	−1199.62	+	1006.60i	−0.0452576	+	0.0379756i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	695.420	−	583.527i	0.0260597	−	0.0218667i
$894$		0			0
$895$	−5604.74	+	31786.0i	−0.209325	+	1.18714i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	19054.4	+	33003.1i	0.706895	+	1.22438i
$900$		0			0
$901$	1164.37	−	2016.75i	0.0430530	−	0.0745700i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−1650.35	−	1384.81i	−0.0606182	−	0.0508647i
$906$		0			0
$907$	−25805.9	−	9392.59i	−0.944731	−	0.343854i	−0.176699	−	0.984265i	$-0.556542\pi$
$907$	−25805.9	−	9392.59i	−0.944731	−	0.343854i	−0.768033	+	0.640411i	$0.778764\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−5849.43	−	33173.7i	−0.212733	−	1.20647i	−0.884797	−	0.465977i	$-0.845703\pi$
$911$	−5849.43	−	33173.7i	−0.212733	−	1.20647i	0.672063	−	0.740494i	$-0.265408\pi$
$912$		0			0
$913$	−34987.7	+	12734.5i	−1.26826	+	0.461609i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	2385.18			0.0858949
$918$		0			0
$919$	−16064.8			−0.576635			−0.288318	−	0.957535i	$-0.593096\pi$
$919$	−16064.8			−0.576635			−0.288318	+	0.957535i	$0.593096\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	4478.90	−	1630.19i	0.159723	−	0.0581346i
$924$		0			0
$925$	−2401.69	−	13620.7i	−0.0853698	−	0.484156i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−34871.8	−	12692.3i	−1.23155	−	0.448247i	−0.357420	−	0.933944i	$-0.616344\pi$
$929$	−34871.8	−	12692.3i	−1.23155	−	0.448247i	−0.874127	+	0.485697i	$0.838566\pi$
$930$		0			0
$931$	−3723.97	−	3124.78i	−0.131094	−	0.110001i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	1864.94	−	3230.17i	0.0652301	−	0.112982i
$936$		0			0
$937$	−11227.4	−	19446.5i	−0.391445	−	0.678003i	0.601195	−	0.799102i	$-0.294691\pi$
$937$	−11227.4	−	19446.5i	−0.391445	−	0.678003i	−0.992640	+	0.121099i	$0.961358\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−1443.08	+	8184.09i	−0.0499925	+	0.283521i	−0.999548	−	0.0300789i	$-0.990424\pi$
$941$	−1443.08	+	8184.09i	−0.0499925	+	0.283521i	0.949555	+	0.313600i	$0.101535\pi$
$942$		0			0
$943$	−29463.4	+	24722.7i	−1.01746	+	0.853746i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−23700.6	+	19887.2i	−0.813270	+	0.682415i	−0.951386	−	0.308001i	$-0.900340\pi$
$947$	−23700.6	+	19887.2i	−0.813270	+	0.682415i	0.138116	+	0.990416i	$0.455895\pi$
$948$		0			0
$949$	−648.324	+	3676.83i	−0.0221765	+	0.125769i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	26600.7	+	46073.8i	0.904178	+	1.56608i	0.822017	+	0.569463i	$0.192849\pi$
$953$	26600.7	+	46073.8i	0.904178	+	1.56608i	0.0821613	+	0.996619i	$0.473818\pi$
$954$		0			0
$955$	−19538.1	+	33841.0i	−0.662031	+	1.14667i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	3474.44	+	2915.40i	0.116992	+	0.0981680i
$960$		0			0
$961$	−40381.3	−	14697.6i	−1.35549	−	0.493357i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	6620.68	+	37547.8i	0.220857	+	1.25254i
$966$		0			0
$967$	20742.7	−	7549.73i	0.689804	−	0.251068i	0.0267528	−	0.999642i	$-0.491483\pi$
$967$	20742.7	−	7549.73i	0.689804	−	0.251068i	0.663051	+	0.748574i	$0.269261\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−2497.51			−0.0825425			−0.0412712	−	0.999148i	$-0.513141\pi$
$971$	−2497.51			−0.0825425			−0.0412712	+	0.999148i	$0.513141\pi$
$972$		0			0
$973$	8224.19			0.270972
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−18538.1	+	6747.30i	−0.607047	+	0.220947i	−0.627211	−	0.778850i	$-0.715804\pi$
$977$	−18538.1	+	6747.30i	−0.607047	+	0.220947i	0.0201636	+	0.999797i	$0.493581\pi$
$978$		0			0
$979$	−3256.49	−	18468.5i	−0.106310	−	0.602916i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	9110.97	+	3316.12i	0.295620	+	0.107597i	0.485573	−	0.874196i	$-0.338611\pi$
$983$	9110.97	+	3316.12i	0.295620	+	0.107597i	−0.189952	+	0.981793i	$0.560833\pi$
$984$		0			0
$985$	−33535.3	−	28139.5i	−1.08480	−	0.910252i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	30255.4	−	52403.9i	0.972767	−	1.68488i
$990$		0			0
$991$	16318.5	+	28264.5i	0.523083	+	0.906006i	0.999639	+	0.0268620i	$0.00855148\pi$
$991$	16318.5	+	28264.5i	0.523083	+	0.906006i	−0.476556	+	0.879144i	$0.658115\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−4527.02	+	25674.0i	−0.144237	+	0.818011i
$996$		0			0
$997$	5862.89	−	4919.55i	0.186238	−	0.156273i	−0.544902	−	0.838500i	$-0.683433\pi$
$997$	5862.89	−	4919.55i	0.186238	−	0.156273i	0.731140	+	0.682227i	$0.238989\pi$
$998$		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.37.9		54
3.2	odd	2		108.4.i.a.49.6	✓	54
27.11	odd	18		108.4.i.a.97.6	yes	54
27.16	even	9	inner	324.4.i.a.289.9		54

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
108.4.i.a.49.6	✓	54	3.2	odd	2
108.4.i.a.97.6	yes	54	27.11	odd	18
324.4.i.a.37.9		54	1.1	even	1	trivial
324.4.i.a.289.9		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)

\(f(q)\)	\(=\)	\(q+(13.0875 - 4.76347i) q^{5} +(-0.689494 - 3.91032i) q^{7} +O(q^{10})\)	\(q+(13.0875 - 4.76347i) q^{5} +(-0.689494 - 3.91032i) q^{7} +(41.0823 + 14.9527i) q^{11} +(4.04425 + 3.39352i) q^{13} +(3.06285 - 5.30501i) q^{17} +(-7.42787 - 12.8654i) q^{19} +(21.3706 - 121.199i) q^{23} +(52.8371 - 44.3356i) q^{25} +(108.223 - 90.8101i) q^{29} +(-46.8412 + 265.650i) q^{31} +(-27.6504 - 47.8920i) q^{35} +(100.261 - 173.657i) q^{37} +(-239.406 - 200.886i) q^{41} +(462.032 + 168.166i) q^{43} +(10.6113 + 60.1798i) q^{47} +(307.499 - 111.921i) q^{49} +380.159 q^{53} +608.892 q^{55} +(154.748 - 56.3238i) q^{59} +(-149.877 - 849.997i) q^{61} +(69.0941 + 25.1482i) q^{65} +(159.359 + 133.718i) q^{67} +(451.411 - 781.866i) q^{71} +(353.597 + 612.447i) q^{73} +(30.1439 - 170.954i) q^{77} +(-847.858 + 711.438i) q^{79} +(-652.401 + 547.430i) q^{83} +(14.8148 - 84.0192i) q^{85} +(-214.477 - 371.485i) q^{89} +(10.4813 - 18.1541i) q^{91} +(-158.497 - 132.994i) q^{95} +(-1166.45 - 424.552i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 54 q - 12 q^{5}+O(q^{10}) \) 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

Varieties

Fields

Representations

Groups

Database

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