LMFDB - Embedded newform 324.4.i.a.181.2

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			181.2
Character	$\chi$	$=$	324.181
Dual form			324.4.i.a.145.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.26368 - 12.8380i) q^{5} +(-3.04053 - 2.55131i) q^{7} +O(q^{10})$	$q+(-2.26368 - 12.8380i) q^{5} +(-3.04053 - 2.55131i) q^{7} +(1.56137 - 8.85499i) q^{11} +(-33.8700 - 12.3277i) q^{13} +(-0.988527 - 1.71218i) q^{17} +(-56.3039 + 97.5212i) q^{19} +(73.1185 - 61.3537i) q^{23} +(-42.2272 + 15.3695i) q^{25} +(-237.316 + 86.3761i) q^{29} +(-154.385 + 129.544i) q^{31} +(-25.8708 + 44.8096i) q^{35} +(112.630 + 195.081i) q^{37} +(-209.778 - 76.3529i) q^{41} +(-10.6717 + 60.5223i) q^{43} +(-140.866 - 118.201i) q^{47} +(-56.8257 - 322.274i) q^{49} -596.365 q^{53} -117.214 q^{55} +(-26.4036 - 149.742i) q^{59} +(-282.206 - 236.799i) q^{61} +(-81.5913 + 462.727i) q^{65} +(869.318 + 316.406i) q^{67} +(-384.966 - 666.780i) q^{71} +(-329.352 + 570.454i) q^{73} +(-27.3392 + 22.9403i) q^{77} +(-1099.10 + 400.040i) q^{79} +(189.025 - 68.7996i) q^{83} +(-19.7432 + 16.5665i) q^{85} +(246.987 - 427.794i) q^{89} +(71.5311 + 123.896i) q^{91} +(1379.43 + 502.070i) q^{95} +(-95.1067 + 539.377i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/324\mathbb{Z}\right)^\times$.

$n$	$163$	$245$
$\chi(n)$	$1$	$e\left(\frac{8}{9}\right)$

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
$5$	−2.26368	−	12.8380i	−0.202469	−	1.14826i	−0.901372	−	0.433045i	$-0.857439\pi$
$5$	−2.26368	−	12.8380i	−0.202469	−	1.14826i	0.698903	−	0.715217i	$-0.253672\pi$
$6$		0			0
$7$	−3.04053	−	2.55131i	−0.164173	−	0.137758i	0.557000	−	0.830513i	$-0.311952\pi$
$7$	−3.04053	−	2.55131i	−0.164173	−	0.137758i	−0.721173	+	0.692755i	$0.756397\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	1.56137	−	8.85499i	0.0427974	−	0.242716i	−0.955903	−	0.293683i	$-0.905119\pi$
$11$	1.56137	−	8.85499i	0.0427974	−	0.242716i	0.998700	+	0.0509665i	$0.0162302\pi$
$12$		0			0
$13$	−33.8700	−	12.3277i	−0.722603	−	0.263006i	−0.0455728	−	0.998961i	$-0.514511\pi$
$13$	−33.8700	−	12.3277i	−0.722603	−	0.263006i	−0.677031	+	0.735955i	$0.736734\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−0.988527	−	1.71218i	−0.0141031	−	0.0244273i	0.858888	−	0.512164i	$-0.171156\pi$
$17$	−0.988527	−	1.71218i	−0.0141031	−	0.0244273i	−0.872991	+	0.487737i	$0.837823\pi$
$18$		0			0
$19$	−56.3039	+	97.5212i	−0.679842	+	1.17752i	0.295186	+	0.955440i	$0.404618\pi$
$19$	−56.3039	+	97.5212i	−0.679842	+	1.17752i	−0.975028	+	0.222081i	$0.928715\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	73.1185	−	61.3537i	0.662881	−	0.556223i	−0.248068	−	0.968743i	$-0.579796\pi$
$23$	73.1185	−	61.3537i	0.662881	−	0.556223i	0.910949	+	0.412520i	$0.135351\pi$
$24$		0			0
$25$	−42.2272	+	15.3695i	−0.337818	+	0.122956i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−237.316	+	86.3761i	−1.51960	+	0.553091i	−0.961049	−	0.276377i	$-0.910866\pi$
$29$	−237.316	+	86.3761i	−1.51960	+	0.553091i	−0.558555	+	0.829467i	$0.688644\pi$
$30$		0			0
$31$	−154.385	+	129.544i	−0.894462	+	0.750543i	−0.969100	−	0.246668i	$-0.920664\pi$
$31$	−154.385	+	129.544i	−0.894462	+	0.750543i	0.0746377	+	0.997211i	$0.476220\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−25.8708	+	44.8096i	−0.124942	+	0.216406i
$36$		0			0
$37$	112.630	+	195.081i	0.500440	+	0.866788i	1.00000		0.000508554i	$0.000161878\pi$
$37$	112.630	+	195.081i	0.500440	+	0.866788i	−0.499560	+	0.866280i	$0.666505\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−209.778	−	76.3529i	−0.799069	−	0.290837i	−0.0899682	−	0.995945i	$-0.528677\pi$
$41$	−209.778	−	76.3529i	−0.799069	−	0.290837i	−0.709101	+	0.705107i	$0.750899\pi$
$42$		0			0
$43$	−10.6717	+	60.5223i	−0.0378470	+	0.214641i	−0.997866	−	0.0652936i	$-0.979202\pi$
$43$	−10.6717	+	60.5223i	−0.0378470	+	0.214641i	0.960019	+	0.279934i	$0.0903127\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−140.866	−	118.201i	−0.437180	−	0.366837i	0.397473	−	0.917614i	$-0.369887\pi$
$47$	−140.866	−	118.201i	−0.437180	−	0.366837i	−0.834653	+	0.550776i	$0.814332\pi$
$48$		0			0
$49$	−56.8257	−	322.274i	−0.165672	−	0.939575i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−596.365			−1.54561			−0.772803	−	0.634647i	$-0.781146\pi$
$53$	−596.365			−1.54561			−0.772803	+	0.634647i	$0.781146\pi$
$54$		0			0
$55$	−117.214			−0.287367
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−26.4036	−	149.742i	−0.0582620	−	0.330420i	0.941720	−	0.336397i	$-0.109209\pi$
$59$	−26.4036	−	149.742i	−0.0582620	−	0.330420i	−0.999982	+	0.00597705i	$0.998097\pi$
$60$		0			0
$61$	−282.206	−	236.799i	−0.592340	−	0.497033i	0.296633	−	0.954992i	$-0.404136\pi$
$61$	−282.206	−	236.799i	−0.592340	−	0.497033i	−0.888973	+	0.457959i	$0.848581\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−81.5913	+	462.727i	−0.155695	+	0.882988i
$66$		0			0
$67$	869.318	+	316.406i	1.58514	+	0.576942i	0.976313	−	0.216364i	$-0.0694197\pi$
$67$	869.318	+	316.406i	1.58514	+	0.576942i	0.608823	+	0.793306i	$0.291642\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−384.966	−	666.780i	−0.643479	−	1.11454i	−0.984651	−	0.174537i	$-0.944157\pi$
$71$	−384.966	−	666.780i	−0.643479	−	1.11454i	0.341171	−	0.940001i	$-0.389176\pi$
$72$		0			0
$73$	−329.352	+	570.454i	−0.528051	+	0.914611i	0.471414	+	0.881912i	$0.343744\pi$
$73$	−329.352	+	570.454i	−0.528051	+	0.914611i	−0.999465	+	0.0326994i	$0.989590\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−27.3392	+	22.9403i	−0.0404623	+	0.0339519i
$78$		0			0
$79$	−1099.10	+	400.040i	−1.56530	+	0.569721i	−0.971942	−	0.235220i	$-0.924419\pi$
$79$	−1099.10	+	400.040i	−1.56530	+	0.569721i	−0.593354	+	0.804941i	$0.702197\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	189.025	−	68.7996i	0.249979	−	0.0909849i	−0.213992	−	0.976835i	$-0.568647\pi$
$83$	189.025	−	68.7996i	0.249979	−	0.0909849i	0.463971	+	0.885851i	$0.346424\pi$
$84$		0			0
$85$	−19.7432	+	16.5665i	−0.0251935	+	0.0211399i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	246.987	−	427.794i	0.294164	−	0.509506i	−0.680626	−	0.732631i	$-0.738292\pi$
$89$	246.987	−	427.794i	0.294164	−	0.509506i	0.974790	+	0.223124i	$0.0716256\pi$
$90$		0			0
$91$	71.5311	+	123.896i	0.0824011	+	0.142723i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	1379.43	+	502.070i	1.48975	+	0.542224i
$96$		0			0
$97$	−95.1067	+	539.377i	−0.0995529	+	0.564592i	0.893704	+	0.448657i	$0.148098\pi$
$97$	−95.1067	+	539.377i	−0.0995529	+	0.564592i	−0.993257	+	0.115935i	$0.963014\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	1429.40	+	1199.41i	1.40822	+	1.18164i	0.957307	+	0.289072i	$0.0933467\pi$
$101$	1429.40	+	1199.41i	1.40822	+	1.18164i	0.450915	+	0.892567i	$0.351098\pi$
$102$		0			0
$103$	−218.788	−	1240.81i	−0.209299	−	1.18699i	−0.890530	−	0.454925i	$-0.849666\pi$
$103$	−218.788	−	1240.81i	−0.209299	−	1.18699i	0.681230	−	0.732069i	$-0.261445\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−440.582			−0.398062			−0.199031	−	0.979993i	$-0.563779\pi$
$107$	−440.582			−0.398062			−0.199031	+	0.979993i	$0.563779\pi$
$108$		0			0
$109$	492.564			0.432836			0.216418	−	0.976301i	$-0.430563\pi$
$109$	492.564			0.432836			0.216418	+	0.976301i	$0.430563\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−17.6894	−	100.321i	−0.0147263	−	0.0835171i	0.976559	−	0.215251i	$-0.0690568\pi$
$113$	−17.6894	−	100.321i	−0.0147263	−	0.0835171i	−0.991285	+	0.131733i	$0.957946\pi$
$114$		0			0
$115$	−953.172	−	799.806i	−0.772902	−	0.648542i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−1.36265	+	7.72797i	−0.00104970	+	0.00595313i
$120$		0			0
$121$	1174.76	+	427.577i	0.882613	+	0.321245i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−521.849	−	903.869i	−0.373405	−	0.646756i
$126$		0			0
$127$	409.633	−	709.506i	0.286213	−	0.495736i	−0.686689	−	0.726951i	$-0.740937\pi$
$127$	409.633	−	709.506i	0.286213	−	0.495736i	0.972903	+	0.231215i	$0.0742701\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	1924.49	−	1614.84i	1.28354	−	1.07702i	0.290795	−	0.956785i	$-0.406080\pi$
$131$	1924.49	−	1614.84i	1.28354	−	1.07702i	0.992746	−	0.120233i	$-0.0383642\pi$
$132$		0			0
$133$	420.001	−	152.868i	0.273825	−	0.0996640i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−2588.91	+	942.285i	−1.61449	+	0.587626i	−0.982321	−	0.187204i	$-0.940057\pi$
$137$	−2588.91	+	942.285i	−1.61449	+	0.587626i	−0.632169	+	0.774831i	$0.717835\pi$
$138$		0			0
$139$	2395.25	−	2009.85i	1.46160	−	1.22643i	0.538073	−	0.842898i	$-0.319152\pi$
$139$	2395.25	−	2009.85i	1.46160	−	1.22643i	0.923528	−	0.383531i	$-0.125292\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−162.045	+	280.670i	−0.0947614	+	0.164132i
$144$		0			0
$145$	1646.10	+	2851.13i	0.942766	+	1.63292i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	1326.33	+	482.746i	0.729245	+	0.265424i	0.679845	−	0.733355i	$-0.262047\pi$
$149$	1326.33	+	482.746i	0.729245	+	0.265424i	0.0494000	+	0.998779i	$0.484269\pi$
$150$		0			0
$151$	597.674	−	3389.58i	0.322106	−	1.82675i	−0.207172	−	0.978305i	$-0.566426\pi$
$151$	597.674	−	3389.58i	0.322106	−	1.82675i	0.529278	−	0.848449i	$-0.322463\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	2012.56	+	1688.74i	1.04292	+	0.875115i
$156$		0			0
$157$	−408.190	−	2314.96i	−0.207497	−	1.17678i	−0.893461	−	0.449141i	$-0.851730\pi$
$157$	−408.190	−	2314.96i	−0.207497	−	1.17678i	0.685964	−	0.727636i	$-0.259381\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−378.851			−0.185451
$162$		0			0
$163$	1291.76			0.620729			0.310364	−	0.950618i	$-0.399549\pi$
$163$	1291.76			0.620729			0.310364	+	0.950618i	$0.399549\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−83.6793	−	474.569i	−0.0387742	−	0.219900i	0.959264	−	0.282512i	$-0.0911678\pi$
$167$	−83.6793	−	474.569i	−0.0387742	−	0.219900i	−0.998038	+	0.0626128i	$0.980057\pi$
$168$		0			0
$169$	−687.795	−	577.129i	−0.313061	−	0.262689i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−313.847	+	1779.91i	−0.137927	+	0.782222i	0.834850	+	0.550478i	$0.185554\pi$
$173$	−313.847	+	1779.91i	−0.137927	+	0.782222i	−0.972777	+	0.231744i	$0.925557\pi$
$174$		0			0
$175$	167.606	+	61.0034i	0.0723988	+	0.0263510i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	692.110	+	1198.77i	0.288998	+	0.500560i	0.973571	−	0.228385i	$-0.0733444\pi$
$179$	692.110	+	1198.77i	0.288998	+	0.500560i	−0.684572	+	0.728945i	$0.740011\pi$
$180$		0			0
$181$	−1360.40	+	2356.28i	−0.558660	+	0.967628i	0.438948	+	0.898512i	$0.355351\pi$
$181$	−1360.40	+	2356.28i	−0.558660	+	0.967628i	−0.997609	+	0.0691156i	$0.977982\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	2249.49	−	1887.54i	0.893975	−	0.750135i
$186$		0			0
$187$	−16.7048	+	6.08004i	−0.00653248	+	0.00237763i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−696.042	+	253.339i	−0.263685	+	0.0959735i	−0.470480	−	0.882411i	$-0.655919\pi$
$191$	−696.042	+	253.339i	−0.263685	+	0.0959735i	0.206795	+	0.978384i	$0.433697\pi$
$192$		0			0
$193$	140.297	−	117.723i	0.0523253	−	0.0439062i	−0.616250	−	0.787551i	$-0.711349\pi$
$193$	140.297	−	117.723i	0.0523253	−	0.0439062i	0.668575	+	0.743644i	$0.266904\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−264.489	+	458.109i	−0.0956552	+	0.165680i	−0.909882	−	0.414867i	$-0.863828\pi$
$197$	−264.489	+	458.109i	−0.0956552	+	0.165680i	0.814227	+	0.580547i	$0.197161\pi$
$198$		0			0
$199$	−129.667	−	224.589i	−0.0461901	−	0.0800036i	0.842006	−	0.539468i	$-0.181375\pi$
$199$	−129.667	−	224.589i	−0.0461901	−	0.0800036i	−0.888196	+	0.459465i	$0.848041\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	941.940	+	342.838i	0.325671	+	0.118535i
$204$		0			0
$205$	−505.346	+	2865.96i	−0.172170	+	0.976426i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	775.638	+	650.837i	0.256708	+	0.215404i
$210$		0			0
$211$	260.176	+	1475.53i	0.0848875	+	0.481421i	0.997381	+	0.0723293i	$0.0230432\pi$
$211$	260.176	+	1475.53i	0.0848875	+	0.481421i	−0.912493	+	0.409092i	$0.865846\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	801.139			0.254127
$216$		0			0
$217$	799.920			0.250240
$218$		0			0
$219$		0			0
$220$		0			0
$221$	12.3742	+	70.1777i	0.00376643	+	0.0213605i
$222$		0			0
$223$	−1309.46	−	1098.77i	−0.393219	−	0.329950i	0.424647	−	0.905359i	$-0.360398\pi$
$223$	−1309.46	−	1098.77i	−0.393219	−	0.329950i	−0.817866	+	0.575409i	$0.804843\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	602.888	−	3419.15i	0.176278	−	0.999722i	−0.760380	−	0.649478i	$-0.774987\pi$
$227$	602.888	−	3419.15i	0.176278	−	0.999722i	0.936658	−	0.350244i	$-0.113901\pi$
$228$		0			0
$229$	−3281.93	−	1194.52i	−0.947056	−	0.344700i	−0.178108	−	0.984011i	$-0.556998\pi$
$229$	−3281.93	−	1194.52i	−0.947056	−	0.344700i	−0.768949	+	0.639311i	$0.779220\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−2934.66	−	5082.98i	−0.825133	−	1.42917i	−0.901817	−	0.432117i	$-0.857767\pi$
$233$	−2934.66	−	5082.98i	−0.825133	−	1.42917i	0.0766840	−	0.997055i	$-0.475567\pi$
$234$		0			0
$235$	−1198.58	+	2076.00i	−0.332710	+	0.576270i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−907.681	+	761.635i	−0.245661	+	0.206134i	−0.757301	−	0.653066i	$-0.773483\pi$
$239$	−907.681	+	761.635i	−0.245661	+	0.206134i	0.511640	+	0.859200i	$0.329038\pi$
$240$		0			0
$241$	−4055.91	+	1476.23i	−1.08408	+	0.394574i	−0.821426	−	0.570315i	$-0.806821\pi$
$241$	−4055.91	+	1476.23i	−1.08408	+	0.394574i	−0.262658	+	0.964889i	$0.584599\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−4008.71	+	1459.05i	−1.04533	+	0.380471i
$246$		0			0
$247$	3109.22	−	2608.95i	0.800951	−	0.672078i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	1097.56	−	1901.04i	0.276006	−	0.478057i	−0.694382	−	0.719606i	$-0.744322\pi$
$251$	1097.56	−	1901.04i	0.276006	−	0.478057i	0.970389	+	0.241549i	$0.0776556\pi$
$252$		0			0
$253$	−429.121	−	743.259i	−0.106635	−	0.184697i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−2279.52	−	829.676i	−0.553277	−	0.201376i	0.0502247	−	0.998738i	$-0.484006\pi$
$257$	−2279.52	−	829.676i	−0.553277	−	0.201376i	−0.603502	+	0.797361i	$0.706228\pi$
$258$		0			0
$259$	155.257	−	880.506i	0.0372479	−	0.211243i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	2454.03	+	2059.17i	0.575368	+	0.482791i	0.883422	−	0.468577i	$-0.155233\pi$
$263$	2454.03	+	2059.17i	0.575368	+	0.482791i	−0.308054	+	0.951369i	$0.599678\pi$
$264$		0			0
$265$	1349.98	+	7656.11i	0.312938	+	1.77476i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−5960.11			−1.35091			−0.675454	−	0.737402i	$-0.736052\pi$
$269$	−5960.11			−1.35091			−0.675454	+	0.737402i	$0.736052\pi$
$270$		0			0
$271$	−5019.58			−1.12516			−0.562579	−	0.826743i	$-0.690191\pi$
$271$	−5019.58			−1.12516			−0.562579	+	0.826743i	$0.690191\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	70.1639	+	397.919i	0.0153856	+	0.0872561i
$276$		0			0
$277$	146.113	+	122.603i	0.0316934	+	0.0265939i	0.658497	−	0.752584i	$-0.271193\pi$
$277$	146.113	+	122.603i	0.0316934	+	0.0265939i	−0.626803	+	0.779177i	$0.715637\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	767.156	−	4350.76i	0.162864	−	0.923646i	−0.788376	−	0.615194i	$-0.789078\pi$
$281$	767.156	−	4350.76i	0.162864	−	0.923646i	0.951240	−	0.308452i	$-0.0998111\pi$
$282$		0			0
$283$	−1405.36	−	511.510i	−0.295195	−	0.107442i	0.190177	−	0.981750i	$-0.439094\pi$
$283$	−1405.36	−	511.510i	−0.295195	−	0.107442i	−0.485372	+	0.874308i	$0.661316\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	443.037	+	767.362i	0.0911207	+	0.157826i
$288$		0			0
$289$	2454.55	−	4251.40i	0.499602	−	0.865336i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−3504.42	+	2940.56i	−0.698739	+	0.586312i	−0.921415	−	0.388581i	$-0.872965\pi$
$293$	−3504.42	+	2940.56i	−0.698739	+	0.586312i	0.222676	+	0.974893i	$0.428521\pi$
$294$		0			0
$295$	−1862.62	+	677.937i	−0.367613	+	0.133800i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−3232.87	+	1176.67i	−0.625290	+	0.227587i
$300$		0			0
$301$	186.859	−	156.793i	0.0357819	−	0.0300246i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−2401.19	+	4158.98i	−0.450792	+	0.780795i
$306$		0			0
$307$	−117.958	−	204.309i	−0.0219290	−	0.0379822i	0.854853	−	0.518871i	$-0.173647\pi$
$307$	−117.958	−	204.309i	−0.0219290	−	0.0379822i	−0.876782	+	0.480889i	$0.840314\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−5819.65	−	2118.18i	−1.06110	−	0.386209i	−0.248258	−	0.968694i	$-0.579858\pi$
$311$	−5819.65	−	2118.18i	−1.06110	−	0.386209i	−0.812841	+	0.582485i	$0.802080\pi$
$312$		0			0
$313$	−1715.58	+	9729.51i	−0.309809	+	1.75701i	0.290150	+	0.956981i	$0.406295\pi$
$313$	−1715.58	+	9729.51i	−0.309809	+	1.75701i	−0.599959	+	0.800031i	$0.704816\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	4080.53	+	3423.97i	0.722982	+	0.606654i	0.928208	−	0.372061i	$-0.121349\pi$
$317$	4080.53	+	3423.97i	0.722982	+	0.606654i	−0.205227	+	0.978714i	$0.565793\pi$
$318$		0			0
$319$	394.320	+	2236.30i	0.0692090	+	0.392504i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	222.632			0.0383516
$324$		0			0
$325$	1619.71			0.276446
$326$		0			0
$327$		0			0
$328$		0			0
$329$	126.742	+	718.787i	0.0212386	+	0.120450i
$330$		0			0
$331$	4838.20	+	4059.73i	0.803419	+	0.674148i	0.949027	−	0.315194i	$-0.102070\pi$
$331$	4838.20	+	4059.73i	0.803419	+	0.674148i	−0.145609	+	0.989342i	$0.546514\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	2094.15	−	11876.5i	0.341539	−	1.93696i
$336$		0			0
$337$	−2402.61	−	874.479i	−0.388364	−	0.141353i	0.140456	−	0.990087i	$-0.455143\pi$
$337$	−2402.61	−	874.479i	−0.388364	−	0.141353i	−0.528820	+	0.848734i	$0.677365\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	906.060	+	1569.34i	0.143888	+	0.249222i
$342$		0			0
$343$	−1330.15	+	2303.88i	−0.209391	+	0.362677i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−8752.25	+	7344.01i	−1.35402	+	1.13616i	−0.376241	+	0.926522i	$0.622784\pi$
$347$	−8752.25	+	7344.01i	−1.35402	+	1.13616i	−0.977779	+	0.209637i	$0.932772\pi$
$348$		0			0
$349$	6887.34	−	2506.79i	1.05636	−	0.384485i	0.245303	−	0.969447i	$-0.421113\pi$
$349$	6887.34	−	2506.79i	1.05636	−	0.384485i	0.811061	+	0.584962i	$0.198890\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−12232.6	+	4452.30i	−1.84441	+	0.671309i	−0.856522	+	0.516110i	$0.827380\pi$
$353$	−12232.6	+	4452.30i	−1.84441	+	0.671309i	−0.987883	+	0.155199i	$0.950398\pi$
$354$		0			0
$355$	−7688.65	+	6451.55i	−1.14950	+	0.964542i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	2710.16	−	4694.13i	0.398431	−	0.690102i	−0.595102	−	0.803650i	$-0.702888\pi$
$359$	2710.16	−	4694.13i	0.398431	−	0.690102i	0.993533	+	0.113548i	$0.0362216\pi$
$360$		0			0
$361$	−2910.76	−	5041.58i	−0.424370	−	0.735031i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	8069.01	+	2936.88i	1.15713	+	0.421160i
$366$		0			0
$367$	2194.97	−	12448.3i	0.312198	−	1.77056i	−0.275323	−	0.961352i	$-0.588785\pi$
$367$	2194.97	−	12448.3i	0.312198	−	1.77056i	0.587520	−	0.809209i	$-0.300104\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	1813.27	+	1521.51i	0.253747	+	0.212919i
$372$		0			0
$373$	−658.319	−	3733.51i	−0.0913847	−	0.518268i	−0.995795	−	0.0916052i	$-0.970800\pi$
$373$	−658.319	−	3733.51i	−0.0913847	−	0.518268i	0.904411	−	0.426663i	$-0.140311\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	9102.71			1.24354
$378$		0			0
$379$	−2493.23			−0.337912			−0.168956	−	0.985624i	$-0.554040\pi$
$379$	−2493.23			−0.337912			−0.168956	+	0.985624i	$0.554040\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−1316.89	−	7468.43i	−0.175691	−	0.996395i	−0.937343	−	0.348408i	$-0.886722\pi$
$383$	−1316.89	−	7468.43i	−0.175691	−	0.996395i	0.761652	−	0.647987i	$-0.224389\pi$
$384$		0			0
$385$	356.394	+	299.050i	0.0471780	+	0.0395870i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	1875.22	−	10634.9i	0.244415	−	1.38615i	−0.577433	−	0.816438i	$-0.695945\pi$
$389$	1875.22	−	10634.9i	0.244415	−	1.38615i	0.821848	−	0.569707i	$-0.192943\pi$
$390$		0			0
$391$	−177.328	−	64.5421i	−0.0229357	−	0.00834792i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	7623.70	+	13204.6i	0.971114	+	1.68202i
$396$		0			0
$397$	3857.11	−	6680.72i	0.487615	−	0.844573i	−0.512284	−	0.858816i	$-0.671200\pi$
$397$	3857.11	−	6680.72i	0.487615	−	0.844573i	0.999899	+	0.0142430i	$0.00453383\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	7720.76	−	6478.49i	0.961487	−	0.806783i	−0.0197074	−	0.999806i	$-0.506273\pi$
$401$	7720.76	−	6478.49i	0.961487	−	0.806783i	0.981194	+	0.193022i	$0.0618290\pi$
$402$		0			0
$403$	6825.99	−	2484.46i	0.843739	−	0.307096i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	1903.30	−	692.744i	0.231801	−	0.0843687i
$408$		0			0
$409$	−4145.48	+	3478.47i	−0.501175	+	0.420536i	−0.858011	−	0.513631i	$-0.828300\pi$
$409$	−4145.48	+	3478.47i	−0.501175	+	0.420536i	0.356836	+	0.934167i	$0.383856\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−301.758	+	522.661i	−0.0359529	+	0.0622723i
$414$		0			0
$415$	−1311.14	−	2270.96i	−0.155087	−	0.268619i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−11675.1	−	4249.40i	−1.36126	−	0.495457i	−0.444815	−	0.895623i	$-0.646730\pi$
$419$	−11675.1	−	4249.40i	−1.36126	−	0.495457i	−0.916443	+	0.400166i	$0.868953\pi$
$420$		0			0
$421$	−428.725	+	2431.42i	−0.0496313	+	0.281473i	−0.999515	−	0.0311294i	$-0.990090\pi$
$421$	−428.725	+	2431.42i	−0.0496313	+	0.281473i	0.949884	+	0.312602i	$0.101201\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	68.0580	+	57.1074i	0.00776776	+	0.00651792i
$426$		0			0
$427$	253.909	+	1439.99i	0.0287764	+	0.163199i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	11653.1			1.30234			0.651172	−	0.758930i	$-0.274278\pi$
$431$	11653.1			1.30234			0.651172	+	0.758930i	$0.274278\pi$
$432$		0			0
$433$	5397.02			0.598994			0.299497	−	0.954097i	$-0.403181\pi$
$433$	5397.02			0.598994			0.299497	+	0.954097i	$0.403181\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	1866.43	+	10585.0i	0.204310	+	1.15870i
$438$		0			0
$439$	7244.36	+	6078.74i	0.787595	+	0.660871i	0.945149	−	0.326640i	$-0.105916\pi$
$439$	7244.36	+	6078.74i	0.787595	+	0.660871i	−0.157554	+	0.987510i	$0.550361\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	2289.99	−	12987.2i	0.245600	−	1.39286i	−0.573496	−	0.819208i	$-0.694413\pi$
$443$	2289.99	−	12987.2i	0.245600	−	1.39286i	0.819096	−	0.573656i	$-0.194476\pi$
$444$		0			0
$445$	−6051.10	−	2202.42i	−0.644606	−	0.234617i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	3593.50	+	6224.12i	0.377701	+	0.654197i	0.990727	−	0.135866i	$-0.0433815\pi$
$449$	3593.50	+	6224.12i	0.377701	+	0.654197i	−0.613027	+	0.790062i	$0.710048\pi$
$450$		0			0
$451$	−1003.65	+	1738.37i	−0.104789	+	0.181500i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	1428.64	−	1198.77i	0.147199	−	0.123515i
$456$		0			0
$457$	10349.3	−	3766.83i	1.05934	−	0.385568i	0.247160	−	0.968975i	$-0.420503\pi$
$457$	10349.3	−	3766.83i	1.05934	−	0.385568i	0.812180	+	0.583406i	$0.198280\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	9343.16	−	3400.63i	0.943936	−	0.343565i	0.176217	−	0.984351i	$-0.443614\pi$
$461$	9343.16	−	3400.63i	0.943936	−	0.343565i	0.767719	+	0.640787i	$0.221392\pi$
$462$		0			0
$463$	−5891.36	+	4943.44i	−0.591349	+	0.496201i	−0.888652	−	0.458582i	$-0.848357\pi$
$463$	−5891.36	+	4943.44i	−0.591349	+	0.496201i	0.297303	+	0.954783i	$0.403913\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−4935.93	+	8549.27i	−0.489095	+	0.847137i	−0.999921	−	0.0125465i	$-0.996006\pi$
$467$	−4935.93	+	8549.27i	−0.489095	+	0.847137i	0.510826	+	0.859684i	$0.329340\pi$
$468$		0			0
$469$	−1835.94	−	3179.94i	−0.180759	−	0.313083i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	519.261	+	188.996i	0.0504771	+	0.0183722i
$474$		0			0
$475$	878.710	−	4983.41i	0.0848799	−	0.481378i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	9244.93	+	7757.42i	0.881861	+	0.739969i	0.966561	−	0.256437i	$-0.0825486\pi$
$479$	9244.93	+	7757.42i	0.881861	+	0.739969i	−0.0846996	+	0.996407i	$0.526993\pi$
$480$		0			0
$481$	−1409.89	−	7995.87i	−0.133649	−	0.757963i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	7139.79			0.668456
$486$		0			0
$487$	−6706.32			−0.624009			−0.312004	−	0.950081i	$-0.601000\pi$
$487$	−6706.32			−0.624009			−0.312004	+	0.950081i	$0.601000\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	1760.17	+	9982.44i	0.161783	+	0.917517i	0.952320	+	0.305102i	$0.0986907\pi$
$491$	1760.17	+	9982.44i	0.161783	+	0.917517i	−0.790536	+	0.612415i	$0.790198\pi$
$492$		0			0
$493$	382.485	+	320.943i	0.0349417	+	0.0293196i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−530.662	+	3009.53i	−0.0478942	+	0.271622i
$498$		0			0
$499$	16918.8	+	6157.95i	1.51782	+	0.552441i	0.960602	−	0.277927i	$-0.0896474\pi$
$499$	16918.8	+	6157.95i	1.51782	+	0.552441i	0.557216	+	0.830368i	$0.311870\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	4061.30	+	7034.37i	0.360009	+	0.623553i	0.987962	−	0.154698i	$-0.0494404\pi$
$503$	4061.30	+	7034.37i	0.360009	+	0.623553i	−0.627953	+	0.778251i	$0.716107\pi$
$504$		0			0
$505$	12162.2	−	21065.6i	1.07171	−	1.85625i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−6075.88	+	5098.27i	−0.529094	+	0.443962i	−0.867788	−	0.496934i	$-0.834459\pi$
$509$	−6075.88	+	5098.27i	−0.529094	+	0.443962i	0.338695	+	0.940896i	$0.390015\pi$
$510$		0			0
$511$	2456.81	−	894.206i	0.212687	−	0.0774117i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−15434.2	+	5617.58i	−1.32060	+	0.480660i
$516$		0			0
$517$	−1266.61	+	1062.81i	−0.107748	+	0.0904110i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	6585.42	−	11406.3i	0.553766	−	0.959151i	−0.444232	−	0.895912i	$-0.646523\pi$
$521$	6585.42	−	11406.3i	0.553766	−	0.959151i	0.997998	−	0.0632397i	$-0.0201433\pi$
$522$		0			0
$523$	−2530.57	−	4383.08i	−0.211576	−	0.366460i	0.740632	−	0.671911i	$-0.234526\pi$
$523$	−2530.57	−	4383.08i	−0.211576	−	0.366460i	−0.952208	+	0.305451i	$0.901193\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	374.416	+	136.276i	0.0309485	+	0.0112643i
$528$		0			0
$529$	−530.742	+	3009.99i	−0.0436214	+	0.247389i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	6163.92	+	5172.15i	0.500918	+	0.420320i
$534$		0			0
$535$	997.335	+	5656.17i	0.0805954	+	0.457079i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−2942.46			−0.235141
$540$		0			0
$541$	6681.41			0.530973			0.265486	−	0.964115i	$-0.414467\pi$
$541$	6681.41			0.530973			0.265486	+	0.964115i	$0.414467\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−1115.01	−	6323.52i	−0.0876360	−	0.497008i
$546$		0			0
$547$	5816.65	+	4880.75i	0.454665	+	0.381510i	0.841164	−	0.540781i	$-0.181871\pi$
$547$	5816.65	+	4880.75i	0.454665	+	0.381510i	−0.386498	+	0.922290i	$0.626315\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	4938.33	−	28006.7i	0.381815	−	2.16538i
$552$		0			0
$553$	4362.48	+	1587.81i	0.335463	+	0.122099i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−11184.2	−	19371.6i	−0.850789	−	1.47361i	−0.880498	−	0.474050i	$-0.842791\pi$
$557$	−11184.2	−	19371.6i	−0.850789	−	1.47361i	0.0297091	−	0.999559i	$-0.490542\pi$
$558$		0			0
$559$	1107.55	−	1918.33i	0.0838002	−	0.145146i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−19967.2	+	16754.5i	−1.49470	+	1.25420i	−0.606213	+	0.795303i	$0.707312\pi$
$563$	−19967.2	+	16754.5i	−1.49470	+	1.25420i	−0.888488	+	0.458900i	$0.848244\pi$
$564$		0			0
$565$	−1247.88	+	454.190i	−0.0929179	+	0.0338193i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−9061.94	+	3298.28i	−0.667656	+	0.243007i	−0.653538	−	0.756893i	$-0.726716\pi$
$569$	−9061.94	+	3298.28i	−0.667656	+	0.243007i	−0.0141178	+	0.999900i	$0.504494\pi$
$570$		0			0
$571$	−515.270	+	432.363i	−0.0377642	+	0.0316879i	−0.661474	−	0.749968i	$-0.730069\pi$
$571$	−515.270	+	432.363i	−0.0377642	+	0.0316879i	0.623710	+	0.781656i	$0.285625\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−2144.62	+	3714.59i	−0.155542	+	0.269407i
$576$		0			0
$577$	−8963.79	−	15525.7i	−0.646738	−	1.12018i	−0.983897	−	0.178735i	$-0.942800\pi$
$577$	−8963.79	−	15525.7i	−0.646738	−	1.12018i	0.337160	−	0.941448i	$-0.390534\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−750.267	−	273.075i	−0.0535737	−	0.0194992i
$582$		0			0
$583$	−931.149	+	5280.81i	−0.0661479	+	0.375144i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	5963.06	+	5003.60i	0.419287	+	0.351824i	0.827892	−	0.560888i	$-0.189540\pi$
$587$	5963.06	+	5003.60i	0.419287	+	0.351824i	−0.408604	+	0.912712i	$0.633984\pi$
$588$		0			0
$589$	−3940.84	−	22349.6i	−0.275687	−	1.56350i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−10084.0			−0.698313			−0.349156	−	0.937064i	$-0.613532\pi$
$593$	−10084.0			−0.698313			−0.349156	+	0.937064i	$0.613532\pi$
$594$		0			0
$595$	102.296			0.00704828
$596$		0			0
$597$		0			0
$598$		0			0
$599$	1532.38	+	8690.56i	0.104526	+	0.592799i	0.991408	+	0.130803i	$0.0417555\pi$
$599$	1532.38	+	8690.56i	0.104526	+	0.592799i	−0.886882	+	0.461996i	$0.847133\pi$
$600$		0			0
$601$	4597.20	+	3857.51i	0.312019	+	0.261815i	0.785326	−	0.619082i	$-0.212495\pi$
$601$	4597.20	+	3857.51i	0.312019	+	0.261815i	−0.473307	+	0.880898i	$0.656940\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	2829.94	−	16049.4i	0.190171	−	1.07851i
$606$		0			0
$607$	8273.70	+	3011.38i	0.553244	+	0.201364i	0.603487	−	0.797373i	$-0.293777\pi$
$607$	8273.70	+	3011.38i	0.553244	+	0.201364i	−0.0502433	+	0.998737i	$0.516000\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	3314.00	+	5740.01i	0.219427	+	0.380059i
$612$		0			0
$613$	−9300.79	+	16109.4i	−0.612814	+	1.06143i	0.377949	+	0.925826i	$0.376629\pi$
$613$	−9300.79	+	16109.4i	−0.612814	+	1.06143i	−0.990764	+	0.135599i	$0.956704\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	13785.7	−	11567.6i	0.899502	−	0.754772i	−0.0705912	−	0.997505i	$-0.522489\pi$
$617$	13785.7	−	11567.6i	0.899502	−	0.754772i	0.970093	+	0.242734i	$0.0780441\pi$
$618$		0			0
$619$	−13208.1	+	4807.34i	−0.857636	+	0.312154i	−0.733149	−	0.680068i	$-0.761950\pi$
$619$	−13208.1	+	4807.34i	−0.857636	+	0.312154i	−0.124486	+	0.992221i	$0.539728\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−1842.41	+	670.581i	−0.118482	+	0.0431240i
$624$		0			0
$625$	−14725.5	+	12356.2i	−0.942433	+	0.790795i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	222.676	−	385.686i	0.0141155	−	0.0244488i
$630$		0			0
$631$	−215.873	−	373.904i	−0.0136193	−	0.0235893i	0.859135	−	0.511748i	$-0.171002\pi$
$631$	−215.873	−	373.904i	−0.0136193	−	0.0235893i	−0.872755	+	0.488159i	$0.837669\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−10035.9	−	3652.76i	−0.627184	−	0.228276i
$636$		0			0
$637$	−2048.21	+	11616.0i	−0.127399	+	0.722513i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	19141.2	+	16061.4i	1.17946	+	0.989683i	0.999982	+	0.00592039i	$0.00188453\pi$
$641$	19141.2	+	16061.4i	1.17946	+	0.989683i	0.179476	+	0.983762i	$0.442560\pi$
$642$		0			0
$643$	4953.92	+	28095.1i	0.303831	+	1.72311i	0.628955	+	0.777441i	$0.283483\pi$
$643$	4953.92	+	28095.1i	0.303831	+	1.72311i	−0.325124	+	0.945671i	$0.605406\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−3112.39			−0.189120			−0.0945601	−	0.995519i	$-0.530144\pi$
$647$	−3112.39			−0.189120			−0.0945601	+	0.995519i	$0.530144\pi$
$648$		0			0
$649$	−1367.19			−0.0826919
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−3977.39	−	22556.9i	−0.238357	−	1.35179i	−0.835427	−	0.549601i	$-0.814780\pi$
$653$	−3977.39	−	22556.9i	−0.238357	−	1.35179i	0.597070	−	0.802189i	$-0.296331\pi$
$654$		0			0
$655$	−25087.7	−	21051.1i	−1.49658	−	1.25578i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−2358.50	+	13375.7i	−0.139415	+	0.790659i	0.832269	+	0.554372i	$0.187042\pi$
$659$	−2358.50	+	13375.7i	−0.139415	+	0.790659i	−0.971683	+	0.236287i	$0.924069\pi$
$660$		0			0
$661$	−18384.6	−	6691.43i	−1.08181	−	0.393747i	−0.261229	−	0.965277i	$-0.584128\pi$
$661$	−18384.6	−	6691.43i	−1.08181	−	0.393747i	−0.820581	+	0.571530i	$0.806350\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−2913.25	−	5045.91i	−0.169881	−	0.294243i
$666$		0			0
$667$	−12052.7	+	20875.9i	−0.699674	+	1.21187i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−2537.48	+	2129.20i	−0.145989	+	0.122499i
$672$		0			0
$673$	−17264.6	+	6283.82i	−0.988861	+	0.359916i	−0.785279	−	0.619142i	$-0.787480\pi$
$673$	−17264.6	+	6283.82i	−0.988861	+	0.359916i	−0.203582	+	0.979058i	$0.565258\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−15478.3	+	5633.63i	−0.878697	+	0.319820i	−0.741684	−	0.670750i	$-0.765973\pi$
$677$	−15478.3	+	5633.63i	−0.878697	+	0.319820i	−0.137013	+	0.990569i	$0.543750\pi$
$678$		0			0
$679$	1665.29	−	1397.35i	0.0941209	−	0.0789768i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−6770.96	+	11727.7i	−0.379332	+	0.657022i	−0.990965	−	0.134119i	$-0.957179\pi$
$683$	−6770.96	+	11727.7i	−0.379332	+	0.657022i	0.611633	+	0.791141i	$0.290513\pi$
$684$		0			0
$685$	17957.5	+	31103.2i	1.00163	+	1.73488i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	20198.9	+	7351.79i	1.11686	+	0.406504i
$690$		0			0
$691$	159.678	−	905.576i	0.00879077	−	0.0498549i	−0.980097	−	0.198520i	$-0.936387\pi$
$691$	159.678	−	905.576i	0.00879077	−	0.0498549i	0.988888	+	0.148665i	$0.0474976\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−31224.5	−	26200.5i	−1.70419	−	1.42999i
$696$		0			0
$697$	76.6413	+	434.654i	0.00416499	+	0.0236208i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	12265.5			0.660857			0.330429	−	0.943831i	$-0.392807\pi$
$701$	12265.5			0.660857			0.330429	+	0.943831i	$0.392807\pi$
$702$		0			0
$703$	−25366.1			−1.36088
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−1286.07	−	7293.68i	−0.0684126	−	0.387987i
$708$		0			0
$709$	6415.08	+	5382.89i	0.339807	+	0.285132i	0.796682	−	0.604399i	$-0.206587\pi$
$709$	6415.08	+	5382.89i	0.339807	+	0.285132i	−0.456875	+	0.889531i	$0.651031\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−3340.37	+	18944.2i	−0.175453	+	0.995041i
$714$		0			0
$715$	3970.05	+	1444.98i	0.207652	+	0.0755793i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−3738.55	−	6475.35i	−0.193914	−	0.335869i	0.752630	−	0.658444i	$-0.228785\pi$
$719$	−3738.55	−	6475.35i	−0.193914	−	0.335869i	−0.946544	+	0.322575i	$0.895452\pi$
$720$		0			0
$721$	−2500.45	+	4330.91i	−0.129156	+	0.223705i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	8693.66	−	7294.84i	0.445344	−	0.373688i
$726$		0			0
$727$	10590.1	−	3854.47i	0.540253	−	0.196636i	−0.0574578	−	0.998348i	$-0.518299\pi$
$727$	10590.1	−	3854.47i	0.540253	−	0.196636i	0.597710	+	0.801712i	$0.296077\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	114.174	−	41.5560i	0.00577686	−	0.00210261i
$732$		0			0
$733$	13571.6	−	11387.9i	0.683871	−	0.573836i	−0.233264	−	0.972414i	$-0.574940\pi$
$733$	13571.6	−	11387.9i	0.683871	−	0.573836i	0.917135	+	0.398578i	$0.130496\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	4159.10	−	7203.77i	0.207873	−	0.360047i
$738$		0			0
$739$	9518.62	+	16486.7i	0.473813	+	0.820669i	0.999551	−	0.0299781i	$-0.00954375\pi$
$739$	9518.62	+	16486.7i	0.473813	+	0.820669i	−0.525737	+	0.850647i	$0.676210\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−10852.2	−	3949.86i	−0.535837	−	0.195029i	0.0599057	−	0.998204i	$-0.480920\pi$
$743$	−10852.2	−	3949.86i	−0.535837	−	0.195029i	−0.595743	+	0.803175i	$0.703142\pi$
$744$		0			0
$745$	3195.08	−	18120.2i	0.157126	−	0.891105i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	1339.60	+	1124.06i	0.0653512	+	0.0548361i
$750$		0			0
$751$	−2308.21	−	13090.5i	−0.112154	−	0.636058i	−0.988120	−	0.153685i	$-0.950886\pi$
$751$	−2308.21	−	13090.5i	−0.112154	−	0.636058i	0.875966	−	0.482374i	$-0.160225\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−44868.2			−2.16281
$756$		0			0
$757$	8467.78			0.406561			0.203280	−	0.979121i	$-0.434840\pi$
$757$	8467.78			0.406561			0.203280	+	0.979121i	$0.434840\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	1320.32	+	7487.93i	0.0628932	+	0.356685i	0.999971	+	0.00756534i	$0.00240815\pi$
$761$	1320.32	+	7487.93i	0.0628932	+	0.356685i	−0.937078	+	0.349119i	$0.886481\pi$
$762$		0			0
$763$	−1497.66	−	1256.68i	−0.0710601	−	0.0596265i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−951.684	+	5397.27i	−0.0448022	+	0.254086i
$768$		0			0
$769$	−26939.6	−	9805.21i	−1.26329	−	0.459799i	−0.378416	−	0.925636i	$-0.623531\pi$
$769$	−26939.6	−	9805.21i	−1.26329	−	0.459799i	−0.884870	+	0.465837i	$0.845753\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−14194.6	−	24585.8i	−0.660473	−	1.14397i	−0.980492	−	0.196561i	$-0.937023\pi$
$773$	−14194.6	−	24585.8i	−0.660473	−	1.14397i	0.320019	−	0.947411i	$-0.396311\pi$
$774$		0			0
$775$	4528.22	−	7843.11i	0.209882	−	0.363526i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	19257.3	−	16158.8i	0.885707	−	0.743197i
$780$		0			0
$781$	−6505.40	+	2367.77i	−0.298056	+	0.108483i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−28795.3	+	10480.6i	−1.30923	+	0.476523i
$786$		0			0
$787$	24754.7	−	20771.6i	1.12123	−	0.940824i	0.122565	−	0.992461i	$-0.460888\pi$
$787$	24754.7	−	20771.6i	1.12123	−	0.940824i	0.998666	+	0.0516363i	$0.0164437\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−202.166	+	350.161i	−0.00908747	+	0.0157400i
$792$		0			0
$793$	6639.13	+	11499.3i	0.297304	+	0.514946i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−721.496	−	262.603i	−0.0320661	−	0.0116711i	0.325937	−	0.945391i	$-0.394320\pi$
$797$	−721.496	−	262.603i	−0.0320661	−	0.0116711i	−0.358003	+	0.933720i	$0.616542\pi$
$798$		0			0
$799$	−63.1308	+	358.033i	−0.00279525	+	0.0158527i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	4537.12	+	3807.10i	0.199392	+	0.167310i
$804$		0			0
$805$	857.597	+	4863.68i	0.0375482	+	0.212947i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	8058.11			0.350195			0.175098	−	0.984551i	$-0.443976\pi$
$809$	8058.11			0.350195			0.175098	+	0.984551i	$0.443976\pi$
$810$		0			0
$811$	21999.7			0.952543			0.476271	−	0.879298i	$-0.341988\pi$
$811$	21999.7			0.952543			0.476271	+	0.879298i	$0.341988\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−2924.14	−	16583.6i	−0.125679	−	0.712759i
$816$		0			0
$817$	−5301.34	−	4448.36i	−0.227014	−	0.190488i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−5711.31	+	32390.4i	−0.242784	+	1.37690i	0.582798	+	0.812617i	$0.301958\pi$
$821$	−5711.31	+	32390.4i	−0.242784	+	1.37690i	−0.825582	+	0.564282i	$0.809153\pi$
$822$		0			0
$823$	−7267.53	−	2645.16i	−0.307813	−	0.112035i	0.183495	−	0.983021i	$-0.441259\pi$
$823$	−7267.53	−	2645.16i	−0.307813	−	0.112035i	−0.491308	+	0.870986i	$0.663481\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	20378.5	+	35296.6i	0.856868	+	1.48414i	0.874901	+	0.484302i	$0.160926\pi$
$827$	20378.5	+	35296.6i	0.856868	+	1.48414i	−0.0180324	+	0.999837i	$0.505740\pi$
$828$		0			0
$829$	2941.40	−	5094.65i	0.123231	−	0.213443i	−0.797809	−	0.602911i	$-0.794008\pi$
$829$	2941.40	−	5094.65i	0.123231	−	0.213443i	0.921040	+	0.389467i	$0.127341\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−495.618	+	415.873i	−0.0206148	+	0.0172979i
$834$		0			0
$835$	−5903.07	+	2148.54i	−0.244652	+	0.0890459i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	8702.18	−	3167.33i	0.358084	−	0.130332i	−0.156713	−	0.987644i	$-0.550090\pi$
$839$	8702.18	−	3167.33i	0.358084	−	0.130332i	0.514797	+	0.857312i	$0.327867\pi$
$840$		0			0
$841$	30175.1	−	25319.9i	1.23724	−	1.03817i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−5852.21	+	10136.3i	−0.238251	+	0.412663i
$846$		0			0
$847$	−2481.01	−	4297.23i	−0.100648	−	0.174327i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	20204.3	+	7353.76i	0.813859	+	0.296221i
$852$		0			0
$853$	−1.63259	+	9.25887i	−6.55320e−5	+	0.000371650i	−0.984841	−	0.173462i	$-0.944505\pi$
$853$	−1.63259	+	9.25887i	−6.55320e−5	+	0.000371650i	0.984775	+	0.173834i	$0.0556156\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	11455.5	+	9612.34i	0.456609	+	0.383140i	0.841881	−	0.539663i	$-0.181448\pi$
$857$	11455.5	+	9612.34i	0.456609	+	0.383140i	−0.385273	+	0.922803i	$0.625893\pi$
$858$		0			0
$859$	2361.07	+	13390.3i	0.0937819	+	0.531864i	0.995114	+	0.0987346i	$0.0314795\pi$
$859$	2361.07	+	13390.3i	0.0937819	+	0.531864i	−0.901332	+	0.433129i	$0.857409\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−10647.0			−0.419965			−0.209982	−	0.977705i	$-0.567341\pi$
$863$	−10647.0			−0.419965			−0.209982	+	0.977705i	$0.567341\pi$
$864$		0			0
$865$	23560.9			0.926121
$866$		0			0
$867$		0			0
$868$		0			0
$869$	1826.24	+	10357.1i	0.0712900	+	0.404306i
$870$		0			0
$871$	−25543.2	−	21433.3i	−0.993685	−	0.833801i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−719.351	+	4079.64i	−0.0277926	+	0.157619i
$876$		0			0
$877$	14775.3	+	5377.75i	0.568899	+	0.207062i	0.610424	−	0.792075i	$-0.290999\pi$
$877$	14775.3	+	5377.75i	0.568899	+	0.207062i	−0.0415243	+	0.999137i	$0.513221\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−25089.4	−	43456.1i	−0.959458	−	1.66183i	−0.723820	−	0.689989i	$-0.757616\pi$
$881$	−25089.4	−	43456.1i	−0.959458	−	1.66183i	−0.235637	−	0.971841i	$-0.575718\pi$
$882$		0			0
$883$	−22336.6	+	38688.1i	−0.851286	+	1.47447i	0.0287625	+	0.999586i	$0.490843\pi$
$883$	−22336.6	+	38688.1i	−0.851286	+	1.47447i	−0.880048	+	0.474884i	$0.842490\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	18077.5	−	15168.9i	0.684312	−	0.574206i	−0.232951	−	0.972488i	$-0.574838\pi$
$887$	18077.5	−	15168.9i	0.684312	−	0.574206i	0.917263	+	0.398283i	$0.130394\pi$
$888$		0			0
$889$	−3055.67	+	1112.17i	−0.115280	+	0.0419585i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	19458.4	−	7082.28i	0.729172	−	0.265397i
$894$		0			0
$895$	13823.0	−	11598.9i	0.516260	−	0.433194i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	25448.5	−	44078.1i	0.944111	−	1.63525i
$900$		0			0
$901$	589.523	+	1021.08i	0.0217978	+	0.0377550i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	33329.3	+	12130.9i	1.22420	+	0.445573i
$906$		0			0
$907$	−1291.82	+	7326.29i	−0.0472924	+	0.268209i	−0.999281	−	0.0379263i	$-0.987925\pi$
$907$	−1291.82	+	7326.29i	−0.0472924	+	0.268209i	0.951988	+	0.306135i	$0.0990359\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	34526.5	+	28971.2i	1.25567	+	1.05363i	0.996129	+	0.0878979i	$0.0280149\pi$
$911$	34526.5	+	28971.2i	1.25567	+	1.05363i	0.259539	+	0.965733i	$0.416430\pi$
$912$		0			0
$913$	−314.081	−	1781.24i	−0.0113851	−	0.0645678i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−9971.45			−0.359091
$918$		0			0
$919$	−16078.1			−0.577114			−0.288557	−	0.957463i	$-0.593176\pi$
$919$	−16078.1			−0.577114			−0.288557	+	0.957463i	$0.593176\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	4818.94	+	27329.6i	0.171850	+	0.974608i
$924$		0			0
$925$	−7754.35	−	6506.68i	−0.275634	−	0.231285i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−2638.13	+	14961.6i	−0.0931692	+	0.528389i	0.902124	+	0.431477i	$0.142008\pi$
$929$	−2638.13	+	14961.6i	−0.0931692	+	0.528389i	−0.995293	+	0.0969115i	$0.969104\pi$
$930$		0			0
$931$	34628.1	+	12603.6i	1.21900	+	0.443680i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	115.870	+	200.692i	0.00405277	+	0.00701960i
$936$		0			0
$937$	5116.61	−	8862.23i	0.178391	−	0.308982i	−0.762939	−	0.646471i	$-0.776244\pi$
$937$	5116.61	−	8862.23i	0.178391	−	0.308982i	0.941330	+	0.337489i	$0.109577\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	25497.0	−	21394.5i	0.883291	−	0.741170i	−0.0835617	−	0.996503i	$-0.526630\pi$
$941$	25497.0	−	21394.5i	0.883291	−	0.741170i	0.966853	+	0.255333i	$0.0821851\pi$
$942$		0			0
$943$	−20023.2	+	7287.84i	−0.691457	+	0.251670i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	42559.0	−	15490.2i	1.46038	−	0.531535i	0.514910	−	0.857244i	$-0.327825\pi$
$947$	42559.0	−	15490.2i	1.46038	−	0.531535i	0.945470	+	0.325709i	$0.105603\pi$
$948$		0			0
$949$	18187.5	−	15261.1i	0.622120	−	0.522021i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−21174.1	+	36674.6i	−0.719723	+	1.24660i	0.241387	+	0.970429i	$0.422398\pi$
$953$	−21174.1	+	36674.6i	−0.719723	+	1.24660i	−0.961110	+	0.276168i	$0.910936\pi$
$954$		0			0
$955$	4827.96	+	8362.28i	0.163591	+	0.283348i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	10275.7	+	3740.05i	0.346006	+	0.125936i
$960$		0			0
$961$	1879.81	−	10660.9i	0.0630999	−	0.357857i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−1828.91	−	1534.64i	−0.0610100	−	0.0511935i
$966$		0			0
$967$	−5422.78	−	30754.1i	−0.180336	−	1.02274i	−0.931803	−	0.362965i	$-0.881764\pi$
$967$	−5422.78	−	30754.1i	−0.180336	−	1.02274i	0.751467	−	0.659771i	$-0.229347\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	11490.0			0.379745			0.189872	−	0.981809i	$-0.439193\pi$
$971$	11490.0			0.379745			0.189872	+	0.981809i	$0.439193\pi$
$972$		0			0
$973$	−12410.6			−0.408906
$974$		0			0
$975$		0			0
$976$		0			0
$977$	3025.14	+	17156.4i	0.0990611	+	0.561803i	0.993427	+	0.114467i	$0.0365161\pi$
$977$	3025.14	+	17156.4i	0.0990611	+	0.561803i	−0.894366	+	0.447336i	$0.852373\pi$
$978$		0			0
$979$	−3402.47	−	2855.01i	−0.111076	−	0.0932039i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−2014.71	+	11426.0i	−0.0653706	+	0.370735i	0.934520	+	0.355912i	$0.115830\pi$
$983$	−2014.71	+	11426.0i	−0.0653706	+	0.370735i	−0.999890	+	0.0148232i	$0.995281\pi$
$984$		0			0
$985$	6479.89	+	2358.49i	0.209611	+	0.0762921i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	2932.96	+	5080.04i	0.0943001	+	0.163333i
$990$		0			0
$991$	19980.1	−	34606.5i	0.640452	−	1.10930i	−0.344880	−	0.938647i	$-0.612080\pi$
$991$	19980.1	−	34606.5i	0.640452	−	1.10930i	0.985332	−	0.170649i	$-0.0545863\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−2589.74	+	2173.05i	−0.0825130	+	0.0692366i
$996$		0			0
$997$	−34508.2	+	12560.0i	−1.09617	+	0.398975i	−0.825905	−	0.563810i	$-0.809335\pi$
$997$	−34508.2	+	12560.0i	−1.09617	+	0.398975i	−0.270270	+	0.962785i	$0.587113\pi$
$998$		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.181.2		54
3.2	odd	2		108.4.i.a.61.5	✓	54
27.4	even	9	inner	324.4.i.a.145.2		54
27.23	odd	18		108.4.i.a.85.5	yes	54

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
108.4.i.a.61.5	✓	54	3.2	odd	2
108.4.i.a.85.5	yes	54	27.23	odd	18
324.4.i.a.145.2		54	27.4	even	9	inner
324.4.i.a.181.2		54	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+(-2.26368 - 12.8380i) q^{5} +(-3.04053 - 2.55131i) q^{7} +O(q^{10})\)	\(q+(-2.26368 - 12.8380i) q^{5} +(-3.04053 - 2.55131i) q^{7} +(1.56137 - 8.85499i) q^{11} +(-33.8700 - 12.3277i) q^{13} +(-0.988527 - 1.71218i) q^{17} +(-56.3039 + 97.5212i) q^{19} +(73.1185 - 61.3537i) q^{23} +(-42.2272 + 15.3695i) q^{25} +(-237.316 + 86.3761i) q^{29} +(-154.385 + 129.544i) q^{31} +(-25.8708 + 44.8096i) q^{35} +(112.630 + 195.081i) q^{37} +(-209.778 - 76.3529i) q^{41} +(-10.6717 + 60.5223i) q^{43} +(-140.866 - 118.201i) q^{47} +(-56.8257 - 322.274i) q^{49} -596.365 q^{53} -117.214 q^{55} +(-26.4036 - 149.742i) q^{59} +(-282.206 - 236.799i) q^{61} +(-81.5913 + 462.727i) q^{65} +(869.318 + 316.406i) q^{67} +(-384.966 - 666.780i) q^{71} +(-329.352 + 570.454i) q^{73} +(-27.3392 + 22.9403i) q^{77} +(-1099.10 + 400.040i) q^{79} +(189.025 - 68.7996i) q^{83} +(-19.7432 + 16.5665i) q^{85} +(246.987 - 427.794i) q^{89} +(71.5311 + 123.896i) q^{91} +(1379.43 + 502.070i) q^{95} +(-95.1067 + 539.377i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 54 q - 12 q^{5}+O(q^{10}) \) 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

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Database

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