LMFDB - Embedded newform 324.4.i.a.37.8

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			37.8
Character	$\chi$	$=$	324.37
Dual form			324.4.i.a.289.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(10.0092 - 3.64305i) q^{5} +(-2.90933 - 16.4997i) q^{7} +O(q^{10})$	$q+(10.0092 - 3.64305i) q^{5} +(-2.90933 - 16.4997i) q^{7} +(-3.53602 - 1.28701i) q^{11} +(-55.0460 - 46.1890i) q^{13} +(14.0443 - 24.3254i) q^{17} +(4.34509 + 7.52591i) q^{19} +(-4.21963 + 23.9307i) q^{23} +(-8.84313 + 7.42027i) q^{25} +(-183.135 + 153.668i) q^{29} +(45.3635 - 257.269i) q^{31} +(-89.2293 - 154.550i) q^{35} +(50.0527 - 86.6937i) q^{37} +(-177.609 - 149.031i) q^{41} +(220.317 + 80.1887i) q^{43} +(-98.9672 - 561.271i) q^{47} +(58.5403 - 21.3069i) q^{49} -368.316 q^{53} -40.0814 q^{55} +(485.116 - 176.568i) q^{59} +(-106.741 - 605.358i) q^{61} +(-719.236 - 261.780i) q^{65} +(-264.013 - 221.533i) q^{67} +(-496.578 + 860.098i) q^{71} +(-180.607 - 312.820i) q^{73} +(-10.9477 + 62.0874i) q^{77} +(127.623 - 107.088i) q^{79} +(718.475 - 602.872i) q^{83} +(51.9533 - 294.642i) q^{85} +(404.414 + 700.465i) q^{89} +(-601.956 + 1042.62i) q^{91} +(70.9082 + 59.4990i) q^{95} +(1723.85 + 627.431i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	10.0092	−	3.64305i	0.895251	−	0.325845i	0.146903	−	0.989151i	$-0.453070\pi$
$5$	10.0092	−	3.64305i	0.895251	−	0.325845i	0.748348	+	0.663306i	$0.230847\pi$
$6$		0			0
$7$	−2.90933	−	16.4997i	−0.157089	−	0.890897i	−0.956851	−	0.290580i	$-0.906152\pi$
$7$	−2.90933	−	16.4997i	−0.157089	−	0.890897i	0.799762	−	0.600318i	$-0.204959\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−3.53602	−	1.28701i	−0.0969227	−	0.0352770i	0.293104	−	0.956081i	$-0.405312\pi$
$11$	−3.53602	−	1.28701i	−0.0969227	−	0.0352770i	−0.390026	+	0.920804i	$0.627534\pi$
$12$		0			0
$13$	−55.0460	−	46.1890i	−1.17438	−	0.985426i	−1.00000		0.000748193i	$-0.999762\pi$
$13$	−55.0460	−	46.1890i	−1.17438	−	0.985426i	−0.174385	−	0.984678i	$-0.555794\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	14.0443	−	24.3254i	0.200367	−	0.347046i	−0.748280	−	0.663383i	$-0.769120\pi$
$17$	14.0443	−	24.3254i	0.200367	−	0.347046i	0.948647	+	0.316338i	$0.102453\pi$
$18$		0			0
$19$	4.34509	+	7.52591i	0.0524648	+	0.0908717i	0.891065	−	0.453876i	$-0.149959\pi$
$19$	4.34509	+	7.52591i	0.0524648	+	0.0908717i	−0.838600	+	0.544747i	$0.816626\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−4.21963	+	23.9307i	−0.0382545	+	0.216952i	−0.997942	−	0.0641155i	$-0.979577\pi$
$23$	−4.21963	+	23.9307i	−0.0382545	+	0.216952i	0.959688	+	0.281068i	$0.0906885\pi$
$24$		0			0
$25$	−8.84313	+	7.42027i	−0.0707451	+	0.0593622i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−183.135	+	153.668i	−1.17267	+	0.983983i	−1.00000		0.000996932i	$-0.999683\pi$
$29$	−183.135	+	153.668i	−1.17267	+	0.983983i	−0.172666	+	0.984980i	$0.555238\pi$
$30$		0			0
$31$	45.3635	−	257.269i	0.262823	−	1.49055i	−0.512339	−	0.858783i	$-0.671221\pi$
$31$	45.3635	−	257.269i	0.262823	−	1.49055i	0.775162	−	0.631762i	$-0.217668\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−89.2293	−	154.550i	−0.430928	−	0.746390i
$36$		0			0
$37$	50.0527	−	86.6937i	0.222395	−	0.385199i	−0.733140	−	0.680078i	$-0.761946\pi$
$37$	50.0527	−	86.6937i	0.222395	−	0.385199i	0.955535	+	0.294879i	$0.0952793\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−177.609	−	149.031i	−0.676531	−	0.567677i	0.238459	−	0.971153i	$-0.423358\pi$
$41$	−177.609	−	149.031i	−0.676531	−	0.567677i	−0.914991	+	0.403475i	$0.867802\pi$
$42$		0			0
$43$	220.317	+	80.1887i	0.781348	+	0.284387i	0.701735	−	0.712438i	$-0.252409\pi$
$43$	220.317	+	80.1887i	0.781348	+	0.284387i	0.0796135	+	0.996826i	$0.474631\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−98.9672	−	561.271i	−0.307146	−	1.74191i	−0.613229	−	0.789905i	$-0.710130\pi$
$47$	−98.9672	−	561.271i	−0.307146	−	1.74191i	0.306083	−	0.952005i	$-0.400981\pi$
$48$		0			0
$49$	58.5403	−	21.3069i	0.170671	−	0.0621193i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−368.316			−0.954567			−0.477283	−	0.878749i	$-0.658378\pi$
$53$	−368.316			−0.954567			−0.477283	+	0.878749i	$0.658378\pi$
$54$		0			0
$55$	−40.0814			−0.0982649
$56$		0			0
$57$		0			0
$58$		0			0
$59$	485.116	−	176.568i	1.07045	−	0.389613i	0.254107	−	0.967176i	$-0.418218\pi$
$59$	485.116	−	176.568i	1.07045	−	0.389613i	0.816346	+	0.577563i	$0.195996\pi$
$60$		0			0
$61$	−106.741	−	605.358i	−0.224046	−	1.27063i	−0.864501	−	0.502631i	$-0.832366\pi$
$61$	−106.741	−	605.358i	−0.224046	−	1.27063i	0.640456	−	0.767995i	$-0.278746\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−719.236	−	261.780i	−1.37246	−	0.499536i
$66$		0			0
$67$	−264.013	−	221.533i	−0.481407	−	0.403949i	0.369528	−	0.929220i	$-0.379520\pi$
$67$	−264.013	−	221.533i	−0.481407	−	0.403949i	−0.850935	+	0.525271i	$0.823964\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−496.578	+	860.098i	−0.830041	+	1.43767i	0.0679634	+	0.997688i	$0.478350\pi$
$71$	−496.578	+	860.098i	−0.830041	+	1.43767i	−0.898005	+	0.439986i	$0.854983\pi$
$72$		0			0
$73$	−180.607	−	312.820i	−0.289568	−	0.501546i	0.684139	−	0.729352i	$-0.260178\pi$
$73$	−180.607	−	312.820i	−0.289568	−	0.501546i	−0.973707	+	0.227806i	$0.926845\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−10.9477	+	62.0874i	−0.0162027	+	0.0918898i
$78$		0			0
$79$	127.623	−	107.088i	0.181756	−	0.152511i	−0.547371	−	0.836890i	$-0.684371\pi$
$79$	127.623	−	107.088i	0.181756	−	0.152511i	0.729126	+	0.684379i	$0.239927\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	718.475	−	602.872i	0.950156	−	0.797275i	−0.0291680	−	0.999575i	$-0.509286\pi$
$83$	718.475	−	602.872i	0.950156	−	0.797275i	0.979324	+	0.202299i	$0.0648413\pi$
$84$		0			0
$85$	51.9533	−	294.642i	0.0662956	−	0.375981i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	404.414	+	700.465i	0.481660	+	0.834260i	0.999778	−	0.0210490i	$-0.00670061\pi$
$89$	404.414	+	700.465i	0.481660	+	0.834260i	−0.518118	+	0.855309i	$0.673367\pi$
$90$		0			0
$91$	−601.956	+	1042.62i	−0.693430	+	1.20106i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	70.9082	+	59.4990i	0.0765792	+	0.0642576i
$96$		0			0
$97$	1723.85	+	627.431i	1.80444	+	0.656762i	0.997842	+	0.0656621i	$0.0209160\pi$
$97$	1723.85	+	627.431i	1.80444	+	0.656762i	0.806598	+	0.591100i	$0.201306\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	104.411	+	592.143i	0.102864	+	0.583370i	0.992052	+	0.125829i	$0.0401589\pi$
$101$	104.411	+	592.143i	0.102864	+	0.583370i	−0.889188	+	0.457542i	$0.848730\pi$
$102$		0			0
$103$	61.8832	−	22.5237i	0.0591994	−	0.0215468i	−0.312251	−	0.950000i	$-0.601083\pi$
$103$	61.8832	−	22.5237i	0.0591994	−	0.0215468i	0.371450	+	0.928453i	$0.378861\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	1123.65			1.01520			0.507602	−	0.861591i	$-0.330532\pi$
$107$	1123.65			1.01520			0.507602	+	0.861591i	$0.330532\pi$
$108$		0			0
$109$	−1790.18			−1.57310			−0.786550	−	0.617527i	$-0.788135\pi$
$109$	−1790.18			−1.57310			−0.786550	+	0.617527i	$0.788135\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	1032.17	−	375.680i	0.859281	−	0.312753i	0.125462	−	0.992098i	$-0.459959\pi$
$113$	1032.17	−	375.680i	0.859281	−	0.312753i	0.733818	+	0.679346i	$0.237736\pi$
$114$		0			0
$115$	44.9457	+	254.900i	0.0364453	+	0.206692i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−442.220	−	160.955i	−0.340657	−	0.123989i
$120$		0			0
$121$	−1008.76	−	846.449i	−0.757895	−	0.635949i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−727.204	+	1259.55i	−0.520345	+	0.901264i
$126$		0			0
$127$	1048.77	+	1816.52i	0.732782	+	1.26922i	0.955690	+	0.294376i	$0.0951118\pi$
$127$	1048.77	+	1816.52i	0.732782	+	1.26922i	−0.222908	+	0.974840i	$0.571555\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	157.676	−	894.228i	0.105162	−	0.596405i	−0.885993	−	0.463698i	$-0.846522\pi$
$131$	157.676	−	894.228i	0.105162	−	0.596405i	0.991155	−	0.132707i	$-0.0423668\pi$
$132$		0			0
$133$	111.534	−	93.5878i	0.0727157	−	0.0610157i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−603.157	+	506.109i	−0.376140	+	0.315619i	−0.811185	−	0.584790i	$-0.801177\pi$
$137$	−603.157	+	506.109i	−0.376140	+	0.315619i	0.435045	+	0.900409i	$0.356733\pi$
$138$		0			0
$139$	175.366	−	994.552i	0.107010	−	0.606883i	−0.883389	−	0.468641i	$-0.844744\pi$
$139$	175.366	−	994.552i	0.107010	−	0.606883i	0.990398	−	0.138242i	$-0.0441452\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	135.198	+	234.170i	0.0790617	+	0.136939i
$144$		0			0
$145$	−1273.21	+	2205.27i	−0.729204	+	1.26302i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	2624.88	+	2202.54i	1.44321	+	1.21100i	0.937352	+	0.348383i	$0.113269\pi$
$149$	2624.88	+	2202.54i	1.44321	+	1.21100i	0.505859	+	0.862616i	$0.331175\pi$
$150$		0			0
$151$	2253.91	+	820.356i	1.21471	+	0.442117i	0.868334	−	0.495980i	$-0.165191\pi$
$151$	2253.91	+	820.356i	1.21471	+	0.442117i	0.346372	+	0.938097i	$0.387413\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−483.193	−	2740.32i	−0.250393	−	1.42005i
$156$		0			0
$157$	−1595.55	+	580.732i	−0.811073	+	0.295207i	−0.714067	−	0.700077i	$-0.753149\pi$
$157$	−1595.55	+	580.732i	−0.811073	+	0.295207i	−0.0970061	+	0.995284i	$0.530927\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	407.125			0.199291
$162$		0			0
$163$	1962.35			0.942966			0.471483	−	0.881875i	$-0.343719\pi$
$163$	1962.35			0.942966			0.471483	+	0.881875i	$0.343719\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	2348.59	−	854.817i	1.08826	−	0.396094i	0.265284	−	0.964170i	$-0.414534\pi$
$167$	2348.59	−	854.817i	1.08826	−	0.396094i	0.822976	+	0.568076i	$0.192312\pi$
$168$		0			0
$169$	515.125	+	2921.42i	0.234467	+	1.32973i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−1234.35	−	449.268i	−0.542463	−	0.197440i	0.0562317	−	0.998418i	$-0.482091\pi$
$173$	−1234.35	−	449.268i	−0.542463	−	0.197440i	−0.598695	+	0.800977i	$0.704314\pi$
$174$		0			0
$175$	148.159	+	124.321i	0.0639989	+	0.0537014i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−1735.29	+	3005.61i	−0.724589	+	1.25503i	0.234554	+	0.972103i	$0.424637\pi$
$179$	−1735.29	+	3005.61i	−0.724589	+	1.25503i	−0.959143	+	0.282922i	$0.908696\pi$
$180$		0			0
$181$	1010.33	+	1749.94i	0.414901	+	0.718630i	0.995418	−	0.0956180i	$-0.0304827\pi$
$181$	1010.33	+	1749.94i	0.414901	+	0.718630i	−0.580517	+	0.814248i	$0.697149\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	185.157	−	1050.08i	0.0735841	−	0.417316i
$186$		0			0
$187$	−80.9677	+	67.9400i	−0.0316628	+	0.0265683i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−1101.83	+	924.547i	−0.417412	+	0.350251i	−0.827178	−	0.561940i	$-0.810055\pi$
$191$	−1101.83	+	924.547i	−0.417412	+	0.350251i	0.409765	+	0.912191i	$0.365611\pi$
$192$		0			0
$193$	165.021	−	935.881i	0.0615465	−	0.349048i	−0.938447	−	0.345423i	$-0.887735\pi$
$193$	165.021	−	935.881i	0.0615465	−	0.349048i	0.999993	−	0.00362415i	$-0.00115361\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−630.410	−	1091.90i	−0.227994	−	0.394898i	0.729219	−	0.684280i	$-0.239883\pi$
$197$	−630.410	−	1091.90i	−0.227994	−	0.394898i	−0.957214	+	0.289382i	$0.906550\pi$
$198$		0			0
$199$	45.3986	−	78.6327i	0.0161720	−	0.0280107i	−0.857826	−	0.513940i	$-0.828185\pi$
$199$	45.3986	−	78.6327i	0.0161720	−	0.0280107i	0.873998	+	0.485929i	$0.161519\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	3068.28	+	2574.59i	1.06084	+	0.890152i
$204$		0			0
$205$	−2320.65	−	844.647i	−0.790640	−	0.287769i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−5.67842	−	32.2039i	−0.00187935	−	0.0106583i
$210$		0			0
$211$	3332.78	−	1213.03i	1.08739	−	0.395776i	0.264734	−	0.964322i	$-0.414716\pi$
$211$	3332.78	−	1213.03i	1.08739	−	0.395776i	0.822652	+	0.568546i	$0.192494\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	2497.33			0.792169
$216$		0			0
$217$	−4376.83			−1.36921
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−1896.65	+	690.323i	−0.577295	+	0.210118i
$222$		0			0
$223$	168.174	+	953.762i	0.0505012	+	0.286406i	0.999591	−	0.0285993i	$-0.00910469\pi$
$223$	168.174	+	953.762i	0.0505012	+	0.286406i	−0.949090	+	0.315006i	$0.897994\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	1558.12	+	567.109i	0.455577	+	0.165816i	0.559608	−	0.828758i	$-0.310952\pi$
$227$	1558.12	+	567.109i	0.455577	+	0.165816i	−0.104031	+	0.994574i	$0.533174\pi$
$228$		0			0
$229$	−3824.88	−	3209.45i	−1.10373	−	0.926142i	−0.106063	−	0.994359i	$-0.533825\pi$
$229$	−3824.88	−	3209.45i	−1.10373	−	0.926142i	−0.997671	+	0.0682170i	$0.978269\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−1892.72	+	3278.28i	−0.532171	+	0.921748i	0.467123	+	0.884192i	$0.345290\pi$
$233$	−1892.72	+	3278.28i	−0.532171	+	0.921748i	−0.999295	+	0.0375555i	$0.988043\pi$
$234$		0			0
$235$	−3035.32	−	5257.33i	−0.842565	−	1.45936i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	280.643	−	1591.60i	0.0759551	−	0.430763i	−0.922989	−	0.384826i	$-0.874262\pi$
$239$	280.643	−	1591.60i	0.0759551	−	0.430763i	0.998944	−	0.0459369i	$-0.0146273\pi$
$240$		0			0
$241$	4034.39	−	3385.25i	1.07833	−	0.904826i	0.0825491	−	0.996587i	$-0.473694\pi$
$241$	4034.39	−	3385.25i	1.07833	−	0.904826i	0.995781	+	0.0917605i	$0.0292494\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	508.320	−	426.531i	0.132553	−	0.111225i
$246$		0			0
$247$	108.435	−	614.966i	0.0279335	−	0.158418i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−2000.32	−	3464.66i	−0.503025	−	0.871264i	−0.999994	−	0.00349620i	$-0.998887\pi$
$251$	−2000.32	−	3464.66i	−0.503025	−	0.871264i	0.496969	−	0.867768i	$-0.334446\pi$
$252$		0			0
$253$	45.7196	−	79.1887i	0.0113611	−	0.0196781i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−5312.76	−	4457.93i	−1.28950	−	1.08202i	−0.991859	−	0.127338i	$-0.959357\pi$
$257$	−5312.76	−	4457.93i	−1.28950	−	1.08202i	−0.297638	−	0.954679i	$-0.596199\pi$
$258$		0			0
$259$	−1576.04	−	573.630i	−0.378109	−	0.137620i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−902.671	−	5119.30i	−0.211639	−	1.20027i	−0.886644	−	0.462452i	$-0.846970\pi$
$263$	−902.671	−	5119.30i	−0.211639	−	1.20027i	0.675005	−	0.737813i	$-0.264141\pi$
$264$		0			0
$265$	−3686.55	+	1341.79i	−0.854577	+	0.311041i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	3202.65			0.725907			0.362953	−	0.931807i	$-0.381768\pi$
$269$	3202.65			0.725907			0.362953	+	0.931807i	$0.381768\pi$
$270$		0			0
$271$	2563.49			0.574616			0.287308	−	0.957838i	$-0.407240\pi$
$271$	2563.49			0.574616			0.287308	+	0.957838i	$0.407240\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	40.8194	−	14.8570i	0.00895092	−	0.00325787i
$276$		0			0
$277$	−934.933	−	5302.27i	−0.202797	−	1.15012i	−0.900869	−	0.434090i	$-0.857070\pi$
$277$	−934.933	−	5302.27i	−0.202797	−	1.15012i	0.698073	−	0.716027i	$-0.254041\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−3415.67	−	1243.20i	−0.725131	−	0.263926i	−0.0470287	−	0.998894i	$-0.514975\pi$
$281$	−3415.67	−	1243.20i	−0.725131	−	0.263926i	−0.678102	+	0.734967i	$0.737197\pi$
$282$		0			0
$283$	5021.18	+	4213.27i	1.05469	+	0.884993i	0.993579	−	0.113138i	$-0.0360902\pi$
$283$	5021.18	+	4213.27i	1.05469	+	0.884993i	0.0611141	+	0.998131i	$0.480535\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−1942.24	+	3364.06i	−0.399466	+	0.691896i
$288$		0			0
$289$	2062.02	+	3571.52i	0.419706	+	0.726953i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−88.6969	+	503.025i	−0.0176851	+	0.100297i	−0.992373	−	0.123274i	$-0.960660\pi$
$293$	−88.6969	+	503.025i	−0.0176851	+	0.100297i	0.974688	+	0.223571i	$0.0717716\pi$
$294$		0			0
$295$	4212.38	−	3534.61i	0.831371	−	0.697603i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	1337.61	−	1122.39i	0.258716	−	0.217088i
$300$		0			0
$301$	682.111	−	3868.44i	0.130619	−	0.740775i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−3273.75	−	5670.29i	−0.614604	−	1.06452i
$306$		0			0
$307$	975.162	−	1689.03i	0.181288	−	0.314000i	−0.761031	−	0.648715i	$-0.775307\pi$
$307$	975.162	−	1689.03i	0.181288	−	0.314000i	0.942319	+	0.334715i	$0.108640\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−4843.13	−	4063.87i	−0.883051	−	0.740968i	0.0837528	−	0.996487i	$-0.473309\pi$
$311$	−4843.13	−	4063.87i	−0.883051	−	0.740968i	−0.966804	+	0.255518i	$0.917754\pi$
$312$		0			0
$313$	472.116	+	171.836i	0.0852575	+	0.0310312i	0.384297	−	0.923210i	$-0.374444\pi$
$313$	472.116	+	171.836i	0.0852575	+	0.0310312i	−0.299039	+	0.954241i	$0.596666\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	1665.94	+	9448.01i	0.295169	+	1.67398i	0.666517	+	0.745490i	$0.267785\pi$
$317$	1665.94	+	9448.01i	0.295169	+	1.67398i	−0.371348	+	0.928494i	$0.621104\pi$
$318$		0			0
$319$	845.341	−	307.679i	0.148370	−	0.0540022i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	244.094			0.0420488
$324$		0			0
$325$	829.514			0.141579
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−8972.84	+	3265.85i	−1.50361	+	0.547271i
$330$		0			0
$331$	862.280	+	4890.23i	0.143188	+	0.812059i	0.968804	+	0.247827i	$0.0797165\pi$
$331$	862.280	+	4890.23i	0.143188	+	0.812059i	−0.825616	+	0.564232i	$0.809172\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−3449.62	−	1255.56i	−0.562605	−	0.204771i
$336$		0			0
$337$	−350.546	−	294.143i	−0.0566632	−	0.0475460i	0.614016	−	0.789293i	$-0.289553\pi$
$337$	−350.546	−	294.143i	−0.0566632	−	0.0475460i	−0.670679	+	0.741747i	$0.733997\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−491.513	+	851.326i	−0.0780555	+	0.135196i
$342$		0			0
$343$	−3395.21	−	5880.68i	−0.534473	−	0.925735i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−645.039	+	3658.20i	−0.0997911	+	0.565943i	0.893382	+	0.449297i	$0.148325\pi$
$347$	−645.039	+	3658.20i	−0.0997911	+	0.565943i	−0.993174	+	0.116646i	$0.962786\pi$
$348$		0			0
$349$	−2640.17	+	2215.37i	−0.404943	+	0.339788i	−0.822401	−	0.568909i	$-0.807366\pi$
$349$	−2640.17	+	2215.37i	−0.404943	+	0.339788i	0.417457	+	0.908696i	$0.362921\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	2239.14	−	1878.86i	0.337613	−	0.283291i	−0.458181	−	0.888859i	$-0.651499\pi$
$353$	2239.14	−	1878.86i	0.337613	−	0.283291i	0.795793	+	0.605569i	$0.207054\pi$
$354$		0			0
$355$	−1836.97	+	10418.0i	−0.274637	+	1.55754i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−6470.47	−	11207.2i	−0.951249	−	1.64761i	−0.742727	−	0.669594i	$-0.766468\pi$
$359$	−6470.47	−	11207.2i	−0.951249	−	1.64761i	−0.208522	−	0.978018i	$-0.566865\pi$
$360$		0			0
$361$	3391.74	−	5874.67i	0.494495	−	0.856490i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−2947.35	−	2473.12i	−0.422662	−	0.354655i
$366$		0			0
$367$	−12298.8	−	4476.41i	−1.74930	−	0.636694i	−0.749620	−	0.661869i	$-0.769764\pi$
$367$	−12298.8	−	4476.41i	−1.74930	−	0.636694i	−0.999683	+	0.0251747i	$0.991986\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	1071.55	+	6077.08i	0.149952	+	0.850421i
$372$		0			0
$373$	8627.89	−	3140.29i	1.19768	−	0.435920i	0.335267	−	0.942123i	$-0.391174\pi$
$373$	8627.89	−	3140.29i	1.19768	−	0.435920i	0.862414	+	0.506203i	$0.168951\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	17178.6			2.34680
$378$		0			0
$379$	2260.05			0.306308			0.153154	−	0.988202i	$-0.451057\pi$
$379$	2260.05			0.306308			0.153154	+	0.988202i	$0.451057\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	9927.52	−	3613.32i	1.32447	−	0.482068i	0.419583	−	0.907717i	$-0.362176\pi$
$383$	9927.52	−	3613.32i	1.32447	−	0.482068i	0.904888	+	0.425649i	$0.139954\pi$
$384$		0			0
$385$	116.610	+	661.329i	0.0154364	+	0.0875440i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	9659.45	+	3515.75i	1.25901	+	0.458241i	0.883436	−	0.468552i	$-0.155224\pi$
$389$	9659.45	+	3515.75i	1.25901	+	0.458241i	0.375571	+	0.926793i	$0.377446\pi$
$390$		0			0
$391$	522.862	+	438.733i	0.0676273	+	0.0567460i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	887.276	−	1536.81i	0.113022	−	0.195760i
$396$		0			0
$397$	1494.01	+	2587.71i	0.188873	+	0.327137i	0.944875	−	0.327432i	$-0.106183\pi$
$397$	1494.01	+	2587.71i	0.188873	+	0.327137i	−0.756002	+	0.654569i	$0.772850\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−1287.47	+	7301.59i	−0.160332	+	0.909287i	0.793416	+	0.608680i	$0.208300\pi$
$401$	−1287.47	+	7301.59i	−0.160332	+	0.909287i	−0.953748	+	0.300607i	$0.902811\pi$
$402$		0			0
$403$	−14380.1	+	12066.3i	−1.77748	+	1.49148i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−288.562	+	242.133i	−0.0351438	+	0.0294891i
$408$		0			0
$409$	−136.025	+	771.437i	−0.0164450	+	0.0932643i	−0.991926	−	0.126821i	$-0.959523\pi$
$409$	−136.025	+	771.437i	−0.0164450	+	0.0932643i	0.975481	+	0.220086i	$0.0706337\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−4324.67	−	7490.55i	−0.515262	−	0.892460i
$414$		0			0
$415$	4995.07	−	8651.72i	0.590840	−	1.02336i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	7526.09	+	6315.14i	0.877502	+	0.736312i	0.965664	−	0.259794i	$-0.0836547\pi$
$419$	7526.09	+	6315.14i	0.877502	+	0.736312i	−0.0881616	+	0.996106i	$0.528099\pi$
$420$		0			0
$421$	1020.41	+	371.400i	0.118128	+	0.0429951i	0.400408	−	0.916337i	$-0.368868\pi$
$421$	1020.41	+	371.400i	0.118128	+	0.0429951i	−0.282280	+	0.959332i	$0.591091\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	56.3056	+	319.325i	0.00642641	+	0.0364460i
$426$		0			0
$427$	−9677.65	+	3522.38i	−1.09680	+	0.399203i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	4258.28			0.475903			0.237951	−	0.971277i	$-0.423524\pi$
$431$	4258.28			0.475903			0.237951	+	0.971277i	$0.423524\pi$
$432$		0			0
$433$	7152.99			0.793882			0.396941	−	0.917844i	$-0.370072\pi$
$433$	7152.99			0.793882			0.396941	+	0.917844i	$0.370072\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−198.435	+	72.2244i	−0.0217218	+	0.00790609i
$438$		0			0
$439$	−1804.32	−	10232.8i	−0.196162	−	1.11249i	−0.910754	−	0.412950i	$-0.864498\pi$
$439$	−1804.32	−	10232.8i	−0.196162	−	1.11249i	0.714591	−	0.699542i	$-0.246613\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	12028.7	+	4378.09i	1.29007	+	0.469548i	0.893750	−	0.448565i	$-0.148065\pi$
$443$	12028.7	+	4378.09i	1.29007	+	0.469548i	0.396321	+	0.918112i	$0.370287\pi$
$444$		0			0
$445$	6599.69	+	5537.80i	0.703046	+	0.589926i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−3046.78	+	5277.17i	−0.320237	+	0.554666i	−0.980537	−	0.196336i	$-0.937096\pi$
$449$	−3046.78	+	5277.17i	−0.320237	+	0.554666i	0.660300	+	0.751002i	$0.270429\pi$
$450$		0			0
$451$	436.223	+	755.560i	0.0455453	+	0.0788868i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−2226.79	+	12628.7i	−0.229436	+	1.30120i
$456$		0			0
$457$	−5464.80	+	4585.51i	−0.559371	+	0.469368i	−0.878099	−	0.478478i	$-0.841189\pi$
$457$	−5464.80	+	4585.51i	−0.559371	+	0.469368i	0.318729	+	0.947846i	$0.396744\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−9715.80	+	8152.53i	−0.981583	+	0.823646i	−0.984328	−	0.176350i	$-0.943571\pi$
$461$	−9715.80	+	8152.53i	−0.981583	+	0.823646i	0.00274406	+	0.999996i	$0.499127\pi$
$462$		0			0
$463$	−2328.16	+	13203.7i	−0.233691	+	1.32533i	0.611663	+	0.791119i	$0.290501\pi$
$463$	−2328.16	+	13203.7i	−0.233691	+	1.32533i	−0.845353	+	0.534208i	$0.820610\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	68.4431	+	118.547i	0.00678195	+	0.0117467i	0.869396	−	0.494115i	$-0.164508\pi$
$467$	68.4431	+	118.547i	0.00678195	+	0.0117467i	−0.862615	+	0.505862i	$0.831175\pi$
$468$		0			0
$469$	−2887.12	+	5000.63i	−0.284253	+	0.492341i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−675.840	−	567.097i	−0.0656980	−	0.0551272i
$474$		0			0
$475$	−94.2685	−	34.3109i	−0.00910597	−	0.00331430i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−599.222	−	3398.35i	−0.0571589	−	0.324164i	0.942799	−	0.333362i	$-0.108183\pi$
$479$	−599.222	−	3398.35i	−0.0571589	−	0.324164i	−0.999958	+	0.00919797i	$0.997072\pi$
$480$		0			0
$481$	−6759.50	+	2460.26i	−0.640762	+	0.233218i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	19540.2			1.82943
$486$		0			0
$487$	8808.83			0.819643			0.409821	−	0.912166i	$-0.365591\pi$
$487$	8808.83			0.819643			0.409821	+	0.912166i	$0.365591\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−11912.6	+	4335.83i	−1.09493	+	0.398520i	−0.825443	−	0.564485i	$-0.809075\pi$
$491$	−11912.6	+	4335.83i	−1.09493	+	0.398520i	−0.269482	+	0.963005i	$0.586853\pi$
$492$		0			0
$493$	1166.05	+	6612.99i	0.106524	+	0.604126i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	15636.0	+	5691.05i	1.41121	+	0.513639i
$498$		0			0
$499$	13503.0	+	11330.4i	1.21138	+	1.01647i	0.999230	+	0.0392306i	$0.0124907\pi$
$499$	13503.0	+	11330.4i	1.21138	+	1.01647i	0.212149	+	0.977237i	$0.431954\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−3751.33	+	6497.50i	−0.332532	+	0.575962i	−0.983008	−	0.183565i	$-0.941236\pi$
$503$	−3751.33	+	6497.50i	−0.332532	+	0.575962i	0.650476	+	0.759527i	$0.274570\pi$
$504$		0			0
$505$	3202.28	+	5546.51i	0.282177	+	0.488745i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	3912.83	−	22190.7i	0.340733	−	1.93239i	−0.0201923	−	0.999796i	$-0.506428\pi$
$509$	3912.83	−	22190.7i	0.340733	−	1.93239i	0.360925	−	0.932595i	$-0.382461\pi$
$510$		0			0
$511$	−4635.98	+	3890.05i	−0.401338	+	0.336762i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	537.347	−	450.888i	0.0459774	−	0.0385796i
$516$		0			0
$517$	−372.409	+	2112.04i	−0.0316799	+	0.179666i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−10309.3	−	17856.2i	−0.866906	−	1.50153i	−0.865142	−	0.501527i	$-0.832772\pi$
$521$	−10309.3	−	17856.2i	−0.866906	−	1.50153i	−0.00176401	−	0.999998i	$-0.500562\pi$
$522$		0			0
$523$	−3362.40	+	5823.85i	−0.281123	+	0.486920i	−0.971662	−	0.236376i	$-0.924040\pi$
$523$	−3362.40	+	5823.85i	−0.281123	+	0.486920i	0.690538	+	0.723296i	$0.257374\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−5621.08	−	4716.64i	−0.464626	−	0.389868i
$528$		0			0
$529$	10878.4	+	3959.40i	0.894088	+	0.325421i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	2893.02	+	16407.1i	0.235104	+	1.33334i
$534$		0			0
$535$	11246.8	−	4093.50i	0.908863	−	0.330799i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−234.422			−0.0187333
$540$		0			0
$541$	−9213.40			−0.732190			−0.366095	−	0.930577i	$-0.619306\pi$
$541$	−9213.40			−0.732190			−0.366095	+	0.930577i	$0.619306\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−17918.3	+	6521.71i	−1.40832	+	0.512586i
$546$		0			0
$547$	−1180.97	−	6697.60i	−0.0923117	−	0.523526i	−0.995538	−	0.0943612i	$-0.969919\pi$
$547$	−1180.97	−	6697.60i	−0.0923117	−	0.523526i	0.903226	−	0.429165i	$-0.141192\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−1952.23	−	710.554i	−0.150940	−	0.0549376i
$552$		0			0
$553$	−2138.22	−	1794.18i	−0.164424	−	0.137968i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	8957.05	−	15514.1i	0.681369	−	1.18017i	−0.293194	−	0.956053i	$-0.594718\pi$
$557$	8957.05	−	15514.1i	0.681369	−	1.18017i	0.974563	−	0.224113i	$-0.0719484\pi$
$558$		0			0
$559$	−8423.70	−	14590.3i	−0.637361	−	1.10394i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−1738.41	+	9859.03i	−0.130134	+	0.738026i	0.847992	+	0.530010i	$0.177812\pi$
$563$	−1738.41	+	9859.03i	−0.130134	+	0.738026i	−0.978125	+	0.208016i	$0.933299\pi$
$564$		0			0
$565$	8962.62	−	7520.53i	0.667363	−	0.559984i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	11756.4	−	9864.81i	0.866177	−	0.726809i	−0.0971124	−	0.995273i	$-0.530961\pi$
$569$	11756.4	−	9864.81i	0.866177	−	0.726809i	0.963290	+	0.268464i	$0.0865162\pi$
$570$		0			0
$571$	661.504	−	3751.58i	0.0484818	−	0.274954i	−0.950924	−	0.309425i	$-0.899864\pi$
$571$	661.504	−	3751.58i	0.0484818	−	0.274954i	0.999406	+	0.0344710i	$0.0109746\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−140.258	−	242.933i	−0.0101724	−	0.0176192i
$576$		0			0
$577$	3496.19	−	6055.58i	0.252250	−	0.436910i	−0.711895	−	0.702286i	$-0.752163\pi$
$577$	3496.19	−	6055.58i	0.252250	−	0.436910i	0.964145	+	0.265376i	$0.0854961\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−12037.5	−	10100.6i	−0.859550	−	0.721248i
$582$		0			0
$583$	1302.37	+	474.024i	0.0925192	+	0.0336742i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−314.256	−	1782.23i	−0.0220966	−	0.125316i	0.971764	−	0.235954i	$-0.0758216\pi$
$587$	−314.256	−	1782.23i	−0.0220966	−	0.125316i	−0.993861	+	0.110638i	$0.964711\pi$
$588$		0			0
$589$	2133.29	−	776.455i	0.149237	−	0.0543180i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	23574.5			1.63253			0.816265	−	0.577677i	$-0.196041\pi$
$593$	23574.5			1.63253			0.816265	+	0.577677i	$0.196041\pi$
$594$		0			0
$595$	−5012.64			−0.345375
$596$		0			0
$597$		0			0
$598$		0			0
$599$	1293.63	−	470.845i	0.0882412	−	0.0321172i	−0.297522	−	0.954715i	$-0.596160\pi$
$599$	1293.63	−	470.845i	0.0882412	−	0.0321172i	0.385764	+	0.922598i	$0.373938\pi$
$600$		0			0
$601$	−2569.25	−	14571.0i	−0.174379	−	0.988955i	−0.938858	−	0.344305i	$-0.888114\pi$
$601$	−2569.25	−	14571.0i	−0.174379	−	0.988955i	0.764478	−	0.644649i	$-0.222997\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−13180.5	−	4797.32i	−0.885727	−	0.322378i
$606$		0			0
$607$	−9051.09	−	7594.77i	−0.605226	−	0.507845i	0.287894	−	0.957662i	$-0.407045\pi$
$607$	−9051.09	−	7594.77i	−0.605226	−	0.507845i	−0.893121	+	0.449817i	$0.851489\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−20476.8	+	35466.9i	−1.35582	+	2.34834i
$612$		0			0
$613$	3225.23	+	5586.27i	0.212506	+	0.368071i	0.952498	−	0.304545i	$-0.0985042\pi$
$613$	3225.23	+	5586.27i	0.212506	+	0.368071i	−0.739992	+	0.672615i	$0.765171\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	2856.84	−	16201.9i	0.186405	−	1.05716i	−0.737732	−	0.675094i	$-0.764103\pi$
$617$	2856.84	−	16201.9i	0.186405	−	1.05716i	0.924137	−	0.382061i	$-0.124786\pi$
$618$		0			0
$619$	−13064.5	+	10962.4i	−0.848316	+	0.711822i	−0.959418	−	0.281987i	$-0.909006\pi$
$619$	−13064.5	+	10962.4i	−0.848316	+	0.711822i	0.111102	+	0.993809i	$0.464562\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	10380.9	−	8710.57i	0.667576	−	0.560163i
$624$		0			0
$625$	−2439.54	+	13835.3i	−0.156131	+	0.885460i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−1405.91	−	2435.10i	−0.0891211	−	0.154362i
$630$		0			0
$631$	7321.24	−	12680.8i	0.461892	−	0.800021i	−0.537163	−	0.843479i	$-0.680504\pi$
$631$	7321.24	−	12680.8i	0.461892	−	0.800021i	0.999055	+	0.0434573i	$0.0138373\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	17115.1	+	14361.2i	1.06959	+	0.897493i
$636$		0			0
$637$	−4206.55	−	1531.06i	−0.261648	−	0.0952321i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−792.456	−	4494.24i	−0.0488301	−	0.276929i	0.950610	−	0.310388i	$-0.100459\pi$
$641$	−792.456	−	4494.24i	−0.0488301	−	0.276929i	−0.999440	+	0.0334585i	$0.989348\pi$
$642$		0			0
$643$	−4259.61	+	1550.37i	−0.261248	+	0.0950865i	−0.469324	−	0.883026i	$-0.655502\pi$
$643$	−4259.61	+	1550.37i	−0.261248	+	0.0950865i	0.208076	+	0.978113i	$0.433280\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−24993.3			−1.51868			−0.759342	−	0.650692i	$-0.774479\pi$
$647$	−24993.3			−1.51868			−0.759342	+	0.650692i	$0.774479\pi$
$648$		0			0
$649$	−1942.62			−0.117496
$650$		0			0
$651$		0			0
$652$		0			0
$653$	2358.00	−	858.241i	0.141310	−	0.0514327i	−0.270397	−	0.962749i	$-0.587155\pi$
$653$	2358.00	−	858.241i	0.141310	−	0.0514327i	0.411707	+	0.911316i	$0.364933\pi$
$654$		0			0
$655$	−1679.50	−	9524.94i	−0.100189	−	0.568199i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−24430.9	−	8892.11i	−1.44415	−	0.525626i	−0.503196	−	0.864172i	$-0.667843\pi$
$659$	−24430.9	−	8892.11i	−1.44415	−	0.525626i	−0.940950	+	0.338546i	$0.890065\pi$
$660$		0			0
$661$	−4539.92	−	3809.44i	−0.267144	−	0.224161i	0.499369	−	0.866390i	$-0.333565\pi$
$661$	−4539.92	−	3809.44i	−0.267144	−	0.224161i	−0.766513	+	0.642229i	$0.778010\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	775.418	−	1343.06i	0.0452171	−	0.0783184i
$666$		0			0
$667$	−2904.63	−	5030.97i	−0.168617	−	0.292054i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−401.661	+	2277.93i	−0.0231087	+	0.131056i
$672$		0			0
$673$	22589.1	−	18954.5i	1.29383	−	1.08565i	0.302649	−	0.953102i	$-0.402129\pi$
$673$	22589.1	−	18954.5i	1.29383	−	1.08565i	0.991177	−	0.132547i	$-0.0423156\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−16020.1	+	13442.5i	−0.909459	+	0.763127i	−0.972016	−	0.234915i	$-0.924519\pi$
$677$	−16020.1	+	13442.5i	−0.909459	+	0.763127i	0.0625569	+	0.998041i	$0.480075\pi$
$678$		0			0
$679$	5337.13	−	30268.4i	0.301650	−	1.71074i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−4479.23	−	7758.25i	−0.250941	−	0.434643i	0.712844	−	0.701323i	$-0.247407\pi$
$683$	−4479.23	−	7758.25i	−0.250941	−	0.434643i	−0.963785	+	0.266680i	$0.914073\pi$
$684$		0			0
$685$	−4193.34	+	7263.08i	−0.233897	+	0.405121i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	20274.3	+	17012.1i	1.12103	+	0.940655i
$690$		0			0
$691$	−23144.5	−	8423.91i	−1.27418	−	0.463764i	−0.385677	−	0.922634i	$-0.626032\pi$
$691$	−23144.5	−	8423.91i	−1.27418	−	0.463764i	−0.888503	+	0.458870i	$0.848254\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−1867.93	−	10593.5i	−0.101949	−	0.578182i
$696$		0			0
$697$	−6119.62	+	2227.36i	−0.332564	+	0.121044i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−13936.1			−0.750869			−0.375435	−	0.926849i	$-0.622507\pi$
$701$	−13936.1			−0.750869			−0.375435	+	0.926849i	$0.622507\pi$
$702$		0			0
$703$	869.932			0.0466716
$704$		0			0
$705$		0			0
$706$		0			0
$707$	9466.38	−	3445.48i	0.503564	−	0.183282i
$708$		0			0
$709$	2364.96	+	13412.3i	0.125272	+	0.710453i	0.981146	+	0.193268i	$0.0619088\pi$
$709$	2364.96	+	13412.3i	0.125272	+	0.710453i	−0.855874	+	0.517185i	$0.826980\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	5965.22	+	2171.16i	0.313323	+	0.114040i
$714$		0			0
$715$	2206.32	+	1851.32i	0.115401	+	0.0968328i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−7771.31	+	13460.3i	−0.403089	+	0.698171i	−0.994097	−	0.108495i	$-0.965397\pi$
$719$	−7771.31	+	13460.3i	−0.403089	+	0.698171i	0.591008	+	0.806666i	$0.298730\pi$
$720$		0			0
$721$	−551.671	−	955.523i	−0.0284956	−	0.0493558i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	479.225	−	2717.82i	0.0245489	−	0.139224i
$726$		0			0
$727$	323.352	−	271.325i	0.0164958	−	0.0138417i	−0.634502	−	0.772921i	$-0.718795\pi$
$727$	323.352	−	271.325i	0.0164958	−	0.0138417i	0.650998	+	0.759079i	$0.274351\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	5044.81	−	4233.10i	0.255252	−	0.214182i
$732$		0			0
$733$	4836.28	−	27427.9i	0.243700	−	1.38209i	−0.579793	−	0.814764i	$-0.696867\pi$
$733$	4836.28	−	27427.9i	0.243700	−	1.38209i	0.823493	−	0.567326i	$-0.192022\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	648.440	+	1123.13i	0.0324092	+	0.0561344i
$738$		0			0
$739$	6838.48	−	11844.6i	0.340403	−	0.589595i	−0.644105	−	0.764937i	$-0.722770\pi$
$739$	6838.48	−	11844.6i	0.340403	−	0.589595i	0.984507	+	0.175343i	$0.0561033\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−25241.3	−	21180.0i	−1.24632	−	1.04578i	−0.997003	−	0.0773619i	$-0.975350\pi$
$743$	−25241.3	−	21180.0i	−1.24632	−	1.04578i	−0.249314	−	0.968423i	$-0.580205\pi$
$744$		0			0
$745$	34296.9	+	12483.1i	1.68663	+	0.613885i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−3269.06	−	18539.8i	−0.159478	−	0.904443i
$750$		0			0
$751$	16216.5	−	5902.33i	0.787948	−	0.286790i	0.0834654	−	0.996511i	$-0.473401\pi$
$751$	16216.5	−	5902.33i	0.787948	−	0.286790i	0.704483	+	0.709721i	$0.251179\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	25548.5			1.23153
$756$		0			0
$757$	11119.9			0.533897			0.266949	−	0.963711i	$-0.413985\pi$
$757$	11119.9			0.533897			0.266949	+	0.963711i	$0.413985\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	15727.2	−	5724.22i	0.749158	−	0.272671i	0.0609065	−	0.998143i	$-0.480601\pi$
$761$	15727.2	−	5724.22i	0.749158	−	0.272671i	0.688251	+	0.725472i	$0.258379\pi$
$762$		0			0
$763$	5208.22	+	29537.3i	0.247117	+	1.40147i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−34859.2	−	12687.7i	−1.64106	−	0.597297i
$768$		0			0
$769$	−29314.2	−	24597.6i	−1.37464	−	1.15346i	−0.971148	−	0.238477i	$-0.923352\pi$
$769$	−29314.2	−	24597.6i	−1.37464	−	1.15346i	−0.403492	−	0.914983i	$-0.632204\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	9959.54	−	17250.4i	0.463415	−	0.802658i	−0.535714	−	0.844400i	$-0.679957\pi$
$773$	9959.54	−	17250.4i	0.463415	−	0.802658i	0.999128	+	0.0417417i	$0.0132907\pi$
$774$		0			0
$775$	1507.85	+	2611.68i	0.0698886	+	0.121051i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	349.871	−	1984.22i	0.0160917	−	0.0912606i
$780$		0			0
$781$	2862.86	−	2402.22i	0.131167	−	0.110062i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−13854.5	+	11625.3i	−0.629923	+	0.528568i
$786$		0			0
$787$	−1236.57	+	7012.96i	−0.0560090	+	0.317643i	−0.999921	−	0.0125537i	$-0.996004\pi$
$787$	−1236.57	+	7012.96i	−0.0560090	+	0.317643i	0.943912	+	0.330197i	$0.107115\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−9201.53	−	15937.5i	−0.413614	−	0.716401i
$792$		0			0
$793$	−22085.3	+	38252.8i	−0.988992	+	1.71298i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	2361.07	+	1981.18i	0.104935	+	0.0880513i	0.693746	−	0.720220i	$-0.255959\pi$
$797$	2361.07	+	1981.18i	0.104935	+	0.0880513i	−0.588810	+	0.808271i	$0.700404\pi$
$798$		0			0
$799$	−15043.1	−	5475.22i	−0.666064	−	0.242427i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	236.028	+	1338.58i	0.0103727	+	0.0588262i
$804$		0			0
$805$	4075.00	−	1483.18i	0.178416	−	0.0649380i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−29146.6			−1.26668			−0.633338	−	0.773875i	$-0.718316\pi$
$809$	−29146.6			−1.26668			−0.633338	+	0.773875i	$0.718316\pi$
$810$		0			0
$811$	−17449.0			−0.755509			−0.377755	−	0.925906i	$-0.623304\pi$
$811$	−17449.0			−0.755509			−0.377755	+	0.925906i	$0.623304\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	19641.6	−	7148.96i	0.844191	−	0.307260i
$816$		0			0
$817$	353.802	+	2006.51i	0.0151505	+	0.0859228i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	3391.72	+	1234.49i	0.144180	+	0.0524773i	0.413102	−	0.910685i	$-0.364445\pi$
$821$	3391.72	+	1234.49i	0.144180	+	0.0524773i	−0.268922	+	0.963162i	$0.586667\pi$
$822$		0			0
$823$	−3559.55	−	2986.82i	−0.150763	−	0.126505i	0.564286	−	0.825579i	$-0.309151\pi$
$823$	−3559.55	−	2986.82i	−0.150763	−	0.126505i	−0.715050	+	0.699074i	$0.753596\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−8056.05	+	13953.5i	−0.338738	+	0.586711i	−0.984196	−	0.177085i	$-0.943333\pi$
$827$	−8056.05	+	13953.5i	−0.338738	+	0.586711i	0.645458	+	0.763796i	$0.276667\pi$
$828$		0			0
$829$	5703.26	+	9878.33i	0.238941	+	0.413858i	0.960411	−	0.278588i	$-0.0898663\pi$
$829$	5703.26	+	9878.33i	0.238941	+	0.413858i	−0.721470	+	0.692446i	$0.756533\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	303.857	−	1723.26i	0.0126387	−	0.0716774i
$834$		0			0
$835$	20393.4	−	17112.1i	0.845201	−	0.709208i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−7637.59	+	6408.70i	−0.314278	+	0.263710i	−0.786257	−	0.617899i	$-0.787984\pi$
$839$	−7637.59	+	6408.70i	−0.314278	+	0.263710i	0.471980	+	0.881609i	$0.343540\pi$
$840$		0			0
$841$	5689.31	−	32265.7i	0.233273	−	1.32296i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	15798.9	+	27364.4i	0.643192	+	1.11404i
$846$		0			0
$847$	−11031.3	+	19106.8i	−0.447508	+	0.775107i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	1863.44	+	1563.61i	0.0750621	+	0.0629846i
$852$		0			0
$853$	31510.1	+	11468.7i	1.26481	+	0.460355i	0.885382	−	0.464865i	$-0.153897\pi$
$853$	31510.1	+	11468.7i	1.26481	+	0.460355i	0.379432	+	0.925220i	$0.376119\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−960.939	−	5449.75i	−0.0383023	−	0.217223i	0.959649	−	0.281200i	$-0.0907325\pi$
$857$	−960.939	−	5449.75i	−0.0383023	−	0.217223i	−0.997951	+	0.0639774i	$0.979621\pi$
$858$		0			0
$859$	−25197.0	+	9170.97i	−1.00083	+	0.364272i	−0.789904	−	0.613231i	$-0.789870\pi$
$859$	−25197.0	+	9170.97i	−1.00083	+	0.364272i	−0.210924	+	0.977502i	$0.567647\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	15485.4			0.610809			0.305404	−	0.952223i	$-0.401208\pi$
$863$	15485.4			0.610809			0.305404	+	0.952223i	$0.401208\pi$
$864$		0			0
$865$	−13991.6			−0.549975
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−589.100	+	214.415i	−0.0229964	+	0.00837000i
$870$		0			0
$871$	4300.44	+	24389.0i	0.167296	+	0.948783i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	22897.9	+	8334.15i	0.884674	+	0.321995i
$876$		0			0
$877$	734.315	+	616.163i	0.0282737	+	0.0237245i	0.656815	−	0.754052i	$-0.271903\pi$
$877$	734.315	+	616.163i	0.0282737	+	0.0237245i	−0.628541	+	0.777776i	$0.716348\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−11851.2	+	20526.9i	−0.453209	+	0.784981i	−0.998583	−	0.0532115i	$-0.983054\pi$
$881$	−11851.2	+	20526.9i	−0.453209	+	0.784981i	0.545374	+	0.838193i	$0.316388\pi$
$882$		0			0
$883$	23503.0	+	40708.3i	0.895739	+	1.55147i	0.832887	+	0.553443i	$0.186686\pi$
$883$	23503.0	+	40708.3i	0.895739	+	1.55147i	0.0628522	+	0.998023i	$0.479980\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−3326.07	+	18863.1i	−0.125906	+	0.714047i	0.854860	+	0.518859i	$0.173643\pi$
$887$	−3326.07	+	18863.1i	−0.125906	+	0.714047i	−0.980766	+	0.195189i	$0.937468\pi$
$888$		0			0
$889$	26920.8	−	22589.2i	1.01563	−	0.852214i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	3794.05	−	3183.59i	0.142176	−	0.119300i
$894$		0			0
$895$	−6419.27	+	36405.5i	−0.239746	+	1.35967i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	31226.5	+	54085.9i	1.15847	+	2.00653i
$900$		0			0
$901$	−5172.72	+	8959.42i	−0.191264	+	0.331278i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	16487.7	+	13834.8i	0.605603	+	0.508161i
$906$		0			0
$907$	10582.7	+	3851.78i	0.387423	+	0.141010i	0.528385	−	0.849005i	$-0.322798\pi$
$907$	10582.7	+	3851.78i	0.387423	+	0.141010i	−0.140963	+	0.990015i	$0.545020\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−5704.27	−	32350.5i	−0.207454	−	1.17653i	−0.893531	−	0.449002i	$-0.851780\pi$
$911$	−5704.27	−	32350.5i	−0.207454	−	1.17653i	0.686077	−	0.727529i	$-0.259331\pi$
$912$		0			0
$913$	−3316.44	+	1207.09i	−0.120217	+	0.0437555i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−15213.2			−0.547855
$918$		0			0
$919$	−9006.48			−0.323282			−0.161641	−	0.986850i	$-0.551679\pi$
$919$	−9006.48			−0.323282			−0.161641	+	0.986850i	$0.551679\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	67061.7	−	24408.5i	2.39151	−	0.870438i
$924$		0			0
$925$	200.669	+	1138.05i	0.00713291	+	0.0404528i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	6268.08	+	2281.40i	0.221366	+	0.0805707i	0.450322	−	0.892866i	$-0.351309\pi$
$929$	6268.08	+	2281.40i	0.221366	+	0.0805707i	−0.228956	+	0.973437i	$0.573531\pi$
$930$		0			0
$931$	414.717	+	347.989i	0.0145991	+	0.0122501i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−562.914	+	974.995i	−0.0196890	+	0.0341024i
$936$		0			0
$937$	−8632.97	−	14952.7i	−0.300989	−	0.521329i	0.675371	−	0.737478i	$-0.263984\pi$
$937$	−8632.97	−	14952.7i	−0.300989	−	0.521329i	−0.976360	+	0.216149i	$0.930650\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	500.873	−	2840.59i	0.0173518	−	0.0984067i	−0.974902	−	0.222635i	$-0.928534\pi$
$941$	500.873	−	2840.59i	0.0173518	−	0.0984067i	0.992254	+	0.124228i	$0.0396454\pi$
$942$		0			0
$943$	4315.86	−	3621.44i	0.149039	−	0.125059i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	9699.06	−	8138.48i	0.332816	−	0.279266i	−0.461030	−	0.887385i	$-0.652520\pi$
$947$	9699.06	−	8138.48i	0.332816	−	0.279266i	0.793846	+	0.608119i	$0.208075\pi$
$948$		0			0
$949$	−4507.19	+	25561.5i	−0.154172	+	0.874355i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	20384.0	+	35306.2i	0.692868	+	1.20008i	0.970894	+	0.239509i	$0.0769864\pi$
$953$	20384.0	+	35306.2i	0.692868	+	1.20008i	−0.278026	+	0.960573i	$0.589680\pi$
$954$		0			0
$955$	−7660.29	+	13268.0i	−0.259562	+	0.449574i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	10105.4	+	8479.44i	0.340272	+	0.285522i
$960$		0			0
$961$	−36135.2	−	13152.1i	−1.21296	−	0.441481i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−1757.73	−	9968.61i	−0.0586357	−	0.332540i
$966$		0			0
$967$	20212.7	−	7356.82i	0.672179	−	0.244653i	0.0166931	−	0.999861i	$-0.494686\pi$
$967$	20212.7	−	7356.82i	0.672179	−	0.244653i	0.655486	+	0.755208i	$0.272464\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−5278.90			−0.174468			−0.0872338	−	0.996188i	$-0.527803\pi$
$971$	−5278.90			−0.174468			−0.0872338	+	0.996188i	$0.527803\pi$
$972$		0			0
$973$	−16920.0			−0.557481
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−27704.0	+	10083.4i	−0.907195	+	0.330192i	−0.753132	−	0.657869i	$-0.771458\pi$
$977$	−27704.0	+	10083.4i	−0.907195	+	0.330192i	−0.154063	+	0.988061i	$0.549236\pi$
$978$		0			0
$979$	−528.512	−	2997.34i	−0.0172536	−	0.0978502i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	20637.6	+	7511.48i	0.669622	+	0.243722i	0.654385	−	0.756162i	$-0.272928\pi$
$983$	20637.6	+	7511.48i	0.669622	+	0.243722i	0.0152366	+	0.999884i	$0.495150\pi$
$984$		0			0
$985$	−10287.8	−	8632.46i	−0.332787	−	0.279242i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−2848.63	+	4933.97i	−0.0915885	+	0.158636i
$990$		0			0
$991$	−14802.0	−	25637.8i	−0.474470	−	0.821807i	0.525102	−	0.851039i	$-0.324027\pi$
$991$	−14802.0	−	25637.8i	−0.474470	−	0.821807i	−0.999573	+	0.0292322i	$0.990694\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	167.941	−	952.441i	0.00535084	−	0.0303461i
$996$		0			0
$997$	−21236.8	+	17819.8i	−0.674599	+	0.566055i	−0.914423	−	0.404761i	$-0.867355\pi$
$997$	−21236.8	+	17819.8i	−0.674599	+	0.566055i	0.239824	+	0.970816i	$0.422910\pi$
$998$		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.37.8		54
3.2	odd	2		108.4.i.a.49.7	✓	54
27.11	odd	18		108.4.i.a.97.7	yes	54
27.16	even	9	inner	324.4.i.a.289.8		54

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
108.4.i.a.49.7	✓	54	3.2	odd	2
108.4.i.a.97.7	yes	54	27.11	odd	18
324.4.i.a.37.8		54	1.1	even	1	trivial
324.4.i.a.289.8		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)

\(f(q)\)	\(=\)	\(q+(10.0092 - 3.64305i) q^{5} +(-2.90933 - 16.4997i) q^{7} +O(q^{10})\)	\(q+(10.0092 - 3.64305i) q^{5} +(-2.90933 - 16.4997i) q^{7} +(-3.53602 - 1.28701i) q^{11} +(-55.0460 - 46.1890i) q^{13} +(14.0443 - 24.3254i) q^{17} +(4.34509 + 7.52591i) q^{19} +(-4.21963 + 23.9307i) q^{23} +(-8.84313 + 7.42027i) q^{25} +(-183.135 + 153.668i) q^{29} +(45.3635 - 257.269i) q^{31} +(-89.2293 - 154.550i) q^{35} +(50.0527 - 86.6937i) q^{37} +(-177.609 - 149.031i) q^{41} +(220.317 + 80.1887i) q^{43} +(-98.9672 - 561.271i) q^{47} +(58.5403 - 21.3069i) q^{49} -368.316 q^{53} -40.0814 q^{55} +(485.116 - 176.568i) q^{59} +(-106.741 - 605.358i) q^{61} +(-719.236 - 261.780i) q^{65} +(-264.013 - 221.533i) q^{67} +(-496.578 + 860.098i) q^{71} +(-180.607 - 312.820i) q^{73} +(-10.9477 + 62.0874i) q^{77} +(127.623 - 107.088i) q^{79} +(718.475 - 602.872i) q^{83} +(51.9533 - 294.642i) q^{85} +(404.414 + 700.465i) q^{89} +(-601.956 + 1042.62i) q^{91} +(70.9082 + 59.4990i) q^{95} +(1723.85 + 627.431i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 54 q - 12 q^{5}+O(q^{10}) \) 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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