LMFDB - Embedded newform 324.4.i.a.37.4

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			37.4
Character	$\chi$	$=$	324.37
Dual form			324.4.i.a.289.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-5.21695 + 1.89881i) q^{5} +(1.58868 + 9.00985i) q^{7} +O(q^{10})$	$q+(-5.21695 + 1.89881i) q^{5} +(1.58868 + 9.00985i) q^{7} +(67.4972 + 24.5670i) q^{11} +(-63.1985 - 53.0298i) q^{13} +(-21.0212 + 36.4097i) q^{17} +(-22.5218 - 39.0089i) q^{19} +(-12.1174 + 68.7214i) q^{23} +(-72.1445 + 60.5364i) q^{25} +(-142.727 + 119.762i) q^{29} +(-46.3221 + 262.706i) q^{31} +(-25.3961 - 43.9873i) q^{35} +(-40.4209 + 70.0111i) q^{37} +(216.792 + 181.910i) q^{41} +(-249.820 - 90.9271i) q^{43} +(10.5496 + 59.8299i) q^{47} +(243.661 - 88.6854i) q^{49} -206.241 q^{53} -398.778 q^{55} +(-558.381 + 203.234i) q^{59} +(124.554 + 706.379i) q^{61} +(430.397 + 156.652i) q^{65} +(23.6459 + 19.8412i) q^{67} +(-155.414 + 269.185i) q^{71} +(-263.889 - 457.070i) q^{73} +(-114.113 + 647.169i) q^{77} +(-500.648 + 420.094i) q^{79} +(883.108 - 741.015i) q^{83} +(40.5310 - 229.863i) q^{85} +(-240.988 - 417.404i) q^{89} +(377.389 - 653.656i) q^{91} +(191.566 + 160.743i) q^{95} +(-858.230 - 312.370i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−5.21695	+	1.89881i	−0.466618	+	0.169835i	−0.564620	−	0.825351i	$-0.690977\pi$
$5$	−5.21695	+	1.89881i	−0.466618	+	0.169835i	0.0980017	+	0.995186i	$0.468755\pi$
$6$		0			0
$7$	1.58868	+	9.00985i	0.0857806	+	0.486486i	0.997185	+	0.0749750i	$0.0238877\pi$
$7$	1.58868	+	9.00985i	0.0857806	+	0.486486i	−0.911405	+	0.411511i	$0.865001\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	67.4972	+	24.5670i	1.85011	+	0.673384i	0.985189	+	0.171472i	$0.0548522\pi$
$11$	67.4972	+	24.5670i	1.85011	+	0.673384i	0.864918	+	0.501912i	$0.167370\pi$
$12$		0			0
$13$	−63.1985	−	53.0298i	−1.34832	−	1.13137i	−0.979405	−	0.201907i	$-0.935286\pi$
$13$	−63.1985	−	53.0298i	−1.34832	−	1.13137i	−0.368911	−	0.929465i	$-0.620269\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−21.0212	+	36.4097i	−0.299905	+	0.519450i	−0.976114	−	0.217259i	$-0.930288\pi$
$17$	−21.0212	+	36.4097i	−0.299905	+	0.519450i	0.676209	+	0.736710i	$0.263622\pi$
$18$		0			0
$19$	−22.5218	−	39.0089i	−0.271940	−	0.471013i	0.697419	−	0.716664i	$-0.254332\pi$
$19$	−22.5218	−	39.0089i	−0.271940	−	0.471013i	−0.969358	+	0.245650i	$0.920998\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−12.1174	+	68.7214i	−0.109855	+	0.623018i	0.879315	+	0.476241i	$0.158001\pi$
$23$	−12.1174	+	68.7214i	−0.109855	+	0.623018i	−0.989170	+	0.146777i	$0.953110\pi$
$24$		0			0
$25$	−72.1445	+	60.5364i	−0.577156	+	0.484291i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−142.727	+	119.762i	−0.913924	+	0.766873i	−0.972861	−	0.231388i	$-0.925673\pi$
$29$	−142.727	+	119.762i	−0.913924	+	0.766873i	0.0589376	+	0.998262i	$0.481229\pi$
$30$		0			0
$31$	−46.3221	+	262.706i	−0.268377	+	1.52204i	0.490866	+	0.871235i	$0.336680\pi$
$31$	−46.3221	+	262.706i	−0.268377	+	1.52204i	−0.759243	+	0.650807i	$0.774431\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−25.3961	−	43.9873i	−0.122649	−	0.212435i
$36$		0			0
$37$	−40.4209	+	70.0111i	−0.179599	+	0.311074i	−0.941743	−	0.336333i	$-0.890813\pi$
$37$	−40.4209	+	70.0111i	−0.179599	+	0.311074i	0.762144	+	0.647407i	$0.224147\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	216.792	+	181.910i	0.825787	+	0.692918i	0.954320	−	0.298788i	$-0.0965822\pi$
$41$	216.792	+	181.910i	0.825787	+	0.692918i	−0.128533	+	0.991705i	$0.541027\pi$
$42$		0			0
$43$	−249.820	−	90.9271i	−0.885982	−	0.322471i	−0.141361	−	0.989958i	$-0.545148\pi$
$43$	−249.820	−	90.9271i	−0.885982	−	0.322471i	−0.744621	+	0.667487i	$0.767370\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	10.5496	+	59.8299i	0.0327409	+	0.185683i	0.996792	−	0.0800335i	$-0.0255027\pi$
$47$	10.5496	+	59.8299i	0.0327409	+	0.185683i	−0.964051	+	0.265716i	$0.914392\pi$
$48$		0			0
$49$	243.661	−	88.6854i	0.710382	−	0.258558i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−206.241			−0.534517			−0.267258	−	0.963625i	$-0.586118\pi$
$53$	−206.241			−0.534517			−0.267258	+	0.963625i	$0.586118\pi$
$54$		0			0
$55$	−398.778			−0.977658
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−558.381	+	203.234i	−1.23212	+	0.448455i	−0.874323	−	0.485345i	$-0.838694\pi$
$59$	−558.381	+	203.234i	−1.23212	+	0.448455i	−0.357797	+	0.933800i	$0.616472\pi$
$60$		0			0
$61$	124.554	+	706.379i	0.261434	+	1.48266i	0.779002	+	0.627022i	$0.215726\pi$
$61$	124.554	+	706.379i	0.261434	+	1.48266i	−0.517568	+	0.855642i	$0.673163\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	430.397	+	156.652i	0.821295	+	0.298927i
$66$		0			0
$67$	23.6459	+	19.8412i	0.0431164	+	0.0361790i	0.664091	−	0.747652i	$-0.268819\pi$
$67$	23.6459	+	19.8412i	0.0431164	+	0.0361790i	−0.620975	+	0.783831i	$0.713263\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−155.414	+	269.185i	−0.259779	+	0.449950i	−0.966182	−	0.257859i	$-0.916983\pi$
$71$	−155.414	+	269.185i	−0.259779	+	0.449950i	0.706404	+	0.707809i	$0.250316\pi$
$72$		0			0
$73$	−263.889	−	457.070i	−0.423094	−	0.732821i	0.573146	−	0.819453i	$-0.305723\pi$
$73$	−263.889	−	457.070i	−0.423094	−	0.732821i	−0.996240	+	0.0866322i	$0.972389\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−114.113	+	647.169i	−0.168889	+	0.957815i
$78$		0			0
$79$	−500.648	+	420.094i	−0.713004	+	0.598281i	−0.925440	−	0.378893i	$-0.876305\pi$
$79$	−500.648	+	420.094i	−0.713004	+	0.598281i	0.212436	+	0.977175i	$0.431860\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	883.108	−	741.015i	1.16788	−	0.979964i	0.167893	−	0.985805i	$-0.446304\pi$
$83$	883.108	−	741.015i	1.16788	−	0.979964i	0.999983	+	0.00584137i	$0.00185938\pi$
$84$		0			0
$85$	40.5310	−	229.863i	0.0517201	−	0.293319i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−240.988	−	417.404i	−0.287019	−	0.497132i	0.686078	−	0.727528i	$-0.259331\pi$
$89$	−240.988	−	417.404i	−0.287019	−	0.497132i	−0.973097	+	0.230397i	$0.925998\pi$
$90$		0			0
$91$	377.389	−	653.656i	0.434737	−	0.752987i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	191.566	+	160.743i	0.206887	+	0.173598i
$96$		0			0
$97$	−858.230	−	312.370i	−0.898351	−	0.326973i	−0.148759	−	0.988873i	$-0.547528\pi$
$97$	−858.230	−	312.370i	−0.898351	−	0.326973i	−0.749592	+	0.661901i	$0.769750\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	92.4446	+	524.279i	0.0910750	+	0.516512i	0.995880	+	0.0906852i	$0.0289057\pi$
$101$	92.4446	+	524.279i	0.0910750	+	0.516512i	−0.904805	+	0.425827i	$0.859983\pi$
$102$		0			0
$103$	303.355	−	110.412i	0.290199	−	0.105624i	−0.192819	−	0.981234i	$-0.561763\pi$
$103$	303.355	−	110.412i	0.290199	−	0.105624i	0.483017	+	0.875611i	$0.339541\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	209.423			0.189212			0.0946062	−	0.995515i	$-0.469841\pi$
$107$	209.423			0.189212			0.0946062	+	0.995515i	$0.469841\pi$
$108$		0			0
$109$	−196.115			−0.172334			−0.0861672	−	0.996281i	$-0.527462\pi$
$109$	−196.115			−0.172334			−0.0861672	+	0.996281i	$0.527462\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−203.446	+	74.0482i	−0.169368	+	0.0616448i	−0.425312	−	0.905047i	$-0.639836\pi$
$113$	−203.446	+	74.0482i	−0.169368	+	0.0616448i	0.255945	+	0.966691i	$0.417614\pi$
$114$		0			0
$115$	−67.2731	−	381.525i	−0.0545500	−	0.309368i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−361.442	−	131.554i	−0.278431	−	0.101341i
$120$		0			0
$121$	2932.73	+	2460.86i	2.20341	+	1.84888i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	608.412	−	1053.80i	0.435344	−	0.754038i
$126$		0			0
$127$	−611.723	−	1059.53i	−0.427414	−	0.740303i	0.569228	−	0.822180i	$-0.307242\pi$
$127$	−611.723	−	1059.53i	−0.427414	−	0.740303i	−0.996642	+	0.0818763i	$0.973909\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−194.741	+	1104.43i	−0.129882	+	0.736599i	0.848405	+	0.529347i	$0.177563\pi$
$131$	−194.741	+	1104.43i	−0.129882	+	0.736599i	−0.978288	+	0.207252i	$0.933548\pi$
$132$		0			0
$133$	315.684	−	264.891i	0.205814	−	0.172699i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	1967.06	−	1650.56i	1.22670	−	1.02932i	0.228252	−	0.973602i	$-0.426699\pi$
$137$	1967.06	−	1650.56i	1.22670	−	1.02932i	0.998446	−	0.0557200i	$-0.0177454\pi$
$138$		0			0
$139$	434.934	−	2466.63i	0.265400	−	1.50516i	−0.502494	−	0.864581i	$-0.667584\pi$
$139$	434.934	−	2466.63i	0.265400	−	1.50516i	0.767894	−	0.640577i	$-0.221305\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−2962.94	−	5131.96i	−1.73268	−	3.00109i
$144$		0			0
$145$	517.194	−	895.807i	0.296211	−	0.513053i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−88.5089	−	74.2678i	−0.0486640	−	0.0408339i	0.618131	−	0.786075i	$-0.287890\pi$
$149$	−88.5089	−	74.2678i	−0.0486640	−	0.0408339i	−0.666795	+	0.745241i	$0.732334\pi$
$150$		0			0
$151$	1464.39	+	532.995i	0.789209	+	0.287249i	0.705007	−	0.709200i	$-0.250944\pi$
$151$	1464.39	+	532.995i	0.789209	+	0.287249i	0.0842019	+	0.996449i	$0.473166\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−257.169	−	1458.48i	−0.133267	−	0.755792i
$156$		0			0
$157$	1712.19	−	623.186i	0.870367	−	0.316788i	0.132052	−	0.991243i	$-0.457844\pi$
$157$	1712.19	−	623.186i	0.870367	−	0.316788i	0.738316	+	0.674455i	$0.235621\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−638.420			−0.312513
$162$		0			0
$163$	1785.32			0.857897			0.428948	−	0.903329i	$-0.358884\pi$
$163$	1785.32			0.857897			0.428948	+	0.903329i	$0.358884\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−1097.06	+	399.298i	−0.508343	+	0.185022i	−0.583442	−	0.812154i	$-0.698295\pi$
$167$	−1097.06	+	399.298i	−0.508343	+	0.185022i	0.0751000	+	0.997176i	$0.476072\pi$
$168$		0			0
$169$	800.382	+	4539.19i	0.364307	+	2.06609i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	3928.51	+	1429.86i	1.72647	+	0.628383i	0.998369	−	0.0570873i	$-0.0181813\pi$
$173$	3928.51	+	1429.86i	1.72647	+	0.628383i	0.728100	+	0.685471i	$0.240404\pi$
$174$		0			0
$175$	−660.038	−	553.838i	−0.285110	−	0.239235i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−918.530	+	1590.94i	−0.383543	+	0.664316i	−0.991566	−	0.129604i	$-0.958629\pi$
$179$	−918.530	+	1590.94i	−0.383543	+	0.664316i	0.608023	+	0.793919i	$0.291963\pi$
$180$		0			0
$181$	−1157.45	−	2004.76i	−0.475318	−	0.823275i	0.524282	−	0.851544i	$-0.324334\pi$
$181$	−1157.45	−	2004.76i	−0.475318	−	0.823275i	−0.999600	+	0.0282697i	$0.991000\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	77.9358	−	441.996i	0.0309727	−	0.175655i
$186$		0			0
$187$	−2313.35	+	1941.13i	−0.904646	+	0.759088i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−1309.35	+	1098.67i	−0.496027	+	0.416216i	−0.856180	−	0.516677i	$-0.827169\pi$
$191$	−1309.35	+	1098.67i	−0.496027	+	0.416216i	0.360153	+	0.932893i	$0.382724\pi$
$192$		0			0
$193$	−538.413	+	3053.49i	−0.200808	+	1.13884i	0.703095	+	0.711096i	$0.251801\pi$
$193$	−538.413	+	3053.49i	−0.200808	+	1.13884i	−0.903902	+	0.427740i	$0.859310\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	269.298	+	466.438i	0.0973943	+	0.168692i	0.910605	−	0.413277i	$-0.135616\pi$
$197$	269.298	+	466.438i	0.0973943	+	0.168692i	−0.813211	+	0.581969i	$0.802283\pi$
$198$		0			0
$199$	1847.26	−	3199.55i	0.658035	−	1.13975i	−0.323089	−	0.946369i	$-0.604721\pi$
$199$	1847.26	−	3199.55i	0.658035	−	1.13975i	0.981124	−	0.193381i	$-0.0619454\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−1305.79	−	1095.69i	−0.451470	−	0.378828i
$204$		0			0
$205$	−1476.41	−	537.369i	−0.503009	−	0.183080i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−561.828	−	3186.28i	−0.185945	−	1.05455i
$210$		0			0
$211$	3344.47	−	1217.29i	1.09120	−	0.397164i	0.267134	−	0.963660i	$-0.413924\pi$
$211$	3344.47	−	1217.29i	1.09120	−	0.397164i	0.824065	+	0.566496i	$0.191701\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	1475.95			0.468182
$216$		0			0
$217$	−2440.53			−0.763474
$218$		0			0
$219$		0			0
$220$		0			0
$221$	3259.31	−	1186.29i	0.992058	−	0.361080i
$222$		0			0
$223$	−932.642	−	5289.28i	−0.280064	−	1.58832i	−0.722401	−	0.691474i	$-0.756962\pi$
$223$	−932.642	−	5289.28i	−0.280064	−	1.58832i	0.442337	−	0.896849i	$-0.354150\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	1788.67	+	651.023i	0.522988	+	0.190352i	0.590005	−	0.807400i	$-0.299126\pi$
$227$	1788.67	+	651.023i	0.522988	+	0.190352i	−0.0670166	+	0.997752i	$0.521348\pi$
$228$		0			0
$229$	−1969.18	−	1652.34i	−0.568240	−	0.476810i	0.312821	−	0.949812i	$-0.398726\pi$
$229$	−1969.18	−	1652.34i	−0.568240	−	0.476810i	−0.881061	+	0.473002i	$0.843170\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−1769.64	+	3065.11i	−0.497567	+	0.861811i	−0.999996	−	0.00280750i	$-0.999106\pi$
$233$	−1769.64	+	3065.11i	−0.497567	+	0.861811i	0.502429	+	0.864618i	$0.332440\pi$
$234$		0			0
$235$	−168.643	−	292.098i	−0.0468129	−	0.0810823i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−1060.55	+	6014.68i	−0.287035	+	1.62786i	0.410890	+	0.911685i	$0.365218\pi$
$239$	−1060.55	+	6014.68i	−0.287035	+	1.62786i	−0.697925	+	0.716171i	$0.745893\pi$
$240$		0			0
$241$	−2459.06	+	2063.40i	−0.657270	+	0.551515i	−0.909267	−	0.416213i	$-0.863357\pi$
$241$	−2459.06	+	2063.40i	−0.657270	+	0.551515i	0.251997	+	0.967728i	$0.418913\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−1102.77	+	925.334i	−0.287565	+	0.241296i
$246$		0			0
$247$	−645.292	+	3659.63i	−0.166230	+	0.942740i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−1804.49	−	3125.47i	−0.453778	−	0.785967i	0.544839	−	0.838541i	$-0.316591\pi$
$251$	−1804.49	−	3125.47i	−0.453778	−	0.785967i	−0.998617	+	0.0525736i	$0.983258\pi$
$252$		0			0
$253$	−2506.17	+	4340.82i	−0.622773	+	1.07868i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	4944.99	+	4149.34i	1.20023	+	1.00712i	0.999623	+	0.0274439i	$0.00873676\pi$
$257$	4944.99	+	4149.34i	1.20023	+	1.00712i	0.200610	+	0.979671i	$0.435708\pi$
$258$		0			0
$259$	−695.005	−	252.961i	−0.166739	−	0.0606882i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	1324.95	+	7514.17i	0.310646	+	1.76176i	0.595657	+	0.803239i	$0.296892\pi$
$263$	1324.95	+	7514.17i	0.310646	+	1.76176i	−0.285011	+	0.958524i	$0.591997\pi$
$264$		0			0
$265$	1075.95	−	391.613i	0.249415	−	0.0907797i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	7418.04			1.68136			0.840681	−	0.541531i	$-0.182155\pi$
$269$	7418.04			1.68136			0.840681	+	0.541531i	$0.182155\pi$
$270$		0			0
$271$	−5946.94			−1.33303			−0.666515	−	0.745492i	$-0.732215\pi$
$271$	−5946.94			−1.33303			−0.666515	+	0.745492i	$0.732215\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−6356.75	+	2313.67i	−1.39391	+	0.507343i
$276$		0			0
$277$	105.510	+	598.379i	0.0228863	+	0.129795i	0.994110	−	0.108373i	$-0.0345642\pi$
$277$	105.510	+	598.379i	0.0228863	+	0.129795i	−0.971224	+	0.238168i	$0.923453\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−6387.32	−	2324.79i	−1.35600	−	0.493543i	−0.441184	−	0.897417i	$-0.645441\pi$
$281$	−6387.32	−	2324.79i	−1.35600	−	0.493543i	−0.914815	+	0.403874i	$0.867664\pi$
$282$		0			0
$283$	−56.9781	−	47.8103i	−0.0119682	−	0.0100425i	0.636784	−	0.771042i	$-0.280264\pi$
$283$	−56.9781	−	47.8103i	−0.0119682	−	0.0100425i	−0.648752	+	0.761000i	$0.724709\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−1294.57	+	2242.26i	−0.266258	+	0.461173i
$288$		0			0
$289$	1572.72	+	2724.03i	0.320114	+	0.554454i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	17.3041	−	98.1364i	0.00345023	−	0.0195672i	−0.983034	−	0.183423i	$-0.941282\pi$
$293$	17.3041	−	98.1364i	0.00345023	−	0.0195672i	0.986484	+	0.163856i	$0.0523932\pi$
$294$		0			0
$295$	2527.14	−	2120.52i	0.498766	−	0.418514i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	4410.09	−	3700.50i	0.852983	−	0.715738i
$300$		0			0
$301$	422.355	−	2395.30i	0.0808776	−	0.458680i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−1991.07	−	3448.64i	−0.373798	−	0.647437i
$306$		0			0
$307$	−2496.99	+	4324.91i	−0.464204	+	0.804024i	−0.999165	−	0.0408521i	$-0.986993\pi$
$307$	−2496.99	+	4324.91i	−0.464204	+	0.804024i	0.534962	+	0.844876i	$0.320326\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−5733.38	−	4810.88i	−1.04537	−	0.877170i	−0.0527712	−	0.998607i	$-0.516805\pi$
$311$	−5733.38	−	4810.88i	−1.04537	−	0.877170i	−0.992599	+	0.121437i	$0.961250\pi$
$312$		0			0
$313$	4171.87	+	1518.44i	0.753380	+	0.274208i	0.690027	−	0.723783i	$-0.257599\pi$
$313$	4171.87	+	1518.44i	0.753380	+	0.274208i	0.0633524	+	0.997991i	$0.479821\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	299.891	+	1700.77i	0.0531342	+	0.301339i	0.999781	−	0.0209343i	$-0.00666407\pi$
$317$	299.891	+	1700.77i	0.0531342	+	0.301339i	−0.946647	+	0.322273i	$0.895553\pi$
$318$		0			0
$319$	−12575.9	+	4577.25i	−2.20726	+	0.803376i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	1893.74			0.326224
$324$		0			0
$325$	7769.66			1.32610
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−522.298	+	190.101i	−0.0875235	+	0.0318559i
$330$		0			0
$331$	−146.572	−	831.251i	−0.0243393	−	0.138035i	0.970217	−	0.242239i	$-0.0778818\pi$
$331$	−146.572	−	831.251i	−0.0243393	−	0.138035i	−0.994556	+	0.104204i	$0.966771\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−161.034	−	58.6116i	−0.0262634	−	0.00955908i
$336$		0			0
$337$	25.8927	+	21.7265i	0.00418536	+	0.00351193i	0.644878	−	0.764286i	$-0.276908\pi$
$337$	25.8927	+	21.7265i	0.00418536	+	0.00351193i	−0.640693	+	0.767798i	$0.721353\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−9580.49	+	16593.9i	−1.52145	+	2.63522i
$342$		0			0
$343$	2755.17	+	4772.09i	0.433717	+	0.751220i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	1785.27	−	10124.8i	0.276191	−	1.56636i	−0.458963	−	0.888455i	$-0.651779\pi$
$347$	1785.27	−	10124.8i	0.276191	−	1.56636i	0.735154	−	0.677900i	$-0.237110\pi$
$348$		0			0
$349$	−5099.40	+	4278.91i	−0.782134	+	0.656289i	−0.943785	−	0.330559i	$-0.892763\pi$
$349$	−5099.40	+	4278.91i	−0.782134	+	0.656289i	0.161651	+	0.986848i	$0.448318\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−6159.61	+	5168.53i	−0.928734	+	0.779300i	−0.975590	−	0.219601i	$-0.929524\pi$
$353$	−6159.61	+	5168.53i	−0.928734	+	0.779300i	0.0468557	+	0.998902i	$0.485080\pi$
$354$		0			0
$355$	299.655	−	1699.43i	0.0448001	−	0.254074i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	1712.40	+	2965.97i	0.251747	+	0.436039i	0.964007	−	0.265877i	$-0.0856615\pi$
$359$	1712.40	+	2965.97i	0.251747	+	0.436039i	−0.712260	+	0.701916i	$0.752328\pi$
$360$		0			0
$361$	2415.04	−	4182.97i	0.352098	−	0.609851i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	2244.59	+	1883.43i	0.321882	+	0.270091i
$366$		0			0
$367$	−3402.99	−	1238.59i	−0.484018	−	0.176168i	0.0884742	−	0.996078i	$-0.471801\pi$
$367$	−3402.99	−	1238.59i	−0.484018	−	0.176168i	−0.572492	+	0.819910i	$0.694023\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−327.651	−	1858.20i	−0.0458512	−	0.260035i
$372$		0			0
$373$	−1812.31	+	659.627i	−0.251576	+	0.0915662i	−0.464730	−	0.885452i	$-0.653849\pi$
$373$	−1812.31	+	659.627i	−0.251576	+	0.0915662i	0.213154	+	0.977019i	$0.431626\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	15371.1			2.09988
$378$		0			0
$379$	4403.13			0.596763			0.298382	−	0.954447i	$-0.403553\pi$
$379$	4403.13			0.596763			0.298382	+	0.954447i	$0.403553\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	10231.3	−	3723.88i	1.36500	−	0.496819i	0.447402	−	0.894333i	$-0.352349\pi$
$383$	10231.3	−	3723.88i	1.36500	−	0.496819i	0.917596	+	0.397515i	$0.130127\pi$
$384$		0			0
$385$	−633.530	−	3592.93i	−0.0838641	−	0.475617i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−7053.47	−	2567.25i	−0.919345	−	0.334614i	−0.161367	−	0.986894i	$-0.551590\pi$
$389$	−7053.47	−	2567.25i	−0.919345	−	0.334614i	−0.757978	+	0.652280i	$0.773813\pi$
$390$		0			0
$391$	−2247.41	−	1885.80i	−0.290681	−	0.243910i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	1814.18	−	3142.25i	0.231091	−	0.400262i
$396$		0			0
$397$	−3635.79	−	6297.37i	−0.459635	−	0.796111i	0.539307	−	0.842110i	$-0.318686\pi$
$397$	−3635.79	−	6297.37i	−0.459635	−	0.796111i	−0.998942	+	0.0459984i	$0.985353\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	2014.16	−	11422.9i	0.250828	−	1.42252i	−0.555729	−	0.831363i	$-0.687561\pi$
$401$	2014.16	−	11422.9i	0.250828	−	1.42252i	0.806558	−	0.591155i	$-0.201328\pi$
$402$		0			0
$403$	16858.7	−	14146.1i	2.08385	−	1.74856i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−4448.26	+	3732.53i	−0.541750	+	0.454582i
$408$		0			0
$409$	1420.37	−	8055.31i	0.171718	−	0.973862i	−0.770146	−	0.637868i	$-0.779817\pi$
$409$	1420.37	−	8055.31i	0.171718	−	0.973862i	0.941864	−	0.335994i	$-0.109072\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−2718.20	−	4708.06i	−0.323859	−	0.560940i
$414$		0			0
$415$	−3200.08	+	5542.70i	−0.378520	+	0.655615i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	6454.20	+	5415.71i	0.752525	+	0.631444i	0.936169	−	0.351549i	$-0.114345\pi$
$419$	6454.20	+	5415.71i	0.752525	+	0.631444i	−0.183644	+	0.982993i	$0.558789\pi$
$420$		0			0
$421$	4401.68	+	1602.08i	0.509560	+	0.185465i	0.583989	−	0.811762i	$-0.301491\pi$
$421$	4401.68	+	1602.08i	0.509560	+	0.185465i	−0.0744290	+	0.997226i	$0.523713\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−687.553	−	3899.31i	−0.0784735	−	0.445045i
$426$		0			0
$427$	−6166.49	+	2244.42i	−0.698869	+	0.254368i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	8784.41			0.981740			0.490870	−	0.871233i	$-0.336679\pi$
$431$	8784.41			0.981740			0.490870	+	0.871233i	$0.336679\pi$
$432$		0			0
$433$	−6469.85			−0.718063			−0.359031	−	0.933325i	$-0.616893\pi$
$433$	−6469.85			−0.718063			−0.359031	+	0.933325i	$0.616893\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	2953.65	−	1075.04i	0.323323	−	0.117680i
$438$		0			0
$439$	−148.990	−	844.963i	−0.0161979	−	0.0918631i	0.975637	−	0.219391i	$-0.0704070\pi$
$439$	−148.990	−	844.963i	−0.0161979	−	0.0918631i	−0.991835	+	0.127528i	$0.959296\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−412.777	−	150.239i	−0.0442701	−	0.0161130i	0.319790	−	0.947488i	$-0.396388\pi$
$443$	−412.777	−	150.239i	−0.0442701	−	0.0161130i	−0.364060	+	0.931375i	$0.618610\pi$
$444$		0			0
$445$	2049.80	+	1719.98i	0.218359	+	0.183225i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−1510.09	+	2615.54i	−0.158720	+	0.274911i	−0.934407	−	0.356206i	$-0.884070\pi$
$449$	−1510.09	+	2615.54i	−0.158720	+	0.274911i	0.775687	+	0.631117i	$0.217403\pi$
$450$		0			0
$451$	10163.9	+	17604.4i	1.06120	+	1.83804i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−727.645	+	4126.68i	−0.0749726	+	0.425191i
$456$		0			0
$457$	5728.39	−	4806.69i	0.586352	−	0.492008i	−0.300674	−	0.953727i	$-0.597212\pi$
$457$	5728.39	−	4806.69i	0.586352	−	0.492008i	0.887026	+	0.461719i	$0.152767\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−4559.80	+	3826.13i	−0.460675	+	0.386552i	−0.843379	−	0.537319i	$-0.819437\pi$
$461$	−4559.80	+	3826.13i	−0.460675	+	0.386552i	0.382705	+	0.923871i	$0.374993\pi$
$462$		0			0
$463$	299.450	−	1698.27i	0.0300575	−	0.170465i	−0.966084	−	0.258229i	$-0.916861\pi$
$463$	299.450	−	1698.27i	0.0300575	−	0.170465i	0.996141	+	0.0877642i	$0.0279722\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−7069.94	−	12245.5i	−0.700552	−	1.21339i	−0.968273	−	0.249895i	$-0.919604\pi$
$467$	−7069.94	−	12245.5i	−0.700552	−	1.21339i	0.267721	−	0.963496i	$-0.413729\pi$
$468$		0			0
$469$	−141.201	+	244.567i	−0.0139020	+	0.0240790i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−14628.4	−	12274.7i	−1.42202	−	1.19321i
$474$		0			0
$475$	3986.28	+	1450.89i	0.385059	+	0.140150i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−467.551	−	2651.62i	−0.0445991	−	0.252934i	0.954354	−	0.298677i	$-0.0965455\pi$
$479$	−467.551	−	2651.62i	−0.0445991	−	0.252934i	−0.998953	+	0.0457434i	$0.985434\pi$
$480$		0			0
$481$	6267.22	−	2281.08i	0.594097	−	0.216234i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	5070.47			0.474718
$486$		0			0
$487$	12687.7			1.18056			0.590280	−	0.807199i	$-0.299017\pi$
$487$	12687.7			1.18056			0.590280	+	0.807199i	$0.299017\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	5094.55	−	1854.26i	0.468256	−	0.170431i	−0.0971061	−	0.995274i	$-0.530959\pi$
$491$	5094.55	−	1854.26i	0.468256	−	0.170431i	0.565362	+	0.824843i	$0.308736\pi$
$492$		0			0
$493$	−1360.22	−	7714.21i	−0.124262	−	0.704727i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−2672.22	−	972.609i	−0.241178	−	0.0877817i
$498$		0			0
$499$	−10451.5	−	8769.82i	−0.937619	−	0.786756i	0.0395506	−	0.999218i	$-0.487407\pi$
$499$	−10451.5	−	8769.82i	−0.937619	−	0.786756i	−0.977169	+	0.212462i	$0.931852\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−3966.78	+	6870.66i	−0.351630	+	0.609041i	−0.986535	−	0.163549i	$-0.947706\pi$
$503$	−3966.78	+	6870.66i	−0.351630	+	0.609041i	0.634905	+	0.772590i	$0.281039\pi$
$504$		0			0
$505$	−1477.79	−	2559.60i	−0.130219	−	0.225546i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	3183.90	−	18056.8i	0.277257	−	1.57240i	−0.454443	−	0.890776i	$-0.650162\pi$
$509$	3183.90	−	18056.8i	0.277257	−	1.57240i	0.731700	−	0.681627i	$-0.238727\pi$
$510$		0			0
$511$	3698.89	−	3103.74i	0.320214	−	0.268691i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−1372.94	+	1152.03i	−0.117473	+	0.0985718i
$516$		0			0
$517$	−757.769	+	4297.52i	−0.0644616	+	0.365580i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	3020.36	+	5231.43i	0.253982	+	0.439910i	0.964618	−	0.263650i	$-0.0849263\pi$
$521$	3020.36	+	5231.43i	0.253982	+	0.439910i	−0.710637	+	0.703559i	$0.751593\pi$
$522$		0			0
$523$	2909.21	−	5038.90i	0.243233	−	0.421292i	−0.718400	−	0.695630i	$-0.755125\pi$
$523$	2909.21	−	5038.90i	0.243233	−	0.421292i	0.961633	+	0.274338i	$0.0884586\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−8591.29	−	7208.95i	−0.710138	−	0.595876i
$528$		0			0
$529$	6857.44	+	2495.90i	0.563610	+	0.205137i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−4054.27	−	22992.9i	−0.329475	−	1.86854i
$534$		0			0
$535$	−1092.55	+	397.656i	−0.0882899	+	0.0321349i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	18625.2			1.48839
$540$		0			0
$541$	6742.32			0.535813			0.267907	−	0.963445i	$-0.413668\pi$
$541$	6742.32			0.535813			0.267907	+	0.963445i	$0.413668\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	1023.12	−	372.387i	0.0804144	−	0.0292684i
$546$		0			0
$547$	996.065	+	5648.96i	0.0778586	+	0.441558i	0.998670	+	0.0515507i	$0.0164164\pi$
$547$	996.065	+	5648.96i	0.0778586	+	0.441558i	−0.920812	+	0.390007i	$0.872473\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	7886.27	+	2870.37i	0.609740	+	0.221927i
$552$		0			0
$553$	−4580.35	−	3843.37i	−0.352217	−	0.295546i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−11916.4	+	20639.9i	−0.906492	+	1.57009i	−0.0875890	+	0.996157i	$0.527916\pi$
$557$	−11916.4	+	20639.9i	−0.906492	+	1.57009i	−0.818902	+	0.573933i	$0.805417\pi$
$558$		0			0
$559$	10966.4	+	18994.4i	0.829749	+	1.43717i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−143.937	+	816.308i	−0.0107748	+	0.0611070i	−0.989721	−	0.143011i	$-0.954322\pi$
$563$	−143.937	+	816.308i	−0.0107748	+	0.0611070i	0.978946	+	0.204118i	$0.0654327\pi$
$564$		0			0
$565$	920.762	−	772.611i	0.0685606	−	0.0575292i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−809.884	+	679.573i	−0.0596698	+	0.0500689i	−0.672135	−	0.740429i	$-0.734622\pi$
$569$	−809.884	+	679.573i	−0.0596698	+	0.0500689i	0.612465	+	0.790498i	$0.290178\pi$
$570$		0			0
$571$	−4496.33	+	25499.9i	−0.329537	+	1.86890i	0.146124	+	0.989266i	$0.453320\pi$
$571$	−4496.33	+	25499.9i	−0.329537	+	1.86890i	−0.475661	+	0.879629i	$0.657791\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−3285.94	−	5691.42i	−0.238319	−	0.412780i
$576$		0			0
$577$	−1489.92	+	2580.62i	−0.107498	+	0.186192i	−0.914756	−	0.404007i	$-0.867617\pi$
$577$	−1489.92	+	2580.62i	−0.107498	+	0.186192i	0.807258	+	0.590199i	$0.200951\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	8079.41	+	6779.43i	0.576920	+	0.484093i
$582$		0			0
$583$	−13920.7	−	5066.72i	−0.988913	−	0.359935i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	3085.89	+	17501.0i	0.216982	+	1.23057i	0.877434	+	0.479698i	$0.159254\pi$
$587$	3085.89	+	17501.0i	0.216982	+	1.23057i	−0.660452	+	0.750868i	$0.729635\pi$
$588$		0			0
$589$	11291.1	−	4109.63i	0.789884	−	0.287494i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	3329.70			0.230581			0.115290	−	0.993332i	$-0.463220\pi$
$593$	3329.70			0.230581			0.115290	+	0.993332i	$0.463220\pi$
$594$		0			0
$595$	2135.42			0.147132
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−12799.7	+	4658.71i	−0.873090	+	0.317779i	−0.739418	−	0.673247i	$-0.764899\pi$
$599$	−12799.7	+	4658.71i	−0.873090	+	0.317779i	−0.133672	+	0.991026i	$0.542677\pi$
$600$		0			0
$601$	447.186	+	2536.12i	0.0303513	+	0.172131i	0.996215	−	0.0869185i	$-0.0277020\pi$
$601$	447.186	+	2536.12i	0.0303513	+	0.172131i	−0.965864	+	0.259049i	$0.916591\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−19972.6	−	7269.45i	−1.34215	−	0.488504i
$606$		0			0
$607$	−8011.80	−	6722.70i	−0.535731	−	0.449532i	0.334344	−	0.942451i	$-0.391485\pi$
$607$	−8011.80	−	6722.70i	−0.535731	−	0.449532i	−0.870075	+	0.492919i	$0.835930\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	2506.05	−	4340.60i	0.165931	−	0.287401i
$612$		0			0
$613$	−12357.4	−	21403.7i	−0.814211	−	1.41025i	−0.909893	−	0.414842i	$-0.863837\pi$
$613$	−12357.4	−	21403.7i	−0.814211	−	1.41025i	0.0956826	−	0.995412i	$-0.469497\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−1578.77	+	8953.67i	−0.103013	+	0.584216i	0.888982	+	0.457942i	$0.151413\pi$
$617$	−1578.77	+	8953.67i	−0.103013	+	0.584216i	−0.991995	+	0.126274i	$0.959698\pi$
$618$		0			0
$619$	7255.84	−	6088.37i	0.471142	−	0.395335i	−0.376069	−	0.926592i	$-0.622725\pi$
$619$	7255.84	−	6088.37i	0.471142	−	0.395335i	0.847211	+	0.531257i	$0.178280\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	3377.89	−	2834.39i	0.217227	−	0.182275i
$624$		0			0
$625$	871.146	−	4940.52i	0.0557534	−	0.316193i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−1699.39	−	2943.43i	−0.107725	−	0.186585i
$630$		0			0
$631$	−7345.25	+	12722.3i	−0.463407	+	0.802644i	−0.999128	−	0.0417505i	$-0.986707\pi$
$631$	−7345.25	+	12722.3i	−0.463407	+	0.802644i	0.535721	+	0.844395i	$0.320040\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	5203.19	+	4365.99i	0.325169	+	0.272849i
$636$		0			0
$637$	−20102.0	−	7316.53i	−1.25035	−	0.455088i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	4526.38	+	25670.4i	0.278910	+	1.58178i	0.726261	+	0.687419i	$0.241256\pi$
$641$	4526.38	+	25670.4i	0.278910	+	1.58178i	−0.447351	+	0.894359i	$0.647632\pi$
$642$		0			0
$643$	2079.34	−	756.817i	0.127529	−	0.0464167i	−0.277467	−	0.960735i	$-0.589495\pi$
$643$	2079.34	−	756.817i	0.127529	−	0.0464167i	0.404996	+	0.914318i	$0.367273\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	13758.7			0.836027			0.418013	−	0.908441i	$-0.362727\pi$
$647$	13758.7			0.836027			0.418013	+	0.908441i	$0.362727\pi$
$648$		0			0
$649$	−42682.0			−2.58154
$650$		0			0
$651$		0			0
$652$		0			0
$653$	15288.2	−	5564.44i	0.916191	−	0.333466i	0.159469	−	0.987203i	$-0.449022\pi$
$653$	15288.2	−	5564.44i	0.916191	−	0.333466i	0.756722	+	0.653737i	$0.226800\pi$
$654$		0			0
$655$	−1081.15	−	6131.53i	−0.0644949	−	0.365769i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	10222.0	+	3720.52i	0.604240	+	0.219925i	0.625981	−	0.779838i	$-0.284699\pi$
$659$	10222.0	+	3720.52i	0.604240	+	0.219925i	−0.0217411	+	0.999764i	$0.506921\pi$
$660$		0			0
$661$	316.290	+	265.399i	0.0186116	+	0.0156170i	0.652046	−	0.758179i	$-0.273911\pi$
$661$	316.290	+	265.399i	0.0186116	+	0.0156170i	−0.633434	+	0.773796i	$0.718355\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−1143.93	+	1981.35i	−0.0667063	+	0.115539i
$666$		0			0
$667$	−6500.75	−	11259.6i	−0.377377	−	0.653635i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−8946.57	+	50738.5i	−0.514722	+	2.91913i
$672$		0			0
$673$	−18608.4	+	15614.3i	−1.06582	+	0.894333i	−0.994668	−	0.103133i	$-0.967113\pi$
$673$	−18608.4	+	15614.3i	−1.06582	+	0.894333i	−0.0711564	+	0.997465i	$0.522669\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	8328.43	−	6988.38i	0.472803	−	0.396729i	−0.375013	−	0.927020i	$-0.622362\pi$
$677$	8328.43	−	6988.38i	0.472803	−	0.396729i	0.847816	+	0.530291i	$0.177917\pi$
$678$		0			0
$679$	1450.95	−	8228.77i	0.0820067	−	0.465083i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−6279.52	−	10876.5i	−0.351800	−	0.609335i	0.634765	−	0.772705i	$-0.281097\pi$
$683$	−6279.52	−	10876.5i	−0.351800	−	0.609335i	−0.986565	+	0.163370i	$0.947764\pi$
$684$		0			0
$685$	−7127.96	+	12346.0i	−0.397585	+	0.688637i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	13034.1	+	10936.9i	0.720698	+	0.604737i
$690$		0			0
$691$	16718.3	+	6084.96i	0.920396	+	0.334997i	0.758396	−	0.651794i	$-0.225983\pi$
$691$	16718.3	+	6084.96i	0.920396	+	0.334997i	0.162000	+	0.986791i	$0.448206\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	2414.65	+	13694.2i	0.131788	+	0.747408i
$696$		0			0
$697$	−11180.5	+	4069.38i	−0.607594	+	0.221146i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−21646.4			−1.16629			−0.583147	−	0.812367i	$-0.698179\pi$
$701$	−21646.4			−1.16629			−0.583147	+	0.812367i	$0.698179\pi$
$702$		0			0
$703$	3641.41			0.195360
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−4576.81	+	1665.82i	−0.243463	+	0.0886135i
$708$		0			0
$709$	3459.28	+	19618.5i	0.183238	+	1.03920i	0.928198	+	0.372086i	$0.121357\pi$
$709$	3459.28	+	19618.5i	0.183238	+	1.03920i	−0.744960	+	0.667109i	$0.767532\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−17492.2	−	6366.64i	−0.918777	−	0.334407i
$714$		0			0
$715$	25202.2	+	21147.1i	1.31819	+	1.10609i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−10932.0	+	18934.8i	−0.567031	+	0.982126i	0.429827	+	0.902911i	$0.358575\pi$
$719$	−10932.0	+	18934.8i	−0.567031	+	0.982126i	−0.996858	+	0.0792148i	$0.974759\pi$
$720$		0			0
$721$	1476.73	+	2557.77i	0.0762779	+	0.132117i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	3047.00	−	17280.4i	0.156087	−	0.885211i
$726$		0			0
$727$	−25330.4	+	21254.7i	−1.29223	+	1.08431i	−0.300799	+	0.953688i	$0.597253\pi$
$727$	−25330.4	+	21254.7i	−1.29223	+	1.08431i	−0.991432	+	0.130623i	$0.958302\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	8562.15	−	7184.49i	0.433218	−	0.363513i
$732$		0			0
$733$	2985.32	−	16930.6i	0.150430	−	0.853131i	−0.812416	−	0.583079i	$-0.801848\pi$
$733$	2985.32	−	16930.6i	0.150430	−	0.853131i	0.962846	−	0.270052i	$-0.0870410\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	1108.59	+	1920.14i	0.0554077	+	0.0959689i
$738$		0			0
$739$	9649.35	−	16713.2i	0.480321	−	0.831940i	−0.519424	−	0.854516i	$-0.673854\pi$
$739$	9649.35	−	16713.2i	0.480321	−	0.831940i	0.999745	+	0.0225766i	$0.00718696\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−20573.9	−	17263.5i	−1.01586	−	0.852405i	−0.0267555	−	0.999642i	$-0.508518\pi$
$743$	−20573.9	−	17263.5i	−1.01586	−	0.852405i	−0.989101	+	0.147237i	$0.952962\pi$
$744$		0			0
$745$	602.767	+	219.389i	0.0296425	+	0.0107890i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	332.707	+	1886.87i	0.0162308	+	0.0920492i
$750$		0			0
$751$	−27645.8	+	10062.2i	−1.34329	+	0.488916i	−0.910846	−	0.412746i	$-0.864570\pi$
$751$	−27645.8	+	10062.2i	−1.34329	+	0.488916i	−0.432440	+	0.901663i	$0.642347\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−8651.72			−0.417044
$756$		0			0
$757$	24191.8			1.16151			0.580756	−	0.814077i	$-0.302757\pi$
$757$	24191.8			1.16151			0.580756	+	0.814077i	$0.302757\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−12610.5	+	4589.85i	−0.600698	+	0.218636i	−0.624428	−	0.781082i	$-0.714668\pi$
$761$	−12610.5	+	4589.85i	−0.600698	+	0.218636i	0.0237304	+	0.999718i	$0.492446\pi$
$762$		0			0
$763$	−311.565	−	1766.97i	−0.0147830	−	0.0838383i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	46066.3	+	16766.8i	2.16866	+	0.789326i
$768$		0			0
$769$	16690.9	+	14005.3i	0.782689	+	0.656754i	0.943924	−	0.330162i	$-0.107103\pi$
$769$	16690.9	+	14005.3i	0.782689	+	0.656754i	−0.161235	+	0.986916i	$0.551548\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	670.497	−	1161.34i	0.0311981	−	0.0540367i	−0.850005	−	0.526775i	$-0.823401\pi$
$773$	670.497	−	1161.34i	0.0311981	−	0.0540367i	0.881203	+	0.472738i	$0.156734\pi$
$774$		0			0
$775$	−12561.4	−	21756.9i	−0.582216	−	1.00843i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	2213.57	−	12553.8i	0.101809	−	0.577389i
$780$		0			0
$781$	−17103.1	+	14351.2i	−0.783607	+	0.657524i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−7749.10	+	6502.26i	−0.352327	+	0.295638i
$786$		0			0
$787$	−508.621	+	2884.53i	−0.0230373	+	0.130651i	−0.994157	−	0.107940i	$-0.965575\pi$
$787$	−508.621	+	2884.53i	−0.0230373	+	0.130651i	0.971120	+	0.238591i	$0.0766856\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−990.373	−	1715.38i	−0.0445178	−	0.0771071i
$792$		0			0
$793$	29587.5	−	51247.1i	1.32495	−	2.29488i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	21377.1	+	17937.5i	0.950082	+	0.797213i	0.979311	−	0.202359i	$-0.0648609\pi$
$797$	21377.1	+	17937.5i	0.950082	+	0.797213i	−0.0292296	+	0.999573i	$0.509305\pi$
$798$		0			0
$799$	−2400.15	−	873.585i	−0.106272	−	0.0386799i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−6582.97	−	37333.9i	−0.289300	−	1.64070i
$804$		0			0
$805$	3330.61	−	1212.24i	0.145824	−	0.0530756i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−12704.1			−0.552104			−0.276052	−	0.961143i	$-0.589026\pi$
$809$	−12704.1			−0.552104			−0.276052	+	0.961143i	$0.589026\pi$
$810$		0			0
$811$	5012.10			0.217014			0.108507	−	0.994096i	$-0.465393\pi$
$811$	5012.10			0.217014			0.108507	+	0.994096i	$0.465393\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−9313.93	+	3389.99i	−0.400310	+	0.145701i
$816$		0			0
$817$	2079.43	+	11793.1i	0.0890455	+	0.505002i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	13568.4	+	4938.51i	0.576787	+	0.209933i	0.613908	−	0.789378i	$-0.289597\pi$
$821$	13568.4	+	4938.51i	0.576787	+	0.209933i	−0.0371213	+	0.999311i	$0.511819\pi$
$822$		0			0
$823$	677.888	+	568.816i	0.0287117	+	0.0240919i	0.657031	−	0.753864i	$-0.271812\pi$
$823$	677.888	+	568.816i	0.0287117	+	0.0240919i	−0.628319	+	0.777956i	$0.716257\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	16997.2	−	29440.1i	0.714693	−	1.23789i	−0.248384	−	0.968662i	$-0.579900\pi$
$827$	16997.2	−	29440.1i	0.714693	−	1.23789i	0.963078	−	0.269224i	$-0.0867671\pi$
$828$		0			0
$829$	2235.23	+	3871.54i	0.0936463	+	0.162200i	0.909043	−	0.416703i	$-0.136814\pi$
$829$	2235.23	+	3871.54i	0.0936463	+	0.162200i	−0.815397	+	0.578903i	$0.803481\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−1893.03	+	10735.9i	−0.0787390	+	0.446551i
$834$		0			0
$835$	4965.12	−	4166.23i	0.205779	−	0.172669i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	4716.06	−	3957.24i	0.194060	−	0.162836i	−0.540580	−	0.841292i	$-0.681795\pi$
$839$	4716.06	−	3957.24i	0.194060	−	0.162836i	0.734640	+	0.678457i	$0.237351\pi$
$840$		0			0
$841$	1792.94	−	10168.3i	0.0735142	−	0.416920i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−12794.6	−	22160.9i	−0.520886	−	0.902201i
$846$		0			0
$847$	−17512.8	+	30333.0i	−0.710444	+	1.23052i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−4321.46	−	3626.14i	−0.174075	−	0.146066i
$852$		0			0
$853$	8837.84	+	3216.71i	0.354750	+	0.129119i	0.513247	−	0.858241i	$-0.328442\pi$
$853$	8837.84	+	3216.71i	0.354750	+	0.129119i	−0.158497	+	0.987360i	$0.550665\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−2250.77	−	12764.8i	−0.0897141	−	0.508794i	−0.996239	−	0.0866426i	$-0.972386\pi$
$857$	−2250.77	−	12764.8i	−0.0897141	−	0.508794i	0.906525	−	0.422151i	$-0.138725\pi$
$858$		0			0
$859$	22836.3	−	8311.73i	0.907059	−	0.330143i	0.153981	−	0.988074i	$-0.450791\pi$
$859$	22836.3	−	8311.73i	0.907059	−	0.330143i	0.753078	+	0.657931i	$0.228568\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−21125.6			−0.833285			−0.416643	−	0.909070i	$-0.636793\pi$
$863$	−21125.6			−0.833285			−0.416643	+	0.909070i	$0.636793\pi$
$864$		0			0
$865$	−23209.9			−0.912323
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−44112.8	+	16055.7i	−1.72201	+	0.626759i
$870$		0			0
$871$	−442.205	−	2507.87i	−0.0172027	−	0.0975614i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	10461.1	+	3807.55i	0.404173	+	0.147107i
$876$		0			0
$877$	22826.9	+	19154.0i	0.878915	+	0.737498i	0.965956	−	0.258707i	$-0.0832965\pi$
$877$	22826.9	+	19154.0i	0.878915	+	0.737498i	−0.0870404	+	0.996205i	$0.527741\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	13196.2	−	22856.4i	0.504643	−	0.874066i	−0.495343	−	0.868697i	$-0.664958\pi$
$881$	13196.2	−	22856.4i	0.504643	−	0.874066i	0.999986	−	0.00536903i	$-0.00170903\pi$
$882$		0			0
$883$	8184.93	+	14176.7i	0.311942	+	0.540299i	0.978783	−	0.204901i	$-0.0656872\pi$
$883$	8184.93	+	14176.7i	0.311942	+	0.540299i	−0.666841	+	0.745200i	$0.732354\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−4816.38	+	27315.0i	−0.182320	+	1.03399i	0.747030	+	0.664791i	$0.231479\pi$
$887$	−4816.38	+	27315.0i	−0.182320	+	1.03399i	−0.929350	+	0.369200i	$0.879632\pi$
$888$		0			0
$889$	8574.42	−	7194.79i	0.323483	−	0.271435i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	2096.30	−	1759.01i	0.0785555	−	0.0659159i
$894$		0			0
$895$	1771.02	−	10044.0i	0.0661439	−	0.375121i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−24850.8	−	43042.9i	−0.921937	−	1.59684i
$900$		0			0
$901$	4335.43	−	7509.18i	0.160304	−	0.277655i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	9845.02	+	8260.96i	0.361613	+	0.303429i
$906$		0			0
$907$	15389.5	+	5601.34i	0.563397	+	0.205060i	0.607989	−	0.793945i	$-0.291976\pi$
$907$	15389.5	+	5601.34i	0.563397	+	0.205060i	−0.0445917	+	0.999005i	$0.514199\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−1424.52	−	8078.83i	−0.0518071	−	0.293813i	0.947885	−	0.318612i	$-0.103217\pi$
$911$	−1424.52	−	8078.83i	−0.0518071	−	0.293813i	−0.999692	+	0.0247988i	$0.992105\pi$
$912$		0			0
$913$	77811.8	−	28321.2i	2.82059	−	1.02661i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−10260.1			−0.369486
$918$		0			0
$919$	−33771.8			−1.21222			−0.606109	−	0.795381i	$-0.707271\pi$
$919$	−33771.8			−1.21222			−0.606109	+	0.795381i	$0.707271\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	24096.8	−	8770.52i	0.859324	−	0.312768i
$924$		0			0
$925$	−1322.07	−	7497.85i	−0.0469941	−	0.266517i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	45473.2	+	16550.9i	1.60595	+	0.584518i	0.980633	−	0.195853i	$-0.0627475\pi$
$929$	45473.2	+	16550.9i	1.60595	+	0.584518i	0.625317	+	0.780371i	$0.284970\pi$
$930$		0			0
$931$	−8947.20	−	7507.60i	−0.314965	−	0.264287i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	8382.77	−	14519.4i	0.293204	−	0.507845i
$936$		0			0
$937$	−20974.0	−	36328.0i	−0.731258	−	1.26658i	−0.956346	−	0.292238i	$-0.905600\pi$
$937$	−20974.0	−	36328.0i	−0.731258	−	1.26658i	0.225087	−	0.974339i	$-0.427733\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	2249.89	−	12759.8i	0.0779431	−	0.442037i	−0.920714	−	0.390237i	$-0.872393\pi$
$941$	2249.89	−	12759.8i	0.0779431	−	0.442037i	0.998657	−	0.0518001i	$-0.0164959\pi$
$942$		0			0
$943$	−15128.1	+	12694.0i	−0.522417	+	0.438360i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	16485.1	−	13832.7i	0.565675	−	0.474658i	−0.314532	−	0.949247i	$-0.601848\pi$
$947$	16485.1	−	13832.7i	0.565675	−	0.474658i	0.880208	+	0.474589i	$0.157403\pi$
$948$		0			0
$949$	−7560.92	+	42880.1i	−0.258628	+	1.46675i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	17377.4	+	30098.5i	0.590669	+	1.02307i	0.994142	+	0.108078i	$0.0344695\pi$
$953$	17377.4	+	30098.5i	0.590669	+	1.02307i	−0.403473	+	0.914991i	$0.632197\pi$
$954$		0			0
$955$	4744.63	−	8217.94i	0.160767	−	0.278457i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	17996.4	+	15100.7i	0.605978	+	0.508476i
$960$		0			0
$961$	−38874.1	−	14149.0i	−1.30489	−	0.474942i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−2989.14	−	16952.3i	−0.0997139	−	0.565506i
$966$		0			0
$967$	17056.1	−	6207.92i	0.567205	−	0.206446i	−0.0424690	−	0.999098i	$-0.513522\pi$
$967$	17056.1	−	6207.92i	0.567205	−	0.206446i	0.609675	+	0.792652i	$0.291300\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	32473.1			1.07324			0.536618	−	0.843825i	$-0.319702\pi$
$971$	32473.1			1.07324			0.536618	+	0.843825i	$0.319702\pi$
$972$		0			0
$973$	22914.9			0.755005
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−2618.73	+	953.139i	−0.0857529	+	0.0312115i	−0.384540	−	0.923108i	$-0.625640\pi$
$977$	−2618.73	+	953.139i	−0.0857529	+	0.0312115i	0.298787	+	0.954320i	$0.403418\pi$
$978$		0			0
$979$	−6011.69	−	34094.0i	−0.196256	−	1.11302i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−54587.2	−	19868.1i	−1.77117	−	0.644654i	−0.999967	−	0.00806713i	$-0.997432\pi$
$983$	−54587.2	−	19868.1i	−1.77117	−	0.644654i	−0.771205	−	0.636587i	$-0.780346\pi$
$984$		0			0
$985$	−2290.59	−	1922.04i	−0.0740958	−	0.0621737i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	9275.82	−	16066.2i	0.298235	−	0.516557i
$990$		0			0
$991$	−16680.9	−	28892.1i	−0.534698	−	0.926124i	−0.999178	−	0.0405401i	$-0.987092\pi$
$991$	−16680.9	−	28892.1i	−0.534698	−	0.926124i	0.464480	−	0.885584i	$-0.346241\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−3561.72	+	20199.5i	−0.113481	+	0.643585i
$996$		0			0
$997$	21033.1	−	17648.8i	0.668128	−	0.560626i	−0.244383	−	0.969679i	$-0.578585\pi$
$997$	21033.1	−	17648.8i	0.668128	−	0.560626i	0.912511	+	0.409053i	$0.134141\pi$
$998$		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.37.4		54
3.2	odd	2		108.4.i.a.49.1	✓	54
27.11	odd	18		108.4.i.a.97.1	yes	54
27.16	even	9	inner	324.4.i.a.289.4		54

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
108.4.i.a.49.1	✓	54	3.2	odd	2
108.4.i.a.97.1	yes	54	27.11	odd	18
324.4.i.a.37.4		54	1.1	even	1	trivial
324.4.i.a.289.4		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)

\(f(q)\)	\(=\)	\(q+(-5.21695 + 1.89881i) q^{5} +(1.58868 + 9.00985i) q^{7} +O(q^{10})\)	\(q+(-5.21695 + 1.89881i) q^{5} +(1.58868 + 9.00985i) q^{7} +(67.4972 + 24.5670i) q^{11} +(-63.1985 - 53.0298i) q^{13} +(-21.0212 + 36.4097i) q^{17} +(-22.5218 - 39.0089i) q^{19} +(-12.1174 + 68.7214i) q^{23} +(-72.1445 + 60.5364i) q^{25} +(-142.727 + 119.762i) q^{29} +(-46.3221 + 262.706i) q^{31} +(-25.3961 - 43.9873i) q^{35} +(-40.4209 + 70.0111i) q^{37} +(216.792 + 181.910i) q^{41} +(-249.820 - 90.9271i) q^{43} +(10.5496 + 59.8299i) q^{47} +(243.661 - 88.6854i) q^{49} -206.241 q^{53} -398.778 q^{55} +(-558.381 + 203.234i) q^{59} +(124.554 + 706.379i) q^{61} +(430.397 + 156.652i) q^{65} +(23.6459 + 19.8412i) q^{67} +(-155.414 + 269.185i) q^{71} +(-263.889 - 457.070i) q^{73} +(-114.113 + 647.169i) q^{77} +(-500.648 + 420.094i) q^{79} +(883.108 - 741.015i) q^{83} +(40.5310 - 229.863i) q^{85} +(-240.988 - 417.404i) q^{89} +(377.389 - 653.656i) q^{91} +(191.566 + 160.743i) q^{95} +(-858.230 - 312.370i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 54 q - 12 q^{5}+O(q^{10}) \) 54 * q - 12 * q^5	\( 54 q - 12 q^{5} + 87 q^{11} - 204 q^{17} - 96 q^{23} - 216 q^{25} - 318 q^{29} - 54 q^{31} - 6 q^{35} - 867 q^{41} - 513 q^{43} + 1548 q^{47} + 594 q^{49} + 1068 q^{53} + 1218 q^{59} - 54 q^{61} - 96 q^{65} - 2997 q^{67} + 120 q^{71} - 216 q^{73} - 3480 q^{77} + 2808 q^{79} - 4464 q^{83} + 2160 q^{85} - 4029 q^{89} + 270 q^{91} + 1650 q^{95} - 3483 q^{97}+O(q^{100}) \) 54 * q - 12 * q^5 + 87 * q^11 - 204 * q^17 - 96 * q^23 - 216 * q^25 - 318 * q^29 - 54 * q^31 - 6 * q^35 - 867 * q^41 - 513 * q^43 + 1548 * q^47 + 594 * q^49 + 1068 * q^53 + 1218 * q^59 - 54 * q^61 - 96 * q^65 - 2997 * q^67 + 120 * q^71 - 216 * q^73 - 3480 * q^77 + 2808 * q^79 - 4464 * q^83 + 2160 * q^85 - 4029 * q^89 + 270 * q^91 + 1650 * q^95 - 3483 * q^97

Introduction

L-functions

Modular forms

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Database

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