LMFDB - Embedded newform 324.3.k.a.197.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [324,3,Mod(17,324)]

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, 11]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("324.17");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 324 = 2^{2} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	324.k (of order $18$, 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:	$8.82836056527$
Analytic rank:	$0$
Dimension:	$36$
Relative dimension:	$6$ over $\Q(\zeta_{18})$
Twist minimal:	no (minimal twist has level 108)
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			197.2
Character	$\chi$	$=$	324.197
Dual form			324.3.k.a.125.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.26650 + 3.47969i) q^{5} +(-0.0728181 + 0.412972i) q^{7} +O(q^{10})$	$q+(-1.26650 + 3.47969i) q^{5} +(-0.0728181 + 0.412972i) q^{7} +(1.69236 + 4.64973i) q^{11} +(3.65864 - 3.06996i) q^{13} +(-20.4937 + 11.8321i) q^{17} +(-13.5371 + 23.4469i) q^{19} +(-20.7690 + 3.66213i) q^{23} +(8.64689 + 7.25561i) q^{25} +(2.12470 - 2.53212i) q^{29} +(3.01492 + 17.0984i) q^{31} +(-1.34479 - 0.776415i) q^{35} +(24.9593 + 43.2309i) q^{37} +(-26.0603 - 31.0575i) q^{41} +(-61.2373 + 22.2885i) q^{43} +(26.9440 + 4.75095i) q^{47} +(45.8797 + 16.6988i) q^{49} +59.1590i q^{53} -18.3230 q^{55} +(16.8092 - 46.1828i) q^{59} +(11.9824 - 67.9556i) q^{61} +(6.04884 + 16.6191i) q^{65} +(56.4874 - 47.3986i) q^{67} +(-88.6346 + 51.1732i) q^{71} +(-3.81627 + 6.60997i) q^{73} +(-2.04344 + 0.360314i) q^{77} +(-99.5742 - 83.5527i) q^{79} +(44.1227 - 52.5834i) q^{83} +(-15.2165 - 86.2972i) q^{85} +(137.829 + 79.5758i) q^{89} +(1.00139 + 1.73447i) q^{91} +(-64.4432 - 76.8005i) q^{95} +(-8.07251 + 2.93815i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 36 q + 9 q^{5}+O(q^{10}) $ 36 * q + 9 * q^5	$ 36 q + 9 q^{5} - 36 q^{11} + 18 q^{23} - 9 q^{25} + 18 q^{29} + 45 q^{31} + 243 q^{35} + 198 q^{41} + 90 q^{43} + 243 q^{47} + 72 q^{49} - 252 q^{59} - 144 q^{61} - 747 q^{65} + 108 q^{67} - 324 q^{71} - 63 q^{73} - 495 q^{77} + 36 q^{79} + 27 q^{83} - 180 q^{85} + 567 q^{89} + 99 q^{91} + 1044 q^{95} - 216 q^{97}+O(q^{100}) $ 36 * q + 9 * q^5 - 36 * q^11 + 18 * q^23 - 9 * q^25 + 18 * q^29 + 45 * q^31 + 243 * q^35 + 198 * q^41 + 90 * q^43 + 243 * q^47 + 72 * q^49 - 252 * q^59 - 144 * q^61 - 747 * q^65 + 108 * q^67 - 324 * q^71 - 63 * q^73 - 495 * q^77 + 36 * q^79 + 27 * q^83 - 180 * q^85 + 567 * q^89 + 99 * q^91 + 1044 * q^95 - 216 * q^97

Display 100 coefficients 10 coefficients

Character values

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

$n$	$163$	$245$
$\chi(n)$	$1$	$e\left(\frac{13}{18}\right)$

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
$5$	−1.26650	+	3.47969i	−0.253301	+	0.695938i	0.746241	+	0.665676i	$0.231857\pi$
$5$	−1.26650	+	3.47969i	−0.253301	+	0.695938i	−0.999542	+	0.0302625i	$0.990366\pi$
$6$		0			0
$7$	−0.0728181	+	0.412972i	−0.0104026	+	0.0589960i	−0.989567	−	0.144072i	$-0.953980\pi$
$7$	−0.0728181	+	0.412972i	−0.0104026	+	0.0589960i	0.979165	+	0.203068i	$0.0650913\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	1.69236	+	4.64973i	0.153851	+	0.422703i	0.992542	−	0.121906i	$-0.0389006\pi$
$11$	1.69236	+	4.64973i	0.153851	+	0.422703i	−0.838691	+	0.544608i	$0.816678\pi$
$12$		0			0
$13$	3.65864	−	3.06996i	0.281434	−	0.236151i	−0.491133	−	0.871085i	$-0.663417\pi$
$13$	3.65864	−	3.06996i	0.281434	−	0.236151i	0.772567	+	0.634934i	$0.218973\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−20.4937	+	11.8321i	−1.20551	+	0.696004i	−0.961776	−	0.273838i	$-0.911707\pi$
$17$	−20.4937	+	11.8321i	−1.20551	+	0.696004i	−0.243738	+	0.969841i	$0.578374\pi$
$18$		0			0
$19$	−13.5371	+	23.4469i	−0.712478	+	1.23405i	0.251446	+	0.967871i	$0.419094\pi$
$19$	−13.5371	+	23.4469i	−0.712478	+	1.23405i	−0.963924	+	0.266177i	$0.914240\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−20.7690	+	3.66213i	−0.902998	+	0.159223i	−0.605822	−	0.795600i	$-0.707156\pi$
$23$	−20.7690	+	3.66213i	−0.902998	+	0.159223i	−0.297176	+	0.954823i	$0.596045\pi$
$24$		0			0
$25$	8.64689	+	7.25561i	0.345876	+	0.290224i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	2.12470	−	2.53212i	0.0732655	−	0.0873144i	−0.728167	−	0.685400i	$-0.759627\pi$
$29$	2.12470	−	2.53212i	0.0732655	−	0.0873144i	0.801432	+	0.598086i	$0.204072\pi$
$30$		0			0
$31$	3.01492	+	17.0984i	0.0972554	+	0.551563i	0.994033	+	0.109081i	$0.0347908\pi$
$31$	3.01492	+	17.0984i	0.0972554	+	0.551563i	−0.896777	+	0.442482i	$0.854098\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−1.34479	−	0.776415i	−0.0384226	−	0.0221833i
$36$		0			0
$37$	24.9593	+	43.2309i	0.674577	+	1.16840i	0.976592	+	0.215098i	$0.0690072\pi$
$37$	24.9593	+	43.2309i	0.674577	+	1.16840i	−0.302016	+	0.953303i	$0.597659\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−26.0603	−	31.0575i	−0.635618	−	0.757500i	0.348053	−	0.937475i	$-0.386843\pi$
$41$	−26.0603	−	31.0575i	−0.635618	−	0.757500i	−0.983671	+	0.179975i	$0.942398\pi$
$42$		0			0
$43$	−61.2373	+	22.2885i	−1.42412	+	0.518338i	−0.935241	−	0.354012i	$-0.884817\pi$
$43$	−61.2373	+	22.2885i	−1.42412	+	0.518338i	−0.488881	+	0.872350i	$0.662595\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	26.9440	+	4.75095i	0.573276	+	0.101084i	0.452768	−	0.891628i	$-0.350436\pi$
$47$	26.9440	+	4.75095i	0.573276	+	0.101084i	0.120508	+	0.992712i	$0.461548\pi$
$48$		0			0
$49$	45.8797	+	16.6988i	0.936320	+	0.340793i
$50$		0			0
$51$		0			0
$52$		0			0
$53$			59.1590i			1.11621i	0.829771	+	0.558104i	$0.188471\pi$
$53$			59.1590i			1.11621i	−0.829771	+	0.558104i	$0.811529\pi$
$54$		0			0
$55$	−18.3230			−0.333145
$56$		0			0
$57$		0			0
$58$		0			0
$59$	16.8092	−	46.1828i	0.284901	−	0.782760i	−0.711858	−	0.702323i	$-0.752146\pi$
$59$	16.8092	−	46.1828i	0.284901	−	0.782760i	0.996760	−	0.0804369i	$-0.0256316\pi$
$60$		0			0
$61$	11.9824	−	67.9556i	0.196433	−	1.11403i	−0.713931	−	0.700216i	$-0.753087\pi$
$61$	11.9824	−	67.9556i	0.196433	−	1.11403i	0.910364	−	0.413809i	$-0.135802\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	6.04884	+	16.6191i	0.0930591	+	0.255678i
$66$		0			0
$67$	56.4874	−	47.3986i	0.843096	−	0.707441i	−0.115162	−	0.993347i	$-0.536739\pi$
$67$	56.4874	−	47.3986i	0.843096	−	0.707441i	0.958258	+	0.285905i	$0.0922943\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−88.6346	+	51.1732i	−1.24837	+	0.720749i	−0.970785	−	0.239951i	$-0.922869\pi$
$71$	−88.6346	+	51.1732i	−1.24837	+	0.720749i	−0.277589	+	0.960700i	$0.589535\pi$
$72$		0			0
$73$	−3.81627	+	6.60997i	−0.0522776	+	0.0905475i	−0.890980	−	0.454043i	$-0.849981\pi$
$73$	−3.81627	+	6.60997i	−0.0522776	+	0.0905475i	0.838702	+	0.544590i	$0.183315\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−2.04344	+	0.360314i	−0.0265382	+	0.00467940i
$78$		0			0
$79$	−99.5742	−	83.5527i	−1.26043	−	1.05763i	−0.995636	−	0.0933246i	$-0.970251\pi$
$79$	−99.5742	−	83.5527i	−1.26043	−	1.05763i	−0.264797	−	0.964304i	$-0.585305\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	44.1227	−	52.5834i	0.531598	−	0.633534i	−0.431684	−	0.902025i	$-0.642080\pi$
$83$	44.1227	−	52.5834i	0.531598	−	0.633534i	0.963282	+	0.268491i	$0.0865248\pi$
$84$		0			0
$85$	−15.2165	−	86.2972i	−0.179018	−	1.01526i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	137.829	+	79.5758i	1.54864	+	0.894110i	0.998246	+	0.0592087i	$0.0188577\pi$
$89$	137.829	+	79.5758i	1.54864	+	0.894110i	0.550399	+	0.834902i	$0.314476\pi$
$90$		0			0
$91$	1.00139	+	1.73447i	0.0110043	+	0.0190601i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−64.4432	−	76.8005i	−0.678350	−	0.808426i
$96$		0			0
$97$	−8.07251	+	2.93815i	−0.0832217	+	0.0302902i	−0.383296	−	0.923626i	$-0.625211\pi$
$97$	−8.07251	+	2.93815i	−0.0832217	+	0.0302902i	0.300074	+	0.953916i	$0.402989\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	177.614	+	31.3181i	1.75855	+	0.310080i	0.957483	−	0.288491i	$-0.0931536\pi$
$101$	177.614	+	31.3181i	1.75855	+	0.310080i	0.801070	+	0.598571i	$0.204265\pi$
$102$		0			0
$103$	53.4283	+	19.4463i	0.518721	+	0.188799i	0.588095	−	0.808792i	$-0.299878\pi$
$103$	53.4283	+	19.4463i	0.518721	+	0.188799i	−0.0693741	+	0.997591i	$0.522100\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	165.438i		−	1.54615i	−0.634315	−	0.773074i	$-0.718718\pi$
$107$		−	165.438i		−	1.54615i	0.634315	−	0.773074i	$-0.281282\pi$
$108$		0			0
$109$	29.6226			0.271767			0.135883	−	0.990725i	$-0.456613\pi$
$109$	29.6226			0.271767			0.135883	+	0.990725i	$0.456613\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	33.8000	−	92.8649i	0.299115	−	0.821813i	−0.695533	−	0.718494i	$-0.744832\pi$
$113$	33.8000	−	92.8649i	0.299115	−	0.821813i	0.994648	−	0.103319i	$-0.0329461\pi$
$114$		0			0
$115$	13.5609	−	76.9077i	0.117921	−	0.668762i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−3.39400	−	9.32493i	−0.0285210	−	0.0783607i
$120$		0			0
$121$	73.9355	−	62.0392i	0.611037	−	0.512721i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−116.371	+	67.1869i	−0.930969	+	0.537495i
$126$		0			0
$127$	−97.3125	+	168.550i	−0.766240	+	1.32717i	0.173348	+	0.984861i	$0.444541\pi$
$127$	−97.3125	+	168.550i	−0.766240	+	1.32717i	−0.939588	+	0.342306i	$0.888792\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	39.4243	−	6.95157i	0.300949	−	0.0530654i	−0.0211346	−	0.999777i	$-0.506728\pi$
$131$	39.4243	−	6.95157i	0.300949	−	0.0530654i	0.322084	+	0.946711i	$0.395617\pi$
$132$		0			0
$133$	−8.69718	−	7.29780i	−0.0653923	−	0.0548707i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	122.095	−	145.507i	0.891201	−	1.06209i	−0.106499	−	0.994313i	$-0.533964\pi$
$137$	122.095	−	145.507i	0.891201	−	1.06209i	0.997700	−	0.0677791i	$-0.0215913\pi$
$138$		0			0
$139$	−9.40172	−	53.3198i	−0.0676383	−	0.383596i	−0.999769	−	0.0214748i	$-0.993164\pi$
$139$	−9.40172	−	53.3198i	−0.0676383	−	0.383596i	0.932131	−	0.362121i	$-0.117947\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	20.4662	+	11.8162i	0.143121	+	0.0826307i
$144$		0			0
$145$	6.12004	+	10.6002i	0.0422072	+	0.0731050i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	20.0605	+	23.9072i	0.134634	+	0.160451i	0.829149	−	0.559028i	$-0.188825\pi$
$149$	20.0605	+	23.9072i	0.134634	+	0.160451i	−0.694515	+	0.719478i	$0.744381\pi$
$150$		0			0
$151$	23.7123	−	8.63059i	0.157035	−	0.0571562i	−0.262306	−	0.964985i	$-0.584483\pi$
$151$	23.7123	−	8.63059i	0.157035	−	0.0571562i	0.419342	+	0.907828i	$0.362261\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−63.3157	−	11.1643i	−0.408488	−	0.0720275i
$156$		0			0
$157$	−74.4482	−	27.0969i	−0.474192	−	0.172592i	0.0938584	−	0.995586i	$-0.470080\pi$
$157$	−74.4482	−	27.0969i	−0.474192	−	0.172592i	−0.568050	+	0.822994i	$0.692302\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	8.84367i		−	0.0549296i
$162$		0			0
$163$	−57.7099			−0.354049			−0.177024	−	0.984206i	$-0.556647\pi$
$163$	−57.7099			−0.354049			−0.177024	+	0.984206i	$0.556647\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−33.8026	+	92.8720i	−0.202411	+	0.556120i	−0.998816	−	0.0486438i	$-0.984510\pi$
$167$	−33.8026	+	92.8720i	−0.202411	+	0.556120i	0.796405	+	0.604763i	$0.206732\pi$
$168$		0			0
$169$	−25.3856	+	143.969i	−0.150210	+	0.851886i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	25.1544	+	69.1111i	0.145401	+	0.399486i	0.990919	−	0.134461i	$-0.0429304\pi$
$173$	25.1544	+	69.1111i	0.145401	+	0.399486i	−0.845518	+	0.533947i	$0.820708\pi$
$174$		0			0
$175$	−3.62601	+	3.04259i	−0.0207201	+	0.0173862i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−149.606	+	86.3751i	−0.835788	+	0.482542i	−0.855830	−	0.517257i	$-0.826953\pi$
$179$	−149.606	+	86.3751i	−0.835788	+	0.482542i	0.0200423	+	0.999799i	$0.493620\pi$
$180$		0			0
$181$	126.841	−	219.695i	0.700778	−	1.21378i	−0.267416	−	0.963581i	$-0.586170\pi$
$181$	126.841	−	219.695i	0.700778	−	1.21378i	0.968194	−	0.250201i	$-0.0804968\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−182.041	+	32.0988i	−0.984006	+	0.173507i
$186$		0			0
$187$	−89.6987	−	75.2661i	−0.479672	−	0.402493i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−197.779	+	235.704i	−1.03549	+	1.23405i	−0.0637623	+	0.997965i	$0.520310\pi$
$191$	−197.779	+	235.704i	−1.03549	+	1.23405i	−0.971732	+	0.236088i	$0.924134\pi$
$192$		0			0
$193$	35.8393	+	203.255i	0.185696	+	1.05313i	0.925058	+	0.379825i	$0.124016\pi$
$193$	35.8393	+	203.255i	0.185696	+	1.05313i	−0.739363	+	0.673307i	$0.764873\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	211.002	+	121.822i	1.07108	+	0.618386i	0.928475	−	0.371394i	$-0.121120\pi$
$197$	211.002	+	121.822i	1.07108	+	0.618386i	0.142601	+	0.989780i	$0.454454\pi$
$198$		0			0
$199$	57.5016	+	99.5957i	0.288953	+	0.500481i	0.973560	−	0.228431i	$-0.0733596\pi$
$199$	57.5016	+	99.5957i	0.288953	+	0.500481i	−0.684607	+	0.728912i	$0.740026\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	0.890977	+	1.06183i	0.00438905	+	0.00523067i
$204$		0			0
$205$	141.076	−	51.3475i	0.688176	−	0.250476i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−131.931	−	23.2631i	−0.631251	−	0.111307i
$210$		0			0
$211$	84.7742	+	30.8553i	0.401773	+	0.146234i	0.535000	−	0.844852i	$-0.320312\pi$
$211$	84.7742	+	30.8553i	0.401773	+	0.146234i	−0.133227	+	0.991086i	$0.542534\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	241.315i		−	1.12240i
$216$		0			0
$217$	−7.28072			−0.0335517
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−38.6552	+	106.204i	−0.174910	+	0.480562i
$222$		0			0
$223$	−33.8615	+	192.038i	−0.151845	+	0.861156i	0.809769	+	0.586749i	$0.199593\pi$
$223$	−33.8615	+	192.038i	−0.151845	+	0.861156i	−0.961614	+	0.274407i	$0.911518\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−11.5741	−	31.7996i	−0.0509873	−	0.140086i	0.911585	−	0.411112i	$-0.134859\pi$
$227$	−11.5741	−	31.7996i	−0.0509873	−	0.140086i	−0.962572	+	0.271026i	$0.912637\pi$
$228$		0			0
$229$	−198.049	+	166.183i	−0.864842	+	0.725689i	−0.963006	−	0.269481i	$-0.913148\pi$
$229$	−198.049	+	166.183i	−0.864842	+	0.725689i	0.0981633	+	0.995170i	$0.468703\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−232.807	+	134.411i	−0.999172	+	0.576872i	−0.908003	−	0.418963i	$-0.862394\pi$
$233$	−232.807	+	134.411i	−0.999172	+	0.576872i	−0.0911690	+	0.995835i	$0.529060\pi$
$234$		0			0
$235$	−50.6565	+	87.7397i	−0.215560	+	0.373360i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	281.988	−	49.7222i	1.17987	−	0.208043i	0.450890	−	0.892580i	$-0.351107\pi$
$239$	281.988	−	49.7222i	1.17987	−	0.208043i	0.728978	+	0.684537i	$0.239996\pi$
$240$		0			0
$241$	203.574	+	170.819i	0.844707	+	0.708793i	0.958617	−	0.284698i	$-0.0918932\pi$
$241$	203.574	+	170.819i	0.844707	+	0.708793i	−0.113910	+	0.993491i	$0.536338\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−116.214	+	138.498i	−0.474341	+	0.565298i
$246$		0			0
$247$	22.4539	+	127.342i	0.0909063	+	0.515555i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	191.060	+	110.308i	0.761195	+	0.439476i	0.829725	−	0.558173i	$-0.188498\pi$
$251$	191.060	+	110.308i	0.761195	+	0.439476i	−0.0685298	+	0.997649i	$0.521831\pi$
$252$		0			0
$253$	−52.1765	−	90.3724i	−0.206231	−	0.357203i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	83.9405	+	100.036i	0.326617	+	0.389246i	0.904217	−	0.427073i	$-0.140455\pi$
$257$	83.9405	+	100.036i	0.326617	+	0.389246i	−0.577601	+	0.816320i	$0.696011\pi$
$258$		0			0
$259$	−19.6706	+	7.15952i	−0.0759484	+	0.0276429i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−367.106	−	64.7307i	−1.39584	−	0.246124i	−0.575408	−	0.817867i	$-0.695157\pi$
$263$	−367.106	−	64.7307i	−1.39584	−	0.246124i	−0.820433	+	0.571742i	$0.806268\pi$
$264$		0			0
$265$	−205.855	−	74.9251i	−0.776811	−	0.282736i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	494.470i		−	1.83818i	−0.394050	−	0.919089i	$-0.628926\pi$
$269$		−	494.470i		−	1.83818i	0.394050	−	0.919089i	$-0.371074\pi$
$270$		0			0
$271$	142.570			0.526089			0.263044	−	0.964784i	$-0.415273\pi$
$271$	142.570			0.526089			0.263044	+	0.964784i	$0.415273\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−19.1029	+	52.4848i	−0.0694651	+	0.190854i
$276$		0			0
$277$	−55.8824	+	316.925i	−0.201742	+	1.14413i	0.700744	+	0.713413i	$0.252852\pi$
$277$	−55.8824	+	316.925i	−0.201742	+	1.14413i	−0.902485	+	0.430721i	$0.858259\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−99.5636	−	273.549i	−0.354319	−	0.973483i	−0.980966	−	0.194180i	$-0.937795\pi$
$281$	−99.5636	−	273.549i	−0.354319	−	0.973483i	0.626647	−	0.779303i	$-0.284427\pi$
$282$		0			0
$283$	−176.969	+	148.495i	−0.625333	+	0.524717i	−0.899475	−	0.436972i	$-0.856051\pi$
$283$	−176.969	+	148.495i	−0.625333	+	0.524717i	0.274142	+	0.961689i	$0.411606\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	14.7236	−	8.50065i	0.0513016	−	0.0296190i
$288$		0			0
$289$	135.495	−	234.685i	0.468842	−	0.812058i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−362.264	+	63.8769i	−1.23640	+	0.218010i	−0.753371	−	0.657596i	$-0.771573\pi$
$293$	−362.264	+	63.8769i	−1.23640	+	0.218010i	−0.483026	+	0.875606i	$0.660462\pi$
$294$		0			0
$295$	139.413	+	116.981i	0.472587	+	0.396547i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−64.7436	+	77.1584i	−0.216534	+	0.258055i
$300$		0			0
$301$	−4.74536	−	26.9123i	−0.0157653	−	0.0894096i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	221.289	+	127.761i	0.725536	+	0.418889i
$306$		0			0
$307$	−194.279	−	336.501i	−0.632831	−	1.09610i	−0.986970	−	0.160903i	$-0.948559\pi$
$307$	−194.279	−	336.501i	−0.632831	−	1.09610i	0.354139	−	0.935193i	$-0.384774\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	298.220	+	355.405i	0.958907	+	1.14278i	0.989686	+	0.143255i	$0.0457569\pi$
$311$	298.220	+	355.405i	0.958907	+	1.14278i	−0.0307784	+	0.999526i	$0.509799\pi$
$312$		0			0
$313$	46.6231	−	16.9694i	0.148956	−	0.0542154i	−0.266466	−	0.963844i	$-0.585856\pi$
$313$	46.6231	−	16.9694i	0.148956	−	0.0542154i	0.415422	+	0.909629i	$0.363634\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	125.557	+	22.1392i	0.396080	+	0.0698396i	0.368141	−	0.929770i	$-0.379994\pi$
$317$	125.557	+	22.1392i	0.396080	+	0.0698396i	0.0279392	+	0.999610i	$0.491106\pi$
$318$		0			0
$319$	15.3694	+	5.59401i	0.0481800	+	0.0175361i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	640.686i		−	1.98355i
$324$		0			0
$325$	53.9103			0.165878
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−3.92402	+	10.7812i	−0.0119271	+	0.0327695i
$330$		0			0
$331$	108.947	−	617.872i	0.329146	−	1.86668i	−0.149626	−	0.988743i	$-0.547807\pi$
$331$	108.947	−	617.872i	0.329146	−	1.86668i	0.478773	−	0.877939i	$-0.341082\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	93.3909	+	256.589i	0.278779	+	0.765938i
$336$		0			0
$337$	−180.611	+	151.551i	−0.535938	+	0.449705i	−0.870146	−	0.492794i	$-0.835976\pi$
$337$	−180.611	+	151.551i	−0.535938	+	0.449705i	0.334208	+	0.942499i	$0.391531\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−74.4008	+	42.9553i	−0.218184	+	0.125969i
$342$		0			0
$343$	−20.5109	+	35.5260i	−0.0597986	+	0.103574i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	669.149	−	117.989i	1.92838	−	0.340026i	0.928847	−	0.370464i	$-0.120801\pi$
$347$	669.149	−	117.989i	1.92838	−	0.340026i	0.999537	+	0.0304385i	$0.00969036\pi$
$348$		0			0
$349$	330.676	+	277.470i	0.947496	+	0.795043i	0.978874	−	0.204464i	$-0.0655452\pi$
$349$	330.676	+	277.470i	0.947496	+	0.795043i	−0.0313783	+	0.999508i	$0.509990\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	198.688	−	236.787i	0.562856	−	0.670786i	−0.407292	−	0.913298i	$-0.633527\pi$
$353$	198.688	−	236.787i	0.562856	−	0.670786i	0.970148	+	0.242512i	$0.0779715\pi$
$354$		0			0
$355$	−65.8109	−	373.232i	−0.185383	−	1.05136i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	110.775	+	63.9560i	0.308566	+	0.178150i	0.646284	−	0.763097i	$-0.276322\pi$
$359$	110.775	+	63.9560i	0.308566	+	0.178150i	−0.337719	+	0.941247i	$0.609655\pi$
$360$		0			0
$361$	−186.005	−	322.170i	−0.515250	−	0.892438i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−18.1673	−	21.6510i	−0.0497735	−	0.0593177i
$366$		0			0
$367$	277.830	−	101.122i	0.757030	−	0.275537i	0.0654694	−	0.997855i	$-0.479146\pi$
$367$	277.830	−	101.122i	0.757030	−	0.275537i	0.691561	+	0.722318i	$0.256923\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−24.4310	−	4.30785i	−0.0658518	−	0.0116114i
$372$		0			0
$373$	−486.381	−	177.028i	−1.30397	−	0.474607i	−0.405683	−	0.914014i	$-0.632966\pi$
$373$	−486.381	−	177.028i	−1.30397	−	0.474607i	−0.898288	+	0.439407i	$0.855189\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	15.7869i		−	0.0418749i
$378$		0			0
$379$	−85.3566			−0.225215			−0.112608	−	0.993640i	$-0.535920\pi$
$379$	−85.3566			−0.225215			−0.112608	+	0.993640i	$0.535920\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−154.048	+	423.243i	−0.402214	+	1.10507i	0.558976	+	0.829184i	$0.311194\pi$
$383$	−154.048	+	423.243i	−0.402214	+	1.10507i	−0.961189	+	0.275889i	$0.911028\pi$
$384$		0			0
$385$	1.33425	−	7.56689i	0.00346558	−	0.0196543i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−77.8458	−	213.880i	−0.200118	−	0.549819i	0.798522	−	0.601966i	$-0.205616\pi$
$389$	−77.8458	−	213.880i	−0.200118	−	0.549819i	−0.998639	+	0.0521470i	$0.983394\pi$
$390$		0			0
$391$	382.303	−	320.790i	0.977757	−	0.820435i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	416.849	−	240.668i	1.05531	−	0.609285i
$396$		0			0
$397$	310.195	−	537.274i	0.781349	−	1.35334i	−0.149808	−	0.988715i	$-0.547865\pi$
$397$	310.195	−	537.274i	0.781349	−	1.35334i	0.931156	−	0.364620i	$-0.118801\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−78.5688	+	13.8538i	−0.195932	+	0.0345481i	−0.270753	−	0.962649i	$-0.587273\pi$
$401$	−78.5688	+	13.8538i	−0.195932	+	0.0345481i	0.0748208	+	0.997197i	$0.476162\pi$
$402$		0			0
$403$	63.5221	+	53.3014i	0.157623	+	0.132262i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−158.771	+	189.216i	−0.390102	+	0.464905i
$408$		0			0
$409$	−50.6875	−	287.463i	−0.123930	−	0.702844i	−0.981937	−	0.189206i	$-0.939409\pi$
$409$	−50.6875	−	287.463i	−0.123930	−	0.702844i	0.858007	−	0.513638i	$-0.171703\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	17.8482	+	10.3047i	0.0432160	+	0.0249508i
$414$		0			0
$415$	127.092	+	220.130i	0.306246	+	0.530434i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	155.180	+	184.937i	0.370359	+	0.441376i	0.918747	−	0.394848i	$-0.129203\pi$
$419$	155.180	+	184.937i	0.370359	+	0.441376i	−0.548388	+	0.836224i	$0.684758\pi$
$420$		0			0
$421$	−227.236	+	82.7073i	−0.539754	+	0.196454i	−0.597488	−	0.801878i	$-0.703835\pi$
$421$	−227.236	+	82.7073i	−0.539754	+	0.196454i	0.0577343	+	0.998332i	$0.481612\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−263.056	−	46.3838i	−0.618955	−	0.109138i
$426$		0			0
$427$	27.1912	+	9.89679i	0.0636797	+	0.0231775i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	173.287i		−	0.402059i	−0.979585	−	0.201029i	$-0.935571\pi$
$431$		−	173.287i		−	0.402059i	0.979585	−	0.201029i	$-0.0644287\pi$
$432$		0			0
$433$	−277.794			−0.641556			−0.320778	−	0.947154i	$-0.603944\pi$
$433$	−277.794			−0.641556			−0.320778	+	0.947154i	$0.603944\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	195.285	−	536.542i	0.446878	−	1.22779i
$438$		0			0
$439$	8.10666	−	45.9751i	0.0184662	−	0.104727i	−0.974182	−	0.225766i	$-0.927512\pi$
$439$	8.10666	−	45.9751i	0.0184662	−	0.104727i	0.992648	+	0.121039i	$0.0386226\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	13.9231	+	38.2535i	0.0314292	+	0.0863511i	0.954415	−	0.298483i	$-0.0964808\pi$
$443$	13.9231	+	38.2535i	0.0314292	+	0.0863511i	−0.922986	+	0.384834i	$0.874259\pi$
$444$		0			0
$445$	−451.461	+	378.821i	−1.01452	+	0.851282i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	59.6506	−	34.4393i	0.132852	−	0.0767022i	−0.432101	−	0.901825i	$-0.642228\pi$
$449$	59.6506	−	34.4393i	0.132852	−	0.0767022i	0.564953	+	0.825123i	$0.308894\pi$
$450$		0			0
$451$	100.305	−	173.734i	0.222407	−	0.385220i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−7.30368	+	1.28784i	−0.0160520	+	0.00283041i
$456$		0			0
$457$	−285.792	−	239.808i	−0.625365	−	0.524744i	0.274120	−	0.961696i	$-0.411614\pi$
$457$	−285.792	−	239.808i	−0.625365	−	0.524744i	−0.899485	+	0.436952i	$0.856058\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	167.368	−	199.462i	0.363055	−	0.432672i	−0.553335	−	0.832959i	$-0.686645\pi$
$461$	167.368	−	199.462i	0.363055	−	0.432672i	0.916390	+	0.400287i	$0.131090\pi$
$462$		0			0
$463$	146.176	+	829.003i	0.315714	+	1.79050i	0.568188	+	0.822899i	$0.307645\pi$
$463$	146.176	+	829.003i	0.315714	+	1.79050i	−0.252474	+	0.967604i	$0.581244\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	43.5511	+	25.1442i	0.0932572	+	0.0538421i	0.545903	−	0.837848i	$-0.316187\pi$
$467$	43.5511	+	25.1442i	0.0932572	+	0.0538421i	−0.452646	+	0.891690i	$0.649520\pi$
$468$		0			0
$469$	15.4610	+	26.7792i	0.0329658	+	0.0570985i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−207.271	−	247.016i	−0.438206	−	0.522233i
$474$		0			0
$475$	−287.175	+	104.523i	−0.604579	+	0.220049i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	285.923	+	50.4159i	0.596917	+	0.105252i	0.463940	−	0.885867i	$-0.346435\pi$
$479$	285.923	+	50.4159i	0.596917	+	0.105252i	0.132977	+	0.991119i	$0.457546\pi$
$480$		0			0
$481$	224.034	+	81.5419i	0.465768	+	0.169526i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	31.8110i		−	0.0655897i
$486$		0			0
$487$	211.322			0.433925			0.216963	−	0.976180i	$-0.430385\pi$
$487$	211.322			0.433925			0.216963	+	0.976180i	$0.430385\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	67.3176	−	184.954i	0.137103	−	0.376688i	−0.852073	−	0.523424i	$-0.824655\pi$
$491$	67.3176	−	184.954i	0.137103	−	0.376688i	0.989176	+	0.146736i	$0.0468768\pi$
$492$		0			0
$493$	−13.5828	+	77.0321i	−0.0275514	+	0.156252i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−14.6789	−	40.3299i	−0.0295350	−	0.0811468i
$498$		0			0
$499$	373.550	−	313.445i	0.748596	−	0.628147i	−0.186535	−	0.982448i	$-0.559726\pi$
$499$	373.550	−	313.445i	0.748596	−	0.628147i	0.935131	+	0.354301i	$0.115281\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−829.430	+	478.871i	−1.64897	+	0.952031i	−0.671481	+	0.741022i	$0.734342\pi$
$503$	−829.430	+	478.871i	−1.64897	+	0.952031i	−0.977484	+	0.211009i	$0.932325\pi$
$504$		0			0
$505$	−333.926	+	578.377i	−0.661239	+	1.14530i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−92.3684	+	16.2870i	−0.181470	+	0.0319981i	−0.263645	−	0.964620i	$-0.584925\pi$
$509$	−92.3684	+	16.2870i	−0.181470	+	0.0319981i	0.0821744	+	0.996618i	$0.473814\pi$
$510$		0			0
$511$	−2.45184	−	2.05734i	−0.00479812	−	0.00402610i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−135.334	+	161.285i	−0.262785	+	0.313175i
$516$		0			0
$517$	23.5084	+	133.323i	0.0454707	+	0.257877i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	463.569	+	267.642i	0.889767	+	0.513707i	0.873866	−	0.486166i	$-0.161605\pi$
$521$	463.569	+	267.642i	0.889767	+	0.513707i	0.0159010	+	0.999874i	$0.494938\pi$
$522$		0			0
$523$	182.495	+	316.090i	0.348939	+	0.604379i	0.986061	−	0.166383i	$-0.0532088\pi$
$523$	182.495	+	316.090i	0.348939	+	0.604379i	−0.637123	+	0.770763i	$0.719875\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−264.097	−	314.738i	−0.501132	−	0.597226i
$528$		0			0
$529$	−79.1589	+	28.8115i	−0.149639	+	0.0544641i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−190.691	−	33.6240i	−0.357769	−	0.0630843i
$534$		0			0
$535$	575.673	+	209.528i	1.07602	+	0.391641i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			241.589i			0.448216i
$540$		0			0
$541$	177.907			0.328848			0.164424	−	0.986390i	$-0.447423\pi$
$541$	177.907			0.328848			0.164424	+	0.986390i	$0.447423\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−37.5171	+	103.077i	−0.0688387	+	0.189133i
$546$		0			0
$547$	−25.3653	+	143.854i	−0.0463717	+	0.262987i	−0.999175	−	0.0405999i	$-0.987073\pi$
$547$	−25.3653	+	143.854i	−0.0463717	+	0.262987i	0.952804	+	0.303587i	$0.0981842\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	30.6081	+	84.0951i	0.0555501	+	0.152623i
$552$		0			0
$553$	41.7557	−	35.0372i	0.0755077	−	0.0633584i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−310.962	+	179.534i	−0.558280	+	0.322323i	−0.752455	−	0.658644i	$-0.771130\pi$
$557$	−310.962	+	179.534i	−0.558280	+	0.322323i	0.194175	+	0.980967i	$0.437797\pi$
$558$		0			0
$559$	−155.620	+	269.542i	−0.278390	+	0.482186i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−88.7692	+	15.6524i	−0.157672	+	0.0278018i	−0.251927	−	0.967746i	$-0.581064\pi$
$563$	−88.7692	+	15.6524i	−0.157672	+	0.0278018i	0.0942550	+	0.995548i	$0.469953\pi$
$564$		0			0
$565$	280.333	+	235.227i	0.496165	+	0.416332i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−55.8825	+	66.5981i	−0.0982117	+	0.117044i	−0.812912	−	0.582387i	$-0.802119\pi$
$569$	−55.8825	+	66.5981i	−0.0982117	+	0.117044i	0.714700	+	0.699431i	$0.246563\pi$
$570$		0			0
$571$	−53.7562	−	304.867i	−0.0941440	−	0.533917i	−0.995006	−	0.0998121i	$-0.968176\pi$
$571$	−53.7562	−	304.867i	−0.0941440	−	0.533917i	0.900862	−	0.434105i	$-0.142935\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−206.158	−	119.025i	−0.358536	−	0.207001i
$576$		0			0
$577$	−0.428109	−	0.741507i	−0.000741957	−	0.00128511i	0.865654	−	0.500642i	$-0.166903\pi$
$577$	−0.428109	−	0.741507i	−0.000741957	−	0.00128511i	−0.866396	+	0.499357i	$0.833570\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	18.5025	+	22.0505i	0.0318460	+	0.0379526i
$582$		0			0
$583$	−275.073	+	100.118i	−0.471824	+	0.171730i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	145.579	+	25.6695i	0.248005	+	0.0437300i	0.296269	−	0.955105i	$-0.404258\pi$
$587$	145.579	+	25.6695i	0.248005	+	0.0437300i	−0.0482634	+	0.998835i	$0.515369\pi$
$588$		0			0
$589$	−441.719	−	160.773i	−0.749947	−	0.272958i
$590$		0			0
$591$		0			0
$592$		0			0
$593$			1100.38i			1.85562i	0.373053	+	0.927810i	$0.378311\pi$
$593$			1100.38i			1.85562i	−0.373053	+	0.927810i	$0.621689\pi$
$594$		0			0
$595$	36.7464			0.0617586
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−278.390	+	764.871i	−0.464758	+	1.27691i	0.457110	+	0.889410i	$0.348884\pi$
$599$	−278.390	+	764.871i	−0.464758	+	1.27691i	−0.921869	+	0.387503i	$0.873338\pi$
$600$		0			0
$601$	29.8172	−	169.102i	0.0496127	−	0.281367i	−0.949901	−	0.312551i	$-0.898817\pi$
$601$	29.8172	−	169.102i	0.0496127	−	0.281367i	0.999514	+	0.0311835i	$0.00992763\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	122.238	+	335.846i	0.202046	+	0.555117i
$606$		0			0
$607$	228.821	−	192.003i	0.376970	−	0.316315i	−0.434542	−	0.900652i	$-0.643090\pi$
$607$	228.821	−	192.003i	0.376970	−	0.316315i	0.811512	+	0.584336i	$0.198645\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	113.164	−	65.3351i	0.185211	−	0.106931i
$612$		0			0
$613$	−531.261	+	920.171i	−0.866657	+	1.50109i	−0.00126544	+	0.999999i	$0.500403\pi$
$613$	−531.261	+	920.171i	−0.866657	+	1.50109i	−0.865392	+	0.501096i	$0.832931\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−1155.37	+	203.722i	−1.87255	+	0.330182i	−0.990116	−	0.140248i	$-0.955210\pi$
$617$	−1155.37	+	203.722i	−1.87255	+	0.330182i	−0.882437	+	0.470430i	$0.844099\pi$
$618$		0			0
$619$	−830.742	−	697.075i	−1.34207	−	1.12613i	−0.981089	−	0.193559i	$-0.937997\pi$
$619$	−830.742	−	697.075i	−1.34207	−	1.12613i	−0.360983	−	0.932572i	$-0.617559\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−42.8991	+	51.1251i	−0.0688589	+	0.0820628i
$624$		0			0
$625$	−37.4029	−	212.122i	−0.0598446	−	0.339395i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−1023.02	−	590.641i	−1.62642	−	0.939016i
$630$		0			0
$631$	224.052	+	388.070i	0.355075	+	0.615008i	0.987131	−	0.159915i	$-0.0511220\pi$
$631$	224.052	+	388.070i	0.355075	+	0.615008i	−0.632056	+	0.774923i	$0.717789\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−463.256	−	552.087i	−0.729537	−	0.869428i
$636$		0			0
$637$	219.122	−	79.7540i	0.343991	−	0.125202i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	1055.80	+	186.166i	1.64711	+	0.290430i	0.918773	−	0.394787i	$-0.129182\pi$
$641$	1055.80	+	186.166i	1.64711	+	0.290430i	0.728339	+	0.685217i	$0.240293\pi$
$642$		0			0
$643$	567.428	+	206.527i	0.882469	+	0.321193i	0.743206	−	0.669063i	$-0.233304\pi$
$643$	567.428	+	206.527i	0.882469	+	0.321193i	0.139263	+	0.990255i	$0.455527\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			666.631i			1.03034i	0.857087	+	0.515171i	$0.172272\pi$
$647$			666.631i			1.03034i	−0.857087	+	0.515171i	$0.827728\pi$
$648$		0			0
$649$	243.185			0.374707
$650$		0			0
$651$		0			0
$652$		0			0
$653$	331.003	−	909.424i	0.506896	−	1.39269i	−0.377526	−	0.925999i	$-0.623225\pi$
$653$	331.003	−	909.424i	0.506896	−	1.39269i	0.884422	−	0.466687i	$-0.154553\pi$
$654$		0			0
$655$	−25.7417	+	145.989i	−0.0393004	+	0.222883i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	80.4022	+	220.903i	0.122006	+	0.335210i	0.985628	−	0.168931i	$-0.0540315\pi$
$659$	80.4022	+	220.903i	0.122006	+	0.335210i	−0.863621	+	0.504141i	$0.831809\pi$
$660$		0			0
$661$	217.996	−	182.920i	0.329797	−	0.276732i	−0.462820	−	0.886452i	$-0.653162\pi$
$661$	217.996	−	182.920i	0.329797	−	0.276732i	0.792617	+	0.609720i	$0.208718\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	36.4091	−	21.0208i	0.0547505	−	0.0316102i
$666$		0			0
$667$	−34.8548	+	60.3703i	−0.0522561	+	0.0905103i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	336.253	−	59.2906i	0.501123	−	0.0883615i
$672$		0			0
$673$	−456.711	−	383.226i	−0.678620	−	0.569430i	0.236983	−	0.971514i	$-0.423841\pi$
$673$	−456.711	−	383.226i	−0.678620	−	0.569430i	−0.915603	+	0.402084i	$0.868286\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	193.259	−	230.317i	0.285464	−	0.340203i	−0.604188	−	0.796842i	$-0.706502\pi$
$677$	193.259	−	230.317i	0.285464	−	0.340203i	0.889652	+	0.456639i	$0.150947\pi$
$678$		0			0
$679$	−0.625550	−	3.54767i	−0.000921282	−	0.00522485i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−690.441	−	398.626i	−1.01089	−	0.583640i	−0.0994413	−	0.995043i	$-0.531706\pi$
$683$	−690.441	−	398.626i	−1.01089	−	0.583640i	−0.911454	+	0.411403i	$0.865039\pi$
$684$		0			0
$685$	351.685	+	609.136i	0.513408	+	0.889249i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	181.616	+	216.442i	0.263594	+	0.314139i
$690$		0			0
$691$	571.492	−	208.006i	0.827050	−	0.301022i	0.106403	−	0.994323i	$-0.466067\pi$
$691$	571.492	−	208.006i	0.827050	−	0.301022i	0.720648	+	0.693301i	$0.243845\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	197.444	+	34.8147i	0.284092	+	0.0500931i
$696$		0			0
$697$	901.548	+	328.137i	1.29347	+	0.470784i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			722.930i			1.03128i	0.856804	+	0.515642i	$0.172446\pi$
$701$			722.930i			1.03128i	−0.856804	+	0.515642i	$0.827554\pi$
$702$		0			0
$703$	−1351.51			−1.92248
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−25.8670	+	71.0690i	−0.0365870	+	0.100522i
$708$		0			0
$709$	−109.152	+	619.033i	−0.153952	+	0.873108i	0.805785	+	0.592208i	$0.201744\pi$
$709$	−109.152	+	619.033i	−0.153952	+	0.873108i	−0.959737	+	0.280899i	$0.909367\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−125.233	−	344.076i	−0.175643	−	0.482575i
$714$		0			0
$715$	−67.0373	+	56.2510i	−0.0937584	+	0.0786727i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	594.383	−	343.167i	0.826680	−	0.477284i	−0.0260347	−	0.999661i	$-0.508288\pi$
$719$	594.383	−	343.167i	0.826680	−	0.477284i	0.852714	+	0.522377i	$0.174955\pi$
$720$		0			0
$721$	−11.9213	+	20.6484i	−0.0165344	+	0.0286385i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	36.7441	−	6.47897i	0.0506815	−	0.00893651i
$726$		0			0
$727$	−178.673	−	149.924i	−0.245767	−	0.206223i	0.511580	−	0.859236i	$-0.329060\pi$
$727$	−178.673	−	149.924i	−0.245767	−	0.206223i	−0.757347	+	0.653012i	$0.773505\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	991.261	−	1181.34i	1.35603	−	1.61606i
$732$		0			0
$733$	−32.0772	−	181.919i	−0.0437615	−	0.248184i	0.955078	−	0.296356i	$-0.0957715\pi$
$733$	−32.0772	−	181.919i	−0.0437615	−	0.248184i	−0.998839	+	0.0481722i	$0.984660\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	315.988	+	182.436i	0.428749	+	0.247538i
$738$		0			0
$739$	117.640	+	203.758i	0.159188	+	0.275721i	0.934576	−	0.355763i	$-0.115779\pi$
$739$	117.640	+	203.758i	0.159188	+	0.275721i	−0.775388	+	0.631485i	$0.782446\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−808.583	−	963.631i	−1.08827	−	1.29695i	−0.951941	−	0.306283i	$-0.900915\pi$
$743$	−808.583	−	963.631i	−1.08827	−	1.29695i	−0.136327	−	0.990664i	$-0.543530\pi$
$744$		0			0
$745$	−108.596	+	39.5258i	−0.145767	+	0.0530547i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	68.3213	+	12.0469i	0.0912166	+	0.0160840i
$750$		0			0
$751$	−611.586	−	222.599i	−0.814363	−	0.296404i	−0.0989379	−	0.995094i	$-0.531545\pi$
$751$	−611.586	−	222.599i	−0.814363	−	0.296404i	−0.715425	+	0.698690i	$0.753767\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			93.4423i			0.123765i
$756$		0			0
$757$	802.442			1.06003			0.530015	−	0.847988i	$-0.322186\pi$
$757$	802.442			1.06003			0.530015	+	0.847988i	$0.322186\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	242.179	−	665.380i	0.318237	−	0.874350i	−0.672687	−	0.739927i	$-0.734860\pi$
$761$	242.179	−	665.380i	0.318237	−	0.874350i	0.990924	−	0.134422i	$-0.0429179\pi$
$762$		0			0
$763$	−2.15706	+	12.2333i	−0.00282708	+	0.0160332i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−80.2809	−	220.570i	−0.104669	−	0.287575i
$768$		0			0
$769$	840.006	−	704.848i	1.09233	−	0.916578i	0.0954490	−	0.995434i	$-0.469571\pi$
$769$	840.006	−	704.848i	1.09233	−	0.916578i	0.996886	+	0.0788564i	$0.0251269\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	218.368	−	126.075i	0.282495	−	0.163098i	−0.352058	−	0.935978i	$-0.614518\pi$
$773$	218.368	−	126.075i	0.282495	−	0.163098i	0.634552	+	0.772880i	$0.281185\pi$
$774$		0			0
$775$	−97.9899	+	169.723i	−0.126439	+	0.218998i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	1080.98	−	190.607i	1.38766	−	0.244681i
$780$		0			0
$781$	−387.943	−	325.523i	−0.496726	−	0.416803i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	188.578	−	224.738i	0.240226	−	0.286291i
$786$		0			0
$787$	−126.861	−	719.465i	−0.161196	−	0.914187i	−0.952901	−	0.303283i	$-0.901917\pi$
$787$	−126.861	−	719.465i	−0.161196	−	0.914187i	0.791705	−	0.610904i	$-0.209194\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	35.8893	+	20.7207i	0.0453721	+	0.0261956i
$792$		0			0
$793$	−164.782	−	285.411i	−0.207796	−	0.359912i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	749.785	+	893.559i	0.940759	+	1.12115i	0.992470	+	0.122491i	$0.0390881\pi$
$797$	749.785	+	893.559i	0.940759	+	1.12115i	−0.0517109	+	0.998662i	$0.516467\pi$
$798$		0			0
$799$	−608.396	+	221.438i	−0.761447	+	0.277144i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−37.1930	−	6.55814i	−0.0463176	−	0.00816705i
$804$		0			0
$805$	30.7732	+	11.2005i	0.0382276	+	0.0139137i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			114.072i			0.141004i	0.997512	+	0.0705020i	$0.0224601\pi$
$809$			114.072i			0.141004i	−0.997512	+	0.0705020i	$0.977540\pi$
$810$		0			0
$811$	−342.119			−0.421849			−0.210924	−	0.977502i	$-0.567647\pi$
$811$	−342.119			−0.421849			−0.210924	+	0.977502i	$0.567647\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	73.0898	−	200.813i	0.0896808	−	0.246396i
$816$		0			0
$817$	306.376	−	1737.55i	0.375002	−	2.12674i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	313.702	+	861.889i	0.382098	+	1.04980i	0.970472	+	0.241216i	$0.0775461\pi$
$821$	313.702	+	861.889i	0.382098	+	1.04980i	−0.588374	+	0.808589i	$0.700232\pi$
$822$		0			0
$823$	−555.765	+	466.342i	−0.675292	+	0.566637i	−0.914626	−	0.404300i	$-0.867515\pi$
$823$	−555.765	+	466.342i	−0.675292	+	0.566637i	0.239335	+	0.970937i	$0.423071\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	407.878	−	235.488i	0.493202	−	0.284750i	−0.232700	−	0.972549i	$-0.574756\pi$
$827$	407.878	−	235.488i	0.493202	−	0.284750i	0.725902	+	0.687798i	$0.241423\pi$
$828$		0			0
$829$	781.648	−	1353.85i	0.942881	−	1.63312i	0.182940	−	0.983124i	$-0.441438\pi$
$829$	781.648	−	1353.85i	0.942881	−	1.63312i	0.759940	−	0.649993i	$-0.225228\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−1137.83	+	200.630i	−1.36594	+	0.240852i
$834$		0			0
$835$	−280.355	−	235.245i	−0.335754	−	0.281731i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−546.909	+	651.781i	−0.651858	+	0.776854i	−0.986193	−	0.165600i	$-0.947044\pi$
$839$	−546.909	+	651.781i	−0.651858	+	0.776854i	0.334335	+	0.942454i	$0.391488\pi$
$840$		0			0
$841$	144.141	+	817.463i	0.171392	+	0.972014i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−468.816	−	270.671i	−0.554811	−	0.320321i
$846$		0			0
$847$	20.2366	+	35.0509i	0.0238921	+	0.0413824i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−676.696	−	806.455i	−0.795178	−	0.947656i
$852$		0			0
$853$	−59.6797	+	21.7216i	−0.0699644	+	0.0254650i	−0.376765	−	0.926309i	$-0.622964\pi$
$853$	−59.6797	+	21.7216i	−0.0699644	+	0.0254650i	0.306801	+	0.951774i	$0.400741\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	134.075	+	23.6410i	0.156447	+	0.0275858i	0.251323	−	0.967903i	$-0.419134\pi$
$857$	134.075	+	23.6410i	0.156447	+	0.0275858i	−0.0948762	+	0.995489i	$0.530246\pi$
$858$		0			0
$859$	−228.573	−	83.1938i	−0.266092	−	0.0968496i	0.205529	−	0.978651i	$-0.434109\pi$
$859$	−228.573	−	83.1938i	−0.266092	−	0.0968496i	−0.471621	+	0.881802i	$0.656331\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			1043.35i			1.20898i	0.796612	+	0.604490i	$0.206623\pi$
$863$			1043.35i			1.20898i	−0.796612	+	0.604490i	$0.793377\pi$
$864$		0			0
$865$	−272.343			−0.314848
$866$		0			0
$867$		0			0
$868$		0			0
$869$	219.982	−	604.394i	0.253143	−	0.695506i
$870$		0			0
$871$	61.1553	−	346.829i	0.0702127	−	0.398196i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−19.2724	−	52.9504i	−0.0220256	−	0.0605148i
$876$		0			0
$877$	1205.65	−	1011.66i	1.37475	−	1.15355i	0.403637	−	0.914919i	$-0.367746\pi$
$877$	1205.65	−	1011.66i	1.37475	−	1.15355i	0.971110	−	0.238631i	$-0.0766986\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	1033.74	−	596.831i	1.17337	−	0.677447i	0.218901	−	0.975747i	$-0.429753\pi$
$881$	1033.74	−	596.831i	1.17337	−	0.677447i	0.954472	+	0.298300i	$0.0964195\pi$
$882$		0			0
$883$	139.976	−	242.446i	0.158523	−	0.274570i	−0.775813	−	0.630963i	$-0.782660\pi$
$883$	139.976	−	242.446i	0.158523	−	0.274570i	0.934336	+	0.356392i	$0.115993\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	279.453	−	49.2751i	0.315054	−	0.0555526i	−0.0138851	−	0.999904i	$-0.504420\pi$
$887$	279.453	−	49.2751i	0.315054	−	0.0555526i	0.328939	+	0.944351i	$0.393309\pi$
$888$		0			0
$889$	−62.5204	−	52.4609i	−0.0703267	−	0.0590111i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−476.138	+	567.439i	−0.533189	+	0.635430i
$894$		0			0
$895$	−111.082	−	629.977i	−0.124114	−	0.703885i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	49.7010	+	28.6949i	0.0552848	+	0.0319187i
$900$		0			0
$901$	−699.973	−	1212.39i	−0.776884	−	1.34560i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	603.825	+	719.611i	0.667210	+	0.795150i
$906$		0			0
$907$	−92.0530	+	33.5046i	−0.101492	+	0.0369400i	−0.392267	−	0.919851i	$-0.628309\pi$
$907$	−92.0530	+	33.5046i	−0.101492	+	0.0369400i	0.290775	+	0.956791i	$0.406087\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	400.721	+	70.6580i	0.439870	+	0.0775609i	0.389197	−	0.921154i	$-0.372752\pi$
$911$	400.721	+	70.6580i	0.439870	+	0.0775609i	0.0506725	+	0.998715i	$0.483864\pi$
$912$		0			0
$913$	319.170	+	116.168i	0.349584	+	0.127238i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			16.7873i			0.0183068i
$918$		0			0
$919$	1484.71			1.61557			0.807784	−	0.589479i	$-0.200667\pi$
$919$	1484.71			1.61557			0.807784	+	0.589479i	$0.200667\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−167.182	+	459.329i	−0.181129	+	0.497648i
$924$		0			0
$925$	−97.8452	+	554.908i	−0.105779	+	0.599900i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−186.754	−	513.101i	−0.201026	−	0.552316i	0.797684	−	0.603075i	$-0.206058\pi$
$929$	−186.754	−	513.101i	−0.201026	−	0.552316i	−0.998711	+	0.0507594i	$0.983836\pi$
$930$		0			0
$931$	−1012.61	+	849.684i	−1.08766	+	0.912657i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	375.507	−	216.799i	0.401611	−	0.231870i
$936$		0			0
$937$	−730.836	+	1265.84i	−0.779974	+	1.35095i	0.151982	+	0.988383i	$0.451435\pi$
$937$	−730.836	+	1265.84i	−0.779974	+	1.35095i	−0.931956	+	0.362572i	$0.881899\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	1339.88	−	236.256i	1.42388	−	0.251069i	0.591964	−	0.805964i	$-0.298353\pi$
$941$	1339.88	−	236.256i	1.42388	−	0.251069i	0.831921	+	0.554895i	$0.187241\pi$
$942$		0			0
$943$	654.983	+	549.596i	0.694574	+	0.582816i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	339.126	−	404.154i	0.358105	−	0.426773i	−0.556672	−	0.830733i	$-0.687922\pi$
$947$	339.126	−	404.154i	0.358105	−	0.426773i	0.914777	+	0.403959i	$0.132366\pi$
$948$		0			0
$949$	6.33001	+	35.8993i	0.00667019	+	0.0378285i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−671.482	−	387.680i	−0.704598	−	0.406800i	0.104460	−	0.994529i	$-0.466689\pi$
$953$	−671.482	−	387.680i	−0.704598	−	0.406800i	−0.809058	+	0.587729i	$0.800022\pi$
$954$		0			0
$955$	−569.690	−	986.731i	−0.596534	−	1.03323i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	51.1995	+	61.0172i	0.0533884	+	0.0636258i
$960$		0			0
$961$	619.778	−	225.581i	0.644930	−	0.234735i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−752.654	−	132.713i	−0.779952	−	0.137527i
$966$		0			0
$967$	−1178.44	−	428.916i	−1.21865	−	0.443553i	−0.348956	−	0.937139i	$-0.613464\pi$
$967$	−1178.44	−	428.916i	−1.21865	−	0.443553i	−0.869697	+	0.493586i	$0.835686\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			1101.11i			1.13400i	0.823719	+	0.566998i	$0.191895\pi$
$971$			1101.11i			1.13400i	−0.823719	+	0.566998i	$0.808105\pi$
$972$		0			0
$973$	22.7042			0.0233342
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−450.997	+	1239.11i	−0.461615	+	1.26828i	0.462656	+	0.886538i	$0.346896\pi$
$977$	−450.997	+	1239.11i	−0.461615	+	1.26828i	−0.924271	+	0.381738i	$0.875326\pi$
$978$		0			0
$979$	−136.749	+	775.540i	−0.139682	+	0.792176i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−545.281	−	1498.15i	−0.554711	−	1.52406i	−0.827206	−	0.561898i	$-0.810071\pi$
$983$	−545.281	−	1498.15i	−0.554711	−	1.52406i	0.272496	−	0.962157i	$-0.412151\pi$
$984$		0			0
$985$	−691.138	+	579.934i	−0.701663	+	0.588765i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	1190.21	−	687.168i	1.20345	−	0.694811i
$990$		0			0
$991$	216.060	−	374.228i	0.218023	−	0.377626i	−0.736181	−	0.676785i	$-0.763373\pi$
$991$	216.060	−	374.228i	0.218023	−	0.377626i	0.954203	+	0.299159i	$0.0967060\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−419.388	+	73.9495i	−0.421496	+	0.0743211i
$996$		0			0
$997$	1122.30	+	941.723i	1.12568	+	0.944556i	0.998877	−	0.0473719i	$-0.0150846\pi$
$997$	1122.30	+	941.723i	1.12568	+	0.944556i	0.126801	+	0.991928i	$0.459529\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	324.3.k.a.197.2		36
3.2	odd	2		108.3.k.a.65.4	yes	36
12.11	even	2		432.3.bc.b.65.3		36
27.5	odd	18	inner	324.3.k.a.125.2		36
27.7	even	9		2916.3.c.b.1457.12		36
27.20	odd	18		2916.3.c.b.1457.25		36
27.22	even	9		108.3.k.a.5.4	✓	36
108.103	odd	18		432.3.bc.b.113.3		36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
108.3.k.a.5.4	✓	36	27.22	even	9
108.3.k.a.65.4	yes	36	3.2	odd	2
324.3.k.a.125.2		36	27.5	odd	18	inner
324.3.k.a.197.2		36	1.1	even	1	trivial
432.3.bc.b.65.3		36	12.11	even	2
432.3.bc.b.113.3		36	108.103	odd	18
2916.3.c.b.1457.12		36	27.7	even	9
2916.3.c.b.1457.25		36	27.20	odd	18

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:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	324.k (of order \(18\), degree \(6\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.26650 + 3.47969i) q^{5} +(-0.0728181 + 0.412972i) q^{7} +O(q^{10})\)	\(q+(-1.26650 + 3.47969i) q^{5} +(-0.0728181 + 0.412972i) q^{7} +(1.69236 + 4.64973i) q^{11} +(3.65864 - 3.06996i) q^{13} +(-20.4937 + 11.8321i) q^{17} +(-13.5371 + 23.4469i) q^{19} +(-20.7690 + 3.66213i) q^{23} +(8.64689 + 7.25561i) q^{25} +(2.12470 - 2.53212i) q^{29} +(3.01492 + 17.0984i) q^{31} +(-1.34479 - 0.776415i) q^{35} +(24.9593 + 43.2309i) q^{37} +(-26.0603 - 31.0575i) q^{41} +(-61.2373 + 22.2885i) q^{43} +(26.9440 + 4.75095i) q^{47} +(45.8797 + 16.6988i) q^{49} +59.1590i q^{53} -18.3230 q^{55} +(16.8092 - 46.1828i) q^{59} +(11.9824 - 67.9556i) q^{61} +(6.04884 + 16.6191i) q^{65} +(56.4874 - 47.3986i) q^{67} +(-88.6346 + 51.1732i) q^{71} +(-3.81627 + 6.60997i) q^{73} +(-2.04344 + 0.360314i) q^{77} +(-99.5742 - 83.5527i) q^{79} +(44.1227 - 52.5834i) q^{83} +(-15.2165 - 86.2972i) q^{85} +(137.829 + 79.5758i) q^{89} +(1.00139 + 1.73447i) q^{91} +(-64.4432 - 76.8005i) q^{95} +(-8.07251 + 2.93815i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q + 9 q^{5}+O(q^{10}) \) 36 * q + 9 * q^5	\( 36 q + 9 q^{5} - 36 q^{11} + 18 q^{23} - 9 q^{25} + 18 q^{29} + 45 q^{31} + 243 q^{35} + 198 q^{41} + 90 q^{43} + 243 q^{47} + 72 q^{49} - 252 q^{59} - 144 q^{61} - 747 q^{65} + 108 q^{67} - 324 q^{71} - 63 q^{73} - 495 q^{77} + 36 q^{79} + 27 q^{83} - 180 q^{85} + 567 q^{89} + 99 q^{91} + 1044 q^{95} - 216 q^{97}+O(q^{100}) \) 36 * q + 9 * q^5 - 36 * q^11 + 18 * q^23 - 9 * q^25 + 18 * q^29 + 45 * q^31 + 243 * q^35 + 198 * q^41 + 90 * q^43 + 243 * q^47 + 72 * q^49 - 252 * q^59 - 144 * q^61 - 747 * q^65 + 108 * q^67 - 324 * q^71 - 63 * q^73 - 495 * q^77 + 36 * q^79 + 27 * q^83 - 180 * q^85 + 567 * q^89 + 99 * q^91 + 1044 * q^95 - 216 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more