LMFDB - Embedded newform 324.4.i.a.37.5

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			37.5
Character	$\chi$	$=$	324.37
Dual form			324.4.i.a.289.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(4.27250 - 1.55506i) q^{5} +(-4.69568 - 26.6305i) q^{7} +O(q^{10})$	$q+(4.27250 - 1.55506i) q^{5} +(-4.69568 - 26.6305i) q^{7} +(-50.9513 - 18.5448i) q^{11} +(42.8395 + 35.9466i) q^{13} +(-49.3862 + 85.5394i) q^{17} +(-68.0936 - 117.942i) q^{19} +(-25.4482 + 144.324i) q^{23} +(-79.9195 + 67.0604i) q^{25} +(37.6146 - 31.5624i) q^{29} +(26.6026 - 150.871i) q^{31} +(-61.4744 - 106.477i) q^{35} +(11.7745 - 20.3941i) q^{37} +(-317.305 - 266.251i) q^{41} +(-269.878 - 98.2276i) q^{43} +(48.6186 + 275.730i) q^{47} +(-364.820 + 132.783i) q^{49} -248.490 q^{53} -246.528 q^{55} +(-243.543 + 88.6422i) q^{59} +(-57.5577 - 326.426i) q^{61} +(238.931 + 86.9637i) q^{65} +(71.5526 + 60.0398i) q^{67} +(333.619 - 577.845i) q^{71} +(-106.851 - 185.072i) q^{73} +(-254.606 + 1443.94i) q^{77} +(-421.142 + 353.380i) q^{79} +(-310.120 + 260.222i) q^{83} +(-77.9834 + 442.266i) q^{85} +(-504.793 - 874.328i) q^{89} +(756.115 - 1309.63i) q^{91} +(-474.337 - 398.016i) q^{95} +(-27.6645 - 10.0690i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	4.27250	−	1.55506i	0.382144	−	0.139089i	−0.143803	−	0.989606i	$-0.545933\pi$
$5$	4.27250	−	1.55506i	0.382144	−	0.139089i	0.525947	+	0.850517i	$0.323711\pi$
$6$		0			0
$7$	−4.69568	−	26.6305i	−0.253543	−	1.43791i	−0.799786	−	0.600285i	$-0.795054\pi$
$7$	−4.69568	−	26.6305i	−0.253543	−	1.43791i	0.546243	−	0.837627i	$-0.316057\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−50.9513	−	18.5448i	−1.39658	−	0.508315i	−0.469420	−	0.882975i	$-0.655537\pi$
$11$	−50.9513	−	18.5448i	−1.39658	−	0.508315i	−0.927162	+	0.374660i	$0.877759\pi$
$12$		0			0
$13$	42.8395	+	35.9466i	0.913964	+	0.766906i	0.972869	−	0.231357i	$-0.0743166\pi$
$13$	42.8395	+	35.9466i	0.913964	+	0.766906i	−0.0589054	+	0.998264i	$0.518761\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−49.3862	+	85.5394i	−0.704583	+	1.22037i	0.262259	+	0.964998i	$0.415533\pi$
$17$	−49.3862	+	85.5394i	−0.704583	+	1.22037i	−0.966842	+	0.255376i	$0.917801\pi$
$18$		0			0
$19$	−68.0936	−	117.942i	−0.822197	−	1.42409i	−0.904043	−	0.427442i	$-0.859415\pi$
$19$	−68.0936	−	117.942i	−0.822197	−	1.42409i	0.0818453	−	0.996645i	$-0.473919\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−25.4482	+	144.324i	−0.230709	+	1.30842i	0.620756	+	0.784004i	$0.286826\pi$
$23$	−25.4482	+	144.324i	−0.230709	+	1.30842i	−0.851465	+	0.524412i	$0.824285\pi$
$24$		0			0
$25$	−79.9195	+	67.0604i	−0.639356	+	0.536483i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	37.6146	−	31.5624i	0.240857	−	0.202103i	−0.514366	−	0.857570i	$-0.671973\pi$
$29$	37.6146	−	31.5624i	0.240857	−	0.202103i	0.755223	+	0.655468i	$0.227528\pi$
$30$		0			0
$31$	26.6026	−	150.871i	0.154128	−	0.874105i	−0.805450	−	0.592663i	$-0.798076\pi$
$31$	26.6026	−	150.871i	0.154128	−	0.874105i	0.959578	−	0.281441i	$-0.0908125\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−61.4744	−	106.477i	−0.296888	−	0.514225i
$36$		0			0
$37$	11.7745	−	20.3941i	0.0523168	−	0.0906154i	−0.838681	−	0.544623i	$-0.816673\pi$
$37$	11.7745	−	20.3941i	0.0523168	−	0.0906154i	0.890998	+	0.454008i	$0.150006\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−317.305	−	266.251i	−1.20865	−	1.01418i	−0.999340	−	0.0363307i	$-0.988433\pi$
$41$	−317.305	−	266.251i	−1.20865	−	1.01418i	−0.209312	−	0.977849i	$-0.567123\pi$
$42$		0			0
$43$	−269.878	−	98.2276i	−0.957117	−	0.348362i	−0.184214	−	0.982886i	$-0.558974\pi$
$43$	−269.878	−	98.2276i	−0.957117	−	0.348362i	−0.772903	+	0.634524i	$0.781196\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	48.6186	+	275.730i	0.150888	+	0.855730i	0.962449	+	0.271462i	$0.0875072\pi$
$47$	48.6186	+	275.730i	0.150888	+	0.855730i	−0.811561	+	0.584268i	$0.801382\pi$
$48$		0			0
$49$	−364.820	+	132.783i	−1.06361	+	0.387124i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−248.490			−0.644015			−0.322007	−	0.946737i	$-0.604358\pi$
$53$	−248.490			−0.644015			−0.322007	+	0.946737i	$0.604358\pi$
$54$		0			0
$55$	−246.528			−0.604397
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−243.543	+	88.6422i	−0.537399	+	0.195597i	−0.596439	−	0.802658i	$-0.703418\pi$
$59$	−243.543	+	88.6422i	−0.537399	+	0.195597i	0.0590401	+	0.998256i	$0.481196\pi$
$60$		0			0
$61$	−57.5577	−	326.426i	−0.120812	−	0.685156i	−0.983708	−	0.179775i	$-0.942463\pi$
$61$	−57.5577	−	326.426i	−0.120812	−	0.685156i	0.862896	−	0.505381i	$-0.168648\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	238.931	+	86.9637i	0.455934	+	0.165946i
$66$		0			0
$67$	71.5526	+	60.0398i	0.130471	+	0.109478i	0.705688	−	0.708523i	$-0.250638\pi$
$67$	71.5526	+	60.0398i	0.130471	+	0.109478i	−0.575217	+	0.818001i	$0.695082\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	333.619	−	577.845i	0.557651	−	0.965881i	−0.440040	−	0.897978i	$-0.645036\pi$
$71$	333.619	−	577.845i	0.557651	−	0.965881i	0.997692	−	0.0679028i	$-0.0216308\pi$
$72$		0			0
$73$	−106.851	−	185.072i	−0.171315	−	0.296726i	0.767565	−	0.640971i	$-0.221468\pi$
$73$	−106.851	−	185.072i	−0.171315	−	0.296726i	−0.938880	+	0.344245i	$0.888135\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−254.606	+	1443.94i	−0.376818	+	2.13704i
$78$		0			0
$79$	−421.142	+	353.380i	−0.599775	+	0.503271i	−0.891373	−	0.453270i	$-0.850257\pi$
$79$	−421.142	+	353.380i	−0.599775	+	0.503271i	0.291598	+	0.956541i	$0.405813\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−310.120	+	260.222i	−0.410122	+	0.344133i	−0.824391	−	0.566021i	$-0.808482\pi$
$83$	−310.120	+	260.222i	−0.410122	+	0.344133i	0.414268	+	0.910155i	$0.364038\pi$
$84$		0			0
$85$	−77.9834	+	442.266i	−0.0995116	+	0.564358i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−504.793	−	874.328i	−0.601213	−	1.04133i	−0.992638	−	0.121122i	$-0.961351\pi$
$89$	−504.793	−	874.328i	−0.601213	−	1.04133i	0.391424	−	0.920210i	$-0.371983\pi$
$90$		0			0
$91$	756.115	−	1309.63i	0.871015	−	1.50864i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−474.337	−	398.016i	−0.512273	−	0.429848i
$96$		0			0
$97$	−27.6645	−	10.0690i	−0.0289578	−	0.0105398i	0.327501	−	0.944851i	$-0.393794\pi$
$97$	−27.6645	−	10.0690i	−0.0289578	−	0.0105398i	−0.356458	+	0.934311i	$0.616016\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	16.0416	+	90.9764i	0.0158039	+	0.0896286i	0.991690	−	0.128654i	$-0.0410657\pi$
$101$	16.0416	+	90.9764i	0.0158039	+	0.0896286i	−0.975886	+	0.218283i	$0.929955\pi$
$102$		0			0
$103$	1917.18	−	697.795i	1.83403	−	0.667532i	0.842330	−	0.538963i	$-0.181184\pi$
$103$	1917.18	−	697.795i	1.83403	−	0.667532i	0.991701	−	0.128569i	$-0.0410385\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	351.086			0.317204			0.158602	−	0.987343i	$-0.449301\pi$
$107$	351.086			0.317204			0.158602	+	0.987343i	$0.449301\pi$
$108$		0			0
$109$	656.929			0.577270			0.288635	−	0.957439i	$-0.406799\pi$
$109$	656.929			0.577270			0.288635	+	0.957439i	$0.406799\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	119.983	−	43.6704i	0.0998857	−	0.0363554i	−0.291594	−	0.956542i	$-0.594186\pi$
$113$	119.983	−	43.6704i	0.0998857	−	0.0363554i	0.391480	+	0.920187i	$0.371963\pi$
$114$		0			0
$115$	115.705	+	656.196i	0.0938223	+	0.532092i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	2509.86	+	913.514i	1.93343	+	0.703711i
$120$		0			0
$121$	1232.53	+	1034.21i	0.926015	+	0.777019i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−521.342	+	902.991i	−0.373042	+	0.646127i
$126$		0			0
$127$	−649.026	−	1124.15i	−0.453478	−	0.785447i	0.545121	−	0.838357i	$-0.316484\pi$
$127$	−649.026	−	1124.15i	−0.453478	−	0.785447i	−0.998599	+	0.0529102i	$0.983150\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	196.126	−	1112.28i	0.130806	−	0.741837i	−0.846883	−	0.531779i	$-0.821524\pi$
$131$	196.126	−	1112.28i	0.130806	−	0.741837i	0.977689	−	0.210058i	$-0.0673653\pi$
$132$		0			0
$133$	−2821.10	+	2367.18i	−1.83925	+	1.54331i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	1978.63	−	1660.27i	1.23391	−	1.03538i	0.235937	−	0.971768i	$-0.424184\pi$
$137$	1978.63	−	1660.27i	1.23391	−	1.03538i	0.997975	−	0.0636072i	$-0.0202605\pi$
$138$		0			0
$139$	−487.444	+	2764.43i	−0.297442	+	1.68688i	0.359666	+	0.933081i	$0.382891\pi$
$139$	−487.444	+	2764.43i	−0.297442	+	1.68688i	−0.657108	+	0.753797i	$0.728220\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−1516.11	−	2625.97i	−0.886596	−	1.53563i
$144$		0			0
$145$	111.627	−	193.343i	0.0639317	−	0.110733i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	2123.02	+	1781.43i	1.16728	+	0.979464i	0.999979	−	0.00644242i	$-0.00205070\pi$
$149$	2123.02	+	1781.43i	1.16728	+	0.979464i	0.167300	+	0.985906i	$0.446495\pi$
$150$		0			0
$151$	−711.587	−	258.996i	−0.383497	−	0.139582i	0.143075	−	0.989712i	$-0.454301\pi$
$151$	−711.587	−	258.996i	−0.383497	−	0.139582i	−0.526573	+	0.850130i	$0.676523\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−120.954	−	685.966i	−0.0626792	−	0.355472i
$156$		0			0
$157$	1105.36	−	402.318i	0.561894	−	0.204513i	−0.0454293	−	0.998968i	$-0.514466\pi$
$157$	1105.36	−	402.318i	0.561894	−	0.204513i	0.607323	+	0.794455i	$0.292243\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	3962.91			1.93988
$162$		0			0
$163$	2323.45			1.11648			0.558240	−	0.829679i	$-0.311477\pi$
$163$	2323.45			1.11648			0.558240	+	0.829679i	$0.311477\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−3377.16	+	1229.19i	−1.56487	+	0.569565i	−0.971845	−	0.235621i	$-0.924287\pi$
$167$	−3377.16	+	1229.19i	−1.56487	+	0.569565i	−0.593022	+	0.805186i	$0.702065\pi$
$168$		0			0
$169$	161.558	+	916.240i	0.0735356	+	0.417041i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	315.163	+	114.710i	0.138505	+	0.0504118i	0.410343	−	0.911931i	$-0.365409\pi$
$173$	315.163	+	114.710i	0.138505	+	0.0504118i	−0.271837	+	0.962343i	$0.587631\pi$
$174$		0			0
$175$	2161.13	+	1813.40i	0.933520	+	0.783316i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	1838.52	−	3184.41i	0.767695	−	1.32969i	−0.171115	−	0.985251i	$-0.554737\pi$
$179$	1838.52	−	3184.41i	0.767695	−	1.32969i	0.938810	−	0.344436i	$-0.111930\pi$
$180$		0			0
$181$	−208.366	−	360.900i	−0.0855675	−	0.148207i	0.820065	−	0.572270i	$-0.193937\pi$
$181$	−208.366	−	360.900i	−0.0855675	−	0.148207i	−0.905633	+	0.424063i	$0.860604\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	18.5926	−	105.444i	0.00738896	−	0.0419049i
$186$		0			0
$187$	4102.60	−	3442.49i	1.60434	−	1.34620i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	1409.52	−	1182.73i	0.533977	−	0.448060i	−0.335495	−	0.942042i	$-0.608904\pi$
$191$	1409.52	−	1182.73i	0.533977	−	0.448060i	0.869472	+	0.493982i	$0.164459\pi$
$192$		0			0
$193$	54.6854	−	310.136i	0.0203956	−	0.115669i	−0.972910	−	0.231183i	$-0.925740\pi$
$193$	54.6854	−	310.136i	0.0203956	−	0.115669i	0.993306	+	0.115514i	$0.0368515\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−839.214	−	1453.56i	−0.303510	−	0.525695i	0.673418	−	0.739262i	$-0.264825\pi$
$197$	−839.214	−	1453.56i	−0.303510	−	0.525695i	−0.976929	+	0.213566i	$0.931492\pi$
$198$		0			0
$199$	−255.716	+	442.913i	−0.0910917	+	0.157775i	−0.907971	−	0.419034i	$-0.862369\pi$
$199$	−255.716	+	442.913i	−0.0910917	+	0.157775i	0.816879	+	0.576809i	$0.195702\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−1017.15	−	853.488i	−0.351674	−	0.295089i
$204$		0			0
$205$	−1769.72	−	644.127i	−0.602941	−	0.219452i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	1282.26	+	7272.07i	0.424382	+	2.40679i
$210$		0			0
$211$	−2331.59	+	848.631i	−0.760728	+	0.276882i	−0.693113	−	0.720829i	$-0.743761\pi$
$211$	−2331.59	+	848.631i	−0.760728	+	0.276882i	−0.0676152	+	0.997711i	$0.521539\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−1305.81			−0.414210
$216$		0			0
$217$	−4142.69			−1.29596
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−5190.53	+	1889.20i	−1.57988	+	0.575028i
$222$		0			0
$223$	−80.0661	−	454.077i	−0.0240432	−	0.136355i	0.970423	−	0.241409i	$-0.0776096\pi$
$223$	−80.0661	−	454.077i	−0.0240432	−	0.136355i	−0.994467	+	0.105054i	$0.966499\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	1280.18	+	465.948i	0.374311	+	0.136238i	0.522324	−	0.852747i	$-0.325065\pi$
$227$	1280.18	+	465.948i	0.374311	+	0.136238i	−0.148012	+	0.988986i	$0.547288\pi$
$228$		0			0
$229$	1253.28	+	1051.62i	0.361654	+	0.303464i	0.805450	−	0.592664i	$-0.201924\pi$
$229$	1253.28	+	1051.62i	0.361654	+	0.303464i	−0.443796	+	0.896128i	$0.646368\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−2293.06	+	3971.70i	−0.644736	+	1.11672i	0.339627	+	0.940560i	$0.389699\pi$
$233$	−2293.06	+	3971.70i	−0.644736	+	1.11672i	−0.984362	+	0.176155i	$0.943634\pi$
$234$		0			0
$235$	636.500	+	1102.45i	0.176684	+	0.306025i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−285.319	+	1618.12i	−0.0772206	+	0.437940i	0.921545	+	0.388271i	$0.126928\pi$
$239$	−285.319	+	1618.12i	−0.0772206	+	0.437940i	−0.998766	+	0.0496686i	$0.984183\pi$
$240$		0			0
$241$	−4699.71	+	3943.52i	−1.25616	+	1.05404i	−0.260080	+	0.965587i	$0.583749\pi$
$241$	−4699.71	+	3943.52i	−1.25616	+	1.05404i	−0.996080	+	0.0884567i	$0.971807\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−1352.21	+	1134.63i	−0.352609	+	0.295874i
$246$		0			0
$247$	1322.50	−	7500.29i	0.340684	−	1.93211i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−2445.01	−	4234.88i	−0.614851	−	1.06495i	−0.990411	−	0.138154i	$-0.955883\pi$
$251$	−2445.01	−	4234.88i	−0.614851	−	1.06495i	0.375560	−	0.926798i	$-0.377450\pi$
$252$		0			0
$253$	3973.07	−	6881.55i	0.987291	−	1.71004i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	904.386	+	758.870i	0.219510	+	0.184191i	0.745911	−	0.666046i	$-0.232014\pi$
$257$	904.386	+	758.870i	0.219510	+	0.184191i	−0.526401	+	0.850236i	$0.676459\pi$
$258$		0			0
$259$	−598.395	−	217.798i	−0.143562	−	0.0522521i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−1187.46	−	6734.40i	−0.278410	−	1.57894i	−0.727919	−	0.685664i	$-0.759512\pi$
$263$	−1187.46	−	6734.40i	−0.278410	−	1.57894i	0.449509	−	0.893276i	$-0.351599\pi$
$264$		0			0
$265$	−1061.68	+	386.418i	−0.246106	+	0.0895754i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−343.601			−0.0778801			−0.0389401	−	0.999242i	$-0.512398\pi$
$269$	−343.601			−0.0778801			−0.0389401	+	0.999242i	$0.512398\pi$
$270$		0			0
$271$	2607.43			0.584466			0.292233	−	0.956347i	$-0.405602\pi$
$271$	2607.43			0.584466			0.292233	+	0.956347i	$0.405602\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	5315.63	−	1934.73i	1.16562	−	0.424249i
$276$		0			0
$277$	0.0627023	+	0.355603i	1.36008e−5	+	7.71339e-5i	0.984815	−	0.173610i	$-0.0555431\pi$
$277$	0.0627023	+	0.355603i	1.36008e−5	+	7.71339e-5i	−0.984801	+	0.173687i	$0.944432\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−4627.50	−	1684.27i	−0.982396	−	0.357563i	−0.199625	−	0.979872i	$-0.563972\pi$
$281$	−4627.50	−	1684.27i	−0.982396	−	0.357563i	−0.782771	+	0.622310i	$0.786194\pi$
$282$		0			0
$283$	−4317.90	−	3623.15i	−0.906971	−	0.761039i	0.0645695	−	0.997913i	$-0.479433\pi$
$283$	−4317.90	−	3623.15i	−0.906971	−	0.761039i	−0.971540	+	0.236874i	$0.923877\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−5600.43	+	9700.22i	−1.15186	+	1.99507i
$288$		0			0
$289$	−2421.49	−	4194.15i	−0.492874	−	0.853684i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	397.381	−	2253.66i	0.0792328	−	0.449352i	−0.919220	−	0.393745i	$-0.871179\pi$
$293$	397.381	−	2253.66i	0.0792328	−	0.449352i	0.998453	−	0.0556072i	$-0.0177094\pi$
$294$		0			0
$295$	−902.692	+	757.448i	−0.178158	+	0.149493i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−6278.12	+	5267.97i	−1.21429	+	1.01891i
$300$		0			0
$301$	−1348.59	+	7648.23i	−0.258244	+	1.46457i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−753.528	−	1305.15i	−0.141465	−	0.245025i
$306$		0			0
$307$	−3242.45	+	5616.10i	−0.602790	+	1.04406i	0.389606	+	0.920982i	$0.372611\pi$
$307$	−3242.45	+	5616.10i	−0.602790	+	1.04406i	−0.992397	+	0.123082i	$0.960722\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−2064.12	−	1732.01i	−0.376353	−	0.315797i	0.434916	−	0.900471i	$-0.356778\pi$
$311$	−2064.12	−	1732.01i	−0.376353	−	0.315797i	−0.811269	+	0.584674i	$0.801223\pi$
$312$		0			0
$313$	−4366.05	−	1589.11i	−0.788446	−	0.286971i	−0.0837560	−	0.996486i	$-0.526692\pi$
$313$	−4366.05	−	1589.11i	−0.788446	−	0.286971i	−0.704690	+	0.709515i	$0.748914\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−860.106	−	4877.90i	−0.152392	−	0.864260i	−0.961131	−	0.276092i	$-0.910961\pi$
$317$	−860.106	−	4877.90i	−0.152392	−	0.864260i	0.808739	−	0.588168i	$-0.200151\pi$
$318$		0			0
$319$	−2501.83	+	910.591i	−0.439108	+	0.159822i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	13451.5			2.31723
$324$		0			0
$325$	−5834.30			−0.995781
$326$		0			0
$327$		0			0
$328$		0			0
$329$	7114.53	−	2589.48i	1.19221	−	0.433928i
$330$		0			0
$331$	−307.323	−	1742.92i	−0.0510333	−	0.289424i	0.948601	−	0.316475i	$-0.102499\pi$
$331$	−307.323	−	1742.92i	−0.0510333	−	0.289424i	−0.999634	+	0.0270512i	$0.991388\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	399.074	+	145.251i	0.0650859	+	0.0236893i
$336$		0			0
$337$	−625.914	−	525.204i	−0.101174	−	0.0848952i	0.590797	−	0.806820i	$-0.298813\pi$
$337$	−625.914	−	525.204i	−0.101174	−	0.0848952i	−0.691972	+	0.721925i	$0.743258\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−4153.31	+	7193.75i	−0.659573	+	1.14241i
$342$		0			0
$343$	611.577	+	1059.28i	0.0962743	+	0.166752i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−1301.07	+	7378.71i	−0.201282	+	1.14153i	0.701902	+	0.712274i	$0.252335\pi$
$347$	−1301.07	+	7378.71i	−0.201282	+	1.14153i	−0.903184	+	0.429254i	$0.858777\pi$
$348$		0			0
$349$	9295.98	−	7800.25i	1.42580	−	1.19638i	0.477648	−	0.878551i	$-0.341489\pi$
$349$	9295.98	−	7800.25i	1.42580	−	1.19638i	0.948147	−	0.317833i	$-0.102955\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	1902.32	−	1596.24i	0.286828	−	0.240678i	−0.488008	−	0.872839i	$-0.662276\pi$
$353$	1902.32	−	1596.24i	0.286828	−	0.240678i	0.774837	+	0.632161i	$0.217832\pi$
$354$		0			0
$355$	526.802	−	2987.64i	0.0787598	−	0.446669i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−5053.00	−	8752.06i	−0.742862	−	1.28667i	−0.951187	−	0.308614i	$-0.900135\pi$
$359$	−5053.00	−	8752.06i	−0.742862	−	1.28667i	0.208326	−	0.978060i	$-0.433199\pi$
$360$		0			0
$361$	−5843.98	+	10122.1i	−0.852017	+	1.47574i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−744.320	−	624.559i	−0.106738	−	0.0895641i
$366$		0			0
$367$	5169.49	+	1881.54i	0.735273	+	0.267618i	0.682395	−	0.730983i	$-0.260938\pi$
$367$	5169.49	+	1881.54i	0.735273	+	0.267618i	0.0528780	+	0.998601i	$0.483161\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	1166.83	+	6617.42i	0.163285	+	0.926036i
$372$		0			0
$373$	8111.81	−	2952.46i	1.12604	−	0.409845i	0.289188	−	0.957272i	$-0.406615\pi$
$373$	8111.81	−	2952.46i	1.12604	−	0.409845i	0.836853	+	0.547427i	$0.184393\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	2745.95			0.375128
$378$		0			0
$379$	1742.39			0.236149			0.118075	−	0.993005i	$-0.462328\pi$
$379$	1742.39			0.236149			0.118075	+	0.993005i	$0.462328\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−2574.27	+	936.958i	−0.343444	+	0.125003i	−0.507983	−	0.861367i	$-0.669609\pi$
$383$	−2574.27	+	936.958i	−0.343444	+	0.125003i	0.164539	+	0.986371i	$0.447386\pi$
$384$		0			0
$385$	1157.62	+	6565.16i	0.153240	+	0.869069i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−6652.69	−	2421.38i	−0.867108	−	0.315601i	−0.130112	−	0.991499i	$-0.541534\pi$
$389$	−6652.69	−	2421.38i	−0.867108	−	0.315601i	−0.736995	+	0.675898i	$0.763756\pi$
$390$		0			0
$391$	−11088.6	−	9304.41i	−1.43420	−	1.20344i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−1249.80	+	2164.72i	−0.159201	+	0.275744i
$396$		0			0
$397$	2624.35	+	4545.52i	0.331770	+	0.574642i	0.982859	−	0.184359i	$-0.0590210\pi$
$397$	2624.35	+	4545.52i	0.331770	+	0.574642i	−0.651089	+	0.759001i	$0.725688\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	949.429	−	5384.48i	0.118235	−	0.670544i	−0.866863	−	0.498547i	$-0.833867\pi$
$401$	949.429	−	5384.48i	0.118235	−	0.670544i	0.985098	−	0.171997i	$-0.0550219\pi$
$402$		0			0
$403$	6562.94	−	5506.96i	0.811224	−	0.680698i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−978.133	+	820.751i	−0.119126	+	0.0999585i
$408$		0			0
$409$	689.848	−	3912.32i	0.0834005	−	0.472987i	−0.914290	−	0.405061i	$-0.867250\pi$
$409$	689.848	−	3912.32i	0.0834005	−	0.472987i	0.997690	−	0.0679269i	$-0.0216385\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	3504.18	+	6069.42i	0.417505	+	0.723140i
$414$		0			0
$415$	−920.329	+	1594.06i	−0.108861	+	0.188552i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−11193.9	−	9392.79i	−1.30515	−	1.09515i	−0.989231	−	0.146363i	$-0.953243\pi$
$419$	−11193.9	−	9392.79i	−1.30515	−	1.09515i	−0.315917	−	0.948787i	$-0.602312\pi$
$420$		0			0
$421$	13068.9	+	4756.68i	1.51292	+	0.550657i	0.959367	−	0.282160i	$-0.0910509\pi$
$421$	13068.9	+	4756.68i	1.51292	+	0.550657i	0.553549	+	0.832816i	$0.313273\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−1789.39	−	10148.1i	−0.204231	−	1.15825i
$426$		0			0
$427$	−8422.61	+	3065.58i	−0.954563	+	0.347433i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	14619.4			1.63385			0.816927	−	0.576741i	$-0.195676\pi$
$431$	14619.4			1.63385			0.816927	+	0.576741i	$0.195676\pi$
$432$		0			0
$433$	14257.6			1.58239			0.791197	−	0.611561i	$-0.209458\pi$
$433$	14257.6			1.58239			0.791197	+	0.611561i	$0.209458\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	18754.6	−	6826.12i	2.05299	−	0.747226i
$438$		0			0
$439$	−2645.69	−	15004.5i	−0.287635	−	1.63126i	−0.695717	−	0.718316i	$-0.744913\pi$
$439$	−2645.69	−	15004.5i	−0.287635	−	1.63126i	0.408081	−	0.912946i	$-0.366198\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	4770.78	+	1736.42i	0.511663	+	0.186230i	0.584932	−	0.811082i	$-0.301121\pi$
$443$	4770.78	+	1736.42i	0.511663	+	0.186230i	−0.0732692	+	0.997312i	$0.523343\pi$
$444$		0			0
$445$	−3516.37	−	2950.58i	−0.374588	−	0.314317i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−7046.08	+	12204.2i	−0.740591	+	1.28274i	0.211636	+	0.977349i	$0.432121\pi$
$449$	−7046.08	+	12204.2i	−0.740591	+	1.28274i	−0.952227	+	0.305392i	$0.901213\pi$
$450$		0			0
$451$	11229.6	+	19450.2i	1.17246	+	2.03076i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	1193.95	−	6771.20i	0.123018	−	0.697668i
$456$		0			0
$457$	−949.074	+	796.368i	−0.0971462	+	0.0815153i	−0.690067	−	0.723745i	$-0.742419\pi$
$457$	−949.074	+	796.368i	−0.0971462	+	0.0815153i	0.592921	+	0.805261i	$0.297975\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	2354.96	−	1976.04i	0.237920	−	0.199639i	−0.516030	−	0.856571i	$-0.672591\pi$
$461$	2354.96	−	1976.04i	0.237920	−	0.199639i	0.753950	+	0.656932i	$0.228146\pi$
$462$		0			0
$463$	−1637.88	+	9288.87i	−0.164403	+	0.932377i	0.785274	+	0.619148i	$0.212522\pi$
$463$	−1637.88	+	9288.87i	−0.164403	+	0.932377i	−0.949678	+	0.313229i	$0.898589\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	5454.27	+	9447.08i	0.540457	+	0.936100i	0.998878	+	0.0473641i	$0.0150821\pi$
$467$	5454.27	+	9447.08i	0.540457	+	0.936100i	−0.458420	+	0.888736i	$0.651585\pi$
$468$		0			0
$469$	1262.90	−	2187.41i	0.124340	−	0.215363i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	11929.0	+	10009.7i	1.15962	+	0.973033i
$474$		0			0
$475$	13351.2	+	4859.45i	1.28968	+	0.469404i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	2009.10	+	11394.2i	0.191645	+	1.08687i	0.917116	+	0.398621i	$0.130511\pi$
$479$	2009.10	+	11394.2i	0.191645	+	1.08687i	−0.725470	+	0.688253i	$0.758378\pi$
$480$		0			0
$481$	1237.51	−	450.418i	0.117309	−	0.0426971i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−133.855			−0.0125320
$486$		0			0
$487$	−15337.4			−1.42711			−0.713554	−	0.700600i	$-0.752916\pi$
$487$	−15337.4			−1.42711			−0.713554	+	0.700600i	$0.752916\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	6647.78	−	2419.59i	0.611018	−	0.222392i	−0.0179304	−	0.999839i	$-0.505708\pi$
$491$	6647.78	−	2419.59i	0.611018	−	0.222392i	0.628949	+	0.777447i	$0.283486\pi$
$492$		0			0
$493$	842.185	+	4776.27i	0.0769374	+	0.436334i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−16954.9	−	6171.06i	−1.53024	−	0.556962i
$498$		0			0
$499$	10522.3	+	8829.24i	0.943972	+	0.792087i	0.978272	−	0.207325i	$-0.0664758\pi$
$499$	10522.3	+	8829.24i	0.943972	+	0.792087i	−0.0343001	+	0.999412i	$0.510920\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−1396.87	+	2419.45i	−0.123824	+	0.214469i	−0.921273	−	0.388918i	$-0.872849\pi$
$503$	−1396.87	+	2419.45i	−0.123824	+	0.214469i	0.797449	+	0.603387i	$0.206182\pi$
$504$		0			0
$505$	210.012	+	363.751i	0.0185057	+	0.0320529i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	1433.78	−	8131.38i	0.124855	−	0.708089i	−0.856539	−	0.516083i	$-0.827390\pi$
$509$	1433.78	−	8131.38i	0.124855	−	0.708089i	0.981394	−	0.192006i	$-0.0614993\pi$
$510$		0			0
$511$	−4426.81	+	3714.54i	−0.383230	+	0.321568i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	7106.03	−	5962.66i	0.608017	−	0.510187i
$516$		0			0
$517$	2636.16	−	14950.4i	0.224252	−	1.27180i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−10192.6	−	17654.1i	−0.857094	−	1.48453i	−0.874689	−	0.484685i	$-0.838934\pi$
$521$	−10192.6	−	17654.1i	−0.857094	−	1.48453i	0.0175944	−	0.999845i	$-0.494399\pi$
$522$		0			0
$523$	−7162.07	+	12405.1i	−0.598806	+	1.03716i	0.394192	+	0.919028i	$0.371025\pi$
$523$	−7162.07	+	12405.1i	−0.598806	+	1.03716i	−0.992998	+	0.118134i	$0.962309\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	11591.6	+	9726.52i	0.958138	+	0.803973i
$528$		0			0
$529$	−8748.46	−	3184.18i	−0.719032	−	0.261706i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−4022.38	−	22812.1i	−0.326883	−	1.85385i
$534$		0			0
$535$	1500.02	−	545.962i	0.121218	−	0.0441196i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	21050.5			1.68221
$540$		0			0
$541$	−9772.64			−0.776633			−0.388317	−	0.921526i	$-0.626943\pi$
$541$	−9772.64			−0.776633			−0.388317	+	0.921526i	$0.626943\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	2806.73	−	1021.57i	0.220600	−	0.0802919i
$546$		0			0
$547$	2487.70	+	14108.5i	0.194454	+	1.10280i	0.913194	+	0.407525i	$0.133608\pi$
$547$	2487.70	+	14108.5i	0.194454	+	1.10280i	−0.718740	+	0.695279i	$0.755281\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−6283.83	−	2287.13i	−0.485844	−	0.176833i
$552$		0			0
$553$	11388.2	+	9555.87i	0.875728	+	0.734823i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−7088.20	+	12277.1i	−0.539204	+	0.933929i	0.459743	+	0.888052i	$0.347942\pi$
$557$	−7088.20	+	12277.1i	−0.539204	+	0.933929i	−0.998947	+	0.0458770i	$0.985392\pi$
$558$		0			0
$559$	−8030.49	−	13909.2i	−0.607609	−	1.05241i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	2412.55	−	13682.2i	0.180598	−	1.02422i	−0.750884	−	0.660435i	$-0.770372\pi$
$563$	2412.55	−	13682.2i	0.180598	−	1.02422i	0.931482	−	0.363788i	$-0.118517\pi$
$564$		0			0
$565$	444.719	−	373.164i	0.0331141	−	0.0277860i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−1899.81	+	1594.13i	−0.139972	+	0.117451i	−0.710086	−	0.704115i	$-0.751344\pi$
$569$	−1899.81	+	1594.13i	−0.139972	+	0.117451i	0.570113	+	0.821566i	$0.306899\pi$
$570$		0			0
$571$	680.576	−	3859.74i	0.0498795	−	0.282881i	−0.949658	−	0.313288i	$-0.898569\pi$
$571$	680.576	−	3859.74i	0.0498795	−	0.282881i	0.999538	+	0.0304075i	$0.00968049\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−7644.60	−	13240.8i	−0.554438	−	0.960315i
$576$		0			0
$577$	9634.98	−	16688.3i	0.695164	−	1.20406i	−0.274962	−	0.961455i	$-0.588665\pi$
$577$	9634.98	−	16688.3i	0.695164	−	1.20406i	0.970125	−	0.242604i	$-0.0780016\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	8386.07	+	7036.74i	0.598817	+	0.502467i
$582$		0			0
$583$	12660.9	+	4608.20i	0.899420	+	0.327362i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	2051.32	+	11633.6i	0.144237	+	0.818007i	0.967977	+	0.251040i	$0.0807725\pi$
$587$	2051.32	+	11633.6i	0.144237	+	0.818007i	−0.823740	+	0.566968i	$0.808116\pi$
$588$		0			0
$589$	−19605.5	+	7135.80i	−1.37153	+	0.499194i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	13127.9			0.909101			0.454551	−	0.890721i	$-0.349800\pi$
$593$	13127.9			0.909101			0.454551	+	0.890721i	$0.349800\pi$
$594$		0			0
$595$	12143.9			0.836728
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−26377.9	+	9600.77i	−1.79928	+	0.654886i	−0.800855	+	0.598858i	$0.795622\pi$
$599$	−26377.9	+	9600.77i	−1.79928	+	0.654886i	−0.998429	+	0.0560284i	$0.982156\pi$
$600$		0			0
$601$	−2662.66	−	15100.7i	−0.180719	−	1.02491i	−0.931333	−	0.364167i	$-0.881354\pi$
$601$	−2662.66	−	15100.7i	−0.180719	−	1.02491i	0.750615	−	0.660740i	$-0.229758\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	6874.24	+	2502.02i	0.461946	+	0.168135i
$606$		0			0
$607$	−20854.2	−	17498.7i	−1.39447	−	1.17010i	−0.963491	−	0.267740i	$-0.913723\pi$
$607$	−20854.2	−	17498.7i	−1.39447	−	1.17010i	−0.430981	−	0.902361i	$-0.641832\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−7828.75	+	13559.8i	−0.518359	+	0.897824i
$612$		0			0
$613$	2651.31	+	4592.20i	0.174690	+	0.302573i	0.940054	−	0.341025i	$-0.110774\pi$
$613$	2651.31	+	4592.20i	0.174690	+	0.302573i	−0.765364	+	0.643598i	$0.777441\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	1200.77	−	6809.92i	0.0783489	−	0.444339i	−0.920246	−	0.391341i	$-0.872011\pi$
$617$	1200.77	−	6809.92i	0.0783489	−	0.444339i	0.998595	−	0.0529978i	$-0.0168776\pi$
$618$		0			0
$619$	1564.51	−	1312.78i	0.101588	−	0.0852424i	−0.590579	−	0.806980i	$-0.701101\pi$
$619$	1564.51	−	1312.78i	0.101588	−	0.0852424i	0.692167	+	0.721737i	$0.256656\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−20913.4	+	17548.5i	−1.34491	+	1.12851i
$624$		0			0
$625$	1441.31	−	8174.07i	0.0922438	−	0.523140i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	1163.00	+	2014.38i	0.0737231	+	0.127692i
$630$		0			0
$631$	−7978.31	+	13818.8i	−0.503346	+	0.871821i	0.496646	+	0.867953i	$0.334565\pi$
$631$	−7978.31	+	13818.8i	−0.503346	+	0.871821i	−0.999993	+	0.00386809i	$0.998769\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−4521.08	−	3793.64i	−0.282541	−	0.237080i
$636$		0			0
$637$	−20401.8	−	7425.64i	−1.26899	−	0.461875i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−95.6010	−	542.180i	−0.00589081	−	0.0334085i	0.981721	−	0.190327i	$-0.0609550\pi$
$641$	−95.6010	−	542.180i	−0.00589081	−	0.0334085i	−0.987611	+	0.156919i	$0.949844\pi$
$642$		0			0
$643$	−10646.4	+	3874.97i	−0.652959	+	0.237658i	−0.647194	−	0.762326i	$-0.724058\pi$
$643$	−10646.4	+	3874.97i	−0.652959	+	0.237658i	−0.00576568	+	0.999983i	$0.501835\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−21233.5			−1.29022			−0.645112	−	0.764088i	$-0.723189\pi$
$647$	−21233.5			−1.29022			−0.645112	+	0.764088i	$0.723189\pi$
$648$		0			0
$649$	14052.7			0.849947
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−21181.2	+	7709.31i	−1.26935	+	0.462004i	−0.886895	−	0.461971i	$-0.847142\pi$
$653$	−21181.2	+	7709.31i	−1.26935	+	0.462004i	−0.382451	+	0.923976i	$0.624920\pi$
$654$		0			0
$655$	−891.725	−	5057.22i	−0.0531948	−	0.301682i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	12762.0	+	4644.99i	0.754381	+	0.274572i	0.690448	−	0.723382i	$-0.257413\pi$
$659$	12762.0	+	4644.99i	0.754381	+	0.274572i	0.0639327	+	0.997954i	$0.479636\pi$
$660$		0			0
$661$	1688.35	+	1416.69i	0.0993481	+	0.0833630i	0.691109	−	0.722751i	$-0.257123\pi$
$661$	1688.35	+	1416.69i	0.0993481	+	0.0833630i	−0.591761	+	0.806114i	$0.701567\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−8372.03	+	14500.8i	−0.488201	+	0.845588i
$666$		0			0
$667$	3597.97	+	6231.87i	0.208867	+	0.361768i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−3120.85	+	17699.2i	−0.179552	+	1.01829i
$672$		0			0
$673$	−468.396	+	393.031i	−0.0268281	+	0.0225115i	−0.656103	−	0.754671i	$-0.727796\pi$
$673$	−468.396	+	393.031i	−0.0268281	+	0.0225115i	0.629275	+	0.777183i	$0.283352\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−10915.1	+	9158.89i	−0.619650	+	0.519948i	−0.897693	−	0.440621i	$-0.854758\pi$
$677$	−10915.1	+	9158.89i	−0.619650	+	0.519948i	0.278044	+	0.960568i	$0.410314\pi$
$678$		0			0
$679$	−138.240	+	784.000i	−0.00781322	+	0.0443110i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−7351.56	−	12733.3i	−0.411859	−	0.713360i	0.583234	−	0.812304i	$-0.301787\pi$
$683$	−7351.56	−	12733.3i	−0.411859	−	0.713360i	−0.995093	+	0.0989436i	$0.968454\pi$
$684$		0			0
$685$	5871.89	−	10170.4i	0.327523	−	0.567286i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−10645.2	−	8932.37i	−0.588606	−	0.493899i
$690$		0			0
$691$	−17454.4	−	6352.88i	−0.960921	−	0.349747i	−0.186527	−	0.982450i	$-0.559723\pi$
$691$	−17454.4	−	6352.88i	−0.960921	−	0.349747i	−0.774394	+	0.632703i	$0.781945\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	2216.26	+	12569.0i	0.120961	+	0.686002i
$696$		0			0
$697$	38445.4	−	13993.0i	2.08927	−	0.760433i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−3749.76			−0.202035			−0.101017	−	0.994885i	$-0.532210\pi$
$701$	−3749.76			−0.202035			−0.101017	+	0.994885i	$0.532210\pi$
$702$		0			0
$703$	−3207.09			−0.172059
$704$		0			0
$705$		0			0
$706$		0			0
$707$	2347.42	−	854.391i	0.124871	−	0.0454494i
$708$		0			0
$709$	101.160	+	573.705i	0.00535843	+	0.0303892i	0.987370	−	0.158433i	$-0.0506442\pi$
$709$	101.160	+	573.705i	0.00535843	+	0.0303892i	−0.982011	+	0.188822i	$0.939533\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	21097.3	+	7678.78i	1.10813	+	0.403328i
$714$		0			0
$715$	−10561.1	−	8861.84i	−0.552397	−	0.463516i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	13076.5	−	22649.2i	0.678264	−	1.17479i	−0.297239	−	0.954803i	$-0.596066\pi$
$719$	13076.5	−	22649.2i	0.678264	−	1.17479i	0.975503	−	0.219985i	$-0.0706008\pi$
$720$		0			0
$721$	−27585.1	−	47778.8i	−1.42486	−	2.46793i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−889.551	+	5044.90i	−0.0455684	+	0.258431i
$726$		0			0
$727$	8156.67	−	6844.26i	0.416113	−	0.349160i	−0.410569	−	0.911829i	$-0.634670\pi$
$727$	8156.67	−	6844.26i	0.416113	−	0.349160i	0.826682	+	0.562669i	$0.190226\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	21730.6	−	18234.1i	1.09950	−	0.922590i
$732$		0			0
$733$	−3183.40	+	18053.9i	−0.160411	+	0.909738i	0.793259	+	0.608884i	$0.208383\pi$
$733$	−3183.40	+	18053.9i	−0.160411	+	0.909738i	−0.953670	+	0.300853i	$0.902729\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−2532.28	−	4386.04i	−0.126564	−	0.219215i
$738$		0			0
$739$	−4580.62	+	7933.86i	−0.228012	+	0.394928i	−0.957219	−	0.289365i	$-0.906556\pi$
$739$	−4580.62	+	7933.86i	−0.228012	+	0.394928i	0.729207	+	0.684293i	$0.239889\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−19612.0	−	16456.4i	−0.968362	−	0.812552i	0.0139313	−	0.999903i	$-0.495565\pi$
$743$	−19612.0	−	16456.4i	−0.968362	−	0.812552i	−0.982293	+	0.187351i	$0.940010\pi$
$744$		0			0
$745$	11840.8	+	4309.71i	0.582302	+	0.211940i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−1648.59	−	9349.61i	−0.0804247	−	0.456111i
$750$		0			0
$751$	−11820.2	+	4302.20i	−0.574334	+	0.209040i	−0.612825	−	0.790218i	$-0.709967\pi$
$751$	−11820.2	+	4302.20i	−0.574334	+	0.209040i	0.0384913	+	0.999259i	$0.487745\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−3443.01			−0.165966
$756$		0			0
$757$	−31497.3			−1.51227			−0.756134	−	0.654416i	$-0.772914\pi$
$757$	−31497.3			−1.51227			−0.756134	+	0.654416i	$0.772914\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−22625.0	+	8234.84i	−1.07774	+	0.392264i	−0.819064	−	0.573702i	$-0.805507\pi$
$761$	−22625.0	+	8234.84i	−1.07774	+	0.392264i	−0.258672	+	0.965965i	$0.583285\pi$
$762$		0			0
$763$	−3084.73	−	17494.4i	−0.146362	−	0.830063i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−13619.6	−	4957.13i	−0.641168	−	0.233366i
$768$		0			0
$769$	17213.9	+	14444.1i	0.807214	+	0.677333i	0.949941	−	0.312429i	$-0.101143\pi$
$769$	17213.9	+	14444.1i	0.807214	+	0.677333i	−0.142727	+	0.989762i	$0.545587\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	16376.9	−	28365.5i	0.762011	−	1.31984i	−0.179801	−	0.983703i	$-0.557545\pi$
$773$	16376.9	−	28365.5i	0.762011	−	1.31984i	0.941812	−	0.336139i	$-0.109121\pi$
$774$		0			0
$775$	7991.41	+	13841.5i	0.370400	+	0.641551i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−9795.57	+	55553.5i	−0.450530	+	2.55508i
$780$		0			0
$781$	−27714.3	+	23255.1i	−1.26978	+	1.06547i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	4097.02	−	3437.81i	0.186279	−	0.156307i
$786$		0			0
$787$	−5068.73	+	28746.2i	−0.229582	+	1.30202i	0.624148	+	0.781306i	$0.285446\pi$
$787$	−5068.73	+	28746.2i	−0.229582	+	1.30202i	−0.853730	+	0.520716i	$0.825665\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−1726.37	−	2990.16i	−0.0776012	−	0.134409i
$792$		0			0
$793$	9268.14	−	16052.9i	0.415033	−	0.718859i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	12766.2	+	10712.1i	0.567381	+	0.476089i	0.880776	−	0.473534i	$-0.157022\pi$
$797$	12766.2	+	10712.1i	0.567381	+	0.476089i	−0.313395	+	0.949623i	$0.601466\pi$
$798$		0			0
$799$	−25986.9	−	9458.44i	−1.15062	−	0.418793i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	2012.10	+	11411.2i	0.0884252	+	0.501485i
$804$		0			0
$805$	16931.5	−	6162.57i	0.741314	−	0.269816i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	5273.69			0.229188			0.114594	−	0.993412i	$-0.463443\pi$
$809$	5273.69			0.229188			0.114594	+	0.993412i	$0.463443\pi$
$810$		0			0
$811$	2370.93			0.102657			0.0513283	−	0.998682i	$-0.483654\pi$
$811$	2370.93			0.102657			0.0513283	+	0.998682i	$0.483654\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	9926.93	−	3613.11i	0.426657	−	0.155290i
$816$		0			0
$817$	6791.86	+	38518.5i	0.290841	+	1.64944i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−9376.22	−	3412.67i	−0.398578	−	0.145070i	0.134951	−	0.990852i	$-0.456912\pi$
$821$	−9376.22	−	3412.67i	−0.398578	−	0.145070i	−0.533529	+	0.845782i	$0.679134\pi$
$822$		0			0
$823$	10088.6	+	8465.38i	0.427300	+	0.358548i	0.830932	−	0.556374i	$-0.187808\pi$
$823$	10088.6	+	8465.38i	0.427300	+	0.358548i	−0.403632	+	0.914922i	$0.632252\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	8360.67	−	14481.1i	0.351547	−	0.608897i	−0.634974	−	0.772533i	$-0.718989\pi$
$827$	8360.67	−	14481.1i	0.351547	−	0.608897i	0.986521	+	0.163637i	$0.0523225\pi$
$828$		0			0
$829$	−1210.13	−	2096.00i	−0.0506990	−	0.0878132i	0.839562	−	0.543264i	$-0.182812\pi$
$829$	−1210.13	−	2096.00i	−0.0506990	−	0.0878132i	−0.890261	+	0.455450i	$0.849478\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	6658.83	−	37764.1i	0.276969	−	1.57077i
$834$		0			0
$835$	−12517.5	+	10503.4i	−0.518784	+	0.435312i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	10927.3	−	9169.08i	0.449645	−	0.377297i	−0.389659	−	0.920959i	$-0.627407\pi$
$839$	10927.3	−	9169.08i	0.449645	−	0.377297i	0.839304	+	0.543662i	$0.182963\pi$
$840$		0			0
$841$	−3816.43	+	21644.1i	−0.156482	+	0.887452i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	2115.07	+	3663.40i	0.0861071	+	0.149142i
$846$		0			0
$847$	21754.0	−	37679.1i	0.882500	−	1.52854i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	2643.71	+	2218.34i	0.106493	+	0.0893580i
$852$		0			0
$853$	−13748.6	−	5004.09i	−0.551869	−	0.200864i	0.0510079	−	0.998698i	$-0.483757\pi$
$853$	−13748.6	−	5004.09i	−0.551869	−	0.200864i	−0.602877	+	0.797834i	$0.705979\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	33.9693	+	192.649i	0.00135399	+	0.00767885i	0.985477	−	0.169807i	$-0.0543146\pi$
$857$	33.9693	+	192.649i	0.00135399	+	0.00767885i	−0.984123	+	0.177486i	$0.943203\pi$
$858$		0			0
$859$	24809.7	−	9030.01i	0.985445	−	0.358673i	0.201490	−	0.979490i	$-0.435422\pi$
$859$	24809.7	−	9030.01i	0.985445	−	0.358673i	0.783955	+	0.620818i	$0.213199\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	2803.15			0.110568			0.0552840	−	0.998471i	$-0.482394\pi$
$863$	2803.15			0.110568			0.0552840	+	0.998471i	$0.482394\pi$
$864$		0			0
$865$	1524.92			0.0599407
$866$		0			0
$867$		0			0
$868$		0			0
$869$	28011.1	−	10195.2i	1.09346	−	0.397985i
$870$		0			0
$871$	907.051	+	5144.14i	0.0352862	+	0.200118i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	26495.1	+	9643.44i	1.02366	+	0.372580i
$876$		0			0
$877$	15359.2	+	12887.9i	0.591385	+	0.496231i	0.888663	−	0.458560i	$-0.151635\pi$
$877$	15359.2	+	12887.9i	0.591385	+	0.496231i	−0.297279	+	0.954791i	$0.596079\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−18708.5	+	32404.1i	−0.715443	+	1.23918i	0.247345	+	0.968927i	$0.420442\pi$
$881$	−18708.5	+	32404.1i	−0.715443	+	1.23918i	−0.962788	+	0.270256i	$0.912892\pi$
$882$		0			0
$883$	3348.08	+	5799.04i	0.127601	+	0.221012i	0.922747	−	0.385407i	$-0.125939\pi$
$883$	3348.08	+	5799.04i	0.127601	+	0.221012i	−0.795146	+	0.606419i	$0.792606\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−5380.75	+	30515.7i	−0.203684	+	1.15515i	0.695813	+	0.718223i	$0.255044\pi$
$887$	−5380.75	+	30515.7i	−0.203684	+	1.15515i	−0.899497	+	0.436927i	$0.856067\pi$
$888$		0			0
$889$	−26888.9	+	22562.5i	−1.01443	+	0.851206i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	29209.4	−	24509.6i	1.09457	−	0.918457i
$894$		0			0
$895$	2903.12	−	16464.4i	0.108425	−	0.614910i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−3761.20	−	6514.59i	−0.139536	−	0.241684i
$900$		0			0
$901$	12272.0	−	21255.7i	0.453762	−	0.785938i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−1451.47	−	1217.93i	−0.0533131	−	0.0447350i
$906$		0			0
$907$	−1562.19	−	568.589i	−0.0571902	−	0.0208155i	0.313267	−	0.949665i	$-0.398577\pi$
$907$	−1562.19	−	568.589i	−0.0571902	−	0.0208155i	−0.370457	+	0.928850i	$0.620799\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	5763.36	+	32685.6i	0.209603	+	1.18872i	0.890030	+	0.455903i	$0.150683\pi$
$911$	5763.36	+	32685.6i	0.209603	+	1.18872i	−0.680426	+	0.732816i	$0.738205\pi$
$912$		0			0
$913$	20626.8	−	7507.55i	0.747698	−	0.272140i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−30541.6			−1.09986
$918$		0			0
$919$	20905.7			0.750397			0.375199	−	0.926944i	$-0.377575\pi$
$919$	20905.7			0.750397			0.375199	+	0.926944i	$0.377575\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	35063.6	−	12762.1i	1.25041	−	0.455113i
$924$		0			0
$925$	426.622	+	2419.49i	0.0151646	+	0.0860027i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	20083.7	+	7309.88i	0.709285	+	0.258159i	0.671370	−	0.741122i	$-0.265706\pi$
$929$	20083.7	+	7309.88i	0.709285	+	0.258159i	0.0379151	+	0.999281i	$0.487928\pi$
$930$		0			0
$931$	40502.6	+	33985.7i	1.42580	+	1.19639i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	12175.1	−	21087.9i	0.425848	−	0.737590i
$936$		0			0
$937$	704.534	+	1220.29i	0.0245636	+	0.0425454i	0.878046	−	0.478576i	$-0.158847\pi$
$937$	704.534	+	1220.29i	0.0245636	+	0.0425454i	−0.853482	+	0.521122i	$0.825514\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−3094.69	+	17550.9i	−0.107209	+	0.608015i	0.883105	+	0.469175i	$0.155449\pi$
$941$	−3094.69	+	17550.9i	−0.107209	+	0.608015i	−0.990315	+	0.138840i	$0.955663\pi$
$942$		0			0
$943$	46501.1	−	39019.0i	1.60582	−	1.34744i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	27667.7	−	23216.0i	0.949399	−	0.796640i	−0.0297975	−	0.999556i	$-0.509486\pi$
$947$	27667.7	−	23216.0i	0.949399	−	0.796640i	0.979196	+	0.202916i	$0.0650418\pi$
$948$		0			0
$949$	2075.25	−	11769.3i	0.0709856	−	0.402579i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−22472.0	−	38922.7i	−0.763841	−	1.32301i	−0.940857	−	0.338804i	$-0.889978\pi$
$953$	−22472.0	−	38922.7i	−0.763841	−	1.32301i	0.177016	−	0.984208i	$-0.443356\pi$
$954$		0			0
$955$	4182.97	−	7245.12i	0.141736	−	0.245494i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−53504.8	−	44895.9i	−1.80163	−	1.51175i
$960$		0			0
$961$	5940.00	+	2161.98i	0.199389	+	0.0725717i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−248.638	−	1410.10i	−0.00829425	−	0.0470390i
$966$		0			0
$967$	−35745.8	+	13010.4i	−1.18873	+	0.432664i	−0.859279	−	0.511507i	$-0.829087\pi$
$967$	−35745.8	+	13010.4i	−1.18873	+	0.432664i	−0.329455	+	0.944171i	$0.606865\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	885.777			0.0292749			0.0146374	−	0.999893i	$-0.495341\pi$
$971$	885.777			0.0292749			0.0146374	+	0.999893i	$0.495341\pi$
$972$		0			0
$973$	75907.1			2.50100
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−48395.1	+	17614.4i	−1.58475	+	0.576801i	−0.976229	−	0.216740i	$-0.930458\pi$
$977$	−48395.1	+	17614.4i	−1.58475	+	0.576801i	−0.608518	+	0.793540i	$0.708236\pi$
$978$		0			0
$979$	9505.69	+	53909.5i	0.310320	+	1.75991i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	17316.6	+	6302.71i	0.561864	+	0.204502i	0.607310	−	0.794465i	$-0.292249\pi$
$983$	17316.6	+	6302.71i	0.561864	+	0.204502i	−0.0454457	+	0.998967i	$0.514471\pi$
$984$		0			0
$985$	−5845.93	−	4905.31i	−0.189103	−	0.158676i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	21044.5	−	36450.1i	0.676618	−	1.17194i
$990$		0			0
$991$	−10476.8	−	18146.4i	−0.335829	−	0.581673i	0.647815	−	0.761798i	$-0.275683\pi$
$991$	−10476.8	−	18146.4i	−0.335829	−	0.581673i	−0.983644	+	0.180125i	$0.942350\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−403.789	+	2290.00i	−0.0128653	+	0.0729628i
$996$		0			0
$997$	1.20726	−	1.01301i	3.83495e−5	−	3.21790e-5i	−0.642768	−	0.766061i	$-0.722214\pi$
$997$	1.20726	−	1.01301i	3.83495e−5	−	3.21790e-5i	0.642807	+	0.766028i	$0.277770\pi$
$998$		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.37.5		54
3.2	odd	2		108.4.i.a.49.2	✓	54
27.11	odd	18		108.4.i.a.97.2	yes	54
27.16	even	9	inner	324.4.i.a.289.5		54

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
108.4.i.a.49.2	✓	54	3.2	odd	2
108.4.i.a.97.2	yes	54	27.11	odd	18
324.4.i.a.37.5		54	1.1	even	1	trivial
324.4.i.a.289.5		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)

\(f(q)\)	\(=\)	\(q+(4.27250 - 1.55506i) q^{5} +(-4.69568 - 26.6305i) q^{7} +O(q^{10})\)	\(q+(4.27250 - 1.55506i) q^{5} +(-4.69568 - 26.6305i) q^{7} +(-50.9513 - 18.5448i) q^{11} +(42.8395 + 35.9466i) q^{13} +(-49.3862 + 85.5394i) q^{17} +(-68.0936 - 117.942i) q^{19} +(-25.4482 + 144.324i) q^{23} +(-79.9195 + 67.0604i) q^{25} +(37.6146 - 31.5624i) q^{29} +(26.6026 - 150.871i) q^{31} +(-61.4744 - 106.477i) q^{35} +(11.7745 - 20.3941i) q^{37} +(-317.305 - 266.251i) q^{41} +(-269.878 - 98.2276i) q^{43} +(48.6186 + 275.730i) q^{47} +(-364.820 + 132.783i) q^{49} -248.490 q^{53} -246.528 q^{55} +(-243.543 + 88.6422i) q^{59} +(-57.5577 - 326.426i) q^{61} +(238.931 + 86.9637i) q^{65} +(71.5526 + 60.0398i) q^{67} +(333.619 - 577.845i) q^{71} +(-106.851 - 185.072i) q^{73} +(-254.606 + 1443.94i) q^{77} +(-421.142 + 353.380i) q^{79} +(-310.120 + 260.222i) q^{83} +(-77.9834 + 442.266i) q^{85} +(-504.793 - 874.328i) q^{89} +(756.115 - 1309.63i) q^{91} +(-474.337 - 398.016i) q^{95} +(-27.6645 - 10.0690i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 54 q - 12 q^{5}+O(q^{10}) \) 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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