LMFDB - Embedded newform 3240.2.q.bd.1081.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3240,2,Mod(1081,3240)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3240, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 2, 0]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("3240.1081");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3240 = 2^{3} \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3240.q (of order $3$, degree $2$, 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:	$25.8715302549$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{-11})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 2x^{2} - 3x + 9 $ x^4 - x^3 - 2x^2 - 3x + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 3 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			1081.1
Root			$-1.18614 - 1.26217i$ of defining polynomial
Character	$\chi$	$=$	3240.1081
Dual form			3240.2.q.bd.2161.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.500000 - 0.866025i) q^{5} +(-1.68614 - 2.92048i) q^{7} +O(q^{10})$	$q+(0.500000 - 0.866025i) q^{5} +(-1.68614 - 2.92048i) q^{7} +(1.18614 + 2.05446i) q^{11} +(1.68614 - 2.92048i) q^{13} +6.74456 q^{17} +1.00000 q^{19} +(2.68614 - 4.65253i) q^{23} +(-0.500000 - 0.866025i) q^{25} +(1.18614 + 2.05446i) q^{29} +(-5.55842 + 9.62747i) q^{31} -3.37228 q^{35} +6.00000 q^{37} +(0.127719 - 0.221215i) q^{41} +(2.37228 + 4.10891i) q^{43} +(-4.68614 - 8.11663i) q^{47} +(-2.18614 + 3.78651i) q^{49} -10.1168 q^{53} +2.37228 q^{55} +(2.50000 - 4.33013i) q^{59} +(-6.37228 - 11.0371i) q^{61} +(-1.68614 - 2.92048i) q^{65} +(0.372281 - 0.644810i) q^{67} +4.37228 q^{71} +14.7446 q^{73} +(4.00000 - 6.92820i) q^{77} +(1.37228 + 2.37686i) q^{79} +(-5.00000 - 8.66025i) q^{83} +(3.37228 - 5.84096i) q^{85} +4.37228 q^{89} -11.3723 q^{91} +(0.500000 - 0.866025i) q^{95} +(2.37228 + 4.10891i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{5} - q^{7}+O(q^{10}) $ 4 * q + 2 * q^5 - q^7	$ 4 q + 2 q^{5} - q^{7} - q^{11} + q^{13} + 4 q^{17} + 4 q^{19} + 5 q^{23} - 2 q^{25} - q^{29} - 5 q^{31} - 2 q^{35} + 24 q^{37} + 12 q^{41} - 2 q^{43} - 13 q^{47} - 3 q^{49} - 6 q^{53} - 2 q^{55} + 10 q^{59} - 14 q^{61} - q^{65} - 10 q^{67} + 6 q^{71} + 36 q^{73} + 16 q^{77} - 6 q^{79} - 20 q^{83} + 2 q^{85} + 6 q^{89} - 34 q^{91} + 2 q^{95} - 2 q^{97}+O(q^{100}) $ 4 * q + 2 * q^5 - q^7 - q^11 + q^13 + 4 * q^17 + 4 * q^19 + 5 * q^23 - 2 * q^25 - q^29 - 5 * q^31 - 2 * q^35 + 24 * q^37 + 12 * q^41 - 2 * q^43 - 13 * q^47 - 3 * q^49 - 6 * q^53 - 2 * q^55 + 10 * q^59 - 14 * q^61 - q^65 - 10 * q^67 + 6 * q^71 + 36 * q^73 + 16 * q^77 - 6 * q^79 - 20 * q^83 + 2 * q^85 + 6 * q^89 - 34 * q^91 + 2 * q^95 - 2 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3240\mathbb{Z}\right)^\times$.

$n$	$1297$	$1621$	$2431$	$3161$
$\chi(n)$	$1$	$1$	$1$	$e\left(\frac{1}{3}\right)$

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
$5$	0.500000	−	0.866025i	0.223607	−	0.387298i
$5$	0.500000	−	0.866025i	0.223607	−	0.387298i
$6$		0			0
$7$	−1.68614	−	2.92048i	−0.637301	−	1.10384i	−0.986023	−	0.166612i	$-0.946717\pi$
$7$	−1.68614	−	2.92048i	−0.637301	−	1.10384i	0.348721	−	0.937226i	$-0.386616\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	1.18614	+	2.05446i	0.357635	+	0.619442i	0.987565	−	0.157210i	$-0.0502499\pi$
$11$	1.18614	+	2.05446i	0.357635	+	0.619442i	−0.629930	+	0.776652i	$0.716917\pi$
$12$		0			0
$13$	1.68614	−	2.92048i	0.467651	−	0.809996i	−0.531666	−	0.846954i	$-0.678434\pi$
$13$	1.68614	−	2.92048i	0.467651	−	0.809996i	0.999317	+	0.0369586i	$0.0117670\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	6.74456			1.63580			0.817898	−	0.575363i	$-0.195139\pi$
$17$	6.74456			1.63580			0.817898	+	0.575363i	$0.195139\pi$
$18$		0			0
$19$	1.00000			0.229416			0.114708	−	0.993399i	$-0.463407\pi$
$19$	1.00000			0.229416			0.114708	+	0.993399i	$0.463407\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	2.68614	−	4.65253i	0.560099	−	0.970120i	−0.437388	−	0.899273i	$-0.644096\pi$
$23$	2.68614	−	4.65253i	0.560099	−	0.970120i	0.997487	−	0.0708472i	$-0.0225703\pi$
$24$		0			0
$25$	−0.500000	−	0.866025i	−0.100000	−	0.173205i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	1.18614	+	2.05446i	0.220261	+	0.381503i	0.954887	−	0.296969i	$-0.0959758\pi$
$29$	1.18614	+	2.05446i	0.220261	+	0.381503i	−0.734626	+	0.678472i	$0.762642\pi$
$30$		0			0
$31$	−5.55842	+	9.62747i	−0.998322	+	1.72914i	−0.449013	+	0.893525i	$0.648224\pi$
$31$	−5.55842	+	9.62747i	−0.998322	+	1.72914i	−0.549309	+	0.835619i	$0.685109\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−3.37228			−0.570020
$36$		0			0
$37$	6.00000			0.986394			0.493197	−	0.869918i	$-0.335828\pi$
$37$	6.00000			0.986394			0.493197	+	0.869918i	$0.335828\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	0.127719	−	0.221215i	0.0199463	−	0.0345480i	−0.855880	−	0.517175i	$-0.826984\pi$
$41$	0.127719	−	0.221215i	0.0199463	−	0.0345480i	0.875826	+	0.482627i	$0.160317\pi$
$42$		0			0
$43$	2.37228	+	4.10891i	0.361770	+	0.626603i	0.988252	−	0.152832i	$-0.0488394\pi$
$43$	2.37228	+	4.10891i	0.361770	+	0.626603i	−0.626483	+	0.779435i	$0.715506\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−4.68614	−	8.11663i	−0.683544	−	1.18393i	−0.973892	−	0.227012i	$-0.927104\pi$
$47$	−4.68614	−	8.11663i	−0.683544	−	1.18393i	0.290348	−	0.956921i	$-0.406229\pi$
$48$		0			0
$49$	−2.18614	+	3.78651i	−0.312306	+	0.540930i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−10.1168			−1.38966			−0.694828	−	0.719176i	$-0.744519\pi$
$53$	−10.1168			−1.38966			−0.694828	+	0.719176i	$0.744519\pi$
$54$		0			0
$55$	2.37228			0.319878
$56$		0			0
$57$		0			0
$58$		0			0
$59$	2.50000	−	4.33013i	0.325472	−	0.563735i	−0.656136	−	0.754643i	$-0.727810\pi$
$59$	2.50000	−	4.33013i	0.325472	−	0.563735i	0.981608	+	0.190909i	$0.0611434\pi$
$60$		0			0
$61$	−6.37228	−	11.0371i	−0.815887	−	1.41316i	−0.908689	−	0.417473i	$-0.862916\pi$
$61$	−6.37228	−	11.0371i	−0.815887	−	1.41316i	0.0928022	−	0.995685i	$-0.470418\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−1.68614	−	2.92048i	−0.209140	−	0.362241i
$66$		0			0
$67$	0.372281	−	0.644810i	0.0454814	−	0.0787761i	−0.842389	−	0.538871i	$-0.818851\pi$
$67$	0.372281	−	0.644810i	0.0454814	−	0.0787761i	0.887870	+	0.460095i	$0.152184\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	4.37228			0.518894			0.259447	−	0.965757i	$-0.416460\pi$
$71$	4.37228			0.518894			0.259447	+	0.965757i	$0.416460\pi$
$72$		0			0
$73$	14.7446			1.72572			0.862860	−	0.505443i	$-0.168671\pi$
$73$	14.7446			1.72572			0.862860	+	0.505443i	$0.168671\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	4.00000	−	6.92820i	0.455842	−	0.789542i
$78$		0			0
$79$	1.37228	+	2.37686i	0.154394	+	0.267418i	0.932838	−	0.360296i	$-0.117324\pi$
$79$	1.37228	+	2.37686i	0.154394	+	0.267418i	−0.778444	+	0.627714i	$0.783991\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−5.00000	−	8.66025i	−0.548821	−	0.950586i	−0.998356	−	0.0573233i	$-0.981743\pi$
$83$	−5.00000	−	8.66025i	−0.548821	−	0.950586i	0.449534	−	0.893263i	$-0.351590\pi$
$84$		0			0
$85$	3.37228	−	5.84096i	0.365775	−	0.633541i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	4.37228			0.463461			0.231730	−	0.972780i	$-0.425561\pi$
$89$	4.37228			0.463461			0.231730	+	0.972780i	$0.425561\pi$
$90$		0			0
$91$	−11.3723			−1.19214
$92$		0			0
$93$		0			0
$94$		0			0
$95$	0.500000	−	0.866025i	0.0512989	−	0.0888523i
$96$		0			0
$97$	2.37228	+	4.10891i	0.240869	+	0.417197i	0.960962	−	0.276680i	$-0.0892344\pi$
$97$	2.37228	+	4.10891i	0.240869	+	0.417197i	−0.720093	+	0.693877i	$0.755901\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−7.93070	−	13.7364i	−0.789134	−	1.36682i	−0.926498	−	0.376300i	$-0.877196\pi$
$101$	−7.93070	−	13.7364i	−0.789134	−	1.36682i	0.137363	−	0.990521i	$-0.456137\pi$
$102$		0			0
$103$	0.0584220	−	0.101190i	0.00575649	−	0.00997053i	−0.863133	−	0.504977i	$-0.831501\pi$
$103$	0.0584220	−	0.101190i	0.00575649	−	0.00997053i	0.868889	+	0.495006i	$0.164834\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	6.00000			0.580042			0.290021	−	0.957020i	$-0.406338\pi$
$107$	6.00000			0.580042			0.290021	+	0.957020i	$0.406338\pi$
$108$		0			0
$109$	0.883156			0.0845910			0.0422955	−	0.999105i	$-0.486533\pi$
$109$	0.883156			0.0845910			0.0422955	+	0.999105i	$0.486533\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	7.37228	−	12.7692i	0.693526	−	1.20122i	−0.277149	−	0.960827i	$-0.589390\pi$
$113$	7.37228	−	12.7692i	0.693526	−	1.20122i	0.970675	−	0.240395i	$-0.0772770\pi$
$114$		0			0
$115$	−2.68614	−	4.65253i	−0.250484	−	0.433851i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−11.3723	−	19.6974i	−1.04250	−	1.80565i
$120$		0			0
$121$	2.68614	−	4.65253i	0.244195	−	0.422957i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−1.00000			−0.0894427
$126$		0			0
$127$	−12.8614			−1.14127			−0.570633	−	0.821205i	$-0.693302\pi$
$127$	−12.8614			−1.14127			−0.570633	+	0.821205i	$0.693302\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	3.12772	−	5.41737i	0.273270	−	0.473318i	−0.696427	−	0.717627i	$-0.745228\pi$
$131$	3.12772	−	5.41737i	0.273270	−	0.473318i	0.969697	+	0.244310i	$0.0785614\pi$
$132$		0			0
$133$	−1.68614	−	2.92048i	−0.146207	−	0.253238i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−2.62772	−	4.55134i	−0.224501	−	0.388847i	0.731669	−	0.681661i	$-0.238742\pi$
$137$	−2.62772	−	4.55134i	−0.224501	−	0.388847i	−0.956170	+	0.292813i	$0.905409\pi$
$138$		0			0
$139$	−7.50000	+	12.9904i	−0.636142	+	1.10183i	0.350130	+	0.936701i	$0.386137\pi$
$139$	−7.50000	+	12.9904i	−0.636142	+	1.10183i	−0.986272	+	0.165129i	$0.947196\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	8.00000			0.668994
$144$		0			0
$145$	2.37228			0.197007
$146$		0			0
$147$		0			0
$148$		0			0
$149$	9.74456	−	16.8781i	0.798306	−	1.38271i	−0.122413	−	0.992479i	$-0.539063\pi$
$149$	9.74456	−	16.8781i	0.798306	−	1.38271i	0.920719	−	0.390227i	$-0.127603\pi$
$150$		0			0
$151$	−12.1861	−	21.1070i	−0.991694	−	1.71766i	−0.607234	−	0.794523i	$-0.707721\pi$
$151$	−12.1861	−	21.1070i	−0.991694	−	1.71766i	−0.384460	−	0.923142i	$-0.625612\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	5.55842	+	9.62747i	0.446463	+	0.773297i
$156$		0			0
$157$	−11.4307	+	19.7986i	−0.912269	+	1.58010i	−0.101419	+	0.994844i	$0.532338\pi$
$157$	−11.4307	+	19.7986i	−0.912269	+	1.58010i	−0.810851	+	0.585253i	$0.800995\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−18.1168			−1.42781
$162$		0			0
$163$	9.48913			0.743246			0.371623	−	0.928384i	$-0.378801\pi$
$163$	9.48913			0.743246			0.371623	+	0.928384i	$0.378801\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−6.00000	+	10.3923i	−0.464294	+	0.804181i	−0.999169	−	0.0407502i	$-0.987025\pi$
$167$	−6.00000	+	10.3923i	−0.464294	+	0.804181i	0.534875	+	0.844931i	$0.320359\pi$
$168$		0			0
$169$	0.813859	+	1.40965i	0.0626046	+	0.108434i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	7.43070	+	12.8704i	0.564946	+	0.978515i	0.997055	+	0.0766935i	$0.0244363\pi$
$173$	7.43070	+	12.8704i	0.564946	+	0.978515i	−0.432109	+	0.901821i	$0.642230\pi$
$174$		0			0
$175$	−1.68614	+	2.92048i	−0.127460	+	0.220768i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	4.25544			0.318066			0.159033	−	0.987273i	$-0.449162\pi$
$179$	4.25544			0.318066			0.159033	+	0.987273i	$0.449162\pi$
$180$		0			0
$181$	−13.8614			−1.03031			−0.515155	−	0.857097i	$-0.672266\pi$
$181$	−13.8614			−1.03031			−0.515155	+	0.857097i	$0.672266\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	3.00000	−	5.19615i	0.220564	−	0.382029i
$186$		0			0
$187$	8.00000	+	13.8564i	0.585018	+	1.01328i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−2.81386	−	4.87375i	−0.203604	−	0.352652i	0.746083	−	0.665853i	$-0.231932\pi$
$191$	−2.81386	−	4.87375i	−0.203604	−	0.352652i	−0.949687	+	0.313201i	$0.898599\pi$
$192$		0			0
$193$	−0.372281	+	0.644810i	−0.0267974	+	0.0464145i	−0.879113	−	0.476613i	$-0.841864\pi$
$193$	−0.372281	+	0.644810i	−0.0267974	+	0.0464145i	0.852316	+	0.523028i	$0.175198\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	0.627719			0.0447231			0.0223616	−	0.999750i	$-0.492882\pi$
$197$	0.627719			0.0447231			0.0223616	+	0.999750i	$0.492882\pi$
$198$		0			0
$199$	13.4891			0.956219			0.478109	−	0.878300i	$-0.341322\pi$
$199$	13.4891			0.956219			0.478109	+	0.878300i	$0.341322\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	4.00000	−	6.92820i	0.280745	−	0.486265i
$204$		0			0
$205$	−0.127719	−	0.221215i	−0.00892026	−	0.0154503i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	1.18614	+	2.05446i	0.0820471	+	0.142110i
$210$		0			0
$211$	−8.50000	+	14.7224i	−0.585164	+	1.01353i	0.409691	+	0.912224i	$0.365637\pi$
$211$	−8.50000	+	14.7224i	−0.585164	+	1.01353i	−0.994855	+	0.101310i	$0.967697\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	4.74456			0.323576
$216$		0			0
$217$	37.4891			2.54493
$218$		0			0
$219$		0			0
$220$		0			0
$221$	11.3723	−	19.6974i	0.764982	−	1.32499i
$222$		0			0
$223$	−12.7446	−	22.0742i	−0.853439	−	1.47820i	−0.878086	−	0.478504i	$-0.841179\pi$
$223$	−12.7446	−	22.0742i	−0.853439	−	1.47820i	0.0246466	−	0.999696i	$-0.492154\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−8.74456	−	15.1460i	−0.580397	−	1.00528i	−0.995432	−	0.0954717i	$-0.969564\pi$
$227$	−8.74456	−	15.1460i	−0.580397	−	1.00528i	0.415035	−	0.909805i	$-0.363769\pi$
$228$		0			0
$229$	8.37228	−	14.5012i	0.553256	−	0.958267i	−0.444781	−	0.895639i	$-0.646718\pi$
$229$	8.37228	−	14.5012i	0.553256	−	0.958267i	0.998037	−	0.0626280i	$-0.0199482\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	14.2337			0.932480			0.466240	−	0.884658i	$-0.345608\pi$
$233$	14.2337			0.932480			0.466240	+	0.884658i	$0.345608\pi$
$234$		0			0
$235$	−9.37228			−0.611380
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−10.1168	+	17.5229i	−0.654404	+	1.13346i	0.327639	+	0.944803i	$0.393747\pi$
$239$	−10.1168	+	17.5229i	−0.654404	+	1.13346i	−0.982043	+	0.188658i	$0.939586\pi$
$240$		0			0
$241$	5.30298	+	9.18504i	0.341595	+	0.591660i	0.984729	−	0.174093i	$-0.0556995\pi$
$241$	5.30298	+	9.18504i	0.341595	+	0.591660i	−0.643134	+	0.765754i	$0.722366\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	2.18614	+	3.78651i	0.139667	+	0.241911i
$246$		0			0
$247$	1.68614	−	2.92048i	0.107287	−	0.185826i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	12.8614			0.811805			0.405902	−	0.913916i	$-0.366957\pi$
$251$	12.8614			0.811805			0.405902	+	0.913916i	$0.366957\pi$
$252$		0			0
$253$	12.7446			0.801244
$254$		0			0
$255$		0			0
$256$		0			0
$257$	0.255437	−	0.442430i	0.0159337	−	0.0275981i	−0.857949	−	0.513736i	$-0.828261\pi$
$257$	0.255437	−	0.442430i	0.0159337	−	0.0275981i	0.873882	+	0.486137i	$0.161595\pi$
$258$		0			0
$259$	−10.1168	−	17.5229i	−0.628630	−	1.08882i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	2.05842	+	3.56529i	0.126928	+	0.219845i	0.922485	−	0.386033i	$-0.126155\pi$
$263$	2.05842	+	3.56529i	0.126928	+	0.219845i	−0.795557	+	0.605879i	$0.792822\pi$
$264$		0			0
$265$	−5.05842	+	8.76144i	−0.310736	+	0.538211i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	5.11684			0.311979			0.155990	−	0.987759i	$-0.450143\pi$
$269$	5.11684			0.311979			0.155990	+	0.987759i	$0.450143\pi$
$270$		0			0
$271$	−9.25544			−0.562228			−0.281114	−	0.959674i	$-0.590704\pi$
$271$	−9.25544			−0.562228			−0.281114	+	0.959674i	$0.590704\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	1.18614	−	2.05446i	0.0715270	−	0.123888i
$276$		0			0
$277$	−0.0584220	−	0.101190i	−0.00351024	−	0.00607991i	0.864265	−	0.503037i	$-0.167784\pi$
$277$	−0.0584220	−	0.101190i	−0.00351024	−	0.00607991i	−0.867775	+	0.496957i	$0.834451\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	8.68614	+	15.0448i	0.518172	+	0.897500i	0.999777	+	0.0211116i	$0.00672052\pi$
$281$	8.68614	+	15.0448i	0.518172	+	0.897500i	−0.481605	+	0.876388i	$0.659946\pi$
$282$		0			0
$283$	−8.37228	+	14.5012i	−0.497680	+	0.862008i	−0.999996	−	0.00267630i	$-0.999148\pi$
$283$	−8.37228	+	14.5012i	−0.497680	+	0.862008i	0.502316	+	0.864684i	$0.332481\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−0.861407			−0.0508472
$288$		0			0
$289$	28.4891			1.67583
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−7.43070	+	12.8704i	−0.434106	+	0.751894i	−0.997222	−	0.0744837i	$-0.976269\pi$
$293$	−7.43070	+	12.8704i	−0.434106	+	0.751894i	0.563116	+	0.826378i	$0.309602\pi$
$294$		0			0
$295$	−2.50000	−	4.33013i	−0.145556	−	0.252110i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−9.05842	−	15.6896i	−0.523862	−	0.907356i
$300$		0			0
$301$	8.00000	−	13.8564i	0.461112	−	0.798670i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−12.7446			−0.729752
$306$		0			0
$307$	0.744563			0.0424944			0.0212472	−	0.999774i	$-0.493236\pi$
$307$	0.744563			0.0424944			0.0212472	+	0.999774i	$0.493236\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	8.93070	−	15.4684i	0.506414	−	0.877134i	−0.493559	−	0.869712i	$-0.664304\pi$
$311$	8.93070	−	15.4684i	0.506414	−	0.877134i	0.999972	−	0.00742183i	$-0.00236246\pi$
$312$		0			0
$313$	2.00000	+	3.46410i	0.113047	+	0.195803i	0.916997	−	0.398894i	$-0.130606\pi$
$313$	2.00000	+	3.46410i	0.113047	+	0.195803i	−0.803951	+	0.594696i	$0.797272\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−5.31386	−	9.20387i	−0.298456	−	0.516941i	0.677327	−	0.735682i	$-0.263138\pi$
$317$	−5.31386	−	9.20387i	−0.298456	−	0.516941i	−0.975783	+	0.218741i	$0.929805\pi$
$318$		0			0
$319$	−2.81386	+	4.87375i	−0.157546	+	0.272877i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	6.74456			0.375278
$324$		0			0
$325$	−3.37228			−0.187061
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−15.8030	+	27.3716i	−0.871247	+	1.50904i
$330$		0			0
$331$	−6.81386	−	11.8020i	−0.374524	−	0.648694i	0.615732	−	0.787956i	$-0.288860\pi$
$331$	−6.81386	−	11.8020i	−0.374524	−	0.648694i	−0.990256	+	0.139262i	$0.955527\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−0.372281	−	0.644810i	−0.0203399	−	0.0352297i
$336$		0			0
$337$	5.37228	−	9.30506i	0.292647	−	0.506879i	−0.681788	−	0.731550i	$-0.738797\pi$
$337$	5.37228	−	9.30506i	0.292647	−	0.506879i	0.974435	+	0.224671i	$0.0721306\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−26.3723			−1.42814
$342$		0			0
$343$	−8.86141			−0.478471
$344$		0			0
$345$		0			0
$346$		0			0
$347$	2.11684	−	3.66648i	0.113638	−	0.196827i	−0.803596	−	0.595175i	$-0.797083\pi$
$347$	2.11684	−	3.66648i	0.113638	−	0.196827i	0.917235	+	0.398348i	$0.130416\pi$
$348$		0			0
$349$	9.81386	+	16.9981i	0.525324	+	0.909888i	0.999565	+	0.0294926i	$0.00938915\pi$
$349$	9.81386	+	16.9981i	0.525324	+	0.909888i	−0.474241	+	0.880395i	$0.657278\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−10.4891	−	18.1677i	−0.558280	−	0.966969i	−0.997640	−	0.0686581i	$-0.978128\pi$
$353$	−10.4891	−	18.1677i	−0.558280	−	0.966969i	0.439360	−	0.898311i	$-0.355205\pi$
$354$		0			0
$355$	2.18614	−	3.78651i	0.116028	−	0.200967i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	16.3723			0.864096			0.432048	−	0.901851i	$-0.357791\pi$
$359$	16.3723			0.864096			0.432048	+	0.901851i	$0.357791\pi$
$360$		0			0
$361$	−18.0000			−0.947368
$362$		0			0
$363$		0			0
$364$		0			0
$365$	7.37228	−	12.7692i	0.385883	−	0.668369i
$366$		0			0
$367$	15.1168	+	26.1831i	0.789093	+	1.36675i	0.926523	+	0.376237i	$0.122782\pi$
$367$	15.1168	+	26.1831i	0.789093	+	1.36675i	−0.137430	+	0.990511i	$0.543884\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	17.0584	+	29.5461i	0.885629	+	1.53395i
$372$		0			0
$373$	5.37228	−	9.30506i	0.278166	−	0.481798i	−0.692763	−	0.721165i	$-0.743607\pi$
$373$	5.37228	−	9.30506i	0.278166	−	0.481798i	0.970929	+	0.239368i	$0.0769401\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	8.00000			0.412021
$378$		0			0
$379$	−26.3505			−1.35354			−0.676768	−	0.736196i	$-0.736620\pi$
$379$	−26.3505			−1.35354			−0.676768	+	0.736196i	$0.736620\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−9.68614	+	16.7769i	−0.494939	+	0.857259i	−0.999983	−	0.00583450i	$-0.998143\pi$
$383$	−9.68614	+	16.7769i	−0.494939	+	0.857259i	0.505044	+	0.863093i	$0.331476\pi$
$384$		0			0
$385$	−4.00000	−	6.92820i	−0.203859	−	0.353094i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	10.3723	+	17.9653i	0.525896	+	0.910878i	0.999545	+	0.0301643i	$0.00960306\pi$
$389$	10.3723	+	17.9653i	0.525896	+	0.910878i	−0.473649	+	0.880713i	$0.657064\pi$
$390$		0			0
$391$	18.1168	−	31.3793i	0.916208	−	1.58692i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	2.74456			0.138094
$396$		0			0
$397$	34.0000			1.70641			0.853206	−	0.521575i	$-0.174655\pi$
$397$	34.0000			1.70641			0.853206	+	0.521575i	$0.174655\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	18.1753	−	31.4805i	0.907629	−	1.57206i	0.0902813	−	0.995916i	$-0.471223\pi$
$401$	18.1753	−	31.4805i	0.907629	−	1.57206i	0.817348	−	0.576144i	$-0.195443\pi$
$402$		0			0
$403$	18.7446	+	32.4665i	0.933733	+	1.61727i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	7.11684	+	12.3267i	0.352769	+	0.611014i
$408$		0			0
$409$	10.6861	−	18.5089i	0.528396	−	0.915208i	−0.471056	−	0.882103i	$-0.656127\pi$
$409$	10.6861	−	18.5089i	0.528396	−	0.915208i	0.999452	−	0.0331049i	$-0.0105396\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−16.8614			−0.829696
$414$		0			0
$415$	−10.0000			−0.490881
$416$		0			0
$417$		0			0
$418$		0			0
$419$	17.4891	−	30.2921i	0.854400	−	1.47986i	−0.0228011	−	0.999740i	$-0.507258\pi$
$419$	17.4891	−	30.2921i	0.854400	−	1.47986i	0.877201	−	0.480124i	$-0.159408\pi$
$420$		0			0
$421$	−14.1861	−	24.5711i	−0.691390	−	1.19752i	−0.971382	−	0.237521i	$-0.923665\pi$
$421$	−14.1861	−	24.5711i	−0.691390	−	1.19752i	0.279992	−	0.960002i	$-0.409668\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−3.37228	−	5.84096i	−0.163580	−	0.283328i
$426$		0			0
$427$	−21.4891	+	37.2203i	−1.03993	+	1.80121i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−9.62772			−0.463751			−0.231875	−	0.972745i	$-0.574486\pi$
$431$	−9.62772			−0.463751			−0.231875	+	0.972745i	$0.574486\pi$
$432$		0			0
$433$	−41.2119			−1.98052			−0.990260	−	0.139233i	$-0.955536\pi$
$433$	−41.2119			−1.98052			−0.990260	+	0.139233i	$0.955536\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	2.68614	−	4.65253i	0.128496	−	0.222561i
$438$		0			0
$439$	16.0475	+	27.7952i	0.765908	+	1.32659i	0.939765	+	0.341820i	$0.111043\pi$
$439$	16.0475	+	27.7952i	0.765908	+	1.32659i	−0.173858	+	0.984771i	$0.555623\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−14.4891	−	25.0959i	−0.688399	−	1.19234i	−0.972356	−	0.233505i	$-0.924980\pi$
$443$	−14.4891	−	25.0959i	−0.688399	−	1.19234i	0.283956	−	0.958837i	$-0.408353\pi$
$444$		0			0
$445$	2.18614	−	3.78651i	0.103633	−	0.179498i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−34.7228			−1.63867			−0.819335	−	0.573314i	$-0.805657\pi$
$449$	−34.7228			−1.63867			−0.819335	+	0.573314i	$0.805657\pi$
$450$		0			0
$451$	0.605969			0.0285340
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−5.68614	+	9.84868i	−0.266570	+	0.461713i
$456$		0			0
$457$	2.88316	+	4.99377i	0.134868	+	0.233599i	0.925547	−	0.378632i	$-0.123606\pi$
$457$	2.88316	+	4.99377i	0.134868	+	0.233599i	−0.790679	+	0.612231i	$0.790272\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	15.5584	+	26.9480i	0.724628	+	1.25509i	0.959127	+	0.282976i	$0.0913217\pi$
$461$	15.5584	+	26.9480i	0.724628	+	1.25509i	−0.234499	+	0.972116i	$0.575345\pi$
$462$		0			0
$463$	−3.43070	+	5.94215i	−0.159438	+	0.276155i	−0.934666	−	0.355526i	$-0.884302\pi$
$463$	−3.43070	+	5.94215i	−0.159438	+	0.276155i	0.775228	+	0.631682i	$0.217635\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−16.2337			−0.751205			−0.375603	−	0.926781i	$-0.622564\pi$
$467$	−16.2337			−0.751205			−0.375603	+	0.926781i	$0.622564\pi$
$468$		0			0
$469$	−2.51087			−0.115941
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−5.62772	+	9.74749i	−0.258763	+	0.448190i
$474$		0			0
$475$	−0.500000	−	0.866025i	−0.0229416	−	0.0397360i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−3.44158	−	5.96099i	−0.157250	−	0.272364i	0.776626	−	0.629962i	$-0.216929\pi$
$479$	−3.44158	−	5.96099i	−0.157250	−	0.272364i	−0.933876	+	0.357597i	$0.883596\pi$
$480$		0			0
$481$	10.1168	−	17.5229i	0.461288	−	0.798975i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	4.74456			0.215439
$486$		0			0
$487$	19.6060			0.888431			0.444216	−	0.895920i	$-0.353482\pi$
$487$	19.6060			0.888431			0.444216	+	0.895920i	$0.353482\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−12.9891	+	22.4978i	−0.586191	+	1.01531i	0.408535	+	0.912743i	$0.366040\pi$
$491$	−12.9891	+	22.4978i	−0.586191	+	1.01531i	−0.994726	+	0.102570i	$0.967293\pi$
$492$		0			0
$493$	8.00000	+	13.8564i	0.360302	+	0.624061i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−7.37228	−	12.7692i	−0.330692	−	0.572775i
$498$		0			0
$499$	−9.87228	+	17.0993i	−0.441944	+	0.765469i	−0.997834	−	0.0657866i	$-0.979044\pi$
$499$	−9.87228	+	17.0993i	−0.441944	+	0.765469i	0.555890	+	0.831256i	$0.312378\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−10.9783			−0.489496			−0.244748	−	0.969587i	$-0.578705\pi$
$503$	−10.9783			−0.489496			−0.244748	+	0.969587i	$0.578705\pi$
$504$		0			0
$505$	−15.8614			−0.705823
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−19.0000	+	32.9090i	−0.842160	+	1.45866i	0.0459045	+	0.998946i	$0.485383\pi$
$509$	−19.0000	+	32.9090i	−0.842160	+	1.45866i	−0.888065	+	0.459718i	$0.847950\pi$
$510$		0			0
$511$	−24.8614	−	43.0612i	−1.09980	−	1.90492i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−0.0584220	−	0.101190i	−0.00257438	−	0.00445896i
$516$		0			0
$517$	11.1168	−	19.2549i	0.488918	−	0.846831i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	0.116844			0.00511903			0.00255951	−	0.999997i	$-0.499185\pi$
$521$	0.116844			0.00511903			0.00255951	+	0.999997i	$0.499185\pi$
$522$		0			0
$523$	−40.2337			−1.75930			−0.879648	−	0.475625i	$-0.842222\pi$
$523$	−40.2337			−1.75930			−0.879648	+	0.475625i	$0.842222\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−37.4891	+	64.9331i	−1.63305	+	2.82853i
$528$		0			0
$529$	−2.93070	−	5.07613i	−0.127422	−	0.220701i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−0.430703	−	0.746000i	−0.0186558	−	0.0323128i
$534$		0			0
$535$	3.00000	−	5.19615i	0.129701	−	0.224649i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−10.3723			−0.446766
$540$		0			0
$541$	13.3505			0.573984			0.286992	−	0.957933i	$-0.407345\pi$
$541$	13.3505			0.573984			0.286992	+	0.957933i	$0.407345\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	0.441578	−	0.764836i	0.0189151	−	0.0327620i
$546$		0			0
$547$	20.0000	+	34.6410i	0.855138	+	1.48114i	0.876517	+	0.481371i	$0.159861\pi$
$547$	20.0000	+	34.6410i	0.855138	+	1.48114i	−0.0213785	+	0.999771i	$0.506805\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	1.18614	+	2.05446i	0.0505313	+	0.0875228i
$552$		0			0
$553$	4.62772	−	8.01544i	0.196791	−	0.340851i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	31.6060			1.33919			0.669594	−	0.742727i	$-0.266468\pi$
$557$	31.6060			1.33919			0.669594	+	0.742727i	$0.266468\pi$
$558$		0			0
$559$	16.0000			0.676728
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−5.11684	+	8.86263i	−0.215649	+	0.373515i	−0.953473	−	0.301478i	$-0.902520\pi$
$563$	−5.11684	+	8.86263i	−0.215649	+	0.373515i	0.737824	+	0.674993i	$0.235853\pi$
$564$		0			0
$565$	−7.37228	−	12.7692i	−0.310154	−	0.537203i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	4.38316	+	7.59185i	0.183751	+	0.318267i	0.943155	−	0.332353i	$-0.107843\pi$
$569$	4.38316	+	7.59185i	0.183751	+	0.318267i	−0.759404	+	0.650620i	$0.774509\pi$
$570$		0			0
$571$	1.55842	−	2.69927i	0.0652179	−	0.112961i	−0.831573	−	0.555416i	$-0.812559\pi$
$571$	1.55842	−	2.69927i	0.0652179	−	0.112961i	0.896791	+	0.442455i	$0.145892\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−5.37228			−0.224040
$576$		0			0
$577$	−9.48913			−0.395037			−0.197519	−	0.980299i	$-0.563288\pi$
$577$	−9.48913			−0.395037			−0.197519	+	0.980299i	$0.563288\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−16.8614	+	29.2048i	−0.699529	+	1.21162i
$582$		0			0
$583$	−12.0000	−	20.7846i	−0.496989	−	0.860811i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	15.0000	+	25.9808i	0.619116	+	1.07234i	0.989647	+	0.143521i	$0.0458424\pi$
$587$	15.0000	+	25.9808i	0.619116	+	1.07234i	−0.370531	+	0.928820i	$0.620824\pi$
$588$		0			0
$589$	−5.55842	+	9.62747i	−0.229031	+	0.396693i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	28.9783			1.18999			0.594997	−	0.803728i	$-0.297153\pi$
$593$	28.9783			1.18999			0.594997	+	0.803728i	$0.297153\pi$
$594$		0			0
$595$	−22.7446			−0.932436
$596$		0			0
$597$		0			0
$598$		0			0
$599$	7.67527	−	13.2940i	0.313603	−	0.543176i	−0.665537	−	0.746365i	$-0.731797\pi$
$599$	7.67527	−	13.2940i	0.313603	−	0.543176i	0.979140	+	0.203189i	$0.0651306\pi$
$600$		0			0
$601$	−6.50000	−	11.2583i	−0.265141	−	0.459237i	0.702460	−	0.711723i	$-0.252085\pi$
$601$	−6.50000	−	11.2583i	−0.265141	−	0.459237i	−0.967600	+	0.252486i	$0.918752\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−2.68614	−	4.65253i	−0.109207	−	0.189152i
$606$		0			0
$607$	22.3723	−	38.7499i	0.908063	−	1.57281i	0.0913107	−	0.995822i	$-0.470894\pi$
$607$	22.3723	−	38.7499i	0.908063	−	1.57281i	0.816752	−	0.576989i	$-0.195772\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−31.6060			−1.27864
$612$		0			0
$613$	18.3505			0.741171			0.370586	−	0.928798i	$-0.379157\pi$
$613$	18.3505			0.741171			0.370586	+	0.928798i	$0.379157\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	6.86141	−	11.8843i	0.276230	−	0.478444i	−0.694215	−	0.719768i	$-0.744248\pi$
$617$	6.86141	−	11.8843i	0.276230	−	0.478444i	0.970445	+	0.241324i	$0.0775816\pi$
$618$		0			0
$619$	16.3139	+	28.2564i	0.655709	+	1.13572i	0.981715	+	0.190354i	$0.0609637\pi$
$619$	16.3139	+	28.2564i	0.655709	+	1.13572i	−0.326006	+	0.945368i	$0.605703\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−7.37228	−	12.7692i	−0.295364	−	0.511586i
$624$		0			0
$625$	−0.500000	+	0.866025i	−0.0200000	+	0.0346410i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	40.4674			1.61354
$630$		0			0
$631$	−42.3723			−1.68681			−0.843407	−	0.537275i	$-0.819454\pi$
$631$	−42.3723			−1.68681			−0.843407	+	0.537275i	$0.819454\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−6.43070	+	11.1383i	−0.255195	+	0.442010i
$636$		0			0
$637$	7.37228	+	12.7692i	0.292100	+	0.505933i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	21.3030	+	36.8979i	0.841417	+	1.45738i	0.888697	+	0.458496i	$0.151612\pi$
$641$	21.3030	+	36.8979i	0.841417	+	1.45738i	−0.0472793	+	0.998882i	$0.515055\pi$
$642$		0			0
$643$	−16.3723	+	28.3576i	−0.645660	+	1.11832i	0.338489	+	0.940970i	$0.390084\pi$
$643$	−16.3723	+	28.3576i	−0.645660	+	1.11832i	−0.984149	+	0.177345i	$0.943249\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	31.7228			1.24715			0.623576	−	0.781763i	$-0.285679\pi$
$647$	31.7228			1.24715			0.623576	+	0.781763i	$0.285679\pi$
$648$		0			0
$649$	11.8614			0.465601
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−6.62772	+	11.4795i	−0.259363	+	0.449229i	−0.966071	−	0.258275i	$-0.916846\pi$
$653$	−6.62772	+	11.4795i	−0.259363	+	0.449229i	0.706709	+	0.707505i	$0.250179\pi$
$654$		0			0
$655$	−3.12772	−	5.41737i	−0.122210	−	0.211674i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	19.2921	+	33.4149i	0.751514	+	1.30166i	0.947089	+	0.320972i	$0.104009\pi$
$659$	19.2921	+	33.4149i	0.751514	+	1.30166i	−0.195575	+	0.980689i	$0.562657\pi$
$660$		0			0
$661$	−9.04755	+	15.6708i	−0.351909	+	0.609524i	−0.986584	−	0.163255i	$-0.947801\pi$
$661$	−9.04755	+	15.6708i	−0.351909	+	0.609524i	0.634675	+	0.772779i	$0.281134\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−3.37228			−0.130771
$666$		0			0
$667$	12.7446			0.493471
$668$		0			0
$669$		0			0
$670$		0			0
$671$	15.1168	−	26.1831i	0.583579	−	1.01079i
$672$		0			0
$673$	11.7446	+	20.3422i	0.452720	+	0.784133i	0.998554	−	0.0537598i	$-0.0171205\pi$
$673$	11.7446	+	20.3422i	0.452720	+	0.784133i	−0.545834	+	0.837893i	$0.683787\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−1.94158	−	3.36291i	−0.0746209	−	0.129247i	0.826300	−	0.563230i	$-0.190441\pi$
$677$	−1.94158	−	3.36291i	−0.0746209	−	0.129247i	−0.900921	+	0.433982i	$0.857108\pi$
$678$		0			0
$679$	8.00000	−	13.8564i	0.307012	−	0.531760i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−3.76631			−0.144114			−0.0720570	−	0.997401i	$-0.522956\pi$
$683$	−3.76631			−0.144114			−0.0720570	+	0.997401i	$0.522956\pi$
$684$		0			0
$685$	−5.25544			−0.200800
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−17.0584	+	29.5461i	−0.649874	+	1.12561i
$690$		0			0
$691$	5.17527	+	8.96382i	0.196876	+	0.341000i	0.947514	−	0.319714i	$-0.103587\pi$
$691$	5.17527	+	8.96382i	0.196876	+	0.341000i	−0.750638	+	0.660714i	$0.770254\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	7.50000	+	12.9904i	0.284491	+	0.492753i
$696$		0			0
$697$	0.861407	−	1.49200i	0.0326281	−	0.0565135i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−9.11684			−0.344338			−0.172169	−	0.985067i	$-0.555078\pi$
$701$	−9.11684			−0.344338			−0.172169	+	0.985067i	$0.555078\pi$
$702$		0			0
$703$	6.00000			0.226294
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−26.7446	+	46.3229i	−1.00583	+	1.74215i
$708$		0			0
$709$	25.2337	+	43.7060i	0.947671	+	1.64141i	0.750313	+	0.661083i	$0.229903\pi$
$709$	25.2337	+	43.7060i	0.947671	+	1.64141i	0.197358	+	0.980331i	$0.436764\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	29.8614	+	51.7215i	1.11832	+	1.93698i
$714$		0			0
$715$	4.00000	−	6.92820i	0.149592	−	0.259100i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	50.6060			1.88728			0.943642	−	0.330968i	$-0.107375\pi$
$719$	50.6060			1.88728			0.943642	+	0.330968i	$0.107375\pi$
$720$		0			0
$721$	−0.394031			−0.0146745
$722$		0			0
$723$		0			0
$724$		0			0
$725$	1.18614	−	2.05446i	0.0440522	−	0.0763006i
$726$		0			0
$727$	14.9416	+	25.8796i	0.554152	+	0.959820i	0.997969	+	0.0637025i	$0.0202909\pi$
$727$	14.9416	+	25.8796i	0.554152	+	0.959820i	−0.443816	+	0.896118i	$0.646376\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	16.0000	+	27.7128i	0.591781	+	1.02500i
$732$		0			0
$733$	−8.48913	+	14.7036i	−0.313553	+	0.543090i	−0.979129	−	0.203240i	$-0.934853\pi$
$733$	−8.48913	+	14.7036i	−0.313553	+	0.543090i	0.665576	+	0.746330i	$0.268186\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	1.76631			0.0650629
$738$		0			0
$739$	15.8614			0.583471			0.291736	−	0.956499i	$-0.405767\pi$
$739$	15.8614			0.583471			0.291736	+	0.956499i	$0.405767\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−3.25544	+	5.63858i	−0.119430	+	0.206860i	−0.919542	−	0.392992i	$-0.871440\pi$
$743$	−3.25544	+	5.63858i	−0.119430	+	0.206860i	0.800112	+	0.599851i	$0.204774\pi$
$744$		0			0
$745$	−9.74456	−	16.8781i	−0.357013	−	0.618365i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−10.1168	−	17.5229i	−0.369661	−	0.640272i
$750$		0			0
$751$	1.88316	−	3.26172i	0.0687173	−	0.119022i	−0.829620	−	0.558329i	$-0.811443\pi$
$751$	1.88316	−	3.26172i	0.0687173	−	0.119022i	0.898337	+	0.439307i	$0.144776\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−24.3723			−0.886998
$756$		0			0
$757$	9.88316			0.359209			0.179605	−	0.983739i	$-0.442518\pi$
$757$	9.88316			0.359209			0.179605	+	0.983739i	$0.442518\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	8.61684	−	14.9248i	0.312360	−	0.541024i	−0.666513	−	0.745494i	$-0.732214\pi$
$761$	8.61684	−	14.9248i	0.312360	−	0.541024i	0.978873	+	0.204470i	$0.0655470\pi$
$762$		0			0
$763$	−1.48913	−	2.57924i	−0.0539100	−	0.0933748i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−8.43070	−	14.6024i	−0.304415	−	0.527262i
$768$		0			0
$769$	−9.67527	+	16.7581i	−0.348899	+	0.604311i	−0.986054	−	0.166425i	$-0.946778\pi$
$769$	−9.67527	+	16.7581i	−0.348899	+	0.604311i	0.637155	+	0.770736i	$0.280111\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−18.0000			−0.647415			−0.323708	−	0.946157i	$-0.604929\pi$
$773$	−18.0000			−0.647415			−0.323708	+	0.946157i	$0.604929\pi$
$774$		0			0
$775$	11.1168			0.399329
$776$		0			0
$777$		0			0
$778$		0			0
$779$	0.127719	−	0.221215i	0.00457600	−	0.00792586i
$780$		0			0
$781$	5.18614	+	8.98266i	0.185575	+	0.321425i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	11.4307	+	19.7986i	0.407979	+	0.706641i
$786$		0			0
$787$	20.8614	−	36.1330i	0.743629	−	1.28800i	−0.207204	−	0.978298i	$-0.566436\pi$
$787$	20.8614	−	36.1330i	0.743629	−	1.28800i	0.950833	−	0.309705i	$-0.100230\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−49.7228			−1.76794
$792$		0			0
$793$	−42.9783			−1.52620
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−10.1168	+	17.5229i	−0.358357	+	0.620693i	−0.987687	−	0.156446i	$-0.949996\pi$
$797$	−10.1168	+	17.5229i	−0.358357	+	0.620693i	0.629330	+	0.777139i	$0.283330\pi$
$798$

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:	\( N \)	\(=\)	\( 3240 = 2^{3} \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3240.q (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.500000 - 0.866025i) q^{5} +(-1.68614 - 2.92048i) q^{7} +O(q^{10})\)	\(q+(0.500000 - 0.866025i) q^{5} +(-1.68614 - 2.92048i) q^{7} +(1.18614 + 2.05446i) q^{11} +(1.68614 - 2.92048i) q^{13} +6.74456 q^{17} +1.00000 q^{19} +(2.68614 - 4.65253i) q^{23} +(-0.500000 - 0.866025i) q^{25} +(1.18614 + 2.05446i) q^{29} +(-5.55842 + 9.62747i) q^{31} -3.37228 q^{35} +6.00000 q^{37} +(0.127719 - 0.221215i) q^{41} +(2.37228 + 4.10891i) q^{43} +(-4.68614 - 8.11663i) q^{47} +(-2.18614 + 3.78651i) q^{49} -10.1168 q^{53} +2.37228 q^{55} +(2.50000 - 4.33013i) q^{59} +(-6.37228 - 11.0371i) q^{61} +(-1.68614 - 2.92048i) q^{65} +(0.372281 - 0.644810i) q^{67} +4.37228 q^{71} +14.7446 q^{73} +(4.00000 - 6.92820i) q^{77} +(1.37228 + 2.37686i) q^{79} +(-5.00000 - 8.66025i) q^{83} +(3.37228 - 5.84096i) q^{85} +4.37228 q^{89} -11.3723 q^{91} +(0.500000 - 0.866025i) q^{95} +(2.37228 + 4.10891i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{5} - q^{7}+O(q^{10}) \) 4 * q + 2 * q^5 - q^7	\( 4 q + 2 q^{5} - q^{7} - q^{11} + q^{13} + 4 q^{17} + 4 q^{19} + 5 q^{23} - 2 q^{25} - q^{29} - 5 q^{31} - 2 q^{35} + 24 q^{37} + 12 q^{41} - 2 q^{43} - 13 q^{47} - 3 q^{49} - 6 q^{53} - 2 q^{55} + 10 q^{59} - 14 q^{61} - q^{65} - 10 q^{67} + 6 q^{71} + 36 q^{73} + 16 q^{77} - 6 q^{79} - 20 q^{83} + 2 q^{85} + 6 q^{89} - 34 q^{91} + 2 q^{95} - 2 q^{97}+O(q^{100}) \) 4 * q + 2 * q^5 - q^7 - q^11 + q^13 + 4 * q^17 + 4 * q^19 + 5 * q^23 - 2 * q^25 - q^29 - 5 * q^31 - 2 * q^35 + 24 * q^37 + 12 * q^41 - 2 * q^43 - 13 * q^47 - 3 * q^49 - 6 * q^53 - 2 * q^55 + 10 * q^59 - 14 * q^61 - q^65 - 10 * q^67 + 6 * q^71 + 36 * q^73 + 16 * q^77 - 6 * q^79 - 20 * q^83 + 2 * q^85 + 6 * q^89 - 34 * q^91 + 2 * q^95 - 2 * q^97

Introduction

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