LMFDB - Embedded newform 3240.2.q.b.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:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1080)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			1081.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	3240.1081
Dual form			3240.2.q.b.2161.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.500000 + 0.866025i) q^{5} +(-1.00000 - 1.73205i) q^{7} +(-2.00000 - 3.46410i) q^{11} +(1.00000 - 1.73205i) q^{13} +5.00000 q^{17} -5.00000 q^{19} +(-0.500000 + 0.866025i) q^{23} +(-0.500000 - 0.866025i) q^{25} +(1.00000 + 1.73205i) q^{29} +(-3.50000 + 6.06218i) q^{31} +2.00000 q^{35} -6.00000 q^{37} +(-2.00000 - 3.46410i) q^{43} +(-2.00000 - 3.46410i) q^{47} +(1.50000 - 2.59808i) q^{49} +9.00000 q^{53} +4.00000 q^{55} +(-7.00000 + 12.1244i) q^{59} +(5.50000 + 9.52628i) q^{61} +(1.00000 + 1.73205i) q^{65} +(-7.00000 + 12.1244i) q^{67} -12.0000 q^{73} +(-4.00000 + 6.92820i) q^{77} +(1.50000 + 2.59808i) q^{79} +(0.500000 + 0.866025i) q^{83} +(-2.50000 + 4.33013i) q^{85} -4.00000 q^{91} +(2.50000 - 4.33013i) q^{95} +(-8.00000 - 13.8564i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{5} - 2 q^{7} - 4 q^{11} + 2 q^{13} + 10 q^{17} - 10 q^{19} - q^{23} - q^{25} + 2 q^{29} - 7 q^{31} + 4 q^{35} - 12 q^{37} - 4 q^{43} - 4 q^{47} + 3 q^{49} + 18 q^{53} + 8 q^{55} - 14 q^{59}+ \cdots - 16 q^{97}+O(q^{100}) $ 2 * q - q^5 - 2 * q^7 - 4 * q^11 + 2 * q^13 + 10 * q^17 - 10 * q^19 - q^23 - q^25 + 2 * q^29 - 7 * q^31 + 4 * q^35 - 12 * q^37 - 4 * q^43 - 4 * q^47 + 3 * q^49 + 18 * q^53 + 8 * q^55 - 14 * q^59 + 11 * q^61 + 2 * q^65 - 14 * q^67 - 24 * q^73 - 8 * q^77 + 3 * q^79 + q^83 - 5 * q^85 - 8 * q^91 + 5 * q^95 - 16 * q^97

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
$7$	−1.00000	−	1.73205i	−0.377964	−	0.654654i	0.612801	−	0.790237i	$-0.290043\pi$
$7$	−1.00000	−	1.73205i	−0.377964	−	0.654654i	−0.990766	+	0.135583i	$0.956709\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−2.00000	−	3.46410i	−0.603023	−	1.04447i	−0.992361	−	0.123371i	$-0.960630\pi$
$11$	−2.00000	−	3.46410i	−0.603023	−	1.04447i	0.389338	−	0.921095i	$-0.372704\pi$
$12$		0			0
$13$	1.00000	−	1.73205i	0.277350	−	0.480384i	−0.693375	−	0.720577i	$-0.743877\pi$
$13$	1.00000	−	1.73205i	0.277350	−	0.480384i	0.970725	+	0.240192i	$0.0772105\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	5.00000			1.21268			0.606339	−	0.795206i	$-0.292637\pi$
$17$	5.00000			1.21268			0.606339	+	0.795206i	$0.292637\pi$
$18$		0			0
$19$	−5.00000			−1.14708			−0.573539	−	0.819178i	$-0.694430\pi$
$19$	−5.00000			−1.14708			−0.573539	+	0.819178i	$0.694430\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−0.500000	+	0.866025i	−0.104257	+	0.180579i	−0.913434	−	0.406986i	$-0.866580\pi$
$23$	−0.500000	+	0.866025i	−0.104257	+	0.180579i	0.809177	+	0.587565i	$0.199913\pi$
$24$		0			0
$25$	−0.500000	−	0.866025i	−0.100000	−	0.173205i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	1.00000	+	1.73205i	0.185695	+	0.321634i	0.943811	−	0.330487i	$-0.107213\pi$
$29$	1.00000	+	1.73205i	0.185695	+	0.321634i	−0.758115	+	0.652121i	$0.773880\pi$
$30$		0			0
$31$	−3.50000	+	6.06218i	−0.628619	+	1.08880i	0.359211	+	0.933257i	$0.383046\pi$
$31$	−3.50000	+	6.06218i	−0.628619	+	1.08880i	−0.987829	+	0.155543i	$0.950287\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	2.00000			0.338062
$36$		0			0
$37$	−6.00000			−0.986394			−0.493197	−	0.869918i	$-0.664172\pi$
$37$	−6.00000			−0.986394			−0.493197	+	0.869918i	$0.664172\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$41$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$42$		0			0
$43$	−2.00000	−	3.46410i	−0.304997	−	0.528271i	0.672264	−	0.740312i	$-0.265322\pi$
$43$	−2.00000	−	3.46410i	−0.304997	−	0.528271i	−0.977261	+	0.212041i	$0.931989\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−2.00000	−	3.46410i	−0.291730	−	0.505291i	0.682489	−	0.730896i	$-0.260898\pi$
$47$	−2.00000	−	3.46410i	−0.291730	−	0.505291i	−0.974219	+	0.225605i	$0.927564\pi$
$48$		0			0
$49$	1.50000	−	2.59808i	0.214286	−	0.371154i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	9.00000			1.23625			0.618123	−	0.786082i	$-0.287894\pi$
$53$	9.00000			1.23625			0.618123	+	0.786082i	$0.287894\pi$
$54$		0			0
$55$	4.00000			0.539360
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−7.00000	+	12.1244i	−0.911322	+	1.57846i	−0.0991242	+	0.995075i	$0.531604\pi$
$59$	−7.00000	+	12.1244i	−0.911322	+	1.57846i	−0.812198	+	0.583382i	$0.801729\pi$
$60$		0			0
$61$	5.50000	+	9.52628i	0.704203	+	1.21972i	0.966978	+	0.254858i	$0.0820288\pi$
$61$	5.50000	+	9.52628i	0.704203	+	1.21972i	−0.262776	+	0.964857i	$0.584638\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	1.00000	+	1.73205i	0.124035	+	0.214834i
$66$		0			0
$67$	−7.00000	+	12.1244i	−0.855186	+	1.48123i	0.0212861	+	0.999773i	$0.493224\pi$
$67$	−7.00000	+	12.1244i	−0.855186	+	1.48123i	−0.876472	+	0.481452i	$0.840109\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$		0			0
$73$	−12.0000			−1.40449			−0.702247	−	0.711934i	$-0.747820\pi$
$73$	−12.0000			−1.40449			−0.702247	+	0.711934i	$0.747820\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−4.00000	+	6.92820i	−0.455842	+	0.789542i
$78$		0			0
$79$	1.50000	+	2.59808i	0.168763	+	0.292306i	0.937985	−	0.346675i	$-0.112689\pi$
$79$	1.50000	+	2.59808i	0.168763	+	0.292306i	−0.769222	+	0.638982i	$0.779356\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	0.500000	+	0.866025i	0.0548821	+	0.0950586i	0.892161	−	0.451717i	$-0.149188\pi$
$83$	0.500000	+	0.866025i	0.0548821	+	0.0950586i	−0.837279	+	0.546776i	$0.815855\pi$
$84$		0			0
$85$	−2.50000	+	4.33013i	−0.271163	+	0.469668i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$		0			0
$91$	−4.00000			−0.419314
$92$		0			0
$93$		0			0
$94$		0			0
$95$	2.50000	−	4.33013i	0.256495	−	0.444262i
$96$		0			0
$97$	−8.00000	−	13.8564i	−0.812277	−	1.40690i	−0.911267	−	0.411816i	$-0.864894\pi$
$97$	−8.00000	−	13.8564i	−0.812277	−	1.40690i	0.0989899	−	0.995088i	$-0.468439\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−6.00000	−	10.3923i	−0.597022	−	1.03407i	−0.993258	−	0.115924i	$-0.963017\pi$
$101$	−6.00000	−	10.3923i	−0.597022	−	1.03407i	0.396236	−	0.918149i	$-0.370316\pi$
$102$		0			0
$103$	−2.00000	+	3.46410i	−0.197066	+	0.341328i	−0.947576	−	0.319531i	$-0.896475\pi$
$103$	−2.00000	+	3.46410i	−0.197066	+	0.341328i	0.750510	+	0.660859i	$0.229808\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−12.0000			−1.16008			−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000			−1.16008			−0.580042	+	0.814587i	$0.696964\pi$
$108$		0			0
$109$	−19.0000			−1.81987			−0.909935	−	0.414751i	$-0.863869\pi$
$109$	−19.0000			−1.81987			−0.909935	+	0.414751i	$0.863869\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−3.00000	+	5.19615i	−0.282216	+	0.488813i	−0.971930	−	0.235269i	$-0.924403\pi$
$113$	−3.00000	+	5.19615i	−0.282216	+	0.488813i	0.689714	+	0.724082i	$0.257736\pi$
$114$		0			0
$115$	−0.500000	−	0.866025i	−0.0466252	−	0.0807573i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−5.00000	−	8.66025i	−0.458349	−	0.793884i
$120$		0			0
$121$	−2.50000	+	4.33013i	−0.227273	+	0.393648i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	1.00000			0.0894427
$126$		0			0
$127$	6.00000			0.532414			0.266207	−	0.963916i	$-0.414230\pi$
$127$	6.00000			0.532414			0.266207	+	0.963916i	$0.414230\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	9.00000	−	15.5885i	0.786334	−	1.36197i	−0.141865	−	0.989886i	$-0.545310\pi$
$131$	9.00000	−	15.5885i	0.786334	−	1.36197i	0.928199	−	0.372084i	$-0.121357\pi$
$132$		0			0
$133$	5.00000	+	8.66025i	0.433555	+	0.750939i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	8.50000	+	14.7224i	0.726204	+	1.25782i	0.958477	+	0.285171i	$0.0920506\pi$
$137$	8.50000	+	14.7224i	0.726204	+	1.25782i	−0.232273	+	0.972651i	$0.574616\pi$
$138$		0			0
$139$	−6.00000	+	10.3923i	−0.508913	+	0.881464i	0.491033	+	0.871141i	$0.336619\pi$
$139$	−6.00000	+	10.3923i	−0.508913	+	0.881464i	−0.999947	+	0.0103230i	$0.996714\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−8.00000			−0.668994
$144$		0			0
$145$	−2.00000			−0.166091
$146$		0			0
$147$		0			0
$148$		0			0
$149$	8.00000	−	13.8564i	0.655386	−	1.13516i	−0.326411	−	0.945228i	$-0.605840\pi$
$149$	8.00000	−	13.8564i	0.655386	−	1.13516i	0.981797	−	0.189933i	$-0.0608272\pi$
$150$		0			0
$151$	−4.00000	−	6.92820i	−0.325515	−	0.563809i	0.656101	−	0.754673i	$-0.272204\pi$
$151$	−4.00000	−	6.92820i	−0.325515	−	0.563809i	−0.981617	+	0.190864i	$0.938871\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−3.50000	−	6.06218i	−0.281127	−	0.486926i
$156$		0			0
$157$	−8.00000	+	13.8564i	−0.638470	+	1.10586i	0.347299	+	0.937754i	$0.387099\pi$
$157$	−8.00000	+	13.8564i	−0.638470	+	1.10586i	−0.985769	+	0.168107i	$0.946235\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	2.00000			0.157622
$162$		0			0
$163$	−14.0000			−1.09656			−0.548282	−	0.836293i	$-0.684718\pi$
$163$	−14.0000			−1.09656			−0.548282	+	0.836293i	$0.684718\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	1.50000	−	2.59808i	0.116073	−	0.201045i	−0.802135	−	0.597143i	$-0.796303\pi$
$167$	1.50000	−	2.59808i	0.116073	−	0.201045i	0.918208	+	0.396098i	$0.129636\pi$
$168$		0			0
$169$	4.50000	+	7.79423i	0.346154	+	0.599556i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	6.50000	+	11.2583i	0.494186	+	0.855955i	0.999978	−	0.00670064i	$-0.00213290\pi$
$173$	6.50000	+	11.2583i	0.494186	+	0.855955i	−0.505792	+	0.862656i	$0.668800\pi$
$174$		0			0
$175$	−1.00000	+	1.73205i	−0.0755929	+	0.130931i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−4.00000			−0.298974			−0.149487	−	0.988764i	$-0.547762\pi$
$179$	−4.00000			−0.298974			−0.149487	+	0.988764i	$0.547762\pi$
$180$		0			0
$181$	−19.0000			−1.41226			−0.706129	−	0.708083i	$-0.749560\pi$
$181$	−19.0000			−1.41226			−0.706129	+	0.708083i	$0.749560\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	3.00000	−	5.19615i	0.220564	−	0.382029i
$186$		0			0
$187$	−10.0000	−	17.3205i	−0.731272	−	1.26660i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−7.00000	−	12.1244i	−0.506502	−	0.877288i	−0.999972	−	0.00752447i	$-0.997605\pi$
$191$	−7.00000	−	12.1244i	−0.506502	−	0.877288i	0.493469	−	0.869763i	$-0.335728\pi$
$192$		0			0
$193$	−5.00000	+	8.66025i	−0.359908	+	0.623379i	−0.987945	−	0.154805i	$-0.950525\pi$
$193$	−5.00000	+	8.66025i	−0.359908	+	0.623379i	0.628037	+	0.778183i	$0.283859\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−5.00000			−0.356235			−0.178118	−	0.984009i	$-0.557001\pi$
$197$	−5.00000			−0.356235			−0.178118	+	0.984009i	$0.557001\pi$
$198$		0			0
$199$	−16.0000			−1.13421			−0.567105	−	0.823646i	$-0.691937\pi$
$199$	−16.0000			−1.13421			−0.567105	+	0.823646i	$0.691937\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	2.00000	−	3.46410i	0.140372	−	0.243132i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	10.0000	+	17.3205i	0.691714	+	1.19808i
$210$		0			0
$211$	9.50000	−	16.4545i	0.654007	−	1.13277i	−0.328135	−	0.944631i	$-0.606420\pi$
$211$	9.50000	−	16.4545i	0.654007	−	1.13277i	0.982142	−	0.188142i	$-0.0602466\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	4.00000			0.272798
$216$		0			0
$217$	14.0000			0.950382
$218$		0			0
$219$		0			0
$220$		0			0
$221$	5.00000	−	8.66025i	0.336336	−	0.582552i
$222$		0			0
$223$	5.00000	+	8.66025i	0.334825	+	0.579934i	0.983451	−	0.181173i	$-0.0579895\pi$
$223$	5.00000	+	8.66025i	0.334825	+	0.579934i	−0.648626	+	0.761107i	$0.724656\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−1.50000	−	2.59808i	−0.0995585	−	0.172440i	0.811943	−	0.583736i	$-0.198410\pi$
$227$	−1.50000	−	2.59808i	−0.0995585	−	0.172440i	−0.911502	+	0.411296i	$0.865076\pi$
$228$		0			0
$229$	14.5000	−	25.1147i	0.958187	−	1.65963i	0.231287	−	0.972886i	$-0.425707\pi$
$229$	14.5000	−	25.1147i	0.958187	−	1.65963i	0.726900	−	0.686743i	$-0.240960\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	6.00000			0.393073			0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000			0.393073			0.196537	+	0.980497i	$0.437031\pi$
$234$		0			0
$235$	4.00000			0.260931
$236$		0			0
$237$		0			0
$238$		0			0
$239$	3.00000	−	5.19615i	0.194054	−	0.336111i	−0.752536	−	0.658551i	$-0.771170\pi$
$239$	3.00000	−	5.19615i	0.194054	−	0.336111i	0.946590	+	0.322440i	$0.104503\pi$
$240$		0			0
$241$	−5.50000	−	9.52628i	−0.354286	−	0.613642i	0.632709	−	0.774389i	$-0.281943\pi$
$241$	−5.50000	−	9.52628i	−0.354286	−	0.613642i	−0.986996	+	0.160748i	$0.948609\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	1.50000	+	2.59808i	0.0958315	+	0.165985i
$246$		0			0
$247$	−5.00000	+	8.66025i	−0.318142	+	0.551039i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−24.0000			−1.51487			−0.757433	−	0.652913i	$-0.773547\pi$
$251$	−24.0000			−1.51487			−0.757433	+	0.652913i	$0.773547\pi$
$252$		0			0
$253$	4.00000			0.251478
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−13.5000	+	23.3827i	−0.842107	+	1.45857i	0.0460033	+	0.998941i	$0.485352\pi$
$257$	−13.5000	+	23.3827i	−0.842107	+	1.45857i	−0.888110	+	0.459631i	$0.847982\pi$
$258$		0			0
$259$	6.00000	+	10.3923i	0.372822	+	0.645746i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	12.0000	+	20.7846i	0.739952	+	1.28163i	0.952517	+	0.304487i	$0.0984850\pi$
$263$	12.0000	+	20.7846i	0.739952	+	1.28163i	−0.212565	+	0.977147i	$0.568182\pi$
$264$		0			0
$265$	−4.50000	+	7.79423i	−0.276433	+	0.478796i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−10.0000			−0.609711			−0.304855	−	0.952399i	$-0.598608\pi$
$269$	−10.0000			−0.609711			−0.304855	+	0.952399i	$0.598608\pi$
$270$		0			0
$271$	−9.00000			−0.546711			−0.273356	−	0.961913i	$-0.588134\pi$
$271$	−9.00000			−0.546711			−0.273356	+	0.961913i	$0.588134\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−2.00000	+	3.46410i	−0.120605	+	0.208893i
$276$		0			0
$277$	2.00000	+	3.46410i	0.120168	+	0.208138i	0.919834	−	0.392308i	$-0.128323\pi$
$277$	2.00000	+	3.46410i	0.120168	+	0.208138i	−0.799666	+	0.600446i	$0.794990\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	7.00000	+	12.1244i	0.417585	+	0.723278i	0.995696	−	0.0926797i	$-0.0295433\pi$
$281$	7.00000	+	12.1244i	0.417585	+	0.723278i	−0.578111	+	0.815958i	$0.696210\pi$
$282$		0			0
$283$	−4.00000	+	6.92820i	−0.237775	+	0.411839i	−0.960076	−	0.279741i	$-0.909752\pi$
$283$	−4.00000	+	6.92820i	−0.237775	+	0.411839i	0.722300	+	0.691580i	$0.243085\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	8.00000			0.470588
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−3.50000	+	6.06218i	−0.204472	+	0.354156i	−0.949964	−	0.312358i	$-0.898881\pi$
$293$	−3.50000	+	6.06218i	−0.204472	+	0.354156i	0.745492	+	0.666514i	$0.232214\pi$
$294$		0			0
$295$	−7.00000	−	12.1244i	−0.407556	−	0.705907i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	1.00000	+	1.73205i	0.0578315	+	0.100167i
$300$		0			0
$301$	−4.00000	+	6.92820i	−0.230556	+	0.399335i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−11.0000			−0.629858
$306$		0			0
$307$	22.0000			1.25561			0.627803	−	0.778372i	$-0.283954\pi$
$307$	22.0000			1.25561			0.627803	+	0.778372i	$0.283954\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	11.0000	−	19.0526i	0.623753	−	1.08037i	−0.365028	−	0.930997i	$-0.618941\pi$
$311$	11.0000	−	19.0526i	0.623753	−	1.08037i	0.988781	−	0.149375i	$-0.0477261\pi$
$312$		0			0
$313$	−4.00000	−	6.92820i	−0.226093	−	0.391605i	0.730554	−	0.682855i	$-0.239262\pi$
$313$	−4.00000	−	6.92820i	−0.226093	−	0.391605i	−0.956647	+	0.291250i	$0.905929\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	13.5000	+	23.3827i	0.758236	+	1.31330i	0.943750	+	0.330661i	$0.107272\pi$
$317$	13.5000	+	23.3827i	0.758236	+	1.31330i	−0.185514	+	0.982642i	$0.559395\pi$
$318$		0			0
$319$	4.00000	−	6.92820i	0.223957	−	0.387905i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−25.0000			−1.39104
$324$		0			0
$325$	−2.00000			−0.110940
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−4.00000	+	6.92820i	−0.220527	+	0.381964i
$330$		0			0
$331$	−6.00000	−	10.3923i	−0.329790	−	0.571213i	0.652680	−	0.757634i	$-0.273645\pi$
$331$	−6.00000	−	10.3923i	−0.329790	−	0.571213i	−0.982470	+	0.186421i	$0.940311\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−7.00000	−	12.1244i	−0.382451	−	0.662424i
$336$		0			0
$337$	−11.0000	+	19.0526i	−0.599208	+	1.03786i	0.393730	+	0.919226i	$0.371184\pi$
$337$	−11.0000	+	19.0526i	−0.599208	+	1.03786i	−0.992938	+	0.118633i	$0.962149\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	28.0000			1.51629
$342$		0			0
$343$	−20.0000			−1.07990
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−10.0000	+	17.3205i	−0.536828	+	0.929814i	0.462244	+	0.886753i	$0.347044\pi$
$347$	−10.0000	+	17.3205i	−0.536828	+	0.929814i	−0.999072	+	0.0430610i	$0.986289\pi$
$348$		0			0
$349$	1.50000	+	2.59808i	0.0802932	+	0.139072i	0.903376	−	0.428850i	$-0.141081\pi$
$349$	1.50000	+	2.59808i	0.0802932	+	0.139072i	−0.823083	+	0.567922i	$0.807748\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−13.0000	−	22.5167i	−0.691920	−	1.19844i	−0.971208	−	0.238233i	$-0.923432\pi$
$353$	−13.0000	−	22.5167i	−0.691920	−	1.19844i	0.279288	−	0.960207i	$-0.409902\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	18.0000			0.950004			0.475002	−	0.879985i	$-0.342447\pi$
$359$	18.0000			0.950004			0.475002	+	0.879985i	$0.342447\pi$
$360$		0			0
$361$	6.00000			0.315789
$362$		0			0
$363$		0			0
$364$		0			0
$365$	6.00000	−	10.3923i	0.314054	−	0.543958i
$366$		0			0
$367$	11.0000	+	19.0526i	0.574195	+	0.994535i	0.996129	+	0.0879086i	$0.0280183\pi$
$367$	11.0000	+	19.0526i	0.574195	+	0.994535i	−0.421933	+	0.906627i	$0.638648\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−9.00000	−	15.5885i	−0.467257	−	0.809312i
$372$		0			0
$373$	16.0000	−	27.7128i	0.828449	−	1.43492i	−0.0708063	−	0.997490i	$-0.522557\pi$
$373$	16.0000	−	27.7128i	0.828449	−	1.43492i	0.899255	−	0.437425i	$-0.144109\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	4.00000			0.206010
$378$		0			0
$379$	−23.0000			−1.18143			−0.590715	−	0.806880i	$-0.701154\pi$
$379$	−23.0000			−1.18143			−0.590715	+	0.806880i	$0.701154\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	13.5000	−	23.3827i	0.689818	−	1.19480i	−0.282079	−	0.959391i	$-0.591024\pi$
$383$	13.5000	−	23.3827i	0.689818	−	1.19480i	0.971897	−	0.235408i	$-0.0756427\pi$
$384$		0			0
$385$	−4.00000	−	6.92820i	−0.203859	−	0.353094i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	18.0000	+	31.1769i	0.912636	+	1.58073i	0.810326	+	0.585980i	$0.199290\pi$
$389$	18.0000	+	31.1769i	0.912636	+	1.58073i	0.102311	+	0.994753i	$0.467376\pi$
$390$		0			0
$391$	−2.50000	+	4.33013i	−0.126430	+	0.218984i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−3.00000			−0.150946
$396$		0			0
$397$	−14.0000			−0.702640			−0.351320	−	0.936255i	$-0.614267\pi$
$397$	−14.0000			−0.702640			−0.351320	+	0.936255i	$0.614267\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	15.0000	−	25.9808i	0.749064	−	1.29742i	−0.199207	−	0.979957i	$-0.563837\pi$
$401$	15.0000	−	25.9808i	0.749064	−	1.29742i	0.948272	−	0.317460i	$-0.102830\pi$
$402$		0			0
$403$	7.00000	+	12.1244i	0.348695	+	0.603957i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	12.0000	+	20.7846i	0.594818	+	1.03025i
$408$		0			0
$409$	−12.5000	+	21.6506i	−0.618085	+	1.07056i	0.371750	+	0.928333i	$0.378758\pi$
$409$	−12.5000	+	21.6506i	−0.618085	+	1.07056i	−0.989835	+	0.142222i	$0.954575\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	28.0000			1.37779
$414$		0			0
$415$	−1.00000			−0.0490881
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−18.0000	+	31.1769i	−0.879358	+	1.52309i	−0.0273103	+	0.999627i	$0.508694\pi$
$419$	−18.0000	+	31.1769i	−0.879358	+	1.52309i	−0.852047	+	0.523465i	$0.824639\pi$
$420$		0			0
$421$	−1.50000	−	2.59808i	−0.0731055	−	0.126622i	0.827155	−	0.561973i	$-0.189958\pi$
$421$	−1.50000	−	2.59808i	−0.0731055	−	0.126622i	−0.900261	+	0.435351i	$0.856624\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−2.50000	−	4.33013i	−0.121268	−	0.210042i
$426$		0			0
$427$	11.0000	−	19.0526i	0.532327	−	0.922018i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	14.0000			0.674356			0.337178	−	0.941441i	$-0.390528\pi$
$431$	14.0000			0.674356			0.337178	+	0.941441i	$0.390528\pi$
$432$		0			0
$433$	26.0000			1.24948			0.624740	−	0.780833i	$-0.285205\pi$
$433$	26.0000			1.24948			0.624740	+	0.780833i	$0.285205\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	2.50000	−	4.33013i	0.119591	−	0.207138i
$438$		0			0
$439$	8.50000	+	14.7224i	0.405683	+	0.702663i	0.994401	−	0.105675i	$-0.0337004\pi$
$439$	8.50000	+	14.7224i	0.405683	+	0.702663i	−0.588718	+	0.808339i	$0.700367\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−19.5000	−	33.7750i	−0.926473	−	1.60470i	−0.789175	−	0.614168i	$-0.789492\pi$
$443$	−19.5000	−	33.7750i	−0.926473	−	1.60470i	−0.137298	−	0.990530i	$-0.543842\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	6.00000			0.283158			0.141579	−	0.989927i	$-0.454782\pi$
$449$	6.00000			0.283158			0.141579	+	0.989927i	$0.454782\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$	2.00000	−	3.46410i	0.0937614	−	0.162400i
$456$		0			0
$457$	−14.0000	−	24.2487i	−0.654892	−	1.13431i	−0.981921	−	0.189292i	$-0.939381\pi$
$457$	−14.0000	−	24.2487i	−0.654892	−	1.13431i	0.327028	−	0.945015i	$-0.393953\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−21.0000	−	36.3731i	−0.978068	−	1.69406i	−0.669417	−	0.742887i	$-0.733456\pi$
$461$	−21.0000	−	36.3731i	−0.978068	−	1.69406i	−0.308651	−	0.951175i	$-0.599877\pi$
$462$		0			0
$463$	9.00000	−	15.5885i	0.418265	−	0.724457i	−0.577500	−	0.816391i	$-0.695972\pi$
$463$	9.00000	−	15.5885i	0.418265	−	0.724457i	0.995765	+	0.0919339i	$0.0293048\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−13.0000			−0.601568			−0.300784	−	0.953692i	$-0.597248\pi$
$467$	−13.0000			−0.601568			−0.300784	+	0.953692i	$0.597248\pi$
$468$		0			0
$469$	28.0000			1.29292
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−8.00000	+	13.8564i	−0.367840	+	0.637118i
$474$		0			0
$475$	2.50000	+	4.33013i	0.114708	+	0.198680i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−14.0000	−	24.2487i	−0.639676	−	1.10795i	−0.985504	−	0.169654i	$-0.945735\pi$
$479$	−14.0000	−	24.2487i	−0.639676	−	1.10795i	0.345827	−	0.938298i	$-0.387598\pi$
$480$		0			0
$481$	−6.00000	+	10.3923i	−0.273576	+	0.473848i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	16.0000			0.726523
$486$		0			0
$487$	40.0000			1.81257			0.906287	−	0.422664i	$-0.138905\pi$
$487$	40.0000			1.81257			0.906287	+	0.422664i	$0.138905\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	3.00000	−	5.19615i	0.135388	−	0.234499i	−0.790358	−	0.612646i	$-0.790105\pi$
$491$	3.00000	−	5.19615i	0.135388	−	0.234499i	0.925746	+	0.378147i	$0.123439\pi$
$492$		0			0
$493$	5.00000	+	8.66025i	0.225189	+	0.390038i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	12.5000	−	21.6506i	0.559577	−	0.969216i	−0.437955	−	0.898997i	$-0.644297\pi$
$499$	12.5000	−	21.6506i	0.559577	−	0.969216i	0.997532	−	0.0702185i	$-0.0223697\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−15.0000			−0.668817			−0.334408	−	0.942428i	$-0.608537\pi$
$503$	−15.0000			−0.668817			−0.334408	+	0.942428i	$0.608537\pi$
$504$		0			0
$505$	12.0000			0.533993
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−14.0000	+	24.2487i	−0.620539	+	1.07481i	0.368846	+	0.929490i	$0.379753\pi$
$509$	−14.0000	+	24.2487i	−0.620539	+	1.07481i	−0.989385	+	0.145315i	$0.953580\pi$
$510$		0			0
$511$	12.0000	+	20.7846i	0.530849	+	0.919457i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−2.00000	−	3.46410i	−0.0881305	−	0.152647i
$516$		0			0
$517$	−8.00000	+	13.8564i	−0.351840	+	0.609404i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	40.0000			1.75243			0.876216	−	0.481919i	$-0.160060\pi$
$521$	40.0000			1.75243			0.876216	+	0.481919i	$0.160060\pi$
$522$		0			0
$523$	10.0000			0.437269			0.218635	−	0.975807i	$-0.429840\pi$
$523$	10.0000			0.437269			0.218635	+	0.975807i	$0.429840\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−17.5000	+	30.3109i	−0.762312	+	1.32036i
$528$		0			0
$529$	11.0000	+	19.0526i	0.478261	+	0.828372i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$	6.00000	−	10.3923i	0.259403	−	0.449299i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−12.0000			−0.516877
$540$		0			0
$541$	22.0000			0.945854			0.472927	−	0.881102i	$-0.343197\pi$
$541$	22.0000			0.945854			0.472927	+	0.881102i	$0.343197\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	9.50000	−	16.4545i	0.406935	−	0.704833i
$546$		0			0
$547$	14.0000	+	24.2487i	0.598597	+	1.03680i	0.993028	+	0.117875i	$0.0376081\pi$
$547$	14.0000	+	24.2487i	0.598597	+	1.03680i	−0.394432	+	0.918925i	$0.629059\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−5.00000	−	8.66025i	−0.213007	−	0.368939i
$552$		0			0
$553$	3.00000	−	5.19615i	0.127573	−	0.220963i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−10.0000			−0.423714			−0.211857	−	0.977301i	$-0.567951\pi$
$557$	−10.0000			−0.423714			−0.211857	+	0.977301i	$0.567951\pi$
$558$		0			0
$559$	−8.00000			−0.338364
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−2.00000	+	3.46410i	−0.0842900	+	0.145994i	−0.905088	−	0.425223i	$-0.860196\pi$
$563$	−2.00000	+	3.46410i	−0.0842900	+	0.145994i	0.820798	+	0.571218i	$0.193529\pi$
$564$		0			0
$565$	−3.00000	−	5.19615i	−0.126211	−	0.218604i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	5.00000	+	8.66025i	0.209611	+	0.363057i	0.951592	−	0.307364i	$-0.0994469\pi$
$569$	5.00000	+	8.66025i	0.209611	+	0.363057i	−0.741981	+	0.670421i	$0.766114\pi$
$570$		0			0
$571$	2.50000	−	4.33013i	0.104622	−	0.181210i	−0.808962	−	0.587861i	$-0.799970\pi$
$571$	2.50000	−	4.33013i	0.104622	−	0.181210i	0.913584	+	0.406651i	$0.133303\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	1.00000			0.0417029
$576$		0			0
$577$	−16.0000			−0.666089			−0.333044	−	0.942911i	$-0.608076\pi$
$577$	−16.0000			−0.666089			−0.333044	+	0.942911i	$0.608076\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	1.00000	−	1.73205i	0.0414870	−	0.0718576i
$582$		0			0
$583$	−18.0000	−	31.1769i	−0.745484	−	1.29122i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	7.50000	+	12.9904i	0.309558	+	0.536170i	0.978266	−	0.207355i	$-0.0664855\pi$
$587$	7.50000	+	12.9904i	0.309558	+	0.536170i	−0.668708	+	0.743525i	$0.733152\pi$
$588$		0			0
$589$	17.5000	−	30.3109i	0.721075	−	1.24894i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−9.00000			−0.369586			−0.184793	−	0.982777i	$-0.559161\pi$
$593$	−9.00000			−0.369586			−0.184793	+	0.982777i	$0.559161\pi$
$594$		0			0
$595$	10.0000			0.409960
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−9.00000	+	15.5885i	−0.367730	+	0.636927i	−0.989210	−	0.146503i	$-0.953198\pi$
$599$	−9.00000	+	15.5885i	−0.367730	+	0.636927i	0.621480	+	0.783430i	$0.286532\pi$
$600$		0			0
$601$	5.50000	+	9.52628i	0.224350	+	0.388585i	0.956124	−	0.292962i	$-0.0946409\pi$
$601$	5.50000	+	9.52628i	0.224350	+	0.388585i	−0.731774	+	0.681547i	$0.761308\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−2.50000	−	4.33013i	−0.101639	−	0.176045i
$606$		0			0
$607$	18.0000	−	31.1769i	0.730597	−	1.26543i	−0.226031	−	0.974120i	$-0.572575\pi$
$607$	18.0000	−	31.1769i	0.730597	−	1.26543i	0.956628	−	0.291312i	$-0.0940917\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−8.00000			−0.323645
$612$		0			0
$613$		0			0			−	1.00000i	$-0.5\pi$
$613$		0			0				1.00000i	$0.5\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−16.5000	+	28.5788i	−0.664265	+	1.15054i	0.315219	+	0.949019i	$0.397922\pi$
$617$	−16.5000	+	28.5788i	−0.664265	+	1.15054i	−0.979484	+	0.201522i	$0.935411\pi$
$618$		0			0
$619$	−16.0000	−	27.7128i	−0.643094	−	1.11387i	−0.984738	−	0.174042i	$-0.944317\pi$
$619$	−16.0000	−	27.7128i	−0.643094	−	1.11387i	0.341644	−	0.939829i	$-0.389016\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−0.500000	+	0.866025i	−0.0200000	+	0.0346410i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−30.0000			−1.19618
$630$		0			0
$631$	13.0000			0.517522			0.258761	−	0.965941i	$-0.416686\pi$
$631$	13.0000			0.517522			0.258761	+	0.965941i	$0.416686\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−3.00000	+	5.19615i	−0.119051	+	0.206203i
$636$		0			0
$637$	−3.00000	−	5.19615i	−0.118864	−	0.205879i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$641$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$642$		0			0
$643$	−3.00000	+	5.19615i	−0.118308	+	0.204916i	−0.919097	−	0.394030i	$-0.871080\pi$
$643$	−3.00000	+	5.19615i	−0.118308	+	0.204916i	0.800789	+	0.598947i	$0.204414\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	33.0000			1.29736			0.648682	−	0.761060i	$-0.275321\pi$
$647$	33.0000			1.29736			0.648682	+	0.761060i	$0.275321\pi$
$648$		0			0
$649$	56.0000			2.19819
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−20.5000	+	35.5070i	−0.802227	+	1.38950i	0.115920	+	0.993259i	$0.463018\pi$
$653$	−20.5000	+	35.5070i	−0.802227	+	1.38950i	−0.918147	+	0.396239i	$0.870315\pi$
$654$		0			0
$655$	9.00000	+	15.5885i	0.351659	+	0.609091i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−9.00000	−	15.5885i	−0.350590	−	0.607240i	0.635763	−	0.771885i	$-0.280686\pi$
$659$	−9.00000	−	15.5885i	−0.350590	−	0.607240i	−0.986353	+	0.164644i	$0.947352\pi$
$660$		0			0
$661$	21.0000	−	36.3731i	0.816805	−	1.41475i	−0.0912190	−	0.995831i	$-0.529076\pi$
$661$	21.0000	−	36.3731i	0.816805	−	1.41475i	0.908024	−	0.418917i	$-0.137590\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−10.0000			−0.387783
$666$		0			0
$667$	−2.00000			−0.0774403
$668$		0			0
$669$		0			0
$670$		0			0
$671$	22.0000	−	38.1051i	0.849301	−	1.47103i
$672$		0			0
$673$	−15.0000	−	25.9808i	−0.578208	−	1.00148i	−0.995685	−	0.0927975i	$-0.970419\pi$
$673$	−15.0000	−	25.9808i	−0.578208	−	1.00148i	0.417477	−	0.908687i	$-0.362914\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	1.00000	+	1.73205i	0.0384331	+	0.0665681i	0.884602	−	0.466347i	$-0.154430\pi$
$677$	1.00000	+	1.73205i	0.0384331	+	0.0665681i	−0.846169	+	0.532915i	$0.821097\pi$
$678$		0			0
$679$	−16.0000	+	27.7128i	−0.614024	+	1.06352i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−9.00000			−0.344375			−0.172188	−	0.985064i	$-0.555084\pi$
$683$	−9.00000			−0.344375			−0.172188	+	0.985064i	$0.555084\pi$
$684$		0			0
$685$	−17.0000			−0.649537
$686$		0			0
$687$		0			0
$688$		0			0
$689$	9.00000	−	15.5885i	0.342873	−	0.593873i
$690$		0			0
$691$	9.50000	+	16.4545i	0.361397	+	0.625958i	0.988191	−	0.153227i	$-0.0489666\pi$
$691$	9.50000	+	16.4545i	0.361397	+	0.625958i	−0.626794	+	0.779185i	$0.715633\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−6.00000	−	10.3923i	−0.227593	−	0.394203i
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−10.0000			−0.377695			−0.188847	−	0.982006i	$-0.560475\pi$
$701$	−10.0000			−0.377695			−0.188847	+	0.982006i	$0.560475\pi$
$702$		0			0
$703$	30.0000			1.13147
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−12.0000	+	20.7846i	−0.451306	+	0.781686i
$708$		0			0
$709$	−13.0000	−	22.5167i	−0.488225	−	0.845631i	0.511683	−	0.859174i	$-0.329022\pi$
$709$	−13.0000	−	22.5167i	−0.488225	−	0.845631i	−0.999908	+	0.0135434i	$0.995689\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−3.50000	−	6.06218i	−0.131076	−	0.227030i
$714$		0			0
$715$	4.00000	−	6.92820i	0.149592	−	0.259100i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	46.0000			1.71551			0.857755	−	0.514058i	$-0.171858\pi$
$719$	46.0000			1.71551			0.857755	+	0.514058i	$0.171858\pi$
$720$		0			0
$721$	8.00000			0.297936
$722$		0			0
$723$		0			0
$724$		0			0
$725$	1.00000	−	1.73205i	0.0371391	−	0.0643268i
$726$		0			0
$727$	−16.0000	−	27.7128i	−0.593407	−	1.02781i	−0.993770	−	0.111454i	$-0.964449\pi$
$727$	−16.0000	−	27.7128i	−0.593407	−	1.02781i	0.400362	−	0.916357i	$-0.368884\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−10.0000	−	17.3205i	−0.369863	−	0.640622i
$732$		0			0
$733$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$733$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	56.0000			2.06279
$738$		0			0
$739$	−15.0000			−0.551784			−0.275892	−	0.961189i	$-0.588973\pi$
$739$	−15.0000			−0.551784			−0.275892	+	0.961189i	$0.588973\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$743$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$744$		0			0
$745$	8.00000	+	13.8564i	0.293097	+	0.507659i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	12.0000	+	20.7846i	0.438470	+	0.759453i
$750$		0			0
$751$	1.50000	−	2.59808i	0.0547358	−	0.0948051i	−0.837359	−	0.546653i	$-0.815902\pi$
$751$	1.50000	−	2.59808i	0.0547358	−	0.0948051i	0.892095	+	0.451848i	$0.149235\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	8.00000			0.291150
$756$		0			0
$757$	2.00000			0.0726912			0.0363456	−	0.999339i	$-0.488428\pi$
$757$	2.00000			0.0726912			0.0363456	+	0.999339i	$0.488428\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	15.0000	−	25.9808i	0.543750	−	0.941802i	−0.454935	−	0.890525i	$-0.650337\pi$
$761$	15.0000	−	25.9808i	0.543750	−	0.941802i	0.998684	−	0.0512772i	$-0.0163292\pi$
$762$		0			0
$763$	19.0000	+	32.9090i	0.687846	+	1.19138i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	14.0000	+	24.2487i	0.505511	+	0.875570i
$768$		0			0
$769$	17.5000	−	30.3109i	0.631066	−	1.09304i	−0.356268	−	0.934384i	$-0.615951\pi$
$769$	17.5000	−	30.3109i	0.631066	−	1.09304i	0.987334	−	0.158655i	$-0.0507157\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−15.0000			−0.539513			−0.269756	−	0.962929i	$-0.586943\pi$
$773$	−15.0000			−0.539513			−0.269756	+	0.962929i	$0.586943\pi$
$774$		0			0
$775$	7.00000			0.251447
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−8.00000	−	13.8564i	−0.285532	−	0.494556i
$786$		0			0
$787$	20.0000	−	34.6410i	0.712923	−	1.23482i	−0.250832	−	0.968031i	$-0.580704\pi$
$787$	20.0000	−	34.6410i	0.712923	−	1.23482i	0.963755	−	0.266788i	$-0.0859624\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	12.0000			0.426671
$792$		0			0
$793$	22.0000			0.781243
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−1.50000	+	2.59808i	−0.0531327	+	0.0920286i	−0.891368	−	0.453279i	$-0.850254\pi$
$797$	−1.50000	+	2.59808i	−0.0531327	+	0.0920286i	0.838236	+	0.545308i	$0.183587\pi$
$798$		0			0
$799$	−10.0000	−	17.3205i	−0.353775	−	0.612756i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	24.0000	+	41.5692i	0.846942	+	1.46695i
$804$		0			0
$805$	−1.00000	+	1.73205i	−0.0352454	+	0.0610468i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−24.0000			−0.843795			−0.421898	−	0.906644i	$-0.638636\pi$
$809$	−24.0000			−0.843795			−0.421898	+	0.906644i	$0.638636\pi$
$810$		0			0
$811$	44.0000			1.54505			0.772524	−	0.634985i	$-0.218994\pi$
$811$	44.0000			1.54505			0.772524	+	0.634985i	$0.218994\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	7.00000	−	12.1244i	0.245199	−	0.424698i
$816$		0			0
$817$	10.0000	+	17.3205i	0.349856	+	0.605968i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−23.0000	−	39.8372i	−0.802706	−	1.39033i	−0.917829	−	0.396976i	$-0.870060\pi$
$821$	−23.0000	−	39.8372i	−0.802706	−	1.39033i	0.115124	−	0.993351i	$-0.463274\pi$
$822$		0			0
$823$	10.0000	−	17.3205i	0.348578	−	0.603755i	−0.637419	−	0.770517i	$-0.719998\pi$
$823$	10.0000	−	17.3205i	0.348578	−	0.603755i	0.985997	+	0.166762i	$0.0533313\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−39.0000			−1.35616			−0.678081	−	0.734987i	$-0.737188\pi$
$827$	−39.0000			−1.35616			−0.678081	+	0.734987i	$0.737188\pi$
$828$		0			0
$829$	−2.00000			−0.0694629			−0.0347314	−	0.999397i	$-0.511058\pi$
$829$	−2.00000			−0.0694629			−0.0347314	+	0.999397i	$0.511058\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	7.50000	−	12.9904i	0.259860	−	0.450090i
$834$		0			0
$835$	1.50000	+	2.59808i	0.0519096	+	0.0899101i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	8.00000	+	13.8564i	0.276191	+	0.478376i	0.970435	−	0.241363i	$-0.0775945\pi$
$839$	8.00000	+	13.8564i	0.276191	+	0.478376i	−0.694244	+	0.719740i	$0.744261\pi$
$840$		0			0
$841$	12.5000	−	21.6506i	0.431034	−	0.746574i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−9.00000			−0.309609
$846$		0			0
$847$	10.0000			0.343604
$848$		0			0
$849$		0			0
$850$		0			0
$851$	3.00000	−	5.19615i	0.102839	−	0.178122i
$852$		0			0
$853$	14.0000	+	24.2487i	0.479351	+	0.830260i	0.999720	−	0.0236816i	$-0.00753881\pi$
$853$	14.0000	+	24.2487i	0.479351	+	0.830260i	−0.520369	+	0.853942i	$0.674205\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−10.5000	−	18.1865i	−0.358673	−	0.621240i	0.629066	−	0.777352i	$-0.283437\pi$
$857$	−10.5000	−	18.1865i	−0.358673	−	0.621240i	−0.987739	+	0.156112i	$0.950104\pi$
$858$		0			0
$859$	−20.5000	+	35.5070i	−0.699451	+	1.21148i	0.269206	+	0.963083i	$0.413239\pi$
$859$	−20.5000	+	35.5070i	−0.699451	+	1.21148i	−0.968657	+	0.248402i	$0.920095\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	33.0000			1.12333			0.561667	−	0.827364i	$-0.310160\pi$
$863$	33.0000			1.12333			0.561667	+	0.827364i	$0.310160\pi$
$864$		0			0
$865$	−13.0000			−0.442013
$866$		0			0
$867$		0			0
$868$		0			0
$869$	6.00000	−	10.3923i	0.203536	−	0.352535i
$870$		0			0
$871$	14.0000	+	24.2487i	0.474372	+	0.821636i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−1.00000	−	1.73205i	−0.0338062	−	0.0585540i
$876$		0			0
$877$	16.0000	−	27.7128i	0.540282	−	0.935795i	−0.458606	−	0.888640i	$-0.651651\pi$
$877$	16.0000	−	27.7128i	0.540282	−	0.935795i	0.998888	−	0.0471555i	$-0.0150156\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−54.0000			−1.81931			−0.909653	−	0.415369i	$-0.863653\pi$
$881$	−54.0000			−1.81931			−0.909653	+	0.415369i	$0.863653\pi$
$882$		0			0
$883$	−22.0000			−0.740359			−0.370179	−	0.928960i	$-0.620704\pi$
$883$	−22.0000			−0.740359			−0.370179	+	0.928960i	$0.620704\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	11.5000	−	19.9186i	0.386132	−	0.668801i	−0.605793	−	0.795622i	$-0.707144\pi$
$887$	11.5000	−	19.9186i	0.386132	−	0.668801i	0.991926	+	0.126821i	$0.0404775\pi$
$888$		0			0
$889$	−6.00000	−	10.3923i	−0.201234	−	0.348547i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	10.0000	+	17.3205i	0.334637	+	0.579609i
$894$		0			0
$895$	2.00000	−	3.46410i	0.0668526	−	0.115792i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−14.0000			−0.466926
$900$		0			0
$901$	45.0000			1.49917
$902$		0			0
$903$		0			0
$904$		0			0
$905$	9.50000	−	16.4545i	0.315791	−	0.546966i
$906$		0			0
$907$	−6.00000	−	10.3923i	−0.199227	−	0.345071i	0.749051	−	0.662512i	$-0.230510\pi$
$907$	−6.00000	−	10.3923i	−0.199227	−	0.345071i	−0.948278	+	0.317441i	$0.897176\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	1.00000	+	1.73205i	0.0331315	+	0.0573854i	0.882116	−	0.471033i	$-0.156119\pi$
$911$	1.00000	+	1.73205i	0.0331315	+	0.0573854i	−0.848984	+	0.528418i	$0.822785\pi$
$912$		0			0
$913$	2.00000	−	3.46410i	0.0661903	−	0.114645i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−36.0000			−1.18882
$918$		0			0
$919$	32.0000			1.05558			0.527791	−	0.849374i	$-0.323020\pi$
$919$	32.0000			1.05558			0.527791	+	0.849374i	$0.323020\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	3.00000	+	5.19615i	0.0986394	+	0.170848i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	4.00000	+	6.92820i	0.131236	+	0.227307i	0.924153	−	0.382022i	$-0.124772\pi$
$929$	4.00000	+	6.92820i	0.131236	+	0.227307i	−0.792917	+	0.609329i	$0.791439\pi$
$930$		0			0
$931$	−7.50000	+	12.9904i	−0.245803	+	0.425743i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	20.0000			0.654070
$936$		0			0
$937$	−56.0000			−1.82944			−0.914720	−	0.404088i	$-0.867589\pi$
$937$	−56.0000			−1.82944			−0.914720	+	0.404088i	$0.867589\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−17.0000	+	29.4449i	−0.554184	+	0.959875i	0.443782	+	0.896135i	$0.353636\pi$
$941$	−17.0000	+	29.4449i	−0.554184	+	0.959875i	−0.997967	+	0.0637405i	$0.979697\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	26.5000	+	45.8993i	0.861134	+	1.49153i	0.870835	+	0.491575i	$0.163579\pi$
$947$	26.5000	+	45.8993i	0.861134	+	1.49153i	−0.00970072	+	0.999953i	$0.503088\pi$
$948$		0			0
$949$	−12.0000	+	20.7846i	−0.389536	+	0.674697i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	6.00000			0.194359			0.0971795	−	0.995267i	$-0.469018\pi$
$953$	6.00000			0.194359			0.0971795	+	0.995267i	$0.469018\pi$
$954$		0			0
$955$	14.0000			0.453029
$956$		0			0
$957$		0			0
$958$		0			0
$959$	17.0000	−	29.4449i	0.548959	−	0.950824i
$960$		0			0
$961$	−9.00000	−	15.5885i	−0.290323	−	0.502853i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−5.00000	−	8.66025i	−0.160956	−	0.278783i
$966$		0			0
$967$	15.0000	−	25.9808i	0.482367	−	0.835485i	−0.517428	−	0.855727i	$-0.673110\pi$
$967$	15.0000	−	25.9808i	0.482367	−	0.835485i	0.999795	+	0.0202420i	$0.00644366\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	14.0000			0.449281			0.224641	−	0.974442i	$-0.427879\pi$
$971$	14.0000			0.449281			0.224641	+	0.974442i	$0.427879\pi$
$972$		0			0
$973$	24.0000			0.769405
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−1.00000	+	1.73205i	−0.0319928	+	0.0554132i	−0.881579	−	0.472037i	$-0.843519\pi$
$977$	−1.00000	+	1.73205i	−0.0319928	+	0.0554132i	0.849586	+	0.527451i	$0.176852\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		0			0
$983$	1.50000	+	2.59808i	0.0478426	+	0.0828658i	0.888955	−	0.457995i	$-0.151432\pi$
$983$	1.50000	+	2.59808i	0.0478426	+	0.0828658i	−0.841112	+	0.540860i	$0.818099\pi$
$984$		0			0
$985$	2.50000	−	4.33013i	0.0796566	−	0.137969i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	4.00000			0.127193
$990$		0			0
$991$	−55.0000			−1.74713			−0.873566	−	0.486705i	$-0.838199\pi$
$991$	−55.0000			−1.74713			−0.873566	+	0.486705i	$0.838199\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	8.00000	−	13.8564i	0.253617	−	0.439278i
$996$		0			0
$997$	24.0000	+	41.5692i	0.760088	+	1.31651i	0.942805	+	0.333345i	$0.108177\pi$
$997$	24.0000	+	41.5692i	0.760088	+	1.31651i	−0.182717	+	0.983165i	$0.558489\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3240.2.q.b.1081.1		2
3.2	odd	2		3240.2.q.p.1081.1		2
9.2	odd	6		3240.2.q.p.2161.1		2
9.4	even	3		1080.2.a.l.1.1	yes	1
9.5	odd	6		1080.2.a.e.1.1	✓	1
9.7	even	3	inner	3240.2.q.b.2161.1		2
36.23	even	6		2160.2.a.e.1.1		1
36.31	odd	6		2160.2.a.m.1.1		1
45.4	even	6		5400.2.a.q.1.1		1
45.13	odd	12		5400.2.f.x.649.1		2
45.14	odd	6		5400.2.a.j.1.1		1
45.22	odd	12		5400.2.f.x.649.2		2
45.23	even	12		5400.2.f.f.649.1		2
45.32	even	12		5400.2.f.f.649.2		2
72.5	odd	6		8640.2.a.cd.1.1		1
72.13	even	6		8640.2.a.t.1.1		1
72.59	even	6		8640.2.a.bi.1.1		1
72.67	odd	6		8640.2.a.k.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1080.2.a.e.1.1	✓	1	9.5	odd	6
1080.2.a.l.1.1	yes	1	9.4	even	3
2160.2.a.e.1.1		1	36.23	even	6
2160.2.a.m.1.1		1	36.31	odd	6
3240.2.q.b.1081.1		2	1.1	even	1	trivial
3240.2.q.b.2161.1		2	9.7	even	3	inner
3240.2.q.p.1081.1		2	3.2	odd	2
3240.2.q.p.2161.1		2	9.2	odd	6
5400.2.a.j.1.1		1	45.14	odd	6
5400.2.a.q.1.1		1	45.4	even	6
5400.2.f.f.649.1		2	45.23	even	12
5400.2.f.f.649.2		2	45.32	even	12
5400.2.f.x.649.1		2	45.13	odd	12
5400.2.f.x.649.2		2	45.22	odd	12
8640.2.a.k.1.1		1	72.67	odd	6
8640.2.a.t.1.1		1	72.13	even	6
8640.2.a.bi.1.1		1	72.59	even	6
8640.2.a.cd.1.1		1	72.5	odd	6

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)

\(f(q)\)	\(=\)	\(q+(-0.500000 + 0.866025i) q^{5} +(-1.00000 - 1.73205i) q^{7} +(-2.00000 - 3.46410i) q^{11} +(1.00000 - 1.73205i) q^{13} +5.00000 q^{17} -5.00000 q^{19} +(-0.500000 + 0.866025i) q^{23} +(-0.500000 - 0.866025i) q^{25} +(1.00000 + 1.73205i) q^{29} +(-3.50000 + 6.06218i) q^{31} +2.00000 q^{35} -6.00000 q^{37} +(-2.00000 - 3.46410i) q^{43} +(-2.00000 - 3.46410i) q^{47} +(1.50000 - 2.59808i) q^{49} +9.00000 q^{53} +4.00000 q^{55} +(-7.00000 + 12.1244i) q^{59} +(5.50000 + 9.52628i) q^{61} +(1.00000 + 1.73205i) q^{65} +(-7.00000 + 12.1244i) q^{67} -12.0000 q^{73} +(-4.00000 + 6.92820i) q^{77} +(1.50000 + 2.59808i) q^{79} +(0.500000 + 0.866025i) q^{83} +(-2.50000 + 4.33013i) q^{85} -4.00000 q^{91} +(2.50000 - 4.33013i) q^{95} +(-8.00000 - 13.8564i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{5} - 2 q^{7} - 4 q^{11} + 2 q^{13} + 10 q^{17} - 10 q^{19} - q^{23} - q^{25} + 2 q^{29} - 7 q^{31} + 4 q^{35} - 12 q^{37} - 4 q^{43} - 4 q^{47} + 3 q^{49} + 18 q^{53} + 8 q^{55} - 14 q^{59}+ \cdots - 16 q^{97}+O(q^{100}) \) 2 * q - q^5 - 2 * q^7 - 4 * q^11 + 2 * q^13 + 10 * q^17 - 10 * q^19 - q^23 - q^25 + 2 * q^29 - 7 * q^31 + 4 * q^35 - 12 * q^37 - 4 * q^43 - 4 * q^47 + 3 * q^49 + 18 * q^53 + 8 * q^55 - 14 * q^59 + 11 * q^61 + 2 * q^65 - 14 * q^67 - 24 * q^73 - 8 * q^77 + 3 * q^79 + q^83 - 5 * q^85 - 8 * q^91 + 5 * q^95 - 16 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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