LMFDB - Embedded newform 1512.2.t.d.289.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1512,2,Mod(289,1512)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1512, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 4, 2]))

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

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

chi := DirichletCharacter("1512.289");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1512 = 2^{3} \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1512.t (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:	$12.0733807856$
Analytic rank:	$0$
Dimension:	$22$
Relative dimension:	$11$ over $\Q(\zeta_{3})$
Twist minimal:	no (minimal twist has level 504)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			289.2
Character	$\chi$	$=$	1512.289
Dual form			1512.2.t.d.361.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.66802 q^{5} +(1.94471 - 1.79391i) q^{7} +O(q^{10})$	$q-2.66802 q^{5} +(1.94471 - 1.79391i) q^{7} -1.36451 q^{11} +(-2.75597 - 4.77348i) q^{13} +(1.23930 + 2.14654i) q^{17} +(-2.19600 + 3.80358i) q^{19} +4.69002 q^{23} +2.11832 q^{25} +(-2.94810 + 5.10625i) q^{29} +(-1.55839 + 2.69921i) q^{31} +(-5.18852 + 4.78617i) q^{35} +(-3.15627 + 5.46681i) q^{37} +(-1.38693 - 2.40224i) q^{41} +(-4.87889 + 8.45048i) q^{43} +(-5.02505 - 8.70364i) q^{47} +(0.563800 - 6.97726i) q^{49} +(1.47823 + 2.56037i) q^{53} +3.64055 q^{55} +(1.77809 - 3.07974i) q^{59} +(-0.663043 - 1.14842i) q^{61} +(7.35297 + 12.7357i) q^{65} +(-4.14937 + 7.18692i) q^{67} -12.3069 q^{71} +(-1.11577 - 1.93257i) q^{73} +(-2.65358 + 2.44781i) q^{77} +(-6.41535 - 11.1117i) q^{79} +(-5.15934 + 8.93625i) q^{83} +(-3.30648 - 5.72700i) q^{85} +(-7.73159 + 13.3915i) q^{89} +(-13.9227 - 4.33908i) q^{91} +(5.85896 - 10.1480i) q^{95} +(-2.55369 + 4.42311i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 22 q + 6 q^{5} + 7 q^{7}+O(q^{10}) $ 22 * q + 6 * q^5 + 7 * q^7	$ 22 q + 6 q^{5} + 7 q^{7} - 6 q^{11} - 3 q^{13} - 7 q^{17} - q^{19} + 4 q^{23} + 20 q^{25} - 9 q^{29} - 4 q^{31} - 14 q^{35} + 2 q^{37} - 16 q^{41} - 5 q^{47} - 15 q^{49} - 11 q^{53} + 22 q^{55} + 19 q^{59} - 13 q^{61} - 13 q^{65} + 26 q^{67} + 48 q^{71} - 35 q^{73} + 4 q^{77} + 10 q^{79} + 28 q^{83} - 20 q^{85} - 6 q^{89} - 37 q^{91} - 12 q^{95} - 29 q^{97}+O(q^{100}) $ 22 * q + 6 * q^5 + 7 * q^7 - 6 * q^11 - 3 * q^13 - 7 * q^17 - q^19 + 4 * q^23 + 20 * q^25 - 9 * q^29 - 4 * q^31 - 14 * q^35 + 2 * q^37 - 16 * q^41 - 5 * q^47 - 15 * q^49 - 11 * q^53 + 22 * q^55 + 19 * q^59 - 13 * q^61 - 13 * q^65 + 26 * q^67 + 48 * q^71 - 35 * q^73 + 4 * q^77 + 10 * q^79 + 28 * q^83 - 20 * q^85 - 6 * q^89 - 37 * q^91 - 12 * q^95 - 29 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1512\mathbb{Z}\right)^\times$.

$n$	$757$	$785$	$1081$	$1135$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{1}{3}\right)$	$1$

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
$5$	−2.66802			−1.19317			−0.596587	−	0.802548i	$-0.703477\pi$
$5$	−2.66802			−1.19317			−0.596587	+	0.802548i	$0.703477\pi$
$6$		0			0
$7$	1.94471	−	1.79391i	0.735032	−	0.678033i
$7$	1.94471	−	1.79391i	0.735032	−	0.678033i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−1.36451			−0.411416			−0.205708	−	0.978613i	$-0.565950\pi$
$11$	−1.36451			−0.411416			−0.205708	+	0.978613i	$0.565950\pi$
$12$		0			0
$13$	−2.75597	−	4.77348i	−0.764368	−	1.32392i	−0.940580	−	0.339572i	$-0.889718\pi$
$13$	−2.75597	−	4.77348i	−0.764368	−	1.32392i	0.176212	−	0.984352i	$-0.443616\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	1.23930	+	2.14654i	0.300575	+	0.520611i	0.976266	−	0.216573i	$-0.0694880\pi$
$17$	1.23930	+	2.14654i	0.300575	+	0.520611i	−0.675691	+	0.737185i	$0.736155\pi$
$18$		0			0
$19$	−2.19600	+	3.80358i	−0.503797	+	0.872601i	0.496194	+	0.868212i	$0.334731\pi$
$19$	−2.19600	+	3.80358i	−0.503797	+	0.872601i	−0.999990	+	0.00438950i	$0.998603\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	4.69002			0.977936			0.488968	−	0.872302i	$-0.337373\pi$
$23$	4.69002			0.977936			0.488968	+	0.872302i	$0.337373\pi$
$24$		0			0
$25$	2.11832			0.423664
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−2.94810	+	5.10625i	−0.547448	+	0.948207i	0.451001	+	0.892524i	$0.351067\pi$
$29$	−2.94810	+	5.10625i	−0.547448	+	0.948207i	−0.998448	+	0.0556837i	$0.982266\pi$
$30$		0			0
$31$	−1.55839	+	2.69921i	−0.279895	+	0.484792i	−0.971358	−	0.237619i	$-0.923633\pi$
$31$	−1.55839	+	2.69921i	−0.279895	+	0.484792i	0.691464	+	0.722411i	$0.256966\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−5.18852	+	4.78617i	−0.877021	+	0.809011i
$36$		0			0
$37$	−3.15627	+	5.46681i	−0.518887	+	0.898739i	0.480872	+	0.876791i	$0.340320\pi$
$37$	−3.15627	+	5.46681i	−0.518887	+	0.898739i	−0.999759	+	0.0219479i	$0.993013\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−1.38693	−	2.40224i	−0.216603	−	0.375167i	0.737164	−	0.675713i	$-0.236164\pi$
$41$	−1.38693	−	2.40224i	−0.216603	−	0.375167i	−0.953767	+	0.300546i	$0.902831\pi$
$42$		0			0
$43$	−4.87889	+	8.45048i	−0.744023	+	1.28869i	0.206626	+	0.978420i	$0.433752\pi$
$43$	−4.87889	+	8.45048i	−0.744023	+	1.28869i	−0.950650	+	0.310267i	$0.899582\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−5.02505	−	8.70364i	−0.732979	−	1.26956i	−0.955605	−	0.294652i	$-0.904796\pi$
$47$	−5.02505	−	8.70364i	−0.732979	−	1.26956i	0.222626	−	0.974904i	$-0.428537\pi$
$48$		0			0
$49$	0.563800	−	6.97726i	0.0805429	−	0.996751i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	1.47823	+	2.56037i	0.203050	+	0.351694i	0.949510	−	0.313737i	$-0.101581\pi$
$53$	1.47823	+	2.56037i	0.203050	+	0.351694i	−0.746459	+	0.665431i	$0.768248\pi$
$54$		0			0
$55$	3.64055			0.490891
$56$		0			0
$57$		0			0
$58$		0			0
$59$	1.77809	−	3.07974i	0.231487	−	0.400948i	−0.726759	−	0.686893i	$-0.758974\pi$
$59$	1.77809	−	3.07974i	0.231487	−	0.400948i	0.958246	+	0.285945i	$0.0923075\pi$
$60$		0			0
$61$	−0.663043	−	1.14842i	−0.0848940	−	0.147041i	0.820452	−	0.571715i	$-0.193722\pi$
$61$	−0.663043	−	1.14842i	−0.0848940	−	0.147041i	−0.905346	+	0.424675i	$0.860388\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	7.35297	+	12.7357i	0.912024	+	1.57967i
$66$		0			0
$67$	−4.14937	+	7.18692i	−0.506926	+	0.878021i	0.493042	+	0.870006i	$0.335885\pi$
$67$	−4.14937	+	7.18692i	−0.506926	+	0.878021i	−0.999968	+	0.00801592i	$0.997448\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−12.3069			−1.46056			−0.730279	−	0.683149i	$-0.760610\pi$
$71$	−12.3069			−1.46056			−0.730279	+	0.683149i	$0.760610\pi$
$72$		0			0
$73$	−1.11577	−	1.93257i	−0.130591	−	0.226190i	0.793314	−	0.608813i	$-0.208354\pi$
$73$	−1.11577	−	1.93257i	−0.130591	−	0.226190i	−0.923905	+	0.382623i	$0.875021\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−2.65358	+	2.44781i	−0.302404	+	0.278954i
$78$		0			0
$79$	−6.41535	−	11.1117i	−0.721783	−	1.25017i	−0.960284	−	0.279023i	$-0.909989\pi$
$79$	−6.41535	−	11.1117i	−0.721783	−	1.25017i	0.238501	−	0.971142i	$-0.423344\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−5.15934	+	8.93625i	−0.566312	+	0.980881i	0.430615	+	0.902536i	$0.358297\pi$
$83$	−5.15934	+	8.93625i	−0.566312	+	0.980881i	−0.996926	+	0.0783447i	$0.975037\pi$
$84$		0			0
$85$	−3.30648	−	5.72700i	−0.358638	−	0.621180i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−7.73159	+	13.3915i	−0.819547	+	1.41950i	0.0864698	+	0.996254i	$0.472441\pi$
$89$	−7.73159	+	13.3915i	−0.819547	+	1.41950i	−0.906017	+	0.423242i	$0.860892\pi$
$90$		0			0
$91$	−13.9227	−	4.33908i	−1.45950	−	0.454860i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	5.85896	−	10.1480i	0.601117	−	1.04117i
$96$		0			0
$97$	−2.55369	+	4.42311i	−0.259288	+	0.449099i	−0.966051	−	0.258351i	$-0.916821\pi$
$97$	−2.55369	+	4.42311i	−0.259288	+	0.449099i	0.706764	+	0.707450i	$0.250154\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	2.69496			0.268159			0.134079	−	0.990971i	$-0.457192\pi$
$101$	2.69496			0.268159			0.134079	+	0.990971i	$0.457192\pi$
$102$		0			0
$103$	−13.0214			−1.28304			−0.641519	−	0.767107i	$-0.721696\pi$
$103$	−13.0214			−1.28304			−0.641519	+	0.767107i	$0.721696\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	1.49753	−	2.59379i	0.144771	−	0.250751i	−0.784516	−	0.620108i	$-0.787089\pi$
$107$	1.49753	−	2.59379i	0.144771	−	0.250751i	0.929288	+	0.369357i	$0.120422\pi$
$108$		0			0
$109$	10.0132	+	17.3434i	0.959093	+	1.66120i	0.724710	+	0.689054i	$0.241974\pi$
$109$	10.0132	+	17.3434i	0.959093	+	1.66120i	0.234383	+	0.972144i	$0.424693\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	6.23211	+	10.7943i	0.586267	+	1.01544i	0.994716	+	0.102664i	$0.0327365\pi$
$113$	6.23211	+	10.7943i	0.586267	+	1.01544i	−0.408449	+	0.912781i	$0.633930\pi$
$114$		0			0
$115$	−12.5130			−1.16685
$116$		0			0
$117$		0			0
$118$		0			0
$119$	6.26077	+	1.95120i	0.573924	+	0.178866i
$120$		0			0
$121$	−9.13810			−0.830737
$122$		0			0
$123$		0			0
$124$		0			0
$125$	7.68837			0.687669
$126$		0			0
$127$	15.0734			1.33754			0.668772	−	0.743467i	$-0.266820\pi$
$127$	15.0734			1.33754			0.668772	+	0.743467i	$0.266820\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−14.8406			−1.29663			−0.648316	−	0.761371i	$-0.724527\pi$
$131$	−14.8406			−1.29663			−0.648316	+	0.761371i	$0.724527\pi$
$132$		0			0
$133$	2.55269	+	11.3363i	0.221346	+	0.982980i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	20.1559			1.72203			0.861017	−	0.508577i	$-0.169828\pi$
$137$	20.1559			1.72203			0.861017	+	0.508577i	$0.169828\pi$
$138$		0			0
$139$	−9.91552	−	17.1742i	−0.841023	−	1.45669i	−0.889031	−	0.457848i	$-0.848620\pi$
$139$	−9.91552	−	17.1742i	−0.841023	−	1.45669i	0.0480074	−	0.998847i	$-0.484713\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	3.76056	+	6.51347i	0.314473	+	0.544684i
$144$		0			0
$145$	7.86557	−	13.6236i	0.653200	−	1.13138i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−17.3033			−1.41754			−0.708772	−	0.705438i	$-0.750750\pi$
$149$	−17.3033			−1.41754			−0.708772	+	0.705438i	$0.750750\pi$
$150$		0			0
$151$	−3.98976			−0.324682			−0.162341	−	0.986735i	$-0.551904\pi$
$151$	−3.98976			−0.324682			−0.162341	+	0.986735i	$0.551904\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	4.15781	−	7.20153i	0.333963	−	0.578441i
$156$		0			0
$157$	12.0994	−	20.9568i	0.965637	−	1.67253i	0.257742	−	0.966214i	$-0.417022\pi$
$157$	12.0994	−	20.9568i	0.965637	−	1.67253i	0.707895	−	0.706318i	$-0.249645\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	9.12073	−	8.41345i	0.718814	−	0.663073i
$162$		0			0
$163$	2.34498	−	4.06162i	0.183673	−	0.318131i	−0.759456	−	0.650559i	$-0.774535\pi$
$163$	2.34498	−	4.06162i	0.183673	−	0.318131i	0.943129	+	0.332428i	$0.107868\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	6.12627	+	10.6110i	0.474065	+	0.821104i	0.999559	−	0.0296928i	$-0.00945290\pi$
$167$	6.12627	+	10.6110i	0.474065	+	0.821104i	−0.525494	+	0.850797i	$0.676120\pi$
$168$		0			0
$169$	−8.69072	+	15.0528i	−0.668517	+	1.15791i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−4.05503	−	7.02352i	−0.308298	−	0.533988i	0.669692	−	0.742639i	$-0.266426\pi$
$173$	−4.05503	−	7.02352i	−0.308298	−	0.533988i	−0.977990	+	0.208651i	$0.933093\pi$
$174$		0			0
$175$	4.11952	−	3.80007i	0.311406	−	0.287258i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−4.91636	−	8.51538i	−0.367466	−	0.636469i	0.621703	−	0.783253i	$-0.286441\pi$
$179$	−4.91636	−	8.51538i	−0.367466	−	0.636469i	−0.989169	+	0.146784i	$0.953108\pi$
$180$		0			0
$181$	15.8876			1.18092			0.590458	−	0.807068i	$-0.298947\pi$
$181$	15.8876			1.18092			0.590458	+	0.807068i	$0.298947\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	8.42097	−	14.5856i	0.619122	−	1.07235i
$186$		0			0
$187$	−1.69105	−	2.92898i	−0.123661	−	0.214188i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−1.10949	−	1.92170i	−0.0802800	−	0.139049i	0.823090	−	0.567911i	$-0.192248\pi$
$191$	−1.10949	−	1.92170i	−0.0802800	−	0.139049i	−0.903370	+	0.428862i	$0.858915\pi$
$192$		0			0
$193$	−2.92084	+	5.05904i	−0.210247	+	0.364158i	−0.951792	−	0.306745i	$-0.900760\pi$
$193$	−2.92084	+	5.05904i	−0.210247	+	0.364158i	0.741545	+	0.670903i	$0.234093\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−1.93695			−0.138002			−0.0690010	−	0.997617i	$-0.521981\pi$
$197$	−1.93695			−0.138002			−0.0690010	+	0.997617i	$0.521981\pi$
$198$		0			0
$199$	1.84540	+	3.19633i	0.130817	+	0.226582i	0.923992	−	0.382412i	$-0.124907\pi$
$199$	1.84540	+	3.19633i	0.130817	+	0.226582i	−0.793175	+	0.608994i	$0.791573\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	3.42694	+	15.2188i	0.240524	+	1.06815i
$204$		0			0
$205$	3.70037	+	6.40922i	0.258445	+	0.447639i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	2.99647	−	5.19004i	0.207270	−	0.359002i
$210$		0			0
$211$	−5.67097	−	9.82241i	−0.390406	−	0.676202i	0.602097	−	0.798423i	$-0.294332\pi$
$211$	−5.67097	−	9.82241i	−0.390406	−	0.676202i	−0.992503	+	0.122220i	$0.960999\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	13.0170	−	22.5460i	0.887749	−	1.53763i
$216$		0			0
$217$	1.81151	+	8.04478i	0.122973	+	0.546115i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	6.83096	−	11.8316i	0.459500	−	0.795878i
$222$		0			0
$223$	−0.965547	+	1.67238i	−0.0646578	+	0.111991i	−0.896542	−	0.442958i	$-0.853929\pi$
$223$	−0.965547	+	1.67238i	−0.0646578	+	0.111991i	0.831884	+	0.554949i	$0.187262\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−19.7283			−1.30941			−0.654705	−	0.755884i	$-0.727207\pi$
$227$	−19.7283			−1.30941			−0.654705	+	0.755884i	$0.727207\pi$
$228$		0			0
$229$	26.4197			1.74586			0.872931	−	0.487844i	$-0.162216\pi$
$229$	26.4197			1.74586			0.872931	+	0.487844i	$0.162216\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	4.24071	−	7.34513i	0.277818	−	0.481196i	−0.693024	−	0.720915i	$-0.743722\pi$
$233$	4.24071	−	7.34513i	0.277818	−	0.481196i	0.970842	+	0.239719i	$0.0770553\pi$
$234$		0			0
$235$	13.4069	+	23.2215i	0.874571	+	1.51480i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	8.08023	+	13.9954i	0.522667	+	0.905286i	0.999652	+	0.0263743i	$0.00839619\pi$
$239$	8.08023	+	13.9954i	0.522667	+	0.905286i	−0.476985	+	0.878911i	$0.658270\pi$
$240$		0			0
$241$	10.9735			0.706868			0.353434	−	0.935460i	$-0.385014\pi$
$241$	10.9735			0.706868			0.353434	+	0.935460i	$0.385014\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−1.50423	+	18.6154i	−0.0961017	+	1.18930i
$246$		0			0
$247$	24.2084			1.54034
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−2.85873			−0.180442			−0.0902208	−	0.995922i	$-0.528757\pi$
$251$	−2.85873			−0.180442			−0.0902208	+	0.995922i	$0.528757\pi$
$252$		0			0
$253$	−6.39959			−0.402339
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−24.5875			−1.53373			−0.766864	−	0.641810i	$-0.778184\pi$
$257$	−24.5875			−1.53373			−0.766864	+	0.641810i	$0.778184\pi$
$258$		0			0
$259$	3.66893	+	16.2934i	0.227976	+	1.01242i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−0.957972			−0.0590711			−0.0295355	−	0.999564i	$-0.509403\pi$
$263$	−0.957972			−0.0590711			−0.0295355	+	0.999564i	$0.509403\pi$
$264$		0			0
$265$	−3.94394	−	6.83111i	−0.242274	−	0.419632i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−8.31005	−	14.3934i	−0.506673	−	0.877583i	−0.999970	−	0.00772245i	$-0.997542\pi$
$269$	−8.31005	−	14.3934i	−0.506673	−	0.877583i	0.493297	−	0.869861i	$-0.335791\pi$
$270$		0			0
$271$	7.21801	−	12.5020i	0.438463	−	0.759440i	−0.559108	−	0.829095i	$-0.688856\pi$
$271$	7.21801	−	12.5020i	0.438463	−	0.759440i	0.997571	+	0.0696545i	$0.0221897\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−2.89048			−0.174302
$276$		0			0
$277$	−4.46642			−0.268361			−0.134181	−	0.990957i	$-0.542840\pi$
$277$	−4.46642			−0.268361			−0.134181	+	0.990957i	$0.542840\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−2.62617	+	4.54867i	−0.156664	+	0.271351i	−0.933664	−	0.358151i	$-0.883407\pi$
$281$	−2.62617	+	4.54867i	−0.156664	+	0.271351i	0.776999	+	0.629501i	$0.216741\pi$
$282$		0			0
$283$	−5.65751	+	9.79909i	−0.336304	+	0.582495i	−0.983734	−	0.179629i	$-0.942510\pi$
$283$	−5.65751	+	9.79909i	−0.336304	+	0.582495i	0.647431	+	0.762124i	$0.275844\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−7.00658	−	2.18363i	−0.413585	−	0.128896i
$288$		0			0
$289$	5.42826	−	9.40201i	0.319309	−	0.553060i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	5.38422	+	9.32574i	0.314549	+	0.544815i	0.979342	−	0.202213i	$-0.0648133\pi$
$293$	5.38422	+	9.32574i	0.314549	+	0.544815i	−0.664792	+	0.747028i	$0.731480\pi$
$294$		0			0
$295$	−4.74397	+	8.21679i	−0.276204	+	0.478400i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−12.9255	−	22.3877i	−0.747503	−	1.29471i
$300$		0			0
$301$	5.67135	+	25.1860i	0.326891	+	1.45170i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	1.76901	+	3.06402i	0.101293	+	0.175445i
$306$		0			0
$307$	9.42151			0.537714			0.268857	−	0.963180i	$-0.413354\pi$
$307$	9.42151			0.537714			0.268857	+	0.963180i	$0.413354\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−5.65754	+	9.79914i	−0.320809	+	0.555658i	−0.980655	−	0.195743i	$-0.937288\pi$
$311$	−5.65754	+	9.79914i	−0.320809	+	0.555658i	0.659846	+	0.751401i	$0.270622\pi$
$312$		0			0
$313$	−10.8431	−	18.7808i	−0.612889	−	1.06156i	−0.990751	−	0.135693i	$-0.956674\pi$
$313$	−10.8431	−	18.7808i	−0.612889	−	1.06156i	0.377862	−	0.925862i	$-0.376660\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−12.6087	−	21.8389i	−0.708174	−	1.22659i	−0.965534	−	0.260278i	$-0.916186\pi$
$317$	−12.6087	−	21.8389i	−0.708174	−	1.22659i	0.257360	−	0.966316i	$-0.417147\pi$
$318$		0			0
$319$	4.02272	−	6.96755i	0.225229	−	0.390108i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−10.8860			−0.605715
$324$		0			0
$325$	−5.83802	−	10.1118i	−0.323835	−	0.560899i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−25.3858	−	7.91160i	−1.39956	−	0.436180i
$330$		0			0
$331$	8.51226	+	14.7437i	0.467876	+	0.810386i	0.999326	−	0.0367042i	$-0.0116859\pi$
$331$	8.51226	+	14.7437i	0.467876	+	0.810386i	−0.531450	+	0.847090i	$0.678353\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	11.0706	−	19.1748i	0.604851	−	1.04763i
$336$		0			0
$337$	6.85166	+	11.8674i	0.373233	+	0.646459i	0.990061	−	0.140639i	$-0.0449157\pi$
$337$	6.85166	+	11.8674i	0.373233	+	0.646459i	−0.616827	+	0.787098i	$0.711582\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	2.12644	−	3.68310i	0.115153	−	0.199451i
$342$		0			0
$343$	−11.4201	−	14.5802i	−0.616628	−	0.787254i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	12.2183	−	21.1627i	0.655912	−	1.13607i	−0.325752	−	0.945455i	$-0.605618\pi$
$347$	12.2183	−	21.1627i	0.655912	−	1.13607i	0.981664	−	0.190618i	$-0.0610491\pi$
$348$		0			0
$349$	11.4881	−	19.8979i	0.614943	−	1.06511i	−0.375451	−	0.926842i	$-0.622512\pi$
$349$	11.4881	−	19.8979i	0.614943	−	1.06511i	0.990394	−	0.138271i	$-0.0441544\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−24.0305			−1.27901			−0.639507	−	0.768785i	$-0.720861\pi$
$353$	−24.0305			−1.27901			−0.639507	+	0.768785i	$0.720861\pi$
$354$		0			0
$355$	32.8350			1.74270
$356$		0			0
$357$		0			0
$358$		0			0
$359$	9.84234	−	17.0474i	0.519459	−	0.899729i	−0.480285	−	0.877112i	$-0.659467\pi$
$359$	9.84234	−	17.0474i	0.519459	−	0.899729i	0.999744	−	0.0226169i	$-0.00719980\pi$
$360$		0			0
$361$	−0.144819	−	0.250833i	−0.00762204	−	0.0132018i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	2.97690	+	5.15613i	0.155818	+	0.269884i
$366$		0			0
$367$	−14.2006			−0.741263			−0.370632	−	0.928780i	$-0.620859\pi$
$367$	−14.2006			−0.741263			−0.370632	+	0.928780i	$0.620859\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	7.46779	+	2.32737i	0.387708	+	0.120831i
$372$		0			0
$373$	−28.4669			−1.47396			−0.736980	−	0.675914i	$-0.763749\pi$
$373$	−28.4669			−1.47396			−0.736980	+	0.675914i	$0.763749\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	32.4994			1.67381
$378$		0			0
$379$	−4.25098			−0.218358			−0.109179	−	0.994022i	$-0.534822\pi$
$379$	−4.25098			−0.218358			−0.109179	+	0.994022i	$0.534822\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	35.8428			1.83148			0.915740	−	0.401772i	$-0.131605\pi$
$383$	35.8428			1.83148			0.915740	+	0.401772i	$0.131605\pi$
$384$		0			0
$385$	7.07981	−	6.53080i	0.360820	−	0.332840i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	31.7944			1.61204			0.806020	−	0.591888i	$-0.201617\pi$
$389$	31.7944			1.61204			0.806020	+	0.591888i	$0.201617\pi$
$390$		0			0
$391$	5.81235	+	10.0673i	0.293943	+	0.509125i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	17.1163	+	29.6462i	0.861213	+	1.49166i
$396$		0			0
$397$	3.07669	−	5.32899i	0.154415	−	0.267454i	−0.778431	−	0.627730i	$-0.783984\pi$
$397$	3.07669	−	5.32899i	0.154415	−	0.267454i	0.932846	+	0.360276i	$0.117317\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	14.8229			0.740222			0.370111	−	0.928988i	$-0.379320\pi$
$401$	14.8229			0.740222			0.370111	+	0.928988i	$0.379320\pi$
$402$		0			0
$403$	17.1795			0.855770
$404$		0			0
$405$		0			0
$406$		0			0
$407$	4.30677	−	7.45954i	0.213479	−	0.369756i
$408$		0			0
$409$	−10.7222	+	18.5713i	−0.530177	+	0.918293i	0.469203	+	0.883090i	$0.344541\pi$
$409$	−10.7222	+	18.5713i	−0.530177	+	0.918293i	−0.999380	+	0.0352032i	$0.988792\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−2.06689	−	9.17892i	−0.101705	−	0.451665i
$414$		0			0
$415$	13.7652	−	23.8421i	0.675708	−	1.17036i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−13.2332	−	22.9205i	−0.646483	−	1.11974i	−0.983957	−	0.178406i	$-0.942906\pi$
$419$	−13.2332	−	22.9205i	−0.646483	−	1.11974i	0.337474	−	0.941335i	$-0.390428\pi$
$420$		0			0
$421$	−8.54824	+	14.8060i	−0.416616	+	0.721600i	−0.995597	−	0.0937415i	$-0.970117\pi$
$421$	−8.54824	+	14.8060i	−0.416616	+	0.721600i	0.578981	+	0.815341i	$0.303451\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	2.62524	+	4.54705i	0.127343	+	0.220564i
$426$		0			0
$427$	−3.34959	−	1.04392i	−0.162098	−	0.0505186i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	12.0292	+	20.8352i	0.579425	+	1.00359i	0.995545	+	0.0942846i	$0.0300564\pi$
$431$	12.0292	+	20.8352i	0.579425	+	1.00359i	−0.416120	+	0.909310i	$0.636610\pi$
$432$		0			0
$433$	−6.58345			−0.316380			−0.158190	−	0.987409i	$-0.550566\pi$
$433$	−6.58345			−0.316380			−0.158190	+	0.987409i	$0.550566\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−10.2993	+	17.8389i	−0.492681	+	0.853348i
$438$		0			0
$439$	−10.6327	−	18.4164i	−0.507472	−	0.878967i	−0.999963	−	0.00864927i	$-0.997247\pi$
$439$	−10.6327	−	18.4164i	−0.507472	−	0.878967i	0.492491	−	0.870318i	$-0.336087\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−0.471724	−	0.817050i	−0.0224123	−	0.0388192i	0.854602	−	0.519284i	$-0.173801\pi$
$443$	−0.471724	−	0.817050i	−0.0224123	−	0.0388192i	−0.877014	+	0.480465i	$0.840468\pi$
$444$		0			0
$445$	20.6280	−	35.7288i	0.977862	−	1.69371i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	17.7959			0.839842			0.419921	−	0.907561i	$-0.362058\pi$
$449$	17.7959			0.839842			0.419921	+	0.907561i	$0.362058\pi$
$450$		0			0
$451$	1.89249	+	3.27789i	0.0891139	+	0.154350i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	37.1461	+	11.5768i	1.74144	+	0.542727i
$456$		0			0
$457$	6.88851	+	11.9313i	0.322231	+	0.558120i	0.980948	−	0.194270i	$-0.0622339\pi$
$457$	6.88851	+	11.9313i	0.322231	+	0.558120i	−0.658717	+	0.752391i	$0.728901\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	2.97576	−	5.15417i	0.138595	−	0.240054i	−0.788370	−	0.615201i	$-0.789075\pi$
$461$	2.97576	−	5.15417i	0.138595	−	0.240054i	0.926965	+	0.375148i	$0.122408\pi$
$462$		0			0
$463$	−17.7618	−	30.7644i	−0.825463	−	1.42974i	−0.901565	−	0.432643i	$-0.857581\pi$
$463$	−17.7618	−	30.7644i	−0.825463	−	1.42974i	0.0761023	−	0.997100i	$-0.475752\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−12.2574	+	21.2305i	−0.567207	+	0.982431i	0.429634	+	0.903003i	$0.358643\pi$
$467$	−12.2574	+	21.2305i	−0.567207	+	0.982431i	−0.996841	+	0.0794277i	$0.974691\pi$
$468$		0			0
$469$	4.82333	+	21.4200i	0.222721	+	0.989086i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	6.65731	−	11.5308i	0.306103	−	0.530187i
$474$		0			0
$475$	−4.65183	+	8.05720i	−0.213440	+	0.369690i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−19.2352			−0.878878			−0.439439	−	0.898272i	$-0.644823\pi$
$479$	−19.2352			−0.878878			−0.439439	+	0.898272i	$0.644823\pi$
$480$		0			0
$481$	34.7943			1.58648
$482$		0			0
$483$		0			0
$484$		0			0
$485$	6.81328	−	11.8009i	0.309375	−	0.535853i
$486$		0			0
$487$	12.3089	+	21.3197i	0.557770	+	0.966086i	0.997682	+	0.0680455i	$0.0216763\pi$
$487$	12.3089	+	21.3197i	0.557770	+	0.966086i	−0.439912	+	0.898041i	$0.644990\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−9.73086	−	16.8543i	−0.439147	−	0.760626i	0.558476	−	0.829520i	$-0.311386\pi$
$491$	−9.73086	−	16.8543i	−0.439147	−	0.760626i	−0.997624	+	0.0688947i	$0.978053\pi$
$492$		0			0
$493$	−14.6143			−0.658197
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−23.9333	+	22.0774i	−1.07356	+	0.990306i
$498$		0			0
$499$	−7.17781			−0.321323			−0.160661	−	0.987010i	$-0.551363\pi$
$499$	−7.17781			−0.321323			−0.160661	+	0.987010i	$0.551363\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−22.1112			−0.985889			−0.492945	−	0.870061i	$-0.664079\pi$
$503$	−22.1112			−0.985889			−0.492945	+	0.870061i	$0.664079\pi$
$504$		0			0
$505$	−7.19021			−0.319960
$506$		0			0
$507$		0			0
$508$		0			0
$509$	31.3575			1.38989			0.694947	−	0.719061i	$-0.255428\pi$
$509$	31.3575			1.38989			0.694947	+	0.719061i	$0.255428\pi$
$510$		0			0
$511$	−5.63670	−	1.75670i	−0.249353	−	0.0777120i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	34.7414			1.53089
$516$		0			0
$517$	6.85675	+	11.8762i	0.301559	+	0.522316i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−1.76588	−	3.05859i	−0.0773645	−	0.133999i	0.824748	−	0.565501i	$-0.191317\pi$
$521$	−1.76588	−	3.05859i	−0.0773645	−	0.133999i	−0.902112	+	0.431502i	$0.857984\pi$
$522$		0			0
$523$	−7.03821	+	12.1905i	−0.307759	+	0.533055i	−0.977872	−	0.209205i	$-0.932912\pi$
$523$	−7.03821	+	12.1905i	−0.307759	+	0.533055i	0.670113	+	0.742259i	$0.266246\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−7.72526			−0.336518
$528$		0			0
$529$	−1.00374			−0.0436409
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−7.64469	+	13.2410i	−0.331128	+	0.573531i
$534$		0			0
$535$	−3.99543	+	6.92029i	−0.172737	+	0.299190i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−0.769313	+	9.52056i	−0.0331367	+	0.410080i
$540$		0			0
$541$	11.5799	−	20.0569i	0.497858	−	0.862315i	−0.502139	−	0.864787i	$-0.667454\pi$
$541$	11.5799	−	20.0569i	0.497858	−	0.862315i	0.999997	+	0.00247207i	$0.000786884\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−26.7155	−	46.2725i	−1.14436	−	1.98210i
$546$		0			0
$547$	−5.76832	+	9.99102i	−0.246635	+	0.427185i	−0.962590	−	0.270962i	$-0.912658\pi$
$547$	−5.76832	+	9.99102i	−0.246635	+	0.427185i	0.715955	+	0.698147i	$0.245992\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−12.9480	−	22.4266i	−0.551605	−	0.955407i
$552$		0			0
$553$	−32.4094	−	10.1005i	−1.37819	−	0.429518i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	1.04108	+	1.80321i	0.0441122	+	0.0764045i	0.887238	−	0.461311i	$-0.152621\pi$
$557$	1.04108	+	1.80321i	0.0441122	+	0.0764045i	−0.843126	+	0.537716i	$0.819287\pi$
$558$		0			0
$559$	53.7842			2.27483
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−3.35403	+	5.80936i	−0.141356	+	0.244835i	−0.928007	−	0.372562i	$-0.878479\pi$
$563$	−3.35403	+	5.80936i	−0.141356	+	0.244835i	0.786652	+	0.617397i	$0.211813\pi$
$564$		0			0
$565$	−16.6274	−	28.7995i	−0.699519	−	1.21160i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−4.01112	−	6.94746i	−0.168155	−	0.291253i	0.769616	−	0.638507i	$-0.220447\pi$
$569$	−4.01112	−	6.94746i	−0.168155	−	0.291253i	−0.937771	+	0.347254i	$0.887114\pi$
$570$		0			0
$571$	−3.34215	+	5.78877i	−0.139865	+	0.242253i	−0.927445	−	0.373959i	$-0.878000\pi$
$571$	−3.34215	+	5.78877i	−0.139865	+	0.242253i	0.787581	+	0.616212i	$0.211333\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	9.93496			0.414316
$576$		0			0
$577$	−14.0088	−	24.2639i	−0.583193	−	1.01012i	−0.995098	−	0.0988925i	$-0.968470\pi$
$577$	−14.0088	−	24.2639i	−0.583193	−	1.01012i	0.411906	−	0.911227i	$-0.364863\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	5.99736	+	26.6338i	0.248812	+	1.10496i
$582$		0			0
$583$	−2.01706	−	3.49366i	−0.0835382	−	0.144692i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	3.35952	−	5.81886i	0.138662	−	0.240170i	−0.788328	−	0.615255i	$-0.789053\pi$
$587$	3.35952	−	5.81886i	0.138662	−	0.240170i	0.926990	+	0.375085i	$0.122386\pi$
$588$		0			0
$589$	−6.84443	−	11.8549i	−0.282020	−	0.488473i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	3.19462	−	5.53325i	0.131187	−	0.227223i	−0.792947	−	0.609290i	$-0.791454\pi$
$593$	3.19462	−	5.53325i	0.131187	−	0.227223i	0.924135	+	0.382067i	$0.124788\pi$
$594$		0			0
$595$	−16.7038	−	5.20583i	−0.684791	−	0.213418i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−2.96098	+	5.12856i	−0.120982	+	0.209547i	−0.920155	−	0.391554i	$-0.871938\pi$
$599$	−2.96098	+	5.12856i	−0.120982	+	0.209547i	0.799173	+	0.601101i	$0.205271\pi$
$600$		0			0
$601$	−1.97104	+	3.41393i	−0.0804002	+	0.139257i	−0.903422	−	0.428753i	$-0.858953\pi$
$601$	−1.97104	+	3.41393i	−0.0804002	+	0.139257i	0.823022	+	0.568010i	$0.192287\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	24.3806			0.991213
$606$		0			0
$607$	−7.09551			−0.287998			−0.143999	−	0.989578i	$-0.545996\pi$
$607$	−7.09551			−0.287998			−0.143999	+	0.989578i	$0.545996\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−27.6978	+	47.9739i	−1.12053	+	1.94082i
$612$		0			0
$613$	6.87000	+	11.8992i	0.277477	+	0.480604i	0.970757	−	0.240064i	$-0.0771685\pi$
$613$	6.87000	+	11.8992i	0.277477	+	0.480604i	−0.693280	+	0.720668i	$0.743835\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	16.3605	+	28.3372i	0.658649	+	1.14081i	0.980966	+	0.194182i	$0.0622051\pi$
$617$	16.3605	+	28.3372i	0.658649	+	1.14081i	−0.322317	+	0.946632i	$0.604462\pi$
$618$		0			0
$619$	22.6180			0.909094			0.454547	−	0.890723i	$-0.349801\pi$
$619$	22.6180			0.909094			0.454547	+	0.890723i	$0.349801\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	8.98740	+	39.9124i	0.360073	+	1.59905i
$624$		0			0
$625$	−31.1043			−1.24417
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−15.6463			−0.623858
$630$		0			0
$631$	−43.9355			−1.74905			−0.874523	−	0.484984i	$-0.838825\pi$
$631$	−43.9355			−1.74905			−0.874523	+	0.484984i	$0.838825\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−40.2160			−1.59592
$636$		0			0
$637$	−34.8596	+	16.5378i	−1.38119	+	0.655252i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−39.8595			−1.57436			−0.787178	−	0.616726i	$-0.788459\pi$
$641$	−39.8595			−1.57436			−0.787178	+	0.616726i	$0.788459\pi$
$642$		0			0
$643$	9.24049	+	16.0050i	0.364410	+	0.631176i	0.988681	−	0.150032i	$-0.0479375\pi$
$643$	9.24049	+	16.0050i	0.364410	+	0.631176i	−0.624272	+	0.781207i	$0.714604\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−8.76068	−	15.1739i	−0.344418	−	0.596549i	0.640830	−	0.767683i	$-0.278590\pi$
$647$	−8.76068	−	15.1739i	−0.344418	−	0.596549i	−0.985248	+	0.171134i	$0.945257\pi$
$648$		0			0
$649$	−2.42622	+	4.20234i	−0.0952376	+	0.164956i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−27.4055			−1.07246			−0.536230	−	0.844072i	$-0.680152\pi$
$653$	−27.4055			−1.07246			−0.536230	+	0.844072i	$0.680152\pi$
$654$		0			0
$655$	39.5951			1.54711
$656$		0			0
$657$		0			0
$658$		0			0
$659$	16.2580	−	28.1597i	0.633322	−	1.09695i	−0.353546	−	0.935417i	$-0.615024\pi$
$659$	16.2580	−	28.1597i	0.633322	−	1.09695i	0.986868	−	0.161529i	$-0.0516424\pi$
$660$		0			0
$661$	−19.4336	+	33.6599i	−0.755878	+	1.30922i	0.189059	+	0.981966i	$0.439456\pi$
$661$	−19.4336	+	33.6599i	−0.755878	+	1.30922i	−0.944937	+	0.327253i	$0.893877\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−6.81061	−	30.2454i	−0.264104	−	1.17287i
$666$		0			0
$667$	−13.8266	+	23.9484i	−0.535369	+	0.927286i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	0.904731	+	1.56704i	0.0349268	+	0.0604949i
$672$		0			0
$673$	−4.50978	+	7.81117i	−0.173839	+	0.301099i	−0.939759	−	0.341838i	$-0.888951\pi$
$673$	−4.50978	+	7.81117i	−0.173839	+	0.301099i	0.765920	+	0.642936i	$0.222284\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	9.41435	+	16.3061i	0.361823	+	0.626695i	0.988261	−	0.152776i	$-0.0488212\pi$
$677$	9.41435	+	16.3061i	0.361823	+	0.626695i	−0.626438	+	0.779471i	$0.715488\pi$
$678$		0			0
$679$	2.96847	+	13.1827i	0.113919	+	0.505907i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−5.35476	−	9.27471i	−0.204894	−	0.354887i	0.745205	−	0.666836i	$-0.232352\pi$
$683$	−5.35476	−	9.27471i	−0.204894	−	0.354887i	−0.950099	+	0.311949i	$0.899018\pi$
$684$		0			0
$685$	−53.7763			−2.05468
$686$		0			0
$687$		0			0
$688$		0			0
$689$	8.14791	−	14.1126i	0.310411	−	0.537647i
$690$		0			0
$691$	−2.52277	−	4.36956i	−0.0959705	−	0.166226i	0.814043	−	0.580805i	$-0.197262\pi$
$691$	−2.52277	−	4.36956i	−0.0959705	−	0.166226i	−0.910013	+	0.414579i	$0.863929\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	26.4548	+	45.8210i	1.00349	+	1.73809i
$696$		0			0
$697$	3.43766	−	5.95421i	0.130211	−	0.225532i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−44.9138			−1.69637			−0.848186	−	0.529698i	$-0.822305\pi$
$701$	−44.9138			−1.69637			−0.848186	+	0.529698i	$0.822305\pi$
$702$		0			0
$703$	−13.8623	−	24.0102i	−0.522827	−	0.905563i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	5.24092	−	4.83451i	0.197105	−	0.181820i
$708$		0			0
$709$	−3.72658	−	6.45463i	−0.139955	−	0.242409i	0.787524	−	0.616283i	$-0.211362\pi$
$709$	−3.72658	−	6.45463i	−0.139955	−	0.242409i	−0.927479	+	0.373875i	$0.878029\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−7.30887	+	12.6593i	−0.273719	+	0.474096i
$714$		0			0
$715$	−10.0332	−	17.3781i	−0.375222	−	0.649903i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−21.5574	+	37.3385i	−0.803954	+	1.39249i	0.113040	+	0.993590i	$0.463941\pi$
$719$	−21.5574	+	37.3385i	−0.803954	+	1.39249i	−0.916995	+	0.398899i	$0.869392\pi$
$720$		0			0
$721$	−25.3229	+	23.3592i	−0.943074	+	0.869943i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−6.24501	+	10.8167i	−0.231934	+	0.401721i
$726$		0			0
$727$	−0.389926	+	0.675372i	−0.0144616	+	0.0250482i	−0.873166	−	0.487424i	$-0.837937\pi$
$727$	−0.389926	+	0.675372i	−0.0144616	+	0.0250482i	0.858704	+	0.512472i	$0.171270\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−24.1857			−0.894540
$732$		0			0
$733$	−15.6772			−0.579050			−0.289525	−	0.957170i	$-0.593497\pi$
$733$	−15.6772			−0.579050			−0.289525	+	0.957170i	$0.593497\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	5.66187	−	9.80664i	0.208558	−	0.361232i
$738$		0			0
$739$	8.87450	+	15.3711i	0.326454	+	0.565434i	0.981805	−	0.189889i	$-0.0608129\pi$
$739$	8.87450	+	15.3711i	0.326454	+	0.565434i	−0.655352	+	0.755324i	$0.727480\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	3.74308	+	6.48321i	0.137320	+	0.237846i	0.926481	−	0.376340i	$-0.122818\pi$
$743$	3.74308	+	6.48321i	0.137320	+	0.237846i	−0.789161	+	0.614186i	$0.789484\pi$
$744$		0			0
$745$	46.1656			1.69138
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−1.74076	−	7.73060i	−0.0636062	−	0.282470i
$750$		0			0
$751$	23.3599			0.852415			0.426208	−	0.904625i	$-0.359849\pi$
$751$	23.3599			0.852415			0.426208	+	0.904625i	$0.359849\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	10.6447			0.387402
$756$		0			0
$757$	31.2350			1.13525			0.567627	−	0.823286i	$-0.307862\pi$
$757$	31.2350			1.13525			0.567627	+	0.823286i	$0.307862\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−16.6147			−0.602282			−0.301141	−	0.953580i	$-0.597367\pi$
$761$	−16.6147			−0.602282			−0.301141	+	0.953580i	$0.597367\pi$
$762$		0			0
$763$	50.5853	+	15.7651i	1.83131	+	0.570736i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−19.6014			−0.707766
$768$		0			0
$769$	18.3794	+	31.8340i	0.662777	+	1.14796i	0.979883	+	0.199573i	$0.0639556\pi$
$769$	18.3794	+	31.8340i	0.662777	+	1.14796i	−0.317106	+	0.948390i	$0.602711\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−4.77690	−	8.27382i	−0.171813	−	0.297589i	0.767241	−	0.641359i	$-0.221629\pi$
$773$	−4.77690	−	8.27382i	−0.171813	−	0.297589i	−0.939054	+	0.343770i	$0.888296\pi$
$774$		0			0
$775$	−3.30116	+	5.71778i	−0.118581	+	0.205389i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	12.1828			0.436495
$780$		0			0
$781$	16.7929			0.600897
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−32.2814	+	55.9130i	−1.15217	+	1.99562i
$786$		0			0
$787$	10.1339	−	17.5524i	0.361233	−	0.625674i	−0.626931	−	0.779075i	$-0.715689\pi$
$787$	10.1339	−	17.5524i	0.361233	−	0.625674i	0.988164	+	0.153401i	$0.0490226\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	31.4837	+	9.81203i	1.11943	+	0.348876i
$792$		0			0
$793$	−3.65465	+	6.33004i	−0.129780	+	0.224786i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−21.4236	−	37.1068i	−0.758863	−	1.31439i	−0.943431	−	0.331570i	$-0.892422\pi$
$797$	−21.4236	−	37.1068i	−0.758863	−	1.31439i	0.184567	−	0.982820i	$-0.440912\pi$
$798$		0			0
$799$	12.4551	−	21.5729i	0.440630	−	0.763194i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	1.52248	+	2.63702i	0.0537273	+	0.0930584i
$804$		0			0
$805$	−24.3343	+	22.4472i	−0.857670	+	0.791161i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	10.4750	+	18.1432i	0.368282	+	0.637883i	0.989297	−	0.145916i	$-0.0466129\pi$
$809$	10.4750	+	18.1432i	0.368282	+	0.637883i	−0.621015	+	0.783798i	$0.713280\pi$
$810$		0			0
$811$	−19.0129			−0.667633			−0.333817	−	0.942638i	$-0.608337\pi$
$811$	−19.0129			−0.667633			−0.333817	+	0.942638i	$0.608337\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−6.25644	+	10.8365i	−0.219154	+	0.379585i
$816$		0			0
$817$	−21.4281	−	37.1145i	−0.749673	−	1.29847i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−16.6953	−	28.9171i	−0.582669	−	1.00921i	−0.995162	−	0.0982515i	$-0.968675\pi$
$821$	−16.6953	−	28.9171i	−0.582669	−	1.00921i	0.412493	−	0.910961i	$-0.364658\pi$
$822$		0			0
$823$	4.52040	−	7.82955i	0.157571	−	0.272921i	−0.776421	−	0.630214i	$-0.782967\pi$
$823$	4.52040	−	7.82955i	0.157571	−	0.272921i	0.933992	+	0.357293i	$0.116300\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−22.4071			−0.779172			−0.389586	−	0.920990i	$-0.627382\pi$
$827$	−22.4071			−0.779172			−0.389586	+	0.920990i	$0.627382\pi$
$828$		0			0
$829$	−11.4090	−	19.7610i	−0.396252	−	0.686328i	0.597008	−	0.802235i	$-0.296356\pi$
$829$	−11.4090	−	19.7610i	−0.396252	−	0.686328i	−0.993260	+	0.115907i	$0.963023\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	15.6757	−	7.43672i	0.543129	−	0.257667i
$834$		0			0
$835$	−16.3450	−	28.3104i	−0.565642	−	0.979720i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	8.05060	−	13.9441i	0.277938	−	0.481402i	−0.692934	−	0.721001i	$-0.743682\pi$
$839$	8.05060	−	13.9441i	0.277938	−	0.481402i	0.970872	+	0.239598i	$0.0770158\pi$
$840$		0			0
$841$	−2.88254	−	4.99271i	−0.0993980	−	0.172162i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	23.1870	−	40.1611i	0.797657	−	1.38158i
$846$		0			0
$847$	−17.7710	+	16.3929i	−0.610618	+	0.563267i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−14.8029	+	25.6394i	−0.507438	+	0.878909i
$852$		0			0
$853$	12.7818	−	22.1387i	0.437639	−	0.758013i	−0.559868	−	0.828582i	$-0.689148\pi$
$853$	12.7818	−	22.1387i	0.437639	−	0.758013i	0.997507	+	0.0705689i	$0.0224815\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	52.6556			1.79868			0.899340	−	0.437250i	$-0.144048\pi$
$857$	52.6556			1.79868			0.899340	+	0.437250i	$0.144048\pi$
$858$		0			0
$859$	−30.8862			−1.05382			−0.526912	−	0.849920i	$-0.676650\pi$
$859$	−30.8862			−1.05382			−0.526912	+	0.849920i	$0.676650\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	0.929596	−	1.61011i	0.0316438	−	0.0548087i	−0.849770	−	0.527154i	$-0.823259\pi$
$863$	0.929596	−	1.61011i	0.0316438	−	0.0548087i	0.881414	+	0.472345i	$0.156592\pi$
$864$		0			0
$865$	10.8189	+	18.7389i	0.367853	+	0.637141i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	8.75383	+	15.1621i	0.296953	+	0.514338i
$870$		0			0
$871$	45.7421			1.54991
$872$		0			0
$873$		0			0
$874$		0			0
$875$	14.9517	−	13.7922i	0.505459	−	0.466262i
$876$		0			0
$877$	−13.3037			−0.449234			−0.224617	−	0.974447i	$-0.572113\pi$
$877$	−13.3037			−0.449234			−0.224617	+	0.974447i	$0.572113\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−21.2210			−0.714954			−0.357477	−	0.933922i	$-0.616363\pi$
$881$	−21.2210			−0.714954			−0.357477	+	0.933922i	$0.616363\pi$
$882$		0			0
$883$	−49.8289			−1.67687			−0.838437	−	0.544998i	$-0.816530\pi$
$883$	−49.8289			−1.67687			−0.838437	+	0.544998i	$0.816530\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	1.77272			0.0595220			0.0297610	−	0.999557i	$-0.490525\pi$
$887$	1.77272			0.0595220			0.0297610	+	0.999557i	$0.490525\pi$
$888$		0			0
$889$	29.3133	−	27.0402i	0.983138	−	0.906899i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	44.1400			1.47709
$894$		0			0
$895$	13.1169	+	22.7192i	0.438450	+	0.759419i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−9.18855	−	15.9150i	−0.306455	−	0.530796i
$900$		0			0
$901$	−3.66395	+	6.34615i	−0.122064	+	0.211421i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−42.3884			−1.40904
$906$		0			0
$907$	−10.5750			−0.351136			−0.175568	−	0.984467i	$-0.556176\pi$
$907$	−10.5750			−0.351136			−0.175568	+	0.984467i	$0.556176\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−15.2693	+	26.4473i	−0.505896	+	0.876237i	0.494081	+	0.869416i	$0.335505\pi$
$911$	−15.2693	+	26.4473i	−0.505896	+	0.876237i	−0.999977	+	0.00682127i	$0.997829\pi$
$912$		0			0
$913$	7.03999	−	12.1936i	0.232990	−	0.403550i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−28.8608	+	26.6227i	−0.953066	+	0.879160i
$918$		0			0
$919$	−0.552490	+	0.956940i	−0.0182249	+	0.0315665i	−0.874994	−	0.484134i	$-0.839135\pi$
$919$	−0.552490	+	0.956940i	−0.0182249	+	0.0315665i	0.856769	+	0.515700i	$0.172468\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	33.9174	+	58.7466i	1.11640	+	1.93367i
$924$		0			0
$925$	−6.68598	+	11.5805i	−0.219834	+	0.380763i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−2.63729	−	4.56792i	−0.0865266	−	0.149869i	0.819514	−	0.573059i	$-0.194243\pi$
$929$	−2.63729	−	4.56792i	−0.0865266	−	0.149869i	−0.906041	+	0.423191i	$0.860910\pi$
$930$		0			0
$931$	25.3005	+	17.4665i	0.829189	+	0.572442i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	4.51174	+	7.81456i	0.147550	+	0.255564i
$936$		0			0
$937$	17.7481			0.579806			0.289903	−	0.957056i	$-0.406377\pi$
$937$	17.7481			0.579806			0.289903	+	0.957056i	$0.406377\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−13.8684	+	24.0207i	−0.452096	+	0.783053i	−0.998516	−	0.0544582i	$-0.982657\pi$
$941$	−13.8684	+	24.0207i	−0.452096	+	0.783053i	0.546420	+	0.837511i	$0.315990\pi$
$942$		0			0
$943$	−6.50474	−	11.2665i	−0.211824	−	0.366889i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	7.63828	+	13.2299i	0.248211	+	0.429914i	0.963029	−	0.269396i	$-0.0868241\pi$
$947$	7.63828	+	13.2299i	0.248211	+	0.429914i	−0.714819	+	0.699310i	$0.753491\pi$
$948$		0			0
$949$	−6.15006	+	10.6522i	−0.199639	+	0.345785i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	25.3569			0.821390			0.410695	−	0.911773i	$-0.365286\pi$
$953$	25.3569			0.821390			0.410695	+	0.911773i	$0.365286\pi$
$954$		0			0
$955$	2.96014	+	5.12712i	0.0957880	+	0.165910i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	39.1974	−	36.1578i	1.26575	−	1.16760i
$960$		0			0
$961$	10.6429	+	18.4340i	0.343318	+	0.594644i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	7.79286	−	13.4976i	0.250861	−	0.434504i
$966$		0			0
$967$	−13.5566	−	23.4808i	−0.435952	−	0.755090i	0.561421	−	0.827530i	$-0.310255\pi$
$967$	−13.5566	−	23.4808i	−0.435952	−	0.755090i	−0.997373	+	0.0724398i	$0.976921\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	21.1555	−	36.6423i	0.678911	−	1.17591i	−0.296398	−	0.955064i	$-0.595786\pi$
$971$	21.1555	−	36.6423i	0.678911	−	1.17591i	0.975309	−	0.220844i	$-0.0708812\pi$
$972$		0			0
$973$	−50.0917	−	15.6113i	−1.60587	−	0.500475i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	1.18908	−	2.05955i	0.0380421	−	0.0658908i	−0.846377	−	0.532583i	$-0.821221\pi$
$977$	1.18908	−	2.05955i	0.0380421	−	0.0658908i	0.884420	+	0.466693i	$0.154555\pi$
$978$		0			0
$979$	10.5499	−	18.2729i	0.337175	−	0.584004i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−3.29283			−0.105025			−0.0525124	−	0.998620i	$-0.516723\pi$
$983$	−3.29283			−0.105025			−0.0525124	+	0.998620i	$0.516723\pi$
$984$		0			0
$985$	5.16782			0.164661
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−22.8821	+	39.6329i	−0.727607	+	1.26025i
$990$		0			0
$991$	−29.5482	−	51.1790i	−0.938630	−	1.62575i	−0.768030	−	0.640414i	$-0.778763\pi$
$991$	−29.5482	−	51.1790i	−0.938630	−	1.62575i	−0.170600	−	0.985340i	$-0.554571\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−4.92356	−	8.52786i	−0.156087	−	0.270351i
$996$		0			0
$997$	−16.9797			−0.537754			−0.268877	−	0.963175i	$-0.586652\pi$
$997$	−16.9797			−0.537754			−0.268877	+	0.963175i	$0.586652\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1512.2.t.d.289.2		22
3.2	odd	2		504.2.t.d.457.9	yes	22
4.3	odd	2		3024.2.t.l.289.2		22
7.4	even	3		1512.2.q.c.1369.10		22
9.4	even	3		1512.2.q.c.793.10		22
9.5	odd	6		504.2.q.d.121.6	yes	22
12.11	even	2		1008.2.t.k.961.3		22
21.11	odd	6		504.2.q.d.25.6	✓	22
28.11	odd	6		3024.2.q.k.2881.10		22
36.23	even	6		1008.2.q.k.625.6		22
36.31	odd	6		3024.2.q.k.2305.10		22
63.4	even	3	inner	1512.2.t.d.361.2		22
63.32	odd	6		504.2.t.d.193.9	yes	22
84.11	even	6		1008.2.q.k.529.6		22
252.67	odd	6		3024.2.t.l.1873.2		22
252.95	even	6		1008.2.t.k.193.3		22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
504.2.q.d.25.6	✓	22	21.11	odd	6
504.2.q.d.121.6	yes	22	9.5	odd	6
504.2.t.d.193.9	yes	22	63.32	odd	6
504.2.t.d.457.9	yes	22	3.2	odd	2
1008.2.q.k.529.6		22	84.11	even	6
1008.2.q.k.625.6		22	36.23	even	6
1008.2.t.k.193.3		22	252.95	even	6
1008.2.t.k.961.3		22	12.11	even	2
1512.2.q.c.793.10		22	9.4	even	3
1512.2.q.c.1369.10		22	7.4	even	3
1512.2.t.d.289.2		22	1.1	even	1	trivial
1512.2.t.d.361.2		22	63.4	even	3	inner
3024.2.q.k.2305.10		22	36.31	odd	6
3024.2.q.k.2881.10		22	28.11	odd	6
3024.2.t.l.289.2		22	4.3	odd	2
3024.2.t.l.1873.2		22	252.67	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:	\( N \)	\(=\)	\( 1512 = 2^{3} \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1512.t (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q-2.66802 q^{5} +(1.94471 - 1.79391i) q^{7} +O(q^{10})\)	\(q-2.66802 q^{5} +(1.94471 - 1.79391i) q^{7} -1.36451 q^{11} +(-2.75597 - 4.77348i) q^{13} +(1.23930 + 2.14654i) q^{17} +(-2.19600 + 3.80358i) q^{19} +4.69002 q^{23} +2.11832 q^{25} +(-2.94810 + 5.10625i) q^{29} +(-1.55839 + 2.69921i) q^{31} +(-5.18852 + 4.78617i) q^{35} +(-3.15627 + 5.46681i) q^{37} +(-1.38693 - 2.40224i) q^{41} +(-4.87889 + 8.45048i) q^{43} +(-5.02505 - 8.70364i) q^{47} +(0.563800 - 6.97726i) q^{49} +(1.47823 + 2.56037i) q^{53} +3.64055 q^{55} +(1.77809 - 3.07974i) q^{59} +(-0.663043 - 1.14842i) q^{61} +(7.35297 + 12.7357i) q^{65} +(-4.14937 + 7.18692i) q^{67} -12.3069 q^{71} +(-1.11577 - 1.93257i) q^{73} +(-2.65358 + 2.44781i) q^{77} +(-6.41535 - 11.1117i) q^{79} +(-5.15934 + 8.93625i) q^{83} +(-3.30648 - 5.72700i) q^{85} +(-7.73159 + 13.3915i) q^{89} +(-13.9227 - 4.33908i) q^{91} +(5.85896 - 10.1480i) q^{95} +(-2.55369 + 4.42311i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q + 6 q^{5} + 7 q^{7}+O(q^{10}) \) 22 * q + 6 * q^5 + 7 * q^7	\( 22 q + 6 q^{5} + 7 q^{7} - 6 q^{11} - 3 q^{13} - 7 q^{17} - q^{19} + 4 q^{23} + 20 q^{25} - 9 q^{29} - 4 q^{31} - 14 q^{35} + 2 q^{37} - 16 q^{41} - 5 q^{47} - 15 q^{49} - 11 q^{53} + 22 q^{55} + 19 q^{59} - 13 q^{61} - 13 q^{65} + 26 q^{67} + 48 q^{71} - 35 q^{73} + 4 q^{77} + 10 q^{79} + 28 q^{83} - 20 q^{85} - 6 q^{89} - 37 q^{91} - 12 q^{95} - 29 q^{97}+O(q^{100}) \) 22 * q + 6 * q^5 + 7 * q^7 - 6 * q^11 - 3 * q^13 - 7 * q^17 - q^19 + 4 * q^23 + 20 * q^25 - 9 * q^29 - 4 * q^31 - 14 * q^35 + 2 * q^37 - 16 * q^41 - 5 * q^47 - 15 * q^49 - 11 * q^53 + 22 * q^55 + 19 * q^59 - 13 * q^61 - 13 * q^65 + 26 * q^67 + 48 * q^71 - 35 * q^73 + 4 * q^77 + 10 * q^79 + 28 * q^83 - 20 * q^85 - 6 * q^89 - 37 * q^91 - 12 * q^95 - 29 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more