LMFDB - Embedded newform 1728.2.bc.d.721.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1728,2,Mod(145,1728)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1728, base_ring=CyclotomicField(12))

chi = DirichletCharacter(H, H._module([0, 9, 4]))

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

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

chi := DirichletCharacter("1728.145");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1728 = 2^{6} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1728.bc (of order $12$, degree $4$, 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:	$13.7981494693$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 144)
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			721.1
Root			$0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	1728.721
Dual form			1728.2.bc.d.1009.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(3.73205 + 1.00000i) q^{5} +(0.633975 + 0.366025i) q^{7} +O(q^{10})$	$q+(3.73205 + 1.00000i) q^{5} +(0.633975 + 0.366025i) q^{7} +(-0.767949 - 2.86603i) q^{11} +(-1.63397 + 6.09808i) q^{13} +2.26795 q^{17} +(0.633975 + 0.633975i) q^{19} +(-1.09808 + 0.633975i) q^{23} +(8.59808 + 4.96410i) q^{25} +(2.36603 - 0.633975i) q^{29} +(3.73205 + 6.46410i) q^{31} +(2.00000 + 2.00000i) q^{35} +(1.26795 - 1.26795i) q^{37} +(2.59808 - 1.50000i) q^{41} +(-0.330127 - 1.23205i) q^{43} +(-4.83013 + 8.36603i) q^{47} +(-3.23205 - 5.59808i) q^{49} +(0.535898 - 0.535898i) q^{53} -11.4641i q^{55} +(4.96410 + 1.33013i) q^{59} +(-3.00000 + 0.803848i) q^{61} +(-12.1962 + 21.1244i) q^{65} +(-1.40192 + 5.23205i) q^{67} -10.9282i q^{71} -9.73205i q^{73} +(0.562178 - 2.09808i) q^{77} +(6.00000 - 10.3923i) q^{79} +(-1.36603 + 0.366025i) q^{83} +(8.46410 + 2.26795i) q^{85} +2.00000i q^{89} +(-3.26795 + 3.26795i) q^{91} +(1.73205 + 3.00000i) q^{95} +(-4.13397 + 7.16025i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8 q^{5} + 6 q^{7}+O(q^{10}) $ 4 * q + 8 * q^5 + 6 * q^7	$ 4 q + 8 q^{5} + 6 q^{7} - 10 q^{11} - 10 q^{13} + 16 q^{17} + 6 q^{19} + 6 q^{23} + 24 q^{25} + 6 q^{29} + 8 q^{31} + 8 q^{35} + 12 q^{37} + 16 q^{43} - 2 q^{47} - 6 q^{49} + 16 q^{53} + 6 q^{59} - 12 q^{61} - 28 q^{65} - 16 q^{67} - 22 q^{77} + 24 q^{79} - 2 q^{83} + 20 q^{85} - 20 q^{91} - 20 q^{97}+O(q^{100}) $ 4 * q + 8 * q^5 + 6 * q^7 - 10 * q^11 - 10 * q^13 + 16 * q^17 + 6 * q^19 + 6 * q^23 + 24 * q^25 + 6 * q^29 + 8 * q^31 + 8 * q^35 + 12 * q^37 + 16 * q^43 - 2 * q^47 - 6 * q^49 + 16 * q^53 + 6 * q^59 - 12 * q^61 - 28 * q^65 - 16 * q^67 - 22 * q^77 + 24 * q^79 - 2 * q^83 + 20 * q^85 - 20 * q^91 - 20 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times$.

$n$	$325$	$703$	$1217$
$\chi(n)$	$e\left(\frac{3}{4}\right)$	$1$	$e\left(\frac{2}{3}\right)$

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
$5$	3.73205	+	1.00000i	1.66902	+	0.447214i	0.964847	−	0.262811i	$-0.0846497\pi$
$5$	3.73205	+	1.00000i	1.66902	+	0.447214i	0.704177	+	0.710025i	$0.251316\pi$
$6$		0			0
$7$	0.633975	+	0.366025i	0.239620	+	0.138345i	0.615002	−	0.788526i	$-0.289155\pi$
$7$	0.633975	+	0.366025i	0.239620	+	0.138345i	−0.375382	+	0.926870i	$0.622489\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−0.767949	−	2.86603i	−0.231545	−	0.864139i	−0.979676	−	0.200587i	$-0.935715\pi$
$11$	−0.767949	−	2.86603i	−0.231545	−	0.864139i	0.748130	−	0.663552i	$-0.230952\pi$
$12$		0			0
$13$	−1.63397	+	6.09808i	−0.453183	+	1.69130i	0.240192	+	0.970725i	$0.422790\pi$
$13$	−1.63397	+	6.09808i	−0.453183	+	1.69130i	−0.693375	+	0.720577i	$0.743877\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	2.26795			0.550058			0.275029	−	0.961436i	$-0.411312\pi$
$17$	2.26795			0.550058			0.275029	+	0.961436i	$0.411312\pi$
$18$		0			0
$19$	0.633975	+	0.633975i	0.145444	+	0.145444i	0.776079	−	0.630635i	$-0.217206\pi$
$19$	0.633975	+	0.633975i	0.145444	+	0.145444i	−0.630635	+	0.776079i	$0.717206\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−1.09808	+	0.633975i	−0.228965	+	0.132193i	−0.610094	−	0.792329i	$-0.708868\pi$
$23$	−1.09808	+	0.633975i	−0.228965	+	0.132193i	0.381130	+	0.924522i	$0.375535\pi$
$24$		0			0
$25$	8.59808	+	4.96410i	1.71962	+	0.992820i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	2.36603	−	0.633975i	0.439360	−	0.117726i	−0.0323566	−	0.999476i	$-0.510301\pi$
$29$	2.36603	−	0.633975i	0.439360	−	0.117726i	0.471717	+	0.881750i	$0.343635\pi$
$30$		0			0
$31$	3.73205	+	6.46410i	0.670296	+	1.16099i	0.977820	+	0.209447i	$0.0671662\pi$
$31$	3.73205	+	6.46410i	0.670296	+	1.16099i	−0.307524	+	0.951540i	$0.599500\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	2.00000	+	2.00000i	0.338062	+	0.338062i
$36$		0			0
$37$	1.26795	−	1.26795i	0.208450	−	0.208450i	−0.595159	−	0.803608i	$-0.702911\pi$
$37$	1.26795	−	1.26795i	0.208450	−	0.208450i	0.803608	+	0.595159i	$0.202911\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	2.59808	−	1.50000i	0.405751	−	0.234261i	−0.283211	−	0.959058i	$-0.591400\pi$
$41$	2.59808	−	1.50000i	0.405751	−	0.234261i	0.688963	+	0.724797i	$0.258066\pi$
$42$		0			0
$43$	−0.330127	−	1.23205i	−0.0503439	−	0.187886i	0.936175	−	0.351535i	$-0.114340\pi$
$43$	−0.330127	−	1.23205i	−0.0503439	−	0.187886i	−0.986519	+	0.163649i	$0.947674\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−4.83013	+	8.36603i	−0.704546	+	1.22031i	0.262309	+	0.964984i	$0.415516\pi$
$47$	−4.83013	+	8.36603i	−0.704546	+	1.22031i	−0.966855	+	0.255326i	$0.917817\pi$
$48$		0			0
$49$	−3.23205	−	5.59808i	−0.461722	−	0.799725i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	0.535898	−	0.535898i	0.0736113	−	0.0736113i	−0.669343	−	0.742954i	$-0.733424\pi$
$53$	0.535898	−	0.535898i	0.0736113	−	0.0736113i	0.742954	+	0.669343i	$0.233424\pi$
$54$		0			0
$55$		−	11.4641i		−	1.54582i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	4.96410	+	1.33013i	0.646271	+	0.173168i	0.567042	−	0.823689i	$-0.308088\pi$
$59$	4.96410	+	1.33013i	0.646271	+	0.173168i	0.0792287	+	0.996856i	$0.474754\pi$
$60$		0			0
$61$	−3.00000	+	0.803848i	−0.384111	+	0.102922i	−0.445707	−	0.895179i	$-0.647048\pi$
$61$	−3.00000	+	0.803848i	−0.384111	+	0.102922i	0.0615961	+	0.998101i	$0.480381\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−12.1962	+	21.1244i	−1.51275	+	2.62015i
$66$		0			0
$67$	−1.40192	+	5.23205i	−0.171272	+	0.639197i	0.825884	+	0.563840i	$0.190676\pi$
$67$	−1.40192	+	5.23205i	−0.171272	+	0.639197i	−0.997157	+	0.0753572i	$0.975990\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	10.9282i		−	1.29694i	−0.761241	−	0.648470i	$-0.775409\pi$
$71$		−	10.9282i		−	1.29694i	0.761241	−	0.648470i	$-0.224591\pi$
$72$		0			0
$73$		−	9.73205i		−	1.13905i	−0.821974	−	0.569525i	$-0.807127\pi$
$73$		−	9.73205i		−	1.13905i	0.821974	−	0.569525i	$-0.192873\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	0.562178	−	2.09808i	0.0640661	−	0.239098i
$78$		0			0
$79$	6.00000	−	10.3923i	0.675053	−	1.16923i	−0.301401	−	0.953498i	$-0.597454\pi$
$79$	6.00000	−	10.3923i	0.675053	−	1.16923i	0.976453	−	0.215728i	$-0.0692125\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−1.36603	+	0.366025i	−0.149941	+	0.0401765i	−0.333009	−	0.942924i	$-0.608064\pi$
$83$	−1.36603	+	0.366025i	−0.149941	+	0.0401765i	0.183068	+	0.983100i	$0.441397\pi$
$84$		0			0
$85$	8.46410	+	2.26795i	0.918061	+	0.245994i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			2.00000i			0.212000i	0.994366	+	0.106000i	$0.0338043\pi$
$89$			2.00000i			0.212000i	−0.994366	+	0.106000i	$0.966196\pi$
$90$		0			0
$91$	−3.26795	+	3.26795i	−0.342574	+	0.342574i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	1.73205	+	3.00000i	0.177705	+	0.307794i
$96$		0			0
$97$	−4.13397	+	7.16025i	−0.419742	+	0.727014i	−0.995913	−	0.0903150i	$-0.971213\pi$
$97$	−4.13397	+	7.16025i	−0.419742	+	0.727014i	0.576172	+	0.817329i	$0.304546\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	2.00000	+	7.46410i	0.199007	+	0.742706i	0.991193	+	0.132426i	$0.0422765\pi$
$101$	2.00000	+	7.46410i	0.199007	+	0.742706i	−0.792186	+	0.610280i	$0.791057\pi$
$102$		0			0
$103$	7.90192	−	4.56218i	0.778600	−	0.449525i	−0.0573341	−	0.998355i	$-0.518260\pi$
$103$	7.90192	−	4.56218i	0.778600	−	0.449525i	0.835934	+	0.548830i	$0.184927\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−13.4904	+	13.4904i	−1.30416	+	1.30416i	−0.378607	+	0.925558i	$0.623597\pi$
$107$	−13.4904	+	13.4904i	−1.30416	+	1.30416i	−0.925558	+	0.378607i	$0.876403\pi$
$108$		0			0
$109$	7.26795	+	7.26795i	0.696143	+	0.696143i	0.963576	−	0.267433i	$-0.0861754\pi$
$109$	7.26795	+	7.26795i	0.696143	+	0.696143i	−0.267433	+	0.963576i	$0.586175\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−6.92820	−	12.0000i	−0.651751	−	1.12887i	−0.982698	−	0.185216i	$-0.940702\pi$
$113$	−6.92820	−	12.0000i	−0.651751	−	1.12887i	0.330947	−	0.943649i	$-0.392632\pi$
$114$		0			0
$115$	−4.73205	+	1.26795i	−0.441266	+	0.118237i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	1.43782	+	0.830127i	0.131805	+	0.0760976i
$120$		0			0
$121$	1.90192	−	1.09808i	0.172902	−	0.0998251i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	13.4641	+	13.4641i	1.20427	+	1.20427i
$126$		0			0
$127$	6.19615			0.549820			0.274910	−	0.961470i	$-0.411352\pi$
$127$	6.19615			0.549820			0.274910	+	0.961470i	$0.411352\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	0.830127	−	3.09808i	0.0725285	−	0.270680i	−0.920133	−	0.391606i	$-0.871920\pi$
$131$	0.830127	−	3.09808i	0.0725285	−	0.270680i	0.992662	+	0.120926i	$0.0385863\pi$
$132$		0			0
$133$	0.169873	+	0.633975i	0.0147299	+	0.0549726i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−14.2583	−	8.23205i	−1.21817	−	0.703312i	−0.253645	−	0.967297i	$-0.581629\pi$
$137$	−14.2583	−	8.23205i	−1.21817	−	0.703312i	−0.964527	+	0.263986i	$0.914963\pi$
$138$		0			0
$139$	9.06218	+	2.42820i	0.768644	+	0.205958i	0.621772	−	0.783198i	$-0.286413\pi$
$139$	9.06218	+	2.42820i	0.768644	+	0.205958i	0.146872	+	0.989156i	$0.453080\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	18.7321			1.56645
$144$		0			0
$145$	9.46410			0.785951
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−3.09808	−	0.830127i	−0.253804	−	0.0680067i	0.129674	−	0.991557i	$-0.458607\pi$
$149$	−3.09808	−	0.830127i	−0.253804	−	0.0680067i	−0.383478	+	0.923550i	$0.625274\pi$
$150$		0			0
$151$	2.36603	+	1.36603i	0.192544	+	0.111166i	0.593173	−	0.805075i	$-0.297875\pi$
$151$	2.36603	+	1.36603i	0.192544	+	0.111166i	−0.400629	+	0.916240i	$0.631208\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	7.46410	+	27.8564i	0.599531	+	2.23748i
$156$		0			0
$157$	−1.26795	+	4.73205i	−0.101193	+	0.377659i	−0.997886	−	0.0649959i	$-0.979297\pi$
$157$	−1.26795	+	4.73205i	−0.101193	+	0.377659i	0.896692	+	0.442655i	$0.145963\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−0.928203			−0.0731527
$162$		0			0
$163$	7.00000	+	7.00000i	0.548282	+	0.548282i	0.925944	−	0.377661i	$-0.123272\pi$
$163$	7.00000	+	7.00000i	0.548282	+	0.548282i	−0.377661	+	0.925944i	$0.623272\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	0.464102	−	0.267949i	0.0359133	−	0.0207345i	−0.481936	−	0.876206i	$-0.660066\pi$
$167$	0.464102	−	0.267949i	0.0359133	−	0.0207345i	0.517849	+	0.855472i	$0.326733\pi$
$168$		0			0
$169$	−23.2583	−	13.4282i	−1.78910	−	1.03294i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	12.5622	−	3.36603i	0.955085	−	0.255914i	0.252566	−	0.967580i	$-0.418725\pi$
$173$	12.5622	−	3.36603i	0.955085	−	0.255914i	0.702519	+	0.711665i	$0.252059\pi$
$174$		0			0
$175$	3.63397	+	6.29423i	0.274703	+	0.475799i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	11.9282	+	11.9282i	0.891556	+	0.891556i	0.994670	−	0.103114i	$-0.0328806\pi$
$179$	11.9282	+	11.9282i	0.891556	+	0.891556i	−0.103114	+	0.994670i	$0.532881\pi$
$180$		0			0
$181$	13.3923	−	13.3923i	0.995442	−	0.995442i	−0.00454748	−	0.999990i	$-0.501448\pi$
$181$	13.3923	−	13.3923i	0.995442	−	0.995442i	0.999990	+	0.00454748i	$0.00144751\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	6.00000	−	3.46410i	0.441129	−	0.254686i
$186$		0			0
$187$	−1.74167	−	6.50000i	−0.127364	−	0.475327i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	7.02628	−	12.1699i	0.508404	−	0.880581i	−0.491549	−	0.870850i	$-0.663569\pi$
$191$	7.02628	−	12.1699i	0.508404	−	0.880581i	0.999953	−	0.00973114i	$-0.00309757\pi$
$192$		0			0
$193$	−9.13397	−	15.8205i	−0.657478	−	1.13879i	−0.981266	−	0.192656i	$-0.938290\pi$
$193$	−9.13397	−	15.8205i	−0.657478	−	1.13879i	0.323789	−	0.946129i	$-0.395043\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	3.66025	−	3.66025i	0.260782	−	0.260782i	−0.564590	−	0.825372i	$-0.690966\pi$
$197$	3.66025	−	3.66025i	0.260782	−	0.260782i	0.825372	+	0.564590i	$0.190966\pi$
$198$		0			0
$199$		−	0.875644i		−	0.0620728i	−0.999518	−	0.0310364i	$-0.990119\pi$
$199$		−	0.875644i		−	0.0620728i	0.999518	−	0.0310364i	$-0.00988078\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	1.73205	+	0.464102i	0.121566	+	0.0325735i
$204$		0			0
$205$	11.1962	−	3.00000i	0.781973	−	0.209529i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	1.33013	−	2.30385i	0.0920068	−	0.159360i
$210$		0			0
$211$	1.09808	−	4.09808i	0.0755947	−	0.282123i	−0.917773	−	0.397106i	$-0.870015\pi$
$211$	1.09808	−	4.09808i	0.0755947	−	0.282123i	0.993367	+	0.114983i	$0.0366812\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	4.92820i		−	0.336101i
$216$		0			0
$217$			5.46410i			0.370927i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−3.70577	+	13.8301i	−0.249277	+	0.930315i
$222$		0			0
$223$	−11.0263	+	19.0981i	−0.738374	+	1.27890i	0.214853	+	0.976646i	$0.431073\pi$
$223$	−11.0263	+	19.0981i	−0.738374	+	1.27890i	−0.953227	+	0.302255i	$0.902260\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−14.4282	+	3.86603i	−0.957633	+	0.256597i	−0.703598	−	0.710598i	$-0.748425\pi$
$227$	−14.4282	+	3.86603i	−0.957633	+	0.256597i	−0.254035	+	0.967195i	$0.581758\pi$
$228$		0			0
$229$	6.83013	+	1.83013i	0.451347	+	0.120938i	0.477330	−	0.878724i	$-0.341605\pi$
$229$	6.83013	+	1.83013i	0.451347	+	0.120938i	−0.0259823	+	0.999662i	$0.508271\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	7.19615i		−	0.471436i	−0.971822	−	0.235718i	$-0.924256\pi$
$233$		−	7.19615i		−	0.471436i	0.971822	−	0.235718i	$-0.0757441\pi$
$234$		0			0
$235$	−26.3923	+	26.3923i	−1.72164	+	1.72164i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−13.0981	−	22.6865i	−0.847244	−	1.46747i	−0.883658	−	0.468133i	$-0.844927\pi$
$239$	−13.0981	−	22.6865i	−0.847244	−	1.46747i	0.0364139	−	0.999337i	$-0.488407\pi$
$240$		0			0
$241$	−6.40192	+	11.0885i	−0.412384	+	0.714270i	−0.995150	−	0.0983699i	$-0.968637\pi$
$241$	−6.40192	+	11.0885i	−0.412384	+	0.714270i	0.582766	+	0.812640i	$0.301971\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−6.46410	−	24.1244i	−0.412976	−	1.54125i
$246$		0			0
$247$	−4.90192	+	2.83013i	−0.311902	+	0.180077i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	2.83013	−	2.83013i	0.178636	−	0.178636i	−0.612125	−	0.790761i	$-0.709685\pi$
$251$	2.83013	−	2.83013i	0.178636	−	0.178636i	0.790761	+	0.612125i	$0.209685\pi$
$252$		0			0
$253$	2.66025	+	2.66025i	0.167249	+	0.167249i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	4.42820	+	7.66987i	0.276224	+	0.478434i	0.970443	−	0.241330i	$-0.0775836\pi$
$257$	4.42820	+	7.66987i	0.276224	+	0.478434i	−0.694219	+	0.719763i	$0.744250\pi$
$258$		0			0
$259$	1.26795	−	0.339746i	0.0787865	−	0.0211108i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−23.4904	−	13.5622i	−1.44848	−	0.836280i	−0.450088	−	0.892984i	$-0.648607\pi$
$263$	−23.4904	−	13.5622i	−1.44848	−	0.836280i	−0.998391	+	0.0567045i	$0.981941\pi$
$264$		0			0
$265$	2.53590	−	1.46410i	0.155779	−	0.0899390i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−4.73205	−	4.73205i	−0.288518	−	0.288518i	0.547976	−	0.836494i	$-0.315399\pi$
$269$	−4.73205	−	4.73205i	−0.288518	−	0.288518i	−0.836494	+	0.547976i	$0.815399\pi$
$270$		0			0
$271$	−20.3923			−1.23874			−0.619372	−	0.785098i	$-0.712613\pi$
$271$	−20.3923			−1.23874			−0.619372	+	0.785098i	$0.712613\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	7.62436	−	28.4545i	0.459766	−	1.71587i
$276$		0			0
$277$	−4.22243	−	15.7583i	−0.253701	−	0.946826i	−0.968808	−	0.247811i	$-0.920289\pi$
$277$	−4.22243	−	15.7583i	−0.253701	−	0.946826i	0.715107	−	0.699015i	$-0.246378\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	8.66025	+	5.00000i	0.516627	+	0.298275i	0.735554	−	0.677466i	$-0.236922\pi$
$281$	8.66025	+	5.00000i	0.516627	+	0.298275i	−0.218926	+	0.975741i	$0.570255\pi$
$282$		0			0
$283$	−27.7583	−	7.43782i	−1.65006	−	0.442133i	−0.690431	−	0.723398i	$-0.742579\pi$
$283$	−27.7583	−	7.43782i	−1.65006	−	0.442133i	−0.959630	+	0.281265i	$0.909246\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	2.19615			0.129635
$288$		0			0
$289$	−11.8564			−0.697436
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−13.5622	−	3.63397i	−0.792311	−	0.212299i	−0.160106	−	0.987100i	$-0.551183\pi$
$293$	−13.5622	−	3.63397i	−0.792311	−	0.212299i	−0.632205	+	0.774801i	$0.717850\pi$
$294$		0			0
$295$	17.1962	+	9.92820i	1.00120	+	0.578042i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−2.07180	−	7.73205i	−0.119815	−	0.447156i
$300$		0			0
$301$	0.241670	−	0.901924i	0.0139296	−	0.0519860i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−12.0000			−0.687118
$306$		0			0
$307$	16.0263	+	16.0263i	0.914668	+	0.914668i	0.996635	−	0.0819670i	$-0.0261202\pi$
$307$	16.0263	+	16.0263i	0.914668	+	0.914668i	−0.0819670	+	0.996635i	$0.526120\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	13.9019	−	8.02628i	0.788306	−	0.455129i	−0.0510600	−	0.998696i	$-0.516260\pi$
$311$	13.9019	−	8.02628i	0.788306	−	0.455129i	0.839366	+	0.543567i	$0.182927\pi$
$312$		0			0
$313$	24.6506	+	14.2321i	1.39334	+	0.804443i	0.993683	−	0.112223i	$-0.0357972\pi$
$313$	24.6506	+	14.2321i	1.39334	+	0.804443i	0.399653	+	0.916666i	$0.369131\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−31.4904	+	8.43782i	−1.76868	+	0.473915i	−0.988445	−	0.151577i	$-0.951565\pi$
$317$	−31.4904	+	8.43782i	−1.76868	+	0.473915i	−0.780231	+	0.625492i	$0.784898\pi$
$318$		0			0
$319$	−3.63397	−	6.29423i	−0.203464	−	0.352409i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	1.43782	+	1.43782i	0.0800026	+	0.0800026i
$324$		0			0
$325$	−44.3205	+	44.3205i	−2.45846	+	2.45846i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−6.12436	+	3.53590i	−0.337647	+	0.194940i
$330$		0			0
$331$	−5.09808	−	19.0263i	−0.280216	−	1.04578i	−0.952265	−	0.305273i	$-0.901252\pi$
$331$	−5.09808	−	19.0263i	−0.280216	−	1.04578i	0.672049	−	0.740506i	$-0.265414\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−10.4641	+	18.1244i	−0.571715	+	0.990239i
$336$		0			0
$337$	−11.8923	−	20.5981i	−0.647815	−	1.12205i	−0.983644	−	0.180126i	$-0.942350\pi$
$337$	−11.8923	−	20.5981i	−0.647815	−	1.12205i	0.335829	−	0.941923i	$-0.390984\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	15.6603	−	15.6603i	0.848050	−	0.848050i
$342$		0			0
$343$		−	9.85641i		−	0.532196i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−24.7224	−	6.62436i	−1.32717	−	0.355614i	−0.475510	−	0.879710i	$-0.657737\pi$
$347$	−24.7224	−	6.62436i	−1.32717	−	0.355614i	−0.851659	+	0.524096i	$0.824403\pi$
$348$		0			0
$349$	−7.73205	+	2.07180i	−0.413887	+	0.110901i	−0.459753	−	0.888047i	$-0.652062\pi$
$349$	−7.73205	+	2.07180i	−0.413887	+	0.110901i	0.0458657	+	0.998948i	$0.485395\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	10.1603	−	17.5981i	0.540776	−	0.936651i	−0.458084	−	0.888909i	$-0.651464\pi$
$353$	10.1603	−	17.5981i	0.540776	−	0.936651i	0.998860	−	0.0477421i	$-0.0152026\pi$
$354$		0			0
$355$	10.9282	−	40.7846i	0.580009	−	2.16462i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			14.7321i			0.777528i	0.921337	+	0.388764i	$0.127098\pi$
$359$			14.7321i			0.777528i	−0.921337	+	0.388764i	$0.872902\pi$
$360$		0			0
$361$		−	18.1962i		−	0.957692i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	9.73205	−	36.3205i	0.509399	−	1.90110i
$366$		0			0
$367$	10.1244	−	17.5359i	0.528487	−	0.915366i	−0.470961	−	0.882154i	$-0.656093\pi$
$367$	10.1244	−	17.5359i	0.528487	−	0.915366i	0.999448	−	0.0332125i	$-0.0105738\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	0.535898	−	0.143594i	0.0278225	−	0.00745501i
$372$		0			0
$373$	−5.63397	−	1.50962i	−0.291716	−	0.0781651i	0.109993	−	0.993932i	$-0.464917\pi$
$373$	−5.63397	−	1.50962i	−0.291716	−	0.0781651i	−0.401709	+	0.915767i	$0.631584\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			15.4641i			0.796442i
$378$		0			0
$379$	18.7583	−	18.7583i	0.963551	−	0.963551i	−0.0358080	−	0.999359i	$-0.511400\pi$
$379$	18.7583	−	18.7583i	0.963551	−	0.963551i	0.999359	+	0.0358080i	$0.0114005\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−3.26795	−	5.66025i	−0.166984	−	0.289225i	0.770374	−	0.637593i	$-0.220070\pi$
$383$	−3.26795	−	5.66025i	−0.166984	−	0.289225i	−0.937358	+	0.348367i	$0.886736\pi$
$384$		0			0
$385$	4.19615	−	7.26795i	0.213856	−	0.370409i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	2.75833	+	10.2942i	0.139853	+	0.521938i	0.999931	+	0.0117752i	$0.00374824\pi$
$389$	2.75833	+	10.2942i	0.139853	+	0.521938i	−0.860078	+	0.510163i	$0.829585\pi$
$390$		0			0
$391$	−2.49038	+	1.43782i	−0.125944	+	0.0727138i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	32.7846	−	32.7846i	1.64957	−	1.64957i
$396$		0			0
$397$	−12.7321	−	12.7321i	−0.639003	−	0.639003i	0.311306	−	0.950310i	$-0.399233\pi$
$397$	−12.7321	−	12.7321i	−0.639003	−	0.639003i	−0.950310	+	0.311306i	$0.899233\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−13.7942	−	23.8923i	−0.688851	−	1.19312i	−0.972210	−	0.234111i	$-0.924782\pi$
$401$	−13.7942	−	23.8923i	−0.688851	−	1.19312i	0.283359	−	0.959014i	$-0.408551\pi$
$402$		0			0
$403$	−45.5167	+	12.1962i	−2.26735	+	0.607534i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−4.60770	−	2.66025i	−0.228395	−	0.131864i
$408$		0			0
$409$	−26.1340	+	15.0885i	−1.29224	+	0.746076i	−0.979051	−	0.203614i	$-0.934731\pi$
$409$	−26.1340	+	15.0885i	−1.29224	+	0.746076i	−0.313191	+	0.949690i	$0.601398\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	2.66025	+	2.66025i	0.130903	+	0.130903i
$414$		0			0
$415$	−5.46410			−0.268222
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−8.36603	+	31.2224i	−0.408707	+	1.52532i	0.388408	+	0.921488i	$0.373025\pi$
$419$	−8.36603	+	31.2224i	−0.408707	+	1.52532i	−0.797115	+	0.603828i	$0.793641\pi$
$420$		0			0
$421$	0.588457	+	2.19615i	0.0286797	+	0.107034i	0.978782	−	0.204905i	$-0.0656884\pi$
$421$	0.588457	+	2.19615i	0.0286797	+	0.107034i	−0.950102	+	0.311938i	$0.899022\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	19.5000	+	11.2583i	0.945889	+	0.546109i
$426$		0			0
$427$	−2.19615	−	0.588457i	−0.106279	−	0.0284774i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−5.80385			−0.279562			−0.139781	−	0.990182i	$-0.544640\pi$
$431$	−5.80385			−0.279562			−0.139781	+	0.990182i	$0.544640\pi$
$432$		0			0
$433$	−2.26795			−0.108991			−0.0544953	−	0.998514i	$-0.517355\pi$
$433$	−2.26795			−0.108991			−0.0544953	+	0.998514i	$0.517355\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−1.09808	−	0.294229i	−0.0525281	−	0.0140749i
$438$		0			0
$439$	4.85641	+	2.80385i	0.231784	+	0.133820i	0.611395	−	0.791326i	$-0.290609\pi$
$439$	4.85641	+	2.80385i	0.231784	+	0.133820i	−0.379611	+	0.925146i	$0.623942\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−5.25833	−	19.6244i	−0.249831	−	0.932381i	−0.970894	−	0.239511i	$-0.923013\pi$
$443$	−5.25833	−	19.6244i	−0.249831	−	0.932381i	0.721063	−	0.692870i	$-0.243654\pi$
$444$		0			0
$445$	−2.00000	+	7.46410i	−0.0948091	+	0.353832i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	20.6603			0.975018			0.487509	−	0.873118i	$-0.337906\pi$
$449$	20.6603			0.975018			0.487509	+	0.873118i	$0.337906\pi$
$450$		0			0
$451$	−6.29423	−	6.29423i	−0.296384	−	0.296384i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−15.4641	+	8.92820i	−0.724968	+	0.418561i
$456$		0			0
$457$	20.2583	+	11.6962i	0.947645	+	0.547123i	0.892348	−	0.451347i	$-0.149056\pi$
$457$	20.2583	+	11.6962i	0.947645	+	0.547123i	0.0552962	+	0.998470i	$0.482390\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−2.56218	+	0.686533i	−0.119333	+	0.0319751i	−0.317991	−	0.948094i	$-0.603008\pi$
$461$	−2.56218	+	0.686533i	−0.119333	+	0.0319751i	0.198659	+	0.980069i	$0.436342\pi$
$462$		0			0
$463$	9.19615	+	15.9282i	0.427381	+	0.740246i	0.996640	−	0.0819125i	$-0.0261028\pi$
$463$	9.19615	+	15.9282i	0.427381	+	0.740246i	−0.569258	+	0.822159i	$0.692769\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	4.36603	+	4.36603i	0.202036	+	0.202036i	0.800872	−	0.598836i	$-0.204370\pi$
$467$	4.36603	+	4.36603i	0.202036	+	0.202036i	−0.598836	+	0.800872i	$0.704370\pi$
$468$		0			0
$469$	−2.80385	+	2.80385i	−0.129470	+	0.129470i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−3.27757	+	1.89230i	−0.150703	+	0.0870083i
$474$		0			0
$475$	2.30385	+	8.59808i	0.105708	+	0.394507i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	12.8301	−	22.2224i	0.586223	−	1.01537i	−0.408498	−	0.912759i	$-0.633947\pi$
$479$	12.8301	−	22.2224i	0.586223	−	1.01537i	0.994722	−	0.102610i	$-0.0327193\pi$
$480$		0			0
$481$	5.66025	+	9.80385i	0.258085	+	0.447017i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−22.5885	+	22.5885i	−1.02569	+	1.02569i
$486$		0			0
$487$		−	16.1962i		−	0.733918i	−0.930237	−	0.366959i	$-0.880399\pi$
$487$		−	16.1962i		−	0.733918i	0.930237	−	0.366959i	$-0.119601\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−25.7224	−	6.89230i	−1.16084	−	0.311045i	−0.373537	−	0.927615i	$-0.621855\pi$
$491$	−25.7224	−	6.89230i	−1.16084	−	0.311045i	−0.787300	+	0.616570i	$0.788522\pi$
$492$		0			0
$493$	5.36603	−	1.43782i	0.241674	−	0.0647563i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	4.00000	−	6.92820i	0.179425	−	0.310772i
$498$		0			0
$499$	1.69615	−	6.33013i	0.0759302	−	0.283375i	−0.917512	−	0.397707i	$-0.869806\pi$
$499$	1.69615	−	6.33013i	0.0759302	−	0.283375i	0.993443	+	0.114332i	$0.0364727\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			27.7128i			1.23565i	0.786314	+	0.617827i	$0.211987\pi$
$503$			27.7128i			1.23565i	−0.786314	+	0.617827i	$0.788013\pi$
$504$		0			0
$505$			29.8564i			1.32859i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	4.53590	−	16.9282i	0.201050	−	0.750329i	−0.789567	−	0.613664i	$-0.789695\pi$
$509$	4.53590	−	16.9282i	0.201050	−	0.750329i	0.990617	−	0.136665i	$-0.0436385\pi$
$510$		0			0
$511$	3.56218	−	6.16987i	0.157581	−	0.272939i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	34.0526	−	9.12436i	1.50054	−	0.402067i
$516$		0			0
$517$	27.6865	+	7.41858i	1.21765	+	0.326269i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	13.0000i		−	0.569540i	−0.958596	−	0.284770i	$-0.908083\pi$
$521$		−	13.0000i		−	0.569540i	0.958596	−	0.284770i	$-0.0919173\pi$
$522$		0			0
$523$	14.4641	−	14.4641i	0.632471	−	0.632471i	−0.316216	−	0.948687i	$-0.602412\pi$
$523$	14.4641	−	14.4641i	0.632471	−	0.632471i	0.948687	+	0.316216i	$0.102412\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	8.46410	+	14.6603i	0.368702	+	0.638611i
$528$		0			0
$529$	−10.6962	+	18.5263i	−0.465050	+	0.805490i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	4.90192	+	18.2942i	0.212326	+	0.792411i
$534$		0			0
$535$	−63.8372	+	36.8564i	−2.75992	+	1.59344i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−13.5622	+	13.5622i	−0.584164	+	0.584164i
$540$		0			0
$541$	−8.19615	−	8.19615i	−0.352380	−	0.352380i	0.508614	−	0.860994i	$-0.330158\pi$
$541$	−8.19615	−	8.19615i	−0.352380	−	0.352380i	−0.860994	+	0.508614i	$0.830158\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	19.8564	+	34.3923i	0.850555	+	1.47320i
$546$		0			0
$547$	31.2583	−	8.37564i	1.33651	−	0.358117i	0.481371	−	0.876517i	$-0.340139\pi$
$547$	31.2583	−	8.37564i	1.33651	−	0.358117i	0.855138	+	0.518400i	$0.173472\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	1.90192	+	1.09808i	0.0810247	+	0.0467796i
$552$		0			0
$553$	7.60770	−	4.39230i	0.323512	−	0.186780i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	25.1962	+	25.1962i	1.06760	+	1.06760i	0.997543	+	0.0700519i	$0.0223165\pi$
$557$	25.1962	+	25.1962i	1.06760	+	1.06760i	0.0700519	+	0.997543i	$0.477684\pi$
$558$		0			0
$559$	8.05256			0.340587
$560$		0			0
$561$		0			0
$562$		0			0
$563$	1.00962	−	3.76795i	0.0425504	−	0.158800i	−0.941382	−	0.337343i	$-0.890472\pi$
$563$	1.00962	−	3.76795i	0.0425504	−	0.158800i	0.983932	+	0.178543i	$0.0571384\pi$
$564$		0			0
$565$	−13.8564	−	51.7128i	−0.582943	−	2.17557i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−23.5981	−	13.6244i	−0.989283	−	0.571163i	−0.0842230	−	0.996447i	$-0.526841\pi$
$569$	−23.5981	−	13.6244i	−0.989283	−	0.571163i	−0.905060	+	0.425284i	$0.860174\pi$
$570$		0			0
$571$	19.8923	+	5.33013i	0.832467	+	0.223059i	0.649790	−	0.760114i	$-0.274857\pi$
$571$	19.8923	+	5.33013i	0.832467	+	0.223059i	0.182677	+	0.983173i	$0.441524\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−12.5885			−0.524975
$576$		0			0
$577$	35.7846			1.48973			0.744866	−	0.667214i	$-0.232513\pi$
$577$	35.7846			1.48973			0.744866	+	0.667214i	$0.232513\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−1.00000	−	0.267949i	−0.0414870	−	0.0111164i
$582$		0			0
$583$	−1.94744	−	1.12436i	−0.0806548	−	0.0465661i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−1.00962	−	3.76795i	−0.0416714	−	0.155520i	0.941955	−	0.335739i	$-0.108986\pi$
$587$	−1.00962	−	3.76795i	−0.0416714	−	0.155520i	−0.983626	+	0.180219i	$0.942319\pi$
$588$		0			0
$589$	−1.73205	+	6.46410i	−0.0713679	+	0.266349i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−10.5359			−0.432657			−0.216329	−	0.976321i	$-0.569408\pi$
$593$	−10.5359			−0.432657			−0.216329	+	0.976321i	$0.569408\pi$
$594$		0			0
$595$	4.53590	+	4.53590i	0.185954	+	0.185954i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	23.3205	−	13.4641i	0.952850	−	0.550128i	0.0588850	−	0.998265i	$-0.481245\pi$
$599$	23.3205	−	13.4641i	0.952850	−	0.550128i	0.893965	+	0.448136i	$0.147912\pi$
$600$		0			0
$601$	−17.5526	−	10.1340i	−0.715984	−	0.413373i	0.0972889	−	0.995256i	$-0.468983\pi$
$601$	−17.5526	−	10.1340i	−0.715984	−	0.413373i	−0.813273	+	0.581883i	$0.802316\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	8.19615	−	2.19615i	0.333221	−	0.0892863i
$606$		0			0
$607$	−22.5885	−	39.1244i	−0.916837	−	1.58801i	−0.804189	−	0.594374i	$-0.797400\pi$
$607$	−22.5885	−	39.1244i	−0.916837	−	1.58801i	−0.112648	−	0.993635i	$-0.535933\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−43.1244	−	43.1244i	−1.74462	−	1.74462i
$612$		0			0
$613$	1.66025	−	1.66025i	0.0670570	−	0.0670570i	−0.672783	−	0.739840i	$-0.734901\pi$
$613$	1.66025	−	1.66025i	0.0670570	−	0.0670570i	0.739840	+	0.672783i	$0.234901\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−3.91154	+	2.25833i	−0.157473	+	0.0909170i	−0.576666	−	0.816980i	$-0.695646\pi$
$617$	−3.91154	+	2.25833i	−0.157473	+	0.0909170i	0.419193	+	0.907897i	$0.362313\pi$
$618$		0			0
$619$	10.4019	+	38.8205i	0.418089	+	1.56033i	0.778568	+	0.627561i	$0.215947\pi$
$619$	10.4019	+	38.8205i	0.418089	+	1.56033i	−0.360479	+	0.932767i	$0.617387\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−0.732051	+	1.26795i	−0.0293290	+	0.0507993i
$624$		0			0
$625$	11.9641	+	20.7224i	0.478564	+	0.828897i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	2.87564	−	2.87564i	0.114659	−	0.114659i
$630$		0			0
$631$		−	38.3923i		−	1.52837i	−0.644995	−	0.764187i	$-0.723141\pi$
$631$		−	38.3923i		−	1.52837i	0.644995	−	0.764187i	$-0.276859\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	23.1244	+	6.19615i	0.917662	+	0.245887i
$636$		0			0
$637$	39.4186	−	10.5622i	1.56182	−	0.418489i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	4.20577	−	7.28461i	0.166118	−	0.287725i	−0.770934	−	0.636915i	$-0.780210\pi$
$641$	4.20577	−	7.28461i	0.166118	−	0.287725i	0.937052	+	0.349191i	$0.113543\pi$
$642$		0			0
$643$	−12.2321	+	45.6506i	−0.482385	+	1.80029i	0.109173	+	0.994023i	$0.465180\pi$
$643$	−12.2321	+	45.6506i	−0.482385	+	1.80029i	−0.591558	+	0.806263i	$0.701487\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			13.2679i			0.521617i	0.965391	+	0.260808i	$0.0839891\pi$
$647$			13.2679i			0.521617i	−0.965391	+	0.260808i	$0.916011\pi$
$648$		0			0
$649$		−	15.2487i		−	0.598564i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	1.50962	−	5.63397i	0.0590760	−	0.220474i	−0.930077	−	0.367365i	$-0.880260\pi$
$653$	1.50962	−	5.63397i	0.0590760	−	0.220474i	0.989153	+	0.146891i	$0.0469266\pi$
$654$		0			0
$655$	6.19615	−	10.7321i	0.242104	−	0.419336i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−15.0263	+	4.02628i	−0.585341	+	0.156842i	−0.539323	−	0.842099i	$-0.681320\pi$
$659$	−15.0263	+	4.02628i	−0.585341	+	0.156842i	−0.0460178	+	0.998941i	$0.514653\pi$
$660$		0			0
$661$	−8.19615	−	2.19615i	−0.318793	−	0.0854204i	0.0958740	−	0.995393i	$-0.469435\pi$
$661$	−8.19615	−	2.19615i	−0.318793	−	0.0854204i	−0.414667	+	0.909973i	$0.636102\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$			2.53590i			0.0983379i
$666$		0			0
$667$	−2.19615	+	2.19615i	−0.0850354	+	0.0850354i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	4.60770	+	7.98076i	0.177878	+	0.308094i
$672$		0			0
$673$	8.80385	−	15.2487i	0.339363	−	0.587795i	−0.644950	−	0.764225i	$-0.723122\pi$
$673$	8.80385	−	15.2487i	0.339363	−	0.587795i	0.984313	+	0.176430i	$0.0564550\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	1.26795	+	4.73205i	0.0487312	+	0.181867i	0.986002	−	0.166736i	$-0.0533227\pi$
$677$	1.26795	+	4.73205i	0.0487312	+	0.181867i	−0.937270	+	0.348603i	$0.886656\pi$
$678$		0			0
$679$	−5.24167	+	3.02628i	−0.201157	+	0.116138i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−4.70577	+	4.70577i	−0.180061	+	0.180061i	−0.791383	−	0.611321i	$-0.790638\pi$
$683$	−4.70577	+	4.70577i	−0.180061	+	0.180061i	0.611321	+	0.791383i	$0.290638\pi$
$684$		0			0
$685$	−44.9808	−	44.9808i	−1.71863	−	1.71863i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	2.39230	+	4.14359i	0.0911396	+	0.157858i
$690$		0			0
$691$	−23.4904	+	6.29423i	−0.893616	+	0.239444i	−0.676273	−	0.736651i	$-0.736406\pi$
$691$	−23.4904	+	6.29423i	−0.893616	+	0.239444i	−0.217344	+	0.976095i	$0.569739\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	31.3923	+	18.1244i	1.19078	+	0.687496i
$696$		0			0
$697$	5.89230	−	3.40192i	0.223187	−	0.128857i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−10.6603	−	10.6603i	−0.402632	−	0.402632i	0.476527	−	0.879160i	$-0.341895\pi$
$701$	−10.6603	−	10.6603i	−0.402632	−	0.402632i	−0.879160	+	0.476527i	$0.841895\pi$
$702$		0			0
$703$	1.60770			0.0606354
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−1.46410	+	5.46410i	−0.0550632	+	0.205499i
$708$		0			0
$709$	5.41154	+	20.1962i	0.203235	+	0.758482i	0.989980	+	0.141205i	$0.0450977\pi$
$709$	5.41154	+	20.1962i	0.203235	+	0.758482i	−0.786746	+	0.617277i	$0.788236\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−8.19615	−	4.73205i	−0.306948	−	0.177217i
$714$		0			0
$715$	69.9090	+	18.7321i	2.61445	+	0.700539i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−16.3923			−0.611330			−0.305665	−	0.952139i	$-0.598879\pi$
$719$	−16.3923			−0.611330			−0.305665	+	0.952139i	$0.598879\pi$
$720$		0			0
$721$	6.67949			0.248757
$722$		0			0
$723$		0			0
$724$		0			0
$725$	23.4904	+	6.29423i	0.872411	+	0.233762i
$726$		0			0
$727$	31.8109	+	18.3660i	1.17980	+	0.681158i	0.955968	−	0.293470i	$-0.0948099\pi$
$727$	31.8109	+	18.3660i	1.17980	+	0.681158i	0.223832	+	0.974628i	$0.428143\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−0.748711	−	2.79423i	−0.0276921	−	0.103348i
$732$		0			0
$733$	8.02628	−	29.9545i	0.296457	−	1.10639i	−0.643596	−	0.765366i	$-0.722558\pi$
$733$	8.02628	−	29.9545i	0.296457	−	1.10639i	0.940053	−	0.341028i	$-0.110775\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	16.0718			0.592012
$738$		0			0
$739$	−21.2224	−	21.2224i	−0.780680	−	0.780680i	0.199266	−	0.979945i	$-0.436144\pi$
$739$	−21.2224	−	21.2224i	−0.780680	−	0.780680i	−0.979945	+	0.199266i	$0.936144\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−2.24167	+	1.29423i	−0.0822389	+	0.0474806i	−0.540556	−	0.841308i	$-0.681786\pi$
$743$	−2.24167	+	1.29423i	−0.0822389	+	0.0474806i	0.458317	+	0.888789i	$0.348453\pi$
$744$		0			0
$745$	−10.7321	−	6.19615i	−0.393192	−	0.227009i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−13.4904	+	3.61474i	−0.492928	+	0.132080i
$750$		0			0
$751$	−18.8564	−	32.6603i	−0.688080	−	1.19179i	−0.972458	−	0.233077i	$-0.925120\pi$
$751$	−18.8564	−	32.6603i	−0.688080	−	1.19179i	0.284378	−	0.958712i	$-0.408213\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	7.46410	+	7.46410i	0.271646	+	0.271646i
$756$		0			0
$757$	−6.07180	+	6.07180i	−0.220683	+	0.220683i	−0.808786	−	0.588103i	$-0.799875\pi$
$757$	−6.07180	+	6.07180i	−0.220683	+	0.220683i	0.588103	+	0.808786i	$0.299875\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−27.3731	+	15.8038i	−0.992273	+	0.572889i	−0.905953	−	0.423378i	$-0.860844\pi$
$761$	−27.3731	+	15.8038i	−0.992273	+	0.572889i	−0.0863200	+	0.996267i	$0.527511\pi$
$762$		0			0
$763$	1.94744	+	7.26795i	0.0705021	+	0.263117i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−16.2224	+	28.0981i	−0.585758	+	1.01456i
$768$		0			0
$769$	10.1244	+	17.5359i	0.365094	+	0.632361i	0.988791	−	0.149305i	$-0.0477036\pi$
$769$	10.1244	+	17.5359i	0.365094	+	0.632361i	−0.623698	+	0.781666i	$0.714370\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−4.41154	+	4.41154i	−0.158672	+	0.158672i	−0.781978	−	0.623306i	$-0.785789\pi$
$773$	−4.41154	+	4.41154i	−0.158672	+	0.158672i	0.623306	+	0.781978i	$0.285789\pi$
$774$		0			0
$775$			74.1051i			2.66193i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	2.59808	+	0.696152i	0.0930857	+	0.0249422i
$780$		0			0
$781$	−31.3205	+	8.39230i	−1.12074	+	0.300300i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−9.46410	+	16.3923i	−0.337788	+	0.585066i
$786$		0			0
$787$	13.3468	−	49.8109i	0.475762	−	1.77557i	−0.142716	−	0.989764i	$-0.545583\pi$
$787$	13.3468	−	49.8109i	0.475762	−	1.77557i	0.618477	−	0.785803i	$-0.287750\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	10.1436i		−	0.360665i
$792$		0			0
$793$		−	19.6077i		−	0.696290i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−14.5167	+	54.1769i	−0.514206	+	1.91904i	−0.146065	+	0.989275i	$0.546661\pi$
$797$	−14.5167	+	54.1769i	−0.514206	+	1.91904i	−0.368142	+	0.929770i	$0.620006\pi$
$798$		0			0
$799$	−10.9545	+	18.9737i	−0.387542	+	0.671242i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−27.8923	+	7.47372i	−0.984298	+	0.263742i
$804$		0			0
$805$	−3.46410	−	0.928203i	−0.122094	−	0.0327149i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	28.3205i		−	0.995696i	−0.867264	−	0.497848i	$-0.834124\pi$
$809$		−	28.3205i		−	0.995696i	0.867264	−	0.497848i	$-0.165876\pi$
$810$		0			0
$811$	−5.02628	+	5.02628i	−0.176497	+	0.176497i	−0.789827	−	0.613330i	$-0.789830\pi$
$811$	−5.02628	+	5.02628i	−0.176497	+	0.176497i	0.613330	+	0.789827i	$0.289830\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	19.1244	+	33.1244i	0.669897	+	1.16030i
$816$		0			0
$817$	0.571797	−	0.990381i	0.0200046	−	0.0346490i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	8.63397	+	32.2224i	0.301328	+	1.12457i	0.936061	+	0.351839i	$0.114444\pi$
$821$	8.63397	+	32.2224i	0.301328	+	1.12457i	−0.634733	+	0.772732i	$0.718890\pi$
$822$		0			0
$823$	10.7321	−	6.19615i	0.374096	−	0.215984i	−0.301151	−	0.953577i	$-0.597371\pi$
$823$	10.7321	−	6.19615i	0.374096	−	0.215984i	0.675246	+	0.737592i	$0.264037\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	24.4641	−	24.4641i	0.850700	−	0.850700i	−0.139519	−	0.990219i	$-0.544556\pi$
$827$	24.4641	−	24.4641i	0.850700	−	0.850700i	0.990219	+	0.139519i	$0.0445557\pi$
$828$		0			0
$829$	24.5167	+	24.5167i	0.851499	+	0.851499i	0.990318	−	0.138819i	$-0.0443306\pi$
$829$	24.5167	+	24.5167i	0.851499	+	0.851499i	−0.138819	+	0.990318i	$0.544331\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−7.33013	−	12.6962i	−0.253974	−	0.439896i
$834$		0			0
$835$	2.00000	−	0.535898i	0.0692129	−	0.0185455i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	35.4449	+	20.4641i	1.22369	+	0.706499i	0.965703	−	0.259649i	$-0.0836067\pi$
$839$	35.4449	+	20.4641i	1.22369	+	0.706499i	0.257989	+	0.966148i	$0.416940\pi$
$840$		0			0
$841$	−19.9186	+	11.5000i	−0.686848	+	0.396552i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−73.3731	−	73.3731i	−2.52411	−	2.52411i
$846$		0			0
$847$	1.60770			0.0552411
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−0.588457	+	2.19615i	−0.0201721	+	0.0752831i
$852$		0			0
$853$	3.36603	+	12.5622i	0.115251	+	0.430121i	0.999306	−	0.0372621i	$-0.0118636\pi$
$853$	3.36603	+	12.5622i	0.115251	+	0.430121i	−0.884055	+	0.467383i	$0.845197\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	20.9090	+	12.0718i	0.714237	+	0.412365i	0.812628	−	0.582783i	$-0.198036\pi$
$857$	20.9090	+	12.0718i	0.714237	+	0.412365i	−0.0983911	+	0.995148i	$0.531370\pi$
$858$		0			0
$859$	−30.8205	−	8.25833i	−1.05158	−	0.281771i	−0.308677	−	0.951167i	$-0.599886\pi$
$859$	−30.8205	−	8.25833i	−1.05158	−	0.281771i	−0.742905	+	0.669396i	$0.766553\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−8.53590			−0.290565			−0.145283	−	0.989390i	$-0.546409\pi$
$863$	−8.53590			−0.290565			−0.145283	+	0.989390i	$0.546409\pi$
$864$		0			0
$865$	50.2487			1.70851
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−34.3923	−	9.21539i	−1.16668	−	0.312611i
$870$		0			0
$871$	−29.6147	−	17.0981i	−1.00346	−	0.579346i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	3.60770	+	13.4641i	0.121962	+	0.455170i
$876$		0			0
$877$	0.411543	−	1.53590i	0.0138968	−	0.0518636i	−0.958629	−	0.284658i	$-0.908120\pi$
$877$	0.411543	−	1.53590i	0.0138968	−	0.0518636i	0.972526	+	0.232794i	$0.0747868\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	7.32051			0.246634			0.123317	−	0.992367i	$-0.460647\pi$
$881$	7.32051			0.246634			0.123317	+	0.992367i	$0.460647\pi$
$882$		0			0
$883$	14.3660	+	14.3660i	0.483455	+	0.483455i	0.906233	−	0.422778i	$-0.138945\pi$
$883$	14.3660	+	14.3660i	0.483455	+	0.483455i	−0.422778	+	0.906233i	$0.638945\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−33.1244	+	19.1244i	−1.11221	+	0.642133i	−0.939400	−	0.342823i	$-0.888617\pi$
$887$	−33.1244	+	19.1244i	−1.11221	+	0.642133i	−0.172807	+	0.984956i	$0.555284\pi$
$888$		0			0
$889$	3.92820	+	2.26795i	0.131748	+	0.0760646i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−8.36603	+	2.24167i	−0.279958	+	0.0750146i
$894$		0			0
$895$	32.5885	+	56.4449i	1.08931	+	1.88674i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	12.9282	+	12.9282i	0.431180	+	0.431180i
$900$		0			0
$901$	1.21539	−	1.21539i	0.0404905	−	0.0404905i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	63.3731	−	36.5885i	2.10659	−	1.21624i
$906$		0			0
$907$	−4.50000	−	16.7942i	−0.149420	−	0.557643i	−0.999519	−	0.0310198i	$-0.990124\pi$
$907$	−4.50000	−	16.7942i	−0.149420	−	0.557643i	0.850099	−	0.526623i	$-0.176542\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−4.46410	+	7.73205i	−0.147902	+	0.256174i	−0.930452	−	0.366414i	$-0.880585\pi$
$911$	−4.46410	+	7.73205i	−0.147902	+	0.256174i	0.782550	+	0.622588i	$0.213919\pi$
$912$		0			0
$913$	2.09808	+	3.63397i	0.0694362	+	0.120267i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	1.66025	−	1.66025i	0.0548264	−	0.0548264i
$918$		0			0
$919$			32.9808i			1.08793i	0.839106	+	0.543967i	$0.183079\pi$
$919$			32.9808i			1.08793i	−0.839106	+	0.543967i	$0.816921\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	66.6410	+	17.8564i	2.19352	+	0.587751i
$924$		0			0
$925$	17.1962	−	4.60770i	0.565406	−	0.151500i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	18.4641	−	31.9808i	0.605788	−	1.04925i	−0.386139	−	0.922441i	$-0.626191\pi$
$929$	18.4641	−	31.9808i	0.605788	−	1.04925i	0.991926	−	0.126814i	$-0.0404752\pi$
$930$		0			0
$931$	1.50000	−	5.59808i	0.0491605	−	0.183470i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	26.0000i		−	0.850291i
$936$		0			0
$937$			51.1769i			1.67188i	0.548823	+	0.835938i	$0.315076\pi$
$937$			51.1769i			1.67188i	−0.548823	+	0.835938i	$0.684924\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−3.26795	+	12.1962i	−0.106532	+	0.397583i	−0.998514	−	0.0544870i	$-0.982648\pi$
$941$	−3.26795	+	12.1962i	−0.106532	+	0.397583i	0.891982	+	0.452070i	$0.149314\pi$
$942$		0			0
$943$	−1.90192	+	3.29423i	−0.0619352	+	0.107275i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	14.9904	−	4.01666i	0.487122	−	0.130524i	−0.00689497	−	0.999976i	$-0.502195\pi$
$947$	14.9904	−	4.01666i	0.487122	−	0.130524i	0.494017	+	0.869452i	$0.335528\pi$
$948$		0			0
$949$	59.3468	+	15.9019i	1.92648	+	0.516198i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			59.1051i			1.91460i	0.289092	+	0.957301i	$0.406647\pi$
$953$			59.1051i			1.91460i	−0.289092	+	0.957301i	$0.593353\pi$
$954$		0			0
$955$	38.3923	−	38.3923i	1.24235	−	1.24235i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−6.02628	−	10.4378i	−0.194599	−	0.337055i
$960$		0			0
$961$	−12.3564	+	21.4019i	−0.398594	+	0.690385i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−18.2679	−	68.1769i	−0.588066	−	2.19469i
$966$		0			0
$967$	9.16987	−	5.29423i	0.294883	−	0.170251i	−0.345259	−	0.938508i	$-0.612209\pi$
$967$	9.16987	−	5.29423i	0.294883	−	0.170251i	0.640142	+	0.768257i	$0.278876\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	22.4641	−	22.4641i	0.720907	−	0.720907i	−0.247883	−	0.968790i	$-0.579735\pi$
$971$	22.4641	−	22.4641i	0.720907	−	0.720907i	0.968790	+	0.247883i	$0.0797348\pi$
$972$		0			0
$973$	4.85641	+	4.85641i	0.155689	+	0.155689i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	9.93782	+	17.2128i	0.317939	+	0.550687i	0.980058	−	0.198712i	$-0.0636759\pi$
$977$	9.93782	+	17.2128i	0.317939	+	0.550687i	−0.662119	+	0.749399i	$0.730343\pi$
$978$		0			0
$979$	5.73205	−	1.53590i	0.183197	−	0.0490875i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	13.8564	+	8.00000i	0.441951	+	0.255160i	0.704425	−	0.709779i	$-0.251205\pi$
$983$	13.8564	+	8.00000i	0.441951	+	0.255160i	−0.262474	+	0.964939i	$0.584538\pi$
$984$		0			0
$985$	17.3205	−	10.0000i	0.551877	−	0.318626i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	1.14359	+	1.14359i	0.0363642	+	0.0363642i
$990$		0			0
$991$	32.6410			1.03688			0.518438	−	0.855115i	$-0.326514\pi$
$991$	32.6410			1.03688			0.518438	+	0.855115i	$0.326514\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	0.875644	−	3.26795i	0.0277598	−	0.103601i
$996$		0			0
$997$	−8.60770	−	32.1244i	−0.272608	−	1.01739i	−0.957427	−	0.288675i	$-0.906785\pi$
$997$	−8.60770	−	32.1244i	−0.272608	−	1.01739i	0.684819	−	0.728713i	$-0.259881\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1728.2.bc.d.721.1		4
3.2	odd	2		576.2.bb.d.529.1		4
4.3	odd	2		432.2.y.c.397.1		4
9.4	even	3		1728.2.bc.a.145.1		4
9.5	odd	6		576.2.bb.c.337.1		4
12.11	even	2		144.2.x.b.61.1	✓	4
16.5	even	4		1728.2.bc.a.1585.1		4
16.11	odd	4		432.2.y.b.181.1		4
36.23	even	6		144.2.x.c.13.1	yes	4
36.31	odd	6		432.2.y.b.253.1		4
48.5	odd	4		576.2.bb.c.241.1		4
48.11	even	4		144.2.x.c.133.1	yes	4
144.5	odd	12		576.2.bb.d.49.1		4
144.59	even	12		144.2.x.b.85.1	yes	4
144.85	even	12	inner	1728.2.bc.d.1009.1		4
144.139	odd	12		432.2.y.c.37.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
144.2.x.b.61.1	✓	4	12.11	even	2
144.2.x.b.85.1	yes	4	144.59	even	12
144.2.x.c.13.1	yes	4	36.23	even	6
144.2.x.c.133.1	yes	4	48.11	even	4
432.2.y.b.181.1		4	16.11	odd	4
432.2.y.b.253.1		4	36.31	odd	6
432.2.y.c.37.1		4	144.139	odd	12
432.2.y.c.397.1		4	4.3	odd	2
576.2.bb.c.241.1		4	48.5	odd	4
576.2.bb.c.337.1		4	9.5	odd	6
576.2.bb.d.49.1		4	144.5	odd	12
576.2.bb.d.529.1		4	3.2	odd	2
1728.2.bc.a.145.1		4	9.4	even	3
1728.2.bc.a.1585.1		4	16.5	even	4
1728.2.bc.d.721.1		4	1.1	even	1	trivial
1728.2.bc.d.1009.1		4	144.85	even	12	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1728 = 2^{6} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1728.bc (of order \(12\), degree \(4\), not minimal)

\(f(q)\)	\(=\)	\(q+(3.73205 + 1.00000i) q^{5} +(0.633975 + 0.366025i) q^{7} +O(q^{10})\)	\(q+(3.73205 + 1.00000i) q^{5} +(0.633975 + 0.366025i) q^{7} +(-0.767949 - 2.86603i) q^{11} +(-1.63397 + 6.09808i) q^{13} +2.26795 q^{17} +(0.633975 + 0.633975i) q^{19} +(-1.09808 + 0.633975i) q^{23} +(8.59808 + 4.96410i) q^{25} +(2.36603 - 0.633975i) q^{29} +(3.73205 + 6.46410i) q^{31} +(2.00000 + 2.00000i) q^{35} +(1.26795 - 1.26795i) q^{37} +(2.59808 - 1.50000i) q^{41} +(-0.330127 - 1.23205i) q^{43} +(-4.83013 + 8.36603i) q^{47} +(-3.23205 - 5.59808i) q^{49} +(0.535898 - 0.535898i) q^{53} -11.4641i q^{55} +(4.96410 + 1.33013i) q^{59} +(-3.00000 + 0.803848i) q^{61} +(-12.1962 + 21.1244i) q^{65} +(-1.40192 + 5.23205i) q^{67} -10.9282i q^{71} -9.73205i q^{73} +(0.562178 - 2.09808i) q^{77} +(6.00000 - 10.3923i) q^{79} +(-1.36603 + 0.366025i) q^{83} +(8.46410 + 2.26795i) q^{85} +2.00000i q^{89} +(-3.26795 + 3.26795i) q^{91} +(1.73205 + 3.00000i) q^{95} +(-4.13397 + 7.16025i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{5} + 6 q^{7}+O(q^{10}) \) 4 * q + 8 * q^5 + 6 * q^7	\( 4 q + 8 q^{5} + 6 q^{7} - 10 q^{11} - 10 q^{13} + 16 q^{17} + 6 q^{19} + 6 q^{23} + 24 q^{25} + 6 q^{29} + 8 q^{31} + 8 q^{35} + 12 q^{37} + 16 q^{43} - 2 q^{47} - 6 q^{49} + 16 q^{53} + 6 q^{59} - 12 q^{61} - 28 q^{65} - 16 q^{67} - 22 q^{77} + 24 q^{79} - 2 q^{83} + 20 q^{85} - 20 q^{91} - 20 q^{97}+O(q^{100}) \) 4 * q + 8 * q^5 + 6 * q^7 - 10 * q^11 - 10 * q^13 + 16 * q^17 + 6 * q^19 + 6 * q^23 + 24 * q^25 + 6 * q^29 + 8 * q^31 + 8 * q^35 + 12 * q^37 + 16 * q^43 - 2 * q^47 - 6 * q^49 + 16 * q^53 + 6 * q^59 - 12 * q^61 - 28 * q^65 - 16 * q^67 - 22 * q^77 + 24 * q^79 - 2 * q^83 + 20 * q^85 - 20 * q^91 - 20 * q^97

Introduction

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