LMFDB - Embedded newform 1728.2.bc.a.1585.1

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			1585.1
Root			$-0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	1728.1585
Dual form			1728.2.bc.a.145.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.00000 + 3.73205i) q^{5} +(-0.633975 - 0.366025i) q^{7} +O(q^{10})$	$q+(-1.00000 + 3.73205i) q^{5} +(-0.633975 - 0.366025i) q^{7} +(2.86603 - 0.767949i) q^{11} +(6.09808 + 1.63397i) q^{13} +2.26795 q^{17} +(0.633975 - 0.633975i) q^{19} +(1.09808 - 0.633975i) q^{23} +(-8.59808 - 4.96410i) q^{25} +(-0.633975 - 2.36603i) q^{29} +(3.73205 + 6.46410i) q^{31} +(2.00000 - 2.00000i) q^{35} +(1.26795 + 1.26795i) q^{37} +(-2.59808 + 1.50000i) q^{41} +(1.23205 - 0.330127i) q^{43} +(-4.83013 + 8.36603i) q^{47} +(-3.23205 - 5.59808i) q^{49} +(0.535898 + 0.535898i) q^{53} +11.4641i q^{55} +(-1.33013 + 4.96410i) q^{59} +(0.803848 + 3.00000i) q^{61} +(-12.1962 + 21.1244i) q^{65} +(5.23205 + 1.40192i) q^{67} +10.9282i q^{71} +9.73205i q^{73} +(-2.09808 - 0.562178i) q^{77} +(6.00000 - 10.3923i) q^{79} +(0.366025 + 1.36603i) q^{83} +(-2.26795 + 8.46410i) q^{85} -2.00000i q^{89} +(-3.26795 - 3.26795i) q^{91} +(1.73205 + 3.00000i) q^{95} +(-4.13397 + 7.16025i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{5} - 6 q^{7}+O(q^{10}) $ 4 * q - 4 * q^5 - 6 * q^7	$ 4 q - 4 q^{5} - 6 q^{7} + 8 q^{11} + 14 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} - 2 q^{43} - 2 q^{47} - 6 q^{49} + 16 q^{53} + 12 q^{59} + 24 q^{61} - 28 q^{65} + 14 q^{67} + 2 q^{77} + 24 q^{79} - 2 q^{83} - 16 q^{85} - 20 q^{91} - 20 q^{97}+O(q^{100}) $ 4 * q - 4 * q^5 - 6 * q^7 + 8 * q^11 + 14 * 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 - 2 * q^43 - 2 * q^47 - 6 * q^49 + 16 * q^53 + 12 * q^59 + 24 * q^61 - 28 * q^65 + 14 * q^67 + 2 * q^77 + 24 * q^79 - 2 * q^83 - 16 * q^85 - 20 * q^91 - 20 * q^97

Display 100 coefficients 10 coefficients

Character values

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

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

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
$5$	−1.00000	+	3.73205i	−0.447214	+	1.66902i	0.262811	+	0.964847i	$0.415350\pi$
$5$	−1.00000	+	3.73205i	−0.447214	+	1.66902i	−0.710025	+	0.704177i	$0.751316\pi$
$6$		0			0
$7$	−0.633975	−	0.366025i	−0.239620	−	0.138345i	0.375382	−	0.926870i	$-0.377511\pi$
$7$	−0.633975	−	0.366025i	−0.239620	−	0.138345i	−0.615002	+	0.788526i	$0.710845\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	2.86603	−	0.767949i	0.864139	−	0.231545i	0.200587	−	0.979676i	$-0.435715\pi$
$11$	2.86603	−	0.767949i	0.864139	−	0.231545i	0.663552	+	0.748130i	$0.269048\pi$
$12$		0			0
$13$	6.09808	+	1.63397i	1.69130	+	0.453183i	0.970725	−	0.240192i	$-0.0772105\pi$
$13$	6.09808	+	1.63397i	1.69130	+	0.453183i	0.720577	+	0.693375i	$0.243877\pi$
$14$		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	−0.630635	−	0.776079i	$-0.717206\pi$
$19$	0.633975	−	0.633975i	0.145444	−	0.145444i	0.776079	+	0.630635i	$0.217206\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	1.09808	−	0.633975i	0.228965	−	0.132193i	−0.381130	−	0.924522i	$-0.624465\pi$
$23$	1.09808	−	0.633975i	0.228965	−	0.132193i	0.610094	+	0.792329i	$0.291132\pi$
$24$		0			0
$25$	−8.59808	−	4.96410i	−1.71962	−	0.992820i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−0.633975	−	2.36603i	−0.117726	−	0.439360i	0.881750	−	0.471717i	$-0.156365\pi$
$29$	−0.633975	−	2.36603i	−0.117726	−	0.439360i	−0.999476	+	0.0323566i	$0.989699\pi$
$30$		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	0.803608	−	0.595159i	$-0.202911\pi$
$37$	1.26795	+	1.26795i	0.208450	+	0.208450i	−0.595159	+	0.803608i	$0.702911\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−2.59808	+	1.50000i	−0.405751	+	0.234261i	−0.688963	−	0.724797i	$-0.741934\pi$
$41$	−2.59808	+	1.50000i	−0.405751	+	0.234261i	0.283211	+	0.959058i	$0.408600\pi$
$42$		0			0
$43$	1.23205	−	0.330127i	0.187886	−	0.0503439i	−0.163649	−	0.986519i	$-0.552326\pi$
$43$	1.23205	−	0.330127i	0.187886	−	0.0503439i	0.351535	+	0.936175i	$0.385660\pi$
$44$		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	0.742954	−	0.669343i	$-0.233424\pi$
$53$	0.535898	+	0.535898i	0.0736113	+	0.0736113i	−0.669343	+	0.742954i	$0.733424\pi$
$54$		0			0
$55$			11.4641i			1.54582i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−1.33013	+	4.96410i	−0.173168	+	0.646271i	0.823689	+	0.567042i	$0.191912\pi$
$59$	−1.33013	+	4.96410i	−0.173168	+	0.646271i	−0.996856	+	0.0792287i	$0.974754\pi$
$60$		0			0
$61$	0.803848	+	3.00000i	0.102922	+	0.384111i	0.998101	−	0.0615961i	$-0.0196191\pi$
$61$	0.803848	+	3.00000i	0.102922	+	0.384111i	−0.895179	+	0.445707i	$0.852952\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−12.1962	+	21.1244i	−1.51275	+	2.62015i
$66$		0			0
$67$	5.23205	+	1.40192i	0.639197	+	0.171272i	0.563840	−	0.825884i	$-0.309324\pi$
$67$	5.23205	+	1.40192i	0.639197	+	0.171272i	0.0753572	+	0.997157i	$0.475990\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			10.9282i			1.29694i	0.761241	+	0.648470i	$0.224591\pi$
$71$			10.9282i			1.29694i	−0.761241	+	0.648470i	$0.775409\pi$
$72$		0			0
$73$			9.73205i			1.13905i	0.821974	+	0.569525i	$0.192873\pi$
$73$			9.73205i			1.13905i	−0.821974	+	0.569525i	$0.807127\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−2.09808	−	0.562178i	−0.239098	−	0.0640661i
$78$		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
$83$	0.366025	+	1.36603i	0.0401765	+	0.149941i	0.983100	−	0.183068i	$-0.0586028\pi$
$83$	0.366025	+	1.36603i	0.0401765	+	0.149941i	−0.942924	+	0.333009i	$0.891936\pi$
$84$		0			0
$85$	−2.26795	+	8.46410i	−0.245994	+	0.918061i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	2.00000i		−	0.212000i	−0.994366	−	0.106000i	$-0.966196\pi$
$89$		−	2.00000i		−	0.212000i	0.994366	−	0.106000i	$-0.0338043\pi$
$90$		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
$101$	−7.46410	+	2.00000i	−0.742706	+	0.199007i	−0.610280	−	0.792186i	$-0.708943\pi$
$101$	−7.46410	+	2.00000i	−0.742706	+	0.199007i	−0.132426	+	0.991193i	$0.542277\pi$
$102$		0			0
$103$	−7.90192	+	4.56218i	−0.778600	+	0.449525i	−0.835934	−	0.548830i	$-0.815073\pi$
$103$	−7.90192	+	4.56218i	−0.778600	+	0.449525i	0.0573341	+	0.998355i	$0.481740\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−13.4904	−	13.4904i	−1.30416	−	1.30416i	−0.925558	−	0.378607i	$-0.876403\pi$
$107$	−13.4904	−	13.4904i	−1.30416	−	1.30416i	−0.378607	−	0.925558i	$-0.623597\pi$
$108$		0			0
$109$	7.26795	−	7.26795i	0.696143	−	0.696143i	−0.267433	−	0.963576i	$-0.586175\pi$
$109$	7.26795	−	7.26795i	0.696143	−	0.696143i	0.963576	+	0.267433i	$0.0861754\pi$
$110$		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
$115$	1.26795	+	4.73205i	0.118237	+	0.441266i
$116$		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
$131$	−3.09808	−	0.830127i	−0.270680	−	0.0725285i	0.120926	−	0.992662i	$-0.461414\pi$
$131$	−3.09808	−	0.830127i	−0.270680	−	0.0725285i	−0.391606	+	0.920133i	$0.628080\pi$
$132$		0			0
$133$	−0.633975	+	0.169873i	−0.0549726	+	0.0147299i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	14.2583	+	8.23205i	1.21817	+	0.703312i	0.964527	−	0.263986i	$-0.0850372\pi$
$137$	14.2583	+	8.23205i	1.21817	+	0.703312i	0.253645	+	0.967297i	$0.418371\pi$
$138$		0			0
$139$	−2.42820	+	9.06218i	−0.205958	+	0.768644i	0.783198	+	0.621772i	$0.213587\pi$
$139$	−2.42820	+	9.06218i	−0.205958	+	0.768644i	−0.989156	+	0.146872i	$0.953080\pi$
$140$		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
$149$	0.830127	−	3.09808i	0.0680067	−	0.253804i	−0.923550	−	0.383478i	$-0.874726\pi$
$149$	0.830127	−	3.09808i	0.0680067	−	0.253804i	0.991557	+	0.129674i	$0.0413929\pi$
$150$		0			0
$151$	−2.36603	−	1.36603i	−0.192544	−	0.111166i	0.400629	−	0.916240i	$-0.368792\pi$
$151$	−2.36603	−	1.36603i	−0.192544	−	0.111166i	−0.593173	+	0.805075i	$0.702125\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−27.8564	+	7.46410i	−2.23748	+	0.599531i
$156$		0			0
$157$	4.73205	+	1.26795i	0.377659	+	0.101193i	0.442655	−	0.896692i	$-0.354037\pi$
$157$	4.73205	+	1.26795i	0.377659	+	0.101193i	−0.0649959	+	0.997886i	$0.520703\pi$
$158$		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	−0.377661	−	0.925944i	$-0.623272\pi$
$163$	7.00000	−	7.00000i	0.548282	−	0.548282i	0.925944	+	0.377661i	$0.123272\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−0.464102	+	0.267949i	−0.0359133	+	0.0207345i	−0.517849	−	0.855472i	$-0.673267\pi$
$167$	−0.464102	+	0.267949i	−0.0359133	+	0.0207345i	0.481936	+	0.876206i	$0.339934\pi$
$168$		0			0
$169$	23.2583	+	13.4282i	1.78910	+	1.03294i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−3.36603	−	12.5622i	−0.255914	−	0.955085i	−0.967580	−	0.252566i	$-0.918725\pi$
$173$	−3.36603	−	12.5622i	−0.255914	−	0.955085i	0.711665	−	0.702519i	$-0.247941\pi$
$174$		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	−0.103114	−	0.994670i	$-0.532881\pi$
$179$	11.9282	−	11.9282i	0.891556	−	0.891556i	0.994670	+	0.103114i	$0.0328806\pi$
$180$		0			0
$181$	13.3923	+	13.3923i	0.995442	+	0.995442i	0.999990	−	0.00454748i	$-0.00144751\pi$
$181$	13.3923	+	13.3923i	0.995442	+	0.995442i	−0.00454748	+	0.999990i	$0.501448\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−6.00000	+	3.46410i	−0.441129	+	0.254686i
$186$		0			0
$187$	6.50000	−	1.74167i	0.475327	−	0.127364i
$188$		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	0.825372	−	0.564590i	$-0.190966\pi$
$197$	3.66025	+	3.66025i	0.260782	+	0.260782i	−0.564590	+	0.825372i	$0.690966\pi$
$198$		0			0
$199$			0.875644i			0.0620728i	0.999518	+	0.0310364i	$0.00988078\pi$
$199$			0.875644i			0.0620728i	−0.999518	+	0.0310364i	$0.990119\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−0.464102	+	1.73205i	−0.0325735	+	0.121566i
$204$		0			0
$205$	−3.00000	−	11.1962i	−0.209529	−	0.781973i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	1.33013	−	2.30385i	0.0920068	−	0.159360i
$210$		0			0
$211$	−4.09808	−	1.09808i	−0.282123	−	0.0755947i	0.114983	−	0.993367i	$-0.463319\pi$
$211$	−4.09808	−	1.09808i	−0.282123	−	0.0755947i	−0.397106	+	0.917773i	$0.629985\pi$
$212$		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
$221$	13.8301	+	3.70577i	0.930315	+	0.249277i
$222$		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
$227$	3.86603	+	14.4282i	0.256597	+	0.957633i	0.967195	+	0.254035i	$0.0817579\pi$
$227$	3.86603	+	14.4282i	0.256597	+	0.957633i	−0.710598	+	0.703598i	$0.751575\pi$
$228$		0			0
$229$	−1.83013	+	6.83013i	−0.120938	+	0.451347i	−0.999662	−	0.0259823i	$-0.991729\pi$
$229$	−1.83013	+	6.83013i	−0.120938	+	0.451347i	0.878724	+	0.477330i	$0.158395\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			7.19615i			0.471436i	0.971822	+	0.235718i	$0.0757441\pi$
$233$			7.19615i			0.471436i	−0.971822	+	0.235718i	$0.924256\pi$
$234$		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
$245$	24.1244	−	6.46410i	1.54125	−	0.412976i
$246$		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	0.790761	−	0.612125i	$-0.209685\pi$
$251$	2.83013	+	2.83013i	0.178636	+	0.178636i	−0.612125	+	0.790761i	$0.709685\pi$
$252$		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
$259$	−0.339746	−	1.26795i	−0.0211108	−	0.0787865i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	23.4904	+	13.5622i	1.44848	+	0.836280i	0.998391	−	0.0567045i	$-0.0180593\pi$
$263$	23.4904	+	13.5622i	1.44848	+	0.836280i	0.450088	+	0.892984i	$0.351393\pi$
$264$		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	−0.836494	−	0.547976i	$-0.815399\pi$
$269$	−4.73205	+	4.73205i	−0.288518	+	0.288518i	0.547976	+	0.836494i	$0.315399\pi$
$270$		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
$275$	−28.4545	−	7.62436i	−1.71587	−	0.459766i
$276$		0			0
$277$	15.7583	−	4.22243i	0.946826	−	0.253701i	0.247811	−	0.968808i	$-0.420289\pi$
$277$	15.7583	−	4.22243i	0.946826	−	0.253701i	0.699015	+	0.715107i	$0.253622\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−8.66025	−	5.00000i	−0.516627	−	0.298275i	0.218926	−	0.975741i	$-0.429745\pi$
$281$	−8.66025	−	5.00000i	−0.516627	−	0.298275i	−0.735554	+	0.677466i	$0.763078\pi$
$282$		0			0
$283$	7.43782	−	27.7583i	0.442133	−	1.65006i	−0.281265	−	0.959630i	$-0.590754\pi$
$283$	7.43782	−	27.7583i	0.442133	−	1.65006i	0.723398	−	0.690431i	$-0.242579\pi$
$284$		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
$293$	3.63397	−	13.5622i	0.212299	−	0.792311i	−0.774801	−	0.632205i	$-0.782150\pi$
$293$	3.63397	−	13.5622i	0.212299	−	0.792311i	0.987100	−	0.160106i	$-0.0511834\pi$
$294$		0			0
$295$	−17.1962	−	9.92820i	−1.00120	−	0.578042i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	7.73205	−	2.07180i	0.447156	−	0.119815i
$300$		0			0
$301$	−0.901924	−	0.241670i	−0.0519860	−	0.0139296i
$302$		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	−0.0819670	−	0.996635i	$-0.526120\pi$
$307$	16.0263	−	16.0263i	0.914668	−	0.914668i	0.996635	+	0.0819670i	$0.0261202\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−13.9019	+	8.02628i	−0.788306	+	0.455129i	−0.839366	−	0.543567i	$-0.817073\pi$
$311$	−13.9019	+	8.02628i	−0.788306	+	0.455129i	0.0510600	+	0.998696i	$0.483740\pi$
$312$		0			0
$313$	−24.6506	−	14.2321i	−1.39334	−	0.804443i	−0.399653	−	0.916666i	$-0.630869\pi$
$313$	−24.6506	−	14.2321i	−1.39334	−	0.804443i	−0.993683	+	0.112223i	$0.964203\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	8.43782	+	31.4904i	0.473915	+	1.76868i	0.625492	+	0.780231i	$0.284898\pi$
$317$	8.43782	+	31.4904i	0.473915	+	1.76868i	−0.151577	+	0.988445i	$0.548435\pi$
$318$		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
$331$	19.0263	−	5.09808i	1.04578	−	0.280216i	0.305273	−	0.952265i	$-0.401252\pi$
$331$	19.0263	−	5.09808i	1.04578	−	0.280216i	0.740506	+	0.672049i	$0.234586\pi$
$332$		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
$347$	6.62436	−	24.7224i	0.355614	−	1.32717i	−0.524096	−	0.851659i	$-0.675597\pi$
$347$	6.62436	−	24.7224i	0.355614	−	1.32717i	0.879710	−	0.475510i	$-0.157737\pi$
$348$		0			0
$349$	2.07180	+	7.73205i	0.110901	+	0.413887i	0.998948	−	0.0458657i	$-0.0146046\pi$
$349$	2.07180	+	7.73205i	0.110901	+	0.413887i	−0.888047	+	0.459753i	$0.847938\pi$
$350$		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
$355$	−40.7846	−	10.9282i	−2.16462	−	0.580009i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	14.7321i		−	0.777528i	−0.921337	−	0.388764i	$-0.872902\pi$
$359$		−	14.7321i		−	0.777528i	0.921337	−	0.388764i	$-0.127098\pi$
$360$		0			0
$361$			18.1962i			0.957692i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−36.3205	−	9.73205i	−1.90110	−	0.509399i
$366$		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
$371$	−0.143594	−	0.535898i	−0.00745501	−	0.0278225i
$372$		0			0
$373$	1.50962	−	5.63397i	0.0781651	−	0.291716i	−0.915767	−	0.401709i	$-0.868416\pi$
$373$	1.50962	−	5.63397i	0.0781651	−	0.291716i	0.993932	+	0.109993i	$0.0350829\pi$
$374$		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	0.999359	−	0.0358080i	$-0.0114005\pi$
$379$	18.7583	+	18.7583i	0.963551	+	0.963551i	−0.0358080	+	0.999359i	$0.511400\pi$
$380$		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
$389$	−10.2942	+	2.75833i	−0.521938	+	0.139853i	−0.510163	−	0.860078i	$-0.670415\pi$
$389$	−10.2942	+	2.75833i	−0.521938	+	0.139853i	−0.0117752	+	0.999931i	$0.503748\pi$
$390$		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	−0.950310	−	0.311306i	$-0.899233\pi$
$397$	−12.7321	+	12.7321i	−0.639003	+	0.639003i	0.311306	+	0.950310i	$0.399233\pi$
$398$		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
$403$	12.1962	+	45.5167i	0.607534	+	2.26735i
$404$		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	0.313191	−	0.949690i	$-0.398602\pi$
$409$	26.1340	−	15.0885i	1.29224	−	0.746076i	0.979051	+	0.203614i	$0.0652688\pi$
$410$		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
$419$	31.2224	+	8.36603i	1.52532	+	0.408707i	0.921488	−	0.388408i	$-0.126975\pi$
$419$	31.2224	+	8.36603i	1.52532	+	0.408707i	0.603828	+	0.797115i	$0.293641\pi$
$420$		0			0
$421$	−2.19615	+	0.588457i	−0.107034	+	0.0286797i	−0.311938	−	0.950102i	$-0.600978\pi$
$421$	−2.19615	+	0.588457i	−0.107034	+	0.0286797i	0.204905	+	0.978782i	$0.434312\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−19.5000	−	11.2583i	−0.945889	−	0.546109i
$426$		0			0
$427$	0.588457	−	2.19615i	0.0284774	−	0.106279i
$428$		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
$437$	0.294229	−	1.09808i	0.0140749	−	0.0525281i
$438$		0			0
$439$	−4.85641	−	2.80385i	−0.231784	−	0.133820i	0.379611	−	0.925146i	$-0.376058\pi$
$439$	−4.85641	−	2.80385i	−0.231784	−	0.133820i	−0.611395	+	0.791326i	$0.709391\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	19.6244	−	5.25833i	0.932381	−	0.249831i	0.239511	−	0.970894i	$-0.423013\pi$
$443$	19.6244	−	5.25833i	0.932381	−	0.249831i	0.692870	+	0.721063i	$0.256346\pi$
$444$		0			0
$445$	7.46410	+	2.00000i	0.353832	+	0.0948091i
$446$		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	−0.0552962	−	0.998470i	$-0.517610\pi$
$457$	−20.2583	−	11.6962i	−0.947645	−	0.547123i	−0.892348	+	0.451347i	$0.850944\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	0.686533	+	2.56218i	0.0319751	+	0.119333i	0.980069	−	0.198659i	$-0.0636585\pi$
$461$	0.686533	+	2.56218i	0.0319751	+	0.119333i	−0.948094	+	0.317991i	$0.896992\pi$
$462$		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	−0.598836	−	0.800872i	$-0.704370\pi$
$467$	4.36603	−	4.36603i	0.202036	−	0.202036i	0.800872	+	0.598836i	$0.204370\pi$
$468$		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
$475$	−8.59808	+	2.30385i	−0.394507	+	0.105708i
$476$		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	$0.119601\pi$
$487$			16.1962i			0.733918i	−0.930237	+	0.366959i	$0.880399\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	6.89230	−	25.7224i	0.311045	−	1.16084i	−0.616570	−	0.787300i	$-0.711478\pi$
$491$	6.89230	−	25.7224i	0.311045	−	1.16084i	0.927615	−	0.373537i	$-0.121855\pi$
$492$		0			0
$493$	−1.43782	−	5.36603i	−0.0647563	−	0.241674i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	4.00000	−	6.92820i	0.179425	−	0.310772i
$498$		0			0
$499$	−6.33013	−	1.69615i	−0.283375	−	0.0759302i	0.114332	−	0.993443i	$-0.463527\pi$
$499$	−6.33013	−	1.69615i	−0.283375	−	0.0759302i	−0.397707	+	0.917512i	$0.630194\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	27.7128i		−	1.23565i	−0.786314	−	0.617827i	$-0.788013\pi$
$503$		−	27.7128i		−	1.23565i	0.786314	−	0.617827i	$-0.211987\pi$
$504$		0			0
$505$		−	29.8564i		−	1.32859i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−16.9282	−	4.53590i	−0.750329	−	0.201050i	−0.136665	−	0.990617i	$-0.543638\pi$
$509$	−16.9282	−	4.53590i	−0.750329	−	0.201050i	−0.613664	+	0.789567i	$0.710305\pi$
$510$		0			0
$511$	3.56218	−	6.16987i	0.157581	−	0.272939i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−9.12436	−	34.0526i	−0.402067	−	1.50054i
$516$		0			0
$517$	−7.41858	+	27.6865i	−0.326269	+	1.21765i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			13.0000i			0.569540i	0.958596	+	0.284770i	$0.0919173\pi$
$521$			13.0000i			0.569540i	−0.958596	+	0.284770i	$0.908083\pi$
$522$		0			0
$523$	14.4641	+	14.4641i	0.632471	+	0.632471i	0.948687	−	0.316216i	$-0.102412\pi$
$523$	14.4641	+	14.4641i	0.632471	+	0.632471i	−0.316216	+	0.948687i	$0.602412\pi$
$524$		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
$533$	−18.2942	+	4.90192i	−0.792411	+	0.212326i
$534$		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	−0.860994	−	0.508614i	$-0.830158\pi$
$541$	−8.19615	+	8.19615i	−0.352380	+	0.352380i	0.508614	+	0.860994i	$0.330158\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	19.8564	+	34.3923i	0.850555	+	1.47320i
$546$		0			0
$547$	−8.37564	−	31.2583i	−0.358117	−	1.33651i	−0.876517	−	0.481371i	$-0.840139\pi$
$547$	−8.37564	−	31.2583i	−0.358117	−	1.33651i	0.518400	−	0.855138i	$-0.326528\pi$
$548$		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	0.0700519	−	0.997543i	$-0.477684\pi$
$557$	25.1962	−	25.1962i	1.06760	−	1.06760i	0.997543	−	0.0700519i	$-0.0223165\pi$
$558$		0			0
$559$	8.05256			0.340587
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−3.76795	−	1.00962i	−0.158800	−	0.0425504i	0.178543	−	0.983932i	$-0.442862\pi$
$563$	−3.76795	−	1.00962i	−0.158800	−	0.0425504i	−0.337343	+	0.941382i	$0.609528\pi$
$564$		0			0
$565$	51.7128	−	13.8564i	2.17557	−	0.582943i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	23.5981	+	13.6244i	0.989283	+	0.571163i	0.905060	−	0.425284i	$-0.139826\pi$
$569$	23.5981	+	13.6244i	0.989283	+	0.571163i	0.0842230	+	0.996447i	$0.473159\pi$
$570$		0			0
$571$	−5.33013	+	19.8923i	−0.223059	+	0.832467i	0.760114	+	0.649790i	$0.225143\pi$
$571$	−5.33013	+	19.8923i	−0.223059	+	0.832467i	−0.983173	+	0.182677i	$0.941524\pi$
$572$		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
$581$	0.267949	−	1.00000i	0.0111164	−	0.0414870i
$582$		0			0
$583$	1.94744	+	1.12436i	0.0806548	+	0.0465661i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	3.76795	−	1.00962i	0.155520	−	0.0416714i	−0.180219	−	0.983626i	$-0.557681\pi$
$587$	3.76795	−	1.00962i	0.155520	−	0.0416714i	0.335739	+	0.941955i	$0.391014\pi$
$588$		0			0
$589$	6.46410	+	1.73205i	0.266349	+	0.0713679i
$590$		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	−0.893965	−	0.448136i	$-0.852088\pi$
$599$	−23.3205	+	13.4641i	−0.952850	+	0.550128i	−0.0588850	+	0.998265i	$0.518755\pi$
$600$		0			0
$601$	17.5526	+	10.1340i	0.715984	+	0.413373i	0.813273	−	0.581883i	$-0.197684\pi$
$601$	17.5526	+	10.1340i	0.715984	+	0.413373i	−0.0972889	+	0.995256i	$0.531017\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−2.19615	−	8.19615i	−0.0892863	−	0.333221i
$606$		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	0.739840	−	0.672783i	$-0.234901\pi$
$613$	1.66025	+	1.66025i	0.0670570	+	0.0670570i	−0.672783	+	0.739840i	$0.734901\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	3.91154	−	2.25833i	0.157473	−	0.0909170i	−0.419193	−	0.907897i	$-0.637687\pi$
$617$	3.91154	−	2.25833i	0.157473	−	0.0909170i	0.576666	+	0.816980i	$0.304354\pi$
$618$		0			0
$619$	−38.8205	+	10.4019i	−1.56033	+	0.418089i	−0.932767	−	0.360479i	$-0.882613\pi$
$619$	−38.8205	+	10.4019i	−1.56033	+	0.418089i	−0.627561	+	0.778568i	$0.715947\pi$
$620$		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	$0.276859\pi$
$631$			38.3923i			1.52837i	−0.644995	+	0.764187i	$0.723141\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−6.19615	+	23.1244i	−0.245887	+	0.917662i
$636$		0			0
$637$	−10.5622	−	39.4186i	−0.418489	−	1.56182i
$638$		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
$643$	45.6506	+	12.2321i	1.80029	+	0.482385i	0.994023	−	0.109173i	$-0.0348202\pi$
$643$	45.6506	+	12.2321i	1.80029	+	0.482385i	0.806263	+	0.591558i	$0.201487\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	13.2679i		−	0.521617i	−0.965391	−	0.260808i	$-0.916011\pi$
$647$		−	13.2679i		−	0.521617i	0.965391	−	0.260808i	$-0.0839891\pi$
$648$		0			0
$649$			15.2487i			0.598564i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−5.63397	−	1.50962i	−0.220474	−	0.0590760i	0.146891	−	0.989153i	$-0.453073\pi$
$653$	−5.63397	−	1.50962i	−0.220474	−	0.0590760i	−0.367365	+	0.930077i	$0.619740\pi$
$654$		0			0
$655$	6.19615	−	10.7321i	0.242104	−	0.419336i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	4.02628	+	15.0263i	0.156842	+	0.585341i	0.998941	+	0.0460178i	$0.0146531\pi$
$659$	4.02628	+	15.0263i	0.156842	+	0.585341i	−0.842099	+	0.539323i	$0.818680\pi$
$660$		0			0
$661$	2.19615	−	8.19615i	0.0854204	−	0.318793i	−0.909973	−	0.414667i	$-0.863898\pi$
$661$	2.19615	−	8.19615i	0.0854204	−	0.318793i	0.995393	+	0.0958740i	$0.0305646\pi$
$662$		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
$677$	−4.73205	+	1.26795i	−0.181867	+	0.0487312i	−0.348603	−	0.937270i	$-0.613344\pi$
$677$	−4.73205	+	1.26795i	−0.181867	+	0.0487312i	0.166736	+	0.986002i	$0.446677\pi$
$678$		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	0.611321	−	0.791383i	$-0.290638\pi$
$683$	−4.70577	−	4.70577i	−0.180061	−	0.180061i	−0.791383	+	0.611321i	$0.790638\pi$
$684$		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
$691$	6.29423	+	23.4904i	0.239444	+	0.893616i	0.976095	+	0.217344i	$0.0697392\pi$
$691$	6.29423	+	23.4904i	0.239444	+	0.893616i	−0.736651	+	0.676273i	$0.763594\pi$
$692$		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	−0.879160	−	0.476527i	$-0.841895\pi$
$701$	−10.6603	+	10.6603i	−0.402632	+	0.402632i	0.476527	+	0.879160i	$0.341895\pi$
$702$		0			0
$703$	1.60770			0.0606354
$704$		0			0
$705$		0			0
$706$		0			0
$707$	5.46410	+	1.46410i	0.205499	+	0.0550632i
$708$		0			0
$709$	−20.1962	+	5.41154i	−0.758482	+	0.203235i	−0.617277	−	0.786746i	$-0.711764\pi$
$709$	−20.1962	+	5.41154i	−0.758482	+	0.203235i	−0.141205	+	0.989980i	$0.545098\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	8.19615	+	4.73205i	0.306948	+	0.177217i
$714$		0			0
$715$	−18.7321	+	69.9090i	−0.700539	+	2.61445i
$716$		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
$725$	−6.29423	+	23.4904i	−0.233762	+	0.872411i
$726$		0			0
$727$	−31.8109	−	18.3660i	−1.17980	−	0.681158i	−0.223832	−	0.974628i	$-0.571857\pi$
$727$	−31.8109	−	18.3660i	−1.17980	−	0.681158i	−0.955968	+	0.293470i	$0.905190\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	2.79423	−	0.748711i	0.103348	−	0.0276921i
$732$		0			0
$733$	−29.9545	−	8.02628i	−1.10639	−	0.296457i	−0.341028	−	0.940053i	$-0.610775\pi$
$733$	−29.9545	−	8.02628i	−1.10639	−	0.296457i	−0.765366	+	0.643596i	$0.777442\pi$
$734$		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	−0.979945	−	0.199266i	$-0.936144\pi$
$739$	−21.2224	+	21.2224i	−0.780680	+	0.780680i	0.199266	+	0.979945i	$0.436144\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	2.24167	−	1.29423i	0.0822389	−	0.0474806i	−0.458317	−	0.888789i	$-0.651547\pi$
$743$	2.24167	−	1.29423i	0.0822389	−	0.0474806i	0.540556	+	0.841308i	$0.318214\pi$
$744$		0			0
$745$	10.7321	+	6.19615i	0.393192	+	0.227009i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	3.61474	+	13.4904i	0.132080	+	0.492928i
$750$		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	0.588103	−	0.808786i	$-0.299875\pi$
$757$	−6.07180	−	6.07180i	−0.220683	−	0.220683i	−0.808786	+	0.588103i	$0.799875\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	27.3731	−	15.8038i	0.992273	−	0.572889i	0.0863200	−	0.996267i	$-0.472489\pi$
$761$	27.3731	−	15.8038i	0.992273	−	0.572889i	0.905953	+	0.423378i	$0.139156\pi$
$762$		0			0
$763$	−7.26795	+	1.94744i	−0.263117	+	0.0705021i
$764$		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	0.623306	−	0.781978i	$-0.285789\pi$
$773$	−4.41154	−	4.41154i	−0.158672	−	0.158672i	−0.781978	+	0.623306i	$0.785789\pi$
$774$		0			0
$775$		−	74.1051i		−	2.66193i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−0.696152	+	2.59808i	−0.0249422	+	0.0930857i
$780$		0			0
$781$	8.39230	+	31.3205i	0.300300	+	1.12074i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−9.46410	+	16.3923i	−0.337788	+	0.585066i
$786$		0			0
$787$	−49.8109	−	13.3468i	−1.77557	−	0.475762i	−0.785803	−	0.618477i	$-0.787750\pi$
$787$	−49.8109	−	13.3468i	−1.77557	−	0.475762i	−0.989764	+	0.142716i	$0.954417\pi$
$788$		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
$797$	54.1769	+	14.5167i	1.91904	+	0.514206i	0.989275	+	0.146065i	$0.0466609\pi$
$797$	54.1769	+	14.5167i	1.91904	+	0.514206i	0.929770	+	0.368142i	$0.120006\pi$
$798$		0			0
$799$	−10.9545	+	18.9737i	−0.387542	+	0.671242i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	7.47372	+	27.8923i	0.263742	+	0.984298i
$804$		0			0
$805$	0.928203	−	3.46410i	0.0327149	−	0.122094i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			28.3205i			0.995696i	0.867264	+	0.497848i	$0.165876\pi$
$809$			28.3205i			0.995696i	−0.867264	+	0.497848i	$0.834124\pi$
$810$		0			0
$811$	−5.02628	−	5.02628i	−0.176497	−	0.176497i	0.613330	−	0.789827i	$-0.289830\pi$
$811$	−5.02628	−	5.02628i	−0.176497	−	0.176497i	−0.789827	+	0.613330i	$0.789830\pi$
$812$		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
$821$	−32.2224	+	8.63397i	−1.12457	+	0.301328i	−0.772732	−	0.634733i	$-0.781110\pi$
$821$	−32.2224	+	8.63397i	−1.12457	+	0.301328i	−0.351839	+	0.936061i	$0.614444\pi$
$822$		0			0
$823$	−10.7321	+	6.19615i	−0.374096	+	0.215984i	−0.675246	−	0.737592i	$-0.735963\pi$
$823$	−10.7321	+	6.19615i	−0.374096	+	0.215984i	0.301151	+	0.953577i	$0.402629\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	24.4641	+	24.4641i	0.850700	+	0.850700i	0.990219	−	0.139519i	$-0.0445557\pi$
$827$	24.4641	+	24.4641i	0.850700	+	0.850700i	−0.139519	+	0.990219i	$0.544556\pi$
$828$		0			0
$829$	24.5167	−	24.5167i	0.851499	−	0.851499i	−0.138819	−	0.990318i	$-0.544331\pi$
$829$	24.5167	−	24.5167i	0.851499	−	0.851499i	0.990318	+	0.138819i	$0.0443306\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−7.33013	−	12.6962i	−0.253974	−	0.439896i
$834$		0			0
$835$	−0.535898	−	2.00000i	−0.0185455	−	0.0692129i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−35.4449	−	20.4641i	−1.22369	−	0.706499i	−0.257989	−	0.966148i	$-0.583060\pi$
$839$	−35.4449	−	20.4641i	−1.22369	−	0.706499i	−0.965703	+	0.259649i	$0.916393\pi$
$840$		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
$851$	2.19615	+	0.588457i	0.0752831	+	0.0201721i
$852$		0			0
$853$	−12.5622	+	3.36603i	−0.430121	+	0.115251i	−0.467383	−	0.884055i	$-0.654803\pi$
$853$	−12.5622	+	3.36603i	−0.430121	+	0.115251i	0.0372621	+	0.999306i	$0.488136\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−20.9090	−	12.0718i	−0.714237	−	0.412365i	0.0983911	−	0.995148i	$-0.468630\pi$
$857$	−20.9090	−	12.0718i	−0.714237	−	0.412365i	−0.812628	+	0.582783i	$0.801964\pi$
$858$		0			0
$859$	8.25833	−	30.8205i	0.281771	−	1.05158i	−0.669396	−	0.742905i	$-0.733447\pi$
$859$	8.25833	−	30.8205i	0.281771	−	1.05158i	0.951167	−	0.308677i	$-0.0998861\pi$
$860$		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
$869$	9.21539	−	34.3923i	0.312611	−	1.16668i
$870$		0			0
$871$	29.6147	+	17.0981i	1.00346	+	0.579346i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−13.4641	+	3.60770i	−0.455170	+	0.121962i
$876$		0			0
$877$	−1.53590	−	0.411543i	−0.0518636	−	0.0138968i	0.232794	−	0.972526i	$-0.425213\pi$
$877$	−1.53590	−	0.411543i	−0.0518636	−	0.0138968i	−0.284658	+	0.958629i	$0.591880\pi$
$878$		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	−0.422778	−	0.906233i	$-0.638945\pi$
$883$	14.3660	−	14.3660i	0.483455	−	0.483455i	0.906233	+	0.422778i	$0.138945\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	33.1244	−	19.1244i	1.11221	−	0.642133i	0.172807	−	0.984956i	$-0.444716\pi$
$887$	33.1244	−	19.1244i	1.11221	−	0.642133i	0.939400	+	0.342823i	$0.111383\pi$
$888$		0			0
$889$	−3.92820	−	2.26795i	−0.131748	−	0.0760646i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	2.24167	+	8.36603i	0.0750146	+	0.279958i
$894$		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
$907$	16.7942	−	4.50000i	0.557643	−	0.149420i	0.0310198	−	0.999519i	$-0.490124\pi$
$907$	16.7942	−	4.50000i	0.557643	−	0.149420i	0.526623	+	0.850099i	$0.323458\pi$
$908$		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	$-0.816921\pi$
$919$		−	32.9808i		−	1.08793i	0.839106	−	0.543967i	$-0.183079\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−17.8564	+	66.6410i	−0.587751	+	2.19352i
$924$		0			0
$925$	−4.60770	−	17.1962i	−0.151500	−	0.565406i
$926$		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
$931$	−5.59808	−	1.50000i	−0.183470	−	0.0491605i
$932$		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	$-0.684924\pi$
$937$		−	51.1769i		−	1.67188i	0.548823	−	0.835938i	$-0.315076\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	12.1962	+	3.26795i	0.397583	+	0.106532i	0.452070	−	0.891982i	$-0.350686\pi$
$941$	12.1962	+	3.26795i	0.397583	+	0.106532i	−0.0544870	+	0.998514i	$0.517352\pi$
$942$		0			0
$943$	−1.90192	+	3.29423i	−0.0619352	+	0.107275i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−4.01666	−	14.9904i	−0.130524	−	0.487122i	0.869452	−	0.494017i	$-0.164472\pi$
$947$	−4.01666	−	14.9904i	−0.130524	−	0.487122i	−0.999976	+	0.00689497i	$0.997805\pi$
$948$		0			0
$949$	−15.9019	+	59.3468i	−0.516198	+	1.92648i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	59.1051i		−	1.91460i	−0.289092	−	0.957301i	$-0.593353\pi$
$953$		−	59.1051i		−	1.91460i	0.289092	−	0.957301i	$-0.406647\pi$
$954$		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
$965$	68.1769	−	18.2679i	2.19469	−	0.588066i
$966$		0			0
$967$	−9.16987	+	5.29423i	−0.294883	+	0.170251i	−0.640142	−	0.768257i	$-0.721124\pi$
$967$	−9.16987	+	5.29423i	−0.294883	+	0.170251i	0.345259	+	0.938508i	$0.387791\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	22.4641	+	22.4641i	0.720907	+	0.720907i	0.968790	−	0.247883i	$-0.0797348\pi$
$971$	22.4641	+	22.4641i	0.720907	+	0.720907i	−0.247883	+	0.968790i	$0.579735\pi$
$972$		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
$979$	−1.53590	−	5.73205i	−0.0490875	−	0.183197i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−13.8564	−	8.00000i	−0.441951	−	0.255160i	0.262474	−	0.964939i	$-0.415462\pi$
$983$	−13.8564	−	8.00000i	−0.441951	−	0.255160i	−0.704425	+	0.709779i	$0.748795\pi$
$984$		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
$995$	−3.26795	−	0.875644i	−0.103601	−	0.0277598i
$996$		0			0
$997$	32.1244	−	8.60770i	1.01739	−	0.272608i	0.288675	−	0.957427i	$-0.406785\pi$
$997$	32.1244	−	8.60770i	1.01739	−	0.272608i	0.728713	+	0.684819i	$0.240119\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1728.2.bc.a.1585.1		4
3.2	odd	2		576.2.bb.c.241.1		4
4.3	odd	2		432.2.y.b.181.1		4
9.4	even	3		1728.2.bc.d.1009.1		4
9.5	odd	6		576.2.bb.d.49.1		4
12.11	even	2		144.2.x.c.133.1	yes	4
16.3	odd	4		432.2.y.c.397.1		4
16.13	even	4		1728.2.bc.d.721.1		4
36.23	even	6		144.2.x.b.85.1	yes	4
36.31	odd	6		432.2.y.c.37.1		4
48.29	odd	4		576.2.bb.d.529.1		4
48.35	even	4		144.2.x.b.61.1	✓	4
144.13	even	12	inner	1728.2.bc.a.145.1		4
144.67	odd	12		432.2.y.b.253.1		4
144.77	odd	12		576.2.bb.c.337.1		4
144.131	even	12		144.2.x.c.13.1	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
144.2.x.b.61.1	✓	4	48.35	even	4
144.2.x.b.85.1	yes	4	36.23	even	6
144.2.x.c.13.1	yes	4	144.131	even	12
144.2.x.c.133.1	yes	4	12.11	even	2
432.2.y.b.181.1		4	4.3	odd	2
432.2.y.b.253.1		4	144.67	odd	12
432.2.y.c.37.1		4	36.31	odd	6
432.2.y.c.397.1		4	16.3	odd	4
576.2.bb.c.241.1		4	3.2	odd	2
576.2.bb.c.337.1		4	144.77	odd	12
576.2.bb.d.49.1		4	9.5	odd	6
576.2.bb.d.529.1		4	48.29	odd	4
1728.2.bc.a.145.1		4	144.13	even	12	inner
1728.2.bc.a.1585.1		4	1.1	even	1	trivial
1728.2.bc.d.721.1		4	16.13	even	4
1728.2.bc.d.1009.1		4	9.4	even	3

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)

\(f(q)\)	\(=\)	\(q+(-1.00000 + 3.73205i) q^{5} +(-0.633975 - 0.366025i) q^{7} +O(q^{10})\)	\(q+(-1.00000 + 3.73205i) q^{5} +(-0.633975 - 0.366025i) q^{7} +(2.86603 - 0.767949i) q^{11} +(6.09808 + 1.63397i) q^{13} +2.26795 q^{17} +(0.633975 - 0.633975i) q^{19} +(1.09808 - 0.633975i) q^{23} +(-8.59808 - 4.96410i) q^{25} +(-0.633975 - 2.36603i) q^{29} +(3.73205 + 6.46410i) q^{31} +(2.00000 - 2.00000i) q^{35} +(1.26795 + 1.26795i) q^{37} +(-2.59808 + 1.50000i) q^{41} +(1.23205 - 0.330127i) q^{43} +(-4.83013 + 8.36603i) q^{47} +(-3.23205 - 5.59808i) q^{49} +(0.535898 + 0.535898i) q^{53} +11.4641i q^{55} +(-1.33013 + 4.96410i) q^{59} +(0.803848 + 3.00000i) q^{61} +(-12.1962 + 21.1244i) q^{65} +(5.23205 + 1.40192i) q^{67} +10.9282i q^{71} +9.73205i q^{73} +(-2.09808 - 0.562178i) q^{77} +(6.00000 - 10.3923i) q^{79} +(0.366025 + 1.36603i) q^{83} +(-2.26795 + 8.46410i) q^{85} -2.00000i q^{89} +(-3.26795 - 3.26795i) q^{91} +(1.73205 + 3.00000i) q^{95} +(-4.13397 + 7.16025i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{5} - 6 q^{7}+O(q^{10}) \) 4 * q - 4 * q^5 - 6 * q^7	\( 4 q - 4 q^{5} - 6 q^{7} + 8 q^{11} + 14 q^{13} + 16 q^{17} + 6 q^{19} - 6 q^{23} - 24 q^{25} - 6 q^{29} + 8 q^{31} + 8 q^{35} + 12 q^{37} - 2 q^{43} - 2 q^{47} - 6 q^{49} + 16 q^{53} + 12 q^{59} + 24 q^{61} - 28 q^{65} + 14 q^{67} + 2 q^{77} + 24 q^{79} - 2 q^{83} - 16 q^{85} - 20 q^{91} - 20 q^{97}+O(q^{100}) \) 4 * q - 4 * q^5 - 6 * q^7 + 8 * q^11 + 14 * 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 - 2 * q^43 - 2 * q^47 - 6 * q^49 + 16 * q^53 + 12 * q^59 + 24 * q^61 - 28 * q^65 + 14 * q^67 + 2 * q^77 + 24 * q^79 - 2 * q^83 - 16 * q^85 - 20 * q^91 - 20 * q^97

Introduction

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