LMFDB - Embedded newform 1728.2.bc.a.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.a.1009.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.00000 - 0.267949i) q^{5} +(-2.36603 - 1.36603i) q^{7} +O(q^{10})$	$q+(-1.00000 - 0.267949i) q^{5} +(-2.36603 - 1.36603i) q^{7} +(1.13397 + 4.23205i) q^{11} +(0.901924 - 3.36603i) q^{13} +5.73205 q^{17} +(2.36603 + 2.36603i) q^{19} +(-4.09808 + 2.36603i) q^{23} +(-3.40192 - 1.96410i) q^{25} +(-2.36603 + 0.633975i) q^{29} +(0.267949 + 0.464102i) q^{31} +(2.00000 + 2.00000i) q^{35} +(4.73205 - 4.73205i) q^{37} +(2.59808 - 1.50000i) q^{41} +(-2.23205 - 8.33013i) q^{43} +(3.83013 - 6.63397i) q^{47} +(0.232051 + 0.401924i) q^{49} +(7.46410 - 7.46410i) q^{53} -4.53590i q^{55} +(7.33013 + 1.96410i) q^{59} +(11.1962 - 3.00000i) q^{61} +(-1.80385 + 3.12436i) q^{65} +(1.76795 - 6.59808i) q^{67} +2.92820i q^{71} -6.26795i q^{73} +(3.09808 - 11.5622i) q^{77} +(6.00000 - 10.3923i) q^{79} +(-1.36603 + 0.366025i) q^{83} +(-5.73205 - 1.53590i) q^{85} +2.00000i q^{89} +(-6.73205 + 6.73205i) q^{91} +(-1.73205 - 3.00000i) q^{95} +(-5.86603 + 10.1603i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times$.

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

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
$5$	−1.00000	−	0.267949i	−0.447214	−	0.119831i	0.0281817	−	0.999603i	$-0.491028\pi$
$5$	−1.00000	−	0.267949i	−0.447214	−	0.119831i	−0.475395	+	0.879772i	$0.657695\pi$
$6$		0			0
$7$	−2.36603	−	1.36603i	−0.894274	−	0.516309i	−0.0189356	−	0.999821i	$-0.506028\pi$
$7$	−2.36603	−	1.36603i	−0.894274	−	0.516309i	−0.875338	+	0.483512i	$0.839361\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	1.13397	+	4.23205i	0.341906	+	1.27601i	0.896185	+	0.443680i	$0.146327\pi$
$11$	1.13397	+	4.23205i	0.341906	+	1.27601i	−0.554279	+	0.832331i	$0.687006\pi$
$12$		0			0
$13$	0.901924	−	3.36603i	0.250149	−	0.933567i	−0.720577	−	0.693375i	$-0.756123\pi$
$13$	0.901924	−	3.36603i	0.250149	−	0.933567i	0.970725	−	0.240192i	$-0.0772105\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	5.73205			1.39023			0.695113	−	0.718900i	$-0.255354\pi$
$17$	5.73205			1.39023			0.695113	+	0.718900i	$0.255354\pi$
$18$		0			0
$19$	2.36603	+	2.36603i	0.542803	+	0.542803i	0.924350	−	0.381546i	$-0.124608\pi$
$19$	2.36603	+	2.36603i	0.542803	+	0.542803i	−0.381546	+	0.924350i	$0.624608\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−4.09808	+	2.36603i	−0.854508	+	0.493350i	−0.862169	−	0.506620i	$-0.830895\pi$
$23$	−4.09808	+	2.36603i	−0.854508	+	0.493350i	0.00766135	+	0.999971i	$0.497561\pi$
$24$		0			0
$25$	−3.40192	−	1.96410i	−0.680385	−	0.392820i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−2.36603	+	0.633975i	−0.439360	+	0.117726i	−0.471717	−	0.881750i	$-0.656365\pi$
$29$	−2.36603	+	0.633975i	−0.439360	+	0.117726i	0.0323566	+	0.999476i	$0.489699\pi$
$30$		0			0
$31$	0.267949	+	0.464102i	0.0481251	+	0.0833551i	0.889085	−	0.457743i	$-0.151342\pi$
$31$	0.267949	+	0.464102i	0.0481251	+	0.0833551i	−0.840959	+	0.541098i	$0.818009\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	2.00000	+	2.00000i	0.338062	+	0.338062i
$36$		0			0
$37$	4.73205	−	4.73205i	0.777944	−	0.777944i	−0.201537	−	0.979481i	$-0.564594\pi$
$37$	4.73205	−	4.73205i	0.777944	−	0.777944i	0.979481	+	0.201537i	$0.0645935\pi$
$38$		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
$43$	−2.23205	−	8.33013i	−0.340385	−	1.27033i	−0.897912	−	0.440174i	$-0.854917\pi$
$43$	−2.23205	−	8.33013i	−0.340385	−	1.27033i	0.557528	−	0.830158i	$-0.311750\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	3.83013	−	6.63397i	0.558681	−	0.967665i	−0.438925	−	0.898523i	$-0.644641\pi$
$47$	3.83013	−	6.63397i	0.558681	−	0.967665i	0.997607	−	0.0691412i	$-0.0220259\pi$
$48$		0			0
$49$	0.232051	+	0.401924i	0.0331501	+	0.0574177i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	7.46410	−	7.46410i	1.02527	−	1.02527i	0.0256010	−	0.999672i	$-0.491850\pi$
$53$	7.46410	−	7.46410i	1.02527	−	1.02527i	0.999672	−	0.0256010i	$-0.00814993\pi$
$54$		0			0
$55$		−	4.53590i		−	0.611620i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	7.33013	+	1.96410i	0.954301	+	0.255704i	0.702186	−	0.711993i	$-0.252207\pi$
$59$	7.33013	+	1.96410i	0.954301	+	0.255704i	0.252115	+	0.967697i	$0.418874\pi$
$60$		0			0
$61$	11.1962	−	3.00000i	1.43352	−	0.384111i	0.543261	−	0.839564i	$-0.317189\pi$
$61$	11.1962	−	3.00000i	1.43352	−	0.384111i	0.890260	+	0.455453i	$0.150523\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−1.80385	+	3.12436i	−0.223740	+	0.387529i
$66$		0			0
$67$	1.76795	−	6.59808i	0.215989	−	0.806083i	−0.769827	−	0.638253i	$-0.779657\pi$
$67$	1.76795	−	6.59808i	0.215989	−	0.806083i	0.985816	−	0.167830i	$-0.0536760\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			2.92820i			0.347514i	0.984789	+	0.173757i	$0.0555907\pi$
$71$			2.92820i			0.347514i	−0.984789	+	0.173757i	$0.944409\pi$
$72$		0			0
$73$		−	6.26795i		−	0.733608i	−0.930298	−	0.366804i	$-0.880452\pi$
$73$		−	6.26795i		−	0.733608i	0.930298	−	0.366804i	$-0.119548\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	3.09808	−	11.5622i	0.353059	−	1.31763i
$78$		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
$85$	−5.73205	−	1.53590i	−0.621728	−	0.166592i
$86$		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
$91$	−6.73205	+	6.73205i	−0.705711	+	0.705711i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−1.73205	−	3.00000i	−0.177705	−	0.307794i
$96$		0			0
$97$	−5.86603	+	10.1603i	−0.595605	+	1.03162i	0.397857	+	0.917448i	$0.369754\pi$
$97$	−5.86603	+	10.1603i	−0.595605	+	1.03162i	−0.993461	+	0.114170i	$0.963579\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−0.535898	−	2.00000i	−0.0533239	−	0.199007i	0.934125	−	0.356946i	$-0.116182\pi$
$101$	−0.535898	−	2.00000i	−0.0533239	−	0.199007i	−0.987449	+	0.157938i	$0.949515\pi$
$102$		0			0
$103$	−13.0981	+	7.56218i	−1.29059	+	0.745124i	−0.978759	−	0.205014i	$-0.934276\pi$
$103$	−13.0981	+	7.56218i	−1.29059	+	0.745124i	−0.311833	+	0.950137i	$0.600943\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	12.4904	−	12.4904i	1.20749	−	1.20749i	0.235654	−	0.971837i	$-0.424277\pi$
$107$	12.4904	−	12.4904i	1.20749	−	1.20749i	0.971837	−	0.235654i	$-0.0757231\pi$
$108$		0			0
$109$	10.7321	+	10.7321i	1.02794	+	1.02794i	0.999598	+	0.0283459i	$0.00902398\pi$
$109$	10.7321	+	10.7321i	1.02794	+	1.02794i	0.0283459	+	0.999598i	$0.490976\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	6.92820	+	12.0000i	0.651751	+	1.12887i	0.982698	+	0.185216i	$0.0592984\pi$
$113$	6.92820	+	12.0000i	0.651751	+	1.12887i	−0.330947	+	0.943649i	$0.607368\pi$
$114$		0			0
$115$	4.73205	−	1.26795i	0.441266	−	0.118237i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−13.5622	−	7.83013i	−1.24324	−	0.717787i
$120$		0			0
$121$	−7.09808	+	4.09808i	−0.645280	+	0.372552i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	6.53590	+	6.53590i	0.584589	+	0.584589i
$126$		0			0
$127$	−4.19615			−0.372348			−0.186174	−	0.982517i	$-0.559609\pi$
$127$	−4.19615			−0.372348			−0.186174	+	0.982517i	$0.559609\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	2.09808	−	7.83013i	0.183310	−	0.684121i	−0.811676	−	0.584108i	$-0.801445\pi$
$131$	2.09808	−	7.83013i	0.183310	−	0.684121i	0.994986	−	0.100014i	$-0.0318887\pi$
$132$		0			0
$133$	−2.36603	−	8.83013i	−0.205160	−	0.765669i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−8.25833	−	4.76795i	−0.705557	−	0.407353i	0.103857	−	0.994592i	$-0.466882\pi$
$137$	−8.25833	−	4.76795i	−0.705557	−	0.407353i	−0.809414	+	0.587239i	$0.800215\pi$
$138$		0			0
$139$	11.4282	+	3.06218i	0.969328	+	0.259731i	0.708544	−	0.705667i	$-0.249352\pi$
$139$	11.4282	+	3.06218i	0.969328	+	0.259731i	0.260784	+	0.965397i	$0.416019\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	15.2679			1.27677
$144$		0			0
$145$	2.53590			0.210595
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−7.83013	−	2.09808i	−0.641469	−	0.171881i	−0.0766003	−	0.997062i	$-0.524407\pi$
$149$	−7.83013	−	2.09808i	−0.641469	−	0.171881i	−0.564869	+	0.825181i	$0.691073\pi$
$150$		0			0
$151$	−0.633975	−	0.366025i	−0.0515921	−	0.0297867i	0.473982	−	0.880534i	$-0.342816\pi$
$151$	−0.633975	−	0.366025i	−0.0515921	−	0.0297867i	−0.525574	+	0.850748i	$0.676149\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−0.143594	−	0.535898i	−0.0115337	−	0.0430444i
$156$		0			0
$157$	1.26795	−	4.73205i	0.101193	−	0.377659i	−0.896692	−	0.442655i	$-0.854037\pi$
$157$	1.26795	−	4.73205i	0.101193	−	0.377659i	0.997886	+	0.0649959i	$0.0207034\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	12.9282			1.01889
$162$		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
$167$	6.46410	−	3.73205i	0.500207	−	0.288795i	−0.228592	−	0.973522i	$-0.573412\pi$
$167$	6.46410	−	3.73205i	0.500207	−	0.288795i	0.728799	+	0.684728i	$0.240079\pi$
$168$		0			0
$169$	0.741670	+	0.428203i	0.0570515	+	0.0329387i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−1.63397	+	0.437822i	−0.124229	+	0.0332870i	−0.320398	−	0.947283i	$-0.603817\pi$
$173$	−1.63397	+	0.437822i	−0.124229	+	0.0332870i	0.196169	+	0.980570i	$0.437150\pi$
$174$		0			0
$175$	5.36603	+	9.29423i	0.405633	+	0.702578i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−1.92820	−	1.92820i	−0.144121	−	0.144121i	0.631365	−	0.775486i	$-0.282495\pi$
$179$	−1.92820	−	1.92820i	−0.144121	−	0.144121i	−0.775486	+	0.631365i	$0.782495\pi$
$180$		0			0
$181$	−7.39230	+	7.39230i	−0.549466	+	0.549466i	−0.926286	−	0.376821i	$-0.877017\pi$
$181$	−7.39230	+	7.39230i	−0.549466	+	0.549466i	0.376821	+	0.926286i	$0.377017\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−6.00000	+	3.46410i	−0.441129	+	0.254686i
$186$		0			0
$187$	6.50000	+	24.2583i	0.475327	+	1.77394i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−12.0263	+	20.8301i	−0.870191	+	1.50722i	−0.00839227	+	0.999965i	$0.502671\pi$
$191$	−12.0263	+	20.8301i	−0.870191	+	1.50722i	−0.861799	+	0.507250i	$0.830662\pi$
$192$		0			0
$193$	−10.8660	−	18.8205i	−0.782154	−	1.35473i	−0.930685	−	0.365822i	$-0.880788\pi$
$193$	−10.8660	−	18.8205i	−0.782154	−	1.35473i	0.148531	−	0.988908i	$-0.452545\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−13.6603	+	13.6603i	−0.973253	+	0.973253i	−0.999651	−	0.0263987i	$-0.991596\pi$
$197$	−13.6603	+	13.6603i	−0.973253	+	0.973253i	0.0263987	+	0.999651i	$0.491596\pi$
$198$		0			0
$199$		−	25.1244i		−	1.78102i	−0.454965	−	0.890509i	$-0.650348\pi$
$199$		−	25.1244i		−	1.78102i	0.454965	−	0.890509i	$-0.349652\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	6.46410	+	1.73205i	0.453691	+	0.121566i
$204$		0			0
$205$	−3.00000	+	0.803848i	−0.209529	+	0.0561432i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−7.33013	+	12.6962i	−0.507035	+	0.878211i
$210$		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
$215$			8.92820i			0.608898i
$216$		0			0
$217$		−	1.46410i		−	0.0993897i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	5.16987	−	19.2942i	0.347763	−	1.29787i
$222$		0			0
$223$	8.02628	−	13.9019i	0.537479	−	0.930942i	−0.461559	−	0.887109i	$-0.652710\pi$
$223$	8.02628	−	13.9019i	0.537479	−	0.930942i	0.999039	−	0.0438324i	$-0.0139568\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	2.13397	−	0.571797i	0.141637	−	0.0379515i	−0.187304	−	0.982302i	$-0.559975\pi$
$227$	2.13397	−	0.571797i	0.141637	−	0.0379515i	0.328941	+	0.944351i	$0.393308\pi$
$228$		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
$233$			3.19615i			0.209387i	0.994505	+	0.104693i	$0.0333861\pi$
$233$			3.19615i			0.209387i	−0.994505	+	0.104693i	$0.966614\pi$
$234$		0			0
$235$	−5.60770	+	5.60770i	−0.365806	+	0.365806i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−7.90192	−	13.6865i	−0.511133	−	0.885308i	−0.999917	−	0.0129033i	$-0.995893\pi$
$239$	−7.90192	−	13.6865i	−0.511133	−	0.885308i	0.488784	−	0.872405i	$-0.337441\pi$
$240$		0			0
$241$	−11.5981	+	20.0885i	−0.747098	+	1.29401i	0.202110	+	0.979363i	$0.435220\pi$
$241$	−11.5981	+	20.0885i	−0.747098	+	1.29401i	−0.949208	+	0.314649i	$0.898113\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−0.124356	−	0.464102i	−0.00794479	−	0.0296504i
$246$		0			0
$247$	10.0981	−	5.83013i	0.642525	−	0.370962i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−5.83013	+	5.83013i	−0.367994	+	0.367994i	−0.866745	−	0.498751i	$-0.833792\pi$
$251$	−5.83013	+	5.83013i	−0.367994	+	0.367994i	0.498751	+	0.866745i	$0.333792\pi$
$252$		0			0
$253$	−14.6603	−	14.6603i	−0.921682	−	0.921682i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−9.42820	−	16.3301i	−0.588115	−	1.01865i	−0.994479	−	0.104934i	$-0.966537\pi$
$257$	−9.42820	−	16.3301i	−0.588115	−	1.01865i	0.406364	−	0.913711i	$-0.366796\pi$
$258$		0			0
$259$	−17.6603	+	4.73205i	−1.09735	+	0.294035i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−2.49038	−	1.43782i	−0.153563	−	0.0886599i	0.421249	−	0.906945i	$-0.361592\pi$
$263$	−2.49038	−	1.43782i	−0.153563	−	0.0886599i	−0.574813	+	0.818285i	$0.694925\pi$
$264$		0			0
$265$	−9.46410	+	5.46410i	−0.581375	+	0.335657i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−1.26795	−	1.26795i	−0.0773082	−	0.0773082i	0.667395	−	0.744704i	$-0.267409\pi$
$269$	−1.26795	−	1.26795i	−0.0773082	−	0.0773082i	−0.744704	+	0.667395i	$0.767409\pi$
$270$		0			0
$271$	0.392305			0.0238308			0.0119154	−	0.999929i	$-0.496207\pi$
$271$	0.392305			0.0238308			0.0119154	+	0.999929i	$0.496207\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	4.45448	−	16.6244i	0.268615	−	1.00249i
$276$		0			0
$277$	−6.75833	−	25.2224i	−0.406069	−	1.51547i	−0.802076	−	0.597222i	$-0.796271\pi$
$277$	−6.75833	−	25.2224i	−0.406069	−	1.51547i	0.396007	−	0.918247i	$-0.370395\pi$
$278$		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
$283$	19.5622	+	5.24167i	1.16285	+	0.311585i	0.788104	−	0.615542i	$-0.211063\pi$
$283$	19.5622	+	5.24167i	1.16285	+	0.311585i	0.374747	+	0.927127i	$0.377730\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−8.19615			−0.483804
$288$		0			0
$289$	15.8564			0.932730
$290$		0			0
$291$		0			0
$292$		0			0
$293$	5.36603	+	1.43782i	0.313487	+	0.0839985i	0.412132	−	0.911124i	$-0.364784\pi$
$293$	5.36603	+	1.43782i	0.313487	+	0.0839985i	−0.0986454	+	0.995123i	$0.531451\pi$
$294$		0			0
$295$	−6.80385	−	3.92820i	−0.396135	−	0.228709i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	4.26795	+	15.9282i	0.246822	+	0.921152i
$300$		0			0
$301$	−6.09808	+	22.7583i	−0.351487	+	1.31177i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−12.0000			−0.687118
$306$		0			0
$307$	−3.02628	−	3.02628i	−0.172719	−	0.172719i	0.615454	−	0.788173i	$-0.288973\pi$
$307$	−3.02628	−	3.02628i	−0.172719	−	0.172719i	−0.788173	+	0.615454i	$0.788973\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−19.0981	+	11.0263i	−1.08295	+	0.625243i	−0.931691	−	0.363251i	$-0.881667\pi$
$311$	−19.0981	+	11.0263i	−1.08295	+	0.625243i	−0.151261	+	0.988494i	$0.548333\pi$
$312$		0			0
$313$	18.6506	+	10.7679i	1.05420	+	0.608640i	0.923821	−	0.382824i	$-0.125049\pi$
$313$	18.6506	+	10.7679i	1.05420	+	0.608640i	0.130375	+	0.991465i	$0.458382\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	20.5622	−	5.50962i	1.15489	−	0.309451i	0.369965	−	0.929046i	$-0.379370\pi$
$317$	20.5622	−	5.50962i	1.15489	−	0.309451i	0.784922	+	0.619595i	$0.212703\pi$
$318$		0			0
$319$	−5.36603	−	9.29423i	−0.300440	−	0.520377i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	13.5622	+	13.5622i	0.754620	+	0.754620i
$324$		0			0
$325$	−9.67949	+	9.67949i	−0.536922	+	0.536922i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−18.1244	+	10.4641i	−0.999228	+	0.576905i
$330$		0			0
$331$	−0.0262794	−	0.0980762i	−0.00144445	−	0.00539076i	0.965200	−	0.261513i	$-0.0842216\pi$
$331$	−0.0262794	−	0.0980762i	−0.00144445	−	0.00539076i	−0.966644	+	0.256123i	$0.917555\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−3.53590	+	6.12436i	−0.193187	+	0.334609i
$336$		0			0
$337$	8.89230	+	15.4019i	0.484395	+	0.838996i	0.999839	−	0.0179267i	$-0.00570654\pi$
$337$	8.89230	+	15.4019i	0.484395	+	0.838996i	−0.515445	+	0.856923i	$0.672373\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−1.66025	+	1.66025i	−0.0899078	+	0.0899078i
$342$		0			0
$343$			17.8564i			0.964155i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−17.6244	−	4.72243i	−0.946125	−	0.253513i	−0.247408	−	0.968911i	$-0.579579\pi$
$347$	−17.6244	−	4.72243i	−0.946125	−	0.253513i	−0.698717	+	0.715398i	$0.746245\pi$
$348$		0			0
$349$	15.9282	−	4.26795i	0.852617	−	0.228458i	0.194061	−	0.980989i	$-0.437834\pi$
$349$	15.9282	−	4.26795i	0.852617	−	0.228458i	0.658556	+	0.752531i	$0.271167\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−7.16025	+	12.4019i	−0.381102	+	0.660088i	−0.991220	−	0.132223i	$-0.957789\pi$
$353$	−7.16025	+	12.4019i	−0.381102	+	0.660088i	0.610118	+	0.792310i	$0.291122\pi$
$354$		0			0
$355$	0.784610	−	2.92820i	0.0416428	−	0.155413i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			11.2679i			0.594700i	0.954769	+	0.297350i	$0.0961028\pi$
$359$			11.2679i			0.594700i	−0.954769	+	0.297350i	$0.903897\pi$
$360$		0			0
$361$		−	7.80385i		−	0.410729i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−1.67949	+	6.26795i	−0.0879086	+	0.328079i
$366$		0			0
$367$	−14.1244	+	24.4641i	−0.737285	+	1.27702i	0.216428	+	0.976299i	$0.430559\pi$
$367$	−14.1244	+	24.4641i	−0.737285	+	1.27702i	−0.953713	+	0.300717i	$0.902774\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−27.8564	+	7.46410i	−1.44623	+	0.387517i
$372$		0			0
$373$	27.4904	+	7.36603i	1.42340	+	0.381398i	0.886688	−	0.462368i	$-0.153000\pi$
$373$	27.4904	+	7.36603i	1.42340	+	0.381398i	0.536710	+	0.843767i	$0.319667\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			8.53590i			0.439621i
$378$		0			0
$379$	−3.75833	+	3.75833i	−0.193052	+	0.193052i	−0.797014	−	0.603961i	$-0.793588\pi$
$379$	−3.75833	+	3.75833i	−0.193052	+	0.193052i	0.603961	+	0.797014i	$0.293588\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−6.73205	−	11.6603i	−0.343992	−	0.595811i	0.641178	−	0.767392i	$-0.278446\pi$
$383$	−6.73205	−	11.6603i	−0.343992	−	0.595811i	−0.985170	+	0.171581i	$0.945113\pi$
$384$		0			0
$385$	−6.19615	+	10.7321i	−0.315785	+	0.546956i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	5.29423	+	19.7583i	0.268428	+	1.00179i	0.960119	+	0.279593i	$0.0901996\pi$
$389$	5.29423	+	19.7583i	0.268428	+	1.00179i	−0.691691	+	0.722194i	$0.743134\pi$
$390$		0			0
$391$	−23.4904	+	13.5622i	−1.18796	+	0.685869i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−8.78461	+	8.78461i	−0.442002	+	0.442002i
$396$		0			0
$397$	−9.26795	−	9.26795i	−0.465145	−	0.465145i	0.435192	−	0.900337i	$-0.356680\pi$
$397$	−9.26795	−	9.26795i	−0.465145	−	0.465145i	−0.900337	+	0.435192i	$0.856680\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	1.79423	+	3.10770i	0.0895995	+	0.155191i	0.907342	−	0.420393i	$-0.138108\pi$
$401$	1.79423	+	3.10770i	0.0895995	+	0.155191i	−0.817742	+	0.575584i	$0.804775\pi$
$402$		0			0
$403$	1.80385	−	0.483340i	0.0898560	−	0.0240769i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	25.3923	+	14.6603i	1.25865	+	0.726682i
$408$		0			0
$409$	27.8660	−	16.0885i	1.37789	−	0.795523i	0.385981	−	0.922507i	$-0.373863\pi$
$409$	27.8660	−	16.0885i	1.37789	−	0.795523i	0.991905	+	0.126984i	$0.0405295\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−14.6603	−	14.6603i	−0.721384	−	0.721384i
$414$		0			0
$415$	1.46410			0.0718699
$416$		0			0
$417$		0			0
$418$		0			0
$419$	1.77757	−	6.63397i	0.0868399	−	0.324091i	−0.908816	−	0.417196i	$-0.863013\pi$
$419$	1.77757	−	6.63397i	0.0868399	−	0.324091i	0.995656	+	0.0931055i	$0.0296794\pi$
$420$		0			0
$421$	8.19615	+	30.5885i	0.399456	+	1.49079i	0.814056	+	0.580786i	$0.197255\pi$
$421$	8.19615	+	30.5885i	0.399456	+	1.49079i	−0.414600	+	0.910004i	$0.636078\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−19.5000	−	11.2583i	−0.945889	−	0.546109i
$426$		0			0
$427$	−30.5885	−	8.19615i	−1.48028	−	0.396640i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−16.1962			−0.780141			−0.390071	−	0.920785i	$-0.627549\pi$
$431$	−16.1962			−0.780141			−0.390071	+	0.920785i	$0.627549\pi$
$432$		0			0
$433$	−5.73205			−0.275465			−0.137732	−	0.990469i	$-0.543981\pi$
$433$	−5.73205			−0.275465			−0.137732	+	0.990469i	$0.543981\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−15.2942	−	4.09808i	−0.731622	−	0.196038i
$438$		0			0
$439$	22.8564	+	13.1962i	1.09088	+	0.629818i	0.933810	−	0.357770i	$-0.116463\pi$
$439$	22.8564	+	13.1962i	1.09088	+	0.629818i	0.157067	+	0.987588i	$0.449796\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−4.62436	−	17.2583i	−0.219710	−	0.819968i	−0.984455	−	0.175636i	$-0.943802\pi$
$443$	−4.62436	−	17.2583i	−0.219710	−	0.819968i	0.764745	−	0.644332i	$-0.222865\pi$
$444$		0			0
$445$	0.535898	−	2.00000i	0.0254040	−	0.0948091i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	3.33975			0.157612			0.0788062	−	0.996890i	$-0.474889\pi$
$449$	3.33975			0.157612			0.0788062	+	0.996890i	$0.474889\pi$
$450$		0			0
$451$	9.29423	+	9.29423i	0.437648	+	0.437648i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	8.53590	−	4.92820i	0.400169	−	0.231038i
$456$		0			0
$457$	2.25833	+	1.30385i	0.105640	+	0.0609914i	0.551889	−	0.833917i	$-0.313907\pi$
$457$	2.25833	+	1.30385i	0.105640	+	0.0609914i	−0.446249	+	0.894909i	$0.647240\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−35.6865	+	9.56218i	−1.66209	+	0.445355i	−0.962961	−	0.269642i	$-0.913094\pi$
$461$	−35.6865	+	9.56218i	−1.66209	+	0.445355i	−0.699127	+	0.714997i	$0.746428\pi$
$462$		0			0
$463$	−1.19615	−	2.07180i	−0.0555899	−	0.0962846i	0.836891	−	0.547369i	$-0.184371\pi$
$463$	−1.19615	−	2.07180i	−0.0555899	−	0.0962846i	−0.892481	+	0.451085i	$0.851037\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	2.63397	+	2.63397i	0.121886	+	0.121886i	0.765419	−	0.643533i	$-0.222532\pi$
$467$	2.63397	+	2.63397i	0.121886	+	0.121886i	−0.643533	+	0.765419i	$0.722532\pi$
$468$		0			0
$469$	−13.1962	+	13.1962i	−0.609342	+	0.609342i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	32.7224	−	18.8923i	1.50458	−	0.868669i
$474$		0			0
$475$	−3.40192	−	12.6962i	−0.156091	−	0.582539i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	4.16987	−	7.22243i	0.190526	−	0.330001i	−0.754898	−	0.655842i	$-0.772314\pi$
$479$	4.16987	−	7.22243i	0.190526	−	0.330001i	0.945425	+	0.325840i	$0.105647\pi$
$480$		0			0
$481$	−11.6603	−	20.1962i	−0.531662	−	0.920865i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	8.58846	−	8.58846i	0.389982	−	0.389982i
$486$		0			0
$487$		−	5.80385i		−	0.262997i	−0.991316	−	0.131499i	$-0.958021\pi$
$487$		−	5.80385i		−	0.262997i	0.991316	−	0.131499i	$-0.0419789\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−13.8923	−	3.72243i	−0.626951	−	0.167991i	−0.0686652	−	0.997640i	$-0.521874\pi$
$491$	−13.8923	−	3.72243i	−0.626951	−	0.167991i	−0.558286	+	0.829649i	$0.688541\pi$
$492$		0			0
$493$	−13.5622	+	3.63397i	−0.610810	+	0.163666i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	4.00000	−	6.92820i	0.179425	−	0.310772i
$498$		0			0
$499$	2.33013	−	8.69615i	0.104311	−	0.389293i	−0.893955	−	0.448156i	$-0.852081\pi$
$499$	2.33013	−	8.69615i	0.104311	−	0.389293i	0.998266	+	0.0588630i	$0.0187475\pi$
$500$		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
$505$			2.14359i			0.0953887i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−3.07180	+	11.4641i	−0.136155	+	0.508137i	0.863835	+	0.503774i	$0.168056\pi$
$509$	−3.07180	+	11.4641i	−0.136155	+	0.508137i	−0.999990	+	0.00436335i	$0.998611\pi$
$510$		0			0
$511$	−8.56218	+	14.8301i	−0.378768	+	0.656046i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	15.1244	−	4.05256i	0.666459	−	0.178577i
$516$		0			0
$517$	32.4186	+	8.68653i	1.42577	+	0.382033i
$518$		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
$523$	7.53590	−	7.53590i	0.329522	−	0.329522i	−0.522883	−	0.852405i	$-0.675143\pi$
$523$	7.53590	−	7.53590i	0.329522	−	0.329522i	0.852405	+	0.522883i	$0.175143\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	1.53590	+	2.66025i	0.0669048	+	0.115882i
$528$		0			0
$529$	−0.303848	+	0.526279i	−0.0132108	+	0.0228817i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−2.70577	−	10.0981i	−0.117200	−	0.437396i
$534$		0			0
$535$	−15.8372	+	9.14359i	−0.684701	+	0.395312i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−1.43782	+	1.43782i	−0.0619314	+	0.0619314i
$540$		0			0
$541$	2.19615	+	2.19615i	0.0944200	+	0.0944200i	0.752739	−	0.658319i	$-0.228732\pi$
$541$	2.19615	+	2.19615i	0.0944200	+	0.0944200i	−0.658319	+	0.752739i	$0.728732\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−7.85641	−	13.6077i	−0.336531	−	0.582890i
$546$		0			0
$547$	−32.6244	+	8.74167i	−1.39492	+	0.373767i	−0.876517	−	0.481371i	$-0.840139\pi$
$547$	−32.6244	+	8.74167i	−1.39492	+	0.373767i	−0.518400	+	0.855138i	$0.673472\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−7.09808	−	4.09808i	−0.302388	−	0.174584i
$552$		0			0
$553$	−28.3923	+	16.3923i	−1.20736	+	0.697072i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	14.8038	+	14.8038i	0.627259	+	0.627259i	0.947378	−	0.320118i	$-0.103723\pi$
$557$	14.8038	+	14.8038i	0.627259	+	0.627259i	−0.320118	+	0.947378i	$0.603723\pi$
$558$		0			0
$559$	−30.0526			−1.27109
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−7.23205	+	26.9904i	−0.304795	+	1.13751i	0.628327	+	0.777949i	$0.283740\pi$
$563$	−7.23205	+	26.9904i	−0.304795	+	1.13751i	−0.933122	+	0.359560i	$0.882927\pi$
$564$		0			0
$565$	−3.71281	−	13.8564i	−0.156199	−	0.582943i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	18.4019	+	10.6244i	0.771449	+	0.445396i	0.833391	−	0.552684i	$-0.186396\pi$
$569$	18.4019	+	10.6244i	0.771449	+	0.445396i	−0.0619424	+	0.998080i	$0.519730\pi$
$570$		0			0
$571$	3.33013	+	0.892305i	0.139361	+	0.0373418i	0.327825	−	0.944738i	$-0.393684\pi$
$571$	3.33013	+	0.892305i	0.139361	+	0.0373418i	−0.188464	+	0.982080i	$0.560351\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	18.5885			0.775192
$576$		0			0
$577$	−5.78461			−0.240816			−0.120408	−	0.992724i	$-0.538420\pi$
$577$	−5.78461			−0.240816			−0.120408	+	0.992724i	$0.538420\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	3.73205	+	1.00000i	0.154832	+	0.0414870i
$582$		0			0
$583$	40.0526	+	23.1244i	1.65881	+	0.957713i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	7.23205	+	26.9904i	0.298499	+	1.11401i	0.938399	+	0.345554i	$0.112309\pi$
$587$	7.23205	+	26.9904i	0.298499	+	1.11401i	−0.639900	+	0.768458i	$0.721024\pi$
$588$		0			0
$589$	−0.464102	+	1.73205i	−0.0191230	+	0.0713679i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−17.4641			−0.717165			−0.358582	−	0.933498i	$-0.616740\pi$
$593$	−17.4641			−0.717165			−0.358582	+	0.933498i	$0.616740\pi$
$594$		0			0
$595$	11.4641	+	11.4641i	0.469982	+	0.469982i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	11.3205	−	6.53590i	0.462543	−	0.267050i	−0.250570	−	0.968099i	$-0.580618\pi$
$599$	11.3205	−	6.53590i	0.462543	−	0.267050i	0.713113	+	0.701049i	$0.247285\pi$
$600$		0			0
$601$	−20.5526	−	11.8660i	−0.838356	−	0.484025i	0.0183488	−	0.999832i	$-0.494159\pi$
$601$	−20.5526	−	11.8660i	−0.838356	−	0.484025i	−0.856705	+	0.515806i	$0.827492\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	8.19615	−	2.19615i	0.333221	−	0.0892863i
$606$		0			0
$607$	8.58846	+	14.8756i	0.348595	+	0.603784i	0.986000	−	0.166745i	$-0.0533256\pi$
$607$	8.58846	+	14.8756i	0.348595	+	0.603784i	−0.637405	+	0.770529i	$0.719992\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−18.8756	−	18.8756i	−0.763627	−	0.763627i
$612$		0			0
$613$	−15.6603	+	15.6603i	−0.632512	+	0.632512i	−0.948697	−	0.316186i	$-0.897598\pi$
$613$	−15.6603	+	15.6603i	−0.632512	+	0.632512i	0.316186	+	0.948697i	$0.397598\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	35.0885	−	20.2583i	1.41261	−	0.815570i	0.416975	−	0.908918i	$-0.363090\pi$
$617$	35.0885	−	20.2583i	1.41261	−	0.815570i	0.995633	+	0.0933485i	$0.0297571\pi$
$618$		0			0
$619$	−4.17949	−	15.5981i	−0.167988	−	0.626940i	−0.997640	−	0.0686590i	$-0.978128\pi$
$619$	−4.17949	−	15.5981i	−0.167988	−	0.626940i	0.829652	−	0.558281i	$-0.188539\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	2.73205	−	4.73205i	0.109457	−	0.189586i
$624$		0			0
$625$	5.03590	+	8.72243i	0.201436	+	0.348897i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	27.1244	−	27.1244i	1.08152	−	1.08152i
$630$		0			0
$631$		−	17.6077i		−	0.700951i	−0.936572	−	0.350476i	$-0.886020\pi$
$631$		−	17.6077i		−	0.700951i	0.936572	−	0.350476i	$-0.113980\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	4.19615	+	1.12436i	0.166519	+	0.0446187i
$636$		0			0
$637$	1.56218	−	0.418584i	0.0618957	−	0.0165849i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	19.7942	−	34.2846i	0.781825	−	1.35416i	−0.149053	−	0.988829i	$-0.547622\pi$
$641$	19.7942	−	34.2846i	0.781825	−	1.35416i	0.930878	−	0.365331i	$-0.119044\pi$
$642$		0			0
$643$	2.34936	−	8.76795i	0.0926499	−	0.345774i	−0.904003	−	0.427527i	$-0.859385\pi$
$643$	2.34936	−	8.76795i	0.0926499	−	0.345774i	0.996653	+	0.0817525i	$0.0260517\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			16.7321i			0.657805i	0.944364	+	0.328902i	$0.106679\pi$
$647$			16.7321i			0.657805i	−0.944364	+	0.328902i	$0.893321\pi$
$648$		0			0
$649$			33.2487i			1.30513i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−7.36603	+	27.4904i	−0.288255	+	1.07578i	0.658173	+	0.752867i	$0.271329\pi$
$653$	−7.36603	+	27.4904i	−0.288255	+	1.07578i	−0.946428	+	0.322915i	$0.895337\pi$
$654$		0			0
$655$	−4.19615	+	7.26795i	−0.163957	+	0.283982i
$656$		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
$665$			9.46410i			0.367002i
$666$		0			0
$667$	8.19615	−	8.19615i	0.317356	−	0.317356i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	25.3923	+	43.9808i	0.980259	+	1.69786i
$672$		0			0
$673$	19.1962	−	33.2487i	0.739957	−	1.28164i	−0.212557	−	0.977149i	$-0.568179\pi$
$673$	19.1962	−	33.2487i	0.739957	−	1.28164i	0.952514	−	0.304495i	$-0.0984877\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−1.26795	−	4.73205i	−0.0487312	−	0.181867i	0.937270	−	0.348603i	$-0.113344\pi$
$677$	−1.26795	−	4.73205i	−0.0487312	−	0.181867i	−0.986002	+	0.166736i	$0.946677\pi$
$678$		0			0
$679$	27.7583	−	16.0263i	1.06527	−	0.615032i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−20.2942	+	20.2942i	−0.776537	+	0.776537i	−0.979240	−	0.202703i	$-0.935027\pi$
$683$	−20.2942	+	20.2942i	−0.776537	+	0.776537i	0.202703	+	0.979240i	$0.435027\pi$
$684$		0			0
$685$	6.98076	+	6.98076i	0.266721	+	0.266721i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−18.3923	−	31.8564i	−0.700691	−	1.21363i
$690$		0			0
$691$	−9.29423	+	2.49038i	−0.353569	+	0.0947386i	−0.431232	−	0.902241i	$-0.641921\pi$
$691$	−9.29423	+	2.49038i	−0.353569	+	0.0947386i	0.0776628	+	0.996980i	$0.475254\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−10.6077	−	6.12436i	−0.402373	−	0.232310i
$696$		0			0
$697$	14.8923	−	8.59808i	0.564086	−	0.325675i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	6.66025	+	6.66025i	0.251554	+	0.251554i	0.821608	−	0.570053i	$-0.193077\pi$
$701$	6.66025	+	6.66025i	0.251554	+	0.251554i	−0.570053	+	0.821608i	$0.693077\pi$
$702$		0			0
$703$	22.3923			0.844542
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−1.46410	+	5.46410i	−0.0550632	+	0.205499i
$708$		0			0
$709$	−9.80385	−	36.5885i	−0.368191	−	1.37411i	−0.863043	−	0.505131i	$-0.831444\pi$
$709$	−9.80385	−	36.5885i	−0.368191	−	1.37411i	0.494852	−	0.868978i	$-0.335222\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−2.19615	−	1.26795i	−0.0822466	−	0.0474851i
$714$		0			0
$715$	−15.2679	−	4.09103i	−0.570989	−	0.152996i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	4.39230			0.163805			0.0819027	−	0.996640i	$-0.473900\pi$
$719$	4.39230			0.163805			0.0819027	+	0.996640i	$0.473900\pi$
$720$		0			0
$721$	41.3205			1.53886
$722$		0			0
$723$		0			0
$724$		0			0
$725$	9.29423	+	2.49038i	0.345179	+	0.0924904i
$726$		0			0
$727$	28.8109	+	16.6340i	1.06854	+	0.616920i	0.927781	−	0.373124i	$-0.121714\pi$
$727$	28.8109	+	16.6340i	1.06854	+	0.616920i	0.140755	+	0.990044i	$0.455047\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−12.7942	−	47.7487i	−0.473212	−	1.76605i
$732$		0			0
$733$	2.95448	−	11.0263i	0.109126	−	0.407265i	−0.889654	−	0.456635i	$-0.849055\pi$
$733$	2.95448	−	11.0263i	0.109126	−	0.407265i	0.998781	+	0.0493698i	$0.0157213\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	29.9282			1.10242
$738$		0			0
$739$	8.22243	+	8.22243i	0.302467	+	0.302467i	0.841978	−	0.539511i	$-0.181391\pi$
$739$	8.22243	+	8.22243i	0.302467	+	0.302467i	−0.539511	+	0.841978i	$0.681391\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	24.7583	−	14.2942i	0.908295	−	0.524404i	0.0284129	−	0.999596i	$-0.490955\pi$
$743$	24.7583	−	14.2942i	0.908295	−	0.524404i	0.879882	+	0.475192i	$0.157621\pi$
$744$		0			0
$745$	7.26795	+	4.19615i	0.266277	+	0.153735i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−46.6147	+	12.4904i	−1.70327	+	0.456389i
$750$		0			0
$751$	8.85641	+	15.3397i	0.323175	+	0.559755i	0.981141	−	0.193292i	$-0.0619165\pi$
$751$	8.85641	+	15.3397i	0.323175	+	0.559755i	−0.657966	+	0.753047i	$0.728583\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	0.535898	+	0.535898i	0.0195033	+	0.0195033i
$756$		0			0
$757$	−19.9282	+	19.9282i	−0.724303	+	0.724303i	−0.969479	−	0.245176i	$-0.921154\pi$
$757$	−19.9282	+	19.9282i	−0.724303	+	0.724303i	0.245176	+	0.969479i	$0.421154\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−45.3731	+	26.1962i	−1.64477	+	0.949610i	−0.665669	+	0.746247i	$0.731854\pi$
$761$	−45.3731	+	26.1962i	−1.64477	+	0.949610i	−0.979104	+	0.203363i	$0.934813\pi$
$762$		0			0
$763$	−10.7321	−	40.0526i	−0.388526	−	1.45000i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	13.2224	−	22.9019i	0.477434	−	0.826941i
$768$		0			0
$769$	−14.1244	−	24.4641i	−0.509337	−	0.882198i	−0.999942	−	0.0108155i	$-0.996557\pi$
$769$	−14.1244	−	24.4641i	−0.509337	−	0.882198i	0.490604	−	0.871383i	$-0.336776\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−35.5885	+	35.5885i	−1.28003	+	1.28003i	−0.339378	+	0.940650i	$0.610216\pi$
$773$	−35.5885	+	35.5885i	−1.28003	+	1.28003i	−0.940650	+	0.339378i	$0.889784\pi$
$774$		0			0
$775$		−	2.10512i		−	0.0756181i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	9.69615	+	2.59808i	0.347401	+	0.0930857i
$780$		0			0
$781$	−12.3923	+	3.32051i	−0.443432	+	0.118817i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−2.53590	+	4.39230i	−0.0905101	+	0.156768i
$786$		0			0
$787$	10.8109	−	40.3468i	0.385367	−	1.43821i	−0.452222	−	0.891906i	$-0.649368\pi$
$787$	10.8109	−	40.3468i	0.385367	−	1.43821i	0.837588	−	0.546302i	$-0.183965\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	37.8564i		−	1.34602i
$792$		0			0
$793$		−	40.3923i		−	1.43437i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−8.17691	+	30.5167i	−0.289641	+	1.08096i	0.655740	+	0.754987i	$0.272357\pi$
$797$	−8.17691	+	30.5167i	−0.289641	+	1.08096i	−0.945381	+	0.325968i	$0.894310\pi$
$798$		0			0
$799$	21.9545	−	38.0263i	0.776694	−	1.34527i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	26.5263	−	7.10770i	0.936092	−	0.250825i
$804$		0			0
$805$	−12.9282	−	3.46410i	−0.455659	−	0.122094i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			6.32051i			0.222217i	0.993808	+	0.111109i	$0.0354401\pi$
$809$			6.32051i			0.222217i	−0.993808	+	0.111109i	$0.964560\pi$
$810$		0			0
$811$	14.0263	−	14.0263i	0.492529	−	0.492529i	−0.416573	−	0.909102i	$-0.636769\pi$
$811$	14.0263	−	14.0263i	0.492529	−	0.492529i	0.909102	+	0.416573i	$0.136769\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−5.12436	−	8.87564i	−0.179498	−	0.310900i
$816$		0			0
$817$	14.4282	−	24.9904i	0.504779	−	0.874303i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−2.77757	−	10.3660i	−0.0969378	−	0.361777i	0.900368	−	0.435129i	$-0.143297\pi$
$821$	−2.77757	−	10.3660i	−0.0969378	−	0.361777i	−0.997306	+	0.0733518i	$0.976630\pi$
$822$		0			0
$823$	−7.26795	+	4.19615i	−0.253345	+	0.146269i	−0.621295	−	0.783577i	$-0.713393\pi$
$823$	−7.26795	+	4.19615i	−0.253345	+	0.146269i	0.367950	+	0.929846i	$0.380060\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	17.5359	−	17.5359i	0.609783	−	0.609783i	−0.333106	−	0.942889i	$-0.608097\pi$
$827$	17.5359	−	17.5359i	0.609783	−	0.609783i	0.942889	+	0.333106i	$0.108097\pi$
$828$		0			0
$829$	−20.5167	−	20.5167i	−0.712573	−	0.712573i	0.254500	−	0.967073i	$-0.418089\pi$
$829$	−20.5167	−	20.5167i	−0.712573	−	0.712573i	−0.967073	+	0.254500i	$0.918089\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	1.33013	+	2.30385i	0.0460862	+	0.0798236i
$834$		0			0
$835$	−7.46410	+	2.00000i	−0.258306	+	0.0692129i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	23.4449	+	13.5359i	0.809407	+	0.467311i	0.846750	−	0.531991i	$-0.178556\pi$
$839$	23.4449	+	13.5359i	0.809407	+	0.467311i	−0.0373432	+	0.999303i	$0.511889\pi$
$840$		0			0
$841$	−19.9186	+	11.5000i	−0.686848	+	0.396552i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−0.626933	−	0.626933i	−0.0215672	−	0.0215672i
$846$		0			0
$847$	22.3923			0.769409
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−8.19615	+	30.5885i	−0.280960	+	1.04856i
$852$		0			0
$853$	−0.437822	−	1.63397i	−0.0149907	−	0.0559462i	0.958025	−	0.286684i	$-0.0925528\pi$
$853$	−0.437822	−	1.63397i	−0.0149907	−	0.0559462i	−0.973016	+	0.230737i	$0.925886\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	44.9090	+	25.9282i	1.53406	+	0.885691i	0.999169	+	0.0407704i	$0.0129812\pi$
$857$	44.9090	+	25.9282i	1.53406	+	0.885691i	0.534892	+	0.844920i	$0.320352\pi$
$858$		0			0
$859$	−14.2583	−	3.82051i	−0.486488	−	0.130354i	0.00723407	−	0.999974i	$-0.497697\pi$
$859$	−14.2583	−	3.82051i	−0.486488	−	0.130354i	−0.493722	+	0.869620i	$0.664364\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−15.4641			−0.526404			−0.263202	−	0.964741i	$-0.584779\pi$
$863$	−15.4641			−0.526404			−0.263202	+	0.964741i	$0.584779\pi$
$864$		0			0
$865$	1.75129			0.0595456
$866$		0			0
$867$		0			0
$868$		0			0
$869$	50.7846	+	13.6077i	1.72275	+	0.461609i
$870$		0			0
$871$	−20.6147	−	11.9019i	−0.698504	−	0.403281i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−6.53590	−	24.3923i	−0.220954	−	0.824610i
$876$		0			0
$877$	−8.46410	+	31.5885i	−0.285812	+	1.06667i	0.662431	+	0.749123i	$0.269525\pi$
$877$	−8.46410	+	31.5885i	−0.285812	+	1.06667i	−0.948244	+	0.317544i	$0.897142\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−27.3205			−0.920451			−0.460226	−	0.887802i	$-0.652231\pi$
$881$	−27.3205			−0.920451			−0.460226	+	0.887802i	$0.652231\pi$
$882$		0			0
$883$	12.6340	+	12.6340i	0.425167	+	0.425167i	0.886978	−	0.461811i	$-0.152800\pi$
$883$	12.6340	+	12.6340i	0.425167	+	0.425167i	−0.461811	+	0.886978i	$0.652800\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	8.87564	−	5.12436i	0.298015	−	0.172059i	−0.343536	−	0.939140i	$-0.611625\pi$
$887$	8.87564	−	5.12436i	0.298015	−	0.172059i	0.641551	+	0.767081i	$0.278291\pi$
$888$		0			0
$889$	9.92820	+	5.73205i	0.332981	+	0.192247i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	24.7583	−	6.63397i	0.828506	−	0.221997i
$894$		0			0
$895$	1.41154	+	2.44486i	0.0471827	+	0.0817228i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−0.928203	−	0.928203i	−0.0309573	−	0.0309573i
$900$		0			0
$901$	42.7846	−	42.7846i	1.42536	−	1.42536i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	9.37307	−	5.41154i	0.311571	−	0.179886i
$906$		0			0
$907$	1.20577	+	4.50000i	0.0400370	+	0.149420i	0.983051	−	0.183334i	$-0.0586889\pi$
$907$	1.20577	+	4.50000i	0.0400370	+	0.149420i	−0.943014	+	0.332754i	$0.892022\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	2.46410	−	4.26795i	0.0816393	−	0.141403i	−0.822315	−	0.569033i	$-0.807318\pi$
$911$	2.46410	−	4.26795i	0.0816393	−	0.141403i	0.903954	+	0.427629i	$0.140651\pi$
$912$		0			0
$913$	−3.09808	−	5.36603i	−0.102531	−	0.177590i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−15.6603	+	15.6603i	−0.517147	+	0.517147i
$918$		0			0
$919$		−	18.9808i		−	0.626118i	−0.949734	−	0.313059i	$-0.898646\pi$
$919$		−	18.9808i		−	0.626118i	0.949734	−	0.313059i	$-0.101354\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	9.85641	+	2.64102i	0.324428	+	0.0869301i
$924$		0			0
$925$	−25.3923	+	6.80385i	−0.834894	+	0.223709i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	11.5359	−	19.9808i	0.378481	−	0.655548i	−0.612361	−	0.790578i	$-0.709780\pi$
$929$	11.5359	−	19.9808i	0.378481	−	0.655548i	0.990841	+	0.135031i	$0.0431134\pi$
$930$		0			0
$931$	−0.401924	+	1.50000i	−0.0131725	+	0.0491605i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	26.0000i		−	0.850291i
$936$		0			0
$937$		−	11.1769i		−	0.365134i	−0.983193	−	0.182567i	$-0.941559\pi$
$937$		−	11.1769i		−	0.365134i	0.983193	−	0.182567i	$-0.0584406\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	1.80385	−	6.73205i	0.0588038	−	0.219459i	−0.930271	−	0.366873i	$-0.880428\pi$
$941$	1.80385	−	6.73205i	0.0588038	−	0.219459i	0.989075	+	0.147414i	$0.0470951\pi$
$942$		0			0
$943$	−7.09808	+	12.2942i	−0.231145	+	0.400355i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	41.0167	−	10.9904i	1.33286	−	0.357139i	0.479081	−	0.877771i	$-0.340970\pi$
$947$	41.0167	−	10.9904i	1.33286	−	0.357139i	0.853782	+	0.520631i	$0.174303\pi$
$948$		0			0
$949$	−21.0981	−	5.65321i	−0.684873	−	0.183511i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	17.1051i		−	0.554089i	−0.960857	−	0.277045i	$-0.910645\pi$
$953$		−	17.1051i		−	0.554089i	0.960857	−	0.277045i	$-0.0893550\pi$
$954$		0			0
$955$	17.6077	−	17.6077i	0.569772	−	0.569772i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	13.0263	+	22.5622i	0.420641	+	0.728571i
$960$		0			0
$961$	15.3564	−	26.5981i	0.495368	−	0.858002i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	5.82309	+	21.7321i	0.187452	+	0.699579i
$966$		0			0
$967$	−17.8301	+	10.2942i	−0.573378	+	0.331040i	−0.758497	−	0.651676i	$-0.774066\pi$
$967$	−17.8301	+	10.2942i	−0.573378	+	0.331040i	0.185119	+	0.982716i	$0.440733\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	15.5359	−	15.5359i	0.498571	−	0.498571i	−0.412422	−	0.910993i	$-0.635317\pi$
$971$	15.5359	−	15.5359i	0.498571	−	0.498571i	0.910993	+	0.412422i	$0.135317\pi$
$972$		0			0
$973$	−22.8564	−	22.8564i	−0.732743	−	0.732743i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	22.0622	+	38.2128i	0.705832	+	1.22254i	0.966391	+	0.257078i	$0.0827598\pi$
$977$	22.0622	+	38.2128i	0.705832	+	1.22254i	−0.260559	+	0.965458i	$0.583907\pi$
$978$		0			0
$979$	−8.46410	+	2.26795i	−0.270514	+	0.0724840i
$980$		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
$989$	28.8564	+	28.8564i	0.917580	+	0.917580i
$990$		0			0
$991$	−36.6410			−1.16394			−0.581970	−	0.813210i	$-0.697718\pi$
$991$	−36.6410			−1.16394			−0.581970	+	0.813210i	$0.697718\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−6.73205	+	25.1244i	−0.213420	+	0.796496i
$996$		0			0
$997$	7.87564	+	29.3923i	0.249424	+	0.930864i	0.971108	+	0.238641i	$0.0767018\pi$
$997$	7.87564	+	29.3923i	0.249424	+	0.930864i	−0.721684	+	0.692223i	$0.756632\pi$
$998$		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.721.1		4
3.2	odd	2		576.2.bb.c.529.1		4
4.3	odd	2		432.2.y.b.397.1		4
9.4	even	3		1728.2.bc.d.145.1		4
9.5	odd	6		576.2.bb.d.337.1		4
12.11	even	2		144.2.x.c.61.1	yes	4
16.5	even	4		1728.2.bc.d.1585.1		4
16.11	odd	4		432.2.y.c.181.1		4
36.23	even	6		144.2.x.b.13.1	✓	4
36.31	odd	6		432.2.y.c.253.1		4
48.5	odd	4		576.2.bb.d.241.1		4
48.11	even	4		144.2.x.b.133.1	yes	4
144.5	odd	12		576.2.bb.c.49.1		4
144.59	even	12		144.2.x.c.85.1	yes	4
144.85	even	12	inner	1728.2.bc.a.1009.1		4
144.139	odd	12		432.2.y.b.37.1		4

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

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)

\(f(q)\)	\(=\)	\(q+(-1.00000 - 0.267949i) q^{5} +(-2.36603 - 1.36603i) q^{7} +O(q^{10})\)	\(q+(-1.00000 - 0.267949i) q^{5} +(-2.36603 - 1.36603i) q^{7} +(1.13397 + 4.23205i) q^{11} +(0.901924 - 3.36603i) q^{13} +5.73205 q^{17} +(2.36603 + 2.36603i) q^{19} +(-4.09808 + 2.36603i) q^{23} +(-3.40192 - 1.96410i) q^{25} +(-2.36603 + 0.633975i) q^{29} +(0.267949 + 0.464102i) q^{31} +(2.00000 + 2.00000i) q^{35} +(4.73205 - 4.73205i) q^{37} +(2.59808 - 1.50000i) q^{41} +(-2.23205 - 8.33013i) q^{43} +(3.83013 - 6.63397i) q^{47} +(0.232051 + 0.401924i) q^{49} +(7.46410 - 7.46410i) q^{53} -4.53590i q^{55} +(7.33013 + 1.96410i) q^{59} +(11.1962 - 3.00000i) q^{61} +(-1.80385 + 3.12436i) q^{65} +(1.76795 - 6.59808i) q^{67} +2.92820i q^{71} -6.26795i q^{73} +(3.09808 - 11.5622i) q^{77} +(6.00000 - 10.3923i) q^{79} +(-1.36603 + 0.366025i) q^{83} +(-5.73205 - 1.53590i) q^{85} +2.00000i q^{89} +(-6.73205 + 6.73205i) q^{91} +(-1.73205 - 3.00000i) q^{95} +(-5.86603 + 10.1603i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{5} - 6 q^{7}+O(q^{10}) \) 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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