LMFDB - Embedded newform 1728.2.bc.d.145.1

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			145.1
Root			$-0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	1728.145
Dual form			1728.2.bc.d.1585.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.267949 + 1.00000i) q^{5} +(2.36603 - 1.36603i) q^{7} +O(q^{10})$	$q+(0.267949 + 1.00000i) q^{5} +(2.36603 - 1.36603i) q^{7} +(-4.23205 - 1.13397i) q^{11} +(-3.36603 + 0.901924i) q^{13} +5.73205 q^{17} +(2.36603 + 2.36603i) q^{19} +(4.09808 + 2.36603i) q^{23} +(3.40192 - 1.96410i) q^{25} +(0.633975 - 2.36603i) q^{29} +(0.267949 - 0.464102i) q^{31} +(2.00000 + 2.00000i) q^{35} +(4.73205 - 4.73205i) q^{37} +(-2.59808 - 1.50000i) q^{41} +(8.33013 + 2.23205i) q^{43} +(3.83013 + 6.63397i) q^{47} +(0.232051 - 0.401924i) q^{49} +(7.46410 - 7.46410i) q^{53} -4.53590i q^{55} +(-1.96410 - 7.33013i) q^{59} +(-3.00000 + 11.1962i) q^{61} +(-1.80385 - 3.12436i) q^{65} +(-6.59808 + 1.76795i) q^{67} +2.92820i q^{71} -6.26795i q^{73} +(-11.5622 + 3.09808i) q^{77} +(6.00000 + 10.3923i) q^{79} +(0.366025 - 1.36603i) q^{83} +(1.53590 + 5.73205i) q^{85} +2.00000i q^{89} +(-6.73205 + 6.73205i) q^{91} +(-1.73205 + 3.00000i) q^{95} +(-5.86603 - 10.1603i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8 q^{5} + 6 q^{7}+O(q^{10}) $ 4 * q + 8 * q^5 + 6 * q^7	$ 4 q + 8 q^{5} + 6 q^{7} - 10 q^{11} - 10 q^{13} + 16 q^{17} + 6 q^{19} + 6 q^{23} + 24 q^{25} + 6 q^{29} + 8 q^{31} + 8 q^{35} + 12 q^{37} + 16 q^{43} - 2 q^{47} - 6 q^{49} + 16 q^{53} + 6 q^{59} - 12 q^{61} - 28 q^{65} - 16 q^{67} - 22 q^{77} + 24 q^{79} - 2 q^{83} + 20 q^{85} - 20 q^{91} - 20 q^{97}+O(q^{100}) $ 4 * q + 8 * q^5 + 6 * q^7 - 10 * q^11 - 10 * q^13 + 16 * q^17 + 6 * q^19 + 6 * q^23 + 24 * q^25 + 6 * q^29 + 8 * q^31 + 8 * q^35 + 12 * q^37 + 16 * q^43 - 2 * q^47 - 6 * q^49 + 16 * q^53 + 6 * q^59 - 12 * q^61 - 28 * q^65 - 16 * q^67 - 22 * q^77 + 24 * q^79 - 2 * q^83 + 20 * q^85 - 20 * q^91 - 20 * q^97

Display 100 coefficients 10 coefficients

Character values

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

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

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	0.267949	+	1.00000i	0.119831	+	0.447214i	0.999603	−	0.0281817i	$-0.00897171\pi$
$5$	0.267949	+	1.00000i	0.119831	+	0.447214i	−0.879772	+	0.475395i	$0.842305\pi$
$6$		0			0
$7$	2.36603	−	1.36603i	0.894274	−	0.516309i	0.0189356	−	0.999821i	$-0.493972\pi$
$7$	2.36603	−	1.36603i	0.894274	−	0.516309i	0.875338	+	0.483512i	$0.160639\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−4.23205	−	1.13397i	−1.27601	−	0.341906i	−0.443680	−	0.896185i	$-0.646327\pi$
$11$	−4.23205	−	1.13397i	−1.27601	−	0.341906i	−0.832331	+	0.554279i	$0.812994\pi$
$12$		0			0
$13$	−3.36603	+	0.901924i	−0.933567	+	0.250149i	−0.693375	−	0.720577i	$-0.743877\pi$
$13$	−3.36603	+	0.901924i	−0.933567	+	0.250149i	−0.240192	+	0.970725i	$0.577210\pi$
$14$		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	$-0.169105\pi$
$23$	4.09808	+	2.36603i	0.854508	+	0.493350i	−0.00766135	+	0.999971i	$0.502439\pi$
$24$		0			0
$25$	3.40192	−	1.96410i	0.680385	−	0.392820i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	0.633975	−	2.36603i	0.117726	−	0.439360i	−0.881750	−	0.471717i	$-0.843635\pi$
$29$	0.633975	−	2.36603i	0.117726	−	0.439360i	0.999476	+	0.0323566i	$0.0103012\pi$
$30$		0			0
$31$	0.267949	−	0.464102i	0.0481251	−	0.0833551i	−0.840959	−	0.541098i	$-0.818009\pi$
$31$	0.267949	−	0.464102i	0.0481251	−	0.0833551i	0.889085	+	0.457743i	$0.151342\pi$
$32$		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	$-0.408600\pi$
$41$	−2.59808	−	1.50000i	−0.405751	−	0.234261i	−0.688963	+	0.724797i	$0.741934\pi$
$42$		0			0
$43$	8.33013	+	2.23205i	1.27033	+	0.340385i	0.830158	−	0.557528i	$-0.188250\pi$
$43$	8.33013	+	2.23205i	1.27033	+	0.340385i	0.440174	+	0.897912i	$0.354917\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	3.83013	+	6.63397i	0.558681	+	0.967665i	0.997607	+	0.0691412i	$0.0220259\pi$
$47$	3.83013	+	6.63397i	0.558681	+	0.967665i	−0.438925	+	0.898523i	$0.644641\pi$
$48$		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
$59$	−1.96410	−	7.33013i	−0.255704	−	0.954301i	−0.967697	−	0.252115i	$-0.918874\pi$
$59$	−1.96410	−	7.33013i	−0.255704	−	0.954301i	0.711993	−	0.702186i	$-0.247793\pi$
$60$		0			0
$61$	−3.00000	+	11.1962i	−0.384111	+	1.43352i	0.455453	+	0.890260i	$0.349477\pi$
$61$	−3.00000	+	11.1962i	−0.384111	+	1.43352i	−0.839564	+	0.543261i	$0.817189\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−1.80385	−	3.12436i	−0.223740	−	0.387529i
$66$		0			0
$67$	−6.59808	+	1.76795i	−0.806083	+	0.215989i	−0.638253	−	0.769827i	$-0.720343\pi$
$67$	−6.59808	+	1.76795i	−0.806083	+	0.215989i	−0.167830	+	0.985816i	$0.553676\pi$
$68$		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
$77$	−11.5622	+	3.09808i	−1.31763	+	0.353059i
$78$		0			0
$79$	6.00000	+	10.3923i	0.675053	+	1.16923i	0.976453	+	0.215728i	$0.0692125\pi$
$79$	6.00000	+	10.3923i	0.675053	+	1.16923i	−0.301401	+	0.953498i	$0.597454\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	0.366025	−	1.36603i	0.0401765	−	0.149941i	−0.942924	−	0.333009i	$-0.891936\pi$
$83$	0.366025	−	1.36603i	0.0401765	−	0.149941i	0.983100	+	0.183068i	$0.0586028\pi$
$84$		0			0
$85$	1.53590	+	5.73205i	0.166592	+	0.621728i
$86$		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	−0.993461	−	0.114170i	$-0.963579\pi$
$97$	−5.86603	−	10.1603i	−0.595605	−	1.03162i	0.397857	−	0.917448i	$-0.369754\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	2.00000	+	0.535898i	0.199007	+	0.0533239i	0.356946	−	0.934125i	$-0.383818\pi$
$101$	2.00000	+	0.535898i	0.199007	+	0.0533239i	−0.157938	+	0.987449i	$0.550485\pi$
$102$		0			0
$103$	13.0981	+	7.56218i	1.29059	+	0.745124i	0.978759	−	0.205014i	$-0.0657238\pi$
$103$	13.0981	+	7.56218i	1.29059	+	0.745124i	0.311833	+	0.950137i	$0.399057\pi$
$104$		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	−0.330947	−	0.943649i	$-0.607368\pi$
$113$	6.92820	−	12.0000i	0.651751	−	1.12887i	0.982698	−	0.185216i	$-0.0592984\pi$
$114$		0			0
$115$	−1.26795	+	4.73205i	−0.118237	+	0.441266i
$116$		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
$131$	−7.83013	+	2.09808i	−0.684121	+	0.183310i	−0.584108	−	0.811676i	$-0.698555\pi$
$131$	−7.83013	+	2.09808i	−0.684121	+	0.183310i	−0.100014	+	0.994986i	$0.531889\pi$
$132$		0			0
$133$	8.83013	+	2.36603i	0.765669	+	0.205160i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	8.25833	−	4.76795i	0.705557	−	0.407353i	−0.103857	−	0.994592i	$-0.533118\pi$
$137$	8.25833	−	4.76795i	0.705557	−	0.407353i	0.809414	+	0.587239i	$0.199785\pi$
$138$		0			0
$139$	−3.06218	−	11.4282i	−0.259731	−	0.969328i	−0.965397	−	0.260784i	$-0.916019\pi$
$139$	−3.06218	−	11.4282i	−0.259731	−	0.969328i	0.705667	−	0.708544i	$-0.250648\pi$
$140$		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
$149$	2.09808	+	7.83013i	0.171881	+	0.641469i	0.997062	+	0.0766003i	$0.0244065\pi$
$149$	2.09808	+	7.83013i	0.171881	+	0.641469i	−0.825181	+	0.564869i	$0.808927\pi$
$150$		0			0
$151$	0.633975	−	0.366025i	0.0515921	−	0.0297867i	−0.473982	−	0.880534i	$-0.657184\pi$
$151$	0.633975	−	0.366025i	0.0515921	−	0.0297867i	0.525574	+	0.850748i	$0.323851\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	0.535898	+	0.143594i	0.0430444	+	0.0115337i
$156$		0			0
$157$	−4.73205	+	1.26795i	−0.377659	+	0.101193i	−0.442655	−	0.896692i	$-0.645963\pi$
$157$	−4.73205	+	1.26795i	−0.377659	+	0.101193i	0.0649959	+	0.997886i	$0.479297\pi$
$158$		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	$-0.426588\pi$
$167$	−6.46410	−	3.73205i	−0.500207	−	0.288795i	−0.728799	+	0.684728i	$0.759921\pi$
$168$		0			0
$169$	−0.741670	+	0.428203i	−0.0570515	+	0.0329387i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	0.437822	−	1.63397i	0.0332870	−	0.124229i	−0.947283	−	0.320398i	$-0.896183\pi$
$173$	0.437822	−	1.63397i	0.0332870	−	0.124229i	0.980570	+	0.196169i	$0.0628501\pi$
$174$		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
$187$	−24.2583	−	6.50000i	−1.77394	−	0.475327i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−12.0263	−	20.8301i	−0.870191	−	1.50722i	−0.861799	−	0.507250i	$-0.830662\pi$
$191$	−12.0263	−	20.8301i	−0.870191	−	1.50722i	−0.00839227	−	0.999965i	$-0.502671\pi$
$192$		0			0
$193$	−10.8660	+	18.8205i	−0.782154	+	1.35473i	0.148531	+	0.988908i	$0.452545\pi$
$193$	−10.8660	+	18.8205i	−0.782154	+	1.35473i	−0.930685	+	0.365822i	$0.880788\pi$
$194$		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
$203$	−1.73205	−	6.46410i	−0.121566	−	0.453691i
$204$		0			0
$205$	0.803848	−	3.00000i	0.0561432	−	0.209529i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−7.33013	−	12.6962i	−0.507035	−	0.878211i
$210$		0			0
$211$	−4.09808	+	1.09808i	−0.282123	+	0.0755947i	−0.397106	−	0.917773i	$-0.629985\pi$
$211$	−4.09808	+	1.09808i	−0.282123	+	0.0755947i	0.114983	+	0.993367i	$0.463319\pi$
$212$		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
$221$	−19.2942	+	5.16987i	−1.29787	+	0.347763i
$222$		0			0
$223$	8.02628	+	13.9019i	0.537479	+	0.930942i	0.999039	+	0.0438324i	$0.0139568\pi$
$223$	8.02628	+	13.9019i	0.537479	+	0.930942i	−0.461559	+	0.887109i	$0.652710\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−0.571797	+	2.13397i	−0.0379515	+	0.141637i	−0.982302	−	0.187304i	$-0.940025\pi$
$227$	−0.571797	+	2.13397i	−0.0379515	+	0.141637i	0.944351	+	0.328941i	$0.106692\pi$
$228$		0			0
$229$	−1.83013	−	6.83013i	−0.120938	−	0.451347i	0.878724	−	0.477330i	$-0.158395\pi$
$229$	−1.83013	−	6.83013i	−0.120938	−	0.451347i	−0.999662	+	0.0259823i	$0.991729\pi$
$230$		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	0.488784	+	0.872405i	$0.337441\pi$
$239$	−7.90192	+	13.6865i	−0.511133	+	0.885308i	−0.999917	+	0.0129033i	$0.995893\pi$
$240$		0			0
$241$	−11.5981	−	20.0885i	−0.747098	−	1.29401i	−0.949208	−	0.314649i	$-0.898113\pi$
$241$	−11.5981	−	20.0885i	−0.747098	−	1.29401i	0.202110	−	0.979363i	$-0.435220\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	0.464102	+	0.124356i	0.0296504	+	0.00794479i
$246$		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	0.406364	+	0.913711i	$0.366796\pi$
$257$	−9.42820	+	16.3301i	−0.588115	+	1.01865i	−0.994479	+	0.104934i	$0.966537\pi$
$258$		0			0
$259$	4.73205	−	17.6603i	0.294035	−	1.09735i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	2.49038	−	1.43782i	0.153563	−	0.0886599i	−0.421249	−	0.906945i	$-0.638408\pi$
$263$	2.49038	−	1.43782i	0.153563	−	0.0886599i	0.574813	+	0.818285i	$0.305075\pi$
$264$		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
$275$	−16.6244	+	4.45448i	−1.00249	+	0.268615i
$276$		0			0
$277$	25.2224	+	6.75833i	1.51547	+	0.406069i	0.918247	−	0.396007i	$-0.129605\pi$
$277$	25.2224	+	6.75833i	1.51547	+	0.406069i	0.597222	+	0.802076i	$0.296271\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−8.66025	+	5.00000i	−0.516627	+	0.298275i	−0.735554	−	0.677466i	$-0.763078\pi$
$281$	−8.66025	+	5.00000i	−0.516627	+	0.298275i	0.218926	+	0.975741i	$0.429745\pi$
$282$		0			0
$283$	−5.24167	−	19.5622i	−0.311585	−	1.16285i	−0.927127	−	0.374747i	$-0.877730\pi$
$283$	−5.24167	−	19.5622i	−0.311585	−	1.16285i	0.615542	−	0.788104i	$-0.288937\pi$
$284$		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
$293$	−1.43782	−	5.36603i	−0.0839985	−	0.313487i	0.911124	−	0.412132i	$-0.135216\pi$
$293$	−1.43782	−	5.36603i	−0.0839985	−	0.313487i	−0.995123	+	0.0986454i	$0.968549\pi$
$294$		0			0
$295$	6.80385	−	3.92820i	0.396135	−	0.228709i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−15.9282	−	4.26795i	−0.921152	−	0.246822i
$300$		0			0
$301$	22.7583	−	6.09808i	1.31177	−	0.351487i
$302$		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	$-0.118333\pi$
$311$	19.0981	+	11.0263i	1.08295	+	0.625243i	0.151261	+	0.988494i	$0.451667\pi$
$312$		0			0
$313$	−18.6506	+	10.7679i	−1.05420	+	0.608640i	−0.923821	−	0.382824i	$-0.874951\pi$
$313$	−18.6506	+	10.7679i	−1.05420	+	0.608640i	−0.130375	+	0.991465i	$0.541618\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−5.50962	+	20.5622i	−0.309451	+	1.15489i	0.619595	+	0.784922i	$0.287297\pi$
$317$	−5.50962	+	20.5622i	−0.309451	+	1.15489i	−0.929046	+	0.369965i	$0.879370\pi$
$318$		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
$331$	0.0980762	+	0.0262794i	0.00539076	+	0.00144445i	0.261513	−	0.965200i	$-0.415778\pi$
$331$	0.0980762	+	0.0262794i	0.00539076	+	0.00144445i	−0.256123	+	0.966644i	$0.582445\pi$
$332$		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	−0.515445	−	0.856923i	$-0.672373\pi$
$337$	8.89230	−	15.4019i	0.484395	−	0.838996i	0.999839	+	0.0179267i	$0.00570654\pi$
$338$		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
$347$	4.72243	+	17.6244i	0.253513	+	0.946125i	0.968911	+	0.247408i	$0.0795787\pi$
$347$	4.72243	+	17.6244i	0.253513	+	0.946125i	−0.715398	+	0.698717i	$0.753755\pi$
$348$		0			0
$349$	−4.26795	+	15.9282i	−0.228458	+	0.852617i	0.752531	+	0.658556i	$0.228833\pi$
$349$	−4.26795	+	15.9282i	−0.228458	+	0.852617i	−0.980989	+	0.194061i	$0.937834\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−7.16025	−	12.4019i	−0.381102	−	0.660088i	0.610118	−	0.792310i	$-0.291122\pi$
$353$	−7.16025	−	12.4019i	−0.381102	−	0.660088i	−0.991220	+	0.132223i	$0.957789\pi$
$354$		0			0
$355$	−2.92820	+	0.784610i	−0.155413	+	0.0416428i
$356$		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
$365$	6.26795	−	1.67949i	0.328079	−	0.0879086i
$366$		0			0
$367$	−14.1244	−	24.4641i	−0.737285	−	1.27702i	−0.953713	−	0.300717i	$-0.902774\pi$
$367$	−14.1244	−	24.4641i	−0.737285	−	1.27702i	0.216428	−	0.976299i	$-0.430559\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	7.46410	−	27.8564i	0.387517	−	1.44623i
$372$		0			0
$373$	−7.36603	−	27.4904i	−0.381398	−	1.42340i	−0.843767	−	0.536710i	$-0.819667\pi$
$373$	−7.36603	−	27.4904i	−0.381398	−	1.42340i	0.462368	−	0.886688i	$-0.347000\pi$
$374$		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	−0.985170	−	0.171581i	$-0.945113\pi$
$383$	−6.73205	+	11.6603i	−0.343992	+	0.595811i	0.641178	+	0.767392i	$0.278446\pi$
$384$		0			0
$385$	−6.19615	−	10.7321i	−0.315785	−	0.546956i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−19.7583	−	5.29423i	−1.00179	−	0.268428i	−0.279593	−	0.960119i	$-0.590200\pi$
$389$	−19.7583	−	5.29423i	−1.00179	−	0.268428i	−0.722194	+	0.691691i	$0.756866\pi$
$390$		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	−0.817742	−	0.575584i	$-0.804775\pi$
$401$	1.79423	−	3.10770i	0.0895995	−	0.155191i	0.907342	+	0.420393i	$0.138108\pi$
$402$		0			0
$403$	−0.483340	+	1.80385i	−0.0240769	+	0.0898560i
$404$		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	$-0.626137\pi$
$409$	−27.8660	−	16.0885i	−1.37789	−	0.795523i	−0.991905	+	0.126984i	$0.959470\pi$
$410$		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
$419$	−6.63397	+	1.77757i	−0.324091	+	0.0868399i	−0.417196	−	0.908816i	$-0.636987\pi$
$419$	−6.63397	+	1.77757i	−0.324091	+	0.0868399i	0.0931055	+	0.995656i	$0.470321\pi$
$420$		0			0
$421$	−30.5885	−	8.19615i	−1.49079	−	0.399456i	−0.580786	−	0.814056i	$-0.697255\pi$
$421$	−30.5885	−	8.19615i	−1.49079	−	0.399456i	−0.910004	+	0.414600i	$0.863922\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	19.5000	−	11.2583i	0.945889	−	0.546109i
$426$		0			0
$427$	8.19615	+	30.5885i	0.396640	+	1.48028i
$428$		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
$437$	4.09808	+	15.2942i	0.196038	+	0.731622i
$438$		0			0
$439$	−22.8564	+	13.1962i	−1.09088	+	0.629818i	−0.933810	−	0.357770i	$-0.883537\pi$
$439$	−22.8564	+	13.1962i	−1.09088	+	0.629818i	−0.157067	+	0.987588i	$0.550204\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	17.2583	+	4.62436i	0.819968	+	0.219710i	0.644332	−	0.764745i	$-0.277135\pi$
$443$	17.2583	+	4.62436i	0.819968	+	0.219710i	0.175636	+	0.984455i	$0.443802\pi$
$444$		0			0
$445$	−2.00000	+	0.535898i	−0.0948091	+	0.0254040i
$446$		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	$-0.686093\pi$
$457$	−2.25833	+	1.30385i	−0.105640	+	0.0609914i	0.446249	+	0.894909i	$0.352760\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	9.56218	−	35.6865i	0.445355	−	1.66209i	−0.269642	−	0.962961i	$-0.586906\pi$
$461$	9.56218	−	35.6865i	0.445355	−	1.66209i	0.714997	−	0.699127i	$-0.246428\pi$
$462$		0			0
$463$	−1.19615	+	2.07180i	−0.0555899	+	0.0962846i	−0.892481	−	0.451085i	$-0.851037\pi$
$463$	−1.19615	+	2.07180i	−0.0555899	+	0.0962846i	0.836891	+	0.547369i	$0.184371\pi$
$464$		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
$475$	12.6962	+	3.40192i	0.582539	+	0.156091i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	4.16987	+	7.22243i	0.190526	+	0.330001i	0.945425	−	0.325840i	$-0.105647\pi$
$479$	4.16987	+	7.22243i	0.190526	+	0.330001i	−0.754898	+	0.655842i	$0.772314\pi$
$480$		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
$491$	3.72243	+	13.8923i	0.167991	+	0.626951i	0.997640	+	0.0686652i	$0.0218740\pi$
$491$	3.72243	+	13.8923i	0.167991	+	0.626951i	−0.829649	+	0.558286i	$0.811459\pi$
$492$		0			0
$493$	3.63397	−	13.5622i	0.163666	−	0.610810i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	4.00000	+	6.92820i	0.179425	+	0.310772i
$498$		0			0
$499$	−8.69615	+	2.33013i	−0.389293	+	0.104311i	−0.448156	−	0.893955i	$-0.647919\pi$
$499$	−8.69615	+	2.33013i	−0.389293	+	0.104311i	0.0588630	+	0.998266i	$0.481252\pi$
$500$		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
$509$	11.4641	−	3.07180i	0.508137	−	0.136155i	0.00436335	−	0.999990i	$-0.498611\pi$
$509$	11.4641	−	3.07180i	0.508137	−	0.136155i	0.503774	+	0.863835i	$0.331944\pi$
$510$		0			0
$511$	−8.56218	−	14.8301i	−0.378768	−	0.656046i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−4.05256	+	15.1244i	−0.178577	+	0.666459i
$516$		0			0
$517$	−8.68653	−	32.4186i	−0.382033	−	1.42577i
$518$		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
$533$	10.0981	+	2.70577i	0.437396	+	0.117200i
$534$		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
$547$	8.74167	−	32.6244i	0.373767	−	1.39492i	−0.481371	−	0.876517i	$-0.659861\pi$
$547$	8.74167	−	32.6244i	0.373767	−	1.39492i	0.855138	−	0.518400i	$-0.173472\pi$
$548$		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
$563$	26.9904	−	7.23205i	1.13751	−	0.304795i	0.359560	−	0.933122i	$-0.382927\pi$
$563$	26.9904	−	7.23205i	1.13751	−	0.304795i	0.777949	+	0.628327i	$0.216260\pi$
$564$		0			0
$565$	13.8564	+	3.71281i	0.582943	+	0.156199i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−18.4019	+	10.6244i	−0.771449	+	0.445396i	−0.833391	−	0.552684i	$-0.813604\pi$
$569$	−18.4019	+	10.6244i	−0.771449	+	0.445396i	0.0619424	+	0.998080i	$0.480270\pi$
$570$		0			0
$571$	−0.892305	−	3.33013i	−0.0373418	−	0.139361i	0.944738	−	0.327825i	$-0.106316\pi$
$571$	−0.892305	−	3.33013i	−0.0373418	−	0.139361i	−0.982080	+	0.188464i	$0.939649\pi$
$572$		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
$581$	−1.00000	−	3.73205i	−0.0414870	−	0.154832i
$582$		0			0
$583$	−40.0526	+	23.1244i	−1.65881	+	0.957713i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−26.9904	−	7.23205i	−1.11401	−	0.298499i	−0.345554	−	0.938399i	$-0.612309\pi$
$587$	−26.9904	−	7.23205i	−1.11401	−	0.298499i	−0.768458	+	0.639900i	$0.778976\pi$
$588$		0			0
$589$	1.73205	−	0.464102i	0.0713679	−	0.0191230i
$590$		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	$-0.419382\pi$
$599$	−11.3205	−	6.53590i	−0.462543	−	0.267050i	−0.713113	+	0.701049i	$0.752715\pi$
$600$		0			0
$601$	20.5526	−	11.8660i	0.838356	−	0.484025i	−0.0183488	−	0.999832i	$-0.505841\pi$
$601$	20.5526	−	11.8660i	0.838356	−	0.484025i	0.856705	+	0.515806i	$0.172508\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−2.19615	+	8.19615i	−0.0892863	+	0.333221i
$606$		0			0
$607$	8.58846	−	14.8756i	0.348595	−	0.603784i	−0.637405	−	0.770529i	$-0.719992\pi$
$607$	8.58846	−	14.8756i	0.348595	−	0.603784i	0.986000	+	0.166745i	$0.0533256\pi$
$608$		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	$-0.636910\pi$
$617$	−35.0885	−	20.2583i	−1.41261	−	0.815570i	−0.995633	+	0.0933485i	$0.970243\pi$
$618$		0			0
$619$	15.5981	+	4.17949i	0.626940	+	0.167988i	0.558281	−	0.829652i	$-0.311461\pi$
$619$	15.5981	+	4.17949i	0.626940	+	0.167988i	0.0686590	+	0.997640i	$0.478128\pi$
$620$		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
$635$	−1.12436	−	4.19615i	−0.0446187	−	0.166519i
$636$		0			0
$637$	−0.418584	+	1.56218i	−0.0165849	+	0.0618957i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	19.7942	+	34.2846i	0.781825	+	1.35416i	0.930878	+	0.365331i	$0.119044\pi$
$641$	19.7942	+	34.2846i	0.781825	+	1.35416i	−0.149053	+	0.988829i	$0.547622\pi$
$642$		0			0
$643$	−8.76795	+	2.34936i	−0.345774	+	0.0926499i	−0.427527	−	0.904003i	$-0.640615\pi$
$643$	−8.76795	+	2.34936i	−0.345774	+	0.0926499i	0.0817525	+	0.996653i	$0.473948\pi$
$644$		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
$653$	27.4904	−	7.36603i	1.07578	−	0.288255i	0.322915	−	0.946428i	$-0.395337\pi$
$653$	27.4904	−	7.36603i	1.07578	−	0.288255i	0.752867	+	0.658173i	$0.228671\pi$
$654$		0			0
$655$	−4.19615	−	7.26795i	−0.163957	−	0.283982i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	4.02628	−	15.0263i	0.156842	−	0.585341i	−0.842099	−	0.539323i	$-0.818680\pi$
$659$	4.02628	−	15.0263i	0.156842	−	0.585341i	0.998941	−	0.0460178i	$-0.0146531\pi$
$660$		0			0
$661$	2.19615	+	8.19615i	0.0854204	+	0.318793i	0.995393	−	0.0958740i	$-0.0305646\pi$
$661$	2.19615	+	8.19615i	0.0854204	+	0.318793i	−0.909973	+	0.414667i	$0.863898\pi$
$662$		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	0.952514	+	0.304495i	$0.0984877\pi$
$673$	19.1962	+	33.2487i	0.739957	+	1.28164i	−0.212557	+	0.977149i	$0.568179\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	4.73205	+	1.26795i	0.181867	+	0.0487312i	0.348603	−	0.937270i	$-0.386656\pi$
$677$	4.73205	+	1.26795i	0.181867	+	0.0487312i	−0.166736	+	0.986002i	$0.553323\pi$
$678$		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
$691$	2.49038	−	9.29423i	0.0947386	−	0.353569i	−0.902241	−	0.431232i	$-0.858079\pi$
$691$	2.49038	−	9.29423i	0.0947386	−	0.353569i	0.996980	+	0.0776628i	$0.0247457\pi$
$692$		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
$707$	5.46410	−	1.46410i	0.205499	−	0.0550632i
$708$		0			0
$709$	36.5885	+	9.80385i	1.37411	+	0.368191i	0.868978	−	0.494852i	$-0.164778\pi$
$709$	36.5885	+	9.80385i	1.37411	+	0.368191i	0.505131	+	0.863043i	$0.331444\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	2.19615	−	1.26795i	0.0822466	−	0.0474851i
$714$		0			0
$715$	4.09103	+	15.2679i	0.152996	+	0.570989i
$716$		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
$725$	−2.49038	−	9.29423i	−0.0924904	−	0.345179i
$726$		0			0
$727$	−28.8109	+	16.6340i	−1.06854	+	0.616920i	−0.927781	−	0.373124i	$-0.878286\pi$
$727$	−28.8109	+	16.6340i	−1.06854	+	0.616920i	−0.140755	+	0.990044i	$0.544953\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	47.7487	+	12.7942i	1.76605	+	0.473212i
$732$		0			0
$733$	−11.0263	+	2.95448i	−0.407265	+	0.109126i	−0.456635	−	0.889654i	$-0.650945\pi$
$733$	−11.0263	+	2.95448i	−0.407265	+	0.109126i	0.0493698	+	0.998781i	$0.484279\pi$
$734$		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	$-0.509045\pi$
$743$	−24.7583	−	14.2942i	−0.908295	−	0.524404i	−0.879882	+	0.475192i	$0.842379\pi$
$744$		0			0
$745$	−7.26795	+	4.19615i	−0.266277	+	0.153735i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	12.4904	−	46.6147i	0.456389	−	1.70327i
$750$		0			0
$751$	8.85641	−	15.3397i	0.323175	−	0.559755i	−0.657966	−	0.753047i	$-0.728583\pi$
$751$	8.85641	−	15.3397i	0.323175	−	0.559755i	0.981141	+	0.193292i	$0.0619165\pi$
$752$		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	0.979104	+	0.203363i	$0.0651870\pi$
$761$	45.3731	+	26.1962i	1.64477	+	0.949610i	0.665669	+	0.746247i	$0.268146\pi$
$762$		0			0
$763$	40.0526	+	10.7321i	1.45000	+	0.388526i
$764$		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	0.490604	+	0.871383i	$0.336776\pi$
$769$	−14.1244	+	24.4641i	−0.509337	+	0.882198i	−0.999942	+	0.0108155i	$0.996557\pi$
$770$		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
$779$	−2.59808	−	9.69615i	−0.0930857	−	0.347401i
$780$		0			0
$781$	3.32051	−	12.3923i	0.118817	−	0.443432i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−2.53590	−	4.39230i	−0.0905101	−	0.156768i
$786$		0			0
$787$	−40.3468	+	10.8109i	−1.43821	+	0.385367i	−0.891906	−	0.452222i	$-0.850632\pi$
$787$	−40.3468	+	10.8109i	−1.43821	+	0.385367i	−0.546302	+	0.837588i	$0.683965\pi$
$788$		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
$797$	30.5167	−	8.17691i	1.08096	−	0.289641i	0.325968	−	0.945381i	$-0.394310\pi$
$797$	30.5167	−	8.17691i	1.08096	−	0.289641i	0.754987	+	0.655740i	$0.227643\pi$
$798$		0			0
$799$	21.9545	+	38.0263i	0.776694	+	1.34527i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−7.10770	+	26.5263i	−0.250825	+	0.936092i
$804$		0			0
$805$	3.46410	+	12.9282i	0.122094	+	0.455659i
$806$		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
$821$	10.3660	+	2.77757i	0.361777	+	0.0969378i	0.435129	−	0.900368i	$-0.356703\pi$
$821$	10.3660	+	2.77757i	0.361777	+	0.0969378i	−0.0733518	+	0.997306i	$0.523370\pi$
$822$		0			0
$823$	7.26795	+	4.19615i	0.253345	+	0.146269i	0.621295	−	0.783577i	$-0.286607\pi$
$823$	7.26795	+	4.19615i	0.253345	+	0.146269i	−0.367950	+	0.929846i	$0.619940\pi$
$824$		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
$835$	2.00000	−	7.46410i	0.0692129	−	0.258306i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−23.4449	+	13.5359i	−0.809407	+	0.467311i	−0.846750	−	0.531991i	$-0.821444\pi$
$839$	−23.4449	+	13.5359i	−0.809407	+	0.467311i	0.0373432	+	0.999303i	$0.488111\pi$
$840$		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
$851$	30.5885	−	8.19615i	1.04856	−	0.280960i
$852$		0			0
$853$	1.63397	+	0.437822i	0.0559462	+	0.0149907i	0.286684	−	0.958025i	$-0.407447\pi$
$853$	1.63397	+	0.437822i	0.0559462	+	0.0149907i	−0.230737	+	0.973016i	$0.574114\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−44.9090	+	25.9282i	−1.53406	+	0.885691i	−0.534892	+	0.844920i	$0.679648\pi$
$857$	−44.9090	+	25.9282i	−1.53406	+	0.885691i	−0.999169	+	0.0407704i	$0.987019\pi$
$858$		0			0
$859$	3.82051	+	14.2583i	0.130354	+	0.486488i	0.999974	−	0.00723407i	$-0.00230270\pi$
$859$	3.82051	+	14.2583i	0.130354	+	0.486488i	−0.869620	+	0.493722i	$0.835636\pi$
$860$		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
$869$	−13.6077	−	50.7846i	−0.461609	−	1.72275i
$870$		0			0
$871$	20.6147	−	11.9019i	0.698504	−	0.403281i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	24.3923	+	6.53590i	0.824610	+	0.220954i
$876$		0			0
$877$	31.5885	−	8.46410i	1.06667	−	0.285812i	0.317544	−	0.948244i	$-0.397142\pi$
$877$	31.5885	−	8.46410i	1.06667	−	0.285812i	0.749123	+	0.662431i	$0.230475\pi$
$878$		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	$-0.388375\pi$
$887$	−8.87564	−	5.12436i	−0.298015	−	0.172059i	−0.641551	+	0.767081i	$0.721709\pi$
$888$		0			0
$889$	−9.92820	+	5.73205i	−0.332981	+	0.192247i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−6.63397	+	24.7583i	−0.221997	+	0.828506i
$894$		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
$907$	−4.50000	−	1.20577i	−0.149420	−	0.0400370i	0.183334	−	0.983051i	$-0.441311\pi$
$907$	−4.50000	−	1.20577i	−0.149420	−	0.0400370i	−0.332754	+	0.943014i	$0.607978\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	2.46410	+	4.26795i	0.0816393	+	0.141403i	0.903954	−	0.427629i	$-0.140651\pi$
$911$	2.46410	+	4.26795i	0.0816393	+	0.141403i	−0.822315	+	0.569033i	$0.807318\pi$
$912$		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
$923$	−2.64102	−	9.85641i	−0.0869301	−	0.324428i
$924$		0			0
$925$	6.80385	−	25.3923i	0.223709	−	0.834894i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	11.5359	+	19.9808i	0.378481	+	0.655548i	0.990841	−	0.135031i	$-0.0431134\pi$
$929$	11.5359	+	19.9808i	0.378481	+	0.655548i	−0.612361	+	0.790578i	$0.709780\pi$
$930$		0			0
$931$	1.50000	−	0.401924i	0.0491605	−	0.0131725i
$932$		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
$941$	−6.73205	+	1.80385i	−0.219459	+	0.0588038i	−0.366873	−	0.930271i	$-0.619572\pi$
$941$	−6.73205	+	1.80385i	−0.219459	+	0.0588038i	0.147414	+	0.989075i	$0.452905\pi$
$942$		0			0
$943$	−7.09808	−	12.2942i	−0.231145	−	0.400355i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−10.9904	+	41.0167i	−0.357139	+	1.33286i	0.520631	+	0.853782i	$0.325697\pi$
$947$	−10.9904	+	41.0167i	−0.357139	+	1.33286i	−0.877771	+	0.479081i	$0.840970\pi$
$948$		0			0
$949$	5.65321	+	21.0981i	0.183511	+	0.684873i
$950$		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
$965$	−21.7321	−	5.82309i	−0.699579	−	0.187452i
$966$		0			0
$967$	17.8301	+	10.2942i	0.573378	+	0.331040i	0.758497	−	0.651676i	$-0.225934\pi$
$967$	17.8301	+	10.2942i	0.573378	+	0.331040i	−0.185119	+	0.982716i	$0.559267\pi$
$968$		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	−0.260559	−	0.965458i	$-0.583907\pi$
$977$	22.0622	−	38.2128i	0.705832	−	1.22254i	0.966391	−	0.257078i	$-0.0827598\pi$
$978$		0			0
$979$	2.26795	−	8.46410i	0.0724840	−	0.270514i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−13.8564	+	8.00000i	−0.441951	+	0.255160i	−0.704425	−	0.709779i	$-0.748795\pi$
$983$	−13.8564	+	8.00000i	−0.441951	+	0.255160i	0.262474	+	0.964939i	$0.415462\pi$
$984$		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
$995$	25.1244	−	6.73205i	0.796496	−	0.213420i
$996$		0			0
$997$	−29.3923	−	7.87564i	−0.930864	−	0.249424i	−0.238641	−	0.971108i	$-0.576702\pi$
$997$	−29.3923	−	7.87564i	−0.930864	−	0.249424i	−0.692223	+	0.721684i	$0.743368\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1728.2.bc.d.145.1		4
3.2	odd	2		576.2.bb.d.337.1		4
4.3	odd	2		432.2.y.c.253.1		4
9.2	odd	6		576.2.bb.c.529.1		4
9.7	even	3		1728.2.bc.a.721.1		4
12.11	even	2		144.2.x.b.13.1	✓	4
16.5	even	4		1728.2.bc.a.1009.1		4
16.11	odd	4		432.2.y.b.37.1		4
36.7	odd	6		432.2.y.b.397.1		4
36.11	even	6		144.2.x.c.61.1	yes	4
48.5	odd	4		576.2.bb.c.49.1		4
48.11	even	4		144.2.x.c.85.1	yes	4
144.11	even	12		144.2.x.b.133.1	yes	4
144.43	odd	12		432.2.y.c.181.1		4
144.101	odd	12		576.2.bb.d.241.1		4
144.133	even	12	inner	1728.2.bc.d.1585.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
144.2.x.b.13.1	✓	4	12.11	even	2
144.2.x.b.133.1	yes	4	144.11	even	12
144.2.x.c.61.1	yes	4	36.11	even	6
144.2.x.c.85.1	yes	4	48.11	even	4
432.2.y.b.37.1		4	16.11	odd	4
432.2.y.b.397.1		4	36.7	odd	6
432.2.y.c.181.1		4	144.43	odd	12
432.2.y.c.253.1		4	4.3	odd	2
576.2.bb.c.49.1		4	48.5	odd	4
576.2.bb.c.529.1		4	9.2	odd	6
576.2.bb.d.241.1		4	144.101	odd	12
576.2.bb.d.337.1		4	3.2	odd	2
1728.2.bc.a.721.1		4	9.7	even	3
1728.2.bc.a.1009.1		4	16.5	even	4
1728.2.bc.d.145.1		4	1.1	even	1	trivial
1728.2.bc.d.1585.1		4	144.133	even	12	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+(0.267949 + 1.00000i) q^{5} +(2.36603 - 1.36603i) q^{7} +O(q^{10})\)	\(q+(0.267949 + 1.00000i) q^{5} +(2.36603 - 1.36603i) q^{7} +(-4.23205 - 1.13397i) q^{11} +(-3.36603 + 0.901924i) q^{13} +5.73205 q^{17} +(2.36603 + 2.36603i) q^{19} +(4.09808 + 2.36603i) q^{23} +(3.40192 - 1.96410i) q^{25} +(0.633975 - 2.36603i) q^{29} +(0.267949 - 0.464102i) q^{31} +(2.00000 + 2.00000i) q^{35} +(4.73205 - 4.73205i) q^{37} +(-2.59808 - 1.50000i) q^{41} +(8.33013 + 2.23205i) q^{43} +(3.83013 + 6.63397i) q^{47} +(0.232051 - 0.401924i) q^{49} +(7.46410 - 7.46410i) q^{53} -4.53590i q^{55} +(-1.96410 - 7.33013i) q^{59} +(-3.00000 + 11.1962i) q^{61} +(-1.80385 - 3.12436i) q^{65} +(-6.59808 + 1.76795i) q^{67} +2.92820i q^{71} -6.26795i q^{73} +(-11.5622 + 3.09808i) q^{77} +(6.00000 + 10.3923i) q^{79} +(0.366025 - 1.36603i) q^{83} +(1.53590 + 5.73205i) q^{85} +2.00000i q^{89} +(-6.73205 + 6.73205i) q^{91} +(-1.73205 + 3.00000i) q^{95} +(-5.86603 - 10.1603i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{5} + 6 q^{7}+O(q^{10}) \) 4 * q + 8 * q^5 + 6 * q^7	\( 4 q + 8 q^{5} + 6 q^{7} - 10 q^{11} - 10 q^{13} + 16 q^{17} + 6 q^{19} + 6 q^{23} + 24 q^{25} + 6 q^{29} + 8 q^{31} + 8 q^{35} + 12 q^{37} + 16 q^{43} - 2 q^{47} - 6 q^{49} + 16 q^{53} + 6 q^{59} - 12 q^{61} - 28 q^{65} - 16 q^{67} - 22 q^{77} + 24 q^{79} - 2 q^{83} + 20 q^{85} - 20 q^{91} - 20 q^{97}+O(q^{100}) \) 4 * q + 8 * q^5 + 6 * q^7 - 10 * q^11 - 10 * q^13 + 16 * q^17 + 6 * q^19 + 6 * q^23 + 24 * q^25 + 6 * q^29 + 8 * q^31 + 8 * q^35 + 12 * q^37 + 16 * q^43 - 2 * q^47 - 6 * q^49 + 16 * q^53 + 6 * q^59 - 12 * q^61 - 28 * q^65 - 16 * q^67 - 22 * q^77 + 24 * q^79 - 2 * q^83 + 20 * q^85 - 20 * q^91 - 20 * q^97

Introduction

L-functions

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