LMFDB - Embedded newform 1520.2.bq.b.1471.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1520,2,Mod(31,1520)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1520.31"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1520, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 0, 0, 5])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1520 = 2^{4} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1520.bq (of order $6$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,-2,0,-1,0,0,0,-1,0,0,0,0,0,-2,0,0,0,-7] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$12.1372611072$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			1471.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	1520.1471
Dual form			1520.2.bq.b.31.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-1.00000 + 1.73205i) q^{3} +(-0.500000 + 0.866025i) q^{5} +1.73205i q^{7} +(-0.500000 - 0.866025i) q^{9} -5.19615i q^{11} +(-1.00000 - 1.73205i) q^{15} +(-3.50000 + 2.59808i) q^{19} +(-3.00000 - 1.73205i) q^{21} +(-7.50000 + 4.33013i) q^{23} +(-0.500000 - 0.866025i) q^{25} -4.00000 q^{27} +(3.00000 - 1.73205i) q^{29} -4.00000 q^{31} +(9.00000 + 5.19615i) q^{33} +(-1.50000 - 0.866025i) q^{35} -1.73205i q^{37} +(-7.50000 - 4.33013i) q^{41} +1.00000 q^{45} +(-3.00000 + 1.73205i) q^{47} +4.00000 q^{49} +(4.50000 - 2.59808i) q^{53} +(4.50000 + 2.59808i) q^{55} +(-1.00000 - 8.66025i) q^{57} +(6.00000 - 10.3923i) q^{59} +(-5.00000 - 8.66025i) q^{61} +(1.50000 - 0.866025i) q^{63} +(2.00000 + 3.46410i) q^{67} -17.3205i q^{69} +(3.00000 - 5.19615i) q^{71} +(1.00000 - 1.73205i) q^{73} +2.00000 q^{75} +9.00000 q^{77} +(-2.00000 + 3.46410i) q^{79} +(5.50000 - 9.52628i) q^{81} -10.3923i q^{83} +6.92820i q^{87} +(-13.5000 + 7.79423i) q^{89} +(4.00000 - 6.92820i) q^{93} +(-0.500000 - 4.33013i) q^{95} +(-4.50000 + 2.59808i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - q^{5} - q^{9} - 2 q^{15} - 7 q^{19} - 6 q^{21} - 15 q^{23} - q^{25} - 8 q^{27} + 6 q^{29} - 8 q^{31} + 18 q^{33} - 3 q^{35} - 15 q^{41} + 2 q^{45} - 6 q^{47} + 8 q^{49} + 9 q^{53} + 9 q^{55}+ \cdots - 9 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 - q^5 - q^9 - 2 * q^15 - 7 * q^19 - 6 * q^21 - 15 * q^23 - q^25 - 8 * q^27 + 6 * q^29 - 8 * q^31 + 18 * q^33 - 3 * q^35 - 15 * q^41 + 2 * q^45 - 6 * q^47 + 8 * q^49 + 9 * q^53 + 9 * q^55 - 2 * q^57 + 12 * q^59 - 10 * q^61 + 3 * q^63 + 4 * q^67 + 6 * q^71 + 2 * q^73 + 4 * q^75 + 18 * q^77 - 4 * q^79 + 11 * q^81 - 27 * q^89 + 8 * q^93 - q^95 - 9 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1520\mathbb{Z}\right)^\times$.

$n$	$191$	$401$	$1141$	$1217$
$\chi(n)$	$-1$	$e\left(\frac{1}{6}\right)$	$1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	−1.00000	+	1.73205i	−0.577350	+	1.00000i	0.418432	+	0.908248i	$0.362580\pi$
$3$	−1.00000	+	1.73205i	−0.577350	+	1.00000i	−0.995782	+	0.0917517i	$0.970753\pi$
$4$		0			0
$5$	−0.500000	+	0.866025i	−0.223607	+	0.387298i
$5$	−0.500000	+	0.866025i	−0.223607	+	0.387298i
$6$		0			0
$7$			1.73205i			0.654654i	0.944911	+	0.327327i	$0.106148\pi$
$7$			1.73205i			0.654654i	−0.944911	+	0.327327i	$0.893852\pi$
$8$		0			0
$9$	−0.500000	−	0.866025i	−0.166667	−	0.288675i
$10$		0			0
$11$		−	5.19615i		−	1.56670i	−0.621582	−	0.783349i	$-0.713510\pi$
$11$		−	5.19615i		−	1.56670i	0.621582	−	0.783349i	$-0.286490\pi$
$12$		0			0
$13$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$13$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$14$		0			0
$15$	−1.00000	−	1.73205i	−0.258199	−	0.447214i
$16$		0			0
$17$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$17$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$18$		0			0
$19$	−3.50000	+	2.59808i	−0.802955	+	0.596040i
$19$	−3.50000	+	2.59808i	−0.802955	+	0.596040i
$20$		0			0
$21$	−3.00000	−	1.73205i	−0.654654	−	0.377964i
$22$		0			0
$23$	−7.50000	+	4.33013i	−1.56386	+	0.902894i	−0.566997	+	0.823720i	$0.691895\pi$
$23$	−7.50000	+	4.33013i	−1.56386	+	0.902894i	−0.996861	+	0.0791743i	$0.974772\pi$
$24$		0			0
$25$	−0.500000	−	0.866025i	−0.100000	−	0.173205i
$26$		0			0
$27$	−4.00000			−0.769800
$28$		0			0
$29$	3.00000	−	1.73205i	0.557086	−	0.321634i	−0.194889	−	0.980825i	$-0.562435\pi$
$29$	3.00000	−	1.73205i	0.557086	−	0.321634i	0.751975	+	0.659192i	$0.229101\pi$
$30$		0			0
$31$	−4.00000			−0.718421			−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000			−0.718421			−0.359211	+	0.933257i	$0.616954\pi$
$32$		0			0
$33$	9.00000	+	5.19615i	1.56670	+	0.904534i
$34$		0			0
$35$	−1.50000	−	0.866025i	−0.253546	−	0.146385i
$36$		0			0
$37$		−	1.73205i		−	0.284747i	−0.989813	−	0.142374i	$-0.954527\pi$
$37$		−	1.73205i		−	0.284747i	0.989813	−	0.142374i	$-0.0454735\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−7.50000	−	4.33013i	−1.17130	−	0.676252i	−0.217317	−	0.976101i	$-0.569730\pi$
$41$	−7.50000	−	4.33013i	−1.17130	−	0.676252i	−0.953987	+	0.299849i	$0.903064\pi$
$42$		0			0
$43$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$43$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$44$		0			0
$45$	1.00000			0.149071
$46$		0			0
$47$	−3.00000	+	1.73205i	−0.437595	+	0.252646i	−0.702577	−	0.711608i	$-0.747967\pi$
$47$	−3.00000	+	1.73205i	−0.437595	+	0.252646i	0.264982	+	0.964253i	$0.414634\pi$
$48$		0			0
$49$	4.00000			0.571429
$50$		0			0
$51$		0			0
$52$		0			0
$53$	4.50000	−	2.59808i	0.618123	−	0.356873i	−0.158015	−	0.987437i	$-0.550509\pi$
$53$	4.50000	−	2.59808i	0.618123	−	0.356873i	0.776138	+	0.630563i	$0.217176\pi$
$54$		0			0
$55$	4.50000	+	2.59808i	0.606780	+	0.350325i
$56$		0			0
$57$	−1.00000	−	8.66025i	−0.132453	−	1.14708i
$58$		0			0
$59$	6.00000	−	10.3923i	0.781133	−	1.35296i	−0.150148	−	0.988663i	$-0.547975\pi$
$59$	6.00000	−	10.3923i	0.781133	−	1.35296i	0.931282	−	0.364299i	$-0.118692\pi$
$60$		0			0
$61$	−5.00000	−	8.66025i	−0.640184	−	1.10883i	−0.985391	−	0.170305i	$-0.945525\pi$
$61$	−5.00000	−	8.66025i	−0.640184	−	1.10883i	0.345207	−	0.938527i	$-0.387809\pi$
$62$		0			0
$63$	1.50000	−	0.866025i	0.188982	−	0.109109i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	2.00000	+	3.46410i	0.244339	+	0.423207i	0.961946	−	0.273241i	$-0.0880957\pi$
$67$	2.00000	+	3.46410i	0.244339	+	0.423207i	−0.717607	+	0.696449i	$0.754762\pi$
$68$		0			0
$69$		−	17.3205i		−	2.08514i
$70$		0			0
$71$	3.00000	−	5.19615i	0.356034	−	0.616670i	−0.631260	−	0.775571i	$-0.717462\pi$
$71$	3.00000	−	5.19615i	0.356034	−	0.616670i	0.987294	+	0.158901i	$0.0507952\pi$
$72$		0			0
$73$	1.00000	−	1.73205i	0.117041	−	0.202721i	−0.801553	−	0.597924i	$-0.795992\pi$
$73$	1.00000	−	1.73205i	0.117041	−	0.202721i	0.918594	+	0.395203i	$0.129326\pi$
$74$		0			0
$75$	2.00000			0.230940
$76$		0			0
$77$	9.00000			1.02565
$78$		0			0
$79$	−2.00000	+	3.46410i	−0.225018	+	0.389742i	−0.956325	−	0.292306i	$-0.905577\pi$
$79$	−2.00000	+	3.46410i	−0.225018	+	0.389742i	0.731307	+	0.682048i	$0.238911\pi$
$80$		0			0
$81$	5.50000	−	9.52628i	0.611111	−	1.05848i
$82$		0			0
$83$		−	10.3923i		−	1.14070i	−0.821401	−	0.570352i	$-0.806807\pi$
$83$		−	10.3923i		−	1.14070i	0.821401	−	0.570352i	$-0.193193\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$			6.92820i			0.742781i
$88$		0			0
$89$	−13.5000	+	7.79423i	−1.43100	+	0.826187i	−0.997197	−	0.0748225i	$-0.976161\pi$
$89$	−13.5000	+	7.79423i	−1.43100	+	0.826187i	−0.433800	+	0.901009i	$0.642828\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	4.00000	−	6.92820i	0.414781	−	0.718421i
$94$		0			0
$95$	−0.500000	−	4.33013i	−0.0512989	−	0.444262i
$96$		0			0
$97$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$97$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$98$		0			0
$99$	−4.50000	+	2.59808i	−0.452267	+	0.261116i
$100$		0			0
$101$	3.00000	+	5.19615i	0.298511	+	0.517036i	0.975796	−	0.218685i	$-0.0701767\pi$
$101$	3.00000	+	5.19615i	0.298511	+	0.517036i	−0.677284	+	0.735721i	$0.736843\pi$
$102$		0			0
$103$	1.00000			0.0985329			0.0492665	−	0.998786i	$-0.484312\pi$
$103$	1.00000			0.0985329			0.0492665	+	0.998786i	$0.484312\pi$
$104$		0			0
$105$	3.00000	−	1.73205i	0.292770	−	0.169031i
$106$		0			0
$107$	−12.0000			−1.16008			−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000			−1.16008			−0.580042	+	0.814587i	$0.696964\pi$
$108$		0			0
$109$	6.00000	+	3.46410i	0.574696	+	0.331801i	0.759023	−	0.651064i	$-0.225677\pi$
$109$	6.00000	+	3.46410i	0.574696	+	0.331801i	−0.184327	+	0.982865i	$0.559010\pi$
$110$		0			0
$111$	3.00000	+	1.73205i	0.284747	+	0.164399i
$112$		0			0
$113$			13.8564i			1.30350i	0.758433	+	0.651751i	$0.225965\pi$
$113$			13.8564i			1.30350i	−0.758433	+	0.651751i	$0.774035\pi$
$114$		0			0
$115$		−	8.66025i		−	0.807573i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−16.0000			−1.45455
$122$		0			0
$123$	15.0000	−	8.66025i	1.35250	−	0.780869i
$124$		0			0
$125$	1.00000			0.0894427
$126$		0			0
$127$	6.50000	+	11.2583i	0.576782	+	0.999015i	0.995846	+	0.0910585i	$0.0290250\pi$
$127$	6.50000	+	11.2583i	0.576782	+	0.999015i	−0.419064	+	0.907957i	$0.637642\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−16.5000	−	9.52628i	−1.44161	−	0.832315i	−0.443654	−	0.896198i	$-0.646318\pi$
$131$	−16.5000	−	9.52628i	−1.44161	−	0.832315i	−0.997957	+	0.0638831i	$0.979652\pi$
$132$		0			0
$133$	−4.50000	−	6.06218i	−0.390199	−	0.525657i
$134$		0			0
$135$	2.00000	−	3.46410i	0.172133	−	0.298142i
$136$		0			0
$137$	−9.00000	−	15.5885i	−0.768922	−	1.33181i	−0.938148	−	0.346235i	$-0.887460\pi$
$137$	−9.00000	−	15.5885i	−0.768922	−	1.33181i	0.169226	−	0.985577i	$-0.445873\pi$
$138$		0			0
$139$	−9.00000	+	5.19615i	−0.763370	+	0.440732i	−0.830504	−	0.557012i	$-0.811948\pi$
$139$	−9.00000	+	5.19615i	−0.763370	+	0.440732i	0.0671344	+	0.997744i	$0.478614\pi$
$140$		0			0
$141$		−	6.92820i		−	0.583460i
$142$		0			0
$143$		0			0
$144$		0			0
$145$			3.46410i			0.287678i
$146$		0			0
$147$	−4.00000	+	6.92820i	−0.329914	+	0.571429i
$148$		0			0
$149$	9.00000	−	15.5885i	0.737309	−	1.27706i	−0.216394	−	0.976306i	$-0.569430\pi$
$149$	9.00000	−	15.5885i	0.737309	−	1.27706i	0.953703	−	0.300750i	$-0.0972370\pi$
$150$		0			0
$151$	−2.00000			−0.162758			−0.0813788	−	0.996683i	$-0.525932\pi$
$151$	−2.00000			−0.162758			−0.0813788	+	0.996683i	$0.525932\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	2.00000	−	3.46410i	0.160644	−	0.278243i
$156$		0			0
$157$	−0.500000	+	0.866025i	−0.0399043	+	0.0691164i	−0.885288	−	0.465044i	$-0.846039\pi$
$157$	−0.500000	+	0.866025i	−0.0399043	+	0.0691164i	0.845383	+	0.534160i	$0.179372\pi$
$158$		0			0
$159$			10.3923i			0.824163i
$160$		0			0
$161$	−7.50000	−	12.9904i	−0.591083	−	1.02379i
$162$		0			0
$163$		−	10.3923i		−	0.813988i	−0.913431	−	0.406994i	$-0.866577\pi$
$163$		−	10.3923i		−	0.813988i	0.913431	−	0.406994i	$-0.133423\pi$
$164$		0			0
$165$	−9.00000	+	5.19615i	−0.700649	+	0.404520i
$166$		0			0
$167$	4.50000	+	7.79423i	0.348220	+	0.603136i	0.985933	−	0.167139i	$-0.0534527\pi$
$167$	4.50000	+	7.79423i	0.348220	+	0.603136i	−0.637713	+	0.770274i	$0.720119\pi$
$168$		0			0
$169$	−6.50000	+	11.2583i	−0.500000	+	0.866025i
$170$		0			0
$171$	4.00000	+	1.73205i	0.305888	+	0.132453i
$172$		0			0
$173$	16.5000	+	9.52628i	1.25447	+	0.724270i	0.971994	−	0.235004i	$-0.0755104\pi$
$173$	16.5000	+	9.52628i	1.25447	+	0.724270i	0.282477	+	0.959274i	$0.408844\pi$
$174$		0			0
$175$	1.50000	−	0.866025i	0.113389	−	0.0654654i
$176$		0			0
$177$	12.0000	+	20.7846i	0.901975	+	1.56227i
$178$		0			0
$179$	21.0000			1.56961			0.784807	−	0.619740i	$-0.212762\pi$
$179$	21.0000			1.56961			0.784807	+	0.619740i	$0.212762\pi$
$180$		0			0
$181$	−12.0000	+	6.92820i	−0.891953	+	0.514969i	−0.874581	−	0.484880i	$-0.838863\pi$
$181$	−12.0000	+	6.92820i	−0.891953	+	0.514969i	−0.0173722	+	0.999849i	$0.505530\pi$
$182$		0			0
$183$	20.0000			1.47844
$184$		0			0
$185$	1.50000	+	0.866025i	0.110282	+	0.0636715i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		−	6.92820i		−	0.503953i
$190$		0			0
$191$		−	3.46410i		−	0.250654i	−0.992116	−	0.125327i	$-0.960002\pi$
$191$		−	3.46410i		−	0.250654i	0.992116	−	0.125327i	$-0.0399979\pi$
$192$		0			0
$193$	−12.0000	−	6.92820i	−0.863779	−	0.498703i	0.00149702	−	0.999999i	$-0.499523\pi$
$193$	−12.0000	−	6.92820i	−0.863779	−	0.498703i	−0.865276	+	0.501296i	$0.832857\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	9.00000			0.641223			0.320612	−	0.947211i	$-0.396112\pi$
$197$	9.00000			0.641223			0.320612	+	0.947211i	$0.396112\pi$
$198$		0			0
$199$	−21.0000	+	12.1244i	−1.48865	+	0.859473i	−0.999916	−	0.0129598i	$-0.995875\pi$
$199$	−21.0000	+	12.1244i	−1.48865	+	0.859473i	−0.488735	+	0.872433i	$0.662541\pi$
$200$		0			0
$201$	−8.00000			−0.564276
$202$		0			0
$203$	3.00000	+	5.19615i	0.210559	+	0.364698i
$204$		0			0
$205$	7.50000	−	4.33013i	0.523823	−	0.302429i
$206$		0			0
$207$	7.50000	+	4.33013i	0.521286	+	0.300965i
$208$		0			0
$209$	13.5000	+	18.1865i	0.933815	+	1.25799i
$210$		0			0
$211$	3.50000	−	6.06218i	0.240950	−	0.417338i	−0.720035	−	0.693938i	$-0.755874\pi$
$211$	3.50000	−	6.06218i	0.240950	−	0.417338i	0.960985	+	0.276600i	$0.0892077\pi$
$212$		0			0
$213$	6.00000	+	10.3923i	0.411113	+	0.712069i
$214$		0			0
$215$		0			0
$216$		0			0
$217$		−	6.92820i		−	0.470317i
$218$		0			0
$219$	2.00000	+	3.46410i	0.135147	+	0.234082i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	8.50000	−	14.7224i	0.569202	−	0.985887i	−0.427443	−	0.904042i	$-0.640586\pi$
$223$	8.50000	−	14.7224i	0.569202	−	0.985887i	0.996645	−	0.0818447i	$-0.0260811\pi$
$224$		0			0
$225$	−0.500000	+	0.866025i	−0.0333333	+	0.0577350i
$226$		0			0
$227$	−24.0000			−1.59294			−0.796468	−	0.604681i	$-0.793301\pi$
$227$	−24.0000			−1.59294			−0.796468	+	0.604681i	$0.793301\pi$
$228$		0			0
$229$	−22.0000			−1.45380			−0.726900	−	0.686743i	$-0.759040\pi$
$229$	−22.0000			−1.45380			−0.726900	+	0.686743i	$0.759040\pi$
$230$		0			0
$231$	−9.00000	+	15.5885i	−0.592157	+	1.02565i
$232$		0			0
$233$	3.00000	−	5.19615i	0.196537	−	0.340411i	−0.750867	−	0.660454i	$-0.770364\pi$
$233$	3.00000	−	5.19615i	0.196537	−	0.340411i	0.947403	+	0.320043i	$0.103697\pi$
$234$		0			0
$235$		−	3.46410i		−	0.225973i
$236$		0			0
$237$	−4.00000	−	6.92820i	−0.259828	−	0.450035i
$238$		0			0
$239$			6.92820i			0.448148i	0.974572	+	0.224074i	$0.0719358\pi$
$239$			6.92820i			0.448148i	−0.974572	+	0.224074i	$0.928064\pi$
$240$		0			0
$241$	−12.0000	+	6.92820i	−0.772988	+	0.446285i	−0.833939	−	0.551856i	$-0.813920\pi$
$241$	−12.0000	+	6.92820i	−0.772988	+	0.446285i	0.0609515	+	0.998141i	$0.480586\pi$
$242$		0			0
$243$	5.00000	+	8.66025i	0.320750	+	0.555556i
$244$		0			0
$245$	−2.00000	+	3.46410i	−0.127775	+	0.221313i
$246$		0			0
$247$		0			0
$248$		0			0
$249$	18.0000	+	10.3923i	1.14070	+	0.658586i
$250$		0			0
$251$	−9.00000	+	5.19615i	−0.568075	+	0.327978i	−0.756380	−	0.654132i	$-0.773034\pi$
$251$	−9.00000	+	5.19615i	−0.568075	+	0.327978i	0.188305	+	0.982111i	$0.439701\pi$
$252$		0			0
$253$	22.5000	+	38.9711i	1.41456	+	2.45009i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	6.00000	−	3.46410i	0.374270	−	0.216085i	−0.301052	−	0.953608i	$-0.597338\pi$
$257$	6.00000	−	3.46410i	0.374270	−	0.216085i	0.675322	+	0.737523i	$0.264005\pi$
$258$		0			0
$259$	3.00000			0.186411
$260$		0			0
$261$	−3.00000	−	1.73205i	−0.185695	−	0.107211i
$262$		0			0
$263$	7.50000	+	4.33013i	0.462470	+	0.267007i	0.713082	−	0.701080i	$-0.247299\pi$
$263$	7.50000	+	4.33013i	0.462470	+	0.267007i	−0.250612	+	0.968088i	$0.580632\pi$
$264$		0			0
$265$			5.19615i			0.319197i
$266$		0			0
$267$		−	31.1769i		−	1.90800i
$268$		0			0
$269$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$269$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$270$		0			0
$271$	6.00000	+	3.46410i	0.364474	+	0.210429i	0.671042	−	0.741420i	$-0.265847\pi$
$271$	6.00000	+	3.46410i	0.364474	+	0.210429i	−0.306568	+	0.951849i	$0.599181\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−4.50000	+	2.59808i	−0.271360	+	0.156670i
$276$		0			0
$277$	−22.0000			−1.32185			−0.660926	−	0.750451i	$-0.729836\pi$
$277$	−22.0000			−1.32185			−0.660926	+	0.750451i	$0.729836\pi$
$278$		0			0
$279$	2.00000	+	3.46410i	0.119737	+	0.207390i
$280$		0			0
$281$	−1.50000	+	0.866025i	−0.0894825	+	0.0516627i	−0.544074	−	0.839038i	$-0.683119\pi$
$281$	−1.50000	+	0.866025i	−0.0894825	+	0.0516627i	0.454591	+	0.890700i	$0.349785\pi$
$282$		0			0
$283$	−24.0000	−	13.8564i	−1.42665	−	0.823678i	−0.429797	−	0.902926i	$-0.641415\pi$
$283$	−24.0000	−	13.8564i	−1.42665	−	0.823678i	−0.996855	+	0.0792477i	$0.974748\pi$
$284$		0			0
$285$	8.00000	+	3.46410i	0.473879	+	0.205196i
$286$		0			0
$287$	7.50000	−	12.9904i	0.442711	−	0.766798i
$288$		0			0
$289$	8.50000	+	14.7224i	0.500000	+	0.866025i
$290$		0			0
$291$		0			0
$292$		0			0
$293$			25.9808i			1.51781i	0.651200	+	0.758906i	$0.274266\pi$
$293$			25.9808i			1.51781i	−0.651200	+	0.758906i	$0.725734\pi$
$294$		0			0
$295$	6.00000	+	10.3923i	0.349334	+	0.605063i
$296$		0			0
$297$			20.7846i			1.20605i
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$	−12.0000			−0.689382
$304$		0			0
$305$	10.0000			0.572598
$306$		0			0
$307$	−8.00000	+	13.8564i	−0.456584	+	0.790827i	−0.998778	−	0.0494267i	$-0.984261\pi$
$307$	−8.00000	+	13.8564i	−0.456584	+	0.790827i	0.542194	+	0.840254i	$0.317594\pi$
$308$		0			0
$309$	−1.00000	+	1.73205i	−0.0568880	+	0.0985329i
$310$		0			0
$311$			13.8564i			0.785725i	0.919597	+	0.392862i	$0.128515\pi$
$311$			13.8564i			0.785725i	−0.919597	+	0.392862i	$0.871485\pi$
$312$		0			0
$313$	4.00000	+	6.92820i	0.226093	+	0.391605i	0.956647	−	0.291250i	$-0.0940712\pi$
$313$	4.00000	+	6.92820i	0.226093	+	0.391605i	−0.730554	+	0.682855i	$0.760738\pi$
$314$		0			0
$315$			1.73205i			0.0975900i
$316$		0			0
$317$	−28.5000	+	16.4545i	−1.60072	+	0.924176i	−0.609376	+	0.792881i	$0.708580\pi$
$317$	−28.5000	+	16.4545i	−1.60072	+	0.924176i	−0.991343	+	0.131294i	$0.958087\pi$
$318$		0			0
$319$	−9.00000	−	15.5885i	−0.503903	−	0.872786i
$320$		0			0
$321$	12.0000	−	20.7846i	0.669775	−	1.16008i
$322$		0			0
$323$		0			0
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−12.0000	+	6.92820i	−0.663602	+	0.383131i
$328$		0			0
$329$	−3.00000	−	5.19615i	−0.165395	−	0.286473i
$330$		0			0
$331$	1.00000			0.0549650			0.0274825	−	0.999622i	$-0.491251\pi$
$331$	1.00000			0.0549650			0.0274825	+	0.999622i	$0.491251\pi$
$332$		0			0
$333$	−1.50000	+	0.866025i	−0.0821995	+	0.0474579i
$334$		0			0
$335$	−4.00000			−0.218543
$336$		0			0
$337$	−15.0000	−	8.66025i	−0.817102	−	0.471754i	0.0323140	−	0.999478i	$-0.489712\pi$
$337$	−15.0000	−	8.66025i	−0.817102	−	0.471754i	−0.849416	+	0.527724i	$0.823046\pi$
$338$		0			0
$339$	−24.0000	−	13.8564i	−1.30350	−	0.752577i
$340$		0			0
$341$			20.7846i			1.12555i
$342$		0			0
$343$			19.0526i			1.02874i
$344$		0			0
$345$	15.0000	+	8.66025i	0.807573	+	0.466252i
$346$		0			0
$347$	27.0000	+	15.5885i	1.44944	+	0.836832i	0.998448	−	0.0556976i	$-0.0177383\pi$
$347$	27.0000	+	15.5885i	1.44944	+	0.836832i	0.450988	+	0.892530i	$0.351072\pi$
$348$		0			0
$349$	−28.0000			−1.49881			−0.749403	−	0.662114i	$-0.769659\pi$
$349$	−28.0000			−1.49881			−0.749403	+	0.662114i	$0.769659\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	18.0000			0.958043			0.479022	−	0.877803i	$-0.340992\pi$
$353$	18.0000			0.958043			0.479022	+	0.877803i	$0.340992\pi$
$354$		0			0
$355$	3.00000	+	5.19615i	0.159223	+	0.275783i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	9.00000	+	5.19615i	0.475002	+	0.274242i	0.718331	−	0.695701i	$-0.244906\pi$
$359$	9.00000	+	5.19615i	0.475002	+	0.274242i	−0.243329	+	0.969944i	$0.578240\pi$
$360$		0			0
$361$	5.50000	−	18.1865i	0.289474	−	0.957186i
$362$		0			0
$363$	16.0000	−	27.7128i	0.839782	−	1.45455i
$364$		0			0
$365$	1.00000	+	1.73205i	0.0523424	+	0.0906597i
$366$		0			0
$367$	9.00000	−	5.19615i	0.469796	−	0.271237i	−0.246358	−	0.969179i	$-0.579234\pi$
$367$	9.00000	−	5.19615i	0.469796	−	0.271237i	0.716154	+	0.697942i	$0.245901\pi$
$368$		0			0
$369$			8.66025i			0.450835i
$370$		0			0
$371$	4.50000	+	7.79423i	0.233628	+	0.404656i
$372$		0			0
$373$			36.3731i			1.88333i	0.336557	+	0.941663i	$0.390737\pi$
$373$			36.3731i			1.88333i	−0.336557	+	0.941663i	$0.609263\pi$
$374$		0			0
$375$	−1.00000	+	1.73205i	−0.0516398	+	0.0894427i
$376$		0			0
$377$		0			0
$378$		0			0
$379$	16.0000			0.821865			0.410932	−	0.911666i	$-0.365203\pi$
$379$	16.0000			0.821865			0.410932	+	0.911666i	$0.365203\pi$
$380$		0			0
$381$	−26.0000			−1.33202
$382$		0			0
$383$	6.00000	−	10.3923i	0.306586	−	0.531022i	−0.671027	−	0.741433i	$-0.734147\pi$
$383$	6.00000	−	10.3923i	0.306586	−	0.531022i	0.977613	+	0.210411i	$0.0674801\pi$
$384$		0			0
$385$	−4.50000	+	7.79423i	−0.229341	+	0.397231i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−9.00000	−	15.5885i	−0.456318	−	0.790366i	0.542445	−	0.840091i	$-0.317499\pi$
$389$	−9.00000	−	15.5885i	−0.456318	−	0.790366i	−0.998763	+	0.0497253i	$0.984165\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$	33.0000	−	19.0526i	1.66463	−	0.961074i
$394$		0			0
$395$	−2.00000	−	3.46410i	−0.100631	−	0.174298i
$396$		0			0
$397$	−5.50000	+	9.52628i	−0.276037	+	0.478110i	−0.970396	−	0.241518i	$-0.922355\pi$
$397$	−5.50000	+	9.52628i	−0.276037	+	0.478110i	0.694359	+	0.719629i	$0.255688\pi$
$398$		0			0
$399$	15.0000	−	1.73205i	0.750939	−	0.0867110i
$400$		0			0
$401$	−6.00000	−	3.46410i	−0.299626	−	0.172989i	0.342649	−	0.939463i	$-0.388676\pi$
$401$	−6.00000	−	3.46410i	−0.299626	−	0.172989i	−0.642275	+	0.766475i	$0.722009\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	5.50000	+	9.52628i	0.273297	+	0.473365i
$406$		0			0
$407$	−9.00000			−0.446113
$408$		0			0
$409$	−16.5000	+	9.52628i	−0.815872	+	0.471044i	−0.848991	−	0.528407i	$-0.822789\pi$
$409$	−16.5000	+	9.52628i	−0.815872	+	0.471044i	0.0331186	+	0.999451i	$0.489456\pi$
$410$		0			0
$411$	36.0000			1.77575
$412$		0			0
$413$	18.0000	+	10.3923i	0.885722	+	0.511372i
$414$		0			0
$415$	9.00000	+	5.19615i	0.441793	+	0.255069i
$416$		0			0
$417$		−	20.7846i		−	1.01783i
$418$		0			0
$419$			1.73205i			0.0846162i	0.999105	+	0.0423081i	$0.0134711\pi$
$419$			1.73205i			0.0846162i	−0.999105	+	0.0423081i	$0.986529\pi$
$420$		0			0
$421$	−6.00000	−	3.46410i	−0.292422	−	0.168830i	0.346612	−	0.938009i	$-0.387332\pi$
$421$	−6.00000	−	3.46410i	−0.292422	−	0.168830i	−0.639034	+	0.769179i	$0.720666\pi$
$422$		0			0
$423$	3.00000	+	1.73205i	0.145865	+	0.0842152i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	15.0000	−	8.66025i	0.725901	−	0.419099i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	9.00000	+	15.5885i	0.433515	+	0.750870i	0.997173	−	0.0751385i	$-0.0239399\pi$
$431$	9.00000	+	15.5885i	0.433515	+	0.750870i	−0.563658	+	0.826008i	$0.690607\pi$
$432$		0			0
$433$	−27.0000	+	15.5885i	−1.29754	+	0.749133i	−0.979978	−	0.199105i	$-0.936197\pi$
$433$	−27.0000	+	15.5885i	−1.29754	+	0.749133i	−0.317559	+	0.948239i	$0.602863\pi$
$434$		0			0
$435$	−6.00000	−	3.46410i	−0.287678	−	0.166091i
$436$		0			0
$437$	15.0000	−	34.6410i	0.717547	−	1.65710i
$438$		0			0
$439$	19.0000	−	32.9090i	0.906821	−	1.57066i	0.0883659	−	0.996088i	$-0.471836\pi$
$439$	19.0000	−	32.9090i	0.906821	−	1.57066i	0.818455	−	0.574571i	$-0.194831\pi$
$440$		0			0
$441$	−2.00000	−	3.46410i	−0.0952381	−	0.164957i
$442$		0			0
$443$	12.0000	−	6.92820i	0.570137	−	0.329169i	−0.187067	−	0.982347i	$-0.559898\pi$
$443$	12.0000	−	6.92820i	0.570137	−	0.329169i	0.757204	+	0.653178i	$0.226565\pi$
$444$		0			0
$445$		−	15.5885i		−	0.738964i
$446$		0			0
$447$	18.0000	+	31.1769i	0.851371	+	1.47462i
$448$		0			0
$449$		−	19.0526i		−	0.899146i	−0.893244	−	0.449573i	$-0.851576\pi$
$449$		−	19.0526i		−	0.899146i	0.893244	−	0.449573i	$-0.148424\pi$
$450$		0			0
$451$	−22.5000	+	38.9711i	−1.05948	+	1.83508i
$452$		0			0
$453$	2.00000	−	3.46410i	0.0939682	−	0.162758i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−26.0000			−1.21623			−0.608114	−	0.793849i	$-0.708074\pi$
$457$	−26.0000			−1.21623			−0.608114	+	0.793849i	$0.708074\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$461$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$462$		0			0
$463$			19.0526i			0.885448i	0.896658	+	0.442724i	$0.145988\pi$
$463$			19.0526i			0.885448i	−0.896658	+	0.442724i	$0.854012\pi$
$464$		0			0
$465$	4.00000	+	6.92820i	0.185496	+	0.321288i
$466$		0			0
$467$		−	38.1051i		−	1.76329i	−0.471909	−	0.881647i	$-0.656435\pi$
$467$		−	38.1051i		−	1.76329i	0.471909	−	0.881647i	$-0.343565\pi$
$468$		0			0
$469$	−6.00000	+	3.46410i	−0.277054	+	0.159957i
$470$		0			0
$471$	−1.00000	−	1.73205i	−0.0460776	−	0.0798087i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	4.00000	+	1.73205i	0.183533	+	0.0794719i
$476$		0			0
$477$	−4.50000	−	2.59808i	−0.206041	−	0.118958i
$478$		0			0
$479$	12.0000	−	6.92820i	0.548294	−	0.316558i	−0.200140	−	0.979767i	$-0.564140\pi$
$479$	12.0000	−	6.92820i	0.548294	−	0.316558i	0.748434	+	0.663210i	$0.230806\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$	30.0000			1.36505
$484$		0			0
$485$		0			0
$486$		0			0
$487$	29.0000			1.31412			0.657058	−	0.753840i	$-0.271801\pi$
$487$	29.0000			1.31412			0.657058	+	0.753840i	$0.271801\pi$
$488$		0			0
$489$	18.0000	+	10.3923i	0.813988	+	0.469956i
$490$		0			0
$491$	−28.5000	−	16.4545i	−1.28619	−	0.742580i	−0.308215	−	0.951317i	$-0.599732\pi$
$491$	−28.5000	−	16.4545i	−1.28619	−	0.742580i	−0.977972	+	0.208736i	$0.933065\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		−	5.19615i		−	0.233550i
$496$		0			0
$497$	9.00000	+	5.19615i	0.403705	+	0.233079i
$498$		0			0
$499$	13.5000	+	7.79423i	0.604343	+	0.348918i	0.770748	−	0.637140i	$-0.219883\pi$
$499$	13.5000	+	7.79423i	0.604343	+	0.348918i	−0.166405	+	0.986057i	$0.553216\pi$
$500$		0			0
$501$	−18.0000			−0.804181
$502$		0			0
$503$	−10.5000	+	6.06218i	−0.468172	+	0.270299i	−0.715474	−	0.698639i	$-0.753789\pi$
$503$	−10.5000	+	6.06218i	−0.468172	+	0.270299i	0.247302	+	0.968938i	$0.420456\pi$
$504$		0			0
$505$	−6.00000			−0.266996
$506$		0			0
$507$	−13.0000	−	22.5167i	−0.577350	−	1.00000i
$508$		0			0
$509$	−3.00000	+	1.73205i	−0.132973	+	0.0767718i	−0.565011	−	0.825084i	$-0.691128\pi$
$509$	−3.00000	+	1.73205i	−0.132973	+	0.0767718i	0.432038	+	0.901855i	$0.357795\pi$
$510$		0			0
$511$	3.00000	+	1.73205i	0.132712	+	0.0766214i
$512$		0			0
$513$	14.0000	−	10.3923i	0.618115	−	0.458831i
$514$		0			0
$515$	−0.500000	+	0.866025i	−0.0220326	+	0.0381616i
$516$		0			0
$517$	9.00000	+	15.5885i	0.395820	+	0.685580i
$518$		0			0
$519$	−33.0000	+	19.0526i	−1.44854	+	0.836315i
$520$		0			0
$521$		−	13.8564i		−	0.607060i	−0.952822	−	0.303530i	$-0.901835\pi$
$521$		−	13.8564i		−	0.607060i	0.952822	−	0.303530i	$-0.0981653\pi$
$522$		0			0
$523$	17.0000	+	29.4449i	0.743358	+	1.28753i	0.950958	+	0.309320i	$0.100101\pi$
$523$	17.0000	+	29.4449i	0.743358	+	1.28753i	−0.207600	+	0.978214i	$0.566565\pi$
$524$		0			0
$525$			3.46410i			0.151186i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	26.0000	−	45.0333i	1.13043	−	1.95797i
$530$		0			0
$531$	−12.0000			−0.520756
$532$		0			0
$533$		0			0
$534$		0			0
$535$	6.00000	−	10.3923i	0.259403	−	0.449299i
$536$		0			0
$537$	−21.0000	+	36.3731i	−0.906217	+	1.56961i
$538$		0			0
$539$		−	20.7846i		−	0.895257i
$540$		0			0
$541$	4.00000	+	6.92820i	0.171973	+	0.297867i	0.939110	−	0.343617i	$-0.111652\pi$
$541$	4.00000	+	6.92820i	0.171973	+	0.297867i	−0.767136	+	0.641484i	$0.778319\pi$
$542$		0			0
$543$		−	27.7128i		−	1.18927i
$544$		0			0
$545$	−6.00000	+	3.46410i	−0.257012	+	0.148386i
$546$		0			0
$547$	−5.00000	−	8.66025i	−0.213785	−	0.370286i	0.739111	−	0.673583i	$-0.235246\pi$
$547$	−5.00000	−	8.66025i	−0.213785	−	0.370286i	−0.952896	+	0.303298i	$0.901912\pi$
$548$		0			0
$549$	−5.00000	+	8.66025i	−0.213395	+	0.369611i
$550$		0			0
$551$	−6.00000	+	13.8564i	−0.255609	+	0.590303i
$552$		0			0
$553$	−6.00000	−	3.46410i	−0.255146	−	0.147309i
$554$		0			0
$555$	−3.00000	+	1.73205i	−0.127343	+	0.0735215i
$556$		0			0
$557$	1.50000	+	2.59808i	0.0635570	+	0.110084i	0.896053	−	0.443947i	$-0.146422\pi$
$557$	1.50000	+	2.59808i	0.0635570	+	0.110084i	−0.832496	+	0.554031i	$0.813089\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−6.00000			−0.252870			−0.126435	−	0.991975i	$-0.540353\pi$
$563$	−6.00000			−0.252870			−0.126435	+	0.991975i	$0.540353\pi$
$564$		0			0
$565$	−12.0000	−	6.92820i	−0.504844	−	0.291472i
$566$		0			0
$567$	16.5000	+	9.52628i	0.692935	+	0.400066i
$568$		0			0
$569$			15.5885i			0.653502i	0.945110	+	0.326751i	$0.105954\pi$
$569$			15.5885i			0.653502i	−0.945110	+	0.326751i	$0.894046\pi$
$570$		0			0
$571$			3.46410i			0.144968i	0.997370	+	0.0724841i	$0.0230926\pi$
$571$			3.46410i			0.144968i	−0.997370	+	0.0724841i	$0.976907\pi$
$572$		0			0
$573$	6.00000	+	3.46410i	0.250654	+	0.144715i
$574$		0			0
$575$	7.50000	+	4.33013i	0.312772	+	0.180579i
$576$		0			0
$577$	4.00000			0.166522			0.0832611	−	0.996528i	$-0.473466\pi$
$577$	4.00000			0.166522			0.0832611	+	0.996528i	$0.473466\pi$
$578$		0			0
$579$	24.0000	−	13.8564i	0.997406	−	0.575853i
$580$		0			0
$581$	18.0000			0.746766
$582$		0			0
$583$	−13.5000	−	23.3827i	−0.559113	−	0.968412i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−30.0000	−	17.3205i	−1.23823	−	0.714894i	−0.269500	−	0.963000i	$-0.586858\pi$
$587$	−30.0000	−	17.3205i	−1.23823	−	0.714894i	−0.968733	+	0.248107i	$0.920192\pi$
$588$		0			0
$589$	14.0000	−	10.3923i	0.576860	−	0.428207i
$590$		0			0
$591$	−9.00000	+	15.5885i	−0.370211	+	0.641223i
$592$		0			0
$593$	−3.00000	−	5.19615i	−0.123195	−	0.213380i	0.797831	−	0.602881i	$-0.205981\pi$
$593$	−3.00000	−	5.19615i	−0.123195	−	0.213380i	−0.921026	+	0.389501i	$0.872647\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		−	48.4974i		−	1.98487i
$598$		0			0
$599$	3.00000	+	5.19615i	0.122577	+	0.212309i	0.920783	−	0.390075i	$-0.127551\pi$
$599$	3.00000	+	5.19615i	0.122577	+	0.212309i	−0.798206	+	0.602384i	$0.794218\pi$
$600$		0			0
$601$			25.9808i			1.05978i	0.848067	+	0.529889i	$0.177766\pi$
$601$			25.9808i			1.05978i	−0.848067	+	0.529889i	$0.822234\pi$
$602$		0			0
$603$	2.00000	−	3.46410i	0.0814463	−	0.141069i
$604$		0			0
$605$	8.00000	−	13.8564i	0.325246	−	0.563343i
$606$		0			0
$607$	−13.0000			−0.527654			−0.263827	−	0.964570i	$-0.584985\pi$
$607$	−13.0000			−0.527654			−0.263827	+	0.964570i	$0.584985\pi$
$608$		0			0
$609$	−12.0000			−0.486265
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−0.500000	+	0.866025i	−0.0201948	+	0.0349784i	−0.875946	−	0.482409i	$-0.839762\pi$
$613$	−0.500000	+	0.866025i	−0.0201948	+	0.0349784i	0.855751	+	0.517387i	$0.173095\pi$
$614$		0			0
$615$			17.3205i			0.698430i
$616$		0			0
$617$	−21.0000	−	36.3731i	−0.845428	−	1.46432i	−0.885249	−	0.465118i	$-0.846012\pi$
$617$	−21.0000	−	36.3731i	−0.845428	−	1.46432i	0.0398207	−	0.999207i	$-0.487321\pi$
$618$		0			0
$619$		−	8.66025i		−	0.348085i	−0.984738	−	0.174042i	$-0.944317\pi$
$619$		−	8.66025i		−	0.348085i	0.984738	−	0.174042i	$-0.0556830\pi$
$620$		0			0
$621$	30.0000	−	17.3205i	1.20386	−	0.695048i
$622$		0			0
$623$	−13.5000	−	23.3827i	−0.540866	−	0.936808i
$624$		0			0
$625$	−0.500000	+	0.866025i	−0.0200000	+	0.0346410i
$626$		0			0
$627$	−45.0000	+	5.19615i	−1.79713	+	0.207514i
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−36.0000	+	20.7846i	−1.43314	+	0.827422i	−0.997359	−	0.0726317i	$-0.976860\pi$
$631$	−36.0000	+	20.7846i	−1.43314	+	0.827422i	−0.435779	+	0.900054i	$0.643527\pi$
$632$		0			0
$633$	7.00000	+	12.1244i	0.278225	+	0.481900i
$634$		0			0
$635$	−13.0000			−0.515889
$636$		0			0
$637$		0			0
$638$		0			0
$639$	−6.00000			−0.237356
$640$		0			0
$641$	30.0000	+	17.3205i	1.18493	+	0.684119i	0.957150	−	0.289594i	$-0.0935202\pi$
$641$	30.0000	+	17.3205i	1.18493	+	0.684119i	0.227779	+	0.973713i	$0.426854\pi$
$642$		0			0
$643$	−3.00000	−	1.73205i	−0.118308	−	0.0683054i	0.439678	−	0.898155i	$-0.355093\pi$
$643$	−3.00000	−	1.73205i	−0.118308	−	0.0683054i	−0.557986	+	0.829850i	$0.688426\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	25.9808i		−	1.02141i	−0.859756	−	0.510705i	$-0.829385\pi$
$647$		−	25.9808i		−	1.02141i	0.859756	−	0.510705i	$-0.170615\pi$
$648$		0			0
$649$	−54.0000	−	31.1769i	−2.11969	−	1.22380i
$650$		0			0
$651$	12.0000	+	6.92820i	0.470317	+	0.271538i
$652$		0			0
$653$	−45.0000			−1.76099			−0.880493	−	0.474059i	$-0.842788\pi$
$653$	−45.0000			−1.76099			−0.880493	+	0.474059i	$0.842788\pi$
$654$		0			0
$655$	16.5000	−	9.52628i	0.644708	−	0.372223i
$656$		0			0
$657$	−2.00000			−0.0780274
$658$		0			0
$659$	10.5000	+	18.1865i	0.409022	+	0.708447i	0.994780	−	0.102039i	$-0.0325366\pi$
$659$	10.5000	+	18.1865i	0.409022	+	0.708447i	−0.585758	+	0.810486i	$0.699203\pi$
$660$		0			0
$661$	21.0000	−	12.1244i	0.816805	−	0.471583i	−0.0325082	−	0.999471i	$-0.510350\pi$
$661$	21.0000	−	12.1244i	0.816805	−	0.471583i	0.849314	+	0.527889i	$0.177016\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	7.50000	−	0.866025i	0.290838	−	0.0335830i
$666$		0			0
$667$	−15.0000	+	25.9808i	−0.580802	+	1.00598i
$668$		0			0
$669$	17.0000	+	29.4449i	0.657258	+	1.13840i
$670$		0			0
$671$	−45.0000	+	25.9808i	−1.73721	+	1.00298i
$672$		0			0
$673$		−	10.3923i		−	0.400594i	−0.979735	−	0.200297i	$-0.935809\pi$
$673$		−	10.3923i		−	0.400594i	0.979735	−	0.200297i	$-0.0641907\pi$
$674$		0			0
$675$	2.00000	+	3.46410i	0.0769800	+	0.133333i
$676$		0			0
$677$			8.66025i			0.332841i	0.986055	+	0.166420i	$0.0532208\pi$
$677$			8.66025i			0.332841i	−0.986055	+	0.166420i	$0.946779\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	24.0000	−	41.5692i	0.919682	−	1.59294i
$682$		0			0
$683$	6.00000			0.229584			0.114792	−	0.993390i	$-0.463380\pi$
$683$	6.00000			0.229584			0.114792	+	0.993390i	$0.463380\pi$
$684$		0			0
$685$	18.0000			0.687745
$686$		0			0
$687$	22.0000	−	38.1051i	0.839352	−	1.45380i
$688$		0			0
$689$		0			0
$690$		0			0
$691$			25.9808i			0.988355i	0.869361	+	0.494177i	$0.164531\pi$
$691$			25.9808i			0.988355i	−0.869361	+	0.494177i	$0.835469\pi$
$692$		0			0
$693$	−4.50000	−	7.79423i	−0.170941	−	0.296078i
$694$		0			0
$695$		−	10.3923i		−	0.394203i
$696$		0			0
$697$		0			0
$698$		0			0
$699$	6.00000	+	10.3923i	0.226941	+	0.393073i
$700$		0			0
$701$	−6.00000	+	10.3923i	−0.226617	+	0.392512i	−0.956803	−	0.290736i	$-0.906100\pi$
$701$	−6.00000	+	10.3923i	−0.226617	+	0.392512i	0.730186	+	0.683248i	$0.239433\pi$
$702$		0			0
$703$	4.50000	+	6.06218i	0.169721	+	0.228639i
$704$		0			0
$705$	6.00000	+	3.46410i	0.225973	+	0.130466i
$706$		0			0
$707$	−9.00000	+	5.19615i	−0.338480	+	0.195421i
$708$		0			0
$709$	−5.00000	−	8.66025i	−0.187779	−	0.325243i	0.756730	−	0.653727i	$-0.226796\pi$
$709$	−5.00000	−	8.66025i	−0.187779	−	0.325243i	−0.944509	+	0.328484i	$0.893462\pi$
$710$		0			0
$711$	4.00000			0.150012
$712$		0			0
$713$	30.0000	−	17.3205i	1.12351	−	0.648658i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−12.0000	−	6.92820i	−0.448148	−	0.258738i
$718$		0			0
$719$	−21.0000	−	12.1244i	−0.783168	−	0.452162i	0.0543839	−	0.998520i	$-0.482681\pi$
$719$	−21.0000	−	12.1244i	−0.783168	−	0.452162i	−0.837552	+	0.546358i	$0.816014\pi$
$720$		0			0
$721$			1.73205i			0.0645049i
$722$		0			0
$723$		−	27.7128i		−	1.03065i
$724$		0			0
$725$	−3.00000	−	1.73205i	−0.111417	−	0.0643268i
$726$		0			0
$727$	3.00000	+	1.73205i	0.111264	+	0.0642382i	0.554599	−	0.832118i	$-0.312872\pi$
$727$	3.00000	+	1.73205i	0.111264	+	0.0642382i	−0.443335	+	0.896356i	$0.646205\pi$
$728$		0			0
$729$	13.0000			0.481481
$730$		0			0
$731$		0			0
$732$		0			0
$733$	49.0000			1.80986			0.904928	−	0.425564i	$-0.139924\pi$
$733$	49.0000			1.80986			0.904928	+	0.425564i	$0.139924\pi$
$734$		0			0
$735$	−4.00000	−	6.92820i	−0.147542	−	0.255551i
$736$		0			0
$737$	18.0000	−	10.3923i	0.663039	−	0.382805i
$738$		0			0
$739$	−16.5000	−	9.52628i	−0.606962	−	0.350430i	0.164813	−	0.986325i	$-0.447298\pi$
$739$	−16.5000	−	9.52628i	−0.606962	−	0.350430i	−0.771776	+	0.635895i	$0.780631\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	13.5000	−	23.3827i	0.495267	−	0.857828i	−0.504718	−	0.863284i	$-0.668404\pi$
$743$	13.5000	−	23.3827i	0.495267	−	0.857828i	0.999985	+	0.00545664i	$0.00173691\pi$
$744$		0			0
$745$	9.00000	+	15.5885i	0.329734	+	0.571117i
$746$		0			0
$747$	−9.00000	+	5.19615i	−0.329293	+	0.190117i
$748$		0			0
$749$		−	20.7846i		−	0.759453i
$750$		0			0
$751$	19.0000	+	32.9090i	0.693320	+	1.20087i	0.970744	+	0.240118i	$0.0771860\pi$
$751$	19.0000	+	32.9090i	0.693320	+	1.20087i	−0.277424	+	0.960748i	$0.589481\pi$
$752$		0			0
$753$		−	20.7846i		−	0.757433i
$754$		0			0
$755$	1.00000	−	1.73205i	0.0363937	−	0.0630358i
$756$		0			0
$757$	−17.5000	+	30.3109i	−0.636048	+	1.10167i	0.350244	+	0.936659i	$0.386099\pi$
$757$	−17.5000	+	30.3109i	−0.636048	+	1.10167i	−0.986292	+	0.165009i	$0.947235\pi$
$758$		0			0
$759$	−90.0000			−3.26679
$760$		0			0
$761$	−9.00000			−0.326250			−0.163125	−	0.986605i	$-0.552157\pi$
$761$	−9.00000			−0.326250			−0.163125	+	0.986605i	$0.552157\pi$
$762$		0			0
$763$	−6.00000	+	10.3923i	−0.217215	+	0.376227i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−5.00000	−	8.66025i	−0.180305	−	0.312297i	0.761680	−	0.647954i	$-0.224375\pi$
$769$	−5.00000	−	8.66025i	−0.180305	−	0.312297i	−0.941984	+	0.335657i	$0.891042\pi$
$770$		0			0
$771$			13.8564i			0.499026i
$772$		0			0
$773$	−10.5000	+	6.06218i	−0.377659	+	0.218041i	−0.676799	−	0.736168i	$-0.736633\pi$
$773$	−10.5000	+	6.06218i	−0.377659	+	0.218041i	0.299140	+	0.954209i	$0.403300\pi$
$774$		0			0
$775$	2.00000	+	3.46410i	0.0718421	+	0.124434i
$776$		0			0
$777$	−3.00000	+	5.19615i	−0.107624	+	0.186411i
$778$		0			0
$779$	37.5000	−	4.33013i	1.34358	−	0.155143i
$780$		0			0
$781$	−27.0000	−	15.5885i	−0.966136	−	0.557799i
$782$		0			0
$783$	−12.0000	+	6.92820i	−0.428845	+	0.247594i
$784$		0			0
$785$	−0.500000	−	0.866025i	−0.0178458	−	0.0309098i
$786$		0			0
$787$	2.00000			0.0712923			0.0356462	−	0.999364i	$-0.488651\pi$
$787$	2.00000			0.0712923			0.0356462	+	0.999364i	$0.488651\pi$
$788$		0			0
$789$	−15.0000	+	8.66025i	−0.534014	+	0.308313i
$790$		0			0
$791$	−24.0000			−0.853342
$792$		0			0
$793$		0			0
$794$		0			0
$795$	−9.00000	−	5.19615i	−0.319197	−	0.184289i
$796$		0			0
$797$		−	22.5167i		−	0.797581i	−0.917042	−	0.398791i	$-0.869430\pi$
$797$		−	22.5167i		−	0.797581i	0.917042	−	0.398791i	$-0.130570\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	13.5000	+	7.79423i	0.476999	+	0.275396i
$802$		0			0
$803$	−9.00000	−	5.19615i	−0.317603	−	0.183368i
$804$		0			0
$805$	15.0000			0.528681
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−6.00000			−0.210949			−0.105474	−	0.994422i	$-0.533636\pi$
$809$	−6.00000			−0.210949			−0.105474	+	0.994422i	$0.533636\pi$
$810$		0			0
$811$	−0.500000	−	0.866025i	−0.0175574	−	0.0304103i	0.857113	−	0.515128i	$-0.172256\pi$
$811$	−0.500000	−	0.866025i	−0.0175574	−	0.0304103i	−0.874671	+	0.484718i	$0.838922\pi$
$812$		0			0
$813$	−12.0000	+	6.92820i	−0.420858	+	0.242983i
$814$		0			0
$815$	9.00000	+	5.19615i	0.315256	+	0.182013i
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−12.0000	−	20.7846i	−0.418803	−	0.725388i	0.577016	−	0.816733i	$-0.304217\pi$
$821$	−12.0000	−	20.7846i	−0.418803	−	0.725388i	−0.995819	+	0.0913446i	$0.970884\pi$
$822$		0			0
$823$	−46.5000	+	26.8468i	−1.62089	+	0.935820i	−0.634205	+	0.773165i	$0.718673\pi$
$823$	−46.5000	+	26.8468i	−1.62089	+	0.935820i	−0.986683	+	0.162655i	$0.947994\pi$
$824$		0			0
$825$		−	10.3923i		−	0.361814i
$826$		0			0
$827$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$827$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$828$		0			0
$829$		−	10.3923i		−	0.360940i	−0.983581	−	0.180470i	$-0.942238\pi$
$829$		−	10.3923i		−	0.360940i	0.983581	−	0.180470i	$-0.0577618\pi$
$830$		0			0
$831$	22.0000	−	38.1051i	0.763172	−	1.32185i
$832$		0			0
$833$		0			0
$834$		0			0
$835$	−9.00000			−0.311458
$836$		0			0
$837$	16.0000			0.553041
$838$		0			0
$839$	−12.0000	+	20.7846i	−0.414286	+	0.717564i	−0.995353	−	0.0962912i	$-0.969302\pi$
$839$	−12.0000	+	20.7846i	−0.414286	+	0.717564i	0.581067	+	0.813856i	$0.302635\pi$
$840$		0			0
$841$	−8.50000	+	14.7224i	−0.293103	+	0.507670i
$842$		0			0
$843$		−	3.46410i		−	0.119310i
$844$		0			0
$845$	−6.50000	−	11.2583i	−0.223607	−	0.387298i
$846$		0			0
$847$		−	27.7128i		−	0.952224i
$848$		0			0
$849$	48.0000	−	27.7128i	1.64736	−	0.951101i
$850$		0			0
$851$	7.50000	+	12.9904i	0.257097	+	0.445305i
$852$		0			0
$853$	5.00000	−	8.66025i	0.171197	−	0.296521i	−0.767642	−	0.640879i	$-0.778570\pi$
$853$	5.00000	−	8.66025i	0.171197	−	0.296521i	0.938839	+	0.344358i	$0.111903\pi$
$854$		0			0
$855$	−3.50000	+	2.59808i	−0.119697	+	0.0888523i
$856$		0			0
$857$	30.0000	+	17.3205i	1.02478	+	0.591657i	0.915485	−	0.402352i	$-0.131807\pi$
$857$	30.0000	+	17.3205i	1.02478	+	0.591657i	0.109295	+	0.994009i	$0.465141\pi$
$858$		0			0
$859$	28.5000	−	16.4545i	0.972407	−	0.561420i	0.0724381	−	0.997373i	$-0.476922\pi$
$859$	28.5000	−	16.4545i	0.972407	−	0.561420i	0.899969	+	0.435953i	$0.143589\pi$
$860$		0			0
$861$	15.0000	+	25.9808i	0.511199	+	0.885422i
$862$		0			0
$863$	21.0000			0.714848			0.357424	−	0.933942i	$-0.383655\pi$
$863$	21.0000			0.714848			0.357424	+	0.933942i	$0.383655\pi$
$864$		0			0
$865$	−16.5000	+	9.52628i	−0.561017	+	0.323903i
$866$		0			0
$867$	−34.0000			−1.15470
$868$		0			0
$869$	18.0000	+	10.3923i	0.610608	+	0.352535i
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$			1.73205i			0.0585540i
$876$		0			0
$877$	10.5000	+	6.06218i	0.354560	+	0.204705i	0.666692	−	0.745334i	$-0.267710\pi$
$877$	10.5000	+	6.06218i	0.354560	+	0.204705i	−0.312132	+	0.950039i	$0.601043\pi$
$878$		0			0
$879$	−45.0000	−	25.9808i	−1.51781	−	0.876309i
$880$		0			0
$881$	−33.0000			−1.11180			−0.555899	−	0.831250i	$-0.687626\pi$
$881$	−33.0000			−1.11180			−0.555899	+	0.831250i	$0.687626\pi$
$882$		0			0
$883$	−42.0000	+	24.2487i	−1.41341	+	0.816034i	−0.995708	−	0.0925489i	$-0.970499\pi$
$883$	−42.0000	+	24.2487i	−1.41341	+	0.816034i	−0.417704	+	0.908583i	$0.637165\pi$
$884$		0			0
$885$	−24.0000			−0.806751
$886$		0			0
$887$	24.0000	+	41.5692i	0.805841	+	1.39576i	0.915722	+	0.401813i	$0.131620\pi$
$887$	24.0000	+	41.5692i	0.805841	+	1.39576i	−0.109881	+	0.993945i	$0.535047\pi$
$888$		0			0
$889$	−19.5000	+	11.2583i	−0.654009	+	0.377592i
$890$		0			0
$891$	−49.5000	−	28.5788i	−1.65831	−	0.957427i
$892$		0			0
$893$	6.00000	−	13.8564i	0.200782	−	0.463687i
$894$		0			0
$895$	−10.5000	+	18.1865i	−0.350976	+	0.607909i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−12.0000	+	6.92820i	−0.400222	+	0.231069i
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	13.8564i		−	0.460603i
$906$		0			0
$907$	−7.00000	+	12.1244i	−0.232431	+	0.402583i	−0.958523	−	0.285015i	$-0.908001\pi$
$907$	−7.00000	+	12.1244i	−0.232431	+	0.402583i	0.726092	+	0.687598i	$0.241335\pi$
$908$		0			0
$909$	3.00000	−	5.19615i	0.0995037	−	0.172345i
$910$		0			0
$911$	18.0000			0.596367			0.298183	−	0.954509i	$-0.403619\pi$
$911$	18.0000			0.596367			0.298183	+	0.954509i	$0.403619\pi$
$912$		0			0
$913$	−54.0000			−1.78714
$914$		0			0
$915$	−10.0000	+	17.3205i	−0.330590	+	0.572598i
$916$		0			0
$917$	16.5000	−	28.5788i	0.544878	−	0.943756i
$918$		0			0
$919$		−	24.2487i		−	0.799891i	−0.916539	−	0.399946i	$-0.869029\pi$
$919$		−	24.2487i		−	0.799891i	0.916539	−	0.399946i	$-0.130971\pi$
$920$		0			0
$921$	−16.0000	−	27.7128i	−0.527218	−	0.913168i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−1.50000	+	0.866025i	−0.0493197	+	0.0284747i
$926$		0			0
$927$	−0.500000	−	0.866025i	−0.0164222	−	0.0284440i
$928$		0			0
$929$	−25.5000	+	44.1673i	−0.836628	+	1.44908i	0.0560703	+	0.998427i	$0.482143\pi$
$929$	−25.5000	+	44.1673i	−0.836628	+	1.44908i	−0.892698	+	0.450655i	$0.851190\pi$
$930$		0			0
$931$	−14.0000	+	10.3923i	−0.458831	+	0.340594i
$932$		0			0
$933$	−24.0000	−	13.8564i	−0.785725	−	0.453638i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	8.00000	+	13.8564i	0.261349	+	0.452669i	0.966601	−	0.256288i	$-0.0824995\pi$
$937$	8.00000	+	13.8564i	0.261349	+	0.452669i	−0.705252	+	0.708957i	$0.749166\pi$
$938$		0			0
$939$	−16.0000			−0.522140
$940$		0			0
$941$	42.0000	−	24.2487i	1.36916	−	0.790485i	0.378340	−	0.925667i	$-0.376495\pi$
$941$	42.0000	−	24.2487i	1.36916	−	0.790485i	0.990821	+	0.135181i	$0.0431617\pi$
$942$		0			0
$943$	75.0000			2.44234
$944$		0			0
$945$	6.00000	+	3.46410i	0.195180	+	0.112687i
$946$		0			0
$947$	6.00000	+	3.46410i	0.194974	+	0.112568i	0.594309	−	0.804237i	$-0.297426\pi$
$947$	6.00000	+	3.46410i	0.194974	+	0.112568i	−0.399335	+	0.916805i	$0.630759\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		−	65.8179i		−	2.13429i
$952$		0			0
$953$	30.0000	+	17.3205i	0.971795	+	0.561066i	0.899783	−	0.436337i	$-0.143725\pi$
$953$	30.0000	+	17.3205i	0.971795	+	0.561066i	0.0720122	+	0.997404i	$0.477058\pi$
$954$		0			0
$955$	3.00000	+	1.73205i	0.0970777	+	0.0560478i
$956$		0			0
$957$	36.0000			1.16371
$958$		0			0
$959$	27.0000	−	15.5885i	0.871875	−	0.503378i
$960$		0			0
$961$	−15.0000			−0.483871
$962$		0			0
$963$	6.00000	+	10.3923i	0.193347	+	0.334887i
$964$		0			0
$965$	12.0000	−	6.92820i	0.386294	−	0.223027i
$966$		0			0
$967$	27.0000	+	15.5885i	0.868261	+	0.501291i	0.866770	−	0.498708i	$-0.166192\pi$
$967$	27.0000	+	15.5885i	0.868261	+	0.501291i	0.00149135	+	0.999999i	$0.499525\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	24.0000	−	41.5692i	0.770197	−	1.33402i	−0.167258	−	0.985913i	$-0.553491\pi$
$971$	24.0000	−	41.5692i	0.770197	−	1.33402i	0.937455	−	0.348107i	$-0.113175\pi$
$972$		0			0
$973$	−9.00000	−	15.5885i	−0.288527	−	0.499743i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	58.8897i		−	1.88405i	−0.335544	−	0.942025i	$-0.608920\pi$
$977$		−	58.8897i		−	1.88405i	0.335544	−	0.942025i	$-0.391080\pi$
$978$		0			0
$979$	40.5000	+	70.1481i	1.29439	+	2.24194i
$980$		0			0
$981$		−	6.92820i		−	0.221201i
$982$		0			0
$983$	−19.5000	+	33.7750i	−0.621953	+	1.07725i	0.367168	+	0.930155i	$0.380327\pi$
$983$	−19.5000	+	33.7750i	−0.621953	+	1.07725i	−0.989122	+	0.147100i	$0.953006\pi$
$984$		0			0
$985$	−4.50000	+	7.79423i	−0.143382	+	0.248345i
$986$		0			0
$987$	12.0000			0.381964
$988$		0			0
$989$		0			0
$990$		0			0
$991$	26.0000	−	45.0333i	0.825917	−	1.43053i	−0.0752991	−	0.997161i	$-0.523991\pi$
$991$	26.0000	−	45.0333i	0.825917	−	1.43053i	0.901216	−	0.433370i	$-0.142676\pi$
$992$		0			0
$993$	−1.00000	+	1.73205i	−0.0317340	+	0.0549650i
$994$		0			0
$995$		−	24.2487i		−	0.768736i
$996$		0			0
$997$	−0.500000	−	0.866025i	−0.0158352	−	0.0274273i	0.857999	−	0.513651i	$-0.171707\pi$
$997$	−0.500000	−	0.866025i	−0.0158352	−	0.0274273i	−0.873834	+	0.486224i	$0.838374\pi$
$998$		0			0
$999$			6.92820i			0.219199i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1520.2.bq.b.1471.1	yes	2
4.3	odd	2		1520.2.bq.h.1471.1	yes	2
19.12	odd	6		1520.2.bq.h.31.1	yes	2
76.31	even	6	inner	1520.2.bq.b.31.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1520.2.bq.b.31.1	✓	2	76.31	even	6	inner
1520.2.bq.b.1471.1	yes	2	1.1	even	1	trivial
1520.2.bq.h.31.1	yes	2	19.12	odd	6
1520.2.bq.h.1471.1	yes	2	4.3	odd	2

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1520 = 2^{4} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1520.bq (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.00000 + 1.73205i) q^{3} +(-0.500000 + 0.866025i) q^{5} +1.73205i q^{7} +(-0.500000 - 0.866025i) q^{9} -5.19615i q^{11} +(-1.00000 - 1.73205i) q^{15} +(-3.50000 + 2.59808i) q^{19} +(-3.00000 - 1.73205i) q^{21} +(-7.50000 + 4.33013i) q^{23} +(-0.500000 - 0.866025i) q^{25} -4.00000 q^{27} +(3.00000 - 1.73205i) q^{29} -4.00000 q^{31} +(9.00000 + 5.19615i) q^{33} +(-1.50000 - 0.866025i) q^{35} -1.73205i q^{37} +(-7.50000 - 4.33013i) q^{41} +1.00000 q^{45} +(-3.00000 + 1.73205i) q^{47} +4.00000 q^{49} +(4.50000 - 2.59808i) q^{53} +(4.50000 + 2.59808i) q^{55} +(-1.00000 - 8.66025i) q^{57} +(6.00000 - 10.3923i) q^{59} +(-5.00000 - 8.66025i) q^{61} +(1.50000 - 0.866025i) q^{63} +(2.00000 + 3.46410i) q^{67} -17.3205i q^{69} +(3.00000 - 5.19615i) q^{71} +(1.00000 - 1.73205i) q^{73} +2.00000 q^{75} +9.00000 q^{77} +(-2.00000 + 3.46410i) q^{79} +(5.50000 - 9.52628i) q^{81} -10.3923i q^{83} +6.92820i q^{87} +(-13.5000 + 7.79423i) q^{89} +(4.00000 - 6.92820i) q^{93} +(-0.500000 - 4.33013i) q^{95} +(-4.50000 + 2.59808i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - q^{5} - q^{9} - 2 q^{15} - 7 q^{19} - 6 q^{21} - 15 q^{23} - q^{25} - 8 q^{27} + 6 q^{29} - 8 q^{31} + 18 q^{33} - 3 q^{35} - 15 q^{41} + 2 q^{45} - 6 q^{47} + 8 q^{49} + 9 q^{53} + 9 q^{55}+ \cdots - 9 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 - q^5 - q^9 - 2 * q^15 - 7 * q^19 - 6 * q^21 - 15 * q^23 - q^25 - 8 * q^27 + 6 * q^29 - 8 * q^31 + 18 * q^33 - 3 * q^35 - 15 * q^41 + 2 * q^45 - 6 * q^47 + 8 * q^49 + 9 * q^53 + 9 * q^55 - 2 * q^57 + 12 * q^59 - 10 * q^61 + 3 * q^63 + 4 * q^67 + 6 * q^71 + 2 * q^73 + 4 * q^75 + 18 * q^77 - 4 * q^79 + 11 * q^81 - 27 * q^89 + 8 * q^93 - q^95 - 9 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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