LMFDB - Embedded newform 1512.2.bl.c.1025.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1512,2,Mod(593,1512)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1512, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 3, 5])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1512.593"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 1512 = 2^{3} \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1512.bl (of order $6$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,-18] 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:	$12.0733807856$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 26x^{14} + 521x^{12} + 3668x^{10} + 19270x^{8} + 25507x^{6} + 25166x^{4} + 8869x^{2} + 2401 $ x^16 + 26x^14 + 521x^12 + 3668x^10 + 19270x^8 + 25507x^6 + 25166x^4 + 8869*x^2 + 2401
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			1025.5
Root			$1.29168 + 2.23726i$ of defining polynomial
Character	$\chi$	$=$	1512.1025
Dual form			1512.2.bl.c.593.5

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.680382 + 1.17846i) q^{5} +(2.57416 + 0.611303i) q^{7} +(-3.45534 - 1.99494i) q^{11} -5.53803i q^{13} +(0.611303 - 1.05881i) q^{17} +(1.83184 - 1.05761i) q^{19} +(3.00903 - 1.73726i) q^{23} +(1.57416 - 2.72653i) q^{25} -4.14832i q^{29} +(-1.79607 - 1.03696i) q^{31} +(1.03102 + 3.44945i) q^{35} +(-0.0588079 - 0.101858i) q^{37} +1.87021 q^{41} +9.25582 q^{43} +(1.29763 + 2.24756i) q^{47} +(6.25262 + 3.14719i) q^{49} +(-4.50201 - 2.59923i) q^{53} -5.42928i q^{55} +(-5.60710 + 9.71179i) q^{59} +(-1.26234 + 0.728810i) q^{61} +(6.52632 - 3.76797i) q^{65} +(0.895706 - 1.55141i) q^{67} +14.9296i q^{71} +(8.57940 + 4.95332i) q^{73} +(-7.67509 - 7.24756i) q^{77} +(1.69887 + 2.94253i) q^{79} -4.08041 q^{83} +1.66368 q^{85} +(7.53074 + 13.0436i) q^{89} +(3.38541 - 14.2558i) q^{91} +(2.49270 + 1.43916i) q^{95} -19.0469i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 4 q^{7} - 18 q^{19} - 12 q^{25} + 24 q^{31} + 16 q^{37} - 52 q^{43} + 44 q^{49} + 6 q^{61} - 4 q^{67} - 12 q^{73} + 34 q^{79} - 68 q^{85} + 102 q^{91}+O(q^{100}) $ 16 * q + 4 * q^7 - 18 * q^19 - 12 * q^25 + 24 * q^31 + 16 * q^37 - 52 * q^43 + 44 * q^49 + 6 * q^61 - 4 * q^67 - 12 * q^73 + 34 * q^79 - 68 * q^85 + 102 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1512\mathbb{Z}\right)^\times$.

$n$	$757$	$785$	$1081$	$1135$
$\chi(n)$	$1$	$-1$	$e\left(\frac{1}{6}\right)$	$1$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	0.680382	+	1.17846i	0.304276	+	0.527021i	0.977100	−	0.212781i	$-0.0682522\pi$
$5$	0.680382	+	1.17846i	0.304276	+	0.527021i	−0.672824	+	0.739803i	$0.734919\pi$
$6$		0			0
$7$	2.57416	+	0.611303i	0.972942	+	0.231051i
$7$	2.57416	+	0.611303i	0.972942	+	0.231051i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−3.45534	−	1.99494i	−1.04182	−	0.601497i	−0.121474	−	0.992595i	$-0.538762\pi$
$11$	−3.45534	−	1.99494i	−1.04182	−	0.601497i	−0.920349	+	0.391097i	$0.872096\pi$
$12$		0			0
$13$		−	5.53803i		−	1.53597i	−0.640467	−	0.767986i	$-0.721259\pi$
$13$		−	5.53803i		−	1.53597i	0.640467	−	0.767986i	$-0.278741\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	0.611303	−	1.05881i	0.148263	−	0.256799i	−0.782323	−	0.622873i	$-0.785965\pi$
$17$	0.611303	−	1.05881i	0.148263	−	0.256799i	0.930585	+	0.366075i	$0.119299\pi$
$18$		0			0
$19$	1.83184	−	1.05761i	0.420253	−	0.242633i	−0.274933	−	0.961463i	$-0.588656\pi$
$19$	1.83184	−	1.05761i	0.420253	−	0.242633i	0.695185	+	0.718830i	$0.255322\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	3.00903	−	1.73726i	0.627426	−	0.362244i	−0.152329	−	0.988330i	$-0.548677\pi$
$23$	3.00903	−	1.73726i	0.627426	−	0.362244i	0.779754	+	0.626085i	$0.215344\pi$
$24$		0			0
$25$	1.57416	−	2.72653i	0.314832	−	0.545306i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	4.14832i		−	0.770324i	−0.922849	−	0.385162i	$-0.874146\pi$
$29$		−	4.14832i		−	0.770324i	0.922849	−	0.385162i	$-0.125854\pi$
$30$		0			0
$31$	−1.79607	−	1.03696i	−0.322584	−	0.186244i	0.329960	−	0.943995i	$-0.392965\pi$
$31$	−1.79607	−	1.03696i	−0.322584	−	0.186244i	−0.652544	+	0.757751i	$0.726298\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	1.03102	+	3.44945i	0.174274	+	0.583064i
$36$		0			0
$37$	−0.0588079	−	0.101858i	−0.00966796	−	0.0167454i	0.861151	−	0.508349i	$-0.169744\pi$
$37$	−0.0588079	−	0.101858i	−0.00966796	−	0.0167454i	−0.870819	+	0.491604i	$0.836411\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	1.87021			0.292077			0.146039	−	0.989279i	$-0.453348\pi$
$41$	1.87021			0.292077			0.146039	+	0.989279i	$0.453348\pi$
$42$		0			0
$43$	9.25582			1.41150			0.705750	−	0.708461i	$-0.250610\pi$
$43$	9.25582			1.41150			0.705750	+	0.708461i	$0.250610\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	1.29763	+	2.24756i	0.189278	+	0.327840i	0.945010	−	0.327042i	$-0.106052\pi$
$47$	1.29763	+	2.24756i	0.189278	+	0.327840i	−0.755731	+	0.654882i	$0.772718\pi$
$48$		0			0
$49$	6.25262	+	3.14719i	0.893231	+	0.449598i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−4.50201	−	2.59923i	−0.618398	−	0.357032i	0.157847	−	0.987464i	$-0.449545\pi$
$53$	−4.50201	−	2.59923i	−0.618398	−	0.357032i	−0.776245	+	0.630431i	$0.782878\pi$
$54$		0			0
$55$		−	5.42928i		−	0.732084i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−5.60710	+	9.71179i	−0.729983	+	1.26437i	0.226907	+	0.973916i	$0.427139\pi$
$59$	−5.60710	+	9.71179i	−0.729983	+	1.26437i	−0.956890	+	0.290451i	$0.906195\pi$
$60$		0			0
$61$	−1.26234	+	0.728810i	−0.161626	+	0.0933145i	−0.578631	−	0.815589i	$-0.696413\pi$
$61$	−1.26234	+	0.728810i	−0.161626	+	0.0933145i	0.417006	+	0.908904i	$0.363080\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	6.52632	−	3.76797i	0.809490	−	0.467359i
$66$		0			0
$67$	0.895706	−	1.55141i	0.109428	−	0.189535i	−0.806111	−	0.591765i	$-0.798431\pi$
$67$	0.895706	−	1.55141i	0.109428	−	0.189535i	0.915539	+	0.402230i	$0.131765\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			14.9296i			1.77182i	0.463858	+	0.885910i	$0.346465\pi$
$71$			14.9296i			1.77182i	−0.463858	+	0.885910i	$0.653535\pi$
$72$		0			0
$73$	8.57940	+	4.95332i	1.00414	+	0.579742i	0.909471	−	0.415767i	$-0.136487\pi$
$73$	8.57940	+	4.95332i	1.00414	+	0.579742i	0.0946711	+	0.995509i	$0.469820\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−7.67509	−	7.24756i	−0.874657	−	0.825936i
$78$		0			0
$79$	1.69887	+	2.94253i	0.191138	+	0.331060i	0.945628	−	0.325252i	$-0.105449\pi$
$79$	1.69887	+	2.94253i	0.191138	+	0.331060i	−0.754490	+	0.656312i	$0.772116\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−4.08041			−0.447883			−0.223941	−	0.974603i	$-0.571892\pi$
$83$	−4.08041			−0.447883			−0.223941	+	0.974603i	$0.571892\pi$
$84$		0			0
$85$	1.66368			0.180451
$86$		0			0
$87$		0			0
$88$		0			0
$89$	7.53074	+	13.0436i	0.798257	+	1.38262i	0.920750	+	0.390153i	$0.127578\pi$
$89$	7.53074	+	13.0436i	0.798257	+	1.38262i	−0.122493	+	0.992469i	$0.539089\pi$
$90$		0			0
$91$	3.38541	−	14.2558i	0.354888	−	1.49441i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	2.49270	+	1.43916i	0.255745	+	0.147655i
$96$		0			0
$97$		−	19.0469i		−	1.93392i	−0.254925	−	0.966961i	$-0.582051\pi$
$97$		−	19.0469i		−	1.93392i	0.254925	−	0.966961i	$-0.417949\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	6.49284	−	11.2459i	0.646062	−	1.11901i	−0.337994	−	0.941148i	$-0.609748\pi$
$101$	6.49284	−	11.2459i	0.646062	−	1.11901i	0.984055	−	0.177863i	$-0.0569184\pi$
$102$		0			0
$103$	10.2829	−	5.93685i	1.01321	−	0.584975i	0.101078	−	0.994879i	$-0.467771\pi$
$103$	10.2829	−	5.93685i	1.01321	−	0.584975i	0.912129	+	0.409903i	$0.134438\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−6.77415	+	3.91106i	−0.654882	+	0.378096i	−0.790324	−	0.612689i	$-0.790088\pi$
$107$	−6.77415	+	3.91106i	−0.654882	+	0.378096i	0.135442	+	0.990785i	$0.456755\pi$
$108$		0			0
$109$	3.98988	−	6.91068i	0.382161	−	0.661923i	−0.609210	−	0.793009i	$-0.708513\pi$
$109$	3.98988	−	6.91068i	0.382161	−	0.661923i	0.991371	+	0.131086i	$0.0418466\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	19.3734i		−	1.82250i	−0.411854	−	0.911250i	$-0.635119\pi$
$113$		−	19.3734i		−	1.82250i	0.411854	−	0.911250i	$-0.364881\pi$
$114$		0			0
$115$	4.09458	+	2.36400i	0.381821	+	0.220445i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	2.22085	−	2.35185i	0.203585	−	0.215594i
$120$		0			0
$121$	2.45957	+	4.26011i	0.223598	+	0.387282i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	11.0879			0.991735
$126$		0			0
$127$	−19.4236			−1.72356			−0.861782	−	0.507279i	$-0.830651\pi$
$127$	−19.4236			−1.72356			−0.861782	+	0.507279i	$0.830651\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−7.04195	−	12.1970i	−0.615258	−	1.06566i	−0.990339	−	0.138666i	$-0.955719\pi$
$131$	−7.04195	−	12.1970i	−0.615258	−	1.06566i	0.375081	−	0.926992i	$-0.377615\pi$
$132$		0			0
$133$	5.36197	−	1.60266i	0.464942	−	0.138968i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−4.56044	−	2.63297i	−0.389624	−	0.224950i	0.292373	−	0.956304i	$-0.405555\pi$
$137$	−4.56044	−	2.63297i	−0.389624	−	0.224950i	−0.681997	+	0.731355i	$0.738888\pi$
$138$		0			0
$139$		−	5.34431i		−	0.453299i	−0.973976	−	0.226649i	$-0.927223\pi$
$139$		−	5.34431i		−	0.453299i	0.973976	−	0.226649i	$-0.0727771\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−11.0480	+	19.1358i	−0.923883	+	1.60021i
$144$		0			0
$145$	4.88861	−	2.82244i	0.405977	−	0.234391i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	6.51725	−	3.76274i	0.533914	−	0.308255i	−0.208695	−	0.977981i	$-0.566922\pi$
$149$	6.51725	−	3.76274i	0.533914	−	0.308255i	0.742609	+	0.669725i	$0.233588\pi$
$150$		0			0
$151$	−3.08952	+	5.35120i	−0.251421	+	0.435474i	−0.963917	−	0.266202i	$-0.914231\pi$
$151$	−3.08952	+	5.35120i	−0.251421	+	0.435474i	0.712496	+	0.701676i	$0.247565\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		−	2.82212i		−	0.226678i
$156$		0			0
$157$	5.65135	+	3.26281i	0.451027	+	0.260401i	0.708264	−	0.705948i	$-0.249479\pi$
$157$	5.65135	+	3.26281i	0.451027	+	0.260401i	−0.257237	+	0.966348i	$0.582812\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	8.80772	−	2.63257i	0.694146	−	0.207475i
$162$		0			0
$163$	11.1479	+	19.3088i	0.873173	+	1.51238i	0.858697	+	0.512484i	$0.171275\pi$
$163$	11.1479	+	19.3088i	0.873173	+	1.51238i	0.0144761	+	0.999895i	$0.495392\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	9.00213			0.696606			0.348303	−	0.937382i	$-0.386758\pi$
$167$	9.00213			0.696606			0.348303	+	0.937382i	$0.386758\pi$
$168$		0			0
$169$	−17.6697			−1.35921
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−10.3651	−	17.9528i	−0.788042	−	1.36493i	−0.927165	−	0.374653i	$-0.877762\pi$
$173$	−10.3651	−	17.9528i	−0.788042	−	1.36493i	0.139123	−	0.990275i	$-0.455572\pi$
$174$		0			0
$175$	5.71888	−	6.05623i	0.432307	−	0.457808i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	6.60158	+	3.81143i	0.493425	+	0.284879i	0.725994	−	0.687701i	$-0.241380\pi$
$179$	6.60158	+	3.81143i	0.493425	+	0.284879i	−0.232569	+	0.972580i	$0.574713\pi$
$180$		0			0
$181$			12.8974i			0.958659i	0.877635	+	0.479329i	$0.159120\pi$
$181$			12.8974i			0.958659i	−0.877635	+	0.479329i	$0.840880\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	0.0800236	−	0.138605i	0.00588345	−	0.0101904i
$186$		0			0
$187$	−4.22452	+	2.43903i	−0.308927	+	0.178359i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−19.0263	+	10.9848i	−1.37669	+	0.794834i	−0.991760	−	0.128111i	$-0.959109\pi$
$191$	−19.0263	+	10.9848i	−1.37669	+	0.794834i	−0.384933	+	0.922945i	$0.625775\pi$
$192$		0			0
$193$	−10.1249	+	17.5368i	−0.728805	+	1.26233i	0.228584	+	0.973524i	$0.426591\pi$
$193$	−10.1249	+	17.5368i	−0.728805	+	1.26233i	−0.957389	+	0.288803i	$0.906743\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			21.6026i			1.53912i	0.638573	+	0.769561i	$0.279525\pi$
$197$			21.6026i			1.53912i	−0.638573	+	0.769561i	$0.720475\pi$
$198$		0			0
$199$	4.66776	+	2.69493i	0.330889	+	0.191039i	0.656236	−	0.754556i	$-0.272148\pi$
$199$	4.66776	+	2.69493i	0.330889	+	0.191039i	−0.325347	+	0.945595i	$0.605481\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	2.53588	−	10.6785i	0.177984	−	0.749481i
$204$		0			0
$205$	1.27246	+	2.20396i	0.0888721	+	0.153931i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−8.43950			−0.583772
$210$		0			0
$211$	−27.8621			−1.91811			−0.959055	−	0.283221i	$-0.908597\pi$
$211$	−27.8621			−1.91811			−0.959055	+	0.283221i	$0.908597\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	6.29749	+	10.9076i	0.429485	+	0.743890i
$216$		0			0
$217$	−3.98948	−	3.76725i	−0.270824	−	0.255738i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−5.86371	−	3.38541i	−0.394436	−	0.227727i
$222$		0			0
$223$			7.53887i			0.504840i	0.967618	+	0.252420i	$0.0812264\pi$
$223$			7.53887i			0.504840i	−0.967618	+	0.252420i	$0.918774\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	11.3246	−	19.6148i	0.751639	−	1.30188i	−0.195388	−	0.980726i	$-0.562597\pi$
$227$	11.3246	−	19.6148i	0.751639	−	1.30188i	0.947028	−	0.321152i	$-0.104070\pi$
$228$		0			0
$229$	−5.47493	+	3.16095i	−0.361793	+	0.208882i	−0.669867	−	0.742481i	$-0.733649\pi$
$229$	−5.47493	+	3.16095i	−0.361793	+	0.208882i	0.308074	+	0.951362i	$0.400316\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−5.06800	+	2.92601i	−0.332016	+	0.191689i	−0.656736	−	0.754121i	$-0.728063\pi$
$233$	−5.06800	+	2.92601i	−0.332016	+	0.191689i	0.324720	+	0.945810i	$0.394730\pi$
$234$		0			0
$235$	−1.76576	+	3.05839i	−0.115186	+	0.199508i
$236$		0			0
$237$		0			0
$238$		0			0
$239$			10.5052i			0.679527i	0.940511	+	0.339764i	$0.110347\pi$
$239$			10.5052i			0.679527i	−0.940511	+	0.339764i	$0.889653\pi$
$240$		0			0
$241$	−10.0626	−	5.80964i	−0.648188	−	0.374232i	0.139574	−	0.990212i	$-0.455427\pi$
$241$	−10.0626	−	5.80964i	−0.648188	−	0.374232i	−0.787762	+	0.615980i	$0.788760\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	0.545347	+	9.50972i	0.0348410	+	0.607554i
$246$		0			0
$247$	−5.85709	−	10.1448i	−0.372677	−	0.645496i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−4.24234			−0.267774			−0.133887	−	0.990997i	$-0.542746\pi$
$251$	−4.24234			−0.267774			−0.133887	+	0.990997i	$0.542746\pi$
$252$		0			0
$253$	−13.8629			−0.871556
$254$		0			0
$255$		0			0
$256$		0			0
$257$	1.98083	+	3.43090i	0.123561	+	0.214014i	0.921169	−	0.389162i	$-0.127235\pi$
$257$	1.98083	+	3.43090i	0.123561	+	0.214014i	−0.797609	+	0.603175i	$0.793902\pi$
$258$		0			0
$259$	−0.0891147	−	0.298149i	−0.00553732	−	0.0185261i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	18.7613	+	10.8318i	1.15687	+	0.667920i	0.950552	−	0.310565i	$-0.100518\pi$
$263$	18.7613	+	10.8318i	1.15687	+	0.667920i	0.206319	+	0.978485i	$0.433852\pi$
$264$		0			0
$265$		−	7.07388i		−	0.434545i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−1.24231	+	2.15175i	−0.0757453	+	0.131195i	−0.901410	−	0.432966i	$-0.857467\pi$
$269$	−1.24231	+	2.15175i	−0.0757453	+	0.131195i	0.825665	+	0.564161i	$0.190800\pi$
$270$		0			0
$271$	−9.97493	+	5.75903i	−0.605934	+	0.349836i	−0.771372	−	0.636384i	$-0.780429\pi$
$271$	−9.97493	+	5.75903i	−0.605934	+	0.349836i	0.165439	+	0.986220i	$0.447096\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−10.8785	+	6.28072i	−0.656000	+	0.378742i
$276$		0			0
$277$	−4.75565	+	8.23702i	−0.285739	+	0.494914i	−0.972788	−	0.231696i	$-0.925572\pi$
$277$	−4.75565	+	8.23702i	−0.285739	+	0.494914i	0.687049	+	0.726611i	$0.258906\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	15.3670i		−	0.916720i	−0.888767	−	0.458360i	$-0.848437\pi$
$281$		−	15.3670i		−	0.916720i	0.888767	−	0.458360i	$-0.151563\pi$
$282$		0			0
$283$	1.86966	+	1.07945i	0.111140	+	0.0641665i	0.554539	−	0.832157i	$-0.312894\pi$
$283$	1.86966	+	1.07945i	0.111140	+	0.0641665i	−0.443400	+	0.896324i	$0.646228\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	4.81422	+	1.14326i	0.284174	+	0.0674847i
$288$		0			0
$289$	7.75262	+	13.4279i	0.456036	+	0.789878i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	6.81382			0.398068			0.199034	−	0.979993i	$-0.436220\pi$
$293$	6.81382			0.398068			0.199034	+	0.979993i	$0.436220\pi$
$294$		0			0
$295$	−15.2599			−0.888465
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−9.62101	−	16.6641i	−0.556397	−	0.963709i
$300$		0			0
$301$	23.8260	+	5.65811i	1.37331	+	0.326128i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−1.71774	−	0.991738i	−0.0983575	−	0.0567867i
$306$		0			0
$307$		−	23.4648i		−	1.33921i	−0.742718	−	0.669604i	$-0.766464\pi$
$307$		−	23.4648i		−	1.33921i	0.742718	−	0.669604i	$-0.233536\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−2.85780	+	4.94986i	−0.162051	+	0.280681i	−0.935604	−	0.353051i	$-0.885144\pi$
$311$	−2.85780	+	4.94986i	−0.162051	+	0.280681i	0.773553	+	0.633731i	$0.218478\pi$
$312$		0			0
$313$	−20.5371	+	11.8571i	−1.16083	+	0.670203i	−0.951502	−	0.307643i	$-0.900460\pi$
$313$	−20.5371	+	11.8571i	−1.16083	+	0.670203i	−0.209324	+	0.977846i	$0.567126\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	23.8780	−	13.7860i	1.34112	−	0.774296i	0.354149	−	0.935189i	$-0.384771\pi$
$317$	23.8780	−	13.7860i	1.34112	−	0.774296i	0.986972	+	0.160893i	$0.0514374\pi$
$318$		0			0
$319$	−8.27566	+	14.3339i	−0.463348	+	0.802542i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	2.58609i		−	0.143894i
$324$		0			0
$325$	−15.0996	−	8.71775i	−0.837574	−	0.483574i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	1.96637	+	6.57882i	0.108409	+	0.362702i
$330$		0			0
$331$	15.5135	+	26.8702i	0.852699	+	1.47692i	0.878763	+	0.477258i	$0.158369\pi$
$331$	15.5135	+	26.8702i	0.852699	+	1.47692i	−0.0260646	+	0.999660i	$0.508298\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	2.43769			0.133185
$336$		0			0
$337$	−5.07153			−0.276264			−0.138132	−	0.990414i	$-0.544110\pi$
$337$	−5.07153			−0.276264			−0.138132	+	0.990414i	$0.544110\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	4.13736	+	7.16611i	0.224050	+	0.388067i
$342$		0			0
$343$	14.1714	+	11.9236i	0.765182	+	0.643814i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−14.6580	−	8.46278i	−0.786880	−	0.454306i	0.0519827	−	0.998648i	$-0.483446\pi$
$347$	−14.6580	−	8.46278i	−0.786880	−	0.454306i	−0.838863	+	0.544342i	$0.816779\pi$
$348$		0			0
$349$		−	32.5007i		−	1.73972i	−0.493295	−	0.869862i	$-0.664208\pi$
$349$		−	32.5007i		−	1.73972i	0.493295	−	0.869862i	$-0.335792\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−1.13006	+	1.95732i	−0.0601469	+	0.104177i	−0.894531	−	0.447006i	$-0.852490\pi$
$353$	−1.13006	+	1.95732i	−0.0601469	+	0.104177i	0.834384	+	0.551183i	$0.185824\pi$
$354$		0			0
$355$	−17.5939	+	10.1578i	−0.933787	+	0.539122i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−3.26915	+	1.88744i	−0.172539	+	0.0996155i	−0.583783	−	0.811910i	$-0.698428\pi$
$359$	−3.26915	+	1.88744i	−0.172539	+	0.0996155i	0.411243	+	0.911526i	$0.365095\pi$
$360$		0			0
$361$	−7.26291	+	12.5797i	−0.382258	+	0.662091i
$362$		0			0
$363$		0			0
$364$		0			0
$365$			13.4806i			0.705606i
$366$		0			0
$367$	18.5602	+	10.7157i	0.968832	+	0.559355i	0.898880	−	0.438195i	$-0.144382\pi$
$367$	18.5602	+	10.7157i	0.968832	+	0.559355i	0.0699521	+	0.997550i	$0.477715\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−9.99997	−	9.44294i	−0.519173	−	0.490253i
$372$		0			0
$373$	9.93974	+	17.2161i	0.514660	+	0.891417i	0.999855	+	0.0170115i	$0.00541518\pi$
$373$	9.93974	+	17.2161i	0.514660	+	0.891417i	−0.485195	+	0.874406i	$0.661251\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−22.9735			−1.18320
$378$		0			0
$379$	−12.0982			−0.621442			−0.310721	−	0.950501i	$-0.600570\pi$
$379$	−12.0982			−0.621442			−0.310721	+	0.950501i	$0.600570\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	4.53360	+	7.85243i	0.231656	+	0.401240i	0.958296	−	0.285779i	$-0.0922522\pi$
$383$	4.53360	+	7.85243i	0.231656	+	0.401240i	−0.726639	+	0.687019i	$0.758919\pi$
$384$		0			0
$385$	3.31894	−	13.9759i	0.169149	−	0.712275i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	14.6818	+	8.47656i	0.744398	+	0.429779i	0.823666	−	0.567075i	$-0.191925\pi$
$389$	14.6818	+	8.47656i	0.744398	+	0.429779i	−0.0792680	+	0.996853i	$0.525258\pi$
$390$		0			0
$391$		−	4.24798i		−	0.214829i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−2.31176	+	4.00408i	−0.116317	+	0.201467i
$396$		0			0
$397$	−26.8809	+	15.5197i	−1.34911	+	0.778910i	−0.988124	−	0.153660i	$-0.950894\pi$
$397$	−26.8809	+	15.5197i	−1.34911	+	0.778910i	−0.360989	+	0.932570i	$0.617561\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	19.0581	−	11.0032i	0.951716	−	0.549474i	0.0581027	−	0.998311i	$-0.481495\pi$
$401$	19.0581	−	11.0032i	0.951716	−	0.549474i	0.893614	+	0.448837i	$0.148162\pi$
$402$		0			0
$403$	−5.74272	+	9.94669i	−0.286065	+	0.495480i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			0.469273i			0.0232610i
$408$		0			0
$409$	−0.0925434	−	0.0534300i	−0.00457598	−	0.00264194i	0.497710	−	0.867343i	$-0.334174\pi$
$409$	−0.0925434	−	0.0534300i	−0.00457598	−	0.00264194i	−0.502286	+	0.864701i	$0.667508\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−20.3704	+	21.5721i	−1.00236	+	1.06149i
$414$		0			0
$415$	−2.77623	−	4.80858i	−0.136280	−	0.236044i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−9.58338			−0.468179			−0.234089	−	0.972215i	$-0.575211\pi$
$419$	−9.58338			−0.468179			−0.234089	+	0.972215i	$0.575211\pi$
$420$		0			0
$421$	−18.7607			−0.914341			−0.457170	−	0.889379i	$-0.651137\pi$
$421$	−18.7607			−0.914341			−0.457170	+	0.889379i	$0.651137\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−1.92458	−	3.33347i	−0.0933558	−	0.161697i
$426$		0			0
$427$	−3.69498	+	1.10440i	−0.178813	+	0.0534459i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	28.1409	+	16.2472i	1.35550	+	0.782598i	0.989013	−	0.147826i	$-0.0472274\pi$
$431$	28.1409	+	16.2472i	1.35550	+	0.782598i	0.366486	+	0.930424i	$0.380561\pi$
$432$		0			0
$433$		−	20.8051i		−	0.999830i	−0.866075	−	0.499915i	$-0.833365\pi$
$433$		−	20.8051i		−	0.999830i	0.866075	−	0.499915i	$-0.166635\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	3.67470	−	6.36477i	0.175785	−	0.304468i
$438$		0			0
$439$	−0.649605	+	0.375050i	−0.0310039	+	0.0179001i	−0.515422	−	0.856937i	$-0.672365\pi$
$439$	−0.649605	+	0.375050i	−0.0310039	+	0.0179001i	0.484418	+	0.874837i	$0.339031\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−16.2819	+	9.40037i	−0.773577	+	0.446625i	−0.834149	−	0.551539i	$-0.814041\pi$
$443$	−16.2819	+	9.40037i	−0.773577	+	0.446625i	0.0605722	+	0.998164i	$0.480707\pi$
$444$		0			0
$445$	−10.2476	+	17.7493i	−0.485781	+	0.841397i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	29.4094i		−	1.38792i	−0.720016	−	0.693958i	$-0.755865\pi$
$449$		−	29.4094i		−	1.38792i	0.720016	−	0.693958i	$-0.244135\pi$
$450$		0			0
$451$	−6.46220	−	3.73095i	−0.304293	−	0.175684i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	19.1032	−	5.70981i	0.895570	−	0.267680i
$456$		0			0
$457$	3.62733	+	6.28273i	0.169680	+	0.293894i	0.938307	−	0.345803i	$-0.112393\pi$
$457$	3.62733	+	6.28273i	0.169680	+	0.293894i	−0.768628	+	0.639696i	$0.779060\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−36.0906			−1.68091			−0.840454	−	0.541883i	$-0.817712\pi$
$461$	−36.0906			−1.68091			−0.840454	+	0.541883i	$0.817712\pi$
$462$		0			0
$463$	−22.9337			−1.06582			−0.532909	−	0.846172i	$-0.678901\pi$
$463$	−22.9337			−1.06582			−0.532909	+	0.846172i	$0.678901\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−11.2752	−	19.5293i	−0.521755	−	0.903706i	−0.999680	−	0.0253053i	$-0.991944\pi$
$467$	−11.2752	−	19.5293i	−0.521755	−	0.903706i	0.477925	−	0.878401i	$-0.341389\pi$
$468$		0			0
$469$	3.25407	−	3.44603i	0.150259	−	0.159123i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−31.9820	−	18.4648i	−1.47053	−	0.849013i
$474$		0			0
$475$		−	6.65941i		−	0.305555i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	19.7007	−	34.1226i	0.900148	−	1.55910i	0.0728476	−	0.997343i	$-0.476791\pi$
$479$	19.7007	−	34.1226i	0.900148	−	1.55910i	0.827301	−	0.561759i	$-0.189875\pi$
$480$		0			0
$481$	−0.564094	+	0.325680i	−0.0257205	+	0.0148497i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	22.4459	−	12.9592i	1.01922	−	0.588446i
$486$		0			0
$487$	−7.01029	+	12.1422i	−0.317667	+	0.550215i	−0.980001	−	0.198994i	$-0.936233\pi$
$487$	−7.01029	+	12.1422i	−0.317667	+	0.550215i	0.662334	+	0.749209i	$0.269566\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	8.76186i		−	0.395417i	−0.980261	−	0.197709i	$-0.936650\pi$
$491$		−	8.76186i		−	0.395417i	0.980261	−	0.197709i	$-0.0633500\pi$
$492$		0			0
$493$	−4.39228	−	2.53588i	−0.197818	−	0.114210i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−9.12652	+	38.4312i	−0.409380	+	1.72388i
$498$		0			0
$499$	−4.32475	−	7.49068i	−0.193602	−	0.335329i	0.752839	−	0.658205i	$-0.228684\pi$
$499$	−4.32475	−	7.49068i	−0.193602	−	0.335329i	−0.946441	+	0.322876i	$0.895350\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	43.4918			1.93920			0.969602	−	0.244686i	$-0.0786850\pi$
$503$	43.4918			1.93920			0.969602	+	0.244686i	$0.0786850\pi$
$504$		0			0
$505$	17.6704			0.786324
$506$		0			0
$507$		0			0
$508$		0			0
$509$	19.2594	+	33.3583i	0.853658	+	1.47858i	0.877885	+	0.478872i	$0.158954\pi$
$509$	19.2594	+	33.3583i	0.853658	+	1.47858i	−0.0242268	+	0.999706i	$0.507712\pi$
$510$		0			0
$511$	19.0568	+	17.9952i	0.843022	+	0.796063i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	13.9926	+	8.07865i	0.616589	+	0.355988i
$516$		0			0
$517$		−	10.3548i		−	0.455402i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−7.28884	+	12.6246i	−0.319330	+	0.553096i	−0.980348	−	0.197274i	$-0.936791\pi$
$521$	−7.28884	+	12.6246i	−0.319330	+	0.553096i	0.661018	+	0.750370i	$0.270125\pi$
$522$		0			0
$523$	−23.4207	+	13.5220i	−1.02412	+	0.591274i	−0.915294	−	0.402787i	$-0.868042\pi$
$523$	−23.4207	+	13.5220i	−1.02412	+	0.591274i	−0.108823	+	0.994061i	$0.534708\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−2.19589	+	1.26780i	−0.0956544	+	0.0552261i
$528$		0			0
$529$	−5.46383	+	9.46363i	−0.237558	+	0.411462i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	10.3573i		−	0.448623i
$534$		0			0
$535$	−9.21802	−	5.32203i	−0.398530	−	0.230091i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−15.3265	−	23.3482i	−0.660157	−	1.00568i
$540$		0			0
$541$	12.4056	+	21.4871i	0.533358	+	0.923804i	0.999241	+	0.0389572i	$0.0124036\pi$
$541$	12.4056	+	21.4871i	0.533358	+	0.923804i	−0.465882	+	0.884847i	$0.654263\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	10.8586			0.465130
$546$		0			0
$547$	2.63297			0.112578			0.0562888	−	0.998415i	$-0.482073\pi$
$547$	2.63297			0.112578			0.0562888	+	0.998415i	$0.482073\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−4.38732	−	7.59906i	−0.186906	−	0.323731i
$552$		0			0
$553$	2.57439	+	8.61307i	0.109474	+	0.366265i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−22.2079	−	12.8217i	−0.940978	−	0.543274i	−0.0507110	−	0.998713i	$-0.516149\pi$
$557$	−22.2079	−	12.8217i	−0.940978	−	0.543274i	−0.890267	+	0.455440i	$0.849482\pi$
$558$		0			0
$559$		−	51.2590i		−	2.16802i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	14.7349	−	25.5215i	0.621000	−	1.07560i	−0.368299	−	0.929707i	$-0.620060\pi$
$563$	14.7349	−	25.5215i	0.621000	−	1.07560i	0.989300	−	0.145897i	$-0.0466068\pi$
$564$		0			0
$565$	22.8307	−	13.1813i	0.960496	−	0.554543i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−19.4963	+	11.2562i	−0.817329	+	0.471885i	−0.849495	−	0.527597i	$-0.823093\pi$
$569$	−19.4963	+	11.2562i	−0.817329	+	0.471885i	0.0321653	+	0.999483i	$0.489760\pi$
$570$		0			0
$571$	17.1255	−	29.6622i	0.716679	−	1.24132i	−0.245629	−	0.969364i	$-0.578995\pi$
$571$	17.1255	−	29.6622i	0.716679	−	1.24132i	0.962308	−	0.271961i	$-0.0876720\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	10.9389i		−	0.456185i
$576$		0			0
$577$	34.8118	+	20.0986i	1.44923	+	0.836715i	0.998436	−	0.0559112i	$-0.0178064\pi$
$577$	34.8118	+	20.0986i	1.44923	+	0.836715i	0.450797	+	0.892626i	$0.351140\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−10.5036	−	2.49436i	−0.435764	−	0.103484i
$582$		0			0
$583$	10.3706	+	17.9625i	0.429508	+	0.743929i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−15.9446			−0.658104			−0.329052	−	0.944312i	$-0.606729\pi$
$587$	−15.9446			−0.658104			−0.329052	+	0.944312i	$0.606729\pi$
$588$		0			0
$589$	−4.38682			−0.180756
$590$		0			0
$591$		0			0
$592$		0			0
$593$	17.0760	+	29.5766i	0.701229	+	1.21456i	0.968035	+	0.250814i	$0.0806982\pi$
$593$	17.0760	+	29.5766i	0.701229	+	1.21456i	−0.266807	+	0.963750i	$0.585968\pi$
$594$		0			0
$595$	4.28257	+	1.01701i	0.175568	+	0.0416934i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−10.5191	−	6.07319i	−0.429797	−	0.248144i	0.269463	−	0.963011i	$-0.413154\pi$
$599$	−10.5191	−	6.07319i	−0.429797	−	0.248144i	−0.699260	+	0.714867i	$0.746487\pi$
$600$		0			0
$601$			9.15890i			0.373599i	0.982398	+	0.186800i	$0.0598115\pi$
$601$			9.15890i			0.373599i	−0.982398	+	0.186800i	$0.940188\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−3.34690	+	5.79700i	−0.136071	+	0.235681i
$606$		0			0
$607$	35.6897	−	20.6055i	1.44860	−	0.836350i	0.450202	−	0.892927i	$-0.351352\pi$
$607$	35.6897	−	20.6055i	1.44860	−	0.836350i	0.998398	+	0.0565772i	$0.0180187\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	12.4470	−	7.18630i	0.503553	−	0.290726i
$612$		0			0
$613$	−6.38396	+	11.0573i	−0.257846	+	0.446602i	−0.965665	−	0.259792i	$-0.916346\pi$
$613$	−6.38396	+	11.0573i	−0.257846	+	0.446602i	0.707819	+	0.706394i	$0.249679\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			16.7469i			0.674204i	0.941468	+	0.337102i	$0.109447\pi$
$617$			16.7469i			0.674204i	−0.941468	+	0.337102i	$0.890553\pi$
$618$		0			0
$619$	14.4869	+	8.36399i	0.582276	+	0.336177i	0.762037	−	0.647533i	$-0.224199\pi$
$619$	14.4869	+	8.36399i	0.582276	+	0.336177i	−0.179762	+	0.983710i	$0.557533\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	11.4117	+	38.1800i	0.457202	+	1.52965i
$624$		0			0
$625$	−0.326779	−	0.565998i	−0.0130712	−	0.0226399i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−0.143798			−0.00573359
$630$		0			0
$631$	15.5056			0.617268			0.308634	−	0.951181i	$-0.400128\pi$
$631$	15.5056			0.617268			0.308634	+	0.951181i	$0.400128\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−13.2154	−	22.8898i	−0.524439	−	0.908355i
$636$		0			0
$637$	17.4292	−	34.6272i	0.690570	−	1.37198i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	31.9557	+	18.4496i	1.26217	+	0.728717i	0.973495	−	0.228709i	$-0.0734505\pi$
$641$	31.9557	+	18.4496i	1.26217	+	0.728717i	0.288679	+	0.957426i	$0.406784\pi$
$642$		0			0
$643$		−	4.05288i		−	0.159830i	−0.996802	−	0.0799149i	$-0.974535\pi$
$643$		−	4.05288i		−	0.159830i	0.996802	−	0.0799149i	$-0.0254649\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−8.79083	+	15.2262i	−0.345603	+	0.598602i	−0.985463	−	0.169889i	$-0.945659\pi$
$647$	−8.79083	+	15.2262i	−0.345603	+	0.598602i	0.639860	+	0.768492i	$0.278992\pi$
$648$		0			0
$649$	38.7489	−	22.3717i	1.52103	−	0.878165i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−12.6252	+	7.28915i	−0.494062	+	0.285247i	−0.726258	−	0.687422i	$-0.758742\pi$
$653$	−12.6252	+	7.28915i	−0.494062	+	0.285247i	0.232196	+	0.972669i	$0.425409\pi$
$654$		0			0
$655$	9.58242	−	16.5972i	0.374416	−	0.648508i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	9.82212i		−	0.382615i	−0.981530	−	0.191308i	$-0.938727\pi$
$659$		−	9.82212i		−	0.382615i	0.981530	−	0.191308i	$-0.0612728\pi$
$660$		0			0
$661$	3.79036	+	2.18836i	0.147428	+	0.0851175i	0.571899	−	0.820324i	$-0.306207\pi$
$661$	3.79036	+	2.18836i	0.147428	+	0.0851175i	−0.424472	+	0.905441i	$0.639540\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	5.53685	+	5.22843i	0.214710	+	0.202750i
$666$		0			0
$667$	−7.20673	−	12.4824i	−0.279046	−	0.483321i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	5.81573			0.224514
$672$		0			0
$673$	−39.7808			−1.53344			−0.766719	−	0.641982i	$-0.778112\pi$
$673$	−39.7808			−1.53344			−0.766719	+	0.641982i	$0.778112\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	21.6256	+	37.4566i	0.831139	+	1.43957i	0.897136	+	0.441755i	$0.145644\pi$
$677$	21.6256	+	37.4566i	0.831139	+	1.43957i	−0.0659972	+	0.997820i	$0.521023\pi$
$678$		0			0
$679$	11.6434	−	49.0298i	0.446834	−	1.88159i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−18.3170	−	10.5753i	−0.700881	−	0.404654i	0.106794	−	0.994281i	$-0.465941\pi$
$683$	−18.3170	−	10.5753i	−0.700881	−	0.404654i	−0.807675	+	0.589627i	$0.799275\pi$
$684$		0			0
$685$		−	7.16570i		−	0.273787i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−14.3946	+	24.9322i	−0.548392	+	0.949842i
$690$		0			0
$691$	−2.03537	+	1.17512i	−0.0774290	+	0.0447036i	−0.538215	−	0.842808i	$-0.680901\pi$
$691$	−2.03537	+	1.17512i	−0.0774290	+	0.0447036i	0.460786	+	0.887511i	$0.347568\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	6.29804	−	3.63617i	0.238898	−	0.137928i
$696$		0			0
$697$	1.14326	−	1.98019i	0.0433042	−	0.0750051i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			35.3730i			1.33602i	0.744153	+	0.668010i	$0.232853\pi$
$701$			35.3730i			1.33602i	−0.744153	+	0.668010i	$0.767147\pi$
$702$		0			0
$703$	−0.215453	−	0.124392i	−0.00812597	−	0.00469153i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	23.5883	−	24.9797i	0.887129	−	0.939460i
$708$		0			0
$709$	−9.49611	−	16.4477i	−0.356634	−	0.617708i	0.630762	−	0.775976i	$-0.282742\pi$
$709$	−9.49611	−	16.4477i	−0.356634	−	0.617708i	−0.987396	+	0.158268i	$0.949409\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−7.20591			−0.269863
$714$		0			0
$715$	−30.0675			−1.12446
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−22.1934	−	38.4401i	−0.827674	−	1.43357i	−0.899858	−	0.436183i	$-0.856330\pi$
$719$	−22.1934	−	38.4401i	−0.827674	−	1.43357i	0.0721838	−	0.997391i	$-0.477003\pi$
$720$		0			0
$721$	30.0991	−	8.99643i	1.12095	−	0.335044i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−11.3105	−	6.53013i	−0.420062	−	0.242523i
$726$		0			0
$727$		−	6.50373i		−	0.241210i	−0.992701	−	0.120605i	$-0.961517\pi$
$727$		−	6.50373i		−	0.241210i	0.992701	−	0.120605i	$-0.0384834\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	5.65811	−	9.80014i	0.209273	−	0.362471i
$732$		0			0
$733$	6.03997	−	3.48718i	0.223091	−	0.128802i	−0.384290	−	0.923213i	$-0.625554\pi$
$733$	6.03997	−	3.48718i	0.223091	−	0.128802i	0.607381	+	0.794411i	$0.292220\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−6.18994	+	3.57376i	−0.228009	+	0.131641i
$738$		0			0
$739$	4.29705	−	7.44271i	0.158069	−	0.273784i	−0.776103	−	0.630606i	$-0.782806\pi$
$739$	4.29705	−	7.44271i	0.158069	−	0.273784i	0.934172	+	0.356822i	$0.116140\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			38.3233i			1.40595i	0.711217	+	0.702973i	$0.248144\pi$
$743$			38.3233i			1.40595i	−0.711217	+	0.702973i	$0.751856\pi$
$744$		0			0
$745$	8.86844	+	5.12019i	0.324914	+	0.187589i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−19.8286	+	5.92664i	−0.724522	+	0.216555i
$750$		0			0
$751$	21.4927	+	37.2264i	0.784279	+	1.35841i	0.929429	+	0.369002i	$0.120300\pi$
$751$	21.4927	+	37.2264i	0.784279	+	1.35841i	−0.145150	+	0.989410i	$0.546366\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−8.40820			−0.306006
$756$		0			0
$757$	−33.0768			−1.20220			−0.601098	−	0.799175i	$-0.705270\pi$
$757$	−33.0768			−1.20220			−0.601098	+	0.799175i	$0.705270\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	8.00334	+	13.8622i	0.290121	+	0.502504i	0.973838	−	0.227243i	$-0.0729711\pi$
$761$	8.00334	+	13.8622i	0.290121	+	0.502504i	−0.683717	+	0.729747i	$0.739638\pi$
$762$		0			0
$763$	14.4951	−	15.3502i	0.524758	−	0.555714i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	53.7841	+	31.0523i	1.94203	+	1.12123i
$768$		0			0
$769$		−	29.5755i		−	1.06652i	−0.845951	−	0.533260i	$-0.820967\pi$
$769$		−	29.5755i		−	1.06652i	0.845951	−	0.533260i	$-0.179033\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	17.8887	−	30.9841i	0.643411	−	1.11442i	−0.341255	−	0.939971i	$-0.610852\pi$
$773$	17.8887	−	30.9841i	0.643411	−	1.11442i	0.984666	−	0.174450i	$-0.0558148\pi$
$774$		0			0
$775$	−5.65461	+	3.26469i	−0.203120	+	0.117271i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	3.42592	−	1.97796i	0.122746	−	0.0708676i
$780$		0			0
$781$	29.7837	−	51.5869i	1.06574	−	1.84592i
$782$		0			0
$783$		0			0
$784$		0			0
$785$			8.87982i			0.316934i
$786$		0			0
$787$	32.8710	+	18.9781i	1.17172	+	0.676496i	0.954086	−	0.299534i	$-0.0968311\pi$
$787$	32.8710	+	18.9781i	1.17172	+	0.676496i	0.217639	+	0.976029i	$0.430164\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	11.8430	−	49.8704i	0.421090	−	1.77319i
$792$		0			0
$793$	4.03617	+	6.99085i	0.143329	+	0.248252i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−24.5926			−0.871114			−0.435557	−	0.900161i	$-0.643449\pi$
$797$	−24.5926			−0.871114			−0.435557	+	0.900161i	$0.643449\pi$
$798$		0			0
$799$	3.17298			0.112252
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−19.7631	−	34.2308i	−0.697426	−	1.20798i
$804$		0			0
$805$	9.09498	+	8.58836i	0.320556	+	0.302700i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−28.2123	−	16.2884i	−0.991892	−	0.572669i	−0.0860527	−	0.996291i	$-0.527425\pi$
$809$	−28.2123	−	16.2884i	−0.991892	−	0.572669i	−0.905839	+	0.423621i	$0.860759\pi$
$810$		0			0
$811$			4.24282i			0.148986i	0.997222	+	0.0744928i	$0.0237338\pi$
$811$			4.24282i			0.148986i	−0.997222	+	0.0744928i	$0.976266\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−15.1697	+	26.2747i	−0.531371	+	0.920361i
$816$		0			0
$817$	16.9552	−	9.78907i	0.593186	−	0.342476i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−1.36483	+	0.787984i	−0.0476328	+	0.0275008i	−0.523627	−	0.851947i	$-0.675422\pi$
$821$	−1.36483	+	0.787984i	−0.0476328	+	0.0275008i	0.475994	+	0.879448i	$0.342088\pi$
$822$		0			0
$823$	−16.4537	+	28.4987i	−0.573540	+	0.993401i	0.422658	+	0.906289i	$0.361097\pi$
$823$	−16.4537	+	28.4987i	−0.573540	+	0.993401i	−0.996199	+	0.0871119i	$0.972236\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	40.7480i		−	1.41695i	−0.705737	−	0.708474i	$-0.749384\pi$
$827$		−	40.7480i		−	1.41695i	0.705737	−	0.708474i	$-0.250616\pi$
$828$		0			0
$829$	−0.386563	−	0.223182i	−0.0134259	−	0.00775145i	0.493272	−	0.869875i	$-0.335801\pi$
$829$	−0.386563	−	0.223182i	−0.0134259	−	0.00775145i	−0.506698	+	0.862124i	$0.669134\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	7.15451	−	4.69644i	0.247889	−	0.162722i
$834$		0			0
$835$	6.12488	+	10.6086i	0.211960	+	0.367126i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−15.3454			−0.529783			−0.264892	−	0.964278i	$-0.585336\pi$
$839$	−15.3454			−0.529783			−0.264892	+	0.964278i	$0.585336\pi$
$840$		0			0
$841$	11.7914			0.406600
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−12.0222	−	20.8230i	−0.413575	−	0.716333i
$846$		0			0
$847$	3.72712	+	12.4697i	0.128066	+	0.428466i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−0.353909	−	0.204330i	−0.0121319	−	0.00700433i
$852$		0			0
$853$			37.9300i			1.29870i	0.760490	+	0.649350i	$0.224959\pi$
$853$			37.9300i			1.29870i	−0.760490	+	0.649350i	$0.775041\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	18.7381	−	32.4553i	0.640080	−	1.10865i	−0.345334	−	0.938480i	$-0.612234\pi$
$857$	18.7381	−	32.4553i	0.640080	−	1.10865i	0.985414	−	0.170172i	$-0.0544322\pi$
$858$		0			0
$859$	24.5565	−	14.1777i	0.837856	−	0.483736i	−0.0186789	−	0.999826i	$-0.505946\pi$
$859$	24.5565	−	14.1777i	0.837856	−	0.483736i	0.856535	+	0.516089i	$0.172613\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−3.57855	+	2.06607i	−0.121815	+	0.0703300i	−0.559669	−	0.828716i	$-0.689072\pi$
$863$	−3.57855	+	2.06607i	−0.121815	+	0.0703300i	0.437854	+	0.899046i	$0.355739\pi$
$864$		0			0
$865$	14.1044	−	24.4296i	0.479564	−	0.830630i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	13.5566i		−	0.459875i
$870$		0			0
$871$	−8.59174	−	4.96044i	−0.291120	−	0.168078i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	28.5421	+	6.77809i	0.964901	+	0.229141i
$876$		0			0
$877$	16.2089	+	28.0747i	0.547337	+	0.948015i	0.998456	+	0.0555515i	$0.0176917\pi$
$877$	16.2089	+	28.0747i	0.547337	+	0.948015i	−0.451119	+	0.892464i	$0.648975\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	43.7257			1.47316			0.736579	−	0.676352i	$-0.236440\pi$
$881$	43.7257			1.47316			0.736579	+	0.676352i	$0.236440\pi$
$882$		0			0
$883$	42.9555			1.44557			0.722783	−	0.691075i	$-0.242863\pi$
$883$	42.9555			1.44557			0.722783	+	0.691075i	$0.242863\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	6.80382	+	11.7846i	0.228450	+	0.395687i	0.957349	−	0.288935i	$-0.0933010\pi$
$887$	6.80382	+	11.7846i	0.228450	+	0.395687i	−0.728899	+	0.684621i	$0.759968\pi$
$888$		0			0
$889$	−49.9994	−	11.8737i	−1.67693	−	0.398231i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	4.75409	+	2.74478i	0.159090	+	0.0918504i
$894$		0			0
$895$			10.3729i			0.346728i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−4.30165	+	7.45068i	−0.143468	+	0.248494i
$900$		0			0
$901$	−5.50418	+	3.17784i	−0.183371	+	0.105869i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−15.1991	+	8.77518i	−0.505234	+	0.291697i
$906$		0			0
$907$	−22.8112	+	39.5102i	−0.757433	+	1.31191i	0.186722	+	0.982413i	$0.440214\pi$
$907$	−22.8112	+	39.5102i	−0.757433	+	1.31191i	−0.944155	+	0.329500i	$0.893120\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			36.9894i			1.22551i	0.790271	+	0.612757i	$0.209940\pi$
$911$			36.9894i			1.22551i	−0.790271	+	0.612757i	$0.790060\pi$
$912$		0			0
$913$	14.0992	+	8.14017i	0.466615	+	0.269400i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−10.6710	−	35.7018i	−0.352389	−	1.17898i
$918$		0			0
$919$	−4.35168	−	7.53732i	−0.143549	−	0.248633i	0.785282	−	0.619138i	$-0.212518\pi$
$919$	−4.35168	−	7.53732i	−0.143549	−	0.248633i	−0.928830	+	0.370505i	$0.879185\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	82.6806			2.72147
$924$		0			0
$925$	−0.370292			−0.0121751
$926$		0			0
$927$		0			0
$928$		0			0
$929$	7.53951	+	13.0588i	0.247363	+	0.428446i	0.962793	−	0.270239i	$-0.0871026\pi$
$929$	7.53951	+	13.0588i	0.247363	+	0.428446i	−0.715430	+	0.698684i	$0.753769\pi$
$930$		0			0
$931$	14.7823	−	0.847710i	0.484470	−	0.0277826i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−5.74857	−	3.31894i	−0.187998	−	0.108541i
$936$		0			0
$937$			2.48603i			0.0812150i	0.999175	+	0.0406075i	$0.0129293\pi$
$937$			2.48603i			0.0812150i	−0.999175	+	0.0406075i	$0.987071\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−21.2383	+	36.7859i	−0.692350	+	1.19919i	0.278715	+	0.960374i	$0.410091\pi$
$941$	−21.2383	+	36.7859i	−0.692350	+	1.19919i	−0.971066	+	0.238812i	$0.923242\pi$
$942$		0			0
$943$	5.62751	−	3.24904i	0.183257	−	0.105803i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−26.8300	+	15.4903i	−0.871856	+	0.503367i	−0.867965	−	0.496626i	$-0.834572\pi$
$947$	−26.8300	+	15.4903i	−0.871856	+	0.503367i	−0.00389169	+	0.999992i	$0.501239\pi$
$948$		0			0
$949$	27.4316	−	47.5129i	0.890467	−	1.54233i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	51.8636i		−	1.68003i	−0.542564	−	0.840014i	$-0.682547\pi$
$953$		−	51.8636i		−	1.68003i	0.542564	−	0.840014i	$-0.317453\pi$
$954$		0			0
$955$	−25.8902	−	14.9477i	−0.837789	−	0.483698i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−10.1298	−	9.56550i	−0.327107	−	0.308886i
$960$		0			0
$961$	−13.3494	−	23.1219i	−0.430626	−	0.745867i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−27.5551			−0.887031
$966$		0			0
$967$	4.31643			0.138807			0.0694036	−	0.997589i	$-0.477890\pi$
$967$	4.31643			0.138807			0.0694036	+	0.997589i	$0.477890\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−19.6891	−	34.1026i	−0.631854	−	1.09440i	−0.987172	−	0.159658i	$-0.948961\pi$
$971$	−19.6891	−	34.1026i	−0.631854	−	1.09440i	0.355319	−	0.934745i	$-0.384372\pi$
$972$		0			0
$973$	3.26699	−	13.7571i	0.104735	−	0.441033i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−7.42640	−	4.28763i	−0.237592	−	0.137174i	0.376478	−	0.926426i	$-0.377135\pi$
$977$	−7.42640	−	4.28763i	−0.237592	−	0.137174i	−0.614069	+	0.789252i	$0.710468\pi$
$978$		0			0
$979$		−	60.0935i		−	1.92060i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−14.0576	+	24.3485i	−0.448368	+	0.776596i	−0.998280	−	0.0586265i	$-0.981328\pi$
$983$	−14.0576	+	24.3485i	−0.448368	+	0.776596i	0.549912	+	0.835223i	$0.314661\pi$
$984$		0			0
$985$	−25.4577	+	14.6980i	−0.811150	+	0.468318i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	27.8510	−	16.0798i	0.885611	−	0.511308i
$990$		0			0
$991$	13.4036	−	23.2157i	0.425778	−	0.737470i	−0.570714	−	0.821149i	$-0.693334\pi$
$991$	13.4036	−	23.2157i	0.425778	−	0.737470i	0.996493	+	0.0836789i	$0.0266670\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			7.33433i			0.232514i
$996$		0			0
$997$	17.6764	+	10.2055i	0.559817	+	0.323211i	0.753072	−	0.657938i	$-0.228571\pi$
$997$	17.6764	+	10.2055i	0.559817	+	0.323211i	−0.193255	+	0.981149i	$0.561904\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1512.2.bl.c.1025.5	yes	16
3.2	odd	2	inner	1512.2.bl.c.1025.4	yes	16
7.5	odd	6	inner	1512.2.bl.c.593.4	✓	16
21.5	even	6	inner	1512.2.bl.c.593.5	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1512.2.bl.c.593.4	✓	16	7.5	odd	6	inner
1512.2.bl.c.593.5	yes	16	21.5	even	6	inner
1512.2.bl.c.1025.4	yes	16	3.2	odd	2	inner
1512.2.bl.c.1025.5	yes	16	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 1512 = 2^{3} \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1512.bl (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(0.680382 + 1.17846i) q^{5} +(2.57416 + 0.611303i) q^{7} +(-3.45534 - 1.99494i) q^{11} -5.53803i q^{13} +(0.611303 - 1.05881i) q^{17} +(1.83184 - 1.05761i) q^{19} +(3.00903 - 1.73726i) q^{23} +(1.57416 - 2.72653i) q^{25} -4.14832i q^{29} +(-1.79607 - 1.03696i) q^{31} +(1.03102 + 3.44945i) q^{35} +(-0.0588079 - 0.101858i) q^{37} +1.87021 q^{41} +9.25582 q^{43} +(1.29763 + 2.24756i) q^{47} +(6.25262 + 3.14719i) q^{49} +(-4.50201 - 2.59923i) q^{53} -5.42928i q^{55} +(-5.60710 + 9.71179i) q^{59} +(-1.26234 + 0.728810i) q^{61} +(6.52632 - 3.76797i) q^{65} +(0.895706 - 1.55141i) q^{67} +14.9296i q^{71} +(8.57940 + 4.95332i) q^{73} +(-7.67509 - 7.24756i) q^{77} +(1.69887 + 2.94253i) q^{79} -4.08041 q^{83} +1.66368 q^{85} +(7.53074 + 13.0436i) q^{89} +(3.38541 - 14.2558i) q^{91} +(2.49270 + 1.43916i) q^{95} -19.0469i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 4 q^{7} - 18 q^{19} - 12 q^{25} + 24 q^{31} + 16 q^{37} - 52 q^{43} + 44 q^{49} + 6 q^{61} - 4 q^{67} - 12 q^{73} + 34 q^{79} - 68 q^{85} + 102 q^{91}+O(q^{100}) \) 16 * q + 4 * q^7 - 18 * q^19 - 12 * q^25 + 24 * q^31 + 16 * q^37 - 52 * q^43 + 44 * q^49 + 6 * q^61 - 4 * q^67 - 12 * q^73 + 34 * q^79 - 68 * q^85 + 102 * q^91

Introduction

L-functions

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