LMFDB - Embedded newform 432.2.s.d.287.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [432,2,Mod(143,432)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(432, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 0, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("432.143");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 432 = 2^{4} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	432.s (of order $6$, degree $2$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$3.44953736732$
Analytic rank:	$0$
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:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 144)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			287.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	432.287
Dual form			432.2.s.d.143.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(3.00000 + 1.73205i) q^{5} +(3.00000 - 1.73205i) q^{7} +(-1.50000 - 2.59808i) q^{11} +(-2.00000 + 3.46410i) q^{13} -1.73205i q^{17} +1.73205i q^{19} +(3.50000 + 6.06218i) q^{25} +(3.00000 - 1.73205i) q^{29} +12.0000 q^{35} +2.00000 q^{37} +(4.50000 + 2.59808i) q^{41} +(-4.50000 + 2.59808i) q^{43} +(-6.00000 - 10.3923i) q^{47} +(2.50000 - 4.33013i) q^{49} -10.3923i q^{55} +(-7.50000 + 12.9904i) q^{59} +(-4.00000 - 6.92820i) q^{61} +(-12.0000 + 6.92820i) q^{65} +(7.50000 + 4.33013i) q^{67} -6.00000 q^{71} -11.0000 q^{73} +(-9.00000 - 5.19615i) q^{77} +(-3.00000 + 1.73205i) q^{79} +(6.00000 + 10.3923i) q^{83} +(3.00000 - 5.19615i) q^{85} -13.8564i q^{89} +13.8564i q^{91} +(-3.00000 + 5.19615i) q^{95} +(-6.50000 - 11.2583i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 6 q^{5} + 6 q^{7} - 3 q^{11} - 4 q^{13} + 7 q^{25} + 6 q^{29} + 24 q^{35} + 4 q^{37} + 9 q^{41} - 9 q^{43} - 12 q^{47} + 5 q^{49} - 15 q^{59} - 8 q^{61} - 24 q^{65} + 15 q^{67} - 12 q^{71} - 22 q^{73}+ \cdots - 13 q^{97}+O(q^{100}) $ 2 * q + 6 * q^5 + 6 * q^7 - 3 * q^11 - 4 * q^13 + 7 * q^25 + 6 * q^29 + 24 * q^35 + 4 * q^37 + 9 * q^41 - 9 * q^43 - 12 * q^47 + 5 * q^49 - 15 * q^59 - 8 * q^61 - 24 * q^65 + 15 * q^67 - 12 * q^71 - 22 * q^73 - 18 * q^77 - 6 * q^79 + 12 * q^83 + 6 * q^85 - 6 * q^95 - 13 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/432\mathbb{Z}\right)^\times$.

$n$	$271$	$325$	$353$
$\chi(n)$	$-1$	$1$	$e\left(\frac{5}{6}\right)$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	3.00000	+	1.73205i	1.34164	+	0.774597i	0.987048	−	0.160424i	$-0.0512862\pi$
$5$	3.00000	+	1.73205i	1.34164	+	0.774597i	0.354593	+	0.935021i	$0.384620\pi$
$6$		0			0
$7$	3.00000	−	1.73205i	1.13389	−	0.654654i	0.188982	−	0.981981i	$-0.439481\pi$
$7$	3.00000	−	1.73205i	1.13389	−	0.654654i	0.944911	+	0.327327i	$0.106148\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−1.50000	−	2.59808i	−0.452267	−	0.783349i	0.546259	−	0.837616i	$-0.316051\pi$
$11$	−1.50000	−	2.59808i	−0.452267	−	0.783349i	−0.998526	+	0.0542666i	$0.982718\pi$
$12$		0			0
$13$	−2.00000	+	3.46410i	−0.554700	+	0.960769i	0.443227	+	0.896410i	$0.353834\pi$
$13$	−2.00000	+	3.46410i	−0.554700	+	0.960769i	−0.997927	+	0.0643593i	$0.979500\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	1.73205i		−	0.420084i	−0.977692	−	0.210042i	$-0.932640\pi$
$17$		−	1.73205i		−	0.420084i	0.977692	−	0.210042i	$-0.0673601\pi$
$18$		0			0
$19$			1.73205i			0.397360i	0.980064	+	0.198680i	$0.0636654\pi$
$19$			1.73205i			0.397360i	−0.980064	+	0.198680i	$0.936335\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$23$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$24$		0			0
$25$	3.50000	+	6.06218i	0.700000	+	1.21244i
$26$		0			0
$27$		0			0
$28$		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
$31$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$31$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	12.0000			2.02837
$36$		0			0
$37$	2.00000			0.328798			0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000			0.328798			0.164399	+	0.986394i	$0.447432\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	4.50000	+	2.59808i	0.702782	+	0.405751i	0.808383	−	0.588657i	$-0.200343\pi$
$41$	4.50000	+	2.59808i	0.702782	+	0.405751i	−0.105601	+	0.994409i	$0.533677\pi$
$42$		0			0
$43$	−4.50000	+	2.59808i	−0.686244	+	0.396203i	−0.802203	−	0.597051i	$-0.796339\pi$
$43$	−4.50000	+	2.59808i	−0.686244	+	0.396203i	0.115960	+	0.993254i	$0.463006\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−6.00000	−	10.3923i	−0.875190	−	1.51587i	−0.856560	−	0.516047i	$-0.827403\pi$
$47$	−6.00000	−	10.3923i	−0.875190	−	1.51587i	−0.0186297	−	0.999826i	$-0.505930\pi$
$48$		0			0
$49$	2.50000	−	4.33013i	0.357143	−	0.618590i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$		0			0
$55$		−	10.3923i		−	1.40130i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−7.50000	+	12.9904i	−0.976417	+	1.69120i	−0.301239	+	0.953549i	$0.597400\pi$
$59$	−7.50000	+	12.9904i	−0.976417	+	1.69120i	−0.675178	+	0.737655i	$0.735933\pi$
$60$		0			0
$61$	−4.00000	−	6.92820i	−0.512148	−	0.887066i	−0.999901	−	0.0140840i	$-0.995517\pi$
$61$	−4.00000	−	6.92820i	−0.512148	−	0.887066i	0.487753	−	0.872982i	$-0.337817\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−12.0000	+	6.92820i	−1.48842	+	0.859338i
$66$		0			0
$67$	7.50000	+	4.33013i	0.916271	+	0.529009i	0.882443	−	0.470418i	$-0.155897\pi$
$67$	7.50000	+	4.33013i	0.916271	+	0.529009i	0.0338274	+	0.999428i	$0.489230\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−6.00000			−0.712069			−0.356034	−	0.934473i	$-0.615871\pi$
$71$	−6.00000			−0.712069			−0.356034	+	0.934473i	$0.615871\pi$
$72$		0			0
$73$	−11.0000			−1.28745			−0.643726	−	0.765256i	$-0.722612\pi$
$73$	−11.0000			−1.28745			−0.643726	+	0.765256i	$0.722612\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−9.00000	−	5.19615i	−1.02565	−	0.592157i
$78$		0			0
$79$	−3.00000	+	1.73205i	−0.337526	+	0.194871i	−0.659178	−	0.751987i	$-0.729095\pi$
$79$	−3.00000	+	1.73205i	−0.337526	+	0.194871i	0.321651	+	0.946858i	$0.395762\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	6.00000	+	10.3923i	0.658586	+	1.14070i	0.980982	+	0.194099i	$0.0621783\pi$
$83$	6.00000	+	10.3923i	0.658586	+	1.14070i	−0.322396	+	0.946605i	$0.604488\pi$
$84$		0			0
$85$	3.00000	−	5.19615i	0.325396	−	0.563602i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	13.8564i		−	1.46878i	−0.678730	−	0.734388i	$-0.737469\pi$
$89$		−	13.8564i		−	1.46878i	0.678730	−	0.734388i	$-0.262531\pi$
$90$		0			0
$91$			13.8564i			1.45255i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−3.00000	+	5.19615i	−0.307794	+	0.533114i
$96$		0			0
$97$	−6.50000	−	11.2583i	−0.659975	−	1.14311i	−0.980622	−	0.195911i	$-0.937234\pi$
$97$	−6.50000	−	11.2583i	−0.659975	−	1.14311i	0.320647	−	0.947199i	$-0.396100\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−9.00000	+	5.19615i	−0.895533	+	0.517036i	−0.875748	−	0.482768i	$-0.839632\pi$
$101$	−9.00000	+	5.19615i	−0.895533	+	0.517036i	−0.0197851	+	0.999804i	$0.506298\pi$
$102$		0			0
$103$	−12.0000	−	6.92820i	−1.18240	−	0.682656i	−0.225828	−	0.974167i	$-0.572509\pi$
$103$	−12.0000	−	6.92820i	−1.18240	−	0.682656i	−0.956567	+	0.291511i	$0.905842\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	3.00000			0.290021			0.145010	−	0.989430i	$-0.453678\pi$
$107$	3.00000			0.290021			0.145010	+	0.989430i	$0.453678\pi$
$108$		0			0
$109$	4.00000			0.383131			0.191565	−	0.981480i	$-0.438644\pi$
$109$	4.00000			0.383131			0.191565	+	0.981480i	$0.438644\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	6.00000	+	3.46410i	0.564433	+	0.325875i	0.754923	−	0.655814i	$-0.227674\pi$
$113$	6.00000	+	3.46410i	0.564433	+	0.325875i	−0.190490	+	0.981689i	$0.561008\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−3.00000	−	5.19615i	−0.275010	−	0.476331i
$120$		0			0
$121$	1.00000	−	1.73205i	0.0909091	−	0.157459i
$122$		0			0
$123$		0			0
$124$		0			0
$125$			6.92820i			0.619677i
$126$		0			0
$127$		0			0		1.00000			$0$
$127$		0			0		−1.00000			$\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	6.00000	−	10.3923i	0.524222	−	0.907980i	−0.475380	−	0.879781i	$-0.657689\pi$
$131$	6.00000	−	10.3923i	0.524222	−	0.907980i	0.999602	−	0.0281993i	$-0.00897729\pi$
$132$		0			0
$133$	3.00000	+	5.19615i	0.260133	+	0.450564i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	1.50000	−	0.866025i	0.128154	−	0.0739895i	−0.434553	−	0.900646i	$-0.643094\pi$
$137$	1.50000	−	0.866025i	0.128154	−	0.0739895i	0.562706	+	0.826657i	$0.309760\pi$
$138$		0			0
$139$	−16.5000	−	9.52628i	−1.39951	−	0.808008i	−0.405170	−	0.914241i	$-0.632788\pi$
$139$	−16.5000	−	9.52628i	−1.39951	−	0.808008i	−0.994341	+	0.106233i	$0.966121\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	12.0000			1.00349
$144$		0			0
$145$	12.0000			0.996546
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−12.0000	−	6.92820i	−0.983078	−	0.567581i	−0.0798802	−	0.996804i	$-0.525454\pi$
$149$	−12.0000	−	6.92820i	−0.983078	−	0.567581i	−0.903198	+	0.429224i	$0.858787\pi$
$150$		0			0
$151$	6.00000	−	3.46410i	0.488273	−	0.281905i	−0.235585	−	0.971854i	$-0.575701\pi$
$151$	6.00000	−	3.46410i	0.488273	−	0.281905i	0.723858	+	0.689949i	$0.242367\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−4.00000	+	6.92820i	−0.319235	+	0.552931i	−0.980329	−	0.197372i	$-0.936759\pi$
$157$	−4.00000	+	6.92820i	−0.319235	+	0.552931i	0.661094	+	0.750303i	$0.270093\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$			3.46410i			0.271329i	0.990755	+	0.135665i	$0.0433170\pi$
$163$			3.46410i			0.271329i	−0.990755	+	0.135665i	$0.956683\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−3.00000	+	5.19615i	−0.232147	+	0.402090i	−0.958440	−	0.285295i	$-0.907908\pi$
$167$	−3.00000	+	5.19615i	−0.232147	+	0.402090i	0.726293	+	0.687386i	$0.241242\pi$
$168$		0			0
$169$	−1.50000	−	2.59808i	−0.115385	−	0.199852i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	21.0000	−	12.1244i	1.59660	−	0.921798i	0.604465	−	0.796632i	$-0.293387\pi$
$173$	21.0000	−	12.1244i	1.59660	−	0.921798i	0.992136	−	0.125166i	$-0.0399462\pi$
$174$		0			0
$175$	21.0000	+	12.1244i	1.58745	+	0.916515i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−12.0000			−0.896922			−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000			−0.896922			−0.448461	+	0.893802i	$0.648028\pi$
$180$		0			0
$181$	−8.00000			−0.594635			−0.297318	−	0.954779i	$-0.596092\pi$
$181$	−8.00000			−0.594635			−0.297318	+	0.954779i	$0.596092\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	6.00000	+	3.46410i	0.441129	+	0.254686i
$186$		0			0
$187$	−4.50000	+	2.59808i	−0.329073	+	0.189990i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	3.00000	+	5.19615i	0.217072	+	0.375980i	0.953912	−	0.300088i	$-0.0970159\pi$
$191$	3.00000	+	5.19615i	0.217072	+	0.375980i	−0.736839	+	0.676068i	$0.763683\pi$
$192$		0			0
$193$	−11.5000	+	19.9186i	−0.827788	+	1.43377i	0.0719816	+	0.997406i	$0.477068\pi$
$193$	−11.5000	+	19.9186i	−0.827788	+	1.43377i	−0.899770	+	0.436365i	$0.856266\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			13.8564i			0.987228i	0.869681	+	0.493614i	$0.164324\pi$
$197$			13.8564i			0.987228i	−0.869681	+	0.493614i	$0.835676\pi$
$198$		0			0
$199$			3.46410i			0.245564i	0.992434	+	0.122782i	$0.0391815\pi$
$199$			3.46410i			0.245564i	−0.992434	+	0.122782i	$0.960818\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	6.00000	−	10.3923i	0.421117	−	0.729397i
$204$		0			0
$205$	9.00000	+	15.5885i	0.628587	+	1.08875i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	4.50000	−	2.59808i	0.311272	−	0.179713i
$210$		0			0
$211$	15.0000	+	8.66025i	1.03264	+	0.596196i	0.917741	−	0.397180i	$-0.130011\pi$
$211$	15.0000	+	8.66025i	1.03264	+	0.596196i	0.114902	+	0.993377i	$0.463345\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−18.0000			−1.22759
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	6.00000	+	3.46410i	0.403604	+	0.233021i
$222$		0			0
$223$	18.0000	−	10.3923i	1.20537	−	0.695920i	0.243625	−	0.969870i	$-0.421663\pi$
$223$	18.0000	−	10.3923i	1.20537	−	0.695920i	0.961744	+	0.273949i	$0.0883300\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−1.50000	−	2.59808i	−0.0995585	−	0.172440i	0.811943	−	0.583736i	$-0.198410\pi$
$227$	−1.50000	−	2.59808i	−0.0995585	−	0.172440i	−0.911502	+	0.411296i	$0.865076\pi$
$228$		0			0
$229$	13.0000	−	22.5167i	0.859064	−	1.48794i	−0.0137585	−	0.999905i	$-0.504380\pi$
$229$	13.0000	−	22.5167i	0.859064	−	1.48794i	0.872823	−	0.488037i	$-0.162287\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			12.1244i			0.794293i	0.917755	+	0.397146i	$0.130000\pi$
$233$			12.1244i			0.794293i	−0.917755	+	0.397146i	$0.870000\pi$
$234$		0			0
$235$		−	41.5692i		−	2.71168i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	12.0000	−	20.7846i	0.776215	−	1.34444i	−0.157893	−	0.987456i	$-0.550470\pi$
$239$	12.0000	−	20.7846i	0.776215	−	1.34444i	0.934109	−	0.356988i	$-0.116196\pi$
$240$		0			0
$241$	8.50000	+	14.7224i	0.547533	+	0.948355i	0.998443	+	0.0557856i	$0.0177663\pi$
$241$	8.50000	+	14.7224i	0.547533	+	0.948355i	−0.450910	+	0.892570i	$0.648900\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	15.0000	−	8.66025i	0.958315	−	0.553283i
$246$		0			0
$247$	−6.00000	−	3.46410i	−0.381771	−	0.220416i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−21.0000			−1.32551			−0.662754	−	0.748837i	$-0.730613\pi$
$251$	−21.0000			−1.32551			−0.662754	+	0.748837i	$0.730613\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$	16.5000	+	9.52628i	1.02924	+	0.594233i	0.916767	−	0.399422i	$-0.130789\pi$
$257$	16.5000	+	9.52628i	1.02924	+	0.594233i	0.112474	+	0.993655i	$0.464122\pi$
$258$		0			0
$259$	6.00000	−	3.46410i	0.372822	−	0.215249i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	9.00000	+	15.5885i	0.554964	+	0.961225i	0.997906	+	0.0646755i	$0.0206012\pi$
$263$	9.00000	+	15.5885i	0.554964	+	0.961225i	−0.442943	+	0.896550i	$0.646065\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$			6.92820i			0.422420i	0.977441	+	0.211210i	$0.0677404\pi$
$269$			6.92820i			0.422420i	−0.977441	+	0.211210i	$0.932260\pi$
$270$		0			0
$271$			6.92820i			0.420858i	0.977609	+	0.210429i	$0.0674861\pi$
$271$			6.92820i			0.420858i	−0.977609	+	0.210429i	$0.932514\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	10.5000	−	18.1865i	0.633174	−	1.09669i
$276$		0			0
$277$	4.00000	+	6.92820i	0.240337	+	0.416275i	0.960810	−	0.277207i	$-0.0894088\pi$
$277$	4.00000	+	6.92820i	0.240337	+	0.416275i	−0.720473	+	0.693482i	$0.756075\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−6.00000	+	3.46410i	−0.357930	+	0.206651i	−0.668172	−	0.744007i	$-0.732923\pi$
$281$	−6.00000	+	3.46410i	−0.357930	+	0.206651i	0.310242	+	0.950657i	$0.399590\pi$
$282$		0			0
$283$	9.00000	+	5.19615i	0.534994	+	0.308879i	0.743048	−	0.669238i	$-0.233379\pi$
$283$	9.00000	+	5.19615i	0.534994	+	0.308879i	−0.208053	+	0.978117i	$0.566713\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	18.0000			1.06251
$288$		0			0
$289$	14.0000			0.823529
$290$		0			0
$291$		0			0
$292$		0			0
$293$	3.00000	+	1.73205i	0.175262	+	0.101187i	0.585065	−	0.810987i	$-0.301069\pi$
$293$	3.00000	+	1.73205i	0.175262	+	0.101187i	−0.409803	+	0.912174i	$0.634402\pi$
$294$		0			0
$295$	−45.0000	+	25.9808i	−2.62000	+	1.51266i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	−9.00000	+	15.5885i	−0.518751	+	0.898504i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	27.7128i		−	1.58683i
$306$		0			0
$307$			25.9808i			1.48280i	0.671063	+	0.741400i	$0.265838\pi$
$307$			25.9808i			1.48280i	−0.671063	+	0.741400i	$0.734162\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	3.00000	−	5.19615i	0.170114	−	0.294647i	−0.768345	−	0.640036i	$-0.778920\pi$
$311$	3.00000	−	5.19615i	0.170114	−	0.294647i	0.938460	+	0.345389i	$0.112253\pi$
$312$		0			0
$313$	0.500000	+	0.866025i	0.0282617	+	0.0489506i	0.879810	−	0.475325i	$-0.157669\pi$
$313$	0.500000	+	0.866025i	0.0282617	+	0.0489506i	−0.851549	+	0.524276i	$0.824336\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	6.00000	−	3.46410i	0.336994	−	0.194563i	−0.321948	−	0.946757i	$-0.604338\pi$
$317$	6.00000	−	3.46410i	0.336994	−	0.194563i	0.658942	+	0.752194i	$0.271004\pi$
$318$		0			0
$319$	−9.00000	−	5.19615i	−0.503903	−	0.290929i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	3.00000			0.166924
$324$		0			0
$325$	−28.0000			−1.55316
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−36.0000	−	20.7846i	−1.98474	−	1.14589i
$330$		0			0
$331$	21.0000	−	12.1244i	1.15426	−	0.666415i	0.204342	−	0.978900i	$-0.434495\pi$
$331$	21.0000	−	12.1244i	1.15426	−	0.666415i	0.949923	+	0.312485i	$0.101161\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	15.0000	+	25.9808i	0.819538	+	1.41948i
$336$		0			0
$337$	5.50000	−	9.52628i	0.299604	−	0.518930i	−0.676441	−	0.736497i	$-0.736479\pi$
$337$	5.50000	−	9.52628i	0.299604	−	0.518930i	0.976045	+	0.217567i	$0.0698121\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$			6.92820i			0.374088i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−13.5000	+	23.3827i	−0.724718	+	1.25525i	0.234372	+	0.972147i	$0.424697\pi$
$347$	−13.5000	+	23.3827i	−0.724718	+	1.25525i	−0.959090	+	0.283101i	$0.908637\pi$
$348$		0			0
$349$	−16.0000	−	27.7128i	−0.856460	−	1.48343i	−0.875284	−	0.483610i	$-0.839325\pi$
$349$	−16.0000	−	27.7128i	−0.856460	−	1.48343i	0.0188232	−	0.999823i	$-0.494008\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−22.5000	+	12.9904i	−1.19755	+	0.691408i	−0.960009	−	0.279968i	$-0.909676\pi$
$353$	−22.5000	+	12.9904i	−1.19755	+	0.691408i	−0.237545	+	0.971377i	$0.576343\pi$
$354$		0			0
$355$	−18.0000	−	10.3923i	−0.955341	−	0.551566i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−30.0000			−1.58334			−0.791670	−	0.610949i	$-0.790788\pi$
$359$	−30.0000			−1.58334			−0.791670	+	0.610949i	$0.790788\pi$
$360$		0			0
$361$	16.0000			0.842105
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−33.0000	−	19.0526i	−1.72730	−	0.997257i
$366$		0			0
$367$	3.00000	−	1.73205i	0.156599	−	0.0904123i	−0.419653	−	0.907685i	$-0.637848\pi$
$367$	3.00000	−	1.73205i	0.156599	−	0.0904123i	0.576252	+	0.817272i	$0.304515\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−10.0000	+	17.3205i	−0.517780	+	0.896822i	0.482006	+	0.876168i	$0.339908\pi$
$373$	−10.0000	+	17.3205i	−0.517780	+	0.896822i	−0.999787	+	0.0206542i	$0.993425\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			13.8564i			0.713641i
$378$		0			0
$379$		−	19.0526i		−	0.978664i	−0.872098	−	0.489332i	$-0.837241\pi$
$379$		−	19.0526i		−	0.978664i	0.872098	−	0.489332i	$-0.162759\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	9.00000	−	15.5885i	0.459879	−	0.796533i	−0.539076	−	0.842257i	$-0.681226\pi$
$383$	9.00000	−	15.5885i	0.459879	−	0.796533i	0.998954	+	0.0457244i	$0.0145596\pi$
$384$		0			0
$385$	−18.0000	−	31.1769i	−0.917365	−	1.58892i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	9.00000	−	5.19615i	0.456318	−	0.263455i	−0.254177	−	0.967158i	$-0.581804\pi$
$389$	9.00000	−	5.19615i	0.456318	−	0.263455i	0.710495	+	0.703702i	$0.248471\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−12.0000			−0.603786
$396$		0			0
$397$	14.0000			0.702640			0.351320	−	0.936255i	$-0.385733\pi$
$397$	14.0000			0.702640			0.351320	+	0.936255i	$0.385733\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	7.50000	+	4.33013i	0.374532	+	0.216236i	0.675437	−	0.737418i	$-0.263955\pi$
$401$	7.50000	+	4.33013i	0.374532	+	0.216236i	−0.300904	+	0.953654i	$0.597289\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−3.00000	−	5.19615i	−0.148704	−	0.257564i
$408$		0			0
$409$	11.5000	−	19.9186i	0.568638	−	0.984911i	−0.428063	−	0.903749i	$-0.640804\pi$
$409$	11.5000	−	19.9186i	0.568638	−	0.984911i	0.996701	−	0.0811615i	$-0.0258630\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			51.9615i			2.55686i
$414$		0			0
$415$			41.5692i			2.04055i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−6.00000	+	10.3923i	−0.293119	+	0.507697i	−0.974546	−	0.224189i	$-0.928027\pi$
$419$	−6.00000	+	10.3923i	−0.293119	+	0.507697i	0.681426	+	0.731887i	$0.261360\pi$
$420$		0			0
$421$	−5.00000	−	8.66025i	−0.243685	−	0.422075i	0.718076	−	0.695965i	$-0.245023\pi$
$421$	−5.00000	−	8.66025i	−0.243685	−	0.422075i	−0.961761	+	0.273890i	$0.911690\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	10.5000	−	6.06218i	0.509325	−	0.294059i
$426$		0			0
$427$	−24.0000	−	13.8564i	−1.16144	−	0.670559i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	18.0000			0.867029			0.433515	−	0.901146i	$-0.357273\pi$
$431$	18.0000			0.867029			0.433515	+	0.901146i	$0.357273\pi$
$432$		0			0
$433$	−31.0000			−1.48976			−0.744882	−	0.667196i	$-0.767494\pi$
$433$	−31.0000			−1.48976			−0.744882	+	0.667196i	$0.767494\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	18.0000	−	10.3923i	0.859093	−	0.495998i	−0.00461537	−	0.999989i	$-0.501469\pi$
$439$	18.0000	−	10.3923i	0.859093	−	0.495998i	0.863708	+	0.503992i	$0.168136\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−4.50000	−	7.79423i	−0.213801	−	0.370315i	0.739100	−	0.673596i	$-0.235251\pi$
$443$	−4.50000	−	7.79423i	−0.213801	−	0.370315i	−0.952901	+	0.303281i	$0.901918\pi$
$444$		0			0
$445$	24.0000	−	41.5692i	1.13771	−	1.97057i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	25.9808i		−	1.22611i	−0.790041	−	0.613054i	$-0.789941\pi$
$449$		−	25.9808i		−	1.22611i	0.790041	−	0.613054i	$-0.210059\pi$
$450$		0			0
$451$		−	15.5885i		−	0.734032i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−24.0000	+	41.5692i	−1.12514	+	1.94880i
$456$		0			0
$457$	5.50000	+	9.52628i	0.257279	+	0.445621i	0.965512	−	0.260358i	$-0.0838407\pi$
$457$	5.50000	+	9.52628i	0.257279	+	0.445621i	−0.708233	+	0.705979i	$0.750507\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−18.0000	+	10.3923i	−0.838344	+	0.484018i	−0.856701	−	0.515814i	$-0.827490\pi$
$461$	−18.0000	+	10.3923i	−0.838344	+	0.484018i	0.0183573	+	0.999831i	$0.494156\pi$
$462$		0			0
$463$	15.0000	+	8.66025i	0.697109	+	0.402476i	0.806270	−	0.591548i	$-0.201483\pi$
$463$	15.0000	+	8.66025i	0.697109	+	0.402476i	−0.109161	+	0.994024i	$0.534816\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	3.00000			0.138823			0.0694117	−	0.997588i	$-0.477888\pi$
$467$	3.00000			0.138823			0.0694117	+	0.997588i	$0.477888\pi$
$468$		0			0
$469$	30.0000			1.38527
$470$		0			0
$471$		0			0
$472$		0			0
$473$	13.5000	+	7.79423i	0.620731	+	0.358379i
$474$		0			0
$475$	−10.5000	+	6.06218i	−0.481773	+	0.278152i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	15.0000	+	25.9808i	0.685367	+	1.18709i	0.973321	+	0.229447i	$0.0736918\pi$
$479$	15.0000	+	25.9808i	0.685367	+	1.18709i	−0.287954	+	0.957644i	$0.592975\pi$
$480$		0			0
$481$	−4.00000	+	6.92820i	−0.182384	+	0.315899i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	45.0333i		−	2.04486i
$486$		0			0
$487$		−	17.3205i		−	0.784867i	−0.919780	−	0.392434i	$-0.871633\pi$
$487$		−	17.3205i		−	0.784867i	0.919780	−	0.392434i	$-0.128367\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	7.50000	−	12.9904i	0.338470	−	0.586248i	−0.645675	−	0.763612i	$-0.723424\pi$
$491$	7.50000	−	12.9904i	0.338470	−	0.586248i	0.984145	+	0.177365i	$0.0567572\pi$
$492$		0			0
$493$	−3.00000	−	5.19615i	−0.135113	−	0.234023i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−18.0000	+	10.3923i	−0.807410	+	0.466159i
$498$		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
$501$		0			0
$502$		0			0
$503$	24.0000			1.07011			0.535054	−	0.844818i	$-0.320291\pi$
$503$	24.0000			1.07011			0.535054	+	0.844818i	$0.320291\pi$
$504$		0			0
$505$	−36.0000			−1.60198
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−18.0000	−	10.3923i	−0.797836	−	0.460631i	0.0448779	−	0.998992i	$-0.485710\pi$
$509$	−18.0000	−	10.3923i	−0.797836	−	0.460631i	−0.842714	+	0.538362i	$0.819043\pi$
$510$		0			0
$511$	−33.0000	+	19.0526i	−1.45983	+	0.842836i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−24.0000	−	41.5692i	−1.05757	−	1.83176i
$516$		0			0
$517$	−18.0000	+	31.1769i	−0.791639	+	1.37116i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			15.5885i			0.682943i	0.939892	+	0.341471i	$0.110925\pi$
$521$			15.5885i			0.682943i	−0.939892	+	0.341471i	$0.889075\pi$
$522$		0			0
$523$			17.3205i			0.757373i	0.925525	+	0.378686i	$0.123624\pi$
$523$			17.3205i			0.757373i	−0.925525	+	0.378686i	$0.876376\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	11.5000	+	19.9186i	0.500000	+	0.866025i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−18.0000	+	10.3923i	−0.779667	+	0.450141i
$534$		0			0
$535$	9.00000	+	5.19615i	0.389104	+	0.224649i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−15.0000			−0.646096
$540$		0			0
$541$	−2.00000			−0.0859867			−0.0429934	−	0.999075i	$-0.513689\pi$
$541$	−2.00000			−0.0859867			−0.0429934	+	0.999075i	$0.513689\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	12.0000	+	6.92820i	0.514024	+	0.296772i
$546$		0			0
$547$	7.50000	−	4.33013i	0.320677	−	0.185143i	−0.331017	−	0.943625i	$-0.607392\pi$
$547$	7.50000	−	4.33013i	0.320677	−	0.185143i	0.651694	+	0.758482i	$0.274059\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	3.00000	+	5.19615i	0.127804	+	0.221364i
$552$		0			0
$553$	−6.00000	+	10.3923i	−0.255146	+	0.441926i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	38.1051i		−	1.61457i	−0.590165	−	0.807283i	$-0.700937\pi$
$557$		−	38.1051i		−	1.61457i	0.590165	−	0.807283i	$-0.299063\pi$
$558$		0			0
$559$		−	20.7846i		−	0.879095i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	4.50000	−	7.79423i	0.189652	−	0.328488i	−0.755482	−	0.655169i	$-0.772597\pi$
$563$	4.50000	−	7.79423i	0.189652	−	0.328488i	0.945134	+	0.326682i	$0.105931\pi$
$564$		0			0
$565$	12.0000	+	20.7846i	0.504844	+	0.874415i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	16.5000	−	9.52628i	0.691716	−	0.399362i	−0.112539	−	0.993647i	$-0.535898\pi$
$569$	16.5000	−	9.52628i	0.691716	−	0.399362i	0.804255	+	0.594285i	$0.202565\pi$
$570$		0			0
$571$	−10.5000	−	6.06218i	−0.439411	−	0.253694i	0.263937	−	0.964540i	$-0.414979\pi$
$571$	−10.5000	−	6.06218i	−0.439411	−	0.253694i	−0.703348	+	0.710846i	$0.748312\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−7.00000			−0.291414			−0.145707	−	0.989328i	$-0.546546\pi$
$577$	−7.00000			−0.291414			−0.145707	+	0.989328i	$0.546546\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	36.0000	+	20.7846i	1.49353	+	0.862291i
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−1.50000	−	2.59808i	−0.0619116	−	0.107234i	0.833408	−	0.552658i	$-0.186386\pi$
$587$	−1.50000	−	2.59808i	−0.0619116	−	0.107234i	−0.895320	+	0.445424i	$0.853053\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$			6.92820i			0.284507i	0.989830	+	0.142254i	$0.0454349\pi$
$593$			6.92820i			0.284507i	−0.989830	+	0.142254i	$0.954565\pi$
$594$		0			0
$595$		−	20.7846i		−	0.852086i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	9.00000	−	15.5885i	0.367730	−	0.636927i	−0.621480	−	0.783430i	$-0.713468\pi$
$599$	9.00000	−	15.5885i	0.367730	−	0.636927i	0.989210	+	0.146503i	$0.0468017\pi$
$600$		0			0
$601$	3.50000	+	6.06218i	0.142768	+	0.247281i	0.928538	−	0.371237i	$-0.121066\pi$
$601$	3.50000	+	6.06218i	0.142768	+	0.247281i	−0.785770	+	0.618519i	$0.787733\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	6.00000	−	3.46410i	0.243935	−	0.140836i
$606$		0			0
$607$	24.0000	+	13.8564i	0.974130	+	0.562414i	0.900493	−	0.434871i	$-0.143206\pi$
$607$	24.0000	+	13.8564i	0.974130	+	0.562414i	0.0736371	+	0.997285i	$0.476539\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	48.0000			1.94187
$612$		0			0
$613$	26.0000			1.05013			0.525065	−	0.851062i	$-0.324041\pi$
$613$	26.0000			1.05013			0.525065	+	0.851062i	$0.324041\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	25.5000	+	14.7224i	1.02659	+	0.592703i	0.916006	−	0.401164i	$-0.131394\pi$
$617$	25.5000	+	14.7224i	1.02659	+	0.592703i	0.110585	+	0.993867i	$0.464728\pi$
$618$		0			0
$619$	−25.5000	+	14.7224i	−1.02493	+	0.591744i	−0.915529	−	0.402253i	$-0.868227\pi$
$619$	−25.5000	+	14.7224i	−1.02493	+	0.591744i	−0.109403	+	0.993997i	$0.534894\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−24.0000	−	41.5692i	−0.961540	−	1.66544i
$624$		0			0
$625$	5.50000	−	9.52628i	0.220000	−	0.381051i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	3.46410i		−	0.138123i
$630$		0			0
$631$		−	3.46410i		−	0.137904i	−0.997620	−	0.0689519i	$-0.978035\pi$
$631$		−	3.46410i		−	0.137904i	0.997620	−	0.0689519i	$-0.0219655\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	10.0000	+	17.3205i	0.396214	+	0.686264i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	37.5000	−	21.6506i	1.48116	−	0.855149i	0.481389	−	0.876507i	$-0.340132\pi$
$641$	37.5000	−	21.6506i	1.48116	−	0.855149i	0.999772	+	0.0213584i	$0.00679909\pi$
$642$		0			0
$643$	22.5000	+	12.9904i	0.887313	+	0.512291i	0.873063	−	0.487608i	$-0.162130\pi$
$643$	22.5000	+	12.9904i	0.887313	+	0.512291i	0.0142506	+	0.999898i	$0.495464\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	42.0000			1.65119			0.825595	−	0.564263i	$-0.190840\pi$
$647$	42.0000			1.65119			0.825595	+	0.564263i	$0.190840\pi$
$648$		0			0
$649$	45.0000			1.76640
$650$		0			0
$651$		0			0
$652$		0			0
$653$	27.0000	+	15.5885i	1.05659	+	0.610023i	0.924487	−	0.381212i	$-0.124493\pi$
$653$	27.0000	+	15.5885i	1.05659	+	0.610023i	0.132104	+	0.991236i	$0.457827\pi$
$654$		0			0
$655$	36.0000	−	20.7846i	1.40664	−	0.812122i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	12.0000	+	20.7846i	0.467454	+	0.809653i	0.999309	−	0.0371821i	$-0.0118382\pi$
$659$	12.0000	+	20.7846i	0.467454	+	0.809653i	−0.531855	+	0.846836i	$0.678505\pi$
$660$		0			0
$661$	−7.00000	+	12.1244i	−0.272268	+	0.471583i	−0.969442	−	0.245319i	$-0.921107\pi$
$661$	−7.00000	+	12.1244i	−0.272268	+	0.471583i	0.697174	+	0.716902i	$0.254441\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$			20.7846i			0.805993i
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−12.0000	+	20.7846i	−0.463255	+	0.802381i
$672$		0			0
$673$	23.0000	+	39.8372i	0.886585	+	1.53561i	0.843886	+	0.536522i	$0.180262\pi$
$673$	23.0000	+	39.8372i	0.886585	+	1.53561i	0.0426985	+	0.999088i	$0.486405\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−18.0000	+	10.3923i	−0.691796	+	0.399409i	−0.804285	−	0.594244i	$-0.797451\pi$
$677$	−18.0000	+	10.3923i	−0.691796	+	0.399409i	0.112488	+	0.993653i	$0.464118\pi$
$678$		0			0
$679$	−39.0000	−	22.5167i	−1.49668	−	0.864110i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−15.0000			−0.573959			−0.286980	−	0.957937i	$-0.592651\pi$
$683$	−15.0000			−0.573959			−0.286980	+	0.957937i	$0.592651\pi$
$684$		0			0
$685$	6.00000			0.229248
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	27.0000	−	15.5885i	1.02713	−	0.593013i	0.110968	−	0.993824i	$-0.464605\pi$
$691$	27.0000	−	15.5885i	1.02713	−	0.593013i	0.916161	+	0.400811i	$0.131272\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−33.0000	−	57.1577i	−1.25176	−	2.16811i
$696$		0			0
$697$	4.50000	−	7.79423i	0.170450	−	0.295227i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			27.7128i			1.04670i	0.852118	+	0.523349i	$0.175318\pi$
$701$			27.7128i			1.04670i	−0.852118	+	0.523349i	$0.824682\pi$
$702$		0			0
$703$			3.46410i			0.130651i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−18.0000	+	31.1769i	−0.676960	+	1.17253i
$708$		0			0
$709$	−14.0000	−	24.2487i	−0.525781	−	0.910679i	−0.999549	−	0.0300298i	$-0.990440\pi$
$709$	−14.0000	−	24.2487i	−0.525781	−	0.910679i	0.473768	−	0.880650i	$-0.342894\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$	36.0000	+	20.7846i	1.34632	+	0.777300i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−42.0000			−1.56634			−0.783168	−	0.621810i	$-0.786397\pi$
$719$	−42.0000			−1.56634			−0.783168	+	0.621810i	$0.786397\pi$
$720$		0			0
$721$	−48.0000			−1.78761
$722$		0			0
$723$		0			0
$724$		0			0
$725$	21.0000	+	12.1244i	0.779920	+	0.450287i
$726$		0			0
$727$	36.0000	−	20.7846i	1.33517	−	0.770859i	0.349080	−	0.937093i	$-0.386494\pi$
$727$	36.0000	−	20.7846i	1.33517	−	0.770859i	0.986086	+	0.166234i	$0.0531609\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	4.50000	+	7.79423i	0.166439	+	0.288280i
$732$		0			0
$733$	−11.0000	+	19.0526i	−0.406294	+	0.703722i	−0.994471	−	0.105010i	$-0.966513\pi$
$733$	−11.0000	+	19.0526i	−0.406294	+	0.703722i	0.588177	+	0.808732i	$0.299846\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	25.9808i		−	0.957014i
$738$		0			0
$739$			25.9808i			0.955718i	0.878437	+	0.477859i	$0.158587\pi$
$739$			25.9808i			0.955718i	−0.878437	+	0.477859i	$0.841413\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−9.00000	+	15.5885i	−0.330178	+	0.571885i	−0.982547	−	0.186017i	$-0.940442\pi$
$743$	−9.00000	+	15.5885i	−0.330178	+	0.571885i	0.652369	+	0.757902i	$0.273775\pi$
$744$		0			0
$745$	−24.0000	−	41.5692i	−0.879292	−	1.52298i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	9.00000	−	5.19615i	0.328853	−	0.189863i
$750$		0			0
$751$	−33.0000	−	19.0526i	−1.20419	−	0.695238i	−0.242704	−	0.970100i	$-0.578034\pi$
$751$	−33.0000	−	19.0526i	−1.20419	−	0.695238i	−0.961483	+	0.274863i	$0.911368\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	24.0000			0.873449
$756$		0			0
$757$	−26.0000			−0.944986			−0.472493	−	0.881334i	$-0.656646\pi$
$757$	−26.0000			−0.944986			−0.472493	+	0.881334i	$0.656646\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−42.0000	−	24.2487i	−1.52250	−	0.879015i	−0.999646	−	0.0265919i	$-0.991535\pi$
$761$	−42.0000	−	24.2487i	−1.52250	−	0.879015i	−0.522852	−	0.852423i	$-0.675132\pi$
$762$		0			0
$763$	12.0000	−	6.92820i	0.434429	−	0.250818i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−30.0000	−	51.9615i	−1.08324	−	1.87622i
$768$		0			0
$769$	−7.00000	+	12.1244i	−0.252426	+	0.437215i	−0.964193	−	0.265200i	$-0.914562\pi$
$769$	−7.00000	+	12.1244i	−0.252426	+	0.437215i	0.711767	+	0.702416i	$0.247895\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	6.92820i		−	0.249190i	−0.992208	−	0.124595i	$-0.960237\pi$
$773$		−	6.92820i		−	0.249190i	0.992208	−	0.124595i	$-0.0397632\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−4.50000	+	7.79423i	−0.161229	+	0.279257i
$780$		0			0
$781$	9.00000	+	15.5885i	0.322045	+	0.557799i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−24.0000	+	13.8564i	−0.856597	+	0.494556i
$786$		0			0
$787$	−15.0000	−	8.66025i	−0.534692	−	0.308705i	0.208233	−	0.978079i	$-0.433229\pi$
$787$	−15.0000	−	8.66025i	−0.534692	−	0.308705i	−0.742925	+	0.669375i	$0.766562\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	24.0000			0.853342
$792$		0			0
$793$	32.0000			1.13635
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−3.00000	−	1.73205i	−0.106265	−	0.0613524i	0.445925	−	0.895070i	$-0.352875\pi$
$797$	−3.00000	−	1.73205i	−0.106265	−	0.0613524i	−0.552191	+	0.833718i	$0.686208\pi$
$798$		0			0
$799$	−18.0000	+	10.3923i	−0.636794	+	0.367653i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	16.5000	+	28.5788i	0.582272	+	1.00853i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	19.0526i		−	0.669852i	−0.942244	−	0.334926i	$-0.891289\pi$
$809$		−	19.0526i		−	0.669852i	0.942244	−	0.334926i	$-0.108711\pi$
$810$		0			0
$811$			36.3731i			1.27723i	0.769526	+	0.638616i	$0.220493\pi$
$811$			36.3731i			1.27723i	−0.769526	+	0.638616i	$0.779507\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−6.00000	+	10.3923i	−0.210171	+	0.364027i
$816$		0			0
$817$	−4.50000	−	7.79423i	−0.157435	−	0.272686i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−3.00000	+	1.73205i	−0.104701	+	0.0604490i	−0.551436	−	0.834217i	$-0.685920\pi$
$821$	−3.00000	+	1.73205i	−0.104701	+	0.0604490i	0.446735	+	0.894666i	$0.352587\pi$
$822$		0			0
$823$	21.0000	+	12.1244i	0.732014	+	0.422628i	0.819159	−	0.573567i	$-0.194441\pi$
$823$	21.0000	+	12.1244i	0.732014	+	0.422628i	−0.0871445	+	0.996196i	$0.527774\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	36.0000			1.25184			0.625921	−	0.779886i	$-0.284723\pi$
$827$	36.0000			1.25184			0.625921	+	0.779886i	$0.284723\pi$
$828$		0			0
$829$	−26.0000			−0.903017			−0.451509	−	0.892267i	$-0.649114\pi$
$829$	−26.0000			−0.903017			−0.451509	+	0.892267i	$0.649114\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−7.50000	−	4.33013i	−0.259860	−	0.150030i
$834$		0			0
$835$	−18.0000	+	10.3923i	−0.622916	+	0.359641i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	18.0000	+	31.1769i	0.621429	+	1.07635i	0.989220	+	0.146438i	$0.0467809\pi$
$839$	18.0000	+	31.1769i	0.621429	+	1.07635i	−0.367791	+	0.929909i	$0.619886\pi$
$840$		0			0
$841$	−8.50000	+	14.7224i	−0.293103	+	0.507670i
$842$		0			0
$843$		0			0
$844$		0			0
$845$		−	10.3923i		−	0.357506i
$846$		0			0
$847$		−	6.92820i		−	0.238056i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−16.0000	−	27.7128i	−0.547830	−	0.948869i	−0.998423	−	0.0561393i	$-0.982121\pi$
$853$	−16.0000	−	27.7128i	−0.547830	−	0.948869i	0.450593	−	0.892729i	$-0.351212\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−30.0000	+	17.3205i	−1.02478	+	0.591657i	−0.915485	−	0.402352i	$-0.868193\pi$
$857$	−30.0000	+	17.3205i	−1.02478	+	0.591657i	−0.109295	+	0.994009i	$0.534859\pi$
$858$		0			0
$859$	−25.5000	−	14.7224i	−0.870049	−	0.502323i	−0.00268433	−	0.999996i	$-0.500854\pi$
$859$	−25.5000	−	14.7224i	−0.870049	−	0.502323i	−0.867364	+	0.497674i	$0.834188\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−30.0000			−1.02121			−0.510606	−	0.859815i	$-0.670579\pi$
$863$	−30.0000			−1.02121			−0.510606	+	0.859815i	$0.670579\pi$
$864$		0			0
$865$	84.0000			2.85609
$866$		0			0
$867$		0			0
$868$		0			0
$869$	9.00000	+	5.19615i	0.305304	+	0.176267i
$870$		0			0
$871$	−30.0000	+	17.3205i	−1.01651	+	0.586883i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	12.0000	+	20.7846i	0.405674	+	0.702648i
$876$		0			0
$877$	20.0000	−	34.6410i	0.675352	−	1.16974i	−0.301014	−	0.953620i	$-0.597325\pi$
$877$	20.0000	−	34.6410i	0.675352	−	1.16974i	0.976366	−	0.216124i	$-0.0693416\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			20.7846i			0.700251i	0.936703	+	0.350126i	$0.113861\pi$
$881$			20.7846i			0.700251i	−0.936703	+	0.350126i	$0.886139\pi$
$882$		0			0
$883$			8.66025i			0.291441i	0.989326	+	0.145720i	$0.0465500\pi$
$883$			8.66025i			0.291441i	−0.989326	+	0.145720i	$0.953450\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	21.0000	−	36.3731i	0.705111	−	1.22129i	−0.261540	−	0.965193i	$-0.584230\pi$
$887$	21.0000	−	36.3731i	0.705111	−	1.22129i	0.966651	−	0.256096i	$-0.0824362\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	18.0000	−	10.3923i	0.602347	−	0.347765i
$894$		0			0
$895$	−36.0000	−	20.7846i	−1.20335	−	0.694753i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−24.0000	−	13.8564i	−0.797787	−	0.460603i
$906$		0			0
$907$	−34.5000	+	19.9186i	−1.14555	+	0.661386i	−0.947800	−	0.318866i	$-0.896698\pi$
$907$	−34.5000	+	19.9186i	−1.14555	+	0.661386i	−0.197754	+	0.980252i	$0.563365\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	12.0000	+	20.7846i	0.397578	+	0.688625i	0.993426	−	0.114472i	$-0.0365176\pi$
$911$	12.0000	+	20.7846i	0.397578	+	0.688625i	−0.595849	+	0.803097i	$0.703184\pi$
$912$		0			0
$913$	18.0000	−	31.1769i	0.595713	−	1.03181i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	41.5692i		−	1.37274i
$918$		0			0
$919$		−	41.5692i		−	1.37124i	−0.727959	−	0.685621i	$-0.759531\pi$
$919$		−	41.5692i		−	1.37124i	0.727959	−	0.685621i	$-0.240469\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	12.0000	−	20.7846i	0.394985	−	0.684134i
$924$		0			0
$925$	7.00000	+	12.1244i	0.230159	+	0.398646i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−6.00000	+	3.46410i	−0.196854	+	0.113653i	−0.595187	−	0.803587i	$-0.702922\pi$
$929$	−6.00000	+	3.46410i	−0.196854	+	0.113653i	0.398333	+	0.917241i	$0.369589\pi$
$930$		0			0
$931$	7.50000	+	4.33013i	0.245803	+	0.141914i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−18.0000			−0.588663
$936$		0			0
$937$	−34.0000			−1.11073			−0.555366	−	0.831606i	$-0.687422\pi$
$937$	−34.0000			−1.11073			−0.555366	+	0.831606i	$0.687422\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−12.0000	−	6.92820i	−0.391189	−	0.225853i	0.291486	−	0.956575i	$-0.405850\pi$
$941$	−12.0000	−	6.92820i	−0.391189	−	0.225853i	−0.682675	+	0.730722i	$0.739184\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	4.50000	+	7.79423i	0.146230	+	0.253278i	0.929831	−	0.367986i	$-0.119953\pi$
$947$	4.50000	+	7.79423i	0.146230	+	0.253278i	−0.783601	+	0.621264i	$0.786619\pi$
$948$		0			0
$949$	22.0000	−	38.1051i	0.714150	−	1.23694i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			12.1244i			0.392746i	0.980529	+	0.196373i	$0.0629164\pi$
$953$			12.1244i			0.392746i	−0.980529	+	0.196373i	$0.937084\pi$
$954$		0			0
$955$			20.7846i			0.672574i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	3.00000	−	5.19615i	0.0968751	−	0.167793i
$960$		0			0
$961$	−15.5000	−	26.8468i	−0.500000	−	0.866025i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−69.0000	+	39.8372i	−2.22119	+	1.28240i
$966$		0			0
$967$	21.0000	+	12.1244i	0.675314	+	0.389893i	0.798087	−	0.602542i	$-0.205845\pi$
$967$	21.0000	+	12.1244i	0.675314	+	0.389893i	−0.122773	+	0.992435i	$0.539179\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−12.0000			−0.385098			−0.192549	−	0.981287i	$-0.561675\pi$
$971$	−12.0000			−0.385098			−0.192549	+	0.981287i	$0.561675\pi$
$972$		0			0
$973$	−66.0000			−2.11586
$974$		0			0
$975$		0			0
$976$		0			0
$977$	13.5000	+	7.79423i	0.431903	+	0.249359i	0.700157	−	0.713989i	$-0.253113\pi$
$977$	13.5000	+	7.79423i	0.431903	+	0.249359i	−0.268254	+	0.963348i	$0.586447\pi$
$978$		0			0
$979$	−36.0000	+	20.7846i	−1.15056	+	0.664279i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−3.00000	−	5.19615i	−0.0956851	−	0.165732i	0.814209	−	0.580572i	$-0.197171\pi$
$983$	−3.00000	−	5.19615i	−0.0956851	−	0.165732i	−0.909894	+	0.414840i	$0.863838\pi$
$984$		0			0
$985$	−24.0000	+	41.5692i	−0.764704	+	1.32451i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		−	48.4974i		−	1.54057i	−0.637699	−	0.770286i	$-0.720114\pi$
$991$		−	48.4974i		−	1.54057i	0.637699	−	0.770286i	$-0.279886\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−6.00000	+	10.3923i	−0.190213	+	0.329458i
$996$		0			0
$997$	−10.0000	−	17.3205i	−0.316703	−	0.548546i	0.663095	−	0.748535i	$-0.269243\pi$
$997$	−10.0000	−	17.3205i	−0.316703	−	0.548546i	−0.979798	+	0.199989i	$0.935909\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	432.2.s.d.287.1		2
3.2	odd	2		144.2.s.d.95.1	yes	2
4.3	odd	2		432.2.s.c.287.1		2
8.3	odd	2		1728.2.s.a.1151.1		2
8.5	even	2		1728.2.s.b.1151.1		2
9.2	odd	6		432.2.s.c.143.1		2
9.4	even	3		1296.2.c.d.1295.1		2
9.5	odd	6		1296.2.c.b.1295.2		2
9.7	even	3		144.2.s.a.47.1	✓	2
12.11	even	2		144.2.s.a.95.1	yes	2
24.5	odd	2		576.2.s.a.383.1		2
24.11	even	2		576.2.s.d.383.1		2
36.7	odd	6		144.2.s.d.47.1	yes	2
36.11	even	6	inner	432.2.s.d.143.1		2
36.23	even	6		1296.2.c.d.1295.2		2
36.31	odd	6		1296.2.c.b.1295.1		2
72.5	odd	6		5184.2.c.c.5183.1		2
72.11	even	6		1728.2.s.b.575.1		2
72.13	even	6		5184.2.c.a.5183.2		2
72.29	odd	6		1728.2.s.a.575.1		2
72.43	odd	6		576.2.s.a.191.1		2
72.59	even	6		5184.2.c.a.5183.1		2
72.61	even	6		576.2.s.d.191.1		2
72.67	odd	6		5184.2.c.c.5183.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
144.2.s.a.47.1	✓	2	9.7	even	3
144.2.s.a.95.1	yes	2	12.11	even	2
144.2.s.d.47.1	yes	2	36.7	odd	6
144.2.s.d.95.1	yes	2	3.2	odd	2
432.2.s.c.143.1		2	9.2	odd	6
432.2.s.c.287.1		2	4.3	odd	2
432.2.s.d.143.1		2	36.11	even	6	inner
432.2.s.d.287.1		2	1.1	even	1	trivial
576.2.s.a.191.1		2	72.43	odd	6
576.2.s.a.383.1		2	24.5	odd	2
576.2.s.d.191.1		2	72.61	even	6
576.2.s.d.383.1		2	24.11	even	2
1296.2.c.b.1295.1		2	36.31	odd	6
1296.2.c.b.1295.2		2	9.5	odd	6
1296.2.c.d.1295.1		2	9.4	even	3
1296.2.c.d.1295.2		2	36.23	even	6
1728.2.s.a.575.1		2	72.29	odd	6
1728.2.s.a.1151.1		2	8.3	odd	2
1728.2.s.b.575.1		2	72.11	even	6
1728.2.s.b.1151.1		2	8.5	even	2
5184.2.c.a.5183.1		2	72.59	even	6
5184.2.c.a.5183.2		2	72.13	even	6
5184.2.c.c.5183.1		2	72.5	odd	6
5184.2.c.c.5183.2		2	72.67	odd	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 432 = 2^{4} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	432.s (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(3.00000 + 1.73205i) q^{5} +(3.00000 - 1.73205i) q^{7} +(-1.50000 - 2.59808i) q^{11} +(-2.00000 + 3.46410i) q^{13} -1.73205i q^{17} +1.73205i q^{19} +(3.50000 + 6.06218i) q^{25} +(3.00000 - 1.73205i) q^{29} +12.0000 q^{35} +2.00000 q^{37} +(4.50000 + 2.59808i) q^{41} +(-4.50000 + 2.59808i) q^{43} +(-6.00000 - 10.3923i) q^{47} +(2.50000 - 4.33013i) q^{49} -10.3923i q^{55} +(-7.50000 + 12.9904i) q^{59} +(-4.00000 - 6.92820i) q^{61} +(-12.0000 + 6.92820i) q^{65} +(7.50000 + 4.33013i) q^{67} -6.00000 q^{71} -11.0000 q^{73} +(-9.00000 - 5.19615i) q^{77} +(-3.00000 + 1.73205i) q^{79} +(6.00000 + 10.3923i) q^{83} +(3.00000 - 5.19615i) q^{85} -13.8564i q^{89} +13.8564i q^{91} +(-3.00000 + 5.19615i) q^{95} +(-6.50000 - 11.2583i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 6 q^{5} + 6 q^{7} - 3 q^{11} - 4 q^{13} + 7 q^{25} + 6 q^{29} + 24 q^{35} + 4 q^{37} + 9 q^{41} - 9 q^{43} - 12 q^{47} + 5 q^{49} - 15 q^{59} - 8 q^{61} - 24 q^{65} + 15 q^{67} - 12 q^{71} - 22 q^{73}+ \cdots - 13 q^{97}+O(q^{100}) \) 2 * q + 6 * q^5 + 6 * q^7 - 3 * q^11 - 4 * q^13 + 7 * q^25 + 6 * q^29 + 24 * q^35 + 4 * q^37 + 9 * q^41 - 9 * q^43 - 12 * q^47 + 5 * q^49 - 15 * q^59 - 8 * q^61 - 24 * q^65 + 15 * q^67 - 12 * q^71 - 22 * q^73 - 18 * q^77 - 6 * q^79 + 12 * q^83 + 6 * q^85 - 6 * q^95 - 13 * q^97

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