LMFDB - Embedded newform 1332.2.j.e.1009.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1332,2,Mod(433,1332)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1332, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 2]))

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

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

chi := DirichletCharacter("1332.433");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1332 = 2^{2} \cdot 3^{2} \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1332.j (of order $3$, degree $2$, 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:	$10.6360735492$
Analytic rank:	$0$
Dimension:	$6$
Relative dimension:	$3$ over $\Q(\zeta_{3})$
Coefficient field:	6.0.27870912.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} + 7x^{4} + 2x^{3} + 38x^{2} - 12x + 4 $ x^6 - x^5 + 7x^4 + 2x^3 + 38x^2 - 12x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 148)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			1009.2
Root			$-1.08870 + 1.88569i$ of defining polynomial
Character	$\chi$	$=$	1332.1009
Dual form			1332.2.j.e.433.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.500000 - 0.866025i) q^{5} +(-0.370556 - 0.641823i) q^{7} +2.91852 q^{11} +(1.54797 + 2.68116i) q^{13} +(-3.41852 + 5.92105i) q^{17} +(-0.0887048 - 0.153641i) q^{19} +3.43630 q^{23} +(2.00000 - 3.46410i) q^{25} +4.61371 q^{29} +2.91852 q^{31} +(-0.370556 + 0.641823i) q^{35} +(-2.04797 - 5.72764i) q^{37} +(0.322590 + 0.558743i) q^{41} -3.48223 q^{43} +9.79112 q^{47} +(3.22538 - 5.58651i) q^{49} +(3.37056 - 5.83798i) q^{53} +(-1.45926 - 2.52751i) q^{55} +(3.72538 - 6.45254i) q^{59} +(3.67741 + 6.36946i) q^{61} +(1.54797 - 2.68116i) q^{65} +(5.18464 + 8.98005i) q^{67} +(2.72538 + 4.72049i) q^{71} +12.3548 q^{73} +(-1.08148 - 1.87317i) q^{77} +(6.26611 + 10.8532i) q^{79} +(-4.18464 + 7.24800i) q^{83} +6.83705 q^{85} +(7.85482 - 13.6049i) q^{89} +(1.14722 - 1.98704i) q^{91} +(-0.0887048 + 0.153641i) q^{95} -16.8056 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 3 q^{5} - q^{7} - 7 q^{13} - 3 q^{17} + 7 q^{19} + 8 q^{23} + 12 q^{25} - q^{35} + 4 q^{37} + 17 q^{41} - 16 q^{43} + 16 q^{47} - 12 q^{49} + 19 q^{53} - 9 q^{59} + 7 q^{61} - 7 q^{65} - 9 q^{67}+ \cdots - 8 q^{97}+O(q^{100}) $ 6 * q - 3 * q^5 - q^7 - 7 * q^13 - 3 * q^17 + 7 * q^19 + 8 * q^23 + 12 * q^25 - q^35 + 4 * q^37 + 17 * q^41 - 16 * q^43 + 16 * q^47 - 12 * q^49 + 19 * q^53 - 9 * q^59 + 7 * q^61 - 7 * q^65 - 9 * q^67 - 15 * q^71 + 44 * q^73 - 24 * q^77 + 15 * q^79 + 15 * q^83 + 6 * q^85 + 17 * q^89 + 13 * q^91 + 7 * q^95 - 8 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1332\mathbb{Z}\right)^\times$.

$n$	$667$	$1037$	$1297$
$\chi(n)$	$1$	$1$	$e\left(\frac{2}{3}\right)$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−0.500000	−	0.866025i	−0.223607	−	0.387298i	0.732294	−	0.680989i	$-0.238450\pi$
$5$	−0.500000	−	0.866025i	−0.223607	−	0.387298i	−0.955901	+	0.293691i	$0.905116\pi$
$6$		0			0
$7$	−0.370556	−	0.641823i	−0.140057	−	0.242586i	0.787461	−	0.616365i	$-0.211395\pi$
$7$	−0.370556	−	0.641823i	−0.140057	−	0.242586i	−0.927518	+	0.373779i	$0.878062\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	2.91852			0.879968			0.439984	−	0.898006i	$-0.354984\pi$
$11$	2.91852			0.879968			0.439984	+	0.898006i	$0.354984\pi$
$12$		0			0
$13$	1.54797	+	2.68116i	0.429329	+	0.743619i	0.996814	−	0.0797647i	$-0.0254169\pi$
$13$	1.54797	+	2.68116i	0.429329	+	0.743619i	−0.567485	+	0.823384i	$0.692084\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−3.41852	+	5.92105i	−0.829114	+	1.43607i	0.0696210	+	0.997574i	$0.477821\pi$
$17$	−3.41852	+	5.92105i	−0.829114	+	1.43607i	−0.898734	+	0.438493i	$0.855512\pi$
$18$		0			0
$19$	−0.0887048	−	0.153641i	−0.0203503	−	0.0352477i	0.855671	−	0.517520i	$-0.173145\pi$
$19$	−0.0887048	−	0.153641i	−0.0203503	−	0.0352477i	−0.876021	+	0.482273i	$0.839811\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	3.43630			0.716517			0.358259	−	0.933622i	$-0.383371\pi$
$23$	3.43630			0.716517			0.358259	+	0.933622i	$0.383371\pi$
$24$		0			0
$25$	2.00000	−	3.46410i	0.400000	−	0.692820i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	4.61371			0.856744			0.428372	−	0.903603i	$-0.359087\pi$
$29$	4.61371			0.856744			0.428372	+	0.903603i	$0.359087\pi$
$30$		0			0
$31$	2.91852			0.524182			0.262091	−	0.965043i	$-0.415588\pi$
$31$	2.91852			0.524182			0.262091	+	0.965043i	$0.415588\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−0.370556	+	0.641823i	−0.0626355	+	0.108488i
$36$		0			0
$37$	−2.04797	−	5.72764i	−0.336684	−	0.941618i
$37$	−2.04797	−	5.72764i	−0.336684	−	0.941618i
$38$		0			0
$39$		0			0
$40$		0			0
$41$	0.322590	+	0.558743i	0.0503801	+	0.0872610i	0.890116	−	0.455735i	$-0.150623\pi$
$41$	0.322590	+	0.558743i	0.0503801	+	0.0872610i	−0.839736	+	0.542996i	$0.817290\pi$
$42$		0			0
$43$	−3.48223			−0.531034			−0.265517	−	0.964106i	$-0.585543\pi$
$43$	−3.48223			−0.531034			−0.265517	+	0.964106i	$0.585543\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	9.79112			1.42818			0.714091	−	0.700053i	$-0.246840\pi$
$47$	9.79112			1.42818			0.714091	+	0.700053i	$0.246840\pi$
$48$		0			0
$49$	3.22538	−	5.58651i	0.460768	−	0.798074i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	3.37056	−	5.83798i	0.462982	−	0.801908i	−0.536126	−	0.844138i	$-0.680113\pi$
$53$	3.37056	−	5.83798i	0.462982	−	0.801908i	0.999108	+	0.0422302i	$0.0134463\pi$
$54$		0			0
$55$	−1.45926	−	2.52751i	−0.196767	−	0.340810i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	3.72538	−	6.45254i	0.485003	−	0.840049i	−0.514849	−	0.857281i	$-0.672152\pi$
$59$	3.72538	−	6.45254i	0.485003	−	0.840049i	0.999852	+	0.0172318i	$0.00548531\pi$
$60$		0			0
$61$	3.67741	+	6.36946i	0.470844	+	0.815526i	0.999444	−	0.0333453i	$-0.0106161\pi$
$61$	3.67741	+	6.36946i	0.470844	+	0.815526i	−0.528600	+	0.848871i	$0.677283\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	1.54797	−	2.68116i	0.192002	−	0.332556i
$66$		0			0
$67$	5.18464	+	8.98005i	0.633404	+	1.09709i	0.986851	+	0.161634i	$0.0516763\pi$
$67$	5.18464	+	8.98005i	0.633404	+	1.09709i	−0.353447	+	0.935455i	$0.614990\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	2.72538	+	4.72049i	0.323443	+	0.560219i	0.981196	−	0.193014i	$-0.0618264\pi$
$71$	2.72538	+	4.72049i	0.323443	+	0.560219i	−0.657753	+	0.753233i	$0.728493\pi$
$72$		0			0
$73$	12.3548			1.44602			0.723011	−	0.690836i	$-0.242758\pi$
$73$	12.3548			1.44602			0.723011	+	0.690836i	$0.242758\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−1.08148	−	1.87317i	−0.123246	−	0.213468i
$78$		0			0
$79$	6.26611	+	10.8532i	0.704993	+	1.22108i	0.966694	+	0.255935i	$0.0823832\pi$
$79$	6.26611	+	10.8532i	0.704993	+	1.22108i	−0.261701	+	0.965149i	$0.584283\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−4.18464	+	7.24800i	−0.459324	+	0.795572i	−0.998925	−	0.0463486i	$-0.985241\pi$
$83$	−4.18464	+	7.24800i	−0.459324	+	0.795572i	0.539602	+	0.841920i	$0.318575\pi$
$84$		0			0
$85$	6.83705			0.741582
$86$		0			0
$87$		0			0
$88$		0			0
$89$	7.85482	−	13.6049i	0.832609	−	1.44212i	−0.0633531	−	0.997991i	$-0.520179\pi$
$89$	7.85482	−	13.6049i	0.832609	−	1.44212i	0.895962	−	0.444130i	$-0.146487\pi$
$90$		0			0
$91$	1.14722	−	1.98704i	0.120261	−	0.208298i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−0.0887048	+	0.153641i	−0.00910092	+	0.0157633i
$96$		0			0
$97$	−16.8056			−1.70635			−0.853174	−	0.521627i	$-0.825325\pi$
$97$	−16.8056			−1.70635			−0.853174	+	0.521627i	$0.825325\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−6.25889			−0.622783			−0.311391	−	0.950282i	$-0.600795\pi$
$101$	−6.25889			−0.622783			−0.311391	+	0.950282i	$0.600795\pi$
$102$		0			0
$103$	−20.1919			−1.98956			−0.994782	−	0.102026i	$-0.967468\pi$
$103$	−20.1919			−1.98956			−0.994782	+	0.102026i	$0.967468\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	6.72538	+	11.6487i	0.650167	+	1.12612i	0.983082	+	0.183165i	$0.0586343\pi$
$107$	6.72538	+	11.6487i	0.650167	+	1.12612i	−0.332916	+	0.942957i	$0.608032\pi$
$108$		0			0
$109$	−3.24111	+	5.61377i	−0.310442	+	0.537702i	−0.978458	−	0.206445i	$-0.933810\pi$
$109$	−3.24111	+	5.61377i	−0.310442	+	0.537702i	0.668016	+	0.744147i	$0.267144\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	4.80685	−	8.32571i	0.452191	−	0.783217i	−0.546331	−	0.837569i	$-0.683976\pi$
$113$	4.80685	−	8.32571i	0.452191	−	0.783217i	0.998522	+	0.0543520i	$0.0173093\pi$
$114$		0			0
$115$	−1.71815	−	2.97592i	−0.160218	−	0.277506i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	5.06702			0.464493
$120$		0			0
$121$	−2.48223			−0.225657
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−9.00000			−0.804984
$126$		0			0
$127$	2.74834	−	4.76027i	0.243876	−	0.422405i	−0.717939	−	0.696106i	$-0.754914\pi$
$127$	2.74834	−	4.76027i	0.243876	−	0.422405i	0.961815	+	0.273701i	$0.0882477\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−0.347592	+	0.602047i	−0.0303692	+	0.0526011i	−0.880811	−	0.473469i	$-0.843002\pi$
$131$	−0.347592	+	0.602047i	−0.0303692	+	0.0526011i	0.850441	+	0.526070i	$0.176335\pi$
$132$		0			0
$133$	−0.0657403	+	0.113866i	−0.00570041	+	0.00987339i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−2.25889			−0.192990			−0.0964949	−	0.995333i	$-0.530763\pi$
$137$	−2.25889			−0.192990			−0.0964949	+	0.995333i	$0.530763\pi$
$138$		0			0
$139$	7.82982	−	13.5616i	0.664116	−	1.15028i	−0.315408	−	0.948956i	$-0.602141\pi$
$139$	7.82982	−	13.5616i	0.664116	−	1.15028i	0.979524	−	0.201327i	$-0.0645254\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	4.51777	+	7.82501i	0.377795	+	0.654361i
$144$		0			0
$145$	−2.30685	−	3.99559i	−0.191574	−	0.331815i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	2.25889			0.185055			0.0925276	−	0.995710i	$-0.470505\pi$
$149$	2.25889			0.185055			0.0925276	+	0.995710i	$0.470505\pi$
$150$		0			0
$151$	8.92575	+	15.4599i	0.726367	+	1.25810i	0.958409	+	0.285399i	$0.0921261\pi$
$151$	8.92575	+	15.4599i	0.726367	+	1.25810i	−0.232042	+	0.972706i	$0.574541\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−1.45926	−	2.52751i	−0.117211	−	0.203015i
$156$		0			0
$157$	−4.67741	+	8.10151i	−0.373298	+	0.646571i	−0.990071	−	0.140570i	$-0.955106\pi$
$157$	−4.67741	+	8.10151i	−0.373298	+	0.646571i	0.616773	+	0.787141i	$0.288440\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−1.27334	−	2.20549i	−0.100353	−	0.173817i
$162$		0			0
$163$	6.72538	−	11.6487i	0.526772	−	0.912396i	−0.472741	−	0.881201i	$-0.656735\pi$
$163$	6.72538	−	11.6487i	0.526772	−	0.912396i	0.999513	−	0.0311947i	$-0.00993120\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	0.984263	+	1.70479i	0.0761646	+	0.131921i	0.901592	−	0.432587i	$-0.142399\pi$
$167$	0.984263	+	1.70479i	0.0761646	+	0.131921i	−0.825428	+	0.564508i	$0.809066\pi$
$168$		0			0
$169$	1.70760	−	2.95765i	0.131354	−	0.227512i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−7.41852	+	12.8493i	−0.564020	+	0.976911i	0.433120	+	0.901336i	$0.357413\pi$
$173$	−7.41852	+	12.8493i	−0.564020	+	0.976911i	−0.997140	+	0.0755749i	$0.975921\pi$
$174$		0			0
$175$	−2.96445			−0.224091
$176$		0			0
$177$		0			0
$178$		0			0
$179$	7.67409			0.573588			0.286794	−	0.957992i	$-0.407410\pi$
$179$	7.67409			0.573588			0.286794	+	0.957992i	$0.407410\pi$
$180$		0			0
$181$	−1.75889	−	3.04648i	−0.130737	−	0.226443i	0.793224	−	0.608930i	$-0.208401\pi$
$181$	−1.75889	−	3.04648i	−0.130737	−	0.226443i	−0.923961	+	0.382487i	$0.875068\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−3.93630	+	4.63741i	−0.289402	+	0.340949i
$186$		0			0
$187$	−9.97704	+	17.2807i	−0.729593	+	1.26369i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−17.2733			−1.24986			−0.624928	−	0.780683i	$-0.714872\pi$
$191$	−17.2733			−1.24986			−0.624928	+	0.780683i	$0.714872\pi$
$192$		0			0
$193$	−4.09593			−0.294832			−0.147416	−	0.989075i	$-0.547096\pi$
$193$	−4.09593			−0.294832			−0.147416	+	0.989075i	$0.547096\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	9.95075	−	17.2352i	0.708962	−	1.22796i	−0.256281	−	0.966602i	$-0.582497\pi$
$197$	9.95075	−	17.2352i	0.708962	−	1.22796i	0.965243	−	0.261355i	$-0.0841694\pi$
$198$		0			0
$199$	8.51777			0.603809			0.301905	−	0.953338i	$-0.402378\pi$
$199$	8.51777			0.603809			0.301905	+	0.953338i	$0.402378\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−1.70964	−	2.96118i	−0.119993	−	0.207834i
$204$		0			0
$205$	0.322590	−	0.558743i	0.0225307	−	0.0390243i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−0.258887	−	0.448406i	−0.0179076	−	0.0310169i
$210$		0			0
$211$	24.1919			1.66544			0.832718	−	0.553697i	$-0.186783\pi$
$211$	24.1919			1.66544			0.832718	+	0.553697i	$0.186783\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	1.74111	+	3.01570i	0.118743	+	0.205669i
$216$		0			0
$217$	−1.08148	−	1.87317i	−0.0734155	−	0.127159i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−21.1670			−1.42385
$222$		0			0
$223$	−15.1104			−1.01187			−0.505933	−	0.862573i	$-0.668852\pi$
$223$	−15.1104			−1.01187			−0.505933	+	0.862573i	$0.668852\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−8.82982	−	15.2937i	−0.586056	−	1.01508i	−0.994743	−	0.102404i	$-0.967347\pi$
$227$	−8.82982	−	15.2937i	−0.586056	−	1.01508i	0.408687	−	0.912674i	$-0.365987\pi$
$228$		0			0
$229$	−13.1282	−	22.7386i	−0.867533	−	1.50261i	−0.864510	−	0.502616i	$-0.832371\pi$
$229$	−13.1282	−	22.7386i	−0.867533	−	1.50261i	−0.00302375	−	0.999995i	$-0.500962\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−21.8411			−1.43086			−0.715430	−	0.698685i	$-0.753769\pi$
$233$	−21.8411			−1.43086			−0.715430	+	0.698685i	$0.753769\pi$
$234$		0			0
$235$	−4.89556	−	8.47936i	−0.319351	−	0.553132i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−8.20760	+	14.2160i	−0.530906	+	0.919556i	0.468444	+	0.883493i	$0.344815\pi$
$239$	−8.20760	+	14.2160i	−0.530906	+	0.919556i	−0.999350	+	0.0360623i	$0.988519\pi$
$240$		0			0
$241$	2.06574	+	3.57797i	0.133066	+	0.230477i	0.924857	−	0.380315i	$-0.124184\pi$
$241$	2.06574	+	3.57797i	0.133066	+	0.230477i	−0.791791	+	0.610792i	$0.790851\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−6.45075			−0.412123
$246$		0			0
$247$	0.274624	−	0.475663i	0.0174739	−	0.0302657i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−11.1563			−0.704180			−0.352090	−	0.935966i	$-0.614529\pi$
$251$	−11.1563			−0.704180			−0.352090	+	0.935966i	$0.614529\pi$
$252$		0			0
$253$	10.0289			0.630512
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−7.51445	+	13.0154i	−0.468739	+	0.811879i	−0.999362	−	0.0357287i	$-0.988625\pi$
$257$	−7.51445	+	13.0154i	−0.468739	+	0.811879i	0.530623	+	0.847608i	$0.321958\pi$
$258$		0			0
$259$	−2.91724	+	3.43684i	−0.181268	+	0.213555i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	12.4665	+	21.5926i	0.768717	+	1.33146i	0.938259	+	0.345933i	$0.112438\pi$
$263$	12.4665	+	21.5926i	0.768717	+	1.33146i	−0.169542	+	0.985523i	$0.554229\pi$
$264$		0			0
$265$	−6.74111			−0.414103
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−14.1919			−0.865293			−0.432647	−	0.901564i	$-0.642420\pi$
$269$	−14.1919			−0.865293			−0.432647	+	0.901564i	$0.642420\pi$
$270$		0			0
$271$	−4.78389	+	8.28594i	−0.290601	+	0.503335i	−0.973952	−	0.226755i	$-0.927188\pi$
$271$	−4.78389	+	8.28594i	−0.290601	+	0.503335i	0.683351	+	0.730090i	$0.260522\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	5.83705	−	10.1101i	0.351987	−	0.609659i
$276$		0			0
$277$	3.41852	+	5.92105i	0.205399	+	0.355762i	0.950260	−	0.311458i	$-0.100817\pi$
$277$	3.41852	+	5.92105i	0.205399	+	0.355762i	−0.744861	+	0.667220i	$0.767484\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	5.64186	−	9.77199i	0.336565	−	0.582948i	−0.647219	−	0.762304i	$-0.724068\pi$
$281$	5.64186	−	9.77199i	0.336565	−	0.582948i	0.983784	+	0.179356i	$0.0574014\pi$
$282$		0			0
$283$	−2.37056	−	4.10592i	−0.140915	−	0.244072i	0.786926	−	0.617047i	$-0.211671\pi$
$283$	−2.37056	−	4.10592i	−0.140915	−	0.244072i	−0.927841	+	0.372975i	$0.878338\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	0.239076	−	0.414092i	0.0141122	−	0.0244430i
$288$		0			0
$289$	−14.8726	−	25.7601i	−0.874858	−	1.51530i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−0.241113	−	0.417620i	−0.0140860	−	0.0243976i	0.858896	−	0.512149i	$-0.171151\pi$
$293$	−0.241113	−	0.417620i	−0.0140860	−	0.0243976i	−0.872982	+	0.487752i	$0.837817\pi$
$294$		0			0
$295$	−7.45075			−0.433800
$296$		0			0
$297$		0			0
$298$		0			0
$299$	5.31927	+	9.21325i	0.307621	+	0.532816i
$300$		0			0
$301$	1.29036	+	2.23497i	0.0743752	+	0.128822i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	3.67741	−	6.36946i	0.210568	−	0.364714i
$306$		0			0
$307$	−3.80814			−0.217342			−0.108671	−	0.994078i	$-0.534659\pi$
$307$	−3.80814			−0.217342			−0.108671	+	0.994078i	$0.534659\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−13.4350	+	23.2701i	−0.761830	+	1.31953i	0.180077	+	0.983653i	$0.442365\pi$
$311$	−13.4350	+	23.2701i	−0.761830	+	1.31953i	−0.941906	+	0.335875i	$0.890968\pi$
$312$		0			0
$313$	−9.38705	+	16.2588i	−0.530587	+	0.919004i	0.468776	+	0.883317i	$0.344695\pi$
$313$	−9.38705	+	16.2588i	−0.530587	+	0.919004i	−0.999363	+	0.0356870i	$0.988638\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	9.90075	−	17.1486i	0.556081	−	0.963161i	−0.441737	−	0.897145i	$-0.645638\pi$
$317$	9.90075	−	17.1486i	0.556081	−	0.963161i	0.997819	−	0.0660167i	$-0.0210291\pi$
$318$		0			0
$319$	13.4652			0.753907
$320$		0			0
$321$		0			0
$322$		0			0
$323$	1.21296			0.0674908
$324$		0			0
$325$	12.3837			0.686926
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−3.62816	−	6.28416i	−0.200027	−	0.346457i
$330$		0			0
$331$	−3.88833	+	6.73479i	−0.213722	+	0.370177i	−0.952876	−	0.303359i	$-0.901892\pi$
$331$	−3.88833	+	6.73479i	−0.213722	+	0.370177i	0.739154	+	0.673536i	$0.235225\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	5.18464	−	8.98005i	0.283267	−	0.490633i
$336$		0			0
$337$	−7.29112	−	12.6286i	−0.397172	−	0.687923i	0.596203	−	0.802833i	$-0.296675\pi$
$337$	−7.29112	−	12.6286i	−0.397172	−	0.687923i	−0.993376	+	0.114911i	$0.963342\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	8.51777			0.461263
$342$		0			0
$343$	−9.96853			−0.538250
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−3.85406			−0.206897			−0.103449	−	0.994635i	$-0.532988\pi$
$347$	−3.85406			−0.206897			−0.103449	+	0.994635i	$0.532988\pi$
$348$		0			0
$349$	−8.11371	+	14.0534i	−0.434317	+	0.752259i	−0.997240	−	0.0742508i	$-0.976343\pi$
$349$	−8.11371	+	14.0534i	−0.434317	+	0.752259i	0.562923	+	0.826509i	$0.309677\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−11.4685	+	19.8641i	−0.610408	+	1.05726i	0.380764	+	0.924672i	$0.375661\pi$
$353$	−11.4685	+	19.8641i	−0.610408	+	1.05726i	−0.991172	+	0.132585i	$0.957672\pi$
$354$		0			0
$355$	2.72538	−	4.72049i	0.144648	−	0.250538i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−37.4193			−1.97491			−0.987457	−	0.157888i	$-0.949531\pi$
$359$	−37.4193			−1.97491			−0.987457	+	0.157888i	$0.949531\pi$
$360$		0			0
$361$	9.48426	−	16.4272i	0.499172	−	0.864591i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−6.17741	−	10.6996i	−0.323340	−	0.560042i
$366$		0			0
$367$	3.88833	+	6.73479i	0.202969	+	0.351553i	0.949484	−	0.313816i	$-0.101608\pi$
$367$	3.88833	+	6.73479i	0.202969	+	0.351553i	−0.746515	+	0.665369i	$0.768274\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−4.99593			−0.259376
$372$		0			0
$373$	−5.69186	−	9.85860i	−0.294714	−	0.510459i	0.680205	−	0.733022i	$-0.261891\pi$
$373$	−5.69186	−	9.85860i	−0.294714	−	0.510459i	−0.974918	+	0.222563i	$0.928558\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	7.14186	+	12.3701i	0.367825	+	0.637091i
$378$		0			0
$379$	−0.756850	+	1.31090i	−0.0388768	+	0.0673365i	−0.884809	−	0.465954i	$-0.845711\pi$
$379$	−0.756850	+	1.31090i	−0.0388768	+	0.0673365i	0.845932	+	0.533290i	$0.179045\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	11.2431	+	19.4737i	0.574498	+	0.995060i	0.996096	+	0.0882770i	$0.0281361\pi$
$383$	11.2431	+	19.4737i	0.574498	+	0.995060i	−0.421598	+	0.906783i	$0.638531\pi$
$384$		0			0
$385$	−1.08148	+	1.87317i	−0.0551172	+	0.0954658i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−10.2241	−	17.7086i	−0.518382	−	0.897864i	−0.999772	−	0.0213575i	$-0.993201\pi$
$389$	−10.2241	−	17.7086i	−0.518382	−	0.897864i	0.481390	−	0.876507i	$-0.340132\pi$
$390$		0			0
$391$	−11.7471	+	20.3465i	−0.594074	+	1.02897i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	6.26611	−	10.8532i	0.315282	−	0.546085i
$396$		0			0
$397$	22.4508			1.12677			0.563385	−	0.826194i	$-0.309499\pi$
$397$	22.4508			1.12677			0.563385	+	0.826194i	$0.309499\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	3.64518			0.182032			0.0910158	−	0.995849i	$-0.470989\pi$
$401$	3.64518			0.182032			0.0910158	+	0.995849i	$0.470989\pi$
$402$		0			0
$403$	4.51777	+	7.82501i	0.225046	+	0.389792i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−5.97704	−	16.7162i	−0.296271	−	0.828593i
$408$		0			0
$409$	−14.8693	+	25.7543i	−0.735238	+	1.27347i	0.219380	+	0.975639i	$0.429596\pi$
$409$	−14.8693	+	25.7543i	−0.735238	+	1.27347i	−0.954619	+	0.297831i	$0.903737\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−5.52185			−0.271712
$414$		0			0
$415$	8.36927			0.410832
$416$		0			0
$417$		0			0
$418$		0			0
$419$	17.9843	−	31.1497i	0.878589	−	1.52176i	0.0256987	−	0.999670i	$-0.491819\pi$
$419$	17.9843	−	31.1497i	0.878589	−	1.52176i	0.852890	−	0.522091i	$-0.174848\pi$
$420$		0			0
$421$	−38.3167			−1.86744			−0.933721	−	0.358002i	$-0.883458\pi$
$421$	−38.3167			−1.86744			−0.933721	+	0.358002i	$0.883458\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	13.6741	+	23.6842i	0.663291	+	1.14885i
$426$		0			0
$427$	2.72538	−	4.72049i	0.131890	−	0.228440i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−5.74834	−	9.95642i	−0.276888	−	0.479584i	0.693722	−	0.720243i	$-0.255970\pi$
$431$	−5.74834	−	9.95642i	−0.276888	−	0.479584i	−0.970610	+	0.240659i	$0.922636\pi$
$432$		0			0
$433$	−5.83297			−0.280315			−0.140157	−	0.990129i	$-0.544761\pi$
$433$	−5.83297			−0.280315			−0.140157	+	0.990129i	$0.544761\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−0.304816	−	0.527957i	−0.0145813	−	0.0252556i
$438$		0			0
$439$	2.24315	+	3.88525i	0.107060	+	0.185433i	0.914578	−	0.404410i	$-0.132523\pi$
$439$	2.24315	+	3.88525i	0.107060	+	0.185433i	−0.807518	+	0.589843i	$0.799190\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−12.0000			−0.570137			−0.285069	−	0.958507i	$-0.592016\pi$
$443$	−12.0000			−0.570137			−0.285069	+	0.958507i	$0.592016\pi$
$444$		0			0
$445$	−15.7096			−0.744708
$446$		0			0
$447$		0			0
$448$		0			0
$449$	1.16167	+	2.01208i	0.0548227	+	0.0949557i	0.892134	−	0.451770i	$-0.149207\pi$
$449$	1.16167	+	2.01208i	0.0548227	+	0.0949557i	−0.837312	+	0.546726i	$0.815874\pi$
$450$		0			0
$451$	0.941487	+	1.63070i	0.0443329	+	0.0767868i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−2.29444			−0.107565
$456$		0			0
$457$	4.19518	+	7.26627i	0.196242	+	0.339902i	0.947307	−	0.320327i	$-0.103793\pi$
$457$	4.19518	+	7.26627i	0.196242	+	0.339902i	−0.751065	+	0.660229i	$0.770459\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	16.9987	−	29.4426i	0.791709	−	1.37128i	−0.133199	−	0.991089i	$-0.542525\pi$
$461$	16.9987	−	29.4426i	0.791709	−	1.37128i	0.924908	−	0.380191i	$-0.124142\pi$
$462$		0			0
$463$	11.1117	+	19.2460i	0.516403	+	0.894436i	0.999819	+	0.0190454i	$0.00606269\pi$
$463$	11.1117	+	19.2460i	0.516403	+	0.894436i	−0.483416	+	0.875391i	$0.660604\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	7.29036			0.337358			0.168679	−	0.985671i	$-0.446050\pi$
$467$	7.29036			0.337358			0.168679	+	0.985671i	$0.446050\pi$
$468$		0			0
$469$	3.84240	−	6.65523i	0.177426	−	0.307310i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−10.1630			−0.467293
$474$		0			0
$475$	−0.709639			−0.0325605
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−0.792398	+	1.37247i	−0.0362056	+	0.0627099i	−0.883560	−	0.468317i	$-0.844860\pi$
$479$	−0.792398	+	1.37247i	−0.0362056	+	0.0627099i	0.847355	+	0.531027i	$0.178194\pi$
$480$		0			0
$481$	12.1865	−	14.3571i	0.555657	−	0.654628i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	8.40279	+	14.5541i	0.381551	+	0.660865i
$486$		0			0
$487$	34.7845			1.57624			0.788118	−	0.615525i	$-0.211056\pi$
$487$	34.7845			1.57624			0.788118	+	0.615525i	$0.211056\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	14.8267			0.669118			0.334559	−	0.942375i	$-0.391413\pi$
$491$	14.8267			0.669118			0.334559	+	0.942375i	$0.391413\pi$
$492$		0			0
$493$	−15.7721	+	27.3180i	−0.710338	+	1.23034i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	2.01981	−	3.49842i	0.0906009	−	0.156925i
$498$		0			0
$499$	5.39352	+	9.34185i	0.241447	+	0.418199i	0.961127	−	0.276108i	$-0.0890446\pi$
$499$	5.39352	+	9.34185i	0.241447	+	0.418199i	−0.719680	+	0.694306i	$0.755711\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−16.2806	+	28.1988i	−0.725915	+	1.25732i	0.232682	+	0.972553i	$0.425250\pi$
$503$	−16.2806	+	28.1988i	−0.725915	+	1.25732i	−0.958596	+	0.284768i	$0.908083\pi$
$504$		0			0
$505$	3.12944	+	5.42036i	0.139258	+	0.241203i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	10.9363	−	18.9422i	0.484743	−	0.839599i	−0.515104	−	0.857128i	$-0.672247\pi$
$509$	10.9363	−	18.9422i	0.484743	−	0.839599i	0.999846	+	0.0175288i	$0.00557986\pi$
$510$		0			0
$511$	−4.57816	−	7.92960i	−0.202526	−	0.350785i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	10.0959	+	17.4867i	0.444880	+	0.770555i
$516$		0			0
$517$	28.5756			1.25675
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−18.8672	−	32.6790i	−0.826589	−	1.43169i	−0.900699	−	0.434444i	$-0.856945\pi$
$521$	−18.8672	−	32.6790i	−0.826589	−	1.43169i	0.0741104	−	0.997250i	$-0.476388\pi$
$522$		0			0
$523$	−21.6669	−	37.5281i	−0.947426	−	1.64099i	−0.750820	−	0.660507i	$-0.770341\pi$
$523$	−21.6669	−	37.5281i	−0.947426	−	1.64099i	−0.196606	−	0.980483i	$-0.562992\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−9.97704	+	17.2807i	−0.434606	+	0.752761i
$528$		0			0
$529$	−11.1919			−0.486603
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−0.998718	+	1.72983i	−0.0432593	+	0.0749273i
$534$		0			0
$535$	6.72538	−	11.6487i	0.290763	−	0.503617i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	9.41333	−	16.3044i	0.405461	−	0.702279i
$540$		0			0
$541$	16.1959			0.696318			0.348159	−	0.937436i	$-0.386807\pi$
$541$	16.1959			0.696318			0.348159	+	0.937436i	$0.386807\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	6.48223			0.277668
$546$		0			0
$547$	−16.0459			−0.686074			−0.343037	−	0.939322i	$-0.611456\pi$
$547$	−16.0459			−0.686074			−0.343037	+	0.939322i	$0.611456\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−0.409258	−	0.708856i	−0.0174350	−	0.0301983i
$552$		0			0
$553$	4.64390	−	8.04347i	0.197479	−	0.342043i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−15.2556	+	26.4234i	−0.646399	+	1.11960i	0.337577	+	0.941298i	$0.390393\pi$
$557$	−15.2556	+	26.4234i	−0.646399	+	1.11960i	−0.983976	+	0.178298i	$0.942941\pi$
$558$		0			0
$559$	−5.39037	−	9.33639i	−0.227988	−	0.394887i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	31.1104			1.31115			0.655573	−	0.755132i	$-0.272427\pi$
$563$	31.1104			1.31115			0.655573	+	0.755132i	$0.272427\pi$
$564$		0			0
$565$	−9.61371			−0.404452
$566$		0			0
$567$		0			0
$568$		0			0
$569$	18.4508			0.773496			0.386748	−	0.922185i	$-0.373598\pi$
$569$	18.4508			0.773496			0.386748	+	0.922185i	$0.373598\pi$
$570$		0			0
$571$	3.07425	−	5.32476i	0.128653	−	0.222834i	−0.794502	−	0.607262i	$-0.792268\pi$
$571$	3.07425	−	5.32476i	0.128653	−	0.222834i	0.923155	+	0.384428i	$0.125601\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	6.87259	−	11.9037i	0.286607	−	0.496418i
$576$		0			0
$577$	−1.72538	+	2.98844i	−0.0718283	+	0.124410i	−0.899703	−	0.436503i	$-0.856217\pi$
$577$	−1.72538	+	2.98844i	−0.0718283	+	0.124410i	0.827874	+	0.560914i	$0.189550\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	6.20258			0.257326
$582$		0			0
$583$	9.83705	−	17.0383i	0.407409	−	0.705653i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	6.91130	+	11.9707i	0.285260	+	0.494084i	0.972672	−	0.232183i	$-0.0745868\pi$
$587$	6.91130	+	11.9707i	0.285260	+	0.494084i	−0.687412	+	0.726267i	$0.741253\pi$
$588$		0			0
$589$	−0.258887	−	0.448406i	−0.0106673	−	0.0184762i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	32.6715			1.34166			0.670829	−	0.741612i	$-0.265938\pi$
$593$	32.6715			1.34166			0.670829	+	0.741612i	$0.265938\pi$
$594$		0			0
$595$	−2.53351	−	4.38817i	−0.103864	−	0.179897i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	7.71687	+	13.3660i	0.315303	+	0.546120i	0.979502	−	0.201436i	$-0.0645607\pi$
$599$	7.71687	+	13.3660i	0.315303	+	0.546120i	−0.664199	+	0.747556i	$0.731227\pi$
$600$		0			0
$601$	−0.744432	+	1.28939i	−0.0303660	+	0.0525955i	−0.880809	−	0.473472i	$-0.843001\pi$
$601$	−0.744432	+	1.28939i	−0.0303660	+	0.0525955i	0.850443	+	0.526067i	$0.176334\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	1.24111	+	2.14967i	0.0504584	+	0.0873965i
$606$		0			0
$607$	3.12425	−	5.41137i	0.126810	−	0.219641i	−0.795629	−	0.605784i	$-0.792860\pi$
$607$	3.12425	−	5.41137i	0.126810	−	0.219641i	0.922439	+	0.386143i	$0.126193\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	15.1563	+	26.2515i	0.613159	+	1.06202i
$612$		0			0
$613$	13.8693	−	24.0223i	0.560175	−	0.970251i	−0.437306	−	0.899313i	$-0.644067\pi$
$613$	13.8693	−	24.0223i	0.560175	−	0.970251i	0.997481	−	0.0709383i	$-0.0225993\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	3.72538	−	6.45254i	0.149978	−	0.259769i	−0.781241	−	0.624229i	$-0.785413\pi$
$617$	3.72538	−	6.45254i	0.149978	−	0.259769i	0.931219	+	0.364460i	$0.118746\pi$
$618$		0			0
$619$	16.1919			0.650806			0.325403	−	0.945575i	$-0.394500\pi$
$619$	16.1919			0.650806			0.325403	+	0.945575i	$0.394500\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−11.6426			−0.466452
$624$		0			0
$625$	−5.50000	−	9.52628i	−0.220000	−	0.381051i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	40.9147	+	7.45394i	1.63137	+	0.297208i
$630$		0			0
$631$	0.393521	−	0.681598i	0.0156658	−	0.0271340i	−0.858086	−	0.513506i	$-0.828347\pi$
$631$	0.393521	−	0.681598i	0.0156658	−	0.0271340i	0.873752	+	0.486372i	$0.161680\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−5.49668			−0.218129
$636$		0			0
$637$	19.9711			0.791283
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−10.1741	+	17.6220i	−0.401852	+	0.696029i	−0.993950	−	0.109838i	$-0.964967\pi$
$641$	−10.1741	+	17.6220i	−0.401852	+	0.696029i	0.592097	+	0.805867i	$0.298300\pi$
$642$		0			0
$643$	0.429658			0.0169441			0.00847203	−	0.999964i	$-0.497303\pi$
$643$	0.429658			0.0169441			0.00847203	+	0.999964i	$0.497303\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	5.40610	+	9.36365i	0.212536	+	0.368123i	0.952507	−	0.304515i	$-0.0984944\pi$
$647$	5.40610	+	9.36365i	0.212536	+	0.368123i	−0.739972	+	0.672638i	$0.765161\pi$
$648$		0			0
$649$	10.8726	−	18.8319i	0.426787	−	0.739216i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−2.12816	−	3.68608i	−0.0832814	−	0.144248i	0.821376	−	0.570387i	$-0.193207\pi$
$653$	−2.12816	−	3.68608i	−0.0832814	−	0.144248i	−0.904658	+	0.426139i	$0.859873\pi$
$654$		0			0
$655$	0.695184			0.0271631
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−22.5980	−	39.1408i	−0.880292	−	1.52471i	−0.851017	−	0.525139i	$-0.824013\pi$
$659$	−22.5980	−	39.1408i	−0.880292	−	1.52471i	−0.0292754	−	0.999571i	$-0.509320\pi$
$660$		0			0
$661$	3.01777	+	5.22694i	0.117378	+	0.203304i	0.918728	−	0.394891i	$-0.129218\pi$
$661$	3.01777	+	5.22694i	0.117378	+	0.203304i	−0.801350	+	0.598196i	$0.795884\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	0.131481			0.00509860
$666$		0			0
$667$	15.8541			0.613872
$668$		0			0
$669$		0			0
$670$		0			0
$671$	10.7326	+	18.5894i	0.414328	+	0.717636i
$672$		0			0
$673$	23.9317	+	41.4509i	0.922499	+	1.59782i	0.795535	+	0.605908i	$0.207190\pi$
$673$	23.9317	+	41.4509i	0.922499	+	1.59782i	0.126964	+	0.991907i	$0.459477\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−45.0893			−1.73292			−0.866461	−	0.499244i	$-0.833611\pi$
$677$	−45.0893			−1.73292			−0.866461	+	0.499244i	$0.833611\pi$
$678$		0			0
$679$	6.22741	+	10.7862i	0.238986	+	0.413936i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−4.01574	+	6.95546i	−0.153658	+	0.266143i	−0.932570	−	0.360990i	$-0.882439\pi$
$683$	−4.01574	+	6.95546i	−0.153658	+	0.266143i	0.778912	+	0.627134i	$0.215772\pi$
$684$		0			0
$685$	1.12944	+	1.95625i	0.0431538	+	0.0747446i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	20.8700			0.795085
$690$		0			0
$691$	−8.78389	+	15.2141i	−0.334155	+	0.578773i	−0.983322	−	0.181872i	$-0.941784\pi$
$691$	−8.78389	+	15.2141i	−0.334155	+	0.578773i	0.649167	+	0.760646i	$0.275118\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−15.6596			−0.594004
$696$		0			0
$697$	−4.41113			−0.167083
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−2.01574	+	3.49136i	−0.0761333	+	0.131867i	−0.901579	−	0.432615i	$-0.857591\pi$
$701$	−2.01574	+	3.49136i	−0.0761333	+	0.131867i	0.825445	+	0.564482i	$0.190924\pi$
$702$		0			0
$703$	−0.698337	+	0.822721i	−0.0263383	+	0.0310295i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	2.31927	+	4.01710i	0.0872252	+	0.151078i
$708$		0			0
$709$	9.15632			0.343873			0.171936	−	0.985108i	$-0.444998\pi$
$709$	9.15632			0.343873			0.171936	+	0.985108i	$0.444998\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	10.0289			0.375586
$714$		0			0
$715$	4.51777	−	7.82501i	0.168955	−	0.292639i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	9.11761	−	15.7922i	0.340030	−	0.588949i	−0.644408	−	0.764682i	$-0.722896\pi$
$719$	9.11761	−	15.7922i	0.340030	−	0.588949i	0.984438	+	0.175733i	$0.0562296\pi$
$720$		0			0
$721$	7.48223	+	12.9596i	0.278653	+	0.482641i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	9.22741	−	15.9823i	0.342698	−	0.593570i
$726$		0			0
$727$	3.88833	+	6.73479i	0.144210	+	0.249779i	0.929078	−	0.369884i	$-0.120603\pi$
$727$	3.88833	+	6.73479i	0.144210	+	0.249779i	−0.784868	+	0.619663i	$0.787269\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	11.9041	−	20.6185i	0.440288	−	0.762601i
$732$		0			0
$733$	−5.41649	−	9.38163i	−0.200062	−	0.346518i	0.748486	−	0.663151i	$-0.230781\pi$
$733$	−5.41649	−	9.38163i	−0.200062	−	0.346518i	−0.948548	+	0.316632i	$0.897448\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	15.1315	+	26.2085i	0.557375	+	0.965402i
$738$		0			0
$739$	−31.9119			−1.17390			−0.586949	−	0.809624i	$-0.699671\pi$
$739$	−31.9119			−1.17390			−0.586949	+	0.809624i	$0.699671\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−13.5395	−	23.4510i	−0.496714	−	0.860335i	0.503278	−	0.864124i	$-0.332127\pi$
$743$	−13.5395	−	23.4510i	−0.496714	−	0.860335i	−0.999993	+	0.00378964i	$0.998794\pi$
$744$		0			0
$745$	−1.12944	−	1.95625i	−0.0413796	−	0.0716716i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	4.98426	−	8.63300i	0.182121	−	0.315443i
$750$		0			0
$751$	−6.77259			−0.247135			−0.123568	−	0.992336i	$-0.539434\pi$
$751$	−6.77259			−0.247135			−0.123568	+	0.992336i	$0.539434\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	8.92575	−	15.4599i	0.324841	−	0.562642i
$756$		0			0
$757$	12.2241	−	21.1728i	0.444292	−	0.769537i	−0.553710	−	0.832709i	$-0.686789\pi$
$757$	12.2241	−	21.1728i	0.444292	−	0.769537i	0.998003	+	0.0631726i	$0.0201219\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−0.595932	+	1.03218i	−0.0216025	+	0.0374167i	−0.876625	−	0.481175i	$-0.840210\pi$
$761$	−0.595932	+	1.03218i	−0.0216025	+	0.0374167i	0.855022	+	0.518592i	$0.173543\pi$
$762$		0			0
$763$	4.80406			0.173919
$764$		0			0
$765$		0			0
$766$		0			0
$767$	23.0670			0.832902
$768$		0			0
$769$	37.7741			1.36217			0.681084	−	0.732205i	$-0.261509\pi$
$769$	37.7741			1.36217			0.681084	+	0.732205i	$0.261509\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	23.6919	+	41.0355i	0.852137	+	1.47594i	0.879276	+	0.476313i	$0.158027\pi$
$773$	23.6919	+	41.0355i	0.852137	+	1.47594i	−0.0271386	+	0.999632i	$0.508640\pi$
$774$		0			0
$775$	5.83705	−	10.1101i	0.209673	−	0.363164i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	0.0572306	−	0.0991264i	0.00205050	−	0.00355157i
$780$		0			0
$781$	7.95407	+	13.7769i	0.284619	+	0.492975i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	9.35482			0.333888
$786$		0			0
$787$	−28.0578			−1.00015			−0.500077	−	0.865981i	$-0.666695\pi$
$787$	−28.0578			−1.00015			−0.500077	+	0.865981i	$0.666695\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−7.12484			−0.253330
$792$		0			0
$793$	−11.3850	+	19.7194i	−0.404294	+	0.700257i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	4.43094	−	7.67461i	0.156952	−	0.271849i	−0.776816	−	0.629728i	$-0.783167\pi$
$797$	4.43094	−	7.67461i	0.156952	−	0.271849i	0.933768	+	0.357879i	$0.116500\pi$
$798$		0			0
$799$	−33.4712	+	57.9737i	−1.18412	+	2.05096i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	36.0578			1.27245
$804$		0			0
$805$	−1.27334	+	2.20549i	−0.0448794	+	0.0777334i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−22.2076	−	38.4647i	−0.780778	−	1.35235i	−0.931489	−	0.363769i	$-0.881490\pi$
$809$	−22.2076	−	38.4647i	−0.780778	−	1.35235i	0.150712	−	0.988578i	$-0.451844\pi$
$810$		0			0
$811$	−18.0487	−	31.2613i	−0.633776	−	1.09773i	−0.986773	−	0.162108i	$-0.948171\pi$
$811$	−18.0487	−	31.2613i	−0.633776	−	1.09773i	0.352997	−	0.935625i	$-0.385163\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−13.4508			−0.471159
$816$		0			0
$817$	0.308890	+	0.535014i	0.0108067	+	0.0187178i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	11.8883	+	20.5912i	0.414906	+	0.718638i	0.995419	−	0.0956134i	$-0.0304813\pi$
$821$	11.8883	+	20.5912i	0.414906	+	0.718638i	−0.580513	+	0.814251i	$0.697148\pi$
$822$		0			0
$823$	22.5395	−	39.0395i	0.785676	−	1.36083i	−0.142918	−	0.989734i	$-0.545649\pi$
$823$	22.5395	−	39.0395i	0.785676	−	1.36083i	0.928594	−	0.371096i	$-0.121018\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−26.5980	−	46.0690i	−0.924902	−	1.60198i	−0.791719	−	0.610886i	$-0.790813\pi$
$827$	−26.5980	−	46.0690i	−0.924902	−	1.60198i	−0.133183	−	0.991091i	$-0.542520\pi$
$828$		0			0
$829$	17.9028	−	31.0085i	0.621789	−	1.07697i	−0.367363	−	0.930078i	$-0.619739\pi$
$829$	17.9028	−	31.0085i	0.621789	−	1.07697i	0.989152	−	0.146893i	$-0.0469274\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	22.0520	+	38.1953i	0.764058	+	1.32339i
$834$		0			0
$835$	0.984263	−	1.70479i	0.0340618	−	0.0589968i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−4.23057	+	7.32756i	−0.146055	+	0.252975i	−0.929766	−	0.368151i	$-0.879991\pi$
$839$	−4.23057	+	7.32756i	−0.146055	+	0.252975i	0.783711	+	0.621126i	$0.213324\pi$
$840$		0			0
$841$	−7.71371			−0.265990
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−3.41520			−0.117487
$846$		0			0
$847$	0.919805	+	1.59315i	0.0316049	+	0.0547412i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−7.03742	−	19.6819i	−0.241240	−	0.674686i
$852$		0			0
$853$	5.01777	−	8.69104i	0.171805	−	0.297576i	−0.767246	−	0.641353i	$-0.778373\pi$
$853$	5.01777	−	8.69104i	0.171805	−	0.297576i	0.939051	+	0.343778i	$0.111707\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−42.6426			−1.45664			−0.728322	−	0.685235i	$-0.759699\pi$
$857$	−42.6426			−1.45664			−0.728322	+	0.685235i	$0.759699\pi$
$858$		0			0
$859$	54.7045			1.86649			0.933247	−	0.359236i	$-0.116963\pi$
$859$	54.7045			1.86649			0.933247	+	0.359236i	$0.116963\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	0.823873	−	1.42699i	0.0280449	−	0.0485753i	−0.851662	−	0.524091i	$-0.824405\pi$
$863$	0.823873	−	1.42699i	0.0280449	−	0.0485753i	0.879707	+	0.475516i	$0.157739\pi$
$864$		0			0
$865$	14.8370			0.504475
$866$		0			0
$867$		0			0
$868$		0			0
$869$	18.2878	+	31.6754i	0.620371	+	1.07451i
$870$		0			0
$871$	−16.0513	+	27.8016i	−0.543877	+	0.942023i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	3.33501	+	5.77640i	0.112744	+	0.195278i
$876$		0			0
$877$	−52.9056			−1.78649			−0.893247	−	0.449566i	$-0.851579\pi$
$877$	−52.9056			−1.78649			−0.893247	+	0.449566i	$0.851579\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−9.11371	−	15.7854i	−0.307049	−	0.531824i	0.670667	−	0.741759i	$-0.266008\pi$
$881$	−9.11371	−	15.7854i	−0.307049	−	0.531824i	−0.977715	+	0.209935i	$0.932675\pi$
$882$		0			0
$883$	−4.65648	−	8.06526i	−0.156703	−	0.271418i	0.776975	−	0.629532i	$-0.216753\pi$
$883$	−4.65648	−	8.06526i	−0.156703	−	0.271418i	−0.933678	+	0.358114i	$0.883420\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−17.6571			−0.592866			−0.296433	−	0.955054i	$-0.595797\pi$
$887$	−17.6571			−0.592866			−0.296433	+	0.955054i	$0.595797\pi$
$888$		0			0
$889$	−4.07366			−0.136626
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−0.868519	−	1.50432i	−0.0290639	−	0.0503401i
$894$		0			0
$895$	−3.83705	−	6.64596i	−0.128258	−	0.222150i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	13.4652			0.449090
$900$		0			0
$901$	23.0446	+	39.9145i	0.767728	+	1.32974i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−1.75889	+	3.04648i	−0.0584674	+	0.101268i
$906$		0			0
$907$	7.37650	+	12.7765i	0.244933	+	0.424236i	0.962113	−	0.272652i	$-0.0879008\pi$
$907$	7.37650	+	12.7765i	0.244933	+	0.424236i	−0.717180	+	0.696888i	$0.754567\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	57.0053			1.88867			0.944334	−	0.328988i	$-0.106708\pi$
$911$	57.0053			1.88867			0.944334	+	0.328988i	$0.106708\pi$
$912$		0			0
$913$	−12.2130	+	21.1535i	−0.404190	+	0.700077i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	0.515210			0.0170137
$918$		0			0
$919$	−14.4007			−0.475037			−0.237518	−	0.971383i	$-0.576334\pi$
$919$	−14.4007			−0.475037			−0.237518	+	0.971383i	$0.576334\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−8.43758	+	14.6143i	−0.277726	+	0.481036i
$924$		0			0
$925$	−23.9371	−	4.36091i	−0.787045	−	0.143386i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	17.1782	+	29.7535i	0.563597	+	0.976179i	0.997179	+	0.0750647i	$0.0239163\pi$
$929$	17.1782	+	29.7535i	0.563597	+	0.976179i	−0.433581	+	0.901114i	$0.642750\pi$
$930$		0			0
$931$	−1.14443			−0.0375070
$932$		0			0
$933$		0			0
$934$		0			0
$935$	19.9541			0.652568
$936$		0			0
$937$	23.9863	−	41.5455i	0.783598	−	1.35723i	−0.146235	−	0.989250i	$-0.546715\pi$
$937$	23.9863	−	41.5455i	0.783598	−	1.35723i	0.929833	−	0.367982i	$-0.119951\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−25.2385	+	43.7144i	−0.822753	+	1.42505i	0.0808712	+	0.996725i	$0.474230\pi$
$941$	−25.2385	+	43.7144i	−0.822753	+	1.42505i	−0.903624	+	0.428326i	$0.859104\pi$
$942$		0			0
$943$	1.10852	+	1.92001i	0.0360982	+	0.0625240i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	14.0802	−	24.3876i	0.457545	−	0.792491i	−0.541286	−	0.840839i	$-0.682062\pi$
$947$	14.0802	−	24.3876i	0.457545	−	0.792491i	0.998831	+	0.0483478i	$0.0153956\pi$
$948$		0			0
$949$	19.1248	+	33.1252i	0.620819	+	1.07529i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−0.917240	+	1.58871i	−0.0297123	+	0.0514633i	−0.880499	−	0.474048i	$-0.842792\pi$
$953$	−0.917240	+	1.58871i	−0.0297123	+	0.0514633i	0.850787	+	0.525511i	$0.176126\pi$
$954$		0			0
$955$	8.63667	+	14.9592i	0.279476	+	0.484067i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	0.837045	+	1.44980i	0.0270296	+	0.0468166i
$960$		0			0
$961$	−22.4822			−0.725233
$962$		0			0
$963$		0			0
$964$		0			0
$965$	2.04797	+	3.54718i	0.0659264	+	0.114188i
$966$		0			0
$967$	−15.8569	−	27.4649i	−0.509922	−	0.883211i	−0.999934	−	0.0114952i	$-0.996341\pi$
$967$	−15.8569	−	27.4649i	−0.509922	−	0.883211i	0.490012	−	0.871716i	$-0.336992\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	4.40798	−	7.63484i	0.141459	−	0.245014i	−0.786587	−	0.617479i	$-0.788154\pi$
$971$	4.40798	−	7.63484i	0.141459	−	0.245014i	0.928046	+	0.372465i	$0.121487\pi$
$972$		0			0
$973$	−11.6056			−0.372057
$974$		0			0
$975$		0			0
$976$		0			0
$977$	14.9698	−	25.9285i	0.478927	−	0.829525i	−0.520781	−	0.853690i	$-0.674359\pi$
$977$	14.9698	−	25.9285i	0.478927	−	0.829525i	0.999708	+	0.0241649i	$0.00769267\pi$
$978$		0			0
$979$	22.9245	−	39.7063i	0.732669	−	1.26902i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−20.7524	+	35.9442i	−0.661899	+	1.14644i	0.318217	+	0.948018i	$0.396916\pi$
$983$	−20.7524	+	35.9442i	−0.661899	+	1.14644i	−0.980116	+	0.198425i	$0.936417\pi$
$984$		0			0
$985$	−19.9015			−0.634115
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−11.9660			−0.380495
$990$		0			0
$991$	−48.0660			−1.52687			−0.763433	−	0.645887i	$-0.776488\pi$
$991$	−48.0660			−1.52687			−0.763433	+	0.645887i	$0.776488\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−4.25889	−	7.37661i	−0.135016	−	0.233854i
$996$		0			0
$997$	−24.3180	+	42.1200i	−0.770158	+	1.33395i	0.167317	+	0.985903i	$0.446490\pi$
$997$	−24.3180	+	42.1200i	−0.770158	+	1.33395i	−0.937476	+	0.348050i	$0.886844\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1332.2.j.e.1009.2		6
3.2	odd	2		148.2.e.a.121.3	✓	6
12.11	even	2		592.2.i.f.417.1		6
37.26	even	3	inner	1332.2.j.e.433.2		6
111.26	odd	6		148.2.e.a.137.3	yes	6
111.47	odd	6		5476.2.a.f.1.1		3
111.101	odd	6		5476.2.a.g.1.1		3
444.359	even	6		592.2.i.f.433.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
148.2.e.a.121.3	✓	6	3.2	odd	2
148.2.e.a.137.3	yes	6	111.26	odd	6
592.2.i.f.417.1		6	12.11	even	2
592.2.i.f.433.1		6	444.359	even	6
1332.2.j.e.433.2		6	37.26	even	3	inner
1332.2.j.e.1009.2		6	1.1	even	1	trivial
5476.2.a.f.1.1		3	111.47	odd	6
5476.2.a.g.1.1		3	111.101	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:	\( N \)	\(=\)	\( 1332 = 2^{2} \cdot 3^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1332.j (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.500000 - 0.866025i) q^{5} +(-0.370556 - 0.641823i) q^{7} +2.91852 q^{11} +(1.54797 + 2.68116i) q^{13} +(-3.41852 + 5.92105i) q^{17} +(-0.0887048 - 0.153641i) q^{19} +3.43630 q^{23} +(2.00000 - 3.46410i) q^{25} +4.61371 q^{29} +2.91852 q^{31} +(-0.370556 + 0.641823i) q^{35} +(-2.04797 - 5.72764i) q^{37} +(0.322590 + 0.558743i) q^{41} -3.48223 q^{43} +9.79112 q^{47} +(3.22538 - 5.58651i) q^{49} +(3.37056 - 5.83798i) q^{53} +(-1.45926 - 2.52751i) q^{55} +(3.72538 - 6.45254i) q^{59} +(3.67741 + 6.36946i) q^{61} +(1.54797 - 2.68116i) q^{65} +(5.18464 + 8.98005i) q^{67} +(2.72538 + 4.72049i) q^{71} +12.3548 q^{73} +(-1.08148 - 1.87317i) q^{77} +(6.26611 + 10.8532i) q^{79} +(-4.18464 + 7.24800i) q^{83} +6.83705 q^{85} +(7.85482 - 13.6049i) q^{89} +(1.14722 - 1.98704i) q^{91} +(-0.0887048 + 0.153641i) q^{95} -16.8056 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 3 q^{5} - q^{7} - 7 q^{13} - 3 q^{17} + 7 q^{19} + 8 q^{23} + 12 q^{25} - q^{35} + 4 q^{37} + 17 q^{41} - 16 q^{43} + 16 q^{47} - 12 q^{49} + 19 q^{53} - 9 q^{59} + 7 q^{61} - 7 q^{65} - 9 q^{67}+ \cdots - 8 q^{97}+O(q^{100}) \) 6 * q - 3 * q^5 - q^7 - 7 * q^13 - 3 * q^17 + 7 * q^19 + 8 * q^23 + 12 * q^25 - q^35 + 4 * q^37 + 17 * q^41 - 16 * q^43 + 16 * q^47 - 12 * q^49 + 19 * q^53 - 9 * q^59 + 7 * q^61 - 7 * q^65 - 9 * q^67 - 15 * q^71 + 44 * q^73 - 24 * q^77 + 15 * q^79 + 15 * q^83 + 6 * q^85 + 17 * q^89 + 13 * q^91 + 7 * q^95 - 8 * q^97

Introduction

L-functions

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