LMFDB - Embedded newform 1332.2.j.e.433.1

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			433.1
Root			$1.42789 + 2.47317i$ of defining polynomial
Character	$\chi$	$=$	1332.433
Dual form			1332.2.j.e.1009.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.500000 + 0.866025i) q^{5} +(-2.07772 + 3.59871i) q^{7} +1.29966 q^{11} +(-1.77805 + 3.07968i) q^{13} +(-1.79966 - 3.11711i) q^{17} +(2.42789 - 4.20522i) q^{19} -5.01121 q^{23} +(2.00000 + 3.46410i) q^{25} -8.86698 q^{29} +1.29966 q^{31} +(-2.07772 - 3.59871i) q^{35} +(1.27805 - 5.94698i) q^{37} +(5.35577 - 9.27647i) q^{41} -10.3109 q^{43} -8.72275 q^{47} +(-5.13383 - 8.89205i) q^{49} +(5.07772 + 8.79487i) q^{53} +(-0.649832 + 1.12554i) q^{55} +(-4.63383 - 8.02602i) q^{59} +(-1.35577 + 2.34827i) q^{61} +(-1.77805 - 3.07968i) q^{65} +(-3.98399 + 6.90048i) q^{67} +(-5.63383 + 9.75807i) q^{71} +2.28845 q^{73} +(-2.70034 + 4.67712i) q^{77} +(-1.28366 + 2.22336i) q^{79} +(4.98399 + 8.63253i) q^{83} +3.59933 q^{85} +(-2.21155 - 3.83051i) q^{89} +(-7.38859 - 12.7974i) q^{91} +(2.42789 + 4.20522i) q^{95} +9.97920 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/1332\mathbb{Z}\right)^\times$.

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

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−0.500000	+	0.866025i	−0.223607	+	0.387298i	−0.955901	−	0.293691i	$-0.905116\pi$
$5$	−0.500000	+	0.866025i	−0.223607	+	0.387298i	0.732294	+	0.680989i	$0.238450\pi$
$6$		0			0
$7$	−2.07772	+	3.59871i	−0.785304	+	1.36019i	0.143514	+	0.989648i	$0.454160\pi$
$7$	−2.07772	+	3.59871i	−0.785304	+	1.36019i	−0.928817	+	0.370538i	$0.879173\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	1.29966			0.391863			0.195932	−	0.980618i	$-0.437227\pi$
$11$	1.29966			0.391863			0.195932	+	0.980618i	$0.437227\pi$
$12$		0			0
$13$	−1.77805	+	3.07968i	−0.493144	+	0.854150i	−0.999969	−	0.00789918i	$-0.997486\pi$
$13$	−1.77805	+	3.07968i	−0.493144	+	0.854150i	0.506825	+	0.862049i	$0.330819\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−1.79966	−	3.11711i	−0.436483	−	0.756010i	0.560933	−	0.827861i	$-0.310443\pi$
$17$	−1.79966	−	3.11711i	−0.436483	−	0.756010i	−0.997415	+	0.0718513i	$0.977109\pi$
$18$		0			0
$19$	2.42789	−	4.20522i	0.556995	−	0.964744i	−0.440750	−	0.897630i	$-0.645287\pi$
$19$	2.42789	−	4.20522i	0.556995	−	0.964744i	0.997745	−	0.0671143i	$-0.0213792\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−5.01121			−1.04491			−0.522455	−	0.852667i	$-0.674984\pi$
$23$	−5.01121			−1.04491			−0.522455	+	0.852667i	$0.674984\pi$
$24$		0			0
$25$	2.00000	+	3.46410i	0.400000	+	0.692820i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−8.86698			−1.64656			−0.823279	−	0.567638i	$-0.807858\pi$
$29$	−8.86698			−1.64656			−0.823279	+	0.567638i	$0.807858\pi$
$30$		0			0
$31$	1.29966			0.233427			0.116713	−	0.993166i	$-0.462764\pi$
$31$	1.29966			0.233427			0.116713	+	0.993166i	$0.462764\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−2.07772	−	3.59871i	−0.351198	−	0.608294i
$36$		0			0
$37$	1.27805	−	5.94698i	0.210111	−	0.977678i
$37$	1.27805	−	5.94698i	0.210111	−	0.977678i
$38$		0			0
$39$		0			0
$40$		0			0
$41$	5.35577	−	9.27647i	0.836431	−	1.44874i	−0.0564287	−	0.998407i	$-0.517971\pi$
$41$	5.35577	−	9.27647i	0.836431	−	1.44874i	0.892860	−	0.450335i	$-0.148695\pi$
$42$		0			0
$43$	−10.3109			−1.57239			−0.786197	−	0.617976i	$-0.787953\pi$
$43$	−10.3109			−1.57239			−0.786197	+	0.617976i	$0.787953\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−8.72275			−1.27234			−0.636172	−	0.771547i	$-0.719483\pi$
$47$	−8.72275			−1.27234			−0.636172	+	0.771547i	$0.719483\pi$
$48$		0			0
$49$	−5.13383	−	8.89205i	−0.733404	−	1.27029i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	5.07772	+	8.79487i	0.697478	+	1.20807i	0.969338	+	0.245731i	$0.0790280\pi$
$53$	5.07772	+	8.79487i	0.697478	+	1.20807i	−0.271860	+	0.962337i	$0.587639\pi$
$54$		0			0
$55$	−0.649832	+	1.12554i	−0.0876233	+	0.151768i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−4.63383	−	8.02602i	−0.603273	−	1.04490i	−0.992322	−	0.123682i	$-0.960530\pi$
$59$	−4.63383	−	8.02602i	−0.603273	−	1.04490i	0.389049	−	0.921217i	$-0.372804\pi$
$60$		0			0
$61$	−1.35577	+	2.34827i	−0.173589	+	0.300665i	−0.939672	−	0.342077i	$-0.888870\pi$
$61$	−1.35577	+	2.34827i	−0.173589	+	0.300665i	0.766083	+	0.642741i	$0.222203\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−1.77805	−	3.07968i	−0.220540	−	0.381987i
$66$		0			0
$67$	−3.98399	+	6.90048i	−0.486722	+	0.843028i	−0.999883	−	0.0152644i	$-0.995141\pi$
$67$	−3.98399	+	6.90048i	−0.486722	+	0.843028i	0.513161	+	0.858292i	$0.328474\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−5.63383	+	9.75807i	−0.668612	+	1.15807i	0.309680	+	0.950841i	$0.399778\pi$
$71$	−5.63383	+	9.75807i	−0.668612	+	1.15807i	−0.978292	+	0.207230i	$0.933555\pi$
$72$		0			0
$73$	2.28845			0.267843			0.133922	−	0.990992i	$-0.457243\pi$
$73$	2.28845			0.267843			0.133922	+	0.990992i	$0.457243\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−2.70034	+	4.67712i	−0.307732	+	0.533007i
$78$		0			0
$79$	−1.28366	+	2.22336i	−0.144423	+	0.250148i	−0.929157	−	0.369684i	$-0.879466\pi$
$79$	−1.28366	+	2.22336i	−0.144423	+	0.250148i	0.784735	+	0.619832i	$0.212799\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	4.98399	+	8.63253i	0.547064	+	0.947543i	0.998474	+	0.0552265i	$0.0175881\pi$
$83$	4.98399	+	8.63253i	0.547064	+	0.947543i	−0.451409	+	0.892317i	$0.649079\pi$
$84$		0			0
$85$	3.59933			0.390402
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−2.21155	−	3.83051i	−0.234423	−	0.406033i	0.724682	−	0.689084i	$-0.241987\pi$
$89$	−2.21155	−	3.83051i	−0.234423	−	0.406033i	−0.959105	+	0.283051i	$0.908654\pi$
$90$		0			0
$91$	−7.38859	−	12.7974i	−0.774535	−	1.34153i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	2.42789	+	4.20522i	0.249096	+	0.431447i
$96$		0			0
$97$	9.97920			1.01323			0.506617	−	0.862171i	$-0.330896\pi$
$97$	9.97920			1.01323			0.506617	+	0.862171i	$0.330896\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−2.84456			−0.283045			−0.141522	−	0.989935i	$-0.545200\pi$
$101$	−2.84456			−0.283045			−0.141522	+	0.989935i	$0.545200\pi$
$102$		0			0
$103$	−6.88778			−0.678673			−0.339337	−	0.940665i	$-0.610203\pi$
$103$	−6.88778			−0.678673			−0.339337	+	0.940665i	$0.610203\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−1.63383	+	2.82987i	−0.157948	+	0.273574i	−0.934129	−	0.356937i	$-0.883821\pi$
$107$	−1.63383	+	2.82987i	−0.157948	+	0.273574i	0.776181	+	0.630511i	$0.217155\pi$
$108$		0			0
$109$	−6.65544	−	11.5276i	−0.637475	−	1.10414i	−0.985985	−	0.166834i	$-0.946646\pi$
$109$	−6.65544	−	11.5276i	−0.637475	−	1.10414i	0.348510	−	0.937305i	$-0.386688\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−1.93349	−	3.34890i	−0.181888	−	0.315038i	0.760636	−	0.649179i	$-0.224887\pi$
$113$	−1.93349	−	3.34890i	−0.181888	−	0.315038i	−0.942523	+	0.334140i	$0.891554\pi$
$114$		0			0
$115$	2.50560	−	4.33983i	0.233649	−	0.404692i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	14.9568			1.37109
$120$		0			0
$121$	−9.31087			−0.846443
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−9.00000			−0.804984
$126$		0			0
$127$	2.02721	+	3.51124i	0.179886	+	0.311572i	0.941841	−	0.336058i	$-0.109094\pi$
$127$	2.02721	+	3.51124i	0.179886	+	0.311572i	−0.761955	+	0.647630i	$0.775760\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	5.58332	+	9.67060i	0.487817	+	0.844924i	0.999902	−	0.0140108i	$-0.00445993\pi$
$131$	5.58332	+	9.67060i	0.487817	+	0.844924i	−0.512085	+	0.858935i	$0.671127\pi$
$132$		0			0
$133$	10.0889	+	17.4745i	0.874821	+	1.51523i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	1.15544			0.0987156			0.0493578	−	0.998781i	$-0.484283\pi$
$137$	1.15544			0.0987156			0.0493578	+	0.998781i	$0.484283\pi$
$138$		0			0
$139$	8.72755	+	15.1166i	0.740261	+	1.28217i	0.952376	+	0.304925i	$0.0986315\pi$
$139$	8.72755	+	15.1166i	0.740261	+	1.28217i	−0.212115	+	0.977245i	$0.568035\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−2.31087	+	4.00255i	−0.193245	+	0.334710i
$144$		0			0
$145$	4.43349	−	7.67903i	0.368181	−	0.637709i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−1.15544			−0.0946571			−0.0473285	−	0.998879i	$-0.515071\pi$
$149$	−1.15544			−0.0946571			−0.0473285	+	0.998879i	$0.515071\pi$
$150$		0			0
$151$	3.17144	−	5.49310i	0.258088	−	0.447022i	−0.707641	−	0.706572i	$-0.750241\pi$
$151$	3.17144	−	5.49310i	0.258088	−	0.447022i	0.965730	+	0.259550i	$0.0835741\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−0.649832	+	1.12554i	−0.0521958	+	0.0904057i
$156$		0			0
$157$	0.355773	+	0.616216i	0.0283937	+	0.0491794i	0.879873	−	0.475209i	$-0.157627\pi$
$157$	0.355773	+	0.616216i	0.0283937	+	0.0491794i	−0.851479	+	0.524388i	$0.824294\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	10.4119	−	18.0339i	0.820571	−	1.42127i
$162$		0			0
$163$	−1.63383	−	2.82987i	−0.127971	−	0.221653i	0.794919	−	0.606715i	$-0.207513\pi$
$163$	−1.63383	−	2.82987i	−0.127971	−	0.221653i	−0.922890	+	0.385063i	$0.874180\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−10.7893	+	18.6876i	−0.834898	+	1.44609i	0.0592150	+	0.998245i	$0.481140\pi$
$167$	−10.7893	+	18.6876i	−0.834898	+	1.44609i	−0.894113	+	0.447841i	$0.852193\pi$
$168$		0			0
$169$	0.177047	+	0.306654i	0.0136190	+	0.0235887i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−5.79966	−	10.0453i	−0.440940	−	0.763731i	0.556819	−	0.830634i	$-0.312022\pi$
$173$	−5.79966	−	10.0453i	−0.440940	−	0.763731i	−0.997760	+	0.0669027i	$0.978688\pi$
$174$		0			0
$175$	−16.6217			−1.25649
$176$		0			0
$177$		0			0
$178$		0			0
$179$	1.19866			0.0895918			0.0447959	−	0.998996i	$-0.485736\pi$
$179$	1.19866			0.0895918			0.0447959	+	0.998996i	$0.485736\pi$
$180$		0			0
$181$	1.65544	−	2.86730i	0.123048	−	0.213125i	−0.797920	−	0.602763i	$-0.794067\pi$
$181$	1.65544	−	2.86730i	0.123048	−	0.213125i	0.920968	+	0.389638i	$0.127400\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	4.51121	+	4.08032i	0.331671	+	0.299991i
$186$		0			0
$187$	−2.33896	−	4.05120i	−0.171042	−	0.296253i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−5.58812			−0.404342			−0.202171	−	0.979350i	$-0.564800\pi$
$191$	−5.58812			−0.404342			−0.202171	+	0.979350i	$0.564800\pi$
$192$		0			0
$193$	2.55611			0.183993			0.0919964	−	0.995759i	$-0.470675\pi$
$193$	2.55611			0.183993			0.0919964	+	0.995759i	$0.470675\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−6.76765	−	11.7219i	−0.482175	−	0.835152i	0.517615	−	0.855613i	$-0.326820\pi$
$197$	−6.76765	−	11.7219i	−0.482175	−	0.835152i	−0.999791	+	0.0204613i	$0.993487\pi$
$198$		0			0
$199$	1.68913			0.119739			0.0598695	−	0.998206i	$-0.480932\pi$
$199$	1.68913			0.119739			0.0598695	+	0.998206i	$0.480932\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	18.4231	−	31.9097i	1.29305	−	2.23962i
$204$		0			0
$205$	5.35577	+	9.27647i	0.374063	+	0.647897i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	3.15544	−	5.46538i	0.218266	−	0.378048i
$210$		0			0
$211$	10.8878			0.749546			0.374773	−	0.927117i	$-0.377721\pi$
$211$	10.8878			0.749546			0.374773	+	0.927117i	$0.377721\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	5.15544	−	8.92948i	0.351598	−	0.608985i
$216$		0			0
$217$	−2.70034	+	4.67712i	−0.183311	+	0.317504i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	12.7996			0.860994
$222$		0			0
$223$	−0.187447			−0.0125524			−0.00627620	−	0.999980i	$-0.501998\pi$
$223$	−0.187447			−0.0125524			−0.00627620	+	0.999980i	$0.501998\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−9.72755	+	16.8486i	−0.645640	+	1.11828i	0.338513	+	0.940962i	$0.390076\pi$
$227$	−9.72755	+	16.8486i	−0.645640	+	1.11828i	−0.984153	+	0.177320i	$0.943257\pi$
$228$		0			0
$229$	8.62343	−	14.9362i	0.569852	−	0.987013i	−0.426728	−	0.904380i	$-0.640334\pi$
$229$	8.62343	−	14.9362i	0.569852	−	0.987013i	0.996580	−	0.0826327i	$-0.0263328\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	18.6009			1.21859			0.609294	−	0.792944i	$-0.291453\pi$
$233$	18.6009			1.21859			0.609294	+	0.792944i	$0.291453\pi$
$234$		0			0
$235$	4.36138	−	7.55413i	0.284505	−	0.492777i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−6.67705	−	11.5650i	−0.431902	−	0.748077i	0.565135	−	0.824999i	$-0.308824\pi$
$239$	−6.67705	−	11.5650i	−0.431902	−	0.748077i	−0.997037	+	0.0769219i	$0.975491\pi$
$240$		0			0
$241$	−8.08893	+	14.0104i	−0.521054	+	0.902491i	0.478647	+	0.878008i	$0.341128\pi$
$241$	−8.08893	+	14.0104i	−0.521054	+	0.902491i	−0.999700	+	0.0244837i	$0.992206\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	10.2677			0.655976
$246$		0			0
$247$	8.63383	+	14.9542i	0.549357	+	0.951515i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−11.5095			−0.726475			−0.363238	−	0.931697i	$-0.618329\pi$
$251$	−11.5095			−0.726475			−0.363238	+	0.931697i	$0.618329\pi$
$252$		0			0
$253$	−6.51289			−0.409462
$254$		0			0
$255$		0			0
$256$		0			0
$257$	0.756444	+	1.31020i	0.0471857	+	0.0817280i	0.888654	−	0.458579i	$-0.151641\pi$
$257$	0.756444	+	1.31020i	0.0471857	+	0.0817280i	−0.841468	+	0.540307i	$0.818308\pi$
$258$		0			0
$259$	18.7460	+	16.9555i	1.16482	+	1.05356i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	7.52161	−	13.0278i	0.463802	−	0.803329i	−0.535344	−	0.844634i	$-0.679818\pi$
$263$	7.52161	−	13.0278i	0.463802	−	0.803329i	0.999147	+	0.0413047i	$0.0131515\pi$
$264$		0			0
$265$	−10.1554			−0.623844
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−0.887783			−0.0541291			−0.0270645	−	0.999634i	$-0.508616\pi$
$269$	−0.887783			−0.0541291			−0.0270645	+	0.999634i	$0.508616\pi$
$270$		0			0
$271$	9.59453	+	16.6182i	0.582826	+	1.00948i	0.995143	+	0.0984440i	$0.0313865\pi$
$271$	9.59453	+	16.6182i	0.582826	+	1.00948i	−0.412316	+	0.911041i	$0.635280\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	2.59933	+	4.50217i	0.156745	+	0.271491i
$276$		0			0
$277$	1.79966	−	3.11711i	0.108131	−	0.187289i	−0.806882	−	0.590713i	$-0.798847\pi$
$277$	1.79966	−	3.11711i	0.108131	−	0.187289i	0.915013	+	0.403424i	$0.132180\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	14.2660	+	24.7094i	0.851037	+	1.47404i	0.880274	+	0.474466i	$0.157359\pi$
$281$	14.2660	+	24.7094i	0.851037	+	1.47404i	−0.0292371	+	0.999573i	$0.509308\pi$
$282$		0			0
$283$	−4.07772	+	7.06282i	−0.242395	+	0.419841i	−0.961396	−	0.275168i	$-0.911266\pi$
$283$	−4.07772	+	7.06282i	−0.242395	+	0.419841i	0.719001	+	0.695009i	$0.244600\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	22.2556	+	38.5478i	1.31370	+	2.27540i
$288$		0			0
$289$	2.02242	−	3.50293i	0.118966	−	0.206055i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−3.65544	+	6.33140i	−0.213553	+	0.369884i	−0.952824	−	0.303524i	$-0.901837\pi$
$293$	−3.65544	+	6.33140i	−0.213553	+	0.369884i	0.739271	+	0.673408i	$0.235170\pi$
$294$		0			0
$295$	9.26765			0.539584
$296$		0			0
$297$		0			0
$298$		0			0
$299$	8.91020	−	15.4329i	0.515290	−	0.892509i
$300$		0			0
$301$	21.4231	−	37.1059i	1.23481	−	2.13875i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−1.35577	−	2.34827i	−0.0776313	−	0.134461i
$306$		0			0
$307$	−17.1122			−0.976646			−0.488323	−	0.872663i	$-0.662391\pi$
$307$	−17.1122			−0.976646			−0.488323	+	0.872663i	$0.662391\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	15.0569	+	26.0793i	0.853799	+	1.47882i	0.877754	+	0.479111i	$0.159041\pi$
$311$	15.0569	+	26.0793i	0.853799	+	1.47882i	−0.0239549	+	0.999713i	$0.507626\pi$
$312$		0			0
$313$	15.7789	+	27.3298i	0.891874	+	1.54477i	0.837626	+	0.546244i	$0.183943\pi$
$313$	15.7789	+	27.3298i	0.891874	+	1.54477i	0.0542477	+	0.998528i	$0.482724\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	15.1105	+	26.1722i	0.848692	+	1.46998i	0.882376	+	0.470546i	$0.155943\pi$
$317$	15.1105	+	26.1722i	0.848692	+	1.46998i	−0.0336833	+	0.999433i	$0.510724\pi$
$318$		0			0
$319$	−11.5241			−0.645226
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−17.4775			−0.972475
$324$		0			0
$325$	−14.2244			−0.789030
$326$		0			0
$327$		0			0
$328$		0			0
$329$	18.1234	−	31.3907i	0.999177	−	1.73063i
$330$		0			0
$331$	1.23315	+	2.13589i	0.0677803	+	0.117399i	0.897924	−	0.440151i	$-0.145075\pi$
$331$	1.23315	+	2.13589i	0.0677803	+	0.117399i	−0.830144	+	0.557550i	$0.811742\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−3.98399	−	6.90048i	−0.217669	−	0.377014i
$336$		0			0
$337$	11.2228	−	19.4384i	0.611342	−	1.05888i	−0.379672	−	0.925121i	$-0.623963\pi$
$337$	11.2228	−	19.4384i	0.611342	−	1.05888i	0.991014	−	0.133755i	$-0.0427035\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	1.68913			0.0914713
$342$		0			0
$343$	13.5785			0.733172
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−32.4343			−1.74116			−0.870582	−	0.492023i	$-0.836258\pi$
$347$	−32.4343			−1.74116			−0.870582	+	0.492023i	$0.836258\pi$
$348$		0			0
$349$	5.36698	+	9.29588i	0.287288	+	0.497597i	0.973161	−	0.230123i	$-0.0739130\pi$
$349$	5.36698	+	9.29588i	0.287288	+	0.497597i	−0.685873	+	0.727721i	$0.740580\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	12.0785	+	20.9206i	0.642875	+	1.11349i	0.984788	+	0.173761i	$0.0555919\pi$
$353$	12.0785	+	20.9206i	0.642875	+	1.11349i	−0.341913	+	0.939732i	$0.611075\pi$
$354$		0			0
$355$	−5.63383	−	9.75807i	−0.299012	−	0.517905i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	2.84618			0.150216			0.0751078	−	0.997175i	$-0.476070\pi$
$359$	2.84618			0.150216			0.0751078	+	0.997175i	$0.476070\pi$
$360$		0			0
$361$	−2.28926	−	3.96512i	−0.120488	−	0.208691i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−1.14423	+	1.98186i	−0.0598916	+	0.103735i
$366$		0			0
$367$	−1.23315	+	2.13589i	−0.0643702	+	0.111492i	−0.896414	−	0.443217i	$-0.853837\pi$
$367$	−1.23315	+	2.13589i	−0.0643702	+	0.111492i	0.832044	+	0.554709i	$0.187171\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−42.2003			−2.19093
$372$		0			0
$373$	7.61222	−	13.1847i	0.394146	−	0.682680i	−0.598846	−	0.800864i	$-0.704374\pi$
$373$	7.61222	−	13.1847i	0.394146	−	0.682680i	0.992992	+	0.118184i	$0.0377072\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	15.7660	−	27.3075i	0.811989	−	1.40641i
$378$		0			0
$379$	−15.9447	−	27.6170i	−0.819024	−	1.41859i	−0.906402	−	0.422416i	$-0.861182\pi$
$379$	−15.9447	−	27.6170i	−0.819024	−	1.41859i	0.0873777	−	0.996175i	$-0.472151\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−3.94470	+	6.83242i	−0.201565	+	0.349120i	−0.949033	−	0.315177i	$-0.897936\pi$
$383$	−3.94470	+	6.83242i	−0.201565	+	0.349120i	0.747468	+	0.664298i	$0.231269\pi$
$384$		0			0
$385$	−2.70034	−	4.67712i	−0.137622	−	0.238368i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	18.1795	−	31.4879i	0.921739	−	1.59650i	0.125015	−	0.992155i	$-0.460102\pi$
$389$	18.1795	−	31.4879i	0.921739	−	1.59650i	0.796724	−	0.604344i	$-0.206565\pi$
$390$		0			0
$391$	9.01849	+	15.6205i	0.456085	+	0.789962i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−1.28366	−	2.22336i	−0.0645879	−	0.111869i
$396$		0			0
$397$	5.73235			0.287698			0.143849	−	0.989600i	$-0.454052\pi$
$397$	5.73235			0.287698			0.143849	+	0.989600i	$0.454052\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	13.7115			0.684722			0.342361	−	0.939569i	$-0.388773\pi$
$401$	13.7115			0.684722			0.342361	+	0.939569i	$0.388773\pi$
$402$		0			0
$403$	−2.31087	+	4.00255i	−0.115113	+	0.199381i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	1.66104	−	7.72908i	0.0823348	−	0.383116i
$408$		0			0
$409$	3.46799	+	6.00673i	0.171481	+	0.297014i	0.938938	−	0.344087i	$-0.111811\pi$
$409$	3.46799	+	6.00673i	0.171481	+	0.297014i	−0.767457	+	0.641101i	$0.778478\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	38.5111			1.89501
$414$		0			0
$415$	−9.96799			−0.489309
$416$		0			0
$417$		0			0
$418$		0			0
$419$	6.21074	+	10.7573i	0.303414	+	0.525529i	0.976907	−	0.213665i	$-0.0685400\pi$
$419$	6.21074	+	10.7573i	0.303414	+	0.525529i	−0.673493	+	0.739194i	$0.735207\pi$
$420$		0			0
$421$	−1.81879			−0.0886422			−0.0443211	−	0.999017i	$-0.514112\pi$
$421$	−1.81879			−0.0886422			−0.0443211	+	0.999017i	$0.514112\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	7.19866	−	12.4684i	0.349186	−	0.604808i
$426$		0			0
$427$	−5.63383	−	9.75807i	−0.272640	−	0.472226i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−5.02721	+	8.70739i	−0.242152	+	0.419420i	−0.961327	−	0.275409i	$-0.911187\pi$
$431$	−5.02721	+	8.70739i	−0.242152	+	0.419420i	0.719175	+	0.694829i	$0.244520\pi$
$432$		0			0
$433$	−39.7996			−1.91265			−0.956323	−	0.292311i	$-0.905576\pi$
$433$	−39.7996			−1.91265			−0.956323	+	0.292311i	$0.905576\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−12.1666	+	21.0732i	−0.582010	+	1.00807i
$438$		0			0
$439$	−12.9447	+	22.4209i	−0.617817	+	1.07009i	0.372066	+	0.928206i	$0.378649\pi$
$439$	−12.9447	+	22.4209i	−0.617817	+	1.07009i	−0.989883	+	0.141884i	$0.954684\pi$
$440$		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
$445$	4.42309			0.209675
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−15.6450	+	27.0980i	−0.738335	+	1.27883i	0.214910	+	0.976634i	$0.431054\pi$
$449$	−15.6450	+	27.0980i	−0.738335	+	1.27883i	−0.953245	+	0.302200i	$0.902279\pi$
$450$		0			0
$451$	6.96071	−	12.0563i	0.327767	−	0.567709i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	14.7772			0.692765
$456$		0			0
$457$	−7.66665	+	13.2790i	−0.358631	+	0.621166i	−0.987732	−	0.156157i	$-0.950090\pi$
$457$	−7.66665	+	13.2790i	−0.358631	+	0.621166i	0.629102	+	0.777323i	$0.283423\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−3.04571	−	5.27532i	−0.141853	−	0.245696i	0.786342	−	0.617792i	$-0.211973\pi$
$461$	−3.04571	−	5.27532i	−0.141853	−	0.245696i	−0.928194	+	0.372096i	$0.878639\pi$
$462$		0			0
$463$	16.2332	−	28.1166i	0.754419	−	1.30669i	−0.191244	−	0.981543i	$-0.561252\pi$
$463$	16.2332	−	28.1166i	0.754419	−	1.30669i	0.945663	−	0.325149i	$-0.105414\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	27.4231			1.26899			0.634495	−	0.772927i	$-0.281208\pi$
$467$	27.4231			1.26899			0.634495	+	0.772927i	$0.281208\pi$
$468$		0			0
$469$	−16.5552	−	28.6745i	−0.764450	−	1.32407i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−13.4007			−0.616164
$474$		0			0
$475$	19.4231			0.891192
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−2.32295	−	4.02347i	−0.106138	−	0.183837i	0.808064	−	0.589094i	$-0.200515\pi$
$479$	−2.32295	−	4.02347i	−0.106138	−	0.183837i	−0.914203	+	0.405257i	$0.867182\pi$
$480$		0			0
$481$	16.0423	+	14.5101i	0.731468	+	0.661601i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−4.98960	+	8.64224i	−0.226566	+	0.392424i
$486$		0			0
$487$	13.3861			0.606582			0.303291	−	0.952898i	$-0.401915\pi$
$487$	13.3861			0.606582			0.303291	+	0.952898i	$0.401915\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−17.3445			−0.782746			−0.391373	−	0.920232i	$-0.628000\pi$
$491$	−17.3445			−0.782746			−0.391373	+	0.920232i	$0.628000\pi$
$492$		0			0
$493$	15.9576	+	27.6394i	0.718694	+	1.24481i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−23.4110	−	40.5491i	−1.05013	−	1.81887i
$498$		0			0
$499$	14.7388	−	25.5283i	0.659797	−	1.14280i	−0.320870	−	0.947123i	$-0.603975\pi$
$499$	14.7388	−	25.5283i	0.659797	−	1.14280i	0.980668	−	0.195680i	$-0.0626913\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−0.459897	−	0.796565i	−0.0205058	−	0.0355171i	0.855590	−	0.517653i	$-0.173194\pi$
$503$	−0.459897	−	0.796565i	−0.0205058	−	0.0355171i	−0.876096	+	0.482136i	$0.839861\pi$
$504$		0			0
$505$	1.42228	−	2.46346i	0.0632907	−	0.109623i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	2.48879	+	4.31071i	0.110314	+	0.191069i	0.915897	−	0.401414i	$-0.131481\pi$
$509$	2.48879	+	4.31071i	0.110314	+	0.191069i	−0.805583	+	0.592483i	$0.798148\pi$
$510$		0			0
$511$	−4.75476	+	8.23549i	−0.210338	+	0.364317i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	3.44389	−	5.96500i	0.151756	−	0.262849i
$516$		0			0
$517$	−11.3367			−0.498585
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−19.1321	+	33.1379i	−0.838195	+	1.45180i	0.0532081	+	0.998583i	$0.483055\pi$
$521$	−19.1321	+	33.1379i	−0.838195	+	1.45180i	−0.891403	+	0.453212i	$0.850278\pi$
$522$		0			0
$523$	−19.3269	+	33.4751i	−0.845105	+	1.46377i	0.0404243	+	0.999183i	$0.487129\pi$
$523$	−19.3269	+	33.4751i	−0.845105	+	1.46377i	−0.885530	+	0.464583i	$0.846204\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−2.33896	−	4.05120i	−0.101887	−	0.176473i
$528$		0			0
$529$	2.11222			0.0918355
$530$		0			0
$531$		0			0
$532$		0			0
$533$	19.0457	+	32.9881i	0.824961	+	1.42887i
$534$		0			0
$535$	−1.63383	−	2.82987i	−0.0706365	−	0.122346i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−6.67225	−	11.5567i	−0.287394	−	0.497781i
$540$		0			0
$541$	−34.3125			−1.47521			−0.737605	−	0.675233i	$-0.764043\pi$
$541$	−34.3125			−1.47521			−0.737605	+	0.675233i	$0.764043\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	13.3109			0.570175
$546$		0			0
$547$	−31.3221			−1.33924			−0.669618	−	0.742706i	$-0.733542\pi$
$547$	−31.3221			−1.33924			−0.669618	+	0.742706i	$0.733542\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−21.5280	+	37.2876i	−0.917125	+	1.58851i
$552$		0			0
$553$	−5.33416	−	9.23904i	−0.226832	−	0.392884i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−10.3990	−	18.0116i	−0.440619	−	0.763175i	0.557116	−	0.830435i	$-0.311908\pi$
$557$	−10.3990	−	18.0116i	−0.440619	−	0.763175i	−0.997736	+	0.0672594i	$0.978575\pi$
$558$		0			0
$559$	18.3333	−	31.7542i	0.775416	−	1.34306i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	16.1874			0.682220			0.341110	−	0.940023i	$-0.389197\pi$
$563$	16.1874			0.682220			0.341110	+	0.940023i	$0.389197\pi$
$564$		0			0
$565$	3.86698			0.162685
$566$		0			0
$567$		0			0
$568$		0			0
$569$	1.73235			0.0726237			0.0363119	−	0.999341i	$-0.488439\pi$
$569$	1.73235			0.0726237			0.0363119	+	0.999341i	$0.488439\pi$
$570$		0			0
$571$	8.82856	+	15.2915i	0.369464	+	0.639930i	0.989482	−	0.144658i	$-0.0462081\pi$
$571$	8.82856	+	15.2915i	0.369464	+	0.639930i	−0.620018	+	0.784588i	$0.712875\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−10.0224	−	17.3593i	−0.417964	−	0.723934i
$576$		0			0
$577$	6.63383	+	11.4901i	0.276170	+	0.478340i	0.970430	−	0.241384i	$-0.0776014\pi$
$577$	6.63383	+	11.4901i	0.276170	+	0.478340i	−0.694260	+	0.719725i	$0.744268\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−41.4213			−1.71845
$582$		0			0
$583$	6.59933	+	11.4304i	0.273316	+	0.473398i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	9.42789	−	16.3296i	0.389131	−	0.673994i	−0.603202	−	0.797588i	$-0.706109\pi$
$587$	9.42789	−	16.3296i	0.389131	−	0.673994i	0.992333	+	0.123594i	$0.0394422\pi$
$588$		0			0
$589$	3.15544	−	5.46538i	0.130018	−	0.225197i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−13.8928			−0.570507			−0.285254	−	0.958452i	$-0.592078\pi$
$593$	−13.8928			−0.570507			−0.285254	+	0.958452i	$0.592078\pi$
$594$		0			0
$595$	−7.47839	+	12.9530i	−0.306584	+	0.531019i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−16.5513	+	28.6677i	−0.676268	+	1.17133i	0.299828	+	0.953993i	$0.403071\pi$
$599$	−16.5513	+	28.6677i	−0.676268	+	1.17133i	−0.976096	+	0.217338i	$0.930263\pi$
$600$		0			0
$601$	−5.60101	−	9.70123i	−0.228470	−	0.395721i	0.728885	−	0.684636i	$-0.240039\pi$
$601$	−5.60101	−	9.70123i	−0.228470	−	0.395721i	−0.957355	+	0.288915i	$0.906706\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	4.65544	−	8.06345i	0.189270	−	0.327826i
$606$		0			0
$607$	−13.0496	−	22.6026i	−0.529668	−	0.917412i	−0.999401	−	0.0346037i	$-0.988983\pi$
$607$	−13.0496	−	22.6026i	−0.529668	−	0.917412i	0.469733	−	0.882809i	$-0.344350\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	15.5095	−	26.8633i	0.627448	−	1.08677i
$612$		0			0
$613$	−4.46799	−	7.73878i	−0.180460	−	0.312567i	0.761577	−	0.648074i	$-0.224425\pi$
$613$	−4.46799	−	7.73878i	−0.180460	−	0.312567i	−0.942037	+	0.335508i	$0.891092\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−4.63383	−	8.02602i	−0.186551	−	0.323116i	0.757547	−	0.652780i	$-0.226398\pi$
$617$	−4.63383	−	8.02602i	−0.186551	−	0.323116i	−0.944098	+	0.329665i	$0.893064\pi$
$618$		0			0
$619$	2.88778			0.116070			0.0580349	−	0.998315i	$-0.481517\pi$
$619$	2.88778			0.116070			0.0580349	+	0.998315i	$0.481517\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	18.3799			0.736374
$624$		0			0
$625$	−5.50000	+	9.52628i	−0.220000	+	0.381051i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−20.8375	+	6.71873i	−0.830844	+	0.267893i
$630$		0			0
$631$	9.73876	+	16.8680i	0.387694	+	0.671506i	0.992139	−	0.125141i	$-0.0399384\pi$
$631$	9.73876	+	16.8680i	0.387694	+	0.671506i	−0.604445	+	0.796647i	$0.706605\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−4.05443			−0.160895
$636$		0			0
$637$	36.5129			1.44669
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−3.69866	−	6.40626i	−0.146088	−	0.253032i	0.783690	−	0.621152i	$-0.213335\pi$
$641$	−3.69866	−	6.40626i	−0.146088	−	0.253032i	−0.929778	+	0.368120i	$0.880002\pi$
$642$		0			0
$643$	−10.9024			−0.429947			−0.214973	−	0.976620i	$-0.568966\pi$
$643$	−10.9024			−0.429947			−0.214973	+	0.976620i	$0.568966\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−6.54403	+	11.3346i	−0.257272	+	0.445609i	−0.965510	−	0.260365i	$-0.916157\pi$
$647$	−6.54403	+	11.3346i	−0.257272	+	0.445609i	0.708238	+	0.705974i	$0.249490\pi$
$648$		0			0
$649$	−6.02242	−	10.4311i	−0.236401	−	0.409458i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	19.6234	−	33.9888i	0.767924	−	1.33008i	−0.170763	−	0.985312i	$-0.554623\pi$
$653$	19.6234	−	33.9888i	0.767924	−	1.33008i	0.938687	−	0.344771i	$-0.112043\pi$
$654$		0			0
$655$	−11.1666			−0.436317
$656$		0			0
$657$		0			0
$658$		0			0
$659$	2.65624	−	4.60075i	0.103473	−	0.179220i	−0.809641	−	0.586926i	$-0.800338\pi$
$659$	2.65624	−	4.60075i	0.103473	−	0.179220i	0.913113	+	0.407706i	$0.133671\pi$
$660$		0			0
$661$	−3.81087	+	6.60063i	−0.148226	+	0.256735i	−0.930572	−	0.366109i	$-0.880690\pi$
$661$	−3.81087	+	6.60063i	−0.148226	+	0.256735i	0.782346	+	0.622844i	$0.214023\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−20.1779			−0.782464
$666$		0			0
$667$	44.4343			1.72050
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−1.76205	+	3.05196i	−0.0680231	+	0.117820i
$672$		0			0
$673$	−6.00249	+	10.3966i	−0.231379	+	0.400760i	−0.958214	−	0.286052i	$-0.907657\pi$
$673$	−6.00249	+	10.3966i	−0.231379	+	0.400760i	0.726835	+	0.686812i	$0.240990\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−35.5527			−1.36640			−0.683202	−	0.730230i	$-0.739413\pi$
$677$	−35.5527			−1.36640			−0.683202	+	0.730230i	$0.739413\pi$
$678$		0			0
$679$	−20.7340	+	35.9123i	−0.795696	+	1.37819i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−15.7893	−	27.3478i	−0.604159	−	1.04643i	−0.992184	−	0.124786i	$-0.960176\pi$
$683$	−15.7893	−	27.3478i	−0.604159	−	1.04643i	0.388024	−	0.921649i	$-0.373158\pi$
$684$		0			0
$685$	−0.577718	+	1.00064i	−0.0220735	+	0.0382324i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−36.1138			−1.37583
$690$		0			0
$691$	5.59453	+	9.69001i	0.212826	+	0.368625i	0.952598	−	0.304232i	$-0.0983999\pi$
$691$	5.59453	+	9.69001i	0.212826	+	0.368625i	−0.739772	+	0.672858i	$0.765067\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−17.4551			−0.662110
$696$		0			0
$697$	−38.5544			−1.46035
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−13.7893	−	23.8837i	−0.520813	−	0.902075i	−0.999707	−	0.0242022i	$-0.992295\pi$
$701$	−13.7893	−	23.8837i	−0.520813	−	0.902075i	0.478894	−	0.877873i	$-0.341038\pi$
$702$		0			0
$703$	−21.9054	−	19.8131i	−0.826178	−	0.747265i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	5.91020	−	10.2368i	0.222276	−	0.384993i
$708$		0			0
$709$	9.50953			0.357138			0.178569	−	0.983927i	$-0.442853\pi$
$709$	9.50953			0.357138			0.178569	+	0.983927i	$0.442853\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−6.51289			−0.243910
$714$		0			0
$715$	−2.31087	−	4.00255i	−0.0864218	−	0.149687i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−9.94077	−	17.2179i	−0.370728	−	0.642120i	0.618949	−	0.785431i	$-0.287559\pi$
$719$	−9.94077	−	17.2179i	−0.370728	−	0.642120i	−0.989678	+	0.143311i	$0.954225\pi$
$720$		0			0
$721$	14.3109	−	24.7872i	0.532965	−	0.923122i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−17.7340	−	30.7161i	−0.658623	−	1.14077i
$726$		0			0
$727$	−1.23315	+	2.13589i	−0.0457352	+	0.0792157i	−0.887987	−	0.459869i	$-0.847896\pi$
$727$	−1.23315	+	2.13589i	−0.0457352	+	0.0792157i	0.842252	+	0.539085i	$0.181230\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	18.5561	+	32.1401i	0.686322	+	1.18875i
$732$		0			0
$733$	−22.3998	+	38.7976i	−0.827356	+	1.43302i	0.0727502	+	0.997350i	$0.476822\pi$
$733$	−22.3998	+	38.7976i	−0.827356	+	1.43302i	−0.900106	+	0.435672i	$0.856511\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−5.17785	+	8.96831i	−0.190729	+	0.330352i
$738$		0			0
$739$	−27.4085			−1.00824			−0.504119	−	0.863634i	$-0.668183\pi$
$739$	−27.4085			−1.00824			−0.504119	+	0.863634i	$0.668183\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	5.69554	−	9.86496i	0.208949	−	0.361910i	−0.742435	−	0.669918i	$-0.766329\pi$
$743$	5.69554	−	9.86496i	0.208949	−	0.361910i	0.951384	+	0.308008i	$0.0996624\pi$
$744$		0			0
$745$	0.577718	−	1.00064i	0.0211660	−	0.0366605i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−6.78926	−	11.7593i	−0.248074	−	0.429677i
$750$		0			0
$751$	−33.7340			−1.23097			−0.615485	−	0.788149i	$-0.711040\pi$
$751$	−33.7340			−1.23097			−0.615485	+	0.788149i	$0.711040\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	3.17144	+	5.49310i	0.115421	+	0.199914i
$756$		0			0
$757$	−16.1795	−	28.0238i	−0.588055	−	1.01854i	−0.994487	−	0.104860i	$-0.966560\pi$
$757$	−16.1795	−	28.0238i	−0.588055	−	1.01854i	0.406432	−	0.913681i	$-0.366773\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	6.05611	+	10.4895i	0.219534	+	0.380244i	0.954666	−	0.297681i	$-0.0962131\pi$
$761$	6.05611	+	10.4895i	0.219534	+	0.380244i	−0.735132	+	0.677924i	$0.762880\pi$
$762$		0			0
$763$	55.3125			2.00245
$764$		0			0
$765$		0			0
$766$		0			0
$767$	32.9568			1.19000
$768$		0			0
$769$	−12.5577			−0.452843			−0.226422	−	0.974029i	$-0.572703\pi$
$769$	−12.5577			−0.452843			−0.226422	+	0.974029i	$0.572703\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	10.3878	−	17.9922i	0.373623	−	0.647133i	−0.616497	−	0.787357i	$-0.711449\pi$
$773$	10.3878	−	17.9922i	0.373623	−	0.647133i	0.990120	+	0.140224i	$0.0447822\pi$
$774$		0			0
$775$	2.59933	+	4.50217i	0.0933706	+	0.161723i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−26.0064	−	45.0444i	−0.931776	−	1.61388i
$780$		0			0
$781$	−7.32208	+	12.6822i	−0.262005	+	0.453805i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−0.711545			−0.0253961
$786$		0			0
$787$	5.02578			0.179150			0.0895748	−	0.995980i	$-0.471449\pi$
$787$	5.02578			0.179150			0.0895748	+	0.995980i	$0.471449\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	16.0690			0.571348
$792$		0			0
$793$	−4.82127	−	8.35069i	−0.171208	−	0.296542i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	13.1434	+	22.7650i	0.465562	+	0.806376i	0.999227	−	0.0393196i	$-0.0125190\pi$
$797$	13.1434	+	22.7650i	0.465562	+	0.806376i	−0.533665	+	0.845696i	$0.679186\pi$
$798$		0			0
$799$	15.6980	+	27.1898i	0.555356	+	0.961905i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	2.97422			0.104958
$804$		0			0
$805$	10.4119	+	18.0339i	0.366971	+	0.635612i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−20.6770	+	35.8137i	−0.726966	+	1.25914i	0.231193	+	0.972908i	$0.425737\pi$
$809$	−20.6770	+	35.8137i	−0.726966	+	1.25914i	−0.958160	+	0.286235i	$0.907596\pi$
$810$		0			0
$811$	23.9239	−	41.4374i	0.840082	−	1.45506i	−0.0497432	−	0.998762i	$-0.515840\pi$
$811$	23.9239	−	41.4374i	0.840082	−	1.45506i	0.889825	−	0.456302i	$-0.150826\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	3.26765			0.114461
$816$		0			0
$817$	−25.0336	+	43.3595i	−0.875816	+	1.51696i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	6.76685	−	11.7205i	0.236165	−	0.409049i	−0.723446	−	0.690381i	$-0.757443\pi$
$821$	6.76685	−	11.7205i	0.236165	−	0.409049i	0.959610	+	0.281332i	$0.0907762\pi$
$822$		0			0
$823$	3.30446	+	5.72349i	0.115186	+	0.199508i	0.917854	−	0.396918i	$-0.129920\pi$
$823$	3.30446	+	5.72349i	0.115186	+	0.199508i	−0.802668	+	0.596426i	$0.796587\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−1.34376	+	2.32745i	−0.0467269	+	0.0809334i	−0.888443	−	0.458987i	$-0.848212\pi$
$827$	−1.34376	+	2.32745i	−0.0467269	+	0.0809334i	0.841716	+	0.539921i	$0.181546\pi$
$828$		0			0
$829$	4.51040	+	7.81224i	0.156653	+	0.271330i	0.933660	−	0.358162i	$-0.116596\pi$
$829$	4.51040	+	7.81224i	0.156653	+	0.271330i	−0.777007	+	0.629492i	$0.783263\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−18.4783	+	32.0054i	−0.640236	+	1.10892i
$834$		0			0
$835$	−10.7893	−	18.6876i	−0.373378	−	0.646709i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−10.3381	−	17.9061i	−0.356910	−	0.618187i	0.630533	−	0.776163i	$-0.282837\pi$
$839$	−10.3381	−	17.9061i	−0.356910	−	0.618187i	−0.987443	+	0.157976i	$0.949503\pi$
$840$		0			0
$841$	49.6234			1.71115
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−0.354093			−0.0121812
$846$		0			0
$847$	19.3454	−	33.5072i	0.664715	−	1.15132i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−6.40460	+	29.8016i	−0.219547	+	1.02158i
$852$		0			0
$853$	−1.81087	−	3.13652i	−0.0620031	−	0.107393i	0.833358	−	0.552734i	$-0.186416\pi$
$853$	−1.81087	−	3.13652i	−0.0620031	−	0.107393i	−0.895361	+	0.445342i	$0.853082\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−12.6201			−0.431095			−0.215548	−	0.976493i	$-0.569154\pi$
$857$	−12.6201			−0.431095			−0.215548	+	0.976493i	$0.569154\pi$
$858$		0			0
$859$	−45.6059			−1.55605			−0.778027	−	0.628231i	$-0.783779\pi$
$859$	−45.6059			−1.55605			−0.778027	+	0.628231i	$0.783779\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	25.9015	+	44.8627i	0.881697	+	1.52714i	0.849454	+	0.527663i	$0.176932\pi$
$863$	25.9015	+	44.8627i	0.881697	+	1.52714i	0.0322430	+	0.999480i	$0.489735\pi$
$864$		0			0
$865$	11.5993			0.394389
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−1.66833	+	2.88962i	−0.0565941	+	0.0980238i
$870$		0			0
$871$	−14.1675	−	24.5389i	−0.480048	−	0.831468i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	18.6995	−	32.3884i	0.632157	−	1.09493i
$876$		0			0
$877$	17.7356			0.598888			0.299444	−	0.954114i	$-0.403199\pi$
$877$	17.7356			0.598888			0.299444	+	0.954114i	$0.403199\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	4.36698	−	7.56383i	0.147127	−	0.254832i	−0.783037	−	0.621975i	$-0.786331\pi$
$881$	4.36698	−	7.56383i	0.147127	−	0.254832i	0.930165	+	0.367143i	$0.119664\pi$
$882$		0			0
$883$	26.6170	−	46.1019i	0.895732	−	1.55145i	0.0628353	−	0.998024i	$-0.479986\pi$
$883$	26.6170	−	46.1019i	0.895732	−	1.55145i	0.832896	−	0.553429i	$-0.186681\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	20.6363			0.692900			0.346450	−	0.938069i	$-0.387387\pi$
$887$	20.6363			0.692900			0.346450	+	0.938069i	$0.387387\pi$
$888$		0			0
$889$	−16.8479			−0.565061
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−21.1779	+	36.6811i	−0.708690	+	1.22749i
$894$		0			0
$895$	−0.599328	+	1.03807i	−0.0200333	+	0.0346987i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−11.5241			−0.384350
$900$		0			0
$901$	18.2764	−	31.6556i	0.608874	−	1.05460i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	1.65544	+	2.86730i	0.0550286	+	0.0953123i
$906$		0			0
$907$	−15.0962	+	26.1474i	−0.501261	+	0.868210i	0.498737	+	0.866753i	$0.333797\pi$
$907$	−15.0962	+	26.1474i	−0.501261	+	0.868210i	−0.999999	+	0.00145724i	$0.999536\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	5.76100			0.190870			0.0954352	−	0.995436i	$-0.469576\pi$
$911$	5.76100			0.190870			0.0954352	+	0.995436i	$0.469576\pi$
$912$		0			0
$913$	6.47752	+	11.2194i	0.214375	+	0.371308i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−46.4023			−1.53234
$918$		0			0
$919$	−19.6105			−0.646892			−0.323446	−	0.946247i	$-0.604841\pi$
$919$	−19.6105			−0.646892			−0.323446	+	0.946247i	$0.604841\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−20.0345	−	34.7008i	−0.659444	−	1.14219i
$924$		0			0
$925$	23.1571	−	7.46665i	0.761399	−	0.245502i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−26.5016	+	45.9021i	−0.869490	+	1.50600i	−0.00697095	+	0.999976i	$0.502219\pi$
$929$	−26.5016	+	45.9021i	−0.869490	+	1.50600i	−0.862519	+	0.506025i	$0.831114\pi$
$930$		0			0
$931$	−49.8574			−1.63401
$932$		0			0
$933$		0			0
$934$		0			0
$935$	4.67792			0.152984
$936$		0			0
$937$	−6.38940	−	11.0668i	−0.208733	−	0.361535i	0.742583	−	0.669754i	$-0.233600\pi$
$937$	−6.38940	−	11.0668i	−0.208733	−	0.361535i	−0.951316	+	0.308219i	$0.900267\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	11.4360	+	19.8077i	0.372802	+	0.645712i	0.989995	−	0.141099i	$-0.0450636\pi$
$941$	11.4360	+	19.8077i	0.372802	+	0.645712i	−0.617193	+	0.786812i	$0.711730\pi$
$942$		0			0
$943$	−26.8389	+	46.4863i	−0.873995	+	1.51380i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−4.34537	−	7.52640i	−0.141206	−	0.244575i	0.786745	−	0.617278i	$-0.211765\pi$
$947$	−4.34537	−	7.52640i	−0.141206	−	0.244575i	−0.927951	+	0.372703i	$0.878431\pi$
$948$		0			0
$949$	−4.06900	+	7.04771i	−0.132085	+	0.228778i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	20.7460	+	35.9332i	0.672030	+	1.16399i	0.977327	+	0.211733i	$0.0679109\pi$
$953$	20.7460	+	35.9332i	0.672030	+	1.16399i	−0.305297	+	0.952257i	$0.598756\pi$
$954$		0			0
$955$	2.79406	−	4.83945i	0.0904136	−	0.156601i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−2.40067	+	4.15809i	−0.0775217	+	0.134272i
$960$		0			0
$961$	−29.3109			−0.945512
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−1.27805	+	2.21365i	−0.0411420	+	0.0712601i
$966$		0			0
$967$	12.8117	−	22.1905i	0.411996	−	0.713598i	−0.583112	−	0.812392i	$-0.698165\pi$
$967$	12.8117	−	22.1905i	0.411996	−	0.713598i	0.995108	+	0.0987939i	$0.0314985\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	5.48232	+	9.49565i	0.175936	+	0.304730i	0.940485	−	0.339836i	$-0.110372\pi$
$971$	5.48232	+	9.49565i	0.175936	+	0.304730i	−0.764549	+	0.644566i	$0.777038\pi$
$972$		0			0
$973$	−72.5336			−2.32532
$974$		0			0
$975$		0			0
$976$		0			0
$977$	11.4672	+	19.8617i	0.366868	+	0.635433i	0.989074	−	0.147420i	$-0.0470969\pi$
$977$	11.4672	+	19.8617i	0.366868	+	0.635433i	−0.622206	+	0.782853i	$0.713764\pi$
$978$		0			0
$979$	−2.87427	−	4.97837i	−0.0918619	−	0.159110i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	17.1731	+	29.7446i	0.547736	+	0.948706i	0.998429	+	0.0560271i	$0.0178433\pi$
$983$	17.1731	+	29.7446i	0.547736	+	0.948706i	−0.450694	+	0.892679i	$0.648823\pi$
$984$		0			0
$985$	13.5353			0.431271
$986$		0			0
$987$		0			0
$988$		0			0
$989$	51.6699			1.64301
$990$		0			0
$991$	59.4263			1.88774			0.943870	−	0.330318i	$-0.107156\pi$
$991$	59.4263			1.88774			0.943870	+	0.330318i	$0.107156\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−0.844563	+	1.46283i	−0.0267745	+	0.0463747i
$996$		0			0
$997$	−7.86449	−	13.6217i	−0.249071	−	0.431404i	0.714197	−	0.699944i	$-0.246792\pi$
$997$	−7.86449	−	13.6217i	−0.249071	−	0.431404i	−0.963268	+	0.268541i	$0.913459\pi$
$998$		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.433.1		6
3.2	odd	2		148.2.e.a.137.1	yes	6
12.11	even	2		592.2.i.f.433.3		6
37.10	even	3	inner	1332.2.j.e.1009.1		6
111.11	odd	6		5476.2.a.g.1.3		3
111.26	odd	6		5476.2.a.f.1.3		3
111.47	odd	6		148.2.e.a.121.1	✓	6
444.47	even	6		592.2.i.f.417.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
148.2.e.a.121.1	✓	6	111.47	odd	6
148.2.e.a.137.1	yes	6	3.2	odd	2
592.2.i.f.417.3		6	444.47	even	6
592.2.i.f.433.3		6	12.11	even	2
1332.2.j.e.433.1		6	1.1	even	1	trivial
1332.2.j.e.1009.1		6	37.10	even	3	inner
5476.2.a.f.1.3		3	111.26	odd	6
5476.2.a.g.1.3		3	111.11	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)

\(f(q)\)	\(=\)	\(q+(-0.500000 + 0.866025i) q^{5} +(-2.07772 + 3.59871i) q^{7} +1.29966 q^{11} +(-1.77805 + 3.07968i) q^{13} +(-1.79966 - 3.11711i) q^{17} +(2.42789 - 4.20522i) q^{19} -5.01121 q^{23} +(2.00000 + 3.46410i) q^{25} -8.86698 q^{29} +1.29966 q^{31} +(-2.07772 - 3.59871i) q^{35} +(1.27805 - 5.94698i) q^{37} +(5.35577 - 9.27647i) q^{41} -10.3109 q^{43} -8.72275 q^{47} +(-5.13383 - 8.89205i) q^{49} +(5.07772 + 8.79487i) q^{53} +(-0.649832 + 1.12554i) q^{55} +(-4.63383 - 8.02602i) q^{59} +(-1.35577 + 2.34827i) q^{61} +(-1.77805 - 3.07968i) q^{65} +(-3.98399 + 6.90048i) q^{67} +(-5.63383 + 9.75807i) q^{71} +2.28845 q^{73} +(-2.70034 + 4.67712i) q^{77} +(-1.28366 + 2.22336i) q^{79} +(4.98399 + 8.63253i) q^{83} +3.59933 q^{85} +(-2.21155 - 3.83051i) q^{89} +(-7.38859 - 12.7974i) q^{91} +(2.42789 + 4.20522i) q^{95} +9.97920 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 3 q^{5} - q^{7} - 7 q^{13} - 3 q^{17} + 7 q^{19} + 8 q^{23} + 12 q^{25} - q^{35} + 4 q^{37} + 17 q^{41} - 16 q^{43} + 16 q^{47} - 12 q^{49} + 19 q^{53} - 9 q^{59} + 7 q^{61} - 7 q^{65} - 9 q^{67}+ \cdots - 8 q^{97}+O(q^{100}) \) 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

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more