LMFDB - Embedded newform 1620.3.o.e.701.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1620,3,Mod(701,1620)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1620, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 5, 0]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1620.701");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1620 = 2^{2} \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1620.o (of order $6$, degree $2$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$44.1418028264$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	8.0.12960000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{6} + 8x^{4} - 3x^{2} + 1 $ x^8 - 3x^6 + 8x^4 - 3*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{4} $
Twist minimal:	no (minimal twist has level 540)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			701.1
Root			$1.40126 - 0.809017i$ of defining polynomial
Character	$\chi$	$=$	1620.701
Dual form			1620.3.o.e.1241.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.93649 - 1.11803i) q^{5} +(-5.85410 - 10.1396i) q^{7} +O(q^{10})$	$q+(-1.93649 - 1.11803i) q^{5} +(-5.85410 - 10.1396i) q^{7} +(-8.40755 + 4.85410i) q^{11} +(0.145898 - 0.252703i) q^{13} -15.0000i q^{17} +12.4164 q^{19} +(-29.8055 - 17.2082i) q^{23} +(2.50000 + 4.33013i) q^{25} +(-3.21140 + 1.85410i) q^{29} +(-3.20820 + 5.55677i) q^{31} +26.1803i q^{35} +24.2492 q^{37} +(21.2531 + 12.2705i) q^{41} +(-37.9787 - 65.7811i) q^{43} +(-22.0113 + 12.7082i) q^{47} +(-44.0410 + 76.2813i) q^{49} +64.0820i q^{53} +21.7082 q^{55} +(61.5957 + 35.5623i) q^{59} +(-39.7492 - 68.8477i) q^{61} +(-0.565061 + 0.326238i) q^{65} +(-7.00000 + 12.1244i) q^{67} +91.9574i q^{71} +13.1246 q^{73} +(98.4373 + 56.8328i) q^{77} +(73.7492 + 127.737i) q^{79} +(-121.530 + 70.1656i) q^{83} +(-16.7705 + 29.0474i) q^{85} +105.039i q^{89} -3.41641 q^{91} +(-24.0443 - 13.8820i) q^{95} +(-7.70820 - 13.3510i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 20 q^{7}+O(q^{10}) $ 8 * q - 20 * q^7	$ 8 q - 20 q^{7} + 28 q^{13} - 8 q^{19} + 20 q^{25} + 28 q^{31} - 128 q^{37} - 116 q^{43} - 84 q^{49} + 120 q^{55} + 4 q^{61} - 56 q^{67} - 56 q^{73} + 268 q^{79} + 80 q^{91} - 8 q^{97}+O(q^{100}) $ 8 * q - 20 * q^7 + 28 * q^13 - 8 * q^19 + 20 * q^25 + 28 * q^31 - 128 * q^37 - 116 * q^43 - 84 * q^49 + 120 * q^55 + 4 * q^61 - 56 * q^67 - 56 * q^73 + 268 * q^79 + 80 * q^91 - 8 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1620\mathbb{Z}\right)^\times$.

$n$	$811$	$1297$	$1541$
$\chi(n)$	$1$	$1$	$e\left(\frac{5}{6}\right)$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−1.93649	−	1.11803i	−0.387298	−	0.223607i
$5$	−1.93649	−	1.11803i	−0.387298	−	0.223607i
$6$		0			0
$7$	−5.85410	−	10.1396i	−0.836300	−	1.44851i	−0.892967	−	0.450121i	$-0.851381\pi$
$7$	−5.85410	−	10.1396i	−0.836300	−	1.44851i	0.0566671	−	0.998393i	$-0.481953\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−8.40755	+	4.85410i	−0.764323	+	0.441282i	−0.830846	−	0.556503i	$-0.812143\pi$
$11$	−8.40755	+	4.85410i	−0.764323	+	0.441282i	0.0665228	+	0.997785i	$0.478809\pi$
$12$		0			0
$13$	0.145898	−	0.252703i	0.0112229	−	0.0194387i	−0.860359	−	0.509688i	$-0.829761\pi$
$13$	0.145898	−	0.252703i	0.0112229	−	0.0194387i	0.871582	+	0.490249i	$0.163094\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	15.0000i		−	0.882353i	−0.897420	−	0.441176i	$-0.854561\pi$
$17$		−	15.0000i		−	0.882353i	0.897420	−	0.441176i	$-0.145439\pi$
$18$		0			0
$19$	12.4164			0.653495			0.326748	−	0.945112i	$-0.394047\pi$
$19$	12.4164			0.653495			0.326748	+	0.945112i	$0.394047\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−29.8055	−	17.2082i	−1.29589	−	0.748183i	−0.316199	−	0.948693i	$-0.602407\pi$
$23$	−29.8055	−	17.2082i	−1.29589	−	0.748183i	−0.979692	+	0.200510i	$0.935740\pi$
$24$		0			0
$25$	2.50000	+	4.33013i	0.100000	+	0.173205i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−3.21140	+	1.85410i	−0.110738	+	0.0639346i	−0.554346	−	0.832286i	$-0.687032\pi$
$29$	−3.21140	+	1.85410i	−0.110738	+	0.0639346i	0.443608	+	0.896221i	$0.353698\pi$
$30$		0			0
$31$	−3.20820	+	5.55677i	−0.103490	+	0.179251i	−0.913120	−	0.407690i	$-0.866334\pi$
$31$	−3.20820	+	5.55677i	−0.103490	+	0.179251i	0.809630	+	0.586941i	$0.199668\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$			26.1803i			0.748010i
$36$		0			0
$37$	24.2492			0.655384			0.327692	−	0.944785i	$-0.393729\pi$
$37$	24.2492			0.655384			0.327692	+	0.944785i	$0.393729\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	21.2531	+	12.2705i	0.518369	+	0.299281i	0.736267	−	0.676691i	$-0.236587\pi$
$41$	21.2531	+	12.2705i	0.518369	+	0.299281i	−0.217898	+	0.975972i	$0.569920\pi$
$42$		0			0
$43$	−37.9787	−	65.7811i	−0.883226	−	1.52979i	−0.847734	−	0.530422i	$-0.822033\pi$
$43$	−37.9787	−	65.7811i	−0.883226	−	1.52979i	−0.0354923	−	0.999370i	$-0.511300\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−22.0113	+	12.7082i	−0.468325	+	0.270387i	−0.715538	−	0.698574i	$-0.753818\pi$
$47$	−22.0113	+	12.7082i	−0.468325	+	0.270387i	0.247214	+	0.968961i	$0.420485\pi$
$48$		0			0
$49$	−44.0410	+	76.2813i	−0.898796	+	1.55676i
$50$		0			0
$51$		0			0
$52$		0			0
$53$			64.0820i			1.20910i	0.796569	+	0.604548i	$0.206646\pi$
$53$			64.0820i			1.20910i	−0.796569	+	0.604548i	$0.793354\pi$
$54$		0			0
$55$	21.7082			0.394695
$56$		0			0
$57$		0			0
$58$		0			0
$59$	61.5957	+	35.5623i	1.04400	+	0.602751i	0.920962	−	0.389652i	$-0.127405\pi$
$59$	61.5957	+	35.5623i	1.04400	+	0.602751i	0.123033	+	0.992403i	$0.460738\pi$
$60$		0			0
$61$	−39.7492	−	68.8477i	−0.651627	−	1.12865i	−0.982728	−	0.185055i	$-0.940754\pi$
$61$	−39.7492	−	68.8477i	−0.651627	−	1.12865i	0.331101	−	0.943595i	$-0.392580\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−0.565061	+	0.326238i	−0.00869324	+	0.00501904i
$66$		0			0
$67$	−7.00000	+	12.1244i	−0.104478	+	0.180961i	−0.913525	−	0.406783i	$-0.866650\pi$
$67$	−7.00000	+	12.1244i	−0.104478	+	0.180961i	0.809047	+	0.587744i	$0.199984\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			91.9574i			1.29518i	0.761991	+	0.647588i	$0.224222\pi$
$71$			91.9574i			1.29518i	−0.761991	+	0.647588i	$0.775778\pi$
$72$		0			0
$73$	13.1246			0.179789			0.0898946	−	0.995951i	$-0.471347\pi$
$73$	13.1246			0.179789			0.0898946	+	0.995951i	$0.471347\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	98.4373	+	56.8328i	1.27841	+	0.738089i
$78$		0			0
$79$	73.7492	+	127.737i	0.933534	+	1.61693i	0.777227	+	0.629221i	$0.216626\pi$
$79$	73.7492	+	127.737i	0.933534	+	1.61693i	0.156308	+	0.987708i	$0.450041\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−121.530	+	70.1656i	−1.46422	+	0.845369i	−0.999202	−	0.0399313i	$-0.987286\pi$
$83$	−121.530	+	70.1656i	−1.46422	+	0.845369i	−0.465020	+	0.885300i	$0.653953\pi$
$84$		0			0
$85$	−16.7705	+	29.0474i	−0.197300	+	0.341734i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			105.039i			1.18022i	0.807323	+	0.590109i	$0.200915\pi$
$89$			105.039i			1.18022i	−0.807323	+	0.590109i	$0.799085\pi$
$90$		0			0
$91$	−3.41641			−0.0375429
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−24.0443	−	13.8820i	−0.253098	−	0.146126i
$96$		0			0
$97$	−7.70820	−	13.3510i	−0.0794660	−	0.137639i	0.823554	−	0.567238i	$-0.191988\pi$
$97$	−7.70820	−	13.3510i	−0.0794660	−	0.137639i	−0.903020	+	0.429599i	$0.858655\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	145.492	−	84.0000i	1.44052	−	0.831683i	0.442634	−	0.896703i	$-0.354044\pi$
$101$	145.492	−	84.0000i	1.44052	−	0.831683i	0.997884	+	0.0650194i	$0.0207109\pi$
$102$		0			0
$103$	90.2492	−	156.316i	0.876206	−	1.51763i	0.0207336	−	0.999785i	$-0.493400\pi$
$103$	90.2492	−	156.316i	0.876206	−	1.51763i	0.855472	−	0.517848i	$-0.173267\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			118.997i			1.11212i	0.831142	+	0.556060i	$0.187688\pi$
$107$			118.997i			1.11212i	−0.831142	+	0.556060i	$0.812312\pi$
$108$		0			0
$109$	102.082			0.936532			0.468266	−	0.883587i	$-0.344879\pi$
$109$	102.082			0.936532			0.468266	+	0.883587i	$0.344879\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	137.264	+	79.2492i	1.21472	+	0.701321i	0.963784	−	0.266682i	$-0.0859275\pi$
$113$	137.264	+	79.2492i	1.21472	+	0.701321i	0.250938	+	0.968003i	$0.419261\pi$
$114$		0			0
$115$	38.4787	+	66.6471i	0.334598	+	0.579540i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−152.094	+	87.8115i	−1.27810	+	0.737912i
$120$		0			0
$121$	−13.3754	+	23.1669i	−0.110540	+	0.191462i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	11.1803i		−	0.0894427i
$126$		0			0
$127$	−136.748			−1.07675			−0.538377	−	0.842704i	$-0.680962\pi$
$127$	−136.748			−1.07675			−0.538377	+	0.842704i	$0.680962\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−205.103	−	118.416i	−1.56567	−	0.903942i	−0.996664	−	0.0816109i	$-0.973994\pi$
$131$	−205.103	−	118.416i	−1.56567	−	0.903942i	−0.569009	−	0.822331i	$-0.692673\pi$
$132$		0			0
$133$	−72.6869	−	125.897i	−0.546518	−	0.946597i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−217.804	+	125.749i	−1.58981	+	0.917878i	−0.596473	+	0.802633i	$0.703432\pi$
$137$	−217.804	+	125.749i	−1.58981	+	0.917878i	−0.993337	+	0.115245i	$0.963235\pi$
$138$		0			0
$139$	−8.75078	+	15.1568i	−0.0629552	+	0.109042i	−0.895785	−	0.444487i	$-0.853386\pi$
$139$	−8.75078	+	15.1568i	−0.0629552	+	0.109042i	0.832830	+	0.553529i	$0.186719\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			2.83282i			0.0198099i
$144$		0			0
$145$	8.29180			0.0571848
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−45.0702	−	26.0213i	−0.302484	−	0.174640i	0.341074	−	0.940036i	$-0.389209\pi$
$149$	−45.0702	−	26.0213i	−0.302484	−	0.174640i	−0.643558	+	0.765397i	$0.722543\pi$
$150$		0			0
$151$	85.4984	+	148.088i	0.566215	+	0.980713i	0.996936	+	0.0782276i	$0.0249261\pi$
$151$	85.4984	+	148.088i	0.566215	+	0.980713i	−0.430721	+	0.902485i	$0.641741\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	12.4253	−	7.17376i	0.0801634	−	0.0462823i
$156$		0			0
$157$	−14.8541	+	25.7281i	−0.0946121	+	0.163873i	−0.909447	−	0.415821i	$-0.863494\pi$
$157$	−14.8541	+	25.7281i	−0.0946121	+	0.163873i	0.814835	+	0.579694i	$0.196828\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			402.954i			2.50282i
$162$		0			0
$163$	−77.8297			−0.477483			−0.238741	−	0.971083i	$-0.576735\pi$
$163$	−77.8297			−0.477483			−0.238741	+	0.971083i	$0.576735\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	64.0148	+	36.9590i	0.383322	+	0.221311i	0.679263	−	0.733895i	$-0.262300\pi$
$167$	64.0148	+	36.9590i	0.383322	+	0.221311i	−0.295940	+	0.955206i	$0.595633\pi$
$168$		0			0
$169$	84.4574	+	146.285i	0.499748	+	0.865589i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−64.6623	+	37.3328i	−0.373771	+	0.215797i	−0.675105	−	0.737722i	$-0.735901\pi$
$173$	−64.6623	+	37.3328i	−0.373771	+	0.215797i	0.301334	+	0.953519i	$0.402568\pi$
$174$		0			0
$175$	29.2705	−	50.6980i	0.167260	−	0.289703i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	150.334i		−	0.839857i	−0.907557	−	0.419928i	$-0.862055\pi$
$179$		−	150.334i		−	0.839857i	0.907557	−	0.419928i	$-0.137945\pi$
$180$		0			0
$181$	−326.082			−1.80156			−0.900779	−	0.434278i	$-0.857004\pi$
$181$	−326.082			−1.80156			−0.900779	+	0.434278i	$0.857004\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−46.9584	−	27.1115i	−0.253829	−	0.146548i
$186$		0			0
$187$	72.8115	+	126.113i	0.389366	+	0.674403i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−43.1538	+	24.9149i	−0.225936	+	0.130444i	−0.608696	−	0.793404i	$-0.708307\pi$
$191$	−43.1538	+	24.9149i	−0.225936	+	0.130444i	0.382760	+	0.923848i	$0.374974\pi$
$192$		0			0
$193$	114.687	−	198.644i	0.594233	−	1.02924i	−0.399422	−	0.916767i	$-0.630789\pi$
$193$	114.687	−	198.644i	0.594233	−	1.02924i	0.993655	−	0.112474i	$-0.0358775\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	201.334i		−	1.02200i	−0.859580	−	0.511001i	$-0.829275\pi$
$197$		−	201.334i		−	1.02200i	0.859580	−	0.511001i	$-0.170725\pi$
$198$		0			0
$199$	−214.413			−1.07745			−0.538727	−	0.842480i	$-0.681095\pi$
$199$	−214.413			−1.07745			−0.538727	+	0.842480i	$0.681095\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	37.5997	+	21.7082i	0.185220	+	0.106937i
$204$		0			0
$205$	−27.4377	−	47.5235i	−0.133842	−	0.231822i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−104.392	+	60.2705i	−0.499481	+	0.288376i
$210$		0			0
$211$	47.8313	−	82.8462i	0.226688	−	0.392636i	−0.730136	−	0.683302i	$-0.760543\pi$
$211$	47.8313	−	82.8462i	0.226688	−	0.392636i	0.956825	+	0.290666i	$0.0938768\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			169.846i			0.789981i
$216$		0			0
$217$	75.1246			0.346196
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−3.79054	−	2.18847i	−0.0171518	−	0.00990258i
$222$		0			0
$223$	33.7902	+	58.5264i	0.151526	+	0.262450i	0.931789	−	0.363002i	$-0.118248\pi$
$223$	33.7902	+	58.5264i	0.151526	+	0.262450i	−0.780263	+	0.625452i	$0.784915\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−197.957	+	114.290i	−0.872055	+	0.503481i	−0.868031	−	0.496511i	$-0.834614\pi$
$227$	−197.957	+	114.290i	−0.872055	+	0.503481i	−0.00402447	+	0.999992i	$0.501281\pi$
$228$		0			0
$229$	30.6672	−	53.1171i	0.133918	−	0.231952i	−0.791266	−	0.611472i	$-0.790578\pi$
$229$	30.6672	−	53.1171i	0.133918	−	0.231952i	0.925184	+	0.379520i	$0.123911\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			254.912i			1.09404i	0.837119	+	0.547021i	$0.184238\pi$
$233$			254.912i			1.09404i	−0.837119	+	0.547021i	$0.815762\pi$
$234$		0			0
$235$	56.8328			0.241842
$236$		0			0
$237$		0			0
$238$		0			0
$239$	220.981	+	127.584i	0.924608	+	0.533823i	0.885102	−	0.465397i	$-0.154088\pi$
$239$	220.981	+	127.584i	0.924608	+	0.533823i	0.0395056	+	0.999219i	$0.487422\pi$
$240$		0			0
$241$	−85.4574	−	148.017i	−0.354595	−	0.614177i	0.632453	−	0.774598i	$-0.282048\pi$
$241$	−85.4574	−	148.017i	−0.354595	−	0.614177i	−0.987049	+	0.160422i	$0.948715\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	170.570	−	98.4787i	0.696205	−	0.401954i
$246$		0			0
$247$	1.81153	−	3.13766i	0.00733413	−	0.0127031i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			298.918i			1.19091i	0.803389	+	0.595454i	$0.203028\pi$
$251$			298.918i			1.19091i	−0.803389	+	0.595454i	$0.796972\pi$
$252$		0			0
$253$	334.122			1.32064
$254$		0			0
$255$		0			0
$256$		0			0
$257$	327.571	+	189.123i	1.27459	+	0.735887i	0.975849	−	0.218446i	$-0.0700986\pi$
$257$	327.571	+	189.123i	1.27459	+	0.735887i	0.298745	+	0.954333i	$0.403432\pi$
$258$		0			0
$259$	−141.957	−	245.877i	−0.548098	−	0.949334i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	397.429	−	229.456i	1.51114	−	0.872456i	0.511222	−	0.859448i	$-0.329193\pi$
$263$	397.429	−	229.456i	1.51114	−	0.872456i	0.999915	−	0.0130073i	$-0.00414047\pi$
$264$		0			0
$265$	71.6459	−	124.094i	0.270362	−	0.468281i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			270.413i			1.00525i	0.864503	+	0.502627i	$0.167633\pi$
$269$			270.413i			1.00525i	−0.864503	+	0.502627i	$0.832367\pi$
$270$		0			0
$271$	194.246			0.716775			0.358388	−	0.933573i	$-0.383327\pi$
$271$	194.246			0.716775			0.358388	+	0.933573i	$0.383327\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−42.0378	−	24.2705i	−0.152865	−	0.0882564i
$276$		0			0
$277$	−20.5197	−	35.5412i	−0.0740785	−	0.128308i	0.826607	−	0.562780i	$-0.190268\pi$
$277$	−20.5197	−	35.5412i	−0.0740785	−	0.128308i	−0.900685	+	0.434472i	$0.856935\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−128.277	+	74.0608i	−0.456502	+	0.263561i	−0.710572	−	0.703624i	$-0.751564\pi$
$281$	−128.277	+	74.0608i	−0.456502	+	0.263561i	0.254070	+	0.967186i	$0.418230\pi$
$282$		0			0
$283$	99.4803	−	172.305i	0.351520	−	0.608851i	−0.634996	−	0.772516i	$-0.718998\pi$
$283$	99.4803	−	172.305i	0.351520	−	0.608851i	0.986516	+	0.163665i	$0.0523315\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	287.331i		−	1.00115i
$288$		0			0
$289$	64.0000			0.221453
$290$		0			0
$291$		0			0
$292$		0			0
$293$	42.3615	+	24.4574i	0.144579	+	0.0834724i	0.570544	−	0.821267i	$-0.306732\pi$
$293$	42.3615	+	24.4574i	0.144579	+	0.0834724i	−0.425966	+	0.904739i	$0.640066\pi$
$294$		0			0
$295$	−79.5197	−	137.732i	−0.269558	−	0.466889i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−8.69712	+	5.02129i	−0.0290874	+	0.0167936i
$300$		0			0
$301$	−444.663	+	770.178i	−1.47728	+	2.55873i
$302$		0			0
$303$		0			0
$304$		0			0
$305$			177.764i			0.582833i
$306$		0			0
$307$	−330.498			−1.07654			−0.538271	−	0.842772i	$-0.680922\pi$
$307$	−330.498			−1.07654			−0.538271	+	0.842772i	$0.680922\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−257.533	−	148.687i	−0.828081	−	0.478093i	0.0251139	−	0.999685i	$-0.492005\pi$
$311$	−257.533	−	148.687i	−0.828081	−	0.478093i	−0.853195	+	0.521592i	$0.825338\pi$
$312$		0			0
$313$	69.4164	+	120.233i	0.221778	+	0.384130i	0.955348	−	0.295484i	$-0.0954807\pi$
$313$	69.4164	+	120.233i	0.221778	+	0.384130i	−0.733570	+	0.679614i	$0.762147\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−89.3481	+	51.5851i	−0.281855	+	0.162729i	−0.634263	−	0.773117i	$-0.718696\pi$
$317$	−89.3481	+	51.5851i	−0.281855	+	0.162729i	0.352408	+	0.935847i	$0.385363\pi$
$318$		0			0
$319$	18.0000	−	31.1769i	0.0564263	−	0.0977333i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	186.246i		−	0.576613i
$324$		0			0
$325$	1.45898			0.00448917
$326$		0			0
$327$		0			0
$328$		0			0
$329$	257.712	+	148.790i	0.783320	+	0.452250i
$330$		0			0
$331$	278.374	+	482.158i	0.841009	+	1.45667i	0.889043	+	0.457824i	$0.151371\pi$
$331$	278.374	+	482.158i	0.841009	+	1.45667i	−0.0480343	+	0.998846i	$0.515296\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	27.1109	−	15.6525i	0.0809280	−	0.0467238i
$336$		0			0
$337$	−79.2705	+	137.301i	−0.235224	+	0.407420i	−0.959338	−	0.282261i	$-0.908916\pi$
$337$	−79.2705	+	137.301i	−0.235224	+	0.407420i	0.724114	+	0.689681i	$0.242249\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	62.2918i		−	0.182674i
$342$		0			0
$343$	457.580			1.33405
$344$		0			0
$345$		0			0
$346$		0			0
$347$	39.9846	+	23.0851i	0.115230	+	0.0665278i	0.556507	−	0.830843i	$-0.312141\pi$
$347$	39.9846	+	23.0851i	0.115230	+	0.0665278i	−0.441278	+	0.897371i	$0.645475\pi$
$348$		0			0
$349$	−232.871	−	403.344i	−0.667251	−	1.15571i	−0.978670	−	0.205440i	$-0.934137\pi$
$349$	−232.871	−	403.344i	−0.667251	−	1.15571i	0.311418	−	0.950273i	$-0.399196\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−195.290	+	112.751i	−0.553230	+	0.319407i	−0.750424	−	0.660957i	$-0.770150\pi$
$353$	−195.290	+	112.751i	−0.553230	+	0.319407i	0.197194	+	0.980365i	$0.436817\pi$
$354$		0			0
$355$	102.812	−	178.075i	0.289610	−	0.501619i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	620.991i		−	1.72978i	−0.501962	−	0.864890i	$-0.667388\pi$
$359$		−	620.991i		−	1.72978i	0.501962	−	0.864890i	$-0.332612\pi$
$360$		0			0
$361$	−206.833			−0.572944
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−25.4157	−	14.6738i	−0.0696321	−	0.0402021i
$366$		0			0
$367$	−107.185	−	185.650i	−0.292058	−	0.505860i	0.682238	−	0.731130i	$-0.261007\pi$
$367$	−107.185	−	185.650i	−0.292058	−	0.505860i	−0.974296	+	0.225270i	$0.927673\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	649.766	−	375.143i	1.75139	−	1.01117i
$372$		0			0
$373$	−136.079	+	235.696i	−0.364823	+	0.631892i	−0.988748	−	0.149593i	$-0.952204\pi$
$373$	−136.079	+	235.696i	−0.364823	+	0.631892i	0.623925	+	0.781484i	$0.285537\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			1.08204i			0.00287013i
$378$		0			0
$379$	−268.836			−0.709330			−0.354665	−	0.934994i	$-0.615405\pi$
$379$	−268.836			−0.709330			−0.354665	+	0.934994i	$0.615405\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−514.990	−	297.330i	−1.34462	−	0.776318i	−0.357140	−	0.934051i	$-0.616248\pi$
$383$	−514.990	−	297.330i	−1.34462	−	0.776318i	−0.987482	+	0.157733i	$0.949581\pi$
$384$		0			0
$385$	−127.082	−	220.113i	−0.330083	−	0.571721i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	59.4320	−	34.3131i	0.152781	−	0.0882084i	−0.421660	−	0.906754i	$-0.638553\pi$
$389$	59.4320	−	34.3131i	0.152781	−	0.0882084i	0.574442	+	0.818545i	$0.305219\pi$
$390$		0			0
$391$	−258.123	+	447.082i	−0.660161	+	1.14343i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		−	329.817i		−	0.834979i
$396$		0			0
$397$	646.529			1.62854			0.814268	−	0.580489i	$-0.197139\pi$
$397$	646.529			1.62854			0.814268	+	0.580489i	$0.197139\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−331.327	−	191.292i	−0.826252	−	0.477037i	0.0263156	−	0.999654i	$-0.491623\pi$
$401$	−331.327	−	191.292i	−0.826252	−	0.477037i	−0.852568	+	0.522617i	$0.824956\pi$
$402$		0			0
$403$	0.936141	+	1.62144i	0.00232293	+	0.00402343i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−203.877	+	117.708i	−0.500925	+	0.289209i
$408$		0			0
$409$	71.6641	−	124.126i	0.175218	−	0.303486i	−0.765019	−	0.644008i	$-0.777270\pi$
$409$	71.6641	−	124.126i	0.175218	−	0.303486i	0.940237	+	0.340522i	$0.110604\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	832.741i		−	2.01632i
$414$		0			0
$415$	313.790			0.756121
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−508.312	−	293.474i	−1.21316	−	0.700415i	−0.249710	−	0.968321i	$-0.580335\pi$
$419$	−508.312	−	293.474i	−1.21316	−	0.700415i	−0.963445	+	0.267905i	$0.913669\pi$
$420$		0			0
$421$	76.8769	+	133.155i	0.182606	+	0.316282i	0.942767	−	0.333452i	$-0.108214\pi$
$421$	76.8769	+	133.155i	0.182606	+	0.316282i	−0.760161	+	0.649734i	$0.774880\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	64.9519	−	37.5000i	0.152828	−	0.0882353i
$426$		0			0
$427$	−465.392	+	806.083i	−1.08991	+	1.88778i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	471.866i		−	1.09482i	−0.836866	−	0.547408i	$-0.815614\pi$
$431$		−	471.866i		−	1.09482i	0.836866	−	0.547408i	$-0.184386\pi$
$432$		0			0
$433$	657.957			1.51953			0.759766	−	0.650197i	$-0.225313\pi$
$433$	657.957			1.51953			0.759766	+	0.650197i	$0.225313\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−370.077	−	213.664i	−0.846858	−	0.488934i
$438$		0			0
$439$	−387.995	−	672.028i	−0.883816	−	1.53081i	−0.847064	−	0.531490i	$-0.821632\pi$
$439$	−387.995	−	672.028i	−0.883816	−	1.53081i	−0.0367519	−	0.999324i	$-0.511701\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−419.654	+	242.287i	−0.947299	+	0.546924i	−0.892241	−	0.451560i	$-0.850868\pi$
$443$	−419.654	+	242.287i	−0.947299	+	0.546924i	−0.0550584	+	0.998483i	$0.517535\pi$
$444$		0			0
$445$	117.438	−	203.408i	0.263905	−	0.457097i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			287.125i			0.639476i	0.947506	+	0.319738i	$0.103595\pi$
$449$			287.125i			0.639476i	−0.947506	+	0.319738i	$0.896405\pi$
$450$		0			0
$451$	−238.249			−0.528269
$452$		0			0
$453$		0			0
$454$		0			0
$455$	6.61585	+	3.81966i	0.0145403	+	0.00839486i
$456$		0			0
$457$	344.620	+	596.899i	0.754092	+	1.30613i	0.945825	+	0.324678i	$0.105256\pi$
$457$	344.620	+	596.899i	0.754092	+	1.30613i	−0.191733	+	0.981447i	$0.561411\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	631.682	−	364.702i	1.37024	−	0.791111i	0.379285	−	0.925280i	$-0.376170\pi$
$461$	631.682	−	364.702i	1.37024	−	0.791111i	0.990958	+	0.134169i	$0.0428365\pi$
$462$		0			0
$463$	248.502	−	430.417i	0.536720	−	0.929627i	−0.462358	−	0.886694i	$-0.652996\pi$
$463$	248.502	−	430.417i	0.536720	−	0.929627i	0.999078	−	0.0429334i	$-0.0136703\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	477.906i		−	1.02335i	−0.859178	−	0.511676i	$-0.829025\pi$
$467$		−	477.906i		−	1.02335i	0.859178	−	0.511676i	$-0.170975\pi$
$468$		0			0
$469$	163.915			0.349499
$470$		0			0
$471$		0			0
$472$		0			0
$473$	638.616	+	368.705i	1.35014	+	0.779503i
$474$		0			0
$475$	31.0410	+	53.7646i	0.0653495	+	0.113189i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	48.5973	−	28.0576i	0.101456	−	0.0585755i	−0.448414	−	0.893826i	$-0.648011\pi$
$479$	48.5973	−	28.0576i	0.101456	−	0.0585755i	0.549869	+	0.835251i	$0.314678\pi$
$480$		0			0
$481$	3.53791	−	6.12785i	0.00735533	−	0.0127398i
$482$		0			0
$483$		0			0
$484$		0			0
$485$			34.4721i			0.0710766i
$486$		0			0
$487$	−535.787			−1.10018			−0.550089	−	0.835106i	$-0.685406\pi$
$487$	−535.787			−1.10018			−0.550089	+	0.835106i	$0.685406\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−484.674	−	279.827i	−0.987116	−	0.569912i	−0.0827049	−	0.996574i	$-0.526356\pi$
$491$	−484.674	−	279.827i	−0.987116	−	0.569912i	−0.904411	+	0.426662i	$0.859689\pi$
$492$		0			0
$493$	27.8115	+	48.1710i	0.0564128	+	0.0977099i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	932.412	−	538.328i	1.87608	−	1.08316i
$498$		0			0
$499$	92.6246	−	160.431i	0.185620	−	0.321504i	−0.758165	−	0.652063i	$-0.773904\pi$
$499$	92.6246	−	160.431i	0.185620	−	0.321504i	0.943785	+	0.330559i	$0.107237\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			234.088i			0.465384i	0.972550	+	0.232692i	$0.0747534\pi$
$503$			234.088i			0.465384i	−0.972550	+	0.232692i	$0.925247\pi$
$504$		0			0
$505$	−375.659			−0.743880
$506$		0			0
$507$		0			0
$508$		0			0
$509$	227.251	+	131.204i	0.446466	+	0.257767i	0.706337	−	0.707876i	$-0.250347\pi$
$509$	227.251	+	131.204i	0.446466	+	0.257767i	−0.259871	+	0.965643i	$0.583680\pi$
$510$		0			0
$511$	−76.8328	−	133.078i	−0.150358	−	0.260427i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−349.534	+	201.803i	−0.678706	+	0.391851i
$516$		0			0
$517$	123.374	−	213.690i	0.238634	−	0.413326i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	328.200i		−	0.629943i	−0.949101	−	0.314972i	$-0.898005\pi$
$521$		−	328.200i		−	0.629943i	0.949101	−	0.314972i	$-0.101995\pi$
$522$		0			0
$523$	289.872			0.554249			0.277125	−	0.960834i	$-0.410619\pi$
$523$	289.872			0.554249			0.277125	+	0.960834i	$0.410619\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	83.3516	+	48.1231i	0.158162	+	0.0913151i
$528$		0			0
$529$	327.745	+	567.670i	0.619555	+	1.07310i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	6.20158	−	3.58049i	0.0116352	−	0.00671761i
$534$		0			0
$535$	133.043	−	230.436i	0.248678	−	0.430722i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	855.118i		−	1.58649i
$540$		0			0
$541$	−689.161			−1.27387			−0.636933	−	0.770920i	$-0.719797\pi$
$541$	−689.161			−1.27387			−0.636933	+	0.770920i	$0.719797\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−197.681	−	114.131i	−0.362717	−	0.209415i
$546$		0			0
$547$	−151.310	−	262.077i	−0.276618	−	0.479116i	0.693924	−	0.720048i	$-0.255880\pi$
$547$	−151.310	−	262.077i	−0.276618	−	0.479116i	−0.970542	+	0.240932i	$0.922547\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−39.8740	+	23.0213i	−0.0723667	+	0.0417809i
$552$		0			0
$553$	863.471	−	1495.58i	1.56143	−	2.70448i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			171.836i			0.308503i	0.988032	+	0.154251i	$0.0492965\pi$
$557$			171.836i			0.308503i	−0.988032	+	0.154251i	$0.950703\pi$
$558$		0			0
$559$	−22.1641			−0.0396495
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−299.350	−	172.830i	−0.531705	−	0.306980i	0.210006	−	0.977700i	$-0.432652\pi$
$563$	−299.350	−	172.830i	−0.531705	−	0.306980i	−0.741710	+	0.670720i	$0.765985\pi$
$564$		0			0
$565$	−177.207	−	306.931i	−0.313640	−	0.543241i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	889.147	−	513.349i	1.56265	−	0.902196i	0.565661	−	0.824638i	$-0.308621\pi$
$569$	889.147	−	513.349i	1.56265	−	0.902196i	0.996988	−	0.0775581i	$-0.0247123\pi$
$570$		0			0
$571$	−420.041	+	727.532i	−0.735624	+	1.27414i	0.218826	+	0.975764i	$0.429777\pi$
$571$	−420.041	+	727.532i	−0.735624	+	1.27414i	−0.954449	+	0.298373i	$0.903556\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	172.082i		−	0.299273i
$576$		0			0
$577$	−1031.86			−1.78832			−0.894159	−	0.447749i	$-0.852226\pi$
$577$	−1031.86			−1.78832			−0.894159	+	0.447749i	$0.852226\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	1422.90	+	821.514i	2.44906	+	1.41396i
$582$		0			0
$583$	−311.061	−	538.773i	−0.533552	−	0.924139i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	123.984	−	71.5820i	0.211216	−	0.121946i	−0.390660	−	0.920535i	$-0.627753\pi$
$587$	123.984	−	71.5820i	0.211216	−	0.121946i	0.601876	+	0.798589i	$0.294420\pi$
$588$		0			0
$589$	−39.8344	+	68.9952i	−0.0676305	+	0.117139i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	8.58670i		−	0.0144801i	−0.999974	−	0.00724005i	$-0.997695\pi$
$593$		−	8.58670i		−	0.0144801i	0.999974	−	0.00724005i	$-0.00230460\pi$
$594$		0			0
$595$	392.705			0.660009
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−660.875	−	381.556i	−1.10330	−	0.636988i	−0.166212	−	0.986090i	$-0.553153\pi$
$599$	−660.875	−	381.556i	−1.10330	−	0.636988i	−0.937085	+	0.349102i	$0.886487\pi$
$600$		0			0
$601$	345.953	+	599.208i	0.575629	+	0.997018i	0.995973	+	0.0896533i	$0.0285759\pi$
$601$	345.953	+	599.208i	0.575629	+	0.997018i	−0.420344	+	0.907365i	$0.638091\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	51.8027	−	29.9083i	0.0856242	−	0.0494352i
$606$		0			0
$607$	−102.690	+	177.864i	−0.169176	+	0.293022i	−0.938131	−	0.346282i	$-0.887444\pi$
$607$	−102.690	+	177.864i	−0.169176	+	0.293022i	0.768954	+	0.639304i	$0.220777\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			7.41641i			0.0121381i
$612$		0			0
$613$	27.2585			0.0444674			0.0222337	−	0.999753i	$-0.492922\pi$
$613$	27.2585			0.0444674			0.0222337	+	0.999753i	$0.492922\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	380.469	+	219.664i	0.616644	+	0.356020i	0.775561	−	0.631272i	$-0.217467\pi$
$617$	380.469	+	219.664i	0.616644	+	0.356020i	−0.158917	+	0.987292i	$0.550800\pi$
$618$		0			0
$619$	271.371	+	470.028i	0.438402	+	0.759334i	0.997566	−	0.0697224i	$-0.0222114\pi$
$619$	271.371	+	470.028i	0.438402	+	0.759334i	−0.559165	+	0.829057i	$0.688878\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	1065.06	−	614.912i	1.70956	−	0.987017i
$624$		0			0
$625$	−12.5000	+	21.6506i	−0.0200000	+	0.0346410i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	363.738i		−	0.578280i
$630$		0			0
$631$	232.155			0.367916			0.183958	−	0.982934i	$-0.441109\pi$
$631$	232.155			0.367916			0.183958	+	0.982934i	$0.441109\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	264.811	+	152.889i	0.417025	+	0.240769i
$636$		0			0
$637$	12.8510	+	22.2586i	0.0201742	+	0.0349428i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−659.264	+	380.626i	−1.02849	+	0.593801i	−0.916553	−	0.399914i	$-0.869040\pi$
$641$	−659.264	+	380.626i	−1.02849	+	0.593801i	−0.111940	+	0.993715i	$0.535707\pi$
$642$		0			0
$643$	365.812	−	633.604i	0.568914	−	0.985387i	−0.427760	−	0.903892i	$-0.640697\pi$
$643$	365.812	−	633.604i	0.568914	−	0.985387i	0.996674	−	0.0814951i	$-0.0259695\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	408.325i		−	0.631105i	−0.948908	−	0.315553i	$-0.897810\pi$
$647$		−	408.325i		−	0.631105i	0.948908	−	0.315553i	$-0.102190\pi$
$648$		0			0
$649$	−690.492			−1.06393
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−222.489	−	128.454i	−0.340719	−	0.196714i	0.319871	−	0.947461i	$-0.396360\pi$
$653$	−222.489	−	128.454i	−0.340719	−	0.196714i	−0.660590	+	0.750747i	$0.729694\pi$
$654$		0			0
$655$	264.787	+	458.625i	0.404255	+	0.700190i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−840.934	+	485.514i	−1.27608	+	0.736743i	−0.976125	−	0.217212i	$-0.930304\pi$
$659$	−840.934	+	485.514i	−1.27608	+	0.736743i	−0.299951	+	0.953954i	$0.596970\pi$
$660$		0			0
$661$	−314.751	+	545.164i	−0.476174	+	0.824757i	−0.999627	−	0.0272972i	$-0.991310\pi$
$661$	−314.751	+	545.164i	−0.476174	+	0.824757i	0.523454	+	0.852054i	$0.324643\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$			325.066i			0.488821i
$666$		0			0
$667$	127.623			0.191339
$668$		0			0
$669$		0			0
$670$		0			0
$671$	668.387	+	385.894i	0.996106	+	0.575102i
$672$		0			0
$673$	−144.316	−	249.963i	−0.214437	−	0.371416i	0.738661	−	0.674077i	$-0.235458\pi$
$673$	−144.316	−	249.963i	−0.214437	−	0.371416i	−0.953098	+	0.302661i	$0.902125\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	485.695	−	280.416i	0.717423	−	0.414204i	−0.0963804	−	0.995345i	$-0.530727\pi$
$677$	485.695	−	280.416i	0.717423	−	0.414204i	0.813804	+	0.581140i	$0.197393\pi$
$678$		0			0
$679$	−90.2492	+	156.316i	−0.132915	+	0.230215i
$680$		0			0
$681$		0			0
$682$		0			0
$683$			195.511i			0.286253i	0.989704	+	0.143127i	$0.0457156\pi$
$683$			195.511i			0.286253i	−0.989704	+	0.143127i	$0.954284\pi$
$684$		0			0
$685$	562.368			0.820975
$686$		0			0
$687$		0			0
$688$		0			0
$689$	16.1937	+	9.34944i	0.0235032	+	0.0135696i
$690$		0			0
$691$	164.251	+	284.491i	0.237700	+	0.411709i	0.960054	−	0.279815i	$-0.0902731\pi$
$691$	164.251	+	284.491i	0.237700	+	0.411709i	−0.722354	+	0.691524i	$0.756940\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	33.8916	−	19.5673i	0.0487649	−	0.0281544i
$696$		0			0
$697$	184.058	−	318.797i	0.264071	−	0.457385i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	580.505i		−	0.828109i	−0.910252	−	0.414055i	$-0.864112\pi$
$701$		−	580.505i		−	0.828109i	0.910252	−	0.414055i	$-0.135888\pi$
$702$		0			0
$703$	301.088			0.428291
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−1703.45	−	983.489i	−2.40941	−	1.39107i
$708$		0			0
$709$	475.164	+	823.008i	0.670189	+	1.16080i	0.977850	+	0.209306i	$0.0671203\pi$
$709$	475.164	+	823.008i	0.670189	+	1.16080i	−0.307661	+	0.951496i	$0.599546\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	191.244	−	110.415i	0.268225	−	0.154860i
$714$		0			0
$715$	3.16718	−	5.48572i	0.00442963	−	0.00767234i
$716$		0			0
$717$		0			0
$718$		0			0
$719$			578.912i			0.805162i	0.915384	+	0.402581i	$0.131887\pi$
$719$			578.912i			0.805162i	−0.915384	+	0.402581i	$0.868113\pi$
$720$		0			0
$721$	−2113.31			−2.93109
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−16.0570	−	9.27051i	−0.0221476	−	0.0127869i
$726$		0			0
$727$	551.502	+	955.229i	0.758599	+	1.31393i	0.943565	+	0.331187i	$0.107449\pi$
$727$	551.502	+	955.229i	0.758599	+	1.31393i	−0.184966	+	0.982745i	$0.559217\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−986.716	+	569.681i	−1.34982	+	0.779317i
$732$		0			0
$733$	567.751	−	983.373i	0.774558	−	1.34157i	−0.160485	−	0.987038i	$-0.551306\pi$
$733$	567.751	−	983.373i	0.774558	−	1.34157i	0.935043	−	0.354535i	$-0.115361\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	135.915i		−	0.184416i
$738$		0			0
$739$	−46.2461			−0.0625793			−0.0312897	−	0.999510i	$-0.509961\pi$
$739$	−46.2461			−0.0625793			−0.0312897	+	0.999510i	$0.509961\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−1175.92	−	678.915i	−1.58266	−	0.913748i	−0.994470	−	0.105024i	$-0.966508\pi$
$743$	−1175.92	−	678.915i	−1.58266	−	0.913748i	−0.588189	−	0.808724i	$-0.700159\pi$
$744$		0			0
$745$	58.1854	+	100.780i	0.0781012	+	0.135275i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	1206.58	−	696.620i	1.61092	−	0.930067i
$750$		0			0
$751$	174.245	−	301.800i	0.232017	−	0.401865i	−0.726385	−	0.687288i	$-0.758801\pi$
$751$	174.245	−	301.800i	0.232017	−	0.401865i	0.958401	+	0.285424i	$0.0921343\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	382.361i		−	0.506438i
$756$		0			0
$757$	80.7477			0.106668			0.0533340	−	0.998577i	$-0.483015\pi$
$757$	80.7477			0.106668			0.0533340	+	0.998577i	$0.483015\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−340.808	−	196.766i	−0.447843	−	0.258562i	0.259076	−	0.965857i	$-0.416582\pi$
$761$	−340.808	−	196.766i	−0.447843	−	0.258562i	−0.706919	+	0.707295i	$0.749915\pi$
$762$		0			0
$763$	−597.599	−	1035.07i	−0.783222	−	1.35658i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	17.9734	−	10.3769i	0.0234334	−	0.0135293i
$768$		0			0
$769$	−544.664	+	943.386i	−0.708276	+	1.22677i	0.257220	+	0.966353i	$0.417193\pi$
$769$	−544.664	+	943.386i	−0.708276	+	1.22677i	−0.965496	+	0.260417i	$0.916140\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	103.091i		−	0.133365i	−0.997774	−	0.0666826i	$-0.978758\pi$
$773$		−	103.091i		−	0.133365i	0.997774	−	0.0666826i	$-0.0212415\pi$
$774$		0			0
$775$	−32.0820			−0.0413962
$776$		0			0
$777$		0			0
$778$		0			0
$779$	263.888	+	152.356i	0.338752	+	0.195579i
$780$		0			0
$781$	−446.371	−	773.137i	−0.571537	−	0.989932i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	57.5297	−	33.2148i	0.0732862	−	0.0423118i
$786$		0			0
$787$	−31.4377	+	54.4517i	−0.0399462	+	0.0691889i	−0.885307	−	0.465006i	$-0.846052\pi$
$787$	−31.4377	+	54.4517i	−0.0399462	+	0.0691889i	0.845361	+	0.534195i	$0.179385\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	1855.73i		−	2.34606i
$792$		0			0
$793$	−23.1973			−0.0292526
$794$		0			0
$795$		0			0
$796$		0			0
$797$	453.131	+	261.615i	0.568546	+	0.328250i	0.756568	−	0.653915i	$-0.226874\pi$
$797$	453.131	+	261.615i	0.568546	+	0.328250i	−0.188023	+	0.982165i	$0.560208\pi$
$798$		0			0
$799$	190.623	+	330.169i	0.238577	+	0.413228i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−110.346	+	63.7082i	−0.137417	+	0.0793377i
$804$		0			0
$805$	450.517	−	780.318i	0.559648	−	0.969339i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			747.787i			0.924335i	0.886793	+	0.462168i	$0.152928\pi$
$809$			747.787i			0.924335i	−0.886793	+	0.462168i	$0.847072\pi$
$810$		0			0
$811$	74.1579			0.0914400			0.0457200	−	0.998954i	$-0.485442\pi$
$811$	74.1579			0.0914400			0.0457200	+	0.998954i	$0.485442\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	150.717	+	87.0163i	0.184928	+	0.106768i
$816$		0			0
$817$	−471.559	−	816.764i	−0.577184	−	0.999712i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−1208.39	+	697.663i	−1.47185	+	0.849772i	−0.999499	−	0.0316403i	$-0.989927\pi$
$821$	−1208.39	+	697.663i	−1.47185	+	0.849772i	−0.472348	+	0.881412i	$0.656594\pi$
$822$		0			0
$823$	338.398	−	586.123i	0.411176	−	0.712179i	−0.583842	−	0.811867i	$-0.698451\pi$
$823$	338.398	−	586.123i	0.411176	−	0.712179i	0.995019	+	0.0996886i	$0.0317847\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			1080.25i			1.30622i	0.757262	+	0.653111i	$0.226537\pi$
$827$			1080.25i			1.30622i	−0.757262	+	0.653111i	$0.773463\pi$
$828$		0			0
$829$	1549.23			1.86880			0.934399	−	0.356228i	$-0.115937\pi$
$829$	1549.23			1.86880			0.934399	+	0.356228i	$0.115937\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	1144.22	+	660.615i	1.37361	+	0.793056i
$834$		0			0
$835$	−82.6428	−	143.142i	−0.0989734	−	0.171427i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−583.843	+	337.082i	−0.695880	+	0.401766i	−0.805811	−	0.592173i	$-0.798270\pi$
$839$	−583.843	+	337.082i	−0.695880	+	0.401766i	0.109931	+	0.993939i	$0.464937\pi$
$840$		0			0
$841$	−413.625	+	716.419i	−0.491825	+	0.851865i
$842$		0			0
$843$		0			0
$844$		0			0
$845$		−	377.705i		−	0.446988i
$846$		0			0
$847$	313.204			0.369780
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−722.760	−	417.286i	−0.849307	−	0.490347i
$852$		0			0
$853$	−376.492	−	652.104i	−0.441374	−	0.764483i	0.556417	−	0.830903i	$-0.312176\pi$
$853$	−376.492	−	652.104i	−0.441374	−	0.764483i	−0.997792	+	0.0664202i	$0.978842\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−246.886	+	142.539i	−0.288081	+	0.166324i	−0.637076	−	0.770801i	$-0.719856\pi$
$857$	−246.886	+	142.539i	−0.288081	+	0.166324i	0.348995	+	0.937125i	$0.386523\pi$
$858$		0			0
$859$	−739.861	+	1281.48i	−0.861305	+	1.49182i	0.00936390	+	0.999956i	$0.497019\pi$
$859$	−739.861	+	1281.48i	−0.861305	+	1.49182i	−0.870669	+	0.491869i	$0.836314\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	17.3406i		−	0.0200934i	−0.999950	−	0.0100467i	$-0.996802\pi$
$863$		−	17.3406i		−	0.0200934i	0.999950	−	0.0100467i	$-0.00319801\pi$
$864$		0			0
$865$	166.957			0.193014
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−1240.10	−	715.973i	−1.42704	−	0.823904i
$870$		0			0
$871$	2.04257	+	3.53784i	0.00234509	+	0.00406181i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−113.364	+	65.4508i	−0.129559	+	0.0748010i
$876$		0			0
$877$	−633.474	+	1097.21i	−0.722319	+	1.25109i	0.237749	+	0.971327i	$0.423591\pi$
$877$	−633.474	+	1097.21i	−0.722319	+	1.25109i	−0.960068	+	0.279767i	$0.909743\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	217.386i		−	0.246749i	−0.992360	−	0.123375i	$-0.960628\pi$
$881$		−	217.386i		−	0.246749i	0.992360	−	0.123375i	$-0.0393717\pi$
$882$		0			0
$883$	1359.53			1.53967			0.769833	−	0.638245i	$-0.220339\pi$
$883$	1359.53			1.53967			0.769833	+	0.638245i	$0.220339\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−595.249	−	343.667i	−0.671081	−	0.387449i	0.125405	−	0.992106i	$-0.459977\pi$
$887$	−595.249	−	343.667i	−0.671081	−	0.387449i	−0.796486	+	0.604657i	$0.793310\pi$
$888$		0			0
$889$	800.535	+	1386.57i	0.900489	+	1.55969i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−273.301	+	157.790i	−0.306048	+	0.176697i
$894$		0			0
$895$	−168.079	+	291.121i	−0.187798	+	0.325275i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	23.7933i		−	0.0264665i
$900$		0			0
$901$	961.231			1.06685
$902$		0			0
$903$		0			0
$904$		0			0
$905$	631.455	+	364.571i	0.697741	+	0.402841i
$906$		0			0
$907$	−174.371	−	302.019i	−0.192250	−	0.332987i	0.753746	−	0.657166i	$-0.228245\pi$
$907$	−174.371	−	302.019i	−0.192250	−	0.332987i	−0.945996	+	0.324180i	$0.894912\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	539.557	−	311.514i	0.592269	−	0.341947i	−0.173725	−	0.984794i	$-0.555580\pi$
$911$	539.557	−	311.514i	0.592269	−	0.341947i	0.765994	+	0.642847i	$0.222247\pi$
$912$		0			0
$913$	681.182	−	1179.84i	0.746092	−	1.29227i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			2772.89i			3.02387i
$918$		0			0
$919$	321.988			0.350367			0.175184	−	0.984536i	$-0.443948\pi$
$919$	321.988			0.350367			0.175184	+	0.984536i	$0.443948\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	23.2379	+	13.4164i	0.0251765	+	0.0145357i
$924$		0			0
$925$	60.6231	+	105.002i	0.0655384	+	0.113516i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−174.600	+	100.805i	−0.187944	+	0.108509i	−0.591020	−	0.806657i	$-0.701274\pi$
$929$	−174.600	+	100.805i	−0.187944	+	0.108509i	0.403076	+	0.915167i	$0.367941\pi$
$930$		0			0
$931$	−546.831	+	947.140i	−0.587359	+	1.01734i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	325.623i		−	0.348260i
$936$		0			0
$937$	−395.672			−0.422275			−0.211138	−	0.977456i	$-0.567717\pi$
$937$	−395.672			−0.422275			−0.211138	+	0.977456i	$0.567717\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	1265.83	+	730.830i	1.34520	+	0.776652i	0.987565	−	0.157209i	$-0.0502496\pi$
$941$	1265.83	+	730.830i	1.34520	+	0.776652i	0.357636	+	0.933861i	$0.383583\pi$
$942$		0			0
$943$	−422.307	−	731.457i	−0.447833	−	0.775670i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	522.366	−	301.588i	0.551601	−	0.318467i	−0.198166	−	0.980168i	$-0.563499\pi$
$947$	522.366	−	301.588i	0.551601	−	0.318467i	0.749767	+	0.661701i	$0.230165\pi$
$948$		0			0
$949$	1.91486	−	3.31663i	0.00201776	−	0.00349486i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	369.246i		−	0.387457i	−0.981055	−	0.193728i	$-0.937942\pi$
$953$		−	369.246i		−	0.387457i	0.981055	−	0.193728i	$-0.0620580\pi$
$954$		0			0
$955$	111.423			0.116673
$956$		0			0
$957$		0			0
$958$		0			0
$959$	2550.09	+	1472.30i	2.65912	+	1.53524i
$960$		0			0
$961$	459.915	+	796.596i	0.478579	+	0.828924i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−444.181	+	256.448i	−0.460291	+	0.265749i
$966$		0			0
$967$	−914.240	+	1583.51i	−0.945439	+	1.63755i	−0.190570	+	0.981674i	$0.561034\pi$
$967$	−914.240	+	1583.51i	−0.945439	+	1.63755i	−0.754869	+	0.655876i	$0.772300\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			1501.22i			1.54606i	0.634371	+	0.773028i	$0.281259\pi$
$971$			1501.22i			1.54606i	−0.634371	+	0.773028i	$0.718741\pi$
$972$		0			0
$973$	204.912			0.210598
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−364.157	−	210.246i	−0.372730	−	0.215196i	0.301921	−	0.953333i	$-0.402372\pi$
$977$	−364.157	−	210.246i	−0.372730	−	0.215196i	−0.674650	+	0.738137i	$0.735706\pi$
$978$		0			0
$979$	−509.872	−	883.125i	−0.520809	−	0.902068i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−421.886	+	243.576i	−0.429182	+	0.247788i	−0.698998	−	0.715124i	$-0.746370\pi$
$983$	−421.886	+	243.576i	−0.429182	+	0.247788i	0.269816	+	0.962912i	$0.413037\pi$
$984$		0			0
$985$	−225.099	+	389.882i	−0.228527	+	0.395820i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			2614.18i			2.64326i
$990$		0			0
$991$	−1520.99			−1.53480			−0.767400	−	0.641168i	$-0.778450\pi$
$991$	−1520.99			−1.53480			−0.767400	+	0.641168i	$0.778450\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	415.210	+	239.721i	0.417296	+	0.240926i
$996$		0			0
$997$	−57.0608	−	98.8321i	−0.0572325	−	0.0991295i	0.835990	−	0.548745i	$-0.184894\pi$
$997$	−57.0608	−	98.8321i	−0.0572325	−	0.0991295i	−0.893222	+	0.449616i	$0.851561\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1620.3.o.e.701.1		8
3.2	odd	2	inner	1620.3.o.e.701.3		8
9.2	odd	6	inner	1620.3.o.e.1241.1		8
9.4	even	3		540.3.g.d.161.4	yes	4
9.5	odd	6		540.3.g.d.161.2	✓	4
9.7	even	3	inner	1620.3.o.e.1241.3		8
36.23	even	6		2160.3.l.e.161.1		4
36.31	odd	6		2160.3.l.e.161.3		4
45.4	even	6		2700.3.g.n.701.1		4
45.13	odd	12		2700.3.b.g.1349.1		4
45.14	odd	6		2700.3.g.n.701.2		4
45.22	odd	12		2700.3.b.l.1349.4		4
45.23	even	12		2700.3.b.l.1349.1		4
45.32	even	12		2700.3.b.g.1349.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
540.3.g.d.161.2	✓	4	9.5	odd	6
540.3.g.d.161.4	yes	4	9.4	even	3
1620.3.o.e.701.1		8	1.1	even	1	trivial
1620.3.o.e.701.3		8	3.2	odd	2	inner
1620.3.o.e.1241.1		8	9.2	odd	6	inner
1620.3.o.e.1241.3		8	9.7	even	3	inner
2160.3.l.e.161.1		4	36.23	even	6
2160.3.l.e.161.3		4	36.31	odd	6
2700.3.b.g.1349.1		4	45.13	odd	12
2700.3.b.g.1349.4		4	45.32	even	12
2700.3.b.l.1349.1		4	45.23	even	12
2700.3.b.l.1349.4		4	45.22	odd	12
2700.3.g.n.701.1		4	45.4	even	6
2700.3.g.n.701.2		4	45.14	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:	\( N \)	\(=\)	\( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1620.o (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.93649 - 1.11803i) q^{5} +(-5.85410 - 10.1396i) q^{7} +O(q^{10})\)	\(q+(-1.93649 - 1.11803i) q^{5} +(-5.85410 - 10.1396i) q^{7} +(-8.40755 + 4.85410i) q^{11} +(0.145898 - 0.252703i) q^{13} -15.0000i q^{17} +12.4164 q^{19} +(-29.8055 - 17.2082i) q^{23} +(2.50000 + 4.33013i) q^{25} +(-3.21140 + 1.85410i) q^{29} +(-3.20820 + 5.55677i) q^{31} +26.1803i q^{35} +24.2492 q^{37} +(21.2531 + 12.2705i) q^{41} +(-37.9787 - 65.7811i) q^{43} +(-22.0113 + 12.7082i) q^{47} +(-44.0410 + 76.2813i) q^{49} +64.0820i q^{53} +21.7082 q^{55} +(61.5957 + 35.5623i) q^{59} +(-39.7492 - 68.8477i) q^{61} +(-0.565061 + 0.326238i) q^{65} +(-7.00000 + 12.1244i) q^{67} +91.9574i q^{71} +13.1246 q^{73} +(98.4373 + 56.8328i) q^{77} +(73.7492 + 127.737i) q^{79} +(-121.530 + 70.1656i) q^{83} +(-16.7705 + 29.0474i) q^{85} +105.039i q^{89} -3.41641 q^{91} +(-24.0443 - 13.8820i) q^{95} +(-7.70820 - 13.3510i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 20 q^{7}+O(q^{10}) \) 8 * q - 20 * q^7	\( 8 q - 20 q^{7} + 28 q^{13} - 8 q^{19} + 20 q^{25} + 28 q^{31} - 128 q^{37} - 116 q^{43} - 84 q^{49} + 120 q^{55} + 4 q^{61} - 56 q^{67} - 56 q^{73} + 268 q^{79} + 80 q^{91} - 8 q^{97}+O(q^{100}) \) 8 * q - 20 * q^7 + 28 * q^13 - 8 * q^19 + 20 * q^25 + 28 * q^31 - 128 * q^37 - 116 * q^43 - 84 * q^49 + 120 * q^55 + 4 * q^61 - 56 * q^67 - 56 * q^73 + 268 * q^79 + 80 * q^91 - 8 * q^97

Introduction

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