LMFDB - Embedded newform 1620.3.o.e.1241.1

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			1241.1
Root			$1.40126 + 0.809017i$ of defining polynomial
Character	$\chi$	$=$	1620.1241
Dual form			1620.3.o.e.701.1

$q$-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 $\left(\mathbb{Z}/1620\mathbb{Z}\right)^\times$.

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

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	0.0566671	+	0.998393i	$0.481953\pi$
$7$	−5.85410	+	10.1396i	−0.836300	+	1.44851i	−0.892967	+	0.450121i	$0.851381\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−8.40755	−	4.85410i	−0.764323	−	0.441282i	0.0665228	−	0.997785i	$-0.478809\pi$
$11$	−8.40755	−	4.85410i	−0.764323	−	0.441282i	−0.830846	+	0.556503i	$0.812143\pi$
$12$		0			0
$13$	0.145898	+	0.252703i	0.0112229	+	0.0194387i	0.871582	−	0.490249i	$-0.163094\pi$
$13$	0.145898	+	0.252703i	0.0112229	+	0.0194387i	−0.860359	+	0.509688i	$0.829761\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			15.0000i			0.882353i	0.897420	+	0.441176i	$0.145439\pi$
$17$			15.0000i			0.882353i	−0.897420	+	0.441176i	$0.854561\pi$
$18$		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	−0.979692	−	0.200510i	$-0.935740\pi$
$23$	−29.8055	+	17.2082i	−1.29589	+	0.748183i	−0.316199	+	0.948693i	$0.602407\pi$
$24$		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	0.443608	−	0.896221i	$-0.353698\pi$
$29$	−3.21140	−	1.85410i	−0.110738	−	0.0639346i	−0.554346	+	0.832286i	$0.687032\pi$
$30$		0			0
$31$	−3.20820	−	5.55677i	−0.103490	−	0.179251i	0.809630	−	0.586941i	$-0.199668\pi$
$31$	−3.20820	−	5.55677i	−0.103490	−	0.179251i	−0.913120	+	0.407690i	$0.866334\pi$
$32$		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	−0.217898	−	0.975972i	$-0.569920\pi$
$41$	21.2531	−	12.2705i	0.518369	−	0.299281i	0.736267	+	0.676691i	$0.236587\pi$
$42$		0			0
$43$	−37.9787	+	65.7811i	−0.883226	+	1.52979i	−0.0354923	+	0.999370i	$0.511300\pi$
$43$	−37.9787	+	65.7811i	−0.883226	+	1.52979i	−0.847734	+	0.530422i	$0.822033\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−22.0113	−	12.7082i	−0.468325	−	0.270387i	0.247214	−	0.968961i	$-0.420485\pi$
$47$	−22.0113	−	12.7082i	−0.468325	−	0.270387i	−0.715538	+	0.698574i	$0.753818\pi$
$48$		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	$-0.793354\pi$
$53$		−	64.0820i		−	1.20910i	0.796569	−	0.604548i	$-0.206646\pi$
$54$		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	0.123033	−	0.992403i	$-0.460738\pi$
$59$	61.5957	−	35.5623i	1.04400	−	0.602751i	0.920962	+	0.389652i	$0.127405\pi$
$60$		0			0
$61$	−39.7492	+	68.8477i	−0.651627	+	1.12865i	0.331101	+	0.943595i	$0.392580\pi$
$61$	−39.7492	+	68.8477i	−0.651627	+	1.12865i	−0.982728	+	0.185055i	$0.940754\pi$
$62$		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	0.809047	−	0.587744i	$-0.199984\pi$
$67$	−7.00000	−	12.1244i	−0.104478	−	0.180961i	−0.913525	+	0.406783i	$0.866650\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	91.9574i		−	1.29518i	−0.761991	−	0.647588i	$-0.775778\pi$
$71$		−	91.9574i		−	1.29518i	0.761991	−	0.647588i	$-0.224222\pi$
$72$		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	0.156308	−	0.987708i	$-0.450041\pi$
$79$	73.7492	−	127.737i	0.933534	−	1.61693i	0.777227	−	0.629221i	$-0.216626\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−121.530	−	70.1656i	−1.46422	−	0.845369i	−0.465020	−	0.885300i	$-0.653953\pi$
$83$	−121.530	−	70.1656i	−1.46422	−	0.845369i	−0.999202	+	0.0399313i	$0.987286\pi$
$84$		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	$-0.799085\pi$
$89$		−	105.039i		−	1.18022i	0.807323	−	0.590109i	$-0.200915\pi$
$90$		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	−0.903020	−	0.429599i	$-0.858655\pi$
$97$	−7.70820	+	13.3510i	−0.0794660	+	0.137639i	0.823554	+	0.567238i	$0.191988\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	145.492	+	84.0000i	1.44052	+	0.831683i	0.997884	−	0.0650194i	$-0.0207109\pi$
$101$	145.492	+	84.0000i	1.44052	+	0.831683i	0.442634	+	0.896703i	$0.354044\pi$
$102$		0			0
$103$	90.2492	+	156.316i	0.876206	+	1.51763i	0.855472	+	0.517848i	$0.173267\pi$
$103$	90.2492	+	156.316i	0.876206	+	1.51763i	0.0207336	+	0.999785i	$0.493400\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	118.997i		−	1.11212i	−0.831142	−	0.556060i	$-0.812312\pi$
$107$		−	118.997i		−	1.11212i	0.831142	−	0.556060i	$-0.187688\pi$
$108$		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	0.250938	−	0.968003i	$-0.419261\pi$
$113$	137.264	−	79.2492i	1.21472	−	0.701321i	0.963784	+	0.266682i	$0.0859275\pi$
$114$		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	−0.569009	+	0.822331i	$0.692673\pi$
$131$	−205.103	+	118.416i	−1.56567	+	0.903942i	−0.996664	+	0.0816109i	$0.973994\pi$
$132$		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	−0.993337	−	0.115245i	$-0.963235\pi$
$137$	−217.804	−	125.749i	−1.58981	−	0.917878i	−0.596473	−	0.802633i	$-0.703432\pi$
$138$		0			0
$139$	−8.75078	−	15.1568i	−0.0629552	−	0.109042i	0.832830	−	0.553529i	$-0.186719\pi$
$139$	−8.75078	−	15.1568i	−0.0629552	−	0.109042i	−0.895785	+	0.444487i	$0.853386\pi$
$140$		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	−0.643558	−	0.765397i	$-0.722543\pi$
$149$	−45.0702	+	26.0213i	−0.302484	+	0.174640i	0.341074	+	0.940036i	$0.389209\pi$
$150$		0			0
$151$	85.4984	−	148.088i	0.566215	−	0.980713i	−0.430721	−	0.902485i	$-0.641741\pi$
$151$	85.4984	−	148.088i	0.566215	−	0.980713i	0.996936	−	0.0782276i	$-0.0249261\pi$
$152$		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	0.814835	−	0.579694i	$-0.196828\pi$
$157$	−14.8541	−	25.7281i	−0.0946121	−	0.163873i	−0.909447	+	0.415821i	$0.863494\pi$
$158$		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	−0.295940	−	0.955206i	$-0.595633\pi$
$167$	64.0148	−	36.9590i	0.383322	−	0.221311i	0.679263	+	0.733895i	$0.262300\pi$
$168$		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	0.301334	−	0.953519i	$-0.402568\pi$
$173$	−64.6623	−	37.3328i	−0.373771	−	0.215797i	−0.675105	+	0.737722i	$0.735901\pi$
$174$		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	$0.137945\pi$
$179$			150.334i			0.839857i	−0.907557	+	0.419928i	$0.862055\pi$
$180$		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	0.382760	−	0.923848i	$-0.374974\pi$
$191$	−43.1538	−	24.9149i	−0.225936	−	0.130444i	−0.608696	+	0.793404i	$0.708307\pi$
$192$		0			0
$193$	114.687	+	198.644i	0.594233	+	1.02924i	0.993655	+	0.112474i	$0.0358775\pi$
$193$	114.687	+	198.644i	0.594233	+	1.02924i	−0.399422	+	0.916767i	$0.630789\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			201.334i			1.02200i	0.859580	+	0.511001i	$0.170725\pi$
$197$			201.334i			1.02200i	−0.859580	+	0.511001i	$0.829275\pi$
$198$		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	0.956825	−	0.290666i	$-0.0938768\pi$
$211$	47.8313	+	82.8462i	0.226688	+	0.392636i	−0.730136	+	0.683302i	$0.760543\pi$
$212$		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	−0.780263	−	0.625452i	$-0.784915\pi$
$223$	33.7902	−	58.5264i	0.151526	−	0.262450i	0.931789	+	0.363002i	$0.118248\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−197.957	−	114.290i	−0.872055	−	0.503481i	−0.00402447	−	0.999992i	$-0.501281\pi$
$227$	−197.957	−	114.290i	−0.872055	−	0.503481i	−0.868031	+	0.496511i	$0.834614\pi$
$228$		0			0
$229$	30.6672	+	53.1171i	0.133918	+	0.231952i	0.925184	−	0.379520i	$-0.123911\pi$
$229$	30.6672	+	53.1171i	0.133918	+	0.231952i	−0.791266	+	0.611472i	$0.790578\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	254.912i		−	1.09404i	−0.837119	−	0.547021i	$-0.815762\pi$
$233$		−	254.912i		−	1.09404i	0.837119	−	0.547021i	$-0.184238\pi$
$234$		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	0.0395056	−	0.999219i	$-0.487422\pi$
$239$	220.981	−	127.584i	0.924608	−	0.533823i	0.885102	+	0.465397i	$0.154088\pi$
$240$		0			0
$241$	−85.4574	+	148.017i	−0.354595	+	0.614177i	−0.987049	−	0.160422i	$-0.948715\pi$
$241$	−85.4574	+	148.017i	−0.354595	+	0.614177i	0.632453	+	0.774598i	$0.282048\pi$
$242$		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	$-0.796972\pi$
$251$		−	298.918i		−	1.19091i	0.803389	−	0.595454i	$-0.203028\pi$
$252$		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	0.298745	−	0.954333i	$-0.403432\pi$
$257$	327.571	−	189.123i	1.27459	−	0.735887i	0.975849	+	0.218446i	$0.0700986\pi$
$258$		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	0.999915	+	0.0130073i	$0.00414047\pi$
$263$	397.429	+	229.456i	1.51114	+	0.872456i	0.511222	+	0.859448i	$0.329193\pi$
$264$		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	$-0.832367\pi$
$269$		−	270.413i		−	1.00525i	0.864503	−	0.502627i	$-0.167633\pi$
$270$		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	−0.900685	−	0.434472i	$-0.856935\pi$
$277$	−20.5197	+	35.5412i	−0.0740785	+	0.128308i	0.826607	+	0.562780i	$0.190268\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−128.277	−	74.0608i	−0.456502	−	0.263561i	0.254070	−	0.967186i	$-0.418230\pi$
$281$	−128.277	−	74.0608i	−0.456502	−	0.263561i	−0.710572	+	0.703624i	$0.751564\pi$
$282$		0			0
$283$	99.4803	+	172.305i	0.351520	+	0.608851i	0.986516	−	0.163665i	$-0.0523315\pi$
$283$	99.4803	+	172.305i	0.351520	+	0.608851i	−0.634996	+	0.772516i	$0.718998\pi$
$284$		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	−0.425966	−	0.904739i	$-0.640066\pi$
$293$	42.3615	−	24.4574i	0.144579	−	0.0834724i	0.570544	+	0.821267i	$0.306732\pi$
$294$		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	−0.853195	−	0.521592i	$-0.825338\pi$
$311$	−257.533	+	148.687i	−0.828081	+	0.478093i	0.0251139	+	0.999685i	$0.492005\pi$
$312$		0			0
$313$	69.4164	−	120.233i	0.221778	−	0.384130i	−0.733570	−	0.679614i	$-0.762147\pi$
$313$	69.4164	−	120.233i	0.221778	−	0.384130i	0.955348	+	0.295484i	$0.0954807\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−89.3481	−	51.5851i	−0.281855	−	0.162729i	0.352408	−	0.935847i	$-0.385363\pi$
$317$	−89.3481	−	51.5851i	−0.281855	−	0.162729i	−0.634263	+	0.773117i	$0.718696\pi$
$318$		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	−0.0480343	−	0.998846i	$-0.515296\pi$
$331$	278.374	−	482.158i	0.841009	−	1.45667i	0.889043	−	0.457824i	$-0.151371\pi$
$332$		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	0.724114	−	0.689681i	$-0.242249\pi$
$337$	−79.2705	−	137.301i	−0.235224	−	0.407420i	−0.959338	+	0.282261i	$0.908916\pi$
$338$		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	−0.441278	−	0.897371i	$-0.645475\pi$
$347$	39.9846	−	23.0851i	0.115230	−	0.0665278i	0.556507	+	0.830843i	$0.312141\pi$
$348$		0			0
$349$	−232.871	+	403.344i	−0.667251	+	1.15571i	0.311418	+	0.950273i	$0.399196\pi$
$349$	−232.871	+	403.344i	−0.667251	+	1.15571i	−0.978670	+	0.205440i	$0.934137\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−195.290	−	112.751i	−0.553230	−	0.319407i	0.197194	−	0.980365i	$-0.436817\pi$
$353$	−195.290	−	112.751i	−0.553230	−	0.319407i	−0.750424	+	0.660957i	$0.770150\pi$
$354$		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	$0.332612\pi$
$359$			620.991i			1.72978i	−0.501962	+	0.864890i	$0.667388\pi$
$360$		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	−0.974296	−	0.225270i	$-0.927673\pi$
$367$	−107.185	+	185.650i	−0.292058	+	0.505860i	0.682238	+	0.731130i	$0.261007\pi$
$368$		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	0.623925	−	0.781484i	$-0.285537\pi$
$373$	−136.079	−	235.696i	−0.364823	−	0.631892i	−0.988748	+	0.149593i	$0.952204\pi$
$374$		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	−0.987482	−	0.157733i	$-0.949581\pi$
$383$	−514.990	+	297.330i	−1.34462	+	0.776318i	−0.357140	+	0.934051i	$0.616248\pi$
$384$		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	0.574442	−	0.818545i	$-0.305219\pi$
$389$	59.4320	+	34.3131i	0.152781	+	0.0882084i	−0.421660	+	0.906754i	$0.638553\pi$
$390$		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	−0.852568	−	0.522617i	$-0.824956\pi$
$401$	−331.327	+	191.292i	−0.826252	+	0.477037i	0.0263156	+	0.999654i	$0.491623\pi$
$402$		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	0.940237	−	0.340522i	$-0.110604\pi$
$409$	71.6641	+	124.126i	0.175218	+	0.303486i	−0.765019	+	0.644008i	$0.777270\pi$
$410$		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	−0.963445	−	0.267905i	$-0.913669\pi$
$419$	−508.312	+	293.474i	−1.21316	+	0.700415i	−0.249710	+	0.968321i	$0.580335\pi$
$420$		0			0
$421$	76.8769	−	133.155i	0.182606	−	0.316282i	−0.760161	−	0.649734i	$-0.774880\pi$
$421$	76.8769	−	133.155i	0.182606	−	0.316282i	0.942767	+	0.333452i	$0.108214\pi$
$422$		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	$0.184386\pi$
$431$			471.866i			1.09482i	−0.836866	+	0.547408i	$0.815614\pi$
$432$		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	−0.0367519	+	0.999324i	$0.511701\pi$
$439$	−387.995	+	672.028i	−0.883816	+	1.53081i	−0.847064	+	0.531490i	$0.821632\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−419.654	−	242.287i	−0.947299	−	0.546924i	−0.0550584	−	0.998483i	$-0.517535\pi$
$443$	−419.654	−	242.287i	−0.947299	−	0.546924i	−0.892241	+	0.451560i	$0.850868\pi$
$444$		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	$-0.896405\pi$
$449$		−	287.125i		−	0.639476i	0.947506	−	0.319738i	$-0.103595\pi$
$450$		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	−0.191733	−	0.981447i	$-0.561411\pi$
$457$	344.620	−	596.899i	0.754092	−	1.30613i	0.945825	−	0.324678i	$-0.105256\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	631.682	+	364.702i	1.37024	+	0.791111i	0.990958	−	0.134169i	$-0.0428365\pi$
$461$	631.682	+	364.702i	1.37024	+	0.791111i	0.379285	+	0.925280i	$0.376170\pi$
$462$		0			0
$463$	248.502	+	430.417i	0.536720	+	0.929627i	0.999078	+	0.0429334i	$0.0136703\pi$
$463$	248.502	+	430.417i	0.536720	+	0.929627i	−0.462358	+	0.886694i	$0.652996\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			477.906i			1.02335i	0.859178	+	0.511676i	$0.170975\pi$
$467$			477.906i			1.02335i	−0.859178	+	0.511676i	$0.829025\pi$
$468$		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	0.549869	−	0.835251i	$-0.314678\pi$
$479$	48.5973	+	28.0576i	0.101456	+	0.0585755i	−0.448414	+	0.893826i	$0.648011\pi$
$480$		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	−0.904411	−	0.426662i	$-0.859689\pi$
$491$	−484.674	+	279.827i	−0.987116	+	0.569912i	−0.0827049	+	0.996574i	$0.526356\pi$
$492$		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	0.943785	−	0.330559i	$-0.107237\pi$
$499$	92.6246	+	160.431i	0.185620	+	0.321504i	−0.758165	+	0.652063i	$0.773904\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	234.088i		−	0.465384i	−0.972550	−	0.232692i	$-0.925247\pi$
$503$		−	234.088i		−	0.465384i	0.972550	−	0.232692i	$-0.0747534\pi$
$504$		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	−0.259871	−	0.965643i	$-0.583680\pi$
$509$	227.251	−	131.204i	0.446466	−	0.257767i	0.706337	+	0.707876i	$0.250347\pi$
$510$		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	$0.101995\pi$
$521$			328.200i			0.629943i	−0.949101	+	0.314972i	$0.898005\pi$
$522$		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	−0.970542	−	0.240932i	$-0.922547\pi$
$547$	−151.310	+	262.077i	−0.276618	+	0.479116i	0.693924	+	0.720048i	$0.255880\pi$
$548$		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	$-0.950703\pi$
$557$		−	171.836i		−	0.308503i	0.988032	−	0.154251i	$-0.0492965\pi$
$558$		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	−0.741710	−	0.670720i	$-0.765985\pi$
$563$	−299.350	+	172.830i	−0.531705	+	0.306980i	0.210006	+	0.977700i	$0.432652\pi$
$564$		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	0.996988	+	0.0775581i	$0.0247123\pi$
$569$	889.147	+	513.349i	1.56265	+	0.902196i	0.565661	+	0.824638i	$0.308621\pi$
$570$		0			0
$571$	−420.041	−	727.532i	−0.735624	−	1.27414i	−0.954449	−	0.298373i	$-0.903556\pi$
$571$	−420.041	−	727.532i	−0.735624	−	1.27414i	0.218826	−	0.975764i	$-0.429777\pi$
$572$		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	0.601876	−	0.798589i	$-0.294420\pi$
$587$	123.984	+	71.5820i	0.211216	+	0.121946i	−0.390660	+	0.920535i	$0.627753\pi$
$588$		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	$0.00230460\pi$
$593$			8.58670i			0.0144801i	−0.999974	+	0.00724005i	$0.997695\pi$
$594$		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	−0.937085	−	0.349102i	$-0.886487\pi$
$599$	−660.875	+	381.556i	−1.10330	+	0.636988i	−0.166212	+	0.986090i	$0.553153\pi$
$600$		0			0
$601$	345.953	−	599.208i	0.575629	−	0.997018i	−0.420344	−	0.907365i	$-0.638091\pi$
$601$	345.953	−	599.208i	0.575629	−	0.997018i	0.995973	−	0.0896533i	$-0.0285759\pi$
$602$		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	0.768954	−	0.639304i	$-0.220777\pi$
$607$	−102.690	−	177.864i	−0.169176	−	0.293022i	−0.938131	+	0.346282i	$0.887444\pi$
$608$		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	−0.158917	−	0.987292i	$-0.550800\pi$
$617$	380.469	−	219.664i	0.616644	−	0.356020i	0.775561	+	0.631272i	$0.217467\pi$
$618$		0			0
$619$	271.371	−	470.028i	0.438402	−	0.759334i	−0.559165	−	0.829057i	$-0.688878\pi$
$619$	271.371	−	470.028i	0.438402	−	0.759334i	0.997566	+	0.0697224i	$0.0222114\pi$
$620$		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	−0.111940	−	0.993715i	$-0.535707\pi$
$641$	−659.264	−	380.626i	−1.02849	−	0.593801i	−0.916553	+	0.399914i	$0.869040\pi$
$642$		0			0
$643$	365.812	+	633.604i	0.568914	+	0.985387i	0.996674	+	0.0814951i	$0.0259695\pi$
$643$	365.812	+	633.604i	0.568914	+	0.985387i	−0.427760	+	0.903892i	$0.640697\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			408.325i			0.631105i	0.948908	+	0.315553i	$0.102190\pi$
$647$			408.325i			0.631105i	−0.948908	+	0.315553i	$0.897810\pi$
$648$		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	−0.660590	−	0.750747i	$-0.729694\pi$
$653$	−222.489	+	128.454i	−0.340719	+	0.196714i	0.319871	+	0.947461i	$0.396360\pi$
$654$		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	−0.299951	−	0.953954i	$-0.596970\pi$
$659$	−840.934	−	485.514i	−1.27608	−	0.736743i	−0.976125	+	0.217212i	$0.930304\pi$
$660$		0			0
$661$	−314.751	−	545.164i	−0.476174	−	0.824757i	0.523454	−	0.852054i	$-0.324643\pi$
$661$	−314.751	−	545.164i	−0.476174	−	0.824757i	−0.999627	+	0.0272972i	$0.991310\pi$
$662$		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	−0.953098	−	0.302661i	$-0.902125\pi$
$673$	−144.316	+	249.963i	−0.214437	+	0.371416i	0.738661	+	0.674077i	$0.235458\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	485.695	+	280.416i	0.717423	+	0.414204i	0.813804	−	0.581140i	$-0.197393\pi$
$677$	485.695	+	280.416i	0.717423	+	0.414204i	−0.0963804	+	0.995345i	$0.530727\pi$
$678$		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	$-0.954284\pi$
$683$		−	195.511i		−	0.286253i	0.989704	−	0.143127i	$-0.0457156\pi$
$684$		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	−0.722354	−	0.691524i	$-0.756940\pi$
$691$	164.251	−	284.491i	0.237700	−	0.411709i	0.960054	+	0.279815i	$0.0902731\pi$
$692$		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	$0.135888\pi$
$701$			580.505i			0.828109i	−0.910252	+	0.414055i	$0.864112\pi$
$702$		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	−0.307661	−	0.951496i	$-0.599546\pi$
$709$	475.164	−	823.008i	0.670189	−	1.16080i	0.977850	−	0.209306i	$-0.0671203\pi$
$710$		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	$-0.868113\pi$
$719$		−	578.912i		−	0.805162i	0.915384	−	0.402581i	$-0.131887\pi$
$720$		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	−0.184966	−	0.982745i	$-0.559217\pi$
$727$	551.502	−	955.229i	0.758599	−	1.31393i	0.943565	−	0.331187i	$-0.107449\pi$
$728$		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	0.935043	+	0.354535i	$0.115361\pi$
$733$	567.751	+	983.373i	0.774558	+	1.34157i	−0.160485	+	0.987038i	$0.551306\pi$
$734$		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	−0.588189	+	0.808724i	$0.700159\pi$
$743$	−1175.92	+	678.915i	−1.58266	+	0.913748i	−0.994470	+	0.105024i	$0.966508\pi$
$744$		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	0.958401	−	0.285424i	$-0.0921343\pi$
$751$	174.245	+	301.800i	0.232017	+	0.401865i	−0.726385	+	0.687288i	$0.758801\pi$
$752$		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	−0.706919	−	0.707295i	$-0.749915\pi$
$761$	−340.808	+	196.766i	−0.447843	+	0.258562i	0.259076	+	0.965857i	$0.416582\pi$
$762$		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	−0.965496	−	0.260417i	$-0.916140\pi$
$769$	−544.664	−	943.386i	−0.708276	−	1.22677i	0.257220	−	0.966353i	$-0.417193\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			103.091i			0.133365i	0.997774	+	0.0666826i	$0.0212415\pi$
$773$			103.091i			0.133365i	−0.997774	+	0.0666826i	$0.978758\pi$
$774$		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	0.845361	−	0.534195i	$-0.179385\pi$
$787$	−31.4377	−	54.4517i	−0.0399462	−	0.0691889i	−0.885307	+	0.465006i	$0.846052\pi$
$788$		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	−0.188023	−	0.982165i	$-0.560208\pi$
$797$	453.131	−	261.615i	0.568546	−	0.328250i	0.756568	+	0.653915i	$0.226874\pi$
$798$		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	$-0.847072\pi$
$809$		−	747.787i		−	0.924335i	0.886793	−	0.462168i	$-0.152928\pi$
$810$		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	−0.472348	−	0.881412i	$-0.656594\pi$
$821$	−1208.39	−	697.663i	−1.47185	−	0.849772i	−0.999499	+	0.0316403i	$0.989927\pi$
$822$		0			0
$823$	338.398	+	586.123i	0.411176	+	0.712179i	0.995019	−	0.0996886i	$-0.0317847\pi$
$823$	338.398	+	586.123i	0.411176	+	0.712179i	−0.583842	+	0.811867i	$0.698451\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	1080.25i		−	1.30622i	−0.757262	−	0.653111i	$-0.773463\pi$
$827$		−	1080.25i		−	1.30622i	0.757262	−	0.653111i	$-0.226537\pi$
$828$		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	0.109931	−	0.993939i	$-0.464937\pi$
$839$	−583.843	−	337.082i	−0.695880	−	0.401766i	−0.805811	+	0.592173i	$0.798270\pi$
$840$		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	−0.997792	−	0.0664202i	$-0.978842\pi$
$853$	−376.492	+	652.104i	−0.441374	+	0.764483i	0.556417	+	0.830903i	$0.312176\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−246.886	−	142.539i	−0.288081	−	0.166324i	0.348995	−	0.937125i	$-0.386523\pi$
$857$	−246.886	−	142.539i	−0.288081	−	0.166324i	−0.637076	+	0.770801i	$0.719856\pi$
$858$		0			0
$859$	−739.861	−	1281.48i	−0.861305	−	1.49182i	−0.870669	−	0.491869i	$-0.836314\pi$
$859$	−739.861	−	1281.48i	−0.861305	−	1.49182i	0.00936390	−	0.999956i	$-0.497019\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			17.3406i			0.0200934i	0.999950	+	0.0100467i	$0.00319801\pi$
$863$			17.3406i			0.0200934i	−0.999950	+	0.0100467i	$0.996802\pi$
$864$		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	−0.960068	−	0.279767i	$-0.909743\pi$
$877$	−633.474	−	1097.21i	−0.722319	−	1.25109i	0.237749	−	0.971327i	$-0.423591\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			217.386i			0.246749i	0.992360	+	0.123375i	$0.0393717\pi$
$881$			217.386i			0.246749i	−0.992360	+	0.123375i	$0.960628\pi$
$882$		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	−0.796486	−	0.604657i	$-0.793310\pi$
$887$	−595.249	+	343.667i	−0.671081	+	0.387449i	0.125405	+	0.992106i	$0.459977\pi$
$888$		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	−0.945996	−	0.324180i	$-0.894912\pi$
$907$	−174.371	+	302.019i	−0.192250	+	0.332987i	0.753746	+	0.657166i	$0.228245\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	539.557	+	311.514i	0.592269	+	0.341947i	0.765994	−	0.642847i	$-0.222247\pi$
$911$	539.557	+	311.514i	0.592269	+	0.341947i	−0.173725	+	0.984794i	$0.555580\pi$
$912$		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	0.403076	−	0.915167i	$-0.367941\pi$
$929$	−174.600	−	100.805i	−0.187944	−	0.108509i	−0.591020	+	0.806657i	$0.701274\pi$
$930$		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	0.357636	−	0.933861i	$-0.383583\pi$
$941$	1265.83	−	730.830i	1.34520	−	0.776652i	0.987565	+	0.157209i	$0.0502496\pi$
$942$		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	0.749767	−	0.661701i	$-0.230165\pi$
$947$	522.366	+	301.588i	0.551601	+	0.318467i	−0.198166	+	0.980168i	$0.563499\pi$
$948$		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	$0.0620580\pi$
$953$			369.246i			0.387457i	−0.981055	+	0.193728i	$0.937942\pi$
$954$		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	−0.754869	−	0.655876i	$-0.772300\pi$
$967$	−914.240	−	1583.51i	−0.945439	−	1.63755i	−0.190570	−	0.981674i	$-0.561034\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	1501.22i		−	1.54606i	−0.634371	−	0.773028i	$-0.718741\pi$
$971$		−	1501.22i		−	1.54606i	0.634371	−	0.773028i	$-0.281259\pi$
$972$		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	−0.674650	−	0.738137i	$-0.735706\pi$
$977$	−364.157	+	210.246i	−0.372730	+	0.215196i	0.301921	+	0.953333i	$0.402372\pi$
$978$		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	0.269816	−	0.962912i	$-0.413037\pi$
$983$	−421.886	−	243.576i	−0.429182	−	0.247788i	−0.698998	+	0.715124i	$0.746370\pi$
$984$		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	−0.893222	−	0.449616i	$-0.851561\pi$
$997$	−57.0608	+	98.8321i	−0.0572325	+	0.0991295i	0.835990	+	0.548745i	$0.184894\pi$
$998$		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.1241.1		8
3.2	odd	2	inner	1620.3.o.e.1241.3		8
9.2	odd	6		540.3.g.d.161.4	yes	4
9.4	even	3	inner	1620.3.o.e.701.3		8
9.5	odd	6	inner	1620.3.o.e.701.1		8
9.7	even	3		540.3.g.d.161.2	✓	4
36.7	odd	6		2160.3.l.e.161.1		4
36.11	even	6		2160.3.l.e.161.3		4
45.2	even	12		2700.3.b.l.1349.4		4
45.7	odd	12		2700.3.b.g.1349.4		4
45.29	odd	6		2700.3.g.n.701.1		4
45.34	even	6		2700.3.g.n.701.2		4
45.38	even	12		2700.3.b.g.1349.1		4
45.43	odd	12		2700.3.b.l.1349.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
540.3.g.d.161.2	✓	4	9.7	even	3
540.3.g.d.161.4	yes	4	9.2	odd	6
1620.3.o.e.701.1		8	9.5	odd	6	inner
1620.3.o.e.701.3		8	9.4	even	3	inner
1620.3.o.e.1241.1		8	1.1	even	1	trivial
1620.3.o.e.1241.3		8	3.2	odd	2	inner
2160.3.l.e.161.1		4	36.7	odd	6
2160.3.l.e.161.3		4	36.11	even	6
2700.3.b.g.1349.1		4	45.38	even	12
2700.3.b.g.1349.4		4	45.7	odd	12
2700.3.b.l.1349.1		4	45.43	odd	12
2700.3.b.l.1349.4		4	45.2	even	12
2700.3.g.n.701.1		4	45.29	odd	6
2700.3.g.n.701.2		4	45.34	even	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

L-functions

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