LMFDB - Embedded newform 3060.2.be.b.1441.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3060,2,Mod(361,3060)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3060, base_ring=CyclotomicField(4))

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

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

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

chi := DirichletCharacter("3060.361");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3060 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3060.be (of order $4$, degree $2$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$24.4342230185$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 6 x^{10} - 16 x^{9} + 9 x^{8} - 72 x^{7} + 114 x^{6} - 144 x^{5} + 391 x^{4} - 484 x^{3} + \cdots + 121 $ x^12 + 6x^10 - 16x^9 + 9x^8 - 72x^7 + 114x^6 - 144x^5 + 391x^4 - 484x^3 + 504x^2 - 396x + 121
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 340)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			1441.6
Root			$-0.411629 + 1.88205i$ of defining polynomial
Character	$\chi$	$=$	3060.1441
Dual form			3060.2.be.b.361.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.707107 + 0.707107i) q^{5} +(1.97356 - 1.97356i) q^{7} +(0.864802 - 0.864802i) q^{11} -3.40173 q^{13} +(-3.98802 + 1.04677i) q^{17} -3.42309i q^{19} +(-6.74584 + 6.74584i) q^{23} +1.00000i q^{25} +(-7.08106 - 7.08106i) q^{29} +(-5.64295 - 5.64295i) q^{31} +2.79103 q^{35} +(-4.04367 - 4.04367i) q^{37} +(8.23755 - 8.23755i) q^{41} +7.49527i q^{43} -8.61396 q^{47} -0.789873i q^{49} -3.58108i q^{53} +1.22301 q^{55} +4.23376i q^{59} +(-7.67312 + 7.67312i) q^{61} +(-2.40539 - 2.40539i) q^{65} +11.1125 q^{67} +(-0.266369 - 0.266369i) q^{71} +(6.97393 + 6.97393i) q^{73} -3.41348i q^{77} +(-10.2529 + 10.2529i) q^{79} +1.52005i q^{83} +(-3.56013 - 2.07978i) q^{85} -4.28461 q^{89} +(-6.71352 + 6.71352i) q^{91} +(2.42049 - 2.42049i) q^{95} +(-1.80991 - 1.80991i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 12 q^{11} + 16 q^{13} + 4 q^{17} - 12 q^{23} - 4 q^{29} - 8 q^{31} + 8 q^{35} - 20 q^{37} - 8 q^{41} + 8 q^{47} + 16 q^{55} - 8 q^{61} - 4 q^{65} + 32 q^{67} - 28 q^{71} - 24 q^{79} - 4 q^{85} + 56 q^{89}+ \cdots - 44 q^{97}+O(q^{100}) $ 12 * q + 12 * q^11 + 16 * q^13 + 4 * q^17 - 12 * q^23 - 4 * q^29 - 8 * q^31 + 8 * q^35 - 20 * q^37 - 8 * q^41 + 8 * q^47 + 16 * q^55 - 8 * q^61 - 4 * q^65 + 32 * q^67 - 28 * q^71 - 24 * q^79 - 4 * q^85 + 56 * q^89 - 4 * q^91 - 8 * q^95 - 44 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3060\mathbb{Z}\right)^\times$.

$n$	$1261$	$1361$	$1531$	$1837$
$\chi(n)$	$e\left(\frac{1}{4}\right)$	$1$	$1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	0.707107	+	0.707107i	0.316228	+	0.316228i
$5$	0.707107	+	0.707107i	0.316228	+	0.316228i
$6$		0			0
$7$	1.97356	−	1.97356i	0.745935	−	0.745935i	−0.227778	−	0.973713i	$-0.573146\pi$
$7$	1.97356	−	1.97356i	0.745935	−	0.745935i	0.973713	+	0.227778i	$0.0731460\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	0.864802	−	0.864802i	0.260748	−	0.260748i	−0.564610	−	0.825358i	$-0.690973\pi$
$11$	0.864802	−	0.864802i	0.260748	−	0.260748i	0.825358	+	0.564610i	$0.190973\pi$
$12$		0			0
$13$	−3.40173			−0.943471			−0.471736	−	0.881740i	$-0.656372\pi$
$13$	−3.40173			−0.943471			−0.471736	+	0.881740i	$0.656372\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−3.98802	+	1.04677i	−0.967236	+	0.253879i
$17$	−3.98802	+	1.04677i	−0.967236	+	0.253879i
$18$		0			0
$19$		−	3.42309i		−	0.785311i	−0.919686	−	0.392656i	$-0.871556\pi$
$19$		−	3.42309i		−	0.785311i	0.919686	−	0.392656i	$-0.128444\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−6.74584	+	6.74584i	−1.40660	+	1.40660i	−0.630047	+	0.776557i	$0.716964\pi$
$23$	−6.74584	+	6.74584i	−1.40660	+	1.40660i	−0.776557	+	0.630047i	$0.783036\pi$
$24$		0			0
$25$			1.00000i			0.200000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−7.08106	−	7.08106i	−1.31492	−	1.31492i	−0.917742	−	0.397178i	$-0.869990\pi$
$29$	−7.08106	−	7.08106i	−1.31492	−	1.31492i	−0.397178	−	0.917742i	$-0.630010\pi$
$30$		0			0
$31$	−5.64295	−	5.64295i	−1.01350	−	1.01350i	−0.999908	−	0.0135959i	$-0.995672\pi$
$31$	−5.64295	−	5.64295i	−1.01350	−	1.01350i	−0.0135959	−	0.999908i	$-0.504328\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	2.79103			0.471771
$36$		0			0
$37$	−4.04367	−	4.04367i	−0.664776	−	0.664776i	0.291726	−	0.956502i	$-0.405770\pi$
$37$	−4.04367	−	4.04367i	−0.664776	−	0.664776i	−0.956502	+	0.291726i	$0.905770\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	8.23755	−	8.23755i	1.28649	−	1.28649i	0.349584	−	0.936905i	$-0.386323\pi$
$41$	8.23755	−	8.23755i	1.28649	−	1.28649i	0.936905	−	0.349584i	$-0.113677\pi$
$42$		0			0
$43$			7.49527i			1.14302i	0.820596	+	0.571509i	$0.193642\pi$
$43$			7.49527i			1.14302i	−0.820596	+	0.571509i	$0.806358\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−8.61396			−1.25648			−0.628238	−	0.778021i	$-0.716223\pi$
$47$	−8.61396			−1.25648			−0.628238	+	0.778021i	$0.716223\pi$
$48$		0			0
$49$		−	0.789873i		−	0.112839i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	3.58108i		−	0.491899i	−0.969283	−	0.245949i	$-0.920900\pi$
$53$		−	3.58108i		−	0.491899i	0.969283	−	0.245949i	$-0.0790997\pi$
$54$		0			0
$55$	1.22301			0.164911
$56$		0			0
$57$		0			0
$58$		0			0
$59$			4.23376i			0.551189i	0.961274	+	0.275594i	$0.0888747\pi$
$59$			4.23376i			0.551189i	−0.961274	+	0.275594i	$0.911125\pi$
$60$		0			0
$61$	−7.67312	+	7.67312i	−0.982443	+	0.982443i	−0.999849	−	0.0174058i	$-0.994459\pi$
$61$	−7.67312	+	7.67312i	−0.982443	+	0.982443i	0.0174058	+	0.999849i	$0.494459\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−2.40539	−	2.40539i	−0.298352	−	0.298352i
$66$		0			0
$67$	11.1125			1.35760			0.678802	−	0.734322i	$-0.262500\pi$
$67$	11.1125			1.35760			0.678802	+	0.734322i	$0.262500\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−0.266369	−	0.266369i	−0.0316122	−	0.0316122i	0.691124	−	0.722736i	$-0.257116\pi$
$71$	−0.266369	−	0.266369i	−0.0316122	−	0.0316122i	−0.722736	+	0.691124i	$0.757116\pi$
$72$		0			0
$73$	6.97393	+	6.97393i	0.816237	+	0.816237i	0.985561	−	0.169324i	$-0.0541583\pi$
$73$	6.97393	+	6.97393i	0.816237	+	0.816237i	−0.169324	+	0.985561i	$0.554158\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	3.41348i		−	0.389002i
$78$		0			0
$79$	−10.2529	+	10.2529i	−1.15355	+	1.15355i	−0.167710	+	0.985836i	$0.553637\pi$
$79$	−10.2529	+	10.2529i	−1.15355	+	1.15355i	−0.985836	+	0.167710i	$0.946363\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			1.52005i			0.166848i	0.996514	+	0.0834238i	$0.0265855\pi$
$83$			1.52005i			0.166848i	−0.996514	+	0.0834238i	$0.973414\pi$
$84$		0			0
$85$	−3.56013	−	2.07978i	−0.386150	−	0.225583i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−4.28461			−0.454168			−0.227084	−	0.973875i	$-0.572919\pi$
$89$	−4.28461			−0.454168			−0.227084	+	0.973875i	$0.572919\pi$
$90$		0			0
$91$	−6.71352	+	6.71352i	−0.703769	+	0.703769i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	2.42049	−	2.42049i	0.248337	−	0.248337i
$96$		0			0
$97$	−1.80991	−	1.80991i	−0.183769	−	0.183769i	0.609227	−	0.792996i	$-0.291480\pi$
$97$	−1.80991	−	1.80991i	−0.183769	−	0.183769i	−0.792996	+	0.609227i	$0.791480\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−2.83260			−0.281855			−0.140927	−	0.990020i	$-0.545008\pi$
$101$	−2.83260			−0.281855			−0.140927	+	0.990020i	$0.545008\pi$
$102$		0			0
$103$	5.02114			0.494748			0.247374	−	0.968920i	$-0.420432\pi$
$103$	5.02114			0.494748			0.247374	+	0.968920i	$0.420432\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−0.442925	−	0.442925i	−0.0428192	−	0.0428192i	0.685373	−	0.728192i	$-0.259639\pi$
$107$	−0.442925	−	0.442925i	−0.0428192	−	0.0428192i	−0.728192	+	0.685373i	$0.759639\pi$
$108$		0			0
$109$	10.0151	−	10.0151i	0.959274	−	0.959274i	−0.0399281	−	0.999203i	$-0.512713\pi$
$109$	10.0151	−	10.0151i	0.959274	−	0.959274i	0.999203	+	0.0399281i	$0.0127129\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	7.12246	−	7.12246i	0.670024	−	0.670024i	−0.287697	−	0.957721i	$-0.592890\pi$
$113$	7.12246	−	7.12246i	0.670024	−	0.670024i	0.957721	+	0.287697i	$0.0928896\pi$
$114$		0			0
$115$	−9.54005			−0.889614
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−5.80473	+	9.93645i	−0.532119	+	0.910873i
$120$		0			0
$121$			9.50424i			0.864021i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−0.707107	+	0.707107i	−0.0632456	+	0.0632456i
$126$		0			0
$127$		−	15.7863i		−	1.40081i	−0.713748	−	0.700403i	$-0.753004\pi$
$127$		−	15.7863i		−	1.40081i	0.713748	−	0.700403i	$-0.246996\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−8.21894	−	8.21894i	−0.718092	−	0.718092i	0.250122	−	0.968214i	$-0.419529\pi$
$131$	−8.21894	−	8.21894i	−0.718092	−	0.718092i	−0.968214	+	0.250122i	$0.919529\pi$
$132$		0			0
$133$	−6.75568	−	6.75568i	−0.585792	−	0.585792i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−3.40173			−0.290630			−0.145315	−	0.989385i	$-0.546420\pi$
$137$	−3.40173			−0.290630			−0.145315	+	0.989385i	$0.546420\pi$
$138$		0			0
$139$	−5.46988	−	5.46988i	−0.463950	−	0.463950i	0.435998	−	0.899948i	$-0.356395\pi$
$139$	−5.46988	−	5.46988i	−0.463950	−	0.463950i	−0.899948	+	0.435998i	$0.856395\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−2.94183	+	2.94183i	−0.246008	+	0.246008i
$144$		0			0
$145$		−	10.0141i		−	0.831628i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−4.56165			−0.373705			−0.186853	−	0.982388i	$-0.559829\pi$
$149$	−4.56165			−0.373705			−0.186853	+	0.982388i	$0.559829\pi$
$150$		0			0
$151$		−	12.7622i		−	1.03857i	−0.854601	−	0.519285i	$-0.826198\pi$
$151$		−	12.7622i		−	1.03857i	0.854601	−	0.519285i	$-0.173802\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		−	7.98033i		−	0.640996i
$156$		0			0
$157$	22.4488			1.79161			0.895803	−	0.444452i	$-0.146601\pi$
$157$	22.4488			1.79161			0.895803	+	0.444452i	$0.146601\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			26.6266i			2.09847i
$162$		0			0
$163$	2.02135	−	2.02135i	0.158324	−	0.158324i	−0.623499	−	0.781824i	$-0.714290\pi$
$163$	2.02135	−	2.02135i	0.158324	−	0.158324i	0.781824	+	0.623499i	$0.214290\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−0.0463087	−	0.0463087i	−0.00358347	−	0.00358347i	0.705313	−	0.708896i	$-0.250807\pi$
$167$	−0.0463087	−	0.0463087i	−0.00358347	−	0.00358347i	−0.708896	+	0.705313i	$0.750807\pi$
$168$		0			0
$169$	−1.42820			−0.109862
$170$		0			0
$171$		0			0
$172$		0			0
$173$	10.2763	+	10.2763i	0.781294	+	0.781294i	0.980049	−	0.198755i	$-0.0636899\pi$
$173$	10.2763	+	10.2763i	0.781294	+	0.781294i	−0.198755	+	0.980049i	$0.563690\pi$
$174$		0			0
$175$	1.97356	+	1.97356i	0.149187	+	0.149187i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	21.5967i		−	1.61421i	−0.590408	−	0.807105i	$-0.701033\pi$
$179$		−	21.5967i		−	1.61421i	0.590408	−	0.807105i	$-0.298967\pi$
$180$		0			0
$181$	−2.58484	+	2.58484i	−0.192130	+	0.192130i	−0.796616	−	0.604486i	$-0.793379\pi$
$181$	−2.58484	+	2.58484i	−0.192130	+	0.192130i	0.604486	+	0.796616i	$0.293379\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		−	5.71862i		−	0.420441i
$186$		0			0
$187$	−2.54360	+	4.35409i	−0.186006	+	0.318403i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−22.8681			−1.65468			−0.827340	−	0.561702i	$-0.810147\pi$
$191$	−22.8681			−1.65468			−0.827340	+	0.561702i	$0.810147\pi$
$192$		0			0
$193$	−5.09713	+	5.09713i	−0.366899	+	0.366899i	−0.866345	−	0.499446i	$-0.833537\pi$
$193$	−5.09713	+	5.09713i	−0.366899	+	0.366899i	0.499446	+	0.866345i	$0.333537\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−11.3698	+	11.3698i	−0.810065	+	0.810065i	−0.984643	−	0.174578i	$-0.944144\pi$
$197$	−11.3698	+	11.3698i	−0.810065	+	0.810065i	0.174578	+	0.984643i	$0.444144\pi$
$198$		0			0
$199$	−7.48293	−	7.48293i	−0.530451	−	0.530451i	0.390255	−	0.920707i	$-0.372387\pi$
$199$	−7.48293	−	7.48293i	−0.530451	−	0.530451i	−0.920707	+	0.390255i	$0.872387\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−27.9498			−1.96169
$204$		0			0
$205$	11.6497			0.813647
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−2.96030	−	2.96030i	−0.204768	−	0.204768i
$210$		0			0
$211$	−3.45096	+	3.45096i	−0.237574	+	0.237574i	−0.815845	−	0.578271i	$-0.803728\pi$
$211$	−3.45096	+	3.45096i	−0.237574	+	0.237574i	0.578271	+	0.815845i	$0.303728\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−5.29996	+	5.29996i	−0.361454	+	0.361454i
$216$		0			0
$217$	−22.2734			−1.51202
$218$		0			0
$219$		0			0
$220$		0			0
$221$	13.5662	−	3.56083i	0.912559	−	0.239527i
$222$		0			0
$223$		−	8.91984i		−	0.597317i	−0.954360	−	0.298658i	$-0.903461\pi$
$223$		−	8.91984i		−	0.597317i	0.954360	−	0.298658i	$-0.0965391\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	9.14513	−	9.14513i	0.606984	−	0.606984i	−0.335173	−	0.942157i	$-0.608795\pi$
$227$	9.14513	−	9.14513i	0.606984	−	0.606984i	0.942157	+	0.335173i	$0.108795\pi$
$228$		0			0
$229$		−	13.9331i		−	0.920726i	−0.887731	−	0.460363i	$-0.847719\pi$
$229$		−	13.9331i		−	0.920726i	0.887731	−	0.460363i	$-0.152281\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−11.0316	−	11.0316i	−0.722702	−	0.722702i	0.246452	−	0.969155i	$-0.420735\pi$
$233$	−11.0316	−	11.0316i	−0.722702	−	0.722702i	−0.969155	+	0.246452i	$0.920735\pi$
$234$		0			0
$235$	−6.09099	−	6.09099i	−0.397332	−	0.397332i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−14.8296			−0.959246			−0.479623	−	0.877475i	$-0.659227\pi$
$239$	−14.8296			−0.959246			−0.479623	+	0.877475i	$0.659227\pi$
$240$		0			0
$241$	0.224326	+	0.224326i	0.0144501	+	0.0144501i	0.714295	−	0.699845i	$-0.246748\pi$
$241$	0.224326	+	0.224326i	0.0144501	+	0.0144501i	−0.699845	+	0.714295i	$0.746748\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	0.558525	−	0.558525i	0.0356828	−	0.0356828i
$246$		0			0
$247$			11.6445i			0.740919i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	6.74154			0.425522			0.212761	−	0.977104i	$-0.431754\pi$
$251$	6.74154			0.425522			0.212761	+	0.977104i	$0.431754\pi$
$252$		0			0
$253$			11.6676i			0.733537i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	4.80913i		−	0.299985i	−0.988687	−	0.149993i	$-0.952075\pi$
$257$		−	4.80913i		−	0.299985i	0.988687	−	0.149993i	$-0.0479250\pi$
$258$		0			0
$259$	−15.9609			−0.991759
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	6.06324i		−	0.373876i	−0.982372	−	0.186938i	$-0.940144\pi$
$263$		−	6.06324i		−	0.373876i	0.982372	−	0.186938i	$-0.0598563\pi$
$264$		0			0
$265$	2.53220	−	2.53220i	0.155552	−	0.155552i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	12.1126	+	12.1126i	0.738517	+	0.738517i	0.972291	−	0.233774i	$-0.0751076\pi$
$269$	12.1126	+	12.1126i	0.738517	+	0.738517i	−0.233774	+	0.972291i	$0.575108\pi$
$270$		0			0
$271$	26.0008			1.57944			0.789719	−	0.613468i	$-0.210226\pi$
$271$	26.0008			1.57944			0.789719	+	0.613468i	$0.210226\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	0.864802	+	0.864802i	0.0521495	+	0.0521495i
$276$		0			0
$277$	−4.51774	−	4.51774i	−0.271445	−	0.271445i	0.558237	−	0.829682i	$-0.311478\pi$
$277$	−4.51774	−	4.51774i	−0.271445	−	0.271445i	−0.829682	+	0.558237i	$0.811478\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	1.36275i		−	0.0812946i	−0.999174	−	0.0406473i	$-0.987058\pi$
$281$		−	1.36275i		−	0.0812946i	0.999174	−	0.0406473i	$-0.0129420\pi$
$282$		0			0
$283$	−17.6020	+	17.6020i	−1.04633	+	1.04633i	−0.0474589	+	0.998873i	$0.515112\pi$
$283$	−17.6020	+	17.6020i	−1.04633	+	1.04633i	−0.998873	+	0.0474589i	$0.984888\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	32.5146i		−	1.91928i
$288$		0			0
$289$	14.8086	−	8.34906i	0.871091	−	0.491121i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	13.2673			0.775081			0.387541	−	0.921853i	$-0.373325\pi$
$293$	13.2673			0.775081			0.387541	+	0.921853i	$0.373325\pi$
$294$		0			0
$295$	−2.99372	+	2.99372i	−0.174301	+	0.174301i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	22.9475	−	22.9475i	1.32709	−	1.32709i
$300$		0			0
$301$	14.7924	+	14.7924i	0.852618	+	0.852618i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−10.8514			−0.621351
$306$		0			0
$307$	−20.9807			−1.19743			−0.598717	−	0.800960i	$-0.704323\pi$
$307$	−20.9807			−1.19743			−0.598717	+	0.800960i	$0.704323\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	0.507265	+	0.507265i	0.0287644	+	0.0287644i	0.721343	−	0.692578i	$-0.243525\pi$
$311$	0.507265	+	0.507265i	0.0287644	+	0.0287644i	−0.692578	+	0.721343i	$0.743525\pi$
$312$		0			0
$313$	7.50076	−	7.50076i	0.423968	−	0.423968i	−0.462599	−	0.886567i	$-0.653083\pi$
$313$	7.50076	−	7.50076i	0.423968	−	0.423968i	0.886567	+	0.462599i	$0.153083\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	15.0951	−	15.0951i	0.847823	−	0.847823i	−0.142038	−	0.989861i	$-0.545365\pi$
$317$	15.0951	−	15.0951i	0.847823	−	0.847823i	0.989861	+	0.142038i	$0.0453655\pi$
$318$		0			0
$319$	−12.2474			−0.685724
$320$		0			0
$321$		0			0
$322$		0			0
$323$	3.58319	+	13.6514i	0.199374	+	0.759582i
$324$		0			0
$325$		−	3.40173i		−	0.188694i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−17.0002	+	17.0002i	−0.937250	+	0.937250i
$330$		0			0
$331$			12.4420i			0.683872i	0.939723	+	0.341936i	$0.111083\pi$
$331$			12.4420i			0.683872i	−0.939723	+	0.341936i	$0.888917\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	7.85770	+	7.85770i	0.429312	+	0.429312i
$336$		0			0
$337$	7.22692	+	7.22692i	0.393675	+	0.393675i	0.875995	−	0.482320i	$-0.160206\pi$
$337$	7.22692	+	7.22692i	0.393675	+	0.393675i	−0.482320	+	0.875995i	$0.660206\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−9.76006			−0.528537
$342$		0			0
$343$	12.2561	+	12.2561i	0.661765	+	0.661765i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−12.3917	+	12.3917i	−0.665223	+	0.665223i	−0.956606	−	0.291383i	$-0.905885\pi$
$347$	−12.3917	+	12.3917i	−0.665223	+	0.665223i	0.291383	+	0.956606i	$0.405885\pi$
$348$		0			0
$349$			20.4572i			1.09505i	0.836789	+	0.547525i	$0.184430\pi$
$349$			20.4572i			1.09505i	−0.836789	+	0.547525i	$0.815570\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	20.0292			1.06605			0.533023	−	0.846101i	$-0.321056\pi$
$353$	20.0292			1.06605			0.533023	+	0.846101i	$0.321056\pi$
$354$		0			0
$355$		−	0.376703i		−	0.0199933i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	19.4670i		−	1.02743i	−0.857961	−	0.513715i	$-0.828269\pi$
$359$		−	19.4670i		−	1.02743i	0.857961	−	0.513715i	$-0.171731\pi$
$360$		0			0
$361$	7.28243			0.383286
$362$		0			0
$363$		0			0
$364$		0			0
$365$			9.86263i			0.516234i
$366$		0			0
$367$	−16.4806	+	16.4806i	−0.860278	+	0.860278i	−0.991370	−	0.131092i	$-0.958152\pi$
$367$	−16.4806	+	16.4806i	−0.860278	+	0.860278i	0.131092	+	0.991370i	$0.458152\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−7.06747	−	7.06747i	−0.366925	−	0.366925i
$372$		0			0
$373$	−15.4630			−0.800642			−0.400321	−	0.916375i	$-0.631101\pi$
$373$	−15.4630			−0.800642			−0.400321	+	0.916375i	$0.631101\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	24.0879	+	24.0879i	1.24059	+	1.24059i
$378$		0			0
$379$	−5.34520	−	5.34520i	−0.274564	−	0.274564i	0.556370	−	0.830935i	$-0.312194\pi$
$379$	−5.34520	−	5.34520i	−0.274564	−	0.274564i	−0.830935	+	0.556370i	$0.812194\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			10.1244i			0.517333i	0.965967	+	0.258666i	$0.0832830\pi$
$383$			10.1244i			0.517333i	−0.965967	+	0.258666i	$0.916717\pi$
$384$		0			0
$385$	2.41369	−	2.41369i	0.123013	−	0.123013i
$386$		0			0
$387$		0			0
$388$		0			0
$389$			12.6889i			0.643355i	0.946849	+	0.321677i	$0.104247\pi$
$389$			12.6889i			0.643355i	−0.946849	+	0.321677i	$0.895753\pi$
$390$		0			0
$391$	19.8412	−	33.9638i	1.00341	−	1.71762i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−14.4999			−0.729567
$396$		0			0
$397$	−16.5441	+	16.5441i	−0.830323	+	0.830323i	−0.987561	−	0.157238i	$-0.949741\pi$
$397$	−16.5441	+	16.5441i	−0.830323	+	0.830323i	0.157238	+	0.987561i	$0.449741\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−7.99397	+	7.99397i	−0.399200	+	0.399200i	−0.877951	−	0.478751i	$-0.841090\pi$
$401$	−7.99397	+	7.99397i	−0.399200	+	0.399200i	0.478751	+	0.877951i	$0.341090\pi$
$402$		0			0
$403$	19.1958	+	19.1958i	0.956211	+	0.956211i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−6.99395			−0.346677
$408$		0			0
$409$	−21.1081			−1.04373			−0.521864	−	0.853029i	$-0.674763\pi$
$409$	−21.1081			−1.04373			−0.521864	+	0.853029i	$0.674763\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	8.35558	+	8.35558i	0.411151	+	0.411151i
$414$		0			0
$415$	−1.07484	+	1.07484i	−0.0527619	+	0.0527619i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	14.2384	−	14.2384i	0.695589	−	0.695589i	−0.267867	−	0.963456i	$-0.586319\pi$
$419$	14.2384	−	14.2384i	0.695589	−	0.695589i	0.963456	+	0.267867i	$0.0863187\pi$
$420$		0			0
$421$	11.6969			0.570072			0.285036	−	0.958517i	$-0.407995\pi$
$421$	11.6969			0.570072			0.285036	+	0.958517i	$0.407995\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−1.04677	−	3.98802i	−0.0507757	−	0.193447i
$426$		0			0
$427$			30.2867i			1.46568i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−2.04736	+	2.04736i	−0.0986177	+	0.0986177i	−0.754694	−	0.656077i	$-0.772215\pi$
$431$	−2.04736	+	2.04736i	−0.0986177	+	0.0986177i	0.656077	+	0.754694i	$0.272215\pi$
$432$		0			0
$433$		−	19.7898i		−	0.951038i	−0.879705	−	0.475519i	$-0.842260\pi$
$433$		−	19.7898i		−	0.951038i	0.879705	−	0.475519i	$-0.157740\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	23.0916	+	23.0916i	1.10462	+	1.10462i
$438$		0			0
$439$	−25.0903	−	25.0903i	−1.19749	−	1.19749i	−0.974914	−	0.222580i	$-0.928552\pi$
$439$	−25.0903	−	25.0903i	−1.19749	−	1.19749i	−0.222580	−	0.974914i	$-0.571448\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	14.7915			0.702763			0.351382	−	0.936232i	$-0.385712\pi$
$443$	14.7915			0.702763			0.351382	+	0.936232i	$0.385712\pi$
$444$		0			0
$445$	−3.02968	−	3.02968i	−0.143621	−	0.143621i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	12.2258	−	12.2258i	0.576973	−	0.576973i	−0.357095	−	0.934068i	$-0.616233\pi$
$449$	12.2258	−	12.2258i	0.576973	−	0.576973i	0.934068	+	0.357095i	$0.116233\pi$
$450$		0			0
$451$		−	14.2477i		−	0.670898i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−9.49436			−0.445102
$456$		0			0
$457$		−	4.94745i		−	0.231432i	−0.993282	−	0.115716i	$-0.963084\pi$
$457$		−	4.94745i		−	0.231432i	0.993282	−	0.115716i	$-0.0369163\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			9.14971i			0.426145i	0.977036	+	0.213072i	$0.0683470\pi$
$461$			9.14971i			0.426145i	−0.977036	+	0.213072i	$0.931653\pi$
$462$		0			0
$463$	9.60773			0.446509			0.223255	−	0.974760i	$-0.428332\pi$
$463$	9.60773			0.446509			0.223255	+	0.974760i	$0.428332\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			15.2164i			0.704132i	0.935975	+	0.352066i	$0.114521\pi$
$467$			15.2164i			0.704132i	−0.935975	+	0.352066i	$0.885479\pi$
$468$		0			0
$469$	21.9311	−	21.9311i	1.01268	−	1.01268i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	6.48192	+	6.48192i	0.298039	+	0.298039i
$474$		0			0
$475$	3.42309			0.157062
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−11.1721	−	11.1721i	−0.510466	−	0.510466i	0.404203	−	0.914669i	$-0.367549\pi$
$479$	−11.1721	−	11.1721i	−0.510466	−	0.510466i	−0.914669	+	0.404203i	$0.867549\pi$
$480$		0			0
$481$	13.7555	+	13.7555i	0.627197	+	0.627197i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	2.55960i		−	0.116225i
$486$		0			0
$487$	8.29075	−	8.29075i	0.375690	−	0.375690i	−0.493855	−	0.869545i	$-0.664412\pi$
$487$	8.29075	−	8.29075i	0.375690	−	0.375690i	0.869545	+	0.493855i	$0.164412\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			6.73370i			0.303888i	0.988389	+	0.151944i	$0.0485533\pi$
$491$			6.73370i			0.303888i	−0.988389	+	0.151944i	$0.951447\pi$
$492$		0			0
$493$	35.6516	+	20.8271i	1.60567	+	0.938008i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−1.05139			−0.0471613
$498$		0			0
$499$	19.5901	−	19.5901i	0.876972	−	0.876972i	−0.116248	−	0.993220i	$-0.537087\pi$
$499$	19.5901	−	19.5901i	0.876972	−	0.876972i	0.993220	+	0.116248i	$0.0370868\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	7.00563	−	7.00563i	0.312366	−	0.312366i	−0.533460	−	0.845825i	$-0.679108\pi$
$503$	7.00563	−	7.00563i	0.312366	−	0.312366i	0.845825	+	0.533460i	$0.179108\pi$
$504$		0			0
$505$	−2.00295	−	2.00295i	−0.0891303	−	0.0891303i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−10.2748			−0.455423			−0.227712	−	0.973729i	$-0.573124\pi$
$509$	−10.2748			−0.455423			−0.227712	+	0.973729i	$0.573124\pi$
$510$		0			0
$511$	27.5269			1.21772
$512$		0			0
$513$		0			0
$514$		0			0
$515$	3.55049	+	3.55049i	0.156453	+	0.156453i
$516$		0			0
$517$	−7.44937	+	7.44937i	−0.327623	+	0.327623i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	6.83069	−	6.83069i	0.299258	−	0.299258i	−0.541465	−	0.840723i	$-0.682130\pi$
$521$	6.83069	−	6.83069i	0.299258	−	0.299258i	0.840723	+	0.541465i	$0.182130\pi$
$522$		0			0
$523$	8.21602			0.359262			0.179631	−	0.983734i	$-0.442510\pi$
$523$	8.21602			0.359262			0.179631	+	0.983734i	$0.442510\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	28.4110	+	16.5973i	1.23760	+	0.722990i
$528$		0			0
$529$		−	68.0126i		−	2.95707i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−28.0220	+	28.0220i	−1.21377	+	1.21377i
$534$		0			0
$535$		−	0.626391i		−	0.0270812i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−0.683084	−	0.683084i	−0.0294225	−	0.0294225i
$540$		0			0
$541$	−24.1876	−	24.1876i	−1.03991	−	1.03991i	−0.999170	−	0.0407372i	$-0.987029\pi$
$541$	−24.1876	−	24.1876i	−1.03991	−	1.03991i	−0.0407372	−	0.999170i	$-0.512971\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	14.1635			0.606698
$546$		0			0
$547$	15.8269	+	15.8269i	0.676711	+	0.676711i	0.959254	−	0.282544i	$-0.0911782\pi$
$547$	15.8269	+	15.8269i	0.676711	+	0.676711i	−0.282544	+	0.959254i	$0.591178\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−24.2391	+	24.2391i	−1.03262	+	1.03262i
$552$		0			0
$553$			40.4696i			1.72094i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	26.3875			1.11807			0.559036	−	0.829143i	$-0.311171\pi$
$557$	26.3875			1.11807			0.559036	+	0.829143i	$0.311171\pi$
$558$		0			0
$559$		−	25.4969i		−	1.07840i
$560$		0			0
$561$		0			0
$562$		0			0
$563$			3.35687i			0.141475i	0.997495	+	0.0707376i	$0.0225353\pi$
$563$			3.35687i			0.141475i	−0.997495	+	0.0707376i	$0.977465\pi$
$564$		0			0
$565$	10.0727			0.423761
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	42.4229i		−	1.77846i	−0.457458	−	0.889231i	$-0.651240\pi$
$569$		−	42.4229i		−	1.77846i	0.457458	−	0.889231i	$-0.348760\pi$
$570$		0			0
$571$	−9.57415	+	9.57415i	−0.400666	+	0.400666i	−0.878468	−	0.477802i	$-0.841434\pi$
$571$	−9.57415	+	9.57415i	−0.400666	+	0.400666i	0.477802	+	0.878468i	$0.341434\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−6.74584	−	6.74584i	−0.281321	−	0.281321i
$576$		0			0
$577$	20.7010			0.861792			0.430896	−	0.902402i	$-0.358198\pi$
$577$	20.7010			0.861792			0.430896	+	0.902402i	$0.358198\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	2.99992	+	2.99992i	0.124458	+	0.124458i
$582$		0			0
$583$	−3.09692	−	3.09692i	−0.128261	−	0.128261i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			33.1051i			1.36639i	0.730235	+	0.683197i	$0.239411\pi$
$587$			33.1051i			1.36639i	−0.730235	+	0.683197i	$0.760589\pi$
$588$		0			0
$589$	−19.3163	+	19.3163i	−0.795916	+	0.795916i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	11.1110i		−	0.456276i	−0.973629	−	0.228138i	$-0.926736\pi$
$593$		−	11.1110i		−	0.456276i	0.973629	−	0.228138i	$-0.0732637\pi$
$594$		0			0
$595$	−11.1307	+	2.92157i	−0.456314	+	0.119773i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	38.6224			1.57807			0.789034	−	0.614349i	$-0.210581\pi$
$599$	38.6224			1.57807			0.789034	+	0.614349i	$0.210581\pi$
$600$		0			0
$601$	−10.5195	+	10.5195i	−0.429099	+	0.429099i	−0.888321	−	0.459223i	$-0.848128\pi$
$601$	−10.5195	+	10.5195i	−0.429099	+	0.429099i	0.459223	+	0.888321i	$0.348128\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−6.72051	+	6.72051i	−0.273228	+	0.273228i
$606$		0			0
$607$	−22.2976	−	22.2976i	−0.905031	−	0.905031i	0.0908348	−	0.995866i	$-0.471046\pi$
$607$	−22.2976	−	22.2976i	−0.905031	−	0.905031i	−0.995866	+	0.0908348i	$0.971046\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	29.3024			1.18545
$612$		0			0
$613$	4.65730			0.188107			0.0940533	−	0.995567i	$-0.470018\pi$
$613$	4.65730			0.188107			0.0940533	+	0.995567i	$0.470018\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−29.1461	−	29.1461i	−1.17338	−	1.17338i	−0.981399	−	0.191978i	$-0.938510\pi$
$617$	−29.1461	−	29.1461i	−1.17338	−	1.17338i	−0.191978	−	0.981399i	$-0.561490\pi$
$618$		0			0
$619$	6.07033	−	6.07033i	0.243987	−	0.243987i	−0.574510	−	0.818497i	$-0.694807\pi$
$619$	6.07033	−	6.07033i	0.243987	−	0.243987i	0.818497	+	0.574510i	$0.194807\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−8.45594	+	8.45594i	−0.338780	+	0.338780i
$624$		0			0
$625$	−1.00000			−0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	20.3590	+	11.8934i	0.811767	+	0.474223i
$630$		0			0
$631$			17.3123i			0.689192i	0.938751	+	0.344596i	$0.111984\pi$
$631$			17.3123i			0.689192i	−0.938751	+	0.344596i	$0.888016\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	11.1626	−	11.1626i	0.442974	−	0.442974i
$636$		0			0
$637$			2.68694i			0.106460i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	11.0134	+	11.0134i	0.435002	+	0.435002i	0.890326	−	0.455324i	$-0.150476\pi$
$641$	11.0134	+	11.0134i	0.435002	+	0.435002i	−0.455324	+	0.890326i	$0.650476\pi$
$642$		0			0
$643$	7.87307	+	7.87307i	0.310484	+	0.310484i	0.845097	−	0.534613i	$-0.179543\pi$
$643$	7.87307	+	7.87307i	0.310484	+	0.310484i	−0.534613	+	0.845097i	$0.679543\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−15.7228			−0.618129			−0.309064	−	0.951041i	$-0.600016\pi$
$647$	−15.7228			−0.618129			−0.309064	+	0.951041i	$0.600016\pi$
$648$		0			0
$649$	3.66136	+	3.66136i	0.143721	+	0.143721i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	18.7020	−	18.7020i	0.731866	−	0.731866i	−0.239123	−	0.970989i	$-0.576860\pi$
$653$	18.7020	−	18.7020i	0.731866	−	0.731866i	0.970989	+	0.239123i	$0.0768600\pi$
$654$		0			0
$655$		−	11.6233i		−	0.454161i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−31.7204			−1.23565			−0.617826	−	0.786315i	$-0.711986\pi$
$659$	−31.7204			−1.23565			−0.617826	+	0.786315i	$0.711986\pi$
$660$		0			0
$661$			41.6769i			1.62105i	0.585707	+	0.810523i	$0.300817\pi$
$661$			41.6769i			1.62105i	−0.585707	+	0.810523i	$0.699183\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		−	9.55397i		−	0.370487i
$666$		0			0
$667$	95.5353			3.69914
$668$		0			0
$669$		0			0
$670$		0			0
$671$			13.2715i			0.512339i
$672$		0			0
$673$	−1.85204	+	1.85204i	−0.0713907	+	0.0713907i	−0.741901	−	0.670510i	$-0.766075\pi$
$673$	−1.85204	+	1.85204i	−0.0713907	+	0.0713907i	0.670510	+	0.741901i	$0.266075\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−8.46007	−	8.46007i	−0.325147	−	0.325147i	0.525591	−	0.850738i	$-0.323844\pi$
$677$	−8.46007	−	8.46007i	−0.325147	−	0.325147i	−0.850738	+	0.525591i	$0.823844\pi$
$678$		0			0
$679$	−7.14393			−0.274159
$680$		0			0
$681$		0			0
$682$		0			0
$683$	14.2066	+	14.2066i	0.543599	+	0.543599i	0.924582	−	0.380983i	$-0.124414\pi$
$683$	14.2066	+	14.2066i	0.543599	+	0.543599i	−0.380983	+	0.924582i	$0.624414\pi$
$684$		0			0
$685$	−2.40539	−	2.40539i	−0.0919052	−	0.0919052i
$686$		0			0
$687$		0			0
$688$		0			0
$689$			12.1819i			0.464092i
$690$		0			0
$691$	−14.9314	+	14.9314i	−0.568017	+	0.568017i	−0.931572	−	0.363556i	$-0.881563\pi$
$691$	−14.9314	+	14.9314i	−0.568017	+	0.568017i	0.363556	+	0.931572i	$0.381563\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	7.73558i		−	0.293427i
$696$		0			0
$697$	−24.2287	+	41.4743i	−0.917727	+	1.57095i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	35.6241			1.34551			0.672753	−	0.739868i	$-0.265112\pi$
$701$	35.6241			1.34551			0.672753	+	0.739868i	$0.265112\pi$
$702$		0			0
$703$	−13.8419	+	13.8419i	−0.522056	+	0.522056i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−5.59031	+	5.59031i	−0.210245	+	0.210245i
$708$		0			0
$709$	27.4579	+	27.4579i	1.03120	+	1.03120i	0.999497	+	0.0317069i	$0.0100943\pi$
$709$	27.4579	+	27.4579i	1.03120	+	1.03120i	0.0317069	+	0.999497i	$0.489906\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	76.1328			2.85120
$714$		0			0
$715$	−4.16037			−0.155589
$716$		0			0
$717$		0			0
$718$		0			0
$719$	7.29926	+	7.29926i	0.272217	+	0.272217i	0.829992	−	0.557775i	$-0.188345\pi$
$719$	7.29926	+	7.29926i	0.272217	+	0.272217i	−0.557775	+	0.829992i	$0.688345\pi$
$720$		0			0
$721$	9.90953	−	9.90953i	0.369050	−	0.369050i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	7.08106	−	7.08106i	0.262984	−	0.262984i
$726$		0			0
$727$	−36.4252			−1.35094			−0.675468	−	0.737389i	$-0.736059\pi$
$727$	−36.4252			−1.35094			−0.675468	+	0.737389i	$0.736059\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−7.84581	−	29.8913i	−0.290188	−	1.10557i
$732$		0			0
$733$			33.2356i			1.22759i	0.789467	+	0.613793i	$0.210357\pi$
$733$			33.2356i			1.22759i	−0.789467	+	0.613793i	$0.789643\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	9.61008	−	9.61008i	0.353992	−	0.353992i
$738$		0			0
$739$			11.8455i			0.435745i	0.975977	+	0.217872i	$0.0699116\pi$
$739$			11.8455i			0.435745i	−0.975977	+	0.217872i	$0.930088\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−12.4200	−	12.4200i	−0.455645	−	0.455645i	0.441578	−	0.897223i	$-0.354419\pi$
$743$	−12.4200	−	12.4200i	−0.455645	−	0.455645i	−0.897223	+	0.441578i	$0.854419\pi$
$744$		0			0
$745$	−3.22558	−	3.22558i	−0.118176	−	0.118176i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−1.74828			−0.0638807
$750$		0			0
$751$	−11.8641	−	11.8641i	−0.432927	−	0.432927i	0.456696	−	0.889623i	$-0.349033\pi$
$751$	−11.8641	−	11.8641i	−0.432927	−	0.432927i	−0.889623	+	0.456696i	$0.849033\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	9.02420	−	9.02420i	0.328424	−	0.328424i
$756$		0			0
$757$		−	45.4038i		−	1.65023i	−0.564966	−	0.825114i	$-0.691111\pi$
$757$		−	45.4038i		−	1.65023i	0.564966	−	0.825114i	$-0.308889\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	8.72830			0.316401			0.158200	−	0.987407i	$-0.449431\pi$
$761$	8.72830			0.316401			0.158200	+	0.987407i	$0.449431\pi$
$762$		0			0
$763$		−	39.5309i		−	1.43111i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	14.4021i		−	0.520031i
$768$		0			0
$769$	−0.280789			−0.0101255			−0.00506276	−	0.999987i	$-0.501612\pi$
$769$	−0.280789			−0.0101255			−0.00506276	+	0.999987i	$0.501612\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			14.6221i			0.525922i	0.964807	+	0.262961i	$0.0846990\pi$
$773$			14.6221i			0.525922i	−0.964807	+	0.262961i	$0.915301\pi$
$774$		0			0
$775$	5.64295	−	5.64295i	0.202701	−	0.202701i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−28.1979	−	28.1979i	−1.01029	−	1.01029i
$780$		0			0
$781$	−0.460713			−0.0164856
$782$		0			0
$783$		0			0
$784$		0			0
$785$	15.8737	+	15.8737i	0.566556	+	0.566556i
$786$		0			0
$787$	−30.0405	−	30.0405i	−1.07083	−	1.07083i	−0.997293	−	0.0735341i	$-0.976572\pi$
$787$	−30.0405	−	30.0405i	−1.07083	−	1.07083i	−0.0735341	−	0.997293i	$-0.523428\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	28.1132i		−	0.999590i
$792$		0			0
$793$	26.1019	−	26.1019i	0.926906	−	0.926906i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	39.4840i		−	1.39860i	−0.714831	−	0.699298i	$-0.753496\pi$
$797$		−	39.4840i		−	1.39860i	0.714831	−	0.699298i	$-0.246504\pi$
$798$		0			0
$799$	34.3526	−	9.01682i	1.21531	−	0.318992i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	12.0621			0.425664
$804$		0			0
$805$	−18.8279	+	18.8279i	−0.663595	+	0.663595i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	22.5276	−	22.5276i	0.792030	−	0.792030i	−0.189794	−	0.981824i	$-0.560782\pi$
$809$	22.5276	−	22.5276i	0.792030	−	0.792030i	0.981824	+	0.189794i	$0.0607820\pi$
$810$		0			0
$811$	−30.7078	−	30.7078i	−1.07830	−	1.07830i	−0.996662	−	0.0816334i	$-0.973986\pi$
$811$	−30.7078	−	30.7078i	−1.07830	−	1.07830i	−0.0816334	−	0.996662i	$-0.526014\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	2.85862			0.100133
$816$		0			0
$817$	25.6570			0.897625
$818$		0			0
$819$		0			0
$820$		0			0
$821$	6.72368	+	6.72368i	0.234658	+	0.234658i	0.814634	−	0.579976i	$-0.196938\pi$
$821$	6.72368	+	6.72368i	0.234658	+	0.234658i	−0.579976	+	0.814634i	$0.696938\pi$
$822$		0			0
$823$	−15.5249	+	15.5249i	−0.541165	+	0.541165i	−0.923870	−	0.382706i	$-0.874992\pi$
$823$	−15.5249	+	15.5249i	−0.541165	+	0.541165i	0.382706	+	0.923870i	$0.374992\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−27.8026	+	27.8026i	−0.966791	+	0.966791i	−0.999466	−	0.0326746i	$-0.989598\pi$
$827$	−27.8026	+	27.8026i	−0.966791	+	0.966791i	0.0326746	+	0.999466i	$0.489598\pi$
$828$		0			0
$829$	−7.75039			−0.269182			−0.134591	−	0.990901i	$-0.542972\pi$
$829$	−7.75039			−0.269182			−0.134591	+	0.990901i	$0.542972\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	0.826814	+	3.15003i	0.0286474	+	0.109142i
$834$		0			0
$835$		−	0.0654903i		−	0.00226639i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−8.10722	+	8.10722i	−0.279892	+	0.279892i	−0.833066	−	0.553174i	$-0.813417\pi$
$839$	−8.10722	+	8.10722i	−0.279892	+	0.279892i	0.553174	+	0.833066i	$0.313417\pi$
$840$		0			0
$841$			71.2827i			2.45803i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−1.00989	−	1.00989i	−0.0347414	−	0.0347414i
$846$		0			0
$847$	18.7572	+	18.7572i	0.644504	+	0.644504i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	54.5559			1.87015
$852$		0			0
$853$	19.5663	+	19.5663i	0.669936	+	0.669936i	0.957701	−	0.287765i	$-0.0929123\pi$
$853$	19.5663	+	19.5663i	0.669936	+	0.669936i	−0.287765	+	0.957701i	$0.592912\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−3.79301	+	3.79301i	−0.129567	+	0.129567i	−0.768916	−	0.639349i	$-0.779204\pi$
$857$	−3.79301	+	3.79301i	−0.129567	+	0.129567i	0.639349	+	0.768916i	$0.279204\pi$
$858$		0			0
$859$		−	8.60568i		−	0.293622i	−0.989165	−	0.146811i	$-0.953099\pi$
$859$		−	8.60568i		−	0.293622i	0.989165	−	0.146811i	$-0.0469009\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−54.1029			−1.84168			−0.920841	−	0.389937i	$-0.872497\pi$
$863$	−54.1029			−1.84168			−0.920841	+	0.389937i	$0.872497\pi$
$864$		0			0
$865$			14.5329i			0.494134i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			17.7335i			0.601569i
$870$		0			0
$871$	−37.8016			−1.28086
$872$		0			0
$873$		0			0
$874$		0			0
$875$			2.79103i			0.0943542i
$876$		0			0
$877$	37.1480	−	37.1480i	1.25440	−	1.25440i	0.300671	−	0.953728i	$-0.402789\pi$
$877$	37.1480	−	37.1480i	1.25440	−	1.25440i	0.953728	−	0.300671i	$-0.0972107\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	21.8952	+	21.8952i	0.737667	+	0.737667i	0.972126	−	0.234459i	$-0.0753318\pi$
$881$	21.8952	+	21.8952i	0.737667	+	0.737667i	−0.234459	+	0.972126i	$0.575332\pi$
$882$		0			0
$883$	38.5522			1.29739			0.648693	−	0.761051i	$-0.275316\pi$
$883$	38.5522			1.29739			0.648693	+	0.761051i	$0.275316\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−14.4598	−	14.4598i	−0.485512	−	0.485512i	0.421375	−	0.906887i	$-0.361548\pi$
$887$	−14.4598	−	14.4598i	−0.485512	−	0.485512i	−0.906887	+	0.421375i	$0.861548\pi$
$888$		0			0
$889$	−31.1552	−	31.1552i	−1.04491	−	1.04491i
$890$		0			0
$891$		0			0
$892$		0			0
$893$			29.4864i			0.986725i
$894$		0			0
$895$	15.2711	−	15.2711i	0.510458	−	0.510458i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			79.9161i			2.66535i
$900$		0			0
$901$	3.74856	+	14.2814i	0.124883	+	0.475782i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−3.65552			−0.121514
$906$		0			0
$907$	2.10137	−	2.10137i	0.0697748	−	0.0697748i	−0.671358	−	0.741133i	$-0.734289\pi$
$907$	2.10137	−	2.10137i	0.0697748	−	0.0697748i	0.741133	+	0.671358i	$0.234289\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	18.3384	−	18.3384i	0.607579	−	0.607579i	−0.334733	−	0.942313i	$-0.608646\pi$
$911$	18.3384	−	18.3384i	0.607579	−	0.607579i	0.942313	+	0.334733i	$0.108646\pi$
$912$		0			0
$913$	1.31455	+	1.31455i	0.0435051	+	0.0435051i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−32.4411			−1.07130
$918$		0			0
$919$	−13.2875			−0.438313			−0.219157	−	0.975690i	$-0.570331\pi$
$919$	−13.2875			−0.438313			−0.219157	+	0.975690i	$0.570331\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	0.906118	+	0.906118i	0.0298252	+	0.0298252i
$924$		0			0
$925$	4.04367	−	4.04367i	0.132955	−	0.132955i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−3.35997	+	3.35997i	−0.110237	+	0.110237i	−0.760074	−	0.649837i	$-0.774837\pi$
$929$	−3.35997	+	3.35997i	−0.110237	+	0.110237i	0.649837	+	0.760074i	$0.274837\pi$
$930$		0			0
$931$	−2.70381			−0.0886138
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−4.87740	+	1.28021i	−0.159508	+	0.0418674i
$936$		0			0
$937$			13.3082i			0.434759i	0.976087	+	0.217380i	$0.0697509\pi$
$937$			13.3082i			0.434759i	−0.976087	+	0.217380i	$0.930249\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	3.04861	−	3.04861i	0.0993817	−	0.0993817i	−0.655668	−	0.755050i	$-0.727613\pi$
$941$	3.04861	−	3.04861i	0.0993817	−	0.0993817i	0.755050	+	0.655668i	$0.227613\pi$
$942$		0			0
$943$			111.138i			3.61916i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	38.5353	+	38.5353i	1.25223	+	1.25223i	0.954719	+	0.297510i	$0.0961563\pi$
$947$	38.5353	+	38.5353i	1.25223	+	1.25223i	0.297510	+	0.954719i	$0.403844\pi$
$948$		0			0
$949$	−23.7235	−	23.7235i	−0.770096	−	0.770096i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−31.4405			−1.01846			−0.509229	−	0.860631i	$-0.670069\pi$
$953$	−31.4405			−1.01846			−0.509229	+	0.860631i	$0.670069\pi$
$954$		0			0
$955$	−16.1702	−	16.1702i	−0.523256	−	0.523256i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−6.71352	+	6.71352i	−0.216791	+	0.216791i
$960$		0			0
$961$			32.6857i			1.05438i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−7.20843			−0.232048
$966$		0			0
$967$			28.4642i			0.915347i	0.889120	+	0.457674i	$0.151317\pi$
$967$			28.4642i			0.915347i	−0.889120	+	0.457674i	$0.848683\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			44.9224i			1.44163i	0.693128	+	0.720815i	$0.256232\pi$
$971$			44.9224i			1.44163i	−0.693128	+	0.720815i	$0.743768\pi$
$972$		0			0
$973$	−21.5903			−0.692153
$974$		0			0
$975$		0			0
$976$		0			0
$977$			37.8272i			1.21020i	0.796150	+	0.605100i	$0.206867\pi$
$977$			37.8272i			1.21020i	−0.796150	+	0.605100i	$0.793133\pi$
$978$		0			0
$979$	−3.70534	+	3.70534i	−0.118423	+	0.118423i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	26.7685	+	26.7685i	0.853783	+	0.853783i	0.990597	−	0.136814i	$-0.0436862\pi$
$983$	26.7685	+	26.7685i	0.853783	+	0.853783i	−0.136814	+	0.990597i	$0.543686\pi$
$984$		0			0
$985$	−16.0793			−0.512330
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−50.5619	−	50.5619i	−1.60777	−	1.60777i
$990$		0			0
$991$	−12.6185	−	12.6185i	−0.400838	−	0.400838i	0.477690	−	0.878528i	$-0.341474\pi$
$991$	−12.6185	−	12.6185i	−0.400838	−	0.400838i	−0.878528	+	0.477690i	$0.841474\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	10.5825i		−	0.335487i
$996$		0			0
$997$	36.3754	−	36.3754i	1.15202	−	1.15202i	0.165872	−	0.986147i	$-0.446956\pi$
$997$	36.3754	−	36.3754i	1.15202	−	1.15202i	0.986147	−	0.165872i	$-0.0530440\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3060.2.be.b.1441.6		12
3.2	odd	2		340.2.o.a.81.3	yes	12
12.11	even	2		1360.2.bt.c.81.4		12
15.2	even	4		1700.2.m.f.149.3		12
15.8	even	4		1700.2.m.c.149.4		12
15.14	odd	2		1700.2.o.d.1101.4		12
17.4	even	4	inner	3060.2.be.b.361.6		12
51.2	odd	8		5780.2.a.n.1.4		6
51.8	odd	8		5780.2.c.h.5201.7		12
51.26	odd	8		5780.2.c.h.5201.6		12
51.32	odd	8		5780.2.a.m.1.3		6
51.38	odd	4		340.2.o.a.21.3	✓	12
204.191	even	4		1360.2.bt.c.1041.4		12
255.38	even	4		1700.2.m.f.1449.3		12
255.89	odd	4		1700.2.o.d.701.4		12
255.242	even	4		1700.2.m.c.1449.4		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
340.2.o.a.21.3	✓	12	51.38	odd	4
340.2.o.a.81.3	yes	12	3.2	odd	2
1360.2.bt.c.81.4		12	12.11	even	2
1360.2.bt.c.1041.4		12	204.191	even	4
1700.2.m.c.149.4		12	15.8	even	4
1700.2.m.c.1449.4		12	255.242	even	4
1700.2.m.f.149.3		12	15.2	even	4
1700.2.m.f.1449.3		12	255.38	even	4
1700.2.o.d.701.4		12	255.89	odd	4
1700.2.o.d.1101.4		12	15.14	odd	2
3060.2.be.b.361.6		12	17.4	even	4	inner
3060.2.be.b.1441.6		12	1.1	even	1	trivial
5780.2.a.m.1.3		6	51.32	odd	8
5780.2.a.n.1.4		6	51.2	odd	8
5780.2.c.h.5201.6		12	51.26	odd	8
5780.2.c.h.5201.7		12	51.8	odd	8

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:	\( N \)	\(=\)	\( 3060 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3060.be (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(0.707107 + 0.707107i) q^{5} +(1.97356 - 1.97356i) q^{7} +(0.864802 - 0.864802i) q^{11} -3.40173 q^{13} +(-3.98802 + 1.04677i) q^{17} -3.42309i q^{19} +(-6.74584 + 6.74584i) q^{23} +1.00000i q^{25} +(-7.08106 - 7.08106i) q^{29} +(-5.64295 - 5.64295i) q^{31} +2.79103 q^{35} +(-4.04367 - 4.04367i) q^{37} +(8.23755 - 8.23755i) q^{41} +7.49527i q^{43} -8.61396 q^{47} -0.789873i q^{49} -3.58108i q^{53} +1.22301 q^{55} +4.23376i q^{59} +(-7.67312 + 7.67312i) q^{61} +(-2.40539 - 2.40539i) q^{65} +11.1125 q^{67} +(-0.266369 - 0.266369i) q^{71} +(6.97393 + 6.97393i) q^{73} -3.41348i q^{77} +(-10.2529 + 10.2529i) q^{79} +1.52005i q^{83} +(-3.56013 - 2.07978i) q^{85} -4.28461 q^{89} +(-6.71352 + 6.71352i) q^{91} +(2.42049 - 2.42049i) q^{95} +(-1.80991 - 1.80991i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 12 q^{11} + 16 q^{13} + 4 q^{17} - 12 q^{23} - 4 q^{29} - 8 q^{31} + 8 q^{35} - 20 q^{37} - 8 q^{41} + 8 q^{47} + 16 q^{55} - 8 q^{61} - 4 q^{65} + 32 q^{67} - 28 q^{71} - 24 q^{79} - 4 q^{85} + 56 q^{89}+ \cdots - 44 q^{97}+O(q^{100}) \) 12 * q + 12 * q^11 + 16 * q^13 + 4 * q^17 - 12 * q^23 - 4 * q^29 - 8 * q^31 + 8 * q^35 - 20 * q^37 - 8 * q^41 + 8 * q^47 + 16 * q^55 - 8 * q^61 - 4 * q^65 + 32 * q^67 - 28 * q^71 - 24 * q^79 - 4 * q^85 + 56 * q^89 - 4 * q^91 - 8 * q^95 - 44 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more