LMFDB - Embedded newform 57.2.j.a.14.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [57,2,Mod(2,57)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(57, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([9, 1]))

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

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

chi := DirichletCharacter("57.2");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 57 = 3 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	57.j (of order $18$, degree $6$, 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:	$0.455147291521$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\Q(\zeta_{18})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{3} + 1 $ x^6 - x^3 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{18}]$

Embedding invariants

Embedding label			14.1
Root			$-0.766044 + 0.642788i$ of defining polynomial
Character	$\chi$	$=$	57.14
Dual form			57.2.j.a.53.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.70574 + 0.300767i) q^{3} +(-1.53209 + 1.28558i) q^{4} +(-2.05303 - 3.55596i) q^{7} +(2.81908 + 1.02606i) q^{9} +O(q^{10})$	$q+(1.70574 + 0.300767i) q^{3} +(-1.53209 + 1.28558i) q^{4} +(-2.05303 - 3.55596i) q^{7} +(2.81908 + 1.02606i) q^{9} +(-3.00000 + 1.73205i) q^{12} +(-3.96064 + 0.698367i) q^{13} +(0.694593 - 3.93923i) q^{16} +(0.500000 + 4.33013i) q^{19} +(-2.43242 - 6.68302i) q^{21} +(0.868241 + 4.92404i) q^{25} +(4.50000 + 2.59808i) q^{27} +(7.71688 + 2.80872i) q^{28} +(9.64203 - 5.56683i) q^{31} +(-5.63816 + 2.05212i) q^{36} +3.09018i q^{37} -6.96585 q^{39} +(-8.48158 - 7.11689i) q^{43} +(2.36959 - 6.51038i) q^{48} +(-4.92989 + 8.53882i) q^{49} +(5.17024 - 6.16166i) q^{52} +(-0.449493 + 7.53644i) q^{57} +(-11.6270 + 9.75622i) q^{61} +(-2.13903 - 12.1311i) q^{63} +(4.00000 + 6.92820i) q^{64} +(5.18345 - 14.2414i) q^{67} +(0.216415 - 1.22735i) q^{73} +8.66025i q^{75} +(-6.33275 - 5.99135i) q^{76} +(0.917404 + 0.161763i) q^{79} +(6.89440 + 5.78509i) q^{81} +(12.3182 + 7.11192i) q^{84} +(10.6147 + 12.6501i) q^{91} +(18.1211 - 6.59553i) q^{93} +(-1.77719 - 4.88279i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q+O(q^{10}) $ 6 * q	$ 6 q - 18 q^{12} - 15 q^{13} + 3 q^{19} + 9 q^{21} + 27 q^{27} + 30 q^{28} - 39 q^{43} - 21 q^{49} - 12 q^{52} - 42 q^{61} - 36 q^{63} + 24 q^{64} + 33 q^{67} + 51 q^{73} + 12 q^{79} + 48 q^{91} + 54 q^{93}+O(q^{100}) $ 6 * q - 18 * q^12 - 15 * q^13 + 3 * q^19 + 9 * q^21 + 27 * q^27 + 30 * q^28 - 39 * q^43 - 21 * q^49 - 12 * q^52 - 42 * q^61 - 36 * q^63 + 24 * q^64 + 33 * q^67 + 51 * q^73 + 12 * q^79 + 48 * q^91 + 54 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/57\mathbb{Z}\right)^\times$.

$n$	$20$	$40$
$\chi(n)$	$-1$	$e\left(\frac{7}{18}\right)$

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		−0.342020	−	0.939693i	$-0.611111\pi$
$2$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$3$	1.70574	+	0.300767i	0.984808	+	0.173648i
$3$	1.70574	+	0.300767i	0.984808	+	0.173648i
$4$	−1.53209	+	1.28558i	−0.766044	+	0.642788i
$5$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$5$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$6$		0			0
$7$	−2.05303	−	3.55596i	−0.775974	−	1.34403i	−0.934246	−	0.356630i	$-0.883926\pi$
$7$	−2.05303	−	3.55596i	−0.775974	−	1.34403i	0.158272	−	0.987396i	$-0.449408\pi$
$8$		0			0
$9$	2.81908	+	1.02606i	0.939693	+	0.342020i
$10$		0			0
$11$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$11$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$12$	−3.00000	+	1.73205i	−0.866025	+	0.500000i
$13$	−3.96064	+	0.698367i	−1.09848	+	0.193692i	−0.693375	−	0.720577i	$-0.743877\pi$
$13$	−3.96064	+	0.698367i	−1.09848	+	0.193692i	−0.405108	+	0.914269i	$0.632766\pi$
$14$		0			0
$15$		0			0
$16$	0.694593	−	3.93923i	0.173648	−	0.984808i
$17$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$17$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$18$		0			0
$19$	0.500000	+	4.33013i	0.114708	+	0.993399i
$19$	0.500000	+	4.33013i	0.114708	+	0.993399i
$20$		0			0
$21$	−2.43242	−	6.68302i	−0.530797	−	1.45835i
$22$		0			0
$23$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$23$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$24$		0			0
$25$	0.868241	+	4.92404i	0.173648	+	0.984808i
$26$		0			0
$27$	4.50000	+	2.59808i	0.866025	+	0.500000i
$28$	7.71688	+	2.80872i	1.45835	+	0.530797i
$29$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$29$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$30$		0			0
$31$	9.64203	−	5.56683i	1.73176	−	0.999832i	0.856702	−	0.515812i	$-0.172510\pi$
$31$	9.64203	−	5.56683i	1.73176	−	0.999832i	0.875057	−	0.484020i	$-0.160824\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−5.63816	+	2.05212i	−0.939693	+	0.342020i
$37$			3.09018i			0.508022i	0.967201	+	0.254011i	$0.0817500\pi$
$37$			3.09018i			0.508022i	−0.967201	+	0.254011i	$0.918250\pi$
$38$		0			0
$39$	−6.96585			−1.11543
$40$		0			0
$41$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$41$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$42$		0			0
$43$	−8.48158	−	7.11689i	−1.29343	−	1.08532i	−0.991241	−	0.132068i	$-0.957838\pi$
$43$	−8.48158	−	7.11689i	−1.29343	−	1.08532i	−0.302188	−	0.953248i	$-0.597717\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$47$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$48$	2.36959	−	6.51038i	0.342020	−	0.939693i
$49$	−4.92989	+	8.53882i	−0.704270	+	1.21983i
$50$		0			0
$51$		0			0
$52$	5.17024	−	6.16166i	0.716984	−	0.854468i
$53$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$53$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−0.449493	+	7.53644i	−0.0595368	+	0.998226i
$58$		0			0
$59$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$59$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$60$		0			0
$61$	−11.6270	+	9.75622i	−1.48869	+	1.24916i	−0.592428	+	0.805623i	$0.701831\pi$
$61$	−11.6270	+	9.75622i	−1.48869	+	1.24916i	−0.896258	+	0.443533i	$0.853725\pi$
$62$		0			0
$63$	−2.13903	−	12.1311i	−0.269493	−	1.52837i
$64$	4.00000	+	6.92820i	0.500000	+	0.866025i
$65$		0			0
$66$		0			0
$67$	5.18345	−	14.2414i	0.633259	−	1.73986i	−0.0386729	−	0.999252i	$-0.512313\pi$
$67$	5.18345	−	14.2414i	0.633259	−	1.73986i	0.671932	−	0.740613i	$-0.265465\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$71$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$72$		0			0
$73$	0.216415	−	1.22735i	0.0253294	−	0.143650i	−0.969520	−	0.245011i	$-0.921208\pi$
$73$	0.216415	−	1.22735i	0.0253294	−	0.143650i	0.994850	+	0.101361i	$0.0323196\pi$
$74$		0			0
$75$			8.66025i			1.00000i
$76$	−6.33275	−	5.99135i	−0.726416	−	0.687255i
$77$		0			0
$78$		0			0
$79$	0.917404	+	0.161763i	0.103216	+	0.0181998i	0.225018	−	0.974355i	$-0.427756\pi$
$79$	0.917404	+	0.161763i	0.103216	+	0.0181998i	−0.121802	+	0.992554i	$0.538867\pi$
$80$		0			0
$81$	6.89440	+	5.78509i	0.766044	+	0.642788i
$82$		0			0
$83$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$83$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$84$	12.3182	+	7.11192i	1.34403	+	0.775974i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$89$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$90$		0			0
$91$	10.6147	+	12.6501i	1.11272	+	1.32609i
$92$		0			0
$93$	18.1211	−	6.59553i	1.87907	−	0.683925i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−1.77719	−	4.88279i	−0.180446	−	0.495772i	0.816185	−	0.577791i	$-0.196085\pi$
$97$	−1.77719	−	4.88279i	−0.180446	−	0.495772i	−0.996631	+	0.0820195i	$0.973863\pi$
$98$		0			0
$99$		0			0
$100$	−7.66044	−	6.42788i	−0.766044	−	0.642788i
$101$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$101$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$102$		0			0
$103$	−8.83527	−	5.10105i	−0.870565	−	0.502621i	−0.00302937	−	0.999995i	$-0.500964\pi$
$103$	−8.83527	−	5.10105i	−0.870565	−	0.502621i	−0.867536	+	0.497374i	$0.834298\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$107$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$108$	−10.2344	+	1.80460i	−0.984808	+	0.173648i
$109$	5.56670	−	6.63414i	0.533194	−	0.635435i	−0.430454	−	0.902613i	$-0.641646\pi$
$109$	5.56670	−	6.63414i	0.533194	−	0.635435i	0.963647	+	0.267177i	$0.0860909\pi$
$110$		0			0
$111$	−0.929426	+	5.27103i	−0.0882172	+	0.500304i
$112$	−15.4338	+	5.61743i	−1.45835	+	0.530797i
$113$		0			0			−	1.00000i	$-0.5\pi$
$113$		0			0				1.00000i	$0.5\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−11.8819	−	2.09510i	−1.09848	−	0.193692i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−5.50000	−	9.52628i	−0.500000	−	0.866025i
$122$		0			0
$123$		0			0
$124$	−7.61587	+	20.9244i	−0.683925	+	1.87907i
$125$		0			0
$126$		0			0
$127$	11.9402	−	2.10537i	1.05952	−	0.186822i	0.383375	−	0.923593i	$-0.374762\pi$
$127$	11.9402	−	2.10537i	1.05952	−	0.186822i	0.676142	+	0.736771i	$0.263650\pi$
$128$		0			0
$129$	−12.3268	−	14.6905i	−1.08532	−	1.29343i
$130$		0			0
$131$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$131$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$132$		0			0
$133$	14.3712	−	10.6679i	1.24614	−	0.925022i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$137$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$138$		0			0
$139$	4.06165	+	23.0348i	0.344505	+	1.95378i	0.296866	+	0.954919i	$0.404058\pi$
$139$	4.06165	+	23.0348i	0.344505	+	1.95378i	0.0476387	+	0.998865i	$0.484830\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	6.00000	−	10.3923i	0.500000	−	0.866025i
$145$		0			0
$146$		0			0
$147$	−10.9773	+	13.0822i	−0.905393	+	1.07900i
$148$	−3.97266	−	4.73443i	−0.326551	−	0.389168i
$149$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$149$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$150$		0			0
$151$		−	15.5885i		−	1.26857i	−0.773099	−	0.634285i	$-0.781294\pi$
$151$		−	15.5885i		−	1.26857i	0.773099	−	0.634285i	$-0.218706\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	10.6723	−	8.95513i	0.854468	−	0.716984i
$157$	13.8177	+	11.5945i	1.10278	+	0.925338i	0.997609	−	0.0691164i	$-0.0220180\pi$
$157$	13.8177	+	11.5945i	1.10278	+	0.925338i	0.105167	+	0.994455i	$0.466462\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−4.72921	+	8.19123i	−0.370420	+	0.641587i	−0.989630	−	0.143639i	$-0.954120\pi$
$163$	−4.72921	+	8.19123i	−0.370420	+	0.641587i	0.619210	+	0.785225i	$0.287453\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$167$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$168$		0			0
$169$	2.98293	−	1.08570i	0.229456	−	0.0835151i
$170$		0			0
$171$	−3.03343	+	12.7200i	−0.231972	+	0.972722i
$172$	22.1438			1.68845
$173$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$173$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$174$		0			0
$175$	15.7271	−	13.1966i	1.18886	−	0.997573i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$179$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$180$		0			0
$181$	−6.51636	+	17.9035i	−0.484357	+	1.33076i	0.421366	+	0.906891i	$0.361551\pi$
$181$	−6.51636	+	17.9035i	−0.484357	+	1.33076i	−0.905723	+	0.423870i	$0.860671\pi$
$182$		0			0
$183$	−22.7670	+	13.1445i	−1.68298	+	0.971671i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		−	21.3357i		−	1.55195i
$190$		0			0
$191$		0			0		1.00000			$0$
$191$		0			0		−1.00000			$\pi$
$192$	4.73917	+	13.0208i	0.342020	+	0.939693i
$193$	24.9722	+	4.40328i	1.79754	+	0.316955i	0.969754	−	0.244086i	$-0.0784878\pi$
$193$	24.9722	+	4.40328i	1.79754	+	0.316955i	0.827788	+	0.561041i	$0.189599\pi$
$194$		0			0
$195$		0			0
$196$	−3.42427	−	19.4200i	−0.244591	−	1.38714i
$197$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$197$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$198$		0			0
$199$	−25.8380	−	9.40425i	−1.83160	−	0.666650i	−0.992434	−	0.122782i	$-0.960818\pi$
$199$	−25.8380	−	9.40425i	−1.83160	−	0.666650i	−0.839171	−	0.543868i	$-0.816959\pi$
$200$		0			0
$201$	13.1250	−	22.7331i	0.925763	−	1.60347i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$			16.0869i			1.11543i
$209$		0			0
$210$		0			0
$211$	5.92174	+	16.2699i	0.407669	+	1.12006i	0.958412	+	0.285388i	$0.0921223\pi$
$211$	5.92174	+	16.2699i	0.407669	+	1.12006i	−0.550743	+	0.834675i	$0.685655\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−39.5908	−	22.8578i	−2.68760	−	1.55169i
$218$		0			0
$219$	0.738293	−	2.02844i	0.0498892	−	0.137069i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−18.8846	+	22.5058i	−1.26461	+	1.50710i	−0.494509	+	0.869172i	$0.664652\pi$
$223$	−18.8846	+	22.5058i	−1.26461	+	1.50710i	−0.770097	+	0.637927i	$0.779792\pi$
$224$		0			0
$225$	−2.60472	+	14.7721i	−0.173648	+	0.984808i
$226$		0			0
$227$		0			0			−	1.00000i	$-0.5\pi$
$227$		0			0				1.00000i	$0.5\pi$
$228$	−9.00000	−	12.1244i	−0.596040	−	0.802955i
$229$	−13.5645			−0.896366			−0.448183	−	0.893942i	$-0.647929\pi$
$229$	−13.5645			−0.896366			−0.448183	+	0.893942i	$0.647929\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$233$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	1.51620	+	0.551851i	0.0984876	+	0.0358465i
$238$		0			0
$239$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$239$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$240$		0			0
$241$	8.83868	−	1.55850i	0.569349	−	0.100392i	0.118440	−	0.992961i	$-0.462211\pi$
$241$	8.83868	−	1.55850i	0.569349	−	0.100392i	0.450910	+	0.892570i	$0.351100\pi$
$242$		0			0
$243$	10.0201	+	11.9415i	0.642788	+	0.766044i
$244$	5.27126	−	29.8948i	0.337458	−	1.91382i
$245$		0			0
$246$		0			0
$247$	−5.00434	−	16.8009i	−0.318418	−	1.06901i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$251$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$252$	18.8726	+	15.8360i	1.18886	+	0.997573i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−15.0351	−	5.47232i	−0.939693	−	0.342020i
$257$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$257$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$258$		0			0
$259$	10.9886	−	6.34424i	0.682795	−	0.394212i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$263$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	10.3669	+	28.4828i	0.633259	+	1.73986i
$269$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$269$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$270$		0			0
$271$	−0.766044	−	0.642788i	−0.0465339	−	0.0390466i	0.619224	−	0.785214i	$-0.287447\pi$
$271$	−0.766044	−	0.642788i	−0.0465339	−	0.0390466i	−0.665758	+	0.746168i	$0.731892\pi$
$272$		0			0
$273$	14.3011	+	24.7703i	0.865544	+	1.49917i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	2.50000	−	4.33013i	0.150210	−	0.260172i	−0.781094	−	0.624413i	$-0.785338\pi$
$277$	2.50000	−	4.33013i	0.150210	−	0.260172i	0.931305	+	0.364241i	$0.118672\pi$
$278$		0			0
$279$	32.8935	−	5.80002i	1.96928	−	0.347238i
$280$		0			0
$281$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$281$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$282$		0			0
$283$	6.57785	−	2.39414i	0.391012	−	0.142317i	−0.139030	−	0.990288i	$-0.544398\pi$
$283$	6.57785	−	2.39414i	0.391012	−	0.142317i	0.530042	+	0.847971i	$0.322176\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	13.0228	−	10.9274i	0.766044	−	0.642788i
$290$		0			0
$291$	−1.56283	−	8.86327i	−0.0916149	−	0.519574i
$292$	1.24628	+	2.15862i	0.0729331	+	0.126324i
$293$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$293$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$	−11.1334	−	13.2683i	−0.642788	−	0.766044i
$301$	−7.89440	+	44.7714i	−0.455026	+	2.58058i
$302$		0			0
$303$		0			0
$304$	17.4047	+	1.03806i	0.998226	+	0.0595368i
$305$		0			0
$306$		0			0
$307$	−28.9975	−	5.11305i	−1.65498	−	0.291817i	−0.733337	−	0.679865i	$-0.762038\pi$
$307$	−28.9975	−	5.11305i	−1.65498	−	0.291817i	−0.921639	+	0.388048i	$0.873149\pi$
$308$		0			0
$309$	−13.5364	−	11.3584i	−0.770060	−	0.646157i
$310$		0			0
$311$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$311$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$312$		0			0
$313$	−12.2160	−	4.44626i	−0.690489	−	0.251318i	−0.0271446	−	0.999632i	$-0.508641\pi$
$313$	−12.2160	−	4.44626i	−0.690489	−	0.251318i	−0.663345	+	0.748314i	$0.730864\pi$
$314$		0			0
$315$		0			0
$316$	−1.61350	+	0.931556i	−0.0907666	+	0.0524041i
$317$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$317$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−18.0000			−1.00000
$325$	−6.87757	−	18.8960i	−0.381499	−	1.04816i
$326$		0			0
$327$	11.4907	−	9.64181i	0.635435	−	0.533194i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	23.5737	+	13.6103i	1.29573	+	0.748090i	0.979663	−	0.200648i	$-0.0643046\pi$
$331$	23.5737	+	13.6103i	1.29573	+	0.748090i	0.316066	+	0.948737i	$0.397638\pi$
$332$		0			0
$333$	−3.17071	+	8.71146i	−0.173754	+	0.477385i
$334$		0			0
$335$		0			0
$336$	−28.0155	+	4.93989i	−1.52837	+	0.269493i
$337$	20.8709	−	24.8730i	1.13691	−	1.35492i	0.210863	−	0.977516i	$-0.432373\pi$
$337$	20.8709	−	24.8730i	1.13691	−	1.35492i	0.926049	−	0.377403i	$-0.123183\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	11.7425			0.634034
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$347$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$348$		0			0
$349$	12.5018	+	21.6538i	0.669207	+	1.15910i	0.978126	+	0.208012i	$0.0666992\pi$
$349$	12.5018	+	21.6538i	0.669207	+	1.15910i	−0.308920	+	0.951088i	$0.599967\pi$
$350$		0			0
$351$	−19.6373	−	7.14738i	−1.04816	−	0.381499i
$352$		0			0
$353$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$353$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$359$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$360$		0			0
$361$	−18.5000	+	4.33013i	−0.973684	+	0.227901i
$362$		0			0
$363$	−6.51636	−	17.9035i	−0.342020	−	0.939693i
$364$	−32.5253	−	5.73508i	−1.70479	−	0.300600i
$365$		0			0
$366$		0			0
$367$	2.91581	+	16.5364i	0.152204	+	0.863192i	0.961298	+	0.275512i	$0.0888475\pi$
$367$	2.91581	+	16.5364i	0.152204	+	0.863192i	−0.809093	+	0.587680i	$0.800041\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	−19.2841	+	33.4010i	−0.999832	+	1.73176i
$373$	−6.00000	+	3.46410i	−0.310668	+	0.179364i	−0.647225	−	0.762299i	$-0.724071\pi$
$373$	−6.00000	+	3.46410i	−0.310668	+	0.179364i	0.336557	+	0.941663i	$0.390737\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$		−	34.3325i		−	1.76354i	−0.471677	−	0.881771i	$-0.656351\pi$
$379$		−	34.3325i		−	1.76354i	0.471677	−	0.881771i	$-0.343649\pi$
$380$		0			0
$381$	21.0000			1.07586
$382$		0			0
$383$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$383$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−16.6079	−	28.7657i	−0.844226	−	1.46224i
$388$	9.00000	+	5.19615i	0.456906	+	0.263795i
$389$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$389$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	11.9204	−	4.33867i	0.598268	−	0.217752i	−0.0250943	−	0.999685i	$-0.507989\pi$
$397$	11.9204	−	4.33867i	0.598268	−	0.217752i	0.623362	+	0.781933i	$0.285766\pi$
$398$		0			0
$399$	27.7221	−	13.8742i	1.38784	−	0.694578i
$400$	20.0000			1.00000
$401$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$401$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$402$		0			0
$403$	−34.3009	+	28.7819i	−1.70865	+	1.43373i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	4.73917	−	13.0208i	0.234337	−	0.643835i	−0.765663	−	0.643242i	$-0.777589\pi$
$409$	4.73917	−	13.0208i	0.234337	−	0.643835i	1.00000		0.000593299i	$-0.000188853\pi$
$410$		0			0
$411$		0			0
$412$	20.0942	−	3.54315i	0.989971	−	0.174559i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$			40.5129i			1.98392i
$418$		0			0
$419$		0			0		1.00000			$0$
$419$		0			0		−1.00000			$\pi$
$420$		0			0
$421$	35.8205	+	6.31612i	1.74578	+	0.307829i	0.953291	−	0.302053i	$-0.0976721\pi$
$421$	35.8205	+	6.31612i	1.74578	+	0.307829i	0.792492	+	0.609882i	$0.208783\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	58.5634	+	21.3153i	2.83408	+	1.03152i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$431$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$432$	13.3601	−	15.9219i	0.642788	−	0.766044i
$433$	3.37387	+	4.02082i	0.162138	+	0.193228i	0.840996	−	0.541041i	$-0.181970\pi$
$433$	3.37387	+	4.02082i	0.162138	+	0.193228i	−0.678859	+	0.734269i	$0.737525\pi$
$434$		0			0
$435$		0			0
$436$			17.3205i			0.829502i
$437$		0			0
$438$		0			0
$439$	2.01274	+	5.52995i	0.0960627	+	0.263930i	0.978412	−	0.206666i	$-0.0662612\pi$
$439$	2.01274	+	5.52995i	0.0960627	+	0.263930i	−0.882349	+	0.470596i	$0.844039\pi$
$440$		0			0
$441$	−22.6591	+	19.0132i	−1.07900	+	0.905393i
$442$		0			0
$443$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$443$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$444$	−5.35235	−	9.27054i	−0.254011	−	0.439960i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	16.4243	−	28.4477i	0.775974	−	1.34403i
$449$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$449$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	4.68850	−	26.5898i	0.220285	−	1.24930i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−42.6742			−1.99621			−0.998107	−	0.0615051i	$-0.980410\pi$
$457$	−42.6742			−1.99621			−0.998107	+	0.0615051i	$0.980410\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$461$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$462$		0			0
$463$	−19.9072	−	34.4803i	−0.925166	−	1.60243i	−0.791294	−	0.611435i	$-0.790592\pi$
$463$	−19.9072	−	34.4803i	−0.925166	−	1.60243i	−0.133871	−	0.990999i	$-0.542741\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$467$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$468$	20.8976	−	12.0652i	0.965990	−	0.557714i
$469$	−61.2836	+	10.8060i	−2.82982	+	0.498973i
$470$		0			0
$471$	20.0822	+	23.9330i	0.925338	+	1.10278i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−20.8876	+	6.22161i	−0.958388	+	0.285467i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$479$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$480$		0			0
$481$	−2.15808	−	12.2391i	−0.0984000	−	0.558054i
$482$		0			0
$483$		0			0
$484$	20.6732	+	7.52444i	0.939693	+	0.342020i
$485$		0			0
$486$		0			0
$487$	−34.5000	+	19.9186i	−1.56334	+	0.902597i	−0.566429	+	0.824110i	$0.691675\pi$
$487$	−34.5000	+	19.9186i	−1.56334	+	0.902597i	−0.996915	+	0.0784867i	$0.974991\pi$
$488$		0			0
$489$	−10.5304	+	12.5497i	−0.476203	+	0.567517i
$490$		0			0
$491$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$491$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−15.2317	−	41.8488i	−0.683925	−	1.87907i
$497$		0			0
$498$		0			0
$499$	−3.42674	−	2.87537i	−0.153402	−	0.128719i	0.562857	−	0.826555i	$-0.309702\pi$
$499$	−3.42674	−	2.87537i	−0.153402	−	0.128719i	−0.716258	+	0.697835i	$0.754147\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$503$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	5.41463	−	0.954745i	0.240472	−	0.0424017i
$508$	−15.5868	+	18.5756i	−0.691551	+	0.824159i
$509$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$509$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$510$		0			0
$511$	−4.80871	+	1.75023i	−0.212725	+	0.0774254i
$512$		0			0
$513$	−9.00000	+	20.7846i	−0.397360	+	0.917663i
$514$		0			0
$515$		0			0
$516$	37.7716	+	6.66015i	1.66280	+	0.293197i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$521$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$522$		0			0
$523$	13.5376	−	37.1943i	0.591958	−	1.62639i	−0.174908	−	0.984585i	$-0.555963\pi$
$523$	13.5376	−	37.1943i	0.591958	−	1.62639i	0.766866	−	0.641807i	$-0.221815\pi$
$524$		0			0
$525$	30.7955	−	17.7798i	1.34403	−	0.775974i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	3.99391	−	22.6506i	0.173648	−	0.984808i
$530$		0			0
$531$		0			0
$532$	−8.30365	+	34.8194i	−0.360009	+	1.50961i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	13.9176	+	5.06558i	0.598363	+	0.217786i	0.623404	−	0.781900i	$-0.285749\pi$
$541$	13.9176	+	5.06558i	0.598363	+	0.217786i	−0.0250408	+	0.999686i	$0.507972\pi$
$542$		0			0
$543$	−16.5000	+	28.5788i	−0.708083	+	1.22644i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−28.0275	−	33.4019i	−1.19837	−	1.42816i	−0.876517	−	0.481371i	$-0.840139\pi$
$547$	−28.0275	−	33.4019i	−1.19837	−	1.42816i	−0.321853	−	0.946790i	$-0.604306\pi$
$548$		0			0
$549$	−42.7879	+	15.5735i	−1.82614	+	0.664662i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−1.30824	−	3.59435i	−0.0556319	−	0.152848i
$554$		0			0
$555$		0			0
$556$	−35.8357	−	30.0698i	−1.51977	−	1.27524i
$557$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$557$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$558$		0			0
$559$	38.5627	+	22.2642i	1.63103	+	0.941674i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$563$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	6.41710	−	36.3932i	0.269493	−	1.52837i
$568$		0			0
$569$		0			0			−	1.00000i	$-0.5\pi$
$569$		0			0				1.00000i	$0.5\pi$
$570$		0			0
$571$	−0.367221			−0.0153677			−0.00768386	−	0.999970i	$-0.502446\pi$
$571$	−0.367221			−0.0153677			−0.00768386	+	0.999970i	$0.502446\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	4.16756	+	23.6354i	0.173648	+	0.984808i
$577$	17.5000	+	30.3109i	0.728535	+	1.26186i	0.957503	+	0.288425i	$0.0931316\pi$
$577$	17.5000	+	30.3109i	0.728535	+	1.26186i	−0.228968	+	0.973434i	$0.573535\pi$
$578$		0			0
$579$	41.2717	+	15.0217i	1.71519	+	0.624280i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$587$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$588$		−	34.1553i		−	1.40854i
$589$	28.9261	+	38.9678i	1.19188	+	1.60564i
$590$		0			0
$591$		0			0
$592$	12.1729	+	2.14642i	0.500304	+	0.0882172i
$593$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$593$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−41.2443	−	23.8124i	−1.68802	−	0.974576i
$598$		0			0
$599$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$599$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$600$		0			0
$601$	42.0510	−	24.2782i	1.71530	−	0.990327i	0.788273	−	0.615325i	$-0.210975\pi$
$601$	42.0510	−	24.2782i	1.71530	−	0.990327i	0.927024	−	0.375002i	$-0.122358\pi$
$602$		0			0
$603$	29.2251	−	34.8291i	1.19014	−	1.41835i
$604$	20.0401	+	23.8829i	0.815422	+	0.971782i
$605$		0			0
$606$		0			0
$607$			27.5161i			1.11684i	0.829557	+	0.558422i	$0.188593\pi$
$607$			27.5161i			1.11684i	−0.829557	+	0.558422i	$0.811407\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	28.3436	+	23.7831i	1.14479	+	0.960592i	0.999585	−	0.0288097i	$-0.00917168\pi$
$613$	28.3436	+	23.7831i	1.14479	+	0.960592i	0.145204	+	0.989402i	$0.453616\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$617$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$618$		0			0
$619$	−15.9847	+	27.6864i	−0.642481	+	1.11281i	0.342396	+	0.939556i	$0.388761\pi$
$619$	−15.9847	+	27.6864i	−0.642481	+	1.11281i	−0.984877	+	0.173254i	$0.944572\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	−4.83843	+	27.4401i	−0.193692	+	1.09848i
$625$	−23.4923	+	8.55050i	−0.939693	+	0.342020i
$626$		0			0
$627$		0			0
$628$	−36.0755			−1.43957
$629$		0			0
$630$		0			0
$631$	38.0265	−	31.9080i	1.51381	−	1.27024i	0.657908	−	0.753098i	$-0.271442\pi$
$631$	38.0265	−	31.9080i	1.51381	−	1.27024i	0.855901	−	0.517139i	$-0.173003\pi$
$632$		0			0
$633$	5.20749	+	29.5332i	0.206979	+	1.17384i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	13.5623	−	37.2621i	0.537357	−	1.47638i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$641$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$642$		0			0
$643$	4.12542	−	23.3964i	0.162691	−	0.922664i	−0.788723	−	0.614749i	$-0.789257\pi$
$643$	4.12542	−	23.3964i	0.162691	−	0.922664i	0.951414	−	0.307916i	$-0.0996315\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		1.00000			$0$
$647$		0			0		−1.00000			$\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	−60.6566	−	50.8970i	−2.37732	−	1.99481i
$652$	−3.28487	−	18.6294i	−0.128646	−	0.729585i
$653$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$653$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	1.86942	−	3.23794i	0.0729331	−	0.126324i
$658$		0			0
$659$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$659$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$660$		0			0
$661$	−22.2668	−	26.5366i	−0.866079	−	1.03215i	−0.999157	−	0.0410470i	$-0.986931\pi$
$661$	−22.2668	−	26.5366i	−0.866079	−	1.03215i	0.133078	−	0.991106i	$-0.457514\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−38.9812	+	32.7091i	−1.50710	+	1.26461i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	44.7272	+	25.8233i	1.72411	+	0.995414i	0.909886	+	0.414859i	$0.136169\pi$
$673$	44.7272	+	25.8233i	1.72411	+	0.995414i	0.814221	+	0.580554i	$0.197164\pi$
$674$		0			0
$675$	−8.88594	+	24.4139i	−0.342020	+	0.939693i
$676$	−3.17436	+	5.49816i	−0.122091	+	0.211468i
$677$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$677$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$678$		0			0
$679$	−13.7144	+	16.3441i	−0.526309	+	0.627230i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0			−	1.00000i	$-0.5\pi$
$683$		0			0				1.00000i	$0.5\pi$
$684$	−11.7050	−	23.3879i	−0.447553	−	0.894258i
$685$		0			0
$686$		0			0
$687$	−23.1374	−	4.07976i	−0.882748	−	0.155652i
$688$	−33.9263	+	28.4676i	−1.29343	+	1.08532i
$689$		0			0
$690$		0			0
$691$	−20.5000	−	35.5070i	−0.779857	−	1.35075i	−0.932024	−	0.362397i	$-0.881959\pi$
$691$	−20.5000	−	35.5070i	−0.779857	−	1.35075i	0.152167	−	0.988355i	$-0.451375\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−7.13011	+	40.4369i	−0.269493	+	1.52837i
$701$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$701$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$702$		0			0
$703$	−13.3809	+	1.54509i	−0.504669	+	0.0582742i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−8.33972	−	47.2969i	−0.313205	−	1.77627i	−0.582115	−	0.813107i	$-0.697775\pi$
$709$	−8.33972	−	47.2969i	−0.313205	−	1.77627i	0.268910	−	0.963165i	$-0.413337\pi$
$710$		0			0
$711$	2.42025	+	1.39733i	0.0907666	+	0.0524041i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$719$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$720$		0			0
$721$			41.8905i			1.56008i
$722$		0			0
$723$	15.5452			0.578132
$724$	−13.0327	−	35.8071i	−0.484357	−	1.33076i
$725$		0			0
$726$		0			0
$727$	41.1719	+	34.5473i	1.52698	+	1.28129i	0.815935	+	0.578144i	$0.196223\pi$
$727$	41.1719	+	34.5473i	1.52698	+	1.28129i	0.711046	+	0.703145i	$0.248222\pi$
$728$		0			0
$729$	13.5000	+	23.3827i	0.500000	+	0.866025i
$730$		0			0
$731$		0			0
$732$	17.9828	−	49.4072i	0.664662	−	1.82614i
$733$	21.5000	−	37.2391i	0.794121	−	1.37546i	−0.129275	−	0.991609i	$-0.541265\pi$
$733$	21.5000	−	37.2391i	0.794121	−	1.37546i	0.923396	−	0.383849i	$-0.125402\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	45.4752	−	16.5516i	1.67283	−	0.608862i	0.680534	−	0.732717i	$-0.261748\pi$
$739$	45.4752	−	16.5516i	1.67283	−	0.608862i	0.992300	+	0.123855i	$0.0395259\pi$
$740$		0			0
$741$	−3.48293	−	30.1630i	−0.127948	−	1.10807i
$742$		0			0
$743$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$743$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−15.9700	+	43.8773i	−0.582755	+	1.60110i	0.200698	+	0.979653i	$0.435679\pi$
$751$	−15.9700	+	43.8773i	−0.582755	+	1.60110i	−0.783452	+	0.621452i	$0.786543\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	27.4287	+	32.6883i	0.997573	+	1.18886i
$757$	1.36225	−	7.72573i	0.0495120	−	0.280796i	−0.949993	−	0.312273i	$-0.898910\pi$
$757$	1.36225	−	7.72573i	0.0495120	−	0.280796i	0.999505	+	0.0314762i	$0.0100208\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		1.00000			$0$
$761$		0			0		−1.00000			$\pi$
$762$		0			0
$763$	−35.0194	−	6.17486i	−1.26779	−	0.223545i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	−24.0000	−	13.8564i	−0.866025	−	0.500000i
$769$	−32.0386	−	11.6611i	−1.15534	−	0.420511i	−0.307912	−	0.951415i	$-0.599630\pi$
$769$	−32.0386	−	11.6611i	−1.15534	−	0.420511i	−0.847432	+	0.530904i	$0.821852\pi$
$770$		0			0
$771$		0			0
$772$	−43.9205	+	25.3575i	−1.58073	+	0.912636i
$773$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$773$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$774$		0			0
$775$	35.7829	+	42.6444i	1.28536	+	1.53183i
$776$		0			0
$777$	20.6517	−	7.51661i	0.740876	−	0.269657i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	30.2121	+	25.3510i	1.07900	+	0.905393i
$785$		0			0
$786$		0			0
$787$	33.4717	+	19.3249i	1.19314	+	0.688858i	0.959017	−	0.283350i	$-0.0914459\pi$
$787$	33.4717	+	19.3249i	1.19314	+	0.688858i	0.234120	+	0.972208i	$0.424779\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	39.2369	−	46.7608i	1.39334	−	1.66052i
$794$		0			0
$795$		0			0
$796$	51.6759	−	18.8085i	1.83160	−	0.666650i
$797$		0			0			−	1.00000i	$-0.5\pi$
$797$		0			0				1.00000i	$0.5\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$	9.11650	+	51.7022i	0.321514	+	1.82340i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$809$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$810$		0			0
$811$	−52.8778	+	9.32379i	−1.85679	+	0.327403i	−0.986324	−	0.164821i	$-0.947295\pi$
$811$	−52.8778	+	9.32379i	−1.85679	+	0.327403i	−0.870469	+	0.492223i	$0.836184\pi$
$812$		0			0
$813$	−1.11334	−	1.32683i	−0.0390466	−	0.0465339i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	26.5763	−	40.2848i	0.929785	−	1.40939i
$818$		0			0
$819$	16.9439	+	46.5529i	0.592067	+	1.62669i
$820$		0			0
$821$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$821$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$822$		0			0
$823$	−8.16146	−	46.2860i	−0.284491	−	1.61343i	−0.707099	−	0.707115i	$-0.749996\pi$
$823$	−8.16146	−	46.2860i	−0.284491	−	1.61343i	0.422608	−	0.906313i	$-0.361115\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$827$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$828$		0			0
$829$	−34.0225	+	19.6429i	−1.18165	+	0.682226i	−0.956396	−	0.292074i	$-0.905655\pi$
$829$	−34.0225	+	19.6429i	−1.18165	+	0.682226i	−0.225255	+	0.974300i	$0.572321\pi$
$830$		0			0
$831$	5.56670	−	6.63414i	0.193107	−	0.230136i
$832$	−20.6810	−	24.6466i	−0.716984	−	0.854468i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	57.8522			1.99966
$838$		0			0
$839$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$839$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$840$		0			0
$841$	−22.2153	−	18.6408i	−0.766044	−	0.642788i
$842$		0			0
$843$		0			0
$844$	−29.9887	−	17.3140i	−1.03226	−	0.595973i
$845$		0			0
$846$		0			0
$847$	−22.5834	+	39.1155i	−0.775974	+	1.34403i
$848$		0			0
$849$	11.9402	−	2.10537i	0.409785	−	0.0722562i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−48.9886	+	17.8304i	−1.67734	+	0.610501i	−0.992941	−	0.118609i	$-0.962157\pi$
$853$	−48.9886	+	17.8304i	−1.67734	+	0.610501i	−0.684397	+	0.729110i	$0.739934\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$857$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$858$		0			0
$859$	5.61746	−	4.71361i	0.191665	−	0.160826i	−0.541905	−	0.840440i	$-0.682297\pi$
$859$	5.61746	−	4.71361i	0.191665	−	0.160826i	0.733571	+	0.679613i	$0.237852\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$863$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	25.5000	−	14.7224i	0.866025	−	0.500000i
$868$	90.0420	−	15.8768i	3.05622	−	0.538895i
$869$		0			0
$870$		0			0
$871$	−10.5840	+	60.0250i	−0.358626	+	2.03387i
$872$		0			0
$873$		−	15.5885i		−	0.527589i
$874$		0			0
$875$		0			0
$876$	1.47659	+	4.05689i	0.0498892	+	0.137069i
$877$	−24.6812	−	4.35197i	−0.833426	−	0.146955i	−0.259377	−	0.965776i	$-0.583517\pi$
$877$	−24.6812	−	4.35197i	−0.833426	−	0.146955i	−0.574049	+	0.818821i	$0.694628\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$881$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$882$		0			0
$883$	−53.1921	−	19.3603i	−1.79006	−	0.651528i	−0.999219	−	0.0395021i	$-0.987423\pi$
$883$	−53.1921	−	19.3603i	−1.79006	−	0.651528i	−0.790838	−	0.612026i	$-0.790355\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$887$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$888$		0			0
$889$	−32.0002	−	38.1363i	−1.07325	−	1.27905i
$890$		0			0
$891$		0			0
$892$		−	58.7584i		−	1.96738i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−15.0000	−	25.9808i	−0.500000	−	0.866025i
$901$		0			0
$902$		0			0
$903$	−26.9315	+	73.9938i	−0.896226	+	2.46236i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−36.7402	+	43.7853i	−1.21994	+	1.45387i	−0.368327	+	0.929696i	$0.620069\pi$
$907$	−36.7402	+	43.7853i	−1.21994	+	1.45387i	−0.851613	+	0.524171i	$0.824375\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0			−	1.00000i	$-0.5\pi$
$911$		0			0				1.00000i	$0.5\pi$
$912$	29.3756	+	7.00541i	0.972722	+	0.231972i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	20.7820	−	17.4382i	0.686656	−	0.576173i
$917$		0			0
$918$		0			0
$919$	19.1004	+	33.0829i	0.630065	+	1.09131i	0.987538	+	0.157382i	$0.0503053\pi$
$919$	19.1004	+	33.0829i	0.630065	+	1.09131i	−0.357472	+	0.933924i	$0.616361\pi$
$920$		0			0
$921$	−47.9243	−	17.4430i	−1.57916	−	0.574767i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−15.2162	+	2.68302i	−0.500304	+	0.0882172i
$926$		0			0
$927$	−19.6733	−	23.4458i	−0.646157	−	0.770060i
$928$		0			0
$929$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$929$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$930$		0			0
$931$	−39.4391	−	17.0776i	−1.29257	−	0.559697i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	10.2623	+	58.2006i	0.335256	+	1.90133i	0.424691	+	0.905338i	$0.360383\pi$
$937$	10.2623	+	58.2006i	0.335256	+	1.90133i	−0.0894356	+	0.995993i	$0.528506\pi$
$938$		0			0
$939$	−19.5000	−	11.2583i	−0.636358	−	0.367402i
$940$		0			0
$941$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$941$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$947$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$948$	−3.03239	+	1.10370i	−0.0984876	+	0.0358465i
$949$			5.01222i			0.162703i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$953$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	46.4791	−	80.5042i	1.49933	−	2.59691i
$962$		0			0
$963$		0			0
$964$	−11.5381	+	13.7505i	−0.371616	+	0.442875i
$965$		0			0
$966$		0			0
$967$	−36.5872	+	13.3167i	−1.17657	+	0.428235i	−0.854988	−	0.518648i	$-0.826436\pi$
$967$	−36.5872	+	13.3167i	−1.17657	+	0.428235i	−0.321578	+	0.946883i	$0.604213\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$971$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$972$	−30.7033	−	5.41381i	−0.984808	−	0.173648i
$973$	73.5720	−	61.7342i	2.35861	−	1.97911i
$974$		0			0
$975$	−6.04804	−	34.3001i	−0.193692	−	1.09848i
$976$	30.3560	+	52.5781i	0.971671	+	1.68298i
$977$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$977$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	22.5000	−	12.9904i	0.718370	−	0.414751i
$982$		0			0
$983$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$983$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	29.2659	+	19.3070i	0.931072	+	0.614237i
$989$		0			0
$990$		0			0
$991$	61.8264	+	10.9017i	1.96398	+	0.346303i	0.995116	+	0.0987109i	$0.0314719\pi$
$991$	61.8264	+	10.9017i	1.96398	+	0.346303i	0.968864	+	0.247592i	$0.0796392\pi$
$992$		0			0
$993$	36.1170	+	30.3058i	1.14614	+	0.961726i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	28.8704	+	10.5080i	0.914333	+	0.332790i	0.755982	−	0.654593i	$-0.227160\pi$
$997$	28.8704	+	10.5080i	0.914333	+	0.332790i	0.158352	+	0.987383i	$0.449382\pi$
$998$		0			0
$999$	−8.02852	+	13.9058i	−0.254011	+	0.439960i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	57.2.j.a.14.1	✓	6
3.2	odd	2	CM	57.2.j.a.14.1	✓	6
4.3	odd	2		912.2.cc.a.641.1		6
12.11	even	2		912.2.cc.a.641.1		6
19.2	odd	18		1083.2.d.a.1082.3		6
19.15	odd	18	inner	57.2.j.a.53.1	yes	6
19.17	even	9		1083.2.d.a.1082.6		6
57.2	even	18		1083.2.d.a.1082.3		6
57.17	odd	18		1083.2.d.a.1082.6		6
57.53	even	18	inner	57.2.j.a.53.1	yes	6
76.15	even	18		912.2.cc.a.737.1		6
228.167	odd	18		912.2.cc.a.737.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
57.2.j.a.14.1	✓	6	1.1	even	1	trivial
57.2.j.a.14.1	✓	6	3.2	odd	2	CM
57.2.j.a.53.1	yes	6	19.15	odd	18	inner
57.2.j.a.53.1	yes	6	57.53	even	18	inner
912.2.cc.a.641.1		6	4.3	odd	2
912.2.cc.a.641.1		6	12.11	even	2
912.2.cc.a.737.1		6	76.15	even	18
912.2.cc.a.737.1		6	228.167	odd	18
1083.2.d.a.1082.3		6	19.2	odd	18
1083.2.d.a.1082.3		6	57.2	even	18
1083.2.d.a.1082.6		6	19.17	even	9
1083.2.d.a.1082.6		6	57.17	odd	18

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:	\( N \)	\(=\)	\( 57 = 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	57.j (of order \(18\), degree \(6\), minimal)

\(f(q)\)	\(=\)	\(q+(1.70574 + 0.300767i) q^{3} +(-1.53209 + 1.28558i) q^{4} +(-2.05303 - 3.55596i) q^{7} +(2.81908 + 1.02606i) q^{9} +O(q^{10})\)	\(q+(1.70574 + 0.300767i) q^{3} +(-1.53209 + 1.28558i) q^{4} +(-2.05303 - 3.55596i) q^{7} +(2.81908 + 1.02606i) q^{9} +(-3.00000 + 1.73205i) q^{12} +(-3.96064 + 0.698367i) q^{13} +(0.694593 - 3.93923i) q^{16} +(0.500000 + 4.33013i) q^{19} +(-2.43242 - 6.68302i) q^{21} +(0.868241 + 4.92404i) q^{25} +(4.50000 + 2.59808i) q^{27} +(7.71688 + 2.80872i) q^{28} +(9.64203 - 5.56683i) q^{31} +(-5.63816 + 2.05212i) q^{36} +3.09018i q^{37} -6.96585 q^{39} +(-8.48158 - 7.11689i) q^{43} +(2.36959 - 6.51038i) q^{48} +(-4.92989 + 8.53882i) q^{49} +(5.17024 - 6.16166i) q^{52} +(-0.449493 + 7.53644i) q^{57} +(-11.6270 + 9.75622i) q^{61} +(-2.13903 - 12.1311i) q^{63} +(4.00000 + 6.92820i) q^{64} +(5.18345 - 14.2414i) q^{67} +(0.216415 - 1.22735i) q^{73} +8.66025i q^{75} +(-6.33275 - 5.99135i) q^{76} +(0.917404 + 0.161763i) q^{79} +(6.89440 + 5.78509i) q^{81} +(12.3182 + 7.11192i) q^{84} +(10.6147 + 12.6501i) q^{91} +(18.1211 - 6.59553i) q^{93} +(-1.77719 - 4.88279i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q+O(q^{10}) \) 6 * q	\( 6 q - 18 q^{12} - 15 q^{13} + 3 q^{19} + 9 q^{21} + 27 q^{27} + 30 q^{28} - 39 q^{43} - 21 q^{49} - 12 q^{52} - 42 q^{61} - 36 q^{63} + 24 q^{64} + 33 q^{67} + 51 q^{73} + 12 q^{79} + 48 q^{91} + 54 q^{93}+O(q^{100}) \) 6 * q - 18 * q^12 - 15 * q^13 + 3 * q^19 + 9 * q^21 + 27 * q^27 + 30 * q^28 - 39 * q^43 - 21 * q^49 - 12 * q^52 - 42 * q^61 - 36 * q^63 + 24 * q^64 + 33 * q^67 + 51 * q^73 + 12 * q^79 + 48 * q^91 + 54 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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