LMFDB - Embedded newform 2385.1.ca.a.1729.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2385,1,Mod(19,2385)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2385, base_ring=CyclotomicField(52))

chi = DirichletCharacter(H, H._module([0, 26, 37]))

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("2385.19");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 2385 = 3^{2} \cdot 5 \cdot 53 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2385.ca (of order $52$, degree $24$, 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:	$1.19027005513$
Analytic rank:	$0$
Dimension:	$48$
Relative dimension:	$2$ over $\Q(\zeta_{52})$
Coefficient field:	$\Q(\zeta_{104})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{48} - x^{44} + x^{40} - x^{36} + x^{32} - x^{28} + x^{24} - x^{20} + x^{16} - x^{12} + x^{8} - x^{4} + 1 $ x^48 - x^44 + x^40 - x^36 + x^32 - x^28 + x^24 - x^20 + x^16 - x^12 + x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{52}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{52} - \cdots)$

Embedding invariants

Embedding label			1729.1
Root			$0.855781 - 0.517338i$ of defining polynomial
Character	$\chi$	$=$	2385.1729
Dual form			2385.1.ca.a.469.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.30452 + 0.239062i) q^{2} +(0.709609 - 0.269119i) q^{4} +(-0.616719 - 0.787183i) q^{5} +(0.273612 - 0.165404i) q^{8} +O(q^{10})$	\(q+(-1.30452 + 0.239062i) q^{2} +(0.709609 - 0.269119i) q^{4} +(-0.616719 - 0.787183i) q^{5} +(0.273612 - 0.165404i) q^{8} +(0.992709 + 0.879463i) q^{10} +(-0.885456 + 0.784446i) q^{16} +(0.117248 - 0.0288990i) q^{17} +(0.783659 + 1.74122i) q^{19} +(-0.649475 - 0.392621i) q^{20} +(-1.25222 + 1.25222i) q^{23} +(-0.239316 + 0.970942i) q^{25} +(0.585260 - 1.87816i) q^{31} +(0.770386 - 0.983325i) q^{32} +(-0.146044 + 0.0657290i) q^{34} +(-1.43856 - 2.08411i) q^{38} +(-0.298946 - 0.113375i) q^{40} +(1.33419 - 1.93291i) q^{46} +(1.81050 - 0.219835i) q^{47} +(0.354605 + 0.935016i) q^{49} +(0.0800767 - 1.32383i) q^{50} +(0.855781 + 0.517338i) q^{53} +(-0.885456 - 1.46472i) q^{61} +(-0.314485 + 2.59002i) q^{62} +(-0.220161 + 0.419482i) q^{64} +(0.0754229 - 0.0520607i) q^{68} +(1.02469 + 1.02469i) q^{76} +(1.21323 + 0.222333i) q^{79} +(1.16358 + 0.213234i) q^{80} +(1.32231 + 1.32231i) q^{83} +(-0.0950579 - 0.0744731i) q^{85} +(-0.551592 + 1.22559i) q^{92} +(-2.30928 + 0.719602i) q^{94} +(0.887362 - 1.69073i) q^{95} +(-0.686117 - 1.13498i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q+O(q^{10}) $ 48 * q	$ 48 q + 4 q^{16} - 4 q^{19} - 4 q^{31} + 4 q^{49} + 4 q^{61} + 4 q^{76} + 4 q^{79} - 4 q^{85}+O(q^{100}) $ 48 * q + 4 * q^16 - 4 * q^19 - 4 * q^31 + 4 * q^49 + 4 * q^61 + 4 * q^76 + 4 * q^79 - 4 * q^85

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2385\mathbb{Z}\right)^\times$.

$n$	$1432$	$1486$	$1856$
$\chi(n)$	$-1$	$e\left(\frac{23}{52}\right)$	$1$

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 $

$2$	−1.30452	+	0.239062i	−1.30452	+	0.239062i	−0.787183	−	0.616719i	$-0.788462\pi$
$2$	−1.30452	+	0.239062i	−1.30452	+	0.239062i	−0.517338	+	0.855781i	$0.673077\pi$
$3$		0			0
$3$		0			0
$4$	0.709609	−	0.269119i	0.709609	−	0.269119i
$5$	−0.616719	−	0.787183i	−0.616719	−	0.787183i
$5$	−0.616719	−	0.787183i	−0.616719	−	0.787183i
$6$		0			0
$7$		0			0		−0.822984	−	0.568065i	$-0.807692\pi$
$7$		0			0		0.822984	+	0.568065i	$0.192308\pi$
$8$	0.273612	−	0.165404i	0.273612	−	0.165404i
$9$		0			0
$10$	0.992709	+	0.879463i	0.992709	+	0.879463i
$11$		0			0		−0.885456	−	0.464723i	$-0.846154\pi$
$11$		0			0		0.885456	+	0.464723i	$0.153846\pi$
$12$		0			0
$13$		0			0		0.354605	−	0.935016i	$-0.384615\pi$
$13$		0			0		−0.354605	+	0.935016i	$0.615385\pi$
$14$		0			0
$15$		0			0
$16$	−0.885456	+	0.784446i	−0.885456	+	0.784446i
$17$	0.117248	−	0.0288990i	0.117248	−	0.0288990i	−0.180255	−	0.983620i	$-0.557692\pi$
$17$	0.117248	−	0.0288990i	0.117248	−	0.0288990i	0.297503	+	0.954721i	$0.403846\pi$
$18$		0			0
$19$	0.783659	+	1.74122i	0.783659	+	1.74122i	0.663123	+	0.748511i	$0.269231\pi$
$19$	0.783659	+	1.74122i	0.783659	+	1.74122i	0.120537	+	0.992709i	$0.461538\pi$
$20$	−0.649475	−	0.392621i	−0.649475	−	0.392621i
$21$		0			0
$22$		0			0
$23$	−1.25222	+	1.25222i	−1.25222	+	1.25222i	−0.297503	+	0.954721i	$0.596154\pi$
$23$	−1.25222	+	1.25222i	−1.25222	+	1.25222i	−0.954721	+	0.297503i	$0.903846\pi$
$24$		0			0
$25$	−0.239316	+	0.970942i	−0.239316	+	0.970942i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		0			0		0.885456	−	0.464723i	$-0.153846\pi$
$29$		0			0		−0.885456	+	0.464723i	$0.846154\pi$
$30$		0			0
$31$	0.585260	−	1.87816i	0.585260	−	1.87816i	0.120537	−	0.992709i	$-0.461538\pi$
$31$	0.585260	−	1.87816i	0.585260	−	1.87816i	0.464723	−	0.885456i	$-0.346154\pi$
$32$	0.770386	−	0.983325i	0.770386	−	0.983325i
$33$		0			0
$34$	−0.146044	+	0.0657290i	−0.146044	+	0.0657290i
$35$		0			0
$36$		0			0
$37$		0			0		−0.663123	−	0.748511i	$-0.730769\pi$
$37$		0			0		0.663123	+	0.748511i	$0.269231\pi$
$38$	−1.43856	−	2.08411i	−1.43856	−	2.08411i
$39$		0			0
$40$	−0.298946	−	0.113375i	−0.298946	−	0.113375i
$41$		0			0		−0.297503	−	0.954721i	$-0.596154\pi$
$41$		0			0		0.297503	+	0.954721i	$0.403846\pi$
$42$		0			0
$43$		0			0		0.663123	−	0.748511i	$-0.269231\pi$
$43$		0			0		−0.663123	+	0.748511i	$0.730769\pi$
$44$		0			0
$45$		0			0
$46$	1.33419	−	1.93291i	1.33419	−	1.93291i
$47$	1.81050	−	0.219835i	1.81050	−	0.219835i	0.855781	−	0.517338i	$-0.173077\pi$
$47$	1.81050	−	0.219835i	1.81050	−	0.219835i	0.954721	+	0.297503i	$0.0961538\pi$
$48$		0			0
$49$	0.354605	+	0.935016i	0.354605	+	0.935016i
$50$	0.0800767	−	1.32383i	0.0800767	−	1.32383i
$51$		0			0
$52$		0			0
$53$	0.855781	+	0.517338i	0.855781	+	0.517338i
$53$	0.855781	+	0.517338i	0.855781	+	0.517338i
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		−0.120537	−	0.992709i	$-0.538462\pi$
$59$		0			0		0.120537	+	0.992709i	$0.461538\pi$
$60$		0			0
$61$	−0.885456	−	1.46472i	−0.885456	−	1.46472i	−0.885456	−	0.464723i	$-0.846154\pi$
$61$	−0.885456	−	1.46472i	−0.885456	−	1.46472i		−	1.00000i	$-0.5\pi$
$62$	−0.314485	+	2.59002i	−0.314485	+	2.59002i
$63$		0			0
$64$	−0.220161	+	0.419482i	−0.220161	+	0.419482i
$65$		0			0
$66$		0			0
$67$		0			0		0.410413	−	0.911900i	$-0.365385\pi$
$67$		0			0		−0.410413	+	0.911900i	$0.634615\pi$
$68$	0.0754229	−	0.0520607i	0.0754229	−	0.0520607i
$69$		0			0
$70$		0			0
$71$		0			0		0.998176	−	0.0603785i	$-0.0192308\pi$
$71$		0			0		−0.998176	+	0.0603785i	$0.980769\pi$
$72$		0			0
$73$		0			0		0.517338	−	0.855781i	$-0.326923\pi$
$73$		0			0		−0.517338	+	0.855781i	$0.673077\pi$
$74$		0			0
$75$		0			0
$76$	1.02469	+	1.02469i	1.02469	+	1.02469i
$77$		0			0
$78$		0			0
$79$	1.21323	+	0.222333i	1.21323	+	0.222333i	0.748511	−	0.663123i	$-0.230769\pi$
$79$	1.21323	+	0.222333i	1.21323	+	0.222333i	0.464723	+	0.885456i	$0.346154\pi$
$80$	1.16358	+	0.213234i	1.16358	+	0.213234i
$81$		0			0
$82$		0			0
$83$	1.32231	+	1.32231i	1.32231	+	1.32231i	0.911900	+	0.410413i	$0.134615\pi$
$83$	1.32231	+	1.32231i	1.32231	+	1.32231i	0.410413	+	0.911900i	$0.365385\pi$
$84$		0			0
$85$	−0.0950579	−	0.0744731i	−0.0950579	−	0.0744731i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		−0.239316	−	0.970942i	$-0.576923\pi$
$89$		0			0		0.239316	+	0.970942i	$0.423077\pi$
$90$		0			0
$91$		0			0
$92$	−0.551592	+	1.22559i	−0.551592	+	1.22559i
$93$		0			0
$94$	−2.30928	+	0.719602i	−2.30928	+	0.719602i
$95$	0.887362	−	1.69073i	0.887362	−	1.69073i
$96$		0			0
$97$		0			0		0.120537	−	0.992709i	$-0.461538\pi$
$97$		0			0		−0.120537	+	0.992709i	$0.538462\pi$
$98$	−0.686117	−	1.13498i	−0.686117	−	1.13498i
$99$		0			0
$100$	0.0914785	+	0.753393i	0.0914785	+	0.753393i
$101$		0			0		0.787183	−	0.616719i	$-0.211538\pi$
$101$		0			0		−0.787183	+	0.616719i	$0.788462\pi$
$102$		0			0
$103$		0			0		−0.998176	−	0.0603785i	$-0.980769\pi$
$103$		0			0		0.998176	+	0.0603785i	$0.0192308\pi$
$104$		0			0
$105$		0			0
$106$	−1.24006	−	0.470293i	−1.24006	−	0.470293i
$107$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$107$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$108$		0			0
$109$	0.0624722	−	1.03279i	0.0624722	−	1.03279i	−0.822984	−	0.568065i	$-0.807692\pi$
$109$	0.0624722	−	1.03279i	0.0624722	−	1.03279i	0.885456	−	0.464723i	$-0.153846\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	0.700673	−	1.01510i	0.700673	−	1.01510i	−0.297503	−	0.954721i	$-0.596154\pi$
$113$	0.700673	−	1.01510i	0.700673	−	1.01510i	0.998176	−	0.0603785i	$-0.0192308\pi$
$114$		0			0
$115$	1.75800	+	0.213460i	1.75800	+	0.213460i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	0.568065	+	0.822984i	0.568065	+	0.822984i
$122$	1.50526	+	1.69908i	1.50526	+	1.69908i
$123$		0			0
$124$	−0.0901445	−	1.49027i	−0.0901445	−	1.49027i
$125$	0.911900	−	0.410413i	0.911900	−	0.410413i
$126$		0			0
$127$		0			0		0.616719	−	0.787183i	$-0.288462\pi$
$127$		0			0		−0.616719	+	0.787183i	$0.711538\pi$
$128$	−0.184709	+	0.592752i	−0.184709	+	0.592752i
$129$		0			0
$130$		0			0
$131$		0			0		−0.970942	−	0.239316i	$-0.923077\pi$
$131$		0			0		0.970942	+	0.239316i	$0.0769231\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$	0.0273005	−	0.0273005i	0.0273005	−	0.0273005i
$137$	−0.677097	−	0.210992i	−0.677097	−	0.210992i	−0.0603785	−	0.998176i	$-0.519231\pi$
$137$	−0.677097	−	0.210992i	−0.677097	−	0.210992i	−0.616719	+	0.787183i	$0.711538\pi$
$138$		0			0
$139$	0.308518	+	0.186505i	0.308518	+	0.186505i	0.663123	−	0.748511i	$-0.269231\pi$
$139$	0.308518	+	0.186505i	0.308518	+	0.186505i	−0.354605	+	0.935016i	$0.615385\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$		0			0		−0.748511	−	0.663123i	$-0.769231\pi$
$149$		0			0		0.748511	+	0.663123i	$0.230769\pi$
$150$		0			0
$151$	0.509195	−	0.307819i	0.509195	−	0.307819i	−0.239316	−	0.970942i	$-0.576923\pi$
$151$	0.509195	−	0.307819i	0.509195	−	0.307819i	0.748511	+	0.663123i	$0.230769\pi$
$152$	0.502425	+	0.346799i	0.502425	+	0.346799i
$153$		0			0
$154$		0			0
$155$	−1.83940	+	0.697593i	−1.83940	+	0.697593i
$156$		0			0
$157$		0			0		0.983620	−	0.180255i	$-0.0576923\pi$
$157$		0			0		−0.983620	+	0.180255i	$0.942308\pi$
$158$	−1.63584			−1.63584
$159$		0			0
$160$	−1.24917			−1.24917
$161$		0			0
$162$		0			0
$163$		0			0		0.935016	−	0.354605i	$-0.115385\pi$
$163$		0			0		−0.935016	+	0.354605i	$0.884615\pi$
$164$		0			0
$165$		0			0
$166$	−2.04110	−	1.40887i	−2.04110	−	1.40887i
$167$	−0.972278	+	0.587763i	−0.972278	+	0.587763i	−0.911900	−	0.410413i	$-0.865385\pi$
$167$	−0.972278	+	0.587763i	−0.972278	+	0.587763i	−0.0603785	+	0.998176i	$0.519231\pi$
$168$		0			0
$169$	−0.748511	−	0.663123i	−0.748511	−	0.663123i
$170$	0.141809	+	0.0744270i	0.141809	+	0.0744270i
$171$		0			0
$172$		0			0
$173$	1.77080	+	0.796974i	1.77080	+	0.796974i	0.983620	+	0.180255i	$0.0576923\pi$
$173$	1.77080	+	0.796974i	1.77080	+	0.796974i	0.787183	+	0.616719i	$0.211538\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		−0.855781	−	0.517338i	$-0.826923\pi$
$179$		0			0		0.855781	+	0.517338i	$0.173077\pi$
$180$		0			0
$181$	−1.74122	−	0.542586i	−1.74122	−	0.542586i	−0.748511	−	0.663123i	$-0.769231\pi$
$181$	−1.74122	−	0.542586i	−1.74122	−	0.542586i	−0.992709	+	0.120537i	$0.961538\pi$
$182$		0			0
$183$		0			0
$184$	−0.135501	+	0.549748i	−0.135501	+	0.549748i
$185$		0			0
$186$		0			0
$187$		0			0
$188$	1.22559	−	0.643237i	1.22559	−	0.643237i
$189$		0			0
$190$	−0.753393	+	2.41772i	−0.753393	+	2.41772i
$191$		0			0		0.616719	−	0.787183i	$-0.288462\pi$
$191$		0			0		−0.616719	+	0.787183i	$0.711538\pi$
$192$		0			0
$193$		0			0		0.911900	−	0.410413i	$-0.134615\pi$
$193$		0			0		−0.911900	+	0.410413i	$0.865385\pi$
$194$		0			0
$195$		0			0
$196$	0.503261	+	0.568065i	0.503261	+	0.568065i
$197$	1.13406	+	1.64296i	1.13406	+	1.64296i	0.616719	+	0.787183i	$0.288462\pi$
$197$	1.13406	+	1.64296i	1.13406	+	1.64296i	0.517338	+	0.855781i	$0.326923\pi$
$198$		0			0
$199$	1.85640	+	0.704039i	1.85640	+	0.704039i	0.970942	+	0.239316i	$0.0769231\pi$
$199$	1.85640	+	0.704039i	1.85640	+	0.704039i	0.885456	+	0.464723i	$0.153846\pi$
$200$	0.0951184	+	0.305246i	0.0951184	+	0.305246i
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$		−	0.709210i		−	0.709210i	−0.935016	−	0.354605i	$-0.884615\pi$
$211$		−	0.709210i		−	0.709210i	0.935016	−	0.354605i	$-0.115385\pi$
$212$	0.746495	+	0.136800i	0.746495	+	0.136800i
$213$		0			0
$214$	−0.338085	−	1.84487i	−0.338085	−	1.84487i
$215$		0			0
$216$		0			0
$217$		0			0
$218$	0.165404	+	1.36223i	0.165404	+	1.36223i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		0.464723	−	0.885456i	$-0.346154\pi$
$223$		0			0		−0.464723	+	0.885456i	$0.653846\pi$
$224$		0			0
$225$		0			0
$226$	−0.671370	+	1.49172i	−0.671370	+	1.49172i
$227$	1.29568	−	0.894342i	1.29568	−	0.894342i	0.297503	−	0.954721i	$-0.403846\pi$
$227$	1.29568	−	0.894342i	1.29568	−	0.894342i	0.998176	+	0.0603785i	$0.0192308\pi$
$228$		0			0
$229$	0.447528	+	1.81569i	0.447528	+	1.81569i	0.568065	+	0.822984i	$0.307692\pi$
$229$	0.447528	+	1.81569i	0.447528	+	1.81569i	−0.120537	+	0.992709i	$0.538462\pi$
$230$	−2.34438	+	0.141809i	−2.34438	+	0.141809i
$231$		0			0
$232$		0			0
$233$	0.376771	+	0.295181i	0.376771	+	0.295181i	0.787183	−	0.616719i	$-0.211538\pi$
$233$	0.376771	+	0.295181i	0.376771	+	0.295181i	−0.410413	+	0.911900i	$0.634615\pi$
$234$		0			0
$235$	−1.28962	−	1.28962i	−1.28962	−	1.28962i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		−0.983620	−	0.180255i	$-0.942308\pi$
$239$		0			0		0.983620	+	0.180255i	$0.0576923\pi$
$240$		0			0
$241$	0.222431	+	0.423807i	0.222431	+	0.423807i	0.970942	−	0.239316i	$-0.0769231\pi$
$241$	0.222431	+	0.423807i	0.222431	+	0.423807i	−0.748511	+	0.663123i	$0.769231\pi$
$242$	−0.937797	−	0.937797i	−0.937797	−	0.937797i
$243$		0			0
$244$	−1.02251	−	0.801087i	−1.02251	−	0.801087i
$245$	0.517338	−	0.855781i	0.517338	−	0.855781i
$246$		0			0
$247$		0			0
$248$	−0.150523	−	0.610694i	−0.150523	−	0.610694i
$249$		0			0
$250$	−1.09148	+	0.753393i	−1.09148	+	0.753393i
$251$		0			0		0.410413	−	0.911900i	$-0.365385\pi$
$251$		0			0		−0.410413	+	0.911900i	$0.634615\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	0.156356	−	1.28771i	0.156356	−	1.28771i
$257$	−0.587763	−	0.972278i	−0.587763	−	0.972278i	−0.998176	−	0.0603785i	$-0.980769\pi$
$257$	−0.587763	−	0.972278i	−0.587763	−	0.972278i	0.410413	−	0.911900i	$-0.365385\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−0.350034	−	1.91008i	−0.350034	−	1.91008i	−0.410413	−	0.911900i	$-0.634615\pi$
$263$	−0.350034	−	1.91008i	−0.350034	−	1.91008i	0.0603785	−	0.998176i	$-0.480769\pi$
$264$		0			0
$265$	−0.120537	−	0.992709i	−0.120537	−	0.992709i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0		−0.354605	−	0.935016i	$-0.615385\pi$
$269$		0			0		0.354605	+	0.935016i	$0.384615\pi$
$270$		0			0
$271$	1.63397	−	0.198399i	1.63397	−	0.198399i	0.748511	−	0.663123i	$-0.230769\pi$
$271$	1.63397	−	0.198399i	1.63397	−	0.198399i	0.885456	+	0.464723i	$0.153846\pi$
$272$	−0.0811482	+	0.117564i	−0.0811482	+	0.117564i
$273$		0			0
$274$	0.933728	+	0.113375i	0.933728	+	0.113375i
$275$		0			0
$276$		0			0
$277$		0			0		−0.297503	−	0.954721i	$-0.596154\pi$
$277$		0			0		0.297503	+	0.954721i	$0.403846\pi$
$278$	−0.447054	−	0.169545i	−0.447054	−	0.169545i
$279$		0			0
$280$		0			0
$281$		0			0		−0.663123	−	0.748511i	$-0.730769\pi$
$281$		0			0		0.663123	+	0.748511i	$0.269231\pi$
$282$		0			0
$283$		0			0		−0.0603785	−	0.998176i	$-0.519231\pi$
$283$		0			0		0.0603785	+	0.998176i	$0.480769\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−0.872544	+	0.457946i	−0.872544	+	0.457946i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	0.196436	−	0.796974i	0.196436	−	0.796974i	−0.787183	−	0.616719i	$-0.788462\pi$
$293$	0.196436	−	0.796974i	0.196436	−	0.796974i	0.983620	−	0.180255i	$-0.0576923\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$	−0.590668	+	0.523286i	−0.590668	+	0.523286i
$303$		0			0
$304$	−2.05979	−	0.927035i	−2.05979	−	0.927035i
$305$	−0.606928	+	1.60034i	−0.606928	+	1.60034i
$306$		0			0
$307$		0			0		−0.885456	−	0.464723i	$-0.846154\pi$
$307$		0			0		0.885456	+	0.464723i	$0.153846\pi$
$308$		0			0
$309$		0			0
$310$	2.23277	−	1.34976i	2.23277	−	1.34976i
$311$		0			0		−0.822984	−	0.568065i	$-0.807692\pi$
$311$		0			0		0.822984	+	0.568065i	$0.192308\pi$
$312$		0			0
$313$		0			0		−0.616719	−	0.787183i	$-0.711538\pi$
$313$		0			0		0.616719	+	0.787183i	$0.288462\pi$
$314$		0			0
$315$		0			0
$316$	0.920756	−	0.168735i	0.920756	−	0.168735i
$317$	−1.03468			−1.03468			−0.517338	−	0.855781i	$-0.673077\pi$
$317$	−1.03468			−1.03468			−0.517338	+	0.855781i	$0.673077\pi$
$318$		0			0
$319$		0			0
$320$	0.465987	−	0.0853953i	0.465987	−	0.0853953i
$321$		0			0
$322$		0			0
$323$	0.142202	+	0.181508i	0.142202	+	0.181508i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−0.169725	+	0.447528i	−0.169725	+	0.447528i	−0.992709	−	0.120537i	$-0.961538\pi$
$331$	−0.169725	+	0.447528i	−0.169725	+	0.447528i	0.822984	+	0.568065i	$0.192308\pi$
$332$	1.29418	+	0.582465i	1.29418	+	0.582465i
$333$		0			0
$334$	1.12785	−	0.999184i	1.12785	−	0.999184i
$335$		0			0
$336$		0			0
$337$		0			0		−0.410413	−	0.911900i	$-0.634615\pi$
$337$		0			0		0.410413	+	0.911900i	$0.365385\pi$
$338$	1.13498	+	0.686117i	1.13498	+	0.686117i
$339$		0			0
$340$	−0.0874961	−	0.0272649i	−0.0874961	−	0.0272649i
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$	−2.50058	−	0.616337i	−2.50058	−	0.616337i
$347$	−0.726805	+	0.381457i	−0.726805	+	0.381457i	−0.787183	−	0.616719i	$-0.788462\pi$
$347$	−0.726805	+	0.381457i	−0.726805	+	0.381457i	0.0603785	+	0.998176i	$0.480769\pi$
$348$		0			0
$349$	−0.468379	+	1.50308i	−0.468379	+	1.50308i	0.354605	+	0.935016i	$0.384615\pi$
$349$	−0.468379	+	1.50308i	−0.468379	+	1.50308i	−0.822984	+	0.568065i	$0.807692\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−0.0903879	−	1.49429i	−0.0903879	−	1.49429i	−0.707107	−	0.707107i	$-0.750000\pi$
$353$	−0.0903879	−	1.49429i	−0.0903879	−	1.49429i	0.616719	−	0.787183i	$-0.288462\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		−0.297503	−	0.954721i	$-0.596154\pi$
$359$		0			0		0.297503	+	0.954721i	$0.403846\pi$
$360$		0			0
$361$	−1.75460	+	1.98054i	−1.75460	+	1.98054i
$362$	2.40117	+	0.291555i	2.40117	+	0.291555i
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		−0.354605	−	0.935016i	$-0.615385\pi$
$367$		0			0		0.354605	+	0.935016i	$0.384615\pi$
$368$	0.126488	−	2.09109i	0.126488	−	2.09109i
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		−0.180255	−	0.983620i	$-0.557692\pi$
$373$		0			0		0.180255	+	0.983620i	$0.442308\pi$
$374$		0			0
$375$		0			0
$376$	0.459014	−	0.359615i	0.459014	−	0.359615i
$377$		0			0
$378$		0			0
$379$	−0.731626	−	1.21026i	−0.731626	−	1.21026i	−0.970942	−	0.239316i	$-0.923077\pi$
$379$	−0.731626	−	1.21026i	−0.731626	−	1.21026i	0.239316	−	0.970942i	$-0.423077\pi$
$380$	0.174673	−	1.43856i	0.174673	−	1.43856i
$381$		0			0
$382$		0			0
$383$	−0.887362	+	0.276513i	−0.887362	+	0.276513i	−0.707107	−	0.707107i	$-0.750000\pi$
$383$	−0.887362	+	0.276513i	−0.887362	+	0.276513i	−0.180255	+	0.983620i	$0.557692\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$		0			0		0.998176	−	0.0603785i	$-0.0192308\pi$
$389$		0			0		−0.998176	+	0.0603785i	$0.980769\pi$
$390$		0			0
$391$	−0.110633	+	0.183009i	−0.110633	+	0.183009i
$392$	0.251680	+	0.197179i	0.251680	+	0.197179i
$393$		0			0
$394$	−1.87217	−	1.87217i	−1.87217	−	1.87217i
$395$	−0.573207	−	1.09215i	−0.573207	−	1.09215i
$396$		0			0
$397$		0			0		−0.983620	−	0.180255i	$-0.942308\pi$
$397$		0			0		0.983620	+	0.180255i	$0.0576923\pi$
$398$	−2.59002	−	0.474639i	−2.59002	−	0.474639i
$399$		0			0
$400$	−0.549748	−	1.04746i	−0.549748	−	1.04746i
$401$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$401$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−1.45743	+	1.00599i	−1.45743	+	1.00599i	−0.464723	+	0.885456i	$0.653846\pi$
$409$	−1.45743	+	1.00599i	−1.45743	+	1.00599i	−0.992709	+	0.120537i	$0.961538\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	0.225408	−	1.85640i	0.225408	−	1.85640i
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		0.787183	−	0.616719i	$-0.211538\pi$
$419$		0			0		−0.787183	+	0.616719i	$0.788462\pi$
$420$		0			0
$421$	−0.359852	−	0.0217671i	−0.359852	−	0.0217671i	−0.120537	−	0.992709i	$-0.538462\pi$
$421$	−0.359852	−	0.0217671i	−0.359852	−	0.0217671i	−0.239316	+	0.970942i	$0.576923\pi$
$422$	0.169545	+	0.925179i	0.169545	+	0.925179i
$423$		0			0
$424$	0.319722			0.319722
$425$			0.120757i			0.120757i
$426$		0			0
$427$		0			0
$428$	0.380592	+	1.00354i	0.380592	+	1.00354i
$429$		0			0
$430$		0			0
$431$		0			0		0.568065	−	0.822984i	$-0.307692\pi$
$431$		0			0		−0.568065	+	0.822984i	$0.692308\pi$
$432$		0			0
$433$		0			0		−0.992709	−	0.120537i	$-0.961538\pi$
$433$		0			0		0.992709	+	0.120537i	$0.0384615\pi$
$434$		0			0
$435$		0			0
$436$	−0.233612	−	0.749688i	−0.233612	−	0.749688i
$437$	−3.16171	−	1.19908i	−3.16171	−	1.19908i
$438$		0			0
$439$	−0.645395	−	0.935016i	−0.645395	−	0.935016i	0.354605	−	0.935016i	$-0.384615\pi$
$439$	−0.645395	−	0.935016i	−0.645395	−	0.935016i	−1.00000			$\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−1.36513	+	0.614397i	−1.36513	+	0.614397i	−0.954721	−	0.297503i	$-0.903846\pi$
$443$	−1.36513	+	0.614397i	−1.36513	+	0.614397i	−0.410413	+	0.911900i	$0.634615\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		0			0		−0.970942	−	0.239316i	$-0.923077\pi$
$449$		0			0		0.970942	+	0.239316i	$0.0769231\pi$
$450$		0			0
$451$		0			0
$452$	0.224021	−	0.908888i	0.224021	−	0.908888i
$453$		0			0
$454$	−1.47644	+	1.47644i	−1.47644	+	1.47644i
$455$		0			0
$456$		0			0
$457$		0			0		−0.855781	−	0.517338i	$-0.826923\pi$
$457$		0			0		0.855781	+	0.517338i	$0.173077\pi$
$458$	−1.01787	−	2.26162i	−1.01787	−	2.26162i
$459$		0			0
$460$	1.30494	−	0.321638i	1.30494	−	0.321638i
$461$		0			0		0.748511	−	0.663123i	$-0.230769\pi$
$461$		0			0		−0.748511	+	0.663123i	$0.769231\pi$
$462$		0			0
$463$		0			0		−0.911900	−	0.410413i	$-0.865385\pi$
$463$		0			0		0.911900	+	0.410413i	$0.134615\pi$
$464$		0			0
$465$		0			0
$466$	−0.562072	−	0.294998i	−0.562072	−	0.294998i
$467$	−0.269846	−	0.239062i	−0.269846	−	0.239062i	0.517338	−	0.855781i	$-0.326923\pi$
$467$	−0.269846	−	0.239062i	−0.269846	−	0.239062i	−0.787183	+	0.616719i	$0.788462\pi$
$468$		0			0
$469$		0			0
$470$	1.99064	+	1.37404i	1.99064	+	1.37404i
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−1.87816	+	0.344186i	−1.87816	+	0.344186i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.983620	−	0.180255i	$-0.0576923\pi$
$479$		0			0		−0.983620	+	0.180255i	$0.942308\pi$
$480$		0			0
$481$		0			0
$482$	−0.391482	−	0.499690i	−0.391482	−	0.499690i
$483$		0			0
$484$	0.624584	+	0.431119i	0.624584	+	0.431119i
$485$		0			0
$486$		0			0
$487$		0			0		−0.748511	−	0.663123i	$-0.769231\pi$
$487$		0			0		0.748511	+	0.663123i	$0.230769\pi$
$488$	−0.484544	−	0.254308i	−0.484544	−	0.254308i
$489$		0			0
$490$	−0.470293	+	1.24006i	−0.470293	+	1.24006i
$491$		0			0		−0.911900	−	0.410413i	$-0.865385\pi$
$491$		0			0		0.911900	+	0.410413i	$0.134615\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	0.955096	+	2.12214i	0.955096	+	2.12214i
$497$		0			0
$498$		0			0
$499$	0.783659	+	0.244198i	0.783659	+	0.244198i	0.663123	−	0.748511i	$-0.269231\pi$
$499$	0.783659	+	0.244198i	0.783659	+	0.244198i	0.120537	+	0.992709i	$0.461538\pi$
$500$	0.536642	−	0.536642i	0.536642	−	0.536642i
$501$		0			0
$502$		0			0
$503$	0.269846	−	1.47250i	0.269846	−	1.47250i	−0.517338	−	0.855781i	$-0.673077\pi$
$503$	0.269846	−	1.47250i	0.269846	−	1.47250i	0.787183	−	0.616719i	$-0.211538\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$		0			0		0.616719	−	0.787183i	$-0.288462\pi$
$509$		0			0		−0.616719	+	0.787183i	$0.711538\pi$
$510$		0			0
$511$		0			0
$512$	0.0663859	+	1.09749i	0.0663859	+	1.09749i
$513$		0			0
$514$	0.999184	+	1.12785i	0.999184	+	1.12785i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		−0.992709	−	0.120537i	$-0.961538\pi$
$521$		0			0		0.992709	+	0.120537i	$0.0384615\pi$
$522$		0			0
$523$		0			0		0.568065	−	0.822984i	$-0.307692\pi$
$523$		0			0		−0.568065	+	0.822984i	$0.692308\pi$
$524$		0			0
$525$		0			0
$526$	0.913254	+	2.40805i	0.913254	+	2.40805i
$527$	0.0143434	−	0.237125i	0.0143434	−	0.237125i
$528$		0			0
$529$		−	2.13613i		−	2.13613i
$530$	0.394562	+	1.26619i	0.394562	+	1.26619i
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$	1.11325	−	0.872172i	1.11325	−	0.872172i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	0.527986	−	1.00599i	0.527986	−	1.00599i	−0.464723	−	0.885456i	$-0.653846\pi$
$541$	0.527986	−	1.00599i	0.527986	−	1.00599i	0.992709	−	0.120537i	$-0.0384615\pi$
$542$	−2.08411	+	0.649436i	−2.08411	+	0.649436i
$543$		0			0
$544$	0.0619091	−	0.137556i	0.0619091	−	0.137556i
$545$	−0.851521	+	0.587763i	−0.851521	+	0.587763i
$546$		0			0
$547$		0			0		−0.239316	−	0.970942i	$-0.576923\pi$
$547$		0			0		0.239316	+	0.970942i	$0.423077\pi$
$548$	−0.537256	+	0.0324980i	−0.537256	+	0.0324980i
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$	0.269119	+	0.0493179i	0.269119	+	0.0493179i
$557$	−1.83940	−	0.337083i	−1.83940	−	0.337083i	−0.855781	−	0.517338i	$-0.826923\pi$
$557$	−1.83940	−	0.337083i	−1.83940	−	0.337083i	−0.983620	+	0.180255i	$0.942308\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−0.480838	+	0.795403i	−0.480838	+	0.795403i	−0.998176	−	0.0603785i	$-0.980769\pi$
$563$	−0.480838	+	0.795403i	−0.480838	+	0.795403i	0.517338	+	0.855781i	$0.326923\pi$
$564$		0			0
$565$	−1.23119	+	0.0744731i	−1.23119	+	0.0744731i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		0.410413	−	0.911900i	$-0.365385\pi$
$569$		0			0		−0.410413	+	0.911900i	$0.634615\pi$
$570$		0			0
$571$	−0.115289	+	0.0359256i	−0.115289	+	0.0359256i	−0.354605	−	0.935016i	$-0.615385\pi$
$571$	−0.115289	+	0.0359256i	−0.115289	+	0.0359256i	0.239316	+	0.970942i	$0.423077\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−0.916160	−	1.51551i	−0.916160	−	1.51551i
$576$		0			0
$577$		0			0		−0.120537	−	0.992709i	$-0.538462\pi$
$577$		0			0		0.120537	+	0.992709i	$0.461538\pi$
$578$	1.02877	−	0.805993i	1.02877	−	0.805993i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$	−0.0657290	+	1.08663i	−0.0657290	+	1.08663i
$587$	0.606928	+	1.60034i	0.606928	+	1.60034i	0.787183	+	0.616719i	$0.211538\pi$
$587$	0.606928	+	1.60034i	0.606928	+	1.60034i	−0.180255	+	0.983620i	$0.557692\pi$
$588$		0			0
$589$	3.72894	−	0.452776i	3.72894	−	0.452776i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−0.937797	+	1.05855i	−0.937797	+	1.05855i	0.0603785	+	0.998176i	$0.480769\pi$
$593$	−0.937797	+	1.05855i	−0.937797	+	1.05855i	−0.998176	+	0.0603785i	$0.980769\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		−0.663123	−	0.748511i	$-0.730769\pi$
$599$		0			0		0.663123	+	0.748511i	$0.269231\pi$
$600$		0			0
$601$	−0.0950579	−	1.57149i	−0.0950579	−	1.57149i	−0.663123	−	0.748511i	$-0.730769\pi$
$601$	−0.0950579	−	1.57149i	−0.0950579	−	1.57149i	0.568065	−	0.822984i	$-0.307692\pi$
$602$		0			0
$603$		0			0
$604$	0.278489	−	0.355465i	0.278489	−	0.355465i
$605$	0.297503	−	0.954721i	0.297503	−	0.954721i
$606$		0			0
$607$		0			0		0.885456	−	0.464723i	$-0.153846\pi$
$607$		0			0		−0.885456	+	0.464723i	$0.846154\pi$
$608$	2.31590	+	0.570819i	2.31590	+	0.570819i
$609$		0			0
$610$	0.409170	−	2.23277i	0.409170	−	2.23277i
$611$		0			0
$612$		0			0
$613$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$613$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	0.0989396	+	0.219835i	0.0989396	+	0.219835i	0.954721	−	0.297503i	$-0.0961538\pi$
$617$	0.0989396	+	0.219835i	0.0989396	+	0.219835i	−0.855781	+	0.517338i	$0.826923\pi$
$618$		0			0
$619$	0.902438	−	0.222431i	0.902438	−	0.222431i	0.239316	−	0.970942i	$-0.423077\pi$
$619$	0.902438	−	0.222431i	0.902438	−	0.222431i	0.663123	+	0.748511i	$0.269231\pi$
$620$	−1.11752	+	0.990036i	−1.11752	+	0.990036i
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−0.885456	−	0.464723i	−0.885456	−	0.464723i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	0.970942	+	1.23932i	0.970942	+	1.23932i	0.970942	+	0.239316i	$0.0769231\pi$
$631$	0.970942	+	1.23932i	0.970942	+	1.23932i			1.00000i	$0.500000\pi$
$632$	0.368731	−	0.139841i	0.368731	−	0.139841i
$633$		0			0
$634$	1.34976	−	0.247352i	1.34976	−	0.247352i
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$	0.580518	−	0.220161i	0.580518	−	0.220161i
$641$		0			0		−0.616719	−	0.787183i	$-0.711538\pi$
$641$		0			0		0.616719	+	0.787183i	$0.288462\pi$
$642$		0			0
$643$		0			0		−0.822984	−	0.568065i	$-0.807692\pi$
$643$		0			0		0.822984	+	0.568065i	$0.192308\pi$
$644$		0			0
$645$		0			0
$646$	−0.228897	−	0.202785i	−0.228897	−	0.202785i
$647$	1.09215	+	0.573207i	1.09215	+	0.573207i	0.911900	−	0.410413i	$-0.134615\pi$
$647$	1.09215	+	0.573207i	1.09215	+	0.573207i	0.180255	+	0.983620i	$0.442308\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−1.66183	+	0.409604i	−1.66183	+	0.409604i	−0.954721	−	0.297503i	$-0.903846\pi$
$653$	−1.66183	+	0.409604i	−1.66183	+	0.409604i	−0.707107	+	0.707107i	$0.750000\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$659$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$660$		0			0
$661$	−0.464723	+	1.88546i	−0.464723	+	1.88546i			1.00000i	$0.5\pi$
$661$	−0.464723	+	1.88546i	−0.464723	+	1.88546i	−0.464723	+	0.885456i	$0.653846\pi$
$662$	0.114423	−	0.624385i	0.114423	−	0.624385i
$663$		0			0
$664$	0.580518	+	0.143085i	0.580518	+	0.143085i
$665$		0			0
$666$		0			0
$667$		0			0
$668$	−0.531759	+	0.678740i	−0.531759	+	0.678740i
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		−0.663123	−	0.748511i	$-0.730769\pi$
$673$		0			0		0.663123	+	0.748511i	$0.269231\pi$
$674$		0			0
$675$		0			0
$676$	−0.709609	−	0.269119i	−0.709609	−	0.269119i
$677$	−0.276513	−	0.887362i	−0.276513	−	0.887362i	−0.983620	−	0.180255i	$-0.942308\pi$
$677$	−0.276513	−	0.887362i	−0.276513	−	0.887362i	0.707107	−	0.707107i	$-0.250000\pi$
$678$		0			0
$679$		0			0
$680$	−0.0383272	−	0.00465377i	−0.0383272	−	0.00465377i
$681$		0			0
$682$		0			0
$683$	1.98180	−	0.240634i	1.98180	−	0.240634i	0.983620	−	0.180255i	$-0.0576923\pi$
$683$	1.98180	−	0.240634i	1.98180	−	0.240634i	0.998176	−	0.0603785i	$-0.0192308\pi$
$684$		0			0
$685$	0.251489	+	0.663123i	0.251489	+	0.663123i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	0.308518	+	1.68353i	0.308518	+	1.68353i	0.663123	+	0.748511i	$0.269231\pi$
$691$	0.308518	+	1.68353i	0.308518	+	1.68353i	−0.354605	+	0.935016i	$0.615385\pi$
$692$	1.47106	+	0.0889826i	1.47106	+	0.0889826i
$693$		0			0
$694$	0.856941	−	0.671370i	0.856941	−	0.671370i
$695$	−0.0434547	−	0.357882i	−0.0434547	−	0.357882i
$696$		0			0
$697$		0			0
$698$	0.251680	−	2.07277i	0.251680	−	2.07277i
$699$		0			0
$700$		0			0
$701$		0			0		0.954721	−	0.297503i	$-0.0961538\pi$
$701$		0			0		−0.954721	+	0.297503i	$0.903846\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$	0.475142	+	1.92773i	0.475142	+	1.92773i
$707$		0			0
$708$		0			0
$709$	−0.731626	+	1.21026i	−0.731626	+	1.21026i	0.239316	+	0.970942i	$0.423077\pi$
$709$	−0.731626	+	1.21026i	−0.731626	+	1.21026i	−0.970942	+	0.239316i	$0.923077\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	1.61901	+	3.08476i	1.61901	+	3.08476i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$719$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$720$		0			0
$721$		0			0
$722$	1.81544	−	3.00311i	1.81544	−	3.00311i
$723$		0			0
$724$	−1.38160	+	0.0835717i	−1.38160	+	0.0835717i
$725$		0			0
$726$		0			0
$727$		0			0		0.822984	−	0.568065i	$-0.192308\pi$
$727$		0			0		−0.822984	+	0.568065i	$0.807692\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		0.120537	−	0.992709i	$-0.461538\pi$
$733$		0			0		−0.120537	+	0.992709i	$0.538462\pi$
$734$		0			0
$735$		0			0
$736$	0.266647	+	2.19604i	0.266647	+	2.19604i
$737$		0			0
$738$		0			0
$739$	−1.03279	−	0.0624722i	−1.03279	−	0.0624722i	−0.464723	−	0.885456i	$-0.653846\pi$
$739$	−1.03279	−	0.0624722i	−1.03279	−	0.0624722i	−0.568065	+	0.822984i	$0.692308\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			1.57437i			1.57437i	0.616719	+	0.787183i	$0.288462\pi$
$743$			1.57437i			1.57437i	−0.616719	+	0.787183i	$0.711538\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	1.92773	+	0.234068i	1.92773	+	0.234068i	0.992709	−	0.120537i	$-0.0384615\pi$
$751$	1.92773	+	0.234068i	1.92773	+	0.234068i	0.935016	+	0.354605i	$0.115385\pi$
$752$	−1.43067	+	1.61489i	−1.43067	+	1.61489i
$753$		0			0
$754$		0			0
$755$	−0.556340	−	0.210992i	−0.556340	−	0.210992i
$756$		0			0
$757$		0			0		−0.568065	−	0.822984i	$-0.692308\pi$
$757$		0			0		0.568065	+	0.822984i	$0.307692\pi$
$758$	1.24375	+	1.40390i	1.24375	+	1.40390i
$759$		0			0
$760$	−0.0368605	−	0.609378i	−0.0368605	−	0.609378i
$761$		0			0		0.911900	−	0.410413i	$-0.134615\pi$
$761$		0			0		−0.911900	+	0.410413i	$0.865385\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$	1.09148	−	0.572852i	1.09148	−	0.572852i
$767$		0			0
$768$		0			0
$769$	0.107253	−	0.585260i	0.107253	−	0.585260i	−0.885456	−	0.464723i	$-0.846154\pi$
$769$	0.107253	−	0.585260i	0.107253	−	0.585260i	0.992709	−	0.120537i	$-0.0384615\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−1.42924	−	0.445368i	−1.42924	−	0.445368i	−0.517338	−	0.855781i	$-0.673077\pi$
$773$	−1.42924	−	0.445368i	−1.42924	−	0.445368i	−0.911900	+	0.410413i	$0.865385\pi$
$774$		0			0
$775$	1.68353	+	1.01773i	1.68353	+	1.01773i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$	0.100572	−	0.265187i	0.100572	−	0.265187i
$783$		0			0
$784$	−1.04746	−	0.549748i	−1.04746	−	0.549748i
$785$		0			0
$786$		0			0
$787$		0			0		0.855781	−	0.517338i	$-0.173077\pi$
$787$		0			0		−0.855781	+	0.517338i	$0.826923\pi$
$788$	1.24689	+	0.860666i	1.24689	+	0.860666i
$789$		0			0
$790$	1.00885	+	1.28771i	1.00885	+	1.28771i
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$	1.50679			1.50679
$797$	0.470791	−	0.0862757i	0.470791	−	0.0862757i	0.0603785	−	0.998176i	$-0.480769\pi$
$797$	0.470791	−	0.0862757i	0.470791	−	0.0862757i	0.410413	+	0.911900i	$0.365385\pi$
$798$		0			0
$799$	0.205925	−	0.0780970i	0.205925	−	0.0780970i
$800$	0.770386	+	0.983325i	0.770386	+	0.983325i
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		−0.911900	−	0.410413i	$-0.865385\pi$
$809$		0			0		0.911900	+	0.410413i	$0.134615\pi$
$810$		0			0
$811$	−0.850405	+	0.753393i	−0.850405	+	0.753393i	−0.970942	−	0.239316i	$-0.923077\pi$
$811$	−0.850405	+	0.753393i	−0.850405	+	0.753393i	0.120537	+	0.992709i	$0.461538\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	1.66076	−	1.66076i	1.66076	−	1.66076i
$819$		0			0
$820$		0			0
$821$		0			0		0.180255	−	0.983620i	$-0.442308\pi$
$821$		0			0		−0.180255	+	0.983620i	$0.557692\pi$
$822$		0			0
$823$		0			0		−0.970942	−	0.239316i	$-0.923077\pi$
$823$		0			0		0.970942	+	0.239316i	$0.0769231\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	0.148674	−	0.189769i	0.148674	−	0.189769i	−0.707107	−	0.707107i	$-0.750000\pi$
$827$	0.148674	−	0.189769i	0.148674	−	0.189769i	0.855781	+	0.517338i	$0.173077\pi$
$828$		0			0
$829$	−1.82047	+	0.819328i	−1.82047	+	0.819328i	−0.885456	+	0.464723i	$0.846154\pi$
$829$	−1.82047	+	0.819328i	−1.82047	+	0.819328i	−0.935016	+	0.354605i	$0.884615\pi$
$830$	0.149746	+	2.47560i	0.149746	+	2.47560i
$831$		0			0
$832$		0			0
$833$	0.0685978	+	0.0993811i	0.0685978	+	0.0993811i
$834$		0			0
$835$	1.06230	+	0.402877i	1.06230	+	0.402877i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		−0.992709	−	0.120537i	$-0.961538\pi$
$839$		0			0		0.992709	+	0.120537i	$0.0384615\pi$
$840$		0			0
$841$	0.568065	−	0.822984i	0.568065	−	0.822984i
$842$	0.474639	−	0.0576316i	0.474639	−	0.0576316i
$843$		0			0
$844$	−0.190862	−	0.503261i	−0.190862	−	0.503261i
$845$	−0.0603785	+	0.998176i	−0.0603785	+	0.998176i
$846$		0			0
$847$		0			0
$848$	−1.16358	+	0.213234i	−1.16358	+	0.213234i
$849$		0			0
$850$	−0.0288685	−	0.157530i	−0.0288685	−	0.157530i
$851$		0			0
$852$		0			0
$853$		0			0		0.787183	−	0.616719i	$-0.211538\pi$
$853$		0			0		−0.787183	+	0.616719i	$0.788462\pi$
$854$		0			0
$855$		0			0
$856$	0.233917	+	0.386946i	0.233917	+	0.386946i
$857$	0.240634	−	1.98180i	0.240634	−	1.98180i	0.0603785	−	0.998176i	$-0.480769\pi$
$857$	0.240634	−	1.98180i	0.240634	−	1.98180i	0.180255	−	0.983620i	$-0.442308\pi$
$858$		0			0
$859$	−0.922670	+	1.75800i	−0.922670	+	1.75800i	−0.354605	+	0.935016i	$0.615385\pi$
$859$	−0.922670	+	1.75800i	−0.922670	+	1.75800i	−0.568065	+	0.822984i	$0.692308\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	0.489680	−	0.338002i	0.489680	−	0.338002i	−0.297503	−	0.954721i	$-0.596154\pi$
$863$	0.489680	−	0.338002i	0.489680	−	0.338002i	0.787183	+	0.616719i	$0.211538\pi$
$864$		0			0
$865$	−0.464723	−	1.88546i	−0.464723	−	1.88546i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$	−0.153735	−	0.292917i	−0.153735	−	0.292917i
$873$		0			0
$874$	4.41118	+	0.808378i	4.41118	+	0.808378i
$875$		0			0
$876$		0			0
$877$		0			0		−0.464723	−	0.885456i	$-0.653846\pi$
$877$		0			0		0.464723	+	0.885456i	$0.346154\pi$
$878$	1.06546	+	1.06546i	1.06546	+	1.06546i
$879$		0			0
$880$		0			0
$881$		0			0		0.517338	−	0.855781i	$-0.326923\pi$
$881$		0			0		−0.517338	+	0.855781i	$0.673077\pi$
$882$		0			0
$883$		0			0		0.998176	−	0.0603785i	$-0.0192308\pi$
$883$		0			0		−0.998176	+	0.0603785i	$0.980769\pi$
$884$		0			0
$885$		0			0
$886$	1.63397	−	1.12785i	1.63397	−	1.12785i
$887$	0.814841	−	1.81050i	0.814841	−	1.81050i	0.297503	−	0.954721i	$-0.403846\pi$
$887$	0.814841	−	1.81050i	0.814841	−	1.81050i	0.517338	−	0.855781i	$-0.326923\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	1.80160	+	2.98021i	1.80160	+	2.98021i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$	0.115289	+	0.0359256i	0.115289	+	0.0359256i
$902$		0			0
$903$		0			0
$904$	0.0238107	−	0.393638i	0.0238107	−	0.393638i
$905$	0.646728	+	1.70528i	0.646728	+	1.70528i
$906$		0			0
$907$		0			0		0.992709	−	0.120537i	$-0.0384615\pi$
$907$		0			0		−0.992709	+	0.120537i	$0.961538\pi$
$908$	0.678740	−	0.983325i	0.678740	−	0.983325i
$909$		0			0
$910$		0			0
$911$		0			0		0.663123	−	0.748511i	$-0.269231\pi$
$911$		0			0		−0.663123	+	0.748511i	$0.730769\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$	0.806207	+	1.16799i	0.806207	+	1.16799i
$917$		0			0
$918$		0			0
$919$	−0.103342	−	1.70844i	−0.103342	−	1.70844i	−0.568065	−	0.822984i	$-0.692308\pi$
$919$	−0.103342	−	1.70844i	−0.103342	−	1.70844i	0.464723	−	0.885456i	$-0.346154\pi$
$920$	0.516318	−	0.232376i	0.516318	−	0.232376i
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$		0			0		0.239316	−	0.970942i	$-0.423077\pi$
$929$		0			0		−0.239316	+	0.970942i	$0.576923\pi$
$930$		0			0
$931$	−1.35018	+	1.35018i	−1.35018	+	1.35018i
$932$	0.346799	+	0.108067i	0.346799	+	0.108067i
$933$		0			0
$934$	0.409170	+	0.247352i	0.409170	+	0.247352i
$935$		0			0
$936$		0			0
$937$		0			0		0.970942	−	0.239316i	$-0.0769231\pi$
$937$		0			0		−0.970942	+	0.239316i	$0.923077\pi$
$938$		0			0
$939$		0			0
$940$	−1.26219	−	0.568065i	−1.26219	−	0.568065i
$941$		0			0		0.354605	−	0.935016i	$-0.384615\pi$
$941$		0			0		−0.354605	+	0.935016i	$0.615385\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	0.851521	+	0.587763i	0.851521	+	0.587763i	0.911900	−	0.410413i	$-0.134615\pi$
$947$	0.851521	+	0.587763i	0.851521	+	0.587763i	−0.0603785	+	0.998176i	$0.519231\pi$
$948$		0			0
$949$		0			0
$950$	2.36782	−	0.897997i	2.36782	−	0.897997i
$951$		0			0
$952$		0			0
$953$	−1.90944			−1.90944			−0.954721	−	0.297503i	$-0.903846\pi$
$953$	−1.90944			−1.90944			−0.954721	+	0.297503i	$0.903846\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−2.36199	−	1.63036i	−2.36199	−	1.63036i
$962$		0			0
$963$		0			0
$964$	0.271894	+	0.240877i	0.271894	+	0.240877i
$965$		0			0
$966$		0			0
$967$		0			0		0.354605	−	0.935016i	$-0.384615\pi$
$967$		0			0		−0.354605	+	0.935016i	$0.615385\pi$
$968$	0.291555	+	0.131218i	0.291555	+	0.131218i
$969$		0			0
$970$		0			0
$971$		0			0		0.970942	−	0.239316i	$-0.0769231\pi$
$971$		0			0		−0.970942	+	0.239316i	$0.923077\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$	1.93303	+	0.602356i	1.93303	+	0.602356i
$977$	1.40390	−	1.40390i	1.40390	−	1.40390i	0.616719	−	0.787183i	$-0.288462\pi$
$977$	1.40390	−	1.40390i	1.40390	−	1.40390i	0.787183	−	0.616719i	$-0.211538\pi$
$978$		0			0
$979$		0			0
$980$	0.136800	−	0.746495i	0.136800	−	0.746495i
$981$		0			0
$982$		0			0
$983$	0.319216	−	0.167537i	0.319216	−	0.167537i	−0.297503	−	0.954721i	$-0.596154\pi$
$983$	0.319216	−	0.167537i	0.319216	−	0.167537i	0.616719	+	0.787183i	$0.288462\pi$
$984$		0			0
$985$	0.593921	−	1.90596i	0.593921	−	1.90596i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−0.616337	−	0.695701i	−0.616337	−	0.695701i	0.354605	−	0.935016i	$-0.384615\pi$
$991$	−0.616337	−	0.695701i	−0.616337	−	0.695701i	−0.970942	+	0.239316i	$0.923077\pi$
$992$	−1.39597	−	2.02241i	−1.39597	−	2.02241i
$993$		0			0
$994$		0			0
$995$	−0.590668	−	1.89552i	−0.590668	−	1.89552i
$996$		0			0
$997$		0			0		0.663123	−	0.748511i	$-0.269231\pi$
$997$		0			0		−0.663123	+	0.748511i	$0.730769\pi$
$998$	−1.08068	−	0.131218i	−1.08068	−	0.131218i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2385.1.ca.a.1729.1	yes	48
3.2	odd	2	inner	2385.1.ca.a.1729.2	yes	48
5.4	even	2	inner	2385.1.ca.a.1729.2	yes	48
15.14	odd	2	CM	2385.1.ca.a.1729.1	yes	48
53.45	odd	52	inner	2385.1.ca.a.469.2	yes	48
159.98	even	52	inner	2385.1.ca.a.469.1	✓	48
265.204	odd	52	inner	2385.1.ca.a.469.1	✓	48
795.734	even	52	inner	2385.1.ca.a.469.2	yes	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2385.1.ca.a.469.1	✓	48	159.98	even	52	inner
2385.1.ca.a.469.1	✓	48	265.204	odd	52	inner
2385.1.ca.a.469.2	yes	48	53.45	odd	52	inner
2385.1.ca.a.469.2	yes	48	795.734	even	52	inner
2385.1.ca.a.1729.1	yes	48	1.1	even	1	trivial
2385.1.ca.a.1729.1	yes	48	15.14	odd	2	CM
2385.1.ca.a.1729.2	yes	48	3.2	odd	2	inner
2385.1.ca.a.1729.2	yes	48	5.4	even	2	inner

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:	\( N \)	\(=\)	\( 2385 = 3^{2} \cdot 5 \cdot 53 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2385.ca (of order \(52\), degree \(24\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.30452 + 0.239062i) q^{2} +(0.709609 - 0.269119i) q^{4} +(-0.616719 - 0.787183i) q^{5} +(0.273612 - 0.165404i) q^{8} +O(q^{10})\)	\(q+(-1.30452 + 0.239062i) q^{2} +(0.709609 - 0.269119i) q^{4} +(-0.616719 - 0.787183i) q^{5} +(0.273612 - 0.165404i) q^{8} +(0.992709 + 0.879463i) q^{10} +(-0.885456 + 0.784446i) q^{16} +(0.117248 - 0.0288990i) q^{17} +(0.783659 + 1.74122i) q^{19} +(-0.649475 - 0.392621i) q^{20} +(-1.25222 + 1.25222i) q^{23} +(-0.239316 + 0.970942i) q^{25} +(0.585260 - 1.87816i) q^{31} +(0.770386 - 0.983325i) q^{32} +(-0.146044 + 0.0657290i) q^{34} +(-1.43856 - 2.08411i) q^{38} +(-0.298946 - 0.113375i) q^{40} +(1.33419 - 1.93291i) q^{46} +(1.81050 - 0.219835i) q^{47} +(0.354605 + 0.935016i) q^{49} +(0.0800767 - 1.32383i) q^{50} +(0.855781 + 0.517338i) q^{53} +(-0.885456 - 1.46472i) q^{61} +(-0.314485 + 2.59002i) q^{62} +(-0.220161 + 0.419482i) q^{64} +(0.0754229 - 0.0520607i) q^{68} +(1.02469 + 1.02469i) q^{76} +(1.21323 + 0.222333i) q^{79} +(1.16358 + 0.213234i) q^{80} +(1.32231 + 1.32231i) q^{83} +(-0.0950579 - 0.0744731i) q^{85} +(-0.551592 + 1.22559i) q^{92} +(-2.30928 + 0.719602i) q^{94} +(0.887362 - 1.69073i) q^{95} +(-0.686117 - 1.13498i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q+O(q^{10}) \) 48 * q	\( 48 q + 4 q^{16} - 4 q^{19} - 4 q^{31} + 4 q^{49} + 4 q^{61} + 4 q^{76} + 4 q^{79} - 4 q^{85}+O(q^{100}) \) 48 * q + 4 * q^16 - 4 * q^19 - 4 * q^31 + 4 * q^49 + 4 * q^61 + 4 * q^76 + 4 * q^79 - 4 * q^85

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