LMFDB - Embedded newform 2385.1.ca.a.2359.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2385,1,Mod(19,2385)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage: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")

magma://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:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)

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			2359.2
Root			$0.180255 + 0.983620i$ of defining polynomial
Character	$\chi$	$=$	2385.2359
Dual form			2385.1.ca.a.1009.2

$q$ -expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.0288990 + 0.477758i) q^{2} +(0.765291 - 0.0929232i) q^{4} +(0.297503 - 0.954721i) q^{5} +(0.152787 + 0.833730i) q^{8} +(0.464723 + 0.114544i) q^{10} +(0.354605 - 0.0874023i) q^{16} +(-0.587763 - 0.851521i) q^{17} +(1.12477 + 1.43566i) q^{19} +(0.138961 - 0.758284i) q^{20} +(0.501487 - 0.501487i) q^{23} +(-0.822984 - 0.568065i) q^{25} +(-0.0495602 - 0.110118i) q^{31} +(0.304173 + 0.976124i) q^{32} +(0.389835 - 0.305417i) q^{34} +(-0.653396 + 0.578858i) q^{38} +(0.841434 + 0.102169i) q^{40} +(0.254082 + 0.225097i) q^{46} +(-0.731645 + 1.39403i) q^{47} +(-0.120537 - 0.992709i) q^{49} +(0.247614 - 0.409604i) q^{50} +(0.180255 - 0.983620i) q^{53} +(0.354605 - 0.0649838i) q^{61} +(0.0511777 - 0.0268601i) q^{62} +(-0.116077 + 0.0440221i) q^{64} +(-0.528936 - 0.597045i) q^{68} +(0.994184 + 0.994184i) q^{76} +(0.0359256 - 0.593921i) q^{79} +(0.0220513 - 0.364551i) q^{80} +(-1.40390 - 1.40390i) q^{83} +(-0.987826 + 0.307819i) q^{85} +(0.337184 - 0.430383i) q^{92} +(-0.687154 - 0.309263i) q^{94} +(1.70528 - 0.646728i) q^{95} +(0.470791 - 0.0862757i) q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$48 q + 4 q^{16} - 4 q^{19} - 4 q^{31} + 4 q^{49} + 4 q^{61} + 4 q^{76} + 4 q^{79} - 4 q^{85}+O(q^{100})$ 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

Character values

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

$n$	$1432$	$1486$	$1856$
$\chi(n)$	$-1$	$e\left(\frac{51}{52}\right)$	$1$

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

)

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

$2$	0.0288990	+	0.477758i	0.0288990	+	0.477758i	0.983620	+	0.180255i	$0.0576923\pi$
$2$	0.0288990	+	0.477758i	0.0288990	+	0.477758i	−0.954721	+	0.297503i	$0.903846\pi$
$3$		0			0
$3$		0			0
$4$	0.765291	−	0.0929232i	0.765291	−	0.0929232i
$5$	0.297503	−	0.954721i	0.297503	−	0.954721i
$5$	0.297503	−	0.954721i	0.297503	−	0.954721i
$6$		0			0
$7$		0			0		0.663123	−	0.748511i	$-0.269231\pi$
$7$		0			0		−0.663123	+	0.748511i	$0.730769\pi$
$8$	0.152787	+	0.833730i	0.152787	+	0.833730i
$9$		0			0
$10$	0.464723	+	0.114544i	0.464723	+	0.114544i
$11$		0			0		−0.354605	−	0.935016i	$-0.615385\pi$
$11$		0			0		0.354605	+	0.935016i	$0.384615\pi$
$12$		0			0
$13$		0			0		0.120537	−	0.992709i	$-0.461538\pi$
$13$		0			0		−0.120537	+	0.992709i	$0.538462\pi$
$14$		0			0
$15$		0			0
$16$	0.354605	−	0.0874023i	0.354605	−	0.0874023i
$17$	−0.587763	−	0.851521i	−0.587763	−	0.851521i	0.410413	−	0.911900i	$-0.365385\pi$
$17$	−0.587763	−	0.851521i	−0.587763	−	0.851521i	−0.998176	+	0.0603785i	$0.980769\pi$
$18$		0			0
$19$	1.12477	+	1.43566i	1.12477	+	1.43566i	0.885456	+	0.464723i	$0.153846\pi$
$19$	1.12477	+	1.43566i	1.12477	+	1.43566i	0.239316	+	0.970942i	$0.423077\pi$
$20$	0.138961	−	0.758284i	0.138961	−	0.758284i
$21$		0			0
$22$		0			0
$23$	0.501487	−	0.501487i	0.501487	−	0.501487i	−0.410413	−	0.911900i	$-0.634615\pi$
$23$	0.501487	−	0.501487i	0.501487	−	0.501487i	0.911900	+	0.410413i	$0.134615\pi$
$24$		0			0
$25$	−0.822984	−	0.568065i	−0.822984	−	0.568065i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		0			0		0.354605	−	0.935016i	$-0.384615\pi$
$29$		0			0		−0.354605	+	0.935016i	$0.615385\pi$
$30$		0			0
$31$	−0.0495602	−	0.110118i	−0.0495602	−	0.110118i	0.885456	−	0.464723i	$-0.153846\pi$
$31$	−0.0495602	−	0.110118i	−0.0495602	−	0.110118i	−0.935016	+	0.354605i	$0.884615\pi$
$32$	0.304173	+	0.976124i	0.304173	+	0.976124i
$33$		0			0
$34$	0.389835	−	0.305417i	0.389835	−	0.305417i
$35$		0			0
$36$		0			0
$37$		0			0		−0.239316	−	0.970942i	$-0.576923\pi$
$37$		0			0		0.239316	+	0.970942i	$0.423077\pi$
$38$	−0.653396	+	0.578858i	−0.653396	+	0.578858i
$39$		0			0
$40$	0.841434	+	0.102169i	0.841434	+	0.102169i
$41$		0			0		0.410413	−	0.911900i	$-0.365385\pi$
$41$		0			0		−0.410413	+	0.911900i	$0.634615\pi$
$42$		0			0
$43$		0			0		0.239316	−	0.970942i	$-0.423077\pi$
$43$		0			0		−0.239316	+	0.970942i	$0.576923\pi$
$44$		0			0
$45$		0			0
$46$	0.254082	+	0.225097i	0.254082	+	0.225097i
$47$	−0.731645	+	1.39403i	−0.731645	+	1.39403i	0.180255	+	0.983620i	$0.442308\pi$
$47$	−0.731645	+	1.39403i	−0.731645	+	1.39403i	−0.911900	+	0.410413i	$0.865385\pi$
$48$		0			0
$49$	−0.120537	−	0.992709i	−0.120537	−	0.992709i
$50$	0.247614	−	0.409604i	0.247614	−	0.409604i
$51$		0			0
$52$		0			0
$53$	0.180255	−	0.983620i	0.180255	−	0.983620i
$53$	0.180255	−	0.983620i	0.180255	−	0.983620i
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		−0.885456	−	0.464723i	$-0.846154\pi$
$59$		0			0		0.885456	+	0.464723i	$0.153846\pi$
$60$		0			0
$61$	0.354605	−	0.0649838i	0.354605	−	0.0649838i		−	1.00000i	$-0.5\pi$
$61$	0.354605	−	0.0649838i	0.354605	−	0.0649838i	0.354605	+	0.935016i	$0.384615\pi$
$62$	0.0511777	−	0.0268601i	0.0511777	−	0.0268601i
$63$		0			0
$64$	−0.116077	+	0.0440221i	−0.116077	+	0.0440221i
$65$		0			0
$66$		0			0
$67$		0			0		0.616719	−	0.787183i	$-0.288462\pi$
$67$		0			0		−0.616719	+	0.787183i	$0.711538\pi$
$68$	−0.528936	−	0.597045i	−0.528936	−	0.597045i
$69$		0			0
$70$		0			0
$71$		0			0		0.855781	−	0.517338i	$-0.173077\pi$
$71$		0			0		−0.855781	+	0.517338i	$0.826923\pi$
$72$		0			0
$73$		0			0		−0.983620	−	0.180255i	$-0.942308\pi$
$73$		0			0		0.983620	+	0.180255i	$0.0576923\pi$
$74$		0			0
$75$		0			0
$76$	0.994184	+	0.994184i	0.994184	+	0.994184i
$77$		0			0
$78$		0			0
$79$	0.0359256	−	0.593921i	0.0359256	−	0.593921i	−0.935016	−	0.354605i	$-0.884615\pi$
$79$	0.0359256	−	0.593921i	0.0359256	−	0.593921i	0.970942	−	0.239316i	$-0.0769231\pi$
$80$	0.0220513	−	0.364551i	0.0220513	−	0.364551i
$81$		0			0
$82$		0			0
$83$	−1.40390	−	1.40390i	−1.40390	−	1.40390i	−0.787183	−	0.616719i	$-0.788462\pi$
$83$	−1.40390	−	1.40390i	−1.40390	−	1.40390i	−0.616719	−	0.787183i	$-0.711538\pi$
$84$		0			0
$85$	−0.987826	+	0.307819i	−0.987826	+	0.307819i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		0.822984	−	0.568065i	$-0.192308\pi$
$89$		0			0		−0.822984	+	0.568065i	$0.807692\pi$
$90$		0			0
$91$		0			0
$92$	0.337184	−	0.430383i	0.337184	−	0.430383i
$93$		0			0
$94$	−0.687154	−	0.309263i	−0.687154	−	0.309263i
$95$	1.70528	−	0.646728i	1.70528	−	0.646728i
$96$		0			0
$97$		0			0		0.885456	−	0.464723i	$-0.153846\pi$
$97$		0			0		−0.885456	+	0.464723i	$0.846154\pi$
$98$	0.470791	−	0.0862757i	0.470791	−	0.0862757i
$99$		0			0
$100$	−0.682609	−	0.358261i	−0.682609	−	0.358261i
$101$		0			0		−0.954721	−	0.297503i	$-0.903846\pi$
$101$		0			0		0.954721	+	0.297503i	$0.0961538\pi$
$102$		0			0
$103$		0			0		−0.855781	−	0.517338i	$-0.826923\pi$
$103$		0			0		0.855781	+	0.517338i	$0.173077\pi$
$104$		0			0
$105$		0			0
$106$	0.475142	+	0.0576926i	0.475142	+	0.0576926i
$107$			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
$109$	−1.01773	+	1.68353i	−1.01773	+	1.68353i	−0.354605	+	0.935016i	$0.615385\pi$
$109$	−1.01773	+	1.68353i	−1.01773	+	1.68353i	−0.663123	+	0.748511i	$0.730769\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	0.445368	+	0.394562i	0.445368	+	0.394562i	0.855781	−	0.517338i	$-0.173077\pi$
$113$	0.445368	+	0.394562i	0.445368	+	0.394562i	−0.410413	+	0.911900i	$0.634615\pi$
$114$		0			0
$115$	−0.329586	−	0.627974i	−0.329586	−	0.627974i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−0.748511	+	0.663123i	−0.748511	+	0.663123i
$122$	0.0412943	+	0.167537i	0.0412943	+	0.167537i
$123$		0			0
$124$	−0.0481605	−	0.0796673i	−0.0481605	−	0.0796673i
$125$	−0.787183	+	0.616719i	−0.787183	+	0.616719i
$126$		0			0
$127$		0			0		−0.297503	−	0.954721i	$-0.596154\pi$
$127$		0			0		0.297503	+	0.954721i	$0.403846\pi$
$128$	0.395227	+	0.878159i	0.395227	+	0.878159i
$129$		0			0
$130$		0			0
$131$		0			0		0.568065	−	0.822984i	$-0.307692\pi$
$131$		0			0		−0.568065	+	0.822984i	$0.692308\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$	0.620137	−	0.620137i	0.620137	−	0.620137i
$137$	−0.219835	+	0.0989396i	−0.219835	+	0.0989396i	−0.517338	−	0.855781i	$-0.673077\pi$
$137$	−0.219835	+	0.0989396i	−0.219835	+	0.0989396i	0.297503	+	0.954721i	$0.403846\pi$
$138$		0			0
$139$	0.359852	−	1.96365i	0.359852	−	1.96365i	0.120537	−	0.992709i	$-0.461538\pi$
$139$	0.359852	−	1.96365i	0.359852	−	1.96365i	0.239316	−	0.970942i	$-0.423077\pi$
$140$		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		−0.970942	−	0.239316i	$-0.923077\pi$
$149$		0			0		0.970942	+	0.239316i	$0.0769231\pi$
$150$		0			0
$151$	0.147958	+	0.807380i	0.147958	+	0.807380i	0.970942	+	0.239316i	$0.0769231\pi$
$151$	0.147958	+	0.807380i	0.147958	+	0.807380i	−0.822984	+	0.568065i	$0.807692\pi$
$152$	−1.02511	+	1.15711i	−1.02511	+	1.15711i
$153$		0			0
$154$		0			0
$155$	−0.119877	+	0.0145556i	−0.119877	+	0.0145556i
$156$		0			0
$157$		0			0		−0.0603785	−	0.998176i	$-0.519231\pi$
$157$		0			0		0.0603785	+	0.998176i	$0.480769\pi$
$158$	0.284789			0.284789
$159$		0			0
$160$	1.02242			1.02242
$161$		0			0
$162$		0			0
$163$		0			0		0.992709	−	0.120537i	$-0.0384615\pi$
$163$		0			0		−0.992709	+	0.120537i	$0.961538\pi$
$164$		0			0
$165$		0			0
$166$	0.630154	−	0.711297i	0.630154	−	0.711297i
$167$	0.269846	+	1.47250i	0.269846	+	1.47250i	0.787183	+	0.616719i	$0.211538\pi$
$167$	0.269846	+	1.47250i	0.269846	+	1.47250i	−0.517338	+	0.855781i	$0.673077\pi$
$168$		0			0
$169$	−0.970942	−	0.239316i	−0.970942	−	0.239316i
$170$	−0.175610	−	0.463046i	−0.175610	−	0.463046i
$171$		0			0
$172$		0			0
$173$	0.894342	+	0.700673i	0.894342	+	0.700673i	0.954721	−	0.297503i	$-0.0961538\pi$
$173$	0.894342	+	0.700673i	0.894342	+	0.700673i	−0.0603785	+	0.998176i	$0.519231\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		0.180255	−	0.983620i	$-0.442308\pi$
$179$		0			0		−0.180255	+	0.983620i	$0.557692\pi$
$180$		0			0
$181$	−1.43566	+	0.646140i	−1.43566	+	0.646140i	−0.970942	−	0.239316i	$-0.923077\pi$
$181$	−1.43566	+	0.646140i	−1.43566	+	0.646140i	−0.464723	+	0.885456i	$0.653846\pi$
$182$		0			0
$183$		0			0
$184$	0.494725	+	0.341484i	0.494725	+	0.341484i
$185$		0			0
$186$		0			0
$187$		0			0
$188$	−0.430383	+	1.13483i	−0.430383	+	1.13483i
$189$		0			0
$190$	0.358261	+	0.796023i	0.358261	+	0.796023i
$191$		0			0		−0.297503	−	0.954721i	$-0.596154\pi$
$191$		0			0		0.297503	+	0.954721i	$0.403846\pi$
$192$		0			0
$193$		0			0		0.787183	−	0.616719i	$-0.211538\pi$
$193$		0			0		−0.787183	+	0.616719i	$0.788462\pi$
$194$		0			0
$195$		0			0
$196$	−0.184491	−	0.748511i	−0.184491	−	0.748511i
$197$	−1.28112	+	1.13498i	−1.28112	+	1.13498i	−0.297503	+	0.954721i	$0.596154\pi$
$197$	−1.28112	+	1.13498i	−1.28112	+	1.13498i	−0.983620	+	0.180255i	$0.942308\pi$
$198$		0			0
$199$	−0.922670	−	0.112032i	−0.922670	−	0.112032i	−0.354605	−	0.935016i	$-0.615385\pi$
$199$	−0.922670	−	0.112032i	−0.922670	−	0.112032i	−0.568065	+	0.822984i	$0.692308\pi$
$200$	0.347872	−	0.772939i	0.347872	−	0.772939i
$201$		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
$211$			0.241073i			0.241073i	0.992709	+	0.120537i	$0.0384615\pi$
$211$			0.241073i			0.241073i	−0.992709	+	0.120537i	$0.961538\pi$
$212$	0.0465465	−	0.769506i	0.0465465	−	0.769506i
$213$		0			0
$214$	−0.675652	+	0.0408694i	−0.675652	+	0.0408694i
$215$		0			0
$216$		0			0
$217$		0			0
$218$	−0.833730	−	0.437575i	−0.833730	−	0.437575i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		0.935016	−	0.354605i	$-0.115385\pi$
$223$		0			0		−0.935016	+	0.354605i	$0.884615\pi$
$224$		0			0
$225$		0			0
$226$	−0.175634	+	0.224181i	−0.175634	+	0.224181i
$227$	1.26619	+	1.42924i	1.26619	+	1.42924i	0.855781	+	0.517338i	$0.173077\pi$
$227$	1.26619	+	1.42924i	1.26619	+	1.42924i	0.410413	+	0.911900i	$0.365385\pi$
$228$		0			0
$229$	−1.63397	+	1.12785i	−1.63397	+	1.12785i	−0.748511	+	0.663123i	$0.769231\pi$
$229$	−1.63397	+	1.12785i	−1.63397	+	1.12785i	−0.885456	+	0.464723i	$0.846154\pi$
$230$	0.290495	−	0.175610i	0.290495	−	0.175610i
$231$		0			0
$232$		0			0
$233$	1.57144	−	0.489680i	1.57144	−	0.489680i	0.616719	−	0.787183i	$-0.288462\pi$
$233$	1.57144	−	0.489680i	1.57144	−	0.489680i	0.954721	+	0.297503i	$0.0961538\pi$
$234$		0			0
$235$	1.11325	+	1.11325i	1.11325	+	1.11325i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		0.0603785	−	0.998176i	$-0.480769\pi$
$239$		0			0		−0.0603785	+	0.998176i	$0.519231\pi$
$240$		0			0
$241$	−1.53901	−	0.583668i	−1.53901	−	0.583668i	−0.568065	−	0.822984i	$-0.692308\pi$
$241$	−1.53901	−	0.583668i	−1.53901	−	0.583668i	−0.970942	+	0.239316i	$0.923077\pi$
$242$	−0.338443	−	0.338443i	−0.338443	−	0.338443i
$243$		0			0
$244$	0.265338	−	0.0826825i	0.265338	−	0.0826825i
$245$	−0.983620	−	0.180255i	−0.983620	−	0.180255i
$246$		0			0
$247$		0			0
$248$	0.0842368	−	0.0581444i	0.0842368	−	0.0581444i
$249$		0			0
$250$	−0.317391	−	0.358261i	−0.317391	−	0.358261i
$251$		0			0		0.616719	−	0.787183i	$-0.288462\pi$
$251$		0			0		−0.616719	+	0.787183i	$0.711538\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−0.518050	+	0.271894i	−0.518050	+	0.271894i
$257$	−1.47250	+	0.269846i	−1.47250	+	0.269846i	−0.855781	−	0.517338i	$-0.826923\pi$
$257$	−1.47250	+	0.269846i	−1.47250	+	0.269846i	−0.616719	+	0.787183i	$0.711538\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	1.13406	−	0.0685978i	1.13406	−	0.0685978i	0.517338	−	0.855781i	$-0.326923\pi$
$263$	1.13406	−	0.0685978i	1.13406	−	0.0685978i	0.616719	+	0.787183i	$0.288462\pi$
$264$		0			0
$265$	−0.885456	−	0.464723i	−0.885456	−	0.464723i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0		−0.120537	−	0.992709i	$-0.538462\pi$
$269$		0			0		0.120537	+	0.992709i	$0.461538\pi$
$270$		0			0
$271$	0.616337	−	1.17433i	0.616337	−	1.17433i	−0.354605	−	0.935016i	$-0.615385\pi$
$271$	0.616337	−	1.17433i	0.616337	−	1.17433i	0.970942	−	0.239316i	$-0.0769231\pi$
$272$	−0.282848	−	0.250582i	−0.282848	−	0.250582i
$273$		0			0
$274$	−0.0536222	−	0.102169i	−0.0536222	−	0.102169i
$275$		0			0
$276$		0			0
$277$		0			0		0.410413	−	0.911900i	$-0.365385\pi$
$277$		0			0		−0.410413	+	0.911900i	$0.634615\pi$
$278$	0.948549	+	0.115175i	0.948549	+	0.115175i
$279$		0			0
$280$		0			0
$281$		0			0		−0.239316	−	0.970942i	$-0.576923\pi$
$281$		0			0		0.239316	+	0.970942i	$0.423077\pi$
$282$		0			0
$283$		0			0		−0.517338	−	0.855781i	$-0.673077\pi$
$283$		0			0		0.517338	+	0.855781i	$0.326923\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−0.0250187	+	0.0659688i	−0.0250187	+	0.0659688i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−1.01510	−	0.700673i	−1.01510	−	0.700673i	−0.0603785	−	0.998176i	$-0.519231\pi$
$293$	−1.01510	−	0.700673i	−1.01510	−	0.700673i	−0.954721	+	0.297503i	$0.903846\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$	−0.381457	+	0.0940206i	−0.381457	+	0.0940206i
$303$		0			0
$304$	0.524330	+	0.410786i	0.524330	+	0.410786i
$305$	0.0434547	−	0.357882i	0.0434547	−	0.357882i
$306$		0			0
$307$		0			0		−0.354605	−	0.935016i	$-0.615385\pi$
$307$		0			0		0.354605	+	0.935016i	$0.384615\pi$
$308$		0			0
$309$		0			0
$310$	−0.0104184	−	0.0568513i	−0.0104184	−	0.0568513i
$311$		0			0		0.663123	−	0.748511i	$-0.269231\pi$
$311$		0			0		−0.663123	+	0.748511i	$0.730769\pi$
$312$		0			0
$313$		0			0		0.297503	−	0.954721i	$-0.403846\pi$
$313$		0			0		−0.297503	+	0.954721i	$0.596154\pi$
$314$		0			0
$315$		0			0
$316$	−0.0276955	−	0.457861i	−0.0276955	−	0.457861i
$317$	1.96724			1.96724			0.983620	−	0.180255i	$-0.0576923\pi$
$317$	1.96724			1.96724			0.983620	+	0.180255i	$0.0576923\pi$
$318$		0			0
$319$		0			0
$320$	0.00749563	+	0.123918i	0.00749563	+	0.123918i
$321$		0			0
$322$		0			0
$323$	0.561400	−	1.80160i	0.561400	−	1.80160i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	0.198399	−	1.63397i	0.198399	−	1.63397i	−0.464723	−	0.885456i	$-0.653846\pi$
$331$	0.198399	−	1.63397i	0.198399	−	1.63397i	0.663123	−	0.748511i	$-0.269231\pi$
$332$	−1.20485	−	0.943939i	−1.20485	−	0.943939i
$333$		0			0
$334$	−0.695701	+	0.171475i	−0.695701	+	0.171475i
$335$		0			0
$336$		0			0
$337$		0			0		−0.616719	−	0.787183i	$-0.711538\pi$
$337$		0			0		0.616719	+	0.787183i	$0.288462\pi$
$338$	0.0862757	−	0.470791i	0.0862757	−	0.470791i
$339$		0			0
$340$	−0.727371	+	0.327363i	−0.727371	+	0.327363i
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$	−0.308906	+	0.447528i	−0.308906	+	0.447528i
$347$	−0.437383	+	1.15328i	−0.437383	+	1.15328i	0.517338	+	0.855781i	$0.326923\pi$
$347$	−0.437383	+	1.15328i	−0.437383	+	1.15328i	−0.954721	+	0.297503i	$0.903846\pi$
$348$		0			0
$349$	−0.783659	−	1.74122i	−0.783659	−	1.74122i	−0.663123	−	0.748511i	$-0.730769\pi$
$349$	−0.783659	−	1.74122i	−0.783659	−	1.74122i	−0.120537	−	0.992709i	$-0.538462\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−1.00461	−	1.66183i	−1.00461	−	1.66183i	−0.707107	−	0.707107i	$-0.750000\pi$
$353$	−1.00461	−	1.66183i	−1.00461	−	1.66183i	−0.297503	−	0.954721i	$-0.596154\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		0.410413	−	0.911900i	$-0.365385\pi$
$359$		0			0		−0.410413	+	0.911900i	$0.634615\pi$
$360$		0			0
$361$	−0.556707	+	2.25865i	−0.556707	+	2.25865i
$362$	−0.350188	−	0.667228i	−0.350188	−	0.667228i
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		−0.120537	−	0.992709i	$-0.538462\pi$
$367$		0			0		0.120537	+	0.992709i	$0.461538\pi$
$368$	0.133999	−	0.221661i	0.133999	−	0.221661i
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		0.998176	−	0.0603785i	$-0.0192308\pi$
$373$		0			0		−0.998176	+	0.0603785i	$0.980769\pi$
$374$		0			0
$375$		0			0
$376$	−1.27403	−	0.397005i	−1.27403	−	0.397005i
$377$		0			0
$378$		0			0
$379$	1.39105	−	0.254919i	1.39105	−	0.254919i	0.568065	−	0.822984i	$-0.307692\pi$
$379$	1.39105	−	0.254919i	1.39105	−	0.254919i	0.822984	+	0.568065i	$0.192308\pi$
$380$	1.24494	−	0.653396i	1.24494	−	0.653396i
$381$		0			0
$382$		0			0
$383$	−1.70528	−	0.767485i	−1.70528	−	0.767485i	−0.998176	−	0.0603785i	$-0.980769\pi$
$383$	−1.70528	−	0.767485i	−1.70528	−	0.767485i	−0.707107	−	0.707107i	$-0.750000\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$		0			0		0.855781	−	0.517338i	$-0.173077\pi$
$389$		0			0		−0.855781	+	0.517338i	$0.826923\pi$
$390$		0			0
$391$	−0.721782	−	0.132272i	−0.721782	−	0.132272i
$392$	0.809235	−	0.252168i	0.809235	−	0.252168i
$393$		0			0
$394$	−0.579267	−	0.579267i	−0.579267	−	0.579267i
$395$	−0.556340	−	0.210992i	−0.556340	−	0.210992i
$396$		0			0
$397$		0			0		0.0603785	−	0.998176i	$-0.480769\pi$
$397$		0			0		−0.0603785	+	0.998176i	$0.519231\pi$
$398$	0.0268601	−	0.444051i	0.0268601	−	0.444051i
$399$		0			0
$400$	−0.341484	−	0.129508i	−0.341484	−	0.129508i
$401$		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
$409$	0.470293	+	0.530851i	0.470293	+	0.530851i	0.935016	−	0.354605i	$-0.115385\pi$
$409$	0.470293	+	0.530851i	0.470293	+	0.530851i	−0.464723	+	0.885456i	$0.653846\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−1.75800	+	0.922670i	−1.75800	+	0.922670i
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		−0.954721	−	0.297503i	$-0.903846\pi$
$419$		0			0		0.954721	+	0.297503i	$0.0961538\pi$
$420$		0			0
$421$	−1.70844	−	1.03279i	−1.70844	−	1.03279i	−0.885456	−	0.464723i	$-0.846154\pi$
$421$	−1.70844	−	1.03279i	−1.70844	−	1.03279i	−0.822984	−	0.568065i	$-0.807692\pi$
$422$	−0.115175	+	0.00696679i	−0.115175	+	0.00696679i
$423$		0			0
$424$	0.847614			0.847614
$425$			1.03468i			1.03468i
$426$		0			0
$427$		0			0
$428$	0.131413	+	1.08229i	0.131413	+	1.08229i
$429$		0			0
$430$		0			0
$431$		0			0		−0.748511	−	0.663123i	$-0.769231\pi$
$431$		0			0		0.748511	+	0.663123i	$0.230769\pi$
$432$		0			0
$433$		0			0		−0.464723	−	0.885456i	$-0.653846\pi$
$433$		0			0		0.464723	+	0.885456i	$0.346154\pi$
$434$		0			0
$435$		0			0
$436$	−0.622419	+	1.38296i	−0.622419	+	1.38296i
$437$	1.28403	+	0.155909i	1.28403	+	0.155909i
$438$		0			0
$439$	−1.12054	+	0.992709i	−1.12054	+	0.992709i	−0.120537	+	0.992709i	$0.538462\pi$
$439$	−1.12054	+	0.992709i	−1.12054	+	0.992709i	−1.00000			$\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	1.52862	−	1.19760i	1.52862	−	1.19760i	0.616719	−	0.787183i	$-0.288462\pi$
$443$	1.52862	−	1.19760i	1.52862	−	1.19760i	0.911900	−	0.410413i	$-0.134615\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		0			0		0.568065	−	0.822984i	$-0.307692\pi$
$449$		0			0		−0.568065	+	0.822984i	$0.692308\pi$
$450$		0			0
$451$		0			0
$452$	0.377501	+	0.260570i	0.377501	+	0.260570i
$453$		0			0
$454$	−0.646238	+	0.646238i	−0.646238	+	0.646238i
$455$		0			0
$456$		0			0
$457$		0			0		0.180255	−	0.983620i	$-0.442308\pi$
$457$		0			0		−0.180255	+	0.983620i	$0.557692\pi$
$458$	−0.586058	−	0.748047i	−0.586058	−	0.748047i
$459$		0			0
$460$	−0.310583	−	0.449957i	−0.310583	−	0.449957i
$461$		0			0		0.970942	−	0.239316i	$-0.0769231\pi$
$461$		0			0		−0.970942	+	0.239316i	$0.923077\pi$
$462$		0			0
$463$		0			0		−0.787183	−	0.616719i	$-0.788462\pi$
$463$		0			0		0.787183	+	0.616719i	$0.211538\pi$
$464$		0			0
$465$		0			0
$466$	0.279362	+	0.736617i	0.279362	+	0.736617i
$467$	−1.93834	−	0.477758i	−1.93834	−	0.477758i	−0.983620	−	0.180255i	$-0.942308\pi$
$467$	−1.93834	−	0.477758i	−1.93834	−	0.477758i	−0.954721	−	0.297503i	$-0.903846\pi$
$468$		0			0
$469$		0			0
$470$	−0.499690	+	0.564034i	−0.499690	+	0.564034i
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−0.110118	−	1.82047i	−0.110118	−	1.82047i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.0603785	−	0.998176i	$-0.519231\pi$
$479$		0			0		0.0603785	+	0.998176i	$0.480769\pi$
$480$		0			0
$481$		0			0
$482$	0.234376	−	0.752140i	0.234376	−	0.752140i
$483$		0			0
$484$	−0.511209	+	0.577036i	−0.511209	+	0.577036i
$485$		0			0
$486$		0			0
$487$		0			0		−0.970942	−	0.239316i	$-0.923077\pi$
$487$		0			0		0.970942	+	0.239316i	$0.0769231\pi$
$488$	0.108358	+	0.285716i	0.108358	+	0.285716i
$489$		0			0
$490$	0.0576926	−	0.475142i	0.0576926	−	0.475142i
$491$		0			0		−0.787183	−	0.616719i	$-0.788462\pi$
$491$		0			0		0.787183	+	0.616719i	$0.211538\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−0.0271989	−	0.0347168i	−0.0271989	−	0.0347168i
$497$		0			0
$498$		0			0
$499$	1.12477	−	0.506219i	1.12477	−	0.506219i	0.239316	−	0.970942i	$-0.423077\pi$
$499$	1.12477	−	0.506219i	1.12477	−	0.506219i	0.885456	+	0.464723i	$0.153846\pi$
$500$	−0.545117	+	0.545117i	−0.545117	+	0.545117i
$501$		0			0
$502$		0			0
$503$	1.93834	+	0.117248i	1.93834	+	0.117248i	0.983620	−	0.180255i	$-0.0576923\pi$
$503$	1.93834	+	0.117248i	1.93834	+	0.117248i	0.954721	+	0.297503i	$0.0961538\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$		0			0		−0.297503	−	0.954721i	$-0.596154\pi$
$509$		0			0		0.297503	+	0.954721i	$0.403846\pi$
$510$		0			0
$511$		0			0
$512$	0.353325	+	0.584472i	0.353325	+	0.584472i
$513$		0			0
$514$	−0.171475	−	0.695701i	−0.171475	−	0.695701i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		−0.464723	−	0.885456i	$-0.653846\pi$
$521$		0			0		0.464723	+	0.885456i	$0.346154\pi$
$522$		0			0
$523$		0			0		−0.748511	−	0.663123i	$-0.769231\pi$
$523$		0			0		0.748511	+	0.663123i	$0.230769\pi$
$524$		0			0
$525$		0			0
$526$	0.0655463	+	0.539822i	0.0655463	+	0.539822i
$527$	−0.0646384	+	0.106925i	−0.0646384	+	0.106925i
$528$		0			0
$529$			0.497021i			0.497021i
$530$	0.196436	−	0.436464i	0.196436	−	0.436464i
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$	1.35018	+	0.420733i	1.35018	+	0.420733i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	1.39974	−	0.530851i	1.39974	−	0.530851i	0.464723	−	0.885456i	$-0.346154\pi$
$541$	1.39974	−	0.530851i	1.39974	−	0.530851i	0.935016	+	0.354605i	$0.115385\pi$
$542$	0.578858	+	0.260523i	0.578858	+	0.260523i
$543$		0			0
$544$	0.652409	−	0.832739i	0.652409	−	0.832739i
$545$	1.30452	+	1.47250i	1.30452	+	1.47250i
$546$		0			0
$547$		0			0		0.822984	−	0.568065i	$-0.192308\pi$
$547$		0			0		−0.822984	+	0.568065i	$0.807692\pi$
$548$	−0.159044	+	0.0961453i	−0.159044	+	0.0961453i
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$	0.0929232	−	1.53620i	0.0929232	−	1.53620i
$557$	−0.119877	+	1.98180i	−0.119877	+	1.98180i	0.0603785	+	0.998176i	$0.480769\pi$
$557$	−0.119877	+	1.98180i	−0.119877	+	1.98180i	−0.180255	+	0.983620i	$0.557692\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−1.83940	−	0.337083i	−1.83940	−	0.337083i	−0.855781	−	0.517338i	$-0.826923\pi$
$563$	−1.83940	−	0.337083i	−1.83940	−	0.337083i	−0.983620	+	0.180255i	$0.942308\pi$
$564$		0			0
$565$	0.509195	−	0.307819i	0.509195	−	0.307819i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		0.616719	−	0.787183i	$-0.288462\pi$
$569$		0			0		−0.616719	+	0.787183i	$0.711538\pi$
$570$		0			0
$571$	0.943521	+	0.424644i	0.943521	+	0.424644i	0.822984	−	0.568065i	$-0.192308\pi$
$571$	0.943521	+	0.424644i	0.943521	+	0.424644i	0.120537	+	0.992709i	$0.461538\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−0.697593	+	0.127839i	−0.697593	+	0.127839i
$576$		0			0
$577$		0			0		−0.885456	−	0.464723i	$-0.846154\pi$
$577$		0			0		0.885456	+	0.464723i	$0.153846\pi$
$578$	−0.0322402	−	0.0100464i	−0.0322402	−	0.0100464i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$	0.305417	−	0.505221i	0.305417	−	0.505221i
$587$	−0.0434547	−	0.357882i	−0.0434547	−	0.357882i	−0.998176	−	0.0603785i	$-0.980769\pi$
$587$	−0.0434547	−	0.357882i	−0.0434547	−	0.357882i	0.954721	−	0.297503i	$-0.0961538\pi$
$588$		0			0
$589$	0.102349	−	0.195010i	0.102349	−	0.195010i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−0.338443	+	1.37312i	−0.338443	+	1.37312i	0.517338	+	0.855781i	$0.326923\pi$
$593$	−0.338443	+	1.37312i	−0.338443	+	1.37312i	−0.855781	+	0.517338i	$0.826923\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		−0.239316	−	0.970942i	$-0.576923\pi$
$599$		0			0		0.239316	+	0.970942i	$0.423077\pi$
$600$		0			0
$601$	−0.987826	−	1.63406i	−0.987826	−	1.63406i	−0.748511	−	0.663123i	$-0.769231\pi$
$601$	−0.987826	−	1.63406i	−0.987826	−	1.63406i	−0.239316	−	0.970942i	$-0.576923\pi$
$602$		0			0
$603$		0			0
$604$	0.188255	+	0.604132i	0.188255	+	0.604132i
$605$	0.410413	+	0.911900i	0.410413	+	0.911900i
$606$		0			0
$607$		0			0		0.354605	−	0.935016i	$-0.384615\pi$
$607$		0			0		−0.354605	+	0.935016i	$0.615385\pi$
$608$	−1.05926	+	1.53461i	−1.05926	+	1.53461i
$609$		0			0
$610$	0.172237	+	0.0104184i	0.172237	+	0.0104184i
$611$		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
$617$	−1.09215	−	1.39403i	−1.09215	−	1.39403i	−0.911900	−	0.410413i	$-0.865385\pi$
$617$	−1.09215	−	1.39403i	−1.09215	−	1.39403i	−0.180255	−	0.983620i	$-0.557692\pi$
$618$		0			0
$619$	1.06230	+	1.53901i	1.06230	+	1.53901i	0.822984	+	0.568065i	$0.192308\pi$
$619$	1.06230	+	1.53901i	1.06230	+	1.53901i	0.239316	+	0.970942i	$0.423077\pi$
$620$	−0.0903879	+	0.0222786i	−0.0903879	+	0.0222786i
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	0.354605	+	0.935016i	0.354605	+	0.935016i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−0.568065	+	1.82298i	−0.568065	+	1.82298i			1.00000i	$0.5\pi$
$631$	−0.568065	+	1.82298i	−0.568065	+	1.82298i	−0.568065	+	0.822984i	$0.692308\pi$
$632$	0.500658	−	0.0607909i	0.500658	−	0.0607909i
$633$		0			0
$634$	0.0568513	+	0.939865i	0.0568513	+	0.939865i
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$	0.955978	−	0.116077i	0.955978	−	0.116077i
$641$		0			0		0.297503	−	0.954721i	$-0.403846\pi$
$641$		0			0		−0.297503	+	0.954721i	$0.596154\pi$
$642$		0			0
$643$		0			0		0.663123	−	0.748511i	$-0.269231\pi$
$643$		0			0		−0.663123	+	0.748511i	$0.730769\pi$
$644$		0			0
$645$		0			0
$646$	0.876952	+	0.216149i	0.876952	+	0.216149i
$647$	0.210992	+	0.556340i	0.210992	+	0.556340i	0.998176	−	0.0603785i	$-0.0192308\pi$
$647$	0.210992	+	0.556340i	0.210992	+	0.556340i	−0.787183	+	0.616719i	$0.788462\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	0.204793	+	0.296694i	0.204793	+	0.296694i	0.911900	−	0.410413i	$-0.134615\pi$
$653$	0.204793	+	0.296694i	0.204793	+	0.296694i	−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
$661$	0.935016	+	0.645395i	0.935016	+	0.645395i	0.935016	−	0.354605i	$-0.115385\pi$
$661$	0.935016	+	0.645395i	0.935016	+	0.645395i			1.00000i	$0.5\pi$
$662$	0.786374	+	0.0475669i	0.786374	+	0.0475669i
$663$		0			0
$664$	0.955978	−	1.38497i	0.955978	−	1.38497i
$665$		0			0
$666$		0			0
$667$		0			0
$668$	0.343340	+	1.10182i	0.343340	+	1.10182i
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		−0.239316	−	0.970942i	$-0.576923\pi$
$673$		0			0		0.239316	+	0.970942i	$0.423077\pi$
$674$		0			0
$675$		0			0
$676$	−0.765291	−	0.0929232i	−0.765291	−	0.0929232i
$677$	0.767485	−	1.70528i	0.767485	−	1.70528i	0.0603785	−	0.998176i	$-0.480769\pi$
$677$	0.767485	−	1.70528i	0.767485	−	1.70528i	0.707107	−	0.707107i	$-0.250000\pi$
$678$		0			0
$679$		0			0
$680$	−0.407565	−	0.776550i	−0.407565	−	0.776550i
$681$		0			0
$682$		0			0
$683$	0.795403	−	1.51551i	0.795403	−	1.51551i	−0.0603785	−	0.998176i	$-0.519231\pi$
$683$	0.795403	−	1.51551i	0.795403	−	1.51551i	0.855781	−	0.517338i	$-0.173077\pi$
$684$		0			0
$685$	0.0290582	+	0.239316i	0.0290582	+	0.239316i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	0.359852	−	0.0217671i	0.359852	−	0.0217671i	0.120537	−	0.992709i	$-0.461538\pi$
$691$	0.359852	−	0.0217671i	0.359852	−	0.0217671i	0.239316	+	0.970942i	$0.423077\pi$
$692$	0.749541	+	0.453113i	0.749541	+	0.453113i
$693$		0			0
$694$	−0.563631	−	0.175634i	−0.563631	−	0.175634i
$695$	−1.76768	−	0.927751i	−1.76768	−	0.927751i
$696$		0			0
$697$		0			0
$698$	0.809235	−	0.424719i	0.809235	−	0.424719i
$699$		0			0
$700$		0			0
$701$		0			0		−0.911900	−	0.410413i	$-0.865385\pi$
$701$		0			0		0.911900	+	0.410413i	$0.134615\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$	0.764919	−	0.527986i	0.764919	−	0.527986i
$707$		0			0
$708$		0			0
$709$	1.39105	+	0.254919i	1.39105	+	0.254919i	0.822984	−	0.568065i	$-0.192308\pi$
$709$	1.39105	+	0.254919i	1.39105	+	0.254919i	0.568065	+	0.822984i	$0.307692\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−0.0800767	−	0.0303691i	−0.0800767	−	0.0303691i
$714$		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
$722$	−1.09518	−	0.200698i	−1.09518	−	0.200698i
$723$		0			0
$724$	−1.03866	+	0.627892i	−1.03866	+	0.627892i
$725$		0			0
$726$		0			0
$727$		0			0		−0.663123	−	0.748511i	$-0.730769\pi$
$727$		0			0		0.663123	+	0.748511i	$0.269231\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		0.885456	−	0.464723i	$-0.153846\pi$
$733$		0			0		−0.885456	+	0.464723i	$0.846154\pi$
$734$		0			0
$735$		0			0
$736$	0.642052	+	0.336975i	0.642052	+	0.336975i
$737$		0			0
$738$		0			0
$739$	1.68353	+	1.01773i	1.68353	+	1.01773i	0.935016	+	0.354605i	$0.115385\pi$
$739$	1.68353	+	1.01773i	1.68353	+	1.01773i	0.748511	+	0.663123i	$0.230769\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			1.90944i			1.90944i	0.297503	+	0.954721i	$0.403846\pi$
$743$			1.90944i			1.90944i	−0.297503	+	0.954721i	$0.596154\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−0.527986	−	1.00599i	−0.527986	−	1.00599i	−0.992709	−	0.120537i	$-0.961538\pi$
$751$	−0.527986	−	1.00599i	−0.527986	−	1.00599i	0.464723	−	0.885456i	$-0.346154\pi$
$752$	−0.137603	+	0.558278i	−0.137603	+	0.558278i
$753$		0			0
$754$		0			0
$755$	0.814841	+	0.0989396i	0.814841	+	0.0989396i
$756$		0			0
$757$		0			0		0.748511	−	0.663123i	$-0.230769\pi$
$757$		0			0		−0.748511	+	0.663123i	$0.769231\pi$
$758$	0.161990	+	0.657218i	0.161990	+	0.657218i
$759$		0			0
$760$	0.799741	+	1.32293i	0.799741	+	1.32293i
$761$		0			0		0.787183	−	0.616719i	$-0.211538\pi$
$761$		0			0		−0.787183	+	0.616719i	$0.788462\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$	0.317391	−	0.836892i	0.317391	−	0.836892i
$767$		0			0
$768$		0			0
$769$	0.819328	+	0.0495602i	0.819328	+	0.0495602i	0.464723	−	0.885456i	$-0.346154\pi$
$769$	0.819328	+	0.0495602i	0.819328	+	0.0495602i	0.354605	+	0.935016i	$0.384615\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	1.77080	−	0.796974i	1.77080	−	0.796974i	0.787183	−	0.616719i	$-0.211538\pi$
$773$	1.77080	−	0.796974i	1.77080	−	0.796974i	0.983620	−	0.180255i	$-0.0576923\pi$
$774$		0			0
$775$	−0.0217671	+	0.118779i	−0.0217671	+	0.118779i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$	0.0423350	−	0.348660i	0.0423350	−	0.348660i
$783$		0			0
$784$	−0.129508	−	0.341484i	−0.129508	−	0.341484i
$785$		0			0
$786$		0			0
$787$		0			0		−0.180255	−	0.983620i	$-0.557692\pi$
$787$		0			0		0.180255	+	0.983620i	$0.442308\pi$
$788$	−0.874967	+	0.987633i	−0.874967	+	0.987633i
$789$		0			0
$790$	0.0847255	−	0.271894i	0.0847255	−	0.271894i
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$	−0.716521			−0.716521
$797$	−0.0993811	−	1.64296i	−0.0993811	−	1.64296i	−0.616719	−	0.787183i	$-0.711538\pi$
$797$	−0.0993811	−	1.64296i	−0.0993811	−	1.64296i	0.517338	−	0.855781i	$-0.326923\pi$
$798$		0			0
$799$	1.61708	−	0.196349i	1.61708	−	0.196349i
$800$	0.304173	−	0.976124i	0.304173	−	0.976124i
$801$		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		−0.787183	−	0.616719i	$-0.788462\pi$
$809$		0			0		0.787183	+	0.616719i	$0.211538\pi$
$810$		0			0
$811$	1.45352	−	0.358261i	1.45352	−	0.358261i	0.568065	−	0.822984i	$-0.307692\pi$
$811$	1.45352	−	0.358261i	1.45352	−	0.358261i	0.885456	+	0.464723i	$0.153846\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	−0.240027	+	0.240027i	−0.240027	+	0.240027i
$819$		0			0
$820$		0			0
$821$		0			0		−0.998176	−	0.0603785i	$-0.980769\pi$
$821$		0			0		0.998176	+	0.0603785i	$0.0192308\pi$
$822$		0			0
$823$		0			0		0.568065	−	0.822984i	$-0.307692\pi$
$823$		0			0		−0.568065	+	0.822984i	$0.692308\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−0.526852	−	1.69073i	−0.526852	−	1.69073i	−0.707107	−	0.707107i	$-0.750000\pi$
$827$	−0.526852	−	1.69073i	−0.526852	−	1.69073i	0.180255	−	0.983620i	$-0.442308\pi$
$828$		0			0
$829$	1.34731	−	1.05555i	1.34731	−	1.05555i	0.354605	−	0.935016i	$-0.384615\pi$
$829$	1.34731	−	1.05555i	1.34731	−	1.05555i	0.992709	−	0.120537i	$-0.0384615\pi$
$830$	−0.491617	−	0.813235i	−0.491617	−	0.813235i
$831$		0			0
$832$		0			0
$833$	−0.774466	+	0.686117i	−0.774466	+	0.686117i
$834$		0			0
$835$	1.48611	+	0.180446i	1.48611	+	0.180446i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		−0.464723	−	0.885456i	$-0.653846\pi$
$839$		0			0		0.464723	+	0.885456i	$0.346154\pi$
$840$		0			0
$841$	−0.748511	−	0.663123i	−0.748511	−	0.663123i
$842$	0.444051	−	0.846068i	0.444051	−	0.846068i
$843$		0			0
$844$	0.0224013	+	0.184491i	0.0224013	+	0.184491i
$845$	−0.517338	+	0.855781i	−0.517338	+	0.855781i
$846$		0			0
$847$		0			0
$848$	−0.0220513	−	0.364551i	−0.0220513	−	0.364551i
$849$		0			0
$850$	−0.494325	+	0.0299011i	−0.494325	+	0.0299011i
$851$		0			0
$852$		0			0
$853$		0			0		−0.954721	−	0.297503i	$-0.903846\pi$
$853$		0			0		0.954721	+	0.297503i	$0.0961538\pi$
$854$		0			0
$855$		0			0
$856$	−1.17907	+	0.216073i	−1.17907	+	0.216073i
$857$	1.51551	−	0.795403i	1.51551	−	0.795403i	0.517338	−	0.855781i	$-0.326923\pi$
$857$	1.51551	−	0.795403i	1.51551	−	0.795403i	0.998176	+	0.0603785i	$0.0192308\pi$
$858$		0			0
$859$	0.869047	−	0.329586i	0.869047	−	0.329586i	0.120537	−	0.992709i	$-0.461538\pi$
$859$	0.869047	−	0.329586i	0.869047	−	0.329586i	0.748511	+	0.663123i	$0.230769\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	0.544308	+	0.614397i	0.544308	+	0.614397i	0.954721	−	0.297503i	$-0.0961538\pi$
$863$	0.544308	+	0.614397i	0.544308	+	0.614397i	−0.410413	+	0.911900i	$0.634615\pi$
$864$		0			0
$865$	0.935016	−	0.645395i	0.935016	−	0.645395i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$	−1.55910	−	0.591289i	−1.55910	−	0.591289i
$873$		0			0
$874$	−0.0373797	+	0.617959i	−0.0373797	+	0.617959i
$875$		0			0
$876$		0			0
$877$		0			0		−0.935016	−	0.354605i	$-0.884615\pi$
$877$		0			0		0.935016	+	0.354605i	$0.115385\pi$
$878$	−0.506657	−	0.506657i	−0.506657	−	0.506657i
$879$		0			0
$880$		0			0
$881$		0			0		−0.983620	−	0.180255i	$-0.942308\pi$
$881$		0			0		0.983620	+	0.180255i	$0.0576923\pi$
$882$		0			0
$883$		0			0		0.855781	−	0.517338i	$-0.173077\pi$
$883$		0			0		−0.855781	+	0.517338i	$0.826923\pi$
$884$		0			0
$885$		0			0
$886$	0.616337	+	0.695701i	0.616337	+	0.695701i
$887$	−0.573207	+	0.731645i	−0.573207	+	0.731645i	−0.983620	−	0.180255i	$-0.942308\pi$
$887$	−0.573207	+	0.731645i	−0.573207	+	0.731645i	0.410413	+	0.911900i	$0.365385\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−2.82430	+	0.517572i	−2.82430	+	0.517572i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$	−0.943521	+	0.424644i	−0.943521	+	0.424644i
$902$		0			0
$903$		0			0
$904$	−0.260912	+	0.431601i	−0.260912	+	0.431601i
$905$	0.189769	+	1.56289i	0.189769	+	1.56289i
$906$		0			0
$907$		0			0		0.464723	−	0.885456i	$-0.346154\pi$
$907$		0			0		−0.464723	+	0.885456i	$0.653846\pi$
$908$	1.10182	+	0.976124i	1.10182	+	0.976124i
$909$		0			0
$910$		0			0
$911$		0			0		0.239316	−	0.970942i	$-0.423077\pi$
$911$		0			0		−0.239316	+	0.970942i	$0.576923\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$	−1.14566	+	1.01496i	−1.14566	+	1.01496i
$917$		0			0
$918$		0			0
$919$	−0.186505	−	0.308518i	−0.186505	−	0.308518i	0.748511	−	0.663123i	$-0.230769\pi$
$919$	−0.186505	−	0.308518i	−0.186505	−	0.308518i	−0.935016	+	0.354605i	$0.884615\pi$
$920$	0.473204	−	0.370732i	0.473204	−	0.370732i
$921$		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		−0.822984	−	0.568065i	$-0.807692\pi$
$929$		0			0		0.822984	+	0.568065i	$0.192308\pi$
$930$		0			0
$931$	1.28962	−	1.28962i	1.28962	−	1.28962i
$932$	1.15711	−	0.520771i	1.15711	−	0.520771i
$933$		0			0
$934$	0.172237	−	0.939865i	0.172237	−	0.939865i
$935$		0			0
$936$		0			0
$937$		0			0		−0.568065	−	0.822984i	$-0.692308\pi$
$937$		0			0		0.568065	+	0.822984i	$0.307692\pi$
$938$		0			0
$939$		0			0
$940$	0.955403	+	0.748511i	0.955403	+	0.748511i
$941$		0			0		0.120537	−	0.992709i	$-0.461538\pi$
$941$		0			0		−0.120537	+	0.992709i	$0.538462\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−1.30452	+	1.47250i	−1.30452	+	1.47250i	−0.517338	+	0.855781i	$0.673077\pi$
$947$	−1.30452	+	1.47250i	−1.30452	+	1.47250i	−0.787183	+	0.616719i	$0.788462\pi$
$948$		0			0
$949$		0			0
$950$	0.866563	−	0.105220i	0.866563	−	0.105220i
$951$		0			0
$952$		0			0
$953$	1.82380			1.82380			0.911900	−	0.410413i	$-0.134615\pi$
$953$	1.82380			1.82380			0.911900	+	0.410413i	$0.134615\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	0.653453	−	0.737596i	0.653453	−	0.737596i
$962$		0			0
$963$		0			0
$964$	−1.23202	−	0.303667i	−1.23202	−	0.303667i
$965$		0			0
$966$		0			0
$967$		0			0		0.120537	−	0.992709i	$-0.461538\pi$
$967$		0			0		−0.120537	+	0.992709i	$0.538462\pi$
$968$	−0.667228	−	0.522740i	−0.667228	−	0.522740i
$969$		0			0
$970$		0			0
$971$		0			0		−0.568065	−	0.822984i	$-0.692308\pi$
$971$		0			0		0.568065	+	0.822984i	$0.307692\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$	0.120065	−	0.0540368i	0.120065	−	0.0540368i
$977$	0.657218	−	0.657218i	0.657218	−	0.657218i	−0.297503	−	0.954721i	$-0.596154\pi$
$977$	0.657218	−	0.657218i	0.657218	−	0.657218i	0.954721	+	0.297503i	$0.0961538\pi$
$978$		0			0
$979$		0			0
$980$	−0.769506	−	0.0465465i	−0.769506	−	0.0465465i
$981$		0			0
$982$		0			0
$983$	−0.707916	+	1.86662i	−0.707916	+	1.86662i	−0.297503	+	0.954721i	$0.596154\pi$
$983$	−0.707916	+	1.86662i	−0.707916	+	1.86662i	−0.410413	+	0.911900i	$0.634615\pi$
$984$		0			0
$985$	0.702447	+	1.56077i	0.702447	+	1.56077i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	0.447528	+	1.81569i	0.447528	+	1.81569i	0.568065	+	0.822984i	$0.307692\pi$
$991$	0.447528	+	1.81569i	0.447528	+	1.81569i	−0.120537	+	0.992709i	$0.538462\pi$
$992$	0.0924143	−	0.0818719i	0.0924143	−	0.0818719i
$993$		0			0
$994$		0			0
$995$	−0.381457	+	0.847562i	−0.381457	+	0.847562i
$996$		0			0
$997$		0			0		0.239316	−	0.970942i	$-0.423077\pi$
$997$		0			0		−0.239316	+	0.970942i	$0.576923\pi$
$998$	0.274355	+	0.522740i	0.274355	+	0.522740i
$999$		0			0

Display

a_p

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p

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a_n

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a_n

instead) (See

a_n

instead) Display

a_n

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n

up to: 50 250 1000 (See only

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) (See only

a_p

) (See only

a_p

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2385.1.ca.a.2359.2	yes	48
3.2	odd	2	inner	2385.1.ca.a.2359.1	yes	48
5.4	even	2	inner	2385.1.ca.a.2359.1	yes	48
15.14	odd	2	CM	2385.1.ca.a.2359.2	yes	48
53.2	odd	52	inner	2385.1.ca.a.1009.1	✓	48
159.2	even	52	inner	2385.1.ca.a.1009.2	yes	48
265.214	odd	52	inner	2385.1.ca.a.1009.2	yes	48
795.479	even	52	inner	2385.1.ca.a.1009.1	✓	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2385.1.ca.a.1009.1	✓	48	53.2	odd	52	inner
2385.1.ca.a.1009.1	✓	48	795.479	even	52	inner
2385.1.ca.a.1009.2	yes	48	159.2	even	52	inner
2385.1.ca.a.1009.2	yes	48	265.214	odd	52	inner
2385.1.ca.a.2359.1	yes	48	3.2	odd	2	inner
2385.1.ca.a.2359.1	yes	48	5.4	even	2	inner
2385.1.ca.a.2359.2	yes	48	1.1	even	1	trivial
2385.1.ca.a.2359.2	yes	48	15.14	odd	2	CM

Overview	Random
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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	2385.1.ca.a.2359.2

Level	$2385$
Weight	$1$
Character	2385.2359
Analytic conductor	$1.190$
Analytic rank	$0$
Dimension	$48$
Projective image	$D_{52}$
CM discriminant	-15
Inner twists	$8$