LMFDB - Embedded newform 1519.1.ce.a.1425.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1519,1,Mod(123,1519)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1519, base_ring=CyclotomicField(42))

chi = DirichletCharacter(H, H._module([16, 21]))

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

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

chi := DirichletCharacter("1519.123");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1519 = 7^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1519.ce (of order $42$, degree $12$, 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:	$0.758079754190$
Analytic rank:	$0$
Dimension:	$36$
Relative dimension:	$3$ over $\Q(\zeta_{42})$
Coefficient field:	$\Q(\zeta_{63})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{36} - x^{33} + x^{27} - x^{24} + x^{18} - x^{12} + x^{9} - x^{3} + 1 $ x^36 - x^33 + x^27 - x^24 + x^18 - x^12 + x^9 - x^3 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{63}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{63} - \cdots)$

Embedding invariants

Embedding label			1425.3
Root			$-0.998757 + 0.0498459i$ of defining polynomial
Character	$\chi$	$=$	1519.1425
Dual form			1519.1.ce.a.1115.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(1.52272 + 1.03817i) q^{2} +(0.875530 + 2.23081i) q^{4} +(-1.63076 + 0.503024i) q^{5} +(-0.853291 - 0.521435i) q^{7} +(-0.572688 + 2.50911i) q^{8} +(0.0747301 + 0.997204i) q^{9} +(-3.00542 - 0.927049i) q^{10} +(-0.757983 - 1.67986i) q^{14} +(-1.72019 + 1.59610i) q^{16} +(-0.921476 + 1.59604i) q^{18} +(0.411287 + 0.712370i) q^{19} +(-2.54994 - 3.19752i) q^{20} +(1.58012 - 1.07731i) q^{25} +(0.416143 - 2.36007i) q^{28} +(-0.500000 + 0.866025i) q^{31} +(-1.73150 + 0.260982i) q^{32} +(1.65381 + 0.421112i) q^{35} +(-2.15915 + 1.03979i) q^{36} +(-0.113288 + 1.51173i) q^{38} +(-0.328223 - 4.37984i) q^{40} +(0.294478 - 1.29019i) q^{41} +(-0.623484 - 1.58861i) q^{45} +(1.57906 + 1.07659i) q^{47} +(0.456211 + 0.889872i) q^{49} +3.52450 q^{50} +(1.79701 - 1.84238i) q^{56} +(0.517616 + 0.159663i) q^{59} +(-1.66044 + 0.799627i) q^{62} +(0.456211 - 0.889872i) q^{63} +(-0.793311 - 0.382038i) q^{64} +(0.988831 - 1.71271i) q^{67} +(2.08110 + 2.35817i) q^{70} +(-0.994008 + 1.24645i) q^{71} +(-2.54489 - 0.383580i) q^{72} +(-1.22907 + 1.54121i) q^{76} +(2.00234 - 3.46816i) q^{80} +(-0.988831 + 0.149042i) q^{81} +(1.78785 - 1.65888i) q^{82} +(0.699861 - 3.06629i) q^{90} +(1.28679 + 3.27868i) q^{94} +(-1.02905 - 0.954820i) q^{95} +1.96034 q^{97} +(-0.229160 + 1.82865i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 36 q + 3 q^{4} + 6 q^{8} + 3 q^{9} - 3 q^{10} + 3 q^{14} + 3 q^{16} - 12 q^{20} + 3 q^{25} + 3 q^{28} - 18 q^{31} - 24 q^{32} + 3 q^{35} - 6 q^{36} - 3 q^{38} - 3 q^{40} - 3 q^{47} + 6 q^{50} + 3 q^{56}+ \cdots - 18 q^{98}+O(q^{100}) $ 36 * q + 3 * q^4 + 6 * q^8 + 3 * q^9 - 3 * q^10 + 3 * q^14 + 3 * q^16 - 12 * q^20 + 3 * q^25 + 3 * q^28 - 18 * q^31 - 24 * q^32 + 3 * q^35 - 6 * q^36 - 3 * q^38 - 3 * q^40 - 3 * q^47 + 6 * q^50 + 3 * q^56 - 3 * q^67 - 6 * q^70 - 24 * q^72 - 15 * q^76 + 24 * q^80 + 3 * q^81 + 6 * q^82 + 6 * q^90 - 3 * q^95 - 18 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1519\mathbb{Z}\right)^\times$.

$n$	$344$	$1179$
$\chi(n)$	$-1$	$e\left(\frac{5}{21}\right)$

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 $

$2$	1.52272	+	1.03817i	1.52272	+	1.03817i	0.980172	+	0.198146i	$0.0634921\pi$
$2$	1.52272	+	1.03817i	1.52272	+	1.03817i	0.542546	+	0.840026i	$0.317460\pi$
$3$		0			0		−0.733052	−	0.680173i	$-0.761905\pi$
$3$		0			0		0.733052	+	0.680173i	$0.238095\pi$
$4$	0.875530	+	2.23081i	0.875530	+	2.23081i
$5$	−1.63076	+	0.503024i	−1.63076	+	0.503024i	−0.969077	−	0.246757i	$-0.920635\pi$
$5$	−1.63076	+	0.503024i	−1.63076	+	0.503024i	−0.661686	+	0.749781i	$0.730159\pi$
$6$		0			0
$7$	−0.853291	−	0.521435i	−0.853291	−	0.521435i
$7$	−0.853291	−	0.521435i	−0.853291	−	0.521435i
$8$	−0.572688	+	2.50911i	−0.572688	+	2.50911i
$9$	0.0747301	+	0.997204i	0.0747301	+	0.997204i
$10$	−3.00542	−	0.927049i	−3.00542	−	0.927049i
$11$		0			0		0.0747301	−	0.997204i	$-0.476190\pi$
$11$		0			0		−0.0747301	+	0.997204i	$0.523810\pi$
$12$		0			0
$13$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$13$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$14$	−0.757983	−	1.67986i	−0.757983	−	1.67986i
$15$		0			0
$16$	−1.72019	+	1.59610i	−1.72019	+	1.59610i
$17$		0			0		0.988831	−	0.149042i	$-0.0476190\pi$
$17$		0			0		−0.988831	+	0.149042i	$0.952381\pi$
$18$	−0.921476	+	1.59604i	−0.921476	+	1.59604i
$19$	0.411287	+	0.712370i	0.411287	+	0.712370i	0.995031	−	0.0995678i	$-0.0317460\pi$
$19$	0.411287	+	0.712370i	0.411287	+	0.712370i	−0.583744	+	0.811938i	$0.698413\pi$
$20$	−2.54994	−	3.19752i	−2.54994	−	3.19752i
$21$		0			0
$22$		0			0
$23$		0			0		−0.988831	−	0.149042i	$-0.952381\pi$
$23$		0			0		0.988831	+	0.149042i	$0.0476190\pi$
$24$		0			0
$25$	1.58012	−	1.07731i	1.58012	−	1.07731i
$26$		0			0
$27$		0			0
$28$	0.416143	−	2.36007i	0.416143	−	2.36007i
$29$		0			0		−0.623490	−	0.781831i	$-0.714286\pi$
$29$		0			0		0.623490	+	0.781831i	$0.285714\pi$
$30$		0			0
$31$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$31$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$32$	−1.73150	+	0.260982i	−1.73150	+	0.260982i
$33$		0			0
$34$		0			0
$35$	1.65381	+	0.421112i	1.65381	+	0.421112i
$36$	−2.15915	+	1.03979i	−2.15915	+	1.03979i
$37$		0			0		0.365341	−	0.930874i	$-0.380952\pi$
$37$		0			0		−0.365341	+	0.930874i	$0.619048\pi$
$38$	−0.113288	+	1.51173i	−0.113288	+	1.51173i
$39$		0			0
$40$	−0.328223	−	4.37984i	−0.328223	−	4.37984i
$41$	0.294478	−	1.29019i	0.294478	−	1.29019i	−0.583744	−	0.811938i	$-0.698413\pi$
$41$	0.294478	−	1.29019i	0.294478	−	1.29019i	0.878222	−	0.478254i	$-0.158730\pi$
$42$		0			0
$43$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$43$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$44$		0			0
$45$	−0.623484	−	1.58861i	−0.623484	−	1.58861i
$46$		0			0
$47$	1.57906	+	1.07659i	1.57906	+	1.07659i	0.955573	+	0.294755i	$0.0952381\pi$
$47$	1.57906	+	1.07659i	1.57906	+	1.07659i	0.623490	+	0.781831i	$0.285714\pi$
$48$		0			0
$49$	0.456211	+	0.889872i	0.456211	+	0.889872i
$50$	3.52450			3.52450
$51$		0			0
$52$		0			0
$53$		0			0		−0.365341	−	0.930874i	$-0.619048\pi$
$53$		0			0		0.365341	+	0.930874i	$0.380952\pi$
$54$		0			0
$55$		0			0
$56$	1.79701	−	1.84238i	1.79701	−	1.84238i
$57$		0			0
$58$		0			0
$59$	0.517616	+	0.159663i	0.517616	+	0.159663i	0.542546	−	0.840026i	$-0.317460\pi$
$59$	0.517616	+	0.159663i	0.517616	+	0.159663i	−0.0249307	+	0.999689i	$0.507937\pi$
$60$		0			0
$61$		0			0		0.365341	−	0.930874i	$-0.380952\pi$
$61$		0			0		−0.365341	+	0.930874i	$0.619048\pi$
$62$	−1.66044	+	0.799627i	−1.66044	+	0.799627i
$63$	0.456211	−	0.889872i	0.456211	−	0.889872i
$64$	−0.793311	−	0.382038i	−0.793311	−	0.382038i
$65$		0			0
$66$		0			0
$67$	0.988831	−	1.71271i	0.988831	−	1.71271i	0.365341	−	0.930874i	$-0.380952\pi$
$67$	0.988831	−	1.71271i	0.988831	−	1.71271i	0.623490	−	0.781831i	$-0.285714\pi$
$68$		0			0
$69$		0			0
$70$	2.08110	+	2.35817i	2.08110	+	2.35817i
$71$	−0.994008	+	1.24645i	−0.994008	+	1.24645i	−0.0249307	+	0.999689i	$0.507937\pi$
$71$	−0.994008	+	1.24645i	−0.994008	+	1.24645i	−0.969077	+	0.246757i	$0.920635\pi$
$72$	−2.54489	−	0.383580i	−2.54489	−	0.383580i
$73$		0			0		0.826239	−	0.563320i	$-0.190476\pi$
$73$		0			0		−0.826239	+	0.563320i	$0.809524\pi$
$74$		0			0
$75$		0			0
$76$	−1.22907	+	1.54121i	−1.22907	+	1.54121i
$77$		0			0
$78$		0			0
$79$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$79$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$80$	2.00234	−	3.46816i	2.00234	−	3.46816i
$81$	−0.988831	+	0.149042i	−0.988831	+	0.149042i
$82$	1.78785	−	1.65888i	1.78785	−	1.65888i
$83$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$83$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		−0.0747301	−	0.997204i	$-0.523810\pi$
$89$		0			0		0.0747301	+	0.997204i	$0.476190\pi$
$90$	0.699861	−	3.06629i	0.699861	−	3.06629i
$91$		0			0
$92$		0			0
$93$		0			0
$94$	1.28679	+	3.27868i	1.28679	+	3.27868i
$95$	−1.02905	−	0.954820i	−1.02905	−	0.954820i
$96$		0			0
$97$	1.96034			1.96034			0.980172	−	0.198146i	$-0.0634921\pi$
$97$	1.96034			1.96034			0.980172	+	0.198146i	$0.0634921\pi$
$98$	−0.229160	+	1.82865i	−0.229160	+	1.82865i
$99$		0			0
$100$	3.78671	+	2.58173i	3.78671	+	2.58173i
$101$	−0.668852	−	0.620604i	−0.668852	−	0.620604i	0.270840	−	0.962624i	$-0.412698\pi$
$101$	−0.668852	−	0.620604i	−0.668852	−	0.620604i	−0.939693	+	0.342020i	$0.888889\pi$
$102$		0			0
$103$	1.67841	−	0.517721i	1.67841	−	0.517721i	0.698237	−	0.715867i	$-0.253968\pi$
$103$	1.67841	−	0.517721i	1.67841	−	0.517721i	0.980172	+	0.198146i	$0.0634921\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	0.146497	+	1.95486i	0.146497	+	1.95486i	0.270840	+	0.962624i	$0.412698\pi$
$107$	0.146497	+	1.95486i	0.146497	+	1.95486i	−0.124344	+	0.992239i	$0.539683\pi$
$108$		0			0
$109$	0.148717	−	1.98450i	0.148717	−	1.98450i	−0.0249307	−	0.999689i	$-0.507937\pi$
$109$	0.148717	−	1.98450i	0.148717	−	1.98450i	0.173648	−	0.984808i	$-0.444444\pi$
$110$		0			0
$111$		0			0
$112$	2.30008	−	0.464972i	2.30008	−	0.464972i
$113$	−0.977635	−	0.470804i	−0.977635	−	0.470804i	−0.124344	−	0.992239i	$-0.539683\pi$
$113$	−0.977635	−	0.470804i	−0.977635	−	0.470804i	−0.853291	+	0.521435i	$0.825397\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$	0.622425	+	0.780496i	0.622425	+	0.780496i
$119$		0			0
$120$		0			0
$121$	−0.988831	−	0.149042i	−0.988831	−	0.149042i
$122$		0			0
$123$		0			0
$124$	−2.36971	−	0.357176i	−2.36971	−	0.357176i
$125$	−0.970849	+	1.21741i	−0.970849	+	1.21741i
$126$	1.61852	−	0.881399i	1.61852	−	0.881399i
$127$		0			0		−0.623490	−	0.781831i	$-0.714286\pi$
$127$		0			0		0.623490	+	0.781831i	$0.285714\pi$
$128$	0.0641625	+	0.111133i	0.0641625	+	0.111133i
$129$		0			0
$130$		0			0
$131$	−1.21135	+	1.12397i	−1.21135	+	1.12397i	−0.222521	+	0.974928i	$0.571429\pi$
$131$	−1.21135	+	1.12397i	−1.21135	+	1.12397i	−0.988831	+	0.149042i	$0.952381\pi$
$132$		0			0
$133$	0.0205073	−	0.822319i	0.0205073	−	0.822319i
$134$	3.28379	−	1.58139i	3.28379	−	1.58139i
$135$		0			0
$136$		0			0
$137$		0			0		−0.955573	−	0.294755i	$-0.904762\pi$
$137$		0			0		0.955573	+	0.294755i	$0.0952381\pi$
$138$		0			0
$139$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$139$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$140$	0.508538	+	4.05804i	0.508538	+	4.05804i
$141$		0			0
$142$	−2.80762	+	0.866036i	−2.80762	+	0.866036i
$143$		0			0
$144$	−1.72019	−	1.59610i	−1.72019	−	1.59610i
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−0.826239	−	0.563320i	−0.826239	−	0.563320i	0.0747301	−	0.997204i	$-0.476190\pi$
$149$	−0.826239	−	0.563320i	−0.826239	−	0.563320i	−0.900969	+	0.433884i	$0.857143\pi$
$150$		0			0
$151$		0			0		−0.365341	−	0.930874i	$-0.619048\pi$
$151$		0			0		0.365341	+	0.930874i	$0.380952\pi$
$152$	−2.02295	+	0.623998i	−2.02295	+	0.623998i
$153$		0			0
$154$		0			0
$155$	0.379750	−	1.66379i	0.379750	−	1.66379i
$156$		0			0
$157$	−1.90877	−	0.588778i	−1.90877	−	0.588778i	−0.969077	−	0.246757i	$-0.920635\pi$
$157$	−1.90877	−	0.588778i	−1.90877	−	0.588778i	−0.939693	−	0.342020i	$-0.888889\pi$
$158$		0			0
$159$		0			0
$160$	2.69239	−	1.29659i	2.69239	−	1.29659i
$161$		0			0
$162$	−1.66044	−	0.799627i	−1.66044	−	0.799627i
$163$	1.37769	−	1.27831i	1.37769	−	1.27831i	0.456211	−	0.889872i	$-0.349206\pi$
$163$	1.37769	−	1.27831i	1.37769	−	1.27831i	0.921476	−	0.388435i	$-0.126984\pi$
$164$	3.13600	−	0.472676i	3.13600	−	0.472676i
$165$		0			0
$166$		0			0
$167$		0			0		−0.623490	−	0.781831i	$-0.714286\pi$
$167$		0			0		0.623490	+	0.781831i	$0.285714\pi$
$168$		0			0
$169$	0.623490	−	0.781831i	0.623490	−	0.781831i
$170$		0			0
$171$	−0.679643	+	0.463373i	−0.679643	+	0.463373i
$172$		0			0
$173$	−1.23305	−	0.185853i	−1.23305	−	0.185853i	−0.500000	−	0.866025i	$-0.666667\pi$
$173$	−1.23305	−	0.185853i	−1.23305	−	0.185853i	−0.733052	+	0.680173i	$0.761905\pi$
$174$		0			0
$175$	−1.91004	+	0.0953263i	−1.91004	+	0.0953263i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		0.988831	−	0.149042i	$-0.0476190\pi$
$179$		0			0		−0.988831	+	0.149042i	$0.952381\pi$
$180$	2.99802	−	2.78176i	2.99802	−	2.78176i
$181$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$181$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$	−1.01915	+	4.46518i	−1.01915	+	4.46518i
$189$		0			0
$190$	−0.575688	−	2.52225i	−0.575688	−	2.52225i
$191$	−1.11562	+	0.344123i	−1.11562	+	0.344123i	−0.797133	−	0.603804i	$-0.793651\pi$
$191$	−1.11562	+	0.344123i	−1.11562	+	0.344123i	−0.318487	+	0.947927i	$0.603175\pi$
$192$		0			0
$193$	1.46428	+	1.35865i	1.46428	+	1.35865i	0.766044	+	0.642788i	$0.222222\pi$
$193$	1.46428	+	1.35865i	1.46428	+	1.35865i	0.698237	+	0.715867i	$0.253968\pi$
$194$	2.98505	+	2.03518i	2.98505	+	2.03518i
$195$		0			0
$196$	−1.58571	+	1.79683i	−1.58571	+	1.79683i
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$		0			0		−0.733052	−	0.680173i	$-0.761905\pi$
$199$		0			0		0.733052	+	0.680173i	$0.238095\pi$
$200$	1.79816	+	4.58164i	1.79816	+	4.58164i
$201$		0			0
$202$	−0.374180	−	1.63939i	−0.374180	−	1.63939i
$203$		0			0
$204$		0			0
$205$	0.168774	+	2.25213i	0.168774	+	2.25213i
$206$	3.09323	+	0.954135i	3.09323	+	0.954135i
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−1.58250	−	0.762092i	−1.58250	−	0.762092i	−0.583744	−	0.811938i	$-0.698413\pi$
$211$	−1.58250	−	0.762092i	−1.58250	−	0.762092i	−0.998757	+	0.0498459i	$0.984127\pi$
$212$		0			0
$213$		0			0
$214$	−1.80641	+	3.12880i	−1.80641	+	3.12880i
$215$		0			0
$216$		0			0
$217$	0.878222	−	0.478254i	0.878222	−	0.478254i
$218$	2.28670	−	2.86744i	2.28670	−	2.86744i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		0.623490	−	0.781831i	$-0.285714\pi$
$223$		0			0		−0.623490	+	0.781831i	$0.714286\pi$
$224$	1.61356	+	0.680173i	1.61356	+	0.680173i
$225$	1.19238	+	1.49519i	1.19238	+	1.49519i
$226$	−0.999887	−	1.73186i	−0.999887	−	1.73186i
$227$	0.900969	−	1.56052i	0.900969	−	1.56052i	0.0747301	−	0.997204i	$-0.476190\pi$
$227$	0.900969	−	1.56052i	0.900969	−	1.56052i	0.826239	−	0.563320i	$-0.190476\pi$
$228$		0			0
$229$		0			0		0.733052	−	0.680173i	$-0.238095\pi$
$229$		0			0		−0.733052	+	0.680173i	$0.761905\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−0.232712	+	0.592942i	−0.232712	+	0.592942i	−0.998757	−	0.0498459i	$-0.984127\pi$
$233$	−0.232712	+	0.592942i	−0.232712	+	0.592942i	0.766044	+	0.642788i	$0.222222\pi$
$234$		0			0
$235$	−3.11663	−	0.961352i	−3.11663	−	0.961352i
$236$	0.0970089	+	1.29449i	0.0970089	+	1.29449i
$237$		0			0
$238$		0			0
$239$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$239$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$240$		0			0
$241$		0			0		−0.365341	−	0.930874i	$-0.619048\pi$
$241$		0			0		0.365341	+	0.930874i	$0.380952\pi$
$242$	−1.35098	−	1.25353i	−1.35098	−	1.25353i
$243$		0			0
$244$		0			0
$245$	−1.19160	−	1.22169i	−1.19160	−	1.22169i
$246$		0			0
$247$		0			0
$248$	−1.88661	−	1.75052i	−1.88661	−	1.75052i
$249$		0			0
$250$	−2.74221	+	0.845859i	−2.74221	+	0.845859i
$251$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$251$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$252$	2.38457	+	0.238612i	2.38457	+	0.238612i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−0.0834739	+	1.11388i	−0.0834739	+	1.11388i
$257$	0.559735	−	1.42618i	0.559735	−	1.42618i	−0.318487	−	0.947927i	$-0.603175\pi$
$257$	0.559735	−	1.42618i	0.559735	−	1.42618i	0.878222	−	0.478254i	$-0.158730\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$	−3.01142	+	0.453899i	−3.01142	+	0.453899i
$263$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$263$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$264$		0			0
$265$		0			0
$266$	0.884935	−	1.23087i	0.884935	−	1.23087i
$267$		0			0
$268$	4.68648	+	0.706373i	4.68648	+	0.706373i
$269$		0			0		0.826239	−	0.563320i	$-0.190476\pi$
$269$		0			0		−0.826239	+	0.563320i	$0.809524\pi$
$270$		0			0
$271$		0			0		−0.988831	−	0.149042i	$-0.952381\pi$
$271$		0			0		0.988831	+	0.149042i	$0.0476190\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$		0			0		0.988831	−	0.149042i	$-0.0476190\pi$
$277$		0			0		−0.988831	+	0.149042i	$0.952381\pi$
$278$		0			0
$279$	−0.900969	−	0.433884i	−0.900969	−	0.433884i
$280$	−2.00373	+	3.90842i	−2.00373	+	3.90842i
$281$	−0.488038	+	0.235027i	−0.488038	+	0.235027i	−0.661686	−	0.749781i	$-0.730159\pi$
$281$	−0.488038	+	0.235027i	−0.488038	+	0.235027i	0.173648	+	0.984808i	$0.444444\pi$
$282$		0			0
$283$	−0.0747301	+	0.997204i	−0.0747301	+	0.997204i	0.826239	+	0.563320i	$0.190476\pi$
$283$	−0.0747301	+	0.997204i	−0.0747301	+	0.997204i	−0.900969	+	0.433884i	$0.857143\pi$
$284$	−3.65087	−	1.12615i	−3.65087	−	1.12615i
$285$		0			0
$286$		0			0
$287$	−0.924027	+	0.947358i	−0.924027	+	0.947358i
$288$	−0.389648	−	1.70716i	−0.389648	−	1.70716i
$289$	0.955573	−	0.294755i	0.955573	−	0.294755i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	0.730682			0.730682			0.365341	−	0.930874i	$-0.380952\pi$
$293$	0.730682			0.730682			0.365341	+	0.930874i	$0.380952\pi$
$294$		0			0
$295$	−0.924423			−0.924423
$296$		0			0
$297$		0			0
$298$	−0.673306	−	1.71556i	−0.673306	−	1.71556i
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$	−1.84451	−	0.568955i	−1.84451	−	0.568955i
$305$		0			0
$306$		0			0
$307$	−1.25818	+	0.605907i	−1.25818	+	0.605907i	−0.939693	−	0.342020i	$-0.888889\pi$
$307$	−1.25818	+	0.605907i	−1.25818	+	0.605907i	−0.318487	+	0.947927i	$0.603175\pi$
$308$		0			0
$309$		0			0
$310$	2.30556	−	2.13924i	2.30556	−	2.13924i
$311$	0.245910	−	0.0370649i	0.245910	−	0.0370649i	−0.0249307	−	0.999689i	$-0.507937\pi$
$311$	0.245910	−	0.0370649i	0.245910	−	0.0370649i	0.270840	+	0.962624i	$0.412698\pi$
$312$		0			0
$313$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$313$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$314$	−2.29527	−	2.87817i	−2.29527	−	2.87817i
$315$	−0.296345	+	1.68065i	−0.296345	+	1.68065i
$316$		0			0
$317$	1.91651	+	0.288867i	1.91651	+	0.288867i	0.995031	−	0.0995678i	$-0.0317460\pi$
$317$	1.91651	+	0.288867i	1.91651	+	0.288867i	0.921476	+	0.388435i	$0.126984\pi$
$318$		0			0
$319$		0			0
$320$	1.48588	+	0.223960i	1.48588	+	0.223960i
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−1.19824	−	2.07541i	−1.19824	−	2.07541i
$325$		0			0
$326$	3.42493	−	0.516225i	3.42493	−	0.516225i
$327$		0			0
$328$	3.06859	+	1.47775i	3.06859	+	1.47775i
$329$	−0.786030	−	1.74202i	−0.786030	−	1.74202i
$330$		0			0
$331$		0			0		0.365341	−	0.930874i	$-0.380952\pi$
$331$		0			0		−0.365341	+	0.930874i	$0.619048\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−0.751017	+	3.29042i	−0.751017	+	3.29042i
$336$		0			0
$337$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$337$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$338$	1.76108	−	0.543220i	1.76108	−	0.543220i
$339$		0			0
$340$		0			0
$341$		0			0
$342$	−1.51597			−1.51597
$343$	0.0747301	−	0.997204i	0.0747301	−	0.997204i
$344$		0			0
$345$		0			0
$346$	−1.68464	−	1.56312i	−1.68464	−	1.56312i
$347$		0			0		−0.365341	−	0.930874i	$-0.619048\pi$
$347$		0			0		0.365341	+	0.930874i	$0.380952\pi$
$348$		0			0
$349$	0.222521	+	0.974928i	0.222521	+	0.974928i	0.955573	+	0.294755i	$0.0952381\pi$
$349$	0.222521	+	0.974928i	0.222521	+	0.974928i	−0.733052	+	0.680173i	$0.761905\pi$
$350$	−3.00742	−	1.83780i	−3.00742	−	1.83780i
$351$		0			0
$352$		0			0
$353$		0			0		−0.955573	−	0.294755i	$-0.904762\pi$
$353$		0			0		0.955573	+	0.294755i	$0.0952381\pi$
$354$		0			0
$355$	0.993999	−	2.53267i	0.993999	−	2.53267i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	0.466934	−	0.433252i	0.466934	−	0.433252i	−0.411287	−	0.911506i	$-0.634921\pi$
$359$	0.466934	−	0.433252i	0.466934	−	0.433252i	0.878222	+	0.478254i	$0.158730\pi$
$360$	4.34306	−	0.654611i	4.34306	−	0.654611i
$361$	0.161686	−	0.280048i	0.161686	−	0.280048i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		0.826239	−	0.563320i	$-0.190476\pi$
$367$		0			0		−0.826239	+	0.563320i	$0.809524\pi$
$368$		0			0
$369$	1.30859	+	0.197238i	1.30859	+	0.197238i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−0.766044	−	1.32683i	−0.766044	−	1.32683i	−0.939693	−	0.342020i	$-0.888889\pi$
$373$	−0.766044	−	1.32683i	−0.766044	−	1.32683i	0.173648	−	0.984808i	$-0.444444\pi$
$374$		0			0
$375$		0			0
$376$	−3.60558	+	3.34549i	−3.60558	+	3.34549i
$377$		0			0
$378$		0			0
$379$	−0.134659	+	0.0648483i	−0.134659	+	0.0648483i	−0.500000	−	0.866025i	$-0.666667\pi$
$379$	−0.134659	+	0.0648483i	−0.134659	+	0.0648483i	0.365341	+	0.930874i	$0.380952\pi$
$380$	1.22906	−	3.13160i	1.22906	−	3.13160i
$381$		0			0
$382$	−2.05603	−	0.634202i	−2.05603	−	0.634202i
$383$		0			0		−0.0747301	−	0.997204i	$-0.523810\pi$
$383$		0			0		0.0747301	+	0.997204i	$0.476190\pi$
$384$		0			0
$385$		0			0
$386$	0.819171	+	3.58902i	0.819171	+	3.58902i
$387$		0			0
$388$	1.71634	+	4.37317i	1.71634	+	4.37317i
$389$		0			0		−0.733052	−	0.680173i	$-0.761905\pi$
$389$		0			0		0.733052	+	0.680173i	$0.238095\pi$
$390$		0			0
$391$		0			0
$392$	−2.49405	+	0.635063i	−2.49405	+	0.635063i
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	1.76108	−	0.543220i	1.76108	−	0.543220i	0.766044	−	0.642788i	$-0.222222\pi$
$397$	1.76108	−	0.543220i	1.76108	−	0.543220i	0.995031	+	0.0995678i	$0.0317460\pi$
$398$		0			0
$399$		0			0
$400$	−0.998610	+	4.37519i	−0.998610	+	4.37519i
$401$		0			0		−0.0747301	−	0.997204i	$-0.523810\pi$
$401$		0			0		0.0747301	+	0.997204i	$0.476190\pi$
$402$		0			0
$403$		0			0
$404$	0.798852	−	2.03544i	0.798852	−	2.03544i
$405$	1.53758	−	0.740458i	1.53758	−	0.740458i
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0		0.988831	−	0.149042i	$-0.0476190\pi$
$409$		0			0		−0.988831	+	0.149042i	$0.952381\pi$
$410$	−2.08110	+	3.60457i	−2.08110	+	3.60457i
$411$		0			0
$412$	2.62444	+	3.29094i	2.62444	+	3.29094i
$413$	−0.358423	−	0.406142i	−0.358423	−	0.406142i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−0.397146	+	0.498006i	−0.397146	+	0.498006i	−0.939693	−	0.342020i	$-0.888889\pi$
$419$	−0.397146	+	0.498006i	−0.397146	+	0.498006i	0.542546	+	0.840026i	$0.317460\pi$
$420$		0			0
$421$	1.09512	+	1.37324i	1.09512	+	1.37324i	0.921476	+	0.388435i	$0.126984\pi$
$421$	1.09512	+	1.37324i	1.09512	+	1.37324i	0.173648	+	0.984808i	$0.444444\pi$
$422$	−1.61852	−	2.80336i	−1.61852	−	2.80336i
$423$	−0.955573	+	1.65510i	−0.955573	+	1.65510i
$424$		0			0
$425$		0			0
$426$		0			0
$427$		0			0
$428$	−4.23268	+	2.03835i	−4.23268	+	2.03835i
$429$		0			0
$430$		0			0
$431$	1.82624	+	0.563320i	1.82624	+	0.563320i	1.00000			$0$
$431$	1.82624	+	0.563320i	1.82624	+	0.563320i	0.826239	+	0.563320i	$0.190476\pi$
$432$		0			0
$433$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$433$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$434$	1.83379	+	0.183499i	1.83379	+	0.183499i
$435$		0			0
$436$	4.55725	−	1.40573i	4.55725	−	1.40573i
$437$		0			0
$438$		0			0
$439$	0.896546	+	0.611254i	0.896546	+	0.611254i	0.921476	−	0.388435i	$-0.126984\pi$
$439$	0.896546	+	0.611254i	0.896546	+	0.611254i	−0.0249307	+	0.999689i	$0.507937\pi$
$440$		0			0
$441$	−0.853291	+	0.521435i	−0.853291	+	0.521435i
$442$		0			0
$443$	0.286950	+	0.195639i	0.286950	+	0.195639i	0.698237	−	0.715867i	$-0.253968\pi$
$443$	0.286950	+	0.195639i	0.286950	+	0.195639i	−0.411287	+	0.911506i	$0.634921\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$	0.477717	+	0.739650i	0.477717	+	0.739650i
$449$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$449$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$450$	0.263386	+	3.51465i	0.263386	+	3.51465i
$451$		0			0
$452$	0.194328	−	2.59312i	0.194328	−	2.59312i
$453$		0			0
$454$	2.99201	−	1.44088i	2.99201	−	1.44088i
$455$		0			0
$456$		0			0
$457$		0			0		0.733052	−	0.680173i	$-0.238095\pi$
$457$		0			0		−0.733052	+	0.680173i	$0.761905\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		−0.623490	−	0.781831i	$-0.714286\pi$
$461$		0			0		0.623490	+	0.781831i	$0.285714\pi$
$462$		0			0
$463$		0			0		0.623490	−	0.781831i	$-0.285714\pi$
$463$		0			0		−0.623490	+	0.781831i	$0.714286\pi$
$464$		0			0
$465$		0			0
$466$	−0.969931	+	0.661288i	−0.969931	+	0.661288i
$467$	−0.343417	−	0.0517618i	−0.343417	−	0.0517618i	−0.0249307	−	0.999689i	$-0.507937\pi$
$467$	−0.343417	−	0.0517618i	−0.343417	−	0.0517618i	−0.318487	+	0.947927i	$0.603175\pi$
$468$		0			0
$469$	−1.73683	+	0.945825i	−1.73683	+	0.945825i
$470$	−3.74770	−	4.69946i	−3.74770	−	4.69946i
$471$		0			0
$472$	−0.697044	+	1.20732i	−0.697044	+	1.20732i
$473$		0			0
$474$		0			0
$475$	1.41732	+	0.682546i	1.41732	+	0.682546i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−0.119140	+	1.58981i	−0.119140	+	1.58981i	0.542546	+	0.840026i	$0.317460\pi$
$479$	−0.119140	+	1.58981i	−0.119140	+	1.58981i	−0.661686	+	0.749781i	$0.730159\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	−0.533266	−	2.33639i	−0.533266	−	2.33639i
$485$	−3.19686	+	0.986100i	−3.19686	+	0.986100i
$486$		0			0
$487$		0			0		−0.733052	−	0.680173i	$-0.761905\pi$
$487$		0			0		0.733052	+	0.680173i	$0.238095\pi$
$488$		0			0
$489$		0			0
$490$	−0.546149	−	3.09737i	−0.546149	−	3.09737i
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−0.522170	−	2.28778i	−0.522170	−	2.28778i
$497$	1.49812	−	0.545271i	1.49812	−	0.545271i
$498$		0			0
$499$		0			0		−0.0747301	−	0.997204i	$-0.523810\pi$
$499$		0			0		0.0747301	+	0.997204i	$0.476190\pi$
$500$	−3.56582	−	1.09991i	−3.56582	−	1.09991i
$501$		0			0
$502$		0			0
$503$	0.0449236	−	0.0216340i	0.0449236	−	0.0216340i	−0.411287	−	0.911506i	$-0.634921\pi$
$503$	0.0449236	−	0.0216340i	0.0449236	−	0.0216340i	0.456211	+	0.889872i	$0.349206\pi$
$504$	1.97152	+	1.65430i	1.97152	+	1.65430i
$505$	1.40292	+	0.675610i	1.40292	+	0.675610i
$506$		0			0
$507$		0			0
$508$		0			0
$509$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$509$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$510$		0			0
$511$		0			0
$512$	−1.20350	+	1.50914i	−1.20350	+	1.50914i
$513$		0			0
$514$	2.33294	−	1.59057i	2.33294	−	1.59057i
$515$	−2.47666	+	1.68856i	−2.47666	+	1.68856i
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	0.733052	−	1.26968i	0.733052	−	1.26968i	−0.222521	−	0.974928i	$-0.571429\pi$
$521$	0.733052	−	1.26968i	0.733052	−	1.26968i	0.955573	−	0.294755i	$-0.0952381\pi$
$522$		0			0
$523$		0			0		0.733052	−	0.680173i	$-0.238095\pi$
$523$		0			0		−0.733052	+	0.680173i	$0.761905\pi$
$524$	−3.56794	−	1.71823i	−3.56794	−	1.71823i
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	0.955573	+	0.294755i	0.955573	+	0.294755i
$530$		0			0
$531$	−0.120535	+	0.528100i	−0.120535	+	0.528100i
$532$	1.85239	−	0.674217i	1.85239	−	0.674217i
$533$		0			0
$534$		0			0
$535$	−1.22224	−	3.11423i	−1.22224	−	3.11423i
$536$	3.73107	+	3.46193i	3.73107	+	3.46193i
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−0.964623	−	0.657669i	−0.964623	−	0.657669i	−0.0249307	−	0.999689i	$-0.507937\pi$
$541$	−0.964623	−	0.657669i	−0.964623	−	0.657669i	−0.939693	+	0.342020i	$0.888889\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	0.755726	+	3.31105i	0.755726	+	3.31105i
$546$		0			0
$547$	−0.340922	+	1.49368i	−0.340922	+	1.49368i	0.456211	+	0.889872i	$0.349206\pi$
$547$	−0.340922	+	1.49368i	−0.340922	+	1.49368i	−0.797133	+	0.603804i	$0.793651\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$557$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$558$	−0.921476	−	1.59604i	−0.921476	−	1.59604i
$559$		0			0
$560$	−3.51700	+	1.91526i	−3.51700	+	1.91526i
$561$		0			0
$562$	−0.987142	−	0.148788i	−0.987142	−	0.148788i
$563$	−1.55282	+	1.05870i	−1.55282	+	1.05870i	−0.583744	+	0.811938i	$0.698413\pi$
$563$	−1.55282	+	1.05870i	−1.55282	+	1.05870i	−0.969077	+	0.246757i	$0.920635\pi$
$564$		0			0
$565$	1.83112	+	0.275996i	1.83112	+	0.275996i
$566$	−1.14906	+	1.44088i	−1.14906	+	1.44088i
$567$	0.921476	+	0.388435i	0.921476	+	0.388435i
$568$	−2.55821	−	3.20790i	−2.55821	−	3.20790i
$569$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$569$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$570$		0			0
$571$		0			0		0.988831	−	0.149042i	$-0.0476190\pi$
$571$		0			0		−0.988831	+	0.149042i	$0.952381\pi$
$572$		0			0
$573$		0			0
$574$	−2.39055	+	0.483261i	−2.39055	+	0.483261i
$575$		0			0
$576$	0.321686	−	0.819642i	0.321686	−	0.819642i
$577$	0.123490	−	1.64786i	0.123490	−	1.64786i	−0.500000	−	0.866025i	$-0.666667\pi$
$577$	0.123490	−	1.64786i	0.123490	−	1.64786i	0.623490	−	0.781831i	$-0.285714\pi$
$578$	1.76108	+	0.543220i	1.76108	+	0.543220i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$	1.11262	+	0.758574i	1.11262	+	0.758574i
$587$		0			0		1.00000			$0$
$587$		0			0		−1.00000			$\pi$
$588$		0			0
$589$	−0.822574			−0.822574
$590$	−1.40764	−	0.959710i	−1.40764	−	0.959710i
$591$		0			0
$592$		0			0
$593$	0.331867	−	0.102367i	0.331867	−	0.102367i	−0.124344	−	0.992239i	$-0.539683\pi$
$593$	0.331867	−	0.102367i	0.331867	−	0.102367i	0.456211	+	0.889872i	$0.349206\pi$
$594$		0			0
$595$		0			0
$596$	0.533266	−	2.33639i	0.533266	−	2.33639i
$597$		0			0
$598$		0			0
$599$	0.114493	−	1.52780i	0.114493	−	1.52780i	−0.583744	−	0.811938i	$-0.698413\pi$
$599$	0.114493	−	1.52780i	0.114493	−	1.52780i	0.698237	−	0.715867i	$-0.253968\pi$
$600$		0			0
$601$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$601$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$602$		0			0
$603$	1.78181	+	0.858075i	1.78181	+	0.858075i
$604$		0			0
$605$	1.68752	−	0.254353i	1.68752	−	0.254353i
$606$		0			0
$607$	−0.0747301	−	0.129436i	−0.0747301	−	0.129436i	0.826239	−	0.563320i	$-0.190476\pi$
$607$	−0.0747301	−	0.129436i	−0.0747301	−	0.129436i	−0.900969	+	0.433884i	$0.857143\pi$
$608$	−0.898060	−	1.12613i	−0.898060	−	1.12613i
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		0.826239	−	0.563320i	$-0.190476\pi$
$613$		0			0		−0.826239	+	0.563320i	$0.809524\pi$
$614$	−2.54489	−	0.383580i	−2.54489	−	0.383580i
$615$		0			0
$616$		0			0
$617$	−0.914101	−	1.14625i	−0.914101	−	1.14625i	−0.988831	−	0.149042i	$-0.952381\pi$
$617$	−0.914101	−	1.14625i	−0.914101	−	1.14625i	0.0747301	−	0.997204i	$-0.476190\pi$
$618$		0			0
$619$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$619$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$620$	4.04410	−	0.609550i	4.04410	−	0.609550i
$621$		0			0
$622$	0.412931	+	0.198857i	0.412931	+	0.198857i
$623$		0			0
$624$		0			0
$625$	0.272155	−	0.693439i	0.272155	−	0.693439i
$626$		0			0
$627$		0			0
$628$	−0.357732	−	4.77360i	−0.357732	−	4.77360i
$629$		0			0
$630$	−2.19606	+	2.25151i	−2.19606	+	2.25151i
$631$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$631$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$632$		0			0
$633$		0			0
$634$	2.61841	+	2.42953i	2.61841	+	2.42953i
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$	−1.31724	−	0.898081i	−1.31724	−	0.898081i
$640$	−0.160536	−	0.148956i	−0.160536	−	0.148956i
$641$		0			0		−0.365341	−	0.930874i	$-0.619048\pi$
$641$		0			0		0.365341	+	0.930874i	$0.380952\pi$
$642$		0			0
$643$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$643$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.955573	−	0.294755i	$-0.904762\pi$
$647$		0			0		0.955573	+	0.294755i	$0.0952381\pi$
$648$	0.192328	−	2.56644i	0.192328	−	2.56644i
$649$		0			0
$650$		0			0
$651$		0			0
$652$	4.05787	+	1.95417i	4.05787	+	1.95417i
$653$	−0.109562	+	0.101659i	−0.109562	+	0.101659i	−0.733052	−	0.680173i	$-0.761905\pi$
$653$	−0.109562	+	0.101659i	−0.109562	+	0.101659i	0.623490	+	0.781831i	$0.285714\pi$
$654$		0			0
$655$	1.41004	−	2.44227i	1.41004	−	2.44227i
$656$	1.55272	+	2.68939i	1.55272	+	2.68939i
$657$		0			0
$658$	0.611615	−	3.46864i	0.611615	−	3.46864i
$659$	0.676544	−	0.848359i	0.676544	−	0.848359i	−0.318487	−	0.947927i	$-0.603175\pi$
$659$	0.676544	−	0.848359i	0.676544	−	0.848359i	0.995031	+	0.0995678i	$0.0317460\pi$
$660$		0			0
$661$	0.286950	−	0.195639i	0.286950	−	0.195639i	−0.411287	−	0.911506i	$-0.634921\pi$
$661$	0.286950	−	0.195639i	0.286950	−	0.195639i	0.698237	+	0.715867i	$0.253968\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	0.380203	+	1.35132i	0.380203	+	1.35132i
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$	−4.55961	+	4.23070i	−4.55961	+	4.23070i
$671$		0			0
$672$		0			0
$673$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$673$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$674$		0			0
$675$		0			0
$676$	2.29001	+	0.706373i	2.29001	+	0.706373i
$677$		0			0		−0.0747301	−	0.997204i	$-0.523810\pi$
$677$		0			0		0.0747301	+	0.997204i	$0.476190\pi$
$678$		0			0
$679$	−1.67274	−	1.02219i	−1.67274	−	1.02219i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−1.45882	−	1.35359i	−1.45882	−	1.35359i	−0.797133	−	0.603804i	$-0.793651\pi$
$683$	−1.45882	−	1.35359i	−1.45882	−	1.35359i	−0.661686	−	0.749781i	$-0.730159\pi$
$684$	−1.62875	−	1.11046i	−1.62875	−	1.11046i
$685$		0			0
$686$	1.14906	−	1.44088i	1.14906	−	1.44088i
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	1.03688	−	0.319837i	1.03688	−	0.319837i	0.270840	−	0.962624i	$-0.412698\pi$
$691$	1.03688	−	0.319837i	1.03688	−	0.319837i	0.766044	+	0.642788i	$0.222222\pi$
$692$	−0.664971	−	2.91343i	−0.664971	−	2.91343i
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$	−0.673306	+	1.71556i	−0.673306	+	1.71556i
$699$		0			0
$700$	−1.88496	−	4.17749i	−1.88496	−	4.17749i
$701$	−1.76621	−	0.850562i	−1.76621	−	0.850562i	−0.969077	−	0.246757i	$-0.920635\pi$
$701$	−1.76621	−	0.850562i	−1.76621	−	0.850562i	−0.797133	−	0.603804i	$-0.793651\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$	0.247121	+	0.878319i	0.247121	+	0.878319i
$708$		0			0
$709$		0			0		−0.988831	−	0.149042i	$-0.952381\pi$
$709$		0			0		0.988831	+	0.149042i	$0.0476190\pi$
$710$	4.14293	−	2.82460i	4.14293	−	2.82460i
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$	1.16080	−	0.174962i	1.16080	−	0.174962i
$719$		0			0		0.733052	−	0.680173i	$-0.238095\pi$
$719$		0			0		−0.733052	+	0.680173i	$0.761905\pi$
$720$	3.60810	+	1.73757i	3.60810	+	1.73757i
$721$	−1.70213	−	0.433415i	−1.70213	−	0.433415i
$722$	0.536940	−	0.258577i	0.536940	−	0.258577i
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	0.141740	−	0.621003i	0.141740	−	0.621003i	−0.853291	−	0.521435i	$-0.825397\pi$
$727$	0.141740	−	0.621003i	0.141740	−	0.621003i	0.995031	−	0.0995678i	$-0.0317460\pi$
$728$		0			0
$729$	−0.222521	−	0.974928i	−0.222521	−	0.974928i
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−1.31724	−	0.898081i	−1.31724	−	0.898081i	−0.318487	−	0.947927i	$-0.603175\pi$
$733$	−1.31724	−	0.898081i	−1.31724	−	0.898081i	−0.998757	+	0.0498459i	$0.984127\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$	1.78785	+	1.65888i	1.78785	+	1.65888i
$739$		0			0		−0.365341	−	0.930874i	$-0.619048\pi$
$739$		0			0		0.365341	+	0.930874i	$0.380952\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$743$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$744$		0			0
$745$	1.63076	+	0.503024i	1.63076	+	0.503024i
$746$	0.211005	−	2.81567i	0.211005	−	2.81567i
$747$		0			0
$748$		0			0
$749$	0.894330	−	1.74446i	0.894330	−	1.74446i
$750$		0			0
$751$	−0.254586	+	0.236222i	−0.254586	+	0.236222i	−0.797133	−	0.603804i	$-0.793651\pi$
$751$	−0.254586	+	0.236222i	−0.254586	+	0.236222i	0.542546	+	0.840026i	$0.317460\pi$
$752$	−4.43463	+	0.668412i	−4.43463	+	0.668412i
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		0.623490	−	0.781831i	$-0.285714\pi$
$757$		0			0		−0.623490	+	0.781831i	$0.714286\pi$
$758$	−0.272371	−	0.0410534i	−0.272371	−	0.0410534i
$759$		0			0
$760$	2.98507	−	2.03519i	2.98507	−	2.03519i
$761$		0			0		−0.988831	−	0.149042i	$-0.952381\pi$
$761$		0			0		0.988831	+	0.149042i	$0.0476190\pi$
$762$		0			0
$763$	−1.16169	+	1.61581i	−1.16169	+	1.61581i
$764$	−1.74443	−	2.18745i	−1.74443	−	2.18745i
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−1.66044	−	0.799627i	−1.66044	−	0.799627i	−0.998757	−	0.0498459i	$-0.984127\pi$
$769$	−1.66044	−	0.799627i	−1.66044	−	0.799627i	−0.661686	−	0.749781i	$-0.730159\pi$
$770$		0			0
$771$		0			0
$772$	−1.74888	+	4.45608i	−1.74888	+	4.45608i
$773$		0			0		0.0747301	−	0.997204i	$-0.476190\pi$
$773$		0			0		−0.0747301	+	0.997204i	$0.523810\pi$
$774$		0			0
$775$	0.142915	+	1.90707i	0.142915	+	1.90707i
$776$	−1.12267	+	4.91872i	−1.12267	+	4.91872i
$777$		0			0
$778$		0			0
$779$	1.04021	−	0.320862i	1.04021	−	0.320862i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−2.20509	−	0.802589i	−2.20509	−	0.802589i
$785$	3.40892			3.40892
$786$		0			0
$787$		0			0		−0.733052	−	0.680173i	$-0.761905\pi$
$787$		0			0		0.733052	+	0.680173i	$0.238095\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	0.588713	+	0.911506i	0.588713	+	0.911506i
$792$		0			0
$793$		0			0
$794$	3.24558	+	1.00113i	3.24558	+	1.00113i
$795$		0			0
$796$		0			0
$797$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$797$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$798$		0			0
$799$		0			0
$800$	−2.45482	+	2.27774i	−2.45482	+	2.27774i
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$	1.94021	−	1.32281i	1.94021	−	1.32281i
$809$		0			0		0.826239	−	0.563320i	$-0.190476\pi$
$809$		0			0		−0.826239	+	0.563320i	$0.809524\pi$
$810$	3.11002	+	0.468760i	3.11002	+	0.468760i
$811$	−0.914101	+	1.14625i	−0.914101	+	1.14625i	0.0747301	+	0.997204i	$0.476190\pi$
$811$	−0.914101	+	1.14625i	−0.914101	+	1.14625i	−0.988831	+	0.149042i	$0.952381\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−1.60366	+	2.77762i	−1.60366	+	2.77762i
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$	−4.87631	+	2.34831i	−4.87631	+	2.34831i
$821$		0			0		0.365341	−	0.930874i	$-0.380952\pi$
$821$		0			0		−0.365341	+	0.930874i	$0.619048\pi$
$822$		0			0
$823$		0			0		−0.955573	−	0.294755i	$-0.904762\pi$
$823$		0			0		0.955573	+	0.294755i	$0.0952381\pi$
$824$	0.337813	+	4.50780i	0.337813	+	4.50780i
$825$		0			0
$826$	−0.124131	−	0.990545i	−0.124131	−	0.990545i
$827$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$827$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$828$		0			0
$829$		0			0		−0.365341	−	0.930874i	$-0.619048\pi$
$829$		0			0		0.365341	+	0.930874i	$0.380952\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$	−1.12176	+	0.346016i	−1.12176	+	0.346016i
$839$	−0.162592	−	0.712362i	−0.162592	−	0.712362i	−0.988831	−	0.149042i	$-0.952381\pi$
$839$	−0.162592	−	0.712362i	−0.162592	−	0.712362i	0.826239	−	0.563320i	$-0.190476\pi$
$840$		0			0
$841$	−0.222521	+	0.974928i	−0.222521	+	0.974928i
$842$	0.241904	+	3.22799i	0.241904	+	3.22799i
$843$		0			0
$844$	0.314559	−	4.19750i	0.314559	−	4.19750i
$845$	−0.623484	+	1.58861i	−0.623484	+	1.58861i
$846$	−3.17335	+	1.52820i	−3.17335	+	1.52820i
$847$	0.766044	+	0.642788i	0.766044	+	0.642788i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−0.914101	−	1.14625i	−0.914101	−	1.14625i	−0.988831	−	0.149042i	$-0.952381\pi$
$853$	−0.914101	−	1.14625i	−0.914101	−	1.14625i	0.0747301	−	0.997204i	$-0.476190\pi$
$854$		0			0
$855$	0.875249	−	1.09753i	0.875249	−	1.09753i
$856$	−4.98886	−	0.751950i	−4.98886	−	0.751950i
$857$	−0.826239	+	0.563320i	−0.826239	+	0.563320i	−0.900969	−	0.433884i	$-0.857143\pi$
$857$	−0.826239	+	0.563320i	−0.826239	+	0.563320i	0.0747301	+	0.997204i	$0.476190\pi$
$858$		0			0
$859$		0			0		−0.988831	−	0.149042i	$-0.952381\pi$
$859$		0			0		0.988831	+	0.149042i	$0.0476190\pi$
$860$		0			0
$861$		0			0
$862$	2.19602	+	2.75373i	2.19602	+	2.75373i
$863$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$863$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$864$		0			0
$865$	2.10430	−	0.317173i	2.10430	−	0.317173i
$866$		0			0
$867$		0			0
$868$	1.83581	+	1.54042i	1.83581	+	1.54042i
$869$		0			0
$870$		0			0
$871$		0			0
$872$	4.89415	+	1.50964i	4.89415	+	1.50964i
$873$	0.146497	+	1.95486i	0.146497	+	1.95486i
$874$		0			0
$875$	1.46322	−	0.532567i	1.46322	−	0.532567i
$876$		0			0
$877$	−1.63076	+	0.503024i	−1.63076	+	0.503024i	−0.969077	−	0.246757i	$-0.920635\pi$
$877$	−1.63076	+	0.503024i	−1.63076	+	0.503024i	−0.661686	+	0.749781i	$0.730159\pi$
$878$	0.730599	+	1.86154i	0.730599	+	1.86154i
$879$		0			0
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$	−1.84066	−	0.0918636i	−1.84066	−	0.0918636i
$883$		0			0		1.00000			$0$
$883$		0			0		−1.00000			$\pi$
$884$		0			0
$885$		0			0
$886$	0.233837	+	0.595806i	0.233837	+	0.595806i
$887$	1.90165	−	0.586581i	1.90165	−	0.586581i	0.921476	−	0.388435i	$-0.126984\pi$
$887$	1.90165	−	0.586581i	1.90165	−	0.586581i	0.980172	−	0.198146i	$-0.0634921\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−0.117480	+	1.56766i	−0.117480	+	1.56766i
$894$		0			0
$895$		0			0
$896$	0.00319923	−	0.128285i	0.00319923	−	0.128285i
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−2.29153	+	3.96905i	−2.29153	+	3.96905i
$901$		0			0
$902$		0			0
$903$		0			0
$904$	1.74118	−	2.18337i	1.74118	−	2.18337i
$905$		0			0
$906$		0			0
$907$	−0.526292	+	0.358820i	−0.526292	+	0.358820i	−0.797133	−	0.603804i	$-0.793651\pi$
$907$	−0.526292	+	0.358820i	−0.526292	+	0.358820i	0.270840	+	0.962624i	$0.412698\pi$
$908$	4.27006	+	0.643609i	4.27006	+	0.643609i
$909$	0.568885	−	0.713360i	0.568885	−	0.713360i
$910$		0			0
$911$		0			0		−0.623490	−	0.781831i	$-0.714286\pi$
$911$		0			0		0.623490	+	0.781831i	$0.285714\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$	1.61971	−	0.327432i	1.61971	−	0.327432i
$918$		0			0
$919$	−0.365341	+	0.930874i	−0.365341	+	0.930874i	0.623490	+	0.781831i	$0.285714\pi$
$919$	−0.365341	+	0.930874i	−0.365341	+	0.930874i	−0.988831	+	0.149042i	$0.952381\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$	0.641701	+	1.63503i	0.641701	+	1.63503i
$928$		0			0
$929$		0			0		−0.826239	−	0.563320i	$-0.809524\pi$
$929$		0			0		0.826239	+	0.563320i	$0.190476\pi$
$930$		0			0
$931$	−0.446285	+	0.690984i	−0.446285	+	0.690984i
$932$	−1.52649			−1.52649
$933$		0			0
$934$	−0.469190	−	0.435345i	−0.469190	−	0.435345i
$935$		0			0
$936$		0			0
$937$	0.0990311	+	0.433884i	0.0990311	+	0.433884i	1.00000			$0$
$937$	0.0990311	+	0.433884i	0.0990311	+	0.433884i	−0.900969	+	0.433884i	$0.857143\pi$
$938$	−3.62662	−	0.362899i	−3.62662	−	0.362899i
$939$		0			0
$940$	−0.584102	−	7.79431i	−0.584102	−	7.79431i
$941$		0			0		−0.955573	−	0.294755i	$-0.904762\pi$
$941$		0			0		0.955573	+	0.294755i	$0.0952381\pi$
$942$		0			0
$943$		0			0
$944$	−1.14524	+	0.551516i	−1.14524	+	0.551516i
$945$		0			0
$946$		0			0
$947$		0			0		0.733052	−	0.680173i	$-0.238095\pi$
$947$		0			0		−0.733052	+	0.680173i	$0.761905\pi$
$948$		0			0
$949$		0			0
$950$	1.44958	+	2.51075i	1.44958	+	2.51075i
$951$		0			0
$952$		0			0
$953$		0			0		0.623490	−	0.781831i	$-0.285714\pi$
$953$		0			0		−0.623490	+	0.781831i	$0.714286\pi$
$954$		0			0
$955$	1.64621	−	1.12237i	1.64621	−	1.12237i
$956$		0			0
$957$		0			0
$958$	−1.83191	+	2.29714i	−1.83191	+	2.29714i
$959$		0			0
$960$		0			0
$961$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$962$		0			0
$963$	−1.93845	+	0.292174i	−1.93845	+	0.292174i
$964$		0			0
$965$	−3.07133	−	1.47908i	−3.07133	−	1.47908i
$966$		0			0
$967$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$967$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$968$	0.940254	−	2.39573i	0.940254	−	2.39573i
$969$		0			0
$970$	−5.89166	−	1.81734i	−5.89166	−	1.81734i
$971$	−0.0332580	−	0.443797i	−0.0332580	−	0.443797i	−0.988831	−	0.149042i	$-0.952381\pi$
$971$	−0.0332580	−	0.443797i	−0.0332580	−	0.443797i	0.955573	−	0.294755i	$-0.0952381\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	0.602990	+	0.559493i	0.602990	+	0.559493i	0.921476	−	0.388435i	$-0.126984\pi$
$977$	0.602990	+	0.559493i	0.602990	+	0.559493i	−0.318487	+	0.947927i	$0.603175\pi$
$978$		0			0
$979$		0			0
$980$	1.68207	−	3.72786i	1.68207	−	3.72786i
$981$	1.99006			1.99006
$982$		0			0
$983$		0			0		−0.733052	−	0.680173i	$-0.761905\pi$
$983$		0			0		0.733052	+	0.680173i	$0.238095\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		0.0747301	−	0.997204i	$-0.476190\pi$
$991$		0			0		−0.0747301	+	0.997204i	$0.523810\pi$
$992$	0.639734	−	1.63002i	0.639734	−	1.63002i
$993$		0			0
$994$	2.84730	+	0.725011i	2.84730	+	0.725011i
$995$		0			0
$996$		0			0
$997$	1.91651	−	0.288867i	1.91651	−	0.288867i	0.921476	−	0.388435i	$-0.126984\pi$
$997$	1.91651	−	0.288867i	1.91651	−	0.288867i	0.995031	+	0.0995678i	$0.0317460\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1519.1.ce.a.1425.3	yes	36
31.30	odd	2	CM	1519.1.ce.a.1425.3	yes	36
49.37	even	21	inner	1519.1.ce.a.1115.3	✓	36
1519.1115	odd	42	inner	1519.1.ce.a.1115.3	✓	36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1519.1.ce.a.1115.3	✓	36	49.37	even	21	inner
1519.1.ce.a.1115.3	✓	36	1519.1115	odd	42	inner
1519.1.ce.a.1425.3	yes	36	1.1	even	1	trivial
1519.1.ce.a.1425.3	yes	36	31.30	odd	2	CM

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:	\( N \)	\(=\)	\( 1519 = 7^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1519.ce (of order \(42\), degree \(12\), minimal)

\(f(q)\)	\(=\)	\(q+(1.52272 + 1.03817i) q^{2} +(0.875530 + 2.23081i) q^{4} +(-1.63076 + 0.503024i) q^{5} +(-0.853291 - 0.521435i) q^{7} +(-0.572688 + 2.50911i) q^{8} +(0.0747301 + 0.997204i) q^{9} +(-3.00542 - 0.927049i) q^{10} +(-0.757983 - 1.67986i) q^{14} +(-1.72019 + 1.59610i) q^{16} +(-0.921476 + 1.59604i) q^{18} +(0.411287 + 0.712370i) q^{19} +(-2.54994 - 3.19752i) q^{20} +(1.58012 - 1.07731i) q^{25} +(0.416143 - 2.36007i) q^{28} +(-0.500000 + 0.866025i) q^{31} +(-1.73150 + 0.260982i) q^{32} +(1.65381 + 0.421112i) q^{35} +(-2.15915 + 1.03979i) q^{36} +(-0.113288 + 1.51173i) q^{38} +(-0.328223 - 4.37984i) q^{40} +(0.294478 - 1.29019i) q^{41} +(-0.623484 - 1.58861i) q^{45} +(1.57906 + 1.07659i) q^{47} +(0.456211 + 0.889872i) q^{49} +3.52450 q^{50} +(1.79701 - 1.84238i) q^{56} +(0.517616 + 0.159663i) q^{59} +(-1.66044 + 0.799627i) q^{62} +(0.456211 - 0.889872i) q^{63} +(-0.793311 - 0.382038i) q^{64} +(0.988831 - 1.71271i) q^{67} +(2.08110 + 2.35817i) q^{70} +(-0.994008 + 1.24645i) q^{71} +(-2.54489 - 0.383580i) q^{72} +(-1.22907 + 1.54121i) q^{76} +(2.00234 - 3.46816i) q^{80} +(-0.988831 + 0.149042i) q^{81} +(1.78785 - 1.65888i) q^{82} +(0.699861 - 3.06629i) q^{90} +(1.28679 + 3.27868i) q^{94} +(-1.02905 - 0.954820i) q^{95} +1.96034 q^{97} +(-0.229160 + 1.82865i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q + 3 q^{4} + 6 q^{8} + 3 q^{9} - 3 q^{10} + 3 q^{14} + 3 q^{16} - 12 q^{20} + 3 q^{25} + 3 q^{28} - 18 q^{31} - 24 q^{32} + 3 q^{35} - 6 q^{36} - 3 q^{38} - 3 q^{40} - 3 q^{47} + 6 q^{50} + 3 q^{56}+ \cdots - 18 q^{98}+O(q^{100}) \) 36 * q + 3 * q^4 + 6 * q^8 + 3 * q^9 - 3 * q^10 + 3 * q^14 + 3 * q^16 - 12 * q^20 + 3 * q^25 + 3 * q^28 - 18 * q^31 - 24 * q^32 + 3 * q^35 - 6 * q^36 - 3 * q^38 - 3 * q^40 - 3 * q^47 + 6 * q^50 + 3 * q^56 - 3 * q^67 - 6 * q^70 - 24 * q^72 - 15 * q^76 + 24 * q^80 + 3 * q^81 + 6 * q^82 + 6 * q^90 - 3 * q^95 - 18 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more