LMFDB - Embedded newform 1519.1.ba.b.92.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1519,1,Mod(92,1519)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1519, base_ring=CyclotomicField(14))

chi = DirichletCharacter(H, H._module([2, 7]))

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

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

chi := DirichletCharacter("1519.92");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1519 = 7^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1519.ba (of order $14$, degree $6$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$0.758079754190$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$2$ over $\Q(\zeta_{14})$
Coefficient field:	$\Q(\zeta_{21})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - x^{11} + x^{9} - x^{8} + x^{6} - x^{4} + x^{3} - x + 1 $ x^12 - x^11 + x^9 - x^8 + x^6 - x^4 + x^3 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{21}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{21} + \cdots)$

Embedding invariants

Embedding label			92.2
Root			$0.826239 + 0.563320i$ of defining polynomial
Character	$\chi$	$=$	1519.92
Dual form			1519.1.ba.b.743.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(1.03030 - 1.29196i) q^{2} +(-0.385113 - 1.68729i) q^{4} +(1.32091 + 0.636119i) q^{5} +(-0.733052 - 0.680173i) q^{7} +(-1.08786 - 0.523887i) q^{8} +(0.623490 + 0.781831i) q^{9} +(2.18278 - 1.05117i) q^{10} +(-1.63402 + 0.246289i) q^{14} +(-0.238377 + 0.114796i) q^{16} +1.65248 q^{18} -1.97766 q^{19} +(0.564616 - 2.47374i) q^{20} +(0.716677 + 0.898684i) q^{25} +(-0.865341 + 1.49881i) q^{28} +1.00000 q^{31} +(0.171391 - 0.750915i) q^{32} +(-0.535628 - 1.36476i) q^{35} +(1.07906 - 1.35310i) q^{36} +(-2.03759 + 2.55506i) q^{38} +(-1.10372 - 1.38402i) q^{40} +(-1.72188 - 0.829215i) q^{41} +(0.326239 + 1.42935i) q^{45} +(-1.12349 + 1.40881i) q^{47} +(0.0747301 + 0.997204i) q^{49} +1.89946 q^{50} +(0.441126 + 1.12397i) q^{56} +(-0.658322 + 0.317031i) q^{59} +(1.03030 - 1.29196i) q^{62} +(0.0747301 - 0.997204i) q^{63} +(-0.958528 - 1.20196i) q^{64} -0.445042 q^{67} +(-2.31507 - 0.714104i) q^{70} +(-0.367711 - 1.61105i) q^{71} +(-0.268680 - 1.17716i) q^{72} +(0.761623 + 3.33689i) q^{76} -0.387899 q^{80} +(-0.222521 + 0.974928i) q^{81} +(-2.84537 + 1.37026i) q^{82} +(2.18278 + 1.05117i) q^{90} +(0.662592 + 2.90301i) q^{94} +(-2.61232 - 1.25803i) q^{95} +1.91115 q^{97} +(1.36534 + 0.930874i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 2 q^{2} + 2 q^{5} + q^{7} - 9 q^{8} - 2 q^{9} - 2 q^{10} - q^{14} + 2 q^{16} + 2 q^{18} + 2 q^{19} - 7 q^{20} - 7 q^{28} + 12 q^{31} + 14 q^{32} - q^{35} - 7 q^{36} - 2 q^{38} - 5 q^{40} + 2 q^{41}+ \cdots + 13 q^{98}+O(q^{100}) $ 12 * q + 2 * q^2 + 2 * q^5 + q^7 - 9 * q^8 - 2 * q^9 - 2 * q^10 - q^14 + 2 * q^16 + 2 * q^18 + 2 * q^19 - 7 * q^20 - 7 * q^28 + 12 * q^31 + 14 * q^32 - q^35 - 7 * q^36 - 2 * q^38 - 5 * q^40 + 2 * q^41 - 5 * q^45 - 4 * q^47 + q^49 - 14 * q^50 + q^56 + 2 * q^59 + 2 * q^62 + q^63 - 9 * q^64 - 4 * q^67 - 6 * q^70 + 2 * q^71 + 12 * q^72 + 14 * q^76 + 26 * q^80 - 2 * q^81 - 2 * q^82 - 2 * q^90 + 4 * q^94 - 9 * q^95 + 2 * q^97 + 13 * q^98

Character values

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

$n$	$344$	$1179$
$\chi(n)$	$-1$	$e\left(\frac{1}{7}\right)$

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 $

$2$	1.03030	−	1.29196i	1.03030	−	1.29196i	0.0747301	−	0.997204i	$-0.476190\pi$
$2$	1.03030	−	1.29196i	1.03030	−	1.29196i	0.955573	−	0.294755i	$-0.0952381\pi$
$3$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$3$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$4$	−0.385113	−	1.68729i	−0.385113	−	1.68729i
$5$	1.32091	+	0.636119i	1.32091	+	0.636119i	0.955573	−	0.294755i	$-0.0952381\pi$
$5$	1.32091	+	0.636119i	1.32091	+	0.636119i	0.365341	+	0.930874i	$0.380952\pi$
$6$		0			0
$7$	−0.733052	−	0.680173i	−0.733052	−	0.680173i
$7$	−0.733052	−	0.680173i	−0.733052	−	0.680173i
$8$	−1.08786	−	0.523887i	−1.08786	−	0.523887i
$9$	0.623490	+	0.781831i	0.623490	+	0.781831i
$10$	2.18278	−	1.05117i	2.18278	−	1.05117i
$11$		0			0		0.623490	−	0.781831i	$-0.285714\pi$
$11$		0			0		−0.623490	+	0.781831i	$0.714286\pi$
$12$		0			0
$13$		0			0		0.623490	−	0.781831i	$-0.285714\pi$
$13$		0			0		−0.623490	+	0.781831i	$0.714286\pi$
$14$	−1.63402	+	0.246289i	−1.63402	+	0.246289i
$15$		0			0
$16$	−0.238377	+	0.114796i	−0.238377	+	0.114796i
$17$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$17$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$18$	1.65248			1.65248
$19$	−1.97766			−1.97766			−0.988831	−	0.149042i	$-0.952381\pi$
$19$	−1.97766			−1.97766			−0.988831	+	0.149042i	$0.952381\pi$
$20$	0.564616	−	2.47374i	0.564616	−	2.47374i
$21$		0			0
$22$		0			0
$23$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$23$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$24$		0			0
$25$	0.716677	+	0.898684i	0.716677	+	0.898684i
$26$		0			0
$27$		0			0
$28$	−0.865341	+	1.49881i	−0.865341	+	1.49881i
$29$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$29$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$30$		0			0
$31$	1.00000			1.00000
$31$	1.00000			1.00000
$32$	0.171391	−	0.750915i	0.171391	−	0.750915i
$33$		0			0
$34$		0			0
$35$	−0.535628	−	1.36476i	−0.535628	−	1.36476i
$36$	1.07906	−	1.35310i	1.07906	−	1.35310i
$37$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$37$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$38$	−2.03759	+	2.55506i	−2.03759	+	2.55506i
$39$		0			0
$40$	−1.10372	−	1.38402i	−1.10372	−	1.38402i
$41$	−1.72188	−	0.829215i	−1.72188	−	0.829215i	−0.988831	−	0.149042i	$-0.952381\pi$
$41$	−1.72188	−	0.829215i	−1.72188	−	0.829215i	−0.733052	−	0.680173i	$-0.761905\pi$
$42$		0			0
$43$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$43$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$44$		0			0
$45$	0.326239	+	1.42935i	0.326239	+	1.42935i
$46$		0			0
$47$	−1.12349	+	1.40881i	−1.12349	+	1.40881i	−0.222521	+	0.974928i	$0.571429\pi$
$47$	−1.12349	+	1.40881i	−1.12349	+	1.40881i	−0.900969	+	0.433884i	$0.857143\pi$
$48$		0			0
$49$	0.0747301	+	0.997204i	0.0747301	+	0.997204i
$50$	1.89946			1.89946
$51$		0			0
$52$		0			0
$53$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$53$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$54$		0			0
$55$		0			0
$56$	0.441126	+	1.12397i	0.441126	+	1.12397i
$57$		0			0
$58$		0			0
$59$	−0.658322	+	0.317031i	−0.658322	+	0.317031i	−0.733052	−	0.680173i	$-0.761905\pi$
$59$	−0.658322	+	0.317031i	−0.658322	+	0.317031i	0.0747301	+	0.997204i	$0.476190\pi$
$60$		0			0
$61$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$61$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$62$	1.03030	−	1.29196i	1.03030	−	1.29196i
$63$	0.0747301	−	0.997204i	0.0747301	−	0.997204i
$64$	−0.958528	−	1.20196i	−0.958528	−	1.20196i
$65$		0			0
$66$		0			0
$67$	−0.445042			−0.445042			−0.222521	−	0.974928i	$-0.571429\pi$
$67$	−0.445042			−0.445042			−0.222521	+	0.974928i	$0.571429\pi$
$68$		0			0
$69$		0			0
$70$	−2.31507	−	0.714104i	−2.31507	−	0.714104i
$71$	−0.367711	−	1.61105i	−0.367711	−	1.61105i	−0.733052	−	0.680173i	$-0.761905\pi$
$71$	−0.367711	−	1.61105i	−0.367711	−	1.61105i	0.365341	−	0.930874i	$-0.380952\pi$
$72$	−0.268680	−	1.17716i	−0.268680	−	1.17716i
$73$		0			0		−0.623490	−	0.781831i	$-0.714286\pi$
$73$		0			0		0.623490	+	0.781831i	$0.285714\pi$
$74$		0			0
$75$		0			0
$76$	0.761623	+	3.33689i	0.761623	+	3.33689i
$77$		0			0
$78$		0			0
$79$		0			0		1.00000			$0$
$79$		0			0		−1.00000			$\pi$
$80$	−0.387899			−0.387899
$81$	−0.222521	+	0.974928i	−0.222521	+	0.974928i
$82$	−2.84537	+	1.37026i	−2.84537	+	1.37026i
$83$		0			0		−0.623490	−	0.781831i	$-0.714286\pi$
$83$		0			0		0.623490	+	0.781831i	$0.285714\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		−0.623490	−	0.781831i	$-0.714286\pi$
$89$		0			0		0.623490	+	0.781831i	$0.285714\pi$
$90$	2.18278	+	1.05117i	2.18278	+	1.05117i
$91$		0			0
$92$		0			0
$93$		0			0
$94$	0.662592	+	2.90301i	0.662592	+	2.90301i
$95$	−2.61232	−	1.25803i	−2.61232	−	1.25803i
$96$		0			0
$97$	1.91115			1.91115			0.955573	−	0.294755i	$-0.0952381\pi$
$97$	1.91115			1.91115			0.955573	+	0.294755i	$0.0952381\pi$
$98$	1.36534	+	0.930874i	1.36534	+	0.930874i
$99$		0			0
$100$	1.24034	−	1.55534i	1.24034	−	1.55534i
$101$	−0.134659	−	0.0648483i	−0.134659	−	0.0648483i	0.365341	−	0.930874i	$-0.380952\pi$
$101$	−0.134659	−	0.0648483i	−0.134659	−	0.0648483i	−0.500000	+	0.866025i	$0.666667\pi$
$102$		0			0
$103$	1.32091	+	0.636119i	1.32091	+	0.636119i	0.955573	−	0.294755i	$-0.0952381\pi$
$103$	1.32091	+	0.636119i	1.32091	+	0.636119i	0.365341	+	0.930874i	$0.380952\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	1.19158	+	1.49419i	1.19158	+	1.49419i	0.826239	+	0.563320i	$0.190476\pi$
$107$	1.19158	+	1.49419i	1.19158	+	1.49419i	0.365341	+	0.930874i	$0.380952\pi$
$108$		0			0
$109$	−1.23305	+	1.54620i	−1.23305	+	1.54620i	−0.500000	+	0.866025i	$0.666667\pi$
$109$	−1.23305	+	1.54620i	−1.23305	+	1.54620i	−0.733052	+	0.680173i	$0.761905\pi$
$110$		0			0
$111$		0			0
$112$	0.252824	+	0.0779858i	0.252824	+	0.0779858i
$113$	0.0931869	+	0.116853i	0.0931869	+	0.116853i	0.826239	−	0.563320i	$-0.190476\pi$
$113$	0.0931869	+	0.116853i	0.0931869	+	0.116853i	−0.733052	+	0.680173i	$0.761905\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$	−0.268680	+	1.17716i	−0.268680	+	1.17716i
$119$		0			0
$120$		0			0
$121$	−0.222521	−	0.974928i	−0.222521	−	0.974928i
$122$		0			0
$123$		0			0
$124$	−0.385113	−	1.68729i	−0.385113	−	1.68729i
$125$	0.0487597	+	0.213630i	0.0487597	+	0.213630i
$126$	−1.21135	−	1.12397i	−1.21135	−	1.12397i
$127$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$127$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$128$	−1.77023			−1.77023
$129$		0			0
$130$		0			0
$131$	−1.12349	+	0.541044i	−1.12349	+	0.541044i	−0.900969	−	0.433884i	$-0.857143\pi$
$131$	−1.12349	+	0.541044i	−1.12349	+	0.541044i	−0.222521	+	0.974928i	$0.571429\pi$
$132$		0			0
$133$	1.44973	+	1.34515i	1.44973	+	1.34515i
$134$	−0.458528	+	0.574976i	−0.458528	+	0.574976i
$135$		0			0
$136$		0			0
$137$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$137$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$138$		0			0
$139$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$139$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$140$	−2.09646	+	1.42935i	−2.09646	+	1.42935i
$141$		0			0
$142$	−2.46026	−	1.18480i	−2.46026	−	1.18480i
$143$		0			0
$144$	−0.238377	−	0.114796i	−0.238377	−	0.114796i
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	1.24698	−	1.56366i	1.24698	−	1.56366i	0.623490	−	0.781831i	$-0.285714\pi$
$149$	1.24698	−	1.56366i	1.24698	−	1.56366i	0.623490	−	0.781831i	$-0.285714\pi$
$150$		0			0
$151$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$151$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$152$	2.15142	+	1.03607i	2.15142	+	1.03607i
$153$		0			0
$154$		0			0
$155$	1.32091	+	0.636119i	1.32091	+	0.636119i
$156$		0			0
$157$	−0.134659	+	0.0648483i	−0.134659	+	0.0648483i	−0.500000	−	0.866025i	$-0.666667\pi$
$157$	−0.134659	+	0.0648483i	−0.134659	+	0.0648483i	0.365341	+	0.930874i	$0.380952\pi$
$158$		0			0
$159$		0			0
$160$	0.704064	−	0.882868i	0.704064	−	0.882868i
$161$		0			0
$162$	1.03030	+	1.29196i	1.03030	+	1.29196i
$163$	0.900969	−	0.433884i	0.900969	−	0.433884i	0.0747301	−	0.997204i	$-0.476190\pi$
$163$	0.900969	−	0.433884i	0.900969	−	0.433884i	0.826239	+	0.563320i	$0.190476\pi$
$164$	−0.736007	+	3.22466i	−0.736007	+	3.22466i
$165$		0			0
$166$		0			0
$167$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$167$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$168$		0			0
$169$	−0.222521	−	0.974928i	−0.222521	−	0.974928i
$170$		0			0
$171$	−1.23305	−	1.54620i	−1.23305	−	1.54620i
$172$		0			0
$173$	0.0990311	+	0.433884i	0.0990311	+	0.433884i	1.00000			$0$
$173$	0.0990311	+	0.433884i	0.0990311	+	0.433884i	−0.900969	+	0.433884i	$0.857143\pi$
$174$		0			0
$175$	0.0858993	−	1.14625i	0.0858993	−	1.14625i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$179$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$180$	2.28608	−	1.10092i	2.28608	−	1.10092i
$181$		0			0		−0.623490	−	0.781831i	$-0.714286\pi$
$181$		0			0		0.623490	+	0.781831i	$0.285714\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$	2.80974	+	1.35310i	2.80974	+	1.35310i
$189$		0			0
$190$	−4.31680	+	2.07886i	−4.31680	+	2.07886i
$191$	1.78181	+	0.858075i	1.78181	+	0.858075i	0.955573	+	0.294755i	$0.0952381\pi$
$191$	1.78181	+	0.858075i	1.78181	+	0.858075i	0.826239	+	0.563320i	$0.190476\pi$
$192$		0			0
$193$	−0.134659	−	0.0648483i	−0.134659	−	0.0648483i	0.365341	−	0.930874i	$-0.380952\pi$
$193$	−0.134659	−	0.0648483i	−0.134659	−	0.0648483i	−0.500000	+	0.866025i	$0.666667\pi$
$194$	1.96906	−	2.46912i	1.96906	−	2.46912i
$195$		0			0
$196$	1.65379	−	0.510127i	1.65379	−	0.510127i
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$199$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$200$	−0.308837	−	1.35310i	−0.308837	−	1.35310i
$201$		0			0
$202$	−0.222521	+	0.107160i	−0.222521	+	0.107160i
$203$		0			0
$204$		0			0
$205$	−1.74698	−	2.19064i	−1.74698	−	2.19064i
$206$	2.18278	−	1.05117i	2.18278	−	1.05117i
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−0.914101	−	1.14625i	−0.914101	−	1.14625i	−0.988831	−	0.149042i	$-0.952381\pi$
$211$	−0.914101	−	1.14625i	−0.914101	−	1.14625i	0.0747301	−	0.997204i	$-0.476190\pi$
$212$		0			0
$213$		0			0
$214$	3.15813			3.15813
$215$		0			0
$216$		0			0
$217$	−0.733052	−	0.680173i	−0.733052	−	0.680173i
$218$	0.727208	+	3.18610i	0.727208	+	3.18610i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$223$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$224$	−0.636391	+	0.433884i	−0.636391	+	0.433884i
$225$	−0.255779	+	1.12064i	−0.255779	+	1.12064i
$226$	0.246980			0.246980
$227$	1.24698			1.24698			0.623490	−	0.781831i	$-0.285714\pi$
$227$	1.24698			1.24698			0.623490	+	0.781831i	$0.285714\pi$
$228$		0			0
$229$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$229$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−0.425270	+	1.86323i	−0.425270	+	1.86323i	0.0747301	+	0.997204i	$0.476190\pi$
$233$	−0.425270	+	1.86323i	−0.425270	+	1.86323i	−0.500000	+	0.866025i	$0.666667\pi$
$234$		0			0
$235$	−2.38020	+	1.14625i	−2.38020	+	1.14625i
$236$	0.788452	+	0.988687i	0.788452	+	0.988687i
$237$		0			0
$238$		0			0
$239$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$239$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$240$		0			0
$241$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$241$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$242$	−1.48883	−	0.716983i	−1.48883	−	0.716983i
$243$		0			0
$244$		0			0
$245$	−0.535628	+	1.36476i	−0.535628	+	1.36476i
$246$		0			0
$247$		0			0
$248$	−1.08786	−	0.523887i	−1.08786	−	0.523887i
$249$		0			0
$250$	0.326239	+	0.157108i	0.326239	+	0.157108i
$251$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$251$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$252$	−1.71135	+	0.257945i	−1.71135	+	0.257945i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−0.865341	+	1.08510i	−0.865341	+	1.08510i
$257$	0.222521	−	0.974928i	0.222521	−	0.974928i	−0.733052	−	0.680173i	$-0.761905\pi$
$257$	0.222521	−	0.974928i	0.222521	−	0.974928i	0.955573	−	0.294755i	$-0.0952381\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$	−0.458528	+	2.00894i	−0.458528	+	2.00894i
$263$		0			0		1.00000			$0$
$263$		0			0		−1.00000			$\pi$
$264$		0			0
$265$		0			0
$266$	3.23154	−	0.487076i	3.23154	−	0.487076i
$267$		0			0
$268$	0.171391	+	0.750915i	0.171391	+	0.750915i
$269$		0			0		−0.623490	−	0.781831i	$-0.714286\pi$
$269$		0			0		0.623490	+	0.781831i	$0.285714\pi$
$270$		0			0
$271$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$271$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$277$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$278$		0			0
$279$	0.623490	+	0.781831i	0.623490	+	0.781831i
$280$	−0.132289	+	1.76528i	−0.132289	+	1.76528i
$281$	0.455573	−	0.571270i	0.455573	−	0.571270i	−0.500000	−	0.866025i	$-0.666667\pi$
$281$	0.455573	−	0.571270i	0.455573	−	0.571270i	0.955573	+	0.294755i	$0.0952381\pi$
$282$		0			0
$283$	1.24698	−	1.56366i	1.24698	−	1.56366i	0.623490	−	0.781831i	$-0.285714\pi$
$283$	1.24698	−	1.56366i	1.24698	−	1.56366i	0.623490	−	0.781831i	$-0.285714\pi$
$284$	−2.57669	+	1.24087i	−2.57669	+	1.24087i
$285$		0			0
$286$		0			0
$287$	0.698220	+	1.77904i	0.698220	+	1.77904i
$288$	0.693950	−	0.334189i	0.693950	−	0.334189i
$289$	−0.900969	−	0.433884i	−0.900969	−	0.433884i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−0.445042			−0.445042			−0.222521	−	0.974928i	$-0.571429\pi$
$293$	−0.445042			−0.445042			−0.222521	+	0.974928i	$0.571429\pi$
$294$		0			0
$295$	−1.07126			−1.07126
$296$		0			0
$297$		0			0
$298$	−0.735422	−	3.22209i	−0.735422	−	3.22209i
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$	0.471429	−	0.227028i	0.471429	−	0.227028i
$305$		0			0
$306$		0			0
$307$	0.455573	−	0.571270i	0.455573	−	0.571270i	−0.500000	−	0.866025i	$-0.666667\pi$
$307$	0.455573	−	0.571270i	0.455573	−	0.571270i	0.955573	+	0.294755i	$0.0952381\pi$
$308$		0			0
$309$		0			0
$310$	2.18278	−	1.05117i	2.18278	−	1.05117i
$311$	−0.367711	+	1.61105i	−0.367711	+	1.61105i	0.365341	+	0.930874i	$0.380952\pi$
$311$	−0.367711	+	1.61105i	−0.367711	+	1.61105i	−0.733052	+	0.680173i	$0.761905\pi$
$312$		0			0
$313$		0			0		1.00000			$0$
$313$		0			0		−1.00000			$\pi$
$314$	−0.0549581	+	0.240787i	−0.0549581	+	0.240787i
$315$	0.733052	−	1.26968i	0.733052	−	1.26968i
$316$		0			0
$317$	−0.162592	−	0.712362i	−0.162592	−	0.712362i	−0.988831	−	0.149042i	$-0.952381\pi$
$317$	−0.162592	−	0.712362i	−0.162592	−	0.712362i	0.826239	−	0.563320i	$-0.190476\pi$
$318$		0			0
$319$		0			0
$320$	−0.501546	−	2.19742i	−0.501546	−	2.19742i
$321$		0			0
$322$		0			0
$323$		0			0
$324$	1.73068			1.73068
$325$		0			0
$326$	0.367711	−	1.61105i	0.367711	−	1.61105i
$327$		0			0
$328$	1.43876	+	1.80414i	1.43876	+	1.80414i
$329$	1.78181	−	0.268565i	1.78181	−	0.268565i
$330$		0			0
$331$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$331$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−0.587862	−	0.283099i	−0.587862	−	0.283099i
$336$		0			0
$337$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$337$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$338$	−1.48883	−	0.716983i	−1.48883	−	0.716983i
$339$		0			0
$340$		0			0
$341$		0			0
$342$	−3.26804			−3.26804
$343$	0.623490	−	0.781831i	0.623490	−	0.781831i
$344$		0			0
$345$		0			0
$346$	0.662592	+	0.319088i	0.662592	+	0.319088i
$347$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$347$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$348$		0			0
$349$	−1.80194	+	0.867767i	−1.80194	+	0.867767i	−0.900969	+	0.433884i	$0.857143\pi$
$349$	−1.80194	+	0.867767i	−1.80194	+	0.867767i	−0.900969	+	0.433884i	$0.857143\pi$
$350$	−1.39240	−	1.29196i	−1.39240	−	1.29196i
$351$		0			0
$352$		0			0
$353$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$353$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$354$		0			0
$355$	0.539102	−	2.36196i	0.539102	−	2.36196i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−1.72188	+	0.829215i	−1.72188	+	0.829215i	−0.733052	+	0.680173i	$0.761905\pi$
$359$	−1.72188	+	0.829215i	−1.72188	+	0.829215i	−0.988831	+	0.149042i	$0.952381\pi$
$360$	0.393912	−	1.72584i	0.393912	−	1.72584i
$361$	2.91115			2.91115
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		−0.623490	−	0.781831i	$-0.714286\pi$
$367$		0			0		0.623490	+	0.781831i	$0.285714\pi$
$368$		0			0
$369$	−0.425270	−	1.86323i	−0.425270	−	1.86323i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$373$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$374$		0			0
$375$		0			0
$376$	1.96026	−	0.944011i	1.96026	−	0.944011i
$377$		0			0
$378$		0			0
$379$	0.777479	−	0.974928i	0.777479	−	0.974928i	−0.222521	−	0.974928i	$-0.571429\pi$
$379$	0.777479	−	0.974928i	0.777479	−	0.974928i	1.00000			$0$
$380$	−1.11662	+	4.89223i	−1.11662	+	4.89223i
$381$		0			0
$382$	2.94440	−	1.41795i	2.94440	−	1.41795i
$383$		0			0		−0.623490	−	0.781831i	$-0.714286\pi$
$383$		0			0		0.623490	+	0.781831i	$0.285714\pi$
$384$		0			0
$385$		0			0
$386$	−0.222521	+	0.107160i	−0.222521	+	0.107160i
$387$		0			0
$388$	−0.736007	−	3.22466i	−0.736007	−	3.22466i
$389$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$389$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$390$		0			0
$391$		0			0
$392$	0.441126	−	1.12397i	0.441126	−	1.12397i
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−1.48883	−	0.716983i	−1.48883	−	0.716983i	−0.500000	−	0.866025i	$-0.666667\pi$
$397$	−1.48883	−	0.716983i	−1.48883	−	0.716983i	−0.988831	+	0.149042i	$0.952381\pi$
$398$		0			0
$399$		0			0
$400$	−0.274005	−	0.131954i	−0.274005	−	0.131954i
$401$		0			0		−0.623490	−	0.781831i	$-0.714286\pi$
$401$		0			0		0.623490	+	0.781831i	$0.285714\pi$
$402$		0			0
$403$		0			0
$404$	−0.0575591	+	0.252183i	−0.0575591	+	0.252183i
$405$	−0.914101	+	1.14625i	−0.914101	+	1.14625i
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$409$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$410$	−4.63014			−4.63014
$411$		0			0
$412$	0.564616	−	2.47374i	0.564616	−	2.47374i
$413$	0.698220	+	0.215372i	0.698220	+	0.215372i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−0.425270	−	1.86323i	−0.425270	−	1.86323i	−0.500000	−	0.866025i	$-0.666667\pi$
$419$	−0.425270	−	1.86323i	−0.425270	−	1.86323i	0.0747301	−	0.997204i	$-0.476190\pi$
$420$		0			0
$421$	0.326239	−	1.42935i	0.326239	−	1.42935i	−0.500000	−	0.866025i	$-0.666667\pi$
$421$	0.326239	−	1.42935i	0.326239	−	1.42935i	0.826239	−	0.563320i	$-0.190476\pi$
$422$	−2.42270			−2.42270
$423$	−1.80194			−1.80194
$424$		0			0
$425$		0			0
$426$		0			0
$427$		0			0
$428$	2.06225	−	2.58597i	2.06225	−	2.58597i
$429$		0			0
$430$		0			0
$431$	1.62349	−	0.781831i	1.62349	−	0.781831i	0.623490	−	0.781831i	$-0.285714\pi$
$431$	1.62349	−	0.781831i	1.62349	−	0.781831i	1.00000			$0$
$432$		0			0
$433$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$433$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$434$	−1.63402	+	0.246289i	−1.63402	+	0.246289i
$435$		0			0
$436$	3.08375	+	1.48506i	3.08375	+	1.48506i
$437$		0			0
$438$		0			0
$439$	0.0931869	−	0.116853i	0.0931869	−	0.116853i	−0.733052	−	0.680173i	$-0.761905\pi$
$439$	0.0931869	−	0.116853i	0.0931869	−	0.116853i	0.826239	+	0.563320i	$0.190476\pi$
$440$		0			0
$441$	−0.733052	+	0.680173i	−0.733052	+	0.680173i
$442$		0			0
$443$	−0.623490	+	0.781831i	−0.623490	+	0.781831i	−0.988831	−	0.149042i	$-0.952381\pi$
$443$	−0.623490	+	0.781831i	−0.623490	+	0.781831i	0.365341	+	0.930874i	$0.380952\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−0.114887	+	1.53306i	−0.114887	+	1.53306i
$449$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$449$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$450$	1.18429	+	1.48506i	1.18429	+	1.48506i
$451$		0			0
$452$	0.161277	−	0.202235i	0.161277	−	0.202235i
$453$		0			0
$454$	1.28477	−	1.61105i	1.28477	−	1.61105i
$455$		0			0
$456$		0			0
$457$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$457$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$461$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$462$		0			0
$463$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$463$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$464$		0			0
$465$		0			0
$466$	1.96906	+	2.46912i	1.96906	+	2.46912i
$467$	0.222521	+	0.974928i	0.222521	+	0.974928i	0.955573	+	0.294755i	$0.0952381\pi$
$467$	0.222521	+	0.974928i	0.222521	+	0.974928i	−0.733052	+	0.680173i	$0.761905\pi$
$468$		0			0
$469$	0.326239	+	0.302705i	0.326239	+	0.302705i
$470$	−0.971429	+	4.25611i	−0.971429	+	4.25611i
$471$		0			0
$472$	0.882252			0.882252
$473$		0			0
$474$		0			0
$475$	−1.41734	−	1.77729i	−1.41734	−	1.77729i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	1.03030	−	1.29196i	1.03030	−	1.29196i	0.0747301	−	0.997204i	$-0.476190\pi$
$479$	1.03030	−	1.29196i	1.03030	−	1.29196i	0.955573	−	0.294755i	$-0.0952381\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	−1.55929	+	0.750915i	−1.55929	+	0.750915i
$485$	2.52446	+	1.21572i	2.52446	+	1.21572i
$486$		0			0
$487$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$487$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$488$		0			0
$489$		0			0
$490$	1.21135	+	2.09812i	1.21135	+	2.09812i
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−0.238377	+	0.114796i	−0.238377	+	0.114796i
$497$	−0.826239	+	1.43109i	−0.826239	+	1.43109i
$498$		0			0
$499$		0			0		−0.623490	−	0.781831i	$-0.714286\pi$
$499$		0			0		0.623490	+	0.781831i	$0.285714\pi$
$500$	0.341678	−	0.164544i	0.341678	−	0.164544i
$501$		0			0
$502$		0			0
$503$	−0.914101	+	1.14625i	−0.914101	+	1.14625i	0.0747301	+	0.997204i	$0.476190\pi$
$503$	−0.914101	+	1.14625i	−0.914101	+	1.14625i	−0.988831	+	0.149042i	$0.952381\pi$
$504$	−0.603718	+	1.04567i	−0.603718	+	1.04567i
$505$	−0.136622	−	0.171318i	−0.136622	−	0.171318i
$506$		0			0
$507$		0			0
$508$		0			0
$509$		0			0		1.00000			$0$
$509$		0			0		−1.00000			$\pi$
$510$		0			0
$511$		0			0
$512$	0.116433	+	0.510127i	0.116433	+	0.510127i
$513$		0			0
$514$	−1.03030	−	1.29196i	−1.03030	−	1.29196i
$515$	1.34017	+	1.68052i	1.34017	+	1.68052i
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−1.80194			−1.80194			−0.900969	−	0.433884i	$-0.857143\pi$
$521$	−1.80194			−1.80194			−0.900969	+	0.433884i	$0.857143\pi$
$522$		0			0
$523$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$523$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$524$	1.34557	+	1.68729i	1.34557	+	1.68729i
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−0.900969	+	0.433884i	−0.900969	+	0.433884i
$530$		0			0
$531$	−0.658322	−	0.317031i	−0.658322	−	0.317031i
$532$	1.71135	−	2.96415i	1.71135	−	2.96415i
$533$		0			0
$534$		0			0
$535$	0.623490	+	2.73169i	0.623490	+	2.73169i
$536$	0.484144	+	0.233152i	0.484144	+	0.233152i
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−1.23305	+	1.54620i	−1.23305	+	1.54620i	−0.500000	+	0.866025i	$0.666667\pi$
$541$	−1.23305	+	1.54620i	−1.23305	+	1.54620i	−0.733052	+	0.680173i	$0.761905\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−2.61232	+	1.25803i	−2.61232	+	1.25803i
$546$		0			0
$547$	0.900969	+	0.433884i	0.900969	+	0.433884i	0.826239	−	0.563320i	$-0.190476\pi$
$547$	0.900969	+	0.433884i	0.900969	+	0.433884i	0.0747301	+	0.997204i	$0.476190\pi$
$548$		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		1.00000			$0$
$557$		0			0		−1.00000			$\pi$
$558$	1.65248			1.65248
$559$		0			0
$560$	0.284350	+	0.263838i	0.284350	+	0.263838i
$561$		0			0
$562$	−0.268680	−	1.17716i	−0.268680	−	1.17716i
$563$	−0.623490	−	0.781831i	−0.623490	−	0.781831i	0.365341	−	0.930874i	$-0.380952\pi$
$563$	−0.623490	−	0.781831i	−0.623490	−	0.781831i	−0.988831	+	0.149042i	$0.952381\pi$
$564$		0			0
$565$	0.0487597	+	0.213630i	0.0487597	+	0.213630i
$566$	−0.735422	−	3.22209i	−0.735422	−	3.22209i
$567$	0.826239	−	0.563320i	0.826239	−	0.563320i
$568$	−0.443987	+	1.94524i	−0.443987	+	1.94524i
$569$		0			0		1.00000			$0$
$569$		0			0		−1.00000			$\pi$
$570$		0			0
$571$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$571$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$572$		0			0
$573$		0			0
$574$	3.01782	+	0.930874i	3.01782	+	0.930874i
$575$		0			0
$576$	0.342095	−	1.49881i	0.342095	−	1.49881i
$577$	0.777479	−	0.974928i	0.777479	−	0.974928i	−0.222521	−	0.974928i	$-0.571429\pi$
$577$	0.777479	−	0.974928i	0.777479	−	0.974928i	1.00000			$0$
$578$	−1.48883	+	0.716983i	−1.48883	+	0.716983i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$	−0.458528	+	0.574976i	−0.458528	+	0.574976i
$587$		0			0		1.00000			$0$
$587$		0			0		−1.00000			$\pi$
$588$		0			0
$589$	−1.97766			−1.97766
$590$	−1.10372	+	1.38402i	−1.10372	+	1.38402i
$591$		0			0
$592$		0			0
$593$	0.900969	+	0.433884i	0.900969	+	0.433884i	0.826239	−	0.563320i	$-0.190476\pi$
$593$	0.900969	+	0.433884i	0.900969	+	0.433884i	0.0747301	+	0.997204i	$0.476190\pi$
$594$		0			0
$595$		0			0
$596$	−3.11858	−	1.50183i	−3.11858	−	1.50183i
$597$		0			0
$598$		0			0
$599$	−0.623490	+	0.781831i	−0.623490	+	0.781831i	−0.988831	−	0.149042i	$-0.952381\pi$
$599$	−0.623490	+	0.781831i	−0.623490	+	0.781831i	0.365341	+	0.930874i	$0.380952\pi$
$600$		0			0
$601$		0			0		0.623490	−	0.781831i	$-0.285714\pi$
$601$		0			0		−0.623490	+	0.781831i	$0.714286\pi$
$602$		0			0
$603$	−0.277479	−	0.347948i	−0.277479	−	0.347948i
$604$		0			0
$605$	0.326239	−	1.42935i	0.326239	−	1.42935i
$606$		0			0
$607$	1.24698			1.24698			0.623490	−	0.781831i	$-0.285714\pi$
$607$	1.24698			1.24698			0.623490	+	0.781831i	$0.285714\pi$
$608$	−0.338954	+	1.48506i	−0.338954	+	1.48506i
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		−0.623490	−	0.781831i	$-0.714286\pi$
$613$		0			0		0.623490	+	0.781831i	$0.285714\pi$
$614$	−0.268680	−	1.17716i	−0.268680	−	1.17716i
$615$		0			0
$616$		0			0
$617$	0.400969	−	1.75676i	0.400969	−	1.75676i	−0.222521	−	0.974928i	$-0.571429\pi$
$617$	0.400969	−	1.75676i	0.400969	−	1.75676i	0.623490	−	0.781831i	$-0.285714\pi$
$618$		0			0
$619$		0			0		1.00000			$0$
$619$		0			0		−1.00000			$\pi$
$620$	0.564616	−	2.47374i	0.564616	−	2.47374i
$621$		0			0
$622$	1.70255	+	2.13493i	1.70255	+	2.13493i
$623$		0			0
$624$		0			0
$625$	0.184292	−	0.807437i	0.184292	−	0.807437i
$626$		0			0
$627$		0			0
$628$	0.161277	+	0.202235i	0.161277	+	0.202235i
$629$		0			0
$630$	−0.885113	−	2.25523i	−0.885113	−	2.25523i
$631$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$631$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$632$		0			0
$633$		0			0
$634$	−1.08786	−	0.523887i	−1.08786	−	0.523887i
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$	1.03030	−	1.29196i	1.03030	−	1.29196i
$640$	−2.33832	−	1.12607i	−2.33832	−	1.12607i
$641$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$641$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$642$		0			0
$643$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$643$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$647$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$648$	0.752824	−	0.944011i	0.752824	−	0.944011i
$649$		0			0
$650$		0			0
$651$		0			0
$652$	−1.07906	−	1.35310i	−1.07906	−	1.35310i
$653$	−1.12349	+	0.541044i	−1.12349	+	0.541044i	−0.900969	−	0.433884i	$-0.857143\pi$
$653$	−1.12349	+	0.541044i	−1.12349	+	0.541044i	−0.222521	+	0.974928i	$0.571429\pi$
$654$		0			0
$655$	−1.82820			−1.82820
$656$	0.505648			0.505648
$657$		0			0
$658$	1.48883	−	2.57873i	1.48883	−	2.57873i
$659$	−0.0332580	−	0.145713i	−0.0332580	−	0.145713i	0.955573	−	0.294755i	$-0.0952381\pi$
$659$	−0.0332580	−	0.145713i	−0.0332580	−	0.145713i	−0.988831	+	0.149042i	$0.952381\pi$
$660$		0			0
$661$	−0.623490	−	0.781831i	−0.623490	−	0.781831i	0.365341	−	0.930874i	$-0.380952\pi$
$661$	−0.623490	−	0.781831i	−0.623490	−	0.781831i	−0.988831	+	0.149042i	$0.952381\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	1.05929	+	2.69903i	1.05929	+	2.69903i
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$	−0.971429	+	0.467815i	−0.971429	+	0.467815i
$671$		0			0
$672$		0			0
$673$		0			0		0.623490	−	0.781831i	$-0.285714\pi$
$673$		0			0		−0.623490	+	0.781831i	$0.714286\pi$
$674$		0			0
$675$		0			0
$676$	−1.55929	+	0.750915i	−1.55929	+	0.750915i
$677$		0			0		−0.623490	−	0.781831i	$-0.714286\pi$
$677$		0			0		0.623490	+	0.781831i	$0.285714\pi$
$678$		0			0
$679$	−1.40097	−	1.29991i	−1.40097	−	1.29991i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	1.78181	+	0.858075i	1.78181	+	0.858075i	0.955573	+	0.294755i	$0.0952381\pi$
$683$	1.78181	+	0.858075i	1.78181	+	0.858075i	0.826239	+	0.563320i	$0.190476\pi$
$684$	−2.13402	+	2.67598i	−2.13402	+	2.67598i
$685$		0			0
$686$	−0.367711	−	1.61105i	−0.367711	−	1.61105i
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−0.134659	−	0.0648483i	−0.134659	−	0.0648483i	0.365341	−	0.930874i	$-0.380952\pi$
$691$	−0.134659	−	0.0648483i	−0.134659	−	0.0648483i	−0.500000	+	0.866025i	$0.666667\pi$
$692$	0.693950	−	0.334189i	0.693950	−	0.334189i
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$	−0.735422	+	3.22209i	−0.735422	+	3.22209i
$699$		0			0
$700$	−1.96713	+	0.296497i	−1.96713	+	0.296497i
$701$	1.19158	+	1.49419i	1.19158	+	1.49419i	0.826239	+	0.563320i	$0.190476\pi$
$701$	1.19158	+	1.49419i	1.19158	+	1.49419i	0.365341	+	0.930874i	$0.380952\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$	0.0546039	+	0.139129i	0.0546039	+	0.139129i
$708$		0			0
$709$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$709$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$710$	−2.49612	−	3.13003i	−2.49612	−	3.13003i
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$	−0.702749	+	3.07894i	−0.702749	+	3.07894i
$719$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$719$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$720$	−0.241851	−	0.303272i	−0.241851	−	0.303272i
$721$	−0.535628	−	1.36476i	−0.535628	−	1.36476i
$722$	2.99936	−	3.76108i	2.99936	−	3.76108i
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−1.72188	−	0.829215i	−1.72188	−	0.829215i	−0.988831	−	0.149042i	$-0.952381\pi$
$727$	−1.72188	−	0.829215i	−1.72188	−	0.829215i	−0.733052	−	0.680173i	$-0.761905\pi$
$728$		0			0
$729$	−0.900969	+	0.433884i	−0.900969	+	0.433884i
$730$		0			0
$731$		0			0
$732$		0			0
$733$	1.03030	−	1.29196i	1.03030	−	1.29196i	0.0747301	−	0.997204i	$-0.476190\pi$
$733$	1.03030	−	1.29196i	1.03030	−	1.29196i	0.955573	−	0.294755i	$-0.0952381\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$	−2.84537	−	1.37026i	−2.84537	−	1.37026i
$739$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$739$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$743$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$744$		0			0
$745$	2.64183	−	1.27224i	2.64183	−	1.27224i
$746$	−1.03030	+	1.29196i	−1.03030	+	1.29196i
$747$		0			0
$748$		0			0
$749$	0.142820	−	1.90580i	0.142820	−	1.90580i
$750$		0			0
$751$	0.900969	−	0.433884i	0.900969	−	0.433884i	0.0747301	−	0.997204i	$-0.476190\pi$
$751$	0.900969	−	0.433884i	0.900969	−	0.433884i	0.826239	+	0.563320i	$0.190476\pi$
$752$	0.106088	−	0.464800i	0.106088	−	0.464800i
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$757$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$758$	−0.458528	−	2.00894i	−0.458528	−	2.00894i
$759$		0			0
$760$	2.18278	+	2.73712i	2.18278	+	2.73712i
$761$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$761$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$762$		0			0
$763$	1.95557	−	0.294755i	1.95557	−	0.294755i
$764$	0.761623	−	3.33689i	0.761623	−	3.33689i
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	1.03030	+	1.29196i	1.03030	+	1.29196i	0.955573	+	0.294755i	$0.0952381\pi$
$769$	1.03030	+	1.29196i	1.03030	+	1.29196i	0.0747301	+	0.997204i	$0.476190\pi$
$770$		0			0
$771$		0			0
$772$	−0.0575591	+	0.252183i	−0.0575591	+	0.252183i
$773$		0			0		0.623490	−	0.781831i	$-0.285714\pi$
$773$		0			0		−0.623490	+	0.781831i	$0.714286\pi$
$774$		0			0
$775$	0.716677	+	0.898684i	0.716677	+	0.898684i
$776$	−2.07906	−	1.00122i	−2.07906	−	1.00122i
$777$		0			0
$778$		0			0
$779$	3.40530	+	1.63991i	3.40530	+	1.63991i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−0.132289	−	0.229132i	−0.132289	−	0.229132i
$785$	−0.219124			−0.219124
$786$		0			0
$787$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$787$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	0.0111692	−	0.149042i	0.0111692	−	0.149042i
$792$		0			0
$793$		0			0
$794$	−2.46026	+	1.18480i	−2.46026	+	1.18480i
$795$		0			0
$796$		0			0
$797$		0			0		0.623490	−	0.781831i	$-0.285714\pi$
$797$		0			0		−0.623490	+	0.781831i	$0.714286\pi$
$798$		0			0
$799$		0			0
$800$	0.797667	−	0.384136i	0.797667	−	0.384136i
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$	0.112517	+	0.141092i	0.112517	+	0.141092i
$809$		0			0		−0.623490	−	0.781831i	$-0.714286\pi$
$809$		0			0		0.623490	+	0.781831i	$0.285714\pi$
$810$	0.539102	+	2.36196i	0.539102	+	2.36196i
$811$	0.400969	+	1.75676i	0.400969	+	1.75676i	0.623490	+	0.781831i	$0.285714\pi$
$811$	0.400969	+	1.75676i	0.400969	+	1.75676i	−0.222521	+	0.974928i	$0.571429\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	1.46610			1.46610
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$	−3.02347	+	3.79131i	−3.02347	+	3.79131i
$821$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$821$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$822$		0			0
$823$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$823$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$824$	−1.10372	−	1.38402i	−1.10372	−	1.38402i
$825$		0			0
$826$	0.997630	−	0.680173i	0.997630	−	0.680173i
$827$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$827$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$828$		0			0
$829$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$829$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$	−2.84537	−	1.37026i	−2.84537	−	1.37026i
$839$	0.400969	−	0.193096i	0.400969	−	0.193096i	−0.222521	−	0.974928i	$-0.571429\pi$
$839$	0.400969	−	0.193096i	0.400969	−	0.193096i	0.623490	+	0.781831i	$0.285714\pi$
$840$		0			0
$841$	−0.900969	−	0.433884i	−0.900969	−	0.433884i
$842$	−1.51053	−	1.89415i	−1.51053	−	1.89415i
$843$		0			0
$844$	−1.58202	+	1.98379i	−1.58202	+	1.98379i
$845$	0.326239	−	1.42935i	0.326239	−	1.42935i
$846$	−1.85654	+	2.32803i	−1.85654	+	2.32803i
$847$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	0.400969	−	1.75676i	0.400969	−	1.75676i	−0.222521	−	0.974928i	$-0.571429\pi$
$853$	0.400969	−	1.75676i	0.400969	−	1.75676i	0.623490	−	0.781831i	$-0.285714\pi$
$854$		0			0
$855$	−0.645190	−	2.82676i	−0.645190	−	2.82676i
$856$	−0.513486	−	2.24973i	−0.513486	−	2.24973i
$857$	1.24698	+	1.56366i	1.24698	+	1.56366i	0.623490	+	0.781831i	$0.285714\pi$
$857$	1.24698	+	1.56366i	1.24698	+	1.56366i	0.623490	+	0.781831i	$0.285714\pi$
$858$		0			0
$859$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$859$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$860$		0			0
$861$		0			0
$862$	0.662592	−	2.90301i	0.662592	−	2.90301i
$863$		0			0		1.00000			$0$
$863$		0			0		−1.00000			$\pi$
$864$		0			0
$865$	−0.145190	+	0.636119i	−0.145190	+	0.636119i
$866$		0			0
$867$		0			0
$868$	−0.865341	+	1.49881i	−0.865341	+	1.49881i
$869$		0			0
$870$		0			0
$871$		0			0
$872$	2.15142	−	1.03607i	2.15142	−	1.03607i
$873$	1.19158	+	1.49419i	1.19158	+	1.49419i
$874$		0			0
$875$	0.109562	−	0.189767i	0.109562	−	0.189767i
$876$		0			0
$877$	1.32091	+	0.636119i	1.32091	+	0.636119i	0.955573	−	0.294755i	$-0.0952381\pi$
$877$	1.32091	+	0.636119i	1.32091	+	0.636119i	0.365341	+	0.930874i	$0.380952\pi$
$878$	−0.0549581	−	0.240787i	−0.0549581	−	0.240787i
$879$		0			0
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$	0.123490	+	1.64786i	0.123490	+	1.64786i
$883$		0			0		1.00000			$0$
$883$		0			0		−1.00000			$\pi$
$884$		0			0
$885$		0			0
$886$	0.367711	+	1.61105i	0.367711	+	1.61105i
$887$	1.78181	+	0.858075i	1.78181	+	0.858075i	0.955573	+	0.294755i	$0.0952381\pi$
$887$	1.78181	+	0.858075i	1.78181	+	0.858075i	0.826239	+	0.563320i	$0.190476\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	2.22188	−	2.78615i	2.22188	−	2.78615i
$894$		0			0
$895$		0			0
$896$	1.29767	+	1.20406i	1.29767	+	1.20406i
$897$		0			0
$898$		0			0
$899$		0			0
$900$	1.98935			1.98935
$901$		0			0
$902$		0			0
$903$		0			0
$904$	−0.0401569	−	0.175939i	−0.0401569	−	0.175939i
$905$		0			0
$906$		0			0
$907$	1.19158	+	1.49419i	1.19158	+	1.49419i	0.826239	+	0.563320i	$0.190476\pi$
$907$	1.19158	+	1.49419i	1.19158	+	1.49419i	0.365341	+	0.930874i	$0.380952\pi$
$908$	−0.480228	−	2.10402i	−0.480228	−	2.10402i
$909$	−0.0332580	−	0.145713i	−0.0332580	−	0.145713i
$910$		0			0
$911$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$911$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$	1.19158	+	0.367554i	1.19158	+	0.367554i
$918$		0			0
$919$	−0.445042	+	1.94986i	−0.445042	+	1.94986i	−0.222521	+	0.974928i	$0.571429\pi$
$919$	−0.445042	+	1.94986i	−0.445042	+	1.94986i	−0.222521	+	0.974928i	$0.571429\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$	0.326239	+	1.42935i	0.326239	+	1.42935i
$928$		0			0
$929$		0			0		0.623490	−	0.781831i	$-0.285714\pi$
$929$		0			0		−0.623490	+	0.781831i	$0.714286\pi$
$930$		0			0
$931$	−0.147791	−	1.97213i	−0.147791	−	1.97213i
$932$	3.30759			3.30759
$933$		0			0
$934$	1.48883	+	0.716983i	1.48883	+	0.716983i
$935$		0			0
$936$		0			0
$937$	1.62349	−	0.781831i	1.62349	−	0.781831i	0.623490	−	0.781831i	$-0.285714\pi$
$937$	1.62349	−	0.781831i	1.62349	−	0.781831i	1.00000			$0$
$938$	0.727208	−	0.109609i	0.727208	−	0.109609i
$939$		0			0
$940$	2.85070	+	3.57466i	2.85070	+	3.57466i
$941$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$941$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$942$		0			0
$943$		0			0
$944$	0.120535	−	0.151146i	0.120535	−	0.151146i
$945$		0			0
$946$		0			0
$947$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$947$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$948$		0			0
$949$		0			0
$950$	−3.75648			−3.75648
$951$		0			0
$952$		0			0
$953$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$953$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$954$		0			0
$955$	1.80778	+	2.26689i	1.80778	+	2.26689i
$956$		0			0
$957$		0			0
$958$	−0.607634	−	2.66222i	−0.607634	−	2.66222i
$959$		0			0
$960$		0			0
$961$	1.00000			1.00000
$962$		0			0
$963$	−0.425270	+	1.86323i	−0.425270	+	1.86323i
$964$		0			0
$965$	−0.136622	−	0.171318i	−0.136622	−	0.171318i
$966$		0			0
$967$		0			0		0.623490	−	0.781831i	$-0.285714\pi$
$967$		0			0		−0.623490	+	0.781831i	$0.714286\pi$
$968$	−0.268680	+	1.17716i	−0.268680	+	1.17716i
$969$		0			0
$970$	4.17161	−	2.00894i	4.17161	−	2.00894i
$971$	−1.12349	−	1.40881i	−1.12349	−	1.40881i	−0.900969	−	0.433884i	$-0.857143\pi$
$971$	−1.12349	−	1.40881i	−1.12349	−	1.40881i	−0.222521	−	0.974928i	$-0.571429\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	1.78181	+	0.858075i	1.78181	+	0.858075i	0.955573	+	0.294755i	$0.0952381\pi$
$977$	1.78181	+	0.858075i	1.78181	+	0.858075i	0.826239	+	0.563320i	$0.190476\pi$
$978$		0			0
$979$		0			0
$980$	2.50902	+	0.378174i	2.50902	+	0.378174i
$981$	−1.97766			−1.97766
$982$		0			0
$983$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$983$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		0.623490	−	0.781831i	$-0.285714\pi$
$991$		0			0		−0.623490	+	0.781831i	$0.714286\pi$
$992$	0.171391	−	0.750915i	0.171391	−	0.750915i
$993$		0			0
$994$	0.997630	+	2.54192i	0.997630	+	2.54192i
$995$		0			0
$996$		0			0
$997$	−0.162592	+	0.712362i	−0.162592	+	0.712362i	0.826239	+	0.563320i	$0.190476\pi$
$997$	−0.162592	+	0.712362i	−0.162592	+	0.712362i	−0.988831	+	0.149042i	$0.952381\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1519.1.ba.b.92.2	✓	12
31.30	odd	2	CM	1519.1.ba.b.92.2	✓	12
49.8	even	7	inner	1519.1.ba.b.743.2	yes	12
1519.743	odd	14	inner	1519.1.ba.b.743.2	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1519.1.ba.b.92.2	✓	12	1.1	even	1	trivial
1519.1.ba.b.92.2	✓	12	31.30	odd	2	CM
1519.1.ba.b.743.2	yes	12	49.8	even	7	inner
1519.1.ba.b.743.2	yes	12	1519.743	odd	14	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1519 = 7^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1519.ba (of order \(14\), degree \(6\), minimal)

\(f(q)\)	\(=\)	\(q+(1.03030 - 1.29196i) q^{2} +(-0.385113 - 1.68729i) q^{4} +(1.32091 + 0.636119i) q^{5} +(-0.733052 - 0.680173i) q^{7} +(-1.08786 - 0.523887i) q^{8} +(0.623490 + 0.781831i) q^{9} +(2.18278 - 1.05117i) q^{10} +(-1.63402 + 0.246289i) q^{14} +(-0.238377 + 0.114796i) q^{16} +1.65248 q^{18} -1.97766 q^{19} +(0.564616 - 2.47374i) q^{20} +(0.716677 + 0.898684i) q^{25} +(-0.865341 + 1.49881i) q^{28} +1.00000 q^{31} +(0.171391 - 0.750915i) q^{32} +(-0.535628 - 1.36476i) q^{35} +(1.07906 - 1.35310i) q^{36} +(-2.03759 + 2.55506i) q^{38} +(-1.10372 - 1.38402i) q^{40} +(-1.72188 - 0.829215i) q^{41} +(0.326239 + 1.42935i) q^{45} +(-1.12349 + 1.40881i) q^{47} +(0.0747301 + 0.997204i) q^{49} +1.89946 q^{50} +(0.441126 + 1.12397i) q^{56} +(-0.658322 + 0.317031i) q^{59} +(1.03030 - 1.29196i) q^{62} +(0.0747301 - 0.997204i) q^{63} +(-0.958528 - 1.20196i) q^{64} -0.445042 q^{67} +(-2.31507 - 0.714104i) q^{70} +(-0.367711 - 1.61105i) q^{71} +(-0.268680 - 1.17716i) q^{72} +(0.761623 + 3.33689i) q^{76} -0.387899 q^{80} +(-0.222521 + 0.974928i) q^{81} +(-2.84537 + 1.37026i) q^{82} +(2.18278 + 1.05117i) q^{90} +(0.662592 + 2.90301i) q^{94} +(-2.61232 - 1.25803i) q^{95} +1.91115 q^{97} +(1.36534 + 0.930874i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 2 q^{2} + 2 q^{5} + q^{7} - 9 q^{8} - 2 q^{9} - 2 q^{10} - q^{14} + 2 q^{16} + 2 q^{18} + 2 q^{19} - 7 q^{20} - 7 q^{28} + 12 q^{31} + 14 q^{32} - q^{35} - 7 q^{36} - 2 q^{38} - 5 q^{40} + 2 q^{41}+ \cdots + 13 q^{98}+O(q^{100}) \) 12 * q + 2 * q^2 + 2 * q^5 + q^7 - 9 * q^8 - 2 * q^9 - 2 * q^10 - q^14 + 2 * q^16 + 2 * q^18 + 2 * q^19 - 7 * q^20 - 7 * q^28 + 12 * q^31 + 14 * q^32 - q^35 - 7 * q^36 - 2 * q^38 - 5 * q^40 + 2 * q^41 - 5 * q^45 - 4 * q^47 + q^49 - 14 * q^50 + q^56 + 2 * q^59 + 2 * q^62 + q^63 - 9 * q^64 - 4 * q^67 - 6 * q^70 + 2 * q^71 + 12 * q^72 + 14 * q^76 + 26 * q^80 - 2 * q^81 - 2 * q^82 - 2 * q^90 + 4 * q^94 - 9 * q^95 + 2 * q^97 + 13 * q^98

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