LMFDB - Embedded newform 2527.1.y.f.2400.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2527,1,Mod(62,2527)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2527, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([9, 16]))

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

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

chi := DirichletCharacter("2527.62");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 2527 = 7 \cdot 19^{2} $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2527.y (of order $18$, degree $6$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$1.26113728692$
Analytic rank:	$0$
Dimension:	$24$
Relative dimension:	$4$ over $\Q(\zeta_{18})$
Coefficient field:	24.0.9606056659007943744000000000000000000.3
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{24} + 50x^{18} + 2375x^{12} + 6250x^{6} + 15625 $ x^24 + 50x^18 + 2375x^12 + 6250*x^6 + 15625
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{10}$
Projective field:	Galois closure of 10.0.774773162367379.1

Embedding invariants

Embedding label			2400.1
Root			$0.900539 + 0.755642i$ of defining polynomial
Character	$\chi$	$=$	2527.2400
Dual form			2527.1.y.f.776.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.45710 + 1.22265i) q^{2} +(0.454617 - 2.57826i) q^{4} +(0.500000 - 0.866025i) q^{7} +(1.53884 + 2.66535i) q^{8} +(0.766044 + 0.642788i) q^{9} +O(q^{10})$	\(q+(-1.45710 + 1.22265i) q^{2} +(0.454617 - 2.57826i) q^{4} +(0.500000 - 0.866025i) q^{7} +(1.53884 + 2.66535i) q^{8} +(0.766044 + 0.642788i) q^{9} +(0.309017 + 0.535233i) q^{11} +(0.330298 + 1.87322i) q^{14} +(-3.04091 - 1.10680i) q^{16} -1.90211 q^{18} +(-1.10467 - 0.402069i) q^{22} +(-0.280969 + 1.59345i) q^{23} +(-0.939693 + 0.342020i) q^{25} +(-2.00553 - 1.68284i) q^{28} +(0.900539 + 0.755642i) q^{29} +(2.89208 - 1.05263i) q^{32} +(2.00553 - 1.68284i) q^{36} +(0.107320 + 0.608645i) q^{43} +(1.52045 - 0.553400i) q^{44} +(-1.53884 - 2.66535i) q^{46} +(-0.500000 - 0.866025i) q^{49} +(0.951057 - 1.64728i) q^{50} +(0.330298 - 1.87322i) q^{53} +3.07768 q^{56} -2.23607 q^{58} +(0.939693 - 0.342020i) q^{63} +(-1.30902 + 2.26728i) q^{64} +(1.45710 + 1.22265i) q^{67} +(-0.534434 + 3.03093i) q^{72} +0.618034 q^{77} +(1.10467 + 0.402069i) q^{79} +(0.173648 + 0.984808i) q^{81} +(-0.900539 - 0.755642i) q^{86} +(-0.951057 + 1.64728i) q^{88} +(3.98060 + 1.44882i) q^{92} +(1.78740 + 0.650561i) q^{98} +(-0.107320 + 0.608645i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 12 q^{7}+O(q^{10}) $ 24 * q + 12 * q^7	$ 24 q + 12 q^{7} - 6 q^{11} - 12 q^{49} - 18 q^{64} - 12 q^{77}+O(q^{100}) $ 24 * q + 12 * q^7 - 6 * q^11 - 12 * q^49 - 18 * q^64 - 12 * q^77

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2527\mathbb{Z}\right)^\times$.

$n$	$1445$	$1807$
$\chi(n)$	$-1$	$e\left(\frac{7}{9}\right)$

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 $

$2$	−1.45710	+	1.22265i	−1.45710	+	1.22265i	−0.529919	+	0.848048i	$0.677778\pi$
$2$	−1.45710	+	1.22265i	−1.45710	+	1.22265i	−0.927184	+	0.374607i	$0.877778\pi$
$3$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$3$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$4$	0.454617	−	2.57826i	0.454617	−	2.57826i
$5$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$5$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$6$		0			0
$7$	0.500000	−	0.866025i	0.500000	−	0.866025i
$7$	0.500000	−	0.866025i	0.500000	−	0.866025i
$8$	1.53884	+	2.66535i	1.53884	+	2.66535i
$9$	0.766044	+	0.642788i	0.766044	+	0.642788i
$10$		0			0
$11$	0.309017	+	0.535233i	0.309017	+	0.535233i	0.978148	−	0.207912i	$-0.0666667\pi$
$11$	0.309017	+	0.535233i	0.309017	+	0.535233i	−0.669131	+	0.743145i	$0.733333\pi$
$12$		0			0
$13$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$13$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$14$	0.330298	+	1.87322i	0.330298	+	1.87322i
$15$		0			0
$16$	−3.04091	−	1.10680i	−3.04091	−	1.10680i
$17$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$17$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$18$	−1.90211			−1.90211
$19$		0			0
$19$		0			0
$20$		0			0
$21$		0			0
$22$	−1.10467	−	0.402069i	−1.10467	−	0.402069i
$23$	−0.280969	+	1.59345i	−0.280969	+	1.59345i	0.438371	+	0.898794i	$0.355556\pi$
$23$	−0.280969	+	1.59345i	−0.280969	+	1.59345i	−0.719340	+	0.694658i	$0.755556\pi$
$24$		0			0
$25$	−0.939693	+	0.342020i	−0.939693	+	0.342020i
$26$		0			0
$27$		0			0
$28$	−2.00553	−	1.68284i	−2.00553	−	1.68284i
$29$	0.900539	+	0.755642i	0.900539	+	0.755642i	0.970296	−	0.241922i	$-0.0777778\pi$
$29$	0.900539	+	0.755642i	0.900539	+	0.755642i	−0.0697565	+	0.997564i	$0.522222\pi$
$30$		0			0
$31$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$31$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$32$	2.89208	−	1.05263i	2.89208	−	1.05263i
$33$		0			0
$34$		0			0
$35$		0			0
$36$	2.00553	−	1.68284i	2.00553	−	1.68284i
$37$		0			0			−	1.00000i	$-0.5\pi$
$37$		0			0				1.00000i	$0.5\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$41$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$42$		0			0
$43$	0.107320	+	0.608645i	0.107320	+	0.608645i	0.990268	+	0.139173i	$0.0444444\pi$
$43$	0.107320	+	0.608645i	0.107320	+	0.608645i	−0.882948	+	0.469472i	$0.844444\pi$
$44$	1.52045	−	0.553400i	1.52045	−	0.553400i
$45$		0			0
$46$	−1.53884	−	2.66535i	−1.53884	−	2.66535i
$47$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$47$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$48$		0			0
$49$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$50$	0.951057	−	1.64728i	0.951057	−	1.64728i
$51$		0			0
$52$		0			0
$53$	0.330298	−	1.87322i	0.330298	−	1.87322i	−0.139173	−	0.990268i	$-0.544444\pi$
$53$	0.330298	−	1.87322i	0.330298	−	1.87322i	0.469472	−	0.882948i	$-0.344444\pi$
$54$		0			0
$55$		0			0
$56$	3.07768			3.07768
$57$		0			0
$58$	−2.23607			−2.23607
$59$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$59$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$60$		0			0
$61$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$61$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$62$		0			0
$63$	0.939693	−	0.342020i	0.939693	−	0.342020i
$64$	−1.30902	+	2.26728i	−1.30902	+	2.26728i
$65$		0			0
$66$		0			0
$67$	1.45710	+	1.22265i	1.45710	+	1.22265i	0.927184	+	0.374607i	$0.122222\pi$
$67$	1.45710	+	1.22265i	1.45710	+	1.22265i	0.529919	+	0.848048i	$0.322222\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$71$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$72$	−0.534434	+	3.03093i	−0.534434	+	3.03093i
$73$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$73$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	0.618034			0.618034
$78$		0			0
$79$	1.10467	+	0.402069i	1.10467	+	0.402069i	0.829038	−	0.559193i	$-0.188889\pi$
$79$	1.10467	+	0.402069i	1.10467	+	0.402069i	0.275637	+	0.961262i	$0.411111\pi$
$80$		0			0
$81$	0.173648	+	0.984808i	0.173648	+	0.984808i
$82$		0			0
$83$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$83$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$84$		0			0
$85$		0			0
$86$	−0.900539	−	0.755642i	−0.900539	−	0.755642i
$87$		0			0
$88$	−0.951057	+	1.64728i	−0.951057	+	1.64728i
$89$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$89$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$90$		0			0
$91$		0			0
$92$	3.98060	+	1.44882i	3.98060	+	1.44882i
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$97$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$98$	1.78740	+	0.650561i	1.78740	+	0.650561i
$99$	−0.107320	+	0.608645i	−0.107320	+	0.608645i
$100$	0.454617	+	2.57826i	0.454617	+	2.57826i
$101$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$101$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$102$		0			0
$103$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$103$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$104$		0			0
$105$		0			0
$106$	1.80902	+	3.13331i	1.80902	+	3.13331i
$107$	−0.587785	+	1.01807i	−0.587785	+	1.01807i	0.406737	+	0.913545i	$0.366667\pi$
$107$	−0.587785	+	1.01807i	−0.587785	+	1.01807i	−0.994522	+	0.104528i	$0.966667\pi$
$108$		0			0
$109$	−0.204136	−	1.15771i	−0.204136	−	1.15771i	−0.898794	−	0.438371i	$-0.855556\pi$
$109$	−0.204136	−	1.15771i	−0.204136	−	1.15771i	0.694658	−	0.719340i	$-0.255556\pi$
$110$		0			0
$111$		0			0
$112$	−2.47897	+	2.08010i	−2.47897	+	2.08010i
$113$	−1.17557			−1.17557			−0.587785	−	0.809017i	$-0.700000\pi$
$113$	−1.17557			−1.17557			−0.587785	+	0.809017i	$0.700000\pi$
$114$		0			0
$115$		0			0
$116$	2.35764	−	1.97830i	2.35764	−	1.97830i
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	0.309017	−	0.535233i	0.309017	−	0.535233i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$	−0.951057	+	1.64728i	−0.951057	+	1.64728i
$127$	−1.10467	+	0.402069i	−1.10467	+	0.402069i	−0.829038	−	0.559193i	$-0.811111\pi$
$127$	−1.10467	+	0.402069i	−1.10467	+	0.402069i	−0.275637	+	0.961262i	$0.588889\pi$
$128$	−0.330298	−	1.87322i	−0.330298	−	1.87322i
$129$		0			0
$130$		0			0
$131$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$131$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$132$		0			0
$133$		0			0
$134$	−3.61803			−3.61803
$135$		0			0
$136$		0			0
$137$	0.107320	−	0.608645i	0.107320	−	0.608645i	−0.882948	−	0.469472i	$-0.844444\pi$
$137$	0.107320	−	0.608645i	0.107320	−	0.608645i	0.990268	−	0.139173i	$-0.0444444\pi$
$138$		0			0
$139$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$139$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	−1.61803	−	2.80252i	−1.61803	−	2.80252i
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	1.52045	+	0.553400i	1.52045	+	0.553400i	0.961262	−	0.275637i	$-0.0888889\pi$
$149$	1.52045	+	0.553400i	1.52045	+	0.553400i	0.559193	+	0.829038i	$0.311111\pi$
$150$		0			0
$151$	1.90211			1.90211			0.951057	−	0.309017i	$-0.100000\pi$
$151$	1.90211			1.90211			0.951057	+	0.309017i	$0.100000\pi$
$152$		0			0
$153$		0			0
$154$	−0.900539	+	0.755642i	−0.900539	+	0.755642i
$155$		0			0
$156$		0			0
$157$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$157$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$158$	−2.10122	+	0.764780i	−2.10122	+	0.764780i
$159$		0			0
$160$		0			0
$161$	1.23949	+	1.04005i	1.23949	+	1.04005i
$162$	−1.45710	−	1.22265i	−1.45710	−	1.22265i
$163$	−0.809017	−	1.40126i	−0.809017	−	1.40126i	−0.913545	−	0.406737i	$-0.866667\pi$
$163$	−0.809017	−	1.40126i	−0.809017	−	1.40126i	0.104528	−	0.994522i	$-0.466667\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$167$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$168$		0			0
$169$	0.766044	−	0.642788i	0.766044	−	0.642788i
$170$		0			0
$171$		0			0
$172$	1.61803			1.61803
$173$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$173$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$174$		0			0
$175$	−0.173648	+	0.984808i	−0.173648	+	0.984808i
$176$	−0.347296	−	1.96962i	−0.347296	−	1.96962i
$177$		0			0
$178$		0			0
$179$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$179$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$180$		0			0
$181$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$181$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$182$		0			0
$183$		0			0
$184$	−4.67948	+	1.70319i	−4.67948	+	1.70319i
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	1.61803			1.61803			0.809017	−	0.587785i	$-0.200000\pi$
$191$	1.61803			1.61803			0.809017	+	0.587785i	$0.200000\pi$
$192$		0			0
$193$	−1.78740	−	0.650561i	−1.78740	−	0.650561i	−0.999391	−	0.0348995i	$-0.988889\pi$
$193$	−1.78740	−	0.650561i	−1.78740	−	0.650561i	−0.788011	−	0.615661i	$-0.788889\pi$
$194$		0			0
$195$		0			0
$196$	−2.46015	+	0.895420i	−2.46015	+	0.895420i
$197$	−0.809017	+	1.40126i	−0.809017	+	1.40126i	0.104528	+	0.994522i	$0.466667\pi$
$197$	−0.809017	+	1.40126i	−0.809017	+	1.40126i	−0.913545	+	0.406737i	$0.866667\pi$
$198$	−0.587785	−	1.01807i	−0.587785	−	1.01807i
$199$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$199$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$200$	−2.35764	−	1.97830i	−2.35764	−	1.97830i
$201$		0			0
$202$		0			0
$203$	1.10467	−	0.402069i	1.10467	−	0.402069i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−1.23949	+	1.04005i	−1.23949	+	1.04005i
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−0.900539	+	0.755642i	−0.900539	+	0.755642i	−0.970296	−	0.241922i	$-0.922222\pi$
$211$	−0.900539	+	0.755642i	−0.900539	+	0.755642i	0.0697565	+	0.997564i	$0.477778\pi$
$212$	−4.67948	−	1.70319i	−4.67948	−	1.70319i
$213$		0			0
$214$	−0.388289	−	2.20210i	−0.388289	−	2.20210i
$215$		0			0
$216$		0			0
$217$		0			0
$218$	1.71293	+	1.43732i	1.71293	+	1.43732i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$223$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$224$	0.534434	−	3.03093i	0.534434	−	3.03093i
$225$	−0.939693	−	0.342020i	−0.939693	−	0.342020i
$226$	1.71293	−	1.43732i	1.71293	−	1.43732i
$227$		0			0		1.00000			$0$
$227$		0			0		−1.00000			$\pi$
$228$		0			0
$229$		0			0		1.00000			$0$
$229$		0			0		−1.00000			$\pi$
$230$		0			0
$231$		0			0
$232$	−0.628265	+	3.56307i	−0.628265	+	3.56307i
$233$	−0.107320	−	0.608645i	−0.107320	−	0.608645i	−0.990268	−	0.139173i	$-0.955556\pi$
$233$	−0.107320	−	0.608645i	−0.107320	−	0.608645i	0.882948	−	0.469472i	$-0.155556\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−0.309017	−	0.535233i	−0.309017	−	0.535233i	0.669131	−	0.743145i	$-0.266667\pi$
$239$	−0.309017	−	0.535233i	−0.309017	−	0.535233i	−0.978148	+	0.207912i	$0.933333\pi$
$240$		0			0
$241$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$241$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$242$	0.204136	+	1.15771i	0.204136	+	1.15771i
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$251$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$252$	−0.454617	−	2.57826i	−0.454617	−	2.57826i
$253$	−0.939693	+	0.342020i	−0.939693	+	0.342020i
$254$	1.11803	−	1.93649i	1.11803	−	1.93649i
$255$		0			0
$256$	0.766044	+	0.642788i	0.766044	+	0.642788i
$257$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$257$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	0.204136	+	1.15771i	0.204136	+	1.15771i
$262$		0			0
$263$	−1.52045	−	0.553400i	−1.52045	−	0.553400i	−0.559193	−	0.829038i	$-0.688889\pi$
$263$	−1.52045	−	0.553400i	−1.52045	−	0.553400i	−0.961262	+	0.275637i	$0.911111\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	3.81475	−	3.20095i	3.81475	−	3.20095i
$269$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$269$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$270$		0			0
$271$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$271$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$272$		0			0
$273$		0			0
$274$	0.587785	+	1.01807i	0.587785	+	1.01807i
$275$	−0.473442	−	0.397265i	−0.473442	−	0.397265i
$276$		0			0
$277$	0.809017	+	1.40126i	0.809017	+	1.40126i	0.913545	+	0.406737i	$0.133333\pi$
$277$	0.809017	+	1.40126i	0.809017	+	1.40126i	−0.104528	+	0.994522i	$0.533333\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$281$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$282$		0			0
$283$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$283$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	2.89208	+	1.05263i	2.89208	+	1.05263i
$289$	0.173648	−	0.984808i	0.173648	−	0.984808i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$293$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$	−2.89208	+	1.05263i	−2.89208	+	1.05263i
$299$		0			0
$300$		0			0
$301$	0.580762	+	0.211380i	0.580762	+	0.211380i
$302$	−2.77157	+	2.32563i	−2.77157	+	2.32563i
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$307$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$308$	0.280969	−	1.59345i	0.280969	−	1.59345i
$309$		0			0
$310$		0			0
$311$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$311$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$312$		0			0
$313$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$313$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$314$		0			0
$315$		0			0
$316$	1.53884	−	2.66535i	1.53884	−	2.66535i
$317$	1.78740	−	0.650561i	1.78740	−	0.650561i	0.788011	−	0.615661i	$-0.211111\pi$
$317$	1.78740	−	0.650561i	1.78740	−	0.650561i	0.999391	−	0.0348995i	$-0.0111111\pi$
$318$		0			0
$319$	−0.126163	+	0.715505i	−0.126163	+	0.715505i
$320$		0			0
$321$		0			0
$322$	−3.07768			−3.07768
$323$		0			0
$324$	2.61803			2.61803
$325$		0			0
$326$	2.89208	+	1.05263i	2.89208	+	1.05263i
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	0.951057	+	1.64728i	0.951057	+	1.64728i	0.743145	+	0.669131i	$0.233333\pi$
$331$	0.951057	+	1.64728i	0.951057	+	1.64728i	0.207912	+	0.978148i	$0.433333\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−0.330298	−	1.87322i	−0.330298	−	1.87322i	−0.469472	−	0.882948i	$-0.655556\pi$
$337$	−0.330298	−	1.87322i	−0.330298	−	1.87322i	0.139173	−	0.990268i	$-0.455556\pi$
$338$	−0.330298	+	1.87322i	−0.330298	+	1.87322i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−1.00000			−1.00000
$344$	−1.45710	+	1.22265i	−1.45710	+	1.22265i
$345$		0			0
$346$		0			0
$347$	0.280969	+	1.59345i	0.280969	+	1.59345i	0.719340	+	0.694658i	$0.244444\pi$
$347$	0.280969	+	1.59345i	0.280969	+	1.59345i	−0.438371	+	0.898794i	$0.644444\pi$
$348$		0			0
$349$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$349$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$350$	−0.951057	−	1.64728i	−0.951057	−	1.64728i
$351$		0			0
$352$	1.45710	+	1.22265i	1.45710	+	1.22265i
$353$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$353$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	0.473442	−	0.397265i	0.473442	−	0.397265i	−0.374607	−	0.927184i	$-0.622222\pi$
$359$	0.473442	−	0.397265i	0.473442	−	0.397265i	0.848048	+	0.529919i	$0.177778\pi$
$360$		0			0
$361$		0			0
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$367$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$368$	2.61803	−	4.53457i	2.61803	−	4.53457i
$369$		0			0
$370$		0			0
$371$	−1.45710	−	1.22265i	−1.45710	−	1.22265i
$372$		0			0
$373$	0.587785	−	1.01807i	0.587785	−	1.01807i	−0.406737	−	0.913545i	$-0.633333\pi$
$373$	0.587785	−	1.01807i	0.587785	−	1.01807i	0.994522	−	0.104528i	$-0.0333333\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	1.17557			1.17557			0.587785	−	0.809017i	$-0.300000\pi$
$379$	1.17557			1.17557			0.587785	+	0.809017i	$0.300000\pi$
$380$		0			0
$381$		0			0
$382$	−2.35764	+	1.97830i	−2.35764	+	1.97830i
$383$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$383$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$384$		0			0
$385$		0			0
$386$	3.39984	−	1.23744i	3.39984	−	1.23744i
$387$	−0.309017	+	0.535233i	−0.309017	+	0.535233i
$388$		0			0
$389$	−1.53209	−	1.28558i	−1.53209	−	1.28558i	−0.766044	−	0.642788i	$-0.777778\pi$
$389$	−1.53209	−	1.28558i	−1.53209	−	1.28558i	−0.766044	−	0.642788i	$-0.777778\pi$
$390$		0			0
$391$		0			0
$392$	1.53884	−	2.66535i	1.53884	−	2.66535i
$393$		0			0
$394$	−0.534434	−	3.03093i	−0.534434	−	3.03093i
$395$		0			0
$396$	1.52045	+	0.553400i	1.52045	+	0.553400i
$397$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$397$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$398$		0			0
$399$		0			0
$400$	3.23607			3.23607
$401$	0.900539	−	0.755642i	0.900539	−	0.755642i	−0.0697565	−	0.997564i	$-0.522222\pi$
$401$	0.900539	−	0.755642i	0.900539	−	0.755642i	0.970296	+	0.241922i	$0.0777778\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$	−1.11803	+	1.93649i	−1.11803	+	1.93649i
$407$		0			0
$408$		0			0
$409$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$409$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$	0.534434	−	3.03093i	0.534434	−	3.03093i
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		1.00000			$0$
$419$		0			0		−1.00000			$\pi$
$420$		0			0
$421$	1.78740	+	0.650561i	1.78740	+	0.650561i	0.999391	+	0.0348995i	$0.0111111\pi$
$421$	1.78740	+	0.650561i	1.78740	+	0.650561i	0.788011	+	0.615661i	$0.211111\pi$
$422$	0.388289	−	2.20210i	0.388289	−	2.20210i
$423$		0			0
$424$	5.50106	−	2.00222i	5.50106	−	2.00222i
$425$		0			0
$426$		0			0
$427$		0			0
$428$	2.35764	+	1.97830i	2.35764	+	1.97830i
$429$		0			0
$430$		0			0
$431$	−1.78740	+	0.650561i	−1.78740	+	0.650561i	−0.788011	+	0.615661i	$0.788889\pi$
$431$	−1.78740	+	0.650561i	−1.78740	+	0.650561i	−0.999391	+	0.0348995i	$0.988889\pi$
$432$		0			0
$433$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$433$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$434$		0			0
$435$		0			0
$436$	−3.07768			−3.07768
$437$		0			0
$438$		0			0
$439$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$439$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$440$		0			0
$441$	0.173648	−	0.984808i	0.173648	−	0.984808i
$442$		0			0
$443$	1.52045	−	0.553400i	1.52045	−	0.553400i	0.559193	−	0.829038i	$-0.311111\pi$
$443$	1.52045	−	0.553400i	1.52045	−	0.553400i	0.961262	+	0.275637i	$0.0888889\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$	1.30902	+	2.26728i	1.30902	+	2.26728i
$449$	−0.951057	+	1.64728i	−0.951057	+	1.64728i	−0.207912	+	0.978148i	$0.566667\pi$
$449$	−0.951057	+	1.64728i	−0.951057	+	1.64728i	−0.743145	+	0.669131i	$0.766667\pi$
$450$	1.78740	−	0.650561i	1.78740	−	0.650561i
$451$		0			0
$452$	−0.534434	+	3.03093i	−0.534434	+	3.03093i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−0.618034			−0.618034			−0.309017	−	0.951057i	$-0.600000\pi$
$457$	−0.618034			−0.618034			−0.309017	+	0.951057i	$0.600000\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$461$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$462$		0			0
$463$	−0.309017	+	0.535233i	−0.309017	+	0.535233i	−0.978148	−	0.207912i	$-0.933333\pi$
$463$	−0.309017	+	0.535233i	−0.309017	+	0.535233i	0.669131	+	0.743145i	$0.266667\pi$
$464$	−1.90211	−	3.29456i	−1.90211	−	3.29456i
$465$		0			0
$466$	0.900539	+	0.755642i	0.900539	+	0.755642i
$467$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$467$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$468$		0			0
$469$	1.78740	−	0.650561i	1.78740	−	0.650561i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−0.292603	+	0.245523i	−0.292603	+	0.245523i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	1.45710	−	1.22265i	1.45710	−	1.22265i
$478$	1.10467	+	0.402069i	1.10467	+	0.402069i
$479$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$479$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	−1.23949	−	1.04005i	−1.23949	−	1.04005i
$485$		0			0
$486$		0			0
$487$	0.951057	−	1.64728i	0.951057	−	1.64728i	0.207912	−	0.978148i	$-0.433333\pi$
$487$	0.951057	−	1.64728i	0.951057	−	1.64728i	0.743145	−	0.669131i	$-0.233333\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−1.52045	−	0.553400i	−1.52045	−	0.553400i	−0.559193	−	0.829038i	$-0.688889\pi$
$491$	−1.52045	−	0.553400i	−1.52045	−	0.553400i	−0.961262	+	0.275637i	$0.911111\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−0.280969	−	1.59345i	−0.280969	−	1.59345i	−0.719340	−	0.694658i	$-0.755556\pi$
$499$	−0.280969	−	1.59345i	−0.280969	−	1.59345i	0.438371	−	0.898794i	$-0.355556\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$503$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$504$	2.35764	+	1.97830i	2.35764	+	1.97830i
$505$		0			0
$506$	0.951057	−	1.64728i	0.951057	−	1.64728i
$507$		0			0
$508$	0.534434	+	3.03093i	0.534434	+	3.03093i
$509$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$509$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$521$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$522$	−1.71293	−	1.43732i	−1.71293	−	1.43732i
$523$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$523$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$524$		0			0
$525$		0			0
$526$	2.89208	−	1.05263i	2.89208	−	1.05263i
$527$		0			0
$528$		0			0
$529$	−1.52045	−	0.553400i	−1.52045	−	0.553400i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$	−1.01655	+	5.76517i	−1.01655	+	5.76517i
$537$		0			0
$538$		0			0
$539$	0.309017	−	0.535233i	0.309017	−	0.535233i
$540$		0			0
$541$	0.473442	+	0.397265i	0.473442	+	0.397265i	0.848048	−	0.529919i	$-0.177778\pi$
$541$	0.473442	+	0.397265i	0.473442	+	0.397265i	−0.374607	+	0.927184i	$0.622222\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−0.330298	+	1.87322i	−0.330298	+	1.87322i	0.139173	+	0.990268i	$0.455556\pi$
$547$	−0.330298	+	1.87322i	−0.330298	+	1.87322i	−0.469472	+	0.882948i	$0.655556\pi$
$548$	−1.52045	−	0.553400i	−1.52045	−	0.553400i
$549$		0			0
$550$	1.17557			1.17557
$551$		0			0
$552$		0			0
$553$	0.900539	−	0.755642i	0.900539	−	0.755642i
$554$	−2.89208	−	1.05263i	−2.89208	−	1.05263i
$555$		0			0
$556$		0			0
$557$	0.580762	−	0.211380i	0.580762	−	0.211380i	−0.0348995	−	0.999391i	$-0.511111\pi$
$557$	0.580762	−	0.211380i	0.580762	−	0.211380i	0.615661	+	0.788011i	$0.288889\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$563$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	0.939693	+	0.342020i	0.939693	+	0.342020i
$568$		0			0
$569$	−1.90211			−1.90211			−0.951057	−	0.309017i	$-0.900000\pi$
$569$	−1.90211			−1.90211			−0.951057	+	0.309017i	$0.900000\pi$
$570$		0			0
$571$	−1.61803			−1.61803			−0.809017	−	0.587785i	$-0.800000\pi$
$571$	−1.61803			−1.61803			−0.809017	+	0.587785i	$0.800000\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−0.280969	−	1.59345i	−0.280969	−	1.59345i
$576$	−2.46015	+	0.895420i	−2.46015	+	0.895420i
$577$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$577$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$578$	0.951057	+	1.64728i	0.951057	+	1.64728i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	1.10467	−	0.402069i	1.10467	−	0.402069i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$587$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$593$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$594$		0			0
$595$		0			0
$596$	2.11803	−	3.66854i	2.11803	−	3.66854i
$597$		0			0
$598$		0			0
$599$	−0.900539	−	0.755642i	−0.900539	−	0.755642i	0.0697565	−	0.997564i	$-0.477778\pi$
$599$	−0.900539	−	0.755642i	−0.900539	−	0.755642i	−0.970296	+	0.241922i	$0.922222\pi$
$600$		0			0
$601$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$601$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$602$	−1.10467	+	0.402069i	−1.10467	+	0.402069i
$603$	0.330298	+	1.87322i	0.330298	+	1.87322i
$604$	0.864733	−	4.90414i	0.864733	−	4.90414i
$605$		0			0
$606$		0			0
$607$		0			0		1.00000			$0$
$607$		0			0		−1.00000			$\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−0.347296	−	1.96962i	−0.347296	−	1.96962i	−0.173648	−	0.984808i	$-0.555556\pi$
$613$	−0.347296	−	1.96962i	−0.347296	−	1.96962i	−0.173648	−	0.984808i	$-0.555556\pi$
$614$		0			0
$615$		0			0
$616$	0.951057	+	1.64728i	0.951057	+	1.64728i
$617$	−1.23949	−	1.04005i	−1.23949	−	1.04005i	−0.997564	−	0.0697565i	$-0.977778\pi$
$617$	−1.23949	−	1.04005i	−1.23949	−	1.04005i	−0.241922	−	0.970296i	$-0.577778\pi$
$618$		0			0
$619$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$619$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	0.766044	−	0.642788i	0.766044	−	0.642788i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	0.280969	−	1.59345i	0.280969	−	1.59345i	−0.438371	−	0.898794i	$-0.644444\pi$
$631$	0.280969	−	1.59345i	0.280969	−	1.59345i	0.719340	−	0.694658i	$-0.244444\pi$
$632$	0.628265	+	3.56307i	0.628265	+	3.56307i
$633$		0			0
$634$	−1.80902	+	3.13331i	−1.80902	+	3.13331i
$635$		0			0
$636$		0			0
$637$		0			0
$638$	−0.690983	−	1.19682i	−0.690983	−	1.19682i
$639$		0			0
$640$		0			0
$641$	−0.330298	−	1.87322i	−0.330298	−	1.87322i	−0.469472	−	0.882948i	$-0.655556\pi$
$641$	−0.330298	−	1.87322i	−0.330298	−	1.87322i	0.139173	−	0.990268i	$-0.455556\pi$
$642$		0			0
$643$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$643$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$644$	3.24502	−	2.72289i	3.24502	−	2.72289i
$645$		0			0
$646$		0			0
$647$		0			0		1.00000			$0$
$647$		0			0		−1.00000			$\pi$
$648$	−2.35764	+	1.97830i	−2.35764	+	1.97830i
$649$		0			0
$650$		0			0
$651$		0			0
$652$	−3.98060	+	1.44882i	−3.98060	+	1.44882i
$653$	−0.809017	+	1.40126i	−0.809017	+	1.40126i	0.104528	+	0.994522i	$0.466667\pi$
$653$	−0.809017	+	1.40126i	−0.809017	+	1.40126i	−0.913545	+	0.406737i	$0.866667\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−1.78740	+	0.650561i	−1.78740	+	0.650561i	−0.788011	+	0.615661i	$0.788889\pi$
$659$	−1.78740	+	0.650561i	−1.78740	+	0.650561i	−0.999391	+	0.0348995i	$0.988889\pi$
$660$		0			0
$661$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$661$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$662$	−3.39984	−	1.23744i	−3.39984	−	1.23744i
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−1.45710	+	1.22265i	−1.45710	+	1.22265i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−0.587785	−	1.01807i	−0.587785	−	1.01807i	−0.994522	−	0.104528i	$-0.966667\pi$
$673$	−0.587785	−	1.01807i	−0.587785	−	1.01807i	0.406737	−	0.913545i	$-0.366667\pi$
$674$	2.77157	+	2.32563i	2.77157	+	2.32563i
$675$		0			0
$676$	−1.30902	−	2.26728i	−1.30902	−	2.26728i
$677$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$677$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	1.90211			1.90211			0.951057	−	0.309017i	$-0.100000\pi$
$683$	1.90211			1.90211			0.951057	+	0.309017i	$0.100000\pi$
$684$		0			0
$685$		0			0
$686$	1.45710	−	1.22265i	1.45710	−	1.22265i
$687$		0			0
$688$	0.347296	−	1.96962i	0.347296	−	1.96962i
$689$		0			0
$690$		0			0
$691$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$691$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$692$		0			0
$693$	0.473442	+	0.397265i	0.473442	+	0.397265i
$694$	−2.35764	−	1.97830i	−2.35764	−	1.97830i
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	2.46015	+	0.895420i	2.46015	+	0.895420i
$701$	−0.473442	+	0.397265i	−0.473442	+	0.397265i	−0.848048	−	0.529919i	$-0.822222\pi$
$701$	−0.473442	+	0.397265i	−0.473442	+	0.397265i	0.374607	+	0.927184i	$0.377778\pi$
$702$		0			0
$703$		0			0
$704$	−1.61803			−1.61803
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−0.580762	+	0.211380i	−0.580762	+	0.211380i	−0.615661	−	0.788011i	$-0.711111\pi$
$709$	−0.580762	+	0.211380i	−0.580762	+	0.211380i	0.0348995	+	0.999391i	$0.488889\pi$
$710$		0			0
$711$	0.587785	+	1.01807i	0.587785	+	1.01807i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$	−0.204136	+	1.15771i	−0.204136	+	1.15771i
$719$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$719$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−1.10467	−	0.402069i	−1.10467	−	0.402069i
$726$		0			0
$727$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$727$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$728$		0			0
$729$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$733$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$734$		0			0
$735$		0			0
$736$	0.864733	+	4.90414i	0.864733	+	4.90414i
$737$	−0.204136	+	1.15771i	−0.204136	+	1.15771i
$738$		0			0
$739$	−1.23949	+	1.04005i	−1.23949	+	1.04005i	−0.241922	+	0.970296i	$0.577778\pi$
$739$	−1.23949	+	1.04005i	−1.23949	+	1.04005i	−0.997564	+	0.0697565i	$0.977778\pi$
$740$		0			0
$741$		0			0
$742$	3.61803			3.61803
$743$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$743$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$744$		0			0
$745$		0			0
$746$	0.388289	+	2.20210i	0.388289	+	2.20210i
$747$		0			0
$748$		0			0
$749$	0.587785	+	1.01807i	0.587785	+	1.01807i
$750$		0			0
$751$	−1.45710	−	1.22265i	−1.45710	−	1.22265i	−0.927184	−	0.374607i	$-0.877778\pi$
$751$	−1.45710	−	1.22265i	−1.45710	−	1.22265i	−0.529919	−	0.848048i	$-0.677778\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	1.52045	+	0.553400i	1.52045	+	0.553400i	0.961262	−	0.275637i	$-0.0888889\pi$
$757$	1.52045	+	0.553400i	1.52045	+	0.553400i	0.559193	+	0.829038i	$0.311111\pi$
$758$	−1.71293	+	1.43732i	−1.71293	+	1.43732i
$759$		0			0
$760$		0			0
$761$		0			0		1.00000			$0$
$761$		0			0		−1.00000			$\pi$
$762$		0			0
$763$	−1.10467	−	0.402069i	−1.10467	−	0.402069i
$764$	0.735585	−	4.17171i	0.735585	−	4.17171i
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$769$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$770$		0			0
$771$		0			0
$772$	−2.48990	+	4.31263i	−2.48990	+	4.31263i
$773$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$773$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$774$	−0.204136	−	1.15771i	−0.204136	−	1.15771i
$775$		0			0
$776$		0			0
$777$		0			0
$778$	3.80423			3.80423
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	0.561937	+	3.18690i	0.561937	+	3.18690i
$785$		0			0
$786$		0			0
$787$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$787$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$788$	3.24502	+	2.72289i	3.24502	+	2.72289i
$789$		0			0
$790$		0			0
$791$	−0.587785	+	1.01807i	−0.587785	+	1.01807i
$792$	−1.78740	+	0.650561i	−1.78740	+	0.650561i
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		1.00000			$0$
$797$		0			0		−1.00000			$\pi$
$798$		0			0
$799$		0			0
$800$	−2.35764	+	1.97830i	−2.35764	+	1.97830i
$801$		0			0
$802$	−0.388289	+	2.20210i	−0.388289	+	2.20210i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$809$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$810$		0			0
$811$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$811$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$812$	−0.534434	−	3.03093i	−0.534434	−	3.03093i
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	0.280969	−	1.59345i	0.280969	−	1.59345i	−0.438371	−	0.898794i	$-0.644444\pi$
$821$	0.280969	−	1.59345i	0.280969	−	1.59345i	0.719340	−	0.694658i	$-0.244444\pi$
$822$		0			0
$823$	−0.580762	+	0.211380i	−0.580762	+	0.211380i	−0.615661	−	0.788011i	$-0.711111\pi$
$823$	−0.580762	+	0.211380i	−0.580762	+	0.211380i	0.0348995	+	0.999391i	$0.488889\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−0.900539	−	0.755642i	−0.900539	−	0.755642i	0.0697565	−	0.997564i	$-0.477778\pi$
$827$	−0.900539	−	0.755642i	−0.900539	−	0.755642i	−0.970296	+	0.241922i	$0.922222\pi$
$828$	2.11803	+	3.66854i	2.11803	+	3.66854i
$829$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$829$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$839$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$840$		0			0
$841$	0.0663277	+	0.376163i	0.0663277	+	0.376163i
$842$	−3.39984	+	1.23744i	−3.39984	+	1.23744i
$843$		0			0
$844$	1.53884	+	2.66535i	1.53884	+	2.66535i
$845$		0			0
$846$		0			0
$847$	−0.309017	−	0.535233i	−0.309017	−	0.535233i
$848$	−3.07768	+	5.33070i	−3.07768	+	5.33070i
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$853$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$854$		0			0
$855$		0			0
$856$	−3.61803			−3.61803
$857$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$857$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$858$		0			0
$859$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$859$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$860$		0			0
$861$		0			0
$862$	1.80902	−	3.13331i	1.80902	−	3.13331i
$863$	−0.587785	−	1.01807i	−0.587785	−	1.01807i	−0.994522	−	0.104528i	$-0.966667\pi$
$863$	−0.587785	−	1.01807i	−0.587785	−	1.01807i	0.406737	−	0.913545i	$-0.366667\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	0.126163	+	0.715505i	0.126163	+	0.715505i
$870$		0			0
$871$		0			0
$872$	2.77157	−	2.32563i	2.77157	−	2.32563i
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	1.78740	+	0.650561i	1.78740	+	0.650561i	0.999391	+	0.0348995i	$0.0111111\pi$
$877$	1.78740	+	0.650561i	1.78740	+	0.650561i	0.788011	+	0.615661i	$0.211111\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$881$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$882$	0.951057	+	1.64728i	0.951057	+	1.64728i
$883$	−1.23949	−	1.04005i	−1.23949	−	1.04005i	−0.997564	−	0.0697565i	$-0.977778\pi$
$883$	−1.23949	−	1.04005i	−1.23949	−	1.04005i	−0.241922	−	0.970296i	$-0.577778\pi$
$884$		0			0
$885$		0			0
$886$	−1.53884	+	2.66535i	−1.53884	+	2.66535i
$887$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$887$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$888$		0			0
$889$	−0.204136	+	1.15771i	−0.204136	+	1.15771i
$890$		0			0
$891$	−0.473442	+	0.397265i	−0.473442	+	0.397265i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$	−1.78740	−	0.650561i	−1.78740	−	0.650561i
$897$		0			0
$898$	−0.628265	−	3.56307i	−0.628265	−	3.56307i
$899$		0			0
$900$	−1.30902	+	2.26728i	−1.30902	+	2.26728i
$901$		0			0
$902$		0			0
$903$		0			0
$904$	−1.80902	−	3.13331i	−1.80902	−	3.13331i
$905$		0			0
$906$		0			0
$907$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$907$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−1.90211			−1.90211			−0.951057	−	0.309017i	$-0.900000\pi$
$911$	−1.90211			−1.90211			−0.951057	+	0.309017i	$0.900000\pi$
$912$		0			0
$913$		0			0
$914$	0.900539	−	0.755642i	0.900539	−	0.755642i
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	0.309017	−	0.535233i	0.309017	−	0.535233i	−0.669131	−	0.743145i	$-0.733333\pi$
$919$	0.309017	−	0.535233i	0.309017	−	0.535233i	0.978148	+	0.207912i	$0.0666667\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$	−0.204136	−	1.15771i	−0.204136	−	1.15771i
$927$		0			0
$928$	3.39984	+	1.23744i	3.39984	+	1.23744i
$929$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$929$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$930$		0			0
$931$		0			0
$932$	−1.61803			−1.61803
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$937$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$938$	−1.80902	+	3.13331i	−1.80902	+	3.13331i
$939$		0			0
$940$		0			0
$941$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$941$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$	0.126163	−	0.715505i	0.126163	−	0.715505i
$947$	−0.580762	−	0.211380i	−0.580762	−	0.211380i	0.0348995	−	0.999391i	$-0.488889\pi$
$947$	−0.580762	−	0.211380i	−0.580762	−	0.211380i	−0.615661	+	0.788011i	$0.711111\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−1.78740	−	0.650561i	−1.78740	−	0.650561i	−0.999391	−	0.0348995i	$-0.988889\pi$
$953$	−1.78740	−	0.650561i	−1.78740	−	0.650561i	−0.788011	−	0.615661i	$-0.788889\pi$
$954$	−0.628265	+	3.56307i	−0.628265	+	3.56307i
$955$		0			0
$956$	−1.52045	+	0.553400i	−1.52045	+	0.553400i
$957$		0			0
$958$		0			0
$959$	−0.473442	−	0.397265i	−0.473442	−	0.397265i
$960$		0			0
$961$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$962$		0			0
$963$	−1.10467	+	0.402069i	−1.10467	+	0.402069i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−0.473442	+	0.397265i	−0.473442	+	0.397265i	−0.848048	−	0.529919i	$-0.822222\pi$
$967$	−0.473442	+	0.397265i	−0.473442	+	0.397265i	0.374607	+	0.927184i	$0.377778\pi$
$968$	1.90211			1.90211
$969$		0			0
$970$		0			0
$971$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$971$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$972$		0			0
$973$		0			0
$974$	0.628265	+	3.56307i	0.628265	+	3.56307i
$975$		0			0
$976$		0			0
$977$	−0.587785	−	1.01807i	−0.587785	−	1.01807i	−0.994522	−	0.104528i	$-0.966667\pi$
$977$	−0.587785	−	1.01807i	−0.587785	−	1.01807i	0.406737	−	0.913545i	$-0.366667\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	0.587785	−	1.01807i	0.587785	−	1.01807i
$982$	2.89208	−	1.05263i	2.89208	−	1.05263i
$983$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$983$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−1.00000			−1.00000
$990$		0			0
$991$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$991$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$997$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$998$	2.35764	+	1.97830i	2.35764	+	1.97830i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2527.1.y.f.2400.1		24
7.6	odd	2	CM	2527.1.y.f.2400.1		24
19.2	odd	18	inner	2527.1.y.f.62.1		24
19.3	odd	18	inner	2527.1.y.f.776.4		24
19.4	even	9		2527.1.d.f.1084.1	✓	4
19.5	even	9	inner	2527.1.y.f.2050.1		24
19.6	even	9		2527.1.m.e.1014.4		8
19.7	even	3	inner	2527.1.y.f.1833.1		24
19.8	odd	6	inner	2527.1.y.f.1182.1		24
19.9	even	9		2527.1.m.e.790.4		8
19.10	odd	18		2527.1.m.e.790.1		8
19.11	even	3	inner	2527.1.y.f.1182.4		24
19.12	odd	6	inner	2527.1.y.f.1833.4		24
19.13	odd	18		2527.1.m.e.1014.1		8
19.14	odd	18	inner	2527.1.y.f.2050.4		24
19.15	odd	18		2527.1.d.f.1084.4	yes	4
19.16	even	9	inner	2527.1.y.f.776.1		24
19.17	even	9	inner	2527.1.y.f.62.4		24
19.18	odd	2	inner	2527.1.y.f.2400.4		24
133.6	odd	18		2527.1.m.e.1014.4		8
133.13	even	18		2527.1.m.e.1014.1		8
133.27	even	6	inner	2527.1.y.f.1182.1		24
133.34	even	18		2527.1.d.f.1084.4	yes	4
133.41	even	18	inner	2527.1.y.f.776.4		24
133.48	even	18		2527.1.m.e.790.1		8
133.55	odd	18	inner	2527.1.y.f.62.4		24
133.62	odd	18	inner	2527.1.y.f.2050.1		24
133.69	even	6	inner	2527.1.y.f.1833.4		24
133.83	odd	6	inner	2527.1.y.f.1833.1		24
133.90	even	18	inner	2527.1.y.f.2050.4		24
133.97	even	18	inner	2527.1.y.f.62.1		24
133.104	odd	18		2527.1.m.e.790.4		8
133.111	odd	18	inner	2527.1.y.f.776.1		24
133.118	odd	18		2527.1.d.f.1084.1	✓	4
133.125	odd	6	inner	2527.1.y.f.1182.4		24
133.132	even	2	inner	2527.1.y.f.2400.4		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2527.1.d.f.1084.1	✓	4	19.4	even	9
2527.1.d.f.1084.1	✓	4	133.118	odd	18
2527.1.d.f.1084.4	yes	4	19.15	odd	18
2527.1.d.f.1084.4	yes	4	133.34	even	18
2527.1.m.e.790.1		8	19.10	odd	18
2527.1.m.e.790.1		8	133.48	even	18
2527.1.m.e.790.4		8	19.9	even	9
2527.1.m.e.790.4		8	133.104	odd	18
2527.1.m.e.1014.1		8	19.13	odd	18
2527.1.m.e.1014.1		8	133.13	even	18
2527.1.m.e.1014.4		8	19.6	even	9
2527.1.m.e.1014.4		8	133.6	odd	18
2527.1.y.f.62.1		24	19.2	odd	18	inner
2527.1.y.f.62.1		24	133.97	even	18	inner
2527.1.y.f.62.4		24	19.17	even	9	inner
2527.1.y.f.62.4		24	133.55	odd	18	inner
2527.1.y.f.776.1		24	19.16	even	9	inner
2527.1.y.f.776.1		24	133.111	odd	18	inner
2527.1.y.f.776.4		24	19.3	odd	18	inner
2527.1.y.f.776.4		24	133.41	even	18	inner
2527.1.y.f.1182.1		24	19.8	odd	6	inner
2527.1.y.f.1182.1		24	133.27	even	6	inner
2527.1.y.f.1182.4		24	19.11	even	3	inner
2527.1.y.f.1182.4		24	133.125	odd	6	inner
2527.1.y.f.1833.1		24	19.7	even	3	inner
2527.1.y.f.1833.1		24	133.83	odd	6	inner
2527.1.y.f.1833.4		24	19.12	odd	6	inner
2527.1.y.f.1833.4		24	133.69	even	6	inner
2527.1.y.f.2050.1		24	19.5	even	9	inner
2527.1.y.f.2050.1		24	133.62	odd	18	inner
2527.1.y.f.2050.4		24	19.14	odd	18	inner
2527.1.y.f.2050.4		24	133.90	even	18	inner
2527.1.y.f.2400.1		24	1.1	even	1	trivial
2527.1.y.f.2400.1		24	7.6	odd	2	CM
2527.1.y.f.2400.4		24	19.18	odd	2	inner
2527.1.y.f.2400.4		24	133.132	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2527 = 7 \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2527.y (of order \(18\), degree \(6\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.45710 + 1.22265i) q^{2} +(0.454617 - 2.57826i) q^{4} +(0.500000 - 0.866025i) q^{7} +(1.53884 + 2.66535i) q^{8} +(0.766044 + 0.642788i) q^{9} +O(q^{10})\)	\(q+(-1.45710 + 1.22265i) q^{2} +(0.454617 - 2.57826i) q^{4} +(0.500000 - 0.866025i) q^{7} +(1.53884 + 2.66535i) q^{8} +(0.766044 + 0.642788i) q^{9} +(0.309017 + 0.535233i) q^{11} +(0.330298 + 1.87322i) q^{14} +(-3.04091 - 1.10680i) q^{16} -1.90211 q^{18} +(-1.10467 - 0.402069i) q^{22} +(-0.280969 + 1.59345i) q^{23} +(-0.939693 + 0.342020i) q^{25} +(-2.00553 - 1.68284i) q^{28} +(0.900539 + 0.755642i) q^{29} +(2.89208 - 1.05263i) q^{32} +(2.00553 - 1.68284i) q^{36} +(0.107320 + 0.608645i) q^{43} +(1.52045 - 0.553400i) q^{44} +(-1.53884 - 2.66535i) q^{46} +(-0.500000 - 0.866025i) q^{49} +(0.951057 - 1.64728i) q^{50} +(0.330298 - 1.87322i) q^{53} +3.07768 q^{56} -2.23607 q^{58} +(0.939693 - 0.342020i) q^{63} +(-1.30902 + 2.26728i) q^{64} +(1.45710 + 1.22265i) q^{67} +(-0.534434 + 3.03093i) q^{72} +0.618034 q^{77} +(1.10467 + 0.402069i) q^{79} +(0.173648 + 0.984808i) q^{81} +(-0.900539 - 0.755642i) q^{86} +(-0.951057 + 1.64728i) q^{88} +(3.98060 + 1.44882i) q^{92} +(1.78740 + 0.650561i) q^{98} +(-0.107320 + 0.608645i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 12 q^{7}+O(q^{10}) \) 24 * q + 12 * q^7	\( 24 q + 12 q^{7} - 6 q^{11} - 12 q^{49} - 18 q^{64} - 12 q^{77}+O(q^{100}) \) 24 * q + 12 * q^7 - 6 * q^11 - 12 * q^49 - 18 * q^64 - 12 * q^77

Introduction

L-functions

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