LMFDB - Embedded newform 2527.1.m.e.790.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2527,1,Mod(790,2527)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2527, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 4]))

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

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

chi := DirichletCharacter("2527.790");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 2527 = 7 \cdot 19^{2} $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2527.m (of order $6$, degree $2$, 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:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	8.0.324000000.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 5x^{6} + 20x^{4} + 25x^{2} + 25 $ x^8 + 5x^6 + 20x^4 + 25*x^2 + 25
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			790.1
Root			$-0.587785 + 1.01807i$ of defining polynomial
Character	$\chi$	$=$	2527.790
Dual form			2527.1.m.e.1014.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.951057 + 1.64728i) q^{2} +(-1.30902 - 2.26728i) q^{4} -1.00000 q^{7} +3.07768 q^{8} +(-0.500000 - 0.866025i) q^{9} +O(q^{10})$	$q+(-0.951057 + 1.64728i) q^{2} +(-1.30902 - 2.26728i) q^{4} -1.00000 q^{7} +3.07768 q^{8} +(-0.500000 - 0.866025i) q^{9} -0.618034 q^{11} +(0.951057 - 1.64728i) q^{14} +(-1.61803 + 2.80252i) q^{16} +1.90211 q^{18} +(0.587785 - 1.01807i) q^{22} +(0.809017 + 1.40126i) q^{23} +(-0.500000 - 0.866025i) q^{25} +(1.30902 + 2.26728i) q^{28} +(0.587785 + 1.01807i) q^{29} +(-1.53884 - 2.66535i) q^{32} +(-1.30902 + 2.26728i) q^{36} +(-0.309017 + 0.535233i) q^{43} +(0.809017 + 1.40126i) q^{44} -3.07768 q^{46} +1.00000 q^{49} +1.90211 q^{50} +(0.951057 + 1.64728i) q^{53} -3.07768 q^{56} -2.23607 q^{58} +(0.500000 + 0.866025i) q^{63} +2.61803 q^{64} +(0.951057 + 1.64728i) q^{67} +(-1.53884 - 2.66535i) q^{72} +0.618034 q^{77} +(-0.587785 + 1.01807i) q^{79} +(-0.500000 + 0.866025i) q^{81} +(-0.587785 - 1.01807i) q^{86} -1.90211 q^{88} +(2.11803 - 3.66854i) q^{92} +(-0.951057 + 1.64728i) q^{98} +(0.309017 + 0.535233i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 6 q^{4} - 8 q^{7} - 4 q^{9}+O(q^{10}) $ 8 * q - 6 * q^4 - 8 * q^7 - 4 * q^9	$ 8 q - 6 q^{4} - 8 q^{7} - 4 q^{9} + 4 q^{11} - 4 q^{16} + 2 q^{23} - 4 q^{25} + 6 q^{28} - 6 q^{36} + 2 q^{43} + 2 q^{44} + 8 q^{49} + 4 q^{63} + 12 q^{64} - 4 q^{77} - 4 q^{81} + 8 q^{92} - 2 q^{99}+O(q^{100}) $ 8 * q - 6 * q^4 - 8 * q^7 - 4 * q^9 + 4 * q^11 - 4 * q^16 + 2 * q^23 - 4 * q^25 + 6 * q^28 - 6 * q^36 + 2 * q^43 + 2 * q^44 + 8 * q^49 + 4 * q^63 + 12 * q^64 - 4 * q^77 - 4 * q^81 + 8 * q^92 - 2 * q^99

Display 100 coefficients 10 coefficients

Character values

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

$n$	$1445$	$1807$
$\chi(n)$	$-1$	$e\left(\frac{2}{3}\right)$

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 $

$2$	−0.951057	+	1.64728i	−0.951057	+	1.64728i	−0.207912	+	0.978148i	$0.566667\pi$
$2$	−0.951057	+	1.64728i	−0.951057	+	1.64728i	−0.743145	+	0.669131i	$0.766667\pi$
$3$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$3$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$4$	−1.30902	−	2.26728i	−1.30902	−	2.26728i
$5$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$5$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$6$		0			0
$7$	−1.00000			−1.00000
$7$	−1.00000			−1.00000
$8$	3.07768			3.07768
$9$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$10$		0			0
$11$	−0.618034			−0.618034			−0.309017	−	0.951057i	$-0.600000\pi$
$11$	−0.618034			−0.618034			−0.309017	+	0.951057i	$0.600000\pi$
$12$		0			0
$13$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$13$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$14$	0.951057	−	1.64728i	0.951057	−	1.64728i
$15$		0			0
$16$	−1.61803	+	2.80252i	−1.61803	+	2.80252i
$17$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$17$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$18$	1.90211			1.90211
$19$		0			0
$19$		0			0
$20$		0			0
$21$		0			0
$22$	0.587785	−	1.01807i	0.587785	−	1.01807i
$23$	0.809017	+	1.40126i	0.809017	+	1.40126i	0.913545	+	0.406737i	$0.133333\pi$
$23$	0.809017	+	1.40126i	0.809017	+	1.40126i	−0.104528	+	0.994522i	$0.533333\pi$
$24$		0			0
$25$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$26$		0			0
$27$		0			0
$28$	1.30902	+	2.26728i	1.30902	+	2.26728i
$29$	0.587785	+	1.01807i	0.587785	+	1.01807i	0.994522	+	0.104528i	$0.0333333\pi$
$29$	0.587785	+	1.01807i	0.587785	+	1.01807i	−0.406737	+	0.913545i	$0.633333\pi$
$30$		0			0
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$	−1.53884	−	2.66535i	−1.53884	−	2.66535i
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−1.30902	+	2.26728i	−1.30902	+	2.26728i
$37$		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		0.500000	−	0.866025i	$-0.333333\pi$
$41$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$42$		0			0
$43$	−0.309017	+	0.535233i	−0.309017	+	0.535233i	−0.978148	−	0.207912i	$-0.933333\pi$
$43$	−0.309017	+	0.535233i	−0.309017	+	0.535233i	0.669131	+	0.743145i	$0.266667\pi$
$44$	0.809017	+	1.40126i	0.809017	+	1.40126i
$45$		0			0
$46$	−3.07768			−3.07768
$47$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$47$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$48$		0			0
$49$	1.00000			1.00000
$50$	1.90211			1.90211
$51$		0			0
$52$		0			0
$53$	0.951057	+	1.64728i	0.951057	+	1.64728i	0.743145	+	0.669131i	$0.233333\pi$
$53$	0.951057	+	1.64728i	0.951057	+	1.64728i	0.207912	+	0.978148i	$0.433333\pi$
$54$		0			0
$55$		0			0
$56$	−3.07768			−3.07768
$57$		0			0
$58$	−2.23607			−2.23607
$59$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$59$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$60$		0			0
$61$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$61$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$62$		0			0
$63$	0.500000	+	0.866025i	0.500000	+	0.866025i
$64$	2.61803			2.61803
$65$		0			0
$66$		0			0
$67$	0.951057	+	1.64728i	0.951057	+	1.64728i	0.743145	+	0.669131i	$0.233333\pi$
$67$	0.951057	+	1.64728i	0.951057	+	1.64728i	0.207912	+	0.978148i	$0.433333\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$71$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$72$	−1.53884	−	2.66535i	−1.53884	−	2.66535i
$73$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$73$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	0.618034			0.618034
$78$		0			0
$79$	−0.587785	+	1.01807i	−0.587785	+	1.01807i	0.406737	+	0.913545i	$0.366667\pi$
$79$	−0.587785	+	1.01807i	−0.587785	+	1.01807i	−0.994522	+	0.104528i	$0.966667\pi$
$80$		0			0
$81$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$82$		0			0
$83$		0			0		1.00000			$0$
$83$		0			0		−1.00000			$\pi$
$84$		0			0
$85$		0			0
$86$	−0.587785	−	1.01807i	−0.587785	−	1.01807i
$87$		0			0
$88$	−1.90211			−1.90211
$89$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$89$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$90$		0			0
$91$		0			0
$92$	2.11803	−	3.66854i	2.11803	−	3.66854i
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$97$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$98$	−0.951057	+	1.64728i	−0.951057	+	1.64728i
$99$	0.309017	+	0.535233i	0.309017	+	0.535233i
$100$	−1.30902	+	2.26728i	−1.30902	+	2.26728i
$101$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$101$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$102$		0			0
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$		0			0
$105$		0			0
$106$	−3.61803			−3.61803
$107$	−1.17557			−1.17557			−0.587785	−	0.809017i	$-0.700000\pi$
$107$	−1.17557			−1.17557			−0.587785	+	0.809017i	$0.700000\pi$
$108$		0			0
$109$	−0.587785	+	1.01807i	−0.587785	+	1.01807i	0.406737	+	0.913545i	$0.366667\pi$
$109$	−0.587785	+	1.01807i	−0.587785	+	1.01807i	−0.994522	+	0.104528i	$0.966667\pi$
$110$		0			0
$111$		0			0
$112$	1.61803	−	2.80252i	1.61803	−	2.80252i
$113$	1.17557			1.17557			0.587785	−	0.809017i	$-0.300000\pi$
$113$	1.17557			1.17557			0.587785	+	0.809017i	$0.300000\pi$
$114$		0			0
$115$		0			0
$116$	1.53884	−	2.66535i	1.53884	−	2.66535i
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−0.618034			−0.618034
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$	−1.90211			−1.90211
$127$	0.587785	+	1.01807i	0.587785	+	1.01807i	0.994522	+	0.104528i	$0.0333333\pi$
$127$	0.587785	+	1.01807i	0.587785	+	1.01807i	−0.406737	+	0.913545i	$0.633333\pi$
$128$	−0.951057	+	1.64728i	−0.951057	+	1.64728i
$129$		0			0
$130$		0			0
$131$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$131$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$132$		0			0
$133$		0			0
$134$	−3.61803			−3.61803
$135$		0			0
$136$		0			0
$137$	−0.309017	−	0.535233i	−0.309017	−	0.535233i	0.669131	−	0.743145i	$-0.266667\pi$
$137$	−0.309017	−	0.535233i	−0.309017	−	0.535233i	−0.978148	+	0.207912i	$0.933333\pi$
$138$		0			0
$139$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$139$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	3.23607			3.23607
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	0.809017	−	1.40126i	0.809017	−	1.40126i	−0.104528	−	0.994522i	$-0.533333\pi$
$149$	0.809017	−	1.40126i	0.809017	−	1.40126i	0.913545	−	0.406737i	$-0.133333\pi$
$150$		0			0
$151$	−1.90211			−1.90211			−0.951057	−	0.309017i	$-0.900000\pi$
$151$	−1.90211			−1.90211			−0.951057	+	0.309017i	$0.900000\pi$
$152$		0			0
$153$		0			0
$154$	−0.587785	+	1.01807i	−0.587785	+	1.01807i
$155$		0			0
$156$		0			0
$157$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$157$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$158$	−1.11803	−	1.93649i	−1.11803	−	1.93649i
$159$		0			0
$160$		0			0
$161$	−0.809017	−	1.40126i	−0.809017	−	1.40126i
$162$	−0.951057	−	1.64728i	−0.951057	−	1.64728i
$163$	1.61803			1.61803			0.809017	−	0.587785i	$-0.200000\pi$
$163$	1.61803			1.61803			0.809017	+	0.587785i	$0.200000\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$167$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$168$		0			0
$169$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$170$		0			0
$171$		0			0
$172$	1.61803			1.61803
$173$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$173$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$174$		0			0
$175$	0.500000	+	0.866025i	0.500000	+	0.866025i
$176$	1.00000	−	1.73205i	1.00000	−	1.73205i
$177$		0			0
$178$		0			0
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$		0			0
$181$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$181$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$182$		0			0
$183$		0			0
$184$	2.48990	+	4.31263i	2.48990	+	4.31263i
$185$		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
$193$	0.951057	−	1.64728i	0.951057	−	1.64728i	0.207912	−	0.978148i	$-0.433333\pi$
$193$	0.951057	−	1.64728i	0.951057	−	1.64728i	0.743145	−	0.669131i	$-0.233333\pi$
$194$		0			0
$195$		0			0
$196$	−1.30902	−	2.26728i	−1.30902	−	2.26728i
$197$	1.61803			1.61803			0.809017	−	0.587785i	$-0.200000\pi$
$197$	1.61803			1.61803			0.809017	+	0.587785i	$0.200000\pi$
$198$	−1.17557			−1.17557
$199$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$199$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$200$	−1.53884	−	2.66535i	−1.53884	−	2.66535i
$201$		0			0
$202$		0			0
$203$	−0.587785	−	1.01807i	−0.587785	−	1.01807i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	0.809017	−	1.40126i	0.809017	−	1.40126i
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−0.587785	+	1.01807i	−0.587785	+	1.01807i	0.406737	+	0.913545i	$0.366667\pi$
$211$	−0.587785	+	1.01807i	−0.587785	+	1.01807i	−0.994522	+	0.104528i	$0.966667\pi$
$212$	2.48990	−	4.31263i	2.48990	−	4.31263i
$213$		0			0
$214$	1.11803	−	1.93649i	1.11803	−	1.93649i
$215$		0			0
$216$		0			0
$217$		0			0
$218$	−1.11803	−	1.93649i	−1.11803	−	1.93649i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$223$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$224$	1.53884	+	2.66535i	1.53884	+	2.66535i
$225$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$226$	−1.11803	+	1.93649i	−1.11803	+	1.93649i
$227$		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
$232$	1.80902	+	3.13331i	1.80902	+	3.13331i
$233$	0.309017	−	0.535233i	0.309017	−	0.535233i	−0.669131	−	0.743145i	$-0.733333\pi$
$233$	0.309017	−	0.535233i	0.309017	−	0.535233i	0.978148	+	0.207912i	$0.0666667\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	0.618034			0.618034			0.309017	−	0.951057i	$-0.400000\pi$
$239$	0.618034			0.618034			0.309017	+	0.951057i	$0.400000\pi$
$240$		0			0
$241$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$241$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$242$	0.587785	−	1.01807i	0.587785	−	1.01807i
$243$		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		−0.500000	−	0.866025i	$-0.666667\pi$
$251$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$252$	1.30902	−	2.26728i	1.30902	−	2.26728i
$253$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$254$	−2.23607			−2.23607
$255$		0			0
$256$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$257$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$257$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	0.587785	−	1.01807i	0.587785	−	1.01807i
$262$		0			0
$263$	−0.809017	+	1.40126i	−0.809017	+	1.40126i	0.104528	+	0.994522i	$0.466667\pi$
$263$	−0.809017	+	1.40126i	−0.809017	+	1.40126i	−0.913545	+	0.406737i	$0.866667\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	2.48990	−	4.31263i	2.48990	−	4.31263i
$269$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$269$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$270$		0			0
$271$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$271$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$272$		0			0
$273$		0			0
$274$	1.17557			1.17557
$275$	0.309017	+	0.535233i	0.309017	+	0.535233i
$276$		0			0
$277$	−1.61803			−1.61803			−0.809017	−	0.587785i	$-0.800000\pi$
$277$	−1.61803			−1.61803			−0.809017	+	0.587785i	$0.800000\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$281$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$282$		0			0
$283$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$283$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−1.53884	+	2.66535i	−1.53884	+	2.66535i
$289$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		0			0		1.00000			$0$
$293$		0			0		−1.00000			$\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$	1.53884	+	2.66535i	1.53884	+	2.66535i
$299$		0			0
$300$		0			0
$301$	0.309017	−	0.535233i	0.309017	−	0.535233i
$302$	1.80902	−	3.13331i	1.80902	−	3.13331i
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$307$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$308$	−0.809017	−	1.40126i	−0.809017	−	1.40126i
$309$		0			0
$310$		0			0
$311$		0			0		1.00000			$0$
$311$		0			0		−1.00000			$\pi$
$312$		0			0
$313$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$313$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$314$		0			0
$315$		0			0
$316$	3.07768			3.07768
$317$	−0.951057	−	1.64728i	−0.951057	−	1.64728i	−0.743145	−	0.669131i	$-0.766667\pi$
$317$	−0.951057	−	1.64728i	−0.951057	−	1.64728i	−0.207912	−	0.978148i	$-0.566667\pi$
$318$		0			0
$319$	−0.363271	−	0.629204i	−0.363271	−	0.629204i
$320$		0			0
$321$		0			0
$322$	3.07768			3.07768
$323$		0			0
$324$	2.61803			2.61803
$325$		0			0
$326$	−1.53884	+	2.66535i	−1.53884	+	2.66535i
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	1.90211			1.90211			0.951057	−	0.309017i	$-0.100000\pi$
$331$	1.90211			1.90211			0.951057	+	0.309017i	$0.100000\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−0.951057	+	1.64728i	−0.951057	+	1.64728i	−0.207912	+	0.978148i	$0.566667\pi$
$337$	−0.951057	+	1.64728i	−0.951057	+	1.64728i	−0.743145	+	0.669131i	$0.766667\pi$
$338$	−0.951057	−	1.64728i	−0.951057	−	1.64728i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−1.00000			−1.00000
$344$	−0.951057	+	1.64728i	−0.951057	+	1.64728i
$345$		0			0
$346$		0			0
$347$	−0.809017	+	1.40126i	−0.809017	+	1.40126i	0.104528	+	0.994522i	$0.466667\pi$
$347$	−0.809017	+	1.40126i	−0.809017	+	1.40126i	−0.913545	+	0.406737i	$0.866667\pi$
$348$		0			0
$349$		0			0		1.00000			$0$
$349$		0			0		−1.00000			$\pi$
$350$	−1.90211			−1.90211
$351$		0			0
$352$	0.951057	+	1.64728i	0.951057	+	1.64728i
$353$		0			0		1.00000			$0$
$353$		0			0		−1.00000			$\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−0.309017	+	0.535233i	−0.309017	+	0.535233i	−0.978148	−	0.207912i	$-0.933333\pi$
$359$	−0.309017	+	0.535233i	−0.309017	+	0.535233i	0.669131	+	0.743145i	$0.266667\pi$
$360$		0			0
$361$		0			0
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$367$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$368$	−5.23607			−5.23607
$369$		0			0
$370$		0			0
$371$	−0.951057	−	1.64728i	−0.951057	−	1.64728i
$372$		0			0
$373$	1.17557			1.17557			0.587785	−	0.809017i	$-0.300000\pi$
$373$	1.17557			1.17557			0.587785	+	0.809017i	$0.300000\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−1.17557			−1.17557			−0.587785	−	0.809017i	$-0.700000\pi$
$379$	−1.17557			−1.17557			−0.587785	+	0.809017i	$0.700000\pi$
$380$		0			0
$381$		0			0
$382$	−1.53884	+	2.66535i	−1.53884	+	2.66535i
$383$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$383$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$384$		0			0
$385$		0			0
$386$	1.80902	+	3.13331i	1.80902	+	3.13331i
$387$	0.618034			0.618034
$388$		0			0
$389$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$389$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$390$		0			0
$391$		0			0
$392$	3.07768			3.07768
$393$		0			0
$394$	−1.53884	+	2.66535i	−1.53884	+	2.66535i
$395$		0			0
$396$	0.809017	−	1.40126i	0.809017	−	1.40126i
$397$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$397$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$398$		0			0
$399$		0			0
$400$	3.23607			3.23607
$401$	0.587785	−	1.01807i	0.587785	−	1.01807i	−0.406737	−	0.913545i	$-0.633333\pi$
$401$	0.587785	−	1.01807i	0.587785	−	1.01807i	0.994522	−	0.104528i	$-0.0333333\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$	2.23607			2.23607
$407$		0			0
$408$		0			0
$409$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$409$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$	1.53884	+	2.66535i	1.53884	+	2.66535i
$415$		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
$421$	−0.951057	+	1.64728i	−0.951057	+	1.64728i	−0.207912	+	0.978148i	$0.566667\pi$
$421$	−0.951057	+	1.64728i	−0.951057	+	1.64728i	−0.743145	+	0.669131i	$0.766667\pi$
$422$	−1.11803	−	1.93649i	−1.11803	−	1.93649i
$423$		0			0
$424$	2.92705	+	5.06980i	2.92705	+	5.06980i
$425$		0			0
$426$		0			0
$427$		0			0
$428$	1.53884	+	2.66535i	1.53884	+	2.66535i
$429$		0			0
$430$		0			0
$431$	0.951057	+	1.64728i	0.951057	+	1.64728i	0.743145	+	0.669131i	$0.233333\pi$
$431$	0.951057	+	1.64728i	0.951057	+	1.64728i	0.207912	+	0.978148i	$0.433333\pi$
$432$		0			0
$433$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$433$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$434$		0			0
$435$		0			0
$436$	3.07768			3.07768
$437$		0			0
$438$		0			0
$439$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$439$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$440$		0			0
$441$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$442$		0			0
$443$	0.809017	+	1.40126i	0.809017	+	1.40126i	0.913545	+	0.406737i	$0.133333\pi$
$443$	0.809017	+	1.40126i	0.809017	+	1.40126i	−0.104528	+	0.994522i	$0.533333\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−2.61803			−2.61803
$449$	−1.90211			−1.90211			−0.951057	−	0.309017i	$-0.900000\pi$
$449$	−1.90211			−1.90211			−0.951057	+	0.309017i	$0.900000\pi$
$450$	−0.951057	−	1.64728i	−0.951057	−	1.64728i
$451$		0			0
$452$	−1.53884	−	2.66535i	−1.53884	−	2.66535i
$453$		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		0.500000	−	0.866025i	$-0.333333\pi$
$461$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$462$		0			0
$463$	0.618034			0.618034			0.309017	−	0.951057i	$-0.400000\pi$
$463$	0.618034			0.618034			0.309017	+	0.951057i	$0.400000\pi$
$464$	−3.80423			−3.80423
$465$		0			0
$466$	0.587785	+	1.01807i	0.587785	+	1.01807i
$467$		0			0		1.00000			$0$
$467$		0			0		−1.00000			$\pi$
$468$		0			0
$469$	−0.951057	−	1.64728i	−0.951057	−	1.64728i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	0.190983	−	0.330792i	0.190983	−	0.330792i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	0.951057	−	1.64728i	0.951057	−	1.64728i
$478$	−0.587785	+	1.01807i	−0.587785	+	1.01807i
$479$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$479$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	0.809017	+	1.40126i	0.809017	+	1.40126i
$485$		0			0
$486$		0			0
$487$	1.90211			1.90211			0.951057	−	0.309017i	$-0.100000\pi$
$487$	1.90211			1.90211			0.951057	+	0.309017i	$0.100000\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−0.809017	+	1.40126i	−0.809017	+	1.40126i	0.104528	+	0.994522i	$0.466667\pi$
$491$	−0.809017	+	1.40126i	−0.809017	+	1.40126i	−0.913545	+	0.406737i	$0.866667\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	0.809017	−	1.40126i	0.809017	−	1.40126i	−0.104528	−	0.994522i	$-0.533333\pi$
$499$	0.809017	−	1.40126i	0.809017	−	1.40126i	0.913545	−	0.406737i	$-0.133333\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$503$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$504$	1.53884	+	2.66535i	1.53884	+	2.66535i
$505$		0			0
$506$	1.90211			1.90211
$507$		0			0
$508$	1.53884	−	2.66535i	1.53884	−	2.66535i
$509$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$509$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$510$		0			0
$511$		0			0
$512$		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		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$	1.11803	+	1.93649i	1.11803	+	1.93649i
$523$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$523$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$524$		0			0
$525$		0			0
$526$	−1.53884	−	2.66535i	−1.53884	−	2.66535i
$527$		0			0
$528$		0			0
$529$	−0.809017	+	1.40126i	−0.809017	+	1.40126i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$	2.92705	+	5.06980i	2.92705	+	5.06980i
$537$		0			0
$538$		0			0
$539$	−0.618034			−0.618034
$540$		0			0
$541$	−0.309017	−	0.535233i	−0.309017	−	0.535233i	0.669131	−	0.743145i	$-0.266667\pi$
$541$	−0.309017	−	0.535233i	−0.309017	−	0.535233i	−0.978148	+	0.207912i	$0.933333\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−0.951057	−	1.64728i	−0.951057	−	1.64728i	−0.743145	−	0.669131i	$-0.766667\pi$
$547$	−0.951057	−	1.64728i	−0.951057	−	1.64728i	−0.207912	−	0.978148i	$-0.566667\pi$
$548$	−0.809017	+	1.40126i	−0.809017	+	1.40126i
$549$		0			0
$550$	−1.17557			−1.17557
$551$		0			0
$552$		0			0
$553$	0.587785	−	1.01807i	0.587785	−	1.01807i
$554$	1.53884	−	2.66535i	1.53884	−	2.66535i
$555$		0			0
$556$		0			0
$557$	0.309017	+	0.535233i	0.309017	+	0.535233i	0.978148	−	0.207912i	$-0.0666667\pi$
$557$	0.309017	+	0.535233i	0.309017	+	0.535233i	−0.669131	+	0.743145i	$0.733333\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		1.00000			$0$
$563$		0			0		−1.00000			$\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	0.500000	−	0.866025i	0.500000	−	0.866025i
$568$		0			0
$569$	1.90211			1.90211			0.951057	−	0.309017i	$-0.100000\pi$
$569$	1.90211			1.90211			0.951057	+	0.309017i	$0.100000\pi$
$570$		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
$575$	0.809017	−	1.40126i	0.809017	−	1.40126i
$576$	−1.30902	−	2.26728i	−1.30902	−	2.26728i
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$	1.90211			1.90211
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−0.587785	−	1.01807i	−0.587785	−	1.01807i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$587$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$593$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$594$		0			0
$595$		0			0
$596$	−4.23607			−4.23607
$597$		0			0
$598$		0			0
$599$	−0.587785	−	1.01807i	−0.587785	−	1.01807i	−0.994522	−	0.104528i	$-0.966667\pi$
$599$	−0.587785	−	1.01807i	−0.587785	−	1.01807i	0.406737	−	0.913545i	$-0.366667\pi$
$600$		0			0
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$	0.587785	+	1.01807i	0.587785	+	1.01807i
$603$	0.951057	−	1.64728i	0.951057	−	1.64728i
$604$	2.48990	+	4.31263i	2.48990	+	4.31263i
$605$		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
$613$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$613$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$614$		0			0
$615$		0			0
$616$	1.90211			1.90211
$617$	0.809017	+	1.40126i	0.809017	+	1.40126i	0.913545	+	0.406737i	$0.133333\pi$
$617$	0.809017	+	1.40126i	0.809017	+	1.40126i	−0.104528	+	0.994522i	$0.533333\pi$
$618$		0			0
$619$		0			0		1.00000			$0$
$619$		0			0		−1.00000			$\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−0.809017	−	1.40126i	−0.809017	−	1.40126i	−0.913545	−	0.406737i	$-0.866667\pi$
$631$	−0.809017	−	1.40126i	−0.809017	−	1.40126i	0.104528	−	0.994522i	$-0.466667\pi$
$632$	−1.80902	+	3.13331i	−1.80902	+	3.13331i
$633$		0			0
$634$	3.61803			3.61803
$635$		0			0
$636$		0			0
$637$		0			0
$638$	1.38197			1.38197
$639$		0			0
$640$		0			0
$641$	−0.951057	+	1.64728i	−0.951057	+	1.64728i	−0.207912	+	0.978148i	$0.566667\pi$
$641$	−0.951057	+	1.64728i	−0.951057	+	1.64728i	−0.743145	+	0.669131i	$0.766667\pi$
$642$		0			0
$643$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$643$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$644$	−2.11803	+	3.66854i	−2.11803	+	3.66854i
$645$		0			0
$646$		0			0
$647$		0			0		1.00000			$0$
$647$		0			0		−1.00000			$\pi$
$648$	−1.53884	+	2.66535i	−1.53884	+	2.66535i
$649$		0			0
$650$		0			0
$651$		0			0
$652$	−2.11803	−	3.66854i	−2.11803	−	3.66854i
$653$	1.61803			1.61803			0.809017	−	0.587785i	$-0.200000\pi$
$653$	1.61803			1.61803			0.809017	+	0.587785i	$0.200000\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	0.951057	+	1.64728i	0.951057	+	1.64728i	0.743145	+	0.669131i	$0.233333\pi$
$659$	0.951057	+	1.64728i	0.951057	+	1.64728i	0.207912	+	0.978148i	$0.433333\pi$
$660$		0			0
$661$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$661$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$662$	−1.80902	+	3.13331i	−1.80902	+	3.13331i
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−0.951057	+	1.64728i	−0.951057	+	1.64728i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−1.17557			−1.17557			−0.587785	−	0.809017i	$-0.700000\pi$
$673$	−1.17557			−1.17557			−0.587785	+	0.809017i	$0.700000\pi$
$674$	−1.80902	−	3.13331i	−1.80902	−	3.13331i
$675$		0			0
$676$	2.61803			2.61803
$677$		0			0		1.00000			$0$
$677$		0			0		−1.00000			$\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−1.90211			−1.90211			−0.951057	−	0.309017i	$-0.900000\pi$
$683$	−1.90211			−1.90211			−0.951057	+	0.309017i	$0.900000\pi$
$684$		0			0
$685$		0			0
$686$	0.951057	−	1.64728i	0.951057	−	1.64728i
$687$		0			0
$688$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$689$		0			0
$690$		0			0
$691$		0			0		1.00000			$0$
$691$		0			0		−1.00000			$\pi$
$692$		0			0
$693$	−0.309017	−	0.535233i	−0.309017	−	0.535233i
$694$	−1.53884	−	2.66535i	−1.53884	−	2.66535i
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	1.30902	−	2.26728i	1.30902	−	2.26728i
$701$	0.309017	−	0.535233i	0.309017	−	0.535233i	−0.669131	−	0.743145i	$-0.733333\pi$
$701$	0.309017	−	0.535233i	0.309017	−	0.535233i	0.978148	+	0.207912i	$0.0666667\pi$
$702$		0			0
$703$		0			0
$704$	−1.61803			−1.61803
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−0.309017	−	0.535233i	−0.309017	−	0.535233i	0.669131	−	0.743145i	$-0.266667\pi$
$709$	−0.309017	−	0.535233i	−0.309017	−	0.535233i	−0.978148	+	0.207912i	$0.933333\pi$
$710$		0			0
$711$	1.17557			1.17557
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$	−0.587785	−	1.01807i	−0.587785	−	1.01807i
$719$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$719$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	0.587785	−	1.01807i	0.587785	−	1.01807i
$726$		0			0
$727$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$727$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$728$		0			0
$729$	1.00000			1.00000
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		1.00000			$0$
$733$		0			0		−1.00000			$\pi$
$734$		0			0
$735$		0			0
$736$	2.48990	−	4.31263i	2.48990	−	4.31263i
$737$	−0.587785	−	1.01807i	−0.587785	−	1.01807i
$738$		0			0
$739$	0.809017	−	1.40126i	0.809017	−	1.40126i	−0.104528	−	0.994522i	$-0.533333\pi$
$739$	0.809017	−	1.40126i	0.809017	−	1.40126i	0.913545	−	0.406737i	$-0.133333\pi$
$740$		0			0
$741$		0			0
$742$	3.61803			3.61803
$743$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$743$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$744$		0			0
$745$		0			0
$746$	−1.11803	+	1.93649i	−1.11803	+	1.93649i
$747$		0			0
$748$		0			0
$749$	1.17557			1.17557
$750$		0			0
$751$	−0.951057	−	1.64728i	−0.951057	−	1.64728i	−0.743145	−	0.669131i	$-0.766667\pi$
$751$	−0.951057	−	1.64728i	−0.951057	−	1.64728i	−0.207912	−	0.978148i	$-0.566667\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	0.809017	−	1.40126i	0.809017	−	1.40126i	−0.104528	−	0.994522i	$-0.533333\pi$
$757$	0.809017	−	1.40126i	0.809017	−	1.40126i	0.913545	−	0.406737i	$-0.133333\pi$
$758$	1.11803	−	1.93649i	1.11803	−	1.93649i
$759$		0			0
$760$		0			0
$761$		0			0		1.00000			$0$
$761$		0			0		−1.00000			$\pi$
$762$		0			0
$763$	0.587785	−	1.01807i	0.587785	−	1.01807i
$764$	−2.11803	−	3.66854i	−2.11803	−	3.66854i
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$769$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$770$		0			0
$771$		0			0
$772$	−4.97980			−4.97980
$773$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$773$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$774$	−0.587785	+	1.01807i	−0.587785	+	1.01807i
$775$		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
$784$	−1.61803	+	2.80252i	−1.61803	+	2.80252i
$785$		0			0
$786$		0			0
$787$		0			0		1.00000			$0$
$787$		0			0		−1.00000			$\pi$
$788$	−2.11803	−	3.66854i	−2.11803	−	3.66854i
$789$		0			0
$790$		0			0
$791$	−1.17557			−1.17557
$792$	0.951057	+	1.64728i	0.951057	+	1.64728i
$793$		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
$800$	−1.53884	+	2.66535i	−1.53884	+	2.66535i
$801$		0			0
$802$	1.11803	+	1.93649i	1.11803	+	1.93649i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−2.00000			−2.00000			−1.00000			$\pi$
$809$	−2.00000			−2.00000			−1.00000			$\pi$
$810$		0			0
$811$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$811$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$812$	−1.53884	+	2.66535i	−1.53884	+	2.66535i
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−0.809017	−	1.40126i	−0.809017	−	1.40126i	−0.913545	−	0.406737i	$-0.866667\pi$
$821$	−0.809017	−	1.40126i	−0.809017	−	1.40126i	0.104528	−	0.994522i	$-0.466667\pi$
$822$		0			0
$823$	−0.309017	−	0.535233i	−0.309017	−	0.535233i	0.669131	−	0.743145i	$-0.266667\pi$
$823$	−0.309017	−	0.535233i	−0.309017	−	0.535233i	−0.978148	+	0.207912i	$0.933333\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−0.587785	−	1.01807i	−0.587785	−	1.01807i	−0.994522	−	0.104528i	$-0.966667\pi$
$827$	−0.587785	−	1.01807i	−0.587785	−	1.01807i	0.406737	−	0.913545i	$-0.366667\pi$
$828$	−4.23607			−4.23607
$829$		0			0		1.00000			$0$
$829$		0			0		−1.00000			$\pi$
$830$		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		0.500000	−	0.866025i	$-0.333333\pi$
$839$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$840$		0			0
$841$	−0.190983	+	0.330792i	−0.190983	+	0.330792i
$842$	−1.80902	−	3.13331i	−1.80902	−	3.13331i
$843$		0			0
$844$	3.07768			3.07768
$845$		0			0
$846$		0			0
$847$	0.618034			0.618034
$848$	−6.15537			−6.15537
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$853$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$854$		0			0
$855$		0			0
$856$	−3.61803			−3.61803
$857$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$857$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$858$		0			0
$859$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$859$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$860$		0			0
$861$		0			0
$862$	−3.61803			−3.61803
$863$	−1.17557			−1.17557			−0.587785	−	0.809017i	$-0.700000\pi$
$863$	−1.17557			−1.17557			−0.587785	+	0.809017i	$0.700000\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	0.363271	−	0.629204i	0.363271	−	0.629204i
$870$		0			0
$871$		0			0
$872$	−1.80902	+	3.13331i	−1.80902	+	3.13331i
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−0.951057	+	1.64728i	−0.951057	+	1.64728i	−0.207912	+	0.978148i	$0.566667\pi$
$877$	−0.951057	+	1.64728i	−0.951057	+	1.64728i	−0.743145	+	0.669131i	$0.766667\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$	1.90211			1.90211
$883$	0.809017	+	1.40126i	0.809017	+	1.40126i	0.913545	+	0.406737i	$0.133333\pi$
$883$	0.809017	+	1.40126i	0.809017	+	1.40126i	−0.104528	+	0.994522i	$0.533333\pi$
$884$		0			0
$885$		0			0
$886$	−3.07768			−3.07768
$887$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$887$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$888$		0			0
$889$	−0.587785	−	1.01807i	−0.587785	−	1.01807i
$890$		0			0
$891$	0.309017	−	0.535233i	0.309017	−	0.535233i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$	0.951057	−	1.64728i	0.951057	−	1.64728i
$897$		0			0
$898$	1.80902	−	3.13331i	1.80902	−	3.13331i
$899$		0			0
$900$	2.61803			2.61803
$901$		0			0
$902$		0			0
$903$		0			0
$904$	3.61803			3.61803
$905$		0			0
$906$		0			0
$907$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$907$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	1.90211			1.90211			0.951057	−	0.309017i	$-0.100000\pi$
$911$	1.90211			1.90211			0.951057	+	0.309017i	$0.100000\pi$
$912$		0			0
$913$		0			0
$914$	0.587785	−	1.01807i	0.587785	−	1.01807i
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−0.618034			−0.618034			−0.309017	−	0.951057i	$-0.600000\pi$
$919$	−0.618034			−0.618034			−0.309017	+	0.951057i	$0.600000\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$	−0.587785	+	1.01807i	−0.587785	+	1.01807i
$927$		0			0
$928$	1.80902	−	3.13331i	1.80902	−	3.13331i
$929$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$929$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$930$		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		−0.500000	−	0.866025i	$-0.666667\pi$
$937$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$938$	3.61803			3.61803
$939$		0			0
$940$		0			0
$941$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$941$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$	0.363271	+	0.629204i	0.363271	+	0.629204i
$947$	−0.309017	+	0.535233i	−0.309017	+	0.535233i	−0.978148	−	0.207912i	$-0.933333\pi$
$947$	−0.309017	+	0.535233i	−0.309017	+	0.535233i	0.669131	+	0.743145i	$0.266667\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	0.951057	−	1.64728i	0.951057	−	1.64728i	0.207912	−	0.978148i	$-0.433333\pi$
$953$	0.951057	−	1.64728i	0.951057	−	1.64728i	0.743145	−	0.669131i	$-0.233333\pi$
$954$	1.80902	+	3.13331i	1.80902	+	3.13331i
$955$		0			0
$956$	−0.809017	−	1.40126i	−0.809017	−	1.40126i
$957$		0			0
$958$		0			0
$959$	0.309017	+	0.535233i	0.309017	+	0.535233i
$960$		0			0
$961$	1.00000			1.00000
$962$		0			0
$963$	0.587785	+	1.01807i	0.587785	+	1.01807i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	0.309017	−	0.535233i	0.309017	−	0.535233i	−0.669131	−	0.743145i	$-0.733333\pi$
$967$	0.309017	−	0.535233i	0.309017	−	0.535233i	0.978148	+	0.207912i	$0.0666667\pi$
$968$	−1.90211			−1.90211
$969$		0			0
$970$		0			0
$971$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$971$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$972$		0			0
$973$		0			0
$974$	−1.80902	+	3.13331i	−1.80902	+	3.13331i
$975$		0			0
$976$		0			0
$977$	−1.17557			−1.17557			−0.587785	−	0.809017i	$-0.700000\pi$
$977$	−1.17557			−1.17557			−0.587785	+	0.809017i	$0.700000\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	1.17557			1.17557
$982$	−1.53884	−	2.66535i	−1.53884	−	2.66535i
$983$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$983$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$984$		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		−0.866025	−	0.500000i	$-0.833333\pi$
$991$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$997$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$998$	1.53884	+	2.66535i	1.53884	+	2.66535i
$999$		0			0

Twists

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

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

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.m (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.951057 + 1.64728i) q^{2} +(-1.30902 - 2.26728i) q^{4} -1.00000 q^{7} +3.07768 q^{8} +(-0.500000 - 0.866025i) q^{9} +O(q^{10})\)	\(q+(-0.951057 + 1.64728i) q^{2} +(-1.30902 - 2.26728i) q^{4} -1.00000 q^{7} +3.07768 q^{8} +(-0.500000 - 0.866025i) q^{9} -0.618034 q^{11} +(0.951057 - 1.64728i) q^{14} +(-1.61803 + 2.80252i) q^{16} +1.90211 q^{18} +(0.587785 - 1.01807i) q^{22} +(0.809017 + 1.40126i) q^{23} +(-0.500000 - 0.866025i) q^{25} +(1.30902 + 2.26728i) q^{28} +(0.587785 + 1.01807i) q^{29} +(-1.53884 - 2.66535i) q^{32} +(-1.30902 + 2.26728i) q^{36} +(-0.309017 + 0.535233i) q^{43} +(0.809017 + 1.40126i) q^{44} -3.07768 q^{46} +1.00000 q^{49} +1.90211 q^{50} +(0.951057 + 1.64728i) q^{53} -3.07768 q^{56} -2.23607 q^{58} +(0.500000 + 0.866025i) q^{63} +2.61803 q^{64} +(0.951057 + 1.64728i) q^{67} +(-1.53884 - 2.66535i) q^{72} +0.618034 q^{77} +(-0.587785 + 1.01807i) q^{79} +(-0.500000 + 0.866025i) q^{81} +(-0.587785 - 1.01807i) q^{86} -1.90211 q^{88} +(2.11803 - 3.66854i) q^{92} +(-0.951057 + 1.64728i) q^{98} +(0.309017 + 0.535233i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 6 q^{4} - 8 q^{7} - 4 q^{9}+O(q^{10}) \) 8 * q - 6 * q^4 - 8 * q^7 - 4 * q^9	\( 8 q - 6 q^{4} - 8 q^{7} - 4 q^{9} + 4 q^{11} - 4 q^{16} + 2 q^{23} - 4 q^{25} + 6 q^{28} - 6 q^{36} + 2 q^{43} + 2 q^{44} + 8 q^{49} + 4 q^{63} + 12 q^{64} - 4 q^{77} - 4 q^{81} + 8 q^{92} - 2 q^{99}+O(q^{100}) \) 8 * q - 6 * q^4 - 8 * q^7 - 4 * q^9 + 4 * q^11 - 4 * q^16 + 2 * q^23 - 4 * q^25 + 6 * q^28 - 6 * q^36 + 2 * q^43 + 2 * q^44 + 8 * q^49 + 4 * q^63 + 12 * q^64 - 4 * q^77 - 4 * q^81 + 8 * q^92 - 2 * q^99

Introduction

L-functions

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