LMFDB - Embedded newform 2541.1.r.e.2393.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2541,1,Mod(524,2541)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2541.524"); S:= CuspForms(chi, 1); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2541, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([5, 5, 7])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]

Level:	$ N $	$=$	$ 2541 = 3 \cdot 7 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2541.r (of order $10$, degree $4$, not minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	no
Analytic conductor:	$1.26812419710$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$4$ over $\Q(\zeta_{10})$
Coefficient field:	$\Q(\zeta_{40})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - x^{12} + x^{8} - x^{4} + 1 $ x^16 - x^12 + x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{4}$
Projective field:	Galois closure of 4.0.195657.1

Embedding invariants

Embedding label			2393.4
Root			$0.156434 - 0.987688i$ of defining polynomial
Character	$\chi$	$=$	2541.2393
Dual form			2541.1.r.e.1322.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.587785 - 0.809017i) q^{3} +(0.809017 - 0.587785i) q^{4} +(0.987688 + 0.156434i) q^{7} +(-0.309017 - 0.951057i) q^{9} -1.00000i q^{12} +(0.437016 + 1.34500i) q^{13} +(0.309017 - 0.951057i) q^{16} +(-1.14412 - 0.831254i) q^{19} +(0.707107 - 0.707107i) q^{21} +(0.809017 + 0.587785i) q^{25} +(-0.951057 - 0.309017i) q^{27} +(0.891007 - 0.453990i) q^{28} +(-1.90211 + 0.618034i) q^{31} +(-0.809017 - 0.587785i) q^{36} +(1.34500 + 0.437016i) q^{39} +1.41421i q^{43} +(-0.587785 - 0.809017i) q^{48} +(0.951057 + 0.309017i) q^{49} +(1.14412 + 0.831254i) q^{52} +(-1.34500 + 0.437016i) q^{57} +(0.437016 - 1.34500i) q^{61} +(-0.156434 - 0.987688i) q^{63} +(-0.309017 - 0.951057i) q^{64} +(-1.14412 + 0.831254i) q^{73} +(0.951057 - 0.309017i) q^{75} -1.41421 q^{76} +(-1.34500 + 0.437016i) q^{79} +(-0.809017 + 0.587785i) q^{81} +(0.156434 - 0.987688i) q^{84} +(0.221232 + 1.39680i) q^{91} +(-0.618034 + 1.90211i) q^{93} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 4 q^{4} + 4 q^{9} - 4 q^{16} + 4 q^{25} - 4 q^{36} + 4 q^{64} - 4 q^{81} + 4 q^{91} + 8 q^{93}+O(q^{100}) $ 16 * q + 4 * q^4 + 4 * q^9 - 4 * q^16 + 4 * q^25 - 4 * q^36 + 4 * q^64 - 4 * q^81 + 4 * q^91 + 8 * q^93

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2541\mathbb{Z}\right)^\times$.

$n$	$848$	$1816$	$2059$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{9}{10}\right)$

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			0		0.951057	−	0.309017i	$-0.100000\pi$
$2$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$3$	0.587785	−	0.809017i	0.587785	−	0.809017i
$3$	0.587785	−	0.809017i	0.587785	−	0.809017i
$4$	0.809017	−	0.587785i	0.809017	−	0.587785i
$5$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$5$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$6$		0			0
$7$	0.987688	+	0.156434i	0.987688	+	0.156434i
$7$	0.987688	+	0.156434i	0.987688	+	0.156434i
$8$		0			0
$9$	−0.309017	−	0.951057i	−0.309017	−	0.951057i
$10$		0			0
$11$		0			0
$11$		0			0
$12$		−	1.00000i		−	1.00000i
$13$	0.437016	+	1.34500i	0.437016	+	1.34500i	0.891007	+	0.453990i	$0.150000\pi$
$13$	0.437016	+	1.34500i	0.437016	+	1.34500i	−0.453990	+	0.891007i	$0.650000\pi$
$14$		0			0
$15$		0			0
$16$	0.309017	−	0.951057i	0.309017	−	0.951057i
$17$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$17$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$18$		0			0
$19$	−1.14412	−	0.831254i	−1.14412	−	0.831254i	−0.156434	−	0.987688i	$-0.550000\pi$
$19$	−1.14412	−	0.831254i	−1.14412	−	0.831254i	−0.987688	+	0.156434i	$0.950000\pi$
$20$		0			0
$21$	0.707107	−	0.707107i	0.707107	−	0.707107i
$22$		0			0
$23$		0			0		1.00000			$0$
$23$		0			0		−1.00000			$\pi$
$24$		0			0
$25$	0.809017	+	0.587785i	0.809017	+	0.587785i
$26$		0			0
$27$	−0.951057	−	0.309017i	−0.951057	−	0.309017i
$28$	0.891007	−	0.453990i	0.891007	−	0.453990i
$29$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$29$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$30$		0			0
$31$	−1.90211	+	0.618034i	−1.90211	+	0.618034i	−0.951057	+	0.309017i	$0.900000\pi$
$31$	−1.90211	+	0.618034i	−1.90211	+	0.618034i	−0.951057	+	0.309017i	$0.900000\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$37$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$37$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$38$		0			0
$39$	1.34500	+	0.437016i	1.34500	+	0.437016i
$40$		0			0
$41$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$41$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$42$		0			0
$43$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$43$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$47$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$48$	−0.587785	−	0.809017i	−0.587785	−	0.809017i
$49$	0.951057	+	0.309017i	0.951057	+	0.309017i
$50$		0			0
$51$		0			0
$52$	1.14412	+	0.831254i	1.14412	+	0.831254i
$53$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$53$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−1.34500	+	0.437016i	−1.34500	+	0.437016i
$58$		0			0
$59$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$59$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$60$		0			0
$61$	0.437016	−	1.34500i	0.437016	−	1.34500i	−0.453990	−	0.891007i	$-0.650000\pi$
$61$	0.437016	−	1.34500i	0.437016	−	1.34500i	0.891007	−	0.453990i	$-0.150000\pi$
$62$		0			0
$63$	−0.156434	−	0.987688i	−0.156434	−	0.987688i
$64$	−0.309017	−	0.951057i	−0.309017	−	0.951057i
$65$		0			0
$66$		0			0
$67$		0			0			−	1.00000i	$-0.5\pi$
$67$		0			0				1.00000i	$0.5\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$71$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$72$		0			0
$73$	−1.14412	+	0.831254i	−1.14412	+	0.831254i	−0.987688	−	0.156434i	$-0.950000\pi$
$73$	−1.14412	+	0.831254i	−1.14412	+	0.831254i	−0.156434	+	0.987688i	$0.550000\pi$
$74$		0			0
$75$	0.951057	−	0.309017i	0.951057	−	0.309017i
$76$	−1.41421			−1.41421
$77$		0			0
$78$		0			0
$79$	−1.34500	+	0.437016i	−1.34500	+	0.437016i	−0.891007	−	0.453990i	$-0.850000\pi$
$79$	−1.34500	+	0.437016i	−1.34500	+	0.437016i	−0.453990	+	0.891007i	$0.650000\pi$
$80$		0			0
$81$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$82$		0			0
$83$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$83$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$84$	0.156434	−	0.987688i	0.156434	−	0.987688i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$		0			0
$91$	0.221232	+	1.39680i	0.221232	+	1.39680i
$92$		0			0
$93$	−0.618034	+	1.90211i	−0.618034	+	1.90211i
$94$		0			0
$95$		0			0
$96$		0			0
$97$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$97$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$98$		0			0
$99$		0			0
$100$	1.00000			1.00000
$101$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$101$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$102$		0			0
$103$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$103$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$107$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$108$	−0.951057	+	0.309017i	−0.951057	+	0.309017i
$109$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$109$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$110$		0			0
$111$		0			0
$112$	0.453990	−	0.891007i	0.453990	−	0.891007i
$113$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$113$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	1.14412	−	0.831254i	1.14412	−	0.831254i
$118$		0			0
$119$		0			0
$120$		0			0
$121$		0			0
$122$		0			0
$123$		0			0
$124$	−1.17557	+	1.61803i	−1.17557	+	1.61803i
$125$		0			0
$126$		0			0
$127$	−1.34500	−	0.437016i	−1.34500	−	0.437016i	−0.453990	−	0.891007i	$-0.650000\pi$
$127$	−1.34500	−	0.437016i	−1.34500	−	0.437016i	−0.891007	+	0.453990i	$0.850000\pi$
$128$		0			0
$129$	1.14412	+	0.831254i	1.14412	+	0.831254i
$130$		0			0
$131$		0			0		1.00000			$0$
$131$		0			0		−1.00000			$\pi$
$132$		0			0
$133$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$137$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$138$		0			0
$139$	−1.14412	+	0.831254i	−1.14412	+	0.831254i	−0.987688	−	0.156434i	$-0.950000\pi$
$139$	−1.14412	+	0.831254i	−1.14412	+	0.831254i	−0.156434	+	0.987688i	$0.550000\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	−1.00000			−1.00000
$145$		0			0
$146$		0			0
$147$	0.809017	−	0.587785i	0.809017	−	0.587785i
$148$		0			0
$149$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$149$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$150$		0			0
$151$	0.831254	−	1.14412i	0.831254	−	1.14412i	−0.156434	−	0.987688i	$-0.550000\pi$
$151$	0.831254	−	1.14412i	0.831254	−	1.14412i	0.987688	−	0.156434i	$-0.0500000\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	1.34500	−	0.437016i	1.34500	−	0.437016i
$157$	1.17557	−	1.61803i	1.17557	−	1.61803i	0.587785	−	0.809017i	$-0.300000\pi$
$157$	1.17557	−	1.61803i	1.17557	−	1.61803i	0.587785	−	0.809017i	$-0.300000\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	0.618034	+	1.90211i	0.618034	+	1.90211i	0.309017	+	0.951057i	$0.400000\pi$
$163$	0.618034	+	1.90211i	0.618034	+	1.90211i	0.309017	+	0.951057i	$0.400000\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$167$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$168$		0			0
$169$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$170$		0			0
$171$	−0.437016	+	1.34500i	−0.437016	+	1.34500i
$172$	0.831254	+	1.14412i	0.831254	+	1.14412i
$173$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$173$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$174$		0			0
$175$	0.707107	+	0.707107i	0.707107	+	0.707107i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$179$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$180$		0			0
$181$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$181$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$182$		0			0
$183$	−0.831254	−	1.14412i	−0.831254	−	1.14412i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	−0.891007	−	0.453990i	−0.891007	−	0.453990i
$190$		0			0
$191$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$191$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$192$	−0.951057	−	0.309017i	−0.951057	−	0.309017i
$193$	−1.34500	−	0.437016i	−1.34500	−	0.437016i	−0.453990	−	0.891007i	$-0.650000\pi$
$193$	−1.34500	−	0.437016i	−1.34500	−	0.437016i	−0.891007	+	0.453990i	$0.850000\pi$
$194$		0			0
$195$		0			0
$196$	0.951057	−	0.309017i	0.951057	−	0.309017i
$197$		0			0			−	1.00000i	$-0.5\pi$
$197$		0			0				1.00000i	$0.5\pi$
$198$		0			0
$199$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$199$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	1.41421			1.41421
$209$		0			0
$210$		0			0
$211$	1.34500	−	0.437016i	1.34500	−	0.437016i	0.453990	−	0.891007i	$-0.350000\pi$
$211$	1.34500	−	0.437016i	1.34500	−	0.437016i	0.891007	+	0.453990i	$0.150000\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−1.97538	+	0.312869i	−1.97538	+	0.312869i
$218$		0			0
$219$			1.41421i			1.41421i
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$223$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$224$		0			0
$225$	0.309017	−	0.951057i	0.309017	−	0.951057i
$226$		0			0
$227$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$227$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$228$	−0.831254	+	1.14412i	−0.831254	+	1.14412i
$229$	1.90211	−	0.618034i	1.90211	−	0.618034i	0.951057	−	0.309017i	$-0.100000\pi$
$229$	1.90211	−	0.618034i	1.90211	−	0.618034i	0.951057	−	0.309017i	$-0.100000\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$233$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−0.437016	+	1.34500i	−0.437016	+	1.34500i
$238$		0			0
$239$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$239$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$240$		0			0
$241$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$241$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$242$		0			0
$243$			1.00000i			1.00000i
$244$	−0.437016	−	1.34500i	−0.437016	−	1.34500i
$245$		0			0
$246$		0			0
$247$	0.618034	−	1.90211i	0.618034	−	1.90211i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$251$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$252$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$257$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$257$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0			−	1.00000i	$-0.5\pi$
$263$		0			0				1.00000i	$0.5\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$269$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$270$		0			0
$271$	1.14412	−	0.831254i	1.14412	−	0.831254i	0.156434	−	0.987688i	$-0.450000\pi$
$271$	1.14412	−	0.831254i	1.14412	−	0.831254i	0.987688	+	0.156434i	$0.0500000\pi$
$272$		0			0
$273$	1.26007	+	0.642040i	1.26007	+	0.642040i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	1.34500	−	0.437016i	1.34500	−	0.437016i	0.453990	−	0.891007i	$-0.350000\pi$
$277$	1.34500	−	0.437016i	1.34500	−	0.437016i	0.891007	+	0.453990i	$0.150000\pi$
$278$		0			0
$279$	1.17557	+	1.61803i	1.17557	+	1.61803i
$280$		0			0
$281$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$281$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$282$		0			0
$283$	1.14412	+	0.831254i	1.14412	+	0.831254i	0.987688	−	0.156434i	$-0.0500000\pi$
$283$	1.14412	+	0.831254i	1.14412	+	0.831254i	0.156434	+	0.987688i	$0.450000\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$290$		0			0
$291$		0			0
$292$	−0.437016	+	1.34500i	−0.437016	+	1.34500i
$293$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$293$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$	0.587785	−	0.809017i	0.587785	−	0.809017i
$301$	−0.221232	+	1.39680i	−0.221232	+	1.39680i
$302$		0			0
$303$		0			0
$304$	−1.14412	+	0.831254i	−1.14412	+	0.831254i
$305$		0			0
$306$		0			0
$307$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$307$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$311$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$312$		0			0
$313$	1.90211	+	0.618034i	1.90211	+	0.618034i	0.951057	+	0.309017i	$0.100000\pi$
$313$	1.90211	+	0.618034i	1.90211	+	0.618034i	0.951057	+	0.309017i	$0.100000\pi$
$314$		0			0
$315$		0			0
$316$	−0.831254	+	1.14412i	−0.831254	+	1.14412i
$317$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$317$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$325$	−0.437016	+	1.34500i	−0.437016	+	1.34500i
$326$		0			0
$327$	−1.14412	−	0.831254i	−1.14412	−	0.831254i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−2.00000			−2.00000			−1.00000			$\pi$
$331$	−2.00000			−2.00000			−1.00000			$\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$	−0.453990	−	0.891007i	−0.453990	−	0.891007i
$337$	0.831254	+	1.14412i	0.831254	+	1.14412i	0.987688	+	0.156434i	$0.0500000\pi$
$337$	0.831254	+	1.14412i	0.831254	+	1.14412i	−0.156434	+	0.987688i	$0.550000\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	0.891007	+	0.453990i	0.891007	+	0.453990i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$347$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$348$		0			0
$349$	1.14412	+	0.831254i	1.14412	+	0.831254i	0.987688	−	0.156434i	$-0.0500000\pi$
$349$	1.14412	+	0.831254i	1.14412	+	0.831254i	0.156434	+	0.987688i	$0.450000\pi$
$350$		0			0
$351$		−	1.41421i		−	1.41421i
$352$		0			0
$353$		0			0			−	1.00000i	$-0.5\pi$
$353$		0			0				1.00000i	$0.5\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$359$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$360$		0			0
$361$	0.309017	+	0.951057i	0.309017	+	0.951057i
$362$		0			0
$363$		0			0
$364$	1.00000	+	1.00000i	1.00000	+	1.00000i
$365$		0			0
$366$		0			0
$367$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$367$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	0.618034	+	1.90211i	0.618034	+	1.90211i
$373$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$373$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−0.618034	+	1.90211i	−0.618034	+	1.90211i	−0.309017	+	0.951057i	$0.600000\pi$
$379$	−0.618034	+	1.90211i	−0.618034	+	1.90211i	−0.309017	+	0.951057i	$0.600000\pi$
$380$		0			0
$381$	−1.14412	+	0.831254i	−1.14412	+	0.831254i
$382$		0			0
$383$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$383$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	1.34500	−	0.437016i	1.34500	−	0.437016i
$388$		0			0
$389$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$389$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$		0			0		1.00000			$0$
$397$		0			0		−1.00000			$\pi$
$398$		0			0
$399$	−1.39680	+	0.221232i	−1.39680	+	0.221232i
$400$	0.809017	−	0.587785i	0.809017	−	0.587785i
$401$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$401$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$402$		0			0
$403$	−1.66251	−	2.28825i	−1.66251	−	2.28825i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	0.437016	+	1.34500i	0.437016	+	1.34500i	0.891007	+	0.453990i	$0.150000\pi$
$409$	0.437016	+	1.34500i	0.437016	+	1.34500i	−0.453990	+	0.891007i	$0.650000\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$			1.41421i			1.41421i
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$421$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	0.642040	−	1.26007i	0.642040	−	1.26007i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$431$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$432$	−0.587785	+	0.809017i	−0.587785	+	0.809017i
$433$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$433$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$434$		0			0
$435$		0			0
$436$	−0.831254	−	1.14412i	−0.831254	−	1.14412i
$437$		0			0
$438$		0			0
$439$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$439$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$440$		0			0
$441$		−	1.00000i		−	1.00000i
$442$		0			0
$443$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$443$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−0.156434	−	0.987688i	−0.156434	−	0.987688i
$449$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$449$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−0.437016	−	1.34500i	−0.437016	−	1.34500i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−1.34500	−	0.437016i	−1.34500	−	0.437016i	−0.453990	−	0.891007i	$-0.650000\pi$
$457$	−1.34500	−	0.437016i	−1.34500	−	0.437016i	−0.891007	+	0.453990i	$0.850000\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$	−2.00000			−2.00000			−1.00000			$\pi$
$463$	−2.00000			−2.00000			−1.00000			$\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$467$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$468$	0.437016	−	1.34500i	0.437016	−	1.34500i
$469$		0			0
$470$		0			0
$471$	−0.618034	−	1.90211i	−0.618034	−	1.90211i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−0.437016	−	1.34500i	−0.437016	−	1.34500i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$479$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$487$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$488$		0			0
$489$	1.90211	+	0.618034i	1.90211	+	0.618034i
$490$		0			0
$491$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$491$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$			2.00000i			2.00000i
$497$		0			0
$498$		0			0
$499$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$499$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$503$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$			1.00000i			1.00000i
$508$	−1.34500	+	0.437016i	−1.34500	+	0.437016i
$509$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$509$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$510$		0			0
$511$	−1.26007	+	0.642040i	−1.26007	+	0.642040i
$512$		0			0
$513$	0.831254	+	1.14412i	0.831254	+	1.14412i
$514$		0			0
$515$		0			0
$516$	1.41421			1.41421
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$521$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$522$		0			0
$523$	0.437016	−	1.34500i	0.437016	−	1.34500i	−0.453990	−	0.891007i	$-0.650000\pi$
$523$	0.437016	−	1.34500i	0.437016	−	1.34500i	0.891007	−	0.453990i	$-0.150000\pi$
$524$		0			0
$525$	0.987688	−	0.156434i	0.987688	−	0.156434i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	1.00000			1.00000
$530$		0			0
$531$		0			0
$532$	−1.39680	−	0.221232i	−1.39680	−	0.221232i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−1.34500	+	0.437016i	−1.34500	+	0.437016i	−0.891007	−	0.453990i	$-0.850000\pi$
$541$	−1.34500	+	0.437016i	−1.34500	+	0.437016i	−0.453990	+	0.891007i	$0.650000\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	0.831254	−	1.14412i	0.831254	−	1.14412i	−0.156434	−	0.987688i	$-0.550000\pi$
$547$	0.831254	−	1.14412i	0.831254	−	1.14412i	0.987688	−	0.156434i	$-0.0500000\pi$
$548$		0			0
$549$	−1.41421			−1.41421
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−1.39680	+	0.221232i	−1.39680	+	0.221232i
$554$		0			0
$555$		0			0
$556$	−0.437016	+	1.34500i	−0.437016	+	1.34500i
$557$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$557$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$558$		0			0
$559$	−1.90211	+	0.618034i	−1.90211	+	0.618034i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$563$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−0.891007	+	0.453990i	−0.891007	+	0.453990i
$568$		0			0
$569$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$569$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$570$		0			0
$571$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$571$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$577$	−1.90211	−	0.618034i	−1.90211	−	0.618034i	−0.951057	−	0.309017i	$-0.900000\pi$
$577$	−1.90211	−	0.618034i	−1.90211	−	0.618034i	−0.951057	−	0.309017i	$-0.900000\pi$
$578$		0			0
$579$	−1.14412	+	0.831254i	−1.14412	+	0.831254i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$587$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$588$	0.309017	−	0.951057i	0.309017	−	0.951057i
$589$	2.68999	+	0.874032i	2.68999	+	0.874032i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0		1.00000			$0$
$593$		0			0		−1.00000			$\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	1.61803	+	1.17557i	1.61803	+	1.17557i
$598$		0			0
$599$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$599$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$600$		0			0
$601$	1.14412	−	0.831254i	1.14412	−	0.831254i	0.156434	−	0.987688i	$-0.450000\pi$
$601$	1.14412	−	0.831254i	1.14412	−	0.831254i	0.987688	+	0.156434i	$0.0500000\pi$
$602$		0			0
$603$		0			0
$604$		−	1.41421i		−	1.41421i
$605$		0			0
$606$		0			0
$607$	−0.437016	−	1.34500i	−0.437016	−	1.34500i	−0.891007	−	0.453990i	$-0.850000\pi$
$607$	−0.437016	−	1.34500i	−0.437016	−	1.34500i	0.453990	−	0.891007i	$-0.350000\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−0.831254	+	1.14412i	−0.831254	+	1.14412i	0.156434	+	0.987688i	$0.450000\pi$
$613$	−0.831254	+	1.14412i	−0.831254	+	1.14412i	−0.987688	+	0.156434i	$0.950000\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		1.00000			$0$
$617$		0			0		−1.00000			$\pi$
$618$		0			0
$619$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$619$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	0.831254	−	1.14412i	0.831254	−	1.14412i
$625$	0.309017	+	0.951057i	0.309017	+	0.951057i
$626$		0			0
$627$		0			0
$628$		−	2.00000i		−	2.00000i
$629$		0			0
$630$		0			0
$631$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$631$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$632$		0			0
$633$	0.437016	−	1.34500i	0.437016	−	1.34500i
$634$		0			0
$635$		0			0
$636$		0			0
$637$			1.41421i			1.41421i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$641$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$642$		0			0
$643$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$643$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$647$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	−0.907981	+	1.78201i	−0.907981	+	1.78201i
$652$	1.61803	+	1.17557i	1.61803	+	1.17557i
$653$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$653$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	1.14412	+	0.831254i	1.14412	+	0.831254i
$658$		0			0
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$661$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	1.34500	−	0.437016i	1.34500	−	0.437016i	0.453990	−	0.891007i	$-0.350000\pi$
$673$	1.34500	−	0.437016i	1.34500	−	0.437016i	0.891007	+	0.453990i	$0.150000\pi$
$674$		0			0
$675$	−0.587785	−	0.809017i	−0.587785	−	0.809017i
$676$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$677$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$677$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		1.00000			$0$
$683$		0			0		−1.00000			$\pi$
$684$	0.437016	+	1.34500i	0.437016	+	1.34500i
$685$		0			0
$686$		0			0
$687$	0.618034	−	1.90211i	0.618034	−	1.90211i
$688$	1.34500	+	0.437016i	1.34500	+	0.437016i
$689$		0			0
$690$		0			0
$691$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$691$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	0.987688	+	0.156434i	0.987688	+	0.156434i
$701$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$701$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$709$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$710$		0			0
$711$	0.831254	+	1.14412i	0.831254	+	1.14412i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$719$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	−0.831254	+	1.14412i	−0.831254	+	1.14412i
$724$		0			0
$725$		0			0
$726$		0			0
$727$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$727$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$728$		0			0
$729$	0.809017	+	0.587785i	0.809017	+	0.587785i
$730$		0			0
$731$		0			0
$732$	−1.34500	−	0.437016i	−1.34500	−	0.437016i
$733$	1.14412	−	0.831254i	1.14412	−	0.831254i	0.156434	−	0.987688i	$-0.450000\pi$
$733$	1.14412	−	0.831254i	1.14412	−	0.831254i	0.987688	+	0.156434i	$0.0500000\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−1.34500	+	0.437016i	−1.34500	+	0.437016i	−0.891007	−	0.453990i	$-0.850000\pi$
$739$	−1.34500	+	0.437016i	−1.34500	+	0.437016i	−0.453990	+	0.891007i	$0.650000\pi$
$740$		0			0
$741$	−1.17557	−	1.61803i	−1.17557	−	1.61803i
$742$		0			0
$743$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$743$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$751$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	−0.987688	+	0.156434i	−0.987688	+	0.156434i
$757$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$757$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$761$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$762$		0			0
$763$	0.221232	−	1.39680i	0.221232	−	1.39680i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	−0.951057	+	0.309017i	−0.951057	+	0.309017i
$769$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$769$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$770$		0			0
$771$		0			0
$772$	−1.34500	+	0.437016i	−1.34500	+	0.437016i
$773$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$773$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$774$		0			0
$775$	−1.90211	−	0.618034i	−1.90211	−	0.618034i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	0.587785	−	0.809017i	0.587785	−	0.809017i
$785$		0			0
$786$		0			0
$787$	0.437016	−	1.34500i	0.437016	−	1.34500i	−0.453990	−	0.891007i	$-0.650000\pi$
$787$	0.437016	−	1.34500i	0.437016	−	1.34500i	0.891007	−	0.453990i	$-0.150000\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	2.00000			2.00000
$794$		0			0
$795$		0			0
$796$	1.17557	+	1.61803i	1.17557	+	1.61803i
$797$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$797$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$809$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$810$		0			0
$811$	−1.14412	−	0.831254i	−1.14412	−	0.831254i	−0.156434	−	0.987688i	$-0.550000\pi$
$811$	−1.14412	−	0.831254i	−1.14412	−	0.831254i	−0.987688	+	0.156434i	$0.950000\pi$
$812$		0			0
$813$		−	1.41421i		−	1.41421i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	1.17557	−	1.61803i	1.17557	−	1.61803i
$818$		0			0
$819$	1.26007	−	0.642040i	1.26007	−	0.642040i
$820$		0			0
$821$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$821$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$822$		0			0
$823$	−0.618034	−	1.90211i	−0.618034	−	1.90211i	−0.309017	−	0.951057i	$-0.600000\pi$
$823$	−0.618034	−	1.90211i	−0.618034	−	1.90211i	−0.309017	−	0.951057i	$-0.600000\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$827$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$828$		0			0
$829$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$829$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$830$		0			0
$831$	0.437016	−	1.34500i	0.437016	−	1.34500i
$832$	1.14412	−	0.831254i	1.14412	−	0.831254i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	2.00000			2.00000
$838$		0			0
$839$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$839$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$840$		0			0
$841$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$842$		0			0
$843$		0			0
$844$	0.831254	−	1.14412i	0.831254	−	1.14412i
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$	1.34500	−	0.437016i	1.34500	−	0.437016i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−0.437016	+	1.34500i	−0.437016	+	1.34500i	0.453990	+	0.891007i	$0.350000\pi$
$853$	−0.437016	+	1.34500i	−0.437016	+	1.34500i	−0.891007	+	0.453990i	$0.850000\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		1.00000			$0$
$857$		0			0		−1.00000			$\pi$
$858$		0			0
$859$		0			0		1.00000			$0$
$859$		0			0		−1.00000			$\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$863$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−0.951057	+	0.309017i	−0.951057	+	0.309017i
$868$	−1.41421	+	1.41421i	−1.41421	+	1.41421i
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$	0.831254	+	1.14412i	0.831254	+	1.14412i
$877$	0.831254	−	1.14412i	0.831254	−	1.14412i	−0.156434	−	0.987688i	$-0.550000\pi$
$877$	0.831254	−	1.14412i	0.831254	−	1.14412i	0.987688	−	0.156434i	$-0.0500000\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0			−	1.00000i	$-0.5\pi$
$881$		0			0				1.00000i	$0.5\pi$
$882$		0			0
$883$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$883$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$887$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$888$		0			0
$889$	−1.26007	−	0.642040i	−1.26007	−	0.642040i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−0.309017	−	0.951057i	−0.309017	−	0.951057i
$901$		0			0
$902$		0			0
$903$	1.00000	+	1.00000i	1.00000	+	1.00000i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	0.618034	−	1.90211i	0.618034	−	1.90211i	0.309017	−	0.951057i	$-0.400000\pi$
$907$	0.618034	−	1.90211i	0.618034	−	1.90211i	0.309017	−	0.951057i	$-0.400000\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$911$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$912$			1.41421i			1.41421i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	1.17557	−	1.61803i	1.17557	−	1.61803i
$917$		0			0
$918$		0			0
$919$	1.34500	+	0.437016i	1.34500	+	0.437016i	0.891007	−	0.453990i	$-0.150000\pi$
$919$	1.34500	+	0.437016i	1.34500	+	0.437016i	0.453990	+	0.891007i	$0.350000\pi$
$920$		0			0
$921$	0.831254	−	1.14412i	0.831254	−	1.14412i
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$929$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$930$		0			0
$931$	−0.831254	−	1.14412i	−0.831254	−	1.14412i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	0.437016	+	1.34500i	0.437016	+	1.34500i	0.891007	+	0.453990i	$0.150000\pi$
$937$	0.437016	+	1.34500i	0.437016	+	1.34500i	−0.453990	+	0.891007i	$0.650000\pi$
$938$		0			0
$939$	1.61803	−	1.17557i	1.61803	−	1.17557i
$940$		0			0
$941$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$941$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		1.00000			$0$
$947$		0			0		−1.00000			$\pi$
$948$	0.437016	+	1.34500i	0.437016	+	1.34500i
$949$	−1.61803	−	1.17557i	−1.61803	−	1.17557i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$953$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	2.42705	−	1.76336i	2.42705	−	1.76336i
$962$		0			0
$963$		0			0
$964$	−1.14412	+	0.831254i	−1.14412	+	0.831254i
$965$		0			0
$966$		0			0
$967$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$967$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$971$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$972$	0.587785	+	0.809017i	0.587785	+	0.809017i
$973$	−1.26007	+	0.642040i	−1.26007	+	0.642040i
$974$		0			0
$975$	0.831254	+	1.14412i	0.831254	+	1.14412i
$976$	−1.14412	−	0.831254i	−1.14412	−	0.831254i
$977$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$977$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−1.34500	+	0.437016i	−1.34500	+	0.437016i
$982$		0			0
$983$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$983$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	−0.618034	−	1.90211i	−0.618034	−	1.90211i
$989$		0			0
$990$		0			0
$991$		0			0			−	1.00000i	$-0.5\pi$
$991$		0			0				1.00000i	$0.5\pi$
$992$		0			0
$993$	−1.17557	+	1.61803i	−1.17557	+	1.61803i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	1.14412	−	0.831254i	1.14412	−	0.831254i	0.156434	−	0.987688i	$-0.450000\pi$
$997$	1.14412	−	0.831254i	1.14412	−	0.831254i	0.987688	+	0.156434i	$0.0500000\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2541.1.r.e.2393.4		16
3.2	odd	2	CM	2541.1.r.e.2393.4		16
7.6	odd	2	inner	2541.1.r.e.2393.1		16
11.2	odd	10	inner	2541.1.r.e.1322.1		16
11.3	even	5		2541.1.h.a.2540.3	yes	4
11.4	even	5	inner	2541.1.r.e.965.1		16
11.5	even	5	inner	2541.1.r.e.524.4		16
11.6	odd	10	inner	2541.1.r.e.524.3		16
11.7	odd	10	inner	2541.1.r.e.965.2		16
11.8	odd	10		2541.1.h.a.2540.4	yes	4
11.9	even	5	inner	2541.1.r.e.1322.2		16
11.10	odd	2	inner	2541.1.r.e.2393.3		16
21.20	even	2	inner	2541.1.r.e.2393.1		16
33.2	even	10	inner	2541.1.r.e.1322.1		16
33.5	odd	10	inner	2541.1.r.e.524.4		16
33.8	even	10		2541.1.h.a.2540.4	yes	4
33.14	odd	10		2541.1.h.a.2540.3	yes	4
33.17	even	10	inner	2541.1.r.e.524.3		16
33.20	odd	10	inner	2541.1.r.e.1322.2		16
33.26	odd	10	inner	2541.1.r.e.965.1		16
33.29	even	10	inner	2541.1.r.e.965.2		16
33.32	even	2	inner	2541.1.r.e.2393.3		16
77.6	even	10	inner	2541.1.r.e.524.1		16
77.13	even	10	inner	2541.1.r.e.1322.4		16
77.20	odd	10	inner	2541.1.r.e.1322.3		16
77.27	odd	10	inner	2541.1.r.e.524.2		16
77.41	even	10		2541.1.h.a.2540.1	✓	4
77.48	odd	10	inner	2541.1.r.e.965.3		16
77.62	even	10	inner	2541.1.r.e.965.4		16
77.69	odd	10		2541.1.h.a.2540.2	yes	4
77.76	even	2	inner	2541.1.r.e.2393.2		16
231.20	even	10	inner	2541.1.r.e.1322.3		16
231.41	odd	10		2541.1.h.a.2540.1	✓	4
231.62	odd	10	inner	2541.1.r.e.965.4		16
231.83	odd	10	inner	2541.1.r.e.524.1		16
231.104	even	10	inner	2541.1.r.e.524.2		16
231.125	even	10	inner	2541.1.r.e.965.3		16
231.146	even	10		2541.1.h.a.2540.2	yes	4
231.167	odd	10	inner	2541.1.r.e.1322.4		16
231.230	odd	2	inner	2541.1.r.e.2393.2		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2541.1.h.a.2540.1	✓	4	77.41	even	10
2541.1.h.a.2540.1	✓	4	231.41	odd	10
2541.1.h.a.2540.2	yes	4	77.69	odd	10
2541.1.h.a.2540.2	yes	4	231.146	even	10
2541.1.h.a.2540.3	yes	4	11.3	even	5
2541.1.h.a.2540.3	yes	4	33.14	odd	10
2541.1.h.a.2540.4	yes	4	11.8	odd	10
2541.1.h.a.2540.4	yes	4	33.8	even	10
2541.1.r.e.524.1		16	77.6	even	10	inner
2541.1.r.e.524.1		16	231.83	odd	10	inner
2541.1.r.e.524.2		16	77.27	odd	10	inner
2541.1.r.e.524.2		16	231.104	even	10	inner
2541.1.r.e.524.3		16	11.6	odd	10	inner
2541.1.r.e.524.3		16	33.17	even	10	inner
2541.1.r.e.524.4		16	11.5	even	5	inner
2541.1.r.e.524.4		16	33.5	odd	10	inner
2541.1.r.e.965.1		16	11.4	even	5	inner
2541.1.r.e.965.1		16	33.26	odd	10	inner
2541.1.r.e.965.2		16	11.7	odd	10	inner
2541.1.r.e.965.2		16	33.29	even	10	inner
2541.1.r.e.965.3		16	77.48	odd	10	inner
2541.1.r.e.965.3		16	231.125	even	10	inner
2541.1.r.e.965.4		16	77.62	even	10	inner
2541.1.r.e.965.4		16	231.62	odd	10	inner
2541.1.r.e.1322.1		16	11.2	odd	10	inner
2541.1.r.e.1322.1		16	33.2	even	10	inner
2541.1.r.e.1322.2		16	11.9	even	5	inner
2541.1.r.e.1322.2		16	33.20	odd	10	inner
2541.1.r.e.1322.3		16	77.20	odd	10	inner
2541.1.r.e.1322.3		16	231.20	even	10	inner
2541.1.r.e.1322.4		16	77.13	even	10	inner
2541.1.r.e.1322.4		16	231.167	odd	10	inner
2541.1.r.e.2393.1		16	7.6	odd	2	inner
2541.1.r.e.2393.1		16	21.20	even	2	inner
2541.1.r.e.2393.2		16	77.76	even	2	inner
2541.1.r.e.2393.2		16	231.230	odd	2	inner
2541.1.r.e.2393.3		16	11.10	odd	2	inner
2541.1.r.e.2393.3		16	33.32	even	2	inner
2541.1.r.e.2393.4		16	1.1	even	1	trivial
2541.1.r.e.2393.4		16	3.2	odd	2	CM

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2541 = 3 \cdot 7 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2541.r (of order \(10\), degree \(4\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.587785 - 0.809017i) q^{3} +(0.809017 - 0.587785i) q^{4} +(0.987688 + 0.156434i) q^{7} +(-0.309017 - 0.951057i) q^{9} -1.00000i q^{12} +(0.437016 + 1.34500i) q^{13} +(0.309017 - 0.951057i) q^{16} +(-1.14412 - 0.831254i) q^{19} +(0.707107 - 0.707107i) q^{21} +(0.809017 + 0.587785i) q^{25} +(-0.951057 - 0.309017i) q^{27} +(0.891007 - 0.453990i) q^{28} +(-1.90211 + 0.618034i) q^{31} +(-0.809017 - 0.587785i) q^{36} +(1.34500 + 0.437016i) q^{39} +1.41421i q^{43} +(-0.587785 - 0.809017i) q^{48} +(0.951057 + 0.309017i) q^{49} +(1.14412 + 0.831254i) q^{52} +(-1.34500 + 0.437016i) q^{57} +(0.437016 - 1.34500i) q^{61} +(-0.156434 - 0.987688i) q^{63} +(-0.309017 - 0.951057i) q^{64} +(-1.14412 + 0.831254i) q^{73} +(0.951057 - 0.309017i) q^{75} -1.41421 q^{76} +(-1.34500 + 0.437016i) q^{79} +(-0.809017 + 0.587785i) q^{81} +(0.156434 - 0.987688i) q^{84} +(0.221232 + 1.39680i) q^{91} +(-0.618034 + 1.90211i) q^{93} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 4 q^{4} + 4 q^{9} - 4 q^{16} + 4 q^{25} - 4 q^{36} + 4 q^{64} - 4 q^{81} + 4 q^{91} + 8 q^{93}+O(q^{100}) \) 16 * q + 4 * q^4 + 4 * q^9 - 4 * q^16 + 4 * q^25 - 4 * q^36 + 4 * q^64 - 4 * q^81 + 4 * q^91 + 8 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more