LMFDB - Newform orbit 507.2.k.j

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [507,2,Mod(80,507)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(507, base_ring=CyclotomicField(12))

chi = DirichletCharacter(H, H._module([6, 1]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("507.80");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 507 = 3 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	507.k (of order $12$, degree $4$, 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:	$4.04841538248$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$2$ over $\Q(\zeta_{12})$
Coefficient field:	$\Q(\zeta_{24})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 39)
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

Coefficients of the $q$-expansion are expressed in terms of a primitive root of unity $\zeta_{24}$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + \zeta_{24}^{7} q^{2} + (\zeta_{24}^{7} + \zeta_{24}^{4} - \zeta_{24}) q^{3} + \zeta_{24}^{2} q^{4} + 2 \zeta_{24}^{3} q^{5} + (\zeta_{24}^{7} - \zeta_{24}^{4} + \cdots + 1) q^{6}+ \cdots + (2 \zeta_{24}^{7} - 2 \zeta_{24}^{5} + \cdots + 1) q^{9}+O(q^{10}) $ q + z^7 * q^2 + (z^7 + z^4 - z) * q^3 + z^2 * q^4 + 2z^3 q^5 + (z^7 - z^4 - z^3 - z^2 + 1) * q^6 + (z^4 - z^2 - 1) * q^7 + (3z^5 - 3z) * q^8 + (2z^7 - 2z^5 - z^4 - 2z^3 + 1) q^9	$ q + \zeta_{24}^{7} q^{2} + (\zeta_{24}^{7} + \zeta_{24}^{4} - \zeta_{24}) q^{3} + \zeta_{24}^{2} q^{4} + 2 \zeta_{24}^{3} q^{5} + (\zeta_{24}^{7} - \zeta_{24}^{4} + \cdots + 1) q^{6}+ \cdots + (8 \zeta_{24}^{6} + 4 \zeta_{24}^{5} + \cdots + 8) q^{99}+O(q^{100}) $ q + z^7 * q^2 + (z^7 + z^4 - z) * q^3 + z^2 * q^4 + 2z^3 q^5 + (z^7 - z^4 - z^3 - z^2 + 1) * q^6 + (z^4 - z^2 - 1) * q^7 + (3z^5 - 3z) * q^8 + (2z^7 - 2z^5 - z^4 - 2z^3 + 1) q^9 + (2z^6 - 2z^2) * q^10 - 4z q^11 + (z^6 + z^5 - z^3 - z) * q^12 + (-z^5 - z^3 + z) * q^14 + (2z^7 + 2z^6 - 2z^4 - 2z^2) * q^15 - z^4 * q^16 + (-2z^6 + z^3 + 2) q^18 + (z^4 + z^2 - 1) * q^19 + 2z^5 q^20 + (-z^6 - 2z^5 + 2z - 1) * q^21 + (-4z^4 + 4) q^22 + (6z^7 + 6z) * q^23 + (-3z^6 - 3z^4 + 3z^2 - 3z) * q^24 - z^6 * q^25 + (-z^5 - z^3 + z + 5) * q^27 + (z^6 - z^4 - z^2) * q^28 + (-2z^7 + 2z) * q^29 + (-2z^7 - 2z^5 + 2z^3 - 2z^2) * q^30 + (5z^6 - 5) q^31 + (5z^7 - 5z^3) * q^32 + (-4z^5 - 4z^4 + 4z^2 + 4) q^33 + (2z^7 - 2z^5 - 2z^3) q^35 + (-2z^7 - z^6 + z^2 - 2z) * q^36 + (-z^6 - z^4 + z^2) * q^37 + (z^5 - z^3 - z) * q^38 - 6 * q^40 - 2z^7 q^41 + (-z^7 + 2z^4 + z) q^42 + 6z^2 q^43 - 4z^3 q^44 + (-2z^7 - 4z^4 + 2z^3 - 4z^2 + 4) * q^45 + (6z^4 - 6z^2 - 6) * q^46 + (-4z^5 + 4z) * q^47 + (-z^7 + z^5 - z^4 + z^3 + 1) * q^48 + (5z^6 - 5z^2) * q^49 + z * q^50 + (-4z^5 - 4z^3 + 4z) q^53 + (5z^7 - z^6 + z^4 + z^2) q^54 - 8z^4 q^55 + (-3z^7 - 3z^5 + 3z^3) q^56 + (z^6 - 2z^3 - 1) q^57 + (2z^4 + 2z^2 - 2) * q^58 + 4z^5 q^59 + (-2z^6 + 2z^5 - 2z - 2) q^60 + (8z^4 - 8) q^61 + (-5z^7 - 5z) * q^62 + (z^6 + z^4 - z^2 + 4z) q^63 - 7z^6 q^64 + (4z^5 + 4z^3 - 4z + 4) q^66 + (-5z^6 + 5z^4 + 5z^2) q^67 + (6z^7 + 6z^5 - 6z^3 - 12z^2) * q^69 + (-2z^6 + 2) q^70 + (-4z^7 + 4z^3) * q^71 + (3z^5 - 6z^4 + 6z^2 + 6) q^72 + (z^6 + 1) * q^73 + (-z^7 + z^5 + z^3) * q^74 + (z^7 - z^6 + z^2 + z) * q^75 + (z^6 + z^4 - z^2) * q^76 + (-4z^5 + 4z^3 + 4z) q^77 - 10 * q^79 - 2z^7 q^80 + (4z^7 + 7z^4 - 4z) q^81 + 2z^2 q^82 + 8z^3 q^83 + (-2z^7 - z^4 + 2z^3 - z^2 + 1) * q^84 + (6z^5 - 6z) * q^86 + (-2z^7 + 2z^5 + 4z^4 + 2z^3 - 4) * q^87 + (-12z^6 + 12z^2) * q^88 + 14z q^89 + (2z^6 - 4z^5 + 4z^3 + 4z) * q^90 + (6z^5 + 6z^3 - 6z) q^92 + (-10z^7 + 5z^6 - 5z^4 - 5z^2) * q^93 + 4z^4 q^94 + (2z^7 + 2z^5 - 2z^3) q^95 + (-5z^6 - 5z^3 + 5) * q^96 + (7z^4 + 7z^2 - 7) * q^97 - 5z^5 q^98 + (8z^6 + 4z^5 - 4z + 8) q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 4 q^{3} + 4 q^{6} - 4 q^{7} + 4 q^{9}+O(q^{10}) $ 8 * q + 4 * q^3 + 4 * q^6 - 4 * q^7 + 4 * q^9	$ 8 q + 4 q^{3} + 4 q^{6} - 4 q^{7} + 4 q^{9} - 8 q^{15} - 4 q^{16} + 16 q^{18} - 4 q^{19} - 8 q^{21} + 16 q^{22} - 12 q^{24} + 40 q^{27} - 4 q^{28} - 40 q^{31} + 16 q^{33} - 4 q^{37} - 48 q^{40} + 8 q^{42} + 16 q^{45} - 24 q^{46} + 4 q^{48} + 4 q^{54} - 32 q^{55} - 8 q^{57} - 8 q^{58} - 16 q^{60} - 32 q^{61} + 4 q^{63} + 32 q^{66} + 20 q^{67} + 16 q^{70} + 24 q^{72} + 8 q^{73} + 4 q^{76} - 80 q^{79} + 28 q^{81} + 4 q^{84} - 16 q^{87} - 20 q^{93} + 16 q^{94} + 40 q^{96} - 28 q^{97} + 64 q^{99}+O(q^{100}) $ 8 * q + 4 * q^3 + 4 * q^6 - 4 * q^7 + 4 * q^9 - 8 * q^15 - 4 * q^16 + 16 * q^18 - 4 * q^19 - 8 * q^21 + 16 * q^22 - 12 * q^24 + 40 * q^27 - 4 * q^28 - 40 * q^31 + 16 * q^33 - 4 * q^37 - 48 * q^40 + 8 * q^42 + 16 * q^45 - 24 * q^46 + 4 * q^48 + 4 * q^54 - 32 * q^55 - 8 * q^57 - 8 * q^58 - 16 * q^60 - 32 * q^61 + 4 * q^63 + 32 * q^66 + 20 * q^67 + 16 * q^70 + 24 * q^72 + 8 * q^73 + 4 * q^76 - 80 * q^79 + 28 * q^81 + 4 * q^84 - 16 * q^87 - 20 * q^93 + 16 * q^94 + 40 * q^96 - 28 * q^97 + 64 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/507\mathbb{Z}\right)^\times$.

$n$	$170$	$340$
$\chi(n)$	$-1$	$\zeta_{24}^{2}$

Embeddings

For each embedding $\iota_m$ of the coefficient field, the values $\iota_m(a_n)$ are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label

$\iota_m(\nu)$

$ a_{2} $

$ a_{3} $

$ a_{4} $

$ a_{5} $

$ a_{6} $

$ a_{7} $

$ a_{8} $

$ a_{9} $

$ a_{10} $

80.1

0.965926	+	0.258819i
−0.965926	−	0.258819i
0.258819	−	0.965926i
−0.258819	+	0.965926i
0.258819	+	0.965926i
−0.258819	−	0.965926i
0.965926	−	0.258819i
−0.965926	+	0.258819i

−0.258819

0.965926i

−0.724745

1.57313i

0.866025

0.500000i

1.41421

1.41421i

−1.33195

−

1.10721i

−1.36603

0.366025i

−2.12132

2.12132i

−1.94949

−

2.28024i

−1.73205

1.00000i

80.2

0.258819

−

0.965926i

1.72474

0.158919i

0.866025

0.500000i

−1.41421

−

1.41421i

0.599900

−

1.62484i

−1.36603

0.366025i

2.12132

−

2.12132i

2.94949

0.548188i

−1.73205

1.00000i

89.1

−0.965926

−

0.258819i

−0.724745

1.57313i

−0.866025

−

0.500000i

−1.41421

1.41421i

1.10721

−

1.33195i

0.366025

1.36603i

2.12132

2.12132i

−1.94949

−

2.28024i

1.73205

−

1.00000i

89.2

0.965926

0.258819i

1.72474

0.158919i

−0.866025

−

0.500000i

1.41421

−

1.41421i

1.62484

0.599900i

0.366025

1.36603i

−2.12132

−

2.12132i

2.94949

0.548188i

1.73205

−

1.00000i

188.1

−0.965926

0.258819i

−0.724745

−

1.57313i

−0.866025

0.500000i

−1.41421

−

1.41421i

1.10721

1.33195i

0.366025

−

1.36603i

2.12132

−

2.12132i

−1.94949

2.28024i

1.73205

1.00000i

188.2

0.965926

−

0.258819i

1.72474

−

0.158919i

−0.866025

0.500000i

1.41421

1.41421i

1.62484

−

0.599900i

0.366025

−

1.36603i

−2.12132

2.12132i

2.94949

−

0.548188i

1.73205

1.00000i

488.1

−0.258819

−

0.965926i

−0.724745

−

1.57313i

0.866025

−

0.500000i

1.41421

−

1.41421i

−1.33195

1.10721i

−1.36603

−

0.366025i

−2.12132

−

2.12132i

−1.94949

2.28024i

−1.73205

−

1.00000i

488.2

0.258819

0.965926i

1.72474

−

0.158919i

0.866025

−

0.500000i

−1.41421

1.41421i

0.599900

1.62484i

−1.36603

−

0.366025i

2.12132

2.12132i

2.94949

−

0.548188i

−1.73205

−

1.00000i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
13.c	even	3	1	inner
13.d	odd	4	1	inner
13.f	odd	12	1	inner
39.f	even	4	1	inner
39.i	odd	6	1	inner
39.k	even	12	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	507.2.k.j		8
3.b	odd	2	1	inner	507.2.k.j		8
13.b	even	2	1		507.2.k.i		8
13.c	even	3	1		39.2.f.a	✓	4
13.c	even	3	1	inner	507.2.k.j		8
13.d	odd	4	1		507.2.k.i		8
13.d	odd	4	1	inner	507.2.k.j		8
13.e	even	6	1		507.2.f.a		4
13.e	even	6	1		507.2.k.i		8
13.f	odd	12	1		39.2.f.a	✓	4
13.f	odd	12	1		507.2.f.a		4
13.f	odd	12	1		507.2.k.i		8
13.f	odd	12	1	inner	507.2.k.j		8
39.d	odd	2	1		507.2.k.i		8
39.f	even	4	1		507.2.k.i		8
39.f	even	4	1	inner	507.2.k.j		8
39.h	odd	6	1		507.2.f.a		4
39.h	odd	6	1		507.2.k.i		8
39.i	odd	6	1		39.2.f.a	✓	4
39.i	odd	6	1	inner	507.2.k.j		8
39.k	even	12	1		39.2.f.a	✓	4
39.k	even	12	1		507.2.f.a		4
39.k	even	12	1		507.2.k.i		8
39.k	even	12	1	inner	507.2.k.j		8
52.j	odd	6	1		624.2.bf.d		4
52.l	even	12	1		624.2.bf.d		4
65.n	even	6	1		975.2.o.j		4
65.o	even	12	1		975.2.n.c		4
65.q	odd	12	1		975.2.n.c		4
65.q	odd	12	1		975.2.n.d		4
65.s	odd	12	1		975.2.o.j		4
65.t	even	12	1		975.2.n.d		4
156.p	even	6	1		624.2.bf.d		4
156.v	odd	12	1		624.2.bf.d		4
195.x	odd	6	1		975.2.o.j		4
195.bc	odd	12	1		975.2.n.d		4
195.bh	even	12	1		975.2.o.j		4
195.bl	even	12	1		975.2.n.c		4
195.bl	even	12	1		975.2.n.d		4
195.bn	odd	12	1		975.2.n.c		4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
39.2.f.a	✓	4	13.c	even	3	1
39.2.f.a	✓	4	13.f	odd	12	1
39.2.f.a	✓	4	39.i	odd	6	1
39.2.f.a	✓	4	39.k	even	12	1
507.2.f.a		4	13.e	even	6	1
507.2.f.a		4	13.f	odd	12	1
507.2.f.a		4	39.h	odd	6	1
507.2.f.a		4	39.k	even	12	1
507.2.k.i		8	13.b	even	2	1
507.2.k.i		8	13.d	odd	4	1
507.2.k.i		8	13.e	even	6	1
507.2.k.i		8	13.f	odd	12	1
507.2.k.i		8	39.d	odd	2	1
507.2.k.i		8	39.f	even	4	1
507.2.k.i		8	39.h	odd	6	1
507.2.k.i		8	39.k	even	12	1
507.2.k.j		8	1.a	even	1	1	trivial
507.2.k.j		8	3.b	odd	2	1	inner
507.2.k.j		8	13.c	even	3	1	inner
507.2.k.j		8	13.d	odd	4	1	inner
507.2.k.j		8	13.f	odd	12	1	inner
507.2.k.j		8	39.f	even	4	1	inner
507.2.k.j		8	39.i	odd	6	1	inner
507.2.k.j		8	39.k	even	12	1	inner
624.2.bf.d		4	52.j	odd	6	1
624.2.bf.d		4	52.l	even	12	1
624.2.bf.d		4	156.p	even	6	1
624.2.bf.d		4	156.v	odd	12	1
975.2.n.c		4	65.o	even	12	1
975.2.n.c		4	65.q	odd	12	1
975.2.n.c		4	195.bl	even	12	1
975.2.n.c		4	195.bn	odd	12	1
975.2.n.d		4	65.q	odd	12	1
975.2.n.d		4	65.t	even	12	1
975.2.n.d		4	195.bc	odd	12	1
975.2.n.d		4	195.bl	even	12	1
975.2.o.j		4	65.n	even	6	1
975.2.o.j		4	65.s	odd	12	1
975.2.o.j		4	195.x	odd	6	1
975.2.o.j		4	195.bh	even	12	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(507, [\chi])$:

$ T_{2}^{8} - T_{2}^{4} + 1 $

$ T_{5}^{4} + 16 $

$ T_{7}^{4} + 2T_{7}^{3} + 2T_{7}^{2} + 4T_{7} + 4 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{8} - T^{4} + 1 $ T^8 - T^4 + 1
$3$	$ (T^{4} - 2 T^{3} + T^{2} + \cdots + 9)^{2} $ (T^4 - 2T^3 + T^2 - 6T + 9)^2
$5$	$ (T^{4} + 16)^{2} $ (T^4 + 16)^2
$7$	$ (T^{4} + 2 T^{3} + 2 T^{2} + \cdots + 4)^{2} $ (T^4 + 2T^3 + 2T^2 + 4*T + 4)^2
$11$	$ T^{8} - 256 T^{4} + 65536 $ T^8 - 256*T^4 + 65536
$13$	$ T^{8} $ T^8
$17$	$ T^{8} $ T^8
$19$	$ (T^{4} + 2 T^{3} + 2 T^{2} + \cdots + 4)^{2} $ (T^4 + 2T^3 + 2T^2 + 4*T + 4)^2
$23$	$ (T^{4} + 72 T^{2} + 5184)^{2} $ (T^4 + 72*T^2 + 5184)^2
$29$	$ (T^{4} - 8 T^{2} + 64)^{2} $ (T^4 - 8*T^2 + 64)^2
$31$	$ (T^{2} + 10 T + 50)^{4} $ (T^2 + 10*T + 50)^4
$37$	$ (T^{4} + 2 T^{3} + 2 T^{2} + \cdots + 4)^{2} $ (T^4 + 2T^3 + 2T^2 + 4*T + 4)^2
$41$	$ T^{8} - 16T^{4} + 256 $ T^8 - 16*T^4 + 256
$43$	$ (T^{4} - 36 T^{2} + 1296)^{2} $ (T^4 - 36*T^2 + 1296)^2
$47$	$ (T^{4} + 256)^{2} $ (T^4 + 256)^2
$53$	$ (T^{2} + 32)^{4} $ (T^2 + 32)^4
$59$	$ T^{8} - 256 T^{4} + 65536 $ T^8 - 256*T^4 + 65536
$61$	$ (T^{2} + 8 T + 64)^{4} $ (T^2 + 8*T + 64)^4
$67$	$ (T^{4} - 10 T^{3} + \cdots + 2500)^{2} $ (T^4 - 10T^3 + 50T^2 - 500*T + 2500)^2
$71$	$ T^{8} - 256 T^{4} + 65536 $ T^8 - 256*T^4 + 65536
$73$	$ (T^{2} - 2 T + 2)^{4} $ (T^2 - 2*T + 2)^4
$79$	$ (T + 10)^{8} $ (T + 10)^8
$83$	$ (T^{4} + 4096)^{2} $ (T^4 + 4096)^2
$89$	$ T^{8} + \cdots + 1475789056 $ T^8 - 38416*T^4 + 1475789056
$97$	$ (T^{4} + 14 T^{3} + \cdots + 9604)^{2} $ (T^4 + 14T^3 + 98T^2 + 1372*T + 9604)^2
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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:	\( N \)	\(=\)	\( 507 = 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	507.k (of order \(12\), degree \(4\), not minimal)

\(f(q)\)	\(=\)	\( q + \zeta_{24}^{7} q^{2} + (\zeta_{24}^{7} + \zeta_{24}^{4} - \zeta_{24}) q^{3} + \zeta_{24}^{2} q^{4} + 2 \zeta_{24}^{3} q^{5} + (\zeta_{24}^{7} - \zeta_{24}^{4} + \cdots + 1) q^{6}+ \cdots + (2 \zeta_{24}^{7} - 2 \zeta_{24}^{5} + \cdots + 1) q^{9}+O(q^{10}) \) q + z^7 * q^2 + (z^7 + z^4 - z) * q^3 + z^2 * q^4 + 2z^3 q^5 + (z^7 - z^4 - z^3 - z^2 + 1) * q^6 + (z^4 - z^2 - 1) * q^7 + (3z^5 - 3z) * q^8 + (2z^7 - 2z^5 - z^4 - 2z^3 + 1) q^9	\( q + \zeta_{24}^{7} q^{2} + (\zeta_{24}^{7} + \zeta_{24}^{4} - \zeta_{24}) q^{3} + \zeta_{24}^{2} q^{4} + 2 \zeta_{24}^{3} q^{5} + (\zeta_{24}^{7} - \zeta_{24}^{4} + \cdots + 1) q^{6}+ \cdots + (8 \zeta_{24}^{6} + 4 \zeta_{24}^{5} + \cdots + 8) q^{99}+O(q^{100}) \) q + z^7 * q^2 + (z^7 + z^4 - z) * q^3 + z^2 * q^4 + 2z^3 q^5 + (z^7 - z^4 - z^3 - z^2 + 1) * q^6 + (z^4 - z^2 - 1) * q^7 + (3z^5 - 3z) * q^8 + (2z^7 - 2z^5 - z^4 - 2z^3 + 1) q^9 + (2z^6 - 2z^2) * q^10 - 4z q^11 + (z^6 + z^5 - z^3 - z) * q^12 + (-z^5 - z^3 + z) * q^14 + (2z^7 + 2z^6 - 2z^4 - 2z^2) * q^15 - z^4 * q^16 + (-2z^6 + z^3 + 2) q^18 + (z^4 + z^2 - 1) * q^19 + 2z^5 q^20 + (-z^6 - 2z^5 + 2z - 1) * q^21 + (-4z^4 + 4) q^22 + (6z^7 + 6z) * q^23 + (-3z^6 - 3z^4 + 3z^2 - 3z) * q^24 - z^6 * q^25 + (-z^5 - z^3 + z + 5) * q^27 + (z^6 - z^4 - z^2) * q^28 + (-2z^7 + 2z) * q^29 + (-2z^7 - 2z^5 + 2z^3 - 2z^2) * q^30 + (5z^6 - 5) q^31 + (5z^7 - 5z^3) * q^32 + (-4z^5 - 4z^4 + 4z^2 + 4) q^33 + (2z^7 - 2z^5 - 2z^3) q^35 + (-2z^7 - z^6 + z^2 - 2z) * q^36 + (-z^6 - z^4 + z^2) * q^37 + (z^5 - z^3 - z) * q^38 - 6 * q^40 - 2z^7 q^41 + (-z^7 + 2z^4 + z) q^42 + 6z^2 q^43 - 4z^3 q^44 + (-2z^7 - 4z^4 + 2z^3 - 4z^2 + 4) * q^45 + (6z^4 - 6z^2 - 6) * q^46 + (-4z^5 + 4z) * q^47 + (-z^7 + z^5 - z^4 + z^3 + 1) * q^48 + (5z^6 - 5z^2) * q^49 + z * q^50 + (-4z^5 - 4z^3 + 4z) q^53 + (5z^7 - z^6 + z^4 + z^2) q^54 - 8z^4 q^55 + (-3z^7 - 3z^5 + 3z^3) q^56 + (z^6 - 2z^3 - 1) q^57 + (2z^4 + 2z^2 - 2) * q^58 + 4z^5 q^59 + (-2z^6 + 2z^5 - 2z - 2) q^60 + (8z^4 - 8) q^61 + (-5z^7 - 5z) * q^62 + (z^6 + z^4 - z^2 + 4z) q^63 - 7z^6 q^64 + (4z^5 + 4z^3 - 4z + 4) q^66 + (-5z^6 + 5z^4 + 5z^2) q^67 + (6z^7 + 6z^5 - 6z^3 - 12z^2) * q^69 + (-2z^6 + 2) q^70 + (-4z^7 + 4z^3) * q^71 + (3z^5 - 6z^4 + 6z^2 + 6) q^72 + (z^6 + 1) * q^73 + (-z^7 + z^5 + z^3) * q^74 + (z^7 - z^6 + z^2 + z) * q^75 + (z^6 + z^4 - z^2) * q^76 + (-4z^5 + 4z^3 + 4z) q^77 - 10 * q^79 - 2z^7 q^80 + (4z^7 + 7z^4 - 4z) q^81 + 2z^2 q^82 + 8z^3 q^83 + (-2z^7 - z^4 + 2z^3 - z^2 + 1) * q^84 + (6z^5 - 6z) * q^86 + (-2z^7 + 2z^5 + 4z^4 + 2z^3 - 4) * q^87 + (-12z^6 + 12z^2) * q^88 + 14z q^89 + (2z^6 - 4z^5 + 4z^3 + 4z) * q^90 + (6z^5 + 6z^3 - 6z) q^92 + (-10z^7 + 5z^6 - 5z^4 - 5z^2) * q^93 + 4z^4 q^94 + (2z^7 + 2z^5 - 2z^3) q^95 + (-5z^6 - 5z^3 + 5) * q^96 + (7z^4 + 7z^2 - 7) * q^97 - 5z^5 q^98 + (8z^6 + 4z^5 - 4z + 8) q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{3} + 4 q^{6} - 4 q^{7} + 4 q^{9}+O(q^{10}) \) 8 * q + 4 * q^3 + 4 * q^6 - 4 * q^7 + 4 * q^9	\( 8 q + 4 q^{3} + 4 q^{6} - 4 q^{7} + 4 q^{9} - 8 q^{15} - 4 q^{16} + 16 q^{18} - 4 q^{19} - 8 q^{21} + 16 q^{22} - 12 q^{24} + 40 q^{27} - 4 q^{28} - 40 q^{31} + 16 q^{33} - 4 q^{37} - 48 q^{40} + 8 q^{42} + 16 q^{45} - 24 q^{46} + 4 q^{48} + 4 q^{54} - 32 q^{55} - 8 q^{57} - 8 q^{58} - 16 q^{60} - 32 q^{61} + 4 q^{63} + 32 q^{66} + 20 q^{67} + 16 q^{70} + 24 q^{72} + 8 q^{73} + 4 q^{76} - 80 q^{79} + 28 q^{81} + 4 q^{84} - 16 q^{87} - 20 q^{93} + 16 q^{94} + 40 q^{96} - 28 q^{97} + 64 q^{99}+O(q^{100}) \) 8 * q + 4 * q^3 + 4 * q^6 - 4 * q^7 + 4 * q^9 - 8 * q^15 - 4 * q^16 + 16 * q^18 - 4 * q^19 - 8 * q^21 + 16 * q^22 - 12 * q^24 + 40 * q^27 - 4 * q^28 - 40 * q^31 + 16 * q^33 - 4 * q^37 - 48 * q^40 + 8 * q^42 + 16 * q^45 - 24 * q^46 + 4 * q^48 + 4 * q^54 - 32 * q^55 - 8 * q^57 - 8 * q^58 - 16 * q^60 - 32 * q^61 + 4 * q^63 + 32 * q^66 + 20 * q^67 + 16 * q^70 + 24 * q^72 + 8 * q^73 + 4 * q^76 - 80 * q^79 + 28 * q^81 + 4 * q^84 - 16 * q^87 - 20 * q^93 + 16 * q^94 + 40 * q^96 - 28 * q^97 + 64 * q^99

Introduction

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Newspace parameters

Newform invariants

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Hecke characteristic polynomials

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	507.2.k.j

Level	$507$
Weight	$2$
Character orbit	507.k
Analytic conductor	$4.048$
Analytic rank	$0$
Dimension	$8$
CM	no
Inner twists	$8$

Self dual:	no
Analytic conductor:	\(4.04841538248\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Relative dimension:	\(2\) over \(\Q(\zeta_{12})\)
Coefficient field:	\(\Q(\zeta_{24})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{8} - x^{4} + 1 \) x^8 - x^4 + 1
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 39)
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 80.2
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{8} - T^{4} + 1 \) T^8 - T^4 + 1
$3$	\( (T^{4} - 2 T^{3} + T^{2} + \cdots + 9)^{2} \) (T^4 - 2T^3 + T^2 - 6T + 9)^2
$5$	\( (T^{4} + 16)^{2} \) (T^4 + 16)^2
$7$	\( (T^{4} + 2 T^{3} + 2 T^{2} + \cdots + 4)^{2} \) (T^4 + 2T^3 + 2T^2 + 4*T + 4)^2
$11$	\( T^{8} - 256 T^{4} + 65536 \) T^8 - 256*T^4 + 65536
$13$	\( T^{8} \) T^8
$17$	\( T^{8} \) T^8
$19$	\( (T^{4} + 2 T^{3} + 2 T^{2} + \cdots + 4)^{2} \) (T^4 + 2T^3 + 2T^2 + 4*T + 4)^2
$23$	\( (T^{4} + 72 T^{2} + 5184)^{2} \) (T^4 + 72*T^2 + 5184)^2
$29$	\( (T^{4} - 8 T^{2} + 64)^{2} \) (T^4 - 8*T^2 + 64)^2
$31$	\( (T^{2} + 10 T + 50)^{4} \) (T^2 + 10*T + 50)^4
$37$	\( (T^{4} + 2 T^{3} + 2 T^{2} + \cdots + 4)^{2} \) (T^4 + 2T^3 + 2T^2 + 4*T + 4)^2
$41$	\( T^{8} - 16T^{4} + 256 \) T^8 - 16*T^4 + 256
$43$	\( (T^{4} - 36 T^{2} + 1296)^{2} \) (T^4 - 36*T^2 + 1296)^2
$47$	\( (T^{4} + 256)^{2} \) (T^4 + 256)^2
$53$	\( (T^{2} + 32)^{4} \) (T^2 + 32)^4
$59$	\( T^{8} - 256 T^{4} + 65536 \) T^8 - 256*T^4 + 65536
$61$	\( (T^{2} + 8 T + 64)^{4} \) (T^2 + 8*T + 64)^4
$67$	\( (T^{4} - 10 T^{3} + \cdots + 2500)^{2} \) (T^4 - 10T^3 + 50T^2 - 500*T + 2500)^2
$71$	\( T^{8} - 256 T^{4} + 65536 \) T^8 - 256*T^4 + 65536
$73$	\( (T^{2} - 2 T + 2)^{4} \) (T^2 - 2*T + 2)^4
$79$	\( (T + 10)^{8} \) (T + 10)^8
$83$	\( (T^{4} + 4096)^{2} \) (T^4 + 4096)^2
$89$	\( T^{8} + \cdots + 1475789056 \) T^8 - 38416*T^4 + 1475789056
$97$	\( (T^{4} + 14 T^{3} + \cdots + 9604)^{2} \) (T^4 + 14T^3 + 98T^2 + 1372*T + 9604)^2
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\(n\)	\(170\)	\(340\)
\(\chi(n)\)	\(-1\)	\(\zeta_{24}^{2}\)