Properties

Label 507.2.j.f
Level $507$
Weight $2$
Character orbit 507.j
Analytic conductor $4.048$
Analytic rank $0$
Dimension $8$
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [507,2,Mod(316,507)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(507, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("507.316");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 507 = 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 507.j (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: \(4.04841538248\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 39)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} - \beta_1 q^{3} + (\beta_{6} - \beta_{4} - \beta_1 + 1) q^{4} + (\beta_{7} - \beta_{5} + \cdots - \beta_{2}) q^{5}+ \cdots + (\beta_1 - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{2} - \beta_1 q^{3} + (\beta_{6} - \beta_{4} - \beta_1 + 1) q^{4} + (\beta_{7} - \beta_{5} + \cdots - \beta_{2}) q^{5}+ \cdots + ( - \beta_{7} + \beta_{5} + \cdots - \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{3} + 4 q^{4} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{3} + 4 q^{4} - 4 q^{9} - 16 q^{10} - 8 q^{12} + 32 q^{14} - 12 q^{16} + 8 q^{17} - 8 q^{22} - 16 q^{23} - 24 q^{25} + 8 q^{27} - 8 q^{29} - 16 q^{30} + 32 q^{35} + 4 q^{36} + 32 q^{38} - 32 q^{40} - 16 q^{42} + 16 q^{43} - 12 q^{48} + 4 q^{49} - 16 q^{51} - 16 q^{53} + 16 q^{56} - 8 q^{61} + 32 q^{62} + 56 q^{64} + 16 q^{66} + 56 q^{68} - 16 q^{69} + 24 q^{74} + 12 q^{75} - 4 q^{81} - 48 q^{82} - 8 q^{87} + 24 q^{88} + 32 q^{90} - 32 q^{92} + 8 q^{94} + 32 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{24}^{4} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -\zeta_{24}^{7} + \zeta_{24}^{5} + \zeta_{24}^{3} - \zeta_{24}^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{24}^{7} + \zeta_{24}^{6} - \zeta_{24}^{2} + \zeta_{24} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -2\zeta_{24}^{5} + 2\zeta_{24}^{3} + 2\zeta_{24} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -\zeta_{24}^{7} + \zeta_{24}^{5} + \zeta_{24}^{3} + \zeta_{24}^{2} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( 2\zeta_{24}^{7} + 2\zeta_{24} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -\zeta_{24}^{7} - \zeta_{24}^{6} + \zeta_{24}^{2} + \zeta_{24} \) Copy content Toggle raw display
\(\zeta_{24}\)\(=\) \( ( \beta_{7} + \beta_{6} + \beta_{3} ) / 4 \) Copy content Toggle raw display
\(\zeta_{24}^{2}\)\(=\) \( ( \beta_{5} - \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{3}\)\(=\) \( ( -\beta_{7} + \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} ) / 4 \) Copy content Toggle raw display
\(\zeta_{24}^{4}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{24}^{5}\)\(=\) \( ( \beta_{6} + \beta_{5} - \beta_{4} + \beta_{2} ) / 4 \) Copy content Toggle raw display
\(\zeta_{24}^{6}\)\(=\) \( ( -\beta_{7} + \beta_{5} + \beta_{3} - \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{7}\)\(=\) \( ( -\beta_{7} + \beta_{6} - \beta_{3} ) / 4 \) Copy content Toggle raw display

Character values

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

\(n\) \(170\) \(340\)
\(\chi(n)\) \(1\) \(1 - \beta_{1}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
316.1
−0.965926 + 0.258819i
−0.258819 0.965926i
0.965926 0.258819i
0.258819 + 0.965926i
−0.965926 0.258819i
−0.258819 + 0.965926i
0.965926 + 0.258819i
0.258819 0.965926i
−2.09077 1.20711i −0.500000 + 0.866025i 1.91421 + 3.31552i 2.82843i 2.09077 1.20711i −2.44949 + 1.41421i 4.41421i −0.500000 0.866025i −3.41421 + 5.91359i
316.2 −0.358719 0.207107i −0.500000 + 0.866025i −0.914214 1.58346i 2.82843i 0.358719 0.207107i −2.44949 + 1.41421i 1.58579i −0.500000 0.866025i −0.585786 + 1.01461i
316.3 0.358719 + 0.207107i −0.500000 + 0.866025i −0.914214 1.58346i 2.82843i −0.358719 + 0.207107i 2.44949 1.41421i 1.58579i −0.500000 0.866025i −0.585786 + 1.01461i
316.4 2.09077 + 1.20711i −0.500000 + 0.866025i 1.91421 + 3.31552i 2.82843i −2.09077 + 1.20711i 2.44949 1.41421i 4.41421i −0.500000 0.866025i −3.41421 + 5.91359i
361.1 −2.09077 + 1.20711i −0.500000 0.866025i 1.91421 3.31552i 2.82843i 2.09077 + 1.20711i −2.44949 1.41421i 4.41421i −0.500000 + 0.866025i −3.41421 5.91359i
361.2 −0.358719 + 0.207107i −0.500000 0.866025i −0.914214 + 1.58346i 2.82843i 0.358719 + 0.207107i −2.44949 1.41421i 1.58579i −0.500000 + 0.866025i −0.585786 1.01461i
361.3 0.358719 0.207107i −0.500000 0.866025i −0.914214 + 1.58346i 2.82843i −0.358719 0.207107i 2.44949 + 1.41421i 1.58579i −0.500000 + 0.866025i −0.585786 1.01461i
361.4 2.09077 1.20711i −0.500000 0.866025i 1.91421 3.31552i 2.82843i −2.09077 1.20711i 2.44949 + 1.41421i 4.41421i −0.500000 + 0.866025i −3.41421 5.91359i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 316.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner
13.c even 3 1 inner
13.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 507.2.j.f 8
13.b even 2 1 inner 507.2.j.f 8
13.c even 3 1 507.2.b.e 4
13.c even 3 1 inner 507.2.j.f 8
13.d odd 4 1 507.2.e.d 4
13.d odd 4 1 507.2.e.h 4
13.e even 6 1 507.2.b.e 4
13.e even 6 1 inner 507.2.j.f 8
13.f odd 12 1 39.2.a.b 2
13.f odd 12 1 507.2.a.h 2
13.f odd 12 1 507.2.e.d 4
13.f odd 12 1 507.2.e.h 4
39.h odd 6 1 1521.2.b.j 4
39.i odd 6 1 1521.2.b.j 4
39.k even 12 1 117.2.a.c 2
39.k even 12 1 1521.2.a.f 2
52.l even 12 1 624.2.a.k 2
52.l even 12 1 8112.2.a.bm 2
65.o even 12 1 975.2.c.h 4
65.s odd 12 1 975.2.a.l 2
65.t even 12 1 975.2.c.h 4
91.bc even 12 1 1911.2.a.h 2
104.u even 12 1 2496.2.a.bi 2
104.x odd 12 1 2496.2.a.bf 2
117.w odd 12 1 1053.2.e.m 4
117.x even 12 1 1053.2.e.e 4
117.bb odd 12 1 1053.2.e.m 4
117.bc even 12 1 1053.2.e.e 4
143.o even 12 1 4719.2.a.p 2
156.v odd 12 1 1872.2.a.w 2
195.bc odd 12 1 2925.2.c.u 4
195.bh even 12 1 2925.2.a.v 2
195.bn odd 12 1 2925.2.c.u 4
273.ca odd 12 1 5733.2.a.u 2
312.bo even 12 1 7488.2.a.cl 2
312.bq odd 12 1 7488.2.a.co 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.a.b 2 13.f odd 12 1
117.2.a.c 2 39.k even 12 1
507.2.a.h 2 13.f odd 12 1
507.2.b.e 4 13.c even 3 1
507.2.b.e 4 13.e even 6 1
507.2.e.d 4 13.d odd 4 1
507.2.e.d 4 13.f odd 12 1
507.2.e.h 4 13.d odd 4 1
507.2.e.h 4 13.f odd 12 1
507.2.j.f 8 1.a even 1 1 trivial
507.2.j.f 8 13.b even 2 1 inner
507.2.j.f 8 13.c even 3 1 inner
507.2.j.f 8 13.e even 6 1 inner
624.2.a.k 2 52.l even 12 1
975.2.a.l 2 65.s odd 12 1
975.2.c.h 4 65.o even 12 1
975.2.c.h 4 65.t even 12 1
1053.2.e.e 4 117.x even 12 1
1053.2.e.e 4 117.bc even 12 1
1053.2.e.m 4 117.w odd 12 1
1053.2.e.m 4 117.bb odd 12 1
1521.2.a.f 2 39.k even 12 1
1521.2.b.j 4 39.h odd 6 1
1521.2.b.j 4 39.i odd 6 1
1872.2.a.w 2 156.v odd 12 1
1911.2.a.h 2 91.bc even 12 1
2496.2.a.bf 2 104.x odd 12 1
2496.2.a.bi 2 104.u even 12 1
2925.2.a.v 2 195.bh even 12 1
2925.2.c.u 4 195.bc odd 12 1
2925.2.c.u 4 195.bn odd 12 1
4719.2.a.p 2 143.o even 12 1
5733.2.a.u 2 273.ca odd 12 1
7488.2.a.cl 2 312.bo even 12 1
7488.2.a.co 2 312.bq odd 12 1
8112.2.a.bm 2 52.l even 12 1

Hecke kernels

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

\( T_{2}^{8} - 6T_{2}^{6} + 35T_{2}^{4} - 6T_{2}^{2} + 1 \) Copy content Toggle raw display
\( T_{5}^{2} + 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 6 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( (T^{2} + T + 1)^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} + 8)^{4} \) Copy content Toggle raw display
$7$ \( (T^{4} - 8 T^{2} + 64)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} - 4 T^{2} + 16)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} \) Copy content Toggle raw display
$17$ \( (T^{4} - 4 T^{3} + \cdots + 784)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} - 8 T^{2} + 64)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 4 T + 16)^{4} \) Copy content Toggle raw display
$29$ \( (T^{2} + 2 T + 4)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} + 48 T^{2} + 64)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} - 72 T^{6} + \cdots + 614656 \) Copy content Toggle raw display
$41$ \( T^{8} - 144 T^{6} + \cdots + 9834496 \) Copy content Toggle raw display
$43$ \( (T^{4} - 8 T^{3} + \cdots + 256)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 136 T^{2} + 16)^{2} \) Copy content Toggle raw display
$53$ \( (T + 2)^{8} \) Copy content Toggle raw display
$59$ \( T^{8} - 72 T^{6} + \cdots + 614656 \) Copy content Toggle raw display
$61$ \( (T^{4} + 4 T^{3} + \cdots + 15376)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} - 48 T^{6} + \cdots + 4096 \) Copy content Toggle raw display
$71$ \( (T^{4} - 4 T^{2} + 16)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 136 T^{2} + 16)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 128)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} + 72 T^{2} + 784)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} - 304 T^{6} + \cdots + 342102016 \) Copy content Toggle raw display
$97$ \( T^{8} - 72 T^{6} + \cdots + 614656 \) Copy content Toggle raw display
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