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Properties

Label 504.2.cj.b
Level $504$
Weight $2$
Character orbit 504.cj
Analytic conductor $4.024$
Analytic rank $0$
Dimension $8$
CM discriminant -24
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [504,2,Mod(37,504)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(504, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("504.37");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 504 = 2^{3} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 504.cj (of order \(6\), degree \(2\), 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: \(4.02446026187\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{5} q^{2} + 2 \beta_1 q^{4} + ( - \beta_{5} - \beta_{4} - \beta_{2}) q^{5} + (\beta_{7} - \beta_{6} - \beta_1) q^{7} + (2 \beta_{5} - 2 \beta_{3}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{5} q^{2} + 2 \beta_1 q^{4} + ( - \beta_{5} - \beta_{4} - \beta_{2}) q^{5} + (\beta_{7} - \beta_{6} - \beta_1) q^{7} + (2 \beta_{5} - 2 \beta_{3}) q^{8} + ( - \beta_{7} - \beta_{6} - 2 \beta_1) q^{10} + ( - 2 \beta_{4} - 2 \beta_{3} + \beta_{2}) q^{11} + ( - \beta_{5} - 2 \beta_{4} + \beta_{3}) q^{14} + (4 \beta_1 - 4) q^{16} + ( - 2 \beta_{5} + 2 \beta_{4} + \cdots - 4 \beta_{2}) q^{20}+ \cdots + (5 \beta_{3} + 4 \beta_{2}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{4} - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{4} - 4 q^{7} - 8 q^{10} - 16 q^{16} - 32 q^{22} + 24 q^{25} + 8 q^{28} - 20 q^{31} + 16 q^{40} + 20 q^{49} + 104 q^{55} - 8 q^{58} - 64 q^{64} + 64 q^{70} - 56 q^{73} - 20 q^{79} - 32 q^{88} - 8 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring

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

Display \(\zeta_{24}^j\) in terms of \(\beta_i\)

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

Display \(\beta_i\) in terms of \(\zeta_{24}^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/504\mathbb{Z}\right)^\times\).

\(n\) \(73\) \(127\) \(253\) \(281\)
\(\chi(n)\) \(-1 + \beta_{1}\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1
−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
−1.22474 − 0.707107i 0 1.00000 + 1.73205i −1.37333 − 0.792893i 0 1.62132 − 2.09077i − 2.82843i 0 1.12132 + 1.94218i
37.2 −1.22474 − 0.707107i 0 1.00000 + 1.73205i 3.82282 + 2.20711i 0 −2.62132 + 0.358719i − 2.82843i 0 −3.12132 − 5.40629i
37.3 1.22474 + 0.707107i 0 1.00000 + 1.73205i −3.82282 − 2.20711i 0 −2.62132 + 0.358719i 2.82843i 0 −3.12132 − 5.40629i
37.4 1.22474 + 0.707107i 0 1.00000 + 1.73205i 1.37333 + 0.792893i 0 1.62132 − 2.09077i 2.82843i 0 1.12132 + 1.94218i
109.1 −1.22474 + 0.707107i 0 1.00000 − 1.73205i −1.37333 + 0.792893i 0 1.62132 + 2.09077i 2.82843i 0 1.12132 − 1.94218i
109.2 −1.22474 + 0.707107i 0 1.00000 − 1.73205i 3.82282 − 2.20711i 0 −2.62132 − 0.358719i 2.82843i 0 −3.12132 + 5.40629i
109.3 1.22474 − 0.707107i 0 1.00000 − 1.73205i −3.82282 + 2.20711i 0 −2.62132 − 0.358719i − 2.82843i 0 −3.12132 + 5.40629i
109.4 1.22474 − 0.707107i 0 1.00000 − 1.73205i 1.37333 − 0.792893i 0 1.62132 + 2.09077i − 2.82843i 0 1.12132 − 1.94218i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 37.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
24.h odd 2 1 CM by \(\Q(\sqrt{-6}) \)
3.b odd 2 1 inner
7.c even 3 1 inner
8.b even 2 1 inner
21.h odd 6 1 inner
56.p even 6 1 inner
168.s odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 504.2.cj.b ✓ 8
3.b odd 2 1 inner 504.2.cj.b ✓ 8
4.b odd 2 1 2016.2.cr.b 8
7.c even 3 1 inner 504.2.cj.b ✓ 8
8.b even 2 1 inner 504.2.cj.b ✓ 8
8.d odd 2 1 2016.2.cr.b 8
12.b even 2 1 2016.2.cr.b 8
21.h odd 6 1 inner 504.2.cj.b ✓ 8
24.f even 2 1 2016.2.cr.b 8
24.h odd 2 1 CM 504.2.cj.b ✓ 8
28.g odd 6 1 2016.2.cr.b 8
56.k odd 6 1 2016.2.cr.b 8
56.p even 6 1 inner 504.2.cj.b ✓ 8
84.n even 6 1 2016.2.cr.b 8
168.s odd 6 1 inner 504.2.cj.b ✓ 8
168.v even 6 1 2016.2.cr.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.2.cj.b ✓ 8 1.a even 1 1 trivial
504.2.cj.b ✓ 8 3.b odd 2 1 inner
504.2.cj.b ✓ 8 7.c even 3 1 inner
504.2.cj.b ✓ 8 8.b even 2 1 inner
504.2.cj.b ✓ 8 21.h odd 6 1 inner
504.2.cj.b ✓ 8 24.h odd 2 1 CM
504.2.cj.b ✓ 8 56.p even 6 1 inner
504.2.cj.b ✓ 8 168.s odd 6 1 inner
2016.2.cr.b 8 4.b odd 2 1
2016.2.cr.b 8 8.d odd 2 1
2016.2.cr.b 8 12.b even 2 1
2016.2.cr.b 8 24.f even 2 1
2016.2.cr.b 8 28.g odd 6 1
2016.2.cr.b 8 56.k odd 6 1
2016.2.cr.b 8 84.n even 6 1
2016.2.cr.b 8 168.v even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} - 22T_{5}^{6} + 435T_{5}^{4} - 1078T_{5}^{2} + 2401 \) acting on \(S_{2}^{\mathrm{new}}(504, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 2 T^{2} + 4)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} - 22 T^{6} + \cdots + 2401 \) Copy content Toggle raw display
$7$ \( (T^{4} + 2 T^{3} - 3 T^{2} + \cdots + 49)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} - 34 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$13$ \( T^{8} \) Copy content Toggle raw display
$17$ \( T^{8} \) Copy content Toggle raw display
$19$ \( T^{8} \) Copy content Toggle raw display
$23$ \( T^{8} \) Copy content Toggle raw display
$29$ \( (T^{4} + 166 T^{2} + 6241)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 10 T^{3} + \cdots + 49)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} \) Copy content Toggle raw display
$41$ \( T^{8} \) Copy content Toggle raw display
$43$ \( T^{8} \) Copy content Toggle raw display
$47$ \( T^{8} \) Copy content Toggle raw display
$53$ \( T^{8} - 118 T^{6} + \cdots + 2825761 \) Copy content Toggle raw display
$59$ \( T^{8} - 226 T^{6} + \cdots + 5764801 \) Copy content Toggle raw display
$61$ \( T^{8} \) Copy content Toggle raw display
$67$ \( T^{8} \) Copy content Toggle raw display
$71$ \( T^{8} \) Copy content Toggle raw display
$73$ \( (T^{2} + 14 T + 196)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} + 10 T^{3} + \cdots + 18769)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 466 T^{2} + 47089)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} \) Copy content Toggle raw display
$97$ \( (T^{2} + 2 T - 287)^{4} \) Copy content Toggle raw display
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