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Properties

Label 504.2.bm.b
Level $504$
Weight $2$
Character orbit 504.bm
Analytic conductor $4.024$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [504,2,Mod(107,504)]
 
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([3, 3, 3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("504.107");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 504 = 2^{3} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 504.bm (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: \( 2^{2} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + ( - \beta_{2} - \beta_1 + 1) q^{2} + (2 \beta_{3} - 2 \beta_1) q^{4} + (2 \beta_{7} - 2 \beta_{4} + \beta_{2} - 1) q^{5} + (2 \beta_{6} - \beta_{5} + \cdots - \beta_1) q^{7}+ \cdots + (2 \beta_{3} + 2) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} - \beta_1 + 1) q^{2} + (2 \beta_{3} - 2 \beta_1) q^{4} + (2 \beta_{7} - 2 \beta_{4} + \beta_{2} - 1) q^{5} + (2 \beta_{6} - \beta_{5} + \cdots - \beta_1) q^{7}+ \cdots + ( - 4 \beta_{7} + 4 \beta_{6} + \cdots + 5 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{2} - 4 q^{5} + 16 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{2} - 4 q^{5} + 16 q^{8} + 4 q^{10} + 8 q^{14} + 16 q^{16} + 16 q^{19} - 24 q^{22} + 8 q^{23} - 16 q^{25} - 16 q^{26} + 8 q^{28} + 24 q^{29} - 16 q^{32} - 16 q^{34} - 16 q^{38} - 8 q^{40} + 80 q^{43} - 24 q^{44} - 8 q^{46} - 16 q^{47} - 20 q^{49} - 32 q^{50} + 32 q^{52} - 36 q^{53} - 8 q^{56} + 12 q^{58} + 24 q^{62} + 8 q^{67} - 16 q^{68} - 52 q^{70} - 32 q^{71} - 16 q^{73} + 8 q^{74} - 36 q^{77} + 16 q^{80} - 24 q^{82} + 40 q^{86} + 24 q^{88} - 8 q^{91} + 16 q^{94} + 32 q^{95} - 104 q^{97} + 20 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring

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

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

\(\zeta_{24}\)\(=\) \( ( \beta_{5} + \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{2}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{24}^{3}\)\(=\) \( ( \beta_{7} + \beta_{6} - \beta_{5} ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{4}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\zeta_{24}^{5}\)\(=\) \( ( -\beta_{7} + \beta_{6} + \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{6}\)\(=\) \( \beta_{3} \) Copy content Toggle raw display
\(\zeta_{24}^{7}\)\(=\) \( ( -\beta_{5} + \beta_{4} ) / 2 \) 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 \(\left(\mathbb{Z}/504\mathbb{Z}\right)^\times\).

\(n\) \(73\) \(127\) \(253\) \(281\)
\(\chi(n)\) \(-1 + \beta_{2}\) \(-1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
107.1
−0.965926 − 0.258819i
0.965926 + 0.258819i
0.258819 − 0.965926i
−0.258819 + 0.965926i
−0.965926 + 0.258819i
0.965926 − 0.258819i
0.258819 + 0.965926i
−0.258819 − 0.965926i
−0.366025 − 1.36603i 0 −1.73205 + 1.00000i −1.91421 + 3.31552i 0 −2.09077 − 1.62132i 2.00000 + 2.00000i 0 5.22973 + 1.40130i
107.2 −0.366025 − 1.36603i 0 −1.73205 + 1.00000i 0.914214 − 1.58346i 0 0.358719 + 2.62132i 2.00000 + 2.00000i 0 −2.49768 − 0.669251i
107.3 1.36603 − 0.366025i 0 1.73205 − 1.00000i −1.91421 + 3.31552i 0 2.09077 + 1.62132i 2.00000 − 2.00000i 0 −1.40130 + 5.22973i
107.4 1.36603 − 0.366025i 0 1.73205 − 1.00000i 0.914214 − 1.58346i 0 −0.358719 − 2.62132i 2.00000 − 2.00000i 0 0.669251 − 2.49768i
179.1 −0.366025 + 1.36603i 0 −1.73205 − 1.00000i −1.91421 − 3.31552i 0 −2.09077 + 1.62132i 2.00000 − 2.00000i 0 5.22973 − 1.40130i
179.2 −0.366025 + 1.36603i 0 −1.73205 − 1.00000i 0.914214 + 1.58346i 0 0.358719 − 2.62132i 2.00000 − 2.00000i 0 −2.49768 + 0.669251i
179.3 1.36603 + 0.366025i 0 1.73205 + 1.00000i −1.91421 − 3.31552i 0 2.09077 − 1.62132i 2.00000 + 2.00000i 0 −1.40130 − 5.22973i
179.4 1.36603 + 0.366025i 0 1.73205 + 1.00000i 0.914214 + 1.58346i 0 −0.358719 + 2.62132i 2.00000 + 2.00000i 0 0.669251 + 2.49768i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 107.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
24.f even 2 1 inner
168.v even 6 1 inner

Twists

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 2T_{5}^{3} + 11T_{5}^{2} - 14T_{5} + 49 \) acting on \(S_{2}^{\mathrm{new}}(504, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 2 T^{3} + 2 T^{2} + \cdots + 4)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T^{4} + 2 T^{3} + 11 T^{2} + \cdots + 49)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} + 10 T^{6} + \cdots + 2401 \) Copy content Toggle raw display
$11$ \( T^{8} - 22 T^{6} + \cdots + 2401 \) Copy content Toggle raw display
$13$ \( (T^{4} + 36 T^{2} + 196)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} - 44 T^{6} + \cdots + 38416 \) Copy content Toggle raw display
$19$ \( (T^{4} - 8 T^{3} + \cdots + 196)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} - 4 T^{3} + 14 T^{2} + \cdots + 4)^{2} \) Copy content Toggle raw display
$29$ \( (T - 3)^{8} \) Copy content Toggle raw display
$31$ \( T^{8} - 22 T^{6} + \cdots + 2401 \) Copy content Toggle raw display
$37$ \( T^{8} - 108 T^{6} + \cdots + 4477456 \) Copy content Toggle raw display
$41$ \( (T^{4} + 88 T^{2} + 784)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 20 T + 98)^{4} \) Copy content Toggle raw display
$47$ \( (T^{4} + 8 T^{3} + \cdots + 196)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 18 T^{3} + \cdots + 5329)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} - 102 T^{6} + \cdots + 5764801 \) Copy content Toggle raw display
$61$ \( T^{8} - 44 T^{6} + \cdots + 38416 \) Copy content Toggle raw display
$67$ \( (T^{4} - 4 T^{3} + \cdots + 784)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 8 T - 82)^{4} \) Copy content Toggle raw display
$73$ \( (T^{4} + 8 T^{3} + \cdots + 12544)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} - 102 T^{6} + \cdots + 5764801 \) Copy content Toggle raw display
$83$ \( (T^{4} + 102 T^{2} + 2401)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} \) Copy content Toggle raw display
$97$ \( (T^{2} + 26 T + 161)^{4} \) Copy content Toggle raw display
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