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Properties

Label 504.2.j.a
Level $504$
Weight $2$
Character orbit 504.j
Analytic conductor $4.024$
Analytic rank $0$
Dimension $24$
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(323,504)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(504, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("504.323");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 504 = 2^{3} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 504.j (of order \(2\), degree \(1\), 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: \(24\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 8 q^{4}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 8 q^{4} + 24 q^{10} + 12 q^{16} + 32 q^{19} + 12 q^{22} + 24 q^{25} + 4 q^{28} - 8 q^{40} - 64 q^{43} - 12 q^{46} - 24 q^{49} - 16 q^{52} - 12 q^{58} + 16 q^{64} + 16 q^{67} + 24 q^{70} + 8 q^{76} + 24 q^{82} - 84 q^{88} - 72 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
323.1 −1.40979 − 0.111820i 0 1.97499 + 0.315285i −0.472391 0 1.00000i −2.74906 − 0.665328i 0 0.665969 + 0.0528227i
323.2 −1.40979 + 0.111820i 0 1.97499 − 0.315285i −0.472391 0 − 1.00000i −2.74906 + 0.665328i 0 0.665969 − 0.0528227i
323.3 −1.25516 − 0.651592i 0 1.15086 + 1.63571i −1.82464 0 1.00000i −0.378695 − 2.80296i 0 2.29021 + 1.18892i
323.4 −1.25516 + 0.651592i 0 1.15086 − 1.63571i −1.82464 0 − 1.00000i −0.378695 + 2.80296i 0 2.29021 − 1.18892i
323.5 −0.937511 − 1.05881i 0 −0.242146 + 1.98529i −4.34020 0 − 1.00000i 2.32905 − 1.60484i 0 4.06898 + 4.59543i
323.6 −0.937511 + 1.05881i 0 −0.242146 − 1.98529i −4.34020 0 1.00000i 2.32905 + 1.60484i 0 4.06898 − 4.59543i
323.7 −0.619645 − 1.27124i 0 −1.23208 + 1.57543i −0.753720 0 − 1.00000i 2.76619 + 0.590057i 0 0.467039 + 0.958156i
323.8 −0.619645 + 1.27124i 0 −1.23208 − 1.57543i −0.753720 0 1.00000i 2.76619 − 0.590057i 0 0.467039 − 0.958156i
323.9 −0.386593 − 1.36035i 0 −1.70109 + 1.05180i 3.11999 0 1.00000i 2.08845 + 1.90746i 0 −1.20617 − 4.24428i
323.10 −0.386593 + 1.36035i 0 −1.70109 − 1.05180i 3.11999 0 − 1.00000i 2.08845 − 1.90746i 0 −1.20617 + 4.24428i
323.11 −0.157273 − 1.40544i 0 −1.95053 + 0.442076i 1.81873 0 − 1.00000i 0.928077 + 2.67183i 0 −0.286037 − 2.55612i
323.12 −0.157273 + 1.40544i 0 −1.95053 − 0.442076i 1.81873 0 1.00000i 0.928077 − 2.67183i 0 −0.286037 + 2.55612i
323.13 0.157273 − 1.40544i 0 −1.95053 − 0.442076i −1.81873 0 1.00000i −0.928077 + 2.67183i 0 −0.286037 + 2.55612i
323.14 0.157273 + 1.40544i 0 −1.95053 + 0.442076i −1.81873 0 − 1.00000i −0.928077 − 2.67183i 0 −0.286037 − 2.55612i
323.15 0.386593 − 1.36035i 0 −1.70109 − 1.05180i −3.11999 0 − 1.00000i −2.08845 + 1.90746i 0 −1.20617 + 4.24428i
323.16 0.386593 + 1.36035i 0 −1.70109 + 1.05180i −3.11999 0 1.00000i −2.08845 − 1.90746i 0 −1.20617 − 4.24428i
323.17 0.619645 − 1.27124i 0 −1.23208 − 1.57543i 0.753720 0 1.00000i −2.76619 + 0.590057i 0 0.467039 − 0.958156i
323.18 0.619645 + 1.27124i 0 −1.23208 + 1.57543i 0.753720 0 − 1.00000i −2.76619 − 0.590057i 0 0.467039 + 0.958156i
323.19 0.937511 − 1.05881i 0 −0.242146 − 1.98529i 4.34020 0 1.00000i −2.32905 − 1.60484i 0 4.06898 − 4.59543i
323.20 0.937511 + 1.05881i 0 −0.242146 + 1.98529i 4.34020 0 − 1.00000i −2.32905 + 1.60484i 0 4.06898 + 4.59543i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 323.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
8.d odd 2 1 inner
24.f even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 504.2.j.a ✓ 24
3.b odd 2 1 inner 504.2.j.a ✓ 24
4.b odd 2 1 2016.2.j.a 24
8.b even 2 1 2016.2.j.a 24
8.d odd 2 1 inner 504.2.j.a ✓ 24
12.b even 2 1 2016.2.j.a 24
24.f even 2 1 inner 504.2.j.a ✓ 24
24.h odd 2 1 2016.2.j.a 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.2.j.a ✓ 24 1.a even 1 1 trivial
504.2.j.a ✓ 24 3.b odd 2 1 inner
504.2.j.a ✓ 24 8.d odd 2 1 inner
504.2.j.a ✓ 24 24.f even 2 1 inner
2016.2.j.a 24 4.b odd 2 1
2016.2.j.a 24 8.b even 2 1
2016.2.j.a 24 12.b even 2 1
2016.2.j.a 24 24.h odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(504, [\chi])\).