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Properties

Label 2394.2.bo.a
Level $2394$
Weight $2$
Character orbit 2394.bo
Analytic conductor $19.116$
Analytic rank $0$
Dimension $96$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2394,2,Mod(125,2394)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2394, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 3, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2394.125");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2394 = 2 \cdot 3^{2} \cdot 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2394.bo (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: \(19.1161862439\)
Analytic rank: \(0\)
Dimension: \(96\)
Relative dimension: \(48\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 96 q + 48 q^{4} - 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 96 q + 48 q^{4} - 8 q^{7} - 48 q^{16} - 8 q^{22} - 8 q^{25} - 4 q^{28} - 96 q^{37} + 40 q^{43} - 32 q^{46} + 56 q^{49} - 32 q^{58} - 96 q^{64} - 24 q^{67} - 16 q^{70} - 40 q^{79} - 8 q^{85} - 16 q^{88} + 44 q^{91}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
125.1 −0.866025 − 0.500000i 0 0.500000 + 0.866025i 1.73151 − 2.99907i 0 −2.40266 − 1.10779i − 1.00000i 0 −2.99907 + 1.73151i
125.2 −0.866025 − 0.500000i 0 0.500000 + 0.866025i 0.897263 − 1.55410i 0 −0.493381 − 2.59934i − 1.00000i 0 −1.55410 + 0.897263i
125.3 −0.866025 − 0.500000i 0 0.500000 + 0.866025i −0.703885 + 1.21916i 0 −2.63516 − 0.236500i − 1.00000i 0 1.21916 − 0.703885i
125.4 −0.866025 − 0.500000i 0 0.500000 + 0.866025i −1.80468 + 3.12580i 0 −1.27697 − 2.31719i − 1.00000i 0 3.12580 − 1.80468i
125.5 −0.866025 − 0.500000i 0 0.500000 + 0.866025i −1.21807 + 2.10976i 0 2.48091 − 0.919296i − 1.00000i 0 2.10976 − 1.21807i
125.6 −0.866025 − 0.500000i 0 0.500000 + 0.866025i 0.336071 − 0.582092i 0 −1.02688 + 2.43834i − 1.00000i 0 −0.582092 + 0.336071i
125.7 −0.866025 − 0.500000i 0 0.500000 + 0.866025i 0.917549 − 1.58924i 0 2.51695 + 0.815460i − 1.00000i 0 −1.58924 + 0.917549i
125.8 −0.866025 − 0.500000i 0 0.500000 + 0.866025i 0.0197535 − 0.0342140i 0 1.14723 − 2.38409i − 1.00000i 0 −0.0342140 + 0.0197535i
125.9 −0.866025 − 0.500000i 0 0.500000 + 0.866025i 1.55422 − 2.69199i 0 2.39869 − 1.11636i − 1.00000i 0 −2.69199 + 1.55422i
125.10 −0.866025 − 0.500000i 0 0.500000 + 0.866025i −0.0197535 + 0.0342140i 0 1.14723 + 2.38409i − 1.00000i 0 0.0342140 − 0.0197535i
125.11 −0.866025 − 0.500000i 0 0.500000 + 0.866025i −0.336071 + 0.582092i 0 −1.02688 − 2.43834i − 1.00000i 0 0.582092 − 0.336071i
125.12 −0.866025 − 0.500000i 0 0.500000 + 0.866025i −0.917549 + 1.58924i 0 2.51695 − 0.815460i − 1.00000i 0 1.58924 − 0.917549i
125.13 −0.866025 − 0.500000i 0 0.500000 + 0.866025i 0.539273 − 0.934048i 0 1.87538 − 1.86626i − 1.00000i 0 −0.934048 + 0.539273i
125.14 −0.866025 − 0.500000i 0 0.500000 + 0.866025i −0.539273 + 0.934048i 0 1.87538 + 1.86626i − 1.00000i 0 0.934048 − 0.539273i
125.15 −0.866025 − 0.500000i 0 0.500000 + 0.866025i 1.43564 − 2.48660i 0 −1.37084 + 2.26292i − 1.00000i 0 −2.48660 + 1.43564i
125.16 −0.866025 − 0.500000i 0 0.500000 + 0.866025i 0.703885 − 1.21916i 0 −2.63516 + 0.236500i − 1.00000i 0 −1.21916 + 0.703885i
125.17 −0.866025 − 0.500000i 0 0.500000 + 0.866025i 0.859091 − 1.48799i 0 −2.21326 − 1.44965i − 1.00000i 0 −1.48799 + 0.859091i
125.18 −0.866025 − 0.500000i 0 0.500000 + 0.866025i −0.897263 + 1.55410i 0 −0.493381 + 2.59934i − 1.00000i 0 1.55410 − 0.897263i
125.19 −0.866025 − 0.500000i 0 0.500000 + 0.866025i −1.43564 + 2.48660i 0 −1.37084 − 2.26292i − 1.00000i 0 2.48660 − 1.43564i
125.20 −0.866025 − 0.500000i 0 0.500000 + 0.866025i −1.73151 + 2.99907i 0 −2.40266 + 1.10779i − 1.00000i 0 2.99907 − 1.73151i
See all 96 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 125.48
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.b odd 2 1 inner
19.c even 3 1 inner
21.c even 2 1 inner
57.h odd 6 1 inner
133.m odd 6 1 inner
399.z even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2394.2.bo.a ✓ 96
3.b odd 2 1 inner 2394.2.bo.a ✓ 96
7.b odd 2 1 inner 2394.2.bo.a ✓ 96
19.c even 3 1 inner 2394.2.bo.a ✓ 96
21.c even 2 1 inner 2394.2.bo.a ✓ 96
57.h odd 6 1 inner 2394.2.bo.a ✓ 96
133.m odd 6 1 inner 2394.2.bo.a ✓ 96
399.z even 6 1 inner 2394.2.bo.a ✓ 96
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2394.2.bo.a ✓ 96 1.a even 1 1 trivial
2394.2.bo.a ✓ 96 3.b odd 2 1 inner
2394.2.bo.a ✓ 96 7.b odd 2 1 inner
2394.2.bo.a ✓ 96 19.c even 3 1 inner
2394.2.bo.a ✓ 96 21.c even 2 1 inner
2394.2.bo.a ✓ 96 57.h odd 6 1 inner
2394.2.bo.a ✓ 96 133.m odd 6 1 inner
2394.2.bo.a ✓ 96 399.z even 6 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(2394, [\chi])\).