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Properties

Label 232.4.e.a
Level $232$
Weight $4$
Character orbit 232.e
Analytic conductor $13.688$
Analytic rank $0$
Dimension $22$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [232,4,Mod(57,232)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(232, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("232.57");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 232 = 2^{3} \cdot 29 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 232.e (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: \(13.6884431213\)
Analytic rank: \(0\)
Dimension: \(22\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 22 q + 18 q^{5} - 12 q^{7} - 152 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 22 q + 18 q^{5} - 12 q^{7} - 152 q^{9} - 58 q^{13} + 76 q^{23} + 980 q^{25} + 38 q^{29} - 6 q^{33} - 668 q^{35} - 1404 q^{45} + 1358 q^{49} - 180 q^{51} - 1002 q^{53} + 720 q^{57} + 1500 q^{59} + 3512 q^{63} - 1386 q^{65} - 656 q^{67} - 3472 q^{71} - 818 q^{81} + 1092 q^{83} - 1356 q^{87} - 692 q^{91} + 942 q^{93}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
57.1 0 − 8.71208i 0 20.8976 0 −30.7679 0 −48.9004 0
57.2 0 − 8.53598i 0 −1.11669 0 12.7954 0 −45.8630 0
57.3 0 − 8.11782i 0 −12.7846 0 −4.35653 0 −38.8990 0
57.4 0 − 7.96461i 0 2.14129 0 −19.9843 0 −36.4351 0
57.5 0 − 5.74564i 0 15.2775 0 15.1993 0 −6.01243 0
57.6 0 − 5.48906i 0 2.51048 0 16.1051 0 −3.12977 0
57.7 0 − 3.89906i 0 −19.8345 0 −19.4786 0 11.7973 0
57.8 0 − 3.20482i 0 −16.3505 0 33.5494 0 16.7291 0
57.9 0 − 1.61403i 0 −4.87333 0 −3.57097 0 24.3949 0
57.10 0 − 1.54921i 0 17.3651 0 18.4685 0 24.5999 0
57.11 0 − 1.13209i 0 5.76762 0 −23.9594 0 25.7184 0
57.12 0 1.13209i 0 5.76762 0 −23.9594 0 25.7184 0
57.13 0 1.54921i 0 17.3651 0 18.4685 0 24.5999 0
57.14 0 1.61403i 0 −4.87333 0 −3.57097 0 24.3949 0
57.15 0 3.20482i 0 −16.3505 0 33.5494 0 16.7291 0
57.16 0 3.89906i 0 −19.8345 0 −19.4786 0 11.7973 0
57.17 0 5.48906i 0 2.51048 0 16.1051 0 −3.12977 0
57.18 0 5.74564i 0 15.2775 0 15.1993 0 −6.01243 0
57.19 0 7.96461i 0 2.14129 0 −19.9843 0 −36.4351 0
57.20 0 8.11782i 0 −12.7846 0 −4.35653 0 −38.8990 0
See all 22 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 57.22
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
29.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 232.4.e.a ✓ 22
4.b odd 2 1 464.4.e.d 22
29.b even 2 1 inner 232.4.e.a ✓ 22
116.d odd 2 1 464.4.e.d 22
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
232.4.e.a ✓ 22 1.a even 1 1 trivial
232.4.e.a ✓ 22 29.b even 2 1 inner
464.4.e.d 22 4.b odd 2 1
464.4.e.d 22 116.d odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(232, [\chi])\).