↑

Properties

Label 1248.4.c.d
Level $1248$
Weight $4$
Character orbit 1248.c
Analytic conductor $73.634$
Analytic rank $0$
Dimension $22$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1248,4,Mod(961,1248)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1248.961"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1248, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 1248 = 2^{5} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1248.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [22,0,66] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(73.6343836872\)
Analytic rank: \(0\)
Dimension: \(22\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 22 q + 66 q^{3} + 198 q^{9} - 38 q^{13} + 12 q^{17} + 184 q^{23} - 794 q^{25} + 594 q^{27} - 412 q^{29} + 72 q^{35} - 114 q^{39} - 504 q^{43} - 806 q^{49} + 36 q^{51} + 756 q^{53} - 272 q^{55} - 1044 q^{61}+ \cdots + 504 q^{95}+O(q^{100}) \) Copy content Toggle raw display

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
961.1 0 3.00000 0 − 21.5545i 0 24.3006i 0 9.00000 0
961.2 0 3.00000 0 − 20.2331i 0 − 8.88843i 0 9.00000 0
961.3 0 3.00000 0 − 14.4426i 0 − 10.7922i 0 9.00000 0
961.4 0 3.00000 0 − 12.7645i 0 35.0891i 0 9.00000 0
961.5 0 3.00000 0 − 11.8996i 0 − 27.2326i 0 9.00000 0
961.6 0 3.00000 0 − 11.8183i 0 − 23.5900i 0 9.00000 0
961.7 0 3.00000 0 − 10.8963i 0 9.78782i 0 9.00000 0
961.8 0 3.00000 0 − 9.65691i 0 − 16.9227i 0 9.00000 0
961.9 0 3.00000 0 − 5.52501i 0 11.2861i 0 9.00000 0
961.10 0 3.00000 0 − 1.55321i 0 4.75339i 0 9.00000 0
961.11 0 3.00000 0 − 0.556701i 0 − 18.1247i 0 9.00000 0
961.12 0 3.00000 0 0.556701i 0 18.1247i 0 9.00000 0
961.13 0 3.00000 0 1.55321i 0 − 4.75339i 0 9.00000 0
961.14 0 3.00000 0 5.52501i 0 − 11.2861i 0 9.00000 0
961.15 0 3.00000 0 9.65691i 0 16.9227i 0 9.00000 0
961.16 0 3.00000 0 10.8963i 0 − 9.78782i 0 9.00000 0
961.17 0 3.00000 0 11.8183i 0 23.5900i 0 9.00000 0
961.18 0 3.00000 0 11.8996i 0 27.2326i 0 9.00000 0
961.19 0 3.00000 0 12.7645i 0 − 35.0891i 0 9.00000 0
961.20 0 3.00000 0 14.4426i 0 10.7922i 0 9.00000 0
See all 22 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 961.22
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1248.4.c.d yes 22
4.b odd 2 1 1248.4.c.c ✓ 22
13.b even 2 1 inner 1248.4.c.d yes 22
52.b odd 2 1 1248.4.c.c ✓ 22
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1248.4.c.c ✓ 22 4.b odd 2 1
1248.4.c.c ✓ 22 52.b odd 2 1
1248.4.c.d yes 22 1.a even 1 1 trivial
1248.4.c.d yes 22 13.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(1248, [\chi])\):

\( T_{5}^{22} + 1772 T_{5}^{20} + 1311600 T_{5}^{18} + 532926976 T_{5}^{16} + 131382720768 T_{5}^{14} + \cdots + 32\!\cdots\!36 \) Copy content Toggle raw display
\( T_{23}^{11} - 92 T_{23}^{10} - 48272 T_{23}^{9} + 5778560 T_{23}^{8} + 295998720 T_{23}^{7} + \cdots - 42\!\cdots\!08 \) Copy content Toggle raw display