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Properties

Label 1248.2.n.e
Level $1248$
Weight $2$
Character orbit 1248.n
Analytic conductor $9.965$
Analytic rank $0$
Dimension $40$
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] = [1248,2,Mod(1247,1248)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1248, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1248.1247");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1248 = 2^{5} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1248.n (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: \(9.96533017226\)
Analytic rank: \(0\)
Dimension: \(40\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 40 q + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q + 16 q^{9} - 40 q^{13} + 40 q^{25} + 88 q^{49} + 80 q^{61} + 144 q^{81}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1247.1 0 −1.71844 − 0.216677i 0 −0.192613 0 −4.11279 0 2.90610 + 0.744696i 0
1247.2 0 −1.71844 − 0.216677i 0 0.192613 0 4.11279 0 2.90610 + 0.744696i 0
1247.3 0 −1.71844 + 0.216677i 0 −0.192613 0 −4.11279 0 2.90610 − 0.744696i 0
1247.4 0 −1.71844 + 0.216677i 0 0.192613 0 4.11279 0 2.90610 − 0.744696i 0
1247.5 0 −1.63494 − 0.571815i 0 −4.09647 0 −1.93097 0 2.34606 + 1.86977i 0
1247.6 0 −1.63494 − 0.571815i 0 4.09647 0 1.93097 0 2.34606 + 1.86977i 0
1247.7 0 −1.63494 + 0.571815i 0 −4.09647 0 −1.93097 0 2.34606 − 1.86977i 0
1247.8 0 −1.63494 + 0.571815i 0 4.09647 0 1.93097 0 2.34606 − 1.86977i 0
1247.9 0 −1.57517 − 0.720296i 0 −2.14968 0 1.51142 0 1.96235 + 2.26918i 0
1247.10 0 −1.57517 − 0.720296i 0 2.14968 0 −1.51142 0 1.96235 + 2.26918i 0
1247.11 0 −1.57517 + 0.720296i 0 −2.14968 0 1.51142 0 1.96235 − 2.26918i 0
1247.12 0 −1.57517 + 0.720296i 0 2.14968 0 −1.51142 0 1.96235 − 2.26918i 0
1247.13 0 −0.557260 − 1.63996i 0 −1.26385 0 −0.277977 0 −2.37892 + 1.82776i 0
1247.14 0 −0.557260 − 1.63996i 0 1.26385 0 0.277977 0 −2.37892 + 1.82776i 0
1247.15 0 −0.557260 + 1.63996i 0 −1.26385 0 −0.277977 0 −2.37892 − 1.82776i 0
1247.16 0 −0.557260 + 1.63996i 0 1.26385 0 0.277977 0 −2.37892 − 1.82776i 0
1247.17 0 −0.286722 − 1.70815i 0 −2.63883 0 −4.79528 0 −2.83558 + 0.979532i 0
1247.18 0 −0.286722 − 1.70815i 0 2.63883 0 4.79528 0 −2.83558 + 0.979532i 0
1247.19 0 −0.286722 + 1.70815i 0 −2.63883 0 −4.79528 0 −2.83558 − 0.979532i 0
1247.20 0 −0.286722 + 1.70815i 0 2.63883 0 4.79528 0 −2.83558 − 0.979532i 0
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1247.40
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner
13.b even 2 1 inner
39.d odd 2 1 inner
52.b odd 2 1 inner
156.h even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1248.2.n.e ✓ 40
3.b odd 2 1 inner 1248.2.n.e ✓ 40
4.b odd 2 1 inner 1248.2.n.e ✓ 40
12.b even 2 1 inner 1248.2.n.e ✓ 40
13.b even 2 1 inner 1248.2.n.e ✓ 40
39.d odd 2 1 inner 1248.2.n.e ✓ 40
52.b odd 2 1 inner 1248.2.n.e ✓ 40
156.h even 2 1 inner 1248.2.n.e ✓ 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1248.2.n.e ✓ 40 1.a even 1 1 trivial
1248.2.n.e ✓ 40 3.b odd 2 1 inner
1248.2.n.e ✓ 40 4.b odd 2 1 inner
1248.2.n.e ✓ 40 12.b even 2 1 inner
1248.2.n.e ✓ 40 13.b even 2 1 inner
1248.2.n.e ✓ 40 39.d odd 2 1 inner
1248.2.n.e ✓ 40 52.b odd 2 1 inner
1248.2.n.e ✓ 40 156.h 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 \(S_{2}^{\mathrm{new}}(1248, [\chi])\):

\( T_{5}^{10} - 30T_{5}^{8} + 273T_{5}^{6} - 912T_{5}^{4} + 896T_{5}^{2} - 32 \) Copy content Toggle raw display
\( T_{7}^{10} - 46T_{7}^{8} + 641T_{7}^{6} - 2728T_{7}^{4} + 3520T_{7}^{2} - 256 \) Copy content Toggle raw display