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Properties

Label 1248.2.ca.b
Level $1248$
Weight $2$
Character orbit 1248.ca
Analytic conductor $9.965$
Analytic rank $0$
Dimension $48$
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] = [1248,2,Mod(49,1248)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1248, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 0, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1248.49");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1248 = 2^{5} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1248.ca (of order \(6\), degree \(2\), not 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: \(48\)
Relative dimension: \(24\) over \(\Q(\zeta_{6})\)
Twist minimal: no (minimal twist has level 312)
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.

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 12 q^{7} + 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q + 12 q^{7} + 24 q^{9} + 12 q^{17} - 20 q^{23} + 48 q^{25} + 12 q^{33} - 28 q^{39} - 12 q^{41} + 16 q^{49} + 68 q^{55} + 12 q^{63} + 12 q^{65} + 12 q^{71} + 192 q^{79} - 24 q^{81} - 48 q^{89} + 20 q^{95} + 144 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1 0 −0.866025 + 0.500000i 0 −4.32865 0 −1.87318 − 1.08148i 0 0.500000 − 0.866025i 0
49.2 0 −0.866025 + 0.500000i 0 −3.46216 0 2.41814 + 1.39611i 0 0.500000 − 0.866025i 0
49.3 0 −0.866025 + 0.500000i 0 3.08386 0 −0.257604 − 0.148728i 0 0.500000 − 0.866025i 0
49.4 0 −0.866025 + 0.500000i 0 2.09848 0 0.271467 + 0.156732i 0 0.500000 − 0.866025i 0
49.5 0 −0.866025 + 0.500000i 0 1.25972 0 −1.81933 − 1.05039i 0 0.500000 − 0.866025i 0
49.6 0 −0.866025 + 0.500000i 0 −0.112463 0 −0.0378844 − 0.0218726i 0 0.500000 − 0.866025i 0
49.7 0 −0.866025 + 0.500000i 0 0.230393 0 2.89019 + 1.66865i 0 0.500000 − 0.866025i 0
49.8 0 −0.866025 + 0.500000i 0 −1.13631 0 −1.76332 − 1.01805i 0 0.500000 − 0.866025i 0
49.9 0 −0.866025 + 0.500000i 0 −1.72209 0 4.22521 + 2.43943i 0 0.500000 − 0.866025i 0
49.10 0 −0.866025 + 0.500000i 0 −1.88587 0 −3.80954 − 2.19944i 0 0.500000 − 0.866025i 0
49.11 0 −0.866025 + 0.500000i 0 2.83647 0 −0.691790 − 0.399405i 0 0.500000 − 0.866025i 0
49.12 0 −0.866025 + 0.500000i 0 3.13862 0 3.44764 + 1.99050i 0 0.500000 − 0.866025i 0
49.13 0 0.866025 − 0.500000i 0 −3.13862 0 3.44764 + 1.99050i 0 0.500000 − 0.866025i 0
49.14 0 0.866025 − 0.500000i 0 −2.83647 0 −0.691790 − 0.399405i 0 0.500000 − 0.866025i 0
49.15 0 0.866025 − 0.500000i 0 1.88587 0 −3.80954 − 2.19944i 0 0.500000 − 0.866025i 0
49.16 0 0.866025 − 0.500000i 0 1.72209 0 4.22521 + 2.43943i 0 0.500000 − 0.866025i 0
49.17 0 0.866025 − 0.500000i 0 1.13631 0 −1.76332 − 1.01805i 0 0.500000 − 0.866025i 0
49.18 0 0.866025 − 0.500000i 0 −0.230393 0 2.89019 + 1.66865i 0 0.500000 − 0.866025i 0
49.19 0 0.866025 − 0.500000i 0 0.112463 0 −0.0378844 − 0.0218726i 0 0.500000 − 0.866025i 0
49.20 0 0.866025 − 0.500000i 0 −1.25972 0 −1.81933 − 1.05039i 0 0.500000 − 0.866025i 0
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner
13.e even 6 1 inner
104.s even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1248.2.ca.b 48
4.b odd 2 1 312.2.bk.b ✓ 48
8.b even 2 1 inner 1248.2.ca.b 48
8.d odd 2 1 312.2.bk.b ✓ 48
12.b even 2 1 936.2.dg.e 48
13.e even 6 1 inner 1248.2.ca.b 48
24.f even 2 1 936.2.dg.e 48
52.i odd 6 1 312.2.bk.b ✓ 48
104.p odd 6 1 312.2.bk.b ✓ 48
104.s even 6 1 inner 1248.2.ca.b 48
156.r even 6 1 936.2.dg.e 48
312.ba even 6 1 936.2.dg.e 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
312.2.bk.b ✓ 48 4.b odd 2 1
312.2.bk.b ✓ 48 8.d odd 2 1
312.2.bk.b ✓ 48 52.i odd 6 1
312.2.bk.b ✓ 48 104.p odd 6 1
936.2.dg.e 48 12.b even 2 1
936.2.dg.e 48 24.f even 2 1
936.2.dg.e 48 156.r even 6 1
936.2.dg.e 48 312.ba even 6 1
1248.2.ca.b 48 1.a even 1 1 trivial
1248.2.ca.b 48 8.b even 2 1 inner
1248.2.ca.b 48 13.e even 6 1 inner
1248.2.ca.b 48 104.s even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{24} - 72 T_{5}^{22} + 2196 T_{5}^{20} - 37296 T_{5}^{18} + 389342 T_{5}^{16} - 2600048 T_{5}^{14} + \cdots + 10816 \) acting on \(S_{2}^{\mathrm{new}}(1248, [\chi])\). Copy content Toggle raw display