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Properties

Label 3696.2.q.g
Level $3696$
Weight $2$
Character orbit 3696.q
Analytic conductor $29.513$
Analytic rank $0$
Dimension $24$
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] = [3696,2,Mod(769,3696)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3696, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 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("3696.769");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3696 = 2^{4} \cdot 3 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3696.q (of order \(2\), degree \(1\), 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: \(29.5127085871\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: no (minimal twist has level 1848)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 4 q^{7} - 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 4 q^{7} - 24 q^{9} - 4 q^{11} + 4 q^{15} + 4 q^{17} - 4 q^{19} + 24 q^{23} - 8 q^{25} + 8 q^{33} - 16 q^{35} + 8 q^{37} - 20 q^{41} + 4 q^{49} - 24 q^{53} + 8 q^{55} + 8 q^{61} - 4 q^{63} + 8 q^{67} + 4 q^{73} - 8 q^{77} + 24 q^{81} + 4 q^{83} + 8 q^{87} + 4 q^{91} + 12 q^{93} + 4 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
769.1 0 − 1.00000i 0 − 3.54741i 0 1.12129 − 2.39640i 0 −1.00000 0
769.2 0 − 1.00000i 0 − 2.64261i 0 1.38114 + 2.25664i 0 −1.00000 0
769.3 0 − 1.00000i 0 − 1.71581i 0 1.06269 − 2.42295i 0 −1.00000 0
769.4 0 − 1.00000i 0 − 1.64449i 0 2.63155 + 0.273742i 0 −1.00000 0
769.5 0 − 1.00000i 0 − 1.09147i 0 −2.61768 − 0.384362i 0 −1.00000 0
769.6 0 − 1.00000i 0 − 0.331450i 0 −1.23509 + 2.33977i 0 −1.00000 0
769.7 0 − 1.00000i 0 0.210416i 0 −2.16457 − 1.52140i 0 −1.00000 0
769.8 0 − 1.00000i 0 0.883273i 0 −0.763345 + 2.53324i 0 −1.00000 0
769.9 0 − 1.00000i 0 2.32410i 0 −0.212212 − 2.63723i 0 −1.00000 0
769.10 0 − 1.00000i 0 2.50231i 0 2.45571 + 0.984620i 0 −1.00000 0
769.11 0 − 1.00000i 0 3.27939i 0 2.61990 − 0.368974i 0 −1.00000 0
769.12 0 − 1.00000i 0 3.77374i 0 −2.27938 + 1.34328i 0 −1.00000 0
769.13 0 1.00000i 0 − 3.77374i 0 −2.27938 − 1.34328i 0 −1.00000 0
769.14 0 1.00000i 0 − 3.27939i 0 2.61990 + 0.368974i 0 −1.00000 0
769.15 0 1.00000i 0 − 2.50231i 0 2.45571 − 0.984620i 0 −1.00000 0
769.16 0 1.00000i 0 − 2.32410i 0 −0.212212 + 2.63723i 0 −1.00000 0
769.17 0 1.00000i 0 − 0.883273i 0 −0.763345 − 2.53324i 0 −1.00000 0
769.18 0 1.00000i 0 − 0.210416i 0 −2.16457 + 1.52140i 0 −1.00000 0
769.19 0 1.00000i 0 0.331450i 0 −1.23509 − 2.33977i 0 −1.00000 0
769.20 0 1.00000i 0 1.09147i 0 −2.61768 + 0.384362i 0 −1.00000 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 769.24
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3696.2.q.g 24
4.b odd 2 1 1848.2.q.a ✓ 24
7.b odd 2 1 3696.2.q.f 24
11.b odd 2 1 3696.2.q.f 24
28.d even 2 1 1848.2.q.b yes 24
44.c even 2 1 1848.2.q.b yes 24
77.b even 2 1 inner 3696.2.q.g 24
308.g odd 2 1 1848.2.q.a ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1848.2.q.a ✓ 24 4.b odd 2 1
1848.2.q.a ✓ 24 308.g odd 2 1
1848.2.q.b yes 24 28.d even 2 1
1848.2.q.b yes 24 44.c even 2 1
3696.2.q.f 24 7.b odd 2 1
3696.2.q.f 24 11.b odd 2 1
3696.2.q.g 24 1.a even 1 1 trivial
3696.2.q.g 24 77.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 \(S_{2}^{\mathrm{new}}(3696, [\chi])\):

\( T_{5}^{24} + 64 T_{5}^{22} + 1742 T_{5}^{20} + 26412 T_{5}^{18} + 245601 T_{5}^{16} + 1454284 T_{5}^{14} + \cdots + 16384 \) Copy content Toggle raw display
\( T_{13}^{12} - 92 T_{13}^{10} + 32 T_{13}^{9} + 2921 T_{13}^{8} - 1384 T_{13}^{7} - 37978 T_{13}^{6} + \cdots + 241664 \) Copy content Toggle raw display