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Properties

Label 3696.2.d.f
Level $3696$
Weight $2$
Character orbit 3696.d
Analytic conductor $29.513$
Analytic rank $0$
Dimension $28$
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(2575,3696)]
 
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([1, 0, 0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3696.2575");
 
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.d (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: \(29.5127085871\)
Analytic rank: \(0\)
Dimension: \(28\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 28 q + 28 q^{3} - 8 q^{7} + 28 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 28 q + 28 q^{3} - 8 q^{7} + 28 q^{9} - 16 q^{19} - 8 q^{21} - 44 q^{25} + 28 q^{27} + 24 q^{29} + 16 q^{31} - 24 q^{35} - 16 q^{37} - 8 q^{47} - 4 q^{49} - 16 q^{57} + 16 q^{59} - 8 q^{63} + 8 q^{65} - 44 q^{75} + 4 q^{77} + 28 q^{81} + 16 q^{83} + 8 q^{85} + 24 q^{87} + 16 q^{91} + 16 q^{93}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2575.1 0 1.00000 0 − 4.25683i 0 −1.43185 − 2.22481i 0 1.00000 0
2575.2 0 1.00000 0 − 4.11098i 0 −2.52592 − 0.787224i 0 1.00000 0
2575.3 0 1.00000 0 − 3.17124i 0 2.43176 − 1.04236i 0 1.00000 0
2575.4 0 1.00000 0 − 3.10821i 0 −0.0783443 − 2.64459i 0 1.00000 0
2575.5 0 1.00000 0 − 2.89216i 0 2.47419 + 0.937228i 0 1.00000 0
2575.6 0 1.00000 0 − 2.57440i 0 2.34334 + 1.22831i 0 1.00000 0
2575.7 0 1.00000 0 − 2.50903i 0 −0.580985 + 2.58117i 0 1.00000 0
2575.8 0 1.00000 0 − 2.39313i 0 −2.02432 + 1.70356i 0 1.00000 0
2575.9 0 1.00000 0 − 2.01879i 0 −1.63712 + 2.07842i 0 1.00000 0
2575.10 0 1.00000 0 − 1.92823i 0 0.412483 − 2.61340i 0 1.00000 0
2575.11 0 1.00000 0 − 1.17764i 0 −2.58423 − 0.567213i 0 1.00000 0
2575.12 0 1.00000 0 − 0.965537i 0 0.953117 − 2.46811i 0 1.00000 0
2575.13 0 1.00000 0 − 0.305668i 0 −2.55380 + 0.691450i 0 1.00000 0
2575.14 0 1.00000 0 − 0.202428i 0 0.801692 − 2.52137i 0 1.00000 0
2575.15 0 1.00000 0 0.202428i 0 0.801692 + 2.52137i 0 1.00000 0
2575.16 0 1.00000 0 0.305668i 0 −2.55380 − 0.691450i 0 1.00000 0
2575.17 0 1.00000 0 0.965537i 0 0.953117 + 2.46811i 0 1.00000 0
2575.18 0 1.00000 0 1.17764i 0 −2.58423 + 0.567213i 0 1.00000 0
2575.19 0 1.00000 0 1.92823i 0 0.412483 + 2.61340i 0 1.00000 0
2575.20 0 1.00000 0 2.01879i 0 −1.63712 − 2.07842i 0 1.00000 0
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2575.28
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
28.d 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.d.f yes 28
4.b odd 2 1 3696.2.d.e ✓ 28
7.b odd 2 1 3696.2.d.e ✓ 28
28.d even 2 1 inner 3696.2.d.f yes 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3696.2.d.e ✓ 28 4.b odd 2 1
3696.2.d.e ✓ 28 7.b odd 2 1
3696.2.d.f yes 28 1.a even 1 1 trivial
3696.2.d.f yes 28 28.d 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 \(S_{2}^{\mathrm{new}}(3696, [\chi])\):

\( T_{5}^{28} + 92 T_{5}^{26} + 3718 T_{5}^{24} + 87204 T_{5}^{22} + 1320633 T_{5}^{20} + 13580984 T_{5}^{18} + \cdots + 4460544 \) Copy content Toggle raw display
\( T_{19}^{14} + 8 T_{19}^{13} - 126 T_{19}^{12} - 1112 T_{19}^{11} + 4725 T_{19}^{10} + 50888 T_{19}^{9} + \cdots + 1487872 \) Copy content Toggle raw display