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Properties

Label 4620.2.f.e
Level $4620$
Weight $2$
Character orbit 4620.f
Analytic conductor $36.891$
Analytic rank $0$
Dimension $40$
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] = [4620,2,Mod(769,4620)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4620, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 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("4620.769");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4620 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4620.f (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: \(36.8908857338\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 40 q - 40 q^{3} + 2 q^{5} + 40 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q - 40 q^{3} + 2 q^{5} + 40 q^{9} - 8 q^{11} - 2 q^{15} - 6 q^{25} - 40 q^{27} + 8 q^{33} + 2 q^{45} + 68 q^{47} - 16 q^{49} - 4 q^{55} - 40 q^{71} + 6 q^{75} + 4 q^{77} + 40 q^{81} - 12 q^{91} + 4 q^{97} - 8 q^{99}+O(q^{100}) \) 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 −1.00000 0 −2.12407 − 0.698811i 0 −2.57540 − 0.606063i 0 1.00000 0
769.2 0 −1.00000 0 −2.12407 − 0.698811i 0 2.57540 + 0.606063i 0 1.00000 0
769.3 0 −1.00000 0 −2.12407 + 0.698811i 0 −2.57540 + 0.606063i 0 1.00000 0
769.4 0 −1.00000 0 −2.12407 + 0.698811i 0 2.57540 − 0.606063i 0 1.00000 0
769.5 0 −1.00000 0 −1.96602 − 1.06526i 0 −0.805532 + 2.52014i 0 1.00000 0
769.6 0 −1.00000 0 −1.96602 − 1.06526i 0 0.805532 − 2.52014i 0 1.00000 0
769.7 0 −1.00000 0 −1.96602 + 1.06526i 0 −0.805532 − 2.52014i 0 1.00000 0
769.8 0 −1.00000 0 −1.96602 + 1.06526i 0 0.805532 + 2.52014i 0 1.00000 0
769.9 0 −1.00000 0 −1.28729 − 1.82835i 0 −1.16982 − 2.37308i 0 1.00000 0
769.10 0 −1.00000 0 −1.28729 − 1.82835i 0 1.16982 + 2.37308i 0 1.00000 0
769.11 0 −1.00000 0 −1.28729 + 1.82835i 0 −1.16982 + 2.37308i 0 1.00000 0
769.12 0 −1.00000 0 −1.28729 + 1.82835i 0 1.16982 − 2.37308i 0 1.00000 0
769.13 0 −1.00000 0 −1.07835 − 1.95887i 0 −2.64573 − 0.0107606i 0 1.00000 0
769.14 0 −1.00000 0 −1.07835 − 1.95887i 0 2.64573 + 0.0107606i 0 1.00000 0
769.15 0 −1.00000 0 −1.07835 + 1.95887i 0 −2.64573 + 0.0107606i 0 1.00000 0
769.16 0 −1.00000 0 −1.07835 + 1.95887i 0 2.64573 − 0.0107606i 0 1.00000 0
769.17 0 −1.00000 0 −0.510039 − 2.17712i 0 −1.78936 − 1.94889i 0 1.00000 0
769.18 0 −1.00000 0 −0.510039 − 2.17712i 0 1.78936 + 1.94889i 0 1.00000 0
769.19 0 −1.00000 0 −0.510039 + 2.17712i 0 −1.78936 + 1.94889i 0 1.00000 0
769.20 0 −1.00000 0 −0.510039 + 2.17712i 0 1.78936 − 1.94889i 0 1.00000 0
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 769.40
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 inner
35.c odd 2 1 inner
385.h even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4620.2.f.e ✓ 40
5.b even 2 1 4620.2.f.f yes 40
7.b odd 2 1 4620.2.f.f yes 40
11.b odd 2 1 inner 4620.2.f.e ✓ 40
35.c odd 2 1 inner 4620.2.f.e ✓ 40
55.d odd 2 1 4620.2.f.f yes 40
77.b even 2 1 4620.2.f.f yes 40
385.h even 2 1 inner 4620.2.f.e ✓ 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4620.2.f.e ✓ 40 1.a even 1 1 trivial
4620.2.f.e ✓ 40 11.b odd 2 1 inner
4620.2.f.e ✓ 40 35.c odd 2 1 inner
4620.2.f.e ✓ 40 385.h even 2 1 inner
4620.2.f.f yes 40 5.b even 2 1
4620.2.f.f yes 40 7.b odd 2 1
4620.2.f.f yes 40 55.d odd 2 1
4620.2.f.f yes 40 77.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(4620, [\chi])\):

\( T_{13}^{20} + 115 T_{13}^{18} + 4960 T_{13}^{16} + 104456 T_{13}^{14} + 1139716 T_{13}^{12} + \cdots + 65536 \) Copy content Toggle raw display
\( T_{19}^{20} - 226 T_{19}^{18} + 20292 T_{19}^{16} - 930070 T_{19}^{14} + 23586923 T_{19}^{12} + \cdots + 3157540864 \) Copy content Toggle raw display
\( T_{43}^{20} - 525 T_{43}^{18} + 114451 T_{43}^{16} - 13556511 T_{43}^{14} + 961462184 T_{43}^{12} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
\( T_{47}^{10} - 17 T_{47}^{9} - 128 T_{47}^{8} + 3868 T_{47}^{7} - 11226 T_{47}^{6} - 196944 T_{47}^{5} + \cdots - 34127872 \) Copy content Toggle raw display