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Properties

Label 4620.2.m.b
Level $4620$
Weight $2$
Character orbit 4620.m
Analytic conductor $36.891$
Analytic rank $0$
Dimension $48$
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] = [4620,2,Mod(1121,4620)]
 
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, 1, 0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4620.1121");
 
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.m (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: \(48\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 4 q^{3} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q - 4 q^{3} + 6 q^{9} + 6 q^{11} + 2 q^{15} - 4 q^{17} - 2 q^{21} - 48 q^{25} - 16 q^{27} - 36 q^{29} - 16 q^{31} - 4 q^{33} + 48 q^{35} - 8 q^{37} + 18 q^{39} - 48 q^{49} + 30 q^{51} + 4 q^{55} - 16 q^{57} + 12 q^{65} + 24 q^{67} + 4 q^{69} + 4 q^{75} - 4 q^{77} - 22 q^{81} - 20 q^{83} - 44 q^{87} - 12 q^{91} - 28 q^{93} - 8 q^{95} - 56 q^{97} - 12 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1121.1 0 −1.72700 − 0.132234i 0 1.00000i 0 − 1.00000i 0 2.96503 + 0.456733i 0
1121.2 0 −1.72700 + 0.132234i 0 − 1.00000i 0 1.00000i 0 2.96503 − 0.456733i 0
1121.3 0 −1.71125 − 0.267648i 0 − 1.00000i 0 1.00000i 0 2.85673 + 0.916025i 0
1121.4 0 −1.71125 + 0.267648i 0 1.00000i 0 − 1.00000i 0 2.85673 − 0.916025i 0
1121.5 0 −1.68394 − 0.405392i 0 − 1.00000i 0 1.00000i 0 2.67131 + 1.36531i 0
1121.6 0 −1.68394 + 0.405392i 0 1.00000i 0 − 1.00000i 0 2.67131 − 1.36531i 0
1121.7 0 −1.63283 − 0.577800i 0 − 1.00000i 0 1.00000i 0 2.33229 + 1.88690i 0
1121.8 0 −1.63283 + 0.577800i 0 1.00000i 0 − 1.00000i 0 2.33229 − 1.88690i 0
1121.9 0 −1.54192 − 0.788974i 0 1.00000i 0 − 1.00000i 0 1.75504 + 2.43307i 0
1121.10 0 −1.54192 + 0.788974i 0 − 1.00000i 0 1.00000i 0 1.75504 − 2.43307i 0
1121.11 0 −1.38915 − 1.03454i 0 1.00000i 0 − 1.00000i 0 0.859455 + 2.87425i 0
1121.12 0 −1.38915 + 1.03454i 0 − 1.00000i 0 1.00000i 0 0.859455 − 2.87425i 0
1121.13 0 −1.34978 − 1.08540i 0 1.00000i 0 − 1.00000i 0 0.643830 + 2.93010i 0
1121.14 0 −1.34978 + 1.08540i 0 − 1.00000i 0 1.00000i 0 0.643830 − 2.93010i 0
1121.15 0 −1.11749 − 1.32334i 0 − 1.00000i 0 1.00000i 0 −0.502432 + 2.95763i 0
1121.16 0 −1.11749 + 1.32334i 0 1.00000i 0 − 1.00000i 0 −0.502432 − 2.95763i 0
1121.17 0 −0.839247 − 1.51514i 0 − 1.00000i 0 1.00000i 0 −1.59133 + 2.54316i 0
1121.18 0 −0.839247 + 1.51514i 0 1.00000i 0 − 1.00000i 0 −1.59133 − 2.54316i 0
1121.19 0 −0.814543 − 1.52857i 0 1.00000i 0 − 1.00000i 0 −1.67304 + 2.49017i 0
1121.20 0 −0.814543 + 1.52857i 0 − 1.00000i 0 1.00000i 0 −1.67304 − 2.49017i 0
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1121.48
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
33.d 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.m.b yes 48
3.b odd 2 1 4620.2.m.a ✓ 48
11.b odd 2 1 4620.2.m.a ✓ 48
33.d even 2 1 inner 4620.2.m.b yes 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4620.2.m.a ✓ 48 3.b odd 2 1
4620.2.m.a ✓ 48 11.b odd 2 1
4620.2.m.b yes 48 1.a even 1 1 trivial
4620.2.m.b yes 48 33.d even 2 1 inner