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Properties

Label 1320.2.f.a
Level $1320$
Weight $2$
Character orbit 1320.f
Analytic conductor $10.540$
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] = [1320,2,Mod(1121,1320)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1320, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 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("1320.1121");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1320 = 2^{3} \cdot 3 \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1320.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: \(10.5402530668\)
Analytic rank: \(0\)
Dimension: \(24\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 2 q^{3} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 2 q^{3} - 6 q^{9} - 6 q^{11} - 12 q^{17} - 8 q^{21} - 24 q^{25} - 2 q^{27} - 4 q^{29} - 4 q^{31} + 14 q^{33} + 4 q^{37} + 24 q^{39} + 8 q^{41} - 20 q^{49} + 12 q^{51} + 2 q^{55} - 30 q^{57} - 30 q^{63} + 12 q^{65} + 12 q^{67} + 32 q^{69} + 2 q^{75} + 4 q^{77} - 14 q^{81} - 34 q^{87} + 16 q^{91} + 16 q^{93} - 12 q^{95} + 40 q^{97} - 10 q^{99}+O(q^{100}) \) 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 −1.73177 − 0.0314311i 0 1.00000i 0 − 3.41348i 0 2.99802 + 0.108863i 0
1121.2 0 −1.73177 + 0.0314311i 0 − 1.00000i 0 3.41348i 0 2.99802 − 0.108863i 0
1121.3 0 −1.65245 − 0.519042i 0 − 1.00000i 0 − 1.91053i 0 2.46119 + 1.71538i 0
1121.4 0 −1.65245 + 0.519042i 0 1.00000i 0 1.91053i 0 2.46119 − 1.71538i 0
1121.5 0 −1.24397 − 1.20522i 0 1.00000i 0 2.71015i 0 0.0949023 + 2.99850i 0
1121.6 0 −1.24397 + 1.20522i 0 − 1.00000i 0 − 2.71015i 0 0.0949023 − 2.99850i 0
1121.7 0 −0.861793 − 1.50244i 0 − 1.00000i 0 − 3.29695i 0 −1.51462 + 2.58958i 0
1121.8 0 −0.861793 + 1.50244i 0 1.00000i 0 3.29695i 0 −1.51462 − 2.58958i 0
1121.9 0 −0.804860 − 1.53369i 0 1.00000i 0 1.44652i 0 −1.70440 + 2.46881i 0
1121.10 0 −0.804860 + 1.53369i 0 − 1.00000i 0 − 1.44652i 0 −1.70440 − 2.46881i 0
1121.11 0 −0.566021 − 1.63695i 0 − 1.00000i 0 1.58806i 0 −2.35924 + 1.85310i 0
1121.12 0 −0.566021 + 1.63695i 0 1.00000i 0 − 1.58806i 0 −2.35924 − 1.85310i 0
1121.13 0 0.100991 − 1.72910i 0 1.00000i 0 − 3.28518i 0 −2.97960 − 0.349247i 0
1121.14 0 0.100991 + 1.72910i 0 − 1.00000i 0 3.28518i 0 −2.97960 + 0.349247i 0
1121.15 0 0.488478 − 1.66174i 0 − 1.00000i 0 3.72161i 0 −2.52278 − 1.62345i 0
1121.16 0 0.488478 + 1.66174i 0 1.00000i 0 − 3.72161i 0 −2.52278 + 1.62345i 0
1121.17 0 0.875092 − 1.49473i 0 1.00000i 0 0.174070i 0 −1.46843 − 2.61605i 0
1121.18 0 0.875092 + 1.49473i 0 − 1.00000i 0 − 0.174070i 0 −1.46843 + 2.61605i 0
1121.19 0 1.32197 − 1.11911i 0 − 1.00000i 0 − 5.22520i 0 0.495201 − 2.95885i 0
1121.20 0 1.32197 + 1.11911i 0 1.00000i 0 5.22520i 0 0.495201 + 2.95885i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1121.24
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 1320.2.f.a ✓ 24
3.b odd 2 1 1320.2.f.b yes 24
4.b odd 2 1 2640.2.f.g 24
11.b odd 2 1 1320.2.f.b yes 24
12.b even 2 1 2640.2.f.f 24
33.d even 2 1 inner 1320.2.f.a ✓ 24
44.c even 2 1 2640.2.f.f 24
132.d odd 2 1 2640.2.f.g 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1320.2.f.a ✓ 24 1.a even 1 1 trivial
1320.2.f.a ✓ 24 33.d even 2 1 inner
1320.2.f.b yes 24 3.b odd 2 1
1320.2.f.b yes 24 11.b odd 2 1
2640.2.f.f 24 12.b even 2 1
2640.2.f.f 24 44.c even 2 1
2640.2.f.g 24 4.b odd 2 1
2640.2.f.g 24 132.d odd 2 1