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Properties

Label 330.2.f.a
Level $330$
Weight $2$
Character orbit 330.f
Analytic conductor $2.635$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [330,2,Mod(329,330)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(330, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("330.329");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 330 = 2 \cdot 3 \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 330.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: \(2.63506326670\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 24 q^{4} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 24 q^{4} + 12 q^{9} - 4 q^{15} + 24 q^{16} - 8 q^{25} + 24 q^{31} - 8 q^{34} - 12 q^{36} - 48 q^{45} - 16 q^{49} - 16 q^{55} + 4 q^{60} - 24 q^{64} + 36 q^{66} + 8 q^{69} - 32 q^{70} - 32 q^{75} + 76 q^{81} + 80 q^{91} - 56 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
329.1 − 1.00000i −1.69665 − 0.348406i −1.00000 −1.46224 + 1.69170i −0.348406 + 1.69665i −1.01891 1.00000i 2.75723 + 1.18224i 1.69170 + 1.46224i
329.2 − 1.00000i −1.69665 + 0.348406i −1.00000 −1.46224 − 1.69170i 0.348406 + 1.69665i 1.01891 1.00000i 2.75723 − 1.18224i −1.69170 + 1.46224i
329.3 − 1.00000i −1.51517 − 0.839210i −1.00000 2.09320 + 0.786447i −0.839210 + 1.51517i 3.51327 1.00000i 1.59145 + 2.54308i 0.786447 − 2.09320i
329.4 − 1.00000i −1.51517 + 0.839210i −1.00000 2.09320 − 0.786447i 0.839210 + 1.51517i −3.51327 1.00000i 1.59145 − 2.54308i −0.786447 − 2.09320i
329.5 − 1.00000i −0.275064 − 1.71007i −1.00000 −0.693067 + 2.12595i −1.71007 + 0.275064i −2.37039 1.00000i −2.84868 + 0.940756i 2.12595 + 0.693067i
329.6 − 1.00000i −0.275064 + 1.71007i −1.00000 −0.693067 − 2.12595i 1.71007 + 0.275064i 2.37039 1.00000i −2.84868 − 0.940756i −2.12595 + 0.693067i
329.7 − 1.00000i 0.275064 − 1.71007i −1.00000 0.693067 − 2.12595i −1.71007 − 0.275064i 2.37039 1.00000i −2.84868 − 0.940756i −2.12595 − 0.693067i
329.8 − 1.00000i 0.275064 + 1.71007i −1.00000 0.693067 + 2.12595i 1.71007 − 0.275064i −2.37039 1.00000i −2.84868 + 0.940756i 2.12595 − 0.693067i
329.9 − 1.00000i 1.51517 − 0.839210i −1.00000 −2.09320 − 0.786447i −0.839210 − 1.51517i −3.51327 1.00000i 1.59145 − 2.54308i −0.786447 + 2.09320i
329.10 − 1.00000i 1.51517 + 0.839210i −1.00000 −2.09320 + 0.786447i 0.839210 − 1.51517i 3.51327 1.00000i 1.59145 + 2.54308i 0.786447 + 2.09320i
329.11 − 1.00000i 1.69665 − 0.348406i −1.00000 1.46224 − 1.69170i −0.348406 − 1.69665i 1.01891 1.00000i 2.75723 − 1.18224i −1.69170 − 1.46224i
329.12 − 1.00000i 1.69665 + 0.348406i −1.00000 1.46224 + 1.69170i 0.348406 − 1.69665i −1.01891 1.00000i 2.75723 + 1.18224i 1.69170 − 1.46224i
329.13 1.00000i −1.69665 − 0.348406i −1.00000 −1.46224 + 1.69170i 0.348406 − 1.69665i 1.01891 − 1.00000i 2.75723 + 1.18224i −1.69170 − 1.46224i
329.14 1.00000i −1.69665 + 0.348406i −1.00000 −1.46224 − 1.69170i −0.348406 − 1.69665i −1.01891 − 1.00000i 2.75723 − 1.18224i 1.69170 − 1.46224i
329.15 1.00000i −1.51517 − 0.839210i −1.00000 2.09320 + 0.786447i 0.839210 − 1.51517i −3.51327 − 1.00000i 1.59145 + 2.54308i −0.786447 + 2.09320i
329.16 1.00000i −1.51517 + 0.839210i −1.00000 2.09320 − 0.786447i −0.839210 − 1.51517i 3.51327 − 1.00000i 1.59145 − 2.54308i 0.786447 + 2.09320i
329.17 1.00000i −0.275064 − 1.71007i −1.00000 −0.693067 + 2.12595i 1.71007 − 0.275064i 2.37039 − 1.00000i −2.84868 + 0.940756i −2.12595 − 0.693067i
329.18 1.00000i −0.275064 + 1.71007i −1.00000 −0.693067 − 2.12595i −1.71007 − 0.275064i −2.37039 − 1.00000i −2.84868 − 0.940756i 2.12595 − 0.693067i
329.19 1.00000i 0.275064 − 1.71007i −1.00000 0.693067 − 2.12595i 1.71007 + 0.275064i −2.37039 − 1.00000i −2.84868 − 0.940756i 2.12595 + 0.693067i
329.20 1.00000i 0.275064 + 1.71007i −1.00000 0.693067 + 2.12595i −1.71007 + 0.275064i 2.37039 − 1.00000i −2.84868 + 0.940756i −2.12595 + 0.693067i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 329.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
11.b odd 2 1 inner
15.d odd 2 1 inner
33.d even 2 1 inner
55.d odd 2 1 inner
165.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 330.2.f.a ✓ 24
3.b odd 2 1 inner 330.2.f.a ✓ 24
5.b even 2 1 inner 330.2.f.a ✓ 24
5.c odd 4 1 1650.2.d.i 12
5.c odd 4 1 1650.2.d.j 12
11.b odd 2 1 inner 330.2.f.a ✓ 24
15.d odd 2 1 inner 330.2.f.a ✓ 24
15.e even 4 1 1650.2.d.i 12
15.e even 4 1 1650.2.d.j 12
33.d even 2 1 inner 330.2.f.a ✓ 24
55.d odd 2 1 inner 330.2.f.a ✓ 24
55.e even 4 1 1650.2.d.i 12
55.e even 4 1 1650.2.d.j 12
165.d even 2 1 inner 330.2.f.a ✓ 24
165.l odd 4 1 1650.2.d.i 12
165.l odd 4 1 1650.2.d.j 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
330.2.f.a ✓ 24 1.a even 1 1 trivial
330.2.f.a ✓ 24 3.b odd 2 1 inner
330.2.f.a ✓ 24 5.b even 2 1 inner
330.2.f.a ✓ 24 11.b odd 2 1 inner
330.2.f.a ✓ 24 15.d odd 2 1 inner
330.2.f.a ✓ 24 33.d even 2 1 inner
330.2.f.a ✓ 24 55.d odd 2 1 inner
330.2.f.a ✓ 24 165.d even 2 1 inner
1650.2.d.i 12 5.c odd 4 1
1650.2.d.i 12 15.e even 4 1
1650.2.d.i 12 55.e even 4 1
1650.2.d.i 12 165.l odd 4 1
1650.2.d.j 12 5.c odd 4 1
1650.2.d.j 12 15.e even 4 1
1650.2.d.j 12 55.e even 4 1
1650.2.d.j 12 165.l odd 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(330, [\chi])\).