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Properties

Label 660.2.bb.a
Level $660$
Weight $2$
Character orbit 660.bb
Analytic conductor $5.270$
Analytic rank $0$
Dimension $96$
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] = [660,2,Mod(29,660)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(660, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 5, 5, 7]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("660.29");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 660 = 2^{2} \cdot 3 \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 660.bb (of order \(10\), degree \(4\), 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: \(5.27012653340\)
Analytic rank: \(0\)
Dimension: \(96\)
Relative dimension: \(24\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 96 q - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 96 q - 4 q^{9} + q^{15} + 4 q^{25} + 32 q^{31} + 20 q^{39} + 30 q^{45} + 24 q^{49} - 20 q^{51} - 6 q^{55} + 20 q^{61} - 18 q^{69} + 57 q^{75} - 20 q^{79} - 28 q^{81} + 60 q^{85} - 92 q^{91} - 70 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
29.1 0 −1.72824 + 0.114875i 0 2.03632 − 0.923789i 0 0.359074 + 1.10511i 0 2.97361 − 0.397062i 0
29.2 0 −1.68752 − 0.390242i 0 0.519201 + 2.17496i 0 −0.939019 − 2.89000i 0 2.69542 + 1.31708i 0
29.3 0 −1.60108 + 0.660723i 0 0.990546 + 2.00470i 0 1.16743 + 3.59298i 0 2.12689 − 2.11574i 0
29.4 0 −1.59461 − 0.676185i 0 −0.519201 − 2.17496i 0 −0.939019 − 2.89000i 0 2.08555 + 2.15650i 0
29.5 0 −1.33065 − 1.10877i 0 −2.03632 + 0.923789i 0 0.359074 + 1.10511i 0 0.541267 + 2.95077i 0
29.6 0 −1.32321 + 1.11764i 0 −1.65903 − 1.49920i 0 −0.293488 − 0.903262i 0 0.501771 − 2.95774i 0
29.7 0 −1.11219 + 1.32779i 0 −1.67505 + 1.48128i 0 −1.15406 − 3.55182i 0 −0.526065 − 2.95352i 0
29.8 0 −1.02125 + 1.39894i 0 0.865669 − 2.06170i 0 0.417126 + 1.28378i 0 −0.914085 − 2.85735i 0
29.9 0 −0.906935 − 1.47563i 0 −0.990546 − 2.00470i 0 1.16743 + 3.59298i 0 −1.35494 + 2.67659i 0
29.10 0 −0.413568 − 1.68195i 0 1.65903 + 1.49920i 0 −0.293488 − 0.903262i 0 −2.65792 + 1.39120i 0
29.11 0 −0.119324 − 1.72794i 0 1.67505 − 1.48128i 0 −1.15406 − 3.55182i 0 −2.97152 + 0.412369i 0
29.12 0 −0.00393258 − 1.73205i 0 −0.865669 + 2.06170i 0 0.417126 + 1.28378i 0 −2.99997 + 0.0136228i 0
29.13 0 0.00393258 + 1.73205i 0 2.22830 − 0.186200i 0 −0.417126 − 1.28378i 0 −2.99997 + 0.0136228i 0
29.14 0 0.119324 + 1.72794i 0 −1.92640 + 1.13533i 0 1.15406 + 3.55182i 0 −2.97152 + 0.412369i 0
29.15 0 0.413568 + 1.68195i 0 0.913157 + 2.04111i 0 0.293488 + 0.903262i 0 −2.65792 + 1.39120i 0
29.16 0 0.906935 + 1.47563i 0 −1.60049 − 1.56155i 0 −1.16743 − 3.59298i 0 −1.35494 + 2.67659i 0
29.17 0 1.02125 − 1.39894i 0 −2.22830 + 0.186200i 0 −0.417126 − 1.28378i 0 −0.914085 − 2.85735i 0
29.18 0 1.11219 − 1.32779i 0 1.92640 − 1.13533i 0 1.15406 + 3.55182i 0 −0.526065 − 2.95352i 0
29.19 0 1.32321 − 1.11764i 0 −0.913157 − 2.04111i 0 0.293488 + 0.903262i 0 0.501771 − 2.95774i 0
29.20 0 1.33065 + 1.10877i 0 1.50783 − 1.65119i 0 −0.359074 − 1.10511i 0 0.541267 + 2.95077i 0
See all 96 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 29.24
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.d odd 10 1 inner
15.d odd 2 1 inner
33.f even 10 1 inner
55.h odd 10 1 inner
165.r even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 660.2.bb.a ✓ 96
3.b odd 2 1 inner 660.2.bb.a ✓ 96
5.b even 2 1 inner 660.2.bb.a ✓ 96
11.d odd 10 1 inner 660.2.bb.a ✓ 96
15.d odd 2 1 inner 660.2.bb.a ✓ 96
33.f even 10 1 inner 660.2.bb.a ✓ 96
55.h odd 10 1 inner 660.2.bb.a ✓ 96
165.r even 10 1 inner 660.2.bb.a ✓ 96
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
660.2.bb.a ✓ 96 1.a even 1 1 trivial
660.2.bb.a ✓ 96 3.b odd 2 1 inner
660.2.bb.a ✓ 96 5.b even 2 1 inner
660.2.bb.a ✓ 96 11.d odd 10 1 inner
660.2.bb.a ✓ 96 15.d odd 2 1 inner
660.2.bb.a ✓ 96 33.f even 10 1 inner
660.2.bb.a ✓ 96 55.h odd 10 1 inner
660.2.bb.a ✓ 96 165.r even 10 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(660, [\chi])\).