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Properties

Label 490.2.l.d
Level $490$
Weight $2$
Character orbit 490.l
Analytic conductor $3.913$
Analytic rank $0$
Dimension $32$
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] = [490,2,Mod(117,490)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(490, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([3, 10]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("490.117");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 490 = 2 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 490.l (of order \(12\), degree \(4\), not 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: \(3.91266969904\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(8\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 16 q^{11} - 48 q^{15} + 16 q^{16} + 16 q^{18} + 32 q^{22} - 16 q^{23} - 32 q^{25} - 40 q^{30} - 96 q^{36} + 48 q^{37} + 32 q^{43} + 16 q^{46} - 64 q^{50} + 80 q^{51} - 32 q^{53} + 96 q^{57} - 16 q^{58} - 40 q^{60} - 32 q^{65} - 16 q^{67} + 32 q^{71} + 16 q^{72} + 112 q^{81} + 48 q^{85} - 48 q^{86} + 16 q^{88} + 32 q^{92} - 64 q^{93} - 40 q^{95}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
117.1 −0.965926 + 0.258819i −0.830751 + 3.10041i 0.866025 − 0.500000i −1.65469 + 1.50399i − 3.20978i 0 −0.707107 + 0.707107i −6.32429 − 3.65133i 1.20905 − 1.88101i
117.2 −0.965926 + 0.258819i −0.550608 + 2.05490i 0.866025 − 0.500000i −1.20905 − 1.88101i − 2.12738i 0 −0.707107 + 0.707107i −1.32135 − 0.762882i 1.65469 + 1.50399i
117.3 −0.965926 + 0.258819i 0.550608 − 2.05490i 0.866025 − 0.500000i 1.20905 + 1.88101i 2.12738i 0 −0.707107 + 0.707107i −1.32135 − 0.762882i −1.65469 − 1.50399i
117.4 −0.965926 + 0.258819i 0.830751 − 3.10041i 0.866025 − 0.500000i 1.65469 − 1.50399i 3.20978i 0 −0.707107 + 0.707107i −6.32429 − 3.65133i −1.20905 + 1.88101i
117.5 0.965926 − 0.258819i −0.777293 + 2.90090i 0.866025 − 0.500000i 0.00103668 + 2.23607i 3.00323i 0 0.707107 − 0.707107i −5.21295 − 3.00970i 0.579738 + 2.15961i
117.6 0.965926 − 0.258819i −0.100966 + 0.376812i 0.866025 − 0.500000i −0.579738 + 2.15961i 0.390104i 0 0.707107 − 0.707107i 2.46628 + 1.42391i −0.00103668 + 2.23607i
117.7 0.965926 − 0.258819i 0.100966 − 0.376812i 0.866025 − 0.500000i 0.579738 − 2.15961i − 0.390104i 0 0.707107 − 0.707107i 2.46628 + 1.42391i 0.00103668 − 2.23607i
117.8 0.965926 − 0.258819i 0.777293 − 2.90090i 0.866025 − 0.500000i −0.00103668 − 2.23607i − 3.00323i 0 0.707107 − 0.707107i −5.21295 − 3.00970i −0.579738 − 2.15961i
227.1 −0.258819 + 0.965926i −2.90090 + 0.777293i −0.866025 − 0.500000i 1.93701 + 1.11714i − 3.00323i 0 0.707107 − 0.707107i 5.21295 − 3.00970i −1.58041 + 1.58187i
227.2 −0.258819 + 0.965926i −0.376812 + 0.100966i −0.866025 − 0.500000i 1.58041 + 1.58187i − 0.390104i 0 0.707107 − 0.707107i −2.46628 + 1.42391i −1.93701 + 1.11714i
227.3 −0.258819 + 0.965926i 0.376812 − 0.100966i −0.866025 − 0.500000i −1.58041 − 1.58187i 0.390104i 0 0.707107 − 0.707107i −2.46628 + 1.42391i 1.93701 − 1.11714i
227.4 −0.258819 + 0.965926i 2.90090 − 0.777293i −0.866025 − 0.500000i −1.93701 − 1.11714i 3.00323i 0 0.707107 − 0.707107i 5.21295 − 3.00970i 1.58041 − 1.58187i
227.5 0.258819 − 0.965926i −3.10041 + 0.830751i −0.866025 − 0.500000i 0.475150 + 2.18500i 3.20978i 0 −0.707107 + 0.707107i 6.32429 − 3.65133i 2.23353 + 0.106560i
227.6 0.258819 − 0.965926i −2.05490 + 0.550608i −0.866025 − 0.500000i −2.23353 + 0.106560i 2.12738i 0 −0.707107 + 0.707107i 1.32135 − 0.762882i −0.475150 + 2.18500i
227.7 0.258819 − 0.965926i 2.05490 − 0.550608i −0.866025 − 0.500000i 2.23353 − 0.106560i − 2.12738i 0 −0.707107 + 0.707107i 1.32135 − 0.762882i 0.475150 − 2.18500i
227.8 0.258819 − 0.965926i 3.10041 − 0.830751i −0.866025 − 0.500000i −0.475150 − 2.18500i − 3.20978i 0 −0.707107 + 0.707107i 6.32429 − 3.65133i −2.23353 − 0.106560i
313.1 −0.258819 − 0.965926i −2.90090 − 0.777293i −0.866025 + 0.500000i 1.93701 − 1.11714i 3.00323i 0 0.707107 + 0.707107i 5.21295 + 3.00970i −1.58041 − 1.58187i
313.2 −0.258819 − 0.965926i −0.376812 − 0.100966i −0.866025 + 0.500000i 1.58041 − 1.58187i 0.390104i 0 0.707107 + 0.707107i −2.46628 − 1.42391i −1.93701 − 1.11714i
313.3 −0.258819 − 0.965926i 0.376812 + 0.100966i −0.866025 + 0.500000i −1.58041 + 1.58187i − 0.390104i 0 0.707107 + 0.707107i −2.46628 − 1.42391i 1.93701 + 1.11714i
313.4 −0.258819 − 0.965926i 2.90090 + 0.777293i −0.866025 + 0.500000i −1.93701 + 1.11714i − 3.00323i 0 0.707107 + 0.707107i 5.21295 + 3.00970i 1.58041 + 1.58187i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 117.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner
35.f even 4 1 inner
35.k even 12 1 inner
35.l odd 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 490.2.l.d 32
5.c odd 4 1 inner 490.2.l.d 32
7.b odd 2 1 inner 490.2.l.d 32
7.c even 3 1 490.2.g.b ✓ 16
7.c even 3 1 inner 490.2.l.d 32
7.d odd 6 1 490.2.g.b ✓ 16
7.d odd 6 1 inner 490.2.l.d 32
35.f even 4 1 inner 490.2.l.d 32
35.k even 12 1 490.2.g.b ✓ 16
35.k even 12 1 inner 490.2.l.d 32
35.l odd 12 1 490.2.g.b ✓ 16
35.l odd 12 1 inner 490.2.l.d 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
490.2.g.b ✓ 16 7.c even 3 1
490.2.g.b ✓ 16 7.d odd 6 1
490.2.g.b ✓ 16 35.k even 12 1
490.2.g.b ✓ 16 35.l odd 12 1
490.2.l.d 32 1.a even 1 1 trivial
490.2.l.d 32 5.c odd 4 1 inner
490.2.l.d 32 7.b odd 2 1 inner
490.2.l.d 32 7.c even 3 1 inner
490.2.l.d 32 7.d odd 6 1 inner
490.2.l.d 32 35.f even 4 1 inner
490.2.l.d 32 35.k even 12 1 inner
490.2.l.d 32 35.l odd 12 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{32} - 208 T_{3}^{28} + 30784 T_{3}^{24} - 2241536 T_{3}^{20} + 118898688 T_{3}^{16} + \cdots + 16777216 \) acting on \(S_{2}^{\mathrm{new}}(490, [\chi])\). Copy content Toggle raw display