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Properties

Label 980.2.c.d
Level $980$
Weight $2$
Character orbit 980.c
Analytic conductor $7.825$
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] = [980,2,Mod(979,980)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(980, 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("980.979");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 980 = 2^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 980.c (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: \(7.82533939809\)
Analytic rank: \(0\)
Dimension: \(32\)
Twist minimal: no (minimal twist has level 140)
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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 12 q^{4} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q - 12 q^{4} - 8 q^{9} - 36 q^{16} + 52 q^{25} + 52 q^{30} - 28 q^{36} + 52 q^{44} + 44 q^{46} + 36 q^{50} - 8 q^{60} + 36 q^{64} + 8 q^{65} - 28 q^{74} - 144 q^{81} + 20 q^{85} - 16 q^{86}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
979.1 −1.29809 − 0.561208i − 1.88009i 1.37009 + 1.45700i −2.11556 − 0.724173i −1.05512 + 2.44053i 0 −0.960828 − 2.66023i −0.534733 2.33978 + 2.12731i
979.2 −1.29809 − 0.561208i 1.88009i 1.37009 + 1.45700i 2.11556 + 0.724173i 1.05512 − 2.44053i 0 −0.960828 − 2.66023i −0.534733 −2.33978 − 2.12731i
979.3 −1.29809 + 0.561208i − 1.88009i 1.37009 − 1.45700i 2.11556 − 0.724173i 1.05512 + 2.44053i 0 −0.960828 + 2.66023i −0.534733 −2.33978 + 2.12731i
979.4 −1.29809 + 0.561208i 1.88009i 1.37009 − 1.45700i −2.11556 + 0.724173i −1.05512 − 2.44053i 0 −0.960828 + 2.66023i −0.534733 2.33978 − 2.12731i
979.5 −0.940044 − 1.05656i − 2.59619i −0.232633 + 1.98642i 1.75874 − 1.38089i −2.74302 + 2.44053i 0 2.31746 − 1.62154i −3.74018 −3.11228 − 0.560116i
979.6 −0.940044 − 1.05656i 2.59619i −0.232633 + 1.98642i −1.75874 + 1.38089i 2.74302 − 2.44053i 0 2.31746 − 1.62154i −3.74018 3.11228 + 0.560116i
979.7 −0.940044 + 1.05656i − 2.59619i −0.232633 − 1.98642i −1.75874 − 1.38089i 2.74302 + 2.44053i 0 2.31746 + 1.62154i −3.74018 3.11228 − 0.560116i
979.8 −0.940044 + 1.05656i 2.59619i −0.232633 − 1.98642i 1.75874 + 1.38089i −2.74302 − 2.44053i 0 2.31746 + 1.62154i −3.74018 −3.11228 + 0.560116i
979.9 −0.739583 − 1.20541i − 0.732905i −0.906034 + 1.78300i −2.18048 − 0.495485i −0.883452 + 0.542044i 0 2.81934 − 0.226536i 2.46285 1.01538 + 2.99483i
979.10 −0.739583 − 1.20541i 0.732905i −0.906034 + 1.78300i 2.18048 + 0.495485i 0.883452 − 0.542044i 0 2.81934 − 0.226536i 2.46285 −1.01538 − 2.99483i
979.11 −0.739583 + 1.20541i − 0.732905i −0.906034 − 1.78300i 2.18048 − 0.495485i 0.883452 + 0.542044i 0 2.81934 + 0.226536i 2.46285 −1.01538 + 2.99483i
979.12 −0.739583 + 1.20541i 0.732905i −0.906034 − 1.78300i −2.18048 + 0.495485i −0.883452 − 0.542044i 0 2.81934 + 0.226536i 2.46285 1.01538 − 2.99483i
979.13 −0.366453 − 1.36591i − 1.47917i −1.73142 + 1.00108i −0.962692 − 2.01822i −2.02041 + 0.542044i 0 2.00188 + 1.99812i 0.812067 −2.40393 + 2.05453i
979.14 −0.366453 − 1.36591i 1.47917i −1.73142 + 1.00108i 0.962692 + 2.01822i 2.02041 − 0.542044i 0 2.00188 + 1.99812i 0.812067 2.40393 − 2.05453i
979.15 −0.366453 + 1.36591i − 1.47917i −1.73142 − 1.00108i 0.962692 − 2.01822i 2.02041 + 0.542044i 0 2.00188 − 1.99812i 0.812067 2.40393 + 2.05453i
979.16 −0.366453 + 1.36591i 1.47917i −1.73142 − 1.00108i −0.962692 + 2.01822i −2.02041 − 0.542044i 0 2.00188 − 1.99812i 0.812067 −2.40393 − 2.05453i
979.17 0.366453 − 1.36591i − 1.47917i −1.73142 − 1.00108i 0.962692 + 2.01822i −2.02041 − 0.542044i 0 −2.00188 + 1.99812i 0.812067 3.10949 − 0.575369i
979.18 0.366453 − 1.36591i 1.47917i −1.73142 − 1.00108i −0.962692 − 2.01822i 2.02041 + 0.542044i 0 −2.00188 + 1.99812i 0.812067 −3.10949 + 0.575369i
979.19 0.366453 + 1.36591i − 1.47917i −1.73142 + 1.00108i −0.962692 + 2.01822i 2.02041 − 0.542044i 0 −2.00188 − 1.99812i 0.812067 −3.10949 − 0.575369i
979.20 0.366453 + 1.36591i 1.47917i −1.73142 + 1.00108i 0.962692 − 2.01822i −2.02041 + 0.542044i 0 −2.00188 − 1.99812i 0.812067 3.10949 + 0.575369i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 979.32
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.b even 2 1 inner
7.b odd 2 1 inner
20.d odd 2 1 inner
28.d even 2 1 inner
35.c odd 2 1 inner
140.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 980.2.c.d 32
4.b odd 2 1 inner 980.2.c.d 32
5.b even 2 1 inner 980.2.c.d 32
7.b odd 2 1 inner 980.2.c.d 32
7.c even 3 1 140.2.s.b ✓ 32
7.c even 3 1 980.2.s.e 32
7.d odd 6 1 140.2.s.b ✓ 32
7.d odd 6 1 980.2.s.e 32
20.d odd 2 1 inner 980.2.c.d 32
28.d even 2 1 inner 980.2.c.d 32
28.f even 6 1 140.2.s.b ✓ 32
28.f even 6 1 980.2.s.e 32
28.g odd 6 1 140.2.s.b ✓ 32
28.g odd 6 1 980.2.s.e 32
35.c odd 2 1 inner 980.2.c.d 32
35.i odd 6 1 140.2.s.b ✓ 32
35.i odd 6 1 980.2.s.e 32
35.j even 6 1 140.2.s.b ✓ 32
35.j even 6 1 980.2.s.e 32
35.k even 12 2 700.2.p.e 32
35.l odd 12 2 700.2.p.e 32
140.c even 2 1 inner 980.2.c.d 32
140.p odd 6 1 140.2.s.b ✓ 32
140.p odd 6 1 980.2.s.e 32
140.s even 6 1 140.2.s.b ✓ 32
140.s even 6 1 980.2.s.e 32
140.w even 12 2 700.2.p.e 32
140.x odd 12 2 700.2.p.e 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.s.b ✓ 32 7.c even 3 1
140.2.s.b ✓ 32 7.d odd 6 1
140.2.s.b ✓ 32 28.f even 6 1
140.2.s.b ✓ 32 28.g odd 6 1
140.2.s.b ✓ 32 35.i odd 6 1
140.2.s.b ✓ 32 35.j even 6 1
140.2.s.b ✓ 32 140.p odd 6 1
140.2.s.b ✓ 32 140.s even 6 1
700.2.p.e 32 35.k even 12 2
700.2.p.e 32 35.l odd 12 2
700.2.p.e 32 140.w even 12 2
700.2.p.e 32 140.x odd 12 2
980.2.c.d 32 1.a even 1 1 trivial
980.2.c.d 32 4.b odd 2 1 inner
980.2.c.d 32 5.b even 2 1 inner
980.2.c.d 32 7.b odd 2 1 inner
980.2.c.d 32 20.d odd 2 1 inner
980.2.c.d 32 28.d even 2 1 inner
980.2.c.d 32 35.c odd 2 1 inner
980.2.c.d 32 140.c even 2 1 inner
980.2.s.e 32 7.c even 3 1
980.2.s.e 32 7.d odd 6 1
980.2.s.e 32 28.f even 6 1
980.2.s.e 32 28.g odd 6 1
980.2.s.e 32 35.i odd 6 1
980.2.s.e 32 35.j even 6 1
980.2.s.e 32 140.p odd 6 1
980.2.s.e 32 140.s even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + 13T_{3}^{6} + 53T_{3}^{4} + 77T_{3}^{2} + 28 \) acting on \(S_{2}^{\mathrm{new}}(980, [\chi])\). Copy content Toggle raw display