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Properties

Label 980.2.s.a
Level $980$
Weight $2$
Character orbit 980.s
Analytic conductor $7.825$
Analytic rank $0$
Dimension $8$
CM discriminant -20
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(19,980)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(980, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 3, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("980.19");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 980 = 2^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 980.s (of order \(6\), degree \(2\), 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: \(7.82533939809\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.3317760000.3
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{6} + 7x^{4} - 36x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 140)
Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{5} q^{2} + ( - \beta_{5} + \beta_1) q^{3} - 2 \beta_{4} q^{4} + \beta_{7} q^{5} + ( - \beta_{7} - 2 \beta_{4} + \beta_{2} + 1) q^{6} - 2 \beta_{3} q^{8} + ( - \beta_{7} - \beta_{4} + 2 \beta_{2} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{5} q^{2} + ( - \beta_{5} + \beta_1) q^{3} - 2 \beta_{4} q^{4} + \beta_{7} q^{5} + ( - \beta_{7} - 2 \beta_{4} + \beta_{2} + 1) q^{6} - 2 \beta_{3} q^{8} + ( - \beta_{7} - \beta_{4} + 2 \beta_{2} + 1) q^{9} + ( - \beta_{5} - \beta_{3} + 2 \beta_1) q^{10} + (2 \beta_{6} - 2 \beta_{5} - 2 \beta_{3}) q^{12} + ( - \beta_{6} + 2 \beta_{3} + \beta_1) q^{15} + (4 \beta_{4} - 4) q^{16} + (4 \beta_{6} - 4 \beta_{5} + \cdots + 2 \beta_1) q^{18}+ \cdots + (4 \beta_{7} + 4 \beta_{4} + 4) q^{96}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{4} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{4} + 4 q^{9} - 16 q^{16} - 24 q^{24} + 20 q^{25} - 24 q^{29} + 20 q^{30} - 16 q^{36} + 60 q^{45} - 4 q^{46} - 36 q^{54} + 48 q^{61} + 64 q^{64} - 16 q^{81} - 36 q^{86} - 36 q^{89} + 48 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{6} + 7x^{4} - 36x^{2} + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{6} + 14\nu^{4} + 7\nu^{2} - 36 ) / 63 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -4\nu^{7} + 7\nu^{5} + 35\nu^{3} + 81\nu ) / 189 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -4\nu^{6} + 7\nu^{4} - 28\nu^{2} + 144 ) / 63 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -5\nu^{7} - 7\nu^{5} - 35\nu^{3} + 180\nu ) / 189 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{7} - 4\nu^{5} + 7\nu^{3} - 36\nu ) / 27 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -8\nu^{6} + 14\nu^{4} + 7\nu^{2} + 162 ) / 63 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} - 2\beta_{4} + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{6} - \beta_{5} + 3\beta_{3} + \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} + 4\beta_{2} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -5\beta_{6} - 7\beta_{5} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -7\beta_{7} + 7\beta_{2} + 22 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -21\beta_{5} - 21\beta_{3} + 29\beta_1 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/980\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(197\) \(491\)
\(\chi(n)\) \(1 - \beta_{4}\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
−1.72286 + 0.178197i
1.01575 − 1.40294i
−1.01575 + 1.40294i
1.72286 − 0.178197i
−1.72286 − 0.178197i
1.01575 + 1.40294i
−1.01575 − 1.40294i
1.72286 + 0.178197i
−0.707107 + 1.22474i −2.42997 + 1.40294i −1.00000 − 1.73205i 1.93649 + 1.11803i − 3.96812i 0 2.82843 2.43649 − 4.22013i −2.73861 + 1.58114i
19.2 −0.707107 + 1.22474i 0.308646 − 0.178197i −1.00000 − 1.73205i −1.93649 − 1.11803i 0.504017i 0 2.82843 −1.43649 + 2.48808i 2.73861 − 1.58114i
19.3 0.707107 − 1.22474i −0.308646 + 0.178197i −1.00000 − 1.73205i −1.93649 − 1.11803i 0.504017i 0 −2.82843 −1.43649 + 2.48808i −2.73861 + 1.58114i
19.4 0.707107 − 1.22474i 2.42997 − 1.40294i −1.00000 − 1.73205i 1.93649 + 1.11803i − 3.96812i 0 −2.82843 2.43649 − 4.22013i 2.73861 − 1.58114i
619.1 −0.707107 − 1.22474i −2.42997 − 1.40294i −1.00000 + 1.73205i 1.93649 − 1.11803i 3.96812i 0 2.82843 2.43649 + 4.22013i −2.73861 − 1.58114i
619.2 −0.707107 − 1.22474i 0.308646 + 0.178197i −1.00000 + 1.73205i −1.93649 + 1.11803i − 0.504017i 0 2.82843 −1.43649 − 2.48808i 2.73861 + 1.58114i
619.3 0.707107 + 1.22474i −0.308646 − 0.178197i −1.00000 + 1.73205i −1.93649 + 1.11803i − 0.504017i 0 −2.82843 −1.43649 − 2.48808i −2.73861 − 1.58114i
619.4 0.707107 + 1.22474i 2.42997 + 1.40294i −1.00000 + 1.73205i 1.93649 − 1.11803i 3.96812i 0 −2.82843 2.43649 + 4.22013i 2.73861 + 1.58114i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
20.d odd 2 1 CM by \(\Q(\sqrt{-5}) \)
4.b odd 2 1 inner
5.b even 2 1 inner
7.d odd 6 1 inner
28.f even 6 1 inner
35.i odd 6 1 inner
140.s even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 980.2.s.a 8
4.b odd 2 1 inner 980.2.s.a 8
5.b even 2 1 inner 980.2.s.a 8
7.b odd 2 1 140.2.s.a ✓ 8
7.c even 3 1 140.2.s.a ✓ 8
7.c even 3 1 980.2.c.b 8
7.d odd 6 1 980.2.c.b 8
7.d odd 6 1 inner 980.2.s.a 8
20.d odd 2 1 CM 980.2.s.a 8
28.d even 2 1 140.2.s.a ✓ 8
28.f even 6 1 980.2.c.b 8
28.f even 6 1 inner 980.2.s.a 8
28.g odd 6 1 140.2.s.a ✓ 8
28.g odd 6 1 980.2.c.b 8
35.c odd 2 1 140.2.s.a ✓ 8
35.f even 4 2 700.2.p.b 8
35.i odd 6 1 980.2.c.b 8
35.i odd 6 1 inner 980.2.s.a 8
35.j even 6 1 140.2.s.a ✓ 8
35.j even 6 1 980.2.c.b 8
35.l odd 12 2 700.2.p.b 8
140.c even 2 1 140.2.s.a ✓ 8
140.j odd 4 2 700.2.p.b 8
140.p odd 6 1 140.2.s.a ✓ 8
140.p odd 6 1 980.2.c.b 8
140.s even 6 1 980.2.c.b 8
140.s even 6 1 inner 980.2.s.a 8
140.w even 12 2 700.2.p.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.s.a ✓ 8 7.b odd 2 1
140.2.s.a ✓ 8 7.c even 3 1
140.2.s.a ✓ 8 28.d even 2 1
140.2.s.a ✓ 8 28.g odd 6 1
140.2.s.a ✓ 8 35.c odd 2 1
140.2.s.a ✓ 8 35.j even 6 1
140.2.s.a ✓ 8 140.c even 2 1
140.2.s.a ✓ 8 140.p odd 6 1
700.2.p.b 8 35.f even 4 2
700.2.p.b 8 35.l odd 12 2
700.2.p.b 8 140.j odd 4 2
700.2.p.b 8 140.w even 12 2
980.2.c.b 8 7.c even 3 1
980.2.c.b 8 7.d odd 6 1
980.2.c.b 8 28.f even 6 1
980.2.c.b 8 28.g odd 6 1
980.2.c.b 8 35.i odd 6 1
980.2.c.b 8 35.j even 6 1
980.2.c.b 8 140.p odd 6 1
980.2.c.b 8 140.s even 6 1
980.2.s.a 8 1.a even 1 1 trivial
980.2.s.a 8 4.b odd 2 1 inner
980.2.s.a 8 5.b even 2 1 inner
980.2.s.a 8 7.d odd 6 1 inner
980.2.s.a 8 20.d odd 2 1 CM
980.2.s.a 8 28.f even 6 1 inner
980.2.s.a 8 35.i odd 6 1 inner
980.2.s.a 8 140.s even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} - 8T_{3}^{6} + 63T_{3}^{4} - 8T_{3}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(980, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 2 T^{2} + 4)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} - 8 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( (T^{4} - 5 T^{2} + 25)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( T^{8} \) Copy content Toggle raw display
$17$ \( T^{8} \) Copy content Toggle raw display
$19$ \( T^{8} \) Copy content Toggle raw display
$23$ \( T^{8} + 136 T^{6} + \cdots + 20151121 \) Copy content Toggle raw display
$29$ \( (T^{2} + 6 T - 51)^{4} \) Copy content Toggle raw display
$31$ \( T^{8} \) Copy content Toggle raw display
$37$ \( T^{8} \) Copy content Toggle raw display
$41$ \( (T^{4} + 226 T^{2} + 10609)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} - 96 T^{2} + 1089)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} - 90 T^{2} + 8100)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} \) Copy content Toggle raw display
$59$ \( T^{8} \) Copy content Toggle raw display
$61$ \( (T^{4} - 24 T^{3} + 195 T^{2} + \cdots + 9)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 1121513121 \) Copy content Toggle raw display
$71$ \( T^{8} \) Copy content Toggle raw display
$73$ \( T^{8} \) Copy content Toggle raw display
$79$ \( T^{8} \) Copy content Toggle raw display
$83$ \( (T^{4} + 408 T^{2} + 25281)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 18 T^{3} + \cdots + 2809)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} \) Copy content Toggle raw display
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