↑

Properties

Label 980.2.q.h
Level $980$
Weight $2$
Character orbit 980.q
Analytic conductor $7.825$
Analytic rank $0$
Dimension $4$
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [980,2,Mod(569,980)]
 
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([0, 3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("980.569");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 980 = 2^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 980.q (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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{2} + 1) q^{3} + (\beta_{3} + \beta_{2} + \beta_1 + 2) q^{5} + (\beta_{2} + 1) q^{11} + (2 \beta_{2} + 1) q^{13} + (\beta_{3} + 2 \beta_1 + 3) q^{15} + ( - 3 \beta_{2} + 3) q^{17} + (2 \beta_{3} + 2 \beta_1) q^{19}+ \cdots + ( - 6 \beta_{2} - 3) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 6 q^{3} + 6 q^{5} + 2 q^{11} + 12 q^{15} + 18 q^{17} + 2 q^{25} + 28 q^{29} + 6 q^{33} + 6 q^{39} - 42 q^{47} + 18 q^{51} - 6 q^{65} - 40 q^{71} + 6 q^{75} + 6 q^{79} + 18 q^{81} + 36 q^{85} + 42 q^{87}+ \cdots - 8 q^{95}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 2 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{3} \) Copy content Toggle raw display

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

Character values

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

\(n\) \(101\) \(197\) \(491\)
\(\chi(n)\) \(\beta_{2}\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
569.1
0.707107 + 1.22474i
−0.707107 − 1.22474i
0.707107 − 1.22474i
−0.707107 + 1.22474i
0 1.50000 − 0.866025i 0 0.792893 + 2.09077i 0 0 0 0 0
569.2 0 1.50000 − 0.866025i 0 2.20711 − 0.358719i 0 0 0 0 0
949.1 0 1.50000 + 0.866025i 0 0.792893 − 2.09077i 0 0 0 0 0
949.2 0 1.50000 + 0.866025i 0 2.20711 + 0.358719i 0 0 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner
35.c odd 2 1 inner
35.j 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.q.h 4
5.b even 2 1 980.2.q.a 4
7.b odd 2 1 980.2.q.a 4
7.c even 3 1 980.2.e.e ✓ 4
7.c even 3 1 980.2.q.a 4
7.d odd 6 1 980.2.e.e ✓ 4
7.d odd 6 1 inner 980.2.q.h 4
35.c odd 2 1 inner 980.2.q.h 4
35.i odd 6 1 980.2.e.e ✓ 4
35.i odd 6 1 980.2.q.a 4
35.j even 6 1 980.2.e.e ✓ 4
35.j even 6 1 inner 980.2.q.h 4
35.k even 12 2 4900.2.a.bh 4
35.l odd 12 2 4900.2.a.bh 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
980.2.e.e ✓ 4 7.c even 3 1
980.2.e.e ✓ 4 7.d odd 6 1
980.2.e.e ✓ 4 35.i odd 6 1
980.2.e.e ✓ 4 35.j even 6 1
980.2.q.a 4 5.b even 2 1
980.2.q.a 4 7.b odd 2 1
980.2.q.a 4 7.c even 3 1
980.2.q.a 4 35.i odd 6 1
980.2.q.h 4 1.a even 1 1 trivial
980.2.q.h 4 7.d odd 6 1 inner
980.2.q.h 4 35.c odd 2 1 inner
980.2.q.h 4 35.j even 6 1 inner
4900.2.a.bh 4 35.k even 12 2
4900.2.a.bh 4 35.l odd 12 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(980, [\chi])\):

\( T_{3}^{2} - 3T_{3} + 3 \) Copy content Toggle raw display
\( T_{11}^{2} - T_{11} + 1 \) Copy content Toggle raw display
\( T_{19}^{4} + 8T_{19}^{2} + 64 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} - 3 T + 3)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} - 6 T^{3} + \cdots + 25 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 3)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 9 T + 27)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} + 8T^{2} + 64 \) Copy content Toggle raw display
$23$ \( T^{4} - 6T^{2} + 36 \) Copy content Toggle raw display
$29$ \( (T - 7)^{4} \) Copy content Toggle raw display
$31$ \( T^{4} + 50T^{2} + 2500 \) Copy content Toggle raw display
$37$ \( T^{4} - 54T^{2} + 2916 \) Copy content Toggle raw display
$41$ \( (T^{2} - 50)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 96)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 21 T + 147)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} - 150 T^{2} + 22500 \) Copy content Toggle raw display
$59$ \( T^{4} + 50T^{2} + 2500 \) Copy content Toggle raw display
$61$ \( T^{4} + 200 T^{2} + 40000 \) Copy content Toggle raw display
$67$ \( T^{4} - 150 T^{2} + 22500 \) Copy content Toggle raw display
$71$ \( (T + 10)^{4} \) Copy content Toggle raw display
$73$ \( T^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} - 3 T + 9)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} \) Copy content Toggle raw display
$89$ \( T^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} + 27)^{2} \) Copy content Toggle raw display
show more
show less