Properties

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

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [980,2,Mod(589,980)]
 
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([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("980.589");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 980 = 2^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 980.e (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: \(4\)
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: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{3} + (\beta_{2} + \beta_1) q^{5}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{3} + (\beta_{2} + \beta_1) q^{5} - q^{11} - \beta_{2} q^{13} + (\beta_{3} + 3) q^{15} - 3 \beta_{2} q^{17} + 2 \beta_1 q^{19} - \beta_{3} q^{23} + ( - 2 \beta_{3} - 1) q^{25} - 3 \beta_{2} q^{27} + 7 q^{29} + 5 \beta_1 q^{31} + \beta_{2} q^{33} - 3 \beta_{3} q^{37} - 3 q^{39} + 5 \beta_1 q^{41} + 4 \beta_{3} q^{43} - 7 \beta_{2} q^{47} - 9 q^{51} - 5 \beta_{3} q^{53} + ( - \beta_{2} - \beta_1) q^{55} + 2 \beta_{3} q^{57} + 5 \beta_1 q^{59} - 10 \beta_1 q^{61} + (\beta_{3} + 3) q^{65} + 5 \beta_{3} q^{67} + 3 \beta_1 q^{69} - 10 q^{71} + (\beta_{2} + 6 \beta_1) q^{75} - 3 q^{79} - 9 q^{81} + (3 \beta_{3} + 9) q^{85} - 7 \beta_{2} q^{87} + 5 \beta_{3} q^{93} + ( - 2 \beta_{3} + 4) q^{95} + 3 \beta_{2} q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{11} + 12 q^{15} - 4 q^{25} + 28 q^{29} - 12 q^{39} - 36 q^{51} + 12 q^{65} - 40 q^{71} - 12 q^{79} - 36 q^{81} + 36 q^{85} + 16 q^{95}+O(q^{100}) \) 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

\(\beta_{1}\)\(=\) \( ( \nu^{3} ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} + 4\nu ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_1 \) Copy content Toggle raw display

Character values

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

\(n\) \(101\) \(197\) \(491\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
589.1
0.707107 + 1.22474i
−0.707107 1.22474i
0.707107 1.22474i
−0.707107 + 1.22474i
0 1.73205i 0 −1.41421 + 1.73205i 0 0 0 0 0
589.2 0 1.73205i 0 1.41421 + 1.73205i 0 0 0 0 0
589.3 0 1.73205i 0 −1.41421 1.73205i 0 0 0 0 0
589.4 0 1.73205i 0 1.41421 1.73205i 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
5.b even 2 1 inner
7.b odd 2 1 inner
35.c odd 2 1 inner

Twists

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

Hecke kernels

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} + 3)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} + 2T^{2} + 25 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( (T + 1)^{4} \) Copy content Toggle raw display
$13$ \( (T^{2} + 3)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 27)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 8)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 6)^{2} \) Copy content Toggle raw display
$29$ \( (T - 7)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} - 50)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 54)^{2} \) 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
$47$ \( (T^{2} + 147)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 150)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 50)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 200)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 150)^{2} \) Copy content Toggle raw display
$71$ \( (T + 10)^{4} \) Copy content Toggle raw display
$73$ \( T^{4} \) Copy content Toggle raw display
$79$ \( (T + 3)^{4} \) 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
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