Properties

Label 980.2.a.j
Level $980$
Weight $2$
Character orbit 980.a
Self dual yes
Analytic conductor $7.825$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [980,2,Mod(1,980)]
 
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, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("980.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 980 = 2^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 980.a (trivial)

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: yes
Analytic conductor: \(7.82533939809\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta - 1) q^{3} + q^{5} - 2 \beta q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (\beta - 1) q^{3} + q^{5} - 2 \beta q^{9} + ( - 2 \beta - 1) q^{11} + (\beta - 5) q^{13} + (\beta - 1) q^{15} + ( - \beta - 5) q^{17} + ( - 4 \beta + 2) q^{19} + ( - \beta + 2) q^{23} + q^{25} + ( - \beta - 1) q^{27} + (4 \beta + 1) q^{29} + (\beta - 6) q^{31} + (\beta - 3) q^{33} + ( - \beta - 2) q^{37} + ( - 6 \beta + 7) q^{39} + (\beta - 2) q^{41} + (4 \beta + 6) q^{43} - 2 \beta q^{45} + (7 \beta - 1) q^{47} + ( - 4 \beta + 3) q^{51} + (3 \beta - 8) q^{53} + ( - 2 \beta - 1) q^{55} + (6 \beta - 10) q^{57} + ( - \beta - 2) q^{59} + (2 \beta - 8) q^{61} + (\beta - 5) q^{65} + ( - 5 \beta - 4) q^{67} + (3 \beta - 4) q^{69} + (6 \beta - 2) q^{71} + (2 \beta - 8) q^{73} + (\beta - 1) q^{75} + (10 \beta - 1) q^{79} + (6 \beta - 1) q^{81} + 8 q^{83} + ( - \beta - 5) q^{85} + ( - 3 \beta + 7) q^{87} - 12 \beta q^{89} + ( - 7 \beta + 8) q^{93} + ( - 4 \beta + 2) q^{95} + ( - 9 \beta - 3) q^{97} + (2 \beta + 8) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} + 2 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} + 2 q^{5} - 2 q^{11} - 10 q^{13} - 2 q^{15} - 10 q^{17} + 4 q^{19} + 4 q^{23} + 2 q^{25} - 2 q^{27} + 2 q^{29} - 12 q^{31} - 6 q^{33} - 4 q^{37} + 14 q^{39} - 4 q^{41} + 12 q^{43} - 2 q^{47} + 6 q^{51} - 16 q^{53} - 2 q^{55} - 20 q^{57} - 4 q^{59} - 16 q^{61} - 10 q^{65} - 8 q^{67} - 8 q^{69} - 4 q^{71} - 16 q^{73} - 2 q^{75} - 2 q^{79} - 2 q^{81} + 16 q^{83} - 10 q^{85} + 14 q^{87} + 16 q^{93} + 4 q^{95} - 6 q^{97} + 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.41421
1.41421
0 −2.41421 0 1.00000 0 0 0 2.82843 0
1.2 0 0.414214 0 1.00000 0 0 0 −2.82843 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(-1\)
\(7\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 980.2.a.j 2
3.b odd 2 1 8820.2.a.bg 2
4.b odd 2 1 3920.2.a.bx 2
5.b even 2 1 4900.2.a.z 2
5.c odd 4 2 4900.2.e.q 4
7.b odd 2 1 980.2.a.k yes 2
7.c even 3 2 980.2.i.l 4
7.d odd 6 2 980.2.i.k 4
21.c even 2 1 8820.2.a.bl 2
28.d even 2 1 3920.2.a.bo 2
35.c odd 2 1 4900.2.a.x 2
35.f even 4 2 4900.2.e.r 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
980.2.a.j 2 1.a even 1 1 trivial
980.2.a.k yes 2 7.b odd 2 1
980.2.i.k 4 7.d odd 6 2
980.2.i.l 4 7.c even 3 2
3920.2.a.bo 2 28.d even 2 1
3920.2.a.bx 2 4.b odd 2 1
4900.2.a.x 2 35.c odd 2 1
4900.2.a.z 2 5.b even 2 1
4900.2.e.q 4 5.c odd 4 2
4900.2.e.r 4 35.f even 4 2
8820.2.a.bg 2 3.b odd 2 1
8820.2.a.bl 2 21.c even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(980))\):

\( T_{3}^{2} + 2T_{3} - 1 \) Copy content Toggle raw display
\( T_{11}^{2} + 2T_{11} - 7 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 2T - 1 \) Copy content Toggle raw display
$5$ \( (T - 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 2T - 7 \) Copy content Toggle raw display
$13$ \( T^{2} + 10T + 23 \) Copy content Toggle raw display
$17$ \( T^{2} + 10T + 23 \) Copy content Toggle raw display
$19$ \( T^{2} - 4T - 28 \) Copy content Toggle raw display
$23$ \( T^{2} - 4T + 2 \) Copy content Toggle raw display
$29$ \( T^{2} - 2T - 31 \) Copy content Toggle raw display
$31$ \( T^{2} + 12T + 34 \) Copy content Toggle raw display
$37$ \( T^{2} + 4T + 2 \) Copy content Toggle raw display
$41$ \( T^{2} + 4T + 2 \) Copy content Toggle raw display
$43$ \( T^{2} - 12T + 4 \) Copy content Toggle raw display
$47$ \( T^{2} + 2T - 97 \) Copy content Toggle raw display
$53$ \( T^{2} + 16T + 46 \) Copy content Toggle raw display
$59$ \( T^{2} + 4T + 2 \) Copy content Toggle raw display
$61$ \( T^{2} + 16T + 56 \) Copy content Toggle raw display
$67$ \( T^{2} + 8T - 34 \) Copy content Toggle raw display
$71$ \( T^{2} + 4T - 68 \) Copy content Toggle raw display
$73$ \( T^{2} + 16T + 56 \) Copy content Toggle raw display
$79$ \( T^{2} + 2T - 199 \) Copy content Toggle raw display
$83$ \( (T - 8)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 288 \) Copy content Toggle raw display
$97$ \( T^{2} + 6T - 153 \) Copy content Toggle raw display
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