Properties

Label 8820.2.a.bg
Level $8820$
Weight $2$
Character orbit 8820.a
Self dual yes
Analytic conductor $70.428$
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] = [8820,2,Mod(1,8820)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8820, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("8820.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8820 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8820.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: \(70.4280545828\)
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: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 980)
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.

\(f(q)\) \(=\) \( q - q^{5}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{5} + (2 \beta + 1) q^{11} + (\beta - 5) q^{13} + (\beta + 5) q^{17} + ( - 4 \beta + 2) q^{19} + (\beta - 2) q^{23} + q^{25} + ( - 4 \beta - 1) q^{29} + (\beta - 6) q^{31} + ( - \beta - 2) q^{37} + ( - \beta + 2) q^{41} + (4 \beta + 6) q^{43} + ( - 7 \beta + 1) q^{47} + ( - 3 \beta + 8) q^{53} + ( - 2 \beta - 1) q^{55} + (\beta + 2) q^{59} + (2 \beta - 8) q^{61} + ( - \beta + 5) q^{65} + ( - 5 \beta - 4) q^{67} + ( - 6 \beta + 2) q^{71} + (2 \beta - 8) q^{73} + (10 \beta - 1) q^{79} - 8 q^{83} + ( - \beta - 5) q^{85} + 12 \beta q^{89} + (4 \beta - 2) q^{95} + ( - 9 \beta - 3) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{5} + 2 q^{11} - 10 q^{13} + 10 q^{17} + 4 q^{19} - 4 q^{23} + 2 q^{25} - 2 q^{29} - 12 q^{31} - 4 q^{37} + 4 q^{41} + 12 q^{43} + 2 q^{47} + 16 q^{53} - 2 q^{55} + 4 q^{59} - 16 q^{61} + 10 q^{65} - 8 q^{67} + 4 q^{71} - 16 q^{73} - 2 q^{79} - 16 q^{83} - 10 q^{85} - 4 q^{95} - 6 q^{97}+O(q^{100}) \) 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 0 0 −1.00000 0 0 0 0 0
1.2 0 0 0 −1.00000 0 0 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(5\) \(1\)
\(7\) \(1\)

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 8820.2.a.bg 2
3.b odd 2 1 980.2.a.j 2
7.b odd 2 1 8820.2.a.bl 2
12.b even 2 1 3920.2.a.bx 2
15.d odd 2 1 4900.2.a.z 2
15.e even 4 2 4900.2.e.q 4
21.c even 2 1 980.2.a.k yes 2
21.g even 6 2 980.2.i.k 4
21.h odd 6 2 980.2.i.l 4
84.h odd 2 1 3920.2.a.bo 2
105.g even 2 1 4900.2.a.x 2
105.k odd 4 2 4900.2.e.r 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
980.2.a.j 2 3.b odd 2 1
980.2.a.k yes 2 21.c even 2 1
980.2.i.k 4 21.g even 6 2
980.2.i.l 4 21.h odd 6 2
3920.2.a.bo 2 84.h odd 2 1
3920.2.a.bx 2 12.b even 2 1
4900.2.a.x 2 105.g even 2 1
4900.2.a.z 2 15.d odd 2 1
4900.2.e.q 4 15.e even 4 2
4900.2.e.r 4 105.k odd 4 2
8820.2.a.bg 2 1.a even 1 1 trivial
8820.2.a.bl 2 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8820))\):

\( T_{11}^{2} - 2T_{11} - 7 \) Copy content Toggle raw display
\( T_{13}^{2} + 10T_{13} + 23 \) Copy content Toggle raw display
\( T_{17}^{2} - 10T_{17} + 23 \) Copy content Toggle raw display
\( T_{31}^{2} + 12T_{31} + 34 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) 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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