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Properties

Label 8820.2.a.bj
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 / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [8820,2,Mod(1,8820)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://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);
 
Copy content sage: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")
 
Level: \( N \) \(=\) \( 8820 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8820.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,2,0,0,0,0,0,-2,0,0,0,0,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(70.4280545828\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{7}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 7 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 420)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{7}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{5} + (\beta - 1) q^{11} + \beta q^{13} + ( - \beta + 1) q^{17} + ( - 2 \beta - 3) q^{19} + ( - \beta + 1) q^{23} + q^{25} + ( - \beta - 5) q^{29} + (2 \beta - 1) q^{31} + (\beta - 2) q^{37} + (3 \beta - 3) q^{41}+ \cdots - 8 q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{5} - 2 q^{11} + 2 q^{17} - 6 q^{19} + 2 q^{23} + 2 q^{25} - 10 q^{29} - 2 q^{31} - 4 q^{37} - 6 q^{41} + 4 q^{43} + 12 q^{47} + 4 q^{53} - 2 q^{55} + 6 q^{59} - 16 q^{61} - 4 q^{67} - 22 q^{71}+ \cdots - 16 q^{97}+O(q^{100}) \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content 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
−2.64575
2.64575
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.bj 2
3.b odd 2 1 2940.2.a.m 2
7.b odd 2 1 8820.2.a.be 2
7.d odd 6 2 1260.2.s.f 4
21.c even 2 1 2940.2.a.s 2
21.g even 6 2 420.2.q.c ✓ 4
21.h odd 6 2 2940.2.q.t 4
84.j odd 6 2 1680.2.bg.q 4
105.p even 6 2 2100.2.q.h 4
105.w odd 12 4 2100.2.bc.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
420.2.q.c ✓ 4 21.g even 6 2
1260.2.s.f 4 7.d odd 6 2
1680.2.bg.q 4 84.j odd 6 2
2100.2.q.h 4 105.p even 6 2
2100.2.bc.e 8 105.w odd 12 4
2940.2.a.m 2 3.b odd 2 1
2940.2.a.s 2 21.c even 2 1
2940.2.q.t 4 21.h odd 6 2
8820.2.a.be 2 7.b odd 2 1
8820.2.a.bj 2 1.a even 1 1 trivial

Hecke kernels

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

\( T_{11}^{2} + 2T_{11} - 6 \) Copy content Toggle raw display
\( T_{13}^{2} - 7 \) Copy content Toggle raw display
\( T_{17}^{2} - 2T_{17} - 6 \) Copy content Toggle raw display
\( T_{31}^{2} + 2T_{31} - 27 \) 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
$11$ \( T^{2} + 2T - 6 \) Copy content Toggle raw display
$13$ \( T^{2} - 7 \) Copy content Toggle raw display
$17$ \( T^{2} - 2T - 6 \) Copy content Toggle raw display
$19$ \( T^{2} + 6T - 19 \) Copy content Toggle raw display
$23$ \( T^{2} - 2T - 6 \) Copy content Toggle raw display
$29$ \( T^{2} + 10T + 18 \) Copy content Toggle raw display
$31$ \( T^{2} + 2T - 27 \) Copy content Toggle raw display
$37$ \( T^{2} + 4T - 3 \) Copy content Toggle raw display
$41$ \( T^{2} + 6T - 54 \) Copy content Toggle raw display
$43$ \( T^{2} - 4T - 59 \) Copy content Toggle raw display
$47$ \( (T - 6)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} - 4T - 24 \) Copy content Toggle raw display
$59$ \( T^{2} - 6T - 54 \) Copy content Toggle raw display
$61$ \( (T + 8)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 4T - 3 \) Copy content Toggle raw display
$71$ \( T^{2} + 22T + 114 \) Copy content Toggle raw display
$73$ \( T^{2} - 175 \) Copy content Toggle raw display
$79$ \( T^{2} + 6T - 19 \) Copy content Toggle raw display
$83$ \( T^{2} + 6T - 54 \) Copy content Toggle raw display
$89$ \( T^{2} + 2T - 174 \) Copy content Toggle raw display
$97$ \( (T + 8)^{2} \) Copy content Toggle raw display
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