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Properties

Label 882.4.a.bf
Level $882$
Weight $4$
Character orbit 882.a
Self dual yes
Analytic conductor $52.040$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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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] = [882,4,Mod(1,882)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(882, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("882.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Level: \( N \) \(=\) \( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 882.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,4,0,8,0,0,0,16,0,0,4,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(52.0396846251\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{58}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 58 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
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 \(\beta = 2\sqrt{58}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 q^{2} + 4 q^{4} + \beta q^{5} + 8 q^{8} + 2 \beta q^{10} + 2 q^{11} - 2 \beta q^{13} + 16 q^{16} - 3 \beta q^{17} + 10 \beta q^{19} + 4 \beta q^{20} + 4 q^{22} + 30 q^{23} + 107 q^{25} - 4 \beta q^{26} + \cdots + 20 \beta q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{2} + 8 q^{4} + 16 q^{8} + 4 q^{11} + 32 q^{16} + 8 q^{22} + 60 q^{23} + 214 q^{25} + 424 q^{29} + 64 q^{32} + 492 q^{37} - 568 q^{43} + 16 q^{44} + 120 q^{46} + 428 q^{50} + 1096 q^{53} + 848 q^{58}+ \cdots + 4640 q^{95}+O(q^{100}) \) 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
−7.61577
7.61577
2.00000 0 4.00000 −15.2315 0 0 8.00000 0 −30.4631
1.2 2.00000 0 4.00000 15.2315 0 0 8.00000 0 30.4631
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 882.4.a.bf yes 2
3.b odd 2 1 882.4.a.x ✓ 2
7.b odd 2 1 inner 882.4.a.bf yes 2
7.c even 3 2 882.4.g.bb 4
7.d odd 6 2 882.4.g.bb 4
21.c even 2 1 882.4.a.x ✓ 2
21.g even 6 2 882.4.g.bh 4
21.h odd 6 2 882.4.g.bh 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
882.4.a.x ✓ 2 3.b odd 2 1
882.4.a.x ✓ 2 21.c even 2 1
882.4.a.bf yes 2 1.a even 1 1 trivial
882.4.a.bf yes 2 7.b odd 2 1 inner
882.4.g.bb 4 7.c even 3 2
882.4.g.bb 4 7.d odd 6 2
882.4.g.bh 4 21.g even 6 2
882.4.g.bh 4 21.h odd 6 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(882))\):

\( T_{5}^{2} - 232 \) Copy content Toggle raw display
\( T_{11} - 2 \) Copy content Toggle raw display
\( T_{13}^{2} - 928 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 2)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 232 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( (T - 2)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 928 \) Copy content Toggle raw display
$17$ \( T^{2} - 2088 \) Copy content Toggle raw display
$19$ \( T^{2} - 23200 \) Copy content Toggle raw display
$23$ \( (T - 30)^{2} \) Copy content Toggle raw display
$29$ \( (T - 212)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 45472 \) Copy content Toggle raw display
$37$ \( (T - 246)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 102312 \) Copy content Toggle raw display
$43$ \( (T + 284)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 3712 \) Copy content Toggle raw display
$53$ \( (T - 548)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 449152 \) Copy content Toggle raw display
$61$ \( T^{2} - 268192 \) Copy content Toggle raw display
$67$ \( (T - 652)^{2} \) Copy content Toggle raw display
$71$ \( (T - 770)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 950272 \) Copy content Toggle raw display
$79$ \( (T - 472)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} - 33408 \) Copy content Toggle raw display
$89$ \( T^{2} - 512488 \) Copy content Toggle raw display
$97$ \( T^{2} - 92800 \) Copy content Toggle raw display
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