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Properties

Label 882.4.a.ba
Level $882$
Weight $4$
Character orbit 882.a
Self dual yes
Analytic conductor $52.040$
Analytic rank $0$
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] = [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,-4,0,8,7,0,0,-16,0,-14,-25,0,-49] 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: \(\Q(\sqrt{193}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 48 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 126)
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 \(\beta = \frac{1}{2}(1 + \sqrt{193})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 2 q^{2} + 4 q^{4} + ( - \beta + 4) q^{5} - 8 q^{8} + (2 \beta - 8) q^{10} + (7 \beta - 16) q^{11} + (5 \beta - 27) q^{13} + 16 q^{16} + (10 \beta + 44) q^{17} + (3 \beta - 61) q^{19} + ( - 4 \beta + 16) q^{20}+ \cdots + ( - 155 \beta + 46) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{2} + 8 q^{4} + 7 q^{5} - 16 q^{8} - 14 q^{10} - 25 q^{11} - 49 q^{13} + 32 q^{16} + 98 q^{17} - 119 q^{19} + 28 q^{20} + 50 q^{22} + 122 q^{23} - 129 q^{25} + 98 q^{26} + 73 q^{29} + 98 q^{31}+ \cdots - 63 q^{97}+O(q^{100}) \) 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
7.44622
−6.44622
−2.00000 0 4.00000 −3.44622 0 0 −8.00000 0 6.89244
1.2 −2.00000 0 4.00000 10.4462 0 0 −8.00000 0 −20.8924
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 882.4.a.ba 2
3.b odd 2 1 882.4.a.bd 2
7.b odd 2 1 882.4.a.u 2
7.c even 3 2 126.4.g.f yes 4
7.d odd 6 2 882.4.g.bj 4
21.c even 2 1 882.4.a.bh 2
21.g even 6 2 882.4.g.z 4
21.h odd 6 2 126.4.g.e ✓ 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.4.g.e ✓ 4 21.h odd 6 2
126.4.g.f yes 4 7.c even 3 2
882.4.a.u 2 7.b odd 2 1
882.4.a.ba 2 1.a even 1 1 trivial
882.4.a.bd 2 3.b odd 2 1
882.4.a.bh 2 21.c even 2 1
882.4.g.z 4 21.g even 6 2
882.4.g.bj 4 7.d odd 6 2

Hecke kernels

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

\( T_{5}^{2} - 7T_{5} - 36 \) Copy content Toggle raw display
\( T_{11}^{2} + 25T_{11} - 2208 \) Copy content Toggle raw display
\( T_{13}^{2} + 49T_{13} - 606 \) 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
$5$ \( T^{2} - 7T - 36 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 25T - 2208 \) Copy content Toggle raw display
$13$ \( T^{2} + 49T - 606 \) Copy content Toggle raw display
$17$ \( T^{2} - 98T - 2424 \) Copy content Toggle raw display
$19$ \( T^{2} + 119T + 3106 \) Copy content Toggle raw display
$23$ \( T^{2} - 122T - 5736 \) Copy content Toggle raw display
$29$ \( T^{2} - 73T - 1032 \) Copy content Toggle raw display
$31$ \( T^{2} - 98T - 35427 \) Copy content Toggle raw display
$37$ \( T^{2} - 289T - 398 \) Copy content Toggle raw display
$41$ \( (T - 168)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} - 307T + 2284 \) Copy content Toggle raw display
$47$ \( T^{2} - 672T + 85104 \) Copy content Toggle raw display
$53$ \( T^{2} + 375T - 156348 \) Copy content Toggle raw display
$59$ \( T^{2} - 763T - 33996 \) Copy content Toggle raw display
$61$ \( T^{2} + 406T - 14568 \) Copy content Toggle raw display
$67$ \( T^{2} + 1041 T + 211814 \) Copy content Toggle raw display
$71$ \( T^{2} - 1652 T + 644448 \) Copy content Toggle raw display
$73$ \( T^{2} + 189T - 445054 \) Copy content Toggle raw display
$79$ \( T^{2} - 524T - 394749 \) Copy content Toggle raw display
$83$ \( T^{2} - 287T - 361596 \) Copy content Toggle raw display
$89$ \( T^{2} - 2394 T + 1292112 \) Copy content Toggle raw display
$97$ \( T^{2} + 63T - 1158214 \) Copy content Toggle raw display
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