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Properties

Label 2352.4.a.bl
Level $2352$
Weight $4$
Character orbit 2352.a
Self dual yes
Analytic conductor $138.772$
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] = [2352,4,Mod(1,2352)] 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("2352.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2352, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2352.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,-6,0,-20,0,0,0,18,0,20] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(138.772492334\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 147)
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 \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 3 q^{3} + (7 \beta - 10) q^{5} + 9 q^{9} + (24 \beta + 10) q^{11} + (25 \beta - 52) q^{13} + ( - 21 \beta + 30) q^{15} + ( - 45 \beta - 58) q^{17} + (22 \beta + 96) q^{19} + ( - 28 \beta - 14) q^{23}+ \cdots + (216 \beta + 90) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 6 q^{3} - 20 q^{5} + 18 q^{9} + 20 q^{11} - 104 q^{13} + 60 q^{15} - 116 q^{17} + 192 q^{19} - 28 q^{23} + 146 q^{25} - 54 q^{27} + 296 q^{29} - 104 q^{31} - 60 q^{33} - 248 q^{37} + 312 q^{39} - 20 q^{41}+ \cdots + 180 q^{99}+O(q^{100}) \) 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.

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
−1.41421
1.41421
0 −3.00000 0 −19.8995 0 0 0 9.00000 0
1.2 0 −3.00000 0 −0.100505 0 0 0 9.00000 0
\(n\): 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 2352.4.a.bl 2
4.b odd 2 1 147.4.a.k yes 2
7.b odd 2 1 2352.4.a.cf 2
12.b even 2 1 441.4.a.o 2
28.d even 2 1 147.4.a.j ✓ 2
28.f even 6 2 147.4.e.k 4
28.g odd 6 2 147.4.e.j 4
84.h odd 2 1 441.4.a.n 2
84.j odd 6 2 441.4.e.v 4
84.n even 6 2 441.4.e.u 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
147.4.a.j ✓ 2 28.d even 2 1
147.4.a.k yes 2 4.b odd 2 1
147.4.e.j 4 28.g odd 6 2
147.4.e.k 4 28.f even 6 2
441.4.a.n 2 84.h odd 2 1
441.4.a.o 2 12.b even 2 1
441.4.e.u 4 84.n even 6 2
441.4.e.v 4 84.j odd 6 2
2352.4.a.bl 2 1.a even 1 1 trivial
2352.4.a.cf 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2352))\):

\( T_{5}^{2} + 20T_{5} + 2 \) Copy content Toggle raw display
\( T_{11}^{2} - 20T_{11} - 1052 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( (T + 3)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 20T + 2 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 20T - 1052 \) Copy content Toggle raw display
$13$ \( T^{2} + 104T + 1454 \) Copy content Toggle raw display
$17$ \( T^{2} + 116T - 686 \) Copy content Toggle raw display
$19$ \( T^{2} - 192T + 8248 \) Copy content Toggle raw display
$23$ \( T^{2} + 28T - 1372 \) Copy content Toggle raw display
$29$ \( T^{2} - 296T + 14216 \) Copy content Toggle raw display
$31$ \( T^{2} + 104T - 2296 \) Copy content Toggle raw display
$37$ \( T^{2} + 248T + 10768 \) Copy content Toggle raw display
$41$ \( T^{2} + 20T - 95822 \) Copy content Toggle raw display
$43$ \( T^{2} - 720T + 109600 \) Copy content Toggle raw display
$47$ \( T^{2} + 96T - 122696 \) Copy content Toggle raw display
$53$ \( T^{2} - 268T - 241244 \) Copy content Toggle raw display
$59$ \( T^{2} + 616T - 7288 \) Copy content Toggle raw display
$61$ \( T^{2} + 16T + 46 \) Copy content Toggle raw display
$67$ \( T^{2} - 144T - 543968 \) Copy content Toggle raw display
$71$ \( T^{2} + 988T + 136388 \) Copy content Toggle raw display
$73$ \( T^{2} + 104T - 318898 \) Copy content Toggle raw display
$79$ \( T^{2} - 944T + 111392 \) Copy content Toggle raw display
$83$ \( T^{2} - 1016 T + 245264 \) Copy content Toggle raw display
$89$ \( T^{2} - 388T - 192206 \) Copy content Toggle raw display
$97$ \( T^{2} + 488T - 658066 \) Copy content Toggle raw display
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