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Properties

Label 2352.2.a.t
Level $2352$
Weight $2$
Character orbit 2352.a
Self dual yes
Analytic conductor $18.781$
Analytic rank $1$
Dimension $1$
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] = [2352,2,Mod(1,2352)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
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, 2, names="a")
 
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, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2352.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,1,0,1,0,0,0,1,0,-5,0,0] 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: \(18.7808145554\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 42)
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)
 
\(f(q)\) \(=\) \( q + q^{3} + q^{5} + q^{9} - 5 q^{11} + q^{15} - 4 q^{17} - 8 q^{19} + 4 q^{23} - 4 q^{25} + q^{27} - 5 q^{29} - 3 q^{31} - 5 q^{33} - 4 q^{37} - 2 q^{43} + q^{45} + 6 q^{47} - 4 q^{51} - 9 q^{53}+ \cdots - 5 q^{99}+O(q^{100}) \) 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
0
0 1.00000 0 1.00000 0 0 0 1.00000 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 2352.2.a.t 1
3.b odd 2 1 7056.2.a.w 1
4.b odd 2 1 294.2.a.e 1
7.b odd 2 1 2352.2.a.f 1
7.c even 3 2 336.2.q.b 2
7.d odd 6 2 2352.2.q.u 2
8.b even 2 1 9408.2.a.q 1
8.d odd 2 1 9408.2.a.ce 1
12.b even 2 1 882.2.a.c 1
20.d odd 2 1 7350.2.a.bl 1
21.c even 2 1 7056.2.a.bl 1
21.h odd 6 2 1008.2.s.k 2
28.d even 2 1 294.2.a.f 1
28.f even 6 2 294.2.e.b 2
28.g odd 6 2 42.2.e.a ✓ 2
56.e even 2 1 9408.2.a.z 1
56.h odd 2 1 9408.2.a.cr 1
56.k odd 6 2 1344.2.q.g 2
56.p even 6 2 1344.2.q.s 2
84.h odd 2 1 882.2.a.d 1
84.j odd 6 2 882.2.g.i 2
84.n even 6 2 126.2.g.c 2
140.c even 2 1 7350.2.a.q 1
140.p odd 6 2 1050.2.i.l 2
140.w even 12 4 1050.2.o.a 4
252.o even 6 2 1134.2.h.l 2
252.u odd 6 2 1134.2.e.l 2
252.bb even 6 2 1134.2.e.e 2
252.bl odd 6 2 1134.2.h.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.2.e.a ✓ 2 28.g odd 6 2
126.2.g.c 2 84.n even 6 2
294.2.a.e 1 4.b odd 2 1
294.2.a.f 1 28.d even 2 1
294.2.e.b 2 28.f even 6 2
336.2.q.b 2 7.c even 3 2
882.2.a.c 1 12.b even 2 1
882.2.a.d 1 84.h odd 2 1
882.2.g.i 2 84.j odd 6 2
1008.2.s.k 2 21.h odd 6 2
1050.2.i.l 2 140.p odd 6 2
1050.2.o.a 4 140.w even 12 4
1134.2.e.e 2 252.bb even 6 2
1134.2.e.l 2 252.u odd 6 2
1134.2.h.e 2 252.bl odd 6 2
1134.2.h.l 2 252.o even 6 2
1344.2.q.g 2 56.k odd 6 2
1344.2.q.s 2 56.p even 6 2
2352.2.a.f 1 7.b odd 2 1
2352.2.a.t 1 1.a even 1 1 trivial
2352.2.q.u 2 7.d odd 6 2
7056.2.a.w 1 3.b odd 2 1
7056.2.a.bl 1 21.c even 2 1
7350.2.a.q 1 140.c even 2 1
7350.2.a.bl 1 20.d odd 2 1
9408.2.a.q 1 8.b even 2 1
9408.2.a.z 1 56.e even 2 1
9408.2.a.ce 1 8.d odd 2 1
9408.2.a.cr 1 56.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2352))\):

\( T_{5} - 1 \) Copy content Toggle raw display
\( T_{11} + 5 \) Copy content Toggle raw display
\( T_{13} \) Copy content Toggle raw display
\( T_{17} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T - 1 \) Copy content Toggle raw display
$5$ \( T - 1 \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T + 5 \) Copy content Toggle raw display
$13$ \( T \) Copy content Toggle raw display
$17$ \( T + 4 \) Copy content Toggle raw display
$19$ \( T + 8 \) Copy content Toggle raw display
$23$ \( T - 4 \) Copy content Toggle raw display
$29$ \( T + 5 \) Copy content Toggle raw display
$31$ \( T + 3 \) Copy content Toggle raw display
$37$ \( T + 4 \) Copy content Toggle raw display
$41$ \( T \) Copy content Toggle raw display
$43$ \( T + 2 \) Copy content Toggle raw display
$47$ \( T - 6 \) Copy content Toggle raw display
$53$ \( T + 9 \) Copy content Toggle raw display
$59$ \( T - 11 \) Copy content Toggle raw display
$61$ \( T + 6 \) Copy content Toggle raw display
$67$ \( T - 2 \) Copy content Toggle raw display
$71$ \( T + 2 \) Copy content Toggle raw display
$73$ \( T - 10 \) Copy content Toggle raw display
$79$ \( T + 3 \) Copy content Toggle raw display
$83$ \( T - 7 \) Copy content Toggle raw display
$89$ \( T + 6 \) Copy content Toggle raw display
$97$ \( T - 7 \) Copy content Toggle raw display
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