Properties

Label 2352.4.a.f
Level $2352$
Weight $4$
Character orbit 2352.a
Self dual yes
Analytic conductor $138.772$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2352,4,Mod(1,2352)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
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")
 
//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);
 
Level: \( N \) \(=\) \( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2352.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(138.772492334\)
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

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q - 3 q^{3} - 6 q^{5} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 3 q^{3} - 6 q^{5} + 9 q^{9} + 30 q^{11} + 53 q^{13} + 18 q^{15} - 84 q^{17} + 97 q^{19} - 84 q^{23} - 89 q^{25} - 27 q^{27} - 180 q^{29} - 179 q^{31} - 90 q^{33} - 145 q^{37} - 159 q^{39} + 126 q^{41} + 325 q^{43} - 54 q^{45} + 366 q^{47} + 252 q^{51} - 768 q^{53} - 180 q^{55} - 291 q^{57} + 264 q^{59} + 818 q^{61} - 318 q^{65} + 523 q^{67} + 252 q^{69} + 342 q^{71} - 43 q^{73} + 267 q^{75} + 1171 q^{79} + 81 q^{81} + 810 q^{83} + 504 q^{85} + 540 q^{87} - 600 q^{89} + 537 q^{93} - 582 q^{95} + 386 q^{97} + 270 q^{99}+O(q^{100}) \) 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.

comment: embeddings in the coefficient field
 
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 −3.00000 0 −6.00000 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.f 1
4.b odd 2 1 294.4.a.d 1
7.b odd 2 1 2352.4.a.bf 1
7.c even 3 2 336.4.q.f 2
12.b even 2 1 882.4.a.o 1
28.d even 2 1 294.4.a.c 1
28.f even 6 2 294.4.e.i 2
28.g odd 6 2 42.4.e.a 2
84.h odd 2 1 882.4.a.l 1
84.j odd 6 2 882.4.g.g 2
84.n even 6 2 126.4.g.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.4.e.a 2 28.g odd 6 2
126.4.g.b 2 84.n even 6 2
294.4.a.c 1 28.d even 2 1
294.4.a.d 1 4.b odd 2 1
294.4.e.i 2 28.f even 6 2
336.4.q.f 2 7.c even 3 2
882.4.a.l 1 84.h odd 2 1
882.4.a.o 1 12.b even 2 1
882.4.g.g 2 84.j odd 6 2
2352.4.a.f 1 1.a even 1 1 trivial
2352.4.a.bf 1 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2352))\):

\( T_{5} + 6 \) Copy content Toggle raw display
\( T_{11} - 30 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T + 3 \) Copy content Toggle raw display
$5$ \( T + 6 \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T - 30 \) Copy content Toggle raw display
$13$ \( T - 53 \) Copy content Toggle raw display
$17$ \( T + 84 \) Copy content Toggle raw display
$19$ \( T - 97 \) Copy content Toggle raw display
$23$ \( T + 84 \) Copy content Toggle raw display
$29$ \( T + 180 \) Copy content Toggle raw display
$31$ \( T + 179 \) Copy content Toggle raw display
$37$ \( T + 145 \) Copy content Toggle raw display
$41$ \( T - 126 \) Copy content Toggle raw display
$43$ \( T - 325 \) Copy content Toggle raw display
$47$ \( T - 366 \) Copy content Toggle raw display
$53$ \( T + 768 \) Copy content Toggle raw display
$59$ \( T - 264 \) Copy content Toggle raw display
$61$ \( T - 818 \) Copy content Toggle raw display
$67$ \( T - 523 \) Copy content Toggle raw display
$71$ \( T - 342 \) Copy content Toggle raw display
$73$ \( T + 43 \) Copy content Toggle raw display
$79$ \( T - 1171 \) Copy content Toggle raw display
$83$ \( T - 810 \) Copy content Toggle raw display
$89$ \( T + 600 \) Copy content Toggle raw display
$97$ \( T - 386 \) Copy content Toggle raw display
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