Properties

Label 2352.4.a.e
Level $2352$
Weight $4$
Character orbit 2352.a
Self dual yes
Analytic conductor $138.772$
Analytic rank $0$
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: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 6)
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 \( q - 3 q^{3} - 6 q^{5} + 9 q^{9} - 12 q^{11} - 38 q^{13} + 18 q^{15} + 126 q^{17} + 20 q^{19} - 168 q^{23} - 89 q^{25} - 27 q^{27} + 30 q^{29} - 88 q^{31} + 36 q^{33} + 254 q^{37} + 114 q^{39} - 42 q^{41} + 52 q^{43} - 54 q^{45} - 96 q^{47} - 378 q^{51} + 198 q^{53} + 72 q^{55} - 60 q^{57} - 660 q^{59} + 538 q^{61} + 228 q^{65} - 884 q^{67} + 504 q^{69} - 792 q^{71} - 218 q^{73} + 267 q^{75} + 520 q^{79} + 81 q^{81} - 492 q^{83} - 756 q^{85} - 90 q^{87} - 810 q^{89} + 264 q^{93} - 120 q^{95} - 1154 q^{97} - 108 q^{99}+O(q^{100}) \) 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.e 1
4.b odd 2 1 294.4.a.e 1
7.b odd 2 1 48.4.a.c 1
12.b even 2 1 882.4.a.n 1
21.c even 2 1 144.4.a.c 1
28.d even 2 1 6.4.a.a 1
28.f even 6 2 294.4.e.h 2
28.g odd 6 2 294.4.e.g 2
35.c odd 2 1 1200.4.a.b 1
35.f even 4 2 1200.4.f.j 2
56.e even 2 1 192.4.a.i 1
56.h odd 2 1 192.4.a.c 1
84.h odd 2 1 18.4.a.a 1
84.j odd 6 2 882.4.g.i 2
84.n even 6 2 882.4.g.f 2
112.j even 4 2 768.4.d.n 2
112.l odd 4 2 768.4.d.c 2
140.c even 2 1 150.4.a.i 1
140.j odd 4 2 150.4.c.d 2
168.e odd 2 1 576.4.a.q 1
168.i even 2 1 576.4.a.r 1
252.s odd 6 2 162.4.c.c 2
252.bi even 6 2 162.4.c.f 2
308.g odd 2 1 726.4.a.f 1
364.h even 2 1 1014.4.a.g 1
364.p odd 4 2 1014.4.b.d 2
420.o odd 2 1 450.4.a.h 1
420.w even 4 2 450.4.c.e 2
476.e even 2 1 1734.4.a.d 1
532.b odd 2 1 2166.4.a.i 1
924.n even 2 1 2178.4.a.e 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.4.a.a 1 28.d even 2 1
18.4.a.a 1 84.h odd 2 1
48.4.a.c 1 7.b odd 2 1
144.4.a.c 1 21.c even 2 1
150.4.a.i 1 140.c even 2 1
150.4.c.d 2 140.j odd 4 2
162.4.c.c 2 252.s odd 6 2
162.4.c.f 2 252.bi even 6 2
192.4.a.c 1 56.h odd 2 1
192.4.a.i 1 56.e even 2 1
294.4.a.e 1 4.b odd 2 1
294.4.e.g 2 28.g odd 6 2
294.4.e.h 2 28.f even 6 2
450.4.a.h 1 420.o odd 2 1
450.4.c.e 2 420.w even 4 2
576.4.a.q 1 168.e odd 2 1
576.4.a.r 1 168.i even 2 1
726.4.a.f 1 308.g odd 2 1
768.4.d.c 2 112.l odd 4 2
768.4.d.n 2 112.j even 4 2
882.4.a.n 1 12.b even 2 1
882.4.g.f 2 84.n even 6 2
882.4.g.i 2 84.j odd 6 2
1014.4.a.g 1 364.h even 2 1
1014.4.b.d 2 364.p odd 4 2
1200.4.a.b 1 35.c odd 2 1
1200.4.f.j 2 35.f even 4 2
1734.4.a.d 1 476.e even 2 1
2166.4.a.i 1 532.b odd 2 1
2178.4.a.e 1 924.n even 2 1
2352.4.a.e 1 1.a even 1 1 trivial

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
\( T_{11} + 12 \) 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
$11$ \( T + 12 \) Copy content Toggle raw display
$13$ \( T + 38 \) Copy content Toggle raw display
$17$ \( T - 126 \) Copy content Toggle raw display
$19$ \( T - 20 \) Copy content Toggle raw display
$23$ \( T + 168 \) Copy content Toggle raw display
$29$ \( T - 30 \) Copy content Toggle raw display
$31$ \( T + 88 \) Copy content Toggle raw display
$37$ \( T - 254 \) Copy content Toggle raw display
$41$ \( T + 42 \) Copy content Toggle raw display
$43$ \( T - 52 \) Copy content Toggle raw display
$47$ \( T + 96 \) Copy content Toggle raw display
$53$ \( T - 198 \) Copy content Toggle raw display
$59$ \( T + 660 \) Copy content Toggle raw display
$61$ \( T - 538 \) Copy content Toggle raw display
$67$ \( T + 884 \) Copy content Toggle raw display
$71$ \( T + 792 \) Copy content Toggle raw display
$73$ \( T + 218 \) Copy content Toggle raw display
$79$ \( T - 520 \) Copy content Toggle raw display
$83$ \( T + 492 \) Copy content Toggle raw display
$89$ \( T + 810 \) Copy content Toggle raw display
$97$ \( T + 1154 \) Copy content Toggle raw display
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