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Properties

Label 4704.2.a.t
Level $4704$
Weight $2$
Character orbit 4704.a
Self dual yes
Analytic conductor $37.562$
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] = [4704,2,Mod(1,4704)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4704, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4704.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4704 = 2^{5} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4704.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: \(37.5616291108\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 96)
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)
 
\(f(q)\) \(=\) \( q + q^{3} - 2 q^{5} + q^{9} - 4 q^{11} + 2 q^{13} - 2 q^{15} + 6 q^{17} - 4 q^{19} - q^{25} + q^{27} + 2 q^{29} + 4 q^{31} - 4 q^{33} - 2 q^{37} + 2 q^{39} - 2 q^{41} - 4 q^{43} - 2 q^{45} + 8 q^{47}+ \cdots - 4 q^{99}+O(q^{100}) \) 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 1.00000 0 −2.00000 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 4704.2.a.t 1
4.b odd 2 1 4704.2.a.e 1
7.b odd 2 1 96.2.a.a ✓ 1
8.b even 2 1 9408.2.a.bj 1
8.d odd 2 1 9408.2.a.ct 1
21.c even 2 1 288.2.a.c 1
28.d even 2 1 96.2.a.b yes 1
35.c odd 2 1 2400.2.a.r 1
35.f even 4 2 2400.2.f.a 2
56.e even 2 1 192.2.a.a 1
56.h odd 2 1 192.2.a.c 1
63.l odd 6 2 2592.2.i.b 2
63.o even 6 2 2592.2.i.q 2
84.h odd 2 1 288.2.a.b 1
105.g even 2 1 7200.2.a.e 1
105.k odd 4 2 7200.2.f.x 2
112.j even 4 2 768.2.d.h 2
112.l odd 4 2 768.2.d.a 2
140.c even 2 1 2400.2.a.q 1
140.j odd 4 2 2400.2.f.r 2
168.e odd 2 1 576.2.a.g 1
168.i even 2 1 576.2.a.h 1
252.s odd 6 2 2592.2.i.w 2
252.bi even 6 2 2592.2.i.h 2
280.c odd 2 1 4800.2.a.f 1
280.n even 2 1 4800.2.a.co 1
280.s even 4 2 4800.2.f.bh 2
280.y odd 4 2 4800.2.f.e 2
336.v odd 4 2 2304.2.d.s 2
336.y even 4 2 2304.2.d.c 2
420.o odd 2 1 7200.2.a.bx 1
420.w even 4 2 7200.2.f.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
96.2.a.a ✓ 1 7.b odd 2 1
96.2.a.b yes 1 28.d even 2 1
192.2.a.a 1 56.e even 2 1
192.2.a.c 1 56.h odd 2 1
288.2.a.b 1 84.h odd 2 1
288.2.a.c 1 21.c even 2 1
576.2.a.g 1 168.e odd 2 1
576.2.a.h 1 168.i even 2 1
768.2.d.a 2 112.l odd 4 2
768.2.d.h 2 112.j even 4 2
2304.2.d.c 2 336.y even 4 2
2304.2.d.s 2 336.v odd 4 2
2400.2.a.q 1 140.c even 2 1
2400.2.a.r 1 35.c odd 2 1
2400.2.f.a 2 35.f even 4 2
2400.2.f.r 2 140.j odd 4 2
2592.2.i.b 2 63.l odd 6 2
2592.2.i.h 2 252.bi even 6 2
2592.2.i.q 2 63.o even 6 2
2592.2.i.w 2 252.s odd 6 2
4704.2.a.e 1 4.b odd 2 1
4704.2.a.t 1 1.a even 1 1 trivial
4800.2.a.f 1 280.c odd 2 1
4800.2.a.co 1 280.n even 2 1
4800.2.f.e 2 280.y odd 4 2
4800.2.f.bh 2 280.s even 4 2
7200.2.a.e 1 105.g even 2 1
7200.2.a.bx 1 420.o odd 2 1
7200.2.f.f 2 420.w even 4 2
7200.2.f.x 2 105.k odd 4 2
9408.2.a.bj 1 8.b even 2 1
9408.2.a.ct 1 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4704))\):

\( T_{5} + 2 \) Copy content Toggle raw display
\( T_{11} + 4 \) Copy content Toggle raw display
\( T_{13} - 2 \) Copy content Toggle raw display
\( T_{19} + 4 \) Copy content Toggle raw display
\( T_{31} - 4 \) 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
$5$ \( T + 2 \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T + 4 \) Copy content Toggle raw display
$13$ \( T - 2 \) Copy content Toggle raw display
$17$ \( T - 6 \) Copy content Toggle raw display
$19$ \( T + 4 \) Copy content Toggle raw display
$23$ \( T \) Copy content Toggle raw display
$29$ \( T - 2 \) Copy content Toggle raw display
$31$ \( T - 4 \) Copy content Toggle raw display
$37$ \( T + 2 \) Copy content Toggle raw display
$41$ \( T + 2 \) Copy content Toggle raw display
$43$ \( T + 4 \) Copy content Toggle raw display
$47$ \( T - 8 \) Copy content Toggle raw display
$53$ \( T - 10 \) Copy content Toggle raw display
$59$ \( T + 4 \) Copy content Toggle raw display
$61$ \( T + 6 \) Copy content Toggle raw display
$67$ \( T + 4 \) Copy content Toggle raw display
$71$ \( T - 16 \) Copy content Toggle raw display
$73$ \( T - 6 \) Copy content Toggle raw display
$79$ \( T + 4 \) Copy content Toggle raw display
$83$ \( T - 12 \) Copy content Toggle raw display
$89$ \( T + 10 \) Copy content Toggle raw display
$97$ \( T - 14 \) Copy content Toggle raw display
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