Properties

Label 2496.4.a.b
Level $2496$
Weight $4$
Character orbit 2496.a
Self dual yes
Analytic conductor $147.269$
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] = [2496,4,Mod(1,2496)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2496, 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("2496.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2496 = 2^{6} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2496.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: \(147.268767374\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 78)
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)
 
\(f(q)\) \(=\) \( q - 3 q^{3} - 6 q^{5} - 20 q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 3 q^{3} - 6 q^{5} - 20 q^{7} + 9 q^{9} + 24 q^{11} - 13 q^{13} + 18 q^{15} - 30 q^{17} - 16 q^{19} + 60 q^{21} + 72 q^{23} - 89 q^{25} - 27 q^{27} + 282 q^{29} - 164 q^{31} - 72 q^{33} + 120 q^{35} - 110 q^{37} + 39 q^{39} - 126 q^{41} + 164 q^{43} - 54 q^{45} + 204 q^{47} + 57 q^{49} + 90 q^{51} + 738 q^{53} - 144 q^{55} + 48 q^{57} + 120 q^{59} - 614 q^{61} - 180 q^{63} + 78 q^{65} + 848 q^{67} - 216 q^{69} - 132 q^{71} + 218 q^{73} + 267 q^{75} - 480 q^{77} + 1096 q^{79} + 81 q^{81} + 552 q^{83} + 180 q^{85} - 846 q^{87} + 210 q^{89} + 260 q^{91} + 492 q^{93} + 96 q^{95} - 1726 q^{97} + 216 q^{99}+O(q^{100}) \) 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 −20.0000 0 9.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(3\) \( +1 \)
\(13\) \( +1 \)

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 2496.4.a.b 1
4.b odd 2 1 2496.4.a.k 1
8.b even 2 1 624.4.a.i 1
8.d odd 2 1 78.4.a.e 1
24.f even 2 1 234.4.a.b 1
24.h odd 2 1 1872.4.a.e 1
40.e odd 2 1 1950.4.a.c 1
104.h odd 2 1 1014.4.a.b 1
104.m even 4 2 1014.4.b.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.4.a.e 1 8.d odd 2 1
234.4.a.b 1 24.f even 2 1
624.4.a.i 1 8.b even 2 1
1014.4.a.b 1 104.h odd 2 1
1014.4.b.c 2 104.m even 4 2
1872.4.a.e 1 24.h odd 2 1
1950.4.a.c 1 40.e odd 2 1
2496.4.a.b 1 1.a even 1 1 trivial
2496.4.a.k 1 4.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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2496))\):

\( T_{5} + 6 \) Copy content Toggle raw display
\( T_{7} + 20 \) Copy content Toggle raw display
\( T_{11} - 24 \) 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
$7$ \( T + 20 \) Copy content Toggle raw display
$11$ \( T - 24 \) Copy content Toggle raw display
$13$ \( T + 13 \) Copy content Toggle raw display
$17$ \( T + 30 \) Copy content Toggle raw display
$19$ \( T + 16 \) Copy content Toggle raw display
$23$ \( T - 72 \) Copy content Toggle raw display
$29$ \( T - 282 \) Copy content Toggle raw display
$31$ \( T + 164 \) Copy content Toggle raw display
$37$ \( T + 110 \) Copy content Toggle raw display
$41$ \( T + 126 \) Copy content Toggle raw display
$43$ \( T - 164 \) Copy content Toggle raw display
$47$ \( T - 204 \) Copy content Toggle raw display
$53$ \( T - 738 \) Copy content Toggle raw display
$59$ \( T - 120 \) Copy content Toggle raw display
$61$ \( T + 614 \) Copy content Toggle raw display
$67$ \( T - 848 \) Copy content Toggle raw display
$71$ \( T + 132 \) Copy content Toggle raw display
$73$ \( T - 218 \) Copy content Toggle raw display
$79$ \( T - 1096 \) Copy content Toggle raw display
$83$ \( T - 552 \) Copy content Toggle raw display
$89$ \( T - 210 \) Copy content Toggle raw display
$97$ \( T + 1726 \) Copy content Toggle raw display
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