Properties

Label 288.4.a.i
Level $288$
Weight $4$
Character orbit 288.a
Self dual yes
Analytic conductor $16.993$
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] = [288,4,Mod(1,288)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(288, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("288.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 288 = 2^{5} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 288.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: \(16.9925500817\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 32)
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)
 
\(f(q)\) \(=\) \( q + 10 q^{5} + 16 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + 10 q^{5} + 16 q^{7} + 40 q^{11} - 50 q^{13} + 30 q^{17} + 40 q^{19} - 48 q^{23} - 25 q^{25} + 34 q^{29} + 320 q^{31} + 160 q^{35} + 310 q^{37} - 410 q^{41} + 152 q^{43} + 416 q^{47} - 87 q^{49} + 410 q^{53} + 400 q^{55} + 200 q^{59} + 30 q^{61} - 500 q^{65} + 776 q^{67} - 400 q^{71} - 630 q^{73} + 640 q^{77} - 1120 q^{79} - 552 q^{83} + 300 q^{85} + 326 q^{89} - 800 q^{91} + 400 q^{95} - 110 q^{97}+O(q^{100}) \) 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 0 0 10.0000 0 16.0000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 288.4.a.i 1
3.b odd 2 1 32.4.a.c yes 1
4.b odd 2 1 288.4.a.h 1
8.b even 2 1 576.4.a.h 1
8.d odd 2 1 576.4.a.g 1
12.b even 2 1 32.4.a.a 1
15.d odd 2 1 800.4.a.a 1
15.e even 4 2 800.4.c.a 2
21.c even 2 1 1568.4.a.c 1
24.f even 2 1 64.4.a.e 1
24.h odd 2 1 64.4.a.a 1
48.i odd 4 2 256.4.b.c 2
48.k even 4 2 256.4.b.e 2
60.h even 2 1 800.4.a.k 1
60.l odd 4 2 800.4.c.b 2
84.h odd 2 1 1568.4.a.o 1
120.i odd 2 1 1600.4.a.bw 1
120.m even 2 1 1600.4.a.e 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
32.4.a.a 1 12.b even 2 1
32.4.a.c yes 1 3.b odd 2 1
64.4.a.a 1 24.h odd 2 1
64.4.a.e 1 24.f even 2 1
256.4.b.c 2 48.i odd 4 2
256.4.b.e 2 48.k even 4 2
288.4.a.h 1 4.b odd 2 1
288.4.a.i 1 1.a even 1 1 trivial
576.4.a.g 1 8.d odd 2 1
576.4.a.h 1 8.b even 2 1
800.4.a.a 1 15.d odd 2 1
800.4.a.k 1 60.h even 2 1
800.4.c.a 2 15.e even 4 2
800.4.c.b 2 60.l odd 4 2
1568.4.a.c 1 21.c even 2 1
1568.4.a.o 1 84.h odd 2 1
1600.4.a.e 1 120.m even 2 1
1600.4.a.bw 1 120.i odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(288))\):

\( T_{5} - 10 \) Copy content Toggle raw display
\( T_{7} - 16 \) Copy content Toggle raw display
\( T_{11} - 40 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T - 10 \) Copy content Toggle raw display
$7$ \( T - 16 \) Copy content Toggle raw display
$11$ \( T - 40 \) Copy content Toggle raw display
$13$ \( T + 50 \) Copy content Toggle raw display
$17$ \( T - 30 \) Copy content Toggle raw display
$19$ \( T - 40 \) Copy content Toggle raw display
$23$ \( T + 48 \) Copy content Toggle raw display
$29$ \( T - 34 \) Copy content Toggle raw display
$31$ \( T - 320 \) Copy content Toggle raw display
$37$ \( T - 310 \) Copy content Toggle raw display
$41$ \( T + 410 \) Copy content Toggle raw display
$43$ \( T - 152 \) Copy content Toggle raw display
$47$ \( T - 416 \) Copy content Toggle raw display
$53$ \( T - 410 \) Copy content Toggle raw display
$59$ \( T - 200 \) Copy content Toggle raw display
$61$ \( T - 30 \) Copy content Toggle raw display
$67$ \( T - 776 \) Copy content Toggle raw display
$71$ \( T + 400 \) Copy content Toggle raw display
$73$ \( T + 630 \) Copy content Toggle raw display
$79$ \( T + 1120 \) Copy content Toggle raw display
$83$ \( T + 552 \) Copy content Toggle raw display
$89$ \( T - 326 \) Copy content Toggle raw display
$97$ \( T + 110 \) Copy content Toggle raw display
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