Properties

Label 2880.2.a.f
Level $2880$
Weight $2$
Character orbit 2880.a
Self dual yes
Analytic conductor $22.997$
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] = [2880,2,Mod(1,2880)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2880, 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("2880.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2880 = 2^{6} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2880.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: \(22.9969157821\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 20)
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)
 
\(f(q)\) \(=\) \( q - q^{5} - 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{5} - 2 q^{7} - 2 q^{13} + 6 q^{17} - 4 q^{19} + 6 q^{23} + q^{25} + 6 q^{29} + 4 q^{31} + 2 q^{35} - 2 q^{37} - 6 q^{41} - 10 q^{43} - 6 q^{47} - 3 q^{49} - 6 q^{53} - 12 q^{59} - 2 q^{61} + 2 q^{65} + 2 q^{67} - 12 q^{71} + 2 q^{73} - 8 q^{79} - 6 q^{83} - 6 q^{85} + 6 q^{89} + 4 q^{91} + 4 q^{95} + 2 q^{97}+O(q^{100}) \) 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 −1.00000 0 −2.00000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(5\) \(1\)

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 2880.2.a.f 1
3.b odd 2 1 320.2.a.a 1
4.b odd 2 1 2880.2.a.m 1
8.b even 2 1 720.2.a.h 1
8.d odd 2 1 180.2.a.a 1
12.b even 2 1 320.2.a.f 1
15.d odd 2 1 1600.2.a.w 1
15.e even 4 2 1600.2.c.e 2
24.f even 2 1 20.2.a.a 1
24.h odd 2 1 80.2.a.b 1
40.e odd 2 1 900.2.a.b 1
40.f even 2 1 3600.2.a.be 1
40.i odd 4 2 3600.2.f.j 2
40.k even 4 2 900.2.d.c 2
48.i odd 4 2 1280.2.d.g 2
48.k even 4 2 1280.2.d.c 2
56.e even 2 1 8820.2.a.g 1
60.h even 2 1 1600.2.a.c 1
60.l odd 4 2 1600.2.c.d 2
72.l even 6 2 1620.2.i.h 2
72.p odd 6 2 1620.2.i.b 2
120.i odd 2 1 400.2.a.c 1
120.m even 2 1 100.2.a.a 1
120.q odd 4 2 100.2.c.a 2
120.w even 4 2 400.2.c.b 2
168.e odd 2 1 980.2.a.h 1
168.i even 2 1 3920.2.a.h 1
168.v even 6 2 980.2.i.i 2
168.be odd 6 2 980.2.i.c 2
264.m even 2 1 9680.2.a.ba 1
264.p odd 2 1 2420.2.a.a 1
312.h even 2 1 3380.2.a.c 1
312.w odd 4 2 3380.2.f.b 2
408.h even 2 1 5780.2.a.f 1
408.q even 4 2 5780.2.c.a 2
456.l odd 2 1 7220.2.a.f 1
840.b odd 2 1 4900.2.a.e 1
840.bm even 4 2 4900.2.e.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.2.a.a 1 24.f even 2 1
80.2.a.b 1 24.h odd 2 1
100.2.a.a 1 120.m even 2 1
100.2.c.a 2 120.q odd 4 2
180.2.a.a 1 8.d odd 2 1
320.2.a.a 1 3.b odd 2 1
320.2.a.f 1 12.b even 2 1
400.2.a.c 1 120.i odd 2 1
400.2.c.b 2 120.w even 4 2
720.2.a.h 1 8.b even 2 1
900.2.a.b 1 40.e odd 2 1
900.2.d.c 2 40.k even 4 2
980.2.a.h 1 168.e odd 2 1
980.2.i.c 2 168.be odd 6 2
980.2.i.i 2 168.v even 6 2
1280.2.d.c 2 48.k even 4 2
1280.2.d.g 2 48.i odd 4 2
1600.2.a.c 1 60.h even 2 1
1600.2.a.w 1 15.d odd 2 1
1600.2.c.d 2 60.l odd 4 2
1600.2.c.e 2 15.e even 4 2
1620.2.i.b 2 72.p odd 6 2
1620.2.i.h 2 72.l even 6 2
2420.2.a.a 1 264.p odd 2 1
2880.2.a.f 1 1.a even 1 1 trivial
2880.2.a.m 1 4.b odd 2 1
3380.2.a.c 1 312.h even 2 1
3380.2.f.b 2 312.w odd 4 2
3600.2.a.be 1 40.f even 2 1
3600.2.f.j 2 40.i odd 4 2
3920.2.a.h 1 168.i even 2 1
4900.2.a.e 1 840.b odd 2 1
4900.2.e.f 2 840.bm even 4 2
5780.2.a.f 1 408.h even 2 1
5780.2.c.a 2 408.q even 4 2
7220.2.a.f 1 456.l odd 2 1
8820.2.a.g 1 56.e even 2 1
9680.2.a.ba 1 264.m even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2880))\):

\( T_{7} + 2 \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display
\( T_{13} + 2 \) Copy content Toggle raw display
\( T_{17} - 6 \) Copy content Toggle raw display
\( T_{19} + 4 \) 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
$5$ \( T + 1 \) Copy content Toggle raw display
$7$ \( T + 2 \) Copy content Toggle raw display
$11$ \( T \) 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
$23$ \( T - 6 \) Copy content Toggle raw display
$29$ \( T - 6 \) Copy content Toggle raw display
$31$ \( T - 4 \) Copy content Toggle raw display
$37$ \( T + 2 \) Copy content Toggle raw display
$41$ \( T + 6 \) Copy content Toggle raw display
$43$ \( T + 10 \) Copy content Toggle raw display
$47$ \( T + 6 \) Copy content Toggle raw display
$53$ \( T + 6 \) Copy content Toggle raw display
$59$ \( T + 12 \) Copy content Toggle raw display
$61$ \( T + 2 \) Copy content Toggle raw display
$67$ \( T - 2 \) Copy content Toggle raw display
$71$ \( T + 12 \) Copy content Toggle raw display
$73$ \( T - 2 \) Copy content Toggle raw display
$79$ \( T + 8 \) Copy content Toggle raw display
$83$ \( T + 6 \) Copy content Toggle raw display
$89$ \( T - 6 \) Copy content Toggle raw display
$97$ \( T - 2 \) Copy content Toggle raw display
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