Properties

Label 2240.2.a.bb
Level $2240$
Weight $2$
Character orbit 2240.a
Self dual yes
Analytic conductor $17.886$
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] = [2240,2,Mod(1,2240)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2240, 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("2240.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2240 = 2^{6} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2240.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: \(17.8864900528\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 140)
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)
 
\(f(q)\) \(=\) \( q + 3 q^{3} + q^{5} + q^{7} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 3 q^{3} + q^{5} + q^{7} + 6 q^{9} - 5 q^{11} + 3 q^{13} + 3 q^{15} - q^{17} + 6 q^{19} + 3 q^{21} - 6 q^{23} + q^{25} + 9 q^{27} + 9 q^{29} + 4 q^{31} - 15 q^{33} + q^{35} - 2 q^{37} + 9 q^{39} - 4 q^{41} + 10 q^{43} + 6 q^{45} + q^{47} + q^{49} - 3 q^{51} - 4 q^{53} - 5 q^{55} + 18 q^{57} - 8 q^{59} + 8 q^{61} + 6 q^{63} + 3 q^{65} + 12 q^{67} - 18 q^{69} - 8 q^{71} + 2 q^{73} + 3 q^{75} - 5 q^{77} - 13 q^{79} + 9 q^{81} - 4 q^{83} - q^{85} + 27 q^{87} + 4 q^{89} + 3 q^{91} + 12 q^{93} + 6 q^{95} - 13 q^{97} - 30 q^{99}+O(q^{100}) \) 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 1.00000 0 1.00000 0 6.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(5\) \( -1 \)
\(7\) \( -1 \)

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 2240.2.a.bb 1
4.b odd 2 1 2240.2.a.c 1
8.b even 2 1 560.2.a.a 1
8.d odd 2 1 140.2.a.b 1
24.f even 2 1 1260.2.a.h 1
24.h odd 2 1 5040.2.a.bd 1
40.e odd 2 1 700.2.a.b 1
40.f even 2 1 2800.2.a.be 1
40.i odd 4 2 2800.2.g.c 2
40.k even 4 2 700.2.e.a 2
56.e even 2 1 980.2.a.b 1
56.h odd 2 1 3920.2.a.bl 1
56.k odd 6 2 980.2.i.b 2
56.m even 6 2 980.2.i.j 2
120.m even 2 1 6300.2.a.bf 1
120.q odd 4 2 6300.2.k.p 2
168.e odd 2 1 8820.2.a.n 1
280.n even 2 1 4900.2.a.u 1
280.y odd 4 2 4900.2.e.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.a.b 1 8.d odd 2 1
560.2.a.a 1 8.b even 2 1
700.2.a.b 1 40.e odd 2 1
700.2.e.a 2 40.k even 4 2
980.2.a.b 1 56.e even 2 1
980.2.i.b 2 56.k odd 6 2
980.2.i.j 2 56.m even 6 2
1260.2.a.h 1 24.f even 2 1
2240.2.a.c 1 4.b odd 2 1
2240.2.a.bb 1 1.a even 1 1 trivial
2800.2.a.be 1 40.f even 2 1
2800.2.g.c 2 40.i odd 4 2
3920.2.a.bl 1 56.h odd 2 1
4900.2.a.u 1 280.n even 2 1
4900.2.e.a 2 280.y odd 4 2
5040.2.a.bd 1 24.h odd 2 1
6300.2.a.bf 1 120.m even 2 1
6300.2.k.p 2 120.q odd 4 2
8820.2.a.n 1 168.e odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2240))\):

\( T_{3} - 3 \) Copy content Toggle raw display
\( T_{11} + 5 \) Copy content Toggle raw display
\( T_{13} - 3 \) Copy content Toggle raw display
\( T_{19} - 6 \) 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
$5$ \( T - 1 \) Copy content Toggle raw display
$7$ \( T - 1 \) Copy content Toggle raw display
$11$ \( T + 5 \) Copy content Toggle raw display
$13$ \( T - 3 \) Copy content Toggle raw display
$17$ \( T + 1 \) Copy content Toggle raw display
$19$ \( T - 6 \) Copy content Toggle raw display
$23$ \( T + 6 \) Copy content Toggle raw display
$29$ \( T - 9 \) Copy content Toggle raw display
$31$ \( T - 4 \) Copy content Toggle raw display
$37$ \( T + 2 \) Copy content Toggle raw display
$41$ \( T + 4 \) Copy content Toggle raw display
$43$ \( T - 10 \) Copy content Toggle raw display
$47$ \( T - 1 \) Copy content Toggle raw display
$53$ \( T + 4 \) Copy content Toggle raw display
$59$ \( T + 8 \) Copy content Toggle raw display
$61$ \( T - 8 \) Copy content Toggle raw display
$67$ \( T - 12 \) Copy content Toggle raw display
$71$ \( T + 8 \) Copy content Toggle raw display
$73$ \( T - 2 \) Copy content Toggle raw display
$79$ \( T + 13 \) Copy content Toggle raw display
$83$ \( T + 4 \) Copy content Toggle raw display
$89$ \( T - 4 \) Copy content Toggle raw display
$97$ \( T + 13 \) Copy content Toggle raw display
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