Properties

Label 240.2.a.d
Level $240$
Weight $2$
Character orbit 240.a
Self dual yes
Analytic conductor $1.916$
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] = [240,2,Mod(1,240)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(240, 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("240.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 240 = 2^{4} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 240.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: \(1.91640964851\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 15)
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)
 
\(f(q)\) \(=\) \( q + q^{3} + q^{5} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} + q^{5} + q^{9} + 4 q^{11} - 2 q^{13} + q^{15} + 2 q^{17} - 4 q^{19} + q^{25} + q^{27} - 2 q^{29} + 4 q^{33} - 10 q^{37} - 2 q^{39} + 10 q^{41} - 4 q^{43} + q^{45} - 8 q^{47} - 7 q^{49} + 2 q^{51} - 10 q^{53} + 4 q^{55} - 4 q^{57} + 4 q^{59} - 2 q^{61} - 2 q^{65} - 12 q^{67} + 8 q^{71} + 10 q^{73} + q^{75} + q^{81} - 12 q^{83} + 2 q^{85} - 2 q^{87} - 6 q^{89} - 4 q^{95} + 2 q^{97} + 4 q^{99}+O(q^{100}) \) 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 1.00000 0 0 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 240.2.a.d 1
3.b odd 2 1 720.2.a.c 1
4.b odd 2 1 15.2.a.a 1
5.b even 2 1 1200.2.a.e 1
5.c odd 4 2 1200.2.f.h 2
8.b even 2 1 960.2.a.a 1
8.d odd 2 1 960.2.a.l 1
12.b even 2 1 45.2.a.a 1
15.d odd 2 1 3600.2.a.u 1
15.e even 4 2 3600.2.f.e 2
16.e even 4 2 3840.2.k.r 2
16.f odd 4 2 3840.2.k.m 2
20.d odd 2 1 75.2.a.b 1
20.e even 4 2 75.2.b.b 2
24.f even 2 1 2880.2.a.y 1
24.h odd 2 1 2880.2.a.bc 1
28.d even 2 1 735.2.a.c 1
28.f even 6 2 735.2.i.d 2
28.g odd 6 2 735.2.i.e 2
36.f odd 6 2 405.2.e.f 2
36.h even 6 2 405.2.e.c 2
40.e odd 2 1 4800.2.a.t 1
40.f even 2 1 4800.2.a.bz 1
40.i odd 4 2 4800.2.f.c 2
40.k even 4 2 4800.2.f.bf 2
44.c even 2 1 1815.2.a.d 1
52.b odd 2 1 2535.2.a.j 1
60.h even 2 1 225.2.a.b 1
60.l odd 4 2 225.2.b.b 2
68.d odd 2 1 4335.2.a.c 1
76.d even 2 1 5415.2.a.j 1
84.h odd 2 1 2205.2.a.i 1
92.b even 2 1 7935.2.a.d 1
132.d odd 2 1 5445.2.a.c 1
140.c even 2 1 3675.2.a.j 1
156.h even 2 1 7605.2.a.g 1
220.g even 2 1 9075.2.a.g 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.2.a.a 1 4.b odd 2 1
45.2.a.a 1 12.b even 2 1
75.2.a.b 1 20.d odd 2 1
75.2.b.b 2 20.e even 4 2
225.2.a.b 1 60.h even 2 1
225.2.b.b 2 60.l odd 4 2
240.2.a.d 1 1.a even 1 1 trivial
405.2.e.c 2 36.h even 6 2
405.2.e.f 2 36.f odd 6 2
720.2.a.c 1 3.b odd 2 1
735.2.a.c 1 28.d even 2 1
735.2.i.d 2 28.f even 6 2
735.2.i.e 2 28.g odd 6 2
960.2.a.a 1 8.b even 2 1
960.2.a.l 1 8.d odd 2 1
1200.2.a.e 1 5.b even 2 1
1200.2.f.h 2 5.c odd 4 2
1815.2.a.d 1 44.c even 2 1
2205.2.a.i 1 84.h odd 2 1
2535.2.a.j 1 52.b odd 2 1
2880.2.a.y 1 24.f even 2 1
2880.2.a.bc 1 24.h odd 2 1
3600.2.a.u 1 15.d odd 2 1
3600.2.f.e 2 15.e even 4 2
3675.2.a.j 1 140.c even 2 1
3840.2.k.m 2 16.f odd 4 2
3840.2.k.r 2 16.e even 4 2
4335.2.a.c 1 68.d odd 2 1
4800.2.a.t 1 40.e odd 2 1
4800.2.a.bz 1 40.f even 2 1
4800.2.f.c 2 40.i odd 4 2
4800.2.f.bf 2 40.k even 4 2
5415.2.a.j 1 76.d even 2 1
5445.2.a.c 1 132.d odd 2 1
7605.2.a.g 1 156.h even 2 1
7935.2.a.d 1 92.b even 2 1
9075.2.a.g 1 220.g even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(240))\):

\( T_{7} \) Copy content Toggle raw display
\( T_{13} + 2 \) 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
$5$ \( T - 1 \) 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
$17$ \( T - 2 \) 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
$31$ \( T \) Copy content Toggle raw display
$37$ \( T + 10 \) Copy content Toggle raw display
$41$ \( T - 10 \) 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
$61$ \( T + 2 \) Copy content Toggle raw display
$67$ \( T + 12 \) Copy content Toggle raw display
$71$ \( T - 8 \) Copy content Toggle raw display
$73$ \( T - 10 \) Copy content Toggle raw display
$79$ \( T \) Copy content Toggle raw display
$83$ \( T + 12 \) 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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