Properties

Label 240.6.a.j
Level $240$
Weight $6$
Character orbit 240.a
Self dual yes
Analytic conductor $38.492$
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] = [240,6,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, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("240.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 240 = 2^{4} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
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: \(38.4921167551\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 120)
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)
 
\(f(q)\) \(=\) \( q + 9 q^{3} - 25 q^{5} + 100 q^{7} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 9 q^{3} - 25 q^{5} + 100 q^{7} + 81 q^{9} + 136 q^{11} + 82 q^{13} - 225 q^{15} + 358 q^{17} - 796 q^{19} + 900 q^{21} - 488 q^{23} + 625 q^{25} + 729 q^{27} + 7466 q^{29} - 2728 q^{31} + 1224 q^{33} - 2500 q^{35} + 7794 q^{37} + 738 q^{39} + 18234 q^{41} + 2444 q^{43} - 2025 q^{45} + 2200 q^{47} - 6807 q^{49} + 3222 q^{51} + 10122 q^{53} - 3400 q^{55} - 7164 q^{57} + 6776 q^{59} + 23398 q^{61} + 8100 q^{63} - 2050 q^{65} + 9676 q^{67} - 4392 q^{69} - 13728 q^{71} - 27390 q^{73} + 5625 q^{75} + 13600 q^{77} + 93288 q^{79} + 6561 q^{81} + 23276 q^{83} - 8950 q^{85} + 67194 q^{87} + 102354 q^{89} + 8200 q^{91} - 24552 q^{93} + 19900 q^{95} - 49502 q^{97} + 11016 q^{99}+O(q^{100}) \) 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 9.00000 0 −25.0000 0 100.000 0 81.0000 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 240.6.a.j 1
3.b odd 2 1 720.6.a.u 1
4.b odd 2 1 120.6.a.a 1
8.b even 2 1 960.6.a.l 1
8.d odd 2 1 960.6.a.w 1
12.b even 2 1 360.6.a.e 1
20.d odd 2 1 600.6.a.h 1
20.e even 4 2 600.6.f.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.6.a.a 1 4.b odd 2 1
240.6.a.j 1 1.a even 1 1 trivial
360.6.a.e 1 12.b even 2 1
600.6.a.h 1 20.d odd 2 1
600.6.f.e 2 20.e even 4 2
720.6.a.u 1 3.b odd 2 1
960.6.a.l 1 8.b even 2 1
960.6.a.w 1 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7} - 100 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(240))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T - 9 \) Copy content Toggle raw display
$5$ \( T + 25 \) Copy content Toggle raw display
$7$ \( T - 100 \) Copy content Toggle raw display
$11$ \( T - 136 \) Copy content Toggle raw display
$13$ \( T - 82 \) Copy content Toggle raw display
$17$ \( T - 358 \) Copy content Toggle raw display
$19$ \( T + 796 \) Copy content Toggle raw display
$23$ \( T + 488 \) Copy content Toggle raw display
$29$ \( T - 7466 \) Copy content Toggle raw display
$31$ \( T + 2728 \) Copy content Toggle raw display
$37$ \( T - 7794 \) Copy content Toggle raw display
$41$ \( T - 18234 \) Copy content Toggle raw display
$43$ \( T - 2444 \) Copy content Toggle raw display
$47$ \( T - 2200 \) Copy content Toggle raw display
$53$ \( T - 10122 \) Copy content Toggle raw display
$59$ \( T - 6776 \) Copy content Toggle raw display
$61$ \( T - 23398 \) Copy content Toggle raw display
$67$ \( T - 9676 \) Copy content Toggle raw display
$71$ \( T + 13728 \) Copy content Toggle raw display
$73$ \( T + 27390 \) Copy content Toggle raw display
$79$ \( T - 93288 \) Copy content Toggle raw display
$83$ \( T - 23276 \) Copy content Toggle raw display
$89$ \( T - 102354 \) Copy content Toggle raw display
$97$ \( T + 49502 \) Copy content Toggle raw display
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