Properties

Label 240.8.a.j
Level $240$
Weight $8$
Character orbit 240.a
Self dual yes
Analytic conductor $74.972$
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] = [240,8,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, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("240.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 240 = 2^{4} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 8 \)
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: \(74.9724061162\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 30)
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)
 
\(f(q)\) \(=\) \( q + 27 q^{3} - 125 q^{5} + 988 q^{7} + 729 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 27 q^{3} - 125 q^{5} + 988 q^{7} + 729 q^{9} + 8040 q^{11} - 3334 q^{13} - 3375 q^{15} + 6582 q^{17} + 27436 q^{19} + 26676 q^{21} - 48600 q^{23} + 15625 q^{25} + 19683 q^{27} - 132414 q^{29} - 254408 q^{31} + 217080 q^{33} - 123500 q^{35} + 519434 q^{37} - 90018 q^{39} + 92394 q^{41} + 234532 q^{43} - 91125 q^{45} + 1277640 q^{47} + 152601 q^{49} + 177714 q^{51} - 835278 q^{53} - 1005000 q^{55} + 740772 q^{57} + 3068760 q^{59} - 1009330 q^{61} + 720252 q^{63} + 416750 q^{65} - 3082172 q^{67} - 1312200 q^{69} + 3666720 q^{71} + 1122866 q^{73} + 421875 q^{75} + 7943520 q^{77} + 4128808 q^{79} + 531441 q^{81} - 4586556 q^{83} - 822750 q^{85} - 3575178 q^{87} - 5763678 q^{89} - 3293992 q^{91} - 6869016 q^{93} - 3429500 q^{95} + 6747554 q^{97} + 5861160 q^{99}+O(q^{100}) \) 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 27.0000 0 −125.000 0 988.000 0 729.000 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 240.8.a.j 1
4.b odd 2 1 30.8.a.d 1
12.b even 2 1 90.8.a.c 1
20.d odd 2 1 150.8.a.i 1
20.e even 4 2 150.8.c.a 2
60.h even 2 1 450.8.a.x 1
60.l odd 4 2 450.8.c.r 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.8.a.d 1 4.b odd 2 1
90.8.a.c 1 12.b even 2 1
150.8.a.i 1 20.d odd 2 1
150.8.c.a 2 20.e even 4 2
240.8.a.j 1 1.a even 1 1 trivial
450.8.a.x 1 60.h even 2 1
450.8.c.r 2 60.l odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7} - 988 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(240))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T - 27 \) Copy content Toggle raw display
$5$ \( T + 125 \) Copy content Toggle raw display
$7$ \( T - 988 \) Copy content Toggle raw display
$11$ \( T - 8040 \) Copy content Toggle raw display
$13$ \( T + 3334 \) Copy content Toggle raw display
$17$ \( T - 6582 \) Copy content Toggle raw display
$19$ \( T - 27436 \) Copy content Toggle raw display
$23$ \( T + 48600 \) Copy content Toggle raw display
$29$ \( T + 132414 \) Copy content Toggle raw display
$31$ \( T + 254408 \) Copy content Toggle raw display
$37$ \( T - 519434 \) Copy content Toggle raw display
$41$ \( T - 92394 \) Copy content Toggle raw display
$43$ \( T - 234532 \) Copy content Toggle raw display
$47$ \( T - 1277640 \) Copy content Toggle raw display
$53$ \( T + 835278 \) Copy content Toggle raw display
$59$ \( T - 3068760 \) Copy content Toggle raw display
$61$ \( T + 1009330 \) Copy content Toggle raw display
$67$ \( T + 3082172 \) Copy content Toggle raw display
$71$ \( T - 3666720 \) Copy content Toggle raw display
$73$ \( T - 1122866 \) Copy content Toggle raw display
$79$ \( T - 4128808 \) Copy content Toggle raw display
$83$ \( T + 4586556 \) Copy content Toggle raw display
$89$ \( T + 5763678 \) Copy content Toggle raw display
$97$ \( T - 6747554 \) Copy content Toggle raw display
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