Properties

Label 960.6.a.n
Level $960$
Weight $6$
Character orbit 960.a
Self dual yes
Analytic conductor $153.968$
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] = [960,6,Mod(1,960)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(960, 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("960.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 960 = 2^{6} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 960.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: \(153.968467020\)
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)
 
\(f(q)\) \(=\) \( q - 9 q^{3} + 25 q^{5} + 164 q^{7} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 9 q^{3} + 25 q^{5} + 164 q^{7} + 81 q^{9} - 720 q^{11} - 698 q^{13} - 225 q^{15} - 2226 q^{17} - 356 q^{19} - 1476 q^{21} - 1800 q^{23} + 625 q^{25} - 729 q^{27} - 714 q^{29} + 848 q^{31} + 6480 q^{33} + 4100 q^{35} + 11302 q^{37} + 6282 q^{39} + 9354 q^{41} + 5956 q^{43} + 2025 q^{45} - 11160 q^{47} + 10089 q^{49} + 20034 q^{51} - 14106 q^{53} - 18000 q^{55} + 3204 q^{57} - 7920 q^{59} + 13450 q^{61} + 13284 q^{63} - 17450 q^{65} + 65476 q^{67} + 16200 q^{69} + 34560 q^{71} + 86258 q^{73} - 5625 q^{75} - 118080 q^{77} - 108832 q^{79} + 6561 q^{81} - 10668 q^{83} - 55650 q^{85} + 6426 q^{87} + 10818 q^{89} - 114472 q^{91} - 7632 q^{93} - 8900 q^{95} + 4418 q^{97} - 58320 q^{99}+O(q^{100}) \) 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 0 164.000 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 960.6.a.n 1
4.b odd 2 1 960.6.a.u 1
8.b even 2 1 30.6.a.a 1
8.d odd 2 1 240.6.a.a 1
24.f even 2 1 720.6.a.m 1
24.h odd 2 1 90.6.a.g 1
40.f even 2 1 150.6.a.h 1
40.i odd 4 2 150.6.c.d 2
120.i odd 2 1 450.6.a.b 1
120.w even 4 2 450.6.c.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.6.a.a 1 8.b even 2 1
90.6.a.g 1 24.h odd 2 1
150.6.a.h 1 40.f even 2 1
150.6.c.d 2 40.i odd 4 2
240.6.a.a 1 8.d odd 2 1
450.6.a.b 1 120.i odd 2 1
450.6.c.b 2 120.w even 4 2
720.6.a.m 1 24.f even 2 1
960.6.a.n 1 1.a even 1 1 trivial
960.6.a.u 1 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7} - 164 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(960))\). 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
$7$ \( T - 164 \) Copy content Toggle raw display
$11$ \( T + 720 \) Copy content Toggle raw display
$13$ \( T + 698 \) Copy content Toggle raw display
$17$ \( T + 2226 \) Copy content Toggle raw display
$19$ \( T + 356 \) Copy content Toggle raw display
$23$ \( T + 1800 \) Copy content Toggle raw display
$29$ \( T + 714 \) Copy content Toggle raw display
$31$ \( T - 848 \) Copy content Toggle raw display
$37$ \( T - 11302 \) Copy content Toggle raw display
$41$ \( T - 9354 \) Copy content Toggle raw display
$43$ \( T - 5956 \) Copy content Toggle raw display
$47$ \( T + 11160 \) Copy content Toggle raw display
$53$ \( T + 14106 \) Copy content Toggle raw display
$59$ \( T + 7920 \) Copy content Toggle raw display
$61$ \( T - 13450 \) Copy content Toggle raw display
$67$ \( T - 65476 \) Copy content Toggle raw display
$71$ \( T - 34560 \) Copy content Toggle raw display
$73$ \( T - 86258 \) Copy content Toggle raw display
$79$ \( T + 108832 \) Copy content Toggle raw display
$83$ \( T + 10668 \) Copy content Toggle raw display
$89$ \( T - 10818 \) Copy content Toggle raw display
$97$ \( T - 4418 \) Copy content Toggle raw display
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