Properties

Label 960.6.a.bb
Level $960$
Weight $6$
Character orbit 960.a
Self dual yes
Analytic conductor $153.968$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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 60)
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)
 
\(f(q)\) \(=\) \( q + 9 q^{3} + 25 q^{5} + 244 q^{7} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 9 q^{3} + 25 q^{5} + 244 q^{7} + 81 q^{9} - 144 q^{11} - 50 q^{13} + 225 q^{15} - 1914 q^{17} + 140 q^{19} + 2196 q^{21} + 624 q^{23} + 625 q^{25} + 729 q^{27} + 3126 q^{29} + 5176 q^{31} - 1296 q^{33} + 6100 q^{35} - 15698 q^{37} - 450 q^{39} + 12570 q^{41} + 11516 q^{43} + 2025 q^{45} + 26736 q^{47} + 42729 q^{49} - 17226 q^{51} + 19158 q^{53} - 3600 q^{55} + 1260 q^{57} + 27984 q^{59} - 22022 q^{61} + 19764 q^{63} - 1250 q^{65} - 12676 q^{67} + 5616 q^{69} + 59520 q^{71} - 67102 q^{73} + 5625 q^{75} - 35136 q^{77} - 11048 q^{79} + 6561 q^{81} - 115284 q^{83} - 47850 q^{85} + 28134 q^{87} + 73650 q^{89} - 12200 q^{91} + 46584 q^{93} + 3500 q^{95} + 35522 q^{97} - 11664 q^{99}+O(q^{100}) \) 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 0 244.000 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.bb 1
4.b odd 2 1 960.6.a.g 1
8.b even 2 1 240.6.a.d 1
8.d odd 2 1 60.6.a.c 1
24.f even 2 1 180.6.a.c 1
24.h odd 2 1 720.6.a.x 1
40.e odd 2 1 300.6.a.c 1
40.k even 4 2 300.6.d.c 2
120.m even 2 1 900.6.a.k 1
120.q odd 4 2 900.6.d.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.6.a.c 1 8.d odd 2 1
180.6.a.c 1 24.f even 2 1
240.6.a.d 1 8.b even 2 1
300.6.a.c 1 40.e odd 2 1
300.6.d.c 2 40.k even 4 2
720.6.a.x 1 24.h odd 2 1
900.6.a.k 1 120.m even 2 1
900.6.d.f 2 120.q odd 4 2
960.6.a.g 1 4.b odd 2 1
960.6.a.bb 1 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7} - 244 \) 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
$7$ \( T - 244 \) Copy content Toggle raw display
$11$ \( T + 144 \) Copy content Toggle raw display
$13$ \( T + 50 \) Copy content Toggle raw display
$17$ \( T + 1914 \) Copy content Toggle raw display
$19$ \( T - 140 \) Copy content Toggle raw display
$23$ \( T - 624 \) Copy content Toggle raw display
$29$ \( T - 3126 \) Copy content Toggle raw display
$31$ \( T - 5176 \) Copy content Toggle raw display
$37$ \( T + 15698 \) Copy content Toggle raw display
$41$ \( T - 12570 \) Copy content Toggle raw display
$43$ \( T - 11516 \) Copy content Toggle raw display
$47$ \( T - 26736 \) Copy content Toggle raw display
$53$ \( T - 19158 \) Copy content Toggle raw display
$59$ \( T - 27984 \) Copy content Toggle raw display
$61$ \( T + 22022 \) Copy content Toggle raw display
$67$ \( T + 12676 \) Copy content Toggle raw display
$71$ \( T - 59520 \) Copy content Toggle raw display
$73$ \( T + 67102 \) Copy content Toggle raw display
$79$ \( T + 11048 \) Copy content Toggle raw display
$83$ \( T + 115284 \) Copy content Toggle raw display
$89$ \( T - 73650 \) Copy content Toggle raw display
$97$ \( T - 35522 \) Copy content Toggle raw display
show more
show less