Properties

Label 960.4.a.j
Level $960$
Weight $4$
Character orbit 960.a
Self dual yes
Analytic conductor $56.642$
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] = [960,4,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, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("960.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 960 = 2^{6} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
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: \(56.6418336055\)
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)
 
\(f(q)\) \(=\) \( q - 3 q^{3} - 5 q^{5} + 32 q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 3 q^{3} - 5 q^{5} + 32 q^{7} + 9 q^{9} + 60 q^{11} + 34 q^{13} + 15 q^{15} + 42 q^{17} + 76 q^{19} - 96 q^{21} + 25 q^{25} - 27 q^{27} - 6 q^{29} - 232 q^{31} - 180 q^{33} - 160 q^{35} - 134 q^{37} - 102 q^{39} + 234 q^{41} + 412 q^{43} - 45 q^{45} - 360 q^{47} + 681 q^{49} - 126 q^{51} - 222 q^{53} - 300 q^{55} - 228 q^{57} - 660 q^{59} + 490 q^{61} + 288 q^{63} - 170 q^{65} - 812 q^{67} + 120 q^{71} + 746 q^{73} - 75 q^{75} + 1920 q^{77} + 152 q^{79} + 81 q^{81} + 804 q^{83} - 210 q^{85} + 18 q^{87} - 678 q^{89} + 1088 q^{91} + 696 q^{93} - 380 q^{95} + 194 q^{97} + 540 q^{99}+O(q^{100}) \) 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 −3.00000 0 −5.00000 0 32.0000 0 9.00000 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 960.4.a.j 1
4.b odd 2 1 960.4.a.s 1
8.b even 2 1 30.4.a.a 1
8.d odd 2 1 240.4.a.c 1
24.f even 2 1 720.4.a.b 1
24.h odd 2 1 90.4.a.d 1
40.e odd 2 1 1200.4.a.bk 1
40.f even 2 1 150.4.a.e 1
40.i odd 4 2 150.4.c.a 2
40.k even 4 2 1200.4.f.u 2
56.h odd 2 1 1470.4.a.a 1
72.j odd 6 2 810.4.e.e 2
72.n even 6 2 810.4.e.m 2
120.i odd 2 1 450.4.a.b 1
120.w even 4 2 450.4.c.k 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.4.a.a 1 8.b even 2 1
90.4.a.d 1 24.h odd 2 1
150.4.a.e 1 40.f even 2 1
150.4.c.a 2 40.i odd 4 2
240.4.a.c 1 8.d odd 2 1
450.4.a.b 1 120.i odd 2 1
450.4.c.k 2 120.w even 4 2
720.4.a.b 1 24.f even 2 1
810.4.e.e 2 72.j odd 6 2
810.4.e.m 2 72.n even 6 2
960.4.a.j 1 1.a even 1 1 trivial
960.4.a.s 1 4.b odd 2 1
1200.4.a.bk 1 40.e odd 2 1
1200.4.f.u 2 40.k even 4 2
1470.4.a.a 1 56.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(960))\):

\( T_{7} - 32 \) Copy content Toggle raw display
\( T_{11} - 60 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T + 3 \) Copy content Toggle raw display
$5$ \( T + 5 \) Copy content Toggle raw display
$7$ \( T - 32 \) Copy content Toggle raw display
$11$ \( T - 60 \) Copy content Toggle raw display
$13$ \( T - 34 \) Copy content Toggle raw display
$17$ \( T - 42 \) Copy content Toggle raw display
$19$ \( T - 76 \) Copy content Toggle raw display
$23$ \( T \) Copy content Toggle raw display
$29$ \( T + 6 \) Copy content Toggle raw display
$31$ \( T + 232 \) Copy content Toggle raw display
$37$ \( T + 134 \) Copy content Toggle raw display
$41$ \( T - 234 \) Copy content Toggle raw display
$43$ \( T - 412 \) Copy content Toggle raw display
$47$ \( T + 360 \) Copy content Toggle raw display
$53$ \( T + 222 \) Copy content Toggle raw display
$59$ \( T + 660 \) Copy content Toggle raw display
$61$ \( T - 490 \) Copy content Toggle raw display
$67$ \( T + 812 \) Copy content Toggle raw display
$71$ \( T - 120 \) Copy content Toggle raw display
$73$ \( T - 746 \) Copy content Toggle raw display
$79$ \( T - 152 \) Copy content Toggle raw display
$83$ \( T - 804 \) Copy content Toggle raw display
$89$ \( T + 678 \) Copy content Toggle raw display
$97$ \( T - 194 \) Copy content Toggle raw display
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