Properties

Label 960.4.a.n
Level $960$
Weight $4$
Character orbit 960.a
Self dual yes
Analytic conductor $56.642$
Analytic rank $1$
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: \(1\)
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)
 
\(f(q)\) \(=\) \( q - 3 q^{3} + 5 q^{5} - 4 q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 3 q^{3} + 5 q^{5} - 4 q^{7} + 9 q^{9} + 48 q^{11} - 2 q^{13} - 15 q^{15} - 114 q^{17} - 140 q^{19} + 12 q^{21} + 72 q^{23} + 25 q^{25} - 27 q^{27} - 210 q^{29} + 272 q^{31} - 144 q^{33} - 20 q^{35} + 334 q^{37} + 6 q^{39} - 198 q^{41} + 268 q^{43} + 45 q^{45} + 216 q^{47} - 327 q^{49} + 342 q^{51} + 78 q^{53} + 240 q^{55} + 420 q^{57} - 240 q^{59} - 302 q^{61} - 36 q^{63} - 10 q^{65} - 596 q^{67} - 216 q^{69} - 768 q^{71} - 478 q^{73} - 75 q^{75} - 192 q^{77} - 640 q^{79} + 81 q^{81} + 348 q^{83} - 570 q^{85} + 630 q^{87} + 210 q^{89} + 8 q^{91} - 816 q^{93} - 700 q^{95} - 1534 q^{97} + 432 q^{99}+O(q^{100}) \) 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 −4.00000 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.n 1
4.b odd 2 1 960.4.a.bg 1
8.b even 2 1 30.4.a.b 1
8.d odd 2 1 240.4.a.b 1
24.f even 2 1 720.4.a.y 1
24.h odd 2 1 90.4.a.c 1
40.e odd 2 1 1200.4.a.ba 1
40.f even 2 1 150.4.a.b 1
40.i odd 4 2 150.4.c.c 2
40.k even 4 2 1200.4.f.r 2
56.h odd 2 1 1470.4.a.r 1
72.j odd 6 2 810.4.e.p 2
72.n even 6 2 810.4.e.i 2
120.i odd 2 1 450.4.a.r 1
120.w even 4 2 450.4.c.j 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.4.a.b 1 8.b even 2 1
90.4.a.c 1 24.h odd 2 1
150.4.a.b 1 40.f even 2 1
150.4.c.c 2 40.i odd 4 2
240.4.a.b 1 8.d odd 2 1
450.4.a.r 1 120.i odd 2 1
450.4.c.j 2 120.w even 4 2
720.4.a.y 1 24.f even 2 1
810.4.e.i 2 72.n even 6 2
810.4.e.p 2 72.j odd 6 2
960.4.a.n 1 1.a even 1 1 trivial
960.4.a.bg 1 4.b odd 2 1
1200.4.a.ba 1 40.e odd 2 1
1200.4.f.r 2 40.k even 4 2
1470.4.a.r 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(960))\):

\( T_{7} + 4 \) Copy content Toggle raw display
\( T_{11} - 48 \) 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
$7$ \( T + 4 \) Copy content Toggle raw display
$11$ \( T - 48 \) Copy content Toggle raw display
$13$ \( T + 2 \) Copy content Toggle raw display
$17$ \( T + 114 \) Copy content Toggle raw display
$19$ \( T + 140 \) Copy content Toggle raw display
$23$ \( T - 72 \) Copy content Toggle raw display
$29$ \( T + 210 \) Copy content Toggle raw display
$31$ \( T - 272 \) Copy content Toggle raw display
$37$ \( T - 334 \) Copy content Toggle raw display
$41$ \( T + 198 \) Copy content Toggle raw display
$43$ \( T - 268 \) Copy content Toggle raw display
$47$ \( T - 216 \) Copy content Toggle raw display
$53$ \( T - 78 \) Copy content Toggle raw display
$59$ \( T + 240 \) Copy content Toggle raw display
$61$ \( T + 302 \) Copy content Toggle raw display
$67$ \( T + 596 \) Copy content Toggle raw display
$71$ \( T + 768 \) Copy content Toggle raw display
$73$ \( T + 478 \) Copy content Toggle raw display
$79$ \( T + 640 \) Copy content Toggle raw display
$83$ \( T - 348 \) Copy content Toggle raw display
$89$ \( T - 210 \) Copy content Toggle raw display
$97$ \( T + 1534 \) Copy content Toggle raw display
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