Properties

Label 960.4.a.e
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 120)
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)
 
\(f(q)\) \(=\) \( q - 3 q^{3} - 5 q^{5} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 3 q^{3} - 5 q^{5} + 9 q^{9} + 4 q^{11} - 54 q^{13} + 15 q^{15} + 114 q^{17} + 44 q^{19} - 96 q^{23} + 25 q^{25} - 27 q^{27} - 134 q^{29} + 272 q^{31} - 12 q^{33} + 98 q^{37} + 162 q^{39} - 6 q^{41} + 12 q^{43} - 45 q^{45} + 200 q^{47} - 343 q^{49} - 342 q^{51} - 654 q^{53} - 20 q^{55} - 132 q^{57} + 36 q^{59} + 442 q^{61} + 270 q^{65} - 188 q^{67} + 288 q^{69} + 632 q^{71} - 390 q^{73} - 75 q^{75} - 688 q^{79} + 81 q^{81} + 1188 q^{83} - 570 q^{85} + 402 q^{87} - 694 q^{89} - 816 q^{93} - 220 q^{95} - 1726 q^{97} + 36 q^{99}+O(q^{100}) \) 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 0 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.e 1
4.b odd 2 1 960.4.a.x 1
8.b even 2 1 240.4.a.k 1
8.d odd 2 1 120.4.a.d 1
24.f even 2 1 360.4.a.c 1
24.h odd 2 1 720.4.a.i 1
40.e odd 2 1 600.4.a.m 1
40.f even 2 1 1200.4.a.j 1
40.i odd 4 2 1200.4.f.l 2
40.k even 4 2 600.4.f.d 2
120.m even 2 1 1800.4.a.s 1
120.q odd 4 2 1800.4.f.m 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.4.a.d 1 8.d odd 2 1
240.4.a.k 1 8.b even 2 1
360.4.a.c 1 24.f even 2 1
600.4.a.m 1 40.e odd 2 1
600.4.f.d 2 40.k even 4 2
720.4.a.i 1 24.h odd 2 1
960.4.a.e 1 1.a even 1 1 trivial
960.4.a.x 1 4.b odd 2 1
1200.4.a.j 1 40.f even 2 1
1200.4.f.l 2 40.i odd 4 2
1800.4.a.s 1 120.m even 2 1
1800.4.f.m 2 120.q odd 4 2

Hecke kernels

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

\( T_{7} \) Copy content Toggle raw display
\( T_{11} - 4 \) 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
$7$ \( T \) Copy content Toggle raw display
$11$ \( T - 4 \) Copy content Toggle raw display
$13$ \( T + 54 \) Copy content Toggle raw display
$17$ \( T - 114 \) Copy content Toggle raw display
$19$ \( T - 44 \) Copy content Toggle raw display
$23$ \( T + 96 \) Copy content Toggle raw display
$29$ \( T + 134 \) Copy content Toggle raw display
$31$ \( T - 272 \) Copy content Toggle raw display
$37$ \( T - 98 \) Copy content Toggle raw display
$41$ \( T + 6 \) Copy content Toggle raw display
$43$ \( T - 12 \) Copy content Toggle raw display
$47$ \( T - 200 \) Copy content Toggle raw display
$53$ \( T + 654 \) Copy content Toggle raw display
$59$ \( T - 36 \) Copy content Toggle raw display
$61$ \( T - 442 \) Copy content Toggle raw display
$67$ \( T + 188 \) Copy content Toggle raw display
$71$ \( T - 632 \) Copy content Toggle raw display
$73$ \( T + 390 \) Copy content Toggle raw display
$79$ \( T + 688 \) Copy content Toggle raw display
$83$ \( T - 1188 \) Copy content Toggle raw display
$89$ \( T + 694 \) Copy content Toggle raw display
$97$ \( T + 1726 \) Copy content Toggle raw display
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