Properties

Label 960.4.a.w
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 480)
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 \( q + 3 q^{3} - 5 q^{5} - 4 q^{7} + 9 q^{9} - 40 q^{11} + 90 q^{13} - 15 q^{15} - 70 q^{17} - 40 q^{19} - 12 q^{21} + 108 q^{23} + 25 q^{25} + 27 q^{27} - 166 q^{29} - 40 q^{31} - 120 q^{33} + 20 q^{35} + 130 q^{37} + 270 q^{39} - 310 q^{41} + 268 q^{43} - 45 q^{45} - 556 q^{47} - 327 q^{49} - 210 q^{51} + 370 q^{53} + 200 q^{55} - 120 q^{57} - 240 q^{59} + 130 q^{61} - 36 q^{63} - 450 q^{65} - 876 q^{67} + 324 q^{69} - 840 q^{71} + 250 q^{73} + 75 q^{75} + 160 q^{77} - 880 q^{79} + 81 q^{81} + 188 q^{83} + 350 q^{85} - 498 q^{87} - 726 q^{89} - 360 q^{91} - 120 q^{93} + 200 q^{95} - 1550 q^{97} - 360 q^{99}+O(q^{100}) \) 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.w 1
4.b odd 2 1 960.4.a.f 1
8.b even 2 1 480.4.a.d 1
8.d odd 2 1 480.4.a.k yes 1
24.f even 2 1 1440.4.a.f 1
24.h odd 2 1 1440.4.a.e 1
40.e odd 2 1 2400.4.a.d 1
40.f even 2 1 2400.4.a.s 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
480.4.a.d 1 8.b even 2 1
480.4.a.k yes 1 8.d odd 2 1
960.4.a.f 1 4.b odd 2 1
960.4.a.w 1 1.a even 1 1 trivial
1440.4.a.e 1 24.h odd 2 1
1440.4.a.f 1 24.f even 2 1
2400.4.a.d 1 40.e odd 2 1
2400.4.a.s 1 40.f even 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
\( T_{11} + 40 \) 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
$11$ \( T + 40 \) Copy content Toggle raw display
$13$ \( T - 90 \) Copy content Toggle raw display
$17$ \( T + 70 \) Copy content Toggle raw display
$19$ \( T + 40 \) Copy content Toggle raw display
$23$ \( T - 108 \) Copy content Toggle raw display
$29$ \( T + 166 \) Copy content Toggle raw display
$31$ \( T + 40 \) Copy content Toggle raw display
$37$ \( T - 130 \) Copy content Toggle raw display
$41$ \( T + 310 \) Copy content Toggle raw display
$43$ \( T - 268 \) Copy content Toggle raw display
$47$ \( T + 556 \) Copy content Toggle raw display
$53$ \( T - 370 \) Copy content Toggle raw display
$59$ \( T + 240 \) Copy content Toggle raw display
$61$ \( T - 130 \) Copy content Toggle raw display
$67$ \( T + 876 \) Copy content Toggle raw display
$71$ \( T + 840 \) Copy content Toggle raw display
$73$ \( T - 250 \) Copy content Toggle raw display
$79$ \( T + 880 \) Copy content Toggle raw display
$83$ \( T - 188 \) Copy content Toggle raw display
$89$ \( T + 726 \) Copy content Toggle raw display
$97$ \( T + 1550 \) Copy content Toggle raw display
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