Properties

Label 960.4.a.u
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)
 
\(f(q)\) \(=\) \( q + 3 q^{3} - 5 q^{5} - 16 q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 3 q^{3} - 5 q^{5} - 16 q^{7} + 9 q^{9} + 28 q^{11} + 26 q^{13} - 15 q^{15} - 62 q^{17} + 68 q^{19} - 48 q^{21} - 208 q^{23} + 25 q^{25} + 27 q^{27} + 58 q^{29} + 160 q^{31} + 84 q^{33} + 80 q^{35} - 270 q^{37} + 78 q^{39} + 282 q^{41} - 76 q^{43} - 45 q^{45} - 280 q^{47} - 87 q^{49} - 186 q^{51} + 210 q^{53} - 140 q^{55} + 204 q^{57} - 196 q^{59} - 742 q^{61} - 144 q^{63} - 130 q^{65} - 836 q^{67} - 624 q^{69} - 504 q^{71} - 1062 q^{73} + 75 q^{75} - 448 q^{77} + 768 q^{79} + 81 q^{81} + 1052 q^{83} + 310 q^{85} + 174 q^{87} - 726 q^{89} - 416 q^{91} + 480 q^{93} - 340 q^{95} - 1406 q^{97} + 252 q^{99}+O(q^{100}) \) 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 −16.0000 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.u 1
4.b odd 2 1 960.4.a.h 1
8.b even 2 1 120.4.a.c 1
8.d odd 2 1 240.4.a.l 1
24.f even 2 1 720.4.a.l 1
24.h odd 2 1 360.4.a.b 1
40.e odd 2 1 1200.4.a.c 1
40.f even 2 1 600.4.a.q 1
40.i odd 4 2 600.4.f.c 2
40.k even 4 2 1200.4.f.o 2
120.i odd 2 1 1800.4.a.bb 1
120.w even 4 2 1800.4.f.r 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.4.a.c 1 8.b even 2 1
240.4.a.l 1 8.d odd 2 1
360.4.a.b 1 24.h odd 2 1
600.4.a.q 1 40.f even 2 1
600.4.f.c 2 40.i odd 4 2
720.4.a.l 1 24.f even 2 1
960.4.a.h 1 4.b odd 2 1
960.4.a.u 1 1.a even 1 1 trivial
1200.4.a.c 1 40.e odd 2 1
1200.4.f.o 2 40.k even 4 2
1800.4.a.bb 1 120.i odd 2 1
1800.4.f.r 2 120.w even 4 2

Hecke kernels

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

\( T_{7} + 16 \) Copy content Toggle raw display
\( T_{11} - 28 \) 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
$7$ \( T + 16 \) Copy content Toggle raw display
$11$ \( T - 28 \) Copy content Toggle raw display
$13$ \( T - 26 \) Copy content Toggle raw display
$17$ \( T + 62 \) Copy content Toggle raw display
$19$ \( T - 68 \) Copy content Toggle raw display
$23$ \( T + 208 \) Copy content Toggle raw display
$29$ \( T - 58 \) Copy content Toggle raw display
$31$ \( T - 160 \) Copy content Toggle raw display
$37$ \( T + 270 \) Copy content Toggle raw display
$41$ \( T - 282 \) Copy content Toggle raw display
$43$ \( T + 76 \) Copy content Toggle raw display
$47$ \( T + 280 \) Copy content Toggle raw display
$53$ \( T - 210 \) Copy content Toggle raw display
$59$ \( T + 196 \) Copy content Toggle raw display
$61$ \( T + 742 \) Copy content Toggle raw display
$67$ \( T + 836 \) Copy content Toggle raw display
$71$ \( T + 504 \) Copy content Toggle raw display
$73$ \( T + 1062 \) Copy content Toggle raw display
$79$ \( T - 768 \) Copy content Toggle raw display
$83$ \( T - 1052 \) Copy content Toggle raw display
$89$ \( T + 726 \) Copy content Toggle raw display
$97$ \( T + 1406 \) Copy content Toggle raw display
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