Properties

Label 960.3.q.a
Level $960$
Weight $3$
Character orbit 960.q
Analytic conductor $26.158$
Analytic rank $0$
Dimension $96$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [960,3,Mod(79,960)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(960, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 1, 0, 2]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("960.79");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 960 = 2^{6} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 960.q (of order \(4\), degree \(2\), not minimal)

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: no
Analytic conductor: \(26.1581053786\)
Analytic rank: \(0\)
Dimension: \(96\)
Relative dimension: \(48\) over \(\Q(i)\)
Twist minimal: no (minimal twist has level 240)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 96 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 96 q - 32 q^{19} - 96 q^{35} - 672 q^{49} + 96 q^{51} - 256 q^{55} - 128 q^{59} - 32 q^{61} + 32 q^{65} - 96 q^{69} - 512 q^{71} + 192 q^{75} - 864 q^{81} + 384 q^{91}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
79.1 0 −1.22474 + 1.22474i 0 −4.72148 + 1.64548i 0 11.4402i 0 3.00000i 0
79.2 0 −1.22474 + 1.22474i 0 4.59527 + 1.97065i 0 11.1879i 0 3.00000i 0
79.3 0 −1.22474 + 1.22474i 0 −0.515045 4.97340i 0 10.8255i 0 3.00000i 0
79.4 0 −1.22474 + 1.22474i 0 −2.87557 + 4.09037i 0 10.3500i 0 3.00000i 0
79.5 0 −1.22474 + 1.22474i 0 −0.367849 4.98645i 0 9.62585i 0 3.00000i 0
79.6 0 −1.22474 + 1.22474i 0 4.99865 0.116331i 0 5.97898i 0 3.00000i 0
79.7 0 −1.22474 + 1.22474i 0 4.17527 2.75084i 0 4.98546i 0 3.00000i 0
79.8 0 −1.22474 + 1.22474i 0 −2.26868 + 4.45568i 0 4.16156i 0 3.00000i 0
79.9 0 −1.22474 + 1.22474i 0 −4.10833 2.84985i 0 3.77669i 0 3.00000i 0
79.10 0 −1.22474 + 1.22474i 0 −4.99932 + 0.0824727i 0 0.574521i 0 3.00000i 0
79.11 0 −1.22474 + 1.22474i 0 −0.437516 + 4.98082i 0 0.666894i 0 3.00000i 0
79.12 0 −1.22474 + 1.22474i 0 2.85432 + 4.10522i 0 1.41651i 0 3.00000i 0
79.13 0 −1.22474 + 1.22474i 0 3.84224 3.19957i 0 1.61537i 0 3.00000i 0
79.14 0 −1.22474 + 1.22474i 0 −1.95649 4.60132i 0 1.97864i 0 3.00000i 0
79.15 0 −1.22474 + 1.22474i 0 4.31221 + 2.53078i 0 2.25392i 0 3.00000i 0
79.16 0 −1.22474 + 1.22474i 0 −4.89181 + 1.03448i 0 4.36400i 0 3.00000i 0
79.17 0 −1.22474 + 1.22474i 0 −3.86518 3.17182i 0 4.51781i 0 3.00000i 0
79.18 0 −1.22474 + 1.22474i 0 1.19637 4.85476i 0 5.60935i 0 3.00000i 0
79.19 0 −1.22474 + 1.22474i 0 −3.11729 + 3.90928i 0 6.76702i 0 3.00000i 0
79.20 0 −1.22474 + 1.22474i 0 −0.338767 + 4.98851i 0 7.33616i 0 3.00000i 0
See all 96 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 79.48
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
16.f odd 4 1 inner
80.k odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 960.3.q.a 96
4.b odd 2 1 240.3.q.a 96
5.b even 2 1 inner 960.3.q.a 96
16.e even 4 1 240.3.q.a 96
16.f odd 4 1 inner 960.3.q.a 96
20.d odd 2 1 240.3.q.a 96
80.k odd 4 1 inner 960.3.q.a 96
80.q even 4 1 240.3.q.a 96
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
240.3.q.a 96 4.b odd 2 1
240.3.q.a 96 16.e even 4 1
240.3.q.a 96 20.d odd 2 1
240.3.q.a 96 80.q even 4 1
960.3.q.a 96 1.a even 1 1 trivial
960.3.q.a 96 5.b even 2 1 inner
960.3.q.a 96 16.f odd 4 1 inner
960.3.q.a 96 80.k odd 4 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(960, [\chi])\).