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Properties

Label 720.2.u.a
Level $720$
Weight $2$
Character orbit 720.u
Analytic conductor $5.749$
Analytic rank $0$
Dimension $96$
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [720,2,Mod(179,720)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(720, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 3, 2, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("720.179");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 720.u (of order \(4\), degree \(2\), 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: \(5.74922894553\)
Analytic rank: \(0\)
Dimension: \(96\)
Relative dimension: \(48\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 96 q - 8 q^{16} - 16 q^{19} + 72 q^{34} + 8 q^{40} + 8 q^{46} - 96 q^{49} + 64 q^{55} - 32 q^{61} + 48 q^{64} + 24 q^{70} + 40 q^{76} - 88 q^{94}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
179.1 −1.41373 + 0.0371259i 0 1.99724 − 0.104972i −1.81012 − 1.31281i 0 1.40695i −2.81966 + 0.222551i 0 2.60775 + 1.78875i
179.2 −1.41373 + 0.0371259i 0 1.99724 − 0.104972i 1.31281 + 1.81012i 0 − 1.40695i −2.81966 + 0.222551i 0 −1.92316 − 2.51027i
179.3 −1.39867 + 0.209116i 0 1.91254 − 0.584968i 1.41522 − 1.73123i 0 − 3.80565i −2.55268 + 1.21812i 0 −1.61739 + 2.71736i
179.4 −1.39867 + 0.209116i 0 1.91254 − 0.584968i 1.73123 − 1.41522i 0 3.80565i −2.55268 + 1.21812i 0 −2.12547 + 2.34145i
179.5 −1.31018 − 0.532382i 0 1.43314 + 1.39503i −1.50698 − 1.65197i 0 − 4.30751i −1.13498 − 2.59072i 0 1.09494 + 2.96667i
179.6 −1.31018 − 0.532382i 0 1.43314 + 1.39503i 1.65197 + 1.50698i 0 4.30751i −1.13498 − 2.59072i 0 −1.36208 − 2.85390i
179.7 −1.23687 + 0.685679i 0 1.05969 − 1.69619i −2.19631 + 0.419806i 0 0.263783i −0.147653 + 2.82457i 0 2.42869 − 2.02521i
179.8 −1.23687 + 0.685679i 0 1.05969 − 1.69619i −0.419806 + 2.19631i 0 − 0.263783i −0.147653 + 2.82457i 0 −0.986717 − 3.00440i
179.9 −1.22941 − 0.698958i 0 1.02291 + 1.71862i −1.91479 + 1.15480i 0 3.02955i −0.0563436 − 2.82787i 0 3.16123 − 0.0813664i
179.10 −1.22941 − 0.698958i 0 1.02291 + 1.71862i −1.15480 + 1.91479i 0 − 3.02955i −0.0563436 − 2.82787i 0 2.75809 − 1.54692i
179.11 −1.04776 − 0.949842i 0 0.195601 + 1.99041i 0.469629 − 2.18619i 0 2.94937i 1.68563 − 2.27126i 0 −2.56860 + 1.84453i
179.12 −1.04776 − 0.949842i 0 0.195601 + 1.99041i 2.18619 − 0.469629i 0 − 2.94937i 1.68563 − 2.27126i 0 −2.73668 − 1.58448i
179.13 −0.956592 + 1.04160i 0 −0.169864 − 1.99277i 1.11222 − 1.93984i 0 − 1.78786i 2.23816 + 1.72934i 0 0.956594 + 3.01412i
179.14 −0.956592 + 1.04160i 0 −0.169864 − 1.99277i 1.93984 − 1.11222i 0 1.78786i 2.23816 + 1.72934i 0 −0.697143 + 3.08448i
179.15 −0.638599 − 1.26182i 0 −1.18438 + 1.61160i −2.14049 − 0.646750i 0 − 0.594230i 2.78989 + 0.465314i 0 0.550835 + 3.11393i
179.16 −0.638599 − 1.26182i 0 −1.18438 + 1.61160i 0.646750 + 2.14049i 0 0.594230i 2.78989 + 0.465314i 0 2.28790 − 2.18300i
179.17 −0.612434 − 1.27473i 0 −1.24985 + 1.56137i −0.263678 − 2.22047i 0 2.95946i 2.75577 + 0.636981i 0 −2.66900 + 1.69601i
179.18 −0.612434 − 1.27473i 0 −1.24985 + 1.56137i 2.22047 + 0.263678i 0 − 2.95946i 2.75577 + 0.636981i 0 −1.02377 − 2.99197i
179.19 −0.581659 + 1.28906i 0 −1.32335 − 1.49958i −2.23581 − 0.0342493i 0 4.97879i 2.70279 − 0.833625i 0 1.34463 − 2.86216i
179.20 −0.581659 + 1.28906i 0 −1.32335 − 1.49958i 0.0342493 + 2.23581i 0 − 4.97879i 2.70279 − 0.833625i 0 −2.90201 − 1.25633i
See all 96 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 179.48
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
15.d odd 2 1 inner
16.f odd 4 1 inner
48.k even 4 1 inner
80.k odd 4 1 inner
240.t even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 720.2.u.a ✓ 96
3.b odd 2 1 inner 720.2.u.a ✓ 96
4.b odd 2 1 2880.2.u.a 96
5.b even 2 1 inner 720.2.u.a ✓ 96
12.b even 2 1 2880.2.u.a 96
15.d odd 2 1 inner 720.2.u.a ✓ 96
16.e even 4 1 2880.2.u.a 96
16.f odd 4 1 inner 720.2.u.a ✓ 96
20.d odd 2 1 2880.2.u.a 96
48.i odd 4 1 2880.2.u.a 96
48.k even 4 1 inner 720.2.u.a ✓ 96
60.h even 2 1 2880.2.u.a 96
80.k odd 4 1 inner 720.2.u.a ✓ 96
80.q even 4 1 2880.2.u.a 96
240.t even 4 1 inner 720.2.u.a ✓ 96
240.bm odd 4 1 2880.2.u.a 96
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
720.2.u.a ✓ 96 1.a even 1 1 trivial
720.2.u.a ✓ 96 3.b odd 2 1 inner
720.2.u.a ✓ 96 5.b even 2 1 inner
720.2.u.a ✓ 96 15.d odd 2 1 inner
720.2.u.a ✓ 96 16.f odd 4 1 inner
720.2.u.a ✓ 96 48.k even 4 1 inner
720.2.u.a ✓ 96 80.k odd 4 1 inner
720.2.u.a ✓ 96 240.t even 4 1 inner
2880.2.u.a 96 4.b odd 2 1
2880.2.u.a 96 12.b even 2 1
2880.2.u.a 96 16.e even 4 1
2880.2.u.a 96 20.d odd 2 1
2880.2.u.a 96 48.i odd 4 1
2880.2.u.a 96 60.h even 2 1
2880.2.u.a 96 80.q even 4 1
2880.2.u.a 96 240.bm odd 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(720, [\chi])\).