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Properties

Label 720.2.t.e
Level $720$
Weight $2$
Character orbit 720.t
Analytic conductor $5.749$
Analytic rank $0$
Dimension $32$
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] = [720,2,Mod(181,720)]
 
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([0, 1, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("720.181");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 720.t (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: \(32\)
Relative dimension: \(16\) over \(\Q(i)\)
Twist minimal: yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 8 q^{4}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q - 8 q^{4} - 4 q^{10} - 4 q^{16} + 8 q^{19} + 32 q^{22} + 64 q^{28} + 44 q^{34} - 32 q^{37} + 16 q^{43} - 68 q^{46} - 32 q^{49} - 56 q^{52} - 48 q^{58} - 16 q^{61} - 32 q^{64} - 16 q^{67} - 16 q^{70} + 60 q^{76} - 16 q^{79} + 40 q^{82} + 16 q^{85} - 72 q^{88} - 80 q^{91} - 12 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
181.1 −1.33609 + 0.463544i 0 1.57025 − 1.23867i −0.707107 − 0.707107i 0 4.06749i −1.52382 + 2.38285i 0 1.27253 + 0.616981i
181.2 −1.32578 − 0.492236i 0 1.51541 + 1.30520i 0.707107 + 0.707107i 0 − 2.05446i −1.36664 − 2.47635i 0 −0.589408 − 1.28553i
181.3 −1.20776 + 0.735736i 0 0.917386 − 1.77719i 0.707107 + 0.707107i 0 − 0.635963i 0.199558 + 2.82138i 0 −1.37426 − 0.333774i
181.4 −1.04637 − 0.951377i 0 0.189763 + 1.99098i −0.707107 − 0.707107i 0 1.69880i 1.69561 − 2.26383i 0 0.0671670 + 1.41262i
181.5 −0.615432 − 1.27328i 0 −1.24249 + 1.56724i 0.707107 + 0.707107i 0 − 0.511707i 2.76020 + 0.617506i 0 0.465169 − 1.33552i
181.6 −0.556055 + 1.30031i 0 −1.38161 − 1.44609i 0.707107 + 0.707107i 0 4.30899i 2.64861 − 0.992411i 0 −1.31265 + 0.526267i
181.7 −0.422271 − 1.34970i 0 −1.64337 + 1.13988i −0.707107 − 0.707107i 0 − 4.35099i 2.23244 + 1.73672i 0 −0.655790 + 1.25297i
181.8 −0.193205 + 1.40095i 0 −1.92534 − 0.541342i −0.707107 − 0.707107i 0 1.47784i 1.13038 − 2.59273i 0 1.12724 − 0.854008i
181.9 0.193205 − 1.40095i 0 −1.92534 − 0.541342i 0.707107 + 0.707107i 0 1.47784i −1.13038 + 2.59273i 0 1.12724 − 0.854008i
181.10 0.422271 + 1.34970i 0 −1.64337 + 1.13988i 0.707107 + 0.707107i 0 − 4.35099i −2.23244 − 1.73672i 0 −0.655790 + 1.25297i
181.11 0.556055 − 1.30031i 0 −1.38161 − 1.44609i −0.707107 − 0.707107i 0 4.30899i −2.64861 + 0.992411i 0 −1.31265 + 0.526267i
181.12 0.615432 + 1.27328i 0 −1.24249 + 1.56724i −0.707107 − 0.707107i 0 − 0.511707i −2.76020 − 0.617506i 0 0.465169 − 1.33552i
181.13 1.04637 + 0.951377i 0 0.189763 + 1.99098i 0.707107 + 0.707107i 0 1.69880i −1.69561 + 2.26383i 0 0.0671670 + 1.41262i
181.14 1.20776 − 0.735736i 0 0.917386 − 1.77719i −0.707107 − 0.707107i 0 − 0.635963i −0.199558 − 2.82138i 0 −1.37426 − 0.333774i
181.15 1.32578 + 0.492236i 0 1.51541 + 1.30520i −0.707107 − 0.707107i 0 − 2.05446i 1.36664 + 2.47635i 0 −0.589408 − 1.28553i
181.16 1.33609 − 0.463544i 0 1.57025 − 1.23867i 0.707107 + 0.707107i 0 4.06749i 1.52382 − 2.38285i 0 1.27253 + 0.616981i
541.1 −1.33609 − 0.463544i 0 1.57025 + 1.23867i −0.707107 + 0.707107i 0 − 4.06749i −1.52382 − 2.38285i 0 1.27253 − 0.616981i
541.2 −1.32578 + 0.492236i 0 1.51541 − 1.30520i 0.707107 − 0.707107i 0 2.05446i −1.36664 + 2.47635i 0 −0.589408 + 1.28553i
541.3 −1.20776 − 0.735736i 0 0.917386 + 1.77719i 0.707107 − 0.707107i 0 0.635963i 0.199558 − 2.82138i 0 −1.37426 + 0.333774i
541.4 −1.04637 + 0.951377i 0 0.189763 − 1.99098i −0.707107 + 0.707107i 0 − 1.69880i 1.69561 + 2.26383i 0 0.0671670 − 1.41262i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 181.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
16.e even 4 1 inner
48.i odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 720.2.t.e ✓ 32
3.b odd 2 1 inner 720.2.t.e ✓ 32
4.b odd 2 1 2880.2.t.e 32
12.b even 2 1 2880.2.t.e 32
16.e even 4 1 inner 720.2.t.e ✓ 32
16.f odd 4 1 2880.2.t.e 32
48.i odd 4 1 inner 720.2.t.e ✓ 32
48.k even 4 1 2880.2.t.e 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
720.2.t.e ✓ 32 1.a even 1 1 trivial
720.2.t.e ✓ 32 3.b odd 2 1 inner
720.2.t.e ✓ 32 16.e even 4 1 inner
720.2.t.e ✓ 32 48.i odd 4 1 inner
2880.2.t.e 32 4.b odd 2 1
2880.2.t.e 32 12.b even 2 1
2880.2.t.e 32 16.f odd 4 1
2880.2.t.e 32 48.k even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{16} + 64 T_{7}^{14} + 1544 T_{7}^{12} + 17376 T_{7}^{10} + 93456 T_{7}^{8} + 243584 T_{7}^{6} + \cdots + 16384 \) acting on \(S_{2}^{\mathrm{new}}(720, [\chi])\). Copy content Toggle raw display