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Properties

Label 572.2.x.a
Level $572$
Weight $2$
Character orbit 572.x
Analytic conductor $4.567$
Analytic rank $0$
Dimension $56$
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] = [572,2,Mod(25,572)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(572, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 8, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("572.25");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 572 = 2^{2} \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 572.x (of order \(10\), degree \(4\), 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: \(4.56744299562\)
Analytic rank: \(0\)
Dimension: \(56\)
Relative dimension: \(14\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 56 q - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 56 q - 2 q^{9} + q^{13} - 10 q^{17} + 12 q^{23} + 2 q^{25} + 12 q^{27} + 44 q^{29} - 42 q^{35} + 15 q^{39} + 48 q^{43} - 2 q^{49} - 12 q^{51} - 22 q^{53} - 40 q^{55} - 4 q^{61} - 6 q^{65} + 8 q^{69} + 20 q^{75} - 2 q^{77} + 48 q^{79} - 130 q^{81} - 20 q^{87} + 47 q^{91} + 12 q^{95}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
25.1 0 −0.798565 − 2.45773i 0 −1.49474 + 2.05733i 0 2.65396 + 0.862324i 0 −2.97568 + 2.16196i 0
25.2 0 −0.798565 − 2.45773i 0 1.49474 − 2.05733i 0 −2.65396 − 0.862324i 0 −2.97568 + 2.16196i 0
25.3 0 −0.741447 − 2.28194i 0 −0.419790 + 0.577792i 0 −2.31756 − 0.753022i 0 −2.23045 + 1.62052i 0
25.4 0 −0.741447 − 2.28194i 0 0.419790 − 0.577792i 0 2.31756 + 0.753022i 0 −2.23045 + 1.62052i 0
25.5 0 −0.129602 − 0.398873i 0 −2.17400 + 2.99226i 0 −1.13565 − 0.368994i 0 2.28475 − 1.65997i 0
25.6 0 −0.129602 − 0.398873i 0 2.17400 − 2.99226i 0 1.13565 + 0.368994i 0 2.28475 − 1.65997i 0
25.7 0 −0.0444965 − 0.136946i 0 −1.21904 + 1.67786i 0 −3.54683 − 1.15243i 0 2.41028 − 1.75117i 0
25.8 0 −0.0444965 − 0.136946i 0 1.21904 − 1.67786i 0 3.54683 + 1.15243i 0 2.41028 − 1.75117i 0
25.9 0 0.268029 + 0.824907i 0 −0.0443935 + 0.0611023i 0 2.23222 + 0.725291i 0 1.81842 − 1.32116i 0
25.10 0 0.268029 + 0.824907i 0 0.0443935 − 0.0611023i 0 −2.23222 − 0.725291i 0 1.81842 − 1.32116i 0
25.11 0 0.579681 + 1.78407i 0 −2.40717 + 3.31319i 0 3.67038 + 1.19258i 0 −0.419842 + 0.305033i 0
25.12 0 0.579681 + 1.78407i 0 2.40717 − 3.31319i 0 −3.67038 − 1.19258i 0 −0.419842 + 0.305033i 0
25.13 0 0.866400 + 2.66651i 0 −0.720035 + 0.991043i 0 −1.55809 − 0.506254i 0 −3.93255 + 2.85717i 0
25.14 0 0.866400 + 2.66651i 0 0.720035 − 0.991043i 0 1.55809 + 0.506254i 0 −3.93255 + 2.85717i 0
181.1 0 −2.67869 − 1.94618i 0 −2.67966 − 0.870675i 0 1.22618 + 1.68769i 0 2.46069 + 7.57323i 0
181.2 0 −2.67869 − 1.94618i 0 2.67966 + 0.870675i 0 −1.22618 − 1.68769i 0 2.46069 + 7.57323i 0
181.3 0 −1.46518 − 1.06452i 0 −0.739289 − 0.240210i 0 0.0268071 + 0.0368969i 0 0.0865070 + 0.266241i 0
181.4 0 −1.46518 − 1.06452i 0 0.739289 + 0.240210i 0 −0.0268071 − 0.0368969i 0 0.0865070 + 0.266241i 0
181.5 0 −0.480503 − 0.349106i 0 −1.45044 − 0.471277i 0 −0.743441 − 1.02326i 0 −0.818043 − 2.51768i 0
181.6 0 −0.480503 − 0.349106i 0 1.45044 + 0.471277i 0 0.743441 + 1.02326i 0 −0.818043 − 2.51768i 0
See all 56 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 25.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner
13.b even 2 1 inner
143.n even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 572.2.x.a ✓ 56
11.c even 5 1 inner 572.2.x.a ✓ 56
13.b even 2 1 inner 572.2.x.a ✓ 56
143.n even 10 1 inner 572.2.x.a ✓ 56
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
572.2.x.a ✓ 56 1.a even 1 1 trivial
572.2.x.a ✓ 56 11.c even 5 1 inner
572.2.x.a ✓ 56 13.b even 2 1 inner
572.2.x.a ✓ 56 143.n even 10 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(572, [\chi])\).