Properties

Label 575.2.e.e
Level $575$
Weight $2$
Character orbit 575.e
Analytic conductor $4.591$
Analytic rank $0$
Dimension $24$
CM discriminant -23
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [575,2,Mod(68,575)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(575, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("575.68");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 575 = 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 575.e (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: \(4.59139811622\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 12 q^{6}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 12 q^{6} - 96 q^{16} - 108 q^{26} + 276 q^{36} - 216 q^{81} - 276 q^{96}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
68.1 −1.97358 + 1.97358i −2.17182 2.17182i 5.79000i 0 8.57249 0 7.47985 + 7.47985i 6.43359i 0
68.2 −1.74373 + 1.74373i 2.39185 + 2.39185i 4.08117i 0 −8.34147 0 3.62899 + 3.62899i 8.44190i 0
68.3 −1.72013 + 1.72013i 0.738445 + 0.738445i 3.91772i 0 −2.54045 0 3.29873 + 3.29873i 1.90940i 0
68.4 −1.26741 + 1.26741i −2.06693 2.06693i 1.21267i 0 5.23931 0 −0.997876 0.997876i 5.54441i 0
68.5 −0.706163 + 0.706163i −0.104887 0.104887i 1.00267i 0 0.148135 0 −2.12037 2.12037i 2.97800i 0
68.6 −0.0235935 + 0.0235935i 1.65341 + 1.65341i 1.99889i 0 −0.0780193 0 −0.0943478 0.0943478i 2.46750i 0
68.7 0.0235935 0.0235935i −1.65341 1.65341i 1.99889i 0 −0.0780193 0 0.0943478 + 0.0943478i 2.46750i 0
68.8 0.706163 0.706163i 0.104887 + 0.104887i 1.00267i 0 0.148135 0 2.12037 + 2.12037i 2.97800i 0
68.9 1.26741 1.26741i 2.06693 + 2.06693i 1.21267i 0 5.23931 0 0.997876 + 0.997876i 5.54441i 0
68.10 1.72013 1.72013i −0.738445 0.738445i 3.91772i 0 −2.54045 0 −3.29873 3.29873i 1.90940i 0
68.11 1.74373 1.74373i −2.39185 2.39185i 4.08117i 0 −8.34147 0 −3.62899 3.62899i 8.44190i 0
68.12 1.97358 1.97358i 2.17182 + 2.17182i 5.79000i 0 8.57249 0 −7.47985 7.47985i 6.43359i 0
482.1 −1.97358 1.97358i −2.17182 + 2.17182i 5.79000i 0 8.57249 0 7.47985 7.47985i 6.43359i 0
482.2 −1.74373 1.74373i 2.39185 2.39185i 4.08117i 0 −8.34147 0 3.62899 3.62899i 8.44190i 0
482.3 −1.72013 1.72013i 0.738445 0.738445i 3.91772i 0 −2.54045 0 3.29873 3.29873i 1.90940i 0
482.4 −1.26741 1.26741i −2.06693 + 2.06693i 1.21267i 0 5.23931 0 −0.997876 + 0.997876i 5.54441i 0
482.5 −0.706163 0.706163i −0.104887 + 0.104887i 1.00267i 0 0.148135 0 −2.12037 + 2.12037i 2.97800i 0
482.6 −0.0235935 0.0235935i 1.65341 1.65341i 1.99889i 0 −0.0780193 0 −0.0943478 + 0.0943478i 2.46750i 0
482.7 0.0235935 + 0.0235935i −1.65341 + 1.65341i 1.99889i 0 −0.0780193 0 0.0943478 0.0943478i 2.46750i 0
482.8 0.706163 + 0.706163i 0.104887 0.104887i 1.00267i 0 0.148135 0 2.12037 2.12037i 2.97800i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 68.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.b odd 2 1 CM by \(\Q(\sqrt{-23}) \)
5.b even 2 1 inner
5.c odd 4 2 inner
115.c odd 2 1 inner
115.e even 4 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 575.2.e.e 24
5.b even 2 1 inner 575.2.e.e 24
5.c odd 4 2 inner 575.2.e.e 24
23.b odd 2 1 CM 575.2.e.e 24
115.c odd 2 1 inner 575.2.e.e 24
115.e even 4 2 inner 575.2.e.e 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
575.2.e.e 24 1.a even 1 1 trivial
575.2.e.e 24 5.b even 2 1 inner
575.2.e.e 24 5.c odd 4 2 inner
575.2.e.e 24 23.b odd 2 1 CM
575.2.e.e 24 115.c odd 2 1 inner
575.2.e.e 24 115.e even 4 2 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{24} + 144T_{2}^{20} + 7176T_{2}^{16} + 144047T_{2}^{12} + 947448T_{2}^{8} + 806808T_{2}^{4} + 1 \) acting on \(S_{2}^{\mathrm{new}}(575, [\chi])\). Copy content Toggle raw display