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Properties

Label 3100.3.d.h
Level $3100$
Weight $3$
Character orbit 3100.d
Analytic conductor $84.469$
Analytic rank $0$
Dimension $24$
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] = [3100,3,Mod(1301,3100)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3100, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3100.1301");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3100 = 2^{2} \cdot 5^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 3100.d (of order \(2\), degree \(1\), 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: \(84.4688819517\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: no (minimal twist has level 620)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 44 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 44 q^{9} - 8 q^{19} - 24 q^{31} + 280 q^{39} - 248 q^{41} + 644 q^{49} - 100 q^{51} - 152 q^{59} + 288 q^{69} - 352 q^{71} - 368 q^{81}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1301.1 0 − 4.99148i 0 0 0 −11.3647 0 −15.9149 0
1301.2 0 4.99148i 0 0 0 −11.3647 0 −15.9149 0
1301.3 0 − 3.03111i 0 0 0 12.6370 0 −0.187623 0
1301.4 0 3.03111i 0 0 0 12.6370 0 −0.187623 0
1301.5 0 − 2.79728i 0 0 0 −9.40252 0 1.17522 0
1301.6 0 2.79728i 0 0 0 −9.40252 0 1.17522 0
1301.7 0 − 0.662053i 0 0 0 8.05015 0 8.56169 0
1301.8 0 0.662053i 0 0 0 8.05015 0 8.56169 0
1301.9 0 − 4.48325i 0 0 0 3.21821 0 −11.0995 0
1301.10 0 4.48325i 0 0 0 3.21821 0 −11.0995 0
1301.11 0 − 1.59213i 0 0 0 1.60644 0 6.46514 0
1301.12 0 1.59213i 0 0 0 1.60644 0 6.46514 0
1301.13 0 − 1.59213i 0 0 0 −1.60644 0 6.46514 0
1301.14 0 1.59213i 0 0 0 −1.60644 0 6.46514 0
1301.15 0 − 4.48325i 0 0 0 −3.21821 0 −11.0995 0
1301.16 0 4.48325i 0 0 0 −3.21821 0 −11.0995 0
1301.17 0 − 0.662053i 0 0 0 −8.05015 0 8.56169 0
1301.18 0 0.662053i 0 0 0 −8.05015 0 8.56169 0
1301.19 0 − 2.79728i 0 0 0 9.40252 0 1.17522 0
1301.20 0 2.79728i 0 0 0 9.40252 0 1.17522 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1301.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
31.b odd 2 1 inner
155.c odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3100.3.d.h 24
5.b even 2 1 inner 3100.3.d.h 24
5.c odd 4 2 620.3.f.c ✓ 24
31.b odd 2 1 inner 3100.3.d.h 24
155.c odd 2 1 inner 3100.3.d.h 24
155.f even 4 2 620.3.f.c ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
620.3.f.c ✓ 24 5.c odd 4 2
620.3.f.c ✓ 24 155.f even 4 2
3100.3.d.h 24 1.a even 1 1 trivial
3100.3.d.h 24 5.b even 2 1 inner
3100.3.d.h 24 31.b odd 2 1 inner
3100.3.d.h 24 155.c odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(3100, [\chi])\):

\( T_{3}^{12} + 65T_{3}^{10} + 1524T_{3}^{8} + 15804T_{3}^{6} + 72440T_{3}^{4} + 120100T_{3}^{2} + 40000 \) Copy content Toggle raw display
\( T_{7}^{12} - 455T_{7}^{10} + 76356T_{7}^{8} - 5740296T_{7}^{6} + 182348816T_{7}^{4} - 1657490800T_{7}^{2} + 3158315776 \) Copy content Toggle raw display