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Properties

Label 1300.2.bs.e
Level $1300$
Weight $2$
Character orbit 1300.bs
Analytic conductor $10.381$
Analytic rank $0$
Dimension $40$
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] = [1300,2,Mod(193,1300)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1300, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([0, 9, 7]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1300.193");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1300 = 2^{2} \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1300.bs (of order \(12\), 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: \(10.3805522628\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(10\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 40 q + 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q + 24 q^{9} - 8 q^{11} - 48 q^{21} - 36 q^{29} - 24 q^{31} + 12 q^{39} - 20 q^{41} + 12 q^{49} + 24 q^{59} + 32 q^{61} + 48 q^{69} + 44 q^{71} + 108 q^{81} - 20 q^{89} + 108 q^{91} - 48 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
193.1 0 −0.792104 + 2.95617i 0 0 0 0.947355 + 0.546956i 0 −5.51344 − 3.18319i 0
193.2 0 −0.614904 + 2.29485i 0 0 0 1.40893 + 0.813445i 0 −2.29017 − 1.32223i 0
193.3 0 −0.287503 + 1.07298i 0 0 0 0.319507 + 0.184467i 0 1.52946 + 0.883033i 0
193.4 0 −0.280383 + 1.04640i 0 0 0 −2.63307 − 1.52020i 0 1.58173 + 0.913214i 0
193.5 0 −0.0887400 + 0.331182i 0 0 0 4.31072 + 2.48880i 0 2.49627 + 1.44122i 0
193.6 0 0.0887400 − 0.331182i 0 0 0 −4.31072 − 2.48880i 0 2.49627 + 1.44122i 0
193.7 0 0.280383 − 1.04640i 0 0 0 2.63307 + 1.52020i 0 1.58173 + 0.913214i 0
193.8 0 0.287503 − 1.07298i 0 0 0 −0.319507 − 0.184467i 0 1.52946 + 0.883033i 0
193.9 0 0.614904 − 2.29485i 0 0 0 −1.40893 − 0.813445i 0 −2.29017 − 1.32223i 0
193.10 0 0.792104 − 2.95617i 0 0 0 −0.947355 − 0.546956i 0 −5.51344 − 3.18319i 0
293.1 0 −3.09190 + 0.828471i 0 0 0 −2.32417 + 1.34186i 0 6.27538 − 3.62309i 0
293.2 0 −3.08578 + 0.826832i 0 0 0 2.97944 − 1.72018i 0 6.24030 − 3.60284i 0
293.3 0 −1.69883 + 0.455201i 0 0 0 3.52668 − 2.03613i 0 0.0807523 − 0.0466224i 0
293.4 0 −0.910042 + 0.243845i 0 0 0 1.31020 − 0.756442i 0 −1.82936 + 1.05618i 0
293.5 0 −0.171016 + 0.0458237i 0 0 0 −0.258937 + 0.149497i 0 −2.57093 + 1.48433i 0
293.6 0 0.171016 − 0.0458237i 0 0 0 0.258937 − 0.149497i 0 −2.57093 + 1.48433i 0
293.7 0 0.910042 − 0.243845i 0 0 0 −1.31020 + 0.756442i 0 −1.82936 + 1.05618i 0
293.8 0 1.69883 − 0.455201i 0 0 0 −3.52668 + 2.03613i 0 0.0807523 − 0.0466224i 0
293.9 0 3.08578 − 0.826832i 0 0 0 −2.97944 + 1.72018i 0 6.24030 − 3.60284i 0
293.10 0 3.09190 − 0.828471i 0 0 0 2.32417 − 1.34186i 0 6.27538 − 3.62309i 0
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 193.10
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
65.o even 12 1 inner
65.t even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1300.2.bs.e yes 40
5.b even 2 1 inner 1300.2.bs.e yes 40
5.c odd 4 2 1300.2.bn.e ✓ 40
13.f odd 12 1 1300.2.bn.e ✓ 40
65.o even 12 1 inner 1300.2.bs.e yes 40
65.s odd 12 1 1300.2.bn.e ✓ 40
65.t even 12 1 inner 1300.2.bs.e yes 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1300.2.bn.e ✓ 40 5.c odd 4 2
1300.2.bn.e ✓ 40 13.f odd 12 1
1300.2.bn.e ✓ 40 65.s odd 12 1
1300.2.bs.e yes 40 1.a even 1 1 trivial
1300.2.bs.e yes 40 5.b even 2 1 inner
1300.2.bs.e yes 40 65.o even 12 1 inner
1300.2.bs.e yes 40 65.t even 12 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{40} - 12 T_{3}^{38} - 99 T_{3}^{36} + 1764 T_{3}^{34} + 10842 T_{3}^{32} - 169692 T_{3}^{30} + \cdots + 6561 \) acting on \(S_{2}^{\mathrm{new}}(1300, [\chi])\). Copy content Toggle raw display