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Properties

Label 1148.2.ba.a
Level $1148$
Weight $2$
Character orbit 1148.ba
Analytic conductor $9.167$
Analytic rank $0$
Dimension $80$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1148,2,Mod(113,1148)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1148, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 0, 7]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1148.113");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1148 = 2^{2} \cdot 7 \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1148.ba (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: \(9.16682615204\)
Analytic rank: \(0\)
Dimension: \(80\)
Relative dimension: \(20\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 80 q - 4 q^{5} - 60 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 80 q - 4 q^{5} - 60 q^{9} + 10 q^{11} + 20 q^{15} - 10 q^{17} - 30 q^{19} - 4 q^{21} - 20 q^{25} + 2 q^{31} + 10 q^{33} + 10 q^{37} + 36 q^{39} - 14 q^{41} + 30 q^{43} + 44 q^{45} - 60 q^{47} + 20 q^{49} - 32 q^{51} + 16 q^{57} - 60 q^{59} + 44 q^{61} - 10 q^{65} - 10 q^{67} - 40 q^{71} - 88 q^{73} - 70 q^{75} - 8 q^{77} - 40 q^{81} + 28 q^{83} - 24 q^{87} + 24 q^{91} - 100 q^{93} + 120 q^{97} - 100 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
113.1 0 − 3.08866i 0 −1.11364 − 3.42744i 0 −0.587785 − 0.809017i 0 −6.53981 0
113.2 0 − 2.81622i 0 1.10836 + 3.41119i 0 0.587785 + 0.809017i 0 −4.93108 0
113.3 0 − 2.47691i 0 0.848862 + 2.61253i 0 −0.587785 − 0.809017i 0 −3.13508 0
113.4 0 − 2.09016i 0 −0.164331 − 0.505760i 0 0.587785 + 0.809017i 0 −1.36876 0
113.5 0 − 1.92602i 0 −0.139283 − 0.428669i 0 0.587785 + 0.809017i 0 −0.709536 0
113.6 0 − 1.80180i 0 −0.0346732 − 0.106713i 0 −0.587785 − 0.809017i 0 −0.246466 0
113.7 0 − 1.15530i 0 −1.04872 − 3.22763i 0 −0.587785 − 0.809017i 0 1.66528 0
113.8 0 − 0.904348i 0 0.129544 + 0.398696i 0 −0.587785 − 0.809017i 0 2.18215 0
113.9 0 0.216695i 0 −1.22916 − 3.78295i 0 0.587785 + 0.809017i 0 2.95304 0
113.10 0 0.268055i 0 0.725265 + 2.23214i 0 0.587785 + 0.809017i 0 2.92815 0
113.11 0 0.304900i 0 −0.420940 − 1.29552i 0 0.587785 + 0.809017i 0 2.90704 0
113.12 0 0.927936i 0 0.718305 + 2.21072i 0 −0.587785 − 0.809017i 0 2.13894 0
113.13 0 1.06924i 0 0.727989 + 2.24052i 0 −0.587785 − 0.809017i 0 1.85672 0
113.14 0 1.48167i 0 −0.0508728 − 0.156570i 0 0.587785 + 0.809017i 0 0.804648 0
113.15 0 1.49455i 0 −0.331199 − 1.01933i 0 −0.587785 − 0.809017i 0 0.766322 0
113.16 0 2.20495i 0 0.330069 + 1.01585i 0 −0.587785 − 0.809017i 0 −1.86181 0
113.17 0 2.28455i 0 1.15282 + 3.54801i 0 0.587785 + 0.809017i 0 −2.21915 0
113.18 0 2.48140i 0 −0.290583 − 0.894324i 0 0.587785 + 0.809017i 0 −3.15734 0
113.19 0 2.90590i 0 −0.726535 − 2.23605i 0 −0.587785 − 0.809017i 0 −5.44428 0
113.20 0 2.97070i 0 −1.19128 − 3.66638i 0 0.587785 + 0.809017i 0 −5.82504 0
See all 80 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 113.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
41.f even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1148.2.ba.a ✓ 80
41.f even 10 1 inner 1148.2.ba.a ✓ 80
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1148.2.ba.a ✓ 80 1.a even 1 1 trivial
1148.2.ba.a ✓ 80 41.f even 10 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(1148, [\chi])\).