↑

Properties

Label 1148.2.i.e
Level $1148$
Weight $2$
Character orbit 1148.i
Analytic conductor $9.167$
Analytic rank $0$
Dimension $30$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1148,2,Mod(165,1148)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1148, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 4, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1148.165");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1148 = 2^{2} \cdot 7 \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1148.i (of order \(3\), 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: \(9.16682615204\)
Analytic rank: \(0\)
Dimension: \(30\)
Relative dimension: \(15\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 30 q + q^{3} - 3 q^{5} + 3 q^{7} - 30 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 30 q + q^{3} - 3 q^{5} + 3 q^{7} - 30 q^{9} - 9 q^{11} + 14 q^{13} + 4 q^{15} - 3 q^{17} - 7 q^{19} - 3 q^{21} + q^{23} - 32 q^{25} + 22 q^{27} + 36 q^{29} - 30 q^{31} + 16 q^{33} - 47 q^{35} - 23 q^{37} - 5 q^{39} - 30 q^{41} + 24 q^{43} + 13 q^{45} + 16 q^{47} - 31 q^{49} - 29 q^{51} - 33 q^{53} + 74 q^{55} + 32 q^{57} + 10 q^{59} - q^{61} - 75 q^{63} - 16 q^{65} - 20 q^{67} + 42 q^{69} + 10 q^{71} + 3 q^{73} + 51 q^{75} - 15 q^{77} - 25 q^{79} - 43 q^{81} + 36 q^{83} + 72 q^{85} + 53 q^{87} + 11 q^{89} - 41 q^{91} - 65 q^{93} + 30 q^{95} + 32 q^{97} - 36 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
165.1 0 −1.62107 + 2.80777i 0 −0.748394 − 1.29626i 0 2.53169 + 0.768468i 0 −3.75571 − 6.50509i 0
165.2 0 −1.45317 + 2.51697i 0 −0.253051 − 0.438298i 0 −2.53278 − 0.764863i 0 −2.72341 − 4.71709i 0
165.3 0 −1.36871 + 2.37067i 0 0.431527 + 0.747427i 0 −2.57022 + 0.627666i 0 −2.24672 − 3.89144i 0
165.4 0 −1.23867 + 2.14545i 0 1.74922 + 3.02973i 0 1.19623 − 2.35988i 0 −1.56862 − 2.71694i 0
165.5 0 −1.05884 + 1.83397i 0 −1.96828 − 3.40916i 0 1.39753 − 2.24653i 0 −0.742303 − 1.28571i 0
165.6 0 −0.370691 + 0.642056i 0 1.24331 + 2.15348i 0 0.670716 + 2.55932i 0 1.22518 + 2.12207i 0
165.7 0 −0.224795 + 0.389356i 0 0.393901 + 0.682257i 0 1.75718 + 1.97796i 0 1.39893 + 2.42303i 0
165.8 0 0.257541 − 0.446074i 0 −1.09982 − 1.90494i 0 −2.48493 + 0.908374i 0 1.36735 + 2.36831i 0
165.9 0 0.637220 − 1.10370i 0 −2.18714 − 3.78824i 0 1.89439 + 1.84696i 0 0.687901 + 1.19148i 0
165.10 0 0.702089 − 1.21605i 0 −1.80784 − 3.13127i 0 0.606375 − 2.57533i 0 0.514141 + 0.890519i 0
165.11 0 0.864978 − 1.49819i 0 0.0691753 + 0.119815i 0 0.329472 − 2.62516i 0 0.00362599 + 0.00628040i 0
165.12 0 0.966125 − 1.67338i 0 0.237733 + 0.411765i 0 −1.29492 + 2.30720i 0 −0.366793 − 0.635305i 0
165.13 0 1.14590 − 1.98477i 0 1.96151 + 3.39744i 0 −0.412096 + 2.61346i 0 −1.12620 − 1.95063i 0
165.14 0 1.55023 − 2.68508i 0 1.65684 + 2.86974i 0 −1.55882 − 2.13778i 0 −3.30645 − 5.72694i 0
165.15 0 1.71186 − 2.96502i 0 −1.17870 − 2.04157i 0 1.97017 − 1.76591i 0 −4.36091 − 7.55332i 0
821.1 0 −1.62107 − 2.80777i 0 −0.748394 + 1.29626i 0 2.53169 − 0.768468i 0 −3.75571 + 6.50509i 0
821.2 0 −1.45317 − 2.51697i 0 −0.253051 + 0.438298i 0 −2.53278 + 0.764863i 0 −2.72341 + 4.71709i 0
821.3 0 −1.36871 − 2.37067i 0 0.431527 − 0.747427i 0 −2.57022 − 0.627666i 0 −2.24672 + 3.89144i 0
821.4 0 −1.23867 − 2.14545i 0 1.74922 − 3.02973i 0 1.19623 + 2.35988i 0 −1.56862 + 2.71694i 0
821.5 0 −1.05884 − 1.83397i 0 −1.96828 + 3.40916i 0 1.39753 + 2.24653i 0 −0.742303 + 1.28571i 0
See all 30 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 165.15
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1148.2.i.e ✓ 30
7.c even 3 1 inner 1148.2.i.e ✓ 30
7.c even 3 1 8036.2.a.q 15
7.d odd 6 1 8036.2.a.r 15
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1148.2.i.e ✓ 30 1.a even 1 1 trivial
1148.2.i.e ✓ 30 7.c even 3 1 inner
8036.2.a.q 15 7.c even 3 1
8036.2.a.r 15 7.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1148, [\chi])\):

\( T_{3}^{30} - T_{3}^{29} + 38 T_{3}^{28} - 41 T_{3}^{27} + 871 T_{3}^{26} - 985 T_{3}^{25} + \cdots + 11431161 \) Copy content Toggle raw display
\( T_{11}^{30} + 9 T_{11}^{29} + 157 T_{11}^{28} + 1092 T_{11}^{27} + 12607 T_{11}^{26} + \cdots + 220991189409 \) Copy content Toggle raw display