Properties

Label 1140.2.bo.c
Level $1140$
Weight $2$
Character orbit 1140.bo
Analytic conductor $9.103$
Analytic rank $0$
Dimension $24$
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] = [1140,2,Mod(61,1140)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1140, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1140.61");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1140 = 2^{2} \cdot 3 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1140.bo (of order \(9\), degree \(6\), 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: \(9.10294583043\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(4\) over \(\Q(\zeta_{9})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 6 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 6 q^{7} - 12 q^{11} - 12 q^{17} + 12 q^{19} - 6 q^{21} + 9 q^{23} - 12 q^{27} + 3 q^{29} - 3 q^{33} + 3 q^{35} - 12 q^{37} + 6 q^{39} - 15 q^{41} - 12 q^{45} + 15 q^{47} - 24 q^{49} - 3 q^{51} + 12 q^{53} + 6 q^{55} + 9 q^{57} + 3 q^{59} + 24 q^{61} + 3 q^{63} - 3 q^{65} - 15 q^{67} - 24 q^{71} + 57 q^{73} + 24 q^{75} + 36 q^{77} - 6 q^{79} - 33 q^{83} - 12 q^{85} - 15 q^{87} - 24 q^{89} - 15 q^{91} + 15 q^{93} - 9 q^{95} + 18 q^{97} - 3 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
61.1 0 0.173648 + 0.984808i 0 0.766044 0.642788i 0 −1.88256 + 3.26069i 0 −0.939693 + 0.342020i 0
61.2 0 0.173648 + 0.984808i 0 0.766044 0.642788i 0 −1.63777 + 2.83669i 0 −0.939693 + 0.342020i 0
61.3 0 0.173648 + 0.984808i 0 0.766044 0.642788i 0 0.671715 1.16344i 0 −0.939693 + 0.342020i 0
61.4 0 0.173648 + 0.984808i 0 0.766044 0.642788i 0 0.908917 1.57429i 0 −0.939693 + 0.342020i 0
301.1 0 −0.939693 + 0.342020i 0 0.173648 0.984808i 0 −1.72562 2.98886i 0 0.766044 0.642788i 0
301.2 0 −0.939693 + 0.342020i 0 0.173648 0.984808i 0 −1.49396 2.58762i 0 0.766044 0.642788i 0
301.3 0 −0.939693 + 0.342020i 0 0.173648 0.984808i 0 0.802800 + 1.39049i 0 0.766044 0.642788i 0
301.4 0 −0.939693 + 0.342020i 0 0.173648 0.984808i 0 2.18283 + 3.78077i 0 0.766044 0.642788i 0
481.1 0 −0.939693 0.342020i 0 0.173648 + 0.984808i 0 −1.72562 + 2.98886i 0 0.766044 + 0.642788i 0
481.2 0 −0.939693 0.342020i 0 0.173648 + 0.984808i 0 −1.49396 + 2.58762i 0 0.766044 + 0.642788i 0
481.3 0 −0.939693 0.342020i 0 0.173648 + 0.984808i 0 0.802800 1.39049i 0 0.766044 + 0.642788i 0
481.4 0 −0.939693 0.342020i 0 0.173648 + 0.984808i 0 2.18283 3.78077i 0 0.766044 + 0.642788i 0
541.1 0 0.766044 0.642788i 0 −0.939693 0.342020i 0 −1.59457 + 2.76187i 0 0.173648 0.984808i 0
541.2 0 0.766044 0.642788i 0 −0.939693 0.342020i 0 −1.35032 + 2.33882i 0 0.173648 0.984808i 0
541.3 0 0.766044 0.642788i 0 −0.939693 0.342020i 0 −0.00547393 + 0.00948113i 0 0.173648 0.984808i 0
541.4 0 0.766044 0.642788i 0 −0.939693 0.342020i 0 2.12401 3.67889i 0 0.173648 0.984808i 0
841.1 0 0.173648 0.984808i 0 0.766044 + 0.642788i 0 −1.88256 3.26069i 0 −0.939693 0.342020i 0
841.2 0 0.173648 0.984808i 0 0.766044 + 0.642788i 0 −1.63777 2.83669i 0 −0.939693 0.342020i 0
841.3 0 0.173648 0.984808i 0 0.766044 + 0.642788i 0 0.671715 + 1.16344i 0 −0.939693 0.342020i 0
841.4 0 0.173648 0.984808i 0 0.766044 + 0.642788i 0 0.908917 + 1.57429i 0 −0.939693 0.342020i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 61.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.e even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1140.2.bo.c 24
19.e even 9 1 inner 1140.2.bo.c 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1140.2.bo.c 24 1.a even 1 1 trivial
1140.2.bo.c 24 19.e even 9 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{24} + 6 T_{7}^{23} + 72 T_{7}^{22} + 336 T_{7}^{21} + 2691 T_{7}^{20} + 11001 T_{7}^{19} + 64658 T_{7}^{18} + 218331 T_{7}^{17} + 1028043 T_{7}^{16} + 2904099 T_{7}^{15} + 11270763 T_{7}^{14} + 24386904 T_{7}^{13} + \cdots + 760384 \) acting on \(S_{2}^{\mathrm{new}}(1140, [\chi])\). Copy content Toggle raw display