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Properties

Label 1160.2.d.d
Level $1160$
Weight $2$
Character orbit 1160.d
Analytic conductor $9.263$
Analytic rank $0$
Dimension $22$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1160,2,Mod(929,1160)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1160.929"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1160, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1160 = 2^{3} \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1160.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [22,0,0,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(9.26264663447\)
Analytic rank: \(0\)
Dimension: \(22\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 22 q + 2 q^{5} - 28 q^{9} - 18 q^{15} - 16 q^{19} + 12 q^{21} + 10 q^{25} + 22 q^{29} - 42 q^{31} - 2 q^{35} + 26 q^{39} - 24 q^{41} + 18 q^{45} - 60 q^{49} - 16 q^{51} + 8 q^{55} - 14 q^{59} + 30 q^{61}+ \cdots - 12 q^{99}+O(q^{100}) \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
929.1 0 − 3.44922i 0 0.522951 + 2.17406i 0 − 1.63244i 0 −8.89713 0
929.2 0 − 3.16481i 0 −1.27652 − 1.83589i 0 5.27690i 0 −7.01603 0
929.3 0 − 2.75417i 0 1.15611 − 1.91401i 0 − 0.126741i 0 −4.58543 0
929.4 0 − 2.34151i 0 −2.22212 + 0.249354i 0 − 3.94606i 0 −2.48268 0
929.5 0 − 2.07176i 0 2.21186 − 0.328108i 0 2.89053i 0 −1.29219 0
929.6 0 − 1.91805i 0 −0.142182 − 2.23154i 0 − 2.14388i 0 −0.678920 0
929.7 0 − 1.72768i 0 −2.22653 + 0.206357i 0 2.91925i 0 0.0151200 0
929.8 0 − 0.895367i 0 2.09199 − 0.789681i 0 − 2.05014i 0 2.19832 0
929.9 0 − 0.375015i 0 −1.68758 − 1.46699i 0 − 4.63080i 0 2.85936 0
929.10 0 − 0.307878i 0 2.12882 + 0.684212i 0 3.70760i 0 2.90521 0
929.11 0 − 0.160119i 0 0.443212 − 2.19170i 0 0.186633i 0 2.97436 0
929.12 0 0.160119i 0 0.443212 + 2.19170i 0 − 0.186633i 0 2.97436 0
929.13 0 0.307878i 0 2.12882 − 0.684212i 0 − 3.70760i 0 2.90521 0
929.14 0 0.375015i 0 −1.68758 + 1.46699i 0 4.63080i 0 2.85936 0
929.15 0 0.895367i 0 2.09199 + 0.789681i 0 2.05014i 0 2.19832 0
929.16 0 1.72768i 0 −2.22653 − 0.206357i 0 − 2.91925i 0 0.0151200 0
929.17 0 1.91805i 0 −0.142182 + 2.23154i 0 2.14388i 0 −0.678920 0
929.18 0 2.07176i 0 2.21186 + 0.328108i 0 − 2.89053i 0 −1.29219 0
929.19 0 2.34151i 0 −2.22212 − 0.249354i 0 3.94606i 0 −2.48268 0
929.20 0 2.75417i 0 1.15611 + 1.91401i 0 0.126741i 0 −4.58543 0
See all 22 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 929.22
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1160.2.d.d ✓ 22
4.b odd 2 1 2320.2.d.k 22
5.b even 2 1 inner 1160.2.d.d ✓ 22
5.c odd 4 1 5800.2.a.bg 11
5.c odd 4 1 5800.2.a.bh 11
20.d odd 2 1 2320.2.d.k 22
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1160.2.d.d ✓ 22 1.a even 1 1 trivial
1160.2.d.d ✓ 22 5.b even 2 1 inner
2320.2.d.k 22 4.b odd 2 1
2320.2.d.k 22 20.d odd 2 1
5800.2.a.bg 11 5.c odd 4 1
5800.2.a.bh 11 5.c odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{22} + 47 T_{3}^{20} + 919 T_{3}^{18} + 9734 T_{3}^{16} + 60899 T_{3}^{14} + 229679 T_{3}^{12} + \cdots + 64 \) acting on \(S_{2}^{\mathrm{new}}(1160, [\chi])\). Copy content Toggle raw display