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Properties

Label 1160.2.j.a
Level $1160$
Weight $2$
Character orbit 1160.j
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(289,1160)] 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("1160.289"); 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, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1160 = 2^{3} \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1160.j (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [22,0,0,0,-1,0,0,0,18,0,0,0,0,0,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(15)] == 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.

\(\operatorname{Tr}(f)(q) = \) \( 22 q - q^{5} + 18 q^{9} - 3 q^{15} - 18 q^{17} + 5 q^{25} + 6 q^{27} - 2 q^{29} - 4 q^{35} + 4 q^{37} + 12 q^{43} - 2 q^{45} + 18 q^{47} - 12 q^{49} - 7 q^{55} - 10 q^{59} + 7 q^{65} - 12 q^{71} + 26 q^{73}+ \cdots + 14 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
289.1 0 −3.08006 0 1.18739 − 1.89476i 0 − 1.33277i 0 6.48679 0
289.2 0 −3.08006 0 1.18739 + 1.89476i 0 1.33277i 0 6.48679 0
289.3 0 −2.60309 0 −2.14698 − 0.624868i 0 0.754603i 0 3.77607 0
289.4 0 −2.60309 0 −2.14698 + 0.624868i 0 − 0.754603i 0 3.77607 0
289.5 0 −1.55819 0 1.55152 − 1.61021i 0 4.44353i 0 −0.572033 0
289.6 0 −1.55819 0 1.55152 + 1.61021i 0 − 4.44353i 0 −0.572033 0
289.7 0 −0.872443 0 −0.408890 − 2.19836i 0 − 4.74919i 0 −2.23884 0
289.8 0 −0.872443 0 −0.408890 + 2.19836i 0 4.74919i 0 −2.23884 0
289.9 0 −0.680621 0 −1.86606 − 1.23200i 0 − 0.588801i 0 −2.53675 0
289.10 0 −0.680621 0 −1.86606 + 1.23200i 0 0.588801i 0 −2.53675 0
289.11 0 −0.668550 0 2.16879 − 0.544377i 0 − 0.883688i 0 −2.55304 0
289.12 0 −0.668550 0 2.16879 + 0.544377i 0 0.883688i 0 −2.55304 0
289.13 0 0.741618 0 0.330897 − 2.21145i 0 1.94517i 0 −2.45000 0
289.14 0 0.741618 0 0.330897 + 2.21145i 0 − 1.94517i 0 −2.45000 0
289.15 0 1.12583 0 −1.91562 − 1.15343i 0 3.28732i 0 −1.73250 0
289.16 0 1.12583 0 −1.91562 + 1.15343i 0 − 3.28732i 0 −1.73250 0
289.17 0 1.94693 0 0.211001 + 2.22609i 0 − 0.687666i 0 0.790520 0
289.18 0 1.94693 0 0.211001 − 2.22609i 0 0.687666i 0 0.790520 0
289.19 0 2.62864 0 2.20289 + 0.383753i 0 3.80695i 0 3.90976 0
289.20 0 2.62864 0 2.20289 − 0.383753i 0 − 3.80695i 0 3.90976 0
See all 22 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 289.22
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
145.d 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.j.a ✓ 22
4.b odd 2 1 2320.2.j.h 22
5.b even 2 1 1160.2.j.b yes 22
20.d odd 2 1 2320.2.j.g 22
29.b even 2 1 1160.2.j.b yes 22
116.d odd 2 1 2320.2.j.g 22
145.d even 2 1 inner 1160.2.j.a ✓ 22
580.e odd 2 1 2320.2.j.h 22
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1160.2.j.a ✓ 22 1.a even 1 1 trivial
1160.2.j.a ✓ 22 145.d even 2 1 inner
1160.2.j.b yes 22 5.b even 2 1
1160.2.j.b yes 22 29.b even 2 1
2320.2.j.g 22 20.d odd 2 1
2320.2.j.g 22 116.d odd 2 1
2320.2.j.h 22 4.b odd 2 1
2320.2.j.h 22 580.e odd 2 1