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Properties

Label 1160.2.s.d
Level $1160$
Weight $2$
Character orbit 1160.s
Analytic conductor $9.263$
Analytic rank $0$
Dimension $42$
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(17,1160)] 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("1160.17"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1160, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 0, 1, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1160 = 2^{3} \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1160.s (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [42,0,0,0,2] 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: \(42\)
Relative dimension: \(21\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 42 q + 2 q^{5} + 4 q^{7} - 34 q^{9} + 2 q^{11} - 4 q^{13} - 10 q^{15} + 8 q^{17} - 8 q^{19} + 4 q^{21} - 20 q^{23} + 4 q^{25} - 20 q^{29} + 2 q^{31} + 10 q^{33} - 16 q^{35} - 10 q^{39} - 30 q^{41} - 32 q^{45}+ \cdots - 60 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1 0 − 3.08485i 0 2.23219 + 0.131647i 0 3.58659 + 3.58659i 0 −6.51632 0
17.2 0 − 3.04972i 0 −0.407762 − 2.19857i 0 −0.312635 − 0.312635i 0 −6.30081 0
17.3 0 − 2.72898i 0 −1.11616 + 1.93757i 0 0.522867 + 0.522867i 0 −4.44732 0
17.4 0 − 1.64255i 0 −2.21353 − 0.316645i 0 1.69441 + 1.69441i 0 0.302036 0
17.5 0 − 1.61930i 0 1.77596 − 1.35866i 0 −1.85656 − 1.85656i 0 0.377864 0
17.6 0 − 1.43143i 0 1.20242 − 1.88526i 0 −1.50596 − 1.50596i 0 0.951000 0
17.7 0 − 0.911666i 0 2.14439 + 0.633727i 0 −0.763192 − 0.763192i 0 2.16886 0
17.8 0 − 0.650627i 0 −0.944182 + 2.02695i 0 −2.68175 − 2.68175i 0 2.57668 0
17.9 0 − 0.206943i 0 −1.77191 − 1.36395i 0 −2.59138 − 2.59138i 0 2.95717 0
17.10 0 − 0.111611i 0 0.657451 − 2.13723i 0 3.03882 + 3.03882i 0 2.98754 0
17.11 0 0.349330i 0 −1.82703 + 1.28916i 0 2.13337 + 2.13337i 0 2.87797 0
17.12 0 0.533118i 0 0.872855 + 2.05867i 0 1.75538 + 1.75538i 0 2.71578 0
17.13 0 0.589224i 0 −1.18636 − 1.89540i 0 −1.11858 − 1.11858i 0 2.65282 0
17.14 0 1.24535i 0 −1.73825 + 1.40659i 0 −1.01848 − 1.01848i 0 1.44910 0
17.15 0 1.70895i 0 1.82020 − 1.29879i 0 1.24431 + 1.24431i 0 0.0795021 0
17.16 0 1.84640i 0 2.19775 + 0.412181i 0 0.563388 + 0.563388i 0 −0.409193 0
17.17 0 1.87952i 0 1.77003 + 1.36638i 0 −3.24588 − 3.24588i 0 −0.532591 0
17.18 0 2.04140i 0 −1.68595 − 1.46887i 0 −0.492226 − 0.492226i 0 −1.16731 0
17.19 0 2.70177i 0 −2.15370 − 0.601320i 0 2.86333 + 2.86333i 0 −4.29955 0
17.20 0 3.17069i 0 1.08411 − 1.95569i 0 −1.73480 − 1.73480i 0 −7.05330 0
See all 42 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 17.21
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
145.j even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1160.2.s.d ✓ 42
5.c odd 4 1 1160.2.bl.c yes 42
29.c odd 4 1 1160.2.bl.c yes 42
145.j even 4 1 inner 1160.2.s.d ✓ 42
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1160.2.s.d ✓ 42 1.a even 1 1 trivial
1160.2.s.d ✓ 42 145.j even 4 1 inner
1160.2.bl.c yes 42 5.c odd 4 1
1160.2.bl.c yes 42 29.c odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1160, [\chi])\):

\( T_{3}^{42} + 80 T_{3}^{40} + 2906 T_{3}^{38} + 63542 T_{3}^{36} + 935049 T_{3}^{34} + 9810648 T_{3}^{32} + \cdots + 4096 \) Copy content Toggle raw display
\( T_{7}^{42} - 4 T_{7}^{41} + 8 T_{7}^{40} + 40 T_{7}^{39} + 1027 T_{7}^{38} - 3704 T_{7}^{37} + \cdots + 8718953480192 \) Copy content Toggle raw display