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Properties

Label 1160.2.bl.c
Level $1160$
Weight $2$
Character orbit 1160.bl
Analytic conductor $9.263$
Analytic rank $0$
Dimension $42$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1160,2,Mod(713,1160)] 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("1160.713"); 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, 3, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1160 = 2^{3} \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1160.bl (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [42,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == 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.

\(\operatorname{Tr}(f)(q) = \) \( 42 q - 8 q^{3} - 2 q^{5} + 4 q^{7} + 34 q^{9} + 2 q^{11} + 4 q^{13} - 4 q^{15} + 8 q^{19} + 4 q^{21} - 20 q^{23} + 4 q^{25} - 8 q^{27} + 20 q^{29} + 2 q^{31} - 10 q^{33} + 16 q^{35} - 12 q^{37} + 10 q^{39}+ \cdots + 60 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
713.1 0 −3.37194 0 −0.287481 + 2.21751i 0 1.91898 − 1.91898i 0 8.36995 0
713.2 0 −3.17069 0 −1.08411 − 1.95569i 0 −1.73480 + 1.73480i 0 7.05330 0
713.3 0 −2.70177 0 2.15370 − 0.601320i 0 2.86333 − 2.86333i 0 4.29955 0
713.4 0 −2.04140 0 1.68595 − 1.46887i 0 −0.492226 + 0.492226i 0 1.16731 0
713.5 0 −1.87952 0 −1.77003 + 1.36638i 0 −3.24588 + 3.24588i 0 0.532591 0
713.6 0 −1.84640 0 −2.19775 + 0.412181i 0 0.563388 − 0.563388i 0 0.409193 0
713.7 0 −1.70895 0 −1.82020 − 1.29879i 0 1.24431 − 1.24431i 0 −0.0795021 0
713.8 0 −1.24535 0 1.73825 + 1.40659i 0 −1.01848 + 1.01848i 0 −1.44910 0
713.9 0 −0.589224 0 1.18636 − 1.89540i 0 −1.11858 + 1.11858i 0 −2.65282 0
713.10 0 −0.533118 0 −0.872855 + 2.05867i 0 1.75538 − 1.75538i 0 −2.71578 0
713.11 0 −0.349330 0 1.82703 + 1.28916i 0 2.13337 − 2.13337i 0 −2.87797 0
713.12 0 0.111611 0 −0.657451 − 2.13723i 0 3.03882 − 3.03882i 0 −2.98754 0
713.13 0 0.206943 0 1.77191 − 1.36395i 0 −2.59138 + 2.59138i 0 −2.95717 0
713.14 0 0.650627 0 0.944182 + 2.02695i 0 −2.68175 + 2.68175i 0 −2.57668 0
713.15 0 0.911666 0 −2.14439 + 0.633727i 0 −0.763192 + 0.763192i 0 −2.16886 0
713.16 0 1.43143 0 −1.20242 − 1.88526i 0 −1.50596 + 1.50596i 0 −0.951000 0
713.17 0 1.61930 0 −1.77596 − 1.35866i 0 −1.85656 + 1.85656i 0 −0.377864 0
713.18 0 1.64255 0 2.21353 − 0.316645i 0 1.69441 − 1.69441i 0 −0.302036 0
713.19 0 2.72898 0 1.11616 + 1.93757i 0 0.522867 − 0.522867i 0 4.44732 0
713.20 0 3.04972 0 0.407762 − 2.19857i 0 −0.312635 + 0.312635i 0 6.30081 0
See all 42 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 713.21
Significant digits:
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Inner twists

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

Hecke kernels

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

\( T_{3}^{21} + 4 T_{3}^{20} - 32 T_{3}^{19} - 140 T_{3}^{18} + 381 T_{3}^{17} + 1950 T_{3}^{16} + \cdots + 64 \) 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