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Properties

Label 1160.2.s.c
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 = [42,0,0,0,0] 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.

\(\operatorname{Tr}(f)(q) = \) \( 42 q - 2 q^{7} - 42 q^{9} + 4 q^{13} + 14 q^{15} - 12 q^{17} + 6 q^{19} - 8 q^{21} + 10 q^{23} + 8 q^{25} + 30 q^{29} - 4 q^{31} + 22 q^{33} - 2 q^{35} + 38 q^{39} + 10 q^{41} + 30 q^{45} + 8 q^{53} - 18 q^{55}+ \cdots + 34 q^{99}+O(q^{100}) \) 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 − 3.29071i 0 −2.04506 + 0.904273i 0 0.188359 + 0.188359i 0 −7.82877 0
17.2 0 − 2.94543i 0 1.96103 − 1.07441i 0 −2.96878 − 2.96878i 0 −5.67558 0
17.3 0 − 2.85988i 0 0.724019 + 2.11561i 0 −0.362395 − 0.362395i 0 −5.17890 0
17.4 0 − 2.30696i 0 −2.19968 − 0.401754i 0 2.04754 + 2.04754i 0 −2.32204 0
17.5 0 − 2.10106i 0 2.18421 − 0.478777i 0 1.94771 + 1.94771i 0 −1.41447 0
17.6 0 − 1.25308i 0 −2.20888 + 0.347625i 0 −2.59119 − 2.59119i 0 1.42978 0
17.7 0 − 1.24863i 0 1.39263 + 1.74945i 0 0.500287 + 0.500287i 0 1.44093 0
17.8 0 − 0.547365i 0 −0.863627 + 2.06256i 0 −1.50053 − 1.50053i 0 2.70039 0
17.9 0 − 0.480342i 0 0.0382906 − 2.23574i 0 0.312444 + 0.312444i 0 2.76927 0
17.10 0 − 0.0812822i 0 −1.03456 + 1.98235i 0 3.14254 + 3.14254i 0 2.99339 0
17.11 0 0.165721i 0 2.23052 + 0.157369i 0 1.32037 + 1.32037i 0 2.97254 0
17.12 0 0.675231i 0 1.13334 − 1.92757i 0 −1.28356 − 1.28356i 0 2.54406 0
17.13 0 0.835579i 0 −2.22308 − 0.240611i 0 −0.255442 − 0.255442i 0 2.30181 0
17.14 0 0.954423i 0 2.16108 − 0.574216i 0 −3.03409 − 3.03409i 0 2.08908 0
17.15 0 1.23433i 0 0.546927 + 2.16815i 0 −2.14760 − 2.14760i 0 1.47643 0
17.16 0 2.17962i 0 −1.87753 + 1.21445i 0 1.34859 + 1.34859i 0 −1.75076 0
17.17 0 2.27256i 0 0.224633 − 2.22476i 0 −0.0252558 − 0.0252558i 0 −2.16453 0
17.18 0 2.37525i 0 1.85773 + 1.24452i 0 2.68376 + 2.68376i 0 −2.64180 0
17.19 0 2.55288i 0 −0.183584 − 2.22852i 0 3.40258 + 3.40258i 0 −3.51718 0
17.20 0 2.92477i 0 −2.09608 − 0.778750i 0 −2.58090 − 2.58090i 0 −5.55427 0
See all 42 embeddings
\(n\): e.g. 2-40 or 990-1000
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.c ✓ 42
5.c odd 4 1 1160.2.bl.d yes 42
29.c odd 4 1 1160.2.bl.d yes 42
145.j even 4 1 inner 1160.2.s.c ✓ 42
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1160.2.s.c ✓ 42 1.a even 1 1 trivial
1160.2.s.c ✓ 42 145.j even 4 1 inner
1160.2.bl.d yes 42 5.c odd 4 1
1160.2.bl.d 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 \(S_{2}^{\mathrm{new}}(1160, [\chi])\):

\( T_{3}^{42} + 84 T_{3}^{40} + 3234 T_{3}^{38} + 75654 T_{3}^{36} + 1201961 T_{3}^{34} + 13732276 T_{3}^{32} + \cdots + 16384 \) Copy content Toggle raw display
\( T_{7}^{42} + 2 T_{7}^{41} + 2 T_{7}^{40} + 8 T_{7}^{39} + 1223 T_{7}^{38} + 2746 T_{7}^{37} + \cdots + 33554432 \) Copy content Toggle raw display