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Properties

Label 1160.2.bl.d
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,-4] 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.

\(\operatorname{Tr}(f)(q) = \) \( 42 q - 4 q^{3} - 2 q^{7} + 42 q^{9} - 4 q^{13} - 4 q^{15} - 6 q^{19} - 8 q^{21} + 10 q^{23} + 8 q^{25} + 32 q^{27} - 30 q^{29} - 4 q^{31} - 22 q^{33} + 2 q^{35} - 32 q^{37} - 38 q^{39} + 10 q^{41} + 30 q^{45}+ \cdots - 34 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
713.1 0 −2.94438 0 −0.277665 + 2.21876i 0 −1.14444 + 1.14444i 0 5.66938 0
713.2 0 −2.92477 0 2.09608 − 0.778750i 0 −2.58090 + 2.58090i 0 5.55427 0
713.3 0 −2.55288 0 0.183584 − 2.22852i 0 3.40258 − 3.40258i 0 3.51718 0
713.4 0 −2.37525 0 −1.85773 + 1.24452i 0 2.68376 − 2.68376i 0 2.64180 0
713.5 0 −2.27256 0 −0.224633 − 2.22476i 0 −0.0252558 + 0.0252558i 0 2.16453 0
713.6 0 −2.17962 0 1.87753 + 1.21445i 0 1.34859 − 1.34859i 0 1.75076 0
713.7 0 −1.23433 0 −0.546927 + 2.16815i 0 −2.14760 + 2.14760i 0 −1.47643 0
713.8 0 −0.954423 0 −2.16108 − 0.574216i 0 −3.03409 + 3.03409i 0 −2.08908 0
713.9 0 −0.835579 0 2.22308 − 0.240611i 0 −0.255442 + 0.255442i 0 −2.30181 0
713.10 0 −0.675231 0 −1.13334 − 1.92757i 0 −1.28356 + 1.28356i 0 −2.54406 0
713.11 0 −0.165721 0 −2.23052 + 0.157369i 0 1.32037 − 1.32037i 0 −2.97254 0
713.12 0 0.0812822 0 1.03456 + 1.98235i 0 3.14254 − 3.14254i 0 −2.99339 0
713.13 0 0.480342 0 −0.0382906 − 2.23574i 0 0.312444 − 0.312444i 0 −2.76927 0
713.14 0 0.547365 0 0.863627 + 2.06256i 0 −1.50053 + 1.50053i 0 −2.70039 0
713.15 0 1.24863 0 −1.39263 + 1.74945i 0 0.500287 − 0.500287i 0 −1.44093 0
713.16 0 1.25308 0 2.20888 + 0.347625i 0 −2.59119 + 2.59119i 0 −1.42978 0
713.17 0 2.10106 0 −2.18421 − 0.478777i 0 1.94771 − 1.94771i 0 1.41447 0
713.18 0 2.30696 0 2.19968 − 0.401754i 0 2.04754 − 2.04754i 0 2.32204 0
713.19 0 2.85988 0 −0.724019 + 2.11561i 0 −0.362395 + 0.362395i 0 5.17890 0
713.20 0 2.94543 0 −1.96103 − 1.07441i 0 −2.96878 + 2.96878i 0 5.67558 0
See all 42 embeddings
\(n\): e.g. 2-40 or 80-90
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.d yes 42
5.c odd 4 1 1160.2.s.c ✓ 42
29.c odd 4 1 1160.2.s.c ✓ 42
145.e even 4 1 inner 1160.2.bl.d yes 42
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1160.2.s.c ✓ 42 5.c odd 4 1
1160.2.s.c ✓ 42 29.c odd 4 1
1160.2.bl.d yes 42 1.a even 1 1 trivial
1160.2.bl.d 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 \(S_{2}^{\mathrm{new}}(1160, [\chi])\):

\( T_{3}^{21} + 2 T_{3}^{20} - 40 T_{3}^{19} - 84 T_{3}^{18} + 649 T_{3}^{17} + 1432 T_{3}^{16} + \cdots + 128 \) 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