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Properties

Label 936.2.s.f
Level $936$
Weight $2$
Character orbit 936.s
Analytic conductor $7.474$
Analytic rank $0$
Dimension $40$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [936,2,Mod(529,936)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(936, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 4, 4])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("936.529"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 936 = 2^{3} \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 936.s (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [40,0,-3,0,1] 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: \(7.47399762919\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(20\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 40 q - 3 q^{3} + q^{5} - 14 q^{7} - 9 q^{9} - 3 q^{13} + 2 q^{15} - q^{17} + 2 q^{19} - 30 q^{21} + 2 q^{23} - 23 q^{25} - 3 q^{27} + 12 q^{29} + 8 q^{31} - 5 q^{33} - 12 q^{35} + 18 q^{37} - 6 q^{39}+ \cdots + 28 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
529.1 0 −1.70704 + 0.293265i 0 2.01584 + 3.49154i 0 −3.11418 0 2.82799 − 1.00123i 0
529.2 0 −1.69808 + 0.341337i 0 0.928740 + 1.60862i 0 3.89194 0 2.76698 − 1.15924i 0
529.3 0 −1.69053 − 0.376974i 0 −0.185033 − 0.320487i 0 0.220829 0 2.71578 + 1.27457i 0
529.4 0 −1.26687 + 1.18112i 0 0.0819095 + 0.141871i 0 −1.18267 0 0.209912 − 2.99265i 0
529.5 0 −1.16552 + 1.28124i 0 −2.17623 − 3.76934i 0 4.59078 0 −0.283128 − 2.98661i 0
529.6 0 −1.11550 − 1.32501i 0 −0.816777 − 1.41470i 0 −0.664936 0 −0.511312 + 2.95611i 0
529.7 0 −0.983237 + 1.42592i 0 −0.702173 − 1.21620i 0 −2.51315 0 −1.06649 − 2.80403i 0
529.8 0 −0.835133 − 1.51742i 0 0.778035 + 1.34760i 0 3.24341 0 −1.60511 + 2.53449i 0
529.9 0 −0.813072 − 1.52935i 0 0.950006 + 1.64546i 0 −2.76473 0 −1.67783 + 2.48694i 0
529.10 0 0.0740856 + 1.73047i 0 0.606444 + 1.05039i 0 3.66978 0 −2.98902 + 0.256405i 0
529.11 0 0.0896694 − 1.72973i 0 −1.74688 − 3.02568i 0 −3.26004 0 −2.98392 − 0.310207i 0
529.12 0 0.348515 − 1.69663i 0 1.65071 + 2.85912i 0 1.86626 0 −2.75708 − 1.18260i 0
529.13 0 0.349029 + 1.69652i 0 −0.527004 − 0.912798i 0 −0.877064 0 −2.75636 + 1.18427i 0
529.14 0 0.685486 + 1.59063i 0 1.97453 + 3.41999i 0 −3.28148 0 −2.06022 + 2.18071i 0
529.15 0 0.809564 − 1.53121i 0 −1.23958 − 2.14702i 0 1.47278 0 −1.68921 − 2.47923i 0
529.16 0 1.21888 + 1.23058i 0 −1.96859 − 3.40969i 0 −3.13764 0 −0.0286516 + 2.99986i 0
529.17 0 1.31464 − 1.12771i 0 0.766914 + 1.32833i 0 −3.51479 0 0.456560 − 2.96506i 0
529.18 0 1.54706 + 0.778844i 0 0.871295 + 1.50913i 0 1.46715 0 1.78680 + 2.40984i 0
529.19 0 1.64850 + 0.531456i 0 −1.09120 − 1.89001i 0 1.62444 0 2.43511 + 1.75221i 0
529.20 0 1.68955 − 0.381325i 0 0.329028 + 0.569893i 0 −4.73667 0 2.70918 − 1.28854i 0
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 529.20
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
117.f even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 936.2.s.f yes 40
3.b odd 2 1 2808.2.s.f 40
9.c even 3 1 936.2.r.f ✓ 40
9.d odd 6 1 2808.2.r.f 40
13.c even 3 1 936.2.r.f ✓ 40
39.i odd 6 1 2808.2.r.f 40
117.f even 3 1 inner 936.2.s.f yes 40
117.u odd 6 1 2808.2.s.f 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
936.2.r.f ✓ 40 9.c even 3 1
936.2.r.f ✓ 40 13.c even 3 1
936.2.s.f yes 40 1.a even 1 1 trivial
936.2.s.f yes 40 117.f even 3 1 inner
2808.2.r.f 40 9.d odd 6 1
2808.2.r.f 40 39.i odd 6 1
2808.2.s.f 40 3.b odd 2 1
2808.2.s.f 40 117.u odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(936, [\chi])\):

\( T_{5}^{40} - T_{5}^{39} + 62 T_{5}^{38} - 81 T_{5}^{37} + 2315 T_{5}^{36} - 3336 T_{5}^{35} + \cdots + 855036081 \) Copy content Toggle raw display
\( T_{7}^{20} + 7 T_{7}^{19} - 57 T_{7}^{18} - 492 T_{7}^{17} + 1031 T_{7}^{16} + 13819 T_{7}^{15} + \cdots - 2565648 \) Copy content Toggle raw display