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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1332,2,Mod(445,1332)] 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("1332.445"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1332, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 2, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1332 = 2^{2} \cdot 3^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1332.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [34,0,1,0,-7] 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: \(10.6360735492\)
Analytic rank: \(0\)
Dimension: \(34\)
Relative dimension: \(17\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 34 q + q^{3} - 7 q^{5} - q^{7} + 13 q^{9} - 4 q^{11} + 3 q^{13} - 5 q^{15} + 12 q^{17} - 2 q^{19} + 7 q^{21} - 13 q^{23} - 22 q^{25} - 17 q^{27} - 13 q^{29} - 7 q^{31} - q^{33} - 8 q^{35} - 34 q^{37}+ \cdots - 55 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
445.1 0 −1.71232 − 0.260669i 0 −1.51829 + 2.62976i 0 2.42785 + 4.20516i 0 2.86410 + 0.892698i 0
445.2 0 −1.63900 + 0.560063i 0 0.355708 − 0.616104i 0 0.565445 + 0.979380i 0 2.37266 − 1.83589i 0
445.3 0 −1.63601 − 0.568754i 0 −0.132836 + 0.230079i 0 1.19699 + 2.07324i 0 2.35304 + 1.86097i 0
445.4 0 −1.62819 + 0.590748i 0 1.50305 − 2.60337i 0 −2.14036 − 3.70722i 0 2.30203 − 1.92370i 0
445.5 0 −1.16045 + 1.28583i 0 −1.56351 + 2.70808i 0 −1.04639 − 1.81240i 0 −0.306720 − 2.98428i 0
445.6 0 −1.13334 − 1.30978i 0 −0.610625 + 1.05763i 0 −1.60197 − 2.77470i 0 −0.431062 + 2.96887i 0
445.7 0 −0.693778 − 1.58703i 0 0.0543817 − 0.0941919i 0 1.96692 + 3.40681i 0 −2.03734 + 2.20210i 0
445.8 0 −0.484698 + 1.66285i 0 1.31030 − 2.26950i 0 0.436643 + 0.756288i 0 −2.53014 − 1.61196i 0
445.9 0 0.0706845 + 1.73061i 0 −1.19957 + 2.07772i 0 −1.81720 − 3.14748i 0 −2.99001 + 0.244654i 0
445.10 0 0.809770 + 1.53110i 0 1.43595 − 2.48713i 0 1.33445 + 2.31134i 0 −1.68855 + 2.47968i 0
445.11 0 1.01242 − 1.40535i 0 −1.19361 + 2.06740i 0 −0.187955 − 0.325548i 0 −0.950029 − 2.84560i 0
445.12 0 1.25739 − 1.19121i 0 1.42800 − 2.47337i 0 0.194299 + 0.336535i 0 0.162049 − 2.99562i 0
445.13 0 1.29025 + 1.15553i 0 −0.674966 + 1.16908i 0 −2.10309 − 3.64267i 0 0.329512 + 2.98185i 0
445.14 0 1.37366 − 1.05501i 0 1.09488 − 1.89639i 0 −1.39883 − 2.42284i 0 0.773904 − 2.89846i 0
445.15 0 1.43663 + 0.967518i 0 0.388876 − 0.673554i 0 0.561895 + 0.973231i 0 1.12782 + 2.77993i 0
445.16 0 1.61098 + 0.636192i 0 −2.02941 + 3.51505i 0 2.11003 + 3.65467i 0 2.19052 + 2.04979i 0
445.17 0 1.72601 − 0.144553i 0 −2.14832 + 3.72099i 0 −0.998723 − 1.72984i 0 2.95821 − 0.499000i 0
889.1 0 −1.71232 + 0.260669i 0 −1.51829 − 2.62976i 0 2.42785 − 4.20516i 0 2.86410 − 0.892698i 0
889.2 0 −1.63900 − 0.560063i 0 0.355708 + 0.616104i 0 0.565445 − 0.979380i 0 2.37266 + 1.83589i 0
889.3 0 −1.63601 + 0.568754i 0 −0.132836 − 0.230079i 0 1.19699 − 2.07324i 0 2.35304 − 1.86097i 0
See all 34 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 445.17
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1332.2.i.c ✓ 34
3.b odd 2 1 3996.2.i.d 34
9.c even 3 1 inner 1332.2.i.c ✓ 34
9.d odd 6 1 3996.2.i.d 34
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1332.2.i.c ✓ 34 1.a even 1 1 trivial
1332.2.i.c ✓ 34 9.c even 3 1 inner
3996.2.i.d 34 3.b odd 2 1
3996.2.i.d 34 9.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{34} + 7 T_{5}^{33} + 78 T_{5}^{32} + 343 T_{5}^{31} + 2509 T_{5}^{30} + 8811 T_{5}^{29} + \cdots + 12510369 \) acting on \(S_{2}^{\mathrm{new}}(1332, [\chi])\). Copy content Toggle raw display