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Properties

Label 1728.2.bc.e
Level $1728$
Weight $2$
Character orbit 1728.bc
Analytic conductor $13.798$
Analytic rank $0$
Dimension $72$
Inner twists $4$

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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] = [1728,2,Mod(145,1728)] 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("1728.145"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1728, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([0, 9, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1728 = 2^{6} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1728.bc (of order \(12\), degree \(4\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [72,0,0,0,-4] 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: \(13.7981494693\)
Analytic rank: \(0\)
Dimension: \(72\)
Relative dimension: \(18\) over \(\Q(\zeta_{12})\)
Twist minimal: no (minimal twist has level 144)
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 72 q - 4 q^{5} - 2 q^{11} - 16 q^{13} + 16 q^{17} - 28 q^{19} - 4 q^{29} - 28 q^{31} - 16 q^{35} + 16 q^{37} + 10 q^{43} - 56 q^{47} + 4 q^{49} + 8 q^{53} - 14 q^{59} - 32 q^{61} + 64 q^{65} + 18 q^{67}+ \cdots + 40 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
145.1 0 0 0 −1.10094 − 4.10876i 0 −1.63313 + 0.942891i 0 0 0
145.2 0 0 0 −0.722679 − 2.69708i 0 −2.89314 + 1.67035i 0 0 0
145.3 0 0 0 −0.646846 − 2.41406i 0 2.82197 − 1.62927i 0 0 0
145.4 0 0 0 −0.592492 − 2.21121i 0 2.67054 − 1.54184i 0 0 0
145.5 0 0 0 −0.531653 − 1.98415i 0 −1.54969 + 0.894715i 0 0 0
145.6 0 0 0 −0.430214 − 1.60558i 0 3.62762 − 2.09441i 0 0 0
145.7 0 0 0 −0.226831 − 0.846545i 0 −0.567074 + 0.327400i 0 0 0
145.8 0 0 0 −0.131764 − 0.491749i 0 −2.40518 + 1.38863i 0 0 0
145.9 0 0 0 −0.0185225 − 0.0691269i 0 1.28192 − 0.740118i 0 0 0
145.10 0 0 0 −0.00302457 − 0.0112878i 0 1.05753 − 0.610563i 0 0 0
145.11 0 0 0 0.0468197 + 0.174734i 0 −4.04791 + 2.33706i 0 0 0
145.12 0 0 0 0.326078 + 1.21694i 0 −0.707732 + 0.408609i 0 0 0
145.13 0 0 0 0.679606 + 2.53632i 0 0.614293 − 0.354662i 0 0 0
145.14 0 0 0 0.733432 + 2.73721i 0 −1.14487 + 0.660988i 0 0 0
145.15 0 0 0 0.777246 + 2.90072i 0 −1.04527 + 0.603486i 0 0 0
145.16 0 0 0 0.798307 + 2.97932i 0 1.78208 − 1.02889i 0 0 0
145.17 0 0 0 0.884514 + 3.30105i 0 2.63210 − 1.51965i 0 0 0
145.18 0 0 0 0.891015 + 3.32531i 0 −3.95817 + 2.28525i 0 0 0
721.1 0 0 0 −3.32531 − 0.891015i 0 3.95817 + 2.28525i 0 0 0
721.2 0 0 0 −3.30105 − 0.884514i 0 −2.63210 − 1.51965i 0 0 0
See all 72 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 145.18
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner
16.e even 4 1 inner
144.x even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1728.2.bc.e 72
3.b odd 2 1 576.2.bb.e 72
4.b odd 2 1 432.2.y.e 72
9.c even 3 1 inner 1728.2.bc.e 72
9.d odd 6 1 576.2.bb.e 72
12.b even 2 1 144.2.x.e ✓ 72
16.e even 4 1 inner 1728.2.bc.e 72
16.f odd 4 1 432.2.y.e 72
36.f odd 6 1 432.2.y.e 72
36.h even 6 1 144.2.x.e ✓ 72
48.i odd 4 1 576.2.bb.e 72
48.k even 4 1 144.2.x.e ✓ 72
144.u even 12 1 144.2.x.e ✓ 72
144.v odd 12 1 432.2.y.e 72
144.w odd 12 1 576.2.bb.e 72
144.x even 12 1 inner 1728.2.bc.e 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
144.2.x.e ✓ 72 12.b even 2 1
144.2.x.e ✓ 72 36.h even 6 1
144.2.x.e ✓ 72 48.k even 4 1
144.2.x.e ✓ 72 144.u even 12 1
432.2.y.e 72 4.b odd 2 1
432.2.y.e 72 16.f odd 4 1
432.2.y.e 72 36.f odd 6 1
432.2.y.e 72 144.v odd 12 1
576.2.bb.e 72 3.b odd 2 1
576.2.bb.e 72 9.d odd 6 1
576.2.bb.e 72 48.i odd 4 1
576.2.bb.e 72 144.w odd 12 1
1728.2.bc.e 72 1.a even 1 1 trivial
1728.2.bc.e 72 9.c even 3 1 inner
1728.2.bc.e 72 16.e even 4 1 inner
1728.2.bc.e 72 144.x even 12 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{72} + 4 T_{5}^{71} + 8 T_{5}^{70} + 48 T_{5}^{69} - 362 T_{5}^{68} - 1784 T_{5}^{67} + \cdots + 65536 \) acting on \(S_{2}^{\mathrm{new}}(1728, [\chi])\). Copy content Toggle raw display