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Properties

Label 576.2.y.a
Level $576$
Weight $2$
Character orbit 576.y
Analytic conductor $4.599$
Analytic rank $0$
Dimension $88$
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] = [576,2,Mod(47,576)] 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("576.47"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(576, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([6, 9, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 576 = 2^{6} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 576.y (of order \(12\), degree \(4\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.59938315643\)
Analytic rank: \(0\)
Dimension: \(88\)
Relative dimension: \(22\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 88 q + 4 q^{3} - 6 q^{5} + 4 q^{7} + 6 q^{11} - 2 q^{13} + 8 q^{19} + 2 q^{21} + 12 q^{23} + 16 q^{27} - 6 q^{29} - 8 q^{33} - 8 q^{37} + 32 q^{39} + 2 q^{43} + 6 q^{45} - 24 q^{49} + 12 q^{51} + 16 q^{55}+ \cdots - 46 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
47.1 0 −1.72804 + 0.117756i 0 −0.961558 + 3.58858i 0 1.29216 − 2.23809i 0 2.97227 − 0.406975i 0
47.2 0 −1.68223 − 0.412428i 0 0.0776974 − 0.289971i 0 0.374023 − 0.647827i 0 2.65981 + 1.38760i 0
47.3 0 −1.53620 − 0.800065i 0 1.00059 − 3.73424i 0 1.68236 − 2.91393i 0 1.71979 + 2.45811i 0
47.4 0 −1.38573 + 1.03912i 0 0.310357 − 1.15827i 0 −0.356047 + 0.616691i 0 0.840471 − 2.87986i 0
47.5 0 −1.33407 + 1.10465i 0 0.178044 − 0.664471i 0 −0.645693 + 1.11837i 0 0.559489 − 2.94737i 0
47.6 0 −1.06447 − 1.36635i 0 0.619079 − 2.31044i 0 −2.51270 + 4.35213i 0 −0.733803 + 2.90887i 0
47.7 0 −1.05993 − 1.36987i 0 −0.746784 + 2.78704i 0 −1.16672 + 2.02082i 0 −0.753089 + 2.90394i 0
47.8 0 −0.819834 + 1.52574i 0 −0.206232 + 0.769670i 0 2.17574 − 3.76849i 0 −1.65574 − 2.50170i 0
47.9 0 −0.632098 + 1.61259i 0 −1.02195 + 3.81396i 0 −1.46715 + 2.54117i 0 −2.20090 − 2.03863i 0
47.10 0 −0.517369 − 1.65298i 0 −0.521033 + 1.94452i 0 0.322227 − 0.558114i 0 −2.46466 + 1.71040i 0
47.11 0 0.256373 − 1.71297i 0 0.546024 − 2.03779i 0 0.0638076 − 0.110518i 0 −2.86855 − 0.878321i 0
47.12 0 0.313101 + 1.70352i 0 0.752772 − 2.80938i 0 1.02581 − 1.77675i 0 −2.80394 + 1.06674i 0
47.13 0 0.377192 − 1.69048i 0 0.282421 − 1.05401i 0 1.93586 − 3.35301i 0 −2.71545 − 1.27527i 0
47.14 0 0.427367 + 1.67850i 0 −0.0458174 + 0.170993i 0 −1.17432 + 2.03397i 0 −2.63471 + 1.43467i 0
47.15 0 0.692394 − 1.58764i 0 −0.759273 + 2.83365i 0 −1.41719 + 2.45465i 0 −2.04118 − 2.19854i 0
47.16 0 1.21512 + 1.23430i 0 −0.726024 + 2.70956i 0 0.00424642 − 0.00735502i 0 −0.0469689 + 2.99963i 0
47.17 0 1.32549 + 1.11494i 0 0.323102 − 1.20583i 0 −0.140266 + 0.242948i 0 0.513832 + 2.95567i 0
47.18 0 1.49488 − 0.874828i 0 0.473330 − 1.76649i 0 −1.40613 + 2.43549i 0 1.46935 − 2.61553i 0
47.19 0 1.59321 − 0.679468i 0 −0.642590 + 2.39818i 0 1.93190 − 3.34616i 0 2.07665 − 2.16507i 0
47.20 0 1.65787 − 0.501476i 0 0.997280 − 3.72190i 0 0.481387 − 0.833787i 0 2.49704 − 1.66276i 0
See all 88 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 47.22
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.d odd 6 1 inner
16.f odd 4 1 inner
144.u even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 576.2.y.a 88
3.b odd 2 1 1728.2.z.a 88
4.b odd 2 1 144.2.u.a ✓ 88
9.c even 3 1 1728.2.z.a 88
9.d odd 6 1 inner 576.2.y.a 88
12.b even 2 1 432.2.v.a 88
16.e even 4 1 144.2.u.a ✓ 88
16.f odd 4 1 inner 576.2.y.a 88
36.f odd 6 1 432.2.v.a 88
36.h even 6 1 144.2.u.a ✓ 88
48.i odd 4 1 432.2.v.a 88
48.k even 4 1 1728.2.z.a 88
144.u even 12 1 inner 576.2.y.a 88
144.v odd 12 1 1728.2.z.a 88
144.w odd 12 1 144.2.u.a ✓ 88
144.x even 12 1 432.2.v.a 88
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
144.2.u.a ✓ 88 4.b odd 2 1
144.2.u.a ✓ 88 16.e even 4 1
144.2.u.a ✓ 88 36.h even 6 1
144.2.u.a ✓ 88 144.w odd 12 1
432.2.v.a 88 12.b even 2 1
432.2.v.a 88 36.f odd 6 1
432.2.v.a 88 48.i odd 4 1
432.2.v.a 88 144.x even 12 1
576.2.y.a 88 1.a even 1 1 trivial
576.2.y.a 88 9.d odd 6 1 inner
576.2.y.a 88 16.f odd 4 1 inner
576.2.y.a 88 144.u even 12 1 inner
1728.2.z.a 88 3.b odd 2 1
1728.2.z.a 88 9.c even 3 1
1728.2.z.a 88 48.k even 4 1
1728.2.z.a 88 144.v odd 12 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(576, [\chi])\).