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Properties

Label 648.3.h.a
Level $648$
Weight $3$
Character orbit 648.h
Analytic conductor $17.657$
Analytic rank $0$
Dimension $44$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [648,3,Mod(485,648)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(648, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("648.485");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 648 = 2^{3} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 648.h (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(17.6567211305\)
Analytic rank: \(0\)
Dimension: \(44\)
Twist minimal: no (minimal twist has level 72)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 44 q + 2 q^{4} + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 44 q + 2 q^{4} + 4 q^{7} + 4 q^{10} + 2 q^{16} - 14 q^{22} + 144 q^{25} + 28 q^{28} + 4 q^{31} - 18 q^{34} + 112 q^{40} + 72 q^{46} + 144 q^{49} + 84 q^{52} + 92 q^{55} + 76 q^{58} + 2 q^{64} + 8 q^{70} - 8 q^{73} + 126 q^{76} + 4 q^{79} + 186 q^{82} + 154 q^{88} + 372 q^{94} + 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
485.1 −1.98982 − 0.201550i 0 3.91876 + 0.802095i 5.81548 0 0.726764 −7.63595 − 2.38585i 0 −11.5718 − 1.17211i
485.2 −1.98982 + 0.201550i 0 3.91876 − 0.802095i 5.81548 0 0.726764 −7.63595 + 2.38585i 0 −11.5718 + 1.17211i
485.3 −1.91787 − 0.567244i 0 3.35647 + 2.17580i −3.79077 0 11.4149 −5.20306 − 6.07685i 0 7.27021 + 2.15029i
485.4 −1.91787 + 0.567244i 0 3.35647 − 2.17580i −3.79077 0 11.4149 −5.20306 + 6.07685i 0 7.27021 − 2.15029i
485.5 −1.89405 − 0.642331i 0 3.17482 + 2.43321i −6.95397 0 −4.59069 −4.45034 − 6.64790i 0 13.1711 + 4.46675i
485.6 −1.89405 + 0.642331i 0 3.17482 − 2.43321i −6.95397 0 −4.59069 −4.45034 + 6.64790i 0 13.1711 − 4.46675i
485.7 −1.86616 − 0.719346i 0 2.96508 + 2.68483i 1.32371 0 −9.78668 −3.60199 − 7.14323i 0 −2.47024 − 0.952203i
485.8 −1.86616 + 0.719346i 0 2.96508 − 2.68483i 1.32371 0 −9.78668 −3.60199 + 7.14323i 0 −2.47024 + 0.952203i
485.9 −1.49306 − 1.33070i 0 0.458456 + 3.97364i 3.06253 0 1.44096 4.60324 − 6.54295i 0 −4.57254 − 4.07532i
485.10 −1.49306 + 1.33070i 0 0.458456 − 3.97364i 3.06253 0 1.44096 4.60324 + 6.54295i 0 −4.57254 + 4.07532i
485.11 −1.34381 − 1.48127i 0 −0.388328 + 3.98111i −7.29297 0 0.974251 6.41894 − 4.77465i 0 9.80039 + 10.8029i
485.12 −1.34381 + 1.48127i 0 −0.388328 − 3.98111i −7.29297 0 0.974251 6.41894 + 4.77465i 0 9.80039 − 10.8029i
485.13 −1.29200 − 1.52668i 0 −0.661489 + 3.94492i 8.56180 0 7.51600 6.87727 − 4.08695i 0 −11.0618 − 13.0711i
485.14 −1.29200 + 1.52668i 0 −0.661489 − 3.94492i 8.56180 0 7.51600 6.87727 + 4.08695i 0 −11.0618 + 13.0711i
485.15 −1.14446 − 1.64019i 0 −1.38043 + 3.75425i −0.689093 0 −6.41304 7.73752 − 2.03243i 0 0.788638 + 1.13024i
485.16 −1.14446 + 1.64019i 0 −1.38043 − 3.75425i −0.689093 0 −6.41304 7.73752 + 2.03243i 0 0.788638 − 1.13024i
485.17 −0.569909 − 1.91708i 0 −3.35041 + 2.18512i −3.28777 0 9.88862 6.09849 + 5.17768i 0 1.87373 + 6.30292i
485.18 −0.569909 + 1.91708i 0 −3.35041 − 2.18512i −3.28777 0 9.88862 6.09849 − 5.17768i 0 1.87373 − 6.30292i
485.19 −0.332311 − 1.97220i 0 −3.77914 + 1.31077i 7.97647 0 −11.2970 3.84094 + 7.01763i 0 −2.65067 − 15.7312i
485.20 −0.332311 + 1.97220i 0 −3.77914 − 1.31077i 7.97647 0 −11.2970 3.84094 − 7.01763i 0 −2.65067 + 15.7312i
See all 44 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 485.44
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
8.b even 2 1 inner
24.h odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 648.3.h.a 44
3.b odd 2 1 inner 648.3.h.a 44
4.b odd 2 1 2592.3.h.a 44
8.b even 2 1 inner 648.3.h.a 44
8.d odd 2 1 2592.3.h.a 44
9.c even 3 1 72.3.j.a ✓ 44
9.c even 3 1 216.3.j.a 44
9.d odd 6 1 72.3.j.a ✓ 44
9.d odd 6 1 216.3.j.a 44
12.b even 2 1 2592.3.h.a 44
24.f even 2 1 2592.3.h.a 44
24.h odd 2 1 inner 648.3.h.a 44
36.f odd 6 1 288.3.n.a 44
36.f odd 6 1 864.3.n.a 44
36.h even 6 1 288.3.n.a 44
36.h even 6 1 864.3.n.a 44
72.j odd 6 1 72.3.j.a ✓ 44
72.j odd 6 1 216.3.j.a 44
72.l even 6 1 288.3.n.a 44
72.l even 6 1 864.3.n.a 44
72.n even 6 1 72.3.j.a ✓ 44
72.n even 6 1 216.3.j.a 44
72.p odd 6 1 288.3.n.a 44
72.p odd 6 1 864.3.n.a 44
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.3.j.a ✓ 44 9.c even 3 1
72.3.j.a ✓ 44 9.d odd 6 1
72.3.j.a ✓ 44 72.j odd 6 1
72.3.j.a ✓ 44 72.n even 6 1
216.3.j.a 44 9.c even 3 1
216.3.j.a 44 9.d odd 6 1
216.3.j.a 44 72.j odd 6 1
216.3.j.a 44 72.n even 6 1
288.3.n.a 44 36.f odd 6 1
288.3.n.a 44 36.h even 6 1
288.3.n.a 44 72.l even 6 1
288.3.n.a 44 72.p odd 6 1
648.3.h.a 44 1.a even 1 1 trivial
648.3.h.a 44 3.b odd 2 1 inner
648.3.h.a 44 8.b even 2 1 inner
648.3.h.a 44 24.h odd 2 1 inner
864.3.n.a 44 36.f odd 6 1
864.3.n.a 44 36.h even 6 1
864.3.n.a 44 72.l even 6 1
864.3.n.a 44 72.p odd 6 1
2592.3.h.a 44 4.b odd 2 1
2592.3.h.a 44 8.d odd 2 1
2592.3.h.a 44 12.b even 2 1
2592.3.h.a 44 24.f even 2 1