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Properties

Label 288.3.x.a
Level $288$
Weight $3$
Character orbit 288.x
Analytic conductor $7.847$
Analytic rank $0$
Dimension $64$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [288,3,Mod(53,288)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(288, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([0, 5, 4]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("288.53");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 288 = 2^{5} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 288.x (of order \(8\), degree \(4\), 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: \(7.84743161358\)
Analytic rank: \(0\)
Dimension: \(64\)
Relative dimension: \(16\) over \(\Q(\zeta_{8})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 64 q - 4 q^{2} + 20 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 64 q - 4 q^{2} + 20 q^{8} - 40 q^{10} + 32 q^{11} + 60 q^{14} + 8 q^{16} + 64 q^{20} + 112 q^{22} + 92 q^{26} - 32 q^{29} + 56 q^{32} + 96 q^{35} - 32 q^{38} + 224 q^{40} - 168 q^{44} + 32 q^{46} - 384 q^{47} + 232 q^{50} + 104 q^{52} - 160 q^{53} + 256 q^{55} + 248 q^{56} - 352 q^{58} - 128 q^{59} - 64 q^{61} - 180 q^{62} - 48 q^{64} - 64 q^{67} - 152 q^{68} - 336 q^{70} - 68 q^{74} - 104 q^{76} + 224 q^{77} - 128 q^{80} - 520 q^{82} - 160 q^{83} + 472 q^{86} - 280 q^{88} + 192 q^{91} + 656 q^{92} - 456 q^{94} - 620 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
53.1 −1.97643 + 0.306155i 0 3.81254 − 1.21019i 3.05280 + 1.26451i 0 −6.57413 − 6.57413i −7.16470 + 3.55908i 0 −6.42077 − 1.56458i
53.2 −1.85043 + 0.758899i 0 2.84815 − 2.80857i −4.62310 − 1.91495i 0 3.34124 + 3.34124i −3.13886 + 7.35850i 0 10.0079 + 0.0350087i
53.3 −1.78948 − 0.893172i 0 2.40449 + 3.19663i 4.02591 + 1.66758i 0 3.15434 + 3.15434i −1.44765 − 7.86793i 0 −5.71485 − 6.57994i
53.4 −1.71363 − 1.03125i 0 1.87307 + 3.53435i −6.91716 − 2.86518i 0 2.33712 + 2.33712i 0.435032 − 7.98816i 0 8.89876 + 12.0432i
53.5 −1.23426 + 1.57372i 0 −0.953195 − 3.88477i 8.50340 + 3.52222i 0 5.30972 + 5.30972i 7.29003 + 3.29476i 0 −16.0384 + 9.03463i
53.6 −1.02913 + 1.71490i 0 −1.88179 − 3.52971i −0.0174241 − 0.00721732i 0 −8.15851 − 8.15851i 7.98972 + 0.405443i 0 0.0303087 − 0.0224532i
53.7 −0.891593 − 1.79027i 0 −2.41013 + 3.19238i 2.45719 + 1.01780i 0 −4.99207 − 4.99207i 7.86407 + 1.46847i 0 −0.368674 − 5.30649i
53.8 −0.268531 + 1.98189i 0 −3.85578 − 1.06440i −1.91277 − 0.792297i 0 5.76529 + 5.76529i 3.14492 − 7.35591i 0 2.08389 − 3.57815i
53.9 0.0880022 − 1.99806i 0 −3.98451 − 0.351668i −6.78731 − 2.81140i 0 −0.402227 − 0.402227i −1.05330 + 7.93036i 0 −6.21465 + 13.3141i
53.10 0.524318 + 1.93005i 0 −3.45018 + 2.02392i −2.81794 − 1.16723i 0 −2.45726 − 2.45726i −5.71526 − 5.59784i 0 0.775311 − 6.05075i
53.11 0.527453 − 1.92919i 0 −3.44359 − 2.03512i 6.57829 + 2.72482i 0 6.20812 + 6.20812i −5.74247 + 5.56992i 0 8.72644 − 11.2536i
53.12 1.35868 − 1.46765i 0 −0.307970 − 3.98813i −0.365829 − 0.151531i 0 −1.83662 − 1.83662i −6.27159 − 4.96660i 0 −0.719440 + 0.331025i
53.13 1.41770 + 1.41072i 0 0.0197520 + 3.99995i 3.22612 + 1.33630i 0 2.13502 + 2.13502i −5.61480 + 5.69860i 0 2.68853 + 6.44563i
53.14 1.85072 + 0.758185i 0 2.85031 + 2.80637i −8.07100 − 3.34312i 0 −8.03674 − 8.03674i 3.14737 + 7.35487i 0 −12.4024 − 12.3065i
53.15 1.99180 − 0.180966i 0 3.93450 − 0.720894i −3.32592 − 1.37764i 0 9.14542 + 9.14542i 7.70627 − 2.14788i 0 −6.87386 − 2.14210i
53.16 1.99481 − 0.143966i 0 3.95855 − 0.574371i 6.99476 + 2.89732i 0 −4.93873 − 4.93873i 7.81387 − 1.71566i 0 14.3703 + 4.77260i
125.1 −1.97643 − 0.306155i 0 3.81254 + 1.21019i 3.05280 − 1.26451i 0 −6.57413 + 6.57413i −7.16470 − 3.55908i 0 −6.42077 + 1.56458i
125.2 −1.85043 − 0.758899i 0 2.84815 + 2.80857i −4.62310 + 1.91495i 0 3.34124 − 3.34124i −3.13886 − 7.35850i 0 10.0079 − 0.0350087i
125.3 −1.78948 + 0.893172i 0 2.40449 − 3.19663i 4.02591 − 1.66758i 0 3.15434 − 3.15434i −1.44765 + 7.86793i 0 −5.71485 + 6.57994i
125.4 −1.71363 + 1.03125i 0 1.87307 − 3.53435i −6.91716 + 2.86518i 0 2.33712 − 2.33712i 0.435032 + 7.98816i 0 8.89876 − 12.0432i
See all 64 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 53.16
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
96.p odd 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 288.3.x.a ✓ 64
3.b odd 2 1 288.3.x.b yes 64
32.g even 8 1 288.3.x.b yes 64
96.p odd 8 1 inner 288.3.x.a ✓ 64
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
288.3.x.a ✓ 64 1.a even 1 1 trivial
288.3.x.a ✓ 64 96.p odd 8 1 inner
288.3.x.b yes 64 3.b odd 2 1
288.3.x.b yes 64 32.g even 8 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{64} - 32 T_{5}^{61} + 11456 T_{5}^{59} - 8192 T_{5}^{58} - 17536 T_{5}^{57} + \cdots + 12\!\cdots\!76 \) acting on \(S_{3}^{\mathrm{new}}(288, [\chi])\). Copy content Toggle raw display