Properties

Label 288.3.x.b
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

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 −1.99481 + 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
53.2 −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
53.3 −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
53.4 −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
53.5 −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
53.6 −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
53.7 −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
53.8 −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
53.9 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
53.10 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
53.11 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
53.12 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
53.13 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
53.14 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
53.15 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
53.16 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
125.1 −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
125.2 −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
125.3 −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
125.4 −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
See all 64 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 53.16
Significant digits:
Format:

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{64} + 32 T_{5}^{61} - 11456 T_{5}^{59} - 8192 T_{5}^{58} + 17536 T_{5}^{57} + 62953216 T_{5}^{56} + 2693888 T_{5}^{55} - 133727232 T_{5}^{54} - 6721296640 T_{5}^{53} + \cdots + 12\!\cdots\!76 \) acting on \(S_{3}^{\mathrm{new}}(288, [\chi])\). Copy content Toggle raw display