Properties

Label 288.3.o.c
Level $288$
Weight $3$
Character orbit 288.o
Analytic conductor $7.847$
Analytic rank $0$
Dimension $24$
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] = [288,3,Mod(31,288)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(288, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 0, 2]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("288.31");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 288 = 2^{5} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 288.o (of order \(6\), degree \(2\), 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: \(24\)
Relative dimension: \(12\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 4 q^{9} - 24 q^{13} - 24 q^{17} + 56 q^{21} - 108 q^{25} - 24 q^{29} + 52 q^{33} + 96 q^{37} - 60 q^{41} - 224 q^{45} - 132 q^{49} + 96 q^{53} + 348 q^{57} - 336 q^{61} + 216 q^{65} + 416 q^{69} + 696 q^{73} - 24 q^{77} - 788 q^{81} - 528 q^{85} - 240 q^{89} + 1040 q^{93} - 444 q^{97}+O(q^{100}) \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1 0 −2.99814 0.105601i 0 −3.20123 + 5.54470i 0 6.73381 3.88777i 0 8.97770 + 0.633212i 0
31.2 0 −2.40825 + 1.78895i 0 4.14774 7.18409i 0 −7.08016 + 4.08773i 0 2.59929 8.61648i 0
31.3 0 −2.10592 2.13661i 0 −1.25241 + 2.16924i 0 0.842935 0.486669i 0 −0.130192 + 8.99906i 0
31.4 0 −1.80500 2.39624i 0 1.69347 2.93317i 0 −8.58804 + 4.95831i 0 −2.48392 + 8.65044i 0
31.5 0 −1.55460 + 2.56578i 0 −3.70620 + 6.41933i 0 0.198210 0.114437i 0 −4.16646 7.97751i 0
31.6 0 −1.26562 + 2.71996i 0 2.31864 4.01601i 0 1.01137 0.583914i 0 −5.79642 6.88488i 0
31.7 0 1.26562 2.71996i 0 2.31864 4.01601i 0 −1.01137 + 0.583914i 0 −5.79642 6.88488i 0
31.8 0 1.55460 2.56578i 0 −3.70620 + 6.41933i 0 −0.198210 + 0.114437i 0 −4.16646 7.97751i 0
31.9 0 1.80500 + 2.39624i 0 1.69347 2.93317i 0 8.58804 4.95831i 0 −2.48392 + 8.65044i 0
31.10 0 2.10592 + 2.13661i 0 −1.25241 + 2.16924i 0 −0.842935 + 0.486669i 0 −0.130192 + 8.99906i 0
31.11 0 2.40825 1.78895i 0 4.14774 7.18409i 0 7.08016 4.08773i 0 2.59929 8.61648i 0
31.12 0 2.99814 + 0.105601i 0 −3.20123 + 5.54470i 0 −6.73381 + 3.88777i 0 8.97770 + 0.633212i 0
223.1 0 −2.99814 + 0.105601i 0 −3.20123 5.54470i 0 6.73381 + 3.88777i 0 8.97770 0.633212i 0
223.2 0 −2.40825 1.78895i 0 4.14774 + 7.18409i 0 −7.08016 4.08773i 0 2.59929 + 8.61648i 0
223.3 0 −2.10592 + 2.13661i 0 −1.25241 2.16924i 0 0.842935 + 0.486669i 0 −0.130192 8.99906i 0
223.4 0 −1.80500 + 2.39624i 0 1.69347 + 2.93317i 0 −8.58804 4.95831i 0 −2.48392 8.65044i 0
223.5 0 −1.55460 2.56578i 0 −3.70620 6.41933i 0 0.198210 + 0.114437i 0 −4.16646 + 7.97751i 0
223.6 0 −1.26562 2.71996i 0 2.31864 + 4.01601i 0 1.01137 + 0.583914i 0 −5.79642 + 6.88488i 0
223.7 0 1.26562 + 2.71996i 0 2.31864 + 4.01601i 0 −1.01137 0.583914i 0 −5.79642 + 6.88488i 0
223.8 0 1.55460 + 2.56578i 0 −3.70620 6.41933i 0 −0.198210 0.114437i 0 −4.16646 + 7.97751i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 31.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
9.c even 3 1 inner
36.f odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 288.3.o.c 24
3.b odd 2 1 864.3.o.c 24
4.b odd 2 1 inner 288.3.o.c 24
8.b even 2 1 576.3.o.i 24
8.d odd 2 1 576.3.o.i 24
9.c even 3 1 inner 288.3.o.c 24
9.c even 3 1 2592.3.g.k 12
9.d odd 6 1 864.3.o.c 24
9.d odd 6 1 2592.3.g.l 12
12.b even 2 1 864.3.o.c 24
24.f even 2 1 1728.3.o.i 24
24.h odd 2 1 1728.3.o.i 24
36.f odd 6 1 inner 288.3.o.c 24
36.f odd 6 1 2592.3.g.k 12
36.h even 6 1 864.3.o.c 24
36.h even 6 1 2592.3.g.l 12
72.j odd 6 1 1728.3.o.i 24
72.l even 6 1 1728.3.o.i 24
72.n even 6 1 576.3.o.i 24
72.p odd 6 1 576.3.o.i 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
288.3.o.c 24 1.a even 1 1 trivial
288.3.o.c 24 4.b odd 2 1 inner
288.3.o.c 24 9.c even 3 1 inner
288.3.o.c 24 36.f odd 6 1 inner
576.3.o.i 24 8.b even 2 1
576.3.o.i 24 8.d odd 2 1
576.3.o.i 24 72.n even 6 1
576.3.o.i 24 72.p odd 6 1
864.3.o.c 24 3.b odd 2 1
864.3.o.c 24 9.d odd 6 1
864.3.o.c 24 12.b even 2 1
864.3.o.c 24 36.h even 6 1
1728.3.o.i 24 24.f even 2 1
1728.3.o.i 24 24.h odd 2 1
1728.3.o.i 24 72.j odd 6 1
1728.3.o.i 24 72.l even 6 1
2592.3.g.k 12 9.c even 3 1
2592.3.g.k 12 36.f odd 6 1
2592.3.g.l 12 9.d odd 6 1
2592.3.g.l 12 36.h even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{12} + 102 T_{5}^{10} + 16 T_{5}^{9} + 7719 T_{5}^{8} - 96 T_{5}^{7} + 242958 T_{5}^{6} - 207528 T_{5}^{5} + 5622153 T_{5}^{4} - 2696528 T_{5}^{3} + 42417024 T_{5}^{2} + 14125056 T_{5} + 239878144 \) acting on \(S_{3}^{\mathrm{new}}(288, [\chi])\). Copy content Toggle raw display