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Properties

Label 288.3.n.a
Level $288$
Weight $3$
Character orbit 288.n
Analytic conductor $7.847$
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] = [288,3,Mod(113,288)]
 
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([0, 3, 5]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("288.113");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 288 = 2^{5} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 288.n (of order \(6\), degree \(2\), not 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: \(44\)
Relative dimension: \(22\) over \(\Q(\zeta_{6})\)
Twist minimal: no (minimal twist has level 72)
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.

\(\operatorname{Tr}(f)(q) = \) \( 44 q + 2 q^{7} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 44 q + 2 q^{7} - 4 q^{9} - 14 q^{15} + 6 q^{23} - 72 q^{25} + 2 q^{31} + 30 q^{33} + 118 q^{39} + 66 q^{41} + 6 q^{47} - 72 q^{49} - 92 q^{55} - 8 q^{57} + 226 q^{63} - 6 q^{65} - 8 q^{73} + 2 q^{79} - 44 q^{81} - 174 q^{87} - 144 q^{95} - 2 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
113.1 0 −2.99204 − 0.218346i 0 2.90774 − 5.03636i 0 0.363382 + 0.629396i 0 8.90465 + 1.30660i 0
113.2 0 −2.97841 + 0.359265i 0 −0.661853 + 1.14636i 0 −4.89334 − 8.47551i 0 8.74186 − 2.14008i 0
113.3 0 −2.84278 − 0.958433i 0 −3.64648 + 6.31589i 0 0.487126 + 0.843726i 0 7.16281 + 5.44923i 0
113.4 0 −2.50633 + 1.64872i 0 1.64388 − 2.84729i 0 4.94431 + 8.56379i 0 3.56342 − 8.26451i 0
113.5 0 −2.21491 + 2.02340i 0 −4.28090 + 7.41474i 0 3.75800 + 6.50904i 0 0.811683 − 8.96332i 0
113.6 0 −2.13383 − 2.10874i 0 −0.693019 + 1.20034i 0 0.562989 + 0.975125i 0 0.106423 + 8.99937i 0
113.7 0 −1.79120 − 2.40657i 0 1.89538 − 3.28290i 0 5.70744 + 9.88558i 0 −2.58320 + 8.62132i 0
113.8 0 −1.69228 + 2.47713i 0 −0.344546 + 0.596772i 0 −3.20652 − 5.55385i 0 −3.27238 − 8.38400i 0
113.9 0 −1.07504 − 2.80077i 0 3.98823 − 6.90782i 0 −5.64852 − 9.78353i 0 −6.68859 + 6.02186i 0
113.10 0 −0.340809 + 2.98058i 0 1.53127 − 2.65223i 0 0.720479 + 1.24791i 0 −8.76770 − 2.03162i 0
113.11 0 −0.102534 + 2.99825i 0 3.47699 − 6.02232i 0 −2.29534 − 3.97565i 0 −8.97897 − 0.614843i 0
113.12 0 0.102534 − 2.99825i 0 −3.47699 + 6.02232i 0 −2.29534 − 3.97565i 0 −8.97897 − 0.614843i 0
113.13 0 0.340809 − 2.98058i 0 −1.53127 + 2.65223i 0 0.720479 + 1.24791i 0 −8.76770 − 2.03162i 0
113.14 0 1.07504 + 2.80077i 0 −3.98823 + 6.90782i 0 −5.64852 − 9.78353i 0 −6.68859 + 6.02186i 0
113.15 0 1.69228 − 2.47713i 0 0.344546 − 0.596772i 0 −3.20652 − 5.55385i 0 −3.27238 − 8.38400i 0
113.16 0 1.79120 + 2.40657i 0 −1.89538 + 3.28290i 0 5.70744 + 9.88558i 0 −2.58320 + 8.62132i 0
113.17 0 2.13383 + 2.10874i 0 0.693019 − 1.20034i 0 0.562989 + 0.975125i 0 0.106423 + 8.99937i 0
113.18 0 2.21491 − 2.02340i 0 4.28090 − 7.41474i 0 3.75800 + 6.50904i 0 0.811683 − 8.96332i 0
113.19 0 2.50633 − 1.64872i 0 −1.64388 + 2.84729i 0 4.94431 + 8.56379i 0 3.56342 − 8.26451i 0
113.20 0 2.84278 + 0.958433i 0 3.64648 − 6.31589i 0 0.487126 + 0.843726i 0 7.16281 + 5.44923i 0
See all 44 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 113.22
Significant digits:
Format:

Inner twists

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

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(288, [\chi])\).