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Properties

Label 576.2.bb.e
Level $576$
Weight $2$
Character orbit 576.bb
Analytic conductor $4.599$
Analytic rank $0$
Dimension $72$
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] = [576,2,Mod(49,576)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(576, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([0, 3, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("576.49");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 576 = 2^{6} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 576.bb (of order \(12\), degree \(4\), 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: \(4.59938315643\)
Analytic rank: \(0\)
Dimension: \(72\)
Relative dimension: \(18\) over \(\Q(\zeta_{12})\)
Twist minimal: no (minimal twist has level 144)
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 72 q - 2 q^{3} + 4 q^{5}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 72 q - 2 q^{3} + 4 q^{5} + 2 q^{11} - 16 q^{13} + 20 q^{15} - 16 q^{17} - 28 q^{19} - 16 q^{21} - 8 q^{27} + 4 q^{29} - 28 q^{31} - 32 q^{33} + 16 q^{35} + 16 q^{37} + 10 q^{43} + 40 q^{45} + 56 q^{47} + 4 q^{49} + 54 q^{51} - 8 q^{53} + 14 q^{59} - 32 q^{61} + 108 q^{63} - 64 q^{65} + 18 q^{67} + 32 q^{69} - 86 q^{75} - 36 q^{77} - 44 q^{79} - 44 q^{81} - 20 q^{83} - 8 q^{85} + 80 q^{91} - 4 q^{93} - 48 q^{95} + 40 q^{97} - 28 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1 0 −1.71513 + 0.241506i 0 −1.98415 + 0.531653i 0 1.54969 − 0.894715i 0 2.88335 − 0.828427i 0
49.2 0 −1.51206 − 0.844796i 0 3.32531 − 0.891015i 0 3.95817 − 2.28525i 0 1.57264 + 2.55476i 0
49.3 0 −1.45211 + 0.944122i 0 −0.491749 + 0.131764i 0 2.40518 − 1.38863i 0 1.21727 − 2.74194i 0
49.4 0 −1.31071 − 1.13227i 0 −0.0112878 + 0.00302457i 0 −1.05753 + 0.610563i 0 0.435936 + 2.96816i 0
49.5 0 −1.08700 + 1.34849i 0 2.90072 − 0.777246i 0 1.04527 − 0.603486i 0 −0.636858 − 2.93162i 0
49.6 0 −1.05287 − 1.37530i 0 −4.10876 + 1.10094i 0 1.63313 − 0.942891i 0 −0.782911 + 2.89604i 0
49.7 0 −0.841300 + 1.51401i 0 −0.0691269 + 0.0185225i 0 −1.28192 + 0.740118i 0 −1.58443 − 2.54747i 0
49.8 0 0.162237 + 1.72444i 0 −2.21121 + 0.592492i 0 −2.67054 + 1.54184i 0 −2.94736 + 0.559536i 0
49.9 0 0.222657 + 1.71768i 0 2.97932 − 0.798307i 0 −1.78208 + 1.02889i 0 −2.90085 + 0.764906i 0
49.10 0 0.409248 − 1.68301i 0 3.30105 − 0.884514i 0 −2.63210 + 1.51965i 0 −2.66503 − 1.37753i 0
49.11 0 0.460673 + 1.66966i 0 −2.69708 + 0.722679i 0 2.89314 − 1.67035i 0 −2.57556 + 1.53834i 0
49.12 0 0.795173 − 1.53873i 0 −2.41406 + 0.646846i 0 −2.82197 + 1.62927i 0 −1.73540 − 2.44712i 0
49.13 0 1.15652 + 1.28937i 0 1.21694 − 0.326078i 0 0.707732 − 0.408609i 0 −0.324925 + 2.98235i 0
49.14 0 1.22967 − 1.21980i 0 0.174734 − 0.0468197i 0 4.04791 − 2.33706i 0 0.0241875 − 2.99990i 0
49.15 0 1.58205 − 0.705067i 0 2.53632 − 0.679606i 0 −0.614293 + 0.354662i 0 2.00576 − 2.23090i 0
49.16 0 1.64464 + 0.543291i 0 −1.60558 + 0.430214i 0 −3.62762 + 2.09441i 0 2.40967 + 1.78703i 0
49.17 0 1.68781 + 0.388965i 0 −0.846545 + 0.226831i 0 0.567074 − 0.327400i 0 2.69741 + 1.31300i 0
49.18 0 1.71859 + 0.215523i 0 2.73721 − 0.733432i 0 1.14487 − 0.660988i 0 2.90710 + 0.740791i 0
241.1 0 −1.72444 − 0.162237i 0 0.592492 − 2.21121i 0 2.67054 + 1.54184i 0 2.94736 + 0.559536i 0
241.2 0 −1.71768 − 0.222657i 0 −0.798307 + 2.97932i 0 1.78208 + 1.02889i 0 2.90085 + 0.764906i 0
See all 72 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.18
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner
16.e even 4 1 inner
144.x even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 576.2.bb.e 72
3.b odd 2 1 1728.2.bc.e 72
4.b odd 2 1 144.2.x.e ✓ 72
9.c even 3 1 inner 576.2.bb.e 72
9.d odd 6 1 1728.2.bc.e 72
12.b even 2 1 432.2.y.e 72
16.e even 4 1 inner 576.2.bb.e 72
16.f odd 4 1 144.2.x.e ✓ 72
36.f odd 6 1 144.2.x.e ✓ 72
36.h even 6 1 432.2.y.e 72
48.i odd 4 1 1728.2.bc.e 72
48.k even 4 1 432.2.y.e 72
144.u even 12 1 432.2.y.e 72
144.v odd 12 1 144.2.x.e ✓ 72
144.w odd 12 1 1728.2.bc.e 72
144.x even 12 1 inner 576.2.bb.e 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
144.2.x.e ✓ 72 4.b odd 2 1
144.2.x.e ✓ 72 16.f odd 4 1
144.2.x.e ✓ 72 36.f odd 6 1
144.2.x.e ✓ 72 144.v odd 12 1
432.2.y.e 72 12.b even 2 1
432.2.y.e 72 36.h even 6 1
432.2.y.e 72 48.k even 4 1
432.2.y.e 72 144.u even 12 1
576.2.bb.e 72 1.a even 1 1 trivial
576.2.bb.e 72 9.c even 3 1 inner
576.2.bb.e 72 16.e even 4 1 inner
576.2.bb.e 72 144.x even 12 1 inner
1728.2.bc.e 72 3.b odd 2 1
1728.2.bc.e 72 9.d odd 6 1
1728.2.bc.e 72 48.i odd 4 1
1728.2.bc.e 72 144.w odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{72} - 4 T_{5}^{71} + 8 T_{5}^{70} - 48 T_{5}^{69} - 362 T_{5}^{68} + 1784 T_{5}^{67} + \cdots + 65536 \) acting on \(S_{2}^{\mathrm{new}}(576, [\chi])\). Copy content Toggle raw display