Properties

Label 864.3.t.b
Level $864$
Weight $3$
Character orbit 864.t
Analytic conductor $23.542$
Analytic rank $0$
Dimension $40$
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] = [864,3,Mod(559,864)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(864, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 3, 4]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("864.559");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 864 = 2^{5} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 864.t (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: \(23.5422948407\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(20\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 40 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q - 16 q^{11} + 4 q^{17} + 76 q^{19} + 118 q^{25} - 108 q^{35} - 20 q^{41} + 16 q^{43} + 166 q^{49} - 64 q^{59} + 102 q^{65} + 64 q^{67} - 292 q^{73} + 554 q^{83} + 688 q^{89} + 204 q^{91} + 92 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
559.1 0 0 0 −8.07964 + 4.66478i 0 4.91220 + 2.83606i 0 0 0
559.2 0 0 0 −6.07342 + 3.50649i 0 8.07247 + 4.66064i 0 0 0
559.3 0 0 0 −5.84790 + 3.37629i 0 3.50808 + 2.02539i 0 0 0
559.4 0 0 0 −5.42020 + 3.12935i 0 −5.96345 3.44300i 0 0 0
559.5 0 0 0 −5.15803 + 2.97799i 0 4.09037 + 2.36158i 0 0 0
559.6 0 0 0 −4.40783 + 2.54486i 0 −10.9609 6.32830i 0 0 0
559.7 0 0 0 −3.84571 + 2.22032i 0 −0.704321 0.406640i 0 0 0
559.8 0 0 0 −1.70411 + 0.983869i 0 −8.69613 5.02071i 0 0 0
559.9 0 0 0 −1.50948 + 0.871501i 0 −7.93804 4.58303i 0 0 0
559.10 0 0 0 −0.0166003 + 0.00958419i 0 4.07208 + 2.35102i 0 0 0
559.11 0 0 0 0.0166003 0.00958419i 0 −4.07208 2.35102i 0 0 0
559.12 0 0 0 1.50948 0.871501i 0 7.93804 + 4.58303i 0 0 0
559.13 0 0 0 1.70411 0.983869i 0 8.69613 + 5.02071i 0 0 0
559.14 0 0 0 3.84571 2.22032i 0 0.704321 + 0.406640i 0 0 0
559.15 0 0 0 4.40783 2.54486i 0 10.9609 + 6.32830i 0 0 0
559.16 0 0 0 5.15803 2.97799i 0 −4.09037 2.36158i 0 0 0
559.17 0 0 0 5.42020 3.12935i 0 5.96345 + 3.44300i 0 0 0
559.18 0 0 0 5.84790 3.37629i 0 −3.50808 2.02539i 0 0 0
559.19 0 0 0 6.07342 3.50649i 0 −8.07247 4.66064i 0 0 0
559.20 0 0 0 8.07964 4.66478i 0 −4.91220 2.83606i 0 0 0
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 559.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner
9.c even 3 1 inner
72.p odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 864.3.t.b 40
3.b odd 2 1 288.3.t.b 40
4.b odd 2 1 216.3.p.b 40
8.b even 2 1 216.3.p.b 40
8.d odd 2 1 inner 864.3.t.b 40
9.c even 3 1 inner 864.3.t.b 40
9.c even 3 1 2592.3.b.f 20
9.d odd 6 1 288.3.t.b 40
9.d odd 6 1 2592.3.b.e 20
12.b even 2 1 72.3.p.b 40
24.f even 2 1 288.3.t.b 40
24.h odd 2 1 72.3.p.b 40
36.f odd 6 1 216.3.p.b 40
36.f odd 6 1 648.3.b.e 20
36.h even 6 1 72.3.p.b 40
36.h even 6 1 648.3.b.f 20
72.j odd 6 1 72.3.p.b 40
72.j odd 6 1 648.3.b.f 20
72.l even 6 1 288.3.t.b 40
72.l even 6 1 2592.3.b.e 20
72.n even 6 1 216.3.p.b 40
72.n even 6 1 648.3.b.e 20
72.p odd 6 1 inner 864.3.t.b 40
72.p odd 6 1 2592.3.b.f 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.3.p.b 40 12.b even 2 1
72.3.p.b 40 24.h odd 2 1
72.3.p.b 40 36.h even 6 1
72.3.p.b 40 72.j odd 6 1
216.3.p.b 40 4.b odd 2 1
216.3.p.b 40 8.b even 2 1
216.3.p.b 40 36.f odd 6 1
216.3.p.b 40 72.n even 6 1
288.3.t.b 40 3.b odd 2 1
288.3.t.b 40 9.d odd 6 1
288.3.t.b 40 24.f even 2 1
288.3.t.b 40 72.l even 6 1
648.3.b.e 20 36.f odd 6 1
648.3.b.e 20 72.n even 6 1
648.3.b.f 20 36.h even 6 1
648.3.b.f 20 72.j odd 6 1
864.3.t.b 40 1.a even 1 1 trivial
864.3.t.b 40 8.d odd 2 1 inner
864.3.t.b 40 9.c even 3 1 inner
864.3.t.b 40 72.p odd 6 1 inner
2592.3.b.e 20 9.d odd 6 1
2592.3.b.e 20 72.l even 6 1
2592.3.b.f 20 9.c even 3 1
2592.3.b.f 20 72.p odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{40} - 309 T_{5}^{38} + 55716 T_{5}^{36} - 6712983 T_{5}^{34} + 603811641 T_{5}^{32} - 41961529638 T_{5}^{30} + 2327328612189 T_{5}^{28} - 104016298280787 T_{5}^{26} + \cdots + 35\!\cdots\!00 \) acting on \(S_{3}^{\mathrm{new}}(864, [\chi])\). Copy content Toggle raw display