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Properties

Label 1176.2.u.c
Level $1176$
Weight $2$
Character orbit 1176.u
Analytic conductor $9.390$
Analytic rank $0$
Dimension $48$
Inner twists $8$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1176,2,Mod(521,1176)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1176.521"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1176, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 3, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1176 = 2^{3} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1176.u (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [48,0,0,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(9.39040727770\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(24\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 32 q^{15} - 8 q^{25} - 16 q^{37} + 64 q^{39} + 32 q^{43} - 48 q^{51} + 96 q^{57} - 16 q^{67} - 80 q^{81} - 128 q^{85} + 32 q^{93} - 112 q^{99}+O(q^{100}) \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
521.1 0 −1.72182 − 0.187979i 0 0.244530 + 0.423539i 0 0 0 2.92933 + 0.647333i 0
521.2 0 −1.70977 − 0.276907i 0 0.655965 + 1.13616i 0 0 0 2.84664 + 0.946897i 0
521.3 0 −1.54460 − 0.783712i 0 −1.67181 − 2.89567i 0 0 0 1.77159 + 2.42105i 0
521.4 0 −1.51046 + 0.847647i 0 −0.496051 − 0.859186i 0 0 0 1.56299 − 2.56068i 0
521.5 0 −1.45102 − 0.945808i 0 1.67181 + 2.89567i 0 0 0 1.21089 + 2.74477i 0
521.6 0 −1.09470 − 1.34225i 0 −0.655965 − 1.13616i 0 0 0 −0.603285 + 2.93872i 0
521.7 0 −1.06416 + 1.36659i 0 −0.661917 − 1.14647i 0 0 0 −0.735138 − 2.90853i 0
521.8 0 −1.02453 + 1.39655i 0 2.00767 + 3.47739i 0 0 0 −0.900685 − 2.86160i 0
521.9 0 −1.02370 − 1.39715i 0 −0.244530 − 0.423539i 0 0 0 −0.904057 + 2.86054i 0
521.10 0 −0.697181 + 1.58554i 0 2.00767 + 3.47739i 0 0 0 −2.02788 − 2.21082i 0
521.11 0 −0.651423 + 1.60488i 0 −0.661917 − 1.14647i 0 0 0 −2.15130 − 2.09092i 0
521.12 0 −0.0211468 − 1.73192i 0 0.496051 + 0.859186i 0 0 0 −2.99911 + 0.0732492i 0
521.13 0 0.0211468 + 1.73192i 0 −0.496051 − 0.859186i 0 0 0 −2.99911 + 0.0732492i 0
521.14 0 0.651423 − 1.60488i 0 0.661917 + 1.14647i 0 0 0 −2.15130 − 2.09092i 0
521.15 0 0.697181 − 1.58554i 0 −2.00767 − 3.47739i 0 0 0 −2.02788 − 2.21082i 0
521.16 0 1.02370 + 1.39715i 0 0.244530 + 0.423539i 0 0 0 −0.904057 + 2.86054i 0
521.17 0 1.02453 − 1.39655i 0 −2.00767 − 3.47739i 0 0 0 −0.900685 − 2.86160i 0
521.18 0 1.06416 − 1.36659i 0 0.661917 + 1.14647i 0 0 0 −0.735138 − 2.90853i 0
521.19 0 1.09470 + 1.34225i 0 0.655965 + 1.13616i 0 0 0 −0.603285 + 2.93872i 0
521.20 0 1.45102 + 0.945808i 0 −1.67181 − 2.89567i 0 0 0 1.21089 + 2.74477i 0
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 521.24
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner
21.c even 2 1 inner
21.g even 6 1 inner
21.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1176.2.u.c 48
3.b odd 2 1 inner 1176.2.u.c 48
7.b odd 2 1 inner 1176.2.u.c 48
7.c even 3 1 1176.2.k.b ✓ 24
7.c even 3 1 inner 1176.2.u.c 48
7.d odd 6 1 1176.2.k.b ✓ 24
7.d odd 6 1 inner 1176.2.u.c 48
21.c even 2 1 inner 1176.2.u.c 48
21.g even 6 1 1176.2.k.b ✓ 24
21.g even 6 1 inner 1176.2.u.c 48
21.h odd 6 1 1176.2.k.b ✓ 24
21.h odd 6 1 inner 1176.2.u.c 48
28.f even 6 1 2352.2.k.j 24
28.g odd 6 1 2352.2.k.j 24
84.j odd 6 1 2352.2.k.j 24
84.n even 6 1 2352.2.k.j 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1176.2.k.b ✓ 24 7.c even 3 1
1176.2.k.b ✓ 24 7.d odd 6 1
1176.2.k.b ✓ 24 21.g even 6 1
1176.2.k.b ✓ 24 21.h odd 6 1
1176.2.u.c 48 1.a even 1 1 trivial
1176.2.u.c 48 3.b odd 2 1 inner
1176.2.u.c 48 7.b odd 2 1 inner
1176.2.u.c 48 7.c even 3 1 inner
1176.2.u.c 48 7.d odd 6 1 inner
1176.2.u.c 48 21.c even 2 1 inner
1176.2.u.c 48 21.g even 6 1 inner
1176.2.u.c 48 21.h odd 6 1 inner
2352.2.k.j 24 28.f even 6 1
2352.2.k.j 24 28.g odd 6 1
2352.2.k.j 24 84.j odd 6 1
2352.2.k.j 24 84.n even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{24} + 32 T_{5}^{22} + 708 T_{5}^{20} + 8000 T_{5}^{18} + 64588 T_{5}^{16} + 240064 T_{5}^{14} + \cdots + 16384 \) acting on \(S_{2}^{\mathrm{new}}(1176, [\chi])\). Copy content Toggle raw display