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Properties

Label 888.2.c.d
Level $888$
Weight $2$
Character orbit 888.c
Analytic conductor $7.091$
Analytic rank $0$
Dimension $96$
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] = [888,2,Mod(443,888)] 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("888.443"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(888, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 888 = 2^{3} \cdot 3 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 888.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [96,0,-4,24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.09071569949\)
Analytic rank: \(0\)
Dimension: \(96\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 96 q - 4 q^{3} + 24 q^{4} - 4 q^{9} - 20 q^{10} - 18 q^{12} - 56 q^{16} + 72 q^{25} - 4 q^{27} - 64 q^{28} - 16 q^{33} - 8 q^{34} + 42 q^{36} + 44 q^{40} - 68 q^{46} - 94 q^{48} - 104 q^{49} - 28 q^{58}+ \cdots - 4 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
443.1 −1.41368 − 0.0387455i −1.13665 − 1.30691i 1.99700 + 0.109548i 3.37843i 1.55623 + 1.89160i − 1.96308i −2.81888 − 0.232240i −0.416034 + 2.97101i 0.130899 − 4.77603i
443.2 −1.41368 − 0.0387455i −1.13665 + 1.30691i 1.99700 + 0.109548i 3.37843i 1.65751 − 1.80352i − 1.96308i −2.81888 − 0.232240i −0.416034 − 2.97101i 0.130899 − 4.77603i
443.3 −1.41368 + 0.0387455i −1.13665 − 1.30691i 1.99700 − 0.109548i − 3.37843i 1.65751 + 1.80352i 1.96308i −2.81888 + 0.232240i −0.416034 + 2.97101i 0.130899 + 4.77603i
443.4 −1.41368 + 0.0387455i −1.13665 + 1.30691i 1.99700 − 0.109548i − 3.37843i 1.55623 − 1.89160i 1.96308i −2.81888 + 0.232240i −0.416034 − 2.97101i 0.130899 + 4.77603i
443.5 −1.39209 − 0.249187i 1.03985 − 1.38518i 1.87581 + 0.693779i − 0.556466i −1.79273 + 1.66917i 3.66824i −2.43841 − 1.43323i −0.837425 − 2.88075i −0.138664 + 0.774649i
443.6 −1.39209 − 0.249187i 1.03985 + 1.38518i 1.87581 + 0.693779i − 0.556466i −1.10239 − 2.18740i 3.66824i −2.43841 − 1.43323i −0.837425 + 2.88075i −0.138664 + 0.774649i
443.7 −1.39209 + 0.249187i 1.03985 − 1.38518i 1.87581 − 0.693779i 0.556466i −1.10239 + 2.18740i − 3.66824i −2.43841 + 1.43323i −0.837425 − 2.88075i −0.138664 − 0.774649i
443.8 −1.39209 + 0.249187i 1.03985 + 1.38518i 1.87581 − 0.693779i 0.556466i −1.79273 − 1.66917i − 3.66824i −2.43841 + 1.43323i −0.837425 + 2.88075i −0.138664 − 0.774649i
443.9 −1.35601 − 0.401533i −1.43419 − 0.971128i 1.67754 + 1.08897i 2.49584i 1.55485 + 1.89274i 2.93987i −1.83751 − 2.15024i 1.11382 + 2.78557i 1.00216 − 3.38439i
443.10 −1.35601 − 0.401533i −1.43419 + 0.971128i 1.67754 + 1.08897i 2.49584i 2.33472 − 0.740986i 2.93987i −1.83751 − 2.15024i 1.11382 − 2.78557i 1.00216 − 3.38439i
443.11 −1.35601 + 0.401533i −1.43419 − 0.971128i 1.67754 − 1.08897i − 2.49584i 2.33472 + 0.740986i − 2.93987i −1.83751 + 2.15024i 1.11382 + 2.78557i 1.00216 + 3.38439i
443.12 −1.35601 + 0.401533i −1.43419 + 0.971128i 1.67754 − 1.08897i − 2.49584i 1.55485 − 1.89274i − 2.93987i −1.83751 + 2.15024i 1.11382 − 2.78557i 1.00216 + 3.38439i
443.13 −1.27104 − 0.620045i 1.50401 − 0.859035i 1.23109 + 1.57620i 2.29587i −2.44430 + 0.159312i − 1.33692i −0.587448 − 2.76675i 1.52412 − 2.58400i 1.42354 − 2.91814i
443.14 −1.27104 − 0.620045i 1.50401 + 0.859035i 1.23109 + 1.57620i 2.29587i −1.37902 − 2.02442i − 1.33692i −0.587448 − 2.76675i 1.52412 + 2.58400i 1.42354 − 2.91814i
443.15 −1.27104 + 0.620045i 1.50401 − 0.859035i 1.23109 − 1.57620i − 2.29587i −1.37902 + 2.02442i 1.33692i −0.587448 + 2.76675i 1.52412 − 2.58400i 1.42354 + 2.91814i
443.16 −1.27104 + 0.620045i 1.50401 + 0.859035i 1.23109 − 1.57620i − 2.29587i −2.44430 − 0.159312i 1.33692i −0.587448 + 2.76675i 1.52412 + 2.58400i 1.42354 + 2.91814i
443.17 −1.24971 − 0.661983i −1.59972 − 0.663998i 1.12356 + 1.65458i − 0.287164i 1.55963 + 1.88879i − 4.18391i −0.308820 − 2.81152i 2.11821 + 2.12442i −0.190098 + 0.358873i
443.18 −1.24971 − 0.661983i −1.59972 + 0.663998i 1.12356 + 1.65458i − 0.287164i 2.43874 + 0.229182i − 4.18391i −0.308820 − 2.81152i 2.11821 − 2.12442i −0.190098 + 0.358873i
443.19 −1.24971 + 0.661983i −1.59972 − 0.663998i 1.12356 − 1.65458i 0.287164i 2.43874 − 0.229182i 4.18391i −0.308820 + 2.81152i 2.11821 + 2.12442i −0.190098 − 0.358873i
443.20 −1.24971 + 0.661983i −1.59972 + 0.663998i 1.12356 − 1.65458i 0.287164i 1.55963 − 1.88879i 4.18391i −0.308820 + 2.81152i 2.11821 − 2.12442i −0.190098 − 0.358873i
See all 96 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 443.96
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
8.d odd 2 1 inner
24.f even 2 1 inner
37.b even 2 1 inner
111.d odd 2 1 inner
296.h odd 2 1 inner
888.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 888.2.c.d ✓ 96
3.b odd 2 1 inner 888.2.c.d ✓ 96
8.d odd 2 1 inner 888.2.c.d ✓ 96
24.f even 2 1 inner 888.2.c.d ✓ 96
37.b even 2 1 inner 888.2.c.d ✓ 96
111.d odd 2 1 inner 888.2.c.d ✓ 96
296.h odd 2 1 inner 888.2.c.d ✓ 96
888.c even 2 1 inner 888.2.c.d ✓ 96
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
888.2.c.d ✓ 96 1.a even 1 1 trivial
888.2.c.d ✓ 96 3.b odd 2 1 inner
888.2.c.d ✓ 96 8.d odd 2 1 inner
888.2.c.d ✓ 96 24.f even 2 1 inner
888.2.c.d ✓ 96 37.b even 2 1 inner
888.2.c.d ✓ 96 111.d odd 2 1 inner
888.2.c.d ✓ 96 296.h odd 2 1 inner
888.2.c.d ✓ 96 888.c even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{24} + 51 T_{5}^{22} + 1097 T_{5}^{20} + 12984 T_{5}^{18} + 92526 T_{5}^{16} + 408132 T_{5}^{14} + \cdots + 256 \) acting on \(S_{2}^{\mathrm{new}}(888, [\chi])\). Copy content Toggle raw display