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Properties

Label 888.2.f.a
Level $888$
Weight $2$
Character orbit 888.f
Analytic conductor $7.091$
Analytic rank $0$
Dimension $28$
Inner twists $2$

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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(445,888)] 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("888.445"); 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([0, 1, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 888 = 2^{3} \cdot 3 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 888.f (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [28] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.09071569949\)
Analytic rank: \(0\)
Dimension: \(28\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 28 q + 2 q^{4} + 2 q^{6} - 6 q^{8} - 28 q^{9} - 4 q^{10} - 4 q^{12} - 2 q^{14} - 4 q^{15} + 10 q^{16} - 20 q^{17} + 8 q^{20} + 4 q^{24} + 4 q^{25} + 8 q^{26} - 2 q^{28} + 4 q^{30} - 4 q^{31} - 10 q^{32}+ \cdots - 98 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
445.1 −1.41028 − 0.105414i − 1.00000i 1.97778 + 0.297326i − 1.06532i −0.105414 + 1.41028i −0.345915 −2.75787 − 0.627797i −1.00000 −0.112299 + 1.50240i
445.2 −1.41028 + 0.105414i 1.00000i 1.97778 − 0.297326i 1.06532i −0.105414 − 1.41028i −0.345915 −2.75787 + 0.627797i −1.00000 −0.112299 − 1.50240i
445.3 −1.33851 − 0.456512i 1.00000i 1.58319 + 1.22209i − 0.548214i 0.456512 − 1.33851i −4.98928 −1.56121 − 2.35852i −1.00000 −0.250267 + 0.733788i
445.4 −1.33851 + 0.456512i − 1.00000i 1.58319 − 1.22209i 0.548214i 0.456512 + 1.33851i −4.98928 −1.56121 + 2.35852i −1.00000 −0.250267 − 0.733788i
445.5 −1.31810 − 0.512467i − 1.00000i 1.47476 + 1.35096i 0.901738i −0.512467 + 1.31810i 4.21046 −1.25155 − 2.53646i −1.00000 0.462111 − 1.18858i
445.6 −1.31810 + 0.512467i 1.00000i 1.47476 − 1.35096i − 0.901738i −0.512467 − 1.31810i 4.21046 −1.25155 + 2.53646i −1.00000 0.462111 + 1.18858i
445.7 −1.10774 − 0.879160i 1.00000i 0.454155 + 1.94775i − 2.37818i 0.879160 − 1.10774i 2.57729 1.20930 − 2.55687i −1.00000 −2.09080 + 2.63440i
445.8 −1.10774 + 0.879160i − 1.00000i 0.454155 − 1.94775i 2.37818i 0.879160 + 1.10774i 2.57729 1.20930 + 2.55687i −1.00000 −2.09080 − 2.63440i
445.9 −0.671054 − 1.24486i 1.00000i −1.09937 + 1.67074i 1.23483i 1.24486 − 0.671054i −2.39160 2.81759 + 0.247413i −1.00000 1.53719 − 0.828637i
445.10 −0.671054 + 1.24486i − 1.00000i −1.09937 − 1.67074i − 1.23483i 1.24486 + 0.671054i −2.39160 2.81759 − 0.247413i −1.00000 1.53719 + 0.828637i
445.11 −0.508803 − 1.31951i − 1.00000i −1.48224 + 1.34275i − 2.92000i −1.31951 + 0.508803i 1.07969 2.52594 + 1.27264i −1.00000 −3.85299 + 1.48571i
445.12 −0.508803 + 1.31951i 1.00000i −1.48224 − 1.34275i 2.92000i −1.31951 − 0.508803i 1.07969 2.52594 − 1.27264i −1.00000 −3.85299 − 1.48571i
445.13 0.0746208 − 1.41224i 1.00000i −1.98886 − 0.210766i 3.46509i 1.41224 + 0.0746208i 3.38204 −0.446063 + 2.79303i −1.00000 4.89355 + 0.258568i
445.14 0.0746208 + 1.41224i − 1.00000i −1.98886 + 0.210766i − 3.46509i 1.41224 − 0.0746208i 3.38204 −0.446063 − 2.79303i −1.00000 4.89355 − 0.258568i
445.15 0.166620 − 1.40436i − 1.00000i −1.94448 − 0.467991i − 2.58075i −1.40436 − 0.166620i −1.84872 −0.981219 + 2.65277i −1.00000 −3.62431 − 0.430005i
445.16 0.166620 + 1.40436i 1.00000i −1.94448 + 0.467991i 2.58075i −1.40436 + 0.166620i −1.84872 −0.981219 − 2.65277i −1.00000 −3.62431 + 0.430005i
445.17 0.417150 − 1.35129i 1.00000i −1.65197 − 1.12738i − 0.754524i 1.35129 + 0.417150i −1.88713 −2.21254 + 1.76200i −1.00000 −1.01958 − 0.314750i
445.18 0.417150 + 1.35129i − 1.00000i −1.65197 + 1.12738i 0.754524i 1.35129 − 0.417150i −1.88713 −2.21254 − 1.76200i −1.00000 −1.01958 + 0.314750i
445.19 0.618328 − 1.27188i − 1.00000i −1.23534 − 1.57287i 3.71221i −1.27188 − 0.618328i 0.189221 −2.76435 + 0.598651i −1.00000 4.72147 + 2.29536i
445.20 0.618328 + 1.27188i 1.00000i −1.23534 + 1.57287i − 3.71221i −1.27188 + 0.618328i 0.189221 −2.76435 − 0.598651i −1.00000 4.72147 − 2.29536i
See all 28 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 445.28
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b 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.f.a ✓ 28
4.b odd 2 1 3552.2.f.a 28
8.b even 2 1 inner 888.2.f.a ✓ 28
8.d odd 2 1 3552.2.f.a 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
888.2.f.a ✓ 28 1.a even 1 1 trivial
888.2.f.a ✓ 28 8.b even 2 1 inner
3552.2.f.a 28 4.b odd 2 1
3552.2.f.a 28 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{28} + 68 T_{5}^{26} + 2004 T_{5}^{24} + 33632 T_{5}^{22} + 355268 T_{5}^{20} + 2466456 T_{5}^{18} + \cdots + 147456 \) acting on \(S_{2}^{\mathrm{new}}(888, [\chi])\). Copy content Toggle raw display