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Properties

Label 888.2.bh.a
Level $888$
Weight $2$
Character orbit 888.bh
Analytic conductor $7.091$
Analytic rank $0$
Dimension $152$
Inner twists $4$

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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(565,888)] 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("888.565"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(888, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 3, 0, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 888 = 2^{3} \cdot 3 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 888.bh (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.09071569949\)
Analytic rank: \(0\)
Dimension: \(152\)
Relative dimension: \(76\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 152 q - 2 q^{2} + 2 q^{4} + 8 q^{7} - 8 q^{8} + 76 q^{9} - 4 q^{14} + 2 q^{16} - 4 q^{17} + 2 q^{18} - 6 q^{22} - 16 q^{23} + 80 q^{25} + 6 q^{28} + 8 q^{30} + 8 q^{32} + 2 q^{34} + 4 q^{36} + 76 q^{38}+ \cdots + 52 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
565.1 −1.41421 − 0.00150368i 0.866025 − 0.500000i 2.00000 + 0.00425304i −1.17926 + 0.680844i −1.22550 + 0.705804i −0.930629 − 1.61190i −2.82841 − 0.00902206i 0.500000 − 0.866025i 1.66874 − 0.961084i
565.2 −1.41300 − 0.0585472i −0.866025 + 0.500000i 1.99314 + 0.165455i 1.69671 − 0.979598i 1.25297 − 0.655797i 1.99537 + 3.45609i −2.80663 − 0.350480i 0.500000 − 0.866025i −2.45481 + 1.28483i
565.3 −1.41004 − 0.108511i −0.866025 + 0.500000i 1.97645 + 0.306011i −2.48652 + 1.43559i 1.27539 − 0.611049i 1.49987 + 2.59785i −2.75368 − 0.645957i 0.500000 − 0.866025i 3.66188 − 1.75443i
565.4 −1.40985 − 0.111003i −0.866025 + 0.500000i 1.97536 + 0.312994i −2.38814 + 1.37880i 1.27647 − 0.608794i −1.72429 − 2.98656i −2.75021 − 0.660544i 0.500000 − 0.866025i 3.51997 − 1.67881i
565.5 −1.40730 + 0.139705i −0.866025 + 0.500000i 1.96096 − 0.393214i 3.03864 − 1.75436i 1.14890 − 0.824636i −0.822280 − 1.42423i −2.70472 + 0.827325i 0.500000 − 0.866025i −4.03118 + 2.89342i
565.6 −1.40654 − 0.147099i 0.866025 − 0.500000i 1.95672 + 0.413803i 2.32201 − 1.34061i −1.29165 + 0.575879i −0.950490 − 1.64630i −2.69134 − 0.869864i 0.500000 − 0.866025i −3.46321 + 1.54406i
565.7 −1.39353 + 0.241013i 0.866025 − 0.500000i 1.88383 − 0.671716i 3.27063 − 1.88830i −1.08632 + 0.905486i 1.78004 + 3.08311i −2.46327 + 1.39008i 0.500000 − 0.866025i −4.10260 + 3.41966i
565.8 −1.38541 + 0.283957i 0.866025 − 0.500000i 1.83874 − 0.786796i 0.598658 − 0.345635i −1.05782 + 0.938621i −0.383557 − 0.664340i −2.32399 + 1.61216i 0.500000 − 0.866025i −0.731242 + 0.648841i
565.9 −1.35378 + 0.408991i 0.866025 − 0.500000i 1.66545 − 1.10737i −3.47310 + 2.00519i −0.967915 + 1.03109i 2.50259 + 4.33461i −1.80176 + 2.18029i 0.500000 − 0.866025i 3.88171 − 4.13506i
565.10 −1.32259 + 0.500752i −0.866025 + 0.500000i 1.49849 − 1.32458i 0.648798 − 0.374584i 0.895021 − 1.09496i −1.94971 − 3.37700i −1.31861 + 2.50225i 0.500000 − 0.866025i −0.670520 + 0.820308i
565.11 −1.27839 − 0.604737i 0.866025 − 0.500000i 1.26859 + 1.54619i 0.543483 − 0.313780i −1.40949 + 0.115480i 2.01273 + 3.48615i −0.686718 − 2.74380i 0.500000 − 0.866025i −0.884540 + 0.0724704i
565.12 −1.23751 − 0.684515i 0.866025 − 0.500000i 1.06288 + 1.69419i −1.80687 + 1.04320i −1.41398 + 0.0259491i 0.240738 + 0.416970i −0.155625 − 2.82414i 0.500000 − 0.866025i 2.95011 − 0.0541400i
565.13 −1.23251 + 0.693489i −0.866025 + 0.500000i 1.03815 − 1.70946i 0.867653 − 0.500940i 0.720638 − 1.21683i 0.764837 + 1.32474i −0.0940330 + 2.82686i 0.500000 − 0.866025i −0.721993 + 1.21912i
565.14 −1.18340 − 0.774312i −0.866025 + 0.500000i 0.800881 + 1.83265i −0.623830 + 0.360168i 1.41201 + 0.0788731i −0.126978 − 0.219933i 0.471276 − 2.78889i 0.500000 − 0.866025i 1.01712 + 0.0568152i
565.15 −1.14199 + 0.834188i 0.866025 − 0.500000i 0.608261 − 1.90526i 0.280614 − 0.162012i −0.571894 + 1.29342i −0.290310 − 0.502832i 0.894720 + 2.68318i 0.500000 − 0.866025i −0.185308 + 0.419100i
565.16 −1.13796 − 0.839665i −0.866025 + 0.500000i 0.589926 + 1.91102i 2.72899 − 1.57558i 1.40534 + 0.158189i 0.116457 + 0.201710i 0.933300 − 2.67001i 0.500000 − 0.866025i −4.42846 − 0.498480i
565.17 −1.08808 − 0.903371i 0.866025 − 0.500000i 0.367840 + 1.96588i −1.28795 + 0.743600i −1.39399 − 0.238302i −2.36259 − 4.09212i 1.37568 − 2.47134i 0.500000 − 0.866025i 2.07314 + 0.354403i
565.18 −1.08179 + 0.910895i 0.866025 − 0.500000i 0.340540 − 1.97079i −0.713620 + 0.412009i −0.481410 + 1.32975i 0.715669 + 1.23957i 1.42680 + 2.44218i 0.500000 − 0.866025i 0.396690 − 1.09574i
565.19 −1.06938 + 0.925428i −0.866025 + 0.500000i 0.287166 − 1.97928i −2.81225 + 1.62365i 0.463400 − 1.33614i 0.617779 + 1.07002i 1.52459 + 2.38236i 0.500000 − 0.866025i 1.50480 − 4.33884i
565.20 −0.989862 − 1.01004i −0.866025 + 0.500000i −0.0403476 + 1.99959i −1.41657 + 0.817855i 1.36226 + 0.379786i 2.19997 + 3.81047i 2.05960 − 1.93857i 0.500000 − 0.866025i 2.22827 + 0.621221i
See next 80 embeddings (of 152 total)
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 565.76
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner
37.c even 3 1 inner
296.s even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 888.2.bh.a ✓ 152
8.b even 2 1 inner 888.2.bh.a ✓ 152
37.c even 3 1 inner 888.2.bh.a ✓ 152
296.s even 6 1 inner 888.2.bh.a ✓ 152
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
888.2.bh.a ✓ 152 1.a even 1 1 trivial
888.2.bh.a ✓ 152 8.b even 2 1 inner
888.2.bh.a ✓ 152 37.c even 3 1 inner
888.2.bh.a ✓ 152 296.s even 6 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(888, [\chi])\).