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Properties

Label 585.2.bm.a
Level $585$
Weight $2$
Character orbit 585.bm
Analytic conductor $4.671$
Analytic rank $0$
Dimension $112$
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] = [585,2,Mod(166,585)] 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("585.166"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(585, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([2, 0, 5])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 585 = 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 585.bm (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: \(4.67124851824\)
Analytic rank: \(0\)
Dimension: \(112\)
Relative dimension: \(56\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 112 q + 56 q^{4} - 2 q^{9} + 12 q^{12} + 2 q^{13} - 56 q^{16} - 16 q^{17} + 24 q^{18} + 6 q^{19} - 6 q^{21} + 48 q^{23} - 60 q^{24} + 56 q^{25} - 12 q^{26} - 24 q^{27} + 10 q^{29} + 8 q^{30} - 24 q^{31}+ \cdots + 6 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
166.1 −2.40478 − 1.38840i 1.49598 − 0.872953i 2.85531 + 4.94554i 0.866025 + 0.500000i −4.80951 + 0.0222406i − 4.33015i − 10.3037i 1.47591 − 2.61184i −1.38840 − 2.40478i
166.2 −2.39541 − 1.38299i −0.747450 − 1.56247i 2.82531 + 4.89358i −0.866025 − 0.500000i −0.370435 + 4.77647i 1.27008i − 10.0975i −1.88264 + 2.33574i 1.38299 + 2.39541i
166.3 −2.30047 − 1.32818i −1.72715 − 0.130141i 2.52811 + 4.37881i 0.866025 + 0.500000i 3.80042 + 2.59335i 2.19308i − 8.11839i 2.96613 + 0.449546i −1.32818 − 2.30047i
166.4 −2.17231 − 1.25419i −0.608087 + 1.62180i 2.14596 + 3.71691i −0.866025 − 0.500000i 3.35499 − 2.76040i 4.72144i − 5.74899i −2.26046 − 1.97239i 1.25419 + 2.17231i
166.5 −2.11253 − 1.21967i 1.43011 − 0.977136i 1.97518 + 3.42112i −0.866025 − 0.500000i −4.21292 + 0.319972i 1.54032i − 4.75760i 1.09041 − 2.79482i 1.21967 + 2.11253i
166.6 −2.02622 − 1.16984i 1.16504 + 1.28167i 1.73705 + 3.00866i −0.866025 − 0.500000i −0.861293 − 3.95986i − 1.04260i − 3.44893i −0.285344 + 2.98640i 1.16984 + 2.02622i
166.7 −1.96046 − 1.13187i −1.70463 − 0.306984i 1.56227 + 2.70593i −0.866025 − 0.500000i 2.99439 + 2.53125i − 5.19617i − 2.54566i 2.81152 + 1.04659i 1.13187 + 1.96046i
166.8 −1.95145 − 1.12667i −0.645306 − 1.60735i 1.53878 + 2.66524i 0.866025 + 0.500000i −0.551674 + 3.86372i − 2.76231i − 2.42809i −2.16716 + 2.07447i −1.12667 − 1.95145i
166.9 −1.83641 − 1.06025i 0.634643 + 1.61159i 1.24827 + 2.16206i 0.866025 + 0.500000i 0.543228 − 3.63242i − 1.20669i − 1.05290i −2.19446 + 2.04557i −1.06025 − 1.83641i
166.10 −1.80668 − 1.04309i −1.49466 + 0.875215i 1.17606 + 2.03699i 0.866025 + 0.500000i 3.61329 − 0.0221753i 0.0661118i − 0.734567i 1.46800 − 2.61629i −1.04309 − 1.80668i
166.11 −1.80536 − 1.04232i −0.0103486 − 1.73202i 1.17288 + 2.03149i 0.866025 + 0.500000i −1.78664 + 3.13770i 2.65800i − 0.720788i −2.99979 + 0.0358480i −1.04232 − 1.80536i
166.12 −1.79316 − 1.03528i 1.71605 + 0.234856i 1.14362 + 1.98080i 0.866025 + 0.500000i −2.83402 − 2.19773i − 0.896475i − 0.594732i 2.88969 + 0.806051i −1.03528 − 1.79316i
166.13 −1.60598 − 0.927212i 1.05501 − 1.37366i 0.719444 + 1.24611i −0.866025 − 0.500000i −2.96800 + 1.22786i − 1.13212i 1.04054i −0.773903 − 2.89846i 0.927212 + 1.60598i
166.14 −1.48455 − 0.857104i −1.03476 − 1.38898i 0.469254 + 0.812771i −0.866025 − 0.500000i 0.345649 + 2.94891i 1.35905i 1.81962i −0.858542 + 2.87453i 0.857104 + 1.48455i
166.15 −1.23047 − 0.710412i −0.0786278 + 1.73027i 0.00936902 + 0.0162276i 0.866025 + 0.500000i 1.32595 − 2.07318i 4.62258i 2.81502i −2.98764 − 0.272094i −0.710412 − 1.23047i
166.16 −1.22396 − 0.706651i 0.526265 + 1.65017i −0.00128788 − 0.00223067i −0.866025 − 0.500000i 0.521966 − 2.39162i − 1.57767i 2.83025i −2.44609 + 1.73685i 0.706651 + 1.22396i
166.17 −1.16492 − 0.672568i −1.06307 + 1.36743i −0.0953059 − 0.165075i −0.866025 − 0.500000i 2.15809 − 0.877966i 0.0753893i 2.94667i −0.739756 − 2.90736i 0.672568 + 1.16492i
166.18 −1.05608 − 0.609725i −1.66528 − 0.476297i −0.256470 − 0.444219i 0.866025 + 0.500000i 1.46825 + 1.51837i − 3.86765i 3.06441i 2.54628 + 1.58633i −0.609725 − 1.05608i
166.19 −0.824990 − 0.476308i 1.56690 + 0.738116i −0.546261 − 0.946151i −0.866025 − 0.500000i −0.941109 − 1.35527i 3.48015i 2.94599i 1.91037 + 2.31311i 0.476308 + 0.824990i
166.20 −0.805664 − 0.465150i 1.35444 − 1.07957i −0.567270 − 0.982541i 0.866025 + 0.500000i −1.59339 + 0.239754i 2.91231i 2.91607i 0.669037 − 2.92445i −0.465150 − 0.805664i
See next 80 embeddings (of 112 total)
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 166.56
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
117.r even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 585.2.bm.a yes 112
9.c even 3 1 585.2.ba.a ✓ 112
13.e even 6 1 585.2.ba.a ✓ 112
117.r even 6 1 inner 585.2.bm.a yes 112
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
585.2.ba.a ✓ 112 9.c even 3 1
585.2.ba.a ✓ 112 13.e even 6 1
585.2.bm.a yes 112 1.a even 1 1 trivial
585.2.bm.a yes 112 117.r even 6 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(585, [\chi])\).