Properties

Label 168.3.be.a
Level $168$
Weight $3$
Character orbit 168.be
Analytic conductor $4.578$
Analytic rank $0$
Dimension $120$
Inner twists $8$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [168,3,Mod(59,168)] 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("168.59"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(168, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 3, 3, 1])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 168 = 2^{3} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 168.be (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: \(4.57766844125\)
Analytic rank: \(0\)
Dimension: \(120\)
Relative dimension: \(60\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 120 q - 6 q^{3} - 2 q^{4} - 2 q^{9} - 6 q^{10} - 18 q^{12} - 10 q^{16} - 34 q^{18} - 12 q^{19} + 44 q^{22} + 24 q^{24} + 216 q^{25} - 82 q^{28} - 8 q^{30} - 6 q^{33} - 52 q^{36} - 186 q^{40} - 188 q^{42}+ \cdots - 332 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
59.1 −1.97785 + 0.296805i −1.08792 + 2.79579i 3.82381 − 1.17408i 3.91482 − 2.26023i 1.32193 − 5.85256i −0.799710 − 6.95417i −7.21447 + 3.45708i −6.63288 − 6.08316i −7.07210 + 5.63234i
59.2 −1.96931 − 0.349002i −1.67664 − 2.48774i 3.75639 + 1.37459i −2.04175 + 1.17880i 2.43361 + 5.48430i 0.799694 − 6.95417i −6.91779 − 4.01799i −3.37774 + 8.34211i 4.43224 − 1.60886i
59.3 −1.96912 + 0.350078i 2.05788 + 2.18292i 3.75489 − 1.37869i −1.35250 + 0.780864i −4.81641 − 3.57802i −6.09469 + 3.44307i −6.91119 + 4.02932i −0.530263 + 8.98437i 2.38987 − 2.01109i
59.4 −1.96463 − 0.374469i −2.61901 − 1.46315i 3.71955 + 1.47139i 7.45533 − 4.30434i 4.59748 + 3.85528i −3.31214 + 6.16682i −6.75654 − 4.28359i 4.71841 + 7.66398i −16.2588 + 5.66464i
59.5 −1.95967 − 0.399601i 2.83538 − 0.980100i 3.68064 + 1.56618i −3.58905 + 2.07214i −5.94808 + 0.787653i 5.63217 − 4.15676i −6.58700 − 4.53998i 7.07881 − 5.55792i 7.86140 − 2.62653i
59.6 −1.92340 + 0.548206i −2.96784 + 0.438066i 3.39894 − 2.10884i −1.76255 + 1.01761i 5.46820 − 2.46957i 6.88289 + 1.27509i −5.38145 + 5.91946i 8.61620 − 2.60022i 2.83223 − 2.92351i
59.7 −1.90324 + 0.614542i 2.84017 − 0.966154i 3.24468 − 2.33925i 6.95650 − 4.01634i −4.81179 + 3.58423i 5.23931 + 4.64215i −4.73785 + 6.44614i 7.13309 − 5.48808i −10.7717 + 11.9191i
59.8 −1.84821 − 0.764271i −2.48908 + 1.67466i 2.83178 + 2.82507i −7.78745 + 4.49609i 5.88024 − 1.19280i −6.95319 + 0.808201i −3.07461 − 7.38558i 3.39101 − 8.33673i 17.8291 − 2.35800i
59.9 −1.84309 + 0.776552i 0.0152337 − 2.99996i 2.79393 − 2.86251i −4.49938 + 2.59772i 2.30155 + 5.54102i 1.91166 + 6.73391i −2.92657 + 7.44548i −8.99954 − 0.0914007i 6.27549 − 8.28183i
59.10 −1.75207 − 0.964487i −0.327520 + 2.98207i 2.13953 + 3.37971i 1.96566 − 1.13487i 3.45001 − 4.90892i 5.01182 + 4.88688i −0.488933 − 7.98505i −8.78546 − 1.95337i −4.53855 + 0.0925307i
59.11 −1.70834 − 1.03999i 1.25749 − 2.72373i 1.83685 + 3.55331i −1.40984 + 0.813973i −4.98088 + 3.34527i −4.74607 + 5.14537i 0.557443 − 7.98055i −5.83741 − 6.85015i 3.25501 + 0.0756784i
59.12 −1.64427 − 1.13859i 2.81135 + 1.04706i 1.40723 + 3.74429i 5.55759 − 3.20868i −3.43043 − 4.92262i −4.78906 − 5.10538i 1.94936 − 7.75887i 6.80733 + 5.88729i −12.7915 − 1.05190i
59.13 −1.59406 + 1.20788i 0.0152337 − 2.99996i 1.08204 − 3.85087i 4.49938 − 2.59772i 3.59932 + 4.80051i −1.91166 − 6.73391i 2.92657 + 7.44548i −8.99954 − 0.0914007i −4.03453 + 9.57564i
59.14 −1.48383 + 1.34099i 2.84017 − 0.966154i 0.403507 − 3.97960i −6.95650 + 4.01634i −2.91873 + 5.24224i −5.23931 − 4.64215i 4.73785 + 6.44614i 7.13309 − 5.48808i 4.93641 − 15.2881i
59.15 −1.43646 + 1.39161i −2.96784 + 0.438066i 0.126837 − 3.99799i 1.76255 − 1.01761i 3.65357 − 4.75935i −6.88289 − 1.27509i 5.38145 + 5.91946i 8.61620 − 2.60022i −1.11572 + 3.91454i
59.16 −1.28774 + 1.53027i 2.05788 + 2.18292i −0.683463 − 3.94118i 1.35250 − 0.780864i −5.99047 + 0.338089i 6.09469 − 3.44307i 6.91119 + 4.02932i −0.530263 + 8.98437i −0.546726 + 3.07523i
59.17 −1.24597 + 1.56447i −1.08792 + 2.79579i −0.895127 − 3.89856i −3.91482 + 2.26023i −3.01842 − 5.18547i 0.799710 + 6.95417i 7.21447 + 3.45708i −6.63288 − 6.08316i 1.34169 − 8.94079i
59.18 −1.24027 − 1.56899i 0.693678 − 2.91870i −0.923485 + 3.89194i 5.61499 − 3.24182i −5.43977 + 2.53158i 6.83130 − 1.52755i 7.25179 − 3.37809i −8.03762 − 4.04928i −12.0505 − 4.78917i
59.19 −1.14156 − 1.64220i −2.95161 − 0.536678i −1.39367 + 3.74936i −1.83075 + 1.05698i 2.48811 + 5.45979i 6.30723 + 3.03625i 7.74817 − 1.99143i 8.42395 + 3.16812i 3.82570 + 1.79985i
59.20 −1.13960 − 1.64356i 1.34360 + 2.68230i −1.40260 + 3.74603i −5.28741 + 3.05269i 2.87737 − 5.26505i 1.01576 − 6.92591i 7.75524 − 1.96372i −5.38950 + 7.20787i 11.0428 + 5.21134i
See next 80 embeddings (of 120 total)
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 59.60
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.d odd 6 1 inner
8.d odd 2 1 inner
21.g even 6 1 inner
24.f even 2 1 inner
56.m even 6 1 inner
168.be odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 168.3.be.a ✓ 120
3.b odd 2 1 inner 168.3.be.a ✓ 120
7.d odd 6 1 inner 168.3.be.a ✓ 120
8.d odd 2 1 inner 168.3.be.a ✓ 120
21.g even 6 1 inner 168.3.be.a ✓ 120
24.f even 2 1 inner 168.3.be.a ✓ 120
56.m even 6 1 inner 168.3.be.a ✓ 120
168.be odd 6 1 inner 168.3.be.a ✓ 120
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.3.be.a ✓ 120 1.a even 1 1 trivial
168.3.be.a ✓ 120 3.b odd 2 1 inner
168.3.be.a ✓ 120 7.d odd 6 1 inner
168.3.be.a ✓ 120 8.d odd 2 1 inner
168.3.be.a ✓ 120 21.g even 6 1 inner
168.3.be.a ✓ 120 24.f even 2 1 inner
168.3.be.a ✓ 120 56.m even 6 1 inner
168.3.be.a ✓ 120 168.be odd 6 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(168, [\chi])\).