Properties

Label 216.2.t.a
Level $216$
Weight $2$
Character orbit 216.t
Analytic conductor $1.725$
Analytic rank $0$
Dimension $204$
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [216,2,Mod(13,216)] 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("216.13"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(216, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([0, 9, 8])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 216 = 2^{3} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 216.t (of order \(18\), degree \(6\), 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: \(1.72476868366\)
Analytic rank: \(0\)
Dimension: \(204\)
Relative dimension: \(34\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 204 q - 6 q^{2} - 6 q^{4} - 6 q^{6} - 12 q^{7} - 3 q^{8} - 12 q^{9} - 3 q^{10} + 3 q^{12} - 21 q^{14} - 12 q^{15} - 6 q^{16} - 6 q^{17} - 27 q^{18} + 15 q^{20} - 6 q^{22} - 12 q^{23} - 12 q^{25} - 30 q^{26}+ \cdots + 54 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1 −1.39803 − 0.213362i −0.479138 − 1.66446i 1.90895 + 0.596572i 2.25773 − 2.69066i 0.314714 + 2.42919i 3.36848 + 1.22603i −2.54148 − 1.24132i −2.54085 + 1.59501i −3.73045 + 3.27990i
13.2 −1.39072 + 0.256713i −1.23413 + 1.21529i 1.86820 − 0.714031i 1.46572 − 1.74677i 1.40435 − 2.00694i −1.21500 − 0.442225i −2.41483 + 1.47261i 0.0461478 − 2.99965i −1.58998 + 2.80554i
13.3 −1.36295 − 0.377313i 0.411299 − 1.68251i 1.71527 + 1.02852i −2.15975 + 2.57389i −1.19541 + 2.13799i −1.78453 − 0.649515i −1.94976 − 2.04901i −2.66167 − 1.38403i 3.91480 − 2.69319i
13.4 −1.34345 + 0.441744i 1.73179 + 0.0302651i 1.60972 − 1.18692i −0.721676 + 0.860059i −2.33994 + 0.724346i 1.31793 + 0.479686i −1.63827 + 2.30566i 2.99817 + 0.104825i 0.589610 − 1.47424i
13.5 −1.32716 − 0.488522i 0.526029 + 1.65024i 1.52269 + 1.29669i −0.114718 + 0.136716i 0.108055 − 2.44711i 2.42085 + 0.881117i −1.38739 − 2.46478i −2.44659 + 1.73615i 0.219037 − 0.125401i
13.6 −1.28352 + 0.593791i −1.30914 − 1.13409i 1.29483 − 1.52428i −0.508621 + 0.606151i 2.35371 + 0.678272i −3.72062 − 1.35420i −0.756827 + 2.72529i 0.427672 + 2.96936i 0.292896 − 1.08002i
13.7 −1.23486 − 0.689282i 1.71350 − 0.252809i 1.04978 + 1.70234i 2.05804 − 2.45268i −2.29020 − 0.868902i −4.54005 − 1.65244i −0.122943 − 2.82575i 2.87218 − 0.866376i −4.23199 + 1.61015i
13.8 −1.01573 − 0.984023i −1.62462 − 0.600509i 0.0633965 + 1.99899i −0.765978 + 0.912857i 1.05925 + 2.20862i 0.820563 + 0.298660i 1.90266 − 2.09281i 2.27878 + 1.95120i 1.67630 − 0.173472i
13.9 −0.938269 + 1.05814i −0.0518114 + 1.73128i −0.239304 − 1.98563i −1.55117 + 1.84861i −1.78331 − 1.67923i −1.49685 − 0.544810i 2.32560 + 1.60984i −2.99463 − 0.179400i −0.500669 − 3.37584i
13.10 −0.808295 − 1.16046i −0.860059 + 1.50343i −0.693319 + 1.87598i 0.0500411 − 0.0596366i 2.43985 − 0.217152i −3.05712 − 1.11270i 2.73740 − 0.711779i −1.52060 − 2.58608i −0.109654 − 0.00986652i
13.11 −0.691259 + 1.23376i 1.38208 + 1.04396i −1.04432 − 1.70569i 1.90914 − 2.27523i −2.24337 + 0.983500i 1.28267 + 0.466852i 2.82631 − 0.109365i 0.820275 + 2.88568i 1.48737 + 3.92819i
13.12 −0.600214 + 1.28052i 0.281905 − 1.70896i −1.27949 − 1.53718i −2.06932 + 2.46612i 2.01916 + 1.38673i 3.77263 + 1.37313i 2.73636 − 0.715779i −2.84106 − 0.963527i −1.91589 − 4.13002i
13.13 −0.573795 − 1.29258i 1.32680 − 1.11338i −1.34152 + 1.48335i 0.561427 − 0.669083i −2.20044 − 1.07614i 3.98166 + 1.44920i 2.68710 + 0.882882i 0.520786 − 2.95445i −1.18699 − 0.341773i
13.14 −0.565401 + 1.29627i −1.67042 + 0.457945i −1.36064 − 1.46583i 1.52309 − 1.81515i 0.350834 − 2.42424i 0.767101 + 0.279202i 2.66942 − 0.934982i 2.58057 − 1.52992i 1.49177 + 3.00063i
13.15 −0.471714 − 1.33322i 1.55492 + 0.763031i −1.55497 + 1.25780i −2.77087 + 3.30219i 0.283814 − 2.43299i −1.82038 − 0.662564i 2.41043 + 1.47981i 1.83557 + 2.37291i 5.70962 + 2.13650i
13.16 −0.0127237 − 1.41416i −1.55492 − 0.763031i −1.99968 + 0.0359866i 2.77087 − 3.30219i −1.05926 + 2.20861i −1.82038 − 0.662564i 0.0763340 + 2.82740i 1.83557 + 2.37291i −4.70507 − 3.87642i
13.17 −0.0120789 + 1.41416i −0.919076 − 1.46809i −1.99971 − 0.0341629i 0.185555 − 0.221136i 2.08722 − 1.28199i −2.62731 − 0.956263i 0.0724661 − 2.82750i −1.31060 + 2.69858i 0.310481 + 0.265076i
13.18 0.0971026 − 1.41088i −1.32680 + 1.11338i −1.98114 − 0.273999i −0.561427 + 0.669083i 1.44200 + 1.98006i 3.98166 + 1.44920i −0.578953 + 2.76854i 0.520786 − 2.95445i 0.889477 + 0.857073i
13.19 0.201435 + 1.39979i 1.40283 − 1.01591i −1.91885 + 0.563935i 1.55785 − 1.85658i 1.70465 + 1.75903i 0.763942 + 0.278052i −1.17592 − 2.57240i 0.935847 − 2.85030i 2.91263 + 1.80670i
13.20 0.289471 + 1.38427i −1.40283 + 1.01591i −1.83241 + 0.801412i −1.55785 + 1.85658i −1.81237 − 1.64782i 0.763942 + 0.278052i −1.63980 − 2.30457i 0.935847 − 2.85030i −3.02096 − 1.61907i
See next 80 embeddings (of 204 total)
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 13.34
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner
27.e even 9 1 inner
216.t even 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 216.2.t.a ✓ 204
3.b odd 2 1 648.2.t.a 204
4.b odd 2 1 864.2.bf.a 204
8.b even 2 1 inner 216.2.t.a ✓ 204
8.d odd 2 1 864.2.bf.a 204
24.h odd 2 1 648.2.t.a 204
27.e even 9 1 inner 216.2.t.a ✓ 204
27.f odd 18 1 648.2.t.a 204
108.j odd 18 1 864.2.bf.a 204
216.r odd 18 1 864.2.bf.a 204
216.t even 18 1 inner 216.2.t.a ✓ 204
216.x odd 18 1 648.2.t.a 204
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
216.2.t.a ✓ 204 1.a even 1 1 trivial
216.2.t.a ✓ 204 8.b even 2 1 inner
216.2.t.a ✓ 204 27.e even 9 1 inner
216.2.t.a ✓ 204 216.t even 18 1 inner
648.2.t.a 204 3.b odd 2 1
648.2.t.a 204 24.h odd 2 1
648.2.t.a 204 27.f odd 18 1
648.2.t.a 204 216.x odd 18 1
864.2.bf.a 204 4.b odd 2 1
864.2.bf.a 204 8.d odd 2 1
864.2.bf.a 204 108.j odd 18 1
864.2.bf.a 204 216.r odd 18 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(216, [\chi])\).