# Properties

 Label 216.2.t.a Level $216$ Weight $2$ Character orbit 216.t Analytic conductor $1.725$ Analytic rank $0$ Dimension $204$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$216 = 2^{3} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 216.t (of order $$18$$, degree $$6$$, minimal)

## Newform invariants

 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 dimension is sufficiently large that we do not compute an algebraic $$q$$-expansion, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$204 q - 6 q^{2} - 6 q^{4} - 6 q^{6} - 12 q^{7} - 3 q^{8} - 12 q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$204 q - 6 q^{2} - 6 q^{4} - 6 q^{6} - 12 q^{7} - 3 q^{8} - 12 q^{9} - 3 q^{10} + 3 q^{12} - 21 q^{14} - 12 q^{15} - 6 q^{16} - 6 q^{17} - 27 q^{18} + 15 q^{20} - 6 q^{22} - 12 q^{23} - 12 q^{25} - 30 q^{26} - 12 q^{28} - 39 q^{30} - 12 q^{31} - 36 q^{32} - 36 q^{36} - 42 q^{38} - 12 q^{39} - 21 q^{40} - 24 q^{41} - 66 q^{42} + 21 q^{44} - 3 q^{46} - 12 q^{47} + 51 q^{48} - 12 q^{49} - 99 q^{50} - 33 q^{52} - 90 q^{54} - 24 q^{55} + 99 q^{56} - 30 q^{57} + 21 q^{58} + 102 q^{60} - 36 q^{62} - 72 q^{63} - 3 q^{64} - 12 q^{65} - 9 q^{66} + 75 q^{68} + 9 q^{70} - 90 q^{71} + 60 q^{72} - 6 q^{73} + 9 q^{74} - 18 q^{76} + 12 q^{78} - 12 q^{79} + 78 q^{80} - 12 q^{81} - 12 q^{82} + 102 q^{84} - 30 q^{86} - 48 q^{87} - 30 q^{88} - 6 q^{89} + 6 q^{90} + 111 q^{92} - 33 q^{94} - 42 q^{95} + 126 q^{96} - 12 q^{97} + 54 q^{98} + O(q^{100})$$

## 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.

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 990-1000 Embeddings: e.g. 1-3 or 205.34 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 $$S_{2}^{\mathrm{new}}(216, [\chi])$$.