# Properties

 Label 216.3.u.a Level $216$ Weight $3$ Character orbit 216.u Analytic conductor $5.886$ Analytic rank $0$ Dimension $108$ CM no Inner twists $2$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$5.88557371018$$ Analytic rank: $$0$$ Dimension: $$108$$ Relative dimension: $$18$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$108q + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$108q - 18q^{11} - 24q^{15} + 48q^{21} + 72q^{23} + 174q^{27} + 108q^{29} + 18q^{33} - 144q^{39} + 90q^{41} - 90q^{43} + 108q^{45} - 72q^{49} + 84q^{51} - 18q^{57} - 252q^{59} + 144q^{61} - 360q^{63} - 216q^{65} + 126q^{67} - 120q^{69} - 252q^{75} - 504q^{77} - 552q^{81} - 180q^{83} - 60q^{87} - 486q^{89} - 360q^{93} - 1116q^{95} + 270q^{97} - 564q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
41.1 0 −2.95959 0.490762i 0 −2.18142 + 2.59971i 0 4.31508 + 1.57056i 0 8.51831 + 2.90490i 0
41.2 0 −2.88041 0.838587i 0 −0.363134 + 0.432766i 0 −2.59307 0.943802i 0 7.59354 + 4.83095i 0
41.3 0 −2.57188 1.54449i 0 5.89076 7.02034i 0 −11.5107 4.18955i 0 4.22913 + 7.94446i 0
41.4 0 −2.17268 + 2.06868i 0 −6.00836 + 7.16049i 0 1.28551 + 0.467886i 0 0.441099 8.98918i 0
41.5 0 −2.11535 + 2.12727i 0 1.88267 2.24368i 0 4.85852 + 1.76836i 0 −0.0505567 8.99986i 0
41.6 0 −1.45288 + 2.62471i 0 4.32440 5.15362i 0 −6.87984 2.50406i 0 −4.77825 7.62682i 0
41.7 0 −1.38102 2.66323i 0 3.96925 4.73037i 0 9.66760 + 3.51872i 0 −5.18559 + 7.35593i 0
41.8 0 −0.838943 2.88031i 0 −3.67098 + 4.37491i 0 0.364439 + 0.132645i 0 −7.59235 + 4.83283i 0
41.9 0 0.111253 + 2.99794i 0 −2.18378 + 2.60253i 0 −6.99436 2.54574i 0 −8.97525 + 0.667061i 0
41.10 0 0.267462 2.98805i 0 −2.05700 + 2.45143i 0 −4.48537 1.63254i 0 −8.85693 1.59838i 0
41.11 0 1.03803 + 2.81469i 0 0.233456 0.278222i 0 12.9607 + 4.71731i 0 −6.84498 + 5.84348i 0
41.12 0 1.42216 2.64149i 0 3.87609 4.61934i 0 8.68461 + 3.16094i 0 −4.95492 7.51324i 0
41.13 0 1.75298 + 2.43456i 0 −1.18353 + 1.41048i 0 −8.11957 2.95528i 0 −2.85415 + 8.53545i 0
41.14 0 2.11815 2.12448i 0 −0.240393 + 0.286489i 0 −11.7075 4.26117i 0 −0.0268600 8.99996i 0
41.15 0 2.61263 + 1.47451i 0 2.45687 2.92798i 0 2.00721 + 0.730566i 0 4.65163 + 7.70469i 0
41.16 0 2.63593 1.43244i 0 −3.31749 + 3.95363i 0 5.17703 + 1.88429i 0 4.89624 7.55161i 0
41.17 0 2.95864 0.496426i 0 4.40241 5.24659i 0 −1.88768 0.687060i 0 8.50712 2.93749i 0
41.18 0 2.98761 + 0.272321i 0 −5.82981 + 6.94770i 0 4.85733 + 1.76792i 0 8.85168 + 1.62718i 0
65.1 0 −2.85348 0.926102i 0 −1.30481 + 3.58493i 0 0.719444 4.08017i 0 7.28467 + 5.28522i 0
65.2 0 −2.83313 0.986589i 0 −0.258656 + 0.710651i 0 0.583550 3.30948i 0 7.05328 + 5.59028i 0
See next 80 embeddings (of 108 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 209.18 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
27.f odd 18 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 216.3.u.a 108
4.b odd 2 1 432.3.bc.d 108
27.f odd 18 1 inner 216.3.u.a 108
108.l even 18 1 432.3.bc.d 108

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
216.3.u.a 108 1.a even 1 1 trivial
216.3.u.a 108 27.f odd 18 1 inner
432.3.bc.d 108 4.b odd 2 1
432.3.bc.d 108 108.l even 18 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{3}^{\mathrm{new}}(216, [\chi])$$.