# Properties

 Label 216.3.j.a Level $216$ Weight $3$ Character orbit 216.j Analytic conductor $5.886$ Analytic rank $0$ Dimension $44$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$216 = 2^{3} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 216.j (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.88557371018$$ Analytic rank: $$0$$ Dimension: $$44$$ Relative dimension: $$22$$ over $$\Q(\zeta_{6})$$ Twist minimal: no (minimal twist has level 72) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$44q + 3q^{2} - q^{4} - 2q^{7} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$44q + 3q^{2} - q^{4} - 2q^{7} + 4q^{10} + 48q^{14} - q^{16} + 66q^{20} + 7q^{22} + 6q^{23} - 72q^{25} + 28q^{28} - 2q^{31} + 93q^{32} + 9q^{34} - 99q^{38} - 56q^{40} - 66q^{41} + 72q^{46} + 6q^{47} - 72q^{49} - 189q^{50} - 42q^{52} + 92q^{55} - 270q^{56} - 38q^{58} + 2q^{64} + 6q^{65} - 387q^{68} - 4q^{70} - 8q^{73} + 432q^{74} - 63q^{76} - 2q^{79} + 186q^{82} + 615q^{86} - 77q^{88} + 624q^{92} - 186q^{94} - 144q^{95} - 2q^{97} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
125.1 −1.99267 0.171037i 0 3.94149 + 0.681643i −0.344546 + 0.596772i 0 3.20652 + 5.55385i −7.73752 2.03243i 0 0.788638 1.13024i
125.2 −1.96814 0.355563i 0 3.74715 + 1.39960i 4.28090 7.41474i 0 −3.75800 6.50904i −6.87727 4.08695i 0 −11.0618 + 13.0711i
125.3 −1.86434 + 0.724045i 0 2.95152 2.69973i −0.693019 + 1.20034i 0 −0.562989 0.975125i −3.54790 + 7.17024i 0 0.422919 2.73963i
125.4 −1.55921 + 1.25254i 0 0.862276 3.90595i 0.693019 1.20034i 0 −0.562989 0.975125i 3.54790 + 7.17024i 0 0.422919 + 2.73963i
125.5 −1.50330 1.31913i 0 0.519809 + 3.96608i −3.47699 + 6.02232i 0 2.29534 + 3.97565i 4.45034 6.64790i 0 13.1711 4.46675i
125.6 −1.45018 1.37730i 0 0.206068 + 3.99469i −1.89538 + 3.28290i 0 −5.70744 9.88558i 5.20306 6.07685i 0 7.27021 2.15029i
125.7 −1.16946 1.62246i 0 −1.26474 + 3.79479i 2.90774 5.03636i 0 −0.363382 0.629396i 7.63595 2.38585i 0 −11.5718 + 1.17211i
125.8 −0.848214 + 1.81122i 0 −2.56107 3.07261i 0.344546 0.596772i 0 3.20652 + 5.55385i 7.73752 2.03243i 0 0.788638 + 1.13024i
125.9 −0.676143 + 1.88224i 0 −3.08566 2.54533i −4.28090 + 7.41474i 0 −3.75800 6.50904i 6.87727 4.08695i 0 −11.0618 13.0711i
125.10 −0.310106 1.97581i 0 −3.80767 + 1.22542i 0.661853 1.14636i 0 4.89334 + 8.47551i 3.60199 + 7.14323i 0 −2.47024 0.952203i
125.11 0.390748 + 1.96146i 0 −3.69463 + 1.53287i 3.47699 6.02232i 0 2.29534 + 3.97565i −4.45034 6.64790i 0 13.1711 + 4.46675i
125.12 0.405893 1.95838i 0 −3.67050 1.58979i 1.53127 2.65223i 0 −0.720479 1.24791i −4.60324 + 6.54295i 0 −4.57254 4.07532i
125.13 0.467688 + 1.94455i 0 −3.56254 + 1.81888i 1.89538 3.28290i 0 −5.70744 9.88558i −5.20306 6.07685i 0 7.27021 + 2.15029i
125.14 0.610911 1.90441i 0 −3.25357 2.32685i −3.64648 + 6.31589i 0 −0.487126 0.843726i −6.41894 + 4.77465i 0 9.80039 + 10.8029i
125.15 0.820362 + 1.82401i 0 −2.65401 + 2.99269i −2.90774 + 5.03636i 0 −0.363382 0.629396i −7.63595 2.38585i 0 −11.5718 1.17211i
125.16 1.37529 1.45210i 0 −0.217170 3.99410i −1.64388 + 2.84729i 0 −4.94431 8.56379i −6.09849 5.17768i 0 1.87373 + 6.30292i
125.17 1.54182 1.27389i 0 0.754413 3.92821i 3.98823 6.90782i 0 5.64852 + 9.78353i −3.84094 7.01763i 0 −2.65067 15.7312i
125.18 1.55605 + 1.25647i 0 0.842586 + 3.91025i −0.661853 + 1.14636i 0 4.89334 + 8.47551i −3.60199 + 7.14323i 0 −2.47024 + 0.952203i
125.19 1.87413 0.698310i 0 3.02473 2.61745i −3.98823 + 6.90782i 0 5.64852 + 9.78353i 3.84094 7.01763i 0 −2.65067 + 15.7312i
125.20 1.89895 + 0.627676i 0 3.21205 + 2.38385i −1.53127 + 2.65223i 0 −0.720479 1.24791i 4.60324 + 6.54295i 0 −4.57254 + 4.07532i
See all 44 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 197.22 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
9.d odd 6 1 inner
72.j odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 216.3.j.a 44
3.b odd 2 1 72.3.j.a 44
4.b odd 2 1 864.3.n.a 44
8.b even 2 1 inner 216.3.j.a 44
8.d odd 2 1 864.3.n.a 44
9.c even 3 1 72.3.j.a 44
9.c even 3 1 648.3.h.a 44
9.d odd 6 1 inner 216.3.j.a 44
9.d odd 6 1 648.3.h.a 44
12.b even 2 1 288.3.n.a 44
24.f even 2 1 288.3.n.a 44
24.h odd 2 1 72.3.j.a 44
36.f odd 6 1 288.3.n.a 44
36.f odd 6 1 2592.3.h.a 44
36.h even 6 1 864.3.n.a 44
36.h even 6 1 2592.3.h.a 44
72.j odd 6 1 inner 216.3.j.a 44
72.j odd 6 1 648.3.h.a 44
72.l even 6 1 864.3.n.a 44
72.l even 6 1 2592.3.h.a 44
72.n even 6 1 72.3.j.a 44
72.n even 6 1 648.3.h.a 44
72.p odd 6 1 288.3.n.a 44
72.p odd 6 1 2592.3.h.a 44

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.3.j.a 44 3.b odd 2 1
72.3.j.a 44 9.c even 3 1
72.3.j.a 44 24.h odd 2 1
72.3.j.a 44 72.n even 6 1
216.3.j.a 44 1.a even 1 1 trivial
216.3.j.a 44 8.b even 2 1 inner
216.3.j.a 44 9.d odd 6 1 inner
216.3.j.a 44 72.j odd 6 1 inner
288.3.n.a 44 12.b even 2 1
288.3.n.a 44 24.f even 2 1
288.3.n.a 44 36.f odd 6 1
288.3.n.a 44 72.p odd 6 1
648.3.h.a 44 9.c even 3 1
648.3.h.a 44 9.d odd 6 1
648.3.h.a 44 72.j odd 6 1
648.3.h.a 44 72.n even 6 1
864.3.n.a 44 4.b odd 2 1
864.3.n.a 44 8.d odd 2 1
864.3.n.a 44 36.h even 6 1
864.3.n.a 44 72.l even 6 1
2592.3.h.a 44 36.f odd 6 1
2592.3.h.a 44 36.h even 6 1
2592.3.h.a 44 72.l even 6 1
2592.3.h.a 44 72.p odd 6 1

## Hecke kernels

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