# Properties

 Label 324.3.j.a Level 324 Weight 3 Character orbit 324.j Analytic conductor 8.828 Analytic rank 0 Dimension 204 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$324 = 2^{2} \cdot 3^{4}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 324.j (of order $$18$$, degree $$6$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$8.82836056527$$ Analytic rank: $$0$$ Dimension: $$204$$ Relative dimension: $$34$$ over $$\Q(\zeta_{18})$$ Twist minimal: no (minimal twist has level 108) 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.

 $$\operatorname{Tr}(f)(q) =$$ $$204q + 6q^{2} - 6q^{4} + 12q^{5} + 3q^{8} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$204q + 6q^{2} - 6q^{4} + 12q^{5} + 3q^{8} - 3q^{10} - 12q^{13} - 39q^{14} - 6q^{16} + 6q^{17} + 69q^{20} - 6q^{22} - 12q^{25} + 174q^{26} - 12q^{28} - 60q^{29} + 96q^{32} + 6q^{34} - 6q^{37} - 72q^{38} + 69q^{40} + 192q^{41} + 219q^{44} - 3q^{46} - 12q^{49} + 165q^{50} + 21q^{52} + 24q^{53} - 99q^{56} - 141q^{58} - 12q^{61} - 294q^{62} - 3q^{64} + 156q^{65} - 375q^{68} - 165q^{70} - 6q^{73} - 447q^{74} - 54q^{76} - 132q^{77} - 798q^{80} - 12q^{82} + 138q^{85} - 606q^{86} - 198q^{88} + 114q^{89} - 723q^{92} - 357q^{94} + 168q^{97} - 510q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
19.1 −1.99738 0.102316i 0 3.97906 + 0.408727i −1.35104 7.66211i 0 −0.787463 + 0.938462i −7.90589 1.22351i 0 1.91458 + 15.4424i
19.2 −1.99036 0.196104i 0 3.92309 + 0.780636i 0.479555 + 2.71969i 0 −6.88504 + 8.20527i −7.65528 2.32308i 0 −0.421146 5.50722i
19.3 −1.97794 + 0.296237i 0 3.82449 1.17188i 0.00321173 + 0.0182146i 0 0.446686 0.532339i −7.21745 + 3.45086i 0 −0.0117484 0.0350759i
19.4 −1.91833 0.565700i 0 3.35997 + 2.17040i 1.66005 + 9.41462i 0 4.14643 4.94152i −5.21773 6.06427i 0 2.14133 18.9994i
19.5 −1.70561 + 1.04446i 0 1.81819 3.56289i 0.00321173 + 0.0182146i 0 −0.446686 + 0.532339i 0.620191 + 7.97592i 0 −0.0245024 0.0277124i
19.6 −1.63904 1.14610i 0 1.37291 + 3.75701i −1.09499 6.21002i 0 5.65814 6.74311i 2.05564 7.73139i 0 −5.32256 + 11.4335i
19.7 −1.48574 1.33887i 0 0.414837 + 3.97843i 0.542224 + 3.07510i 0 2.98812 3.56110i 4.71027 6.46632i 0 3.31157 5.29477i
19.8 −1.46432 + 1.36227i 0 0.288439 3.98959i −1.35104 7.66211i 0 0.787463 0.938462i 5.01253 + 6.23495i 0 12.4162 + 9.37927i
19.9 −1.43960 1.38836i 0 0.144916 + 3.99737i 0.133306 + 0.756014i 0 −3.08121 + 3.67204i 5.34117 5.95583i 0 0.857712 1.27344i
19.10 −1.39865 + 1.42960i 0 −0.0875399 3.99904i 0.479555 + 2.71969i 0 6.88504 8.20527i 5.83949 + 5.46812i 0 −4.55882 3.11833i
19.11 −1.10590 + 1.66643i 0 −1.55397 3.68581i 1.66005 + 9.41462i 0 −4.14643 + 4.94152i 7.86067 + 1.48655i 0 −17.5246 7.64527i
19.12 −0.747520 1.85505i 0 −2.88243 + 2.77337i −0.463207 2.62698i 0 −4.34885 + 5.18275i 7.29942 + 3.27390i 0 −4.52692 + 2.82299i
19.13 −0.518880 + 1.93152i 0 −3.46153 2.00445i −1.09499 6.21002i 0 −5.65814 + 6.74311i 5.66776 5.64593i 0 12.5629 + 1.10726i
19.14 −0.376263 1.96429i 0 −3.71685 + 1.47818i 1.26933 + 7.19876i 0 2.67622 3.18940i 4.30208 + 6.74478i 0 13.6628 5.20197i
19.15 −0.343802 1.97023i 0 −3.76360 + 1.35474i −0.608002 3.44815i 0 −5.71739 + 6.81372i 3.96308 + 6.94939i 0 −6.58462 + 2.38339i
19.16 −0.277531 + 1.98065i 0 −3.84595 1.09938i 0.542224 + 3.07510i 0 −2.98812 + 3.56110i 3.24486 7.31238i 0 −6.24119 + 0.220520i
19.17 −0.210380 + 1.98890i 0 −3.91148 0.836851i 0.133306 + 0.756014i 0 3.08121 3.67204i 2.48731 7.60350i 0 −1.53168 + 0.106082i
19.18 −0.103865 1.99730i 0 −3.97842 + 0.414901i −1.50670 8.54492i 0 7.53806 8.98351i 1.24190 + 7.90302i 0 −16.9103 + 3.89685i
19.19 0.443343 1.95024i 0 −3.60689 1.72925i 0.890579 + 5.05073i 0 −0.0265129 + 0.0315969i −4.97155 + 6.26767i 0 10.2450 + 0.502357i
19.20 0.619770 + 1.90155i 0 −3.23177 + 2.35705i −0.463207 2.62698i 0 4.34885 5.18275i −6.48499 4.68454i 0 4.70825 2.50894i
See next 80 embeddings (of 204 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 307.34 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
27.e even 9 1 inner
108.j odd 18 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 324.3.j.a 204
3.b odd 2 1 108.3.j.a 204
4.b odd 2 1 inner 324.3.j.a 204
12.b even 2 1 108.3.j.a 204
27.e even 9 1 inner 324.3.j.a 204
27.f odd 18 1 108.3.j.a 204
108.j odd 18 1 inner 324.3.j.a 204
108.l even 18 1 108.3.j.a 204

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.3.j.a 204 3.b odd 2 1
108.3.j.a 204 12.b even 2 1
108.3.j.a 204 27.f odd 18 1
108.3.j.a 204 108.l even 18 1
324.3.j.a 204 1.a even 1 1 trivial
324.3.j.a 204 4.b odd 2 1 inner
324.3.j.a 204 27.e even 9 1 inner
324.3.j.a 204 108.j odd 18 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{3}^{\mathrm{new}}(324, [\chi])$$.

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database