# Properties

 Label 324.3.k.a Level 324 Weight 3 Character orbit 324.k Analytic conductor 8.828 Analytic rank 0 Dimension 36 CM no Inner twists 2

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$8.82836056527$$ Analytic rank: $$0$$ Dimension: $$36$$ Relative dimension: $$6$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$36q + 9q^{5} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$36q + 9q^{5} - 36q^{11} + 18q^{23} - 9q^{25} + 18q^{29} + 45q^{31} + 243q^{35} + 198q^{41} + 90q^{43} + 243q^{47} + 72q^{49} - 252q^{59} - 144q^{61} - 747q^{65} + 108q^{67} - 324q^{71} - 63q^{73} - 495q^{77} + 36q^{79} + 27q^{83} - 180q^{85} + 567q^{89} + 99q^{91} + 1044q^{95} - 216q^{97} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
17.1 0 0 0 −7.10446 1.25271i 0 −3.36440 + 2.82306i 0 0 0
17.2 0 0 0 −3.29223 0.580508i 0 3.84473 3.22611i 0 0 0
17.3 0 0 0 −2.92426 0.515626i 0 0.715829 0.600652i 0 0 0
17.4 0 0 0 4.32861 + 0.763252i 0 −2.73772 + 2.29722i 0 0 0
17.5 0 0 0 5.13754 + 0.905886i 0 −8.93347 + 7.49607i 0 0 0
17.6 0 0 0 7.65293 + 1.34942i 0 10.4750 8.78960i 0 0 0
89.1 0 0 0 −5.64904 6.73227i 0 4.05297 1.47516i 0 0 0
89.2 0 0 0 −2.68656 3.20172i 0 −4.88621 + 1.77844i 0 0 0
89.3 0 0 0 −0.980262 1.16823i 0 3.23920 1.17897i 0 0 0
89.4 0 0 0 0.298552 + 0.355800i 0 −10.1488 + 3.69384i 0 0 0
89.5 0 0 0 2.69546 + 3.21232i 0 11.1367 4.05342i 0 0 0
89.6 0 0 0 5.00278 + 5.96208i 0 −3.39388 + 1.23527i 0 0 0
125.1 0 0 0 −1.65461 4.54600i 0 1.68621 + 9.56295i 0 0 0
125.2 0 0 0 −1.26650 3.47969i 0 −0.0728181 0.412972i 0 0 0
125.3 0 0 0 −0.740753 2.03520i 0 1.08248 + 6.13906i 0 0 0
125.4 0 0 0 −0.0686711 0.188672i 0 −1.47862 8.38565i 0 0 0
125.5 0 0 0 2.50118 + 6.87194i 0 −1.62729 9.22884i 0 0 0
125.6 0 0 0 3.25030 + 8.93012i 0 0.410040 + 2.32545i 0 0 0
197.1 0 0 0 −1.65461 + 4.54600i 0 1.68621 9.56295i 0 0 0
197.2 0 0 0 −1.26650 + 3.47969i 0 −0.0728181 + 0.412972i 0 0 0
See all 36 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 305.6 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 324.3.k.a 36
3.b odd 2 1 108.3.k.a 36
12.b even 2 1 432.3.bc.b 36
27.e even 9 1 108.3.k.a 36
27.e even 9 1 2916.3.c.b 36
27.f odd 18 1 inner 324.3.k.a 36
27.f odd 18 1 2916.3.c.b 36
108.j odd 18 1 432.3.bc.b 36

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.3.k.a 36 3.b odd 2 1
108.3.k.a 36 27.e even 9 1
324.3.k.a 36 1.a even 1 1 trivial
324.3.k.a 36 27.f odd 18 1 inner
432.3.bc.b 36 12.b even 2 1
432.3.bc.b 36 108.j odd 18 1
2916.3.c.b 36 27.e even 9 1
2916.3.c.b 36 27.f odd 18 1

## 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