# Properties

 Label 432.3.bc.b Level 432 Weight 3 Character orbit 432.bc Analytic conductor 11.771 Analytic rank 0 Dimension 36 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$432 = 2^{4} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 432.bc (of order $$18$$, degree $$6$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$11.7711474204$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$36q - 9q^{5} + 6q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$36q - 9q^{5} + 6q^{9} - 36q^{11} - 45q^{15} + 42q^{21} + 18q^{23} - 9q^{25} - 18q^{29} - 45q^{31} - 153q^{33} + 243q^{35} + 123q^{39} - 198q^{41} - 90q^{43} - 333q^{45} + 243q^{47} + 72q^{49} + 99q^{51} + 243q^{57} - 252q^{59} - 144q^{61} - 381q^{63} + 747q^{65} - 108q^{67} + 585q^{69} - 324q^{71} - 63q^{73} - 597q^{75} + 495q^{77} - 36q^{79} - 54q^{81} + 27q^{83} - 180q^{85} + 441q^{87} - 567q^{89} - 99q^{91} - 699q^{93} + 1044q^{95} - 216q^{97} + 945q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
65.1 0 −2.99363 0.195330i 0 −2.50118 + 6.87194i 0 1.62729 9.22884i 0 8.92369 + 1.16949i 0
65.2 0 −2.70588 1.29546i 0 1.65461 4.54600i 0 −1.68621 + 9.56295i 0 5.64356 + 7.01072i 0
65.3 0 −1.02038 + 2.82114i 0 1.26650 3.47969i 0 0.0728181 0.412972i 0 −6.91763 5.75729i 0
65.4 0 0.727525 2.91045i 0 0.0686711 0.188672i 0 1.47862 8.38565i 0 −7.94141 4.23485i 0
65.5 0 1.95183 + 2.27824i 0 −3.25030 + 8.93012i 0 −0.410040 + 2.32545i 0 −1.38071 + 8.89346i 0
65.6 0 2.92720 0.656882i 0 0.740753 2.03520i 0 −1.08248 + 6.13906i 0 8.13701 3.84565i 0
113.1 0 −2.99363 + 0.195330i 0 −2.50118 6.87194i 0 1.62729 + 9.22884i 0 8.92369 1.16949i 0
113.2 0 −2.70588 + 1.29546i 0 1.65461 + 4.54600i 0 −1.68621 9.56295i 0 5.64356 7.01072i 0
113.3 0 −1.02038 2.82114i 0 1.26650 + 3.47969i 0 0.0728181 + 0.412972i 0 −6.91763 + 5.75729i 0
113.4 0 0.727525 + 2.91045i 0 0.0686711 + 0.188672i 0 1.47862 + 8.38565i 0 −7.94141 + 4.23485i 0
113.5 0 1.95183 2.27824i 0 −3.25030 8.93012i 0 −0.410040 2.32545i 0 −1.38071 8.89346i 0
113.6 0 2.92720 + 0.656882i 0 0.740753 + 2.03520i 0 −1.08248 6.13906i 0 8.13701 + 3.84565i 0
209.1 0 −2.36501 1.84574i 0 −7.65293 + 1.34942i 0 −10.4750 8.78960i 0 2.18650 + 8.73036i 0
209.2 0 −2.15561 + 2.08647i 0 2.92426 0.515626i 0 −0.715829 0.600652i 0 0.293266 8.99522i 0
209.3 0 −1.86083 2.35315i 0 7.10446 1.25271i 0 3.36440 + 2.82306i 0 −2.07464 + 8.75762i 0
209.4 0 1.31935 2.69431i 0 −5.13754 + 0.905886i 0 8.93347 + 7.49607i 0 −5.51862 7.10949i 0
209.5 0 1.47205 + 2.61402i 0 −4.32861 + 0.763252i 0 2.73772 + 2.29722i 0 −4.66616 + 7.69591i 0
209.6 0 2.99764 0.118924i 0 3.29223 0.580508i 0 −3.84473 3.22611i 0 8.97171 0.712982i 0
257.1 0 −2.88785 + 0.812609i 0 0.980262 1.16823i 0 −3.23920 1.17897i 0 7.67933 4.69338i 0
257.2 0 −1.49775 2.59937i 0 −5.00278 + 5.96208i 0 3.39388 + 1.23527i 0 −4.51349 + 7.78642i 0
See all 36 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 401.6 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 432.3.bc.b 36
4.b odd 2 1 108.3.k.a 36
12.b even 2 1 324.3.k.a 36
27.f odd 18 1 inner 432.3.bc.b 36
108.j odd 18 1 324.3.k.a 36
108.j odd 18 1 2916.3.c.b 36
108.l even 18 1 108.3.k.a 36
108.l even 18 1 2916.3.c.b 36

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.3.k.a 36 4.b odd 2 1
108.3.k.a 36 108.l even 18 1
324.3.k.a 36 12.b even 2 1
324.3.k.a 36 108.j odd 18 1
432.3.bc.b 36 1.a even 1 1 trivial
432.3.bc.b 36 27.f odd 18 1 inner
2916.3.c.b 36 108.j odd 18 1
2916.3.c.b 36 108.l even 18 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{36} + \cdots$$ acting on $$S_{3}^{\mathrm{new}}(432, [\chi])$$.

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database