# Properties

 Label 432.2.be.a Level 432 Weight 2 Character orbit 432.be Analytic conductor 3.450 Analytic rank 0 Dimension 36 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$432 = 2^{4} \cdot 3^{3}$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 432.be (of order $$18$$, degree $$6$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.44953736732$$ Analytic rank: $$0$$ Dimension: $$36$$ Relative dimension: $$6$$ over $$\Q(\zeta_{18})$$ Coefficient ring index: multiple of None 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.

 $$\operatorname{Tr}(f)(q) =$$ $$36q - 6q^{5} - 12q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$36q - 6q^{5} - 12q^{9} - 36q^{21} + 18q^{25} - 24q^{29} - 36q^{33} - 18q^{41} - 18q^{45} + 42q^{65} + 54q^{69} - 18q^{73} + 90q^{77} + 36q^{81} + 72q^{85} + 72q^{89} + 54q^{93} + 54q^{97} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
47.1 0 −1.72868 + 0.107984i 0 0.902279 0.159096i 0 −2.30964 + 2.75253i 0 2.97668 0.373339i 0
47.2 0 −0.991247 + 1.42036i 0 −3.85028 + 0.678908i 0 1.90230 2.26708i 0 −1.03486 2.81586i 0
47.3 0 −0.340461 1.69826i 0 1.68195 0.296574i 0 0.981328 1.16950i 0 −2.76817 + 1.15638i 0
47.4 0 0.340461 + 1.69826i 0 1.68195 0.296574i 0 −0.981328 + 1.16950i 0 −2.76817 + 1.15638i 0
47.5 0 0.991247 1.42036i 0 −3.85028 + 0.678908i 0 −1.90230 + 2.26708i 0 −1.03486 2.81586i 0
47.6 0 1.72868 0.107984i 0 0.902279 0.159096i 0 2.30964 2.75253i 0 2.97668 0.373339i 0
95.1 0 −1.64304 + 0.548109i 0 −0.0121515 + 0.0144816i 0 −0.279932 + 0.769107i 0 2.39915 1.80113i 0
95.2 0 −1.16393 1.28268i 0 −1.92280 + 2.29150i 0 −0.0588716 + 0.161748i 0 −0.290520 + 2.98590i 0
95.3 0 −0.573329 1.63441i 0 2.37464 2.82999i 0 1.65095 4.53595i 0 −2.34259 + 1.87411i 0
95.4 0 0.573329 + 1.63441i 0 2.37464 2.82999i 0 −1.65095 + 4.53595i 0 −2.34259 + 1.87411i 0
95.5 0 1.16393 + 1.28268i 0 −1.92280 + 2.29150i 0 0.0588716 0.161748i 0 −0.290520 + 2.98590i 0
95.6 0 1.64304 0.548109i 0 −0.0121515 + 0.0144816i 0 0.279932 0.769107i 0 2.39915 1.80113i 0
191.1 0 −1.64304 0.548109i 0 −0.0121515 0.0144816i 0 −0.279932 0.769107i 0 2.39915 + 1.80113i 0
191.2 0 −1.16393 + 1.28268i 0 −1.92280 2.29150i 0 −0.0588716 0.161748i 0 −0.290520 2.98590i 0
191.3 0 −0.573329 + 1.63441i 0 2.37464 + 2.82999i 0 1.65095 + 4.53595i 0 −2.34259 1.87411i 0
191.4 0 0.573329 1.63441i 0 2.37464 + 2.82999i 0 −1.65095 4.53595i 0 −2.34259 1.87411i 0
191.5 0 1.16393 1.28268i 0 −1.92280 2.29150i 0 0.0588716 + 0.161748i 0 −0.290520 2.98590i 0
191.6 0 1.64304 + 0.548109i 0 −0.0121515 0.0144816i 0 0.279932 + 0.769107i 0 2.39915 + 1.80113i 0
239.1 0 −1.72868 0.107984i 0 0.902279 + 0.159096i 0 −2.30964 2.75253i 0 2.97668 + 0.373339i 0
239.2 0 −0.991247 1.42036i 0 −3.85028 0.678908i 0 1.90230 + 2.26708i 0 −1.03486 + 2.81586i 0
See all 36 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 383.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
27.f odd 18 1 inner
108.l even 18 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 432.2.be.a 36
4.b odd 2 1 inner 432.2.be.a 36
27.f odd 18 1 inner 432.2.be.a 36
108.l even 18 1 inner 432.2.be.a 36

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
432.2.be.a 36 1.a even 1 1 trivial
432.2.be.a 36 4.b odd 2 1 inner
432.2.be.a 36 27.f odd 18 1 inner
432.2.be.a 36 108.l even 18 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(432, [\chi])$$:

 $$T_{5}^{18} + \cdots$$ $$T_{7}^{36} - \cdots$$

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database