# 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 $$\Q(\zeta_{18})$$ 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.

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

## 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 $$S_{2}^{\mathrm{new}}(432, [\chi])$$:

 $$T_{5}^{18} + 3 T_{5}^{17} + 24 T_{5}^{15} + 117 T_{5}^{14} + 459 T_{5}^{13} + 126 T_{5}^{12} - 2790 T_{5}^{11} - 54 T_{5}^{10} - 10413 T_{5}^{9} + 51948 T_{5}^{8} - 155169 T_{5}^{7} + 299916 T_{5}^{6} - 258849 T_{5}^{5} + \cdots + 27$$ T5^18 + 3*T5^17 + 24*T5^15 + 117*T5^14 + 459*T5^13 + 126*T5^12 - 2790*T5^11 - 54*T5^10 - 10413*T5^9 + 51948*T5^8 - 155169*T5^7 + 299916*T5^6 - 258849*T5^5 + 153252*T5^4 - 147609*T5^3 + 71928*T5^2 + 1782*T5 + 27 $$T_{7}^{36} - 81 T_{7}^{32} - 6318 T_{7}^{30} + 125226 T_{7}^{28} - 1908765 T_{7}^{26} + 38537127 T_{7}^{24} - 410825034 T_{7}^{22} + 2742799806 T_{7}^{20} - 22827141279 T_{7}^{18} + \cdots + 531441$$ T7^36 - 81*T7^32 - 6318*T7^30 + 125226*T7^28 - 1908765*T7^26 + 38537127*T7^24 - 410825034*T7^22 + 2742799806*T7^20 - 22827141279*T7^18 + 114961730463*T7^16 + 101831949117*T7^14 + 804654640953*T7^12 + 754075334268*T7^10 + 319868314902*T7^8 - 12217297149*T7^6 - 316738836*T7^4 + 4782969*T7^2 + 531441