# Properties

 Label 432.2.u.f Level 432 Weight 2 Character orbit 432.u Analytic conductor 3.450 Analytic rank 0 Dimension 30 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$432 = 2^{4} \cdot 3^{3}$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 432.u (of order $$9$$, degree $$6$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.44953736732$$ Analytic rank: $$0$$ Dimension: $$30$$ Relative dimension: $$5$$ over $$\Q(\zeta_{9})$$ Coefficient ring index: multiple of None Twist minimal: no (minimal twist has level 216) Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

## $q$-expansion

The dimension is sufficiently large that we do not compute an algebraic $$q$$-expansion, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$30q - 3q^{7} - 6q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$30q - 3q^{7} - 6q^{9} + 3q^{11} - 12q^{13} - 15q^{15} + 6q^{17} + 9q^{19} + 30q^{21} + 12q^{23} + 24q^{25} + 15q^{27} - 9q^{29} - 27q^{31} - 30q^{33} + 18q^{35} - 15q^{37} + 21q^{39} - 15q^{41} + 30q^{43} + 15q^{45} + 18q^{47} + 15q^{49} + 6q^{51} - 18q^{53} - 54q^{55} - 72q^{57} + 12q^{59} + 6q^{61} + 54q^{63} - 54q^{65} + 45q^{67} + 9q^{69} - 36q^{73} - 69q^{75} + 12q^{77} - 45q^{79} - 30q^{81} + 3q^{83} + 57q^{85} + 60q^{87} + 36q^{89} + 39q^{91} + 30q^{93} - 51q^{95} - 84q^{97} - 162q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1 0 −1.51177 0.845305i 0 −3.37598 + 1.22876i 0 −0.462643 2.62378i 0 1.57092 + 2.55582i 0
49.2 0 −1.10062 1.33740i 0 3.74067 1.36149i 0 0.452652 + 2.56712i 0 −0.577279 + 2.94393i 0
49.3 0 −0.997080 + 1.41627i 0 1.95510 0.711598i 0 −0.739573 4.19433i 0 −1.01166 2.82428i 0
49.4 0 0.968921 + 1.43569i 0 −2.72954 + 0.993471i 0 0.186943 + 1.06021i 0 −1.12238 + 2.78213i 0
49.5 0 1.70086 0.327233i 0 1.34943 0.491154i 0 −0.111026 0.629660i 0 2.78584 1.11316i 0
97.1 0 −1.51177 + 0.845305i 0 −3.37598 1.22876i 0 −0.462643 + 2.62378i 0 1.57092 2.55582i 0
97.2 0 −1.10062 + 1.33740i 0 3.74067 + 1.36149i 0 0.452652 2.56712i 0 −0.577279 2.94393i 0
97.3 0 −0.997080 1.41627i 0 1.95510 + 0.711598i 0 −0.739573 + 4.19433i 0 −1.01166 + 2.82428i 0
97.4 0 0.968921 1.43569i 0 −2.72954 0.993471i 0 0.186943 1.06021i 0 −1.12238 2.78213i 0
97.5 0 1.70086 + 0.327233i 0 1.34943 + 0.491154i 0 −0.111026 + 0.629660i 0 2.78584 + 1.11316i 0
193.1 0 −1.51338 + 0.842423i 0 −0.672282 + 3.81270i 0 −2.31237 + 1.94030i 0 1.58065 2.54981i 0
193.2 0 −0.824312 1.52332i 0 −0.265520 + 1.50584i 0 3.27915 2.75153i 0 −1.64102 + 2.51139i 0
193.3 0 0.158806 1.72476i 0 0.536441 3.04231i 0 −3.96790 + 3.32946i 0 −2.94956 0.547802i 0
193.4 0 0.631172 + 1.61295i 0 0.156760 0.889029i 0 −1.02544 + 0.860449i 0 −2.20324 + 2.03610i 0
193.5 0 1.72136 0.192108i 0 0.0709538 0.402399i 0 2.76051 2.31635i 0 2.92619 0.661376i 0
241.1 0 −1.59356 0.678655i 0 1.47310 1.23608i 0 0.495883 0.180487i 0 2.07886 + 2.16295i 0
241.2 0 −0.485722 + 1.66255i 0 −0.582934 + 0.489140i 0 −3.39455 + 1.23552i 0 −2.52815 1.61507i 0
241.3 0 0.198479 1.72064i 0 −2.79870 + 2.34839i 0 −0.843226 + 0.306909i 0 −2.92121 0.683023i 0
241.4 0 1.28382 + 1.16267i 0 −1.03831 + 0.871247i 0 4.32665 1.57477i 0 0.296376 + 2.98532i 0
241.5 0 1.36303 1.06872i 0 2.18080 1.82991i 0 −0.145059 + 0.0527971i 0 0.715688 2.91338i 0
See all 30 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 385.5 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.e even 9 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 432.2.u.f 30
4.b odd 2 1 216.2.q.b 30
12.b even 2 1 648.2.q.b 30
27.e even 9 1 inner 432.2.u.f 30
108.j odd 18 1 216.2.q.b 30
108.j odd 18 1 5832.2.a.k 15
108.l even 18 1 648.2.q.b 30
108.l even 18 1 5832.2.a.l 15

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
216.2.q.b 30 4.b odd 2 1
216.2.q.b 30 108.j odd 18 1
432.2.u.f 30 1.a even 1 1 trivial
432.2.u.f 30 27.e even 9 1 inner
648.2.q.b 30 12.b even 2 1
648.2.q.b 30 108.l even 18 1
5832.2.a.k 15 108.j odd 18 1
5832.2.a.l 15 108.l even 18 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{30} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(432, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database