# Properties

 Label 216.2.q.a Level 216 Weight 2 Character orbit 216.q Analytic conductor 1.725 Analytic rank 0 Dimension 24 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$216 = 2^{3} \cdot 3^{3}$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 216.q (of order $$9$$, degree $$6$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.72476868366$$ Analytic rank: $$0$$ Dimension: $$24$$ Relative dimension: $$4$$ over $$\Q(\zeta_{9})$$ Coefficient ring index: multiple of None Twist minimal: yes 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.

 $$\operatorname{Tr}(f)(q) =$$ $$24q - 3q^{7} + 6q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$24q - 3q^{7} + 6q^{9} + 6q^{11} + 12q^{13} - 3q^{15} + 6q^{17} + 9q^{19} - 18q^{21} + 24q^{23} - 24q^{25} - 9q^{29} - 27q^{31} + 21q^{33} - 18q^{35} + 15q^{37} - 15q^{39} - 6q^{41} + 39q^{43} - 69q^{45} - 36q^{47} + 3q^{49} - 36q^{51} - 18q^{53} - 54q^{55} + 27q^{57} - 30q^{59} + 12q^{61} + 18q^{63} - 18q^{65} + 54q^{67} - 57q^{69} + 36q^{73} - 51q^{75} - 24q^{77} - 45q^{79} + 18q^{81} + 33q^{83} - 57q^{85} + 90q^{87} + 9q^{89} + 39q^{91} + 42q^{93} + 87q^{95} + 57q^{97} - 24q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
25.1 0 −1.66239 0.486282i 0 −2.11828 + 1.77745i 0 4.34143 1.58015i 0 2.52706 + 1.61678i 0
25.2 0 −0.962247 1.44017i 0 2.75433 2.31115i 0 −1.28512 + 0.467744i 0 −1.14816 + 2.77159i 0
25.3 0 1.05517 + 1.37354i 0 −1.80911 + 1.51802i 0 −3.12406 + 1.13706i 0 −0.773215 + 2.89864i 0
25.4 0 1.56946 0.732664i 0 0.407020 0.341530i 0 0.507439 0.184693i 0 1.92641 2.29977i 0
49.1 0 −1.57237 + 0.726388i 0 −1.78916 + 0.651202i 0 −0.623407 3.53552i 0 1.94472 2.28431i 0
49.2 0 −0.278117 1.70958i 0 2.42978 0.884366i 0 −0.245784 1.39391i 0 −2.84530 + 0.950925i 0
49.3 0 0.119504 + 1.72792i 0 0.307563 0.111944i 0 0.551939 + 3.13020i 0 −2.97144 + 0.412989i 0
49.4 0 1.73099 0.0606946i 0 −0.00848388 + 0.00308788i 0 −0.356397 2.02123i 0 2.99263 0.210123i 0
97.1 0 −1.57237 0.726388i 0 −1.78916 0.651202i 0 −0.623407 + 3.53552i 0 1.94472 + 2.28431i 0
97.2 0 −0.278117 + 1.70958i 0 2.42978 + 0.884366i 0 −0.245784 + 1.39391i 0 −2.84530 0.950925i 0
97.3 0 0.119504 1.72792i 0 0.307563 + 0.111944i 0 0.551939 3.13020i 0 −2.97144 0.412989i 0
97.4 0 1.73099 + 0.0606946i 0 −0.00848388 0.00308788i 0 −0.356397 + 2.02123i 0 2.99263 + 0.210123i 0
121.1 0 −1.66239 + 0.486282i 0 −2.11828 1.77745i 0 4.34143 + 1.58015i 0 2.52706 1.61678i 0
121.2 0 −0.962247 + 1.44017i 0 2.75433 + 2.31115i 0 −1.28512 0.467744i 0 −1.14816 2.77159i 0
121.3 0 1.05517 1.37354i 0 −1.80911 1.51802i 0 −3.12406 1.13706i 0 −0.773215 2.89864i 0
121.4 0 1.56946 + 0.732664i 0 0.407020 + 0.341530i 0 0.507439 + 0.184693i 0 1.92641 + 2.29977i 0
169.1 0 −1.70050 0.329088i 0 0.198034 + 1.12311i 0 0.914338 + 0.767221i 0 2.78340 + 1.11923i 0
169.2 0 −0.747543 + 1.56243i 0 −0.738874 4.19036i 0 −2.50342 2.10062i 0 −1.88236 2.33596i 0
169.3 0 0.887369 + 1.48747i 0 0.444259 + 2.51952i 0 −0.612199 0.513696i 0 −1.42515 + 2.63988i 0
169.4 0 1.56067 0.751197i 0 −0.0770674 0.437071i 0 0.935232 + 0.784753i 0 1.87141 2.34475i 0
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 193.4 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 216.2.q.a 24
3.b odd 2 1 648.2.q.a 24
4.b odd 2 1 432.2.u.e 24
27.e even 9 1 inner 216.2.q.a 24
27.e even 9 1 5832.2.a.h 12
27.f odd 18 1 648.2.q.a 24
27.f odd 18 1 5832.2.a.i 12
108.j odd 18 1 432.2.u.e 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
216.2.q.a 24 1.a even 1 1 trivial
216.2.q.a 24 27.e even 9 1 inner
432.2.u.e 24 4.b odd 2 1
432.2.u.e 24 108.j odd 18 1
648.2.q.a 24 3.b odd 2 1
648.2.q.a 24 27.f odd 18 1
5832.2.a.h 12 27.e even 9 1
5832.2.a.i 12 27.f odd 18 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{24} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(216, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database