# Properties

 Label 216.2.q.b Level 216 Weight 2 Character orbit 216.q Analytic conductor 1.725 Analytic rank 0 Dimension 30 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: $$30$$ Relative dimension: $$5$$ 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.

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

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
25.1 0 −1.36303 + 1.06872i 0 2.18080 1.82991i 0 0.145059 0.0527971i 0 0.715688 2.91338i 0
25.2 0 −1.28382 1.16267i 0 −1.03831 + 0.871247i 0 −4.32665 + 1.57477i 0 0.296376 + 2.98532i 0
25.3 0 −0.198479 + 1.72064i 0 −2.79870 + 2.34839i 0 0.843226 0.306909i 0 −2.92121 0.683023i 0
25.4 0 0.485722 1.66255i 0 −0.582934 + 0.489140i 0 3.39455 1.23552i 0 −2.52815 1.61507i 0
25.5 0 1.59356 + 0.678655i 0 1.47310 1.23608i 0 −0.495883 + 0.180487i 0 2.07886 + 2.16295i 0
49.1 0 −1.70086 + 0.327233i 0 1.34943 0.491154i 0 0.111026 + 0.629660i 0 2.78584 1.11316i 0
49.2 0 −0.968921 1.43569i 0 −2.72954 + 0.993471i 0 −0.186943 1.06021i 0 −1.12238 + 2.78213i 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 1.10062 + 1.33740i 0 3.74067 1.36149i 0 −0.452652 2.56712i 0 −0.577279 + 2.94393i 0
49.5 0 1.51177 + 0.845305i 0 −3.37598 + 1.22876i 0 0.462643 + 2.62378i 0 1.57092 + 2.55582i 0
97.1 0 −1.70086 0.327233i 0 1.34943 + 0.491154i 0 0.111026 0.629660i 0 2.78584 + 1.11316i 0
97.2 0 −0.968921 + 1.43569i 0 −2.72954 0.993471i 0 −0.186943 + 1.06021i 0 −1.12238 2.78213i 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 1.10062 1.33740i 0 3.74067 + 1.36149i 0 −0.452652 + 2.56712i 0 −0.577279 2.94393i 0
97.5 0 1.51177 0.845305i 0 −3.37598 1.22876i 0 0.462643 2.62378i 0 1.57092 2.55582i 0
121.1 0 −1.36303 1.06872i 0 2.18080 + 1.82991i 0 0.145059 + 0.0527971i 0 0.715688 + 2.91338i 0
121.2 0 −1.28382 + 1.16267i 0 −1.03831 0.871247i 0 −4.32665 1.57477i 0 0.296376 2.98532i 0
121.3 0 −0.198479 1.72064i 0 −2.79870 2.34839i 0 0.843226 + 0.306909i 0 −2.92121 + 0.683023i 0
121.4 0 0.485722 + 1.66255i 0 −0.582934 0.489140i 0 3.39455 + 1.23552i 0 −2.52815 + 1.61507i 0
121.5 0 1.59356 0.678655i 0 1.47310 + 1.23608i 0 −0.495883 0.180487i 0 2.07886 2.16295i 0
See all 30 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 193.5 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 216.2.q.b 30
3.b odd 2 1 648.2.q.b 30
4.b odd 2 1 432.2.u.f 30
27.e even 9 1 inner 216.2.q.b 30
27.e even 9 1 5832.2.a.k 15
27.f odd 18 1 648.2.q.b 30
27.f odd 18 1 5832.2.a.l 15
108.j odd 18 1 432.2.u.f 30

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
216.2.q.b 30 1.a even 1 1 trivial
216.2.q.b 30 27.e even 9 1 inner
432.2.u.f 30 4.b odd 2 1
432.2.u.f 30 108.j odd 18 1
648.2.q.b 30 3.b odd 2 1
648.2.q.b 30 27.f odd 18 1
5832.2.a.k 15 27.e even 9 1
5832.2.a.l 15 27.f odd 18 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{30} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(216, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database