# Properties

 Label 864.2.y.d Level 864 Weight 2 Character orbit 864.y Analytic conductor 6.899 Analytic rank 0 Dimension 60 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$864 = 2^{5} \cdot 3^{3}$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 864.y (of order $$9$$, degree $$6$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.89907473464$$ Analytic rank: $$0$$ Dimension: $$60$$ Relative dimension: $$10$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$60q + 12q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$60q + 12q^{9} - 12q^{17} + 24q^{21} - 24q^{25} + 6q^{29} - 12q^{33} - 30q^{37} - 30q^{41} - 90q^{45} + 42q^{49} - 36q^{53} - 60q^{57} + 48q^{61} + 12q^{65} + 78q^{69} - 48q^{73} - 12q^{77} + 12q^{81} - 102q^{85} - 12q^{89} - 36q^{93} + 12q^{97} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
97.1 0 −1.71652 + 0.231448i 0 −3.16823 1.15314i 0 0.327751 1.85877i 0 2.89286 0.794570i 0
97.2 0 −1.65691 + 0.504627i 0 1.76220 + 0.641388i 0 −0.498467 + 2.82694i 0 2.49070 1.67224i 0
97.3 0 −1.01321 1.40478i 0 −1.03120 0.375327i 0 −0.257635 + 1.46112i 0 −0.946819 + 2.84667i 0
97.4 0 −0.790581 1.54110i 0 3.49481 + 1.27201i 0 0.718530 4.07499i 0 −1.74996 + 2.43673i 0
97.5 0 −0.523054 + 1.65119i 0 −0.117882 0.0429055i 0 0.455772 2.58481i 0 −2.45283 1.72732i 0
97.6 0 0.523054 1.65119i 0 −0.117882 0.0429055i 0 −0.455772 + 2.58481i 0 −2.45283 1.72732i 0
97.7 0 0.790581 + 1.54110i 0 3.49481 + 1.27201i 0 −0.718530 + 4.07499i 0 −1.74996 + 2.43673i 0
97.8 0 1.01321 + 1.40478i 0 −1.03120 0.375327i 0 0.257635 1.46112i 0 −0.946819 + 2.84667i 0
97.9 0 1.65691 0.504627i 0 1.76220 + 0.641388i 0 0.498467 2.82694i 0 2.49070 1.67224i 0
97.10 0 1.71652 0.231448i 0 −3.16823 1.15314i 0 −0.327751 + 1.85877i 0 2.89286 0.794570i 0
193.1 0 −1.72168 0.189239i 0 0.679279 3.85238i 0 −2.91962 + 2.44985i 0 2.92838 + 0.651618i 0
193.2 0 −1.68152 + 0.415309i 0 −0.564773 + 3.20298i 0 −2.06064 + 1.72908i 0 2.65504 1.39670i 0
193.3 0 −1.28924 + 1.15666i 0 −0.110022 + 0.623966i 0 3.44796 2.89318i 0 0.324271 2.98242i 0
193.4 0 −0.998006 1.41562i 0 −0.476400 + 2.70180i 0 0.930891 0.781110i 0 −1.00797 + 2.82560i 0
193.5 0 −0.141380 + 1.72627i 0 0.298268 1.69156i 0 −0.0848018 + 0.0711572i 0 −2.96002 0.488119i 0
193.6 0 0.141380 1.72627i 0 0.298268 1.69156i 0 0.0848018 0.0711572i 0 −2.96002 0.488119i 0
193.7 0 0.998006 + 1.41562i 0 −0.476400 + 2.70180i 0 −0.930891 + 0.781110i 0 −1.00797 + 2.82560i 0
193.8 0 1.28924 1.15666i 0 −0.110022 + 0.623966i 0 −3.44796 + 2.89318i 0 0.324271 2.98242i 0
193.9 0 1.68152 0.415309i 0 −0.564773 + 3.20298i 0 2.06064 1.72908i 0 2.65504 1.39670i 0
193.10 0 1.72168 + 0.189239i 0 0.679279 3.85238i 0 2.91962 2.44985i 0 2.92838 + 0.651618i 0
See all 60 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 769.10 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.e even 9 1 inner
108.j odd 18 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 864.2.y.d 60
4.b odd 2 1 inner 864.2.y.d 60
27.e even 9 1 inner 864.2.y.d 60
108.j odd 18 1 inner 864.2.y.d 60

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
864.2.y.d 60 1.a even 1 1 trivial
864.2.y.d 60 4.b odd 2 1 inner
864.2.y.d 60 27.e even 9 1 inner
864.2.y.d 60 108.j odd 18 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(864, [\chi])$$:

 $$T_{5}^{30} + \cdots$$ $$T_{7}^{60} - \cdots$$

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database