# Properties

 Label 864.2.y.b Level 864 Weight 2 Character orbit 864.y Analytic conductor 6.899 Analytic rank 0 Dimension 54 CM no Inner twists 2

# 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: $$54$$ Relative dimension: $$9$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$54q + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$54q - 9q^{11} + 12q^{17} + 18q^{19} + 12q^{21} - 21q^{27} + 6q^{29} + 36q^{31} - 9q^{33} + 24q^{39} + 3q^{41} - 21q^{43} + 42q^{45} - 18q^{49} + 24q^{51} + 36q^{53} - 72q^{55} + 39q^{57} + 18q^{59} - 18q^{61} - 30q^{63} + 48q^{65} - 27q^{67} + 24q^{69} - 84q^{75} + 36q^{77} + 72q^{79} + 36q^{81} + 6q^{87} + 33q^{89} + 36q^{91} + 72q^{93} + 36q^{95} + 9q^{97} + 120q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
97.1 0 −1.68626 0.395655i 0 −2.65901 0.967801i 0 −0.872194 + 4.94646i 0 2.68691 + 1.33435i 0
97.2 0 −1.59135 + 0.683824i 0 2.16034 + 0.786299i 0 0.181183 1.02754i 0 2.06477 2.17640i 0
97.3 0 −1.06169 1.36851i 0 0.217711 + 0.0792403i 0 0.399809 2.26743i 0 −0.745646 + 2.90586i 0
97.4 0 −1.00024 + 1.41404i 0 −3.61346 1.31519i 0 0.621731 3.52601i 0 −0.999043 2.82877i 0
97.5 0 −0.146186 1.72587i 0 2.31520 + 0.842664i 0 −0.690140 + 3.91398i 0 −2.95726 + 0.504595i 0
97.6 0 0.256084 + 1.71302i 0 −1.06964 0.389317i 0 −0.738542 + 4.18848i 0 −2.86884 + 0.877351i 0
97.7 0 1.11120 1.32862i 0 −0.779464 0.283702i 0 0.260701 1.47851i 0 −0.530450 2.95273i 0
97.8 0 1.53630 + 0.799856i 0 −1.92089 0.699146i 0 0.00661950 0.0375410i 0 1.72046 + 2.45764i 0
97.9 0 1.64243 + 0.549936i 0 3.46982 + 1.26291i 0 0.136240 0.772655i 0 2.39514 + 1.80646i 0
193.1 0 −1.72098 0.195544i 0 0.359864 2.04089i 0 2.05212 1.72193i 0 2.92353 + 0.673052i 0
193.2 0 −1.67597 + 0.437160i 0 −0.158606 + 0.899497i 0 −1.25116 + 1.04985i 0 2.61778 1.46534i 0
193.3 0 −0.661519 1.60075i 0 0.197119 1.11792i 0 −0.270013 + 0.226568i 0 −2.12478 + 2.11785i 0
193.4 0 −0.523926 + 1.65091i 0 −0.649143 + 3.68147i 0 1.60562 1.34727i 0 −2.45100 1.72991i 0
193.5 0 −0.0599758 + 1.73101i 0 0.392840 2.22791i 0 −3.55853 + 2.98596i 0 −2.99281 0.207638i 0
193.6 0 0.527750 1.64969i 0 −0.220114 + 1.24833i 0 −2.06659 + 1.73408i 0 −2.44296 1.74125i 0
193.7 0 1.17779 + 1.26996i 0 0.191385 1.08540i 0 2.62195 2.20007i 0 −0.225616 + 2.99150i 0
193.8 0 1.41286 1.00191i 0 0.727972 4.12853i 0 −0.397562 + 0.333594i 0 0.992364 2.83112i 0
193.9 0 1.69762 + 0.343652i 0 −0.494020 + 2.80173i 0 −1.80000 + 1.51038i 0 2.76381 + 1.16678i 0
385.1 0 −1.72098 + 0.195544i 0 0.359864 + 2.04089i 0 2.05212 + 1.72193i 0 2.92353 0.673052i 0
385.2 0 −1.67597 0.437160i 0 −0.158606 0.899497i 0 −1.25116 1.04985i 0 2.61778 + 1.46534i 0
See all 54 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 769.9 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 864.2.y.b 54
4.b odd 2 1 864.2.y.c yes 54
27.e even 9 1 inner 864.2.y.b 54
108.j odd 18 1 864.2.y.c yes 54

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
864.2.y.b 54 1.a even 1 1 trivial
864.2.y.b 54 27.e even 9 1 inner
864.2.y.c yes 54 4.b odd 2 1
864.2.y.c yes 54 108.j odd 18 1

## Hecke kernels

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

 $$T_{5}^{54} + \cdots$$ $$T_{7}^{54} + \cdots$$

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database