# Properties

 Label 864.2.bi.a Level 864 Weight 2 Character orbit 864.bi Analytic conductor 6.899 Analytic rank 0 Dimension 216 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$864 = 2^{5} \cdot 3^{3}$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 864.bi (of order $$18$$, degree $$6$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.89907473464$$ Analytic rank: $$0$$ Dimension: $$216$$ Relative dimension: $$36$$ over $$\Q(\zeta_{18})$$ Coefficient ring index: multiple of None Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$216q + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$216q - 24q^{29} + 36q^{33} + 36q^{41} - 24q^{45} + 12q^{57} + 48q^{65} + 48q^{69} + 48q^{77} + 48q^{81} + 36q^{89} - 144q^{93} - 36q^{97} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
95.1 0 −1.72799 + 0.118502i 0 1.85973 2.21633i 0 −0.235484 + 0.646986i 0 2.97191 0.409541i 0
95.2 0 −1.70911 0.280941i 0 −1.54632 + 1.84283i 0 0.573007 1.57432i 0 2.84214 + 0.960320i 0
95.3 0 −1.69887 + 0.337423i 0 −0.948003 + 1.12979i 0 −0.391368 + 1.07528i 0 2.77229 1.14647i 0
95.4 0 −1.68666 0.393943i 0 2.16279 2.57751i 0 0.669796 1.84025i 0 2.68962 + 1.32889i 0
95.5 0 −1.59297 0.680026i 0 −0.802661 + 0.956575i 0 −1.53564 + 4.21915i 0 2.07513 + 2.16653i 0
95.6 0 −1.54765 + 0.777684i 0 −1.51205 + 1.80199i 0 0.535936 1.47247i 0 1.79042 2.40716i 0
95.7 0 −1.46065 + 0.930864i 0 0.902634 1.07572i 0 1.52652 4.19407i 0 1.26699 2.71933i 0
95.8 0 −1.37511 + 1.05313i 0 −2.70735 + 3.22649i 0 −1.09514 + 3.00888i 0 0.781832 2.89633i 0
95.9 0 −1.29799 1.14683i 0 −0.251563 + 0.299801i 0 0.212073 0.582665i 0 0.369555 + 2.97715i 0
95.10 0 −1.23859 + 1.21074i 0 1.02326 1.21948i 0 −0.575400 + 1.58090i 0 0.0682160 2.99922i 0
95.11 0 −1.18010 1.26782i 0 1.50278 1.79094i 0 −1.53224 + 4.20980i 0 −0.214731 + 2.99231i 0
95.12 0 −0.948095 1.44952i 0 0.639391 0.761997i 0 1.09080 2.99694i 0 −1.20223 + 2.74857i 0
95.13 0 −0.639313 + 1.60974i 0 1.25880 1.50018i 0 −0.917368 + 2.52045i 0 −2.18256 2.05826i 0
95.14 0 −0.633406 1.61208i 0 −2.10172 + 2.50473i 0 1.57289 4.32148i 0 −2.19759 + 2.04220i 0
95.15 0 −0.413006 + 1.68209i 0 −2.09970 + 2.50233i 0 0.771995 2.12104i 0 −2.65885 1.38943i 0
95.16 0 −0.246052 + 1.71448i 0 0.0151145 0.0180127i 0 1.23843 3.40255i 0 −2.87892 0.843705i 0
95.17 0 −0.152734 + 1.72530i 0 −0.204383 + 0.243574i 0 0.388373 1.06705i 0 −2.95334 0.527025i 0
95.18 0 −0.0609182 1.73098i 0 2.80925 3.34793i 0 −0.417699 + 1.14762i 0 −2.99258 + 0.210896i 0
95.19 0 0.0609182 + 1.73098i 0 2.80925 3.34793i 0 0.417699 1.14762i 0 −2.99258 + 0.210896i 0
95.20 0 0.152734 1.72530i 0 −0.204383 + 0.243574i 0 −0.388373 + 1.06705i 0 −2.95334 0.527025i 0
See next 80 embeddings (of 216 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 767.36 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.f odd 18 1 inner
108.l even 18 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 864.2.bi.a 216
4.b odd 2 1 inner 864.2.bi.a 216
27.f odd 18 1 inner 864.2.bi.a 216
108.l even 18 1 inner 864.2.bi.a 216

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
864.2.bi.a 216 1.a even 1 1 trivial
864.2.bi.a 216 4.b odd 2 1 inner
864.2.bi.a 216 27.f odd 18 1 inner
864.2.bi.a 216 108.l even 18 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(864, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database