# Properties

 Label 864.2.bh.b Level 864 Weight 2 Character orbit 864.bh Analytic conductor 6.899 Analytic rank 0 Dimension 192 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$864 = 2^{5} \cdot 3^{3}$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 864.bh (of order $$18$$, degree $$6$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.89907473464$$ Analytic rank: $$0$$ Dimension: $$192$$ Relative dimension: $$32$$ over $$\Q(\zeta_{18})$$ Coefficient ring index: multiple of None Twist minimal: no (minimal twist has level 216) 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.

 $$\operatorname{Tr}(f)(q) =$$ $$192q + 12q^{3} - 12q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$192q + 12q^{3} - 12q^{9} + 30q^{11} - 18q^{17} + 6q^{19} - 12q^{25} - 18q^{27} - 30q^{33} + 18q^{35} + 18q^{41} + 42q^{43} - 12q^{49} + 18q^{51} - 36q^{57} + 84q^{59} - 12q^{65} - 30q^{67} - 6q^{73} + 96q^{75} - 12q^{81} + 72q^{83} + 144q^{89} + 6q^{91} - 42q^{97} + 12q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
47.1 0 −1.71829 0.217923i 0 −0.693543 3.93328i 0 2.05979 2.45476i 0 2.90502 + 0.748909i 0
47.2 0 −1.71829 0.217923i 0 0.693543 + 3.93328i 0 −2.05979 + 2.45476i 0 2.90502 + 0.748909i 0
47.3 0 −1.67094 + 0.456011i 0 −0.197395 1.11948i 0 −1.00340 + 1.19580i 0 2.58411 1.52394i 0
47.4 0 −1.67094 + 0.456011i 0 0.197395 + 1.11948i 0 1.00340 1.19580i 0 2.58411 1.52394i 0
47.5 0 −1.49214 0.879503i 0 −0.189066 1.07225i 0 −2.50853 + 2.98955i 0 1.45295 + 2.62468i 0
47.6 0 −1.49214 0.879503i 0 0.189066 + 1.07225i 0 2.50853 2.98955i 0 1.45295 + 2.62468i 0
47.7 0 −1.23020 + 1.21926i 0 −0.470553 2.66864i 0 −1.85741 + 2.21358i 0 0.0267915 2.99988i 0
47.8 0 −1.23020 + 1.21926i 0 0.470553 + 2.66864i 0 1.85741 2.21358i 0 0.0267915 2.99988i 0
47.9 0 −0.967099 + 1.43691i 0 −0.586059 3.32370i 0 1.89356 2.25666i 0 −1.12944 2.77927i 0
47.10 0 −0.967099 + 1.43691i 0 0.586059 + 3.32370i 0 −1.89356 + 2.25666i 0 −1.12944 2.77927i 0
47.11 0 −0.608611 1.62160i 0 −0.398052 2.25746i 0 −0.655732 + 0.781471i 0 −2.25918 + 1.97385i 0
47.12 0 −0.608611 1.62160i 0 0.398052 + 2.25746i 0 0.655732 0.781471i 0 −2.25918 + 1.97385i 0
47.13 0 −0.459797 + 1.66991i 0 −0.221850 1.25817i 0 0.0241545 0.0287862i 0 −2.57717 1.53564i 0
47.14 0 −0.459797 + 1.66991i 0 0.221850 + 1.25817i 0 −0.0241545 + 0.0287862i 0 −2.57717 1.53564i 0
47.15 0 0.0100749 1.73202i 0 −0.710827 4.03130i 0 −0.183548 + 0.218744i 0 −2.99980 0.0348998i 0
47.16 0 0.0100749 1.73202i 0 0.710827 + 4.03130i 0 0.183548 0.218744i 0 −2.99980 0.0348998i 0
47.17 0 0.166515 1.72403i 0 −0.135936 0.770931i 0 2.51012 2.99144i 0 −2.94455 0.574153i 0
47.18 0 0.166515 1.72403i 0 0.135936 + 0.770931i 0 −2.51012 + 2.99144i 0 −2.94455 0.574153i 0
47.19 0 0.604794 + 1.62303i 0 −0.408435 2.31635i 0 −1.58241 + 1.88584i 0 −2.26845 + 1.96320i 0
47.20 0 0.604794 + 1.62303i 0 0.408435 + 2.31635i 0 1.58241 1.88584i 0 −2.26845 + 1.96320i 0
See next 80 embeddings (of 192 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 815.32 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner
27.f odd 18 1 inner
216.v 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.bh.b 192
4.b odd 2 1 216.2.v.b 192
8.b even 2 1 216.2.v.b 192
8.d odd 2 1 inner 864.2.bh.b 192
12.b even 2 1 648.2.v.b 192
24.h odd 2 1 648.2.v.b 192
27.f odd 18 1 inner 864.2.bh.b 192
108.j odd 18 1 648.2.v.b 192
108.l even 18 1 216.2.v.b 192
216.t even 18 1 648.2.v.b 192
216.v even 18 1 inner 864.2.bh.b 192
216.x odd 18 1 216.2.v.b 192

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
216.2.v.b 192 4.b odd 2 1
216.2.v.b 192 8.b even 2 1
216.2.v.b 192 108.l even 18 1
216.2.v.b 192 216.x odd 18 1
648.2.v.b 192 12.b even 2 1
648.2.v.b 192 24.h odd 2 1
648.2.v.b 192 108.j odd 18 1
648.2.v.b 192 216.t even 18 1
864.2.bh.b 192 1.a even 1 1 trivial
864.2.bh.b 192 8.d odd 2 1 inner
864.2.bh.b 192 27.f odd 18 1 inner
864.2.bh.b 192 216.v even 18 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{192} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(864, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database