# Properties

 Label 864.2.v.a Level 864 Weight 2 Character orbit 864.v Analytic conductor 6.899 Analytic rank 0 Dimension 128 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$864 = 2^{5} \cdot 3^{3}$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 864.v (of order $$8$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.89907473464$$ Analytic rank: $$0$$ Dimension: $$128$$ Relative dimension: $$32$$ over $$\Q(\zeta_{8})$$ Coefficient ring index: multiple of None Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$128q + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$128q - 8q^{10} - 32q^{16} + 32q^{22} + 64q^{40} + 64q^{46} + 88q^{52} - 64q^{55} + 64q^{58} - 32q^{61} - 96q^{64} + 64q^{67} + 48q^{70} + 32q^{76} + 40q^{82} + 40q^{88} - 48q^{91} + 24q^{94} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
109.1 −1.39805 0.213220i 0 1.90907 + 0.596184i 0.0181233 0.00750691i 0 −1.64798 + 1.64798i −2.54186 1.24055i 0 −0.0269378 + 0.00663076i
109.2 −1.39716 + 0.218946i 0 1.90413 0.611807i −2.13352 + 0.883735i 0 0.426205 0.426205i −2.52642 + 1.27169i 0 2.78739 1.70185i
109.3 −1.38295 + 0.295740i 0 1.82508 0.817983i 3.70533 1.53480i 0 1.70420 1.70420i −2.28207 + 1.67097i 0 −4.67037 + 3.21836i
109.4 −1.33208 0.474937i 0 1.54887 + 1.26531i −0.0485815 + 0.0201231i 0 2.80669 2.80669i −1.46228 2.42110i 0 0.0742716 0.00373246i
109.5 −1.29834 0.560644i 0 1.37136 + 1.45581i 1.49059 0.617423i 0 −3.02902 + 3.02902i −0.964291 2.65897i 0 −2.28144 0.0340682i
109.6 −1.18499 + 0.771885i 0 0.808388 1.82935i −0.796479 + 0.329912i 0 3.41414 3.41414i 0.454114 + 2.79173i 0 0.689163 1.00573i
109.7 −1.17722 + 0.783679i 0 0.771696 1.84512i −3.84162 + 1.59125i 0 −2.86540 + 2.86540i 0.537529 + 2.77688i 0 3.27540 4.88384i
109.8 −1.01339 0.986432i 0 0.0539048 + 1.99927i −2.55155 + 1.05689i 0 1.06497 1.06497i 1.91752 2.07921i 0 3.62825 + 1.44589i
109.9 −0.957543 1.04073i 0 −0.166221 + 1.99308i 3.57149 1.47936i 0 1.49427 1.49427i 2.23341 1.73547i 0 −4.95946 2.30039i
109.10 −0.934820 + 1.06118i 0 −0.252222 1.98403i 0.930451 0.385405i 0 −0.938500 + 0.938500i 2.34121 + 1.58706i 0 −0.460818 + 1.34766i
109.11 −0.791979 + 1.17165i 0 −0.745540 1.85585i 2.17386 0.900442i 0 −1.13840 + 1.13840i 2.76486 + 0.596278i 0 −0.666644 + 3.26014i
109.12 −0.694736 1.23180i 0 −1.03468 + 1.71156i −1.00178 + 0.414949i 0 −1.78507 + 1.78507i 2.82714 + 0.0854454i 0 1.20711 + 0.945711i
109.13 −0.567462 1.29537i 0 −1.35597 + 1.47015i 2.16011 0.894745i 0 1.46314 1.46314i 2.67385 + 0.922239i 0 −2.38480 2.29041i
109.14 −0.139754 + 1.40729i 0 −1.96094 0.393350i −0.345772 + 0.143223i 0 0.299071 0.299071i 0.827608 2.70464i 0 −0.153234 0.506617i
109.15 −0.114947 + 1.40953i 0 −1.97357 0.324044i −2.86147 + 1.18526i 0 −2.81522 + 2.81522i 0.683608 2.74457i 0 −1.34175 4.16959i
109.16 −0.0434074 + 1.41355i 0 −1.99623 0.122717i 2.68483 1.11209i 0 1.54691 1.54691i 0.260117 2.81644i 0 1.45545 + 3.84340i
109.17 0.0434074 1.41355i 0 −1.99623 0.122717i −2.68483 + 1.11209i 0 1.54691 1.54691i −0.260117 + 2.81644i 0 1.45545 + 3.84340i
109.18 0.114947 1.40953i 0 −1.97357 0.324044i 2.86147 1.18526i 0 −2.81522 + 2.81522i −0.683608 + 2.74457i 0 −1.34175 4.16959i
109.19 0.139754 1.40729i 0 −1.96094 0.393350i 0.345772 0.143223i 0 0.299071 0.299071i −0.827608 + 2.70464i 0 −0.153234 0.506617i
109.20 0.567462 + 1.29537i 0 −1.35597 + 1.47015i −2.16011 + 0.894745i 0 1.46314 1.46314i −2.67385 0.922239i 0 −2.38480 2.29041i
See next 80 embeddings (of 128 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 757.32 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
32.g even 8 1 inner
96.p odd 8 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 864.2.v.a 128
3.b odd 2 1 inner 864.2.v.a 128
32.g even 8 1 inner 864.2.v.a 128
96.p odd 8 1 inner 864.2.v.a 128

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
864.2.v.a 128 1.a even 1 1 trivial
864.2.v.a 128 3.b odd 2 1 inner
864.2.v.a 128 32.g even 8 1 inner
864.2.v.a 128 96.p odd 8 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{128} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(864, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database