# Properties

 Label 864.2.w.b Level 864 Weight 2 Character orbit 864.w 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.w (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.

 $$\operatorname{Tr}(f)(q) =$$ $$128q + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$128q + 8q^{10} + 32q^{16} - 32q^{22} + 64q^{40} + 64q^{46} + 40q^{52} + 64q^{55} + 64q^{58} + 32q^{61} + 96q^{64} - 64q^{67} - 48q^{70} - 32q^{76} - 32q^{79} + 40q^{82} + 40q^{88} - 48q^{91} + 72q^{94} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
107.1 −1.41338 0.0486157i 0 1.99527 + 0.137425i 0.817684 + 0.338696i 0 1.96438 1.96438i −2.81339 0.291234i 0 −1.13923 0.518457i
107.2 −1.40430 + 0.167138i 0 1.94413 0.469426i −3.49346 1.44704i 0 −2.54256 + 2.54256i −2.65169 + 0.984154i 0 5.14773 + 1.44819i
107.3 −1.35869 0.392392i 0 1.69206 + 1.06627i 2.69281 + 1.11540i 0 −1.39266 + 1.39266i −1.88058 2.11268i 0 −3.22102 2.57212i
107.4 −1.31895 + 0.510265i 0 1.47926 1.34603i 1.03442 + 0.428472i 0 −0.461188 + 0.461188i −1.26424 + 2.53016i 0 −1.58299 0.0373037i
107.5 −1.31642 0.516757i 0 1.46592 + 1.36054i −2.69312 1.11553i 0 0.0380585 0.0380585i −1.22670 2.54857i 0 2.96882 + 2.86019i
107.6 −1.25848 + 0.645153i 0 1.16756 1.62383i −2.13183 0.883034i 0 3.32511 3.32511i −0.421732 + 2.79681i 0 3.25256 0.264075i
107.7 −1.23132 + 0.695593i 0 1.03230 1.71300i 2.52121 + 1.04432i 0 −2.75999 + 2.75999i −0.0795431 + 2.82731i 0 −3.83084 + 0.467845i
107.8 −0.984674 1.01509i 0 −0.0608353 + 1.99907i 2.42976 + 1.00644i 0 3.62344 3.62344i 2.08915 1.90668i 0 −1.37089 3.45745i
107.9 −0.919792 1.07424i 0 −0.307965 + 1.97615i −1.92195 0.796096i 0 0.893559 0.893559i 2.40611 1.48682i 0 0.912596 + 2.79687i
107.10 −0.856765 + 1.12515i 0 −0.531906 1.92797i −2.84859 1.17993i 0 1.33794 1.33794i 2.62497 + 1.05335i 0 3.76817 2.19416i
107.11 −0.679670 1.24018i 0 −1.07610 + 1.68583i −0.301020 0.124687i 0 −2.88637 + 2.88637i 2.82212 + 0.188749i 0 0.0499604 + 0.458065i
107.12 −0.654624 + 1.25358i 0 −1.14293 1.64125i −0.0422264 0.0174907i 0 −1.54323 + 1.54323i 2.80563 0.358360i 0 0.0495684 0.0414843i
107.13 −0.535739 + 1.30881i 0 −1.42597 1.40236i −0.682524 0.282711i 0 −1.19295 + 1.19295i 2.59937 1.11502i 0 0.735670 0.741835i
107.14 −0.485421 1.32829i 0 −1.52873 + 1.28957i 3.81307 + 1.57942i 0 −1.13033 + 1.13033i 2.45500 + 1.40462i 0 0.246996 5.83156i
107.15 −0.0483362 + 1.41339i 0 −1.99533 0.136636i 3.11062 + 1.28846i 0 1.36902 1.36902i 0.289566 2.81357i 0 −1.97145 + 4.33423i
107.16 −0.0136252 1.41415i 0 −1.99963 + 0.0385360i −0.327253 0.135553i 0 1.35777 1.35777i 0.0817409 + 2.82725i 0 −0.187232 + 0.464631i
107.17 0.0136252 + 1.41415i 0 −1.99963 + 0.0385360i 0.327253 + 0.135553i 0 1.35777 1.35777i −0.0817409 2.82725i 0 −0.187232 + 0.464631i
107.18 0.0483362 1.41339i 0 −1.99533 0.136636i −3.11062 1.28846i 0 1.36902 1.36902i −0.289566 + 2.81357i 0 −1.97145 + 4.33423i
107.19 0.485421 + 1.32829i 0 −1.52873 + 1.28957i −3.81307 1.57942i 0 −1.13033 + 1.13033i −2.45500 1.40462i 0 0.246996 5.83156i
107.20 0.535739 1.30881i 0 −1.42597 1.40236i 0.682524 + 0.282711i 0 −1.19295 + 1.19295i −2.59937 + 1.11502i 0 0.735670 0.741835i
See next 80 embeddings (of 128 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 755.32 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.h odd 8 1 inner
96.o even 8 1 inner

## Twists

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

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