# Properties

 Label 288.2.v.d Level 288 Weight 2 Character orbit 288.v Analytic conductor 2.300 Analytic rank 0 Dimension 32 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$288 = 2^{5} \cdot 3^{2}$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 288.v (of order $$8$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.29969157821$$ Analytic rank: $$0$$ Dimension: $$32$$ Relative dimension: $$8$$ over $$\Q(\zeta_{8})$$ Coefficient ring index: multiple of None Twist minimal: no (minimal twist has level 96) 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.

 $$\operatorname{Tr}(f)(q) =$$ $$32q + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$32q - 8q^{10} + 32q^{14} - 8q^{16} + 32q^{20} - 8q^{22} + 16q^{23} - 40q^{26} + 40q^{28} - 48q^{31} - 40q^{32} + 40q^{34} + 48q^{35} - 40q^{38} + 8q^{40} - 16q^{43} - 8q^{44} - 32q^{46} - 24q^{50} - 8q^{52} + 32q^{53} + 32q^{55} - 56q^{56} - 32q^{58} - 64q^{59} - 32q^{61} - 48q^{62} + 24q^{64} + 16q^{67} + 8q^{68} - 24q^{70} - 64q^{71} + 32q^{74} - 56q^{76} + 32q^{77} + 56q^{80} - 40q^{82} + 64q^{86} - 48q^{88} - 48q^{91} + 80q^{92} - 32q^{94} + 80q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
37.1 −1.40823 + 0.129991i 0 1.96620 0.366115i 2.51374 + 1.04122i 0 2.01027 + 2.01027i −2.72127 + 0.771162i 0 −3.67526 1.13951i
37.2 −1.34827 0.426820i 0 1.63565 + 1.15093i 1.46213 + 0.605634i 0 −3.54889 3.54889i −1.71405 2.24989i 0 −1.71285 1.44062i
37.3 −0.884039 1.10385i 0 −0.436951 + 1.95168i −2.14986 0.890503i 0 −1.10001 1.10001i 2.54064 1.24304i 0 0.917585 + 3.16036i
37.4 −0.605567 + 1.27800i 0 −1.26658 1.54783i −1.60930 0.666593i 0 −0.589445 0.589445i 2.74513 0.681373i 0 1.82644 1.65302i
37.5 −0.333592 1.37431i 0 −1.77743 + 0.916914i 1.20409 + 0.498752i 0 2.59422 + 2.59422i 1.85306 + 2.13686i 0 0.283762 1.82117i
37.6 0.603367 + 1.27904i 0 −1.27190 + 1.54346i −3.68816 1.52768i 0 −1.63704 1.63704i −2.74158 0.695531i 0 −0.271341 5.63906i
37.7 1.26685 0.628571i 0 1.20980 1.59260i 3.09318 + 1.28124i 0 −1.73503 1.73503i 0.531562 2.77803i 0 4.72394 0.321153i
37.8 1.29526 0.567706i 0 1.35542 1.47066i −0.825824 0.342068i 0 1.17750 + 1.17750i 0.920723 2.67437i 0 −1.26385 + 0.0257578i
109.1 −1.40823 0.129991i 0 1.96620 + 0.366115i 2.51374 1.04122i 0 2.01027 2.01027i −2.72127 0.771162i 0 −3.67526 + 1.13951i
109.2 −1.34827 + 0.426820i 0 1.63565 1.15093i 1.46213 0.605634i 0 −3.54889 + 3.54889i −1.71405 + 2.24989i 0 −1.71285 + 1.44062i
109.3 −0.884039 + 1.10385i 0 −0.436951 1.95168i −2.14986 + 0.890503i 0 −1.10001 + 1.10001i 2.54064 + 1.24304i 0 0.917585 3.16036i
109.4 −0.605567 1.27800i 0 −1.26658 + 1.54783i −1.60930 + 0.666593i 0 −0.589445 + 0.589445i 2.74513 + 0.681373i 0 1.82644 + 1.65302i
109.5 −0.333592 + 1.37431i 0 −1.77743 0.916914i 1.20409 0.498752i 0 2.59422 2.59422i 1.85306 2.13686i 0 0.283762 + 1.82117i
109.6 0.603367 1.27904i 0 −1.27190 1.54346i −3.68816 + 1.52768i 0 −1.63704 + 1.63704i −2.74158 + 0.695531i 0 −0.271341 + 5.63906i
109.7 1.26685 + 0.628571i 0 1.20980 + 1.59260i 3.09318 1.28124i 0 −1.73503 + 1.73503i 0.531562 + 2.77803i 0 4.72394 + 0.321153i
109.8 1.29526 + 0.567706i 0 1.35542 + 1.47066i −0.825824 + 0.342068i 0 1.17750 1.17750i 0.920723 + 2.67437i 0 −1.26385 0.0257578i
181.1 −1.34416 0.439595i 0 1.61351 + 1.18177i −0.184062 + 0.444366i 0 −0.134531 0.134531i −1.64931 2.29777i 0 0.442749 0.516384i
181.2 −0.890433 1.09869i 0 −0.414259 + 1.95663i 0.00259461 0.00626394i 0 −2.41880 2.41880i 2.51860 1.28710i 0 −0.00919248 + 0.00272694i
181.3 −0.525864 + 1.31281i 0 −1.44693 1.38072i 0.155637 0.375742i 0 0.709092 + 0.709092i 2.57351 1.17348i 0 0.411433 + 0.401911i
181.4 0.126318 + 1.40856i 0 −1.96809 + 0.355853i −1.36206 + 3.28830i 0 −2.73097 2.73097i −0.749846 2.72722i 0 −4.80383 1.50317i
See all 32 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 253.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
32.g even 8 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 288.2.v.d 32
3.b odd 2 1 96.2.n.a 32
4.b odd 2 1 1152.2.v.c 32
12.b even 2 1 384.2.n.a 32
24.f even 2 1 768.2.n.b 32
24.h odd 2 1 768.2.n.a 32
32.g even 8 1 inner 288.2.v.d 32
32.h odd 8 1 1152.2.v.c 32
96.o even 8 1 384.2.n.a 32
96.o even 8 1 768.2.n.b 32
96.p odd 8 1 96.2.n.a 32
96.p odd 8 1 768.2.n.a 32

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
96.2.n.a 32 3.b odd 2 1
96.2.n.a 32 96.p odd 8 1
288.2.v.d 32 1.a even 1 1 trivial
288.2.v.d 32 32.g even 8 1 inner
384.2.n.a 32 12.b even 2 1
384.2.n.a 32 96.o even 8 1
768.2.n.a 32 24.h odd 2 1
768.2.n.a 32 96.p odd 8 1
768.2.n.b 32 24.f even 2 1
768.2.n.b 32 96.o even 8 1
1152.2.v.c 32 4.b odd 2 1
1152.2.v.c 32 32.h odd 8 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{32} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(288, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database