# Properties

 Label 1152.2.w.a Level $1152$ Weight $2$ Character orbit 1152.w Analytic conductor $9.199$ Analytic rank $0$ Dimension $32$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1152 = 2^{7} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1152.w (of order $$8$$, degree $$4$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$9.19876631285$$ Analytic rank: $$0$$ Dimension: $$32$$ Relative dimension: $$8$$ over $$\Q(\zeta_{8})$$ Twist minimal: no (minimal twist has level 288) 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.

 $$\operatorname{Tr}(f)(q) =$$ $$32 q + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$32 q - 8 q^{11} + 16 q^{29} + 24 q^{35} + 16 q^{53} + 32 q^{55} + 32 q^{59} + 32 q^{61} + 16 q^{67} + 16 q^{71} + 16 q^{77} + 32 q^{79} - 40 q^{83} + 48 q^{91} - 80 q^{95} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
143.1 0 0 0 −1.32268 + 3.19322i 0 2.32913 2.32913i 0 0 0
143.2 0 0 0 −1.11994 + 2.70378i 0 −1.57144 + 1.57144i 0 0 0
143.3 0 0 0 −0.366958 + 0.885915i 0 −1.21471 + 1.21471i 0 0 0
143.4 0 0 0 −0.352978 + 0.852163i 0 3.43393 3.43393i 0 0 0
143.5 0 0 0 −0.0963530 + 0.232617i 0 −0.617536 + 0.617536i 0 0 0
143.6 0 0 0 0.739921 1.78633i 0 −0.385417 + 0.385417i 0 0 0
143.7 0 0 0 1.12344 2.71222i 0 −3.03150 + 3.03150i 0 0 0
143.8 0 0 0 1.39555 3.36915i 0 1.05755 1.05755i 0 0 0
431.1 0 0 0 −4.01952 + 1.66494i 0 −2.72280 2.72280i 0 0 0
431.2 0 0 0 −1.78959 + 0.741273i 0 2.33709 + 2.33709i 0 0 0
431.3 0 0 0 −1.65625 + 0.686041i 0 0.456585 + 0.456585i 0 0 0
431.4 0 0 0 0.0505677 0.0209458i 0 −1.44150 1.44150i 0 0 0
431.5 0 0 0 0.0913223 0.0378270i 0 3.05457 + 3.05457i 0 0 0
431.6 0 0 0 1.64859 0.682869i 0 −2.51270 2.51270i 0 0 0
431.7 0 0 0 2.70076 1.11869i 0 1.06647 + 1.06647i 0 0 0
431.8 0 0 0 2.97412 1.23192i 0 −0.237717 0.237717i 0 0 0
719.1 0 0 0 −4.01952 1.66494i 0 −2.72280 + 2.72280i 0 0 0
719.2 0 0 0 −1.78959 0.741273i 0 2.33709 2.33709i 0 0 0
719.3 0 0 0 −1.65625 0.686041i 0 0.456585 0.456585i 0 0 0
719.4 0 0 0 0.0505677 + 0.0209458i 0 −1.44150 + 1.44150i 0 0 0
See all 32 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1007.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
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 1152.2.w.a 32
3.b odd 2 1 1152.2.w.b 32
4.b odd 2 1 288.2.w.b yes 32
12.b even 2 1 288.2.w.a 32
32.g even 8 1 288.2.w.a 32
32.h odd 8 1 1152.2.w.b 32
96.o even 8 1 inner 1152.2.w.a 32
96.p odd 8 1 288.2.w.b yes 32

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
288.2.w.a 32 12.b even 2 1
288.2.w.a 32 32.g even 8 1
288.2.w.b yes 32 4.b odd 2 1
288.2.w.b yes 32 96.p odd 8 1
1152.2.w.a 32 1.a even 1 1 trivial
1152.2.w.a 32 96.o even 8 1 inner
1152.2.w.b 32 3.b odd 2 1
1152.2.w.b 32 32.h odd 8 1