# Properties

 Label 1152.3.j.b Level $1152$ Weight $3$ Character orbit 1152.j Analytic conductor $31.390$ Analytic rank $0$ Dimension $32$ CM no Inner twists $4$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$31.3897264543$$ Analytic rank: $$0$$ Dimension: $$32$$ Relative dimension: $$16$$ over $$\Q(i)$$ Twist minimal: no (minimal twist has level 144) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$32q + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$32q + 64q^{19} + 128q^{43} - 224q^{49} - 64q^{61} - 64q^{67} + 512q^{79} - 320q^{85} - 192q^{91} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
161.1 0 0 0 −6.08688 6.08688i 0 9.40026i 0 0 0
161.2 0 0 0 −5.52702 5.52702i 0 7.79421i 0 0 0
161.3 0 0 0 −5.14405 5.14405i 0 7.48880i 0 0 0
161.4 0 0 0 −3.90343 3.90343i 0 0.778757i 0 0 0
161.5 0 0 0 −1.92848 1.92848i 0 2.43162i 0 0 0
161.6 0 0 0 −1.84570 1.84570i 0 0.226665i 0 0 0
161.7 0 0 0 −1.66372 1.66372i 0 13.3224i 0 0 0
161.8 0 0 0 −0.900179 0.900179i 0 7.66460i 0 0 0
161.9 0 0 0 0.900179 + 0.900179i 0 7.66460i 0 0 0
161.10 0 0 0 1.66372 + 1.66372i 0 13.3224i 0 0 0
161.11 0 0 0 1.84570 + 1.84570i 0 0.226665i 0 0 0
161.12 0 0 0 1.92848 + 1.92848i 0 2.43162i 0 0 0
161.13 0 0 0 3.90343 + 3.90343i 0 0.778757i 0 0 0
161.14 0 0 0 5.14405 + 5.14405i 0 7.48880i 0 0 0
161.15 0 0 0 5.52702 + 5.52702i 0 7.79421i 0 0 0
161.16 0 0 0 6.08688 + 6.08688i 0 9.40026i 0 0 0
737.1 0 0 0 −6.08688 + 6.08688i 0 9.40026i 0 0 0
737.2 0 0 0 −5.52702 + 5.52702i 0 7.79421i 0 0 0
737.3 0 0 0 −5.14405 + 5.14405i 0 7.48880i 0 0 0
737.4 0 0 0 −3.90343 + 3.90343i 0 0.778757i 0 0 0
See all 32 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 737.16 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
16.e even 4 1 inner
48.i odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1152.3.j.b 32
3.b odd 2 1 inner 1152.3.j.b 32
4.b odd 2 1 1152.3.j.a 32
8.b even 2 1 576.3.j.a 32
8.d odd 2 1 144.3.j.a 32
12.b even 2 1 1152.3.j.a 32
16.e even 4 1 576.3.j.a 32
16.e even 4 1 inner 1152.3.j.b 32
16.f odd 4 1 144.3.j.a 32
16.f odd 4 1 1152.3.j.a 32
24.f even 2 1 144.3.j.a 32
24.h odd 2 1 576.3.j.a 32
48.i odd 4 1 576.3.j.a 32
48.i odd 4 1 inner 1152.3.j.b 32
48.k even 4 1 144.3.j.a 32
48.k even 4 1 1152.3.j.a 32

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
144.3.j.a 32 8.d odd 2 1
144.3.j.a 32 16.f odd 4 1
144.3.j.a 32 24.f even 2 1
144.3.j.a 32 48.k even 4 1
576.3.j.a 32 8.b even 2 1
576.3.j.a 32 16.e even 4 1
576.3.j.a 32 24.h odd 2 1
576.3.j.a 32 48.i odd 4 1
1152.3.j.a 32 4.b odd 2 1
1152.3.j.a 32 12.b even 2 1
1152.3.j.a 32 16.f odd 4 1
1152.3.j.a 32 48.k even 4 1
1152.3.j.b 32 1.a even 1 1 trivial
1152.3.j.b 32 3.b odd 2 1 inner
1152.3.j.b 32 16.e even 4 1 inner
1152.3.j.b 32 48.i odd 4 1 inner