# Properties

 Label 384.2.n.a Level $384$ Weight $2$ Character orbit 384.n Analytic conductor $3.066$ Analytic rank $0$ Dimension $32$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$384 = 2^{7} \cdot 3$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 384.n (of order $$8$$, degree $$4$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.06625543762$$ Analytic rank: $$0$$ Dimension: $$32$$ Relative dimension: $$8$$ over $$\Q(\zeta_{8})$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$32q + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$32q + 16q^{23} + 48q^{31} + 48q^{35} + 16q^{43} - 16q^{51} - 32q^{53} - 32q^{55} - 64q^{59} - 32q^{61} - 16q^{63} - 16q^{67} - 32q^{69} - 64q^{71} - 32q^{75} - 32q^{77} + 48q^{91} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1 0 −0.923880 + 0.382683i 0 −1.35803 + 3.27858i 0 −2.48546 2.48546i 0 0.707107 0.707107i 0
49.2 0 −0.923880 + 0.382683i 0 −0.155637 + 0.375742i 0 −0.709092 0.709092i 0 0.707107 0.707107i 0
49.3 0 −0.923880 + 0.382683i 0 −0.00259461 + 0.00626394i 0 2.41880 + 2.41880i 0 0.707107 0.707107i 0
49.4 0 −0.923880 + 0.382683i 0 0.750897 1.81283i 0 −0.638460 0.638460i 0 0.707107 0.707107i 0
49.5 0 0.923880 0.382683i 0 −1.48656 + 3.58888i 0 −1.03821 1.03821i 0 0.707107 0.707107i 0
49.6 0 0.923880 0.382683i 0 0.184062 0.444366i 0 0.134531 + 0.134531i 0 0.707107 0.707107i 0
49.7 0 0.923880 0.382683i 0 0.705805 1.70396i 0 −3.24150 3.24150i 0 0.707107 0.707107i 0
49.8 0 0.923880 0.382683i 0 1.36206 3.28830i 0 2.73097 + 2.73097i 0 0.707107 0.707107i 0
145.1 0 −0.382683 + 0.923880i 0 −1.46213 + 0.605634i 0 3.54889 3.54889i 0 −0.707107 0.707107i 0
145.2 0 −0.382683 + 0.923880i 0 −1.20409 + 0.498752i 0 −2.59422 + 2.59422i 0 −0.707107 0.707107i 0
145.3 0 −0.382683 + 0.923880i 0 0.825824 0.342068i 0 −1.17750 + 1.17750i 0 −0.707107 0.707107i 0
145.4 0 −0.382683 + 0.923880i 0 3.68816 1.52768i 0 1.63704 1.63704i 0 −0.707107 0.707107i 0
145.5 0 0.382683 0.923880i 0 −3.09318 + 1.28124i 0 1.73503 1.73503i 0 −0.707107 0.707107i 0
145.6 0 0.382683 0.923880i 0 −2.51374 + 1.04122i 0 −2.01027 + 2.01027i 0 −0.707107 0.707107i 0
145.7 0 0.382683 0.923880i 0 1.60930 0.666593i 0 0.589445 0.589445i 0 −0.707107 0.707107i 0
145.8 0 0.382683 0.923880i 0 2.14986 0.890503i 0 1.10001 1.10001i 0 −0.707107 0.707107i 0
241.1 0 −0.382683 0.923880i 0 −1.46213 0.605634i 0 3.54889 + 3.54889i 0 −0.707107 + 0.707107i 0
241.2 0 −0.382683 0.923880i 0 −1.20409 0.498752i 0 −2.59422 2.59422i 0 −0.707107 + 0.707107i 0
241.3 0 −0.382683 0.923880i 0 0.825824 + 0.342068i 0 −1.17750 1.17750i 0 −0.707107 + 0.707107i 0
241.4 0 −0.382683 0.923880i 0 3.68816 + 1.52768i 0 1.63704 + 1.63704i 0 −0.707107 + 0.707107i 0
See all 32 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 337.8 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 384.2.n.a 32
3.b odd 2 1 1152.2.v.c 32
4.b odd 2 1 96.2.n.a 32
8.b even 2 1 768.2.n.b 32
8.d odd 2 1 768.2.n.a 32
12.b even 2 1 288.2.v.d 32
32.g even 8 1 inner 384.2.n.a 32
32.g even 8 1 768.2.n.b 32
32.h odd 8 1 96.2.n.a 32
32.h odd 8 1 768.2.n.a 32
96.o even 8 1 288.2.v.d 32
96.p odd 8 1 1152.2.v.c 32

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

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(384, [\chi])$$.