Properties

 Label 768.2.n.b Level $768$ Weight $2$ Character orbit 768.n Analytic conductor $6.133$ Analytic rank $0$ Dimension $32$ CM no Inner twists $2$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$768 = 2^{8} \cdot 3$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 768.n (of order $$8$$, degree $$4$$, not minimal)

Newform invariants

 Self dual: no Analytic conductor: $$6.13251087523$$ 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
97.1 0 −0.923880 + 0.382683i 0 −1.36206 + 3.28830i 0 2.73097 + 2.73097i 0 0.707107 0.707107i 0
97.2 0 −0.923880 + 0.382683i 0 −0.705805 + 1.70396i 0 −3.24150 3.24150i 0 0.707107 0.707107i 0
97.3 0 −0.923880 + 0.382683i 0 −0.184062 + 0.444366i 0 0.134531 + 0.134531i 0 0.707107 0.707107i 0
97.4 0 −0.923880 + 0.382683i 0 1.48656 3.58888i 0 −1.03821 1.03821i 0 0.707107 0.707107i 0
97.5 0 0.923880 0.382683i 0 −0.750897 + 1.81283i 0 −0.638460 0.638460i 0 0.707107 0.707107i 0
97.6 0 0.923880 0.382683i 0 0.00259461 0.00626394i 0 2.41880 + 2.41880i 0 0.707107 0.707107i 0
97.7 0 0.923880 0.382683i 0 0.155637 0.375742i 0 −0.709092 0.709092i 0 0.707107 0.707107i 0
97.8 0 0.923880 0.382683i 0 1.35803 3.27858i 0 −2.48546 2.48546i 0 0.707107 0.707107i 0
289.1 0 −0.382683 + 0.923880i 0 −2.14986 + 0.890503i 0 1.10001 1.10001i 0 −0.707107 0.707107i 0
289.2 0 −0.382683 + 0.923880i 0 −1.60930 + 0.666593i 0 0.589445 0.589445i 0 −0.707107 0.707107i 0
289.3 0 −0.382683 + 0.923880i 0 2.51374 1.04122i 0 −2.01027 + 2.01027i 0 −0.707107 0.707107i 0
289.4 0 −0.382683 + 0.923880i 0 3.09318 1.28124i 0 1.73503 1.73503i 0 −0.707107 0.707107i 0
289.5 0 0.382683 0.923880i 0 −3.68816 + 1.52768i 0 1.63704 1.63704i 0 −0.707107 0.707107i 0
289.6 0 0.382683 0.923880i 0 −0.825824 + 0.342068i 0 −1.17750 + 1.17750i 0 −0.707107 0.707107i 0
289.7 0 0.382683 0.923880i 0 1.20409 0.498752i 0 −2.59422 + 2.59422i 0 −0.707107 0.707107i 0
289.8 0 0.382683 0.923880i 0 1.46213 0.605634i 0 3.54889 3.54889i 0 −0.707107 0.707107i 0
481.1 0 −0.382683 0.923880i 0 −2.14986 0.890503i 0 1.10001 + 1.10001i 0 −0.707107 + 0.707107i 0
481.2 0 −0.382683 0.923880i 0 −1.60930 0.666593i 0 0.589445 + 0.589445i 0 −0.707107 + 0.707107i 0
481.3 0 −0.382683 0.923880i 0 2.51374 + 1.04122i 0 −2.01027 2.01027i 0 −0.707107 + 0.707107i 0
481.4 0 −0.382683 0.923880i 0 3.09318 + 1.28124i 0 1.73503 + 1.73503i 0 −0.707107 + 0.707107i 0
See all 32 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 673.8 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 768.2.n.b 32
4.b odd 2 1 768.2.n.a 32
8.b even 2 1 384.2.n.a 32
8.d odd 2 1 96.2.n.a 32
24.f even 2 1 288.2.v.d 32
24.h odd 2 1 1152.2.v.c 32
32.g even 8 1 384.2.n.a 32
32.g even 8 1 inner 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 8.d odd 2 1
96.2.n.a 32 32.h odd 8 1
288.2.v.d 32 24.f even 2 1
288.2.v.d 32 96.o even 8 1
384.2.n.a 32 8.b even 2 1
384.2.n.a 32 32.g even 8 1
768.2.n.a 32 4.b odd 2 1
768.2.n.a 32 32.h odd 8 1
768.2.n.b 32 1.a even 1 1 trivial
768.2.n.b 32 32.g even 8 1 inner
1152.2.v.c 32 24.h odd 2 1
1152.2.v.c 32 96.p odd 8 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{32} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(768, [\chi])$$.