# Properties

 Label 768.2.o.b Level 768 Weight 2 Character orbit 768.o Analytic conductor 6.133 Analytic rank 0 Dimension 56 CM no Inner twists 4

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$6.13251087523$$ Analytic rank: $$0$$ Dimension: $$56$$ Relative dimension: $$14$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$56q + 4q^{3} - 8q^{7} - 4q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$56q + 4q^{3} - 8q^{7} - 4q^{9} + 8q^{13} - 8q^{15} + 8q^{19} + 4q^{21} - 8q^{25} + 28q^{27} - 8q^{33} + 8q^{37} - 28q^{39} + 8q^{43} + 4q^{45} + 16q^{51} + 24q^{55} - 4q^{57} + 40q^{61} - 56q^{67} + 4q^{69} - 8q^{73} - 16q^{75} + 16q^{79} + 48q^{85} + 52q^{87} - 40q^{91} - 8q^{93} - 16q^{97} - 60q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
95.1 0 −1.72928 0.0979076i 0 0.180206 0.0746437i 0 −0.289055 0.289055i 0 2.98083 + 0.338620i 0
95.2 0 −1.47792 0.903198i 0 2.81491 1.16597i 0 −0.543879 0.543879i 0 1.36847 + 2.66970i 0
95.3 0 −1.25135 + 1.19755i 0 2.18808 0.906333i 0 −1.93241 1.93241i 0 0.131733 2.99711i 0
95.4 0 −1.18890 + 1.25957i 0 −1.06973 + 0.443098i 0 2.37247 + 2.37247i 0 −0.173046 2.99501i 0
95.5 0 −0.988287 1.42242i 0 −2.70066 + 1.11865i 0 −3.28204 3.28204i 0 −1.04658 + 2.81152i 0
95.6 0 −0.602068 1.62404i 0 −0.378520 + 0.156788i 0 2.01144 + 2.01144i 0 −2.27503 + 1.95557i 0
95.7 0 −0.0499748 + 1.73133i 0 1.06973 0.443098i 0 2.37247 + 2.37247i 0 −2.99501 0.173046i 0
95.8 0 0.0380372 + 1.73163i 0 −2.18808 + 0.906333i 0 −1.93241 1.93241i 0 −2.99711 + 0.131733i 0
95.9 0 0.266294 1.71146i 0 3.14689 1.30348i 0 0.663471 + 0.663471i 0 −2.85817 0.911503i 0
95.10 0 1.02188 1.39848i 0 −3.14689 + 1.30348i 0 0.663471 + 0.663471i 0 −0.911503 2.85817i 0
95.11 0 1.29202 + 1.15356i 0 −0.180206 + 0.0746437i 0 −0.289055 0.289055i 0 0.338620 + 2.98083i 0
95.12 0 1.57410 0.722645i 0 0.378520 0.156788i 0 2.01144 + 2.01144i 0 1.95557 2.27503i 0
95.13 0 1.68370 + 0.406387i 0 −2.81491 + 1.16597i 0 −0.543879 0.543879i 0 2.66970 + 1.36847i 0
95.14 0 1.70463 0.306981i 0 2.70066 1.11865i 0 −3.28204 3.28204i 0 2.81152 1.04658i 0
287.1 0 −1.69392 + 0.361451i 0 0.348970 0.842489i 0 −0.471834 + 0.471834i 0 2.73871 1.22453i 0
287.2 0 −1.45336 + 0.942196i 0 −0.348970 + 0.842489i 0 −0.471834 + 0.471834i 0 1.22453 2.73871i 0
287.3 0 −1.44423 0.956131i 0 −1.56013 + 3.76650i 0 −0.838552 + 0.838552i 0 1.17163 + 2.76175i 0
287.4 0 −1.14895 1.29612i 0 0.970873 2.34389i 0 −1.60572 + 1.60572i 0 −0.359830 + 2.97834i 0
287.5 0 −0.684042 1.59125i 0 0.296199 0.715088i 0 2.77714 2.77714i 0 −2.06417 + 2.17697i 0
287.6 0 −0.345141 + 1.69731i 0 1.56013 3.76650i 0 −0.838552 + 0.838552i 0 −2.76175 1.17163i 0
See all 56 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 671.14 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
32.h odd 8 1 inner
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 768.2.o.b 56
3.b odd 2 1 inner 768.2.o.b 56
4.b odd 2 1 768.2.o.a 56
8.b even 2 1 96.2.o.a 56
8.d odd 2 1 384.2.o.a 56
12.b even 2 1 768.2.o.a 56
24.f even 2 1 384.2.o.a 56
24.h odd 2 1 96.2.o.a 56
32.g even 8 1 384.2.o.a 56
32.g even 8 1 768.2.o.a 56
32.h odd 8 1 96.2.o.a 56
32.h odd 8 1 inner 768.2.o.b 56
96.o even 8 1 96.2.o.a 56
96.o even 8 1 inner 768.2.o.b 56
96.p odd 8 1 384.2.o.a 56
96.p odd 8 1 768.2.o.a 56

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
96.2.o.a 56 8.b even 2 1
96.2.o.a 56 24.h odd 2 1
96.2.o.a 56 32.h odd 8 1
96.2.o.a 56 96.o even 8 1
384.2.o.a 56 8.d odd 2 1
384.2.o.a 56 24.f even 2 1
384.2.o.a 56 32.g even 8 1
384.2.o.a 56 96.p odd 8 1
768.2.o.a 56 4.b odd 2 1
768.2.o.a 56 12.b even 2 1
768.2.o.a 56 32.g even 8 1
768.2.o.a 56 96.p odd 8 1
768.2.o.b 56 1.a even 1 1 trivial
768.2.o.b 56 3.b odd 2 1 inner
768.2.o.b 56 32.h odd 8 1 inner
768.2.o.b 56 96.o even 8 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{28} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(768, [\chi])$$.

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database