# Properties

 Label 96.2.o.a Level 96 Weight 2 Character orbit 96.o Analytic conductor 0.767 Analytic rank 0 Dimension 56 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$96 = 2^{5} \cdot 3$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 96.o (of order $$8$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.766563859404$$ Analytic rank: $$0$$ Dimension: $$56$$ Relative dimension: $$14$$ over $$\Q(\zeta_{8})$$ Coefficient ring index: multiple of None Twist minimal: yes 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.

 $$\operatorname{Tr}(f)(q) =$$ $$56q - 4q^{3} - 8q^{4} - 4q^{6} - 8q^{7} - 4q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$56q - 4q^{3} - 8q^{4} - 4q^{6} - 8q^{7} - 4q^{9} - 16q^{10} - 4q^{12} - 8q^{13} - 8q^{15} - 40q^{16} - 4q^{18} - 8q^{19} - 4q^{21} - 24q^{22} + 16q^{24} - 8q^{25} - 28q^{27} - 8q^{28} + 44q^{30} - 8q^{33} - 24q^{34} + 48q^{36} - 8q^{37} - 28q^{39} + 24q^{40} + 56q^{42} - 8q^{43} - 4q^{45} + 24q^{46} + 48q^{48} - 16q^{51} + 48q^{52} + 40q^{54} + 24q^{55} - 4q^{57} + 96q^{58} + 8q^{60} - 40q^{61} + 40q^{64} - 28q^{66} + 56q^{67} - 4q^{69} + 40q^{70} - 64q^{72} - 8q^{73} + 16q^{75} + 48q^{76} - 92q^{78} + 16q^{79} - 48q^{82} - 136q^{84} - 48q^{85} + 52q^{87} - 48q^{88} - 136q^{90} + 40q^{91} + 8q^{93} - 64q^{94} - 104q^{96} - 16q^{97} + 60q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
11.1 −1.41060 0.101040i 1.72928 0.0979076i 1.97958 + 0.285053i −0.180206 0.0746437i −2.44922 0.0366175i −0.289055 + 0.289055i −2.76360 0.602112i 2.98083 0.338620i 0.246656 + 0.123500i
11.2 −1.12377 + 0.858568i −0.0380372 + 1.73163i 0.525721 1.92967i 2.18808 + 0.906333i −1.44398 1.97862i −1.93241 + 1.93241i 1.06596 + 2.61987i −2.99711 0.131733i −3.23705 + 0.860107i
11.3 −1.09490 0.895090i −1.68370 + 0.406387i 0.397627 + 1.96007i 2.81491 + 1.16597i 2.20724 + 1.06211i −0.543879 + 0.543879i 1.31908 2.50200i 2.66970 1.36847i −2.03840 3.79622i
11.4 −0.838298 + 1.13897i 0.602068 1.62404i −0.594512 1.90960i 0.378520 + 0.156788i 1.34503 + 2.04717i 2.01144 2.01144i 2.67335 + 0.923679i −2.27503 1.95557i −0.495890 + 0.299689i
11.5 −0.816393 1.15477i −0.266294 1.71146i −0.667005 + 1.88550i −3.14689 1.30348i −1.75895 + 1.70473i 0.663471 0.663471i 2.72186 0.769068i −2.85817 + 0.911503i 1.06387 + 4.69810i
11.6 −0.478581 1.33077i 1.18890 + 1.25957i −1.54192 + 1.27377i 1.06973 + 0.443098i 1.10722 2.18496i 2.37247 2.37247i 2.43303 + 1.44235i −0.173046 + 2.99501i 0.0777099 1.63563i
11.7 −0.310967 + 1.37960i −1.70463 0.306981i −1.80660 0.858022i −2.70066 1.11865i 0.953596 2.25625i −3.28204 + 3.28204i 1.74552 2.22557i 2.81152 + 1.04658i 2.38311 3.37797i
11.8 0.310967 1.37960i 0.988287 1.42242i −1.80660 0.858022i 2.70066 + 1.11865i −1.65505 1.80577i −3.28204 + 3.28204i −1.74552 + 2.22557i −1.04658 2.81152i 2.38311 3.37797i
11.9 0.478581 + 1.33077i 0.0499748 + 1.73133i −1.54192 + 1.27377i −1.06973 0.443098i −2.28009 + 0.895086i 2.37247 2.37247i −2.43303 1.44235i −2.99501 + 0.173046i 0.0777099 1.63563i
11.10 0.816393 + 1.15477i −1.02188 1.39848i −0.667005 + 1.88550i 3.14689 + 1.30348i 0.780671 2.32176i 0.663471 0.663471i −2.72186 + 0.769068i −0.911503 + 2.85817i 1.06387 + 4.69810i
11.11 0.838298 1.13897i −1.57410 0.722645i −0.594512 1.90960i −0.378520 0.156788i −2.14264 + 1.18706i 2.01144 2.01144i −2.67335 0.923679i 1.95557 + 2.27503i −0.495890 + 0.299689i
11.12 1.09490 + 0.895090i 1.47792 0.903198i 0.397627 + 1.96007i −2.81491 1.16597i 2.42662 + 0.333953i −0.543879 + 0.543879i −1.31908 + 2.50200i 1.36847 2.66970i −2.03840 3.79622i
11.13 1.12377 0.858568i 1.25135 + 1.19755i 0.525721 1.92967i −2.18808 0.906333i 2.43441 + 0.271409i −1.93241 + 1.93241i −1.06596 2.61987i 0.131733 + 2.99711i −3.23705 + 0.860107i
11.14 1.41060 + 0.101040i −1.29202 + 1.15356i 1.97958 + 0.285053i 0.180206 + 0.0746437i −1.93907 + 1.49666i −0.289055 + 0.289055i 2.76360 + 0.602112i 0.338620 2.98083i 0.246656 + 0.123500i
35.1 −1.41060 + 0.101040i 1.72928 + 0.0979076i 1.97958 0.285053i −0.180206 + 0.0746437i −2.44922 + 0.0366175i −0.289055 0.289055i −2.76360 + 0.602112i 2.98083 + 0.338620i 0.246656 0.123500i
35.2 −1.12377 0.858568i −0.0380372 1.73163i 0.525721 + 1.92967i 2.18808 0.906333i −1.44398 + 1.97862i −1.93241 1.93241i 1.06596 2.61987i −2.99711 + 0.131733i −3.23705 0.860107i
35.3 −1.09490 + 0.895090i −1.68370 0.406387i 0.397627 1.96007i 2.81491 1.16597i 2.20724 1.06211i −0.543879 0.543879i 1.31908 + 2.50200i 2.66970 + 1.36847i −2.03840 + 3.79622i
35.4 −0.838298 1.13897i 0.602068 + 1.62404i −0.594512 + 1.90960i 0.378520 0.156788i 1.34503 2.04717i 2.01144 + 2.01144i 2.67335 0.923679i −2.27503 + 1.95557i −0.495890 0.299689i
35.5 −0.816393 + 1.15477i −0.266294 + 1.71146i −0.667005 1.88550i −3.14689 + 1.30348i −1.75895 1.70473i 0.663471 + 0.663471i 2.72186 + 0.769068i −2.85817 0.911503i 1.06387 4.69810i
35.6 −0.478581 + 1.33077i 1.18890 1.25957i −1.54192 1.27377i 1.06973 0.443098i 1.10722 + 2.18496i 2.37247 + 2.37247i 2.43303 1.44235i −0.173046 2.99501i 0.0777099 + 1.63563i
See all 56 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 83.14 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 96.2.o.a 56
3.b odd 2 1 inner 96.2.o.a 56
4.b odd 2 1 384.2.o.a 56
8.b even 2 1 768.2.o.b 56
8.d odd 2 1 768.2.o.a 56
12.b even 2 1 384.2.o.a 56
24.f even 2 1 768.2.o.a 56
24.h odd 2 1 768.2.o.b 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 inner 96.2.o.a 56
32.h odd 8 1 768.2.o.b 56
96.o even 8 1 inner 96.2.o.a 56
96.o even 8 1 768.2.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 1.a even 1 1 trivial
96.2.o.a 56 3.b odd 2 1 inner
96.2.o.a 56 32.h odd 8 1 inner
96.2.o.a 56 96.o even 8 1 inner
384.2.o.a 56 4.b odd 2 1
384.2.o.a 56 12.b 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 8.d odd 2 1
768.2.o.a 56 24.f 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 8.b even 2 1
768.2.o.b 56 24.h odd 2 1
768.2.o.b 56 32.h odd 8 1
768.2.o.b 56 96.o even 8 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(96, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database