# Properties

 Label 96.9.h.c Level $96$ Weight $9$ Character orbit 96.h Analytic conductor $39.108$ Analytic rank $0$ Dimension $28$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [96,9,Mod(17,96)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(96, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 1]))

N = Newforms(chi, 9, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("96.17");

S:= CuspForms(chi, 9);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$96 = 2^{5} \cdot 3$$ Weight: $$k$$ $$=$$ $$9$$ Character orbit: $$[\chi]$$ $$=$$ 96.h (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$39.1083465659$$ Analytic rank: $$0$$ Dimension: $$28$$ Twist minimal: no (minimal twist has level 24) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$28 q - 9592 q^{7} - 13124 q^{9}+O(q^{10})$$ 28 * q - 9592 * q^7 - 13124 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$28 q - 9592 q^{7} - 13124 q^{9} + 153416 q^{15} + 1000084 q^{25} - 2991608 q^{31} + 637656 q^{33} - 3838144 q^{39} - 25755372 q^{49} + 8933520 q^{55} + 3542240 q^{57} - 10817272 q^{63} + 11921592 q^{73} - 41160184 q^{79} - 78940132 q^{81} - 63134136 q^{87} - 194894984 q^{97}+O(q^{100})$$ 28 * q - 9592 * q^7 - 13124 * q^9 + 153416 * q^15 + 1000084 * q^25 - 2991608 * q^31 + 637656 * q^33 - 3838144 * q^39 - 25755372 * q^49 + 8933520 * q^55 + 3542240 * q^57 - 10817272 * q^63 + 11921592 * q^73 - 41160184 * q^79 - 78940132 * q^81 - 63134136 * q^87 - 194894984 * q^97

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
17.1 0 −80.2470 11.0190i 0 414.987 0 −1996.89 0 6318.16 + 1768.49i 0
17.2 0 −80.2470 + 11.0190i 0 414.987 0 −1996.89 0 6318.16 1768.49i 0
17.3 0 −71.4947 38.0725i 0 −867.471 0 1776.55 0 3661.97 + 5443.96i 0
17.4 0 −71.4947 + 38.0725i 0 −867.471 0 1776.55 0 3661.97 5443.96i 0
17.5 0 −67.2364 45.1693i 0 −629.567 0 123.057 0 2480.47 + 6074.04i 0
17.6 0 −67.2364 + 45.1693i 0 −629.567 0 123.057 0 2480.47 6074.04i 0
17.7 0 −49.3241 64.2506i 0 530.205 0 1274.95 0 −1695.27 + 6338.20i 0
17.8 0 −49.3241 + 64.2506i 0 530.205 0 1274.95 0 −1695.27 6338.20i 0
17.9 0 −48.8926 64.5795i 0 −274.212 0 −3670.87 0 −1780.03 + 6314.92i 0
17.10 0 −48.8926 + 64.5795i 0 −274.212 0 −3670.87 0 −1780.03 6314.92i 0
17.11 0 −16.8860 79.2203i 0 1141.88 0 −2365.96 0 −5990.73 + 2675.43i 0
17.12 0 −16.8860 + 79.2203i 0 1141.88 0 −2365.96 0 −5990.73 2675.43i 0
17.13 0 −11.9463 80.1142i 0 55.8511 0 2461.17 0 −6275.57 + 1914.14i 0
17.14 0 −11.9463 + 80.1142i 0 55.8511 0 2461.17 0 −6275.57 1914.14i 0
17.15 0 11.9463 80.1142i 0 −55.8511 0 2461.17 0 −6275.57 1914.14i 0
17.16 0 11.9463 + 80.1142i 0 −55.8511 0 2461.17 0 −6275.57 + 1914.14i 0
17.17 0 16.8860 79.2203i 0 −1141.88 0 −2365.96 0 −5990.73 2675.43i 0
17.18 0 16.8860 + 79.2203i 0 −1141.88 0 −2365.96 0 −5990.73 + 2675.43i 0
17.19 0 48.8926 64.5795i 0 274.212 0 −3670.87 0 −1780.03 6314.92i 0
17.20 0 48.8926 + 64.5795i 0 274.212 0 −3670.87 0 −1780.03 + 6314.92i 0
See all 28 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 17.28 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
8.b even 2 1 inner
24.h odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 96.9.h.c 28
3.b odd 2 1 inner 96.9.h.c 28
4.b odd 2 1 24.9.h.c 28
8.b even 2 1 inner 96.9.h.c 28
8.d odd 2 1 24.9.h.c 28
12.b even 2 1 24.9.h.c 28
24.f even 2 1 24.9.h.c 28
24.h odd 2 1 inner 96.9.h.c 28

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.9.h.c 28 4.b odd 2 1
24.9.h.c 28 8.d odd 2 1
24.9.h.c 28 12.b even 2 1
24.9.h.c 28 24.f even 2 1
96.9.h.c 28 1.a even 1 1 trivial
96.9.h.c 28 3.b odd 2 1 inner
96.9.h.c 28 8.b even 2 1 inner
96.9.h.c 28 24.h odd 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{14} - 2984396 T_{5}^{12} + 3184387077168 T_{5}^{10} + \cdots - 44\!\cdots\!00$$ acting on $$S_{9}^{\mathrm{new}}(96, [\chi])$$.