# Properties

 Label 96.9.h.a Level $96$ Weight $9$ Character orbit 96.h Self dual yes Analytic conductor $39.108$ Analytic rank $0$ Dimension $1$ CM discriminant -24 Inner twists $2$

# 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: yes Analytic conductor: $$39.1083465659$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 24) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q - 81 q^{3} + 866 q^{5} + 4798 q^{7} + 6561 q^{9}+O(q^{10})$$ q - 81 * q^3 + 866 * q^5 + 4798 * q^7 + 6561 * q^9 $$q - 81 q^{3} + 866 q^{5} + 4798 q^{7} + 6561 q^{9} + 9118 q^{11} - 70146 q^{15} - 388638 q^{21} + 359331 q^{25} - 531441 q^{27} - 745438 q^{29} + 1618558 q^{31} - 738558 q^{33} + 4155068 q^{35} + 5681826 q^{45} + 17256003 q^{49} - 5425438 q^{53} + 7896188 q^{55} - 22852322 q^{59} + 31479678 q^{63} + 9756482 q^{73} - 29105811 q^{75} + 43748164 q^{77} - 5237762 q^{79} + 43046721 q^{81} + 77460958 q^{83} + 60380478 q^{87} - 131103198 q^{93} + 121608962 q^{97} + 59823198 q^{99}+O(q^{100})$$ q - 81 * q^3 + 866 * q^5 + 4798 * q^7 + 6561 * q^9 + 9118 * q^11 - 70146 * q^15 - 388638 * q^21 + 359331 * q^25 - 531441 * q^27 - 745438 * q^29 + 1618558 * q^31 - 738558 * q^33 + 4155068 * q^35 + 5681826 * q^45 + 17256003 * q^49 - 5425438 * q^53 + 7896188 * q^55 - 22852322 * q^59 + 31479678 * q^63 + 9756482 * q^73 - 29105811 * q^75 + 43748164 * q^77 - 5237762 * q^79 + 43046721 * q^81 + 77460958 * q^83 + 60380478 * q^87 - 131103198 * q^93 + 121608962 * q^97 + 59823198 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/96\mathbb{Z}\right)^\times$$.

 $$n$$ $$31$$ $$37$$ $$65$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
17.1
 0
0 −81.0000 0 866.000 0 4798.00 0 6561.00 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
24.h odd 2 1 CM by $$\Q(\sqrt{-6})$$

## Twists

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

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5} - 866$$ acting on $$S_{9}^{\mathrm{new}}(96, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 81$$
$5$ $$T - 866$$
$7$ $$T - 4798$$
$11$ $$T - 9118$$
$13$ $$T$$
$17$ $$T$$
$19$ $$T$$
$23$ $$T$$
$29$ $$T + 745438$$
$31$ $$T - 1618558$$
$37$ $$T$$
$41$ $$T$$
$43$ $$T$$
$47$ $$T$$
$53$ $$T + 5425438$$
$59$ $$T + 22852322$$
$61$ $$T$$
$67$ $$T$$
$71$ $$T$$
$73$ $$T - 9756482$$
$79$ $$T + 5237762$$
$83$ $$T - 77460958$$
$89$ $$T$$
$97$ $$T - 121608962$$