# Properties

 Label 96.3.h.a Level $96$ Weight $3$ Character orbit 96.h Self dual yes Analytic conductor $2.616$ Analytic rank $0$ Dimension $1$ CM discriminant -24 Inner twists $2$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: yes Analytic conductor: $$2.61581053786$$ 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

 $$f(q)$$ $$=$$ $$q - 3q^{3} + 2q^{5} + 10q^{7} + 9q^{9} + O(q^{10})$$ $$q - 3q^{3} + 2q^{5} + 10q^{7} + 9q^{9} + 10q^{11} - 6q^{15} - 30q^{21} - 21q^{25} - 27q^{27} + 50q^{29} - 38q^{31} - 30q^{33} + 20q^{35} + 18q^{45} + 51q^{49} - 94q^{53} + 20q^{55} + 10q^{59} + 90q^{63} + 50q^{73} + 63q^{75} + 100q^{77} + 58q^{79} + 81q^{81} - 134q^{83} - 150q^{87} + 114q^{93} - 190q^{97} + 90q^{99} + O(q^{100})$$

## 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.

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 −3.00000 0 2.00000 0 10.0000 0 9.00000 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 96.3.h.a 1
3.b odd 2 1 96.3.h.b 1
4.b odd 2 1 24.3.h.a 1
8.b even 2 1 96.3.h.b 1
8.d odd 2 1 24.3.h.b yes 1
12.b even 2 1 24.3.h.b yes 1
16.e even 4 2 768.3.e.c 2
16.f odd 4 2 768.3.e.d 2
24.f even 2 1 24.3.h.a 1
24.h odd 2 1 CM 96.3.h.a 1
48.i odd 4 2 768.3.e.c 2
48.k even 4 2 768.3.e.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.3.h.a 1 4.b odd 2 1
24.3.h.a 1 24.f even 2 1
24.3.h.b yes 1 8.d odd 2 1
24.3.h.b yes 1 12.b even 2 1
96.3.h.a 1 1.a even 1 1 trivial
96.3.h.a 1 24.h odd 2 1 CM
96.3.h.b 1 3.b odd 2 1
96.3.h.b 1 8.b even 2 1
768.3.e.c 2 16.e even 4 2
768.3.e.c 2 48.i odd 4 2
768.3.e.d 2 16.f odd 4 2
768.3.e.d 2 48.k even 4 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5} - 2$$ acting on $$S_{3}^{\mathrm{new}}(96, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$3 + T$$
$5$ $$-2 + T$$
$7$ $$-10 + T$$
$11$ $$-10 + T$$
$13$ $$T$$
$17$ $$T$$
$19$ $$T$$
$23$ $$T$$
$29$ $$-50 + T$$
$31$ $$38 + T$$
$37$ $$T$$
$41$ $$T$$
$43$ $$T$$
$47$ $$T$$
$53$ $$94 + T$$
$59$ $$-10 + T$$
$61$ $$T$$
$67$ $$T$$
$71$ $$T$$
$73$ $$-50 + T$$
$79$ $$-58 + T$$
$83$ $$134 + T$$
$89$ $$T$$
$97$ $$190 + T$$