# Properties

 Label 192.2.f.a Level $192$ Weight $2$ Character orbit 192.f Analytic conductor $1.533$ Analytic rank $0$ Dimension $4$ CM discriminant -24 Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$192 = 2^{6} \cdot 3$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 192.f (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.53312771881$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{3} - \beta_{3} q^{5} + \beta_1 q^{7} - 3 q^{9}+O(q^{10})$$ q + b2 * q^3 - b3 * q^5 + b1 * q^7 - 3 * q^9 $$q + \beta_{2} q^{3} - \beta_{3} q^{5} + \beta_1 q^{7} - 3 q^{9} + 2 \beta_{2} q^{11} - 3 \beta_1 q^{15} - \beta_{3} q^{21} + 7 q^{25} - 3 \beta_{2} q^{27} + 3 \beta_{3} q^{29} + 5 \beta_1 q^{31} - 6 q^{33} - 4 \beta_{2} q^{35} + 3 \beta_{3} q^{45} + 3 q^{49} - \beta_{3} q^{53} - 6 \beta_1 q^{55} - 6 \beta_{2} q^{59} - 3 \beta_1 q^{63} - 14 q^{73} + 7 \beta_{2} q^{75} - 2 \beta_{3} q^{77} + 5 \beta_1 q^{79} + 9 q^{81} + 10 \beta_{2} q^{83} + 9 \beta_1 q^{87} - 5 \beta_{3} q^{93} + 2 q^{97} - 6 \beta_{2} q^{99}+O(q^{100})$$ q + b2 * q^3 - b3 * q^5 + b1 * q^7 - 3 * q^9 + 2*b2 * q^11 - 3*b1 * q^15 - b3 * q^21 + 7 * q^25 - 3*b2 * q^27 + 3*b3 * q^29 + 5*b1 * q^31 - 6 * q^33 - 4*b2 * q^35 + 3*b3 * q^45 + 3 * q^49 - b3 * q^53 - 6*b1 * q^55 - 6*b2 * q^59 - 3*b1 * q^63 - 14 * q^73 + 7*b2 * q^75 - 2*b3 * q^77 + 5*b1 * q^79 + 9 * q^81 + 10*b2 * q^83 + 9*b1 * q^87 - 5*b3 * q^93 + 2 * q^97 - 6*b2 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 12 q^{9}+O(q^{10})$$ 4 * q - 12 * q^9 $$4 q - 12 q^{9} + 28 q^{25} - 24 q^{33} + 12 q^{49} - 56 q^{73} + 36 q^{81} + 8 q^{97}+O(q^{100})$$ 4 * q - 12 * q^9 + 28 * q^25 - 24 * q^33 + 12 * q^49 - 56 * q^73 + 36 * q^81 + 8 * q^97

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$2\zeta_{12}^{3}$$ 2*v^3 $$\beta_{2}$$ $$=$$ $$2\zeta_{12}^{2} - 1$$ 2*v^2 - 1 $$\beta_{3}$$ $$=$$ $$-2\zeta_{12}^{3} + 4\zeta_{12}$$ -2*v^3 + 4*v
 $$\zeta_{12}$$ $$=$$ $$( \beta_{3} + \beta_1 ) / 4$$ (b3 + b1) / 4 $$\zeta_{12}^{2}$$ $$=$$ $$( \beta_{2} + 1 ) / 2$$ (b2 + 1) / 2 $$\zeta_{12}^{3}$$ $$=$$ $$( \beta_1 ) / 2$$ (b1) / 2

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/192\mathbb{Z}\right)^\times$$.

 $$n$$ $$65$$ $$127$$ $$133$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
95.1
 0.866025 − 0.500000i −0.866025 + 0.500000i 0.866025 + 0.500000i −0.866025 − 0.500000i
0 1.73205i 0 −3.46410 0 2.00000i 0 −3.00000 0
95.2 0 1.73205i 0 3.46410 0 2.00000i 0 −3.00000 0
95.3 0 1.73205i 0 −3.46410 0 2.00000i 0 −3.00000 0
95.4 0 1.73205i 0 3.46410 0 2.00000i 0 −3.00000 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 $$\Q(\sqrt{-6})$$
3.b odd 2 1 inner
4.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner
12.b even 2 1 inner
24.f even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 192.2.f.a 4
3.b odd 2 1 inner 192.2.f.a 4
4.b odd 2 1 inner 192.2.f.a 4
8.b even 2 1 inner 192.2.f.a 4
8.d odd 2 1 inner 192.2.f.a 4
12.b even 2 1 inner 192.2.f.a 4
16.e even 4 2 768.2.c.i 4
16.f odd 4 2 768.2.c.i 4
24.f even 2 1 inner 192.2.f.a 4
24.h odd 2 1 CM 192.2.f.a 4
48.i odd 4 2 768.2.c.i 4
48.k even 4 2 768.2.c.i 4

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} - 12$$ acting on $$S_{2}^{\mathrm{new}}(192, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$(T^{2} + 3)^{2}$$
$5$ $$(T^{2} - 12)^{2}$$
$7$ $$(T^{2} + 4)^{2}$$
$11$ $$(T^{2} + 12)^{2}$$
$13$ $$T^{4}$$
$17$ $$T^{4}$$
$19$ $$T^{4}$$
$23$ $$T^{4}$$
$29$ $$(T^{2} - 108)^{2}$$
$31$ $$(T^{2} + 100)^{2}$$
$37$ $$T^{4}$$
$41$ $$T^{4}$$
$43$ $$T^{4}$$
$47$ $$T^{4}$$
$53$ $$(T^{2} - 12)^{2}$$
$59$ $$(T^{2} + 108)^{2}$$
$61$ $$T^{4}$$
$67$ $$T^{4}$$
$71$ $$T^{4}$$
$73$ $$(T + 14)^{4}$$
$79$ $$(T^{2} + 100)^{2}$$
$83$ $$(T^{2} + 300)^{2}$$
$89$ $$T^{4}$$
$97$ $$(T - 2)^{4}$$