# Properties

 Label 192.2.c.b Level $192$ Weight $2$ Character orbit 192.c Analytic conductor $1.533$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$192 = 2^{6} \cdot 3$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 192.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.53312771881$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ x^4 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: no (minimal twist has level 96) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{3} - \beta_{3} q^{5} + ( - \beta_{2} + \beta_1) q^{7} + ( - \beta_{3} + 1) q^{9}+O(q^{10})$$ q - b1 * q^3 - b3 * q^5 + (-b2 + b1) * q^7 + (-b3 + 1) * q^9 $$q - \beta_1 q^{3} - \beta_{3} q^{5} + ( - \beta_{2} + \beta_1) q^{7} + ( - \beta_{3} + 1) q^{9} + (\beta_{2} + \beta_1) q^{11} + 2 q^{13} + (3 \beta_{2} - \beta_1) q^{15} + ( - 3 \beta_{2} + 3 \beta_1) q^{19} + (\beta_{3} + 2) q^{21} + ( - 2 \beta_{2} - 2 \beta_1) q^{23} - 3 q^{25} + (3 \beta_{2} - 2 \beta_1) q^{27} + \beta_{3} q^{29} + ( - \beta_{2} + \beta_1) q^{31} + (\beta_{3} - 4) q^{33} + ( - 2 \beta_{2} - 2 \beta_1) q^{35} - 6 q^{37} - 2 \beta_1 q^{39} + 2 \beta_{3} q^{41} + (\beta_{2} - \beta_1) q^{43} + ( - \beta_{3} - 8) q^{45} + (4 \beta_{2} + 4 \beta_1) q^{47} + 3 q^{49} + 3 \beta_{3} q^{53} + ( - 4 \beta_{2} + 4 \beta_1) q^{55} + (3 \beta_{3} + 6) q^{57} + (\beta_{2} + \beta_1) q^{59} + 2 q^{61} + ( - 3 \beta_{2} - \beta_1) q^{63} - 2 \beta_{3} q^{65} + (\beta_{2} - \beta_1) q^{67} + ( - 2 \beta_{3} + 8) q^{69} + (2 \beta_{2} + 2 \beta_1) q^{71} - 6 q^{73} + 3 \beta_1 q^{75} - 2 \beta_{3} q^{77} + (7 \beta_{2} - 7 \beta_1) q^{79} + ( - 2 \beta_{3} - 7) q^{81} + ( - \beta_{2} - \beta_1) q^{83} + ( - 3 \beta_{2} + \beta_1) q^{87} - 6 \beta_{3} q^{89} + ( - 2 \beta_{2} + 2 \beta_1) q^{91} + (\beta_{3} + 2) q^{93} + ( - 6 \beta_{2} - 6 \beta_1) q^{95} + 10 q^{97} + ( - 3 \beta_{2} + 5 \beta_1) q^{99}+O(q^{100})$$ q - b1 * q^3 - b3 * q^5 + (-b2 + b1) * q^7 + (-b3 + 1) * q^9 + (b2 + b1) * q^11 + 2 * q^13 + (3*b2 - b1) * q^15 + (-3*b2 + 3*b1) * q^19 + (b3 + 2) * q^21 + (-2*b2 - 2*b1) * q^23 - 3 * q^25 + (3*b2 - 2*b1) * q^27 + b3 * q^29 + (-b2 + b1) * q^31 + (b3 - 4) * q^33 + (-2*b2 - 2*b1) * q^35 - 6 * q^37 - 2*b1 * q^39 + 2*b3 * q^41 + (b2 - b1) * q^43 + (-b3 - 8) * q^45 + (4*b2 + 4*b1) * q^47 + 3 * q^49 + 3*b3 * q^53 + (-4*b2 + 4*b1) * q^55 + (3*b3 + 6) * q^57 + (b2 + b1) * q^59 + 2 * q^61 + (-3*b2 - b1) * q^63 - 2*b3 * q^65 + (b2 - b1) * q^67 + (-2*b3 + 8) * q^69 + (2*b2 + 2*b1) * q^71 - 6 * q^73 + 3*b1 * q^75 - 2*b3 * q^77 + (7*b2 - 7*b1) * q^79 + (-2*b3 - 7) * q^81 + (-b2 - b1) * q^83 + (-3*b2 + b1) * q^87 - 6*b3 * q^89 + (-2*b2 + 2*b1) * q^91 + (b3 + 2) * q^93 + (-6*b2 - 6*b1) * q^95 + 10 * q^97 + (-3*b2 + 5*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{9}+O(q^{10})$$ 4 * q + 4 * q^9 $$4 q + 4 q^{9} + 8 q^{13} + 8 q^{21} - 12 q^{25} - 16 q^{33} - 24 q^{37} - 32 q^{45} + 12 q^{49} + 24 q^{57} + 8 q^{61} + 32 q^{69} - 24 q^{73} - 28 q^{81} + 8 q^{93} + 40 q^{97}+O(q^{100})$$ 4 * q + 4 * q^9 + 8 * q^13 + 8 * q^21 - 12 * q^25 - 16 * q^33 - 24 * q^37 - 32 * q^45 + 12 * q^49 + 24 * q^57 + 8 * q^61 + 32 * q^69 - 24 * q^73 - 28 * q^81 + 8 * q^93 + 40 * q^97

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$-\zeta_{8}^{3} - \zeta_{8}^{2} + \zeta_{8}$$ -v^3 - v^2 + v $$\beta_{2}$$ $$=$$ $$-\zeta_{8}^{3} + \zeta_{8}^{2} + \zeta_{8}$$ -v^3 + v^2 + v $$\beta_{3}$$ $$=$$ $$2\zeta_{8}^{3} + 2\zeta_{8}$$ 2*v^3 + 2*v
 $$\zeta_{8}$$ $$=$$ $$( \beta_{3} + \beta_{2} + \beta_1 ) / 4$$ (b3 + b2 + b1) / 4 $$\zeta_{8}^{2}$$ $$=$$ $$( \beta_{2} - \beta_1 ) / 2$$ (b2 - b1) / 2 $$\zeta_{8}^{3}$$ $$=$$ $$( \beta_{3} - \beta_{2} - \beta_1 ) / 4$$ (b3 - b2 - b1) / 4

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
191.1
 0.707107 − 0.707107i 0.707107 + 0.707107i −0.707107 + 0.707107i −0.707107 − 0.707107i
0 −1.41421 1.00000i 0 2.82843i 0 2.00000i 0 1.00000 + 2.82843i 0
191.2 0 −1.41421 + 1.00000i 0 2.82843i 0 2.00000i 0 1.00000 2.82843i 0
191.3 0 1.41421 1.00000i 0 2.82843i 0 2.00000i 0 1.00000 2.82843i 0
191.4 0 1.41421 + 1.00000i 0 2.82843i 0 2.00000i 0 1.00000 + 2.82843i 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
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b 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.c.b 4
3.b odd 2 1 inner 192.2.c.b 4
4.b odd 2 1 inner 192.2.c.b 4
8.b even 2 1 96.2.c.a 4
8.d odd 2 1 96.2.c.a 4
12.b even 2 1 inner 192.2.c.b 4
16.e even 4 1 768.2.f.a 4
16.e even 4 1 768.2.f.g 4
16.f odd 4 1 768.2.f.a 4
16.f odd 4 1 768.2.f.g 4
24.f even 2 1 96.2.c.a 4
24.h odd 2 1 96.2.c.a 4
40.e odd 2 1 2400.2.h.c 4
40.f even 2 1 2400.2.h.c 4
40.i odd 4 1 2400.2.o.a 4
40.i odd 4 1 2400.2.o.h 4
40.k even 4 1 2400.2.o.a 4
40.k even 4 1 2400.2.o.h 4
48.i odd 4 1 768.2.f.a 4
48.i odd 4 1 768.2.f.g 4
48.k even 4 1 768.2.f.a 4
48.k even 4 1 768.2.f.g 4
72.j odd 6 2 2592.2.s.e 8
72.l even 6 2 2592.2.s.e 8
72.n even 6 2 2592.2.s.e 8
72.p odd 6 2 2592.2.s.e 8
120.i odd 2 1 2400.2.h.c 4
120.m even 2 1 2400.2.h.c 4
120.q odd 4 1 2400.2.o.a 4
120.q odd 4 1 2400.2.o.h 4
120.w even 4 1 2400.2.o.a 4
120.w even 4 1 2400.2.o.h 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
96.2.c.a 4 8.b even 2 1
96.2.c.a 4 8.d odd 2 1
96.2.c.a 4 24.f even 2 1
96.2.c.a 4 24.h odd 2 1
192.2.c.b 4 1.a even 1 1 trivial
192.2.c.b 4 3.b odd 2 1 inner
192.2.c.b 4 4.b odd 2 1 inner
192.2.c.b 4 12.b even 2 1 inner
768.2.f.a 4 16.e even 4 1
768.2.f.a 4 16.f odd 4 1
768.2.f.a 4 48.i odd 4 1
768.2.f.a 4 48.k even 4 1
768.2.f.g 4 16.e even 4 1
768.2.f.g 4 16.f odd 4 1
768.2.f.g 4 48.i odd 4 1
768.2.f.g 4 48.k even 4 1
2400.2.h.c 4 40.e odd 2 1
2400.2.h.c 4 40.f even 2 1
2400.2.h.c 4 120.i odd 2 1
2400.2.h.c 4 120.m even 2 1
2400.2.o.a 4 40.i odd 4 1
2400.2.o.a 4 40.k even 4 1
2400.2.o.a 4 120.q odd 4 1
2400.2.o.a 4 120.w even 4 1
2400.2.o.h 4 40.i odd 4 1
2400.2.o.h 4 40.k even 4 1
2400.2.o.h 4 120.q odd 4 1
2400.2.o.h 4 120.w even 4 1
2592.2.s.e 8 72.j odd 6 2
2592.2.s.e 8 72.l even 6 2
2592.2.s.e 8 72.n even 6 2
2592.2.s.e 8 72.p odd 6 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} + 8$$ acting on $$S_{2}^{\mathrm{new}}(192, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} - 2T^{2} + 9$$
$5$ $$(T^{2} + 8)^{2}$$
$7$ $$(T^{2} + 4)^{2}$$
$11$ $$(T^{2} - 8)^{2}$$
$13$ $$(T - 2)^{4}$$
$17$ $$T^{4}$$
$19$ $$(T^{2} + 36)^{2}$$
$23$ $$(T^{2} - 32)^{2}$$
$29$ $$(T^{2} + 8)^{2}$$
$31$ $$(T^{2} + 4)^{2}$$
$37$ $$(T + 6)^{4}$$
$41$ $$(T^{2} + 32)^{2}$$
$43$ $$(T^{2} + 4)^{2}$$
$47$ $$(T^{2} - 128)^{2}$$
$53$ $$(T^{2} + 72)^{2}$$
$59$ $$(T^{2} - 8)^{2}$$
$61$ $$(T - 2)^{4}$$
$67$ $$(T^{2} + 4)^{2}$$
$71$ $$(T^{2} - 32)^{2}$$
$73$ $$(T + 6)^{4}$$
$79$ $$(T^{2} + 196)^{2}$$
$83$ $$(T^{2} - 8)^{2}$$
$89$ $$(T^{2} + 288)^{2}$$
$97$ $$(T - 10)^{4}$$