# Properties

 Label 192.3.h.c Level $192$ Weight $3$ Character orbit 192.h Analytic conductor $5.232$ Analytic rank $0$ Dimension $8$ CM no Inner twists $8$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$5.23162107572$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.12960000.1 Defining polynomial: $$x^{8} - 3 x^{6} + 8 x^{4} - 3 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2^{18}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{3} -\beta_{4} q^{5} -\beta_{1} q^{7} + ( 1 - \beta_{5} ) q^{9} +O(q^{10})$$ $$q + \beta_{2} q^{3} -\beta_{4} q^{5} -\beta_{1} q^{7} + ( 1 - \beta_{5} ) q^{9} + ( -3 \beta_{2} - 3 \beta_{3} ) q^{11} + 3 \beta_{6} q^{13} + ( -5 \beta_{1} + \beta_{7} ) q^{15} + ( \beta_{2} - \beta_{3} ) q^{19} + ( -\beta_{4} - \beta_{6} ) q^{21} + 2 \beta_{7} q^{23} + 35 q^{25} + ( 2 \beta_{2} - 9 \beta_{3} ) q^{27} -\beta_{4} q^{29} + 7 \beta_{1} q^{31} + ( -30 + 3 \beta_{5} ) q^{33} + ( 6 \beta_{2} + 6 \beta_{3} ) q^{35} + 5 \beta_{6} q^{37} + ( -12 \beta_{1} - 3 \beta_{7} ) q^{39} -6 \beta_{5} q^{41} + ( -13 \beta_{2} + 13 \beta_{3} ) q^{43} + ( -\beta_{4} - 10 \beta_{6} ) q^{45} -4 \beta_{7} q^{47} -37 q^{49} + 7 \beta_{4} q^{53} + 30 \beta_{1} q^{55} + ( -8 - \beta_{5} ) q^{57} + ( 9 \beta_{2} + 9 \beta_{3} ) q^{59} -\beta_{6} q^{61} + ( -\beta_{1} + 2 \beta_{7} ) q^{63} + 18 \beta_{5} q^{65} + ( -7 \beta_{2} + 7 \beta_{3} ) q^{67} + ( 8 \beta_{4} - 10 \beta_{6} ) q^{69} -2 \beta_{7} q^{71} + 74 q^{73} + 35 \beta_{2} q^{75} + 6 \beta_{4} q^{77} + 15 \beta_{1} q^{79} + ( -79 - 2 \beta_{5} ) q^{81} + ( -27 \beta_{2} - 27 \beta_{3} ) q^{83} + ( -5 \beta_{1} + \beta_{7} ) q^{87} -6 \beta_{5} q^{89} + ( -18 \beta_{2} + 18 \beta_{3} ) q^{91} + ( 7 \beta_{4} + 7 \beta_{6} ) q^{93} + 2 \beta_{7} q^{95} -62 q^{97} + ( -33 \beta_{2} + 27 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 8q^{9} + O(q^{10})$$ $$8q + 8q^{9} + 280q^{25} - 240q^{33} - 296q^{49} - 64q^{57} + 592q^{73} - 632q^{81} - 496q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 3 x^{6} + 8 x^{4} - 3 x^{2} + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{7} - 4 \nu^{5} + 10 \nu^{3} - 7 \nu$$ $$\beta_{2}$$ $$=$$ $$($$$$6 \nu^{7} - \nu^{6} - 16 \nu^{5} + 40 \nu^{3} - 2 \nu - 9$$$$)/4$$ $$\beta_{3}$$ $$=$$ $$($$$$-6 \nu^{7} - \nu^{6} + 16 \nu^{5} - 40 \nu^{3} + 2 \nu - 9$$$$)/4$$ $$\beta_{4}$$ $$=$$ $$($$$$-5 \nu^{7} + 16 \nu^{5} - 44 \nu^{3} + 31 \nu$$$$)/2$$ $$\beta_{5}$$ $$=$$ $$6 \nu^{7} - 16 \nu^{5} + 44 \nu^{3} - 2 \nu$$ $$\beta_{6}$$ $$=$$ $$-3 \nu^{6} + 8 \nu^{4} - 24 \nu^{2} + 5$$ $$\beta_{7}$$ $$=$$ $$-8 \nu^{6} + 24 \nu^{4} - 56 \nu^{2} + 12$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{5} + 2 \beta_{4} + \beta_{3} - \beta_{2} + 2 \beta_{1}$$$$)/16$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{7} - 3 \beta_{6} + 2 \beta_{3} + 2 \beta_{2} + 12$$$$)/16$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{5} + 2 \beta_{3} - 2 \beta_{2}$$$$)/4$$ $$\nu^{4}$$ $$=$$ $$($$$$3 \beta_{7} - 7 \beta_{6} - 6 \beta_{3} - 6 \beta_{2} - 28$$$$)/16$$ $$\nu^{5}$$ $$=$$ $$($$$$5 \beta_{5} - 10 \beta_{4} + 11 \beta_{3} - 11 \beta_{2} - 22 \beta_{1}$$$$)/16$$ $$\nu^{6}$$ $$=$$ $$-2 \beta_{3} - 2 \beta_{2} - 9$$ $$\nu^{7}$$ $$=$$ $$($$$$-13 \beta_{5} - 26 \beta_{4} - 29 \beta_{3} + 29 \beta_{2} - 58 \beta_{1}$$$$)/16$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
161.1
 0.535233 − 0.309017i −0.535233 − 0.309017i 0.535233 + 0.309017i −0.535233 + 0.309017i 1.40126 + 0.809017i −1.40126 + 0.809017i 1.40126 − 0.809017i −1.40126 − 0.809017i
0 −2.23607 2.00000i 0 −7.74597 0 3.46410 0 1.00000 + 8.94427i 0
161.2 0 −2.23607 2.00000i 0 7.74597 0 −3.46410 0 1.00000 + 8.94427i 0
161.3 0 −2.23607 + 2.00000i 0 −7.74597 0 3.46410 0 1.00000 8.94427i 0
161.4 0 −2.23607 + 2.00000i 0 7.74597 0 −3.46410 0 1.00000 8.94427i 0
161.5 0 2.23607 2.00000i 0 −7.74597 0 −3.46410 0 1.00000 8.94427i 0
161.6 0 2.23607 2.00000i 0 7.74597 0 3.46410 0 1.00000 8.94427i 0
161.7 0 2.23607 + 2.00000i 0 −7.74597 0 −3.46410 0 1.00000 + 8.94427i 0
161.8 0 2.23607 + 2.00000i 0 7.74597 0 3.46410 0 1.00000 + 8.94427i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 161.8 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
8.b even 2 1 inner
8.d odd 2 1 inner
12.b even 2 1 inner
24.f 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 192.3.h.c 8
3.b odd 2 1 inner 192.3.h.c 8
4.b odd 2 1 inner 192.3.h.c 8
8.b even 2 1 inner 192.3.h.c 8
8.d odd 2 1 inner 192.3.h.c 8
12.b even 2 1 inner 192.3.h.c 8
16.e even 4 1 768.3.e.g 4
16.e even 4 1 768.3.e.n 4
16.f odd 4 1 768.3.e.g 4
16.f odd 4 1 768.3.e.n 4
24.f even 2 1 inner 192.3.h.c 8
24.h odd 2 1 inner 192.3.h.c 8
48.i odd 4 1 768.3.e.g 4
48.i odd 4 1 768.3.e.n 4
48.k even 4 1 768.3.e.g 4
48.k even 4 1 768.3.e.n 4

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

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(192, [\chi])$$:

 $$T_{5}^{2} - 60$$ $$T_{7}^{2} - 12$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$( 81 - 2 T^{2} + T^{4} )^{2}$$
$5$ $$( -60 + T^{2} )^{4}$$
$7$ $$( -12 + T^{2} )^{4}$$
$11$ $$( -180 + T^{2} )^{4}$$
$13$ $$( 432 + T^{2} )^{4}$$
$17$ $$T^{8}$$
$19$ $$( 16 + T^{2} )^{4}$$
$23$ $$( 960 + T^{2} )^{4}$$
$29$ $$( -60 + T^{2} )^{4}$$
$31$ $$( -588 + T^{2} )^{4}$$
$37$ $$( 1200 + T^{2} )^{4}$$
$41$ $$( 2880 + T^{2} )^{4}$$
$43$ $$( 2704 + T^{2} )^{4}$$
$47$ $$( 3840 + T^{2} )^{4}$$
$53$ $$( -2940 + T^{2} )^{4}$$
$59$ $$( -1620 + T^{2} )^{4}$$
$61$ $$( 48 + T^{2} )^{4}$$
$67$ $$( 784 + T^{2} )^{4}$$
$71$ $$( 960 + T^{2} )^{4}$$
$73$ $$( -74 + T )^{8}$$
$79$ $$( -2700 + T^{2} )^{4}$$
$83$ $$( -14580 + T^{2} )^{4}$$
$89$ $$( 2880 + T^{2} )^{4}$$
$97$ $$( 62 + T )^{8}$$