# Properties

 Label 768.3.h.f Level $768$ Weight $3$ Character orbit 768.h Analytic conductor $20.926$ Analytic rank $0$ Dimension $8$ CM no Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$768 = 2^{8} \cdot 3$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 768.h (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$20.9264843029$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\Q(\zeta_{24})$$ Defining polynomial: $$x^{8} - x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2^{12}\cdot 3^{2}$$ 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 primitive root of unity $$\zeta_{24}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 + \zeta_{24} + \zeta_{24}^{3} - 2 \zeta_{24}^{4} + \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{3} + ( -2 \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} ) q^{5} + ( 12 \zeta_{24}^{2} - 6 \zeta_{24}^{6} ) q^{7} + ( 3 + 6 \zeta_{24} - 6 \zeta_{24}^{3} - 6 \zeta_{24}^{5} ) q^{9} +O(q^{10})$$ $$q + ( 1 + \zeta_{24} + \zeta_{24}^{3} - 2 \zeta_{24}^{4} + \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{3} + ( -2 \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} ) q^{5} + ( 12 \zeta_{24}^{2} - 6 \zeta_{24}^{6} ) q^{7} + ( 3 + 6 \zeta_{24} - 6 \zeta_{24}^{3} - 6 \zeta_{24}^{5} ) q^{9} + ( -6 \zeta_{24} - 6 \zeta_{24}^{3} - 6 \zeta_{24}^{5} + 12 \zeta_{24}^{7} ) q^{11} -6 \zeta_{24}^{6} q^{13} + ( 2 \zeta_{24} - 8 \zeta_{24}^{2} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + 4 \zeta_{24}^{6} + 4 \zeta_{24}^{7} ) q^{15} + ( 16 \zeta_{24} - 16 \zeta_{24}^{3} - 16 \zeta_{24}^{5} ) q^{17} + ( -6 + 12 \zeta_{24}^{4} ) q^{19} + ( 18 \zeta_{24} + 18 \zeta_{24}^{3} - 18 \zeta_{24}^{5} - 18 \zeta_{24}^{6} ) q^{21} + ( -12 \zeta_{24} + 12 \zeta_{24}^{3} - 12 \zeta_{24}^{5} - 24 \zeta_{24}^{7} ) q^{23} -17 q^{25} + ( 15 - 3 \zeta_{24} - 3 \zeta_{24}^{3} - 30 \zeta_{24}^{4} - 3 \zeta_{24}^{5} + 6 \zeta_{24}^{7} ) q^{27} + ( 22 \zeta_{24} + 22 \zeta_{24}^{3} - 22 \zeta_{24}^{5} ) q^{29} + ( 36 \zeta_{24}^{2} - 18 \zeta_{24}^{6} ) q^{31} + ( -36 - 18 \zeta_{24} + 18 \zeta_{24}^{3} + 18 \zeta_{24}^{5} ) q^{33} + ( -12 \zeta_{24} - 12 \zeta_{24}^{3} - 12 \zeta_{24}^{5} + 24 \zeta_{24}^{7} ) q^{35} + 38 \zeta_{24}^{6} q^{37} + ( -6 \zeta_{24} - 12 \zeta_{24}^{2} + 6 \zeta_{24}^{3} - 6 \zeta_{24}^{5} + 6 \zeta_{24}^{6} - 12 \zeta_{24}^{7} ) q^{39} + ( 4 \zeta_{24} - 4 \zeta_{24}^{3} - 4 \zeta_{24}^{5} ) q^{41} + ( 6 - 12 \zeta_{24}^{4} ) q^{43} + ( -6 \zeta_{24} - 6 \zeta_{24}^{3} + 6 \zeta_{24}^{5} + 24 \zeta_{24}^{6} ) q^{45} + ( -24 \zeta_{24} + 24 \zeta_{24}^{3} - 24 \zeta_{24}^{5} - 48 \zeta_{24}^{7} ) q^{47} + 59 q^{49} + ( 32 - 16 \zeta_{24} - 16 \zeta_{24}^{3} - 64 \zeta_{24}^{4} - 16 \zeta_{24}^{5} + 32 \zeta_{24}^{7} ) q^{51} + ( -10 \zeta_{24} - 10 \zeta_{24}^{3} + 10 \zeta_{24}^{5} ) q^{53} + ( 48 \zeta_{24}^{2} - 24 \zeta_{24}^{6} ) q^{55} + ( 18 - 18 \zeta_{24} + 18 \zeta_{24}^{3} + 18 \zeta_{24}^{5} ) q^{57} + ( -6 \zeta_{24} - 6 \zeta_{24}^{3} - 6 \zeta_{24}^{5} + 12 \zeta_{24}^{7} ) q^{59} -22 \zeta_{24}^{6} q^{61} + ( -36 \zeta_{24} + 36 \zeta_{24}^{2} + 36 \zeta_{24}^{3} - 36 \zeta_{24}^{5} - 18 \zeta_{24}^{6} - 72 \zeta_{24}^{7} ) q^{63} + ( -12 \zeta_{24} + 12 \zeta_{24}^{3} + 12 \zeta_{24}^{5} ) q^{65} + ( 66 - 132 \zeta_{24}^{4} ) q^{67} + ( -36 \zeta_{24} - 36 \zeta_{24}^{3} + 36 \zeta_{24}^{5} - 72 \zeta_{24}^{6} ) q^{69} + ( 12 \zeta_{24} - 12 \zeta_{24}^{3} + 12 \zeta_{24}^{5} + 24 \zeta_{24}^{7} ) q^{71} + 30 q^{73} + ( -17 - 17 \zeta_{24} - 17 \zeta_{24}^{3} + 34 \zeta_{24}^{4} - 17 \zeta_{24}^{5} + 34 \zeta_{24}^{7} ) q^{75} + ( -108 \zeta_{24} - 108 \zeta_{24}^{3} + 108 \zeta_{24}^{5} ) q^{77} + ( 36 \zeta_{24}^{2} - 18 \zeta_{24}^{6} ) q^{79} + ( -63 + 36 \zeta_{24} - 36 \zeta_{24}^{3} - 36 \zeta_{24}^{5} ) q^{81} + ( -30 \zeta_{24} - 30 \zeta_{24}^{3} - 30 \zeta_{24}^{5} + 60 \zeta_{24}^{7} ) q^{83} + 64 \zeta_{24}^{6} q^{85} + ( -22 \zeta_{24} + 88 \zeta_{24}^{2} + 22 \zeta_{24}^{3} - 22 \zeta_{24}^{5} - 44 \zeta_{24}^{6} - 44 \zeta_{24}^{7} ) q^{87} + ( 4 \zeta_{24} - 4 \zeta_{24}^{3} - 4 \zeta_{24}^{5} ) q^{89} + ( 36 - 72 \zeta_{24}^{4} ) q^{91} + ( 54 \zeta_{24} + 54 \zeta_{24}^{3} - 54 \zeta_{24}^{5} - 54 \zeta_{24}^{6} ) q^{93} + ( -12 \zeta_{24} + 12 \zeta_{24}^{3} - 12 \zeta_{24}^{5} - 24 \zeta_{24}^{7} ) q^{95} + 90 q^{97} + ( -72 - 18 \zeta_{24} - 18 \zeta_{24}^{3} + 144 \zeta_{24}^{4} - 18 \zeta_{24}^{5} + 36 \zeta_{24}^{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 24q^{9} + O(q^{10})$$ $$8q + 24q^{9} - 136q^{25} - 288q^{33} + 472q^{49} + 144q^{57} + 240q^{73} - 504q^{81} + 720q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/768\mathbb{Z}\right)^\times$$.

 $$n$$ $$257$$ $$511$$ $$517$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
641.1
 −0.258819 + 0.965926i −0.965926 − 0.258819i −0.258819 − 0.965926i −0.965926 + 0.258819i 0.965926 + 0.258819i 0.258819 − 0.965926i 0.965926 − 0.258819i 0.258819 + 0.965926i
0 −2.44949 1.73205i 0 −2.82843 0 −10.3923 0 3.00000 + 8.48528i 0
641.2 0 −2.44949 1.73205i 0 2.82843 0 10.3923 0 3.00000 + 8.48528i 0
641.3 0 −2.44949 + 1.73205i 0 −2.82843 0 −10.3923 0 3.00000 8.48528i 0
641.4 0 −2.44949 + 1.73205i 0 2.82843 0 10.3923 0 3.00000 8.48528i 0
641.5 0 2.44949 1.73205i 0 −2.82843 0 10.3923 0 3.00000 8.48528i 0
641.6 0 2.44949 1.73205i 0 2.82843 0 −10.3923 0 3.00000 8.48528i 0
641.7 0 2.44949 + 1.73205i 0 −2.82843 0 10.3923 0 3.00000 + 8.48528i 0
641.8 0 2.44949 + 1.73205i 0 2.82843 0 −10.3923 0 3.00000 + 8.48528i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 641.8 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 768.3.h.f 8
3.b odd 2 1 inner 768.3.h.f 8
4.b odd 2 1 inner 768.3.h.f 8
8.b even 2 1 inner 768.3.h.f 8
8.d odd 2 1 inner 768.3.h.f 8
12.b even 2 1 inner 768.3.h.f 8
16.e even 4 1 96.3.e.a 4
16.e even 4 1 192.3.e.e 4
16.f odd 4 1 96.3.e.a 4
16.f odd 4 1 192.3.e.e 4
24.f even 2 1 inner 768.3.h.f 8
24.h odd 2 1 inner 768.3.h.f 8
48.i odd 4 1 96.3.e.a 4
48.i odd 4 1 192.3.e.e 4
48.k even 4 1 96.3.e.a 4
48.k even 4 1 192.3.e.e 4

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

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(768, [\chi])$$:

 $$T_{5}^{2} - 8$$ $$T_{7}^{2} - 108$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$( 81 - 6 T^{2} + T^{4} )^{2}$$
$5$ $$( -8 + T^{2} )^{4}$$
$7$ $$( -108 + T^{2} )^{4}$$
$11$ $$( -216 + T^{2} )^{4}$$
$13$ $$( 36 + T^{2} )^{4}$$
$17$ $$( 512 + T^{2} )^{4}$$
$19$ $$( 108 + T^{2} )^{4}$$
$23$ $$( 864 + T^{2} )^{4}$$
$29$ $$( -968 + T^{2} )^{4}$$
$31$ $$( -972 + T^{2} )^{4}$$
$37$ $$( 1444 + T^{2} )^{4}$$
$41$ $$( 32 + T^{2} )^{4}$$
$43$ $$( 108 + T^{2} )^{4}$$
$47$ $$( 3456 + T^{2} )^{4}$$
$53$ $$( -200 + T^{2} )^{4}$$
$59$ $$( -216 + T^{2} )^{4}$$
$61$ $$( 484 + T^{2} )^{4}$$
$67$ $$( 13068 + T^{2} )^{4}$$
$71$ $$( 864 + T^{2} )^{4}$$
$73$ $$( -30 + T )^{8}$$
$79$ $$( -972 + T^{2} )^{4}$$
$83$ $$( -5400 + T^{2} )^{4}$$
$89$ $$( 32 + T^{2} )^{4}$$
$97$ $$( -90 + T )^{8}$$