# Properties

 Label 768.3.g.g Level $768$ Weight $3$ Character orbit 768.g Analytic conductor $20.926$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$20.9264843029$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} - 2 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: no (minimal twist has level 384) 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.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{3} + ( 4 + \beta_{3} ) q^{5} -\beta_{1} q^{7} -3 q^{9} +O(q^{10})$$ $$q + \beta_{2} q^{3} + ( 4 + \beta_{3} ) q^{5} -\beta_{1} q^{7} -3 q^{9} + ( 4 \beta_{1} + 4 \beta_{2} ) q^{11} + ( 4 - 2 \beta_{3} ) q^{13} + ( 3 \beta_{1} + 4 \beta_{2} ) q^{15} + ( -2 - 4 \beta_{3} ) q^{17} + ( 4 \beta_{1} + 4 \beta_{2} ) q^{19} + \beta_{3} q^{21} + ( 2 \beta_{1} + 16 \beta_{2} ) q^{23} + ( 15 + 8 \beta_{3} ) q^{25} -3 \beta_{2} q^{27} + ( 20 - 5 \beta_{3} ) q^{29} + ( -9 \beta_{1} + 16 \beta_{2} ) q^{31} + ( -12 - 4 \beta_{3} ) q^{33} + ( -4 \beta_{1} - 8 \beta_{2} ) q^{35} + ( 4 + 8 \beta_{3} ) q^{37} + ( -6 \beta_{1} + 4 \beta_{2} ) q^{39} + ( -18 + 4 \beta_{3} ) q^{41} + ( -20 \beta_{1} - 4 \beta_{2} ) q^{43} + ( -12 - 3 \beta_{3} ) q^{45} + ( 6 \beta_{1} + 32 \beta_{2} ) q^{47} + 41 q^{49} + ( -12 \beta_{1} - 2 \beta_{2} ) q^{51} + ( 36 + 7 \beta_{3} ) q^{53} + ( 28 \beta_{1} + 48 \beta_{2} ) q^{55} + ( -12 - 4 \beta_{3} ) q^{57} -20 \beta_{2} q^{59} + ( -44 - 4 \beta_{3} ) q^{61} + 3 \beta_{1} q^{63} + ( -32 - 4 \beta_{3} ) q^{65} + ( 16 \beta_{1} - 28 \beta_{2} ) q^{67} + ( -48 - 2 \beta_{3} ) q^{69} + ( -34 \beta_{1} + 16 \beta_{2} ) q^{71} + 10 q^{73} + ( 24 \beta_{1} + 15 \beta_{2} ) q^{75} + ( 32 + 4 \beta_{3} ) q^{77} + ( -17 \beta_{1} + 48 \beta_{2} ) q^{79} + 9 q^{81} + ( -12 \beta_{1} + 44 \beta_{2} ) q^{83} + ( -104 - 18 \beta_{3} ) q^{85} + ( -15 \beta_{1} + 20 \beta_{2} ) q^{87} + ( 34 - 8 \beta_{3} ) q^{89} + ( -4 \beta_{1} + 16 \beta_{2} ) q^{91} + ( -48 + 9 \beta_{3} ) q^{93} + ( 28 \beta_{1} + 48 \beta_{2} ) q^{95} + ( -66 + 8 \beta_{3} ) q^{97} + ( -12 \beta_{1} - 12 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 16 q^{5} - 12 q^{9} + O(q^{10})$$ $$4 q + 16 q^{5} - 12 q^{9} + 16 q^{13} - 8 q^{17} + 60 q^{25} + 80 q^{29} - 48 q^{33} + 16 q^{37} - 72 q^{41} - 48 q^{45} + 164 q^{49} + 144 q^{53} - 48 q^{57} - 176 q^{61} - 128 q^{65} - 192 q^{69} + 40 q^{73} + 128 q^{77} + 36 q^{81} - 416 q^{85} + 136 q^{89} - 192 q^{93} - 264 q^{97} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{3}$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 1$$ $$\beta_{3}$$ $$=$$ $$-\nu^{3} + 4 \nu$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + \beta_{1}$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 1$$ $$\nu^{3}$$ $$=$$ $$\beta_{1}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
511.1
 −1.22474 + 0.707107i 1.22474 − 0.707107i −1.22474 − 0.707107i 1.22474 + 0.707107i
0 1.73205i 0 −0.898979 0 2.82843i 0 −3.00000 0
511.2 0 1.73205i 0 8.89898 0 2.82843i 0 −3.00000 0
511.3 0 1.73205i 0 −0.898979 0 2.82843i 0 −3.00000 0
511.4 0 1.73205i 0 8.89898 0 2.82843i 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
4.b 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.g.g 4
3.b odd 2 1 2304.3.g.o 4
4.b odd 2 1 inner 768.3.g.g 4
8.b even 2 1 768.3.g.c 4
8.d odd 2 1 768.3.g.c 4
12.b even 2 1 2304.3.g.o 4
16.e even 4 2 384.3.b.c 8
16.f odd 4 2 384.3.b.c 8
24.f even 2 1 2304.3.g.x 4
24.h odd 2 1 2304.3.g.x 4
48.i odd 4 2 1152.3.b.j 8
48.k even 4 2 1152.3.b.j 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.3.b.c 8 16.e even 4 2
384.3.b.c 8 16.f odd 4 2
768.3.g.c 4 8.b even 2 1
768.3.g.c 4 8.d odd 2 1
768.3.g.g 4 1.a even 1 1 trivial
768.3.g.g 4 4.b odd 2 1 inner
1152.3.b.j 8 48.i odd 4 2
1152.3.b.j 8 48.k even 4 2
2304.3.g.o 4 3.b odd 2 1
2304.3.g.o 4 12.b even 2 1
2304.3.g.x 4 24.f even 2 1
2304.3.g.x 4 24.h odd 2 1

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( 3 + T^{2} )^{2}$$
$5$ $$( -8 - 8 T + T^{2} )^{2}$$
$7$ $$( 8 + T^{2} )^{2}$$
$11$ $$6400 + 352 T^{2} + T^{4}$$
$13$ $$( -80 - 8 T + T^{2} )^{2}$$
$17$ $$( -380 + 4 T + T^{2} )^{2}$$
$19$ $$6400 + 352 T^{2} + T^{4}$$
$23$ $$541696 + 1600 T^{2} + T^{4}$$
$29$ $$( -200 - 40 T + T^{2} )^{2}$$
$31$ $$14400 + 2832 T^{2} + T^{4}$$
$37$ $$( -1520 - 8 T + T^{2} )^{2}$$
$41$ $$( -60 + 36 T + T^{2} )^{2}$$
$43$ $$9935104 + 6496 T^{2} + T^{4}$$
$47$ $$7750656 + 6720 T^{2} + T^{4}$$
$53$ $$( 120 - 72 T + T^{2} )^{2}$$
$59$ $$( 1200 + T^{2} )^{2}$$
$61$ $$( 1552 + 88 T + T^{2} )^{2}$$
$67$ $$92416 + 8800 T^{2} + T^{4}$$
$71$ $$71910400 + 20032 T^{2} + T^{4}$$
$73$ $$( -10 + T )^{4}$$
$79$ $$21160000 + 18448 T^{2} + T^{4}$$
$83$ $$21678336 + 13920 T^{2} + T^{4}$$
$89$ $$( -380 - 68 T + T^{2} )^{2}$$
$97$ $$( 2820 + 132 T + T^{2} )^{2}$$