# Properties

 Label 768.2.f.f Level $768$ Weight $2$ Character orbit 768.f Analytic conductor $6.133$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$6.13251087523$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{5})$$ Defining polynomial: $$x^{4} + 3 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}$$ 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.

 $$f(q)$$ $$=$$ $$q + ( 1 + \beta_{1} ) q^{3} + ( 2 + \beta_{1} - \beta_{3} ) q^{5} + ( \beta_{1} + \beta_{3} ) q^{7} + ( 1 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{9} +O(q^{10})$$ $$q + ( 1 + \beta_{1} ) q^{3} + ( 2 + \beta_{1} - \beta_{3} ) q^{5} + ( \beta_{1} + \beta_{3} ) q^{7} + ( 1 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{9} + ( -\beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{11} + ( 2 \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{13} + ( 4 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{15} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{17} + ( -2 + \beta_{1} - \beta_{3} ) q^{19} + ( -2 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{21} + ( -2 \beta_{1} + 2 \beta_{3} ) q^{23} + ( 3 + 2 \beta_{1} - 2 \beta_{3} ) q^{25} + ( 3 + 2 \beta_{2} + \beta_{3} ) q^{27} + ( -6 + \beta_{1} - \beta_{3} ) q^{29} + ( -\beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{31} + ( 2 - \beta_{1} - 3 \beta_{2} - 3 \beta_{3} ) q^{33} + 2 \beta_{2} q^{35} + ( 2 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} ) q^{37} + ( -4 + 2 \beta_{1} + 3 \beta_{2} ) q^{39} + ( -2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{41} + ( -6 - \beta_{1} + \beta_{3} ) q^{43} + ( 6 + 3 \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{45} -8 q^{47} + ( 3 + 2 \beta_{1} - 2 \beta_{3} ) q^{49} + ( 4 - 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{51} + ( 2 + \beta_{1} - \beta_{3} ) q^{53} + ( -4 \beta_{1} - 6 \beta_{2} - 4 \beta_{3} ) q^{55} + ( -3 \beta_{1} + \beta_{2} - \beta_{3} ) q^{57} + ( \beta_{1} + \beta_{3} ) q^{59} + ( -2 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{61} + ( -3 \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{63} + ( 2 \beta_{1} + 6 \beta_{2} + 2 \beta_{3} ) q^{65} + ( -6 - 3 \beta_{1} + 3 \beta_{3} ) q^{67} + ( -4 + 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{69} + ( 8 + 6 \beta_{1} - 6 \beta_{3} ) q^{71} + 2 q^{73} + ( 7 + \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{75} + ( 4 + 2 \beta_{1} - 2 \beta_{3} ) q^{77} + ( 3 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} ) q^{79} + ( 1 + 4 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} ) q^{81} + ( -\beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{83} -4 \beta_{2} q^{85} + ( -4 - 7 \beta_{1} + \beta_{2} - \beta_{3} ) q^{87} -2 \beta_{2} q^{89} + ( -8 + 2 \beta_{1} - 2 \beta_{3} ) q^{91} + ( 2 - \beta_{1} - 3 \beta_{2} - 3 \beta_{3} ) q^{93} + ( -2 \beta_{1} + 2 \beta_{3} ) q^{95} + ( -2 + 2 \beta_{1} - 2 \beta_{3} ) q^{97} + ( 8 - \beta_{1} - 4 \beta_{2} - 5 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{3} + 4q^{5} + O(q^{10})$$ $$4q + 2q^{3} + 4q^{5} + 12q^{15} - 12q^{19} - 12q^{21} + 8q^{23} + 4q^{25} + 14q^{27} - 28q^{29} + 4q^{33} - 20q^{39} - 20q^{43} + 20q^{45} - 32q^{47} + 4q^{49} + 24q^{51} + 4q^{53} + 4q^{57} + 8q^{63} - 12q^{67} - 16q^{69} + 8q^{71} + 8q^{73} + 22q^{75} + 8q^{77} + 4q^{81} - 4q^{87} - 40q^{91} + 4q^{93} + 8q^{95} - 16q^{97} + 24q^{99} + O(q^{100})$$

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
383.1
 − 1.61803i 1.61803i − 0.618034i 0.618034i
0 −0.618034 1.61803i 0 −1.23607 0 3.23607i 0 −2.23607 + 2.00000i 0
383.2 0 −0.618034 + 1.61803i 0 −1.23607 0 3.23607i 0 −2.23607 2.00000i 0
383.3 0 1.61803 0.618034i 0 3.23607 0 1.23607i 0 2.23607 2.00000i 0
383.4 0 1.61803 + 0.618034i 0 3.23607 0 1.23607i 0 2.23607 + 2.00000i 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.f even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 768.2.f.f 4
3.b odd 2 1 768.2.f.e 4
4.b odd 2 1 768.2.f.c 4
8.b even 2 1 768.2.f.b 4
8.d odd 2 1 768.2.f.e 4
12.b even 2 1 768.2.f.b 4
16.e even 4 1 384.2.c.b yes 4
16.e even 4 1 384.2.c.d yes 4
16.f odd 4 1 384.2.c.a 4
16.f odd 4 1 384.2.c.c yes 4
24.f even 2 1 inner 768.2.f.f 4
24.h odd 2 1 768.2.f.c 4
48.i odd 4 1 384.2.c.a 4
48.i odd 4 1 384.2.c.c yes 4
48.k even 4 1 384.2.c.b yes 4
48.k even 4 1 384.2.c.d yes 4

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

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(768, [\chi])$$:

 $$T_{5}^{2} - 2 T_{5} - 4$$ $$T_{19}^{2} + 6 T_{19} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$9 - 6 T + 2 T^{2} - 2 T^{3} + T^{4}$$
$5$ $$( -4 - 2 T + T^{2} )^{2}$$
$7$ $$16 + 12 T^{2} + T^{4}$$
$11$ $$16 + 28 T^{2} + T^{4}$$
$13$ $$( 20 + T^{2} )^{2}$$
$17$ $$256 + 48 T^{2} + T^{4}$$
$19$ $$( 4 + 6 T + T^{2} )^{2}$$
$23$ $$( -16 - 4 T + T^{2} )^{2}$$
$29$ $$( 44 + 14 T + T^{2} )^{2}$$
$31$ $$16 + 28 T^{2} + T^{4}$$
$37$ $$16 + 72 T^{2} + T^{4}$$
$41$ $$256 + 48 T^{2} + T^{4}$$
$43$ $$( 20 + 10 T + T^{2} )^{2}$$
$47$ $$( 8 + T )^{4}$$
$53$ $$( -4 - 2 T + T^{2} )^{2}$$
$59$ $$16 + 12 T^{2} + T^{4}$$
$61$ $$16 + 72 T^{2} + T^{4}$$
$67$ $$( -36 + 6 T + T^{2} )^{2}$$
$71$ $$( -176 - 4 T + T^{2} )^{2}$$
$73$ $$( -2 + T )^{4}$$
$79$ $$16 + 188 T^{2} + T^{4}$$
$83$ $$400 + 60 T^{2} + T^{4}$$
$89$ $$( 16 + T^{2} )^{2}$$
$97$ $$( -4 + 8 T + T^{2} )^{2}$$