# Properties

 Label 768.2.a.i Level $768$ Weight $2$ Character orbit 768.a Self dual yes Analytic conductor $6.133$ Analytic rank $1$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$768 = 2^{8} \cdot 3$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 768.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$6.13251087523$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 384) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 2\sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −1.41421 1.41421
0 −1.00000 0 −2.82843 0 2.82843 0 1.00000 0
1.2 0 −1.00000 0 2.82843 0 −2.82843 0 1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$3$$ $$1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 768.2.a.i 2
3.b odd 2 1 2304.2.a.x 2
4.b odd 2 1 768.2.a.l 2
8.b even 2 1 768.2.a.l 2
8.d odd 2 1 inner 768.2.a.i 2
12.b even 2 1 2304.2.a.r 2
16.e even 4 2 384.2.d.c 4
16.f odd 4 2 384.2.d.c 4
24.f even 2 1 2304.2.a.x 2
24.h odd 2 1 2304.2.a.r 2
48.i odd 4 2 1152.2.d.h 4
48.k even 4 2 1152.2.d.h 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.2.d.c 4 16.e even 4 2
384.2.d.c 4 16.f odd 4 2
768.2.a.i 2 1.a even 1 1 trivial
768.2.a.i 2 8.d odd 2 1 inner
768.2.a.l 2 4.b odd 2 1
768.2.a.l 2 8.b even 2 1
1152.2.d.h 4 48.i odd 4 2
1152.2.d.h 4 48.k even 4 2
2304.2.a.r 2 12.b even 2 1
2304.2.a.r 2 24.h odd 2 1
2304.2.a.x 2 3.b odd 2 1
2304.2.a.x 2 24.f even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(768))$$:

 $$T_{5}^{2} - 8$$ $$T_{7}^{2} - 8$$ $$T_{11} + 4$$ $$T_{19} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$( 1 + T )^{2}$$
$5$ $$-8 + T^{2}$$
$7$ $$-8 + T^{2}$$
$11$ $$( 4 + T )^{2}$$
$13$ $$-32 + T^{2}$$
$17$ $$( 2 + T )^{2}$$
$19$ $$( 4 + T )^{2}$$
$23$ $$-32 + T^{2}$$
$29$ $$-8 + T^{2}$$
$31$ $$-72 + T^{2}$$
$37$ $$T^{2}$$
$41$ $$( 10 + T )^{2}$$
$43$ $$( 12 + T )^{2}$$
$47$ $$-32 + T^{2}$$
$53$ $$-8 + T^{2}$$
$59$ $$( -4 + T )^{2}$$
$61$ $$-128 + T^{2}$$
$67$ $$( 4 + T )^{2}$$
$71$ $$-32 + T^{2}$$
$73$ $$( -2 + T )^{2}$$
$79$ $$-72 + T^{2}$$
$83$ $$( -4 + T )^{2}$$
$89$ $$( 6 + T )^{2}$$
$97$ $$( -14 + T )^{2}$$