# Properties

 Label 768.2.c.a Level $768$ Weight $2$ Character orbit 768.c 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.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.13251087523$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-2})$$ Defining polynomial: $$x^{2} + 2$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ 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 $$\beta = \sqrt{-2}$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
767.1
 − 1.41421i 1.41421i
0 −1.00000 1.41421i 0 2.82843i 0 2.82843i 0 −1.00000 + 2.82843i 0
767.2 0 −1.00000 + 1.41421i 0 2.82843i 0 2.82843i 0 −1.00000 2.82843i 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
12.b 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.c.a 2
3.b odd 2 1 768.2.c.e 2
4.b odd 2 1 768.2.c.e 2
8.b even 2 1 768.2.c.f 2
8.d odd 2 1 768.2.c.b 2
12.b even 2 1 inner 768.2.c.a 2
16.e even 4 2 384.2.f.b 4
16.f odd 4 2 384.2.f.d yes 4
24.f even 2 1 768.2.c.f 2
24.h odd 2 1 768.2.c.b 2
48.i odd 4 2 384.2.f.d yes 4
48.k even 4 2 384.2.f.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.2.f.b 4 16.e even 4 2
384.2.f.b 4 48.k even 4 2
384.2.f.d yes 4 16.f odd 4 2
384.2.f.d yes 4 48.i odd 4 2
768.2.c.a 2 1.a even 1 1 trivial
768.2.c.a 2 12.b even 2 1 inner
768.2.c.b 2 8.d odd 2 1
768.2.c.b 2 24.h odd 2 1
768.2.c.e 2 3.b odd 2 1
768.2.c.e 2 4.b odd 2 1
768.2.c.f 2 8.b even 2 1
768.2.c.f 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 $$S_{2}^{\mathrm{new}}(768, [\chi])$$:

 $$T_{5}^{2} + 8$$ $$T_{7}^{2} + 8$$ $$T_{11} + 2$$ $$T_{13} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$3 + 2 T + T^{2}$$
$5$ $$8 + T^{2}$$
$7$ $$8 + T^{2}$$
$11$ $$( 2 + T )^{2}$$
$13$ $$( 4 + T )^{2}$$
$17$ $$32 + T^{2}$$
$19$ $$8 + T^{2}$$
$23$ $$( 8 + T )^{2}$$
$29$ $$8 + T^{2}$$
$31$ $$72 + T^{2}$$
$37$ $$( 4 + T )^{2}$$
$41$ $$T^{2}$$
$43$ $$8 + T^{2}$$
$47$ $$T^{2}$$
$53$ $$72 + T^{2}$$
$59$ $$( 6 + T )^{2}$$
$61$ $$( 4 + T )^{2}$$
$67$ $$200 + T^{2}$$
$71$ $$( 8 + T )^{2}$$
$73$ $$( -10 + T )^{2}$$
$79$ $$8 + T^{2}$$
$83$ $$( 6 + T )^{2}$$
$89$ $$32 + T^{2}$$
$97$ $$( 6 + T )^{2}$$