# Properties

 Label 768.2.c.d Level $768$ Weight $2$ Character orbit 768.c Analytic conductor $6.133$ Analytic rank $0$ Dimension $2$ CM discriminant -8 Inner twists $4$

# 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: $$0$$ 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: $\mathrm{U}(1)[D_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( 1 + \beta ) q^{3} + ( -1 + 2 \beta ) q^{9} +O(q^{10})$$ $$q + ( 1 + \beta ) q^{3} + ( -1 + 2 \beta ) q^{9} -6 q^{11} + 4 \beta q^{17} + 6 \beta q^{19} + 5 q^{25} + ( -5 + \beta ) q^{27} + ( -6 - 6 \beta ) q^{33} -8 \beta q^{41} + 6 \beta q^{43} + 7 q^{49} + ( -8 + 4 \beta ) q^{51} + ( -12 + 6 \beta ) q^{57} + 6 q^{59} -6 \beta q^{67} + 2 q^{73} + ( 5 + 5 \beta ) q^{75} + ( -7 - 4 \beta ) q^{81} -18 q^{83} -4 \beta q^{89} + 10 q^{97} + ( 6 - 12 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{3} - 2q^{9} + O(q^{10})$$ $$2q + 2q^{3} - 2q^{9} - 12q^{11} + 10q^{25} - 10q^{27} - 12q^{33} + 14q^{49} - 16q^{51} - 24q^{57} + 12q^{59} + 4q^{73} + 10q^{75} - 14q^{81} - 36q^{83} + 20q^{97} + 12q^{99} + O(q^{100})$$

## 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 0 0 0 0 −1.00000 2.82843i 0
767.2 0 1.00000 + 1.41421i 0 0 0 0 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
8.d odd 2 1 CM by $$\Q(\sqrt{-2})$$
12.b 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 768.2.c.d 2
3.b odd 2 1 768.2.c.c 2
4.b odd 2 1 768.2.c.c 2
8.b even 2 1 768.2.c.c 2
8.d odd 2 1 CM 768.2.c.d 2
12.b even 2 1 inner 768.2.c.d 2
16.e even 4 2 384.2.f.c 4
16.f odd 4 2 384.2.f.c 4
24.f even 2 1 768.2.c.c 2
24.h odd 2 1 inner 768.2.c.d 2
48.i odd 4 2 384.2.f.c 4
48.k even 4 2 384.2.f.c 4

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

 $$T_{5}$$ $$T_{7}$$ $$T_{11} + 6$$ $$T_{13}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$3 - 2 T + T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$( 6 + T )^{2}$$
$13$ $$T^{2}$$
$17$ $$32 + T^{2}$$
$19$ $$72 + T^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$T^{2}$$
$37$ $$T^{2}$$
$41$ $$128 + T^{2}$$
$43$ $$72 + T^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$( -6 + T )^{2}$$
$61$ $$T^{2}$$
$67$ $$72 + T^{2}$$
$71$ $$T^{2}$$
$73$ $$( -2 + T )^{2}$$
$79$ $$T^{2}$$
$83$ $$( 18 + T )^{2}$$
$89$ $$32 + T^{2}$$
$97$ $$( -10 + T )^{2}$$