# Properties

 Label 768.2.d.f Level $768$ Weight $2$ Character orbit 768.d Analytic conductor $6.133$ Analytic rank $0$ 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.d (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.13251087523$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ 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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + i q^{3} + 4 i q^{5} + 2 q^{7} - q^{9} +O(q^{10})$$ $$q + i q^{3} + 4 i q^{5} + 2 q^{7} - q^{9} + 4 i q^{11} + 2 i q^{13} -4 q^{15} -2 q^{17} -8 i q^{19} + 2 i q^{21} + 4 q^{23} -11 q^{25} -i q^{27} + 6 q^{31} -4 q^{33} + 8 i q^{35} + 2 i q^{37} -2 q^{39} -6 q^{41} -4 i q^{45} -4 q^{47} -3 q^{49} -2 i q^{51} -16 q^{55} + 8 q^{57} -4 i q^{59} + 14 i q^{61} -2 q^{63} -8 q^{65} -4 i q^{67} + 4 i q^{69} + 12 q^{71} + 10 q^{73} -11 i q^{75} + 8 i q^{77} -10 q^{79} + q^{81} + 12 i q^{83} -8 i q^{85} + 14 q^{89} + 4 i q^{91} + 6 i q^{93} + 32 q^{95} + 10 q^{97} -4 i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 4q^{7} - 2q^{9} + O(q^{10})$$ $$2q + 4q^{7} - 2q^{9} - 8q^{15} - 4q^{17} + 8q^{23} - 22q^{25} + 12q^{31} - 8q^{33} - 4q^{39} - 12q^{41} - 8q^{47} - 6q^{49} - 32q^{55} + 16q^{57} - 4q^{63} - 16q^{65} + 24q^{71} + 20q^{73} - 20q^{79} + 2q^{81} + 28q^{89} + 64q^{95} + 20q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
385.1
 − 1.00000i 1.00000i
0 1.00000i 0 4.00000i 0 2.00000 0 −1.00000 0
385.2 0 1.00000i 0 4.00000i 0 2.00000 0 −1.00000 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
8.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.d.f 2
3.b odd 2 1 2304.2.d.o 2
4.b odd 2 1 768.2.d.c 2
8.b even 2 1 inner 768.2.d.f 2
8.d odd 2 1 768.2.d.c 2
12.b even 2 1 2304.2.d.f 2
16.e even 4 1 384.2.a.d yes 1
16.e even 4 1 384.2.a.e yes 1
16.f odd 4 1 384.2.a.a 1
16.f odd 4 1 384.2.a.h yes 1
24.f even 2 1 2304.2.d.f 2
24.h odd 2 1 2304.2.d.o 2
48.i odd 4 1 1152.2.a.a 1
48.i odd 4 1 1152.2.a.s 1
48.k even 4 1 1152.2.a.b 1
48.k even 4 1 1152.2.a.t 1
80.k odd 4 1 9600.2.a.e 1
80.k odd 4 1 9600.2.a.bk 1
80.q even 4 1 9600.2.a.t 1
80.q even 4 1 9600.2.a.bz 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.2.a.a 1 16.f odd 4 1
384.2.a.d yes 1 16.e even 4 1
384.2.a.e yes 1 16.e even 4 1
384.2.a.h yes 1 16.f odd 4 1
768.2.d.c 2 4.b odd 2 1
768.2.d.c 2 8.d odd 2 1
768.2.d.f 2 1.a even 1 1 trivial
768.2.d.f 2 8.b even 2 1 inner
1152.2.a.a 1 48.i odd 4 1
1152.2.a.b 1 48.k even 4 1
1152.2.a.s 1 48.i odd 4 1
1152.2.a.t 1 48.k even 4 1
2304.2.d.f 2 12.b even 2 1
2304.2.d.f 2 24.f even 2 1
2304.2.d.o 2 3.b odd 2 1
2304.2.d.o 2 24.h odd 2 1
9600.2.a.e 1 80.k odd 4 1
9600.2.a.t 1 80.q even 4 1
9600.2.a.bk 1 80.k odd 4 1
9600.2.a.bz 1 80.q even 4 1

## Hecke kernels

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

 $$T_{5}^{2} + 16$$ $$T_{7} - 2$$ $$T_{23} - 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$1 + T^{2}$$
$5$ $$16 + T^{2}$$
$7$ $$( -2 + T )^{2}$$
$11$ $$16 + T^{2}$$
$13$ $$4 + T^{2}$$
$17$ $$( 2 + T )^{2}$$
$19$ $$64 + T^{2}$$
$23$ $$( -4 + T )^{2}$$
$29$ $$T^{2}$$
$31$ $$( -6 + T )^{2}$$
$37$ $$4 + T^{2}$$
$41$ $$( 6 + T )^{2}$$
$43$ $$T^{2}$$
$47$ $$( 4 + T )^{2}$$
$53$ $$T^{2}$$
$59$ $$16 + T^{2}$$
$61$ $$196 + T^{2}$$
$67$ $$16 + T^{2}$$
$71$ $$( -12 + T )^{2}$$
$73$ $$( -10 + T )^{2}$$
$79$ $$( 10 + T )^{2}$$
$83$ $$144 + T^{2}$$
$89$ $$( -14 + T )^{2}$$
$97$ $$( -10 + T )^{2}$$