# Properties

 Label 768.2.d.g 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.

 $$f(q)$$ $$=$$ $$q -i q^{3} + 2 q^{7} - q^{9} +O(q^{10})$$ $$q -i q^{3} + 2 q^{7} - q^{9} + 4 i q^{11} -6 i q^{13} + 6 q^{17} -2 i q^{21} + 4 q^{23} + 5 q^{25} + i q^{27} -4 i q^{29} -10 q^{31} + 4 q^{33} + 2 i q^{37} -6 q^{39} + 2 q^{41} -8 i q^{43} + 12 q^{47} -3 q^{49} -6 i q^{51} -12 i q^{53} + 4 i q^{59} -2 i q^{61} -2 q^{63} + 4 i q^{67} -4 i q^{69} -4 q^{71} + 10 q^{73} -5 i q^{75} + 8 i q^{77} + 6 q^{79} + q^{81} + 12 i q^{83} -4 q^{87} -2 q^{89} -12 i q^{91} + 10 i q^{93} -6 q^{97} -4 i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 4q^{7} - 2q^{9} + O(q^{10})$$ $$2q + 4q^{7} - 2q^{9} + 12q^{17} + 8q^{23} + 10q^{25} - 20q^{31} + 8q^{33} - 12q^{39} + 4q^{41} + 24q^{47} - 6q^{49} - 4q^{63} - 8q^{71} + 20q^{73} + 12q^{79} + 2q^{81} - 8q^{87} - 4q^{89} - 12q^{97} + O(q^{100})$$

## 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 0 0 2.00000 0 −1.00000 0
385.2 0 1.00000i 0 0 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.g 2
3.b odd 2 1 2304.2.d.m 2
4.b odd 2 1 768.2.d.b 2
8.b even 2 1 inner 768.2.d.g 2
8.d odd 2 1 768.2.d.b 2
12.b even 2 1 2304.2.d.d 2
16.e even 4 1 384.2.a.b 1
16.e even 4 1 384.2.a.f yes 1
16.f odd 4 1 384.2.a.c yes 1
16.f odd 4 1 384.2.a.g yes 1
24.f even 2 1 2304.2.d.d 2
24.h odd 2 1 2304.2.d.m 2
48.i odd 4 1 1152.2.a.i 1
48.i odd 4 1 1152.2.a.j 1
48.k even 4 1 1152.2.a.k 1
48.k even 4 1 1152.2.a.l 1
80.k odd 4 1 9600.2.a.h 1
80.k odd 4 1 9600.2.a.bh 1
80.q even 4 1 9600.2.a.w 1
80.q even 4 1 9600.2.a.bw 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.2.a.b 1 16.e even 4 1
384.2.a.c yes 1 16.f odd 4 1
384.2.a.f yes 1 16.e even 4 1
384.2.a.g yes 1 16.f odd 4 1
768.2.d.b 2 4.b odd 2 1
768.2.d.b 2 8.d odd 2 1
768.2.d.g 2 1.a even 1 1 trivial
768.2.d.g 2 8.b even 2 1 inner
1152.2.a.i 1 48.i odd 4 1
1152.2.a.j 1 48.i odd 4 1
1152.2.a.k 1 48.k even 4 1
1152.2.a.l 1 48.k even 4 1
2304.2.d.d 2 12.b even 2 1
2304.2.d.d 2 24.f even 2 1
2304.2.d.m 2 3.b odd 2 1
2304.2.d.m 2 24.h odd 2 1
9600.2.a.h 1 80.k odd 4 1
9600.2.a.w 1 80.q even 4 1
9600.2.a.bh 1 80.k odd 4 1
9600.2.a.bw 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 $$S_{2}^{\mathrm{new}}(768, [\chi])$$:

 $$T_{5}$$ $$T_{7} - 2$$ $$T_{23} - 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$1 + T^{2}$$
$5$ $$T^{2}$$
$7$ $$( -2 + T )^{2}$$
$11$ $$16 + T^{2}$$
$13$ $$36 + T^{2}$$
$17$ $$( -6 + T )^{2}$$
$19$ $$T^{2}$$
$23$ $$( -4 + T )^{2}$$
$29$ $$16 + T^{2}$$
$31$ $$( 10 + T )^{2}$$
$37$ $$4 + T^{2}$$
$41$ $$( -2 + T )^{2}$$
$43$ $$64 + T^{2}$$
$47$ $$( -12 + T )^{2}$$
$53$ $$144 + T^{2}$$
$59$ $$16 + T^{2}$$
$61$ $$4 + T^{2}$$
$67$ $$16 + T^{2}$$
$71$ $$( 4 + T )^{2}$$
$73$ $$( -10 + T )^{2}$$
$79$ $$( -6 + T )^{2}$$
$83$ $$144 + T^{2}$$
$89$ $$( 2 + T )^{2}$$
$97$ $$( 6 + T )^{2}$$