# 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: $$x^{2} + 1$$ 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 $$q - i q^{3} + 2 q^{7} - q^{9} + 4 i q^{11} - 6 i q^{13} + 6 q^{17} - 2 i q^{21} + 4 q^{23} + 5 q^{25} + i q^{27} - 4 i q^{29} - 10 q^{31} + 4 q^{33} + 2 i q^{37} - 6 q^{39} + 2 q^{41} - 8 i q^{43} + 12 q^{47} - 3 q^{49} - 6 i q^{51} - 12 i q^{53} + 4 i q^{59} - 2 i q^{61} - 2 q^{63} + 4 i q^{67} - 4 i q^{69} - 4 q^{71} + 10 q^{73} - 5 i q^{75} + 8 i q^{77} + 6 q^{79} + q^{81} + 12 i q^{83} - 4 q^{87} - 2 q^{89} - 12 i q^{91} + 10 i q^{93} - 6 q^{97} - 4 i q^{99} +O(q^{100})$$ 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 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{7} - 2 q^{9}+O(q^{10})$$ 2 * q + 4 * q^7 - 2 * q^9 $$2 q + 4 q^{7} - 2 q^{9} + 12 q^{17} + 8 q^{23} + 10 q^{25} - 20 q^{31} + 8 q^{33} - 12 q^{39} + 4 q^{41} + 24 q^{47} - 6 q^{49} - 4 q^{63} - 8 q^{71} + 20 q^{73} + 12 q^{79} + 2 q^{81} - 8 q^{87} - 4 q^{89} - 12 q^{97}+O(q^{100})$$ 2 * q + 4 * q^7 - 2 * q^9 + 12 * q^17 + 8 * q^23 + 10 * q^25 - 20 * q^31 + 8 * q^33 - 12 * q^39 + 4 * q^41 + 24 * q^47 - 6 * q^49 - 4 * q^63 - 8 * q^71 + 20 * q^73 + 12 * q^79 + 2 * q^81 - 8 * q^87 - 4 * q^89 - 12 * q^97

## 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 $$S_{2}^{\mathrm{new}}(768, [\chi])$$:

 $$T_{5}$$ T5 $$T_{7} - 2$$ T7 - 2 $$T_{23} - 4$$ T23 - 4

## Hecke characteristic polynomials

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