# Properties

 Label 768.1.e.c Level $768$ Weight $1$ Character orbit 768.e Analytic conductor $0.383$ Analytic rank $0$ Dimension $2$ Projective image $D_{2}$ CM/RM discs -8, -24, 12 Inner twists $8$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$768 = 2^{8} \cdot 3$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 768.e (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.383281929702$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ 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) Projective image $$D_{2}$$ Projective field Galois closure of $$\Q(\sqrt{-2}, \sqrt{3})$$ Artin image $D_4:C_2$ Artin field Galois closure of 8.0.150994944.1

## $q$-expansion

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q -i q^{3} - q^{9} +O(q^{10})$$ $$q -i q^{3} - q^{9} -2 i q^{11} + q^{25} + i q^{27} -2 q^{33} - q^{49} + 2 i q^{59} + 2 q^{73} -i q^{75} + q^{81} + 2 i q^{83} + 2 q^{97} + 2 i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{9} + O(q^{10})$$ $$2q - 2q^{9} + 2q^{25} - 4q^{33} - 2q^{49} + 4q^{73} + 2q^{81} + 4q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
257.1
 1.00000i − 1.00000i
0 1.00000i 0 0 0 0 0 −1.00000 0
257.2 0 1.00000i 0 0 0 0 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.d odd 2 1 CM by $$\Q(\sqrt{-2})$$
12.b even 2 1 RM by $$\Q(\sqrt{3})$$
24.h odd 2 1 CM by $$\Q(\sqrt{-6})$$
3.b odd 2 1 inner
4.b odd 2 1 inner
8.b even 2 1 inner
24.f even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 768.1.e.c 2
3.b odd 2 1 inner 768.1.e.c 2
4.b odd 2 1 inner 768.1.e.c 2
8.b even 2 1 inner 768.1.e.c 2
8.d odd 2 1 CM 768.1.e.c 2
12.b even 2 1 RM 768.1.e.c 2
16.e even 4 1 384.1.h.a 1
16.e even 4 1 384.1.h.b yes 1
16.f odd 4 1 384.1.h.a 1
16.f odd 4 1 384.1.h.b yes 1
24.f even 2 1 inner 768.1.e.c 2
24.h odd 2 1 CM 768.1.e.c 2
32.g even 8 4 3072.1.i.f 4
32.h odd 8 4 3072.1.i.f 4
48.i odd 4 1 384.1.h.a 1
48.i odd 4 1 384.1.h.b yes 1
48.k even 4 1 384.1.h.a 1
48.k even 4 1 384.1.h.b yes 1
96.o even 8 4 3072.1.i.f 4
96.p odd 8 4 3072.1.i.f 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.1.h.a 1 16.e even 4 1
384.1.h.a 1 16.f odd 4 1
384.1.h.a 1 48.i odd 4 1
384.1.h.a 1 48.k even 4 1
384.1.h.b yes 1 16.e even 4 1
384.1.h.b yes 1 16.f odd 4 1
384.1.h.b yes 1 48.i odd 4 1
384.1.h.b yes 1 48.k even 4 1
768.1.e.c 2 1.a even 1 1 trivial
768.1.e.c 2 3.b odd 2 1 inner
768.1.e.c 2 4.b odd 2 1 inner
768.1.e.c 2 8.b even 2 1 inner
768.1.e.c 2 8.d odd 2 1 CM
768.1.e.c 2 12.b even 2 1 RM
768.1.e.c 2 24.f even 2 1 inner
768.1.e.c 2 24.h odd 2 1 CM
3072.1.i.f 4 32.g even 8 4
3072.1.i.f 4 32.h odd 8 4
3072.1.i.f 4 96.o even 8 4
3072.1.i.f 4 96.p odd 8 4

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{1}^{\mathrm{new}}(768, [\chi])$$:

 $$T_{11}^{2} + 4$$ $$T_{19}$$

## Hecke characteristic polynomials

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