# Properties

 Label 768.3.e.k Level $768$ Weight $3$ Character orbit 768.e Analytic conductor $20.926$ Analytic rank $0$ Dimension $4$ CM discriminant -8 Inner twists $8$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$20.9264843029$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{5}$$ 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 a primitive root of unity $$\zeta_{8}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 2 \zeta_{8} + \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{3} + ( 7 + 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{9} +O(q^{10})$$ $$q + ( 2 \zeta_{8} + \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{3} + ( 7 + 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{9} -14 \zeta_{8}^{2} q^{11} + ( -24 \zeta_{8} - 24 \zeta_{8}^{3} ) q^{17} + ( 12 \zeta_{8} - 12 \zeta_{8}^{3} ) q^{19} + 25 q^{25} + ( 10 \zeta_{8} + 23 \zeta_{8}^{2} - 10 \zeta_{8}^{3} ) q^{27} + ( 14 - 28 \zeta_{8} - 28 \zeta_{8}^{3} ) q^{33} + ( 48 \zeta_{8} + 48 \zeta_{8}^{3} ) q^{41} + ( 60 \zeta_{8} - 60 \zeta_{8}^{3} ) q^{43} -49 q^{49} + ( 24 \zeta_{8} - 96 \zeta_{8}^{2} - 24 \zeta_{8}^{3} ) q^{51} + ( 48 + 12 \zeta_{8} + 12 \zeta_{8}^{3} ) q^{57} -82 \zeta_{8}^{2} q^{59} + ( 84 \zeta_{8} - 84 \zeta_{8}^{3} ) q^{67} -142 q^{73} + ( 50 \zeta_{8} + 25 \zeta_{8}^{2} - 50 \zeta_{8}^{3} ) q^{75} + ( 17 + 56 \zeta_{8} + 56 \zeta_{8}^{3} ) q^{81} + 158 \zeta_{8}^{2} q^{83} + ( -72 \zeta_{8} - 72 \zeta_{8}^{3} ) q^{89} -94 q^{97} + ( 56 \zeta_{8} - 98 \zeta_{8}^{2} - 56 \zeta_{8}^{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 28q^{9} + O(q^{10})$$ $$4q + 28q^{9} + 100q^{25} + 56q^{33} - 196q^{49} + 192q^{57} - 568q^{73} + 68q^{81} - 376q^{97} + O(q^{100})$$

## 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
 −0.707107 + 0.707107i −0.707107 − 0.707107i 0.707107 − 0.707107i 0.707107 + 0.707107i
0 −2.82843 1.00000i 0 0 0 0 0 7.00000 + 5.65685i 0
257.2 0 −2.82843 + 1.00000i 0 0 0 0 0 7.00000 5.65685i 0
257.3 0 2.82843 1.00000i 0 0 0 0 0 7.00000 5.65685i 0
257.4 0 2.82843 + 1.00000i 0 0 0 0 0 7.00000 + 5.65685i 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 $$\Q(\sqrt{-2})$$
3.b odd 2 1 inner
4.b odd 2 1 inner
8.b even 2 1 inner
12.b even 2 1 inner
24.f 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 768.3.e.k 4
3.b odd 2 1 inner 768.3.e.k 4
4.b odd 2 1 inner 768.3.e.k 4
8.b even 2 1 inner 768.3.e.k 4
8.d odd 2 1 CM 768.3.e.k 4
12.b even 2 1 inner 768.3.e.k 4
16.e even 4 1 384.3.h.b 2
16.e even 4 1 384.3.h.c yes 2
16.f odd 4 1 384.3.h.b 2
16.f odd 4 1 384.3.h.c yes 2
24.f even 2 1 inner 768.3.e.k 4
24.h odd 2 1 inner 768.3.e.k 4
48.i odd 4 1 384.3.h.b 2
48.i odd 4 1 384.3.h.c yes 2
48.k even 4 1 384.3.h.b 2
48.k even 4 1 384.3.h.c yes 2

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

 $$T_{5}$$ $$T_{7}$$ $$T_{11}^{2} + 196$$ $$T_{19}^{2} - 288$$ $$T_{37}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$81 - 14 T^{2} + T^{4}$$
$5$ $$T^{4}$$
$7$ $$T^{4}$$
$11$ $$( 196 + T^{2} )^{2}$$
$13$ $$T^{4}$$
$17$ $$( 1152 + T^{2} )^{2}$$
$19$ $$( -288 + T^{2} )^{2}$$
$23$ $$T^{4}$$
$29$ $$T^{4}$$
$31$ $$T^{4}$$
$37$ $$T^{4}$$
$41$ $$( 4608 + T^{2} )^{2}$$
$43$ $$( -7200 + T^{2} )^{2}$$
$47$ $$T^{4}$$
$53$ $$T^{4}$$
$59$ $$( 6724 + T^{2} )^{2}$$
$61$ $$T^{4}$$
$67$ $$( -14112 + T^{2} )^{2}$$
$71$ $$T^{4}$$
$73$ $$( 142 + T )^{4}$$
$79$ $$T^{4}$$
$83$ $$( 24964 + T^{2} )^{2}$$
$89$ $$( 10368 + T^{2} )^{2}$$
$97$ $$( 94 + T )^{4}$$