# Properties

 Label 768.3.e.j Level $768$ Weight $3$ Character orbit 768.e Analytic conductor $20.926$ Analytic rank $0$ Dimension $4$ CM discriminant -24 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: $$\Q(i, \sqrt{6})$$ Defining polynomial: $$x^{4} + 9$$ 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 basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 3 \beta_{1} q^{3} + \beta_{2} q^{5} -\beta_{3} q^{7} -9 q^{9} +O(q^{10})$$ $$q + 3 \beta_{1} q^{3} + \beta_{2} q^{5} -\beta_{3} q^{7} -9 q^{9} -10 \beta_{1} q^{11} -3 \beta_{3} q^{15} -3 \beta_{2} q^{21} -71 q^{25} -27 \beta_{1} q^{27} + 3 \beta_{2} q^{29} + 5 \beta_{3} q^{31} + 30 q^{33} -96 \beta_{1} q^{35} -9 \beta_{2} q^{45} + 47 q^{49} + 5 \beta_{2} q^{53} + 10 \beta_{3} q^{55} + 10 \beta_{1} q^{59} + 9 \beta_{3} q^{63} + 50 q^{73} -213 \beta_{1} q^{75} + 10 \beta_{2} q^{77} -15 \beta_{3} q^{79} + 81 q^{81} -134 \beta_{1} q^{83} -9 \beta_{3} q^{87} + 15 \beta_{2} q^{93} -190 q^{97} + 90 \beta_{1} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 36q^{9} + O(q^{10})$$ $$4q - 36q^{9} - 284q^{25} + 120q^{33} + 188q^{49} + 200q^{73} + 324q^{81} - 760q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{2}$$$$/3$$ $$\beta_{2}$$ $$=$$ $$($$$$4 \nu^{3} + 12 \nu$$$$)/3$$ $$\beta_{3}$$ $$=$$ $$($$$$-4 \nu^{3} + 12 \nu$$$$)/3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + \beta_{2}$$$$)/8$$ $$\nu^{2}$$ $$=$$ $$3 \beta_{1}$$ $$\nu^{3}$$ $$=$$ $$($$$$-3 \beta_{3} + 3 \beta_{2}$$$$)/8$$

## 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
 1.22474 − 1.22474i −1.22474 + 1.22474i −1.22474 − 1.22474i 1.22474 + 1.22474i
0 3.00000i 0 9.79796i 0 −9.79796 0 −9.00000 0
257.2 0 3.00000i 0 9.79796i 0 9.79796 0 −9.00000 0
257.3 0 3.00000i 0 9.79796i 0 9.79796 0 −9.00000 0
257.4 0 3.00000i 0 9.79796i 0 −9.79796 0 −9.00000 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
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
8.d odd 2 1 inner
12.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 768.3.e.j 4
3.b odd 2 1 inner 768.3.e.j 4
4.b odd 2 1 inner 768.3.e.j 4
8.b even 2 1 inner 768.3.e.j 4
8.d odd 2 1 inner 768.3.e.j 4
12.b even 2 1 inner 768.3.e.j 4
16.e even 4 1 384.3.h.a 2
16.e even 4 1 384.3.h.d yes 2
16.f odd 4 1 384.3.h.a 2
16.f odd 4 1 384.3.h.d yes 2
24.f even 2 1 inner 768.3.e.j 4
24.h odd 2 1 CM 768.3.e.j 4
48.i odd 4 1 384.3.h.a 2
48.i odd 4 1 384.3.h.d yes 2
48.k even 4 1 384.3.h.a 2
48.k even 4 1 384.3.h.d yes 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.3.h.a 2 16.e even 4 1
384.3.h.a 2 16.f odd 4 1
384.3.h.a 2 48.i odd 4 1
384.3.h.a 2 48.k even 4 1
384.3.h.d yes 2 16.e even 4 1
384.3.h.d yes 2 16.f odd 4 1
384.3.h.d yes 2 48.i odd 4 1
384.3.h.d yes 2 48.k even 4 1
768.3.e.j 4 1.a even 1 1 trivial
768.3.e.j 4 3.b odd 2 1 inner
768.3.e.j 4 4.b odd 2 1 inner
768.3.e.j 4 8.b even 2 1 inner
768.3.e.j 4 8.d odd 2 1 inner
768.3.e.j 4 12.b even 2 1 inner
768.3.e.j 4 24.f even 2 1 inner
768.3.e.j 4 24.h odd 2 1 CM

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(768, [\chi])$$:

 $$T_{5}^{2} + 96$$ $$T_{7}^{2} - 96$$ $$T_{11}^{2} + 100$$ $$T_{19}$$ $$T_{37}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( 9 + T^{2} )^{2}$$
$5$ $$( 96 + T^{2} )^{2}$$
$7$ $$( -96 + T^{2} )^{2}$$
$11$ $$( 100 + T^{2} )^{2}$$
$13$ $$T^{4}$$
$17$ $$T^{4}$$
$19$ $$T^{4}$$
$23$ $$T^{4}$$
$29$ $$( 864 + T^{2} )^{2}$$
$31$ $$( -2400 + T^{2} )^{2}$$
$37$ $$T^{4}$$
$41$ $$T^{4}$$
$43$ $$T^{4}$$
$47$ $$T^{4}$$
$53$ $$( 2400 + T^{2} )^{2}$$
$59$ $$( 100 + T^{2} )^{2}$$
$61$ $$T^{4}$$
$67$ $$T^{4}$$
$71$ $$T^{4}$$
$73$ $$( -50 + T )^{4}$$
$79$ $$( -21600 + T^{2} )^{2}$$
$83$ $$( 17956 + T^{2} )^{2}$$
$89$ $$T^{4}$$
$97$ $$( 190 + T )^{4}$$