# Properties

 Label 768.3.e.m Level $768$ Weight $3$ Character orbit 768.e Analytic conductor $20.926$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# 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: $$\Q(i, \sqrt{5})$$ Defining polynomial: $$x^{4} + 3 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{6}$$ 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 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.

 $$f(q)$$ $$=$$ $$q + ( 2 + \beta_{1} ) q^{3} -\beta_{2} q^{5} + \beta_{3} q^{7} + ( -1 + 4 \beta_{1} ) q^{9} +O(q^{10})$$ $$q + ( 2 + \beta_{1} ) q^{3} -\beta_{2} q^{5} + \beta_{3} q^{7} + ( -1 + 4 \beta_{1} ) q^{9} + 2 \beta_{1} q^{11} -2 \beta_{3} q^{13} + ( -2 \beta_{2} - \beta_{3} ) q^{15} + 8 \beta_{1} q^{17} -20 q^{19} + ( -5 \beta_{2} + 2 \beta_{3} ) q^{21} -4 \beta_{2} q^{23} + 9 q^{25} + ( -22 + 7 \beta_{1} ) q^{27} + 13 \beta_{2} q^{29} + 3 \beta_{3} q^{31} + ( -10 + 4 \beta_{1} ) q^{33} + 16 \beta_{1} q^{35} + 6 \beta_{3} q^{37} + ( 10 \beta_{2} - 4 \beta_{3} ) q^{39} + 16 \beta_{1} q^{41} -36 q^{43} + ( \beta_{2} - 4 \beta_{3} ) q^{45} + 16 \beta_{2} q^{47} + 31 q^{49} + ( -40 + 16 \beta_{1} ) q^{51} -5 \beta_{2} q^{53} -2 \beta_{3} q^{55} + ( -40 - 20 \beta_{1} ) q^{57} + 46 \beta_{1} q^{59} -2 \beta_{3} q^{61} + ( -20 \beta_{2} - \beta_{3} ) q^{63} -32 \beta_{1} q^{65} -44 q^{67} + ( -8 \beta_{2} - 4 \beta_{3} ) q^{69} -20 \beta_{2} q^{71} + 50 q^{73} + ( 18 + 9 \beta_{1} ) q^{75} -10 \beta_{2} q^{77} -9 \beta_{3} q^{79} + ( -79 - 8 \beta_{1} ) q^{81} + 46 \beta_{1} q^{83} -8 \beta_{3} q^{85} + ( 26 \beta_{2} + 13 \beta_{3} ) q^{87} -72 \beta_{1} q^{89} -160 q^{91} + ( -15 \beta_{2} + 6 \beta_{3} ) q^{93} + 20 \beta_{2} q^{95} + 50 q^{97} + ( -40 - 2 \beta_{1} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 8q^{3} - 4q^{9} + O(q^{10})$$ $$4q + 8q^{3} - 4q^{9} - 80q^{19} + 36q^{25} - 88q^{27} - 40q^{33} - 144q^{43} + 124q^{49} - 160q^{51} - 160q^{57} - 176q^{67} + 200q^{73} + 72q^{75} - 316q^{81} - 640q^{91} + 200q^{97} - 160q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 3 x^{2} + 1$$:

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

## 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
 − 0.618034i − 1.61803i 1.61803i 0.618034i
0 2.00000 2.23607i 0 4.00000i 0 8.94427 0 −1.00000 8.94427i 0
257.2 0 2.00000 2.23607i 0 4.00000i 0 −8.94427 0 −1.00000 8.94427i 0
257.3 0 2.00000 + 2.23607i 0 4.00000i 0 −8.94427 0 −1.00000 + 8.94427i 0
257.4 0 2.00000 + 2.23607i 0 4.00000i 0 8.94427 0 −1.00000 + 8.94427i 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
3.b odd 2 1 inner
8.d odd 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.m 4
3.b odd 2 1 inner 768.3.e.m 4
4.b odd 2 1 768.3.e.h 4
8.b even 2 1 768.3.e.h 4
8.d odd 2 1 inner 768.3.e.m 4
12.b even 2 1 768.3.e.h 4
16.e even 4 1 384.3.h.e 4
16.e even 4 1 384.3.h.f yes 4
16.f odd 4 1 384.3.h.e 4
16.f odd 4 1 384.3.h.f yes 4
24.f even 2 1 inner 768.3.e.m 4
24.h odd 2 1 768.3.e.h 4
48.i odd 4 1 384.3.h.e 4
48.i odd 4 1 384.3.h.f yes 4
48.k even 4 1 384.3.h.e 4
48.k even 4 1 384.3.h.f yes 4

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

## Hecke kernels

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

 $$T_{5}^{2} + 16$$ $$T_{7}^{2} - 80$$ $$T_{11}^{2} + 20$$ $$T_{19} + 20$$ $$T_{37}^{2} - 2880$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( 9 - 4 T + T^{2} )^{2}$$
$5$ $$( 16 + T^{2} )^{2}$$
$7$ $$( -80 + T^{2} )^{2}$$
$11$ $$( 20 + T^{2} )^{2}$$
$13$ $$( -320 + T^{2} )^{2}$$
$17$ $$( 320 + T^{2} )^{2}$$
$19$ $$( 20 + T )^{4}$$
$23$ $$( 256 + T^{2} )^{2}$$
$29$ $$( 2704 + T^{2} )^{2}$$
$31$ $$( -720 + T^{2} )^{2}$$
$37$ $$( -2880 + T^{2} )^{2}$$
$41$ $$( 1280 + T^{2} )^{2}$$
$43$ $$( 36 + T )^{4}$$
$47$ $$( 4096 + T^{2} )^{2}$$
$53$ $$( 400 + T^{2} )^{2}$$
$59$ $$( 10580 + T^{2} )^{2}$$
$61$ $$( -320 + T^{2} )^{2}$$
$67$ $$( 44 + T )^{4}$$
$71$ $$( 6400 + T^{2} )^{2}$$
$73$ $$( -50 + T )^{4}$$
$79$ $$( -6480 + T^{2} )^{2}$$
$83$ $$( 10580 + T^{2} )^{2}$$
$89$ $$( 25920 + T^{2} )^{2}$$
$97$ $$( -50 + T )^{4}$$