# Properties

 Label 768.3.e.l 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: $$\Q(\sqrt{-2}, \sqrt{7})$$ Defining polynomial: $$x^{4} + 8 x^{2} + 9$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: no (minimal twist has level 24) 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.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{3} + 2 \beta_{1} q^{5} + 4 q^{7} + ( 5 + \beta_{2} ) q^{9} +O(q^{10})$$ $$q + \beta_{3} q^{3} + 2 \beta_{1} q^{5} + 4 q^{7} + ( 5 + \beta_{2} ) q^{9} + 3 \beta_{1} q^{11} + ( 2 \beta_{1} + 4 \beta_{3} ) q^{13} + ( 8 - 2 \beta_{2} ) q^{15} + 2 \beta_{2} q^{17} + ( -\beta_{1} - 2 \beta_{3} ) q^{19} + 4 \beta_{3} q^{21} -4 \beta_{2} q^{23} -7 q^{25} + ( -9 \beta_{1} + \beta_{3} ) q^{27} + 6 \beta_{1} q^{29} + 4 q^{31} + ( 12 - 3 \beta_{2} ) q^{33} + 8 \beta_{1} q^{35} + ( 10 \beta_{1} + 20 \beta_{3} ) q^{37} + ( 28 + 2 \beta_{2} ) q^{39} -4 \beta_{2} q^{41} + ( -\beta_{1} - 2 \beta_{3} ) q^{43} + ( 18 \beta_{1} + 16 \beta_{3} ) q^{45} -33 q^{49} + ( -18 \beta_{1} - 8 \beta_{3} ) q^{51} + 18 \beta_{1} q^{53} -48 q^{55} + ( -14 - \beta_{2} ) q^{57} + 17 \beta_{1} q^{59} + ( 18 \beta_{1} + 36 \beta_{3} ) q^{61} + ( 20 + 4 \beta_{2} ) q^{63} -8 \beta_{2} q^{65} + ( 9 \beta_{1} + 18 \beta_{3} ) q^{67} + ( 36 \beta_{1} + 16 \beta_{3} ) q^{69} + 12 \beta_{2} q^{71} + 6 q^{73} -7 \beta_{3} q^{75} + 12 \beta_{1} q^{77} -124 q^{79} + ( -31 + 10 \beta_{2} ) q^{81} + \beta_{1} q^{83} + ( 16 \beta_{1} + 32 \beta_{3} ) q^{85} + ( 24 - 6 \beta_{2} ) q^{87} + 14 \beta_{2} q^{89} + ( 8 \beta_{1} + 16 \beta_{3} ) q^{91} + 4 \beta_{3} q^{93} + 4 \beta_{2} q^{95} + 118 q^{97} + ( 27 \beta_{1} + 24 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 16q^{7} + 20q^{9} + O(q^{10})$$ $$4q + 16q^{7} + 20q^{9} + 32q^{15} - 28q^{25} + 16q^{31} + 48q^{33} + 112q^{39} - 132q^{49} - 192q^{55} - 56q^{57} + 80q^{63} + 24q^{73} - 496q^{79} - 124q^{81} + 96q^{87} + 472q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 8 x^{2} + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-2 \nu^{3} - 10 \nu$$$$)/3$$ $$\beta_{2}$$ $$=$$ $$($$$$2 \nu^{3} + 22 \nu$$$$)/3$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{3} + 3 \nu^{2} + 5 \nu + 12$$$$)/3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} + \beta_{1}$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$2 \beta_{3} + \beta_{1} - 8$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$-5 \beta_{2} - 11 \beta_{1}$$$$)/4$$

## 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
 2.57794i − 2.57794i − 1.16372i 1.16372i
0 −2.64575 1.41421i 0 5.65685i 0 4.00000 0 5.00000 + 7.48331i 0
257.2 0 −2.64575 + 1.41421i 0 5.65685i 0 4.00000 0 5.00000 7.48331i 0
257.3 0 2.64575 1.41421i 0 5.65685i 0 4.00000 0 5.00000 7.48331i 0
257.4 0 2.64575 + 1.41421i 0 5.65685i 0 4.00000 0 5.00000 + 7.48331i 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.b 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.l 4
3.b odd 2 1 inner 768.3.e.l 4
4.b odd 2 1 768.3.e.i 4
8.b even 2 1 inner 768.3.e.l 4
8.d odd 2 1 768.3.e.i 4
12.b even 2 1 768.3.e.i 4
16.e even 4 2 96.3.h.c 4
16.f odd 4 2 24.3.h.c 4
24.f even 2 1 768.3.e.i 4
24.h odd 2 1 inner 768.3.e.l 4
48.i odd 4 2 96.3.h.c 4
48.k even 4 2 24.3.h.c 4

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

 $$T_{5}^{2} + 32$$ $$T_{7} - 4$$ $$T_{11}^{2} + 72$$ $$T_{19}^{2} - 28$$ $$T_{37}^{2} - 2800$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 - 10 T^{2} + 81 T^{4}$$
$5$ $$( 1 - 18 T^{2} + 625 T^{4} )^{2}$$
$7$ $$( 1 - 4 T + 49 T^{2} )^{4}$$
$11$ $$( 1 - 170 T^{2} + 14641 T^{4} )^{2}$$
$13$ $$( 1 + 226 T^{2} + 28561 T^{4} )^{2}$$
$17$ $$( 1 - 354 T^{2} + 83521 T^{4} )^{2}$$
$19$ $$( 1 + 694 T^{2} + 130321 T^{4} )^{2}$$
$23$ $$( 1 - 162 T^{2} + 279841 T^{4} )^{2}$$
$29$ $$( 1 - 1394 T^{2} + 707281 T^{4} )^{2}$$
$31$ $$( 1 - 4 T + 961 T^{2} )^{4}$$
$37$ $$( 1 - 62 T^{2} + 1874161 T^{4} )^{2}$$
$41$ $$( 1 - 2466 T^{2} + 2825761 T^{4} )^{2}$$
$43$ $$( 1 + 3670 T^{2} + 3418801 T^{4} )^{2}$$
$47$ $$( 1 - 47 T )^{4}( 1 + 47 T )^{4}$$
$53$ $$( 1 - 3026 T^{2} + 7890481 T^{4} )^{2}$$
$59$ $$( 1 - 4650 T^{2} + 12117361 T^{4} )^{2}$$
$61$ $$( 1 - 1630 T^{2} + 13845841 T^{4} )^{2}$$
$67$ $$( 1 + 6710 T^{2} + 20151121 T^{4} )^{2}$$
$71$ $$( 1 - 110 T + 5041 T^{2} )^{2}( 1 + 110 T + 5041 T^{2} )^{2}$$
$73$ $$( 1 - 6 T + 5329 T^{2} )^{4}$$
$79$ $$( 1 + 124 T + 6241 T^{2} )^{4}$$
$83$ $$( 1 - 13770 T^{2} + 47458321 T^{4} )^{2}$$
$89$ $$( 1 - 4866 T^{2} + 62742241 T^{4} )^{2}$$
$97$ $$( 1 - 118 T + 9409 T^{2} )^{4}$$