# Properties

 Label 768.4.a.u Level $768$ Weight $4$ Character orbit 768.a Self dual yes Analytic conductor $45.313$ Analytic rank $1$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$768 = 2^{8} \cdot 3$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 768.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$45.3134668844$$ Analytic rank: $$1$$ Dimension: $$4$$ Coefficient field: 4.4.9792.1 Defining polynomial: $$x^{4} - 2 x^{3} - 7 x^{2} + 2 x + 7$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{10}$$ Twist minimal: no (minimal twist has level 384) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-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.

 $$f(q)$$ $$=$$ $$q -3 q^{3} + \beta_{1} q^{5} + ( -\beta_{1} + \beta_{3} ) q^{7} + 9 q^{9} +O(q^{10})$$ $$q -3 q^{3} + \beta_{1} q^{5} + ( -\beta_{1} + \beta_{3} ) q^{7} + 9 q^{9} + ( -12 - \beta_{2} ) q^{11} -3 \beta_{3} q^{13} -3 \beta_{1} q^{15} + ( 30 + 3 \beta_{2} ) q^{17} + ( -12 + 3 \beta_{2} ) q^{19} + ( 3 \beta_{1} - 3 \beta_{3} ) q^{21} + ( -6 \beta_{1} - 6 \beta_{3} ) q^{23} + ( 43 - 6 \beta_{2} ) q^{25} -27 q^{27} + ( 3 \beta_{1} + 6 \beta_{3} ) q^{29} + ( -13 \beta_{1} + \beta_{3} ) q^{31} + ( 36 + 3 \beta_{2} ) q^{33} + ( -120 - 3 \beta_{2} ) q^{35} + ( -18 \beta_{1} + 3 \beta_{3} ) q^{37} + 9 \beta_{3} q^{39} + ( 102 - 15 \beta_{2} ) q^{41} + ( -132 - 3 \beta_{2} ) q^{43} + 9 \beta_{1} q^{45} + ( 30 \beta_{1} - 6 \beta_{3} ) q^{47} + ( 209 + 24 \beta_{2} ) q^{49} + ( -90 - 9 \beta_{2} ) q^{51} + ( 11 \beta_{1} + 6 \beta_{3} ) q^{53} + ( 4 \beta_{1} + 8 \beta_{3} ) q^{55} + ( 36 - 9 \beta_{2} ) q^{57} + ( -468 + 16 \beta_{2} ) q^{59} + ( -54 \beta_{1} + 15 \beta_{3} ) q^{61} + ( -9 \beta_{1} + 9 \beta_{3} ) q^{63} + ( -144 + 27 \beta_{2} ) q^{65} + ( -588 + 12 \beta_{2} ) q^{67} + ( 18 \beta_{1} + 18 \beta_{3} ) q^{69} + ( 54 \beta_{1} - 18 \beta_{3} ) q^{71} + ( 242 - 24 \beta_{2} ) q^{73} + ( -129 + 18 \beta_{2} ) q^{75} + ( 28 \beta_{1} - 36 \beta_{3} ) q^{77} + ( 51 \beta_{1} + 9 \beta_{3} ) q^{79} + 81 q^{81} + ( -852 - 17 \beta_{2} ) q^{83} + ( -18 \beta_{1} - 24 \beta_{3} ) q^{85} + ( -9 \beta_{1} - 18 \beta_{3} ) q^{87} + ( -918 - 18 \beta_{2} ) q^{89} + ( -1296 - 63 \beta_{2} ) q^{91} + ( 39 \beta_{1} - 3 \beta_{3} ) q^{93} + ( -60 \beta_{1} - 24 \beta_{3} ) q^{95} + ( -370 - 6 \beta_{2} ) q^{97} + ( -108 - 9 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 12 q^{3} + 36 q^{9} + O(q^{10})$$ $$4 q - 12 q^{3} + 36 q^{9} - 48 q^{11} + 120 q^{17} - 48 q^{19} + 172 q^{25} - 108 q^{27} + 144 q^{33} - 480 q^{35} + 408 q^{41} - 528 q^{43} + 836 q^{49} - 360 q^{51} + 144 q^{57} - 1872 q^{59} - 576 q^{65} - 2352 q^{67} + 968 q^{73} - 516 q^{75} + 324 q^{81} - 3408 q^{83} - 3672 q^{89} - 5184 q^{91} - 1480 q^{97} - 432 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2 x^{3} - 7 x^{2} + 2 x + 7$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$-2 \nu^{3} + 10 \nu^{2} - 6 \nu - 20$$ $$\beta_{2}$$ $$=$$ $$16 \nu^{3} - 48 \nu^{2} - 48 \nu + 64$$ $$\beta_{3}$$ $$=$$ $$-12 \nu^{3} + 44 \nu^{2} + 28 \nu - 80$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$2 \beta_{3} + \beta_{2} - 4 \beta_{1} + 16$$$$)/32$$ $$\nu^{2}$$ $$=$$ $$($$$$3 \beta_{3} + 2 \beta_{2} - 2 \beta_{1} + 72$$$$)/16$$ $$\nu^{3}$$ $$=$$ $$($$$$24 \beta_{3} + 17 \beta_{2} - 24 \beta_{1} + 352$$$$)/32$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 1.06909 3.63019 −1.21597 −1.48330
0 −3.00000 0 −17.4288 0 2.99032 0 9.00000 0
1.2 0 −3.00000 0 −5.67763 0 33.0917 0 9.00000 0
1.3 0 −3.00000 0 5.67763 0 −33.0917 0 9.00000 0
1.4 0 −3.00000 0 17.4288 0 −2.99032 0 9.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 768.4.a.u 4
3.b odd 2 1 2304.4.a.cb 4
4.b odd 2 1 768.4.a.v 4
8.b even 2 1 768.4.a.v 4
8.d odd 2 1 inner 768.4.a.u 4
12.b even 2 1 2304.4.a.by 4
16.e even 4 2 384.4.d.f 8
16.f odd 4 2 384.4.d.f 8
24.f even 2 1 2304.4.a.cb 4
24.h odd 2 1 2304.4.a.by 4
48.i odd 4 2 1152.4.d.p 8
48.k even 4 2 1152.4.d.p 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.4.d.f 8 16.e even 4 2
384.4.d.f 8 16.f odd 4 2
768.4.a.u 4 1.a even 1 1 trivial
768.4.a.u 4 8.d odd 2 1 inner
768.4.a.v 4 4.b odd 2 1
768.4.a.v 4 8.b even 2 1
1152.4.d.p 8 48.i odd 4 2
1152.4.d.p 8 48.k even 4 2
2304.4.a.by 4 12.b even 2 1
2304.4.a.by 4 24.h odd 2 1
2304.4.a.cb 4 3.b odd 2 1
2304.4.a.cb 4 24.f even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(768))$$:

 $$T_{5}^{4} - 336 T_{5}^{2} + 9792$$ $$T_{7}^{4} - 1104 T_{7}^{2} + 9792$$ $$T_{11}^{2} + 24 T_{11} - 368$$ $$T_{19}^{2} + 24 T_{19} - 4464$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( 3 + T )^{4}$$
$5$ $$9792 - 336 T^{2} + T^{4}$$
$7$ $$9792 - 1104 T^{2} + T^{4}$$
$11$ $$( -368 + 24 T + T^{2} )^{2}$$
$13$ $$12690432 - 8640 T^{2} + T^{4}$$
$17$ $$( -3708 - 60 T + T^{2} )^{2}$$
$19$ $$( -4464 + 24 T + T^{2} )^{2}$$
$23$ $$621831168 - 53568 T^{2} + T^{4}$$
$29$ $$419577408 - 41040 T^{2} + T^{4}$$
$31$ $$461095488 - 55248 T^{2} + T^{4}$$
$37$ $$2487324672 - 107136 T^{2} + T^{4}$$
$41$ $$( -104796 - 204 T + T^{2} )^{2}$$
$43$ $$( 12816 + 264 T + T^{2} )^{2}$$
$47$ $$21332616192 - 302400 T^{2} + T^{4}$$
$53$ $$806557248 - 87888 T^{2} + T^{4}$$
$59$ $$( 87952 + 936 T + T^{2} )^{2}$$
$61$ $$270509248512 - 1040256 T^{2} + T^{4}$$
$67$ $$( 272016 + 1176 T + T^{2} )^{2}$$
$71$ $$297070322688 - 1104192 T^{2} + T^{4}$$
$73$ $$( -236348 - 484 T + T^{2} )^{2}$$
$79$ $$1904357952 - 1039824 T^{2} + T^{4}$$
$83$ $$( 577936 + 1704 T + T^{2} )^{2}$$
$89$ $$( 676836 + 1836 T + T^{2} )^{2}$$
$97$ $$( 118468 + 740 T + T^{2} )^{2}$$