# Properties

 Label 768.5.g.c Level $768$ Weight $5$ Character orbit 768.g Analytic conductor $79.388$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$768 = 2^{8} \cdot 3$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 768.g (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$79.3881316484$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{7})$$ Defining polynomial: $$x^{4} + 7 x^{2} + 49$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{6}\cdot 3^{2}$$ 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.

 $$f(q)$$ $$=$$ $$q -\beta_{2} q^{3} + ( -8 - \beta_{3} ) q^{5} + ( \beta_{1} + 5 \beta_{2} ) q^{7} -27 q^{9} +O(q^{10})$$ $$q -\beta_{2} q^{3} + ( -8 - \beta_{3} ) q^{5} + ( \beta_{1} + 5 \beta_{2} ) q^{7} -27 q^{9} + ( -8 \beta_{1} - 4 \beta_{2} ) q^{11} + ( -120 - 2 \beta_{3} ) q^{13} + ( 9 \beta_{1} + 5 \beta_{2} ) q^{15} + ( 174 + 8 \beta_{3} ) q^{17} + ( 24 \beta_{1} + 36 \beta_{2} ) q^{19} + ( 144 + 3 \beta_{3} ) q^{21} + ( 30 \beta_{1} + 54 \beta_{2} ) q^{23} + ( 447 + 16 \beta_{3} ) q^{25} + 27 \beta_{2} q^{27} + ( -712 - 3 \beta_{3} ) q^{29} + ( 41 \beta_{1} - 83 \beta_{2} ) q^{31} + ( -180 - 24 \beta_{3} ) q^{33} + ( -56 \beta_{1} - 136 \beta_{2} ) q^{35} + ( 168 - 20 \beta_{3} ) q^{37} + ( 18 \beta_{1} + 114 \beta_{2} ) q^{39} + ( 126 - 72 \beta_{3} ) q^{41} + ( 72 \beta_{1} - 180 \beta_{2} ) q^{43} + ( 216 + 27 \beta_{3} ) q^{45} + ( 42 \beta_{1} - 558 \beta_{2} ) q^{47} + ( 1297 - 32 \beta_{3} ) q^{49} + ( -72 \beta_{1} - 150 \beta_{2} ) q^{51} + ( -40 + 25 \beta_{3} ) q^{53} + ( 124 \beta_{1} + 908 \beta_{2} ) q^{55} + ( 1188 + 72 \beta_{3} ) q^{57} -252 \beta_{2} q^{59} + ( 1992 - 88 \beta_{3} ) q^{61} + ( -27 \beta_{1} - 135 \beta_{2} ) q^{63} + ( 2976 + 136 \beta_{3} ) q^{65} + ( 96 \beta_{1} - 468 \beta_{2} ) q^{67} + ( 1728 + 90 \beta_{3} ) q^{69} + ( -318 \beta_{1} + 810 \beta_{2} ) q^{71} + ( 1282 + 32 \beta_{3} ) q^{73} + ( -144 \beta_{1} - 399 \beta_{2} ) q^{75} + ( 3648 + 148 \beta_{3} ) q^{77} + ( -135 \beta_{1} - 1539 \beta_{2} ) q^{79} + 729 q^{81} + ( 280 \beta_{1} - 508 \beta_{2} ) q^{83} + ( -9456 - 238 \beta_{3} ) q^{85} + ( 27 \beta_{1} + 703 \beta_{2} ) q^{87} + ( 3954 + 80 \beta_{3} ) q^{89} + ( -216 \beta_{1} - 792 \beta_{2} ) q^{91} + ( -1872 + 123 \beta_{3} ) q^{93} + ( -588 \beta_{1} - 2844 \beta_{2} ) q^{95} + ( -5650 - 80 \beta_{3} ) q^{97} + ( 216 \beta_{1} + 108 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 32 q^{5} - 108 q^{9} + O(q^{10})$$ $$4 q - 32 q^{5} - 108 q^{9} - 480 q^{13} + 696 q^{17} + 576 q^{21} + 1788 q^{25} - 2848 q^{29} - 720 q^{33} + 672 q^{37} + 504 q^{41} + 864 q^{45} + 5188 q^{49} - 160 q^{53} + 4752 q^{57} + 7968 q^{61} + 11904 q^{65} + 6912 q^{69} + 5128 q^{73} + 14592 q^{77} + 2916 q^{81} - 37824 q^{85} + 15816 q^{89} - 7488 q^{93} - 22600 q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 7 x^{2} + 49$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$4 \nu^{3} + 2 \nu^{2} + 56 \nu + 7$$$$)/7$$ $$\beta_{2}$$ $$=$$ $$($$$$6 \nu^{2} + 21$$$$)/7$$ $$\beta_{3}$$ $$=$$ $$($$$$-12 \nu^{3}$$$$)/7$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} - \beta_{2} + 3 \beta_{1}$$$$)/24$$ $$\nu^{2}$$ $$=$$ $$($$$$7 \beta_{2} - 21$$$$)/6$$ $$\nu^{3}$$ $$=$$ $$($$$$-7 \beta_{3}$$$$)/12$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
511.1
 1.32288 + 2.29129i −1.32288 − 2.29129i 1.32288 − 2.29129i −1.32288 + 2.29129i
0 5.19615i 0 −39.7490 0 46.0431i 0 −27.0000 0
511.2 0 5.19615i 0 23.7490 0 9.38251i 0 −27.0000 0
511.3 0 5.19615i 0 −39.7490 0 46.0431i 0 −27.0000 0
511.4 0 5.19615i 0 23.7490 0 9.38251i 0 −27.0000 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
4.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 768.5.g.c 4
4.b odd 2 1 inner 768.5.g.c 4
8.b even 2 1 768.5.g.g 4
8.d odd 2 1 768.5.g.g 4
16.e even 4 2 384.5.b.c 8
16.f odd 4 2 384.5.b.c 8
48.i odd 4 2 1152.5.b.k 8
48.k even 4 2 1152.5.b.k 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.5.b.c 8 16.e even 4 2
384.5.b.c 8 16.f odd 4 2
768.5.g.c 4 1.a even 1 1 trivial
768.5.g.c 4 4.b odd 2 1 inner
768.5.g.g 4 8.b even 2 1
768.5.g.g 4 8.d odd 2 1
1152.5.b.k 8 48.i odd 4 2
1152.5.b.k 8 48.k even 4 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} + 16 T_{5} - 944$$ acting on $$S_{5}^{\mathrm{new}}(768, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( 27 + T^{2} )^{2}$$
$5$ $$( -944 + 16 T + T^{2} )^{2}$$
$7$ $$186624 + 2208 T^{2} + T^{4}$$
$11$ $$412252416 + 45408 T^{2} + T^{4}$$
$13$ $$( 10368 + 240 T + T^{2} )^{2}$$
$17$ $$( -34236 - 348 T + T^{2} )^{2}$$
$19$ $$19955517696 + 491616 T^{2} + T^{4}$$
$23$ $$36790308864 + 825984 T^{2} + T^{4}$$
$29$ $$( 497872 + 1424 T + T^{2} )^{2}$$
$31$ $$189245880576 + 1389216 T^{2} + T^{4}$$
$37$ $$( -374976 - 336 T + T^{2} )^{2}$$
$41$ $$( -5209596 - 252 T + T^{2} )^{2}$$
$43$ $$1176686901504 + 4797792 T^{2} + T^{4}$$
$47$ $$54724012314624 + 17165952 T^{2} + T^{4}$$
$53$ $$( -628400 + 80 T + T^{2} )^{2}$$
$59$ $$( 1714608 + T^{2} )^{2}$$
$61$ $$( -3837888 - 3984 T + T^{2} )^{2}$$
$67$ $$4145361152256 + 16458336 T^{2} + T^{4}$$
$71$ $$424196534145024 + 94718592 T^{2} + T^{4}$$
$73$ $$( 611332 - 2564 T + T^{2} )^{2}$$
$79$ $$3797136795711744 + 147736224 T^{2} + T^{4}$$
$83$ $$470880972843264 + 61970016 T^{2} + T^{4}$$
$89$ $$( 9182916 - 7908 T + T^{2} )^{2}$$
$97$ $$( 25471300 + 11300 T + T^{2} )^{2}$$