# Properties

 Label 784.5.c.a Level $784$ Weight $5$ Character orbit 784.c Analytic conductor $81.042$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$784 = 2^{4} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 784.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$81.0420510577$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.0.2048.2 Defining polynomial: $$x^{4} + 4 x^{2} + 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 98) 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.

 $$f(q)$$ $$=$$ $$q + ( -4 \beta_{1} - 7 \beta_{3} ) q^{3} + ( -5 \beta_{1} - 13 \beta_{3} ) q^{5} + ( -49 - 89 \beta_{2} ) q^{9} +O(q^{10})$$ $$q + ( -4 \beta_{1} - 7 \beta_{3} ) q^{3} + ( -5 \beta_{1} - 13 \beta_{3} ) q^{5} + ( -49 - 89 \beta_{2} ) q^{9} + ( -6 + 103 \beta_{2} ) q^{11} + ( 101 \beta_{1} - 97 \beta_{3} ) q^{13} + ( -222 - 158 \beta_{2} ) q^{15} + ( -282 \beta_{1} - 177 \beta_{3} ) q^{17} + ( -207 \beta_{1} + 10 \beta_{3} ) q^{19} + ( -26 - 144 \beta_{2} ) q^{23} + ( 237 - 274 \beta_{2} ) q^{25} + ( 139 \beta_{1} + 755 \beta_{3} ) q^{27} + ( -352 - 22 \beta_{2} ) q^{29} + ( 698 \beta_{1} - 74 \beta_{3} ) q^{31} + ( -285 \beta_{1} - 1091 \beta_{3} ) q^{33} + ( -848 - 36 \beta_{2} ) q^{37} + ( -550 - 764 \beta_{2} ) q^{39} + ( 251 \beta_{1} - 460 \beta_{3} ) q^{41} + ( 506 - 1175 \beta_{2} ) q^{43} + ( 957 \beta_{1} + 2239 \beta_{3} ) q^{45} + ( 1732 \beta_{1} - 1418 \beta_{3} ) q^{47} + ( -4734 - 2793 \beta_{2} ) q^{51} + ( -4170 - 706 \beta_{2} ) q^{53} + ( -794 \beta_{1} - 1776 \beta_{3} ) q^{55} + ( -1516 - 511 \beta_{2} ) q^{57} + ( -1379 \beta_{1} + 2820 \beta_{3} ) q^{59} + ( -1523 \beta_{1} - 365 \beta_{3} ) q^{61} + ( -1512 - 938 \beta_{2} ) q^{65} + ( 5204 + 786 \beta_{2} ) q^{67} + ( 536 \beta_{1} + 1766 \beta_{3} ) q^{69} + ( 496 - 1244 \beta_{2} ) q^{71} + ( -1385 \beta_{1} + 2736 \beta_{3} ) q^{73} + ( -126 \beta_{1} + 1355 \beta_{3} ) q^{75} + ( 7404 + 2270 \beta_{2} ) q^{79} + ( 7713 + 1513 \beta_{2} ) q^{81} + ( 4703 \beta_{1} + 1164 \beta_{3} ) q^{83} + ( -7422 - 5442 \beta_{2} ) q^{85} + ( 1474 \beta_{1} + 2706 \beta_{3} ) q^{87} + ( 3573 \beta_{1} - 3746 \beta_{3} ) q^{89} + ( 4548 + 1280 \beta_{2} ) q^{93} + ( -1810 - 1476 \beta_{2} ) q^{95} + ( 5324 \beta_{1} - 141 \beta_{3} ) q^{97} + ( -18040 - 4513 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 196q^{9} + O(q^{10})$$ $$4q - 196q^{9} - 24q^{11} - 888q^{15} - 104q^{23} + 948q^{25} - 1408q^{29} - 3392q^{37} - 2200q^{39} + 2024q^{43} - 18936q^{51} - 16680q^{53} - 6064q^{57} - 6048q^{65} + 20816q^{67} + 1984q^{71} + 29616q^{79} + 30852q^{81} - 29688q^{85} + 18192q^{93} - 7240q^{95} - 72160q^{99} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} + 2$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} + 3 \nu$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} - 2$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} - 3 \beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/784\mathbb{Z}\right)^\times$$.

 $$n$$ $$197$$ $$687$$ $$689$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
97.1
 0.765367i 1.84776i − 1.84776i − 0.765367i
0 15.9958i 0 27.8477i 0 0 0 −174.865 0
97.2 0 2.03347i 0 0.710974i 0 0 0 76.8650 0
97.3 0 2.03347i 0 0.710974i 0 0 0 76.8650 0
97.4 0 15.9958i 0 27.8477i 0 0 0 −174.865 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
7.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 784.5.c.a 4
4.b odd 2 1 98.5.b.a 4
7.b odd 2 1 inner 784.5.c.a 4
12.b even 2 1 882.5.c.c 4
28.d even 2 1 98.5.b.a 4
28.f even 6 2 98.5.d.c 8
28.g odd 6 2 98.5.d.c 8
84.h odd 2 1 882.5.c.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
98.5.b.a 4 4.b odd 2 1
98.5.b.a 4 28.d even 2 1
98.5.d.c 8 28.f even 6 2
98.5.d.c 8 28.g odd 6 2
784.5.c.a 4 1.a even 1 1 trivial
784.5.c.a 4 7.b odd 2 1 inner
882.5.c.c 4 12.b even 2 1
882.5.c.c 4 84.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} + 260 T_{3}^{2} + 1058$$ acting on $$S_{5}^{\mathrm{new}}(784, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$1058 + 260 T^{2} + T^{4}$$
$5$ $$392 + 776 T^{2} + T^{4}$$
$7$ $$T^{4}$$
$11$ $$( -21182 + 12 T + T^{2} )^{2}$$
$13$ $$707030408 + 78440 T^{2} + T^{4}$$
$17$ $$43821617058 + 443412 T^{2} + T^{4}$$
$19$ $$2981309762 + 171796 T^{2} + T^{4}$$
$23$ $$( -40796 + 52 T + T^{2} )^{2}$$
$29$ $$( 122936 + 704 T + T^{2} )^{2}$$
$31$ $$286409447552 + 1970720 T^{2} + T^{4}$$
$37$ $$( 716512 + 1696 T + T^{2} )^{2}$$
$41$ $$288069342722 + 1098404 T^{2} + T^{4}$$
$43$ $$( -2505214 - 1012 T + T^{2} )^{2}$$
$47$ $$30777535627808 + 20042192 T^{2} + T^{4}$$
$53$ $$( 16392028 + 8340 T + T^{2} )^{2}$$
$59$ $$382444812731522 + 39416164 T^{2} + T^{4}$$
$61$ $$21754848065672 + 9811016 T^{2} + T^{4}$$
$67$ $$( 25846024 - 10408 T + T^{2} )^{2}$$
$71$ $$( -2849056 - 992 T + T^{2} )^{2}$$
$73$ $$345644675616962 + 37615684 T^{2} + T^{4}$$
$79$ $$( 44513416 - 14808 T + T^{2} )^{2}$$
$83$ $$2011288822677218 + 93892420 T^{2} + T^{4}$$
$89$ $$1571934000441218 + 107195380 T^{2} + T^{4}$$
$97$ $$1439024660341058 + 113459428 T^{2} + T^{4}$$