# Properties

 Label 784.6.a.v Level 784 Weight 6 Character orbit 784.a Self dual yes Analytic conductor 125.741 Analytic rank 1 Dimension 2 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$784 = 2^{4} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 784.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$125.740914733$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{57})$$ Defining polynomial: $$x^{2} - x - 14$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 7) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{57}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -3 - 3 \beta ) q^{3} + ( 9 - 5 \beta ) q^{5} + ( 279 + 18 \beta ) q^{9} +O(q^{10})$$ $$q + ( -3 - 3 \beta ) q^{3} + ( 9 - 5 \beta ) q^{5} + ( 279 + 18 \beta ) q^{9} + ( -198 + 62 \beta ) q^{11} + ( 175 - 63 \beta ) q^{13} + ( 828 - 12 \beta ) q^{15} + ( -900 + 38 \beta ) q^{17} + ( -1633 - 9 \beta ) q^{19} + ( -1044 - 284 \beta ) q^{23} + ( -1619 - 90 \beta ) q^{25} + ( -3186 - 162 \beta ) q^{27} + ( 3348 + 126 \beta ) q^{29} + ( -10 + 270 \beta ) q^{31} + ( -10008 + 408 \beta ) q^{33} + ( 3116 + 270 \beta ) q^{37} + ( 10248 - 336 \beta ) q^{39} + ( 3024 + 546 \beta ) q^{41} + ( 1510 + 2394 \beta ) q^{43} + ( -2619 - 1233 \beta ) q^{45} + ( 5850 + 1874 \beta ) q^{47} + ( -3798 + 2586 \beta ) q^{51} + ( 4734 - 104 \beta ) q^{53} + ( -19452 + 1548 \beta ) q^{55} + ( 6438 + 4926 \beta ) q^{57} + ( -21969 - 1025 \beta ) q^{59} + ( 32377 + 2403 \beta ) q^{61} + ( 19530 - 1442 \beta ) q^{65} + ( -12392 + 972 \beta ) q^{67} + ( 51696 + 3984 \beta ) q^{69} + ( -48708 + 2100 \beta ) q^{71} + ( -8726 + 2628 \beta ) q^{73} + ( 20247 + 5127 \beta ) q^{75} + ( -25628 - 7452 \beta ) q^{79} + ( -30537 + 5670 \beta ) q^{81} + ( 58779 + 7875 \beta ) q^{83} + ( -18930 + 4842 \beta ) q^{85} + ( -31590 - 10422 \beta ) q^{87} + ( -42138 + 11104 \beta ) q^{89} + ( -46140 - 780 \beta ) q^{93} + ( -12132 + 8084 \beta ) q^{95} + ( -10388 - 4410 \beta ) q^{97} + ( 8370 + 13734 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 6q^{3} + 18q^{5} + 558q^{9} + O(q^{10})$$ $$2q - 6q^{3} + 18q^{5} + 558q^{9} - 396q^{11} + 350q^{13} + 1656q^{15} - 1800q^{17} - 3266q^{19} - 2088q^{23} - 3238q^{25} - 6372q^{27} + 6696q^{29} - 20q^{31} - 20016q^{33} + 6232q^{37} + 20496q^{39} + 6048q^{41} + 3020q^{43} - 5238q^{45} + 11700q^{47} - 7596q^{51} + 9468q^{53} - 38904q^{55} + 12876q^{57} - 43938q^{59} + 64754q^{61} + 39060q^{65} - 24784q^{67} + 103392q^{69} - 97416q^{71} - 17452q^{73} + 40494q^{75} - 51256q^{79} - 61074q^{81} + 117558q^{83} - 37860q^{85} - 63180q^{87} - 84276q^{89} - 92280q^{93} - 24264q^{95} - 20776q^{97} + 16740q^{99} + O(q^{100})$$

## 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
 4.27492 −3.27492
0 −25.6495 0 −28.7492 0 0 0 414.897 0
1.2 0 19.6495 0 46.7492 0 0 0 143.103 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 784.6.a.v 2
4.b odd 2 1 49.6.a.f 2
7.b odd 2 1 112.6.a.h 2
12.b even 2 1 441.6.a.l 2
21.c even 2 1 1008.6.a.bq 2
28.d even 2 1 7.6.a.b 2
28.f even 6 2 49.6.c.e 4
28.g odd 6 2 49.6.c.d 4
56.e even 2 1 448.6.a.w 2
56.h odd 2 1 448.6.a.u 2
84.h odd 2 1 63.6.a.f 2
140.c even 2 1 175.6.a.c 2
140.j odd 4 2 175.6.b.c 4
308.g odd 2 1 847.6.a.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.6.a.b 2 28.d even 2 1
49.6.a.f 2 4.b odd 2 1
49.6.c.d 4 28.g odd 6 2
49.6.c.e 4 28.f even 6 2
63.6.a.f 2 84.h odd 2 1
112.6.a.h 2 7.b odd 2 1
175.6.a.c 2 140.c even 2 1
175.6.b.c 4 140.j odd 4 2
441.6.a.l 2 12.b even 2 1
448.6.a.u 2 56.h odd 2 1
448.6.a.w 2 56.e even 2 1
784.6.a.v 2 1.a even 1 1 trivial
847.6.a.c 2 308.g odd 2 1
1008.6.a.bq 2 21.c even 2 1

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$7$$ $$-1$$

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + 6 T_{3} - 504$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(784))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 + 6 T - 18 T^{2} + 1458 T^{3} + 59049 T^{4}$$
$5$ $$1 - 18 T + 4906 T^{2} - 56250 T^{3} + 9765625 T^{4}$$
$7$ 1
$11$ $$1 + 396 T + 142198 T^{2} + 63776196 T^{3} + 25937424601 T^{4}$$
$13$ $$1 - 350 T + 546978 T^{2} - 129952550 T^{3} + 137858491849 T^{4}$$
$17$ $$1 + 1800 T + 3567406 T^{2} + 2555742600 T^{3} + 2015993900449 T^{4}$$
$19$ $$1 + 3266 T + 7614270 T^{2} + 8086939334 T^{3} + 6131066257801 T^{4}$$
$23$ $$1 + 2088 T + 9365230 T^{2} + 13439084184 T^{3} + 41426511213649 T^{4}$$
$29$ $$1 - 6696 T + 51326470 T^{2} - 137342653704 T^{3} + 420707233300201 T^{4}$$
$31$ $$1 + 20 T + 53103102 T^{2} + 572583020 T^{3} + 819628286980801 T^{4}$$
$37$ $$1 - 6232 T + 144242070 T^{2} - 432151540024 T^{3} + 4808584372417849 T^{4}$$
$41$ $$1 - 6048 T + 223864366 T^{2} - 700698303648 T^{3} + 13422659310152401 T^{4}$$
$43$ $$1 - 3020 T - 30383466 T^{2} - 443965497860 T^{3} + 21611482313284249 T^{4}$$
$47$ $$1 - 11700 T + 292735582 T^{2} - 2683336581900 T^{3} + 52599132235830049 T^{4}$$
$53$ $$1 - 9468 T + 858185230 T^{2} - 3959474927724 T^{3} + 174887470365513049 T^{4}$$
$59$ $$1 + 43938 T + 1852599934 T^{2} + 31412343849462 T^{3} + 511116753300641401 T^{4}$$
$61$ $$1 - 64754 T + 2408321418 T^{2} - 54690988874954 T^{3} + 713342911662882601 T^{4}$$
$67$ $$1 + 24784 T + 2799959190 T^{2} + 33461500651888 T^{3} + 1822837804551761449 T^{4}$$
$71$ $$1 + 97416 T + 5729557966 T^{2} + 175760806457016 T^{3} + 3255243551009881201 T^{4}$$
$73$ $$1 + 17452 T + 3828622374 T^{2} + 36179245441036 T^{3} + 4297625829703557649 T^{4}$$
$79$ $$1 + 51256 T + 3645565854 T^{2} + 157717602787144 T^{3} + 9468276082626847201 T^{4}$$
$83$ $$1 - 117558 T + 7798161502 T^{2} - 463065739909794 T^{3} + 15516041187205853449 T^{4}$$
$89$ $$1 + 84276 T + 5915697430 T^{2} + 470602194123924 T^{3} + 31181719929966183601 T^{4}$$
$97$ $$1 + 20776 T + 16174049358 T^{2} + 178410581179432 T^{3} + 73742412689492826049 T^{4}$$