# Properties

 Label 784.4.a.t Level $784$ Weight $4$ Character orbit 784.a Self dual yes Analytic conductor $46.257$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: yes Analytic conductor: $$46.2574974445$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{57})$$ Defining polynomial: $$x^{2} - x - 14$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 56) 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.

 $$f(q)$$ $$=$$ $$q + ( -1 - \beta ) q^{3} + ( -11 - \beta ) q^{5} + ( 31 + 2 \beta ) q^{9} +O(q^{10})$$ $$q + ( -1 - \beta ) q^{3} + ( -11 - \beta ) q^{5} + ( 31 + 2 \beta ) q^{9} + ( -18 - 6 \beta ) q^{11} + ( -21 + 5 \beta ) q^{13} + ( 68 + 12 \beta ) q^{15} + ( -4 + 6 \beta ) q^{17} + ( -59 + 13 \beta ) q^{19} + ( 52 - 4 \beta ) q^{23} + ( 53 + 22 \beta ) q^{25} + ( -118 - 6 \beta ) q^{27} + ( -28 + 22 \beta ) q^{29} + ( 10 - 14 \beta ) q^{31} + ( 360 + 24 \beta ) q^{33} + ( 252 - 10 \beta ) q^{37} + ( -264 + 16 \beta ) q^{39} + ( 272 + 18 \beta ) q^{41} + ( -206 + 14 \beta ) q^{43} + ( -455 - 53 \beta ) q^{45} + ( -250 + 14 \beta ) q^{47} + ( -338 - 2 \beta ) q^{51} + ( 134 - 72 \beta ) q^{53} + ( 540 + 84 \beta ) q^{55} + ( -682 + 46 \beta ) q^{57} + ( -99 + 13 \beta ) q^{59} + ( 173 + 31 \beta ) q^{61} + ( -54 - 34 \beta ) q^{65} + ( -504 + 68 \beta ) q^{67} + ( 176 - 48 \beta ) q^{69} + ( 612 + 44 \beta ) q^{71} + ( -358 + 36 \beta ) q^{73} + ( -1307 - 75 \beta ) q^{75} + ( -292 - 36 \beta ) q^{79} + ( -377 + 70 \beta ) q^{81} + ( -615 - 47 \beta ) q^{83} + ( -298 - 62 \beta ) q^{85} + ( -1226 + 6 \beta ) q^{87} + ( -298 - 160 \beta ) q^{89} + ( 788 + 4 \beta ) q^{93} + ( -92 - 84 \beta ) q^{95} + ( 428 + 70 \beta ) q^{97} + ( -1242 - 222 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{3} - 22q^{5} + 62q^{9} + O(q^{10})$$ $$2q - 2q^{3} - 22q^{5} + 62q^{9} - 36q^{11} - 42q^{13} + 136q^{15} - 8q^{17} - 118q^{19} + 104q^{23} + 106q^{25} - 236q^{27} - 56q^{29} + 20q^{31} + 720q^{33} + 504q^{37} - 528q^{39} + 544q^{41} - 412q^{43} - 910q^{45} - 500q^{47} - 676q^{51} + 268q^{53} + 1080q^{55} - 1364q^{57} - 198q^{59} + 346q^{61} - 108q^{65} - 1008q^{67} + 352q^{69} + 1224q^{71} - 716q^{73} - 2614q^{75} - 584q^{79} - 754q^{81} - 1230q^{83} - 596q^{85} - 2452q^{87} - 596q^{89} + 1576q^{93} - 184q^{95} + 856q^{97} - 2484q^{99} + O(q^{100})$$

## 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 −8.54983 0 −18.5498 0 0 0 46.0997 0
1.2 0 6.54983 0 −3.45017 0 0 0 15.9003 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
$$2$$ $$1$$
$$7$$ $$-1$$

## 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 784.4.a.t 2
4.b odd 2 1 392.4.a.h 2
7.b odd 2 1 112.4.a.h 2
21.c even 2 1 1008.4.a.x 2
28.d even 2 1 56.4.a.c 2
28.f even 6 2 392.4.i.l 4
28.g odd 6 2 392.4.i.i 4
56.e even 2 1 448.4.a.s 2
56.h odd 2 1 448.4.a.r 2
84.h odd 2 1 504.4.a.i 2
140.c even 2 1 1400.4.a.i 2
140.j odd 4 2 1400.4.g.h 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.4.a.c 2 28.d even 2 1
112.4.a.h 2 7.b odd 2 1
392.4.a.h 2 4.b odd 2 1
392.4.i.i 4 28.g odd 6 2
392.4.i.l 4 28.f even 6 2
448.4.a.r 2 56.h odd 2 1
448.4.a.s 2 56.e even 2 1
504.4.a.i 2 84.h odd 2 1
784.4.a.t 2 1.a even 1 1 trivial
1008.4.a.x 2 21.c even 2 1
1400.4.a.i 2 140.c even 2 1
1400.4.g.h 4 140.j odd 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(784))$$:

 $$T_{3}^{2} + 2 T_{3} - 56$$ $$T_{5}^{2} + 22 T_{5} + 64$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$-56 + 2 T + T^{2}$$
$5$ $$64 + 22 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$-1728 + 36 T + T^{2}$$
$13$ $$-984 + 42 T + T^{2}$$
$17$ $$-2036 + 8 T + T^{2}$$
$19$ $$-6152 + 118 T + T^{2}$$
$23$ $$1792 - 104 T + T^{2}$$
$29$ $$-26804 + 56 T + T^{2}$$
$31$ $$-11072 - 20 T + T^{2}$$
$37$ $$57804 - 504 T + T^{2}$$
$41$ $$55516 - 544 T + T^{2}$$
$43$ $$31264 + 412 T + T^{2}$$
$47$ $$51328 + 500 T + T^{2}$$
$53$ $$-277532 - 268 T + T^{2}$$
$59$ $$168 + 198 T + T^{2}$$
$61$ $$-24848 - 346 T + T^{2}$$
$67$ $$-9552 + 1008 T + T^{2}$$
$71$ $$264192 - 1224 T + T^{2}$$
$73$ $$54292 + 716 T + T^{2}$$
$79$ $$11392 + 584 T + T^{2}$$
$83$ $$252312 + 1230 T + T^{2}$$
$89$ $$-1370396 + 596 T + T^{2}$$
$97$ $$-96116 - 856 T + T^{2}$$