Properties

 Label 784.4.a.bh Level $784$ Weight $4$ Character orbit 784.a Self dual yes Analytic conductor $46.257$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

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: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{113})$$ Defining polynomial: $$x^{4} - 2 x^{3} - 59 x^{2} + 60 x + 674$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 392) 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.

 $$f(q)$$ $$=$$ $$q + ( 2 \beta_{1} + \beta_{3} ) q^{3} + ( 9 \beta_{1} - \beta_{3} ) q^{5} + ( 34 - 3 \beta_{2} ) q^{9} +O(q^{10})$$ $$q + ( 2 \beta_{1} + \beta_{3} ) q^{3} + ( 9 \beta_{1} - \beta_{3} ) q^{5} + ( 34 - 3 \beta_{2} ) q^{9} + ( 29 + \beta_{2} ) q^{11} + ( -21 \beta_{1} - 9 \beta_{3} ) q^{13} + ( -28 - 8 \beta_{2} ) q^{15} + 65 \beta_{1} q^{17} + ( 6 \beta_{1} + 17 \beta_{3} ) q^{19} + ( 20 + 8 \beta_{2} ) q^{23} + ( 112 + 19 \beta_{2} ) q^{25} + ( 188 \beta_{1} + 16 \beta_{3} ) q^{27} + ( 9 - 7 \beta_{2} ) q^{29} + ( 88 \beta_{1} + 6 \beta_{3} ) q^{31} + 26 \beta_{3} q^{33} + ( 109 - 19 \beta_{2} ) q^{37} + ( -558 + 30 \beta_{2} ) q^{39} + ( -45 \beta_{1} + 36 \beta_{3} ) q^{41} + ( -93 - \beta_{2} ) q^{43} + ( 165 \beta_{1} + 23 \beta_{3} ) q^{45} + ( -212 \beta_{1} - 10 \beta_{3} ) q^{47} + ( 195 - 65 \beta_{2} ) q^{51} + ( 244 + 18 \beta_{2} ) q^{53} + ( 308 \beta_{1} - 48 \beta_{3} ) q^{55} + ( 953 - 23 \beta_{2} ) q^{57} + ( -498 \beta_{1} + 21 \beta_{3} ) q^{59} + ( 75 \beta_{1} + 25 \beta_{3} ) q^{61} + ( 195 + 69 \beta_{2} ) q^{65} + ( 294 + 22 \beta_{2} ) q^{67} + ( -424 \beta_{1} - 4 \beta_{3} ) q^{69} + ( 602 + 42 \beta_{2} ) q^{71} + ( 315 \beta_{1} + 42 \beta_{3} ) q^{73} + ( -878 \beta_{1} + 55 \beta_{3} ) q^{75} + ( -14 + 26 \beta_{2} ) q^{79} + ( 526 - 123 \beta_{2} ) q^{81} + ( 26 \beta_{1} + \beta_{3} ) q^{83} + ( 1235 + 65 \beta_{2} ) q^{85} + ( 424 \beta_{1} + 30 \beta_{3} ) q^{87} + ( -355 \beta_{1} + 16 \beta_{3} ) q^{89} + ( 594 - 94 \beta_{2} ) q^{93} + ( -1008 - 164 \beta_{2} ) q^{95} + ( 263 \beta_{1} + 22 \beta_{3} ) q^{97} + ( 647 - 53 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 136q^{9} + O(q^{10})$$ $$4q + 136q^{9} + 116q^{11} - 112q^{15} + 80q^{23} + 448q^{25} + 36q^{29} + 436q^{37} - 2232q^{39} - 372q^{43} + 780q^{51} + 976q^{53} + 3812q^{57} + 780q^{65} + 1176q^{67} + 2408q^{71} - 56q^{79} + 2104q^{81} + 4940q^{85} + 2376q^{93} - 4032q^{95} + 2588q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2 x^{3} - 59 x^{2} + 60 x + 674$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-2 \nu^{3} + 3 \nu^{2} + 67 \nu - 34$$$$)/105$$ $$\beta_{2}$$ $$=$$ $$($$$$-4 \nu^{3} + 6 \nu^{2} + 344 \nu - 173$$$$)/105$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{3} + 51 \nu^{2} - 86 \nu - 1558$$$$)/105$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} - 2 \beta_{1} + 1$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$4 \beta_{3} + \beta_{2} + 61$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$6 \beta_{3} + 35 \beta_{2} - 172 \beta_{1} + 91$$$$)/2$$

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
 −3.40086 4.40086 7.22929 −6.22929
0 −9.63797 0 −5.91838 0 0 0 65.8904 0
1.2 0 −5.39533 0 20.9517 0 0 0 2.10956 0
1.3 0 5.39533 0 −20.9517 0 0 0 2.10956 0
1.4 0 9.63797 0 5.91838 0 0 0 65.8904 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

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 784.4.a.bh 4
4.b odd 2 1 392.4.a.n 4
7.b odd 2 1 inner 784.4.a.bh 4
28.d even 2 1 392.4.a.n 4
28.f even 6 2 392.4.i.o 8
28.g odd 6 2 392.4.i.o 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
392.4.a.n 4 4.b odd 2 1
392.4.a.n 4 28.d even 2 1
392.4.i.o 8 28.f even 6 2
392.4.i.o 8 28.g odd 6 2
784.4.a.bh 4 1.a even 1 1 trivial
784.4.a.bh 4 7.b odd 2 1 inner

Hecke kernels

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

 $$T_{3}^{4} - 122 T_{3}^{2} + 2704$$ $$T_{5}^{4} - 474 T_{5}^{2} + 15376$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$2704 - 122 T^{2} + T^{4}$$
$5$ $$15376 - 474 T^{2} + T^{4}$$
$7$ $$T^{4}$$
$11$ $$( 728 - 58 T + T^{2} )^{2}$$
$13$ $$16257024 - 10242 T^{2} + T^{4}$$
$17$ $$( -8450 + T^{2} )^{2}$$
$19$ $$266211856 - 32682 T^{2} + T^{4}$$
$23$ $$( -6832 - 40 T + T^{2} )^{2}$$
$29$ $$( -5456 - 18 T + T^{2} )^{2}$$
$31$ $$154157056 - 32968 T^{2} + T^{4}$$
$37$ $$( -28912 - 218 T + T^{2} )^{2}$$
$41$ $$4262261796 - 162324 T^{2} + T^{4}$$
$43$ $$( 8536 + 186 T + T^{2} )^{2}$$
$47$ $$6407682304 - 182696 T^{2} + T^{4}$$
$53$ $$( 22924 - 488 T + T^{2} )^{2}$$
$59$ $$242288403984 - 1084122 T^{2} + T^{4}$$
$61$ $$756250000 - 86250 T^{2} + T^{4}$$
$67$ $$( 31744 - 588 T + T^{2} )^{2}$$
$71$ $$( 163072 - 1204 T + T^{2} )^{2}$$
$73$ $$5359118436 - 545076 T^{2} + T^{4}$$
$79$ $$( -76192 + 28 T + T^{2} )^{2}$$
$83$ $$1547536 - 2714 T^{2} + T^{4}$$
$89$ $$62037857476 - 556004 T^{2} + T^{4}$$
$97$ $$9932514244 - 308708 T^{2} + T^{4}$$