# Properties

 Label 784.4.a.bb Level $784$ Weight $4$ Character orbit 784.a Self dual yes Analytic conductor $46.257$ Analytic rank $1$ Dimension $3$ 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: $$3$$ Coefficient field: 3.3.1929.1 Defining polynomial: $$x^{3} - x^{2} - 10 x + 13$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{3}$$ 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 a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -2 + \beta_{1} ) q^{3} + ( 1 - \beta_{1} + \beta_{2} ) q^{5} + ( 4 - 7 \beta_{1} + \beta_{2} ) q^{9} +O(q^{10})$$ $$q + ( -2 + \beta_{1} ) q^{3} + ( 1 - \beta_{1} + \beta_{2} ) q^{5} + ( 4 - 7 \beta_{1} + \beta_{2} ) q^{9} -3 \beta_{2} q^{11} + ( 8 + \beta_{1} - 3 \beta_{2} ) q^{13} + ( -38 + 14 \beta_{1} - \beta_{2} ) q^{15} + ( 15 + 13 \beta_{1} + \beta_{2} ) q^{17} + ( -28 + 2 \beta_{1} + 3 \beta_{2} ) q^{19} + ( 64 - 7 \beta_{1} - 2 \beta_{2} ) q^{23} + ( 92 - 14 \beta_{1} - 10 \beta_{2} ) q^{25} + ( -152 + 20 \beta_{1} - 7 \beta_{2} ) q^{27} + ( 60 - 7 \beta_{1} - 3 \beta_{2} ) q^{29} + ( -122 - 26 \beta_{1} - \beta_{2} ) q^{31} + ( 27 - 24 \beta_{1} ) q^{33} + ( -187 - 14 \beta_{1} - 12 \beta_{2} ) q^{37} + ( 38 - 21 \beta_{1} + \beta_{2} ) q^{39} + ( -8 + 23 \beta_{1} + 11 \beta_{2} ) q^{41} + ( -108 - 56 \beta_{1} ) q^{43} + ( 436 - 89 \beta_{1} - 13 \beta_{2} ) q^{45} + ( -86 - 50 \beta_{1} - 3 \beta_{2} ) q^{47} + ( 312 - 42 \beta_{1} + 13 \beta_{2} ) q^{51} + ( 273 + 28 \beta_{1} + 34 \beta_{2} ) q^{53} + ( -540 + 3 \beta_{1} + 30 \beta_{2} ) q^{55} + ( 83 - 14 \beta_{1} + 2 \beta_{2} ) q^{57} + ( -590 + 29 \beta_{1} ) q^{59} + ( -185 + 66 \beta_{1} + 4 \beta_{2} ) q^{61} + ( -568 + 7 \beta_{1} + 39 \beta_{2} ) q^{65} + ( 174 - 7 \beta_{1} + 34 \beta_{2} ) q^{67} + ( -299 + 83 \beta_{1} - 7 \beta_{2} ) q^{69} + ( -220 - 14 \beta_{1} - 6 \beta_{2} ) q^{71} + ( 193 + 98 \beta_{1} + 38 \beta_{2} ) q^{73} + ( -472 + 82 \beta_{1} - 14 \beta_{2} ) q^{75} + ( -44 - 7 \beta_{1} - 46 \beta_{2} ) q^{79} + ( 799 - 119 \beta_{1} - 7 \beta_{2} ) q^{81} + ( -800 + 2 \beta_{1} - 30 \beta_{2} ) q^{83} + ( -273 + 140 \beta_{1} + 18 \beta_{2} ) q^{85} + ( -282 + 71 \beta_{1} - 7 \beta_{2} ) q^{87} + ( -315 - 72 \beta_{1} - 52 \beta_{2} ) q^{89} + ( -449 - 26 \beta_{2} ) q^{93} + ( 440 + 49 \beta_{1} - 56 \beta_{2} ) q^{95} + ( 20 - 263 \beta_{1} - 19 \beta_{2} ) q^{97} + ( -702 + 147 \beta_{1} + 57 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q - 7q^{3} + 3q^{5} + 18q^{9} + O(q^{10})$$ $$3q - 7q^{3} + 3q^{5} + 18q^{9} + 3q^{11} + 26q^{13} - 127q^{15} + 31q^{17} - 89q^{19} + 201q^{23} + 300q^{25} - 469q^{27} + 190q^{29} - 339q^{31} + 105q^{33} - 535q^{37} + 134q^{39} - 58q^{41} - 268q^{43} + 1410q^{45} - 205q^{47} + 965q^{51} + 757q^{53} - 1653q^{55} + 261q^{57} - 1799q^{59} - 625q^{61} - 1750q^{65} + 495q^{67} - 973q^{69} - 640q^{71} + 443q^{73} - 1484q^{75} - 79q^{79} + 2523q^{81} - 2372q^{83} - 977q^{85} - 910q^{87} - 821q^{89} - 1321q^{93} + 1327q^{95} + 342q^{97} - 2310q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 10 x + 13$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$2 \nu - 1$$ $$\beta_{2}$$ $$=$$ $$4 \nu^{2} + 2 \nu - 29$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{1} + 1$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{2} - \beta_{1} + 28$$$$)/4$$

## 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
 −3.27144 1.36922 2.90222
0 −9.54289 0 15.8094 0 0 0 64.0667 0
1.2 0 −0.261560 0 −19.5009 0 0 0 −26.9316 0
1.3 0 2.80445 0 6.69159 0 0 0 −19.1351 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.bb 3
4.b odd 2 1 392.4.a.l 3
7.b odd 2 1 784.4.a.be 3
7.d odd 6 2 112.4.i.e 6
28.d even 2 1 392.4.a.i 3
28.f even 6 2 56.4.i.b 6
28.g odd 6 2 392.4.i.m 6
56.j odd 6 2 448.4.i.m 6
56.m even 6 2 448.4.i.j 6
84.j odd 6 2 504.4.s.h 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.4.i.b 6 28.f even 6 2
112.4.i.e 6 7.d odd 6 2
392.4.a.i 3 28.d even 2 1
392.4.a.l 3 4.b odd 2 1
392.4.i.m 6 28.g odd 6 2
448.4.i.j 6 56.m even 6 2
448.4.i.m 6 56.j odd 6 2
504.4.s.h 6 84.j odd 6 2
784.4.a.bb 3 1.a even 1 1 trivial
784.4.a.be 3 7.b odd 2 1

## Hecke kernels

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

 $$T_{3}^{3} + 7 T_{3}^{2} - 25 T_{3} - 7$$ $$T_{5}^{3} - 3 T_{5}^{2} - 333 T_{5} + 2063$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$-7 - 25 T + 7 T^{2} + T^{3}$$
$5$ $$2063 - 333 T - 3 T^{2} + T^{3}$$
$7$ $$T^{3}$$
$11$ $$-38637 - 2313 T - 3 T^{2} + T^{3}$$
$13$ $$-26328 - 2244 T - 26 T^{2} + T^{3}$$
$17$ $$125571 - 6437 T - 31 T^{2} + T^{3}$$
$19$ $$-22529 + 383 T + 89 T^{2} + T^{3}$$
$23$ $$-85687 + 10935 T - 201 T^{2} + T^{3}$$
$29$ $$48504 + 8476 T - 190 T^{2} + T^{3}$$
$31$ $$-2554243 + 11079 T + 339 T^{2} + T^{3}$$
$37$ $$-885387 + 56523 T + 535 T^{2} + T^{3}$$
$41$ $$-3860392 - 42436 T + 58 T^{2} + T^{3}$$
$43$ $$-24343488 - 105680 T + 268 T^{2} + T^{3}$$
$47$ $$-11237437 - 86041 T + 205 T^{2} + T^{3}$$
$53$ $$74662809 - 103325 T - 757 T^{2} + T^{3}$$
$59$ $$196666617 + 1044039 T + 1799 T^{2} + T^{3}$$
$61$ $$-16537453 - 44101 T + 625 T^{2} + T^{3}$$
$67$ $$112177071 - 226713 T - 495 T^{2} + T^{3}$$
$71$ $$7291392 + 122304 T + 640 T^{2} + T^{3}$$
$73$ $$-100339897 - 564109 T - 443 T^{2} + T^{3}$$
$79$ $$-147181471 - 532441 T + 79 T^{2} + T^{3}$$
$83$ $$266787264 + 1641456 T + 2372 T^{2} + T^{3}$$
$89$ $$-96776649 - 545645 T + 821 T^{2} + T^{3}$$
$97$ $$-217321448 - 2726340 T - 342 T^{2} + T^{3}$$