# Properties

 Label 784.4.a.bd 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: $$\Z[a_1, \ldots, a_{5}]$$ 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.

 $$f(q)$$ $$=$$ $$q -\beta_{2} q^{3} + ( -4 - \beta_{1} ) q^{5} + ( 15 + 3 \beta_{1} - 2 \beta_{2} ) q^{9} +O(q^{10})$$ $$q -\beta_{2} q^{3} + ( -4 - \beta_{1} ) q^{5} + ( 15 + 3 \beta_{1} - 2 \beta_{2} ) q^{9} + ( 5 - \beta_{1} + 3 \beta_{2} ) q^{11} + ( -25 - \beta_{1} - 6 \beta_{2} ) q^{13} + ( 17 + \beta_{1} + 11 \beta_{2} ) q^{15} + ( -30 + 3 \beta_{1} + 10 \beta_{2} ) q^{17} + ( -21 - 7 \beta_{1} + 11 \beta_{2} ) q^{19} + ( -60 + 6 \beta_{1} + 17 \beta_{2} ) q^{23} + ( 6 + 2 \beta_{1} + 8 \beta_{2} ) q^{25} + ( 33 + 3 \beta_{1} - 13 \beta_{2} ) q^{27} + ( -61 + 7 \beta_{1} - 14 \beta_{2} ) q^{29} + ( 197 + 9 \beta_{1} + 3 \beta_{2} ) q^{31} + ( -109 - 8 \beta_{1} + 8 \beta_{2} ) q^{33} + ( 89 - 16 \beta_{1} + 34 \beta_{2} ) q^{37} + ( 269 + 19 \beta_{1} + 20 \beta_{2} ) q^{39} + ( -247 + 17 \beta_{1} - 26 \beta_{2} ) q^{41} + ( 92 + 8 \beta_{1} - 24 \beta_{2} ) q^{43} + ( -371 - 7 \beta_{1} - 2 \beta_{2} ) q^{45} + ( 31 + 27 \beta_{1} - 19 \beta_{2} ) q^{47} + ( -471 - 33 \beta_{1} + 29 \beta_{2} ) q^{51} + ( 71 + 10 \beta_{1} + 26 \beta_{2} ) q^{53} + ( 44 - 10 \beta_{1} - 25 \beta_{2} ) q^{55} + ( -343 - 26 \beta_{1} + 92 \beta_{2} ) q^{57} + ( 148 - 40 \beta_{1} + 51 \beta_{2} ) q^{59} + ( -165 + 48 \beta_{1} + 2 \beta_{2} ) q^{61} + ( 317 + 29 \beta_{1} + 74 \beta_{2} ) q^{65} + ( 186 - 50 \beta_{1} - 11 \beta_{2} ) q^{67} + ( -816 - 57 \beta_{1} + 52 \beta_{2} ) q^{69} + ( -70 - 42 \beta_{1} - 28 \beta_{2} ) q^{71} + ( -581 - 14 \beta_{1} - 56 \beta_{2} ) q^{73} + ( -370 - 26 \beta_{1} - 4 \beta_{2} ) q^{75} + ( -428 + 58 \beta_{1} - 83 \beta_{2} ) q^{79} + ( 90 - 45 \beta_{1} - 26 \beta_{2} ) q^{81} + ( -458 + 22 \beta_{1} + 28 \beta_{2} ) q^{83} + ( -395 + 26 \beta_{1} - 134 \beta_{2} ) q^{85} + ( 469 + 35 \beta_{1} - 16 \beta_{2} ) q^{87} + ( -551 - 52 \beta_{1} + 44 \beta_{2} ) q^{89} + ( -279 - 18 \beta_{1} - 254 \beta_{2} ) q^{93} + ( 702 - 4 \beta_{1} - 65 \beta_{2} ) q^{95} + ( -877 + 31 \beta_{1} + 2 \beta_{2} ) q^{97} + ( -335 + 11 \beta_{1} + 100 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q + q^{3} - 13q^{5} + 50q^{9} + O(q^{10})$$ $$3q + q^{3} - 13q^{5} + 50q^{9} + 11q^{11} - 70q^{13} + 41q^{15} - 97q^{17} - 81q^{19} - 191q^{23} + 12q^{25} + 115q^{27} - 162q^{29} + 597q^{31} - 343q^{33} + 217q^{37} + 806q^{39} - 698q^{41} + 308q^{43} - 1118q^{45} + 139q^{47} - 1475q^{51} + 197q^{53} + 147q^{55} - 1147q^{57} + 353q^{59} - 449q^{61} + 906q^{65} + 519q^{67} - 2557q^{69} - 224q^{71} - 1701q^{73} - 1132q^{75} - 1143q^{79} + 251q^{81} - 1380q^{83} - 1025q^{85} + 1458q^{87} - 1749q^{89} - 601q^{93} + 2167q^{95} - 2602q^{97} - 1094q^{99} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$4 \nu - 1$$ $$\beta_{2}$$ $$=$$ $$2 \nu^{2} + 2 \nu - 15$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{1} + 1$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$2 \beta_{2} - \beta_{1} + 29$$$$)/4$$

## 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
 2.90222 −3.27144 1.36922
0 −7.65024 0 −14.6089 0 0 0 31.5262 0
1.2 0 0.138208 0 10.0858 0 0 0 −26.9809 0
1.3 0 8.51203 0 −8.47688 0 0 0 45.4547 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.bd 3
4.b odd 2 1 392.4.a.j 3
7.b odd 2 1 784.4.a.bc 3
7.d odd 6 2 112.4.i.f 6
28.d even 2 1 392.4.a.k 3
28.f even 6 2 56.4.i.a 6
28.g odd 6 2 392.4.i.n 6
56.j odd 6 2 448.4.i.k 6
56.m even 6 2 448.4.i.l 6
84.j odd 6 2 504.4.s.i 6

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

## Hecke kernels

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

 $$T_{3}^{3} - T_{3}^{2} - 65 T_{3} + 9$$ $$T_{5}^{3} + 13 T_{5}^{2} - 109 T_{5} - 1249$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$9 - 65 T - T^{2} + T^{3}$$
$5$ $$-1249 - 109 T + 13 T^{2} + T^{3}$$
$7$ $$T^{3}$$
$11$ $$8099 - 577 T - 11 T^{2} + T^{3}$$
$13$ $$-17752 - 1156 T + 70 T^{2} + T^{3}$$
$17$ $$-586557 - 6245 T + 97 T^{2} + T^{3}$$
$19$ $$-123377 - 10329 T + 81 T^{2} + T^{3}$$
$23$ $$-3492279 - 17297 T + 191 T^{2} + T^{3}$$
$29$ $$-1324296 - 7716 T + 162 T^{2} + T^{3}$$
$31$ $$-4663139 + 103599 T - 597 T^{2} + T^{3}$$
$37$ $$15110373 - 77493 T - 217 T^{2} + T^{3}$$
$41$ $$-6465192 + 90492 T + 698 T^{2} + T^{3}$$
$43$ $$-38848 - 7888 T - 308 T^{2} + T^{3}$$
$47$ $$18709731 - 114417 T - 139 T^{2} + T^{3}$$
$53$ $$-2914839 - 59549 T - 197 T^{2} + T^{3}$$
$59$ $$37291113 - 300449 T - 353 T^{2} + T^{3}$$
$61$ $$9942147 - 318341 T + 449 T^{2} + T^{3}$$
$67$ $$21323807 - 356385 T - 519 T^{2} + T^{3}$$
$71$ $$-7551488 - 379456 T + 224 T^{2} + T^{3}$$
$73$ $$72721831 + 691635 T + 1701 T^{2} + T^{3}$$
$79$ $$-297161663 - 352545 T + 1143 T^{2} + T^{3}$$
$83$ $$4797376 + 475632 T + 1380 T^{2} + T^{3}$$
$89$ $$-155620521 + 549843 T + 1749 T^{2} + T^{3}$$
$97$ $$528741144 + 2094844 T + 2602 T^{2} + T^{3}$$