# Properties

 Label 384.8.a.l Level $384$ Weight $8$ Character orbit 384.a Self dual yes Analytic conductor $119.956$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$384 = 2^{7} \cdot 3$$ Weight: $$k$$ $$=$$ $$8$$ Character orbit: $$[\chi]$$ $$=$$ 384.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$119.955849786$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: $$\mathbb{Q}[x]/(x^{3} - \cdots)$$ Defining polynomial: $$x^{3} - 286x - 1680$$ x^3 - 286*x - 1680 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{8}\cdot 3$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + 27 q^{3} + (\beta_1 + 103) q^{5} + (\beta_{2} + 2) q^{7} + 729 q^{9}+O(q^{10})$$ q + 27 * q^3 + (b1 + 103) * q^5 + (b2 + 2) * q^7 + 729 * q^9 $$q + 27 q^{3} + (\beta_1 + 103) q^{5} + (\beta_{2} + 2) q^{7} + 729 q^{9} + (2 \beta_{2} + 8 \beta_1 + 492) q^{11} + (3 \beta_{2} + 5 \beta_1 + 1025) q^{13} + (27 \beta_1 + 2781) q^{15} + (16 \beta_{2} + 50 \beta_1 - 1692) q^{17} + (8 \beta_{2} - 2 \beta_1 + 4722) q^{19} + (27 \beta_{2} + 54) q^{21} + ( - 34 \beta_{2} - 4 \beta_1 + 19992) q^{23} + (30 \beta_{2} + 58 \beta_1 + 10529) q^{25} + 19683 q^{27} + (2 \beta_{2} - 383 \beta_1 + 23867) q^{29} + ( - 103 \beta_{2} + 448 \beta_1 + 59306) q^{31} + (54 \beta_{2} + 216 \beta_1 + 13284) q^{33} + (250 \beta_{2} + 580 \beta_1 - 28060) q^{35} + (173 \beta_{2} - 567 \beta_1 - 61527) q^{37} + (81 \beta_{2} + 135 \beta_1 + 27675) q^{39} + (36 \beta_{2} + 146 \beta_1 + 219132) q^{41} + ( - 284 \beta_{2} + 1342 \beta_1 - 39734) q^{43} + (729 \beta_1 + 75087) q^{45} + ( - 258 \beta_{2} + 448 \beta_1 + 221860) q^{47} + ( - 574 \beta_{2} + 1890 \beta_1 + 402179) q^{49} + (432 \beta_{2} + 1350 \beta_1 - 45684) q^{51} + ( - 814 \beta_{2} - 3603 \beta_1 - 38497) q^{53} + (740 \beta_{2} + 1288 \beta_1 + 618504) q^{55} + (216 \beta_{2} - 54 \beta_1 + 127494) q^{57} + ( - 1364 \beta_{2} - 3636 \beta_1 - 345648) q^{59} + ( - 115 \beta_{2} + 2629 \beta_1 + 1195645) q^{61} + (729 \beta_{2} + 1458) q^{63} + (900 \beta_{2} + 2534 \beta_1 + 411002) q^{65} + ( - 1380 \beta_{2} - 1988 \beta_1 - 782296) q^{67} + ( - 918 \beta_{2} - 108 \beta_1 + 539784) q^{69} + (2054 \beta_{2} + 1460 \beta_1 + 2104328) q^{71} + ( - 2938 \beta_{2} - 9690 \beta_1 - 707288) q^{73} + (810 \beta_{2} + 1566 \beta_1 + 284283) q^{75} + (516 \beta_{2} + 8420 \beta_1 + 2226292) q^{77} + ( - 4627 \beta_{2} - 11692 \beta_1 + 669046) q^{79} + 531441 q^{81} + (2838 \beta_{2} - 13452 \beta_1 - 2481312) q^{83} + (5500 \beta_{2} + 5306 \beta_1 + 3275718) q^{85} + (54 \beta_{2} - 10341 \beta_1 + 644409) q^{87} + (2720 \beta_{2} - 27228 \beta_1 - 4442546) q^{89} + (32 \beta_{2} + 8570 \beta_1 + 3537874) q^{91} + ( - 2781 \beta_{2} + 12096 \beta_1 + 1601262) q^{93} + (1940 \beta_{2} + 9436 \beta_1 + 104148) q^{95} + ( - 204 \beta_{2} + 22864 \beta_1 + 994426) q^{97} + (1458 \beta_{2} + 5832 \beta_1 + 358668) q^{99}+O(q^{100})$$ q + 27 * q^3 + (b1 + 103) * q^5 + (b2 + 2) * q^7 + 729 * q^9 + (2*b2 + 8*b1 + 492) * q^11 + (3*b2 + 5*b1 + 1025) * q^13 + (27*b1 + 2781) * q^15 + (16*b2 + 50*b1 - 1692) * q^17 + (8*b2 - 2*b1 + 4722) * q^19 + (27*b2 + 54) * q^21 + (-34*b2 - 4*b1 + 19992) * q^23 + (30*b2 + 58*b1 + 10529) * q^25 + 19683 * q^27 + (2*b2 - 383*b1 + 23867) * q^29 + (-103*b2 + 448*b1 + 59306) * q^31 + (54*b2 + 216*b1 + 13284) * q^33 + (250*b2 + 580*b1 - 28060) * q^35 + (173*b2 - 567*b1 - 61527) * q^37 + (81*b2 + 135*b1 + 27675) * q^39 + (36*b2 + 146*b1 + 219132) * q^41 + (-284*b2 + 1342*b1 - 39734) * q^43 + (729*b1 + 75087) * q^45 + (-258*b2 + 448*b1 + 221860) * q^47 + (-574*b2 + 1890*b1 + 402179) * q^49 + (432*b2 + 1350*b1 - 45684) * q^51 + (-814*b2 - 3603*b1 - 38497) * q^53 + (740*b2 + 1288*b1 + 618504) * q^55 + (216*b2 - 54*b1 + 127494) * q^57 + (-1364*b2 - 3636*b1 - 345648) * q^59 + (-115*b2 + 2629*b1 + 1195645) * q^61 + (729*b2 + 1458) * q^63 + (900*b2 + 2534*b1 + 411002) * q^65 + (-1380*b2 - 1988*b1 - 782296) * q^67 + (-918*b2 - 108*b1 + 539784) * q^69 + (2054*b2 + 1460*b1 + 2104328) * q^71 + (-2938*b2 - 9690*b1 - 707288) * q^73 + (810*b2 + 1566*b1 + 284283) * q^75 + (516*b2 + 8420*b1 + 2226292) * q^77 + (-4627*b2 - 11692*b1 + 669046) * q^79 + 531441 * q^81 + (2838*b2 - 13452*b1 - 2481312) * q^83 + (5500*b2 + 5306*b1 + 3275718) * q^85 + (54*b2 - 10341*b1 + 644409) * q^87 + (2720*b2 - 27228*b1 - 4442546) * q^89 + (32*b2 + 8570*b1 + 3537874) * q^91 + (-2781*b2 + 12096*b1 + 1601262) * q^93 + (1940*b2 + 9436*b1 + 104148) * q^95 + (-204*b2 + 22864*b1 + 994426) * q^97 + (1458*b2 + 5832*b1 + 358668) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 81 q^{3} + 308 q^{5} + 6 q^{7} + 2187 q^{9}+O(q^{10})$$ 3 * q + 81 * q^3 + 308 * q^5 + 6 * q^7 + 2187 * q^9 $$3 q + 81 q^{3} + 308 q^{5} + 6 q^{7} + 2187 q^{9} + 1468 q^{11} + 3070 q^{13} + 8316 q^{15} - 5126 q^{17} + 14168 q^{19} + 162 q^{21} + 59980 q^{23} + 31529 q^{25} + 59049 q^{27} + 71984 q^{29} + 177470 q^{31} + 39636 q^{33} - 84760 q^{35} - 184014 q^{37} + 82890 q^{39} + 657250 q^{41} - 120544 q^{43} + 224532 q^{45} + 665132 q^{47} + 1204647 q^{49} - 138402 q^{51} - 111888 q^{53} + 1854224 q^{55} + 382536 q^{57} - 1033308 q^{59} + 3584306 q^{61} + 4374 q^{63} + 1230472 q^{65} - 2344900 q^{67} + 1619460 q^{69} + 6311524 q^{71} - 2112174 q^{73} + 851283 q^{75} + 6670456 q^{77} + 2018830 q^{79} + 1594323 q^{81} - 7430484 q^{83} + 9821848 q^{85} + 1943568 q^{87} - 13300410 q^{89} + 10605052 q^{91} + 4791690 q^{93} + 303008 q^{95} + 2960414 q^{97} + 1070172 q^{99}+O(q^{100})$$ 3 * q + 81 * q^3 + 308 * q^5 + 6 * q^7 + 2187 * q^9 + 1468 * q^11 + 3070 * q^13 + 8316 * q^15 - 5126 * q^17 + 14168 * q^19 + 162 * q^21 + 59980 * q^23 + 31529 * q^25 + 59049 * q^27 + 71984 * q^29 + 177470 * q^31 + 39636 * q^33 - 84760 * q^35 - 184014 * q^37 + 82890 * q^39 + 657250 * q^41 - 120544 * q^43 + 224532 * q^45 + 665132 * q^47 + 1204647 * q^49 - 138402 * q^51 - 111888 * q^53 + 1854224 * q^55 + 382536 * q^57 - 1033308 * q^59 + 3584306 * q^61 + 4374 * q^63 + 1230472 * q^65 - 2344900 * q^67 + 1619460 * q^69 + 6311524 * q^71 - 2112174 * q^73 + 851283 * q^75 + 6670456 * q^77 + 2018830 * q^79 + 1594323 * q^81 - 7430484 * q^83 + 9821848 * q^85 + 1943568 * q^87 - 13300410 * q^89 + 10605052 * q^91 + 4791690 * q^93 + 303008 * q^95 + 2960414 * q^97 + 1070172 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - 286x - 1680$$ :

 $$\beta_{1}$$ $$=$$ $$4\nu^{2} - 24\nu - 763$$ 4*v^2 - 24*v - 763 $$\beta_{2}$$ $$=$$ $$-12\nu^{2} + 168\nu + 2288$$ -12*v^2 + 168*v + 2288
 $$\nu$$ $$=$$ $$( \beta_{2} + 3\beta _1 + 1 ) / 96$$ (b2 + 3*b1 + 1) / 96 $$\nu^{2}$$ $$=$$ $$( \beta_{2} + 7\beta _1 + 3053 ) / 16$$ (b2 + 7*b1 + 3053) / 16

## 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
 −7.15472 −12.1582 19.3129
0 27.0000 0 −283.526 0 473.726 0 729.000 0
1.2 0 27.0000 0 223.082 0 −1526.43 0 729.000 0
1.3 0 27.0000 0 368.444 0 1058.71 0 729.000 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
$$2$$ $$-1$$
$$3$$ $$-1$$

## 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 384.8.a.l yes 3
4.b odd 2 1 384.8.a.j yes 3
8.b even 2 1 384.8.a.i 3
8.d odd 2 1 384.8.a.k yes 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.8.a.i 3 8.b even 2 1
384.8.a.j yes 3 4.b odd 2 1
384.8.a.k yes 3 8.d odd 2 1
384.8.a.l yes 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 $$S_{8}^{\mathrm{new}}(\Gamma_0(384))$$:

 $$T_{5}^{3} - 308T_{5}^{2} - 85520T_{5} + 23304000$$ T5^3 - 308*T5^2 - 85520*T5 + 23304000 $$T_{7}^{3} - 6T_{7}^{2} - 1837620T_{7} + 765563000$$ T7^3 - 6*T7^2 - 1837620*T7 + 765563000

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$(T - 27)^{3}$$
$5$ $$T^{3} - 308 T^{2} + \cdots + 23304000$$
$7$ $$T^{3} - 6 T^{2} - 1837620 T + 765563000$$
$11$ $$T^{3} - 1468 T^{2} + \cdots - 12568576832$$
$13$ $$T^{3} - 3070 T^{2} + \cdots + 8296093848$$
$17$ $$T^{3} + 5126 T^{2} + \cdots - 7729742082552$$
$19$ $$T^{3} - 14168 T^{2} + \cdots + 907808232960$$
$23$ $$T^{3} - 59980 T^{2} + \cdots + 6681673416640$$
$29$ $$T^{3} + \cdots - 329426101684224$$
$31$ $$T^{3} - 177470 T^{2} + \cdots + 30\!\cdots\!60$$
$37$ $$T^{3} + 184014 T^{2} + \cdots - 27\!\cdots\!16$$
$41$ $$T^{3} - 657250 T^{2} + \cdots - 96\!\cdots\!60$$
$43$ $$T^{3} + 120544 T^{2} + \cdots + 63\!\cdots\!12$$
$47$ $$T^{3} - 665132 T^{2} + \cdots + 33\!\cdots\!52$$
$53$ $$T^{3} + 111888 T^{2} + \cdots + 13\!\cdots\!40$$
$59$ $$T^{3} + 1033308 T^{2} + \cdots + 14\!\cdots\!20$$
$61$ $$T^{3} - 3584306 T^{2} + \cdots - 37\!\cdots\!20$$
$67$ $$T^{3} + 2344900 T^{2} + \cdots - 20\!\cdots\!16$$
$71$ $$T^{3} - 6311524 T^{2} + \cdots + 10\!\cdots\!00$$
$73$ $$T^{3} + 2112174 T^{2} + \cdots + 27\!\cdots\!20$$
$79$ $$T^{3} - 2018830 T^{2} + \cdots + 13\!\cdots\!00$$
$83$ $$T^{3} + 7430484 T^{2} + \cdots - 10\!\cdots\!56$$
$89$ $$T^{3} + 13300410 T^{2} + \cdots - 76\!\cdots\!96$$
$97$ $$T^{3} - 2960414 T^{2} + \cdots + 22\!\cdots\!96$$