Properties

 Label 6384.2.a.ce Level $6384$ Weight $2$ Character orbit 6384.a Self dual yes Analytic conductor $50.976$ Analytic rank $0$ Dimension $5$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$6384 = 2^{4} \cdot 3 \cdot 7 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6384.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$50.9764966504$$ Analytic rank: $$0$$ Dimension: $$5$$ Coefficient field: 5.5.135076.1 Defining polynomial: $$x^{5} - x^{4} - 5x^{3} + 4x^{2} + 4x - 2$$ x^5 - x^4 - 5*x^3 + 4*x^2 + 4*x - 2 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{5}$$ Twist minimal: no (minimal twist has level 3192) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3,\beta_4$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{3} + \beta_1 q^{5} - q^{7} + q^{9}+O(q^{10})$$ q + q^3 + b1 * q^5 - q^7 + q^9 $$q + q^{3} + \beta_1 q^{5} - q^{7} + q^{9} + \beta_{4} q^{11} + ( - \beta_{3} + 2) q^{13} + \beta_1 q^{15} - \beta_{2} q^{17} - q^{19} - q^{21} - \beta_{3} q^{23} + ( - \beta_{4} - \beta_{3} + \beta_{2} + \beta_1 + 3) q^{25} + q^{27} + (\beta_{3} + \beta_1) q^{29} + ( - \beta_{4} + \beta_{3}) q^{31} + \beta_{4} q^{33} - \beta_1 q^{35} + (\beta_{4} + \beta_{3} + 2) q^{37} + ( - \beta_{3} + 2) q^{39} + (2 \beta_1 - 2) q^{41} + \beta_1 q^{45} + (2 \beta_{4} + \beta_{3} - \beta_{2} - 2 \beta_1 + 2) q^{47} + q^{49} - \beta_{2} q^{51} + ( - \beta_{3} - \beta_1 + 4) q^{53} + (\beta_{4} - \beta_{3} + \beta_{2} - \beta_1 + 4) q^{55} - q^{57} + ( - 3 \beta_{4} - \beta_{3} + \beta_{2} + 3 \beta_1) q^{59} + 6 q^{61} - q^{63} + (2 \beta_{4} + 2 \beta_{3} + 2 \beta_1) q^{65} + ( - 2 \beta_{4} - \beta_{3} - \beta_{2} + \beta_1 - 4) q^{67} - \beta_{3} q^{69} + (\beta_{4} + 2 \beta_{3} - \beta_{2} + 2) q^{71} + (\beta_{4} + \beta_{3} + \beta_{2} + \beta_1 + 2) q^{73} + ( - \beta_{4} - \beta_{3} + \beta_{2} + \beta_1 + 3) q^{75} - \beta_{4} q^{77} + (\beta_{4} + \beta_{2} + \beta_1) q^{79} + q^{81} + (2 \beta_{4} + \beta_{3} - \beta_1 + 2) q^{83} + ( - 2 \beta_{4} - 2 \beta_{2} - 4 \beta_1 + 4) q^{85} + (\beta_{3} + \beta_1) q^{87} + (\beta_{4} - \beta_{3} + \beta_{2} + \beta_1 + 2) q^{89} + (\beta_{3} - 2) q^{91} + ( - \beta_{4} + \beta_{3}) q^{93} - \beta_1 q^{95} + (\beta_{4} + 2 \beta_{3} - 2 \beta_1 + 2) q^{97} + \beta_{4} q^{99}+O(q^{100})$$ q + q^3 + b1 * q^5 - q^7 + q^9 + b4 * q^11 + (-b3 + 2) * q^13 + b1 * q^15 - b2 * q^17 - q^19 - q^21 - b3 * q^23 + (-b4 - b3 + b2 + b1 + 3) * q^25 + q^27 + (b3 + b1) * q^29 + (-b4 + b3) * q^31 + b4 * q^33 - b1 * q^35 + (b4 + b3 + 2) * q^37 + (-b3 + 2) * q^39 + (2*b1 - 2) * q^41 + b1 * q^45 + (2*b4 + b3 - b2 - 2*b1 + 2) * q^47 + q^49 - b2 * q^51 + (-b3 - b1 + 4) * q^53 + (b4 - b3 + b2 - b1 + 4) * q^55 - q^57 + (-3*b4 - b3 + b2 + 3*b1) * q^59 + 6 * q^61 - q^63 + (2*b4 + 2*b3 + 2*b1) * q^65 + (-2*b4 - b3 - b2 + b1 - 4) * q^67 - b3 * q^69 + (b4 + 2*b3 - b2 + 2) * q^71 + (b4 + b3 + b2 + b1 + 2) * q^73 + (-b4 - b3 + b2 + b1 + 3) * q^75 - b4 * q^77 + (b4 + b2 + b1) * q^79 + q^81 + (2*b4 + b3 - b1 + 2) * q^83 + (-2*b4 - 2*b2 - 4*b1 + 4) * q^85 + (b3 + b1) * q^87 + (b4 - b3 + b2 + b1 + 2) * q^89 + (b3 - 2) * q^91 + (-b4 + b3) * q^93 - b1 * q^95 + (b4 + 2*b3 - 2*b1 + 2) * q^97 + b4 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5 q + 5 q^{3} + 2 q^{5} - 5 q^{7} + 5 q^{9}+O(q^{10})$$ 5 * q + 5 * q^3 + 2 * q^5 - 5 * q^7 + 5 * q^9 $$5 q + 5 q^{3} + 2 q^{5} - 5 q^{7} + 5 q^{9} - 2 q^{11} + 8 q^{13} + 2 q^{15} - 2 q^{17} - 5 q^{19} - 5 q^{21} - 2 q^{23} + 19 q^{25} + 5 q^{27} + 4 q^{29} + 4 q^{31} - 2 q^{33} - 2 q^{35} + 10 q^{37} + 8 q^{39} - 6 q^{41} + 2 q^{45} + 2 q^{47} + 5 q^{49} - 2 q^{51} + 16 q^{53} + 16 q^{55} - 5 q^{57} + 12 q^{59} + 30 q^{61} - 5 q^{63} + 4 q^{65} - 18 q^{67} - 2 q^{69} + 10 q^{71} + 14 q^{73} + 19 q^{75} + 2 q^{77} + 2 q^{79} + 5 q^{81} + 6 q^{83} + 12 q^{85} + 4 q^{87} + 10 q^{89} - 8 q^{91} + 4 q^{93} - 2 q^{95} + 8 q^{97} - 2 q^{99}+O(q^{100})$$ 5 * q + 5 * q^3 + 2 * q^5 - 5 * q^7 + 5 * q^9 - 2 * q^11 + 8 * q^13 + 2 * q^15 - 2 * q^17 - 5 * q^19 - 5 * q^21 - 2 * q^23 + 19 * q^25 + 5 * q^27 + 4 * q^29 + 4 * q^31 - 2 * q^33 - 2 * q^35 + 10 * q^37 + 8 * q^39 - 6 * q^41 + 2 * q^45 + 2 * q^47 + 5 * q^49 - 2 * q^51 + 16 * q^53 + 16 * q^55 - 5 * q^57 + 12 * q^59 + 30 * q^61 - 5 * q^63 + 4 * q^65 - 18 * q^67 - 2 * q^69 + 10 * q^71 + 14 * q^73 + 19 * q^75 + 2 * q^77 + 2 * q^79 + 5 * q^81 + 6 * q^83 + 12 * q^85 + 4 * q^87 + 10 * q^89 - 8 * q^91 + 4 * q^93 - 2 * q^95 + 8 * q^97 - 2 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{5} - x^{4} - 5x^{3} + 4x^{2} + 4x - 2$$ :

 $$\beta_{1}$$ $$=$$ $$2\nu$$ 2*v $$\beta_{2}$$ $$=$$ $$2\nu^{3} + 2\nu^{2} - 8\nu - 4$$ 2*v^3 + 2*v^2 - 8*v - 4 $$\beta_{3}$$ $$=$$ $$-2\nu^{4} + 2\nu^{3} + 8\nu^{2} - 6\nu - 2$$ -2*v^4 + 2*v^3 + 8*v^2 - 6*v - 2 $$\beta_{4}$$ $$=$$ $$2\nu^{4} - 10\nu^{2} + 6$$ 2*v^4 - 10*v^2 + 6
 $$\nu$$ $$=$$ $$( \beta_1 ) / 2$$ (b1) / 2 $$\nu^{2}$$ $$=$$ $$( -\beta_{4} - \beta_{3} + \beta_{2} + \beta _1 + 8 ) / 4$$ (-b4 - b3 + b2 + b1 + 8) / 4 $$\nu^{3}$$ $$=$$ $$( \beta_{4} + \beta_{3} + \beta_{2} + 7\beta_1 ) / 4$$ (b4 + b3 + b2 + 7*b1) / 4 $$\nu^{4}$$ $$=$$ $$( -3\beta_{4} - 5\beta_{3} + 5\beta_{2} + 5\beta _1 + 28 ) / 4$$ (-3*b4 - 5*b3 + 5*b2 + 5*b1 + 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
 −1.94486 −0.895793 0.420632 1.26848 2.15154
0 1.00000 0 −3.88971 0 −1.00000 0 1.00000 0
1.2 0 1.00000 0 −1.79159 0 −1.00000 0 1.00000 0
1.3 0 1.00000 0 0.841263 0 −1.00000 0 1.00000 0
1.4 0 1.00000 0 2.53696 0 −1.00000 0 1.00000 0
1.5 0 1.00000 0 4.30308 0 −1.00000 0 1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.5 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$3$$ $$-1$$
$$7$$ $$1$$
$$19$$ $$1$$

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 6384.2.a.ce 5
4.b odd 2 1 3192.2.a.ba 5
12.b even 2 1 9576.2.a.co 5

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3192.2.a.ba 5 4.b odd 2 1
6384.2.a.ce 5 1.a even 1 1 trivial
9576.2.a.co 5 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(6384))$$:

 $$T_{5}^{5} - 2T_{5}^{4} - 20T_{5}^{3} + 32T_{5}^{2} + 64T_{5} - 64$$ T5^5 - 2*T5^4 - 20*T5^3 + 32*T5^2 + 64*T5 - 64 $$T_{11}^{5} + 2T_{11}^{4} - 28T_{11}^{3} - 40T_{11}^{2} + 160T_{11} + 128$$ T11^5 + 2*T11^4 - 28*T11^3 - 40*T11^2 + 160*T11 + 128 $$T_{13}^{5} - 8T_{13}^{4} - 12T_{13}^{3} + 208T_{13}^{2} - 352T_{13} - 64$$ T13^5 - 8*T13^4 - 12*T13^3 + 208*T13^2 - 352*T13 - 64 $$T_{17}^{5} + 2T_{17}^{4} - 92T_{17}^{3} - 192T_{17}^{2} + 1984T_{17} + 5504$$ T17^5 + 2*T17^4 - 92*T17^3 - 192*T17^2 + 1984*T17 + 5504 $$T_{23}^{5} + 2T_{23}^{4} - 36T_{23}^{3} + 24T_{23}^{2} + 160T_{23} - 128$$ T23^5 + 2*T23^4 - 36*T23^3 + 24*T23^2 + 160*T23 - 128

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{5}$$
$3$ $$(T - 1)^{5}$$
$5$ $$T^{5} - 2 T^{4} - 20 T^{3} + 32 T^{2} + \cdots - 64$$
$7$ $$(T + 1)^{5}$$
$11$ $$T^{5} + 2 T^{4} - 28 T^{3} - 40 T^{2} + \cdots + 128$$
$13$ $$T^{5} - 8 T^{4} - 12 T^{3} + 208 T^{2} + \cdots - 64$$
$17$ $$T^{5} + 2 T^{4} - 92 T^{3} + \cdots + 5504$$
$19$ $$(T + 1)^{5}$$
$23$ $$T^{5} + 2 T^{4} - 36 T^{3} + 24 T^{2} + \cdots - 128$$
$29$ $$T^{5} - 4 T^{4} - 52 T^{3} + \cdots - 1376$$
$31$ $$T^{5} - 4 T^{4} - 80 T^{3} + 208 T^{2} + \cdots + 256$$
$37$ $$T^{5} - 10 T^{4} - 8 T^{3} + 224 T^{2} + \cdots - 352$$
$41$ $$T^{5} + 6 T^{4} - 72 T^{3} - 240 T^{2} + \cdots + 352$$
$43$ $$T^{5}$$
$47$ $$T^{5} - 2 T^{4} - 180 T^{3} + \cdots + 17728$$
$53$ $$T^{5} - 16 T^{4} + 44 T^{3} + 104 T^{2} + \cdots - 32$$
$59$ $$T^{5} - 12 T^{4} - 256 T^{3} + \cdots - 105472$$
$61$ $$(T - 6)^{5}$$
$67$ $$T^{5} + 18 T^{4} - 108 T^{3} + \cdots - 10624$$
$71$ $$T^{5} - 10 T^{4} - 172 T^{3} + \cdots - 44288$$
$73$ $$T^{5} - 14 T^{4} - 104 T^{3} + \cdots + 8992$$
$79$ $$T^{5} - 2 T^{4} - 156 T^{3} + \cdots + 512$$
$83$ $$T^{5} - 6 T^{4} - 92 T^{3} + \cdots - 4096$$
$89$ $$T^{5} - 10 T^{4} - 168 T^{3} + \cdots - 6304$$
$97$ $$T^{5} - 8 T^{4} - 172 T^{3} + \cdots - 128$$