Properties

 Label 8280.2.a.bq Level $8280$ Weight $2$ Character orbit 8280.a Self dual yes Analytic conductor $66.116$ Analytic rank $1$ Dimension $4$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$8280 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8280.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$66.1161328736$$ Analytic rank: $$1$$ Dimension: $$4$$ Coefficient field: 4.4.54764.1 Defining polynomial: $$x^{4} - x^{3} - 9x^{2} + 3x + 2$$ x^4 - x^3 - 9*x^2 + 3*x + 2 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 2760) 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.

 $$f(q)$$ $$=$$ $$q + q^{5} - \beta_1 q^{7}+O(q^{10})$$ q + q^5 - b1 * q^7 $$q + q^{5} - \beta_1 q^{7} + \beta_{3} q^{11} + ( - \beta_{2} + \beta_1 - 1) q^{13} + ( - 2 \beta_{3} - \beta_{2} - 1) q^{17} + (\beta_{2} + \beta_1 - 1) q^{19} - q^{23} + q^{25} + (\beta_{3} + 2 \beta_{2} - \beta_1) q^{29} + (\beta_{3} + \beta_{2} - 1) q^{31} - \beta_1 q^{35} + (2 \beta_{2} + \beta_1) q^{37} + ( - \beta_{3} + \beta_1 - 2) q^{41} + ( - 2 \beta_{2} - 6) q^{43} + (\beta_{2} + \beta_1 - 1) q^{47} + ( - 3 \beta_{3} - \beta_{2} + 2) q^{49} + (\beta_{3} - 2 \beta_{2} - \beta_1) q^{53} + \beta_{3} q^{55} + (\beta_{3} + 2 \beta_{2} - \beta_1 + 2) q^{59} + (\beta_{3} + 2 \beta_{2}) q^{61} + ( - \beta_{2} + \beta_1 - 1) q^{65} + ( - \beta_{3} - \beta_{2} - 2 \beta_1 - 7) q^{67} + ( - \beta_{3} + \beta_1) q^{71} + ( - \beta_{2} + \beta_1 - 5) q^{73} + (\beta_{2} + 3 \beta_1 - 1) q^{77} + (\beta_{3} + 3 \beta_{2} + \beta_1 - 3) q^{79} + ( - 2 \beta_{3} + \beta_{2} + 7) q^{83} + ( - 2 \beta_{3} - \beta_{2} - 1) q^{85} + (\beta_{3} + \beta_{2} + \beta_1 - 3) q^{89} + (2 \beta_{3} + \beta_{2} + \beta_1 - 9) q^{91} + (\beta_{2} + \beta_1 - 1) q^{95} + ( - 2 \beta_{3} - 2 \beta_{2} - 4) q^{97}+O(q^{100})$$ q + q^5 - b1 * q^7 + b3 * q^11 + (-b2 + b1 - 1) * q^13 + (-2*b3 - b2 - 1) * q^17 + (b2 + b1 - 1) * q^19 - q^23 + q^25 + (b3 + 2*b2 - b1) * q^29 + (b3 + b2 - 1) * q^31 - b1 * q^35 + (2*b2 + b1) * q^37 + (-b3 + b1 - 2) * q^41 + (-2*b2 - 6) * q^43 + (b2 + b1 - 1) * q^47 + (-3*b3 - b2 + 2) * q^49 + (b3 - 2*b2 - b1) * q^53 + b3 * q^55 + (b3 + 2*b2 - b1 + 2) * q^59 + (b3 + 2*b2) * q^61 + (-b2 + b1 - 1) * q^65 + (-b3 - b2 - 2*b1 - 7) * q^67 + (-b3 + b1) * q^71 + (-b2 + b1 - 5) * q^73 + (b2 + 3*b1 - 1) * q^77 + (b3 + 3*b2 + b1 - 3) * q^79 + (-2*b3 + b2 + 7) * q^83 + (-2*b3 - b2 - 1) * q^85 + (b3 + b2 + b1 - 3) * q^89 + (2*b3 + b2 + b1 - 9) * q^91 + (b2 + b1 - 1) * q^95 + (-2*b3 - 2*b2 - 4) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{5}+O(q^{10})$$ 4 * q + 4 * q^5 $$4 q + 4 q^{5} - 2 q^{13} - 2 q^{17} - 6 q^{19} - 4 q^{23} + 4 q^{25} - 4 q^{29} - 6 q^{31} - 4 q^{37} - 8 q^{41} - 20 q^{43} - 6 q^{47} + 10 q^{49} + 4 q^{53} + 4 q^{59} - 4 q^{61} - 2 q^{65} - 26 q^{67} - 18 q^{73} - 6 q^{77} - 18 q^{79} + 26 q^{83} - 2 q^{85} - 14 q^{89} - 38 q^{91} - 6 q^{95} - 12 q^{97}+O(q^{100})$$ 4 * q + 4 * q^5 - 2 * q^13 - 2 * q^17 - 6 * q^19 - 4 * q^23 + 4 * q^25 - 4 * q^29 - 6 * q^31 - 4 * q^37 - 8 * q^41 - 20 * q^43 - 6 * q^47 + 10 * q^49 + 4 * q^53 + 4 * q^59 - 4 * q^61 - 2 * q^65 - 26 * q^67 - 18 * q^73 - 6 * q^77 - 18 * q^79 + 26 * q^83 - 2 * q^85 - 14 * q^89 - 38 * q^91 - 6 * q^95 - 12 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 9x^{2} + 3x + 2$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{3} - 11\nu - 2 ) / 2$$ (v^3 - 11*v - 2) / 2 $$\beta_{2}$$ $$=$$ $$( -\nu^{3} + 2\nu^{2} + 9\nu - 8 ) / 2$$ (-v^3 + 2*v^2 + 9*v - 8) / 2 $$\beta_{3}$$ $$=$$ $$\nu^{3} - \nu^{2} - 8\nu + 2$$ v^3 - v^2 - 8*v + 2
 $$\nu$$ $$=$$ $$( \beta_{3} + \beta_{2} - \beta _1 + 1 ) / 2$$ (b3 + b2 - b1 + 1) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{3} + 3\beta_{2} + \beta _1 + 11 ) / 2$$ (b3 + 3*b2 + b1 + 11) / 2 $$\nu^{3}$$ $$=$$ $$( 11\beta_{3} + 11\beta_{2} - 7\beta _1 + 15 ) / 2$$ (11*b3 + 11*b2 - 7*b1 + 15) / 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
 −2.67673 −0.339102 3.36007 0.655762
0 0 0 1.00000 0 −4.13277 0 0 0
1.2 0 0 0 1.00000 0 −0.845563 0 0 0
1.3 0 0 0 1.00000 0 0.512641 0 0 0
1.4 0 0 0 1.00000 0 4.46569 0 0 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
$$2$$ $$1$$
$$3$$ $$-1$$
$$5$$ $$-1$$
$$23$$ $$1$$

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 8280.2.a.bq 4
3.b odd 2 1 2760.2.a.v 4
12.b even 2 1 5520.2.a.cb 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2760.2.a.v 4 3.b odd 2 1
5520.2.a.cb 4 12.b even 2 1
8280.2.a.bq 4 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(8280))$$:

 $$T_{7}^{4} - 19T_{7}^{2} - 6T_{7} + 8$$ T7^4 - 19*T7^2 - 6*T7 + 8 $$T_{11}^{4} - 22T_{11}^{2} - 12T_{11} + 80$$ T11^4 - 22*T11^2 - 12*T11 + 80 $$T_{13}^{4} + 2T_{13}^{3} - 38T_{13}^{2} - 56T_{13} + 296$$ T13^4 + 2*T13^3 - 38*T13^2 - 56*T13 + 296 $$T_{17}^{4} + 2T_{17}^{3} - 71T_{17}^{2} - 58T_{17} + 1024$$ T17^4 + 2*T17^3 - 71*T17^2 - 58*T17 + 1024

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$(T - 1)^{4}$$
$7$ $$T^{4} - 19 T^{2} - 6 T + 8$$
$11$ $$T^{4} - 22 T^{2} - 12 T + 80$$
$13$ $$T^{4} + 2 T^{3} - 38 T^{2} - 56 T + 296$$
$17$ $$T^{4} + 2 T^{3} - 71 T^{2} + \cdots + 1024$$
$19$ $$T^{4} + 6 T^{3} - 26 T^{2} - 112 T + 256$$
$23$ $$(T + 1)^{4}$$
$29$ $$T^{4} + 4 T^{3} - 75 T^{2} - 400 T - 164$$
$31$ $$T^{4} + 6 T^{3} - 11 T^{2} - 104 T - 128$$
$37$ $$T^{4} + 4 T^{3} - 95 T^{2} + \cdots + 2104$$
$41$ $$T^{4} + 8 T^{3} - 11 T^{2} - 204 T - 380$$
$43$ $$T^{4} + 20 T^{3} + 68 T^{2} + \cdots - 2048$$
$47$ $$T^{4} + 6 T^{3} - 26 T^{2} - 112 T + 256$$
$53$ $$T^{4} - 4 T^{3} - 147 T^{2} + \cdots + 1468$$
$59$ $$T^{4} - 4 T^{3} - 75 T^{2} - 84 T + 320$$
$61$ $$T^{4} + 4 T^{3} - 62 T^{2} - 364 T - 400$$
$67$ $$T^{4} + 26 T^{3} + 141 T^{2} + \cdots - 6976$$
$71$ $$T^{4} - 35 T^{2} - 96 T - 64$$
$73$ $$T^{4} + 18 T^{3} + 82 T^{2} + \cdots - 152$$
$79$ $$T^{4} + 18 T^{3} - 56 T^{2} + \cdots + 512$$
$83$ $$T^{4} - 26 T^{3} + 109 T^{2} + \cdots - 9176$$
$89$ $$T^{4} + 14 T^{3} + 24 T^{2} - 48 T - 64$$
$97$ $$T^{4} + 12 T^{3} - 44 T^{2} - 128 T + 64$$