# Properties

 Label 7280.2.a.bn Level $7280$ Weight $2$ Character orbit 7280.a Self dual yes Analytic conductor $58.131$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$7280 = 2^{4} \cdot 5 \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7280.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$58.1310926715$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.1101.1 Defining polynomial: $$x^{3} - x^{2} - 9x + 12$$ x^3 - x^2 - 9*x + 12 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 3640) 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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{3} - q^{5} - q^{7} + (\beta_{2} - \beta_1 + 4) q^{9}+O(q^{10})$$ q + b1 * q^3 - q^5 - q^7 + (b2 - b1 + 4) * q^9 $$q + \beta_1 q^{3} - q^{5} - q^{7} + (\beta_{2} - \beta_1 + 4) q^{9} - q^{13} - \beta_1 q^{15} + (\beta_{2} - \beta_1 + 1) q^{17} + ( - \beta_{2} + \beta_1 + 1) q^{19} - \beta_1 q^{21} + ( - 2 \beta_{2} - 2) q^{23} + q^{25} + (\beta_{2} + 2 \beta_1 - 5) q^{27} + ( - 2 \beta_{2} + \beta_1 - 4) q^{29} + (\beta_1 - 4) q^{31} + q^{35} + ( - 2 \beta_{2} - \beta_1) q^{37} - \beta_1 q^{39} + (\beta_{2} - \beta_1 + 1) q^{41} + ( - 2 \beta_1 + 4) q^{43} + ( - \beta_{2} + \beta_1 - 4) q^{45} + (2 \beta_{2} - 2) q^{47} + q^{49} + (\beta_{2} + 2 \beta_1 - 5) q^{51} + 2 q^{53} + ( - \beta_{2} + 5) q^{57} + (3 \beta_1 - 4) q^{59} - 10 q^{61} + ( - \beta_{2} + \beta_1 - 4) q^{63} + q^{65} + ( - 3 \beta_{2} - \beta_1 - 1) q^{67} + ( - 4 \beta_{2} - 2 \beta_1 - 4) q^{69} - 2 \beta_1 q^{71} + (2 \beta_{2} - 4 \beta_1 + 4) q^{73} + \beta_1 q^{75} + (\beta_{2} - \beta_1 - 1) q^{79} + (\beta_{2} - 4 \beta_1 + 4) q^{81} + (2 \beta_{2} + 2) q^{83} + ( - \beta_{2} + \beta_1 - 1) q^{85} + ( - 3 \beta_{2} - 5 \beta_1 + 3) q^{87} + ( - 2 \beta_{2} - 3 \beta_1) q^{89} + q^{91} + (\beta_{2} - 5 \beta_1 + 7) q^{93} + (\beta_{2} - \beta_1 - 1) q^{95} + (4 \beta_{2} + 2 \beta_1 + 2) q^{97}+O(q^{100})$$ q + b1 * q^3 - q^5 - q^7 + (b2 - b1 + 4) * q^9 - q^13 - b1 * q^15 + (b2 - b1 + 1) * q^17 + (-b2 + b1 + 1) * q^19 - b1 * q^21 + (-2*b2 - 2) * q^23 + q^25 + (b2 + 2*b1 - 5) * q^27 + (-2*b2 + b1 - 4) * q^29 + (b1 - 4) * q^31 + q^35 + (-2*b2 - b1) * q^37 - b1 * q^39 + (b2 - b1 + 1) * q^41 + (-2*b1 + 4) * q^43 + (-b2 + b1 - 4) * q^45 + (2*b2 - 2) * q^47 + q^49 + (b2 + 2*b1 - 5) * q^51 + 2 * q^53 + (-b2 + 5) * q^57 + (3*b1 - 4) * q^59 - 10 * q^61 + (-b2 + b1 - 4) * q^63 + q^65 + (-3*b2 - b1 - 1) * q^67 + (-4*b2 - 2*b1 - 4) * q^69 - 2*b1 * q^71 + (2*b2 - 4*b1 + 4) * q^73 + b1 * q^75 + (b2 - b1 - 1) * q^79 + (b2 - 4*b1 + 4) * q^81 + (2*b2 + 2) * q^83 + (-b2 + b1 - 1) * q^85 + (-3*b2 - 5*b1 + 3) * q^87 + (-2*b2 - 3*b1) * q^89 + q^91 + (b2 - 5*b1 + 7) * q^93 + (b2 - b1 - 1) * q^95 + (4*b2 + 2*b1 + 2) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + q^{3} - 3 q^{5} - 3 q^{7} + 10 q^{9}+O(q^{10})$$ 3 * q + q^3 - 3 * q^5 - 3 * q^7 + 10 * q^9 $$3 q + q^{3} - 3 q^{5} - 3 q^{7} + 10 q^{9} - 3 q^{13} - q^{15} + q^{17} + 5 q^{19} - q^{21} - 4 q^{23} + 3 q^{25} - 14 q^{27} - 9 q^{29} - 11 q^{31} + 3 q^{35} + q^{37} - q^{39} + q^{41} + 10 q^{43} - 10 q^{45} - 8 q^{47} + 3 q^{49} - 14 q^{51} + 6 q^{53} + 16 q^{57} - 9 q^{59} - 30 q^{61} - 10 q^{63} + 3 q^{65} - q^{67} - 10 q^{69} - 2 q^{71} + 6 q^{73} + q^{75} - 5 q^{79} + 7 q^{81} + 4 q^{83} - q^{85} + 7 q^{87} - q^{89} + 3 q^{91} + 15 q^{93} - 5 q^{95} + 4 q^{97}+O(q^{100})$$ 3 * q + q^3 - 3 * q^5 - 3 * q^7 + 10 * q^9 - 3 * q^13 - q^15 + q^17 + 5 * q^19 - q^21 - 4 * q^23 + 3 * q^25 - 14 * q^27 - 9 * q^29 - 11 * q^31 + 3 * q^35 + q^37 - q^39 + q^41 + 10 * q^43 - 10 * q^45 - 8 * q^47 + 3 * q^49 - 14 * q^51 + 6 * q^53 + 16 * q^57 - 9 * q^59 - 30 * q^61 - 10 * q^63 + 3 * q^65 - q^67 - 10 * q^69 - 2 * q^71 + 6 * q^73 + q^75 - 5 * q^79 + 7 * q^81 + 4 * q^83 - q^85 + 7 * q^87 - q^89 + 3 * q^91 + 15 * q^93 - 5 * q^95 + 4 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 9x + 12$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} + \nu - 7$$ v^2 + v - 7
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} - \beta _1 + 7$$ b2 - b1 + 7

## 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
 −3.11903 1.43163 2.68740
0 −3.11903 0 −1.00000 0 −1.00000 0 6.72833 0
1.2 0 1.43163 0 −1.00000 0 −1.00000 0 −0.950444 0
1.3 0 2.68740 0 −1.00000 0 −1.00000 0 4.22212 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
$$2$$ $$1$$
$$5$$ $$1$$
$$7$$ $$1$$
$$13$$ $$1$$

## 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 7280.2.a.bn 3
4.b odd 2 1 3640.2.a.p 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3640.2.a.p 3 4.b odd 2 1
7280.2.a.bn 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(7280))$$:

 $$T_{3}^{3} - T_{3}^{2} - 9T_{3} + 12$$ T3^3 - T3^2 - 9*T3 + 12 $$T_{11}$$ T11 $$T_{17}^{3} - T_{17}^{2} - 15T_{17} + 18$$ T17^3 - T17^2 - 15*T17 + 18 $$T_{19}^{3} - 5T_{19}^{2} - 7T_{19} + 8$$ T19^3 - 5*T19^2 - 7*T19 + 8 $$T_{23}^{3} + 4T_{23}^{2} - 36T_{23} - 48$$ T23^3 + 4*T23^2 - 36*T23 - 48

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$T^{3} - T^{2} - 9T + 12$$
$5$ $$(T + 1)^{3}$$
$7$ $$(T + 1)^{3}$$
$11$ $$T^{3}$$
$13$ $$(T + 1)^{3}$$
$17$ $$T^{3} - T^{2} - 15 T + 18$$
$19$ $$T^{3} - 5 T^{2} - 7 T + 8$$
$23$ $$T^{3} + 4 T^{2} - 36 T - 48$$
$29$ $$T^{3} + 9 T^{2} - 15 T - 202$$
$31$ $$T^{3} + 11 T^{2} + 31 T + 24$$
$37$ $$T^{3} - T^{2} - 59 T + 186$$
$41$ $$T^{3} - T^{2} - 15 T + 18$$
$43$ $$T^{3} - 10 T^{2} - 4 T + 16$$
$47$ $$T^{3} + 8 T^{2} - 20 T - 96$$
$53$ $$(T - 2)^{3}$$
$59$ $$T^{3} + 9 T^{2} - 57 T + 16$$
$61$ $$(T + 10)^{3}$$
$67$ $$T^{3} + T^{2} - 115 T + 332$$
$71$ $$T^{3} + 2 T^{2} - 36 T - 96$$
$73$ $$T^{3} - 6 T^{2} - 144 T - 128$$
$79$ $$T^{3} + 5 T^{2} - 7 T - 8$$
$83$ $$T^{3} - 4 T^{2} - 36 T + 48$$
$89$ $$T^{3} + T^{2} - 151 T + 386$$
$97$ $$T^{3} - 4 T^{2} - 232 T - 1016$$