# Properties

 Label 7280.2.a.ca Level $7280$ Weight $2$ Character orbit 7280.a Self dual yes Analytic conductor $58.131$ Analytic rank $0$ Dimension $4$ 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: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.2225.1 Defining polynomial: $$x^{4} - x^{3} - 5x^{2} + 2x + 4$$ x^4 - x^3 - 5*x^2 + 2*x + 4 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ 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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{3} + q^{5} + q^{7} + (\beta_{2} + \beta_1) q^{9}+O(q^{10})$$ q + b1 * q^3 + q^5 + q^7 + (b2 + b1) * q^9 $$q + \beta_1 q^{3} + q^{5} + q^{7} + (\beta_{2} + \beta_1) q^{9} + ( - \beta_{3} + 1) q^{11} + q^{13} + \beta_1 q^{15} + ( - \beta_{2} + \beta_1 - 1) q^{17} + (\beta_{3} + \beta_{2} + \beta_1 + 2) q^{19} + \beta_1 q^{21} + (\beta_{3} - 2 \beta_{2} - 2 \beta_1 + 1) q^{23} + q^{25} + (\beta_{3} + \beta_{2} - 2 \beta_1 + 2) q^{27} + (\beta_{3} - 2 \beta_{2} + \beta_1 - 1) q^{29} + (2 \beta_{3} + 2 \beta_{2} - \beta_1 + 4) q^{31} + ( - 2 \beta_{2} - 2) q^{33} + q^{35} + ( - \beta_{3} + \beta_1 - 1) q^{37} + \beta_1 q^{39} + ( - 2 \beta_{3} + \beta_{2} + \beta_1 - 1) q^{41} + ( - \beta_{3} + 2 \beta_1 + 1) q^{43} + (\beta_{2} + \beta_1) q^{45} + ( - 4 \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 2) q^{47} + q^{49} + ( - \beta_{3} + \beta_{2} + 4) q^{51} + (2 \beta_{2} + 2 \beta_1) q^{53} + ( - \beta_{3} + 1) q^{55} + (\beta_{3} + 3 \beta_{2} + 4 \beta_1 + 4) q^{57} + ( - 6 \beta_{2} - 3 \beta_1 - 2) q^{59} + 2 \beta_{2} q^{61} + (\beta_{2} + \beta_1) q^{63} + q^{65} + (2 \beta_{3} - \beta_{2} - 3 \beta_1 + 3) q^{67} + ( - 2 \beta_{3} - 2) q^{69} + ( - \beta_{3} - 2 \beta_{2} + 3) q^{71} + (4 \beta_{2} + 2 \beta_1 + 2) q^{73} + \beta_1 q^{75} + ( - \beta_{3} + 1) q^{77} + ( - 4 \beta_{3} + \beta_{2} + 5 \beta_1 + 3) q^{79} + (\beta_{3} - 3 \beta_{2} - 2 \beta_1 - 5) q^{81} + ( - 2 \beta_{3} + 4 \beta_{2} + 2 \beta_1 + 2) q^{83} + ( - \beta_{2} + \beta_1 - 1) q^{85} + ( - 2 \beta_{3} + 3 \beta_{2} + \beta_1 + 7) q^{87} + (3 \beta_{3} - 6 \beta_{2} - 3 \beta_1 - 3) q^{89} + q^{91} + (2 \beta_{3} + 3 \beta_{2} + 5 \beta_1 - 1) q^{93} + (\beta_{3} + \beta_{2} + \beta_1 + 2) q^{95} + (2 \beta_{3} - 2 \beta_{2} - 4 \beta_1 + 2) q^{97} + (\beta_{3} - 2 \beta_1 - 1) q^{99}+O(q^{100})$$ q + b1 * q^3 + q^5 + q^7 + (b2 + b1) * q^9 + (-b3 + 1) * q^11 + q^13 + b1 * q^15 + (-b2 + b1 - 1) * q^17 + (b3 + b2 + b1 + 2) * q^19 + b1 * q^21 + (b3 - 2*b2 - 2*b1 + 1) * q^23 + q^25 + (b3 + b2 - 2*b1 + 2) * q^27 + (b3 - 2*b2 + b1 - 1) * q^29 + (2*b3 + 2*b2 - b1 + 4) * q^31 + (-2*b2 - 2) * q^33 + q^35 + (-b3 + b1 - 1) * q^37 + b1 * q^39 + (-2*b3 + b2 + b1 - 1) * q^41 + (-b3 + 2*b1 + 1) * q^43 + (b2 + b1) * q^45 + (-4*b3 + 2*b2 + 2*b1 + 2) * q^47 + q^49 + (-b3 + b2 + 4) * q^51 + (2*b2 + 2*b1) * q^53 + (-b3 + 1) * q^55 + (b3 + 3*b2 + 4*b1 + 4) * q^57 + (-6*b2 - 3*b1 - 2) * q^59 + 2*b2 * q^61 + (b2 + b1) * q^63 + q^65 + (2*b3 - b2 - 3*b1 + 3) * q^67 + (-2*b3 - 2) * q^69 + (-b3 - 2*b2 + 3) * q^71 + (4*b2 + 2*b1 + 2) * q^73 + b1 * q^75 + (-b3 + 1) * q^77 + (-4*b3 + b2 + 5*b1 + 3) * q^79 + (b3 - 3*b2 - 2*b1 - 5) * q^81 + (-2*b3 + 4*b2 + 2*b1 + 2) * q^83 + (-b2 + b1 - 1) * q^85 + (-2*b3 + 3*b2 + b1 + 7) * q^87 + (3*b3 - 6*b2 - 3*b1 - 3) * q^89 + q^91 + (2*b3 + 3*b2 + 5*b1 - 1) * q^93 + (b3 + b2 + b1 + 2) * q^95 + (2*b3 - 2*b2 - 4*b1 + 2) * q^97 + (b3 - 2*b1 - 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + q^{3} + 4 q^{5} + 4 q^{7} - q^{9}+O(q^{10})$$ 4 * q + q^3 + 4 * q^5 + 4 * q^7 - q^9 $$4 q + q^{3} + 4 q^{5} + 4 q^{7} - q^{9} + 4 q^{11} + 4 q^{13} + q^{15} - q^{17} + 7 q^{19} + q^{21} + 6 q^{23} + 4 q^{25} + 4 q^{27} + q^{29} + 11 q^{31} - 4 q^{33} + 4 q^{35} - 3 q^{37} + q^{39} - 5 q^{41} + 6 q^{43} - q^{45} + 6 q^{47} + 4 q^{49} + 14 q^{51} - 2 q^{53} + 4 q^{55} + 14 q^{57} + q^{59} - 4 q^{61} - q^{63} + 4 q^{65} + 11 q^{67} - 8 q^{69} + 16 q^{71} + 2 q^{73} + q^{75} + 4 q^{77} + 15 q^{79} - 16 q^{81} + 2 q^{83} - q^{85} + 23 q^{87} - 3 q^{89} + 4 q^{91} - 5 q^{93} + 7 q^{95} + 8 q^{97} - 6 q^{99}+O(q^{100})$$ 4 * q + q^3 + 4 * q^5 + 4 * q^7 - q^9 + 4 * q^11 + 4 * q^13 + q^15 - q^17 + 7 * q^19 + q^21 + 6 * q^23 + 4 * q^25 + 4 * q^27 + q^29 + 11 * q^31 - 4 * q^33 + 4 * q^35 - 3 * q^37 + q^39 - 5 * q^41 + 6 * q^43 - q^45 + 6 * q^47 + 4 * q^49 + 14 * q^51 - 2 * q^53 + 4 * q^55 + 14 * q^57 + q^59 - 4 * q^61 - q^63 + 4 * q^65 + 11 * q^67 - 8 * q^69 + 16 * q^71 + 2 * q^73 + q^75 + 4 * q^77 + 15 * q^79 - 16 * q^81 + 2 * q^83 - q^85 + 23 * q^87 - 3 * q^89 + 4 * q^91 - 5 * q^93 + 7 * q^95 + 8 * q^97 - 6 * q^99

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

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 3$$ v^2 - v - 3 $$\beta_{3}$$ $$=$$ $$\nu^{3} - \nu^{2} - 3\nu + 1$$ v^3 - v^2 - 3*v + 1
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta _1 + 3$$ b2 + b1 + 3 $$\nu^{3}$$ $$=$$ $$\beta_{3} + \beta_{2} + 4\beta _1 + 2$$ b3 + b2 + 4*b1 + 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
 −1.75660 −0.820249 1.13856 2.43828
0 −1.75660 0 1.00000 0 1.00000 0 0.0856374 0
1.2 0 −0.820249 0 1.00000 0 1.00000 0 −2.32719 0
1.3 0 1.13856 0 1.00000 0 1.00000 0 −1.70367 0
1.4 0 2.43828 0 1.00000 0 1.00000 0 2.94523 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.ca 4
4.b odd 2 1 3640.2.a.s 4

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

 $$T_{3}^{4} - T_{3}^{3} - 5T_{3}^{2} + 2T_{3} + 4$$ T3^4 - T3^3 - 5*T3^2 + 2*T3 + 4 $$T_{11}^{2} - 2T_{11} - 4$$ T11^2 - 2*T11 - 4 $$T_{17}^{4} + T_{17}^{3} - 15T_{17}^{2} + 8T_{17} + 4$$ T17^4 + T17^3 - 15*T17^2 + 8*T17 + 4 $$T_{19}^{4} - 7T_{19}^{3} - 5T_{19}^{2} + 26T_{19} + 4$$ T19^4 - 7*T19^3 - 5*T19^2 + 26*T19 + 4 $$T_{23}^{4} - 6T_{23}^{3} - 20T_{23}^{2} + 32T_{23} + 64$$ T23^4 - 6*T23^3 - 20*T23^2 + 32*T23 + 64

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} - T^{3} - 5 T^{2} + 2 T + 4$$
$5$ $$(T - 1)^{4}$$
$7$ $$(T - 1)^{4}$$
$11$ $$(T^{2} - 2 T - 4)^{2}$$
$13$ $$(T - 1)^{4}$$
$17$ $$T^{4} + T^{3} - 15 T^{2} + 8 T + 4$$
$19$ $$T^{4} - 7 T^{3} - 5 T^{2} + 26 T + 4$$
$23$ $$T^{4} - 6 T^{3} - 20 T^{2} + 32 T + 64$$
$29$ $$T^{4} - T^{3} - 53 T^{2} + 236 T - 284$$
$31$ $$T^{4} - 11 T^{3} - 23 T^{2} + \cdots - 1604$$
$37$ $$T^{4} + 3 T^{3} - 7 T^{2} - 12 T - 4$$
$41$ $$T^{4} + 5 T^{3} - 29 T^{2} - 40 T + 124$$
$43$ $$T^{4} - 6 T^{3} - 8 T^{2} + 56 T + 16$$
$47$ $$T^{4} - 6 T^{3} - 140 T^{2} + \cdots + 1024$$
$53$ $$T^{4} + 2 T^{3} - 32 T^{2} - 88 T + 16$$
$59$ $$T^{4} - T^{3} - 219 T^{2} + \cdots + 10636$$
$61$ $$T^{4} + 4 T^{3} - 20 T^{2} - 48 T + 64$$
$67$ $$T^{4} - 11 T^{3} - 9 T^{2} + 44 T + 16$$
$71$ $$T^{4} - 16 T^{3} + 60 T^{2} - 48 T - 16$$
$73$ $$T^{4} - 2 T^{3} - 96 T^{2} + \cdots + 2096$$
$79$ $$T^{4} - 15 T^{3} - 99 T^{2} + \cdots - 5776$$
$83$ $$T^{4} - 2 T^{3} - 116 T^{2} + \cdots + 1216$$
$89$ $$T^{4} + 3 T^{3} - 261 T^{2} + \cdots + 6156$$
$97$ $$T^{4} - 8 T^{3} - 60 T^{2} - 16 T + 64$$