# Properties

 Label 7280.2.a.bh Level $7280$ Weight $2$ Character orbit 7280.a Self dual yes Analytic conductor $58.131$ Analytic rank $0$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{13})$$ Defining polynomial: $$x^{2} - x - 3$$ x^2 - x - 3 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 $$\beta = \frac{1}{2}(1 + \sqrt{13})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{3} + q^{5} - q^{7} + \beta q^{9}+O(q^{10})$$ q + b * q^3 + q^5 - q^7 + b * q^9 $$q + \beta q^{3} + q^{5} - q^{7} + \beta q^{9} + ( - 2 \beta + 2) q^{11} - q^{13} + \beta q^{15} + ( - \beta + 5) q^{17} + ( - \beta + 5) q^{19} - \beta q^{21} + 8 q^{23} + q^{25} + ( - 2 \beta + 3) q^{27} + ( - 3 \beta - 2) q^{29} + (\beta + 2) q^{31} - 6 q^{33} - q^{35} + 3 \beta q^{37} - \beta q^{39} + (\beta - 3) q^{41} + 2 q^{43} + \beta q^{45} + ( - 2 \beta + 2) q^{47} + q^{49} + (4 \beta - 3) q^{51} + 2 \beta q^{53} + ( - 2 \beta + 2) q^{55} + (4 \beta - 3) q^{57} + (7 \beta - 2) q^{59} + (4 \beta + 4) q^{61} - \beta q^{63} - q^{65} + (3 \beta - 5) q^{67} + 8 \beta q^{69} - 2 \beta q^{71} + (2 \beta - 6) q^{73} + \beta q^{75} + (2 \beta - 2) q^{77} + (5 \beta + 3) q^{79} + ( - 2 \beta - 6) q^{81} + ( - 6 \beta + 4) q^{83} + ( - \beta + 5) q^{85} + ( - 5 \beta - 9) q^{87} + (\beta - 6) q^{89} + q^{91} + (3 \beta + 3) q^{93} + ( - \beta + 5) q^{95} + (4 \beta - 8) q^{97} - 6 q^{99} +O(q^{100})$$ q + b * q^3 + q^5 - q^7 + b * q^9 + (-2*b + 2) * q^11 - q^13 + b * q^15 + (-b + 5) * q^17 + (-b + 5) * q^19 - b * q^21 + 8 * q^23 + q^25 + (-2*b + 3) * q^27 + (-3*b - 2) * q^29 + (b + 2) * q^31 - 6 * q^33 - q^35 + 3*b * q^37 - b * q^39 + (b - 3) * q^41 + 2 * q^43 + b * q^45 + (-2*b + 2) * q^47 + q^49 + (4*b - 3) * q^51 + 2*b * q^53 + (-2*b + 2) * q^55 + (4*b - 3) * q^57 + (7*b - 2) * q^59 + (4*b + 4) * q^61 - b * q^63 - q^65 + (3*b - 5) * q^67 + 8*b * q^69 - 2*b * q^71 + (2*b - 6) * q^73 + b * q^75 + (2*b - 2) * q^77 + (5*b + 3) * q^79 + (-2*b - 6) * q^81 + (-6*b + 4) * q^83 + (-b + 5) * q^85 + (-5*b - 9) * q^87 + (b - 6) * q^89 + q^91 + (3*b + 3) * q^93 + (-b + 5) * q^95 + (4*b - 8) * q^97 - 6 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{3} + 2 q^{5} - 2 q^{7} + q^{9}+O(q^{10})$$ 2 * q + q^3 + 2 * q^5 - 2 * q^7 + q^9 $$2 q + q^{3} + 2 q^{5} - 2 q^{7} + q^{9} + 2 q^{11} - 2 q^{13} + q^{15} + 9 q^{17} + 9 q^{19} - q^{21} + 16 q^{23} + 2 q^{25} + 4 q^{27} - 7 q^{29} + 5 q^{31} - 12 q^{33} - 2 q^{35} + 3 q^{37} - q^{39} - 5 q^{41} + 4 q^{43} + q^{45} + 2 q^{47} + 2 q^{49} - 2 q^{51} + 2 q^{53} + 2 q^{55} - 2 q^{57} + 3 q^{59} + 12 q^{61} - q^{63} - 2 q^{65} - 7 q^{67} + 8 q^{69} - 2 q^{71} - 10 q^{73} + q^{75} - 2 q^{77} + 11 q^{79} - 14 q^{81} + 2 q^{83} + 9 q^{85} - 23 q^{87} - 11 q^{89} + 2 q^{91} + 9 q^{93} + 9 q^{95} - 12 q^{97} - 12 q^{99}+O(q^{100})$$ 2 * q + q^3 + 2 * q^5 - 2 * q^7 + q^9 + 2 * q^11 - 2 * q^13 + q^15 + 9 * q^17 + 9 * q^19 - q^21 + 16 * q^23 + 2 * q^25 + 4 * q^27 - 7 * q^29 + 5 * q^31 - 12 * q^33 - 2 * q^35 + 3 * q^37 - q^39 - 5 * q^41 + 4 * q^43 + q^45 + 2 * q^47 + 2 * q^49 - 2 * q^51 + 2 * q^53 + 2 * q^55 - 2 * q^57 + 3 * q^59 + 12 * q^61 - q^63 - 2 * q^65 - 7 * q^67 + 8 * q^69 - 2 * q^71 - 10 * q^73 + q^75 - 2 * q^77 + 11 * q^79 - 14 * q^81 + 2 * q^83 + 9 * q^85 - 23 * q^87 - 11 * q^89 + 2 * q^91 + 9 * q^93 + 9 * q^95 - 12 * q^97 - 12 * q^99

## 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
 −1.30278 2.30278
0 −1.30278 0 1.00000 0 −1.00000 0 −1.30278 0
1.2 0 2.30278 0 1.00000 0 −1.00000 0 2.30278 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.bh 2
4.b odd 2 1 3640.2.a.l 2

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

 $$T_{3}^{2} - T_{3} - 3$$ T3^2 - T3 - 3 $$T_{11}^{2} - 2T_{11} - 12$$ T11^2 - 2*T11 - 12 $$T_{17}^{2} - 9T_{17} + 17$$ T17^2 - 9*T17 + 17 $$T_{19}^{2} - 9T_{19} + 17$$ T19^2 - 9*T19 + 17 $$T_{23} - 8$$ T23 - 8

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - T - 3$$
$5$ $$(T - 1)^{2}$$
$7$ $$(T + 1)^{2}$$
$11$ $$T^{2} - 2T - 12$$
$13$ $$(T + 1)^{2}$$
$17$ $$T^{2} - 9T + 17$$
$19$ $$T^{2} - 9T + 17$$
$23$ $$(T - 8)^{2}$$
$29$ $$T^{2} + 7T - 17$$
$31$ $$T^{2} - 5T + 3$$
$37$ $$T^{2} - 3T - 27$$
$41$ $$T^{2} + 5T + 3$$
$43$ $$(T - 2)^{2}$$
$47$ $$T^{2} - 2T - 12$$
$53$ $$T^{2} - 2T - 12$$
$59$ $$T^{2} - 3T - 157$$
$61$ $$T^{2} - 12T - 16$$
$67$ $$T^{2} + 7T - 17$$
$71$ $$T^{2} + 2T - 12$$
$73$ $$T^{2} + 10T + 12$$
$79$ $$T^{2} - 11T - 51$$
$83$ $$T^{2} - 2T - 116$$
$89$ $$T^{2} + 11T + 27$$
$97$ $$T^{2} + 12T - 16$$