# Properties

 Label 7280.2.a.bc 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: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ x^2 - 2 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 910) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 2\sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{3} + q^{5} - q^{7} + 5 q^{9}+O(q^{10})$$ q + b * q^3 + q^5 - q^7 + 5 * q^9 $$q + \beta q^{3} + q^{5} - q^{7} + 5 q^{9} - 4 q^{11} + q^{13} + \beta q^{15} + ( - \beta - 2) q^{17} + ( - \beta + 4) q^{19} - \beta q^{21} + ( - \beta + 4) q^{23} + q^{25} + 2 \beta q^{27} + 6 q^{29} + (2 \beta + 4) q^{31} - 4 \beta q^{33} - q^{35} + (2 \beta + 2) q^{37} + \beta q^{39} + (3 \beta - 2) q^{41} + 4 \beta q^{43} + 5 q^{45} - 4 \beta q^{47} + q^{49} + ( - 2 \beta - 8) q^{51} + ( - \beta + 6) q^{53} - 4 q^{55} + (4 \beta - 8) q^{57} + (\beta - 4) q^{59} + (2 \beta - 6) q^{61} - 5 q^{63} + q^{65} + ( - 2 \beta + 4) q^{67} + (4 \beta - 8) q^{69} + (\beta - 8) q^{71} + (2 \beta + 6) q^{73} + \beta q^{75} + 4 q^{77} - 4 \beta q^{79} + q^{81} + (2 \beta + 8) q^{83} + ( - \beta - 2) q^{85} + 6 \beta q^{87} + ( - \beta + 14) q^{89} - q^{91} + (4 \beta + 16) q^{93} + ( - \beta + 4) q^{95} + (4 \beta + 6) q^{97} - 20 q^{99} +O(q^{100})$$ q + b * q^3 + q^5 - q^7 + 5 * q^9 - 4 * q^11 + q^13 + b * q^15 + (-b - 2) * q^17 + (-b + 4) * q^19 - b * q^21 + (-b + 4) * q^23 + q^25 + 2*b * q^27 + 6 * q^29 + (2*b + 4) * q^31 - 4*b * q^33 - q^35 + (2*b + 2) * q^37 + b * q^39 + (3*b - 2) * q^41 + 4*b * q^43 + 5 * q^45 - 4*b * q^47 + q^49 + (-2*b - 8) * q^51 + (-b + 6) * q^53 - 4 * q^55 + (4*b - 8) * q^57 + (b - 4) * q^59 + (2*b - 6) * q^61 - 5 * q^63 + q^65 + (-2*b + 4) * q^67 + (4*b - 8) * q^69 + (b - 8) * q^71 + (2*b + 6) * q^73 + b * q^75 + 4 * q^77 - 4*b * q^79 + q^81 + (2*b + 8) * q^83 + (-b - 2) * q^85 + 6*b * q^87 + (-b + 14) * q^89 - q^91 + (4*b + 16) * q^93 + (-b + 4) * q^95 + (4*b + 6) * q^97 - 20 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{5} - 2 q^{7} + 10 q^{9}+O(q^{10})$$ 2 * q + 2 * q^5 - 2 * q^7 + 10 * q^9 $$2 q + 2 q^{5} - 2 q^{7} + 10 q^{9} - 8 q^{11} + 2 q^{13} - 4 q^{17} + 8 q^{19} + 8 q^{23} + 2 q^{25} + 12 q^{29} + 8 q^{31} - 2 q^{35} + 4 q^{37} - 4 q^{41} + 10 q^{45} + 2 q^{49} - 16 q^{51} + 12 q^{53} - 8 q^{55} - 16 q^{57} - 8 q^{59} - 12 q^{61} - 10 q^{63} + 2 q^{65} + 8 q^{67} - 16 q^{69} - 16 q^{71} + 12 q^{73} + 8 q^{77} + 2 q^{81} + 16 q^{83} - 4 q^{85} + 28 q^{89} - 2 q^{91} + 32 q^{93} + 8 q^{95} + 12 q^{97} - 40 q^{99}+O(q^{100})$$ 2 * q + 2 * q^5 - 2 * q^7 + 10 * q^9 - 8 * q^11 + 2 * q^13 - 4 * q^17 + 8 * q^19 + 8 * q^23 + 2 * q^25 + 12 * q^29 + 8 * q^31 - 2 * q^35 + 4 * q^37 - 4 * q^41 + 10 * q^45 + 2 * q^49 - 16 * q^51 + 12 * q^53 - 8 * q^55 - 16 * q^57 - 8 * q^59 - 12 * q^61 - 10 * q^63 + 2 * q^65 + 8 * q^67 - 16 * q^69 - 16 * q^71 + 12 * q^73 + 8 * q^77 + 2 * q^81 + 16 * q^83 - 4 * q^85 + 28 * q^89 - 2 * q^91 + 32 * q^93 + 8 * q^95 + 12 * q^97 - 40 * 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
 −1.41421 1.41421
0 −2.82843 0 1.00000 0 −1.00000 0 5.00000 0
1.2 0 2.82843 0 1.00000 0 −1.00000 0 5.00000 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.bc 2
4.b odd 2 1 910.2.a.m 2
12.b even 2 1 8190.2.a.ck 2
20.d odd 2 1 4550.2.a.bn 2
28.d even 2 1 6370.2.a.bd 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
910.2.a.m 2 4.b odd 2 1
4550.2.a.bn 2 20.d odd 2 1
6370.2.a.bd 2 28.d even 2 1
7280.2.a.bc 2 1.a even 1 1 trivial
8190.2.a.ck 2 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(7280))$$:

 $$T_{3}^{2} - 8$$ T3^2 - 8 $$T_{11} + 4$$ T11 + 4 $$T_{17}^{2} + 4T_{17} - 4$$ T17^2 + 4*T17 - 4 $$T_{19}^{2} - 8T_{19} + 8$$ T19^2 - 8*T19 + 8 $$T_{23}^{2} - 8T_{23} + 8$$ T23^2 - 8*T23 + 8

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - 8$$
$5$ $$(T - 1)^{2}$$
$7$ $$(T + 1)^{2}$$
$11$ $$(T + 4)^{2}$$
$13$ $$(T - 1)^{2}$$
$17$ $$T^{2} + 4T - 4$$
$19$ $$T^{2} - 8T + 8$$
$23$ $$T^{2} - 8T + 8$$
$29$ $$(T - 6)^{2}$$
$31$ $$T^{2} - 8T - 16$$
$37$ $$T^{2} - 4T - 28$$
$41$ $$T^{2} + 4T - 68$$
$43$ $$T^{2} - 128$$
$47$ $$T^{2} - 128$$
$53$ $$T^{2} - 12T + 28$$
$59$ $$T^{2} + 8T + 8$$
$61$ $$T^{2} + 12T + 4$$
$67$ $$T^{2} - 8T - 16$$
$71$ $$T^{2} + 16T + 56$$
$73$ $$T^{2} - 12T + 4$$
$79$ $$T^{2} - 128$$
$83$ $$T^{2} - 16T + 32$$
$89$ $$T^{2} - 28T + 188$$
$97$ $$T^{2} - 12T - 92$$