# Properties

 Label 7280.2.a.bq 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: 4.4.1957.1 Defining polynomial: $$x^{4} - 4x^{2} - x + 1$$ x^4 - 4*x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 455) 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.

 $$f(q)$$ $$=$$ $$q + (\beta_{2} - 1) q^{3} + q^{5} + q^{7} + ( - \beta_{3} - \beta_{2} - \beta_1 + 2) q^{9}+O(q^{10})$$ q + (b2 - 1) * q^3 + q^5 + q^7 + (-b3 - b2 - b1 + 2) * q^9 $$q + (\beta_{2} - 1) q^{3} + q^{5} + q^{7} + ( - \beta_{3} - \beta_{2} - \beta_1 + 2) q^{9} + ( - \beta_{3} + \beta_1 + 1) q^{11} - q^{13} + (\beta_{2} - 1) q^{15} + ( - \beta_1 + 4) q^{17} + (\beta_{3} + \beta_{2} - \beta_1 - 1) q^{19} + (\beta_{2} - 1) q^{21} + ( - 3 \beta_{2} + \beta_1) q^{23} + q^{25} + (3 \beta_{3} + \beta_{2} + 2 \beta_1 - 4) q^{27} + ( - \beta_{3} - \beta_{2} - \beta_1) q^{29} + (\beta_{3} - \beta_{2} - \beta_1) q^{31} + (2 \beta_{3} + \beta_{2} - \beta_1) q^{33} + q^{35} + ( - 2 \beta_{3} + 3 \beta_{2} + 1) q^{37} + ( - \beta_{2} + 1) q^{39} + (\beta_{2} + 2 \beta_1 + 2) q^{41} + ( - \beta_{3} + 2 \beta_{2} - 3 \beta_1 + 3) q^{43} + ( - \beta_{3} - \beta_{2} - \beta_1 + 2) q^{45} + ( - 3 \beta_{3} - 3 \beta_{2} - 1) q^{47} + q^{49} + (5 \beta_{2} + \beta_1 - 5) q^{51} + ( - 3 \beta_{3} - 2 \beta_{2} + \beta_1 + 5) q^{53} + ( - \beta_{3} + \beta_1 + 1) q^{55} + ( - 3 \beta_{3} - \beta_{2} + 4) q^{57} + ( - \beta_{3} + \beta_{2} + \beta_1 + 4) q^{59} + (2 \beta_{3} + 3 \beta_{2} + \beta_1 + 6) q^{61} + ( - \beta_{3} - \beta_{2} - \beta_1 + 2) q^{63} - q^{65} + (4 \beta_{2} + \beta_1 + 2) q^{67} + (3 \beta_{3} - \beta_{2} + 2 \beta_1 - 11) q^{69} + ( - 4 \beta_{3} - 2 \beta_1 + 6) q^{71} + ( - \beta_{3} - 3 \beta_1 + 9) q^{73} + (\beta_{2} - 1) q^{75} + ( - \beta_{3} + \beta_1 + 1) q^{77} + (2 \beta_{3} + \beta_{2} + 2 \beta_1 + 6) q^{79} + ( - 4 \beta_{3} - 6 \beta_{2} + 4) q^{81} + (\beta_{3} - 2 \beta_{2} + \beta_1 + 1) q^{83} + ( - \beta_1 + 4) q^{85} + (3 \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 5) q^{87} + (2 \beta_{3} + 2 \beta_{2} + \beta_1 - 7) q^{89} - q^{91} + ( - \beta_{3} + 2 \beta_1 - 5) q^{93} + (\beta_{3} + \beta_{2} - \beta_1 - 1) q^{95} + (\beta_{3} + 4 \beta_{2} - 3 \beta_1 - 1) q^{97} + ( - 2 \beta_{3} - \beta_{2} - 3 \beta_1) q^{99}+O(q^{100})$$ q + (b2 - 1) * q^3 + q^5 + q^7 + (-b3 - b2 - b1 + 2) * q^9 + (-b3 + b1 + 1) * q^11 - q^13 + (b2 - 1) * q^15 + (-b1 + 4) * q^17 + (b3 + b2 - b1 - 1) * q^19 + (b2 - 1) * q^21 + (-3*b2 + b1) * q^23 + q^25 + (3*b3 + b2 + 2*b1 - 4) * q^27 + (-b3 - b2 - b1) * q^29 + (b3 - b2 - b1) * q^31 + (2*b3 + b2 - b1) * q^33 + q^35 + (-2*b3 + 3*b2 + 1) * q^37 + (-b2 + 1) * q^39 + (b2 + 2*b1 + 2) * q^41 + (-b3 + 2*b2 - 3*b1 + 3) * q^43 + (-b3 - b2 - b1 + 2) * q^45 + (-3*b3 - 3*b2 - 1) * q^47 + q^49 + (5*b2 + b1 - 5) * q^51 + (-3*b3 - 2*b2 + b1 + 5) * q^53 + (-b3 + b1 + 1) * q^55 + (-3*b3 - b2 + 4) * q^57 + (-b3 + b2 + b1 + 4) * q^59 + (2*b3 + 3*b2 + b1 + 6) * q^61 + (-b3 - b2 - b1 + 2) * q^63 - q^65 + (4*b2 + b1 + 2) * q^67 + (3*b3 - b2 + 2*b1 - 11) * q^69 + (-4*b3 - 2*b1 + 6) * q^71 + (-b3 - 3*b1 + 9) * q^73 + (b2 - 1) * q^75 + (-b3 + b1 + 1) * q^77 + (2*b3 + b2 + 2*b1 + 6) * q^79 + (-4*b3 - 6*b2 + 4) * q^81 + (b3 - 2*b2 + b1 + 1) * q^83 + (-b1 + 4) * q^85 + (3*b3 + 2*b2 + 2*b1 - 5) * q^87 + (2*b3 + 2*b2 + b1 - 7) * q^89 - q^91 + (-b3 + 2*b1 - 5) * q^93 + (b3 + b2 - b1 - 1) * q^95 + (b3 + 4*b2 - 3*b1 - 1) * q^97 + (-2*b3 - b2 - 3*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{3} + 4 q^{5} + 4 q^{7} + 6 q^{9}+O(q^{10})$$ 4 * q - 4 * q^3 + 4 * q^5 + 4 * q^7 + 6 * q^9 $$4 q - 4 q^{3} + 4 q^{5} + 4 q^{7} + 6 q^{9} + 2 q^{11} - 4 q^{13} - 4 q^{15} + 16 q^{17} - 2 q^{19} - 4 q^{21} + 4 q^{25} - 10 q^{27} - 2 q^{29} + 2 q^{31} + 4 q^{33} + 4 q^{35} + 4 q^{39} + 8 q^{41} + 10 q^{43} + 6 q^{45} - 10 q^{47} + 4 q^{49} - 20 q^{51} + 14 q^{53} + 2 q^{55} + 10 q^{57} + 14 q^{59} + 28 q^{61} + 6 q^{63} - 4 q^{65} + 8 q^{67} - 38 q^{69} + 16 q^{71} + 34 q^{73} - 4 q^{75} + 2 q^{77} + 28 q^{79} + 8 q^{81} + 6 q^{83} + 16 q^{85} - 14 q^{87} - 24 q^{89} - 4 q^{91} - 22 q^{93} - 2 q^{95} - 2 q^{97} - 4 q^{99}+O(q^{100})$$ 4 * q - 4 * q^3 + 4 * q^5 + 4 * q^7 + 6 * q^9 + 2 * q^11 - 4 * q^13 - 4 * q^15 + 16 * q^17 - 2 * q^19 - 4 * q^21 + 4 * q^25 - 10 * q^27 - 2 * q^29 + 2 * q^31 + 4 * q^33 + 4 * q^35 + 4 * q^39 + 8 * q^41 + 10 * q^43 + 6 * q^45 - 10 * q^47 + 4 * q^49 - 20 * q^51 + 14 * q^53 + 2 * q^55 + 10 * q^57 + 14 * q^59 + 28 * q^61 + 6 * q^63 - 4 * q^65 + 8 * q^67 - 38 * q^69 + 16 * q^71 + 34 * q^73 - 4 * q^75 + 2 * q^77 + 28 * q^79 + 8 * q^81 + 6 * q^83 + 16 * q^85 - 14 * q^87 - 24 * q^89 - 4 * q^91 - 22 * q^93 - 2 * q^95 - 2 * q^97 - 4 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 4x^{2} - x + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu^{2} + \nu - 2$$ v^2 + v - 2 $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 2$$ v^2 - v - 2 $$\beta_{3}$$ $$=$$ $$2\nu^{3} - \nu^{2} - 7\nu + 1$$ 2*v^3 - v^2 - 7*v + 1
 $$\nu$$ $$=$$ $$( -\beta_{2} + \beta_1 ) / 2$$ (-b2 + b1) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{2} + \beta _1 + 4 ) / 2$$ (b2 + b1 + 4) / 2 $$\nu^{3}$$ $$=$$ $$( \beta_{3} - 3\beta_{2} + 4\beta _1 + 1 ) / 2$$ (b3 - 3*b2 + 4*b1 + 1) / 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
 0.396339 −0.693822 2.06150 −1.76401
0 −3.23925 0 1.00000 0 1.00000 0 7.49277 0
1.2 0 −1.82479 0 1.00000 0 1.00000 0 0.329851 0
1.3 0 −0.811721 0 1.00000 0 1.00000 0 −2.34111 0
1.4 0 1.87576 0 1.00000 0 1.00000 0 0.518489 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.bq 4
4.b odd 2 1 455.2.a.d 4
12.b even 2 1 4095.2.a.bc 4
20.d odd 2 1 2275.2.a.o 4
28.d even 2 1 3185.2.a.q 4
52.b odd 2 1 5915.2.a.m 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
455.2.a.d 4 4.b odd 2 1
2275.2.a.o 4 20.d odd 2 1
3185.2.a.q 4 28.d even 2 1
4095.2.a.bc 4 12.b even 2 1
5915.2.a.m 4 52.b odd 2 1
7280.2.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(7280))$$:

 $$T_{3}^{4} + 4T_{3}^{3} - T_{3}^{2} - 14T_{3} - 9$$ T3^4 + 4*T3^3 - T3^2 - 14*T3 - 9 $$T_{11}^{4} - 2T_{11}^{3} - 32T_{11}^{2} + 80T_{11} - 48$$ T11^4 - 2*T11^3 - 32*T11^2 + 80*T11 - 48 $$T_{17}^{4} - 16T_{17}^{3} + 83T_{17}^{2} - 130T_{17} - 49$$ T17^4 - 16*T17^3 + 83*T17^2 - 130*T17 - 49 $$T_{19}^{4} + 2T_{19}^{3} - 33T_{19}^{2} - 50T_{19} + 173$$ T19^4 + 2*T19^3 - 33*T19^2 - 50*T19 + 173 $$T_{23}^{4} - 64T_{23}^{2} + 200T_{23} - 48$$ T23^4 - 64*T23^2 + 200*T23 - 48

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} + 4 T^{3} - T^{2} - 14 T - 9$$
$5$ $$(T - 1)^{4}$$
$7$ $$(T - 1)^{4}$$
$11$ $$T^{4} - 2 T^{3} - 32 T^{2} + 80 T - 48$$
$13$ $$(T + 1)^{4}$$
$17$ $$T^{4} - 16 T^{3} + 83 T^{2} - 130 T - 49$$
$19$ $$T^{4} + 2 T^{3} - 33 T^{2} - 50 T + 173$$
$23$ $$T^{4} - 64 T^{2} + 200 T - 48$$
$29$ $$T^{4} + 2 T^{3} - 25 T^{2} - 78 T - 59$$
$31$ $$T^{4} - 2 T^{3} - 45 T^{2} - 22 T + 201$$
$37$ $$T^{4} - 125 T^{2} - 26 T + 479$$
$41$ $$T^{4} - 8 T^{3} - 43 T^{2} + 100 T + 393$$
$43$ $$T^{4} - 10 T^{3} - 76 T^{2} + \cdots - 1648$$
$47$ $$T^{4} + 10 T^{3} - 120 T^{2} + \cdots - 1136$$
$53$ $$T^{4} - 14 T^{3} - 84 T^{2} + \cdots - 1008$$
$59$ $$T^{4} - 14 T^{3} + 27 T^{2} + \cdots - 479$$
$61$ $$T^{4} - 28 T^{3} + 184 T^{2} + \cdots - 8176$$
$67$ $$T^{4} - 8 T^{3} - 117 T^{2} + \cdots + 2679$$
$71$ $$T^{4} - 16 T^{3} - 92 T^{2} + \cdots + 2768$$
$73$ $$T^{4} - 34 T^{3} + 328 T^{2} + \cdots - 7312$$
$79$ $$T^{4} - 28 T^{3} + 213 T^{2} + \cdots - 2463$$
$83$ $$T^{4} - 6 T^{3} - 28 T^{2} + 232 T - 336$$
$89$ $$T^{4} + 24 T^{3} + 141 T^{2} + \cdots + 213$$
$97$ $$T^{4} + 2 T^{3} - 208 T^{2} + \cdots + 8176$$