# Properties

 Label 7360.2.a.cl Level $7360$ Weight $2$ Character orbit 7360.a Self dual yes Analytic conductor $58.770$ Analytic rank $1$ Dimension $5$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$7360 = 2^{6} \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7360.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$58.7698958877$$ Analytic rank: $$1$$ Dimension: $$5$$ Coefficient field: 5.5.876604.1 Defining polynomial: $$x^{5} - x^{4} - 9x^{3} + 8x^{2} + 18x - 16$$ x^5 - x^4 - 9*x^3 + 8*x^2 + 18*x - 16 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 3680) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3,\beta_4$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{3} q^{3} - q^{5} + (\beta_{4} + \beta_{3} + \beta_{2}) q^{7} + (\beta_{4} - \beta_{2} - \beta_1 + 1) q^{9}+O(q^{10})$$ q - b3 * q^3 - q^5 + (b4 + b3 + b2) * q^7 + (b4 - b2 - b1 + 1) * q^9 $$q - \beta_{3} q^{3} - q^{5} + (\beta_{4} + \beta_{3} + \beta_{2}) q^{7} + (\beta_{4} - \beta_{2} - \beta_1 + 1) q^{9} + ( - 2 \beta_{4} - \beta_1) q^{11} + (\beta_{3} - \beta_{2} + \beta_1 - 2) q^{13} + \beta_{3} q^{15} + (\beta_{4} - \beta_{3} - \beta_1 + 2) q^{17} + ( - \beta_{4} - \beta_{3} + \beta_{2} + 2 \beta_1) q^{19} + (\beta_{2} + \beta_1 - 4) q^{21} - q^{23} + q^{25} + ( - \beta_{3} + \beta_{2} - 2 \beta_1) q^{27} + ( - \beta_{4} - 2 \beta_{2} - \beta_1 + 2) q^{29} + (\beta_{3} + \beta_{2} + \beta_1 + 4) q^{31} + (\beta_{4} + \beta_{3} - 3 \beta_{2} + 2 \beta_1) q^{33} + ( - \beta_{4} - \beta_{3} - \beta_{2}) q^{35} + (3 \beta_{4} - \beta_{3} - 2) q^{37} + ( - 3 \beta_{4} + 2 \beta_{3} + 3 \beta_{2} - 4) q^{39} + ( - 2 \beta_{4} + \beta_{3} + 2 \beta_{2} - 2) q^{41} + ( - \beta_{4} + \beta_{3} - 2 \beta_{2} + 4 \beta_1) q^{43} + ( - \beta_{4} + \beta_{2} + \beta_1 - 1) q^{45} + (\beta_{4} - 2 \beta_{3} + \beta_1) q^{47} + ( - 3 \beta_{4} - \beta_{3} + 3 \beta_1 + 5) q^{49} + (2 \beta_{4} - 4 \beta_{3} - \beta_{2} - 2 \beta_1 + 4) q^{51} + ( - 2 \beta_{4} - 2 \beta_{3} + 2) q^{53} + (2 \beta_{4} + \beta_1) q^{55} + (4 \beta_{3} - \beta_{2} + \beta_1 + 4) q^{57} + (2 \beta_{4} - 4) q^{59} + (2 \beta_{3} - \beta_1 + 2) q^{61} + ( - 3 \beta_{4} + 3 \beta_{3} - 3 \beta_{2} + \beta_1) q^{63} + ( - \beta_{3} + \beta_{2} - \beta_1 + 2) q^{65} + (\beta_{4} + \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 8) q^{67} + \beta_{3} q^{69} + (\beta_{4} - 2 \beta_1 - 4) q^{71} + (\beta_{3} + 2 \beta_{2} - \beta_1 - 2) q^{73} - \beta_{3} q^{75} + (\beta_{4} + \beta_{3} + \beta_{2} - 7 \beta_1 - 8) q^{77} + ( - 3 \beta_{4} + 3 \beta_{3} - 2 \beta_{2} - 4) q^{79} + (\beta_{4} - \beta_{3} - \beta_{2} + 3 \beta_1 + 1) q^{81} + (2 \beta_{4} + 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1 - 4) q^{83} + ( - \beta_{4} + \beta_{3} + \beta_1 - 2) q^{85} + ( - \beta_{4} - 4 \beta_{3} - \beta_1) q^{87} + ( - 3 \beta_{4} + \beta_{3} - 4 \beta_1 - 2) q^{89} + (2 \beta_{4} - 2 \beta_{3} - 4 \beta_{2} - \beta_1) q^{91} + ( - \beta_{4} - 2 \beta_{3} + \beta_{2} + 2 \beta_1 - 4) q^{93} + (\beta_{4} + \beta_{3} - \beta_{2} - 2 \beta_1) q^{95} + (2 \beta_{3} + 2 \beta_{2} - \beta_1 + 6) q^{97} + ( - 2 \beta_{3} + 7 \beta_{2} - 4) q^{99}+O(q^{100})$$ q - b3 * q^3 - q^5 + (b4 + b3 + b2) * q^7 + (b4 - b2 - b1 + 1) * q^9 + (-2*b4 - b1) * q^11 + (b3 - b2 + b1 - 2) * q^13 + b3 * q^15 + (b4 - b3 - b1 + 2) * q^17 + (-b4 - b3 + b2 + 2*b1) * q^19 + (b2 + b1 - 4) * q^21 - q^23 + q^25 + (-b3 + b2 - 2*b1) * q^27 + (-b4 - 2*b2 - b1 + 2) * q^29 + (b3 + b2 + b1 + 4) * q^31 + (b4 + b3 - 3*b2 + 2*b1) * q^33 + (-b4 - b3 - b2) * q^35 + (3*b4 - b3 - 2) * q^37 + (-3*b4 + 2*b3 + 3*b2 - 4) * q^39 + (-2*b4 + b3 + 2*b2 - 2) * q^41 + (-b4 + b3 - 2*b2 + 4*b1) * q^43 + (-b4 + b2 + b1 - 1) * q^45 + (b4 - 2*b3 + b1) * q^47 + (-3*b4 - b3 + 3*b1 + 5) * q^49 + (2*b4 - 4*b3 - b2 - 2*b1 + 4) * q^51 + (-2*b4 - 2*b3 + 2) * q^53 + (2*b4 + b1) * q^55 + (4*b3 - b2 + b1 + 4) * q^57 + (2*b4 - 4) * q^59 + (2*b3 - b1 + 2) * q^61 + (-3*b4 + 3*b3 - 3*b2 + b1) * q^63 + (-b3 + b2 - b1 + 2) * q^65 + (b4 + b3 + 2*b2 + 2*b1 - 8) * q^67 + b3 * q^69 + (b4 - 2*b1 - 4) * q^71 + (b3 + 2*b2 - b1 - 2) * q^73 - b3 * q^75 + (b4 + b3 + b2 - 7*b1 - 8) * q^77 + (-3*b4 + 3*b3 - 2*b2 - 4) * q^79 + (b4 - b3 - b2 + 3*b1 + 1) * q^81 + (2*b4 + 2*b3 + 2*b2 - 2*b1 - 4) * q^83 + (-b4 + b3 + b1 - 2) * q^85 + (-b4 - 4*b3 - b1) * q^87 + (-3*b4 + b3 - 4*b1 - 2) * q^89 + (2*b4 - 2*b3 - 4*b2 - b1) * q^91 + (-b4 - 2*b3 + b2 + 2*b1 - 4) * q^93 + (b4 + b3 - b2 - 2*b1) * q^95 + (2*b3 + 2*b2 - b1 + 6) * q^97 + (-2*b3 + 7*b2 - 4) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5 q - q^{3} - 5 q^{5} + q^{7} + 6 q^{9}+O(q^{10})$$ 5 * q - q^3 - 5 * q^5 + q^7 + 6 * q^9 $$5 q - q^{3} - 5 q^{5} + q^{7} + 6 q^{9} - 3 q^{11} - 7 q^{13} + q^{15} + 9 q^{17} - q^{19} - 20 q^{21} - 5 q^{23} + 5 q^{25} - 4 q^{27} + 10 q^{29} + 21 q^{31} + 7 q^{33} - q^{35} - 8 q^{37} - 24 q^{39} - 13 q^{41} + 6 q^{43} - 6 q^{45} + 24 q^{49} + 17 q^{51} + 6 q^{53} + 3 q^{55} + 26 q^{57} - 18 q^{59} + 11 q^{61} + 4 q^{63} + 7 q^{65} - 38 q^{67} + q^{69} - 21 q^{71} - 12 q^{73} - q^{75} - 46 q^{77} - 18 q^{79} + 9 q^{81} - 20 q^{83} - 9 q^{85} - 6 q^{87} - 16 q^{89} + 3 q^{91} - 22 q^{93} + q^{95} + 29 q^{97} - 29 q^{99}+O(q^{100})$$ 5 * q - q^3 - 5 * q^5 + q^7 + 6 * q^9 - 3 * q^11 - 7 * q^13 + q^15 + 9 * q^17 - q^19 - 20 * q^21 - 5 * q^23 + 5 * q^25 - 4 * q^27 + 10 * q^29 + 21 * q^31 + 7 * q^33 - q^35 - 8 * q^37 - 24 * q^39 - 13 * q^41 + 6 * q^43 - 6 * q^45 + 24 * q^49 + 17 * q^51 + 6 * q^53 + 3 * q^55 + 26 * q^57 - 18 * q^59 + 11 * q^61 + 4 * q^63 + 7 * q^65 - 38 * q^67 + q^69 - 21 * q^71 - 12 * q^73 - q^75 - 46 * q^77 - 18 * q^79 + 9 * q^81 - 20 * q^83 - 9 * q^85 - 6 * q^87 - 16 * q^89 + 3 * q^91 - 22 * q^93 + q^95 + 29 * q^97 - 29 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{5} - x^{4} - 9x^{3} + 8x^{2} + 18x - 16$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 4$$ v^2 - 4 $$\beta_{3}$$ $$=$$ $$( \nu^{4} - \nu^{3} - 7\nu^{2} + 4\nu + 8 ) / 2$$ (v^4 - v^3 - 7*v^2 + 4*v + 8) / 2 $$\beta_{4}$$ $$=$$ $$( \nu^{4} + \nu^{3} - 9\nu^{2} - 6\nu + 16 ) / 2$$ (v^4 + v^3 - 9*v^2 - 6*v + 16) / 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 4$$ b2 + 4 $$\nu^{3}$$ $$=$$ $$\beta_{4} - \beta_{3} + \beta_{2} + 5\beta_1$$ b4 - b3 + b2 + 5*b1 $$\nu^{4}$$ $$=$$ $$\beta_{4} + \beta_{3} + 8\beta_{2} + \beta _1 + 20$$ b4 + b3 + 8*b2 + b1 + 20

## 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
 0.895130 −2.36386 2.60527 1.63675 −1.77328
0 −2.94825 0 −1.00000 0 2.13809 0 5.69219 0
1.2 0 −1.93128 0 −1.00000 0 2.47297 0 0.729857 0
1.3 0 0.352411 0 −1.00000 0 3.95190 0 −2.87581 0
1.4 0 0.706809 0 −1.00000 0 −5.21261 0 −2.50042 0
1.5 0 2.82032 0 −1.00000 0 −2.35036 0 4.95418 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.5 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$5$$ $$1$$
$$23$$ $$1$$

## 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 7360.2.a.cl 5
4.b odd 2 1 7360.2.a.cq 5
8.b even 2 1 3680.2.a.ba yes 5
8.d odd 2 1 3680.2.a.x 5

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3680.2.a.x 5 8.d odd 2 1
3680.2.a.ba yes 5 8.b even 2 1
7360.2.a.cl 5 1.a even 1 1 trivial
7360.2.a.cq 5 4.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(7360))$$:

 $$T_{3}^{5} + T_{3}^{4} - 10T_{3}^{3} - 7T_{3}^{2} + 15T_{3} - 4$$ T3^5 + T3^4 - 10*T3^3 - 7*T3^2 + 15*T3 - 4 $$T_{7}^{5} - T_{7}^{4} - 29T_{7}^{3} + 52T_{7}^{2} + 130T_{7} - 256$$ T7^5 - T7^4 - 29*T7^3 + 52*T7^2 + 130*T7 - 256 $$T_{11}^{5} + 3T_{11}^{4} - 49T_{11}^{3} - 100T_{11}^{2} + 626T_{11} + 592$$ T11^5 + 3*T11^4 - 49*T11^3 - 100*T11^2 + 626*T11 + 592 $$T_{13}^{5} + 7T_{13}^{4} - 16T_{13}^{3} - 179T_{13}^{2} - 249T_{13} + 74$$ T13^5 + 7*T13^4 - 16*T13^3 - 179*T13^2 - 249*T13 + 74 $$T_{17}^{5} - 9T_{17}^{4} + 5T_{17}^{3} + 34T_{17}^{2} - 48T_{17} + 16$$ T17^5 - 9*T17^4 + 5*T17^3 + 34*T17^2 - 48*T17 + 16

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{5}$$
$3$ $$T^{5} + T^{4} - 10 T^{3} - 7 T^{2} + \cdots - 4$$
$5$ $$(T + 1)^{5}$$
$7$ $$T^{5} - T^{4} - 29 T^{3} + 52 T^{2} + \cdots - 256$$
$11$ $$T^{5} + 3 T^{4} - 49 T^{3} - 100 T^{2} + \cdots + 592$$
$13$ $$T^{5} + 7 T^{4} - 16 T^{3} - 179 T^{2} + \cdots + 74$$
$17$ $$T^{5} - 9 T^{4} + 5 T^{3} + 34 T^{2} + \cdots + 16$$
$19$ $$T^{5} + T^{4} - 75 T^{3} - 116 T^{2} + \cdots + 3152$$
$23$ $$(T + 1)^{5}$$
$29$ $$T^{5} - 10 T^{4} - 23 T^{3} + \cdots + 2264$$
$31$ $$T^{5} - 21 T^{4} + 148 T^{3} + \cdots + 1132$$
$37$ $$T^{5} + 8 T^{4} - 70 T^{3} + \cdots - 2368$$
$41$ $$T^{5} + 13 T^{4} - 24 T^{3} - 477 T^{2} + \cdots - 94$$
$43$ $$T^{5} - 6 T^{4} - 202 T^{3} + \cdots - 33664$$
$47$ $$T^{5} - 55 T^{3} + 50 T^{2} + 484 T + 64$$
$53$ $$T^{5} - 6 T^{4} - 80 T^{3} + 416 T^{2} + \cdots - 32$$
$59$ $$T^{5} + 18 T^{4} + 88 T^{3} + 88 T^{2} + \cdots - 64$$
$61$ $$T^{5} - 11 T^{4} - T^{3} + 158 T^{2} + \cdots + 148$$
$67$ $$T^{5} + 38 T^{4} + 478 T^{3} + \cdots - 41344$$
$71$ $$T^{5} + 21 T^{4} + 130 T^{3} + \cdots - 76$$
$73$ $$T^{5} + 12 T^{4} + T^{3} - 384 T^{2} + \cdots + 712$$
$79$ $$T^{5} + 18 T^{4} - 90 T^{3} + \cdots - 9728$$
$83$ $$T^{5} + 20 T^{4} + 12 T^{3} + \cdots - 1024$$
$89$ $$T^{5} + 16 T^{4} - 150 T^{3} + \cdots + 1088$$
$97$ $$T^{5} - 29 T^{4} + 257 T^{3} + \cdots - 76$$