# Properties

 Label 7360.2.a.ca Level $7360$ Weight $2$ Character orbit 7360.a Self dual yes Analytic conductor $58.770$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

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## 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: $$3$$ Coefficient field: 3.3.621.1 Defining polynomial: $$x^{3} - 6x - 3$$ x^3 - 6*x - 3 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 920) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{3} + q^{5} + ( - \beta_1 - 1) q^{7} + (\beta_{2} + \beta_1 + 1) q^{9}+O(q^{10})$$ q + b1 * q^3 + q^5 + (-b1 - 1) * q^7 + (b2 + b1 + 1) * q^9 $$q + \beta_1 q^{3} + q^{5} + ( - \beta_1 - 1) q^{7} + (\beta_{2} + \beta_1 + 1) q^{9} + (\beta_{2} - \beta_1 + 1) q^{11} + (\beta_{2} - \beta_1) q^{13} + \beta_1 q^{15} + ( - \beta_{2} + \beta_1 - 1) q^{17} + ( - 2 \beta_{2} - \beta_1 + 1) q^{19} + ( - \beta_{2} - 2 \beta_1 - 4) q^{21} - q^{23} + q^{25} + 3 q^{27} + ( - \beta_{2} - 1) q^{29} + (\beta_{2} + 3 \beta_1 - 2) q^{31} + ( - 2 \beta_{2} + \beta_1 - 5) q^{33} + ( - \beta_1 - 1) q^{35} - 4 q^{37} + ( - 2 \beta_{2} - 5) q^{39} + ( - 2 \beta_{2} - 3 \beta_1 - 2) q^{41} + (2 \beta_{2} + 6) q^{43} + (\beta_{2} + \beta_1 + 1) q^{45} + ( - \beta_{2} - 5) q^{47} + (\beta_{2} + 3 \beta_1 - 2) q^{49} + (2 \beta_{2} - \beta_1 + 5) q^{51} + ( - 4 \beta_{2} - 2) q^{53} + (\beta_{2} - \beta_1 + 1) q^{55} + (\beta_{2} - 2 \beta_1 - 2) q^{57} + ( - 2 \beta_{2} - 2 \beta_1 + 2) q^{59} + ( - \beta_{2} + \beta_1 - 9) q^{61} + ( - \beta_{2} - 4 \beta_1 - 4) q^{63} + (\beta_{2} - \beta_1) q^{65} + ( - 4 \beta_{2} + 4 \beta_1 + 4) q^{67} - \beta_1 q^{69} + (2 \beta_{2} + 3 \beta_1 - 2) q^{71} + (\beta_{2} - 4 \beta_1 - 5) q^{73} + \beta_1 q^{75} + (\beta_{2} + 4) q^{77} + ( - 3 \beta_{2} - 3) q^{81} + ( - 2 \beta_{2} - 4 \beta_1 - 4) q^{83} + ( - \beta_{2} + \beta_1 - 1) q^{85} + (\beta_{2} - 2 \beta_1 + 1) q^{87} + ( - 2 \beta_{2} - 10) q^{89} + (\beta_{2} + \beta_1 + 5) q^{91} + (2 \beta_{2} + 2 \beta_1 + 11) q^{93} + ( - 2 \beta_{2} - \beta_1 + 1) q^{95} + (3 \beta_{2} - \beta_1 + 3) q^{97} + ( - 3 \beta_1 + 3) q^{99}+O(q^{100})$$ q + b1 * q^3 + q^5 + (-b1 - 1) * q^7 + (b2 + b1 + 1) * q^9 + (b2 - b1 + 1) * q^11 + (b2 - b1) * q^13 + b1 * q^15 + (-b2 + b1 - 1) * q^17 + (-2*b2 - b1 + 1) * q^19 + (-b2 - 2*b1 - 4) * q^21 - q^23 + q^25 + 3 * q^27 + (-b2 - 1) * q^29 + (b2 + 3*b1 - 2) * q^31 + (-2*b2 + b1 - 5) * q^33 + (-b1 - 1) * q^35 - 4 * q^37 + (-2*b2 - 5) * q^39 + (-2*b2 - 3*b1 - 2) * q^41 + (2*b2 + 6) * q^43 + (b2 + b1 + 1) * q^45 + (-b2 - 5) * q^47 + (b2 + 3*b1 - 2) * q^49 + (2*b2 - b1 + 5) * q^51 + (-4*b2 - 2) * q^53 + (b2 - b1 + 1) * q^55 + (b2 - 2*b1 - 2) * q^57 + (-2*b2 - 2*b1 + 2) * q^59 + (-b2 + b1 - 9) * q^61 + (-b2 - 4*b1 - 4) * q^63 + (b2 - b1) * q^65 + (-4*b2 + 4*b1 + 4) * q^67 - b1 * q^69 + (2*b2 + 3*b1 - 2) * q^71 + (b2 - 4*b1 - 5) * q^73 + b1 * q^75 + (b2 + 4) * q^77 + (-3*b2 - 3) * q^81 + (-2*b2 - 4*b1 - 4) * q^83 + (-b2 + b1 - 1) * q^85 + (b2 - 2*b1 + 1) * q^87 + (-2*b2 - 10) * q^89 + (b2 + b1 + 5) * q^91 + (2*b2 + 2*b1 + 11) * q^93 + (-2*b2 - b1 + 1) * q^95 + (3*b2 - b1 + 3) * q^97 + (-3*b1 + 3) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 3 q^{5} - 3 q^{7} + 3 q^{9}+O(q^{10})$$ 3 * q + 3 * q^5 - 3 * q^7 + 3 * q^9 $$3 q + 3 q^{5} - 3 q^{7} + 3 q^{9} + 3 q^{11} - 3 q^{17} + 3 q^{19} - 12 q^{21} - 3 q^{23} + 3 q^{25} + 9 q^{27} - 3 q^{29} - 6 q^{31} - 15 q^{33} - 3 q^{35} - 12 q^{37} - 15 q^{39} - 6 q^{41} + 18 q^{43} + 3 q^{45} - 15 q^{47} - 6 q^{49} + 15 q^{51} - 6 q^{53} + 3 q^{55} - 6 q^{57} + 6 q^{59} - 27 q^{61} - 12 q^{63} + 12 q^{67} - 6 q^{71} - 15 q^{73} + 12 q^{77} - 9 q^{81} - 12 q^{83} - 3 q^{85} + 3 q^{87} - 30 q^{89} + 15 q^{91} + 33 q^{93} + 3 q^{95} + 9 q^{97} + 9 q^{99}+O(q^{100})$$ 3 * q + 3 * q^5 - 3 * q^7 + 3 * q^9 + 3 * q^11 - 3 * q^17 + 3 * q^19 - 12 * q^21 - 3 * q^23 + 3 * q^25 + 9 * q^27 - 3 * q^29 - 6 * q^31 - 15 * q^33 - 3 * q^35 - 12 * q^37 - 15 * q^39 - 6 * q^41 + 18 * q^43 + 3 * q^45 - 15 * q^47 - 6 * q^49 + 15 * q^51 - 6 * q^53 + 3 * q^55 - 6 * q^57 + 6 * q^59 - 27 * q^61 - 12 * q^63 + 12 * q^67 - 6 * q^71 - 15 * q^73 + 12 * q^77 - 9 * q^81 - 12 * q^83 - 3 * q^85 + 3 * q^87 - 30 * q^89 + 15 * q^91 + 33 * q^93 + 3 * q^95 + 9 * q^97 + 9 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - 6x - 3$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 4$$ v^2 - v - 4
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta _1 + 4$$ b2 + b1 + 4

## 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
 −2.14510 −0.523976 2.66908
0 −2.14510 0 1.00000 0 1.14510 0 1.60147 0
1.2 0 −0.523976 0 1.00000 0 −0.476024 0 −2.72545 0
1.3 0 2.66908 0 1.00000 0 −3.66908 0 4.12398 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
$$2$$ $$-1$$
$$5$$ $$-1$$
$$23$$ $$1$$

## 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.ca 3
4.b odd 2 1 7360.2.a.cb 3
8.b even 2 1 1840.2.a.t 3
8.d odd 2 1 920.2.a.g 3
24.f even 2 1 8280.2.a.bo 3
40.e odd 2 1 4600.2.a.y 3
40.f even 2 1 9200.2.a.cd 3
40.k even 4 2 4600.2.e.r 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
920.2.a.g 3 8.d odd 2 1
1840.2.a.t 3 8.b even 2 1
4600.2.a.y 3 40.e odd 2 1
4600.2.e.r 6 40.k even 4 2
7360.2.a.ca 3 1.a even 1 1 trivial
7360.2.a.cb 3 4.b odd 2 1
8280.2.a.bo 3 24.f even 2 1
9200.2.a.cd 3 40.f even 2 1

## Hecke kernels

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

 $$T_{3}^{3} - 6T_{3} - 3$$ T3^3 - 6*T3 - 3 $$T_{7}^{3} + 3T_{7}^{2} - 3T_{7} - 2$$ T7^3 + 3*T7^2 - 3*T7 - 2 $$T_{11}^{3} - 3T_{11}^{2} - 15T_{11} - 12$$ T11^3 - 3*T11^2 - 15*T11 - 12 $$T_{13}^{3} - 18T_{13} - 29$$ T13^3 - 18*T13 - 29 $$T_{17}^{3} + 3T_{17}^{2} - 15T_{17} + 12$$ T17^3 + 3*T17^2 - 15*T17 + 12

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$T^{3} - 6T - 3$$
$5$ $$(T - 1)^{3}$$
$7$ $$T^{3} + 3 T^{2} - 3 T - 2$$
$11$ $$T^{3} - 3 T^{2} - 15 T - 12$$
$13$ $$T^{3} - 18T - 29$$
$17$ $$T^{3} + 3 T^{2} - 15 T + 12$$
$19$ $$T^{3} - 3 T^{2} - 33 T - 48$$
$23$ $$(T + 1)^{3}$$
$29$ $$T^{3} + 3 T^{2} - 6 T - 12$$
$31$ $$T^{3} + 6 T^{2} - 42 T - 249$$
$37$ $$(T + 4)^{3}$$
$41$ $$T^{3} + 6 T^{2} - 60 T - 69$$
$43$ $$T^{3} - 18 T^{2} + 72 T + 32$$
$47$ $$T^{3} + 15 T^{2} + 66 T + 76$$
$53$ $$T^{3} + 6 T^{2} - 132 T - 536$$
$59$ $$T^{3} - 6 T^{2} - 36 T + 32$$
$61$ $$T^{3} + 27 T^{2} + 225 T + 596$$
$67$ $$T^{3} - 12 T^{2} - 240 T + 2944$$
$71$ $$T^{3} + 6 T^{2} - 60 T - 203$$
$73$ $$T^{3} + 15 T^{2} - 42 T - 588$$
$79$ $$T^{3}$$
$83$ $$T^{3} + 12 T^{2} - 60 T - 64$$
$89$ $$T^{3} + 30 T^{2} + 264 T + 608$$
$97$ $$T^{3} - 9 T^{2} - 69 T + 138$$
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