# Properties

 Label 9360.2.a.ct Level $9360$ Weight $2$ Character orbit 9360.a Self dual yes Analytic conductor $74.740$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$9360 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9360.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$74.7399762919$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{41})$$ Defining polynomial: $$x^{2} - x - 10$$ x^2 - x - 10 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1560) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{41})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{5} + \beta q^{7}+O(q^{10})$$ q + q^5 + b * q^7 $$q + q^{5} + \beta q^{7} + ( - \beta - 2) q^{11} - q^{13} + ( - \beta - 2) q^{17} + 2 q^{19} + ( - \beta + 4) q^{23} + q^{25} + (2 \beta - 4) q^{29} + ( - 2 \beta + 4) q^{31} + \beta q^{35} + (\beta - 4) q^{37} + (\beta - 8) q^{41} - 4 q^{43} + (2 \beta + 4) q^{47} + (\beta + 3) q^{49} + (\beta - 8) q^{53} + ( - \beta - 2) q^{55} + (4 \beta - 4) q^{59} - \beta q^{61} - q^{65} + 4 q^{67} + ( - 5 \beta + 2) q^{71} + ( - 2 \beta + 8) q^{73} + ( - 3 \beta - 10) q^{77} + (\beta - 2) q^{79} + 4 q^{83} + ( - \beta - 2) q^{85} + 3 \beta q^{89} - \beta q^{91} + 2 q^{95} + ( - 3 \beta + 10) q^{97} +O(q^{100})$$ q + q^5 + b * q^7 + (-b - 2) * q^11 - q^13 + (-b - 2) * q^17 + 2 * q^19 + (-b + 4) * q^23 + q^25 + (2*b - 4) * q^29 + (-2*b + 4) * q^31 + b * q^35 + (b - 4) * q^37 + (b - 8) * q^41 - 4 * q^43 + (2*b + 4) * q^47 + (b + 3) * q^49 + (b - 8) * q^53 + (-b - 2) * q^55 + (4*b - 4) * q^59 - b * q^61 - q^65 + 4 * q^67 + (-5*b + 2) * q^71 + (-2*b + 8) * q^73 + (-3*b - 10) * q^77 + (b - 2) * q^79 + 4 * q^83 + (-b - 2) * q^85 + 3*b * q^89 - b * q^91 + 2 * q^95 + (-3*b + 10) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{5} + q^{7}+O(q^{10})$$ 2 * q + 2 * q^5 + q^7 $$2 q + 2 q^{5} + q^{7} - 5 q^{11} - 2 q^{13} - 5 q^{17} + 4 q^{19} + 7 q^{23} + 2 q^{25} - 6 q^{29} + 6 q^{31} + q^{35} - 7 q^{37} - 15 q^{41} - 8 q^{43} + 10 q^{47} + 7 q^{49} - 15 q^{53} - 5 q^{55} - 4 q^{59} - q^{61} - 2 q^{65} + 8 q^{67} - q^{71} + 14 q^{73} - 23 q^{77} - 3 q^{79} + 8 q^{83} - 5 q^{85} + 3 q^{89} - q^{91} + 4 q^{95} + 17 q^{97}+O(q^{100})$$ 2 * q + 2 * q^5 + q^7 - 5 * q^11 - 2 * q^13 - 5 * q^17 + 4 * q^19 + 7 * q^23 + 2 * q^25 - 6 * q^29 + 6 * q^31 + q^35 - 7 * q^37 - 15 * q^41 - 8 * q^43 + 10 * q^47 + 7 * q^49 - 15 * q^53 - 5 * q^55 - 4 * q^59 - q^61 - 2 * q^65 + 8 * q^67 - q^71 + 14 * q^73 - 23 * q^77 - 3 * q^79 + 8 * q^83 - 5 * q^85 + 3 * q^89 - q^91 + 4 * q^95 + 17 * q^97

## 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
 −2.70156 3.70156
0 0 0 1.00000 0 −2.70156 0 0 0
1.2 0 0 0 1.00000 0 3.70156 0 0 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
$$2$$ $$1$$
$$3$$ $$-1$$
$$5$$ $$-1$$
$$13$$ $$1$$

## 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 9360.2.a.ct 2
3.b odd 2 1 3120.2.a.bf 2
4.b odd 2 1 4680.2.a.bb 2
12.b even 2 1 1560.2.a.m 2
60.h even 2 1 7800.2.a.be 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1560.2.a.m 2 12.b even 2 1
3120.2.a.bf 2 3.b odd 2 1
4680.2.a.bb 2 4.b odd 2 1
7800.2.a.be 2 60.h even 2 1
9360.2.a.ct 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(9360))$$:

 $$T_{7}^{2} - T_{7} - 10$$ T7^2 - T7 - 10 $$T_{11}^{2} + 5T_{11} - 4$$ T11^2 + 5*T11 - 4 $$T_{17}^{2} + 5T_{17} - 4$$ T17^2 + 5*T17 - 4 $$T_{19} - 2$$ T19 - 2 $$T_{31}^{2} - 6T_{31} - 32$$ T31^2 - 6*T31 - 32

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$(T - 1)^{2}$$
$7$ $$T^{2} - T - 10$$
$11$ $$T^{2} + 5T - 4$$
$13$ $$(T + 1)^{2}$$
$17$ $$T^{2} + 5T - 4$$
$19$ $$(T - 2)^{2}$$
$23$ $$T^{2} - 7T + 2$$
$29$ $$T^{2} + 6T - 32$$
$31$ $$T^{2} - 6T - 32$$
$37$ $$T^{2} + 7T + 2$$
$41$ $$T^{2} + 15T + 46$$
$43$ $$(T + 4)^{2}$$
$47$ $$T^{2} - 10T - 16$$
$53$ $$T^{2} + 15T + 46$$
$59$ $$T^{2} + 4T - 160$$
$61$ $$T^{2} + T - 10$$
$67$ $$(T - 4)^{2}$$
$71$ $$T^{2} + T - 256$$
$73$ $$T^{2} - 14T + 8$$
$79$ $$T^{2} + 3T - 8$$
$83$ $$(T - 4)^{2}$$
$89$ $$T^{2} - 3T - 90$$
$97$ $$T^{2} - 17T - 20$$