# Properties

 Label 9360.2.a.ci 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: $$\Q(\sqrt{17})$$ Defining polynomial: $$x^{2} - x - 4$$ x^2 - x - 4 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 $$\beta = \frac{1}{2}(1 + \sqrt{17})$$. 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 $$q - q^{5} + \beta q^{7} + (\beta - 4) q^{11} + q^{13} + ( - 3 \beta + 2) q^{17} + 2 \beta q^{19} + (\beta - 4) q^{23} + q^{25} + 2 q^{29} + ( - 2 \beta + 4) q^{31} - \beta q^{35} + ( - 5 \beta + 2) q^{37} + (3 \beta + 2) q^{41} + ( - 4 \beta + 4) q^{43} + ( - 2 \beta + 4) q^{47} + (\beta - 3) q^{49} + (\beta - 2) q^{53} + ( - \beta + 4) q^{55} - 8 q^{59} + (3 \beta - 6) q^{61} - q^{65} + ( - 4 \beta + 8) q^{67} - 3 \beta q^{71} + ( - 4 \beta + 10) q^{73} + ( - 3 \beta + 4) q^{77} + (5 \beta - 4) q^{79} - 8 q^{83} + (3 \beta - 2) q^{85} + ( - 3 \beta + 10) q^{89} + \beta q^{91} - 2 \beta q^{95} + (5 \beta - 10) q^{97} +O(q^{100})$$ q - q^5 + b * q^7 + (b - 4) * q^11 + q^13 + (-3*b + 2) * q^17 + 2*b * q^19 + (b - 4) * q^23 + q^25 + 2 * q^29 + (-2*b + 4) * q^31 - b * q^35 + (-5*b + 2) * q^37 + (3*b + 2) * q^41 + (-4*b + 4) * q^43 + (-2*b + 4) * q^47 + (b - 3) * q^49 + (b - 2) * q^53 + (-b + 4) * q^55 - 8 * q^59 + (3*b - 6) * q^61 - q^65 + (-4*b + 8) * q^67 - 3*b * q^71 + (-4*b + 10) * q^73 + (-3*b + 4) * q^77 + (5*b - 4) * q^79 - 8 * q^83 + (3*b - 2) * q^85 + (-3*b + 10) * q^89 + b * q^91 - 2*b * q^95 + (5*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 $$2 q - 2 q^{5} + q^{7} - 7 q^{11} + 2 q^{13} + q^{17} + 2 q^{19} - 7 q^{23} + 2 q^{25} + 4 q^{29} + 6 q^{31} - q^{35} - q^{37} + 7 q^{41} + 4 q^{43} + 6 q^{47} - 5 q^{49} - 3 q^{53} + 7 q^{55} - 16 q^{59} - 9 q^{61} - 2 q^{65} + 12 q^{67} - 3 q^{71} + 16 q^{73} + 5 q^{77} - 3 q^{79} - 16 q^{83} - q^{85} + 17 q^{89} + q^{91} - 2 q^{95} - 15 q^{97}+O(q^{100})$$ 2 * q - 2 * q^5 + q^7 - 7 * q^11 + 2 * q^13 + q^17 + 2 * q^19 - 7 * q^23 + 2 * q^25 + 4 * q^29 + 6 * q^31 - q^35 - q^37 + 7 * q^41 + 4 * q^43 + 6 * q^47 - 5 * q^49 - 3 * q^53 + 7 * q^55 - 16 * q^59 - 9 * q^61 - 2 * q^65 + 12 * q^67 - 3 * q^71 + 16 * q^73 + 5 * q^77 - 3 * q^79 - 16 * q^83 - q^85 + 17 * q^89 + q^91 - 2 * q^95 - 15 * 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
 −1.56155 2.56155
0 0 0 −1.00000 0 −1.56155 0 0 0
1.2 0 0 0 −1.00000 0 2.56155 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.ci 2
3.b odd 2 1 3120.2.a.bg 2
4.b odd 2 1 4680.2.a.y 2
12.b even 2 1 1560.2.a.o 2
60.h even 2 1 7800.2.a.bd 2

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

 $$T_{7}^{2} - T_{7} - 4$$ T7^2 - T7 - 4 $$T_{11}^{2} + 7T_{11} + 8$$ T11^2 + 7*T11 + 8 $$T_{17}^{2} - T_{17} - 38$$ T17^2 - T17 - 38 $$T_{19}^{2} - 2T_{19} - 16$$ T19^2 - 2*T19 - 16 $$T_{31}^{2} - 6T_{31} - 8$$ T31^2 - 6*T31 - 8

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$(T + 1)^{2}$$
$7$ $$T^{2} - T - 4$$
$11$ $$T^{2} + 7T + 8$$
$13$ $$(T - 1)^{2}$$
$17$ $$T^{2} - T - 38$$
$19$ $$T^{2} - 2T - 16$$
$23$ $$T^{2} + 7T + 8$$
$29$ $$(T - 2)^{2}$$
$31$ $$T^{2} - 6T - 8$$
$37$ $$T^{2} + T - 106$$
$41$ $$T^{2} - 7T - 26$$
$43$ $$T^{2} - 4T - 64$$
$47$ $$T^{2} - 6T - 8$$
$53$ $$T^{2} + 3T - 2$$
$59$ $$(T + 8)^{2}$$
$61$ $$T^{2} + 9T - 18$$
$67$ $$T^{2} - 12T - 32$$
$71$ $$T^{2} + 3T - 36$$
$73$ $$T^{2} - 16T - 4$$
$79$ $$T^{2} + 3T - 104$$
$83$ $$(T + 8)^{2}$$
$89$ $$T^{2} - 17T + 34$$
$97$ $$T^{2} + 15T - 50$$