# Properties

 Label 9360.2.a.dd Level $9360$ Weight $2$ Character orbit 9360.a Self dual yes Analytic conductor $74.740$ Analytic rank $1$ Dimension $3$ 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: $$3$$ Coefficient field: 3.3.316.1 Defining polynomial: $$x^{3} - x^{2} - 4x + 2$$ x^3 - x^2 - 4*x + 2 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 195) 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.

 $$f(q)$$ $$=$$ $$q + q^{5} + \beta_1 q^{7}+O(q^{10})$$ q + q^5 + b1 * q^7 $$q + q^{5} + \beta_1 q^{7} - \beta_1 q^{11} + q^{13} + ( - \beta_{2} - \beta_1) q^{17} + ( - \beta_{2} - 2) q^{19} + (\beta_{2} + \beta_1 - 2) q^{23} + q^{25} - 6 q^{29} + (\beta_{2} - 2) q^{31} + \beta_1 q^{35} + (\beta_{2} - \beta_1 + 4) q^{37} + (\beta_{2} + \beta_1) q^{41} + 2 \beta_{2} q^{43} + (\beta_{2} - 6) q^{47} + ( - \beta_{2} - 3 \beta_1 + 3) q^{49} + ( - \beta_{2} - \beta_1 - 4) q^{53} - \beta_1 q^{55} + (\beta_{2} + 2 \beta_1 - 2) q^{59} + (\beta_{2} + 3 \beta_1 + 4) q^{61} + q^{65} + ( - \beta_{2} - 2 \beta_1 - 2) q^{67} + ( - \beta_1 - 4) q^{71} + ( - 2 \beta_{2} - 2) q^{73} + (\beta_{2} + 3 \beta_1 - 10) q^{77} + (\beta_{2} - \beta_1 - 2) q^{79} + ( - \beta_{2} - 2 \beta_1 + 2) q^{83} + ( - \beta_{2} - \beta_1) q^{85} + ( - \beta_{2} - \beta_1 - 4) q^{89} + \beta_1 q^{91} + ( - \beta_{2} - 2) q^{95} + (\beta_{2} + \beta_1 - 8) q^{97}+O(q^{100})$$ q + q^5 + b1 * q^7 - b1 * q^11 + q^13 + (-b2 - b1) * q^17 + (-b2 - 2) * q^19 + (b2 + b1 - 2) * q^23 + q^25 - 6 * q^29 + (b2 - 2) * q^31 + b1 * q^35 + (b2 - b1 + 4) * q^37 + (b2 + b1) * q^41 + 2*b2 * q^43 + (b2 - 6) * q^47 + (-b2 - 3*b1 + 3) * q^49 + (-b2 - b1 - 4) * q^53 - b1 * q^55 + (b2 + 2*b1 - 2) * q^59 + (b2 + 3*b1 + 4) * q^61 + q^65 + (-b2 - 2*b1 - 2) * q^67 + (-b1 - 4) * q^71 + (-2*b2 - 2) * q^73 + (b2 + 3*b1 - 10) * q^77 + (b2 - b1 - 2) * q^79 + (-b2 - 2*b1 + 2) * q^83 + (-b2 - b1) * q^85 + (-b2 - b1 - 4) * q^89 + b1 * q^91 + (-b2 - 2) * q^95 + (b2 + b1 - 8) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 3 q^{5} - q^{7}+O(q^{10})$$ 3 * q + 3 * q^5 - q^7 $$3 q + 3 q^{5} - q^{7} + q^{11} + 3 q^{13} + q^{17} - 6 q^{19} - 7 q^{23} + 3 q^{25} - 18 q^{29} - 6 q^{31} - q^{35} + 13 q^{37} - q^{41} - 18 q^{47} + 12 q^{49} - 11 q^{53} + q^{55} - 8 q^{59} + 9 q^{61} + 3 q^{65} - 4 q^{67} - 11 q^{71} - 6 q^{73} - 33 q^{77} - 5 q^{79} + 8 q^{83} + q^{85} - 11 q^{89} - q^{91} - 6 q^{95} - 25 q^{97}+O(q^{100})$$ 3 * q + 3 * q^5 - q^7 + q^11 + 3 * q^13 + q^17 - 6 * q^19 - 7 * q^23 + 3 * q^25 - 18 * q^29 - 6 * q^31 - q^35 + 13 * q^37 - q^41 - 18 * q^47 + 12 * q^49 - 11 * q^53 + q^55 - 8 * q^59 + 9 * q^61 + 3 * q^65 - 4 * q^67 - 11 * q^71 - 6 * q^73 - 33 * q^77 - 5 * q^79 + 8 * q^83 + q^85 - 11 * q^89 - q^91 - 6 * q^95 - 25 * q^97

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

 $$\beta_{1}$$ $$=$$ $$-\nu^{2} + 2\nu + 2$$ -v^2 + 2*v + 2 $$\beta_{2}$$ $$=$$ $$2\nu^{2} - 6$$ 2*v^2 - 6
 $$\nu$$ $$=$$ $$( \beta_{2} + 2\beta _1 + 2 ) / 4$$ (b2 + 2*b1 + 2) / 4 $$\nu^{2}$$ $$=$$ $$( \beta_{2} + 6 ) / 2$$ (b2 + 6) / 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
 −1.81361 2.34292 0.470683
0 0 0 1.00000 0 −4.91638 0 0 0
1.2 0 0 0 1.00000 0 1.19656 0 0 0
1.3 0 0 0 1.00000 0 2.71982 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.dd 3
3.b odd 2 1 3120.2.a.bj 3
4.b odd 2 1 585.2.a.n 3
12.b even 2 1 195.2.a.e 3
20.d odd 2 1 2925.2.a.bh 3
20.e even 4 2 2925.2.c.w 6
52.b odd 2 1 7605.2.a.bx 3
60.h even 2 1 975.2.a.o 3
60.l odd 4 2 975.2.c.i 6
84.h odd 2 1 9555.2.a.bq 3
156.h even 2 1 2535.2.a.bc 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
195.2.a.e 3 12.b even 2 1
585.2.a.n 3 4.b odd 2 1
975.2.a.o 3 60.h even 2 1
975.2.c.i 6 60.l odd 4 2
2535.2.a.bc 3 156.h even 2 1
2925.2.a.bh 3 20.d odd 2 1
2925.2.c.w 6 20.e even 4 2
3120.2.a.bj 3 3.b odd 2 1
7605.2.a.bx 3 52.b odd 2 1
9360.2.a.dd 3 1.a even 1 1 trivial
9555.2.a.bq 3 84.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(9360))$$:

 $$T_{7}^{3} + T_{7}^{2} - 16T_{7} + 16$$ T7^3 + T7^2 - 16*T7 + 16 $$T_{11}^{3} - T_{11}^{2} - 16T_{11} - 16$$ T11^3 - T11^2 - 16*T11 - 16 $$T_{17}^{3} - T_{17}^{2} - 32T_{17} + 76$$ T17^3 - T17^2 - 32*T17 + 76 $$T_{19}^{3} + 6T_{19}^{2} - 16T_{19} - 64$$ T19^3 + 6*T19^2 - 16*T19 - 64 $$T_{31}^{3} + 6T_{31}^{2} - 16T_{31} - 32$$ T31^3 + 6*T31^2 - 16*T31 - 32

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$T^{3}$$
$5$ $$(T - 1)^{3}$$
$7$ $$T^{3} + T^{2} - 16 T + 16$$
$11$ $$T^{3} - T^{2} - 16 T - 16$$
$13$ $$(T - 1)^{3}$$
$17$ $$T^{3} - T^{2} - 32 T + 76$$
$19$ $$T^{3} + 6 T^{2} - 16 T - 64$$
$23$ $$T^{3} + 7 T^{2} - 16 T - 128$$
$29$ $$(T + 6)^{3}$$
$31$ $$T^{3} + 6 T^{2} - 16 T - 32$$
$37$ $$T^{3} - 13T^{2} + 316$$
$41$ $$T^{3} + T^{2} - 32 T - 76$$
$43$ $$T^{3} - 112T + 128$$
$47$ $$T^{3} + 18 T^{2} + 80 T + 64$$
$53$ $$T^{3} + 11 T^{2} + 8 T - 4$$
$59$ $$T^{3} + 8 T^{2} - 48 T - 128$$
$61$ $$T^{3} - 9 T^{2} - 112 T + 844$$
$67$ $$T^{3} + 4 T^{2} - 64 T - 128$$
$71$ $$T^{3} + 11 T^{2} + 24 T - 32$$
$73$ $$T^{3} + 6 T^{2} - 100 T - 344$$
$79$ $$T^{3} + 5 T^{2} - 48 T + 64$$
$83$ $$T^{3} - 8 T^{2} - 48 T + 128$$
$89$ $$T^{3} + 11 T^{2} + 8 T - 4$$
$97$ $$T^{3} + 25 T^{2} + 176 T + 244$$