# Properties

 Label 9408.2.a.ej Level $9408$ Weight $2$ Character orbit 9408.a Self dual yes Analytic conductor $75.123$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$9408 = 2^{6} \cdot 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9408.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$75.1232582216$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.621.1 Defining polynomial: $$x^{3} - 6x - 3$$ x^3 - 6*x - 3 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 672) 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.

 $$f(q)$$ $$=$$ $$q + q^{3} - \beta_{2} q^{5} + q^{9}+O(q^{10})$$ q + q^3 - b2 * q^5 + q^9 $$q + q^{3} - \beta_{2} q^{5} + q^{9} + ( - \beta_{2} - \beta_1) q^{11} + ( - \beta_{2} + \beta_1 + 1) q^{13} - \beta_{2} q^{15} + (2 \beta_{2} + 2) q^{17} + ( - \beta_{2} + \beta_1 + 1) q^{19} + (2 \beta_{2} + 2) q^{23} + ( - \beta_{2} - \beta_1 + 1) q^{25} + q^{27} + (\beta_{2} - 4) q^{29} + (\beta_1 - 1) q^{31} + ( - \beta_{2} - \beta_1) q^{33} + ( - 3 \beta_{2} - \beta_1 - 1) q^{37} + ( - \beta_{2} + \beta_1 + 1) q^{39} + (2 \beta_{2} + 2 \beta_1 - 2) q^{41} + (\beta_{2} + \beta_1 + 5) q^{43} - \beta_{2} q^{45} + ( - 2 \beta_{2} - 4) q^{47} + (2 \beta_{2} + 2) q^{51} + ( - \beta_{2} - 2 \beta_1 - 2) q^{53} + ( - 3 \beta_{2} + 4) q^{55} + ( - \beta_{2} + \beta_1 + 1) q^{57} + ( - \beta_{2} - \beta_1 + 4) q^{59} + ( - 2 \beta_1 + 6) q^{61} + ( - 2 \beta_1 + 8) q^{65} + ( - \beta_{2} + \beta_1 + 3) q^{67} + (2 \beta_{2} + 2) q^{69} + 2 \beta_{2} q^{71} + ( - \beta_{2} - \beta_1 + 11) q^{73} + ( - \beta_{2} - \beta_1 + 1) q^{75} + ( - 2 \beta_{2} - \beta_1 + 9) q^{79} + q^{81} + ( - 3 \beta_{2} + \beta_1 + 6) q^{83} + (2 \beta_1 - 12) q^{85} + (\beta_{2} - 4) q^{87} + ( - 2 \beta_{2} - 4) q^{89} + (\beta_1 - 1) q^{93} + ( - 2 \beta_1 + 8) q^{95} + ( - \beta_{2} - \beta_1) q^{97} + ( - \beta_{2} - \beta_1) q^{99}+O(q^{100})$$ q + q^3 - b2 * q^5 + q^9 + (-b2 - b1) * q^11 + (-b2 + b1 + 1) * q^13 - b2 * q^15 + (2*b2 + 2) * q^17 + (-b2 + b1 + 1) * q^19 + (2*b2 + 2) * q^23 + (-b2 - b1 + 1) * q^25 + q^27 + (b2 - 4) * q^29 + (b1 - 1) * q^31 + (-b2 - b1) * q^33 + (-3*b2 - b1 - 1) * q^37 + (-b2 + b1 + 1) * q^39 + (2*b2 + 2*b1 - 2) * q^41 + (b2 + b1 + 5) * q^43 - b2 * q^45 + (-2*b2 - 4) * q^47 + (2*b2 + 2) * q^51 + (-b2 - 2*b1 - 2) * q^53 + (-3*b2 + 4) * q^55 + (-b2 + b1 + 1) * q^57 + (-b2 - b1 + 4) * q^59 + (-2*b1 + 6) * q^61 + (-2*b1 + 8) * q^65 + (-b2 + b1 + 3) * q^67 + (2*b2 + 2) * q^69 + 2*b2 * q^71 + (-b2 - b1 + 11) * q^73 + (-b2 - b1 + 1) * q^75 + (-2*b2 - b1 + 9) * q^79 + q^81 + (-3*b2 + b1 + 6) * q^83 + (2*b1 - 12) * q^85 + (b2 - 4) * q^87 + (-2*b2 - 4) * q^89 + (b1 - 1) * q^93 + (-2*b1 + 8) * q^95 + (-b2 - b1) * q^97 + (-b2 - b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 3 q^{3} + 3 q^{9}+O(q^{10})$$ 3 * q + 3 * q^3 + 3 * q^9 $$3 q + 3 q^{3} + 3 q^{9} + 3 q^{13} + 6 q^{17} + 3 q^{19} + 6 q^{23} + 3 q^{25} + 3 q^{27} - 12 q^{29} - 3 q^{31} - 3 q^{37} + 3 q^{39} - 6 q^{41} + 15 q^{43} - 12 q^{47} + 6 q^{51} - 6 q^{53} + 12 q^{55} + 3 q^{57} + 12 q^{59} + 18 q^{61} + 24 q^{65} + 9 q^{67} + 6 q^{69} + 33 q^{73} + 3 q^{75} + 27 q^{79} + 3 q^{81} + 18 q^{83} - 36 q^{85} - 12 q^{87} - 12 q^{89} - 3 q^{93} + 24 q^{95}+O(q^{100})$$ 3 * q + 3 * q^3 + 3 * q^9 + 3 * q^13 + 6 * q^17 + 3 * q^19 + 6 * q^23 + 3 * q^25 + 3 * q^27 - 12 * q^29 - 3 * q^31 - 3 * q^37 + 3 * q^39 - 6 * q^41 + 15 * q^43 - 12 * q^47 + 6 * q^51 - 6 * q^53 + 12 * q^55 + 3 * q^57 + 12 * q^59 + 18 * q^61 + 24 * q^65 + 9 * q^67 + 6 * q^69 + 33 * q^73 + 3 * q^75 + 27 * q^79 + 3 * q^81 + 18 * q^83 - 36 * q^85 - 12 * q^87 - 12 * q^89 - 3 * q^93 + 24 * q^95

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

 $$\beta_{1}$$ $$=$$ $$2\nu$$ 2*v $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 4$$ v^2 - v - 4
 $$\nu$$ $$=$$ $$( \beta_1 ) / 2$$ (b1) / 2 $$\nu^{2}$$ $$=$$ $$( 2\beta_{2} + \beta _1 + 8 ) / 2$$ (2*b2 + b1 + 8) / 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
 −2.14510 2.66908 −0.523976
0 1.00000 0 −2.74657 0 0 0 1.00000 0
1.2 0 1.00000 0 −0.454904 0 0 0 1.00000 0
1.3 0 1.00000 0 3.20147 0 0 0 1.00000 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
$$2$$ $$1$$
$$3$$ $$-1$$
$$7$$ $$1$$

## 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 9408.2.a.ej 3
4.b odd 2 1 9408.2.a.eh 3
7.b odd 2 1 9408.2.a.eg 3
7.c even 3 2 1344.2.q.y 6
8.b even 2 1 4704.2.a.bs 3
8.d odd 2 1 4704.2.a.bu 3
28.d even 2 1 9408.2.a.ei 3
28.g odd 6 2 1344.2.q.z 6
56.e even 2 1 4704.2.a.bt 3
56.h odd 2 1 4704.2.a.bv 3
56.k odd 6 2 672.2.q.k 6
56.p even 6 2 672.2.q.l yes 6
168.s odd 6 2 2016.2.s.v 6
168.v even 6 2 2016.2.s.u 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
672.2.q.k 6 56.k odd 6 2
672.2.q.l yes 6 56.p even 6 2
1344.2.q.y 6 7.c even 3 2
1344.2.q.z 6 28.g odd 6 2
2016.2.s.u 6 168.v even 6 2
2016.2.s.v 6 168.s odd 6 2
4704.2.a.bs 3 8.b even 2 1
4704.2.a.bt 3 56.e even 2 1
4704.2.a.bu 3 8.d odd 2 1
4704.2.a.bv 3 56.h odd 2 1
9408.2.a.eg 3 7.b odd 2 1
9408.2.a.eh 3 4.b odd 2 1
9408.2.a.ei 3 28.d even 2 1
9408.2.a.ej 3 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(9408))$$:

 $$T_{5}^{3} - 9T_{5} - 4$$ T5^3 - 9*T5 - 4 $$T_{11}^{3} - 27T_{11} + 38$$ T11^3 - 27*T11 + 38 $$T_{13}^{3} - 3T_{13}^{2} - 36T_{13} + 112$$ T13^3 - 3*T13^2 - 36*T13 + 112 $$T_{17}^{3} - 6T_{17}^{2} - 24T_{17} + 96$$ T17^3 - 6*T17^2 - 24*T17 + 96 $$T_{19}^{3} - 3T_{19}^{2} - 36T_{19} + 112$$ T19^3 - 3*T19^2 - 36*T19 + 112 $$T_{31}^{3} + 3T_{31}^{2} - 21T_{31} - 47$$ T31^3 + 3*T31^2 - 21*T31 - 47

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$(T - 1)^{3}$$
$5$ $$T^{3} - 9T - 4$$
$7$ $$T^{3}$$
$11$ $$T^{3} - 27T + 38$$
$13$ $$T^{3} - 3 T^{2} - 36 T + 112$$
$17$ $$T^{3} - 6 T^{2} - 24 T + 96$$
$19$ $$T^{3} - 3 T^{2} - 36 T + 112$$
$23$ $$T^{3} - 6 T^{2} - 24 T + 96$$
$29$ $$T^{3} + 12 T^{2} + 39 T + 32$$
$31$ $$T^{3} + 3 T^{2} - 21 T - 47$$
$37$ $$T^{3} + 3 T^{2} - 84 T - 368$$
$41$ $$T^{3} + 6 T^{2} - 96 T - 512$$
$43$ $$T^{3} - 15 T^{2} + 48 T - 28$$
$47$ $$T^{3} + 12 T^{2} + 12 T - 112$$
$53$ $$T^{3} + 6 T^{2} - 81 T + 166$$
$59$ $$T^{3} - 12 T^{2} + 21 T + 82$$
$61$ $$T^{3} - 18 T^{2} + 12 T + 552$$
$67$ $$T^{3} - 9 T^{2} - 12 T + 164$$
$71$ $$T^{3} - 36T + 32$$
$73$ $$T^{3} - 33 T^{2} + 336 T - 996$$
$79$ $$T^{3} - 27 T^{2} + 195 T - 353$$
$83$ $$T^{3} - 18 T^{2} - 15 T + 948$$
$89$ $$T^{3} + 12 T^{2} + 12 T - 112$$
$97$ $$T^{3} - 27T + 38$$