# Properties

 Label 9408.2.a.eg 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: $$x^{3} - 6 x - 3$$ 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.

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

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$2 \nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$$$/2$$ $$\nu^{2}$$ $$=$$ $$($$$$2 \beta_{2} + \beta_{1} + 8$$$$)/2$$

## 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
 −0.523976 2.66908 −2.14510
0 −1.00000 0 −3.20147 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 2.74657 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.eg 3
4.b odd 2 1 9408.2.a.ei 3
7.b odd 2 1 9408.2.a.ej 3
7.d odd 6 2 1344.2.q.y 6
8.b even 2 1 4704.2.a.bv 3
8.d odd 2 1 4704.2.a.bt 3
28.d even 2 1 9408.2.a.eh 3
28.f even 6 2 1344.2.q.z 6
56.e even 2 1 4704.2.a.bu 3
56.h odd 2 1 4704.2.a.bs 3
56.j odd 6 2 672.2.q.l yes 6
56.m even 6 2 672.2.q.k 6
168.ba even 6 2 2016.2.s.v 6
168.be odd 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.m even 6 2
672.2.q.l yes 6 56.j odd 6 2
1344.2.q.y 6 7.d odd 6 2
1344.2.q.z 6 28.f even 6 2
2016.2.s.u 6 168.be odd 6 2
2016.2.s.v 6 168.ba even 6 2
4704.2.a.bs 3 56.h odd 2 1
4704.2.a.bt 3 8.d odd 2 1
4704.2.a.bu 3 56.e even 2 1
4704.2.a.bv 3 8.b even 2 1
9408.2.a.eg 3 1.a even 1 1 trivial
9408.2.a.eh 3 28.d even 2 1
9408.2.a.ei 3 4.b odd 2 1
9408.2.a.ej 3 7.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(9408))$$:

 $$T_{5}^{3} - 9 T_{5} + 4$$ $$T_{11}^{3} - 27 T_{11} + 38$$ $$T_{13}^{3} + 3 T_{13}^{2} - 36 T_{13} - 112$$ $$T_{17}^{3} + 6 T_{17}^{2} - 24 T_{17} - 96$$ $$T_{19}^{3} + 3 T_{19}^{2} - 36 T_{19} - 112$$ $$T_{31}^{3} - 3 T_{31}^{2} - 21 T_{31} + 47$$

## Hecke characteristic polynomials

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