# Properties

 Label 9408.2.a.ek Level $9408$ Weight $2$ Character orbit 9408.a Self dual yes Analytic conductor $75.123$ Analytic rank $1$ Dimension $4$ 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: $$1$$ Dimension: $$4$$ Coefficient field: 4.4.2624.1 Defining polynomial: $$x^{4} - 2x^{3} - 3x^{2} + 2x + 1$$ x^4 - 2*x^3 - 3*x^2 + 2*x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 4704) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{3} + (\beta_{3} - 1) q^{5} + q^{9}+O(q^{10})$$ q - q^3 + (b3 - 1) * q^5 + q^9 $$q - q^{3} + (\beta_{3} - 1) q^{5} + q^{9} + (\beta_{2} - 1) q^{11} + ( - \beta_{3} - \beta_{2} - 2) q^{13} + ( - \beta_{3} + 1) q^{15} + ( - \beta_{2} + \beta_1 + 1) q^{17} + (\beta_{3} + \beta_{2} - \beta_1 - 2) q^{19} + (\beta_{2} + 3) q^{23} + ( - 2 \beta_{3} + 2 \beta_1 + 3) q^{25} - q^{27} + (\beta_{3} - \beta_{2} - \beta_1) q^{29} + ( - \beta_{3} - \beta_{2} - 3 \beta_1 - 2) q^{31} + ( - \beta_{2} + 1) q^{33} + 4 \beta_1 q^{37} + (\beta_{3} + \beta_{2} + 2) q^{39} + (2 \beta_{3} + \beta_{2} - 3 \beta_1 + 5) q^{41} + ( - 2 \beta_{3} - 2 \beta_1 + 2) q^{43} + (\beta_{3} - 1) q^{45} + (3 \beta_{3} + \beta_{2} + \beta_1 - 4) q^{47} + (\beta_{2} - \beta_1 - 1) q^{51} + (2 \beta_{3} - 2 \beta_1) q^{53} + ( - \beta_{3} - \beta_{2} + 5 \beta_1 - 2) q^{55} + ( - \beta_{3} - \beta_{2} + \beta_1 + 2) q^{57} + ( - 3 \beta_{3} - \beta_{2} + 3 \beta_1) q^{59} + ( - \beta_{3} + \beta_{2} - 4) q^{61} + ( - \beta_{3} + \beta_{2} - 7 \beta_1 - 2) q^{65} + ( - 2 \beta_{2} + 4 \beta_1 + 2) q^{67} + ( - \beta_{2} - 3) q^{69} + (\beta_{2} - 4 \beta_1 + 3) q^{71} + ( - 2 \beta_{3} + \beta_1 + 2) q^{73} + (2 \beta_{3} - 2 \beta_1 - 3) q^{75} + ( - 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 4) q^{79} + q^{81} + ( - 2 \beta_{3} - 2 \beta_{2} + 2 \beta_1) q^{83} + (2 \beta_{3} + 2 \beta_{2} - 6 \beta_1 + 2) q^{85} + ( - \beta_{3} + \beta_{2} + \beta_1) q^{87} + (\beta_{2} + 5 \beta_1 + 7) q^{89} + (\beta_{3} + \beta_{2} + 3 \beta_1 + 2) q^{93} + ( - 4 \beta_{3} - 2 \beta_{2} + 8 \beta_1 + 6) q^{95} + ( - 2 \beta_{3} - 3 \beta_1 + 10) q^{97} + (\beta_{2} - 1) q^{99}+O(q^{100})$$ q - q^3 + (b3 - 1) * q^5 + q^9 + (b2 - 1) * q^11 + (-b3 - b2 - 2) * q^13 + (-b3 + 1) * q^15 + (-b2 + b1 + 1) * q^17 + (b3 + b2 - b1 - 2) * q^19 + (b2 + 3) * q^23 + (-2*b3 + 2*b1 + 3) * q^25 - q^27 + (b3 - b2 - b1) * q^29 + (-b3 - b2 - 3*b1 - 2) * q^31 + (-b2 + 1) * q^33 + 4*b1 * q^37 + (b3 + b2 + 2) * q^39 + (2*b3 + b2 - 3*b1 + 5) * q^41 + (-2*b3 - 2*b1 + 2) * q^43 + (b3 - 1) * q^45 + (3*b3 + b2 + b1 - 4) * q^47 + (b2 - b1 - 1) * q^51 + (2*b3 - 2*b1) * q^53 + (-b3 - b2 + 5*b1 - 2) * q^55 + (-b3 - b2 + b1 + 2) * q^57 + (-3*b3 - b2 + 3*b1) * q^59 + (-b3 + b2 - 4) * q^61 + (-b3 + b2 - 7*b1 - 2) * q^65 + (-2*b2 + 4*b1 + 2) * q^67 + (-b2 - 3) * q^69 + (b2 - 4*b1 + 3) * q^71 + (-2*b3 + b1 + 2) * q^73 + (2*b3 - 2*b1 - 3) * q^75 + (-2*b3 - 2*b2 - 2*b1 + 4) * q^79 + q^81 + (-2*b3 - 2*b2 + 2*b1) * q^83 + (2*b3 + 2*b2 - 6*b1 + 2) * q^85 + (-b3 + b2 + b1) * q^87 + (b2 + 5*b1 + 7) * q^89 + (b3 + b2 + 3*b1 + 2) * q^93 + (-4*b3 - 2*b2 + 8*b1 + 6) * q^95 + (-2*b3 - 3*b1 + 10) * q^97 + (b2 - 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{3} - 4 q^{5} + 4 q^{9}+O(q^{10})$$ 4 * q - 4 * q^3 - 4 * q^5 + 4 * q^9 $$4 q - 4 q^{3} - 4 q^{5} + 4 q^{9} - 4 q^{11} - 8 q^{13} + 4 q^{15} + 4 q^{17} - 8 q^{19} + 12 q^{23} + 12 q^{25} - 4 q^{27} - 8 q^{31} + 4 q^{33} + 8 q^{39} + 20 q^{41} + 8 q^{43} - 4 q^{45} - 16 q^{47} - 4 q^{51} - 8 q^{55} + 8 q^{57} - 16 q^{61} - 8 q^{65} + 8 q^{67} - 12 q^{69} + 12 q^{71} + 8 q^{73} - 12 q^{75} + 16 q^{79} + 4 q^{81} + 8 q^{85} + 28 q^{89} + 8 q^{93} + 24 q^{95} + 40 q^{97} - 4 q^{99}+O(q^{100})$$ 4 * q - 4 * q^3 - 4 * q^5 + 4 * q^9 - 4 * q^11 - 8 * q^13 + 4 * q^15 + 4 * q^17 - 8 * q^19 + 12 * q^23 + 12 * q^25 - 4 * q^27 - 8 * q^31 + 4 * q^33 + 8 * q^39 + 20 * q^41 + 8 * q^43 - 4 * q^45 - 16 * q^47 - 4 * q^51 - 8 * q^55 + 8 * q^57 - 16 * q^61 - 8 * q^65 + 8 * q^67 - 12 * q^69 + 12 * q^71 + 8 * q^73 - 12 * q^75 + 16 * q^79 + 4 * q^81 + 8 * q^85 + 28 * q^89 + 8 * q^93 + 24 * q^95 + 40 * q^97 - 4 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2x^{3} - 3x^{2} + 2x + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu^{3} - 2\nu^{2} - 2\nu + 1$$ v^3 - 2*v^2 - 2*v + 1 $$\beta_{2}$$ $$=$$ $$2\nu^{2} - 4\nu - 3$$ 2*v^2 - 4*v - 3 $$\beta_{3}$$ $$=$$ $$-\nu^{3} + 2\nu^{2} + 4\nu - 2$$ -v^3 + 2*v^2 + 4*v - 2
 $$\nu$$ $$=$$ $$( \beta_{3} + \beta _1 + 1 ) / 2$$ (b3 + b1 + 1) / 2 $$\nu^{2}$$ $$=$$ $$( 2\beta_{3} + \beta_{2} + 2\beta _1 + 5 ) / 2$$ (2*b3 + b2 + 2*b1 + 5) / 2 $$\nu^{3}$$ $$=$$ $$3\beta_{3} + \beta_{2} + 4\beta _1 + 5$$ 3*b3 + b2 + 4*b1 + 5

## 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
 −0.360409 −1.22833 0.814115 2.77462
0 −1.00000 0 −4.13503 0 0 0 1.00000 0
1.2 0 −1.00000 0 −3.04244 0 0 0 1.00000 0
1.3 0 −1.00000 0 1.04244 0 0 0 1.00000 0
1.4 0 −1.00000 0 2.13503 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.ek 4
4.b odd 2 1 9408.2.a.em 4
7.b odd 2 1 9408.2.a.en 4
8.b even 2 1 4704.2.a.bz yes 4
8.d odd 2 1 4704.2.a.bx yes 4
28.d even 2 1 9408.2.a.el 4
56.e even 2 1 4704.2.a.by yes 4
56.h odd 2 1 4704.2.a.bw 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4704.2.a.bw 4 56.h odd 2 1
4704.2.a.bx yes 4 8.d odd 2 1
4704.2.a.by yes 4 56.e even 2 1
4704.2.a.bz yes 4 8.b even 2 1
9408.2.a.ek 4 1.a even 1 1 trivial
9408.2.a.el 4 28.d even 2 1
9408.2.a.em 4 4.b odd 2 1
9408.2.a.en 4 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(9408))$$:

 $$T_{5}^{4} + 4T_{5}^{3} - 8T_{5}^{2} - 24T_{5} + 28$$ T5^4 + 4*T5^3 - 8*T5^2 - 24*T5 + 28 $$T_{11}^{4} + 4T_{11}^{3} - 20T_{11}^{2} - 48T_{11} + 16$$ T11^4 + 4*T11^3 - 20*T11^2 - 48*T11 + 16 $$T_{13}^{4} + 8T_{13}^{3} - 4T_{13}^{2} - 80T_{13} + 68$$ T13^4 + 8*T13^3 - 4*T13^2 - 80*T13 + 68 $$T_{17}^{4} - 4T_{17}^{3} - 24T_{17}^{2} + 120T_{17} - 100$$ T17^4 - 4*T17^3 - 24*T17^2 + 120*T17 - 100 $$T_{19}^{4} + 8T_{19}^{3} - 8T_{19}^{2} - 64T_{19} + 64$$ T19^4 + 8*T19^3 - 8*T19^2 - 64*T19 + 64 $$T_{31}^{4} + 8T_{31}^{3} - 40T_{31}^{2} - 128T_{31} - 64$$ T31^4 + 8*T31^3 - 40*T31^2 - 128*T31 - 64

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$(T + 1)^{4}$$
$5$ $$T^{4} + 4 T^{3} - 8 T^{2} - 24 T + 28$$
$7$ $$T^{4}$$
$11$ $$T^{4} + 4 T^{3} - 20 T^{2} - 48 T + 16$$
$13$ $$T^{4} + 8 T^{3} - 4 T^{2} - 80 T + 68$$
$17$ $$T^{4} - 4 T^{3} - 24 T^{2} + 120 T - 100$$
$19$ $$T^{4} + 8 T^{3} - 8 T^{2} - 64 T + 64$$
$23$ $$T^{4} - 12 T^{3} + 28 T^{2} + \cdots - 112$$
$29$ $$T^{4} - 56 T^{2} - 128 T + 64$$
$31$ $$T^{4} + 8 T^{3} - 40 T^{2} - 128 T - 64$$
$37$ $$(T^{2} - 32)^{2}$$
$41$ $$T^{4} - 20 T^{3} + 56 T^{2} + \cdots - 4804$$
$43$ $$T^{4} - 8 T^{3} - 48 T^{2} + 384 T - 256$$
$47$ $$T^{4} + 16 T^{3} - 24 T^{2} + \cdots - 3008$$
$53$ $$T^{4} - 72 T^{2} + 128 T + 272$$
$59$ $$T^{4} - 152 T^{2} - 960 T - 1600$$
$61$ $$T^{4} + 16 T^{3} + 44 T^{2} + \cdots - 412$$
$67$ $$T^{4} - 8 T^{3} - 144 T^{2} + \cdots - 4352$$
$71$ $$T^{4} - 12 T^{3} - 36 T^{2} + \cdots + 272$$
$73$ $$T^{4} - 8 T^{3} - 36 T^{2} + 144 T + 452$$
$79$ $$T^{4} - 16 T^{3} - 32 T^{2} + \cdots - 1024$$
$83$ $$T^{4} - 128 T^{2} - 256 T + 1792$$
$89$ $$T^{4} - 28 T^{3} + 168 T^{2} + \cdots - 4772$$
$97$ $$T^{4} - 40 T^{3} + 508 T^{2} + \cdots - 1148$$