# Properties

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

# Learn more about

## 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: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ 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 $$\beta = 2\sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{3} + 2 \beta q^{5} + q^{9} +O(q^{10})$$ $$q - q^{3} + 2 \beta q^{5} + q^{9} -2 \beta q^{11} -2 \beta q^{15} + 2 \beta q^{17} -4 q^{19} -6 \beta q^{23} + 3 q^{25} - q^{27} -2 q^{29} + 2 \beta q^{33} + 6 q^{37} + 6 \beta q^{41} + 8 \beta q^{43} + 2 \beta q^{45} + 8 q^{47} -2 \beta q^{51} -6 q^{53} -8 q^{55} + 4 q^{57} -12 q^{59} -4 \beta q^{61} -4 \beta q^{67} + 6 \beta q^{69} -2 \beta q^{71} -4 \beta q^{73} -3 q^{75} -4 \beta q^{79} + q^{81} + 4 q^{83} + 8 q^{85} + 2 q^{87} -2 \beta q^{89} -8 \beta q^{95} -12 \beta q^{97} -2 \beta q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{3} + 2q^{9} + O(q^{10})$$ $$2q - 2q^{3} + 2q^{9} - 8q^{19} + 6q^{25} - 2q^{27} - 4q^{29} + 12q^{37} + 16q^{47} - 12q^{53} - 16q^{55} + 8q^{57} - 24q^{59} - 6q^{75} + 2q^{81} + 8q^{83} + 16q^{85} + 4q^{87} + O(q^{100})$$

## 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
 −1.41421 1.41421
0 −1.00000 0 −2.82843 0 0 0 1.00000 0
1.2 0 −1.00000 0 2.82843 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

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
28.d even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 9408.2.a.dk 2
4.b odd 2 1 9408.2.a.eb 2
7.b odd 2 1 9408.2.a.eb 2
8.b even 2 1 4704.2.a.br yes 2
8.d odd 2 1 4704.2.a.bi 2
28.d even 2 1 inner 9408.2.a.dk 2
56.e even 2 1 4704.2.a.br yes 2
56.h odd 2 1 4704.2.a.bi 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4704.2.a.bi 2 8.d odd 2 1
4704.2.a.bi 2 56.h odd 2 1
4704.2.a.br yes 2 8.b even 2 1
4704.2.a.br yes 2 56.e even 2 1
9408.2.a.dk 2 1.a even 1 1 trivial
9408.2.a.dk 2 28.d even 2 1 inner
9408.2.a.eb 2 4.b odd 2 1
9408.2.a.eb 2 7.b odd 2 1

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$1$$
$$7$$ $$-1$$

## Hecke kernels

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

 $$T_{5}^{2} - 8$$ $$T_{11}^{2} - 8$$ $$T_{13}$$ $$T_{17}^{2} - 8$$ $$T_{19} + 4$$ $$T_{31}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$( 1 + T )^{2}$$
$5$ $$1 + 2 T^{2} + 25 T^{4}$$
$7$ 1
$11$ $$1 + 14 T^{2} + 121 T^{4}$$
$13$ $$( 1 + 13 T^{2} )^{2}$$
$17$ $$1 + 26 T^{2} + 289 T^{4}$$
$19$ $$( 1 + 4 T + 19 T^{2} )^{2}$$
$23$ $$1 - 26 T^{2} + 529 T^{4}$$
$29$ $$( 1 + 2 T + 29 T^{2} )^{2}$$
$31$ $$( 1 + 31 T^{2} )^{2}$$
$37$ $$( 1 - 6 T + 37 T^{2} )^{2}$$
$41$ $$1 + 10 T^{2} + 1681 T^{4}$$
$43$ $$1 - 42 T^{2} + 1849 T^{4}$$
$47$ $$( 1 - 8 T + 47 T^{2} )^{2}$$
$53$ $$( 1 + 6 T + 53 T^{2} )^{2}$$
$59$ $$( 1 + 12 T + 59 T^{2} )^{2}$$
$61$ $$1 + 90 T^{2} + 3721 T^{4}$$
$67$ $$1 + 102 T^{2} + 4489 T^{4}$$
$71$ $$1 + 134 T^{2} + 5041 T^{4}$$
$73$ $$1 + 114 T^{2} + 5329 T^{4}$$
$79$ $$1 + 126 T^{2} + 6241 T^{4}$$
$83$ $$( 1 - 4 T + 83 T^{2} )^{2}$$
$89$ $$1 + 170 T^{2} + 7921 T^{4}$$
$97$ $$1 - 94 T^{2} + 9409 T^{4}$$
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