# Properties

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

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## 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: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 672) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{3} - 3q^{5} + q^{9} + O(q^{10})$$ $$q + q^{3} - 3q^{5} + q^{9} - q^{11} + 4q^{13} - 3q^{15} + 4q^{17} - 8q^{23} + 4q^{25} + q^{27} + 7q^{29} - 11q^{31} - q^{33} - 4q^{37} + 4q^{39} - 4q^{41} - 2q^{43} - 3q^{45} + 2q^{47} + 4q^{51} + 11q^{53} + 3q^{55} + 7q^{59} - 10q^{61} - 12q^{65} + 10q^{67} - 8q^{69} - 6q^{71} - 6q^{73} + 4q^{75} - 11q^{79} + q^{81} + 11q^{83} - 12q^{85} + 7q^{87} + 6q^{89} - 11q^{93} + 7q^{97} - q^{99} + O(q^{100})$$

## 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
 0
0 1.00000 0 −3.00000 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.bt 1
4.b odd 2 1 9408.2.a.e 1
7.b odd 2 1 9408.2.a.bl 1
7.c even 3 2 1344.2.q.k 2
8.b even 2 1 4704.2.a.o 1
8.d odd 2 1 4704.2.a.bf 1
28.d even 2 1 9408.2.a.dc 1
28.g odd 6 2 1344.2.q.u 2
56.e even 2 1 4704.2.a.b 1
56.h odd 2 1 4704.2.a.s 1
56.k odd 6 2 672.2.q.a 2
56.p even 6 2 672.2.q.f yes 2
168.s odd 6 2 2016.2.s.k 2
168.v even 6 2 2016.2.s.n 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
672.2.q.a 2 56.k odd 6 2
672.2.q.f yes 2 56.p even 6 2
1344.2.q.k 2 7.c even 3 2
1344.2.q.u 2 28.g odd 6 2
2016.2.s.k 2 168.s odd 6 2
2016.2.s.n 2 168.v even 6 2
4704.2.a.b 1 56.e even 2 1
4704.2.a.o 1 8.b even 2 1
4704.2.a.s 1 56.h odd 2 1
4704.2.a.bf 1 8.d odd 2 1
9408.2.a.e 1 4.b odd 2 1
9408.2.a.bl 1 7.b odd 2 1
9408.2.a.bt 1 1.a even 1 1 trivial
9408.2.a.dc 1 28.d even 2 1

## Hecke kernels

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

 $$T_{5} + 3$$ $$T_{11} + 1$$ $$T_{13} - 4$$ $$T_{17} - 4$$ $$T_{19}$$ $$T_{31} + 11$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 - T$$
$5$ $$1 + 3 T + 5 T^{2}$$
$7$ 1
$11$ $$1 + T + 11 T^{2}$$
$13$ $$1 - 4 T + 13 T^{2}$$
$17$ $$1 - 4 T + 17 T^{2}$$
$19$ $$1 + 19 T^{2}$$
$23$ $$1 + 8 T + 23 T^{2}$$
$29$ $$1 - 7 T + 29 T^{2}$$
$31$ $$1 + 11 T + 31 T^{2}$$
$37$ $$1 + 4 T + 37 T^{2}$$
$41$ $$1 + 4 T + 41 T^{2}$$
$43$ $$1 + 2 T + 43 T^{2}$$
$47$ $$1 - 2 T + 47 T^{2}$$
$53$ $$1 - 11 T + 53 T^{2}$$
$59$ $$1 - 7 T + 59 T^{2}$$
$61$ $$1 + 10 T + 61 T^{2}$$
$67$ $$1 - 10 T + 67 T^{2}$$
$71$ $$1 + 6 T + 71 T^{2}$$
$73$ $$1 + 6 T + 73 T^{2}$$
$79$ $$1 + 11 T + 79 T^{2}$$
$83$ $$1 - 11 T + 83 T^{2}$$
$89$ $$1 - 6 T + 89 T^{2}$$
$97$ $$1 - 7 T + 97 T^{2}$$
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