# Properties

 Label 9408.2.a.do Level 9408 Weight 2 Character orbit 9408.a Self dual yes Analytic conductor 75.123 Analytic rank 1 Dimension 2 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: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ Defining polynomial: $$x^{2} - 3$$ 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 $$\beta = 2\sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{3} + 2 \beta q^{5} + q^{9} +O(q^{10})$$ $$q - q^{3} + 2 \beta q^{5} + q^{9} + ( 2 - 2 \beta ) q^{11} + 2 q^{13} -2 \beta q^{15} + ( -4 + 2 \beta ) q^{17} -4 \beta q^{19} + ( -2 + 2 \beta ) q^{23} + 7 q^{25} - q^{27} + ( -2 + 4 \beta ) q^{29} + ( -4 - 4 \beta ) q^{31} + ( -2 + 2 \beta ) q^{33} + 2 q^{37} -2 q^{39} + ( -8 - 2 \beta ) q^{41} -8 q^{43} + 2 \beta q^{45} + ( 4 + 4 \beta ) q^{47} + ( 4 - 2 \beta ) q^{51} + 2 q^{53} + ( -12 + 4 \beta ) q^{55} + 4 \beta q^{57} + ( 8 - 4 \beta ) q^{59} + ( -2 - 4 \beta ) q^{61} + 4 \beta q^{65} + ( 4 - 4 \beta ) q^{67} + ( 2 - 2 \beta ) q^{69} + ( -6 - 2 \beta ) q^{71} + ( -6 - 4 \beta ) q^{73} -7 q^{75} + ( 4 + 4 \beta ) q^{79} + q^{81} + 4 q^{83} + ( 12 - 8 \beta ) q^{85} + ( 2 - 4 \beta ) q^{87} -2 \beta q^{89} + ( 4 + 4 \beta ) q^{93} -24 q^{95} + ( 2 + 4 \beta ) q^{97} + ( 2 - 2 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{3} + 2q^{9} + O(q^{10})$$ $$2q - 2q^{3} + 2q^{9} + 4q^{11} + 4q^{13} - 8q^{17} - 4q^{23} + 14q^{25} - 2q^{27} - 4q^{29} - 8q^{31} - 4q^{33} + 4q^{37} - 4q^{39} - 16q^{41} - 16q^{43} + 8q^{47} + 8q^{51} + 4q^{53} - 24q^{55} + 16q^{59} - 4q^{61} + 8q^{67} + 4q^{69} - 12q^{71} - 12q^{73} - 14q^{75} + 8q^{79} + 2q^{81} + 8q^{83} + 24q^{85} + 4q^{87} + 8q^{93} - 48q^{95} + 4q^{97} + 4q^{99} + O(q^{100})$$

## 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
 −1.73205 1.73205
0 −1.00000 0 −3.46410 0 0 0 1.00000 0
1.2 0 −1.00000 0 3.46410 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

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.do 2
4.b odd 2 1 9408.2.a.dx 2
7.b odd 2 1 1344.2.a.v 2
8.b even 2 1 4704.2.a.bn 2
8.d odd 2 1 4704.2.a.bm 2
21.c even 2 1 4032.2.a.bs 2
28.d even 2 1 1344.2.a.u 2
56.e even 2 1 672.2.a.j yes 2
56.h odd 2 1 672.2.a.i 2
84.h odd 2 1 4032.2.a.br 2
112.j even 4 2 5376.2.c.bn 4
112.l odd 4 2 5376.2.c.bh 4
168.e odd 2 1 2016.2.a.s 2
168.i even 2 1 2016.2.a.t 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
672.2.a.i 2 56.h odd 2 1
672.2.a.j yes 2 56.e even 2 1
1344.2.a.u 2 28.d even 2 1
1344.2.a.v 2 7.b odd 2 1
2016.2.a.s 2 168.e odd 2 1
2016.2.a.t 2 168.i even 2 1
4032.2.a.br 2 84.h odd 2 1
4032.2.a.bs 2 21.c even 2 1
4704.2.a.bm 2 8.d odd 2 1
4704.2.a.bn 2 8.b even 2 1
5376.2.c.bh 4 112.l odd 4 2
5376.2.c.bn 4 112.j even 4 2
9408.2.a.do 2 1.a even 1 1 trivial
9408.2.a.dx 2 4.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(9408))$$:

 $$T_{5}^{2} - 12$$ $$T_{11}^{2} - 4 T_{11} - 8$$ $$T_{13} - 2$$ $$T_{17}^{2} + 8 T_{17} + 4$$ $$T_{19}^{2} - 48$$ $$T_{31}^{2} + 8 T_{31} - 32$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$( 1 + T )^{2}$$
$5$ $$1 - 2 T^{2} + 25 T^{4}$$
$7$ 1
$11$ $$1 - 4 T + 14 T^{2} - 44 T^{3} + 121 T^{4}$$
$13$ $$( 1 - 2 T + 13 T^{2} )^{2}$$
$17$ $$1 + 8 T + 38 T^{2} + 136 T^{3} + 289 T^{4}$$
$19$ $$1 - 10 T^{2} + 361 T^{4}$$
$23$ $$1 + 4 T + 38 T^{2} + 92 T^{3} + 529 T^{4}$$
$29$ $$1 + 4 T + 14 T^{2} + 116 T^{3} + 841 T^{4}$$
$31$ $$1 + 8 T + 30 T^{2} + 248 T^{3} + 961 T^{4}$$
$37$ $$( 1 - 2 T + 37 T^{2} )^{2}$$
$41$ $$1 + 16 T + 134 T^{2} + 656 T^{3} + 1681 T^{4}$$
$43$ $$( 1 + 8 T + 43 T^{2} )^{2}$$
$47$ $$1 - 8 T + 62 T^{2} - 376 T^{3} + 2209 T^{4}$$
$53$ $$( 1 - 2 T + 53 T^{2} )^{2}$$
$59$ $$1 - 16 T + 134 T^{2} - 944 T^{3} + 3481 T^{4}$$
$61$ $$1 + 4 T + 78 T^{2} + 244 T^{3} + 3721 T^{4}$$
$67$ $$1 - 8 T + 102 T^{2} - 536 T^{3} + 4489 T^{4}$$
$71$ $$1 + 12 T + 166 T^{2} + 852 T^{3} + 5041 T^{4}$$
$73$ $$1 + 12 T + 134 T^{2} + 876 T^{3} + 5329 T^{4}$$
$79$ $$1 - 8 T + 126 T^{2} - 632 T^{3} + 6241 T^{4}$$
$83$ $$( 1 - 4 T + 83 T^{2} )^{2}$$
$89$ $$1 + 166 T^{2} + 7921 T^{4}$$
$97$ $$1 - 4 T + 150 T^{2} - 388 T^{3} + 9409 T^{4}$$