# Properties

 Label 9408.2.a.ef Level $9408$ Weight $2$ Character orbit 9408.a Self dual yes Analytic conductor $75.123$ Analytic rank $0$ 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: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 147) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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
 −1.41421 1.41421
0 1.00000 0 0.585786 0 0 0 1.00000 0
1.2 0 1.00000 0 3.41421 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.ef 2
4.b odd 2 1 9408.2.a.dq 2
7.b odd 2 1 9408.2.a.di 2
8.b even 2 1 147.2.a.d 2
8.d odd 2 1 2352.2.a.be 2
24.f even 2 1 7056.2.a.cv 2
24.h odd 2 1 441.2.a.j 2
28.d even 2 1 9408.2.a.dt 2
40.f even 2 1 3675.2.a.bf 2
56.e even 2 1 2352.2.a.bc 2
56.h odd 2 1 147.2.a.e yes 2
56.j odd 6 2 147.2.e.d 4
56.k odd 6 2 2352.2.q.bb 4
56.m even 6 2 2352.2.q.bd 4
56.p even 6 2 147.2.e.e 4
168.e odd 2 1 7056.2.a.cf 2
168.i even 2 1 441.2.a.i 2
168.s odd 6 2 441.2.e.f 4
168.ba even 6 2 441.2.e.g 4
280.c odd 2 1 3675.2.a.bd 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
147.2.a.d 2 8.b even 2 1
147.2.a.e yes 2 56.h odd 2 1
147.2.e.d 4 56.j odd 6 2
147.2.e.e 4 56.p even 6 2
441.2.a.i 2 168.i even 2 1
441.2.a.j 2 24.h odd 2 1
441.2.e.f 4 168.s odd 6 2
441.2.e.g 4 168.ba even 6 2
2352.2.a.bc 2 56.e even 2 1
2352.2.a.be 2 8.d odd 2 1
2352.2.q.bb 4 56.k odd 6 2
2352.2.q.bd 4 56.m even 6 2
3675.2.a.bd 2 280.c odd 2 1
3675.2.a.bf 2 40.f even 2 1
7056.2.a.cf 2 168.e odd 2 1
7056.2.a.cv 2 24.f even 2 1
9408.2.a.di 2 7.b odd 2 1
9408.2.a.dq 2 4.b odd 2 1
9408.2.a.dt 2 28.d even 2 1
9408.2.a.ef 2 1.a even 1 1 trivial

## Hecke kernels

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

 $$T_{5}^{2} - 4 T_{5} + 2$$ $$T_{11} - 2$$ $$T_{13}^{2} - 8 T_{13} + 14$$ $$T_{17}^{2} + 4 T_{17} - 14$$ $$T_{19}^{2} - 8$$ $$T_{31}^{2} - 8 T_{31} + 8$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$( 1 - T )^{2}$$
$5$ $$1 - 4 T + 12 T^{2} - 20 T^{3} + 25 T^{4}$$
$7$ 1
$11$ $$( 1 - 2 T + 11 T^{2} )^{2}$$
$13$ $$1 - 8 T + 40 T^{2} - 104 T^{3} + 169 T^{4}$$
$17$ $$1 + 4 T + 20 T^{2} + 68 T^{3} + 289 T^{4}$$
$19$ $$1 + 30 T^{2} + 361 T^{4}$$
$23$ $$1 + 4 T + 18 T^{2} + 92 T^{3} + 529 T^{4}$$
$29$ $$1 - 8 T + 66 T^{2} - 232 T^{3} + 841 T^{4}$$
$31$ $$1 - 8 T + 70 T^{2} - 248 T^{3} + 961 T^{4}$$
$37$ $$( 1 - 4 T + 37 T^{2} )^{2}$$
$41$ $$1 + 4 T + 68 T^{2} + 164 T^{3} + 1681 T^{4}$$
$43$ $$1 + 54 T^{2} + 1849 T^{4}$$
$47$ $$1 + 86 T^{2} + 2209 T^{4}$$
$53$ $$( 1 - 2 T + 53 T^{2} )^{2}$$
$59$ $$1 + 8 T + 126 T^{2} + 472 T^{3} + 3481 T^{4}$$
$61$ $$1 - 16 T + 168 T^{2} - 976 T^{3} + 3721 T^{4}$$
$67$ $$1 + 102 T^{2} + 4489 T^{4}$$
$71$ $$1 + 4 T + 18 T^{2} + 284 T^{3} + 5041 T^{4}$$
$73$ $$1 + 8 T + 64 T^{2} + 584 T^{3} + 5329 T^{4}$$
$79$ $$1 - 16 T + 190 T^{2} - 1264 T^{3} + 6241 T^{4}$$
$83$ $$1 + 8 T + 54 T^{2} + 664 T^{3} + 6889 T^{4}$$
$89$ $$1 - 20 T + 260 T^{2} - 1780 T^{3} + 7921 T^{4}$$
$97$ $$1 + 8 T + 208 T^{2} + 776 T^{3} + 9409 T^{4}$$