# Properties

 Label 9408.2.a.dw 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: $$\Q(\sqrt{57})$$ Defining polynomial: $$x^{2} - x - 14$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 168) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{57})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{3} -\beta q^{5} + q^{9} +O(q^{10})$$ $$q + q^{3} -\beta q^{5} + q^{9} -\beta q^{11} + ( 3 - \beta ) q^{13} -\beta q^{15} + 4 q^{17} + ( 3 - \beta ) q^{19} -4 q^{23} + ( 9 + \beta ) q^{25} + q^{27} + ( -2 + \beta ) q^{29} - q^{31} -\beta q^{33} + ( -1 - \beta ) q^{37} + ( 3 - \beta ) q^{39} + ( -2 - 2 \beta ) q^{41} + ( -3 - \beta ) q^{43} -\beta q^{45} -6 q^{47} + 4 q^{51} + ( -6 + \beta ) q^{53} + ( 14 + \beta ) q^{55} + ( 3 - \beta ) q^{57} + ( -2 - \beta ) q^{59} + 10 q^{61} + ( 14 - 2 \beta ) q^{65} + ( 3 + \beta ) q^{67} -4 q^{69} -2 q^{71} + ( 1 - \beta ) q^{73} + ( 9 + \beta ) q^{75} + ( -5 + 2 \beta ) q^{79} + q^{81} + ( -4 + \beta ) q^{83} -4 \beta q^{85} + ( -2 + \beta ) q^{87} + ( 4 - 2 \beta ) q^{89} - q^{93} + ( 14 - 2 \beta ) q^{95} + ( -12 - \beta ) q^{97} -\beta q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{3} - q^{5} + 2q^{9} + O(q^{10})$$ $$2q + 2q^{3} - q^{5} + 2q^{9} - q^{11} + 5q^{13} - q^{15} + 8q^{17} + 5q^{19} - 8q^{23} + 19q^{25} + 2q^{27} - 3q^{29} - 2q^{31} - q^{33} - 3q^{37} + 5q^{39} - 6q^{41} - 7q^{43} - q^{45} - 12q^{47} + 8q^{51} - 11q^{53} + 29q^{55} + 5q^{57} - 5q^{59} + 20q^{61} + 26q^{65} + 7q^{67} - 8q^{69} - 4q^{71} + q^{73} + 19q^{75} - 8q^{79} + 2q^{81} - 7q^{83} - 4q^{85} - 3q^{87} + 6q^{89} - 2q^{93} + 26q^{95} - 25q^{97} - q^{99} + O(q^{100})$$

## 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
 4.27492 −3.27492
0 1.00000 0 −4.27492 0 0 0 1.00000 0
1.2 0 1.00000 0 3.27492 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.dw 2
4.b odd 2 1 9408.2.a.dj 2
7.b odd 2 1 9408.2.a.dp 2
7.d odd 6 2 1344.2.q.x 4
8.b even 2 1 2352.2.a.ba 2
8.d odd 2 1 1176.2.a.n 2
24.f even 2 1 3528.2.a.bd 2
24.h odd 2 1 7056.2.a.ch 2
28.d even 2 1 9408.2.a.ec 2
28.f even 6 2 1344.2.q.w 4
56.e even 2 1 1176.2.a.k 2
56.h odd 2 1 2352.2.a.bf 2
56.j odd 6 2 336.2.q.g 4
56.k odd 6 2 1176.2.q.l 4
56.m even 6 2 168.2.q.c 4
56.p even 6 2 2352.2.q.bf 4
168.e odd 2 1 3528.2.a.bk 2
168.i even 2 1 7056.2.a.cu 2
168.v even 6 2 3528.2.s.bk 4
168.ba even 6 2 1008.2.s.r 4
168.be odd 6 2 504.2.s.i 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.2.q.c 4 56.m even 6 2
336.2.q.g 4 56.j odd 6 2
504.2.s.i 4 168.be odd 6 2
1008.2.s.r 4 168.ba even 6 2
1176.2.a.k 2 56.e even 2 1
1176.2.a.n 2 8.d odd 2 1
1176.2.q.l 4 56.k odd 6 2
1344.2.q.w 4 28.f even 6 2
1344.2.q.x 4 7.d odd 6 2
2352.2.a.ba 2 8.b even 2 1
2352.2.a.bf 2 56.h odd 2 1
2352.2.q.bf 4 56.p even 6 2
3528.2.a.bd 2 24.f even 2 1
3528.2.a.bk 2 168.e odd 2 1
3528.2.s.bk 4 168.v even 6 2
7056.2.a.ch 2 24.h odd 2 1
7056.2.a.cu 2 168.i even 2 1
9408.2.a.dj 2 4.b odd 2 1
9408.2.a.dp 2 7.b odd 2 1
9408.2.a.dw 2 1.a even 1 1 trivial
9408.2.a.ec 2 28.d even 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(9408))$$:

 $$T_{5}^{2} + T_{5} - 14$$ $$T_{11}^{2} + T_{11} - 14$$ $$T_{13}^{2} - 5 T_{13} - 8$$ $$T_{17} - 4$$ $$T_{19}^{2} - 5 T_{19} - 8$$ $$T_{31} + 1$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$( 1 - T )^{2}$$
$5$ $$1 + T - 4 T^{2} + 5 T^{3} + 25 T^{4}$$
$7$ 1
$11$ $$1 + T + 8 T^{2} + 11 T^{3} + 121 T^{4}$$
$13$ $$1 - 5 T + 18 T^{2} - 65 T^{3} + 169 T^{4}$$
$17$ $$( 1 - 4 T + 17 T^{2} )^{2}$$
$19$ $$1 - 5 T + 30 T^{2} - 95 T^{3} + 361 T^{4}$$
$23$ $$( 1 + 4 T + 23 T^{2} )^{2}$$
$29$ $$1 + 3 T + 46 T^{2} + 87 T^{3} + 841 T^{4}$$
$31$ $$( 1 + T + 31 T^{2} )^{2}$$
$37$ $$1 + 3 T + 62 T^{2} + 111 T^{3} + 1369 T^{4}$$
$41$ $$1 + 6 T + 34 T^{2} + 246 T^{3} + 1681 T^{4}$$
$43$ $$1 + 7 T + 84 T^{2} + 301 T^{3} + 1849 T^{4}$$
$47$ $$( 1 + 6 T + 47 T^{2} )^{2}$$
$53$ $$1 + 11 T + 122 T^{2} + 583 T^{3} + 2809 T^{4}$$
$59$ $$1 + 5 T + 110 T^{2} + 295 T^{3} + 3481 T^{4}$$
$61$ $$( 1 - 10 T + 61 T^{2} )^{2}$$
$67$ $$1 - 7 T + 132 T^{2} - 469 T^{3} + 4489 T^{4}$$
$71$ $$( 1 + 2 T + 71 T^{2} )^{2}$$
$73$ $$1 - T + 132 T^{2} - 73 T^{3} + 5329 T^{4}$$
$79$ $$1 + 8 T + 117 T^{2} + 632 T^{3} + 6241 T^{4}$$
$83$ $$1 + 7 T + 164 T^{2} + 581 T^{3} + 6889 T^{4}$$
$89$ $$1 - 6 T + 130 T^{2} - 534 T^{3} + 7921 T^{4}$$
$97$ $$1 + 25 T + 336 T^{2} + 2425 T^{3} + 9409 T^{4}$$