# Properties

 Label 9072.2.a.by Level $9072$ Weight $2$ Character orbit 9072.a Self dual yes Analytic conductor $72.440$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$9072 = 2^{4} \cdot 3^{4} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9072.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$72.4402847137$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.321.1 Defining polynomial: $$x^{3} - x^{2} - 4 x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 252) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{5} + q^{7} +O(q^{10})$$ $$q + \beta_{1} q^{5} + q^{7} + ( -1 - \beta_{1} - 2 \beta_{2} ) q^{11} + ( 1 + 2 \beta_{1} + 2 \beta_{2} ) q^{13} + ( 1 - 2 \beta_{1} - \beta_{2} ) q^{17} + ( -2 \beta_{1} + \beta_{2} ) q^{19} + ( -4 - \beta_{1} + \beta_{2} ) q^{23} + ( -2 + \beta_{1} + \beta_{2} ) q^{25} + ( 1 - 3 \beta_{1} - \beta_{2} ) q^{29} + 3 \beta_{1} q^{31} + \beta_{1} q^{35} + ( -2 + \beta_{1} - 2 \beta_{2} ) q^{37} + ( -1 + \beta_{1} - 2 \beta_{2} ) q^{41} + ( \beta_{1} + 4 \beta_{2} ) q^{43} + ( -8 + 2 \beta_{1} - \beta_{2} ) q^{47} + q^{49} + ( 2 + \beta_{1} + \beta_{2} ) q^{53} + ( -1 - 4 \beta_{1} - \beta_{2} ) q^{55} + ( -11 + \beta_{1} - \beta_{2} ) q^{59} + ( 4 - 7 \beta_{1} - \beta_{2} ) q^{61} + ( 4 + 5 \beta_{1} + 2 \beta_{2} ) q^{65} + ( -3 + 2 \beta_{1} - \beta_{2} ) q^{67} + ( -3 - 5 \beta_{1} + 3 \beta_{2} ) q^{71} + ( -2 \beta_{1} + \beta_{2} ) q^{73} + ( -1 - \beta_{1} - 2 \beta_{2} ) q^{77} + ( 4 - 4 \beta_{1} - \beta_{2} ) q^{79} + ( -7 + 2 \beta_{1} + \beta_{2} ) q^{83} + ( -5 - 2 \beta_{1} - 2 \beta_{2} ) q^{85} + ( 5 + \beta_{1} + 4 \beta_{2} ) q^{89} + ( 1 + 2 \beta_{1} + 2 \beta_{2} ) q^{91} + ( -7 - \beta_{1} - 2 \beta_{2} ) q^{95} + ( -1 - \beta_{1} + 5 \beta_{2} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q + q^{5} + 3q^{7} + O(q^{10})$$ $$3q + q^{5} + 3q^{7} - 2q^{11} + 3q^{13} + 2q^{17} - 3q^{19} - 14q^{23} - 6q^{25} + q^{29} + 3q^{31} + q^{35} - 3q^{37} - 3q^{43} - 21q^{47} + 3q^{49} + 6q^{53} - 6q^{55} - 31q^{59} + 6q^{61} + 15q^{65} - 6q^{67} - 17q^{71} - 3q^{73} - 2q^{77} + 9q^{79} - 20q^{83} - 15q^{85} + 12q^{89} + 3q^{91} - 20q^{95} - 9q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 4 x + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta_{1} + 3$$

## 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
 −1.69963 0.239123 2.46050
0 0 0 −1.69963 0 1.00000 0 0 0
1.2 0 0 0 0.239123 0 1.00000 0 0 0
1.3 0 0 0 2.46050 0 1.00000 0 0 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 9072.2.a.by 3
3.b odd 2 1 9072.2.a.bv 3
4.b odd 2 1 2268.2.a.i 3
9.c even 3 2 1008.2.r.j 6
9.d odd 6 2 3024.2.r.j 6
12.b even 2 1 2268.2.a.h 3
36.f odd 6 2 252.2.j.a 6
36.h even 6 2 756.2.j.b 6
252.n even 6 2 1764.2.l.f 6
252.o even 6 2 5292.2.l.e 6
252.r odd 6 2 5292.2.i.e 6
252.s odd 6 2 5292.2.j.d 6
252.u odd 6 2 1764.2.i.g 6
252.bb even 6 2 5292.2.i.f 6
252.bi even 6 2 1764.2.j.e 6
252.bj even 6 2 1764.2.i.d 6
252.bl odd 6 2 1764.2.l.e 6
252.bn odd 6 2 5292.2.l.f 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.2.j.a 6 36.f odd 6 2
756.2.j.b 6 36.h even 6 2
1008.2.r.j 6 9.c even 3 2
1764.2.i.d 6 252.bj even 6 2
1764.2.i.g 6 252.u odd 6 2
1764.2.j.e 6 252.bi even 6 2
1764.2.l.e 6 252.bl odd 6 2
1764.2.l.f 6 252.n even 6 2
2268.2.a.h 3 12.b even 2 1
2268.2.a.i 3 4.b odd 2 1
3024.2.r.j 6 9.d odd 6 2
5292.2.i.e 6 252.r odd 6 2
5292.2.i.f 6 252.bb even 6 2
5292.2.j.d 6 252.s odd 6 2
5292.2.l.e 6 252.o even 6 2
5292.2.l.f 6 252.bn odd 6 2
9072.2.a.bv 3 3.b odd 2 1
9072.2.a.by 3 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(9072))$$:

 $$T_{5}^{3} - T_{5}^{2} - 4 T_{5} + 1$$ $$T_{11}^{3} + 2 T_{11}^{2} - 25 T_{11} - 59$$ $$T_{13}^{3} - 3 T_{13}^{2} - 33 T_{13} + 27$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$1 - T + 11 T^{2} - 9 T^{3} + 55 T^{4} - 25 T^{5} + 125 T^{6}$$
$7$ $$( 1 - T )^{3}$$
$11$ $$1 + 2 T + 8 T^{2} - 15 T^{3} + 88 T^{4} + 242 T^{5} + 1331 T^{6}$$
$13$ $$1 - 3 T + 6 T^{2} - 51 T^{3} + 78 T^{4} - 507 T^{5} + 2197 T^{6}$$
$17$ $$1 - 2 T + 32 T^{2} - 21 T^{3} + 544 T^{4} - 578 T^{5} + 4913 T^{6}$$
$19$ $$1 + 3 T + 33 T^{2} + 35 T^{3} + 627 T^{4} + 1083 T^{5} + 6859 T^{6}$$
$23$ $$1 + 14 T + 122 T^{2} + 675 T^{3} + 2806 T^{4} + 7406 T^{5} + 12167 T^{6}$$
$29$ $$1 - T + 47 T^{2} + 51 T^{3} + 1363 T^{4} - 841 T^{5} + 24389 T^{6}$$
$31$ $$1 - 3 T + 57 T^{2} - 159 T^{3} + 1767 T^{4} - 2883 T^{5} + 29791 T^{6}$$
$37$ $$1 + 3 T + 81 T^{2} + 199 T^{3} + 2997 T^{4} + 4107 T^{5} + 50653 T^{6}$$
$41$ $$1 + 90 T^{2} + 9 T^{3} + 3690 T^{4} + 68921 T^{6}$$
$43$ $$1 + 3 T + 33 T^{2} + 539 T^{3} + 1419 T^{4} + 5547 T^{5} + 79507 T^{6}$$
$47$ $$1 + 21 T + 261 T^{2} + 2181 T^{3} + 12267 T^{4} + 46389 T^{5} + 103823 T^{6}$$
$53$ $$1 - 6 T + 162 T^{2} - 627 T^{3} + 8586 T^{4} - 16854 T^{5} + 148877 T^{6}$$
$59$ $$1 + 31 T + 485 T^{2} + 4647 T^{3} + 28615 T^{4} + 107911 T^{5} + 205379 T^{6}$$
$61$ $$1 - 6 T - 12 T^{2} + 357 T^{3} - 732 T^{4} - 22326 T^{5} + 226981 T^{6}$$
$67$ $$1 + 6 T + 186 T^{2} + 811 T^{3} + 12462 T^{4} + 26934 T^{5} + 300763 T^{6}$$
$71$ $$1 + 17 T + 119 T^{2} + 507 T^{3} + 8449 T^{4} + 85697 T^{5} + 357911 T^{6}$$
$73$ $$1 + 3 T + 195 T^{2} + 359 T^{3} + 14235 T^{4} + 15987 T^{5} + 389017 T^{6}$$
$79$ $$1 - 9 T + 195 T^{2} - 1053 T^{3} + 15405 T^{4} - 56169 T^{5} + 493039 T^{6}$$
$83$ $$1 + 20 T + 362 T^{2} + 3447 T^{3} + 30046 T^{4} + 137780 T^{5} + 571787 T^{6}$$
$89$ $$1 - 12 T + 216 T^{2} - 1425 T^{3} + 19224 T^{4} - 95052 T^{5} + 704969 T^{6}$$
$97$ $$1 + 9 T + 147 T^{2} + 1673 T^{3} + 14259 T^{4} + 84681 T^{5} + 912673 T^{6}$$