# Properties

 Label 9072.2.a.bk Level $9072$ Weight $2$ Character orbit 9072.a Self dual yes Analytic conductor $72.440$ Analytic rank $1$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{6})$$ Defining polynomial: $$x^{2} - 6$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 126) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 + \beta ) q^{5} + q^{7} +O(q^{10})$$ $$q + ( 1 + \beta ) q^{5} + q^{7} -2 q^{11} -2 \beta q^{13} + 2 q^{17} + ( -5 - \beta ) q^{19} + q^{23} + ( 2 + 2 \beta ) q^{25} + ( -2 + 2 \beta ) q^{29} -6 q^{31} + ( 1 + \beta ) q^{35} + ( 2 - 4 \beta ) q^{37} -4 \beta q^{41} + ( -2 + 2 \beta ) q^{43} + 4 \beta q^{47} + q^{49} + ( -6 + 2 \beta ) q^{53} + ( -2 - 2 \beta ) q^{55} + 2 q^{59} + ( 9 + \beta ) q^{61} + ( -12 - 2 \beta ) q^{65} + ( 8 - 2 \beta ) q^{67} + ( -5 - 2 \beta ) q^{71} + ( -2 + 2 \beta ) q^{73} -2 q^{77} + ( -3 - 2 \beta ) q^{79} -2 q^{83} + ( 2 + 2 \beta ) q^{85} + ( -12 + 2 \beta ) q^{89} -2 \beta q^{91} + ( -11 - 6 \beta ) q^{95} + ( -2 - 2 \beta ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{5} + 2q^{7} + O(q^{10})$$ $$2q + 2q^{5} + 2q^{7} - 4q^{11} + 4q^{17} - 10q^{19} + 2q^{23} + 4q^{25} - 4q^{29} - 12q^{31} + 2q^{35} + 4q^{37} - 4q^{43} + 2q^{49} - 12q^{53} - 4q^{55} + 4q^{59} + 18q^{61} - 24q^{65} + 16q^{67} - 10q^{71} - 4q^{73} - 4q^{77} - 6q^{79} - 4q^{83} + 4q^{85} - 24q^{89} - 22q^{95} - 4q^{97} + O(q^{100})$$

## 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
 −2.44949 2.44949
0 0 0 −1.44949 0 1.00000 0 0 0
1.2 0 0 0 3.44949 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.bk 2
3.b odd 2 1 9072.2.a.bd 2
4.b odd 2 1 1134.2.a.p 2
9.c even 3 2 1008.2.r.e 4
9.d odd 6 2 3024.2.r.e 4
12.b even 2 1 1134.2.a.i 2
28.d even 2 1 7938.2.a.bn 2
36.f odd 6 2 126.2.f.c 4
36.h even 6 2 378.2.f.d 4
84.h odd 2 1 7938.2.a.bm 2
252.n even 6 2 882.2.h.l 4
252.o even 6 2 2646.2.h.m 4
252.r odd 6 2 2646.2.e.k 4
252.s odd 6 2 2646.2.f.k 4
252.u odd 6 2 882.2.e.m 4
252.bb even 6 2 2646.2.e.l 4
252.bi even 6 2 882.2.f.j 4
252.bj even 6 2 882.2.e.n 4
252.bl odd 6 2 882.2.h.k 4
252.bn odd 6 2 2646.2.h.n 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.f.c 4 36.f odd 6 2
378.2.f.d 4 36.h even 6 2
882.2.e.m 4 252.u odd 6 2
882.2.e.n 4 252.bj even 6 2
882.2.f.j 4 252.bi even 6 2
882.2.h.k 4 252.bl odd 6 2
882.2.h.l 4 252.n even 6 2
1008.2.r.e 4 9.c even 3 2
1134.2.a.i 2 12.b even 2 1
1134.2.a.p 2 4.b odd 2 1
2646.2.e.k 4 252.r odd 6 2
2646.2.e.l 4 252.bb even 6 2
2646.2.f.k 4 252.s odd 6 2
2646.2.h.m 4 252.o even 6 2
2646.2.h.n 4 252.bn odd 6 2
3024.2.r.e 4 9.d odd 6 2
7938.2.a.bm 2 84.h odd 2 1
7938.2.a.bn 2 28.d even 2 1
9072.2.a.bd 2 3.b odd 2 1
9072.2.a.bk 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(9072))$$:

 $$T_{5}^{2} - 2 T_{5} - 5$$ $$T_{11} + 2$$ $$T_{13}^{2} - 24$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$1 - 2 T + 5 T^{2} - 10 T^{3} + 25 T^{4}$$
$7$ $$( 1 - T )^{2}$$
$11$ $$( 1 + 2 T + 11 T^{2} )^{2}$$
$13$ $$1 + 2 T^{2} + 169 T^{4}$$
$17$ $$( 1 - 2 T + 17 T^{2} )^{2}$$
$19$ $$1 + 10 T + 57 T^{2} + 190 T^{3} + 361 T^{4}$$
$23$ $$( 1 - T + 23 T^{2} )^{2}$$
$29$ $$1 + 4 T + 38 T^{2} + 116 T^{3} + 841 T^{4}$$
$31$ $$( 1 + 6 T + 31 T^{2} )^{2}$$
$37$ $$1 - 4 T - 18 T^{2} - 148 T^{3} + 1369 T^{4}$$
$41$ $$1 - 14 T^{2} + 1681 T^{4}$$
$43$ $$1 + 4 T + 66 T^{2} + 172 T^{3} + 1849 T^{4}$$
$47$ $$1 - 2 T^{2} + 2209 T^{4}$$
$53$ $$1 + 12 T + 118 T^{2} + 636 T^{3} + 2809 T^{4}$$
$59$ $$( 1 - 2 T + 59 T^{2} )^{2}$$
$61$ $$1 - 18 T + 197 T^{2} - 1098 T^{3} + 3721 T^{4}$$
$67$ $$1 - 16 T + 174 T^{2} - 1072 T^{3} + 4489 T^{4}$$
$71$ $$1 + 10 T + 143 T^{2} + 710 T^{3} + 5041 T^{4}$$
$73$ $$1 + 4 T + 126 T^{2} + 292 T^{3} + 5329 T^{4}$$
$79$ $$1 + 6 T + 143 T^{2} + 474 T^{3} + 6241 T^{4}$$
$83$ $$( 1 + 2 T + 83 T^{2} )^{2}$$
$89$ $$1 + 24 T + 298 T^{2} + 2136 T^{3} + 7921 T^{4}$$
$97$ $$1 + 4 T + 174 T^{2} + 388 T^{3} + 9409 T^{4}$$