# Properties

 Label 9072.2.a.bm Level $9072$ Weight $2$ Character orbit 9072.a Self dual yes Analytic conductor $72.440$ Analytic rank $0$ 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: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{33})$$ Defining polynomial: $$x^{2} - x - 8$$ 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 $$\beta = \frac{1}{2}(1 + \sqrt{33})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 + \beta ) q^{5} - q^{7} +O(q^{10})$$ $$q + ( 1 + \beta ) q^{5} - q^{7} + ( -2 + \beta ) q^{11} + 2 q^{13} + ( -2 + \beta ) q^{17} -5 q^{19} + ( 5 - \beta ) q^{23} + ( 4 + 3 \beta ) q^{25} + ( -2 - 2 \beta ) q^{29} -2 q^{31} + ( -1 - \beta ) q^{35} + 2 q^{37} + ( 8 - \beta ) q^{41} + ( -2 + 3 \beta ) q^{43} + q^{49} + ( -2 - 2 \beta ) q^{53} + 6 q^{55} + 3 \beta q^{59} + ( -7 + 3 \beta ) q^{61} + ( 2 + 2 \beta ) q^{65} + ( -8 + 3 \beta ) q^{67} + ( -3 + 3 \beta ) q^{71} + ( 2 + 3 \beta ) q^{73} + ( 2 - \beta ) q^{77} + ( -5 + 3 \beta ) q^{79} + ( -4 - 4 \beta ) q^{83} + 6 q^{85} + ( 8 + 2 \beta ) q^{89} -2 q^{91} + ( -5 - 5 \beta ) q^{95} + ( 2 - 3 \beta ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 3q^{5} - 2q^{7} + O(q^{10})$$ $$2q + 3q^{5} - 2q^{7} - 3q^{11} + 4q^{13} - 3q^{17} - 10q^{19} + 9q^{23} + 11q^{25} - 6q^{29} - 4q^{31} - 3q^{35} + 4q^{37} + 15q^{41} - q^{43} + 2q^{49} - 6q^{53} + 12q^{55} + 3q^{59} - 11q^{61} + 6q^{65} - 13q^{67} - 3q^{71} + 7q^{73} + 3q^{77} - 7q^{79} - 12q^{83} + 12q^{85} + 18q^{89} - 4q^{91} - 15q^{95} + q^{97} + O(q^{100})$$

## 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
 −2.37228 3.37228
0 0 0 −1.37228 0 −1.00000 0 0 0
1.2 0 0 0 4.37228 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.bm 2
3.b odd 2 1 9072.2.a.bb 2
4.b odd 2 1 1134.2.a.k 2
9.c even 3 2 1008.2.r.f 4
9.d odd 6 2 3024.2.r.f 4
12.b even 2 1 1134.2.a.n 2
28.d even 2 1 7938.2.a.bh 2
36.f odd 6 2 126.2.f.d 4
36.h even 6 2 378.2.f.c 4
84.h odd 2 1 7938.2.a.bs 2
252.n even 6 2 882.2.h.n 4
252.o even 6 2 2646.2.h.k 4
252.r odd 6 2 2646.2.e.m 4
252.s odd 6 2 2646.2.f.j 4
252.u odd 6 2 882.2.e.l 4
252.bb even 6 2 2646.2.e.n 4
252.bi even 6 2 882.2.f.k 4
252.bj even 6 2 882.2.e.k 4
252.bl odd 6 2 882.2.h.m 4
252.bn odd 6 2 2646.2.h.l 4

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

 $$T_{5}^{2} - 3 T_{5} - 6$$ $$T_{11}^{2} + 3 T_{11} - 6$$ $$T_{13} - 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$-6 - 3 T + T^{2}$$
$7$ $$( 1 + T )^{2}$$
$11$ $$-6 + 3 T + T^{2}$$
$13$ $$( -2 + T )^{2}$$
$17$ $$-6 + 3 T + T^{2}$$
$19$ $$( 5 + T )^{2}$$
$23$ $$12 - 9 T + T^{2}$$
$29$ $$-24 + 6 T + T^{2}$$
$31$ $$( 2 + T )^{2}$$
$37$ $$( -2 + T )^{2}$$
$41$ $$48 - 15 T + T^{2}$$
$43$ $$-74 + T + T^{2}$$
$47$ $$T^{2}$$
$53$ $$-24 + 6 T + T^{2}$$
$59$ $$-72 - 3 T + T^{2}$$
$61$ $$-44 + 11 T + T^{2}$$
$67$ $$-32 + 13 T + T^{2}$$
$71$ $$-72 + 3 T + T^{2}$$
$73$ $$-62 - 7 T + T^{2}$$
$79$ $$-62 + 7 T + T^{2}$$
$83$ $$-96 + 12 T + T^{2}$$
$89$ $$48 - 18 T + T^{2}$$
$97$ $$-74 - T + T^{2}$$