# Properties

 Label 5472.2.a.bf Level $5472$ Weight $2$ Character orbit 5472.a Self dual yes Analytic conductor $43.694$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$5472 = 2^{5} \cdot 3^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5472.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$43.6941399860$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ Defining polynomial: $$x^{2} - x - 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 608) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{17})$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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
 −1.56155 2.56155
0 0 0 −1.56155 0 3.00000 0 0 0
1.2 0 0 0 2.56155 0 3.00000 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
$$2$$ $$-1$$
$$3$$ $$-1$$
$$19$$ $$-1$$

## 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 5472.2.a.bf 2
3.b odd 2 1 608.2.a.h yes 2
4.b odd 2 1 5472.2.a.bc 2
12.b even 2 1 608.2.a.g 2
24.f even 2 1 1216.2.a.t 2
24.h odd 2 1 1216.2.a.s 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
608.2.a.g 2 12.b even 2 1
608.2.a.h yes 2 3.b odd 2 1
1216.2.a.s 2 24.h odd 2 1
1216.2.a.t 2 24.f even 2 1
5472.2.a.bc 2 4.b odd 2 1
5472.2.a.bf 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(5472))$$:

 $$T_{5}^{2} - T_{5} - 4$$ $$T_{7} - 3$$ $$T_{11}^{2} - 3 T_{11} - 2$$ $$T_{13}^{2} - T_{13} - 4$$ $$T_{23}^{2} + 7 T_{23} + 8$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$-4 - T + T^{2}$$
$7$ $$( -3 + T )^{2}$$
$11$ $$-2 - 3 T + T^{2}$$
$13$ $$-4 - T + T^{2}$$
$17$ $$-1 - 8 T + T^{2}$$
$19$ $$( -1 + T )^{2}$$
$23$ $$8 + 7 T + T^{2}$$
$29$ $$-36 - 3 T + T^{2}$$
$31$ $$8 - 10 T + T^{2}$$
$37$ $$-8 - 6 T + T^{2}$$
$41$ $$( 4 + T )^{2}$$
$43$ $$52 - 15 T + T^{2}$$
$47$ $$-32 + 5 T + T^{2}$$
$53$ $$38 - 13 T + T^{2}$$
$59$ $$38 - 13 T + T^{2}$$
$61$ $$-86 - 9 T + T^{2}$$
$67$ $$2 + 5 T + T^{2}$$
$71$ $$-52 + 8 T + T^{2}$$
$73$ $$-67 + 2 T + T^{2}$$
$79$ $$( -10 + T )^{2}$$
$83$ $$-128 - 10 T + T^{2}$$
$89$ $$-144 - 6 T + T^{2}$$
$97$ $$-8 - 6 T + T^{2}$$