# Properties

 Label 5445.2.a.q Level $5445$ Weight $2$ Character orbit 5445.a Self dual yes Analytic conductor $43.479$ Analytic rank $1$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$5445 = 3^{2} \cdot 5 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5445.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$43.4785439006$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ Defining polynomial: $$x^{2} - 3$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1815) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -2 q^{4} + q^{5} -\beta q^{7} +O(q^{10})$$ $$q -2 q^{4} + q^{5} -\beta q^{7} + 4 q^{16} -2 \beta q^{17} + 3 \beta q^{19} -2 q^{20} -6 q^{23} + q^{25} + 2 \beta q^{28} + 4 \beta q^{29} + q^{31} -\beta q^{35} -5 q^{37} + 2 \beta q^{41} -6 \beta q^{43} + 12 q^{47} -4 q^{49} -6 q^{53} + 7 \beta q^{61} -8 q^{64} -5 q^{67} + 4 \beta q^{68} + 6 q^{71} + \beta q^{73} -6 \beta q^{76} + 9 \beta q^{79} + 4 q^{80} -4 \beta q^{83} -2 \beta q^{85} + 6 q^{89} + 12 q^{92} + 3 \beta q^{95} -13 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 4q^{4} + 2q^{5} + O(q^{10})$$ $$2q - 4q^{4} + 2q^{5} + 8q^{16} - 4q^{20} - 12q^{23} + 2q^{25} + 2q^{31} - 10q^{37} + 24q^{47} - 8q^{49} - 12q^{53} - 16q^{64} - 10q^{67} + 12q^{71} + 8q^{80} + 12q^{89} + 24q^{92} - 26q^{97} + O(q^{100})$$

## 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
 1.73205 −1.73205
0 0 −2.00000 1.00000 0 −1.73205 0 0 0
1.2 0 0 −2.00000 1.00000 0 1.73205 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
$$3$$ $$-1$$
$$5$$ $$-1$$
$$11$$ $$1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5445.2.a.q 2
3.b odd 2 1 1815.2.a.g 2
11.b odd 2 1 inner 5445.2.a.q 2
15.d odd 2 1 9075.2.a.bl 2
33.d even 2 1 1815.2.a.g 2
165.d even 2 1 9075.2.a.bl 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1815.2.a.g 2 3.b odd 2 1
1815.2.a.g 2 33.d even 2 1
5445.2.a.q 2 1.a even 1 1 trivial
5445.2.a.q 2 11.b odd 2 1 inner
9075.2.a.bl 2 15.d odd 2 1
9075.2.a.bl 2 165.d even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(5445))$$:

 $$T_{2}$$ $$T_{7}^{2} - 3$$ $$T_{23} + 6$$ $$T_{53} + 6$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$( -1 + T )^{2}$$
$7$ $$-3 + T^{2}$$
$11$ $$T^{2}$$
$13$ $$T^{2}$$
$17$ $$-12 + T^{2}$$
$19$ $$-27 + T^{2}$$
$23$ $$( 6 + T )^{2}$$
$29$ $$-48 + T^{2}$$
$31$ $$( -1 + T )^{2}$$
$37$ $$( 5 + T )^{2}$$
$41$ $$-12 + T^{2}$$
$43$ $$-108 + T^{2}$$
$47$ $$( -12 + T )^{2}$$
$53$ $$( 6 + T )^{2}$$
$59$ $$T^{2}$$
$61$ $$-147 + T^{2}$$
$67$ $$( 5 + T )^{2}$$
$71$ $$( -6 + T )^{2}$$
$73$ $$-3 + T^{2}$$
$79$ $$-243 + T^{2}$$
$83$ $$-48 + T^{2}$$
$89$ $$( -6 + T )^{2}$$
$97$ $$( 13 + T )^{2}$$