# Properties

 Label 1815.2.a.g Level $1815$ Weight $2$ Character orbit 1815.a Self dual yes Analytic conductor $14.493$ Analytic rank $1$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1815 = 3 \cdot 5 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1815.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$14.4928479669$$ 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: yes 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.

 $$f(q)$$ $$=$$ $$q - q^{3} -2 q^{4} - q^{5} -\beta q^{7} + q^{9} +O(q^{10})$$ $$q - q^{3} -2 q^{4} - q^{5} -\beta q^{7} + q^{9} + 2 q^{12} + q^{15} + 4 q^{16} + 2 \beta q^{17} + 3 \beta q^{19} + 2 q^{20} + \beta q^{21} + 6 q^{23} + q^{25} - q^{27} + 2 \beta q^{28} -4 \beta q^{29} + q^{31} + \beta q^{35} -2 q^{36} -5 q^{37} -2 \beta q^{41} -6 \beta q^{43} - q^{45} -12 q^{47} -4 q^{48} -4 q^{49} -2 \beta q^{51} + 6 q^{53} -3 \beta q^{57} -2 q^{60} + 7 \beta q^{61} -\beta q^{63} -8 q^{64} -5 q^{67} -4 \beta q^{68} -6 q^{69} -6 q^{71} + \beta q^{73} - q^{75} -6 \beta q^{76} + 9 \beta q^{79} -4 q^{80} + q^{81} + 4 \beta q^{83} -2 \beta q^{84} -2 \beta q^{85} + 4 \beta q^{87} -6 q^{89} -12 q^{92} - q^{93} -3 \beta q^{95} -13 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{3} - 4q^{4} - 2q^{5} + 2q^{9} + O(q^{10})$$ $$2q - 2q^{3} - 4q^{4} - 2q^{5} + 2q^{9} + 4q^{12} + 2q^{15} + 8q^{16} + 4q^{20} + 12q^{23} + 2q^{25} - 2q^{27} + 2q^{31} - 4q^{36} - 10q^{37} - 2q^{45} - 24q^{47} - 8q^{48} - 8q^{49} + 12q^{53} - 4q^{60} - 16q^{64} - 10q^{67} - 12q^{69} - 12q^{71} - 2q^{75} - 8q^{80} + 2q^{81} - 12q^{89} - 24q^{92} - 2q^{93} - 26q^{97} + O(q^{100})$$

## 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 −1.00000 −2.00000 −1.00000 0 −1.73205 0 1.00000 0
1.2 0 −1.00000 −2.00000 −1.00000 0 1.73205 0 1.00000 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 1815.2.a.g 2
3.b odd 2 1 5445.2.a.q 2
5.b even 2 1 9075.2.a.bl 2
11.b odd 2 1 inner 1815.2.a.g 2
33.d even 2 1 5445.2.a.q 2
55.d odd 2 1 9075.2.a.bl 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1815.2.a.g 2 1.a even 1 1 trivial
1815.2.a.g 2 11.b odd 2 1 inner
5445.2.a.q 2 3.b odd 2 1
5445.2.a.q 2 33.d even 2 1
9075.2.a.bl 2 5.b even 2 1
9075.2.a.bl 2 55.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1815))$$:

 $$T_{2}$$ $$T_{7}^{2} - 3$$

## Hecke characteristic polynomials

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