# Properties

 Label 1815.1.g.g Level $1815$ Weight $1$ Character orbit 1815.g Self dual yes Analytic conductor $0.906$ Analytic rank $0$ Dimension $2$ Projective image $D_{6}$ CM discriminant -15 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1815 = 3 \cdot 5 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1815.g (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$0.905802997929$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ Defining polynomial: $$x^{2} - 3$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{6}$$ Projective field: Galois closure of 6.0.36236475.1

## $q$-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.

 $$f(q)$$ $$=$$ $$q -\beta q^{2} - q^{3} + 2 q^{4} + q^{5} + \beta q^{6} -\beta q^{8} + q^{9} +O(q^{10})$$ $$q -\beta q^{2} - q^{3} + 2 q^{4} + q^{5} + \beta q^{6} -\beta q^{8} + q^{9} -\beta q^{10} -2 q^{12} - q^{15} + q^{16} + \beta q^{17} -\beta q^{18} + 2 q^{20} + q^{23} + \beta q^{24} + q^{25} - q^{27} + \beta q^{30} - q^{31} -3 q^{34} + 2 q^{36} -\beta q^{40} + q^{45} -\beta q^{46} - q^{47} - q^{48} + q^{49} -\beta q^{50} -\beta q^{51} - q^{53} + \beta q^{54} -2 q^{60} -\beta q^{61} + \beta q^{62} - q^{64} + 2 \beta q^{68} - q^{69} -\beta q^{72} - q^{75} + \beta q^{79} + q^{80} + q^{81} + \beta q^{85} -\beta q^{90} + 2 q^{92} + q^{93} + \beta q^{94} -\beta q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{3} + 4q^{4} + 2q^{5} + 2q^{9} + O(q^{10})$$ $$2q - 2q^{3} + 4q^{4} + 2q^{5} + 2q^{9} - 4q^{12} - 2q^{15} + 2q^{16} + 4q^{20} + 2q^{23} + 2q^{25} - 2q^{27} - 2q^{31} - 6q^{34} + 4q^{36} + 2q^{45} - 2q^{47} - 2q^{48} + 2q^{49} - 2q^{53} - 4q^{60} - 2q^{64} - 2q^{69} - 2q^{75} + 2q^{80} + 2q^{81} + 4q^{92} + 2q^{93} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1815\mathbb{Z}\right)^\times$$.

 $$n$$ $$727$$ $$1211$$ $$1696$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1574.1
 1.73205 −1.73205
−1.73205 −1.00000 2.00000 1.00000 1.73205 0 −1.73205 1.00000 −1.73205
1574.2 1.73205 −1.00000 2.00000 1.00000 −1.73205 0 1.73205 1.00000 1.73205
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
15.d odd 2 1 CM by $$\Q(\sqrt{-15})$$
11.b odd 2 1 inner
165.d even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1815.1.g.g 2
3.b odd 2 1 1815.1.g.h yes 2
5.b even 2 1 1815.1.g.h yes 2
11.b odd 2 1 inner 1815.1.g.g 2
11.c even 5 4 1815.1.o.h 8
11.d odd 10 4 1815.1.o.h 8
15.d odd 2 1 CM 1815.1.g.g 2
33.d even 2 1 1815.1.g.h yes 2
33.f even 10 4 1815.1.o.g 8
33.h odd 10 4 1815.1.o.g 8
55.d odd 2 1 1815.1.g.h yes 2
55.h odd 10 4 1815.1.o.g 8
55.j even 10 4 1815.1.o.g 8
165.d even 2 1 inner 1815.1.g.g 2
165.o odd 10 4 1815.1.o.h 8
165.r even 10 4 1815.1.o.h 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1815.1.g.g 2 1.a even 1 1 trivial
1815.1.g.g 2 11.b odd 2 1 inner
1815.1.g.g 2 15.d odd 2 1 CM
1815.1.g.g 2 165.d even 2 1 inner
1815.1.g.h yes 2 3.b odd 2 1
1815.1.g.h yes 2 5.b even 2 1
1815.1.g.h yes 2 33.d even 2 1
1815.1.g.h yes 2 55.d odd 2 1
1815.1.o.g 8 33.f even 10 4
1815.1.o.g 8 33.h odd 10 4
1815.1.o.g 8 55.h odd 10 4
1815.1.o.g 8 55.j even 10 4
1815.1.o.h 8 11.c even 5 4
1815.1.o.h 8 11.d odd 10 4
1815.1.o.h 8 165.o odd 10 4
1815.1.o.h 8 165.r even 10 4

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{1}^{\mathrm{new}}(1815, [\chi])$$:

 $$T_{2}^{2} - 3$$ $$T_{19}$$ $$T_{23} - 1$$

## Hecke characteristic polynomials

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