# Properties

 Label 1520.2.g.a Level $1520$ Weight $2$ Character orbit 1520.g Analytic conductor $12.137$ Analytic rank $0$ Dimension $2$ CM discriminant -19 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1520 = 2^{4} \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1520.g (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$12.1372611072$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-19})$$ Defining polynomial: $$x^{2} - x + 5$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{-19})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta q^{5} -3 q^{7} + 3 q^{9} +O(q^{10})$$ $$q -\beta q^{5} -3 q^{7} + 3 q^{9} + ( 1 - 2 \beta ) q^{11} + ( -1 + 2 \beta ) q^{17} + ( 1 - 2 \beta ) q^{19} -4 q^{23} + ( -5 + \beta ) q^{25} + 3 \beta q^{35} + q^{43} -3 \beta q^{45} -13 q^{47} + 2 q^{49} + ( -10 + \beta ) q^{55} -15 q^{61} -9 q^{63} + ( 3 - 6 \beta ) q^{73} + ( -3 + 6 \beta ) q^{77} + 9 q^{81} -16 q^{83} + ( 10 - \beta ) q^{85} + ( -10 + \beta ) q^{95} + ( 3 - 6 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{5} - 6 q^{7} + 6 q^{9} + O(q^{10})$$ $$2 q - q^{5} - 6 q^{7} + 6 q^{9} - 8 q^{23} - 9 q^{25} + 3 q^{35} + 2 q^{43} - 3 q^{45} - 26 q^{47} + 4 q^{49} - 19 q^{55} - 30 q^{61} - 18 q^{63} + 18 q^{81} - 32 q^{83} + 19 q^{85} - 19 q^{95} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1520\mathbb{Z}\right)^\times$$.

 $$n$$ $$191$$ $$401$$ $$1141$$ $$1217$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1519.1
 0.5 + 2.17945i 0.5 − 2.17945i
0 0 0 −0.500000 2.17945i 0 −3.00000 0 3.00000 0
1519.2 0 0 0 −0.500000 + 2.17945i 0 −3.00000 0 3.00000 0
 $$n$$: 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
19.b odd 2 1 CM by $$\Q(\sqrt{-19})$$
20.d odd 2 1 inner
380.d even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1520.2.g.a 2
4.b odd 2 1 1520.2.g.b yes 2
5.b even 2 1 1520.2.g.b yes 2
19.b odd 2 1 CM 1520.2.g.a 2
20.d odd 2 1 inner 1520.2.g.a 2
76.d even 2 1 1520.2.g.b yes 2
95.d odd 2 1 1520.2.g.b yes 2
380.d even 2 1 inner 1520.2.g.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1520.2.g.a 2 1.a even 1 1 trivial
1520.2.g.a 2 19.b odd 2 1 CM
1520.2.g.a 2 20.d odd 2 1 inner
1520.2.g.a 2 380.d even 2 1 inner
1520.2.g.b yes 2 4.b odd 2 1
1520.2.g.b yes 2 5.b even 2 1
1520.2.g.b yes 2 76.d even 2 1
1520.2.g.b yes 2 95.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 $$S_{2}^{\mathrm{new}}(1520, [\chi])$$:

 $$T_{3}$$ $$T_{7} + 3$$ $$T_{31}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$5 + T + T^{2}$$
$7$ $$( 3 + T )^{2}$$
$11$ $$19 + T^{2}$$
$13$ $$T^{2}$$
$17$ $$19 + T^{2}$$
$19$ $$19 + T^{2}$$
$23$ $$( 4 + T )^{2}$$
$29$ $$T^{2}$$
$31$ $$T^{2}$$
$37$ $$T^{2}$$
$41$ $$T^{2}$$
$43$ $$( -1 + T )^{2}$$
$47$ $$( 13 + T )^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$( 15 + T )^{2}$$
$67$ $$T^{2}$$
$71$ $$T^{2}$$
$73$ $$171 + T^{2}$$
$79$ $$T^{2}$$
$83$ $$( 16 + T )^{2}$$
$89$ $$T^{2}$$
$97$ $$T^{2}$$