# Properties

 Label 1520.2.a.k Level $1520$ Weight $2$ Character orbit 1520.a Self dual yes Analytic conductor $12.137$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1520 = 2^{4} \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1520.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$12.1372611072$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ Defining polynomial: $$x^{2} - 3$$ x^2 - 3 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 760) 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.

 $$f(q)$$ $$=$$ $$q + (\beta - 1) q^{3} - q^{5} + ( - 2 \beta + 1) q^{9}+O(q^{10})$$ q + (b - 1) * q^3 - q^5 + (-2*b + 1) * q^9 $$q + (\beta - 1) q^{3} - q^{5} + ( - 2 \beta + 1) q^{9} - 2 q^{11} + (\beta + 1) q^{13} + ( - \beta + 1) q^{15} + ( - 2 \beta + 4) q^{17} - q^{19} + ( - 2 \beta - 2) q^{23} + q^{25} - 4 q^{27} + 2 \beta q^{29} - 4 q^{31} + ( - 2 \beta + 2) q^{33} + ( - 5 \beta - 1) q^{37} + 2 q^{39} + (2 \beta + 4) q^{41} + ( - 4 \beta - 4) q^{43} + (2 \beta - 1) q^{45} + ( - 4 \beta - 4) q^{47} - 7 q^{49} + (6 \beta - 10) q^{51} + (5 \beta - 3) q^{53} + 2 q^{55} + ( - \beta + 1) q^{57} + (2 \beta + 2) q^{59} + ( - 2 \beta - 10) q^{61} + ( - \beta - 1) q^{65} + ( - 3 \beta - 1) q^{67} - 4 q^{69} + (4 \beta - 4) q^{71} + 6 \beta q^{73} + (\beta - 1) q^{75} + ( - 6 \beta - 2) q^{79} + (2 \beta + 1) q^{81} + ( - 2 \beta + 2) q^{83} + (2 \beta - 4) q^{85} + ( - 2 \beta + 6) q^{87} + 2 \beta q^{89} + ( - 4 \beta + 4) q^{93} + q^{95} + (5 \beta - 7) q^{97} + (4 \beta - 2) q^{99}+O(q^{100})$$ q + (b - 1) * q^3 - q^5 + (-2*b + 1) * q^9 - 2 * q^11 + (b + 1) * q^13 + (-b + 1) * q^15 + (-2*b + 4) * q^17 - q^19 + (-2*b - 2) * q^23 + q^25 - 4 * q^27 + 2*b * q^29 - 4 * q^31 + (-2*b + 2) * q^33 + (-5*b - 1) * q^37 + 2 * q^39 + (2*b + 4) * q^41 + (-4*b - 4) * q^43 + (2*b - 1) * q^45 + (-4*b - 4) * q^47 - 7 * q^49 + (6*b - 10) * q^51 + (5*b - 3) * q^53 + 2 * q^55 + (-b + 1) * q^57 + (2*b + 2) * q^59 + (-2*b - 10) * q^61 + (-b - 1) * q^65 + (-3*b - 1) * q^67 - 4 * q^69 + (4*b - 4) * q^71 + 6*b * q^73 + (b - 1) * q^75 + (-6*b - 2) * q^79 + (2*b + 1) * q^81 + (-2*b + 2) * q^83 + (2*b - 4) * q^85 + (-2*b + 6) * q^87 + 2*b * q^89 + (-4*b + 4) * q^93 + q^95 + (5*b - 7) * q^97 + (4*b - 2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} - 2 q^{5} + 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^3 - 2 * q^5 + 2 * q^9 $$2 q - 2 q^{3} - 2 q^{5} + 2 q^{9} - 4 q^{11} + 2 q^{13} + 2 q^{15} + 8 q^{17} - 2 q^{19} - 4 q^{23} + 2 q^{25} - 8 q^{27} - 8 q^{31} + 4 q^{33} - 2 q^{37} + 4 q^{39} + 8 q^{41} - 8 q^{43} - 2 q^{45} - 8 q^{47} - 14 q^{49} - 20 q^{51} - 6 q^{53} + 4 q^{55} + 2 q^{57} + 4 q^{59} - 20 q^{61} - 2 q^{65} - 2 q^{67} - 8 q^{69} - 8 q^{71} - 2 q^{75} - 4 q^{79} + 2 q^{81} + 4 q^{83} - 8 q^{85} + 12 q^{87} + 8 q^{93} + 2 q^{95} - 14 q^{97} - 4 q^{99}+O(q^{100})$$ 2 * q - 2 * q^3 - 2 * q^5 + 2 * q^9 - 4 * q^11 + 2 * q^13 + 2 * q^15 + 8 * q^17 - 2 * q^19 - 4 * q^23 + 2 * q^25 - 8 * q^27 - 8 * q^31 + 4 * q^33 - 2 * q^37 + 4 * q^39 + 8 * q^41 - 8 * q^43 - 2 * q^45 - 8 * q^47 - 14 * q^49 - 20 * q^51 - 6 * q^53 + 4 * q^55 + 2 * q^57 + 4 * q^59 - 20 * q^61 - 2 * q^65 - 2 * q^67 - 8 * q^69 - 8 * q^71 - 2 * q^75 - 4 * q^79 + 2 * q^81 + 4 * q^83 - 8 * q^85 + 12 * q^87 + 8 * q^93 + 2 * q^95 - 14 * q^97 - 4 * q^99

## 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 −2.73205 0 −1.00000 0 0 0 4.46410 0
1.2 0 0.732051 0 −1.00000 0 0 0 −2.46410 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
$$2$$ $$1$$
$$5$$ $$1$$
$$19$$ $$1$$

## 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 1520.2.a.k 2
4.b odd 2 1 760.2.a.g 2
5.b even 2 1 7600.2.a.bd 2
8.b even 2 1 6080.2.a.bk 2
8.d odd 2 1 6080.2.a.ba 2
12.b even 2 1 6840.2.a.y 2
20.d odd 2 1 3800.2.a.j 2
20.e even 4 2 3800.2.d.h 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
760.2.a.g 2 4.b odd 2 1
1520.2.a.k 2 1.a even 1 1 trivial
3800.2.a.j 2 20.d odd 2 1
3800.2.d.h 4 20.e even 4 2
6080.2.a.ba 2 8.d odd 2 1
6080.2.a.bk 2 8.b even 2 1
6840.2.a.y 2 12.b even 2 1
7600.2.a.bd 2 5.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1520))$$:

 $$T_{3}^{2} + 2T_{3} - 2$$ T3^2 + 2*T3 - 2 $$T_{7}$$ T7

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 2T - 2$$
$5$ $$(T + 1)^{2}$$
$7$ $$T^{2}$$
$11$ $$(T + 2)^{2}$$
$13$ $$T^{2} - 2T - 2$$
$17$ $$T^{2} - 8T + 4$$
$19$ $$(T + 1)^{2}$$
$23$ $$T^{2} + 4T - 8$$
$29$ $$T^{2} - 12$$
$31$ $$(T + 4)^{2}$$
$37$ $$T^{2} + 2T - 74$$
$41$ $$T^{2} - 8T + 4$$
$43$ $$T^{2} + 8T - 32$$
$47$ $$T^{2} + 8T - 32$$
$53$ $$T^{2} + 6T - 66$$
$59$ $$T^{2} - 4T - 8$$
$61$ $$T^{2} + 20T + 88$$
$67$ $$T^{2} + 2T - 26$$
$71$ $$T^{2} + 8T - 32$$
$73$ $$T^{2} - 108$$
$79$ $$T^{2} + 4T - 104$$
$83$ $$T^{2} - 4T - 8$$
$89$ $$T^{2} - 12$$
$97$ $$T^{2} + 14T - 26$$