# Properties

 Label 1520.2.a.r Level $1520$ Weight $2$ Character orbit 1520.a Self dual yes Analytic conductor $12.137$ Analytic rank $0$ Dimension $3$ 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: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.568.1 Defining polynomial: $$x^{3} - x^{2} - 6x - 2$$ x^3 - x^2 - 6*x - 2 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 a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{3} - q^{5} + (\beta_{2} + 2) q^{7} + (\beta_{2} + 2 \beta_1 + 1) q^{9}+O(q^{10})$$ q + b1 * q^3 - q^5 + (b2 + 2) * q^7 + (b2 + 2*b1 + 1) * q^9 $$q + \beta_1 q^{3} - q^{5} + (\beta_{2} + 2) q^{7} + (\beta_{2} + 2 \beta_1 + 1) q^{9} + ( - 2 \beta_1 + 2) q^{11} + (2 \beta_{2} + \beta_1 + 2) q^{13} - \beta_1 q^{15} + ( - \beta_{2} - 4) q^{17} + q^{19} + ( - \beta_{2} + 2 \beta_1 - 2) q^{21} + ( - \beta_{2} + 2 \beta_1 + 2) q^{23} + q^{25} + (\beta_{2} + 2 \beta_1 + 6) q^{27} - 3 \beta_{2} q^{29} + ( - 2 \beta_{2} + 4) q^{31} + ( - 2 \beta_{2} - 2 \beta_1 - 8) q^{33} + ( - \beta_{2} - 2) q^{35} + ( - \beta_{2} + \beta_1 + 4) q^{37} + ( - \beta_{2} + 4 \beta_1) q^{39} + (2 \beta_{2} - 2 \beta_1 - 2) q^{41} + (4 \beta_{2} + 2 \beta_1 + 4) q^{43} + ( - \beta_{2} - 2 \beta_1 - 1) q^{45} + ( - 2 \beta_{2} - 2 \beta_1) q^{47} + (3 \beta_{2} - 2 \beta_1 + 3) q^{49} + (\beta_{2} - 4 \beta_1 + 2) q^{51} + (2 \beta_{2} + \beta_1 - 2) q^{53} + (2 \beta_1 - 2) q^{55} + \beta_1 q^{57} + ( - \beta_{2} + 2) q^{59} + ( - 4 \beta_1 + 8) q^{61} + (2 \beta_1 + 4) q^{63} + ( - 2 \beta_{2} - \beta_1 - 2) q^{65} + ( - 4 \beta_{2} - 3 \beta_1) q^{67} + (3 \beta_{2} + 6 \beta_1 + 10) q^{69} + ( - 4 \beta_1 + 4) q^{71} + ( - 3 \beta_{2} - 4 \beta_1 - 4) q^{73} + \beta_1 q^{75} + (4 \beta_{2} - 4 \beta_1 + 8) q^{77} + (4 \beta_{2} + 2 \beta_1 - 4) q^{79} + ( - 2 \beta_{2} + 4 \beta_1 + 3) q^{81} + ( - 2 \beta_{2} + 8) q^{83} + (\beta_{2} + 4) q^{85} + (3 \beta_{2} + 6) q^{87} + ( - 2 \beta_{2} - 2 \beta_1 + 6) q^{89} + (3 \beta_{2} - 2 \beta_1 + 14) q^{91} + (2 \beta_{2} + 4 \beta_1 + 4) q^{93} - q^{95} + (\beta_{2} + 7 \beta_1 - 4) q^{97} + ( - 6 \beta_1 - 10) q^{99}+O(q^{100})$$ q + b1 * q^3 - q^5 + (b2 + 2) * q^7 + (b2 + 2*b1 + 1) * q^9 + (-2*b1 + 2) * q^11 + (2*b2 + b1 + 2) * q^13 - b1 * q^15 + (-b2 - 4) * q^17 + q^19 + (-b2 + 2*b1 - 2) * q^21 + (-b2 + 2*b1 + 2) * q^23 + q^25 + (b2 + 2*b1 + 6) * q^27 - 3*b2 * q^29 + (-2*b2 + 4) * q^31 + (-2*b2 - 2*b1 - 8) * q^33 + (-b2 - 2) * q^35 + (-b2 + b1 + 4) * q^37 + (-b2 + 4*b1) * q^39 + (2*b2 - 2*b1 - 2) * q^41 + (4*b2 + 2*b1 + 4) * q^43 + (-b2 - 2*b1 - 1) * q^45 + (-2*b2 - 2*b1) * q^47 + (3*b2 - 2*b1 + 3) * q^49 + (b2 - 4*b1 + 2) * q^51 + (2*b2 + b1 - 2) * q^53 + (2*b1 - 2) * q^55 + b1 * q^57 + (-b2 + 2) * q^59 + (-4*b1 + 8) * q^61 + (2*b1 + 4) * q^63 + (-2*b2 - b1 - 2) * q^65 + (-4*b2 - 3*b1) * q^67 + (3*b2 + 6*b1 + 10) * q^69 + (-4*b1 + 4) * q^71 + (-3*b2 - 4*b1 - 4) * q^73 + b1 * q^75 + (4*b2 - 4*b1 + 8) * q^77 + (4*b2 + 2*b1 - 4) * q^79 + (-2*b2 + 4*b1 + 3) * q^81 + (-2*b2 + 8) * q^83 + (b2 + 4) * q^85 + (3*b2 + 6) * q^87 + (-2*b2 - 2*b1 + 6) * q^89 + (3*b2 - 2*b1 + 14) * q^91 + (2*b2 + 4*b1 + 4) * q^93 - q^95 + (b2 + 7*b1 - 4) * q^97 + (-6*b1 - 10) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + q^{3} - 3 q^{5} + 5 q^{7} + 4 q^{9}+O(q^{10})$$ 3 * q + q^3 - 3 * q^5 + 5 * q^7 + 4 * q^9 $$3 q + q^{3} - 3 q^{5} + 5 q^{7} + 4 q^{9} + 4 q^{11} + 5 q^{13} - q^{15} - 11 q^{17} + 3 q^{19} - 3 q^{21} + 9 q^{23} + 3 q^{25} + 19 q^{27} + 3 q^{29} + 14 q^{31} - 24 q^{33} - 5 q^{35} + 14 q^{37} + 5 q^{39} - 10 q^{41} + 10 q^{43} - 4 q^{45} + 4 q^{49} + q^{51} - 7 q^{53} - 4 q^{55} + q^{57} + 7 q^{59} + 20 q^{61} + 14 q^{63} - 5 q^{65} + q^{67} + 33 q^{69} + 8 q^{71} - 13 q^{73} + q^{75} + 16 q^{77} - 14 q^{79} + 15 q^{81} + 26 q^{83} + 11 q^{85} + 15 q^{87} + 18 q^{89} + 37 q^{91} + 14 q^{93} - 3 q^{95} - 6 q^{97} - 36 q^{99}+O(q^{100})$$ 3 * q + q^3 - 3 * q^5 + 5 * q^7 + 4 * q^9 + 4 * q^11 + 5 * q^13 - q^15 - 11 * q^17 + 3 * q^19 - 3 * q^21 + 9 * q^23 + 3 * q^25 + 19 * q^27 + 3 * q^29 + 14 * q^31 - 24 * q^33 - 5 * q^35 + 14 * q^37 + 5 * q^39 - 10 * q^41 + 10 * q^43 - 4 * q^45 + 4 * q^49 + q^51 - 7 * q^53 - 4 * q^55 + q^57 + 7 * q^59 + 20 * q^61 + 14 * q^63 - 5 * q^65 + q^67 + 33 * q^69 + 8 * q^71 - 13 * q^73 + q^75 + 16 * q^77 - 14 * q^79 + 15 * q^81 + 26 * q^83 + 11 * q^85 + 15 * q^87 + 18 * q^89 + 37 * q^91 + 14 * q^93 - 3 * q^95 - 6 * q^97 - 36 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 6x - 2$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 2\nu - 4$$ v^2 - 2*v - 4
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 2\beta _1 + 4$$ b2 + 2*b1 + 4

## 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
 −1.76156 −0.363328 3.12489
0 −1.76156 0 −1.00000 0 4.62620 0 0.103084 0
1.2 0 −0.363328 0 −1.00000 0 −1.14134 0 −2.86799 0
1.3 0 3.12489 0 −1.00000 0 1.51514 0 6.76491 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.r 3
4.b odd 2 1 760.2.a.h 3
5.b even 2 1 7600.2.a.bo 3
8.b even 2 1 6080.2.a.bs 3
8.d odd 2 1 6080.2.a.bw 3
12.b even 2 1 6840.2.a.bj 3
20.d odd 2 1 3800.2.a.y 3
20.e even 4 2 3800.2.d.k 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
760.2.a.h 3 4.b odd 2 1
1520.2.a.r 3 1.a even 1 1 trivial
3800.2.a.y 3 20.d odd 2 1
3800.2.d.k 6 20.e even 4 2
6080.2.a.bs 3 8.b even 2 1
6080.2.a.bw 3 8.d odd 2 1
6840.2.a.bj 3 12.b even 2 1
7600.2.a.bo 3 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(1520))$$:

 $$T_{3}^{3} - T_{3}^{2} - 6T_{3} - 2$$ T3^3 - T3^2 - 6*T3 - 2 $$T_{7}^{3} - 5T_{7}^{2} + 8$$ T7^3 - 5*T7^2 + 8

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$T^{3} - T^{2} - 6T - 2$$
$5$ $$(T + 1)^{3}$$
$7$ $$T^{3} - 5T^{2} + 8$$
$11$ $$T^{3} - 4 T^{2} - 20 T + 64$$
$13$ $$T^{3} - 5 T^{2} - 22 T + 106$$
$17$ $$T^{3} + 11 T^{2} + 32 T + 20$$
$19$ $$(T - 1)^{3}$$
$23$ $$T^{3} - 9 T^{2} - 16 T + 160$$
$29$ $$T^{3} - 3 T^{2} - 72 T + 108$$
$31$ $$T^{3} - 14 T^{2} + 32 T + 64$$
$37$ $$T^{3} - 14 T^{2} + 46 T + 20$$
$41$ $$T^{3} + 10 T^{2} - 44 T - 472$$
$43$ $$T^{3} - 10 T^{2} - 88 T + 848$$
$47$ $$T^{3} - 40T - 64$$
$53$ $$T^{3} + 7 T^{2} - 14 T + 2$$
$59$ $$T^{3} - 7 T^{2} + 8 T + 8$$
$61$ $$T^{3} - 20 T^{2} + 32 T + 640$$
$67$ $$T^{3} - T^{2} - 134 T - 530$$
$71$ $$T^{3} - 8 T^{2} - 80 T + 512$$
$73$ $$T^{3} + 13 T^{2} - 64 T - 500$$
$79$ $$T^{3} + 14 T^{2} - 56 T + 16$$
$83$ $$T^{3} - 26 T^{2} + 192 T - 352$$
$89$ $$T^{3} - 18 T^{2} + 68 T - 40$$
$97$ $$T^{3} + 6 T^{2} - 274 T - 2308$$