# Properties

 Label 1520.2.a.p 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: 3.3.148.1 Defining polynomial: $$x^{3} - x^{2} - 3x + 1$$ x^3 - x^2 - 3*x + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 95) 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.

 $$f(q)$$ $$=$$ $$q + (\beta_1 - 1) q^{3} + q^{5} + (\beta_{2} - \beta_1) q^{7} + (\beta_{2} - \beta_1 + 1) q^{9}+O(q^{10})$$ q + (b1 - 1) * q^3 + q^5 + (b2 - b1) * q^7 + (b2 - b1 + 1) * q^9 $$q + (\beta_1 - 1) q^{3} + q^{5} + (\beta_{2} - \beta_1) q^{7} + (\beta_{2} - \beta_1 + 1) q^{9} + (\beta_{2} + \beta_1 + 2) q^{11} + ( - \beta_{2} + 3) q^{13} + (\beta_1 - 1) q^{15} + 2 \beta_{2} q^{17} + q^{19} + ( - 2 \beta_{2} + 2 \beta_1 - 4) q^{21} + ( - \beta_{2} - \beta_1 + 2) q^{23} + q^{25} + ( - 2 \beta_{2} - 2) q^{27} + (2 \beta_1 - 4) q^{29} + ( - 2 \beta_{2} - 2 \beta_1) q^{31} + 4 \beta_1 q^{33} + (\beta_{2} - \beta_1) q^{35} + ( - \beta_{2} + 7) q^{37} + (\beta_{2} + \beta_1 - 2) q^{39} - 2 \beta_{2} q^{41} + ( - \beta_{2} + 5 \beta_1) q^{43} + (\beta_{2} - \beta_1 + 1) q^{45} + (\beta_{2} - \beta_1) q^{47} + ( - 4 \beta_1 + 5) q^{49} + ( - 2 \beta_{2} + 4 \beta_1 - 2) q^{51} + (\beta_{2} + 5) q^{53} + (\beta_{2} + \beta_1 + 2) q^{55} + (\beta_1 - 1) q^{57} + (2 \beta_1 + 6) q^{59} + (3 \beta_{2} + \beta_1 - 2) q^{61} + (\beta_{2} - 5 \beta_1 + 12) q^{63} + ( - \beta_{2} + 3) q^{65} + ( - 2 \beta_{2} - 3 \beta_1 + 1) q^{67} - 4 q^{69} + 4 \beta_1 q^{71} + 2 \beta_1 q^{73} + (\beta_1 - 1) q^{75} + ( - 4 \beta_1 + 4) q^{77} + ( - 6 \beta_1 + 2) q^{79} + ( - \beta_{2} - 3 \beta_1 + 1) q^{81} + (\beta_{2} + \beta_1 + 10) q^{83} + 2 \beta_{2} q^{85} + (2 \beta_{2} - 4 \beta_1 + 10) q^{87} + ( - 2 \beta_{2} + 4 \beta_1) q^{89} + (4 \beta_{2} - 8) q^{91} + ( - 4 \beta_1 - 4) q^{93} + q^{95} + ( - \beta_{2} - 6 \beta_1 + 9) q^{97} + (\beta_{2} - 3 \beta_1 + 6) q^{99}+O(q^{100})$$ q + (b1 - 1) * q^3 + q^5 + (b2 - b1) * q^7 + (b2 - b1 + 1) * q^9 + (b2 + b1 + 2) * q^11 + (-b2 + 3) * q^13 + (b1 - 1) * q^15 + 2*b2 * q^17 + q^19 + (-2*b2 + 2*b1 - 4) * q^21 + (-b2 - b1 + 2) * q^23 + q^25 + (-2*b2 - 2) * q^27 + (2*b1 - 4) * q^29 + (-2*b2 - 2*b1) * q^31 + 4*b1 * q^33 + (b2 - b1) * q^35 + (-b2 + 7) * q^37 + (b2 + b1 - 2) * q^39 - 2*b2 * q^41 + (-b2 + 5*b1) * q^43 + (b2 - b1 + 1) * q^45 + (b2 - b1) * q^47 + (-4*b1 + 5) * q^49 + (-2*b2 + 4*b1 - 2) * q^51 + (b2 + 5) * q^53 + (b2 + b1 + 2) * q^55 + (b1 - 1) * q^57 + (2*b1 + 6) * q^59 + (3*b2 + b1 - 2) * q^61 + (b2 - 5*b1 + 12) * q^63 + (-b2 + 3) * q^65 + (-2*b2 - 3*b1 + 1) * q^67 - 4 * q^69 + 4*b1 * q^71 + 2*b1 * q^73 + (b1 - 1) * q^75 + (-4*b1 + 4) * q^77 + (-6*b1 + 2) * q^79 + (-b2 - 3*b1 + 1) * q^81 + (b2 + b1 + 10) * q^83 + 2*b2 * q^85 + (2*b2 - 4*b1 + 10) * q^87 + (-2*b2 + 4*b1) * q^89 + (4*b2 - 8) * q^91 + (-4*b1 - 4) * q^93 + q^95 + (-b2 - 6*b1 + 9) * q^97 + (b2 - 3*b1 + 6) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 2 q^{3} + 3 q^{5} + 3 q^{9}+O(q^{10})$$ 3 * q - 2 * q^3 + 3 * q^5 + 3 * q^9 $$3 q - 2 q^{3} + 3 q^{5} + 3 q^{9} + 8 q^{11} + 8 q^{13} - 2 q^{15} + 2 q^{17} + 3 q^{19} - 12 q^{21} + 4 q^{23} + 3 q^{25} - 8 q^{27} - 10 q^{29} - 4 q^{31} + 4 q^{33} + 20 q^{37} - 4 q^{39} - 2 q^{41} + 4 q^{43} + 3 q^{45} + 11 q^{49} - 4 q^{51} + 16 q^{53} + 8 q^{55} - 2 q^{57} + 20 q^{59} - 2 q^{61} + 32 q^{63} + 8 q^{65} - 2 q^{67} - 12 q^{69} + 4 q^{71} + 2 q^{73} - 2 q^{75} + 8 q^{77} - q^{81} + 32 q^{83} + 2 q^{85} + 28 q^{87} + 2 q^{89} - 20 q^{91} - 16 q^{93} + 3 q^{95} + 20 q^{97} + 16 q^{99}+O(q^{100})$$ 3 * q - 2 * q^3 + 3 * q^5 + 3 * q^9 + 8 * q^11 + 8 * q^13 - 2 * q^15 + 2 * q^17 + 3 * q^19 - 12 * q^21 + 4 * q^23 + 3 * q^25 - 8 * q^27 - 10 * q^29 - 4 * q^31 + 4 * q^33 + 20 * q^37 - 4 * q^39 - 2 * q^41 + 4 * q^43 + 3 * q^45 + 11 * q^49 - 4 * q^51 + 16 * q^53 + 8 * q^55 - 2 * q^57 + 20 * q^59 - 2 * q^61 + 32 * q^63 + 8 * q^65 - 2 * q^67 - 12 * q^69 + 4 * q^71 + 2 * q^73 - 2 * q^75 + 8 * q^77 - q^81 + 32 * q^83 + 2 * q^85 + 28 * q^87 + 2 * q^89 - 20 * q^91 - 16 * q^93 + 3 * q^95 + 20 * q^97 + 16 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 3x + 1$$ :

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

## 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
 0.311108 −1.48119 2.17009
0 −2.90321 0 1.00000 0 4.42864 0 5.42864 0
1.2 0 −0.806063 0 1.00000 0 −3.35026 0 −2.35026 0
1.3 0 1.70928 0 1.00000 0 −1.07838 0 −0.0783777 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.p 3
4.b odd 2 1 95.2.a.a 3
5.b even 2 1 7600.2.a.bx 3
8.b even 2 1 6080.2.a.by 3
8.d odd 2 1 6080.2.a.bo 3
12.b even 2 1 855.2.a.i 3
20.d odd 2 1 475.2.a.f 3
20.e even 4 2 475.2.b.d 6
28.d even 2 1 4655.2.a.u 3
60.h even 2 1 4275.2.a.bk 3
76.d even 2 1 1805.2.a.f 3
380.d even 2 1 9025.2.a.bb 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.2.a.a 3 4.b odd 2 1
475.2.a.f 3 20.d odd 2 1
475.2.b.d 6 20.e even 4 2
855.2.a.i 3 12.b even 2 1
1520.2.a.p 3 1.a even 1 1 trivial
1805.2.a.f 3 76.d even 2 1
4275.2.a.bk 3 60.h even 2 1
4655.2.a.u 3 28.d even 2 1
6080.2.a.bo 3 8.d odd 2 1
6080.2.a.by 3 8.b even 2 1
7600.2.a.bx 3 5.b even 2 1
9025.2.a.bb 3 380.d even 2 1

## Hecke kernels

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

 $$T_{3}^{3} + 2T_{3}^{2} - 4T_{3} - 4$$ T3^3 + 2*T3^2 - 4*T3 - 4 $$T_{7}^{3} - 16T_{7} - 16$$ T7^3 - 16*T7 - 16

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$T^{3} + 2 T^{2} - 4 T - 4$$
$5$ $$(T - 1)^{3}$$
$7$ $$T^{3} - 16T - 16$$
$11$ $$T^{3} - 8 T^{2} + 8 T + 16$$
$13$ $$T^{3} - 8 T^{2} + 12 T - 4$$
$17$ $$T^{3} - 2 T^{2} - 36 T + 104$$
$19$ $$(T - 1)^{3}$$
$23$ $$T^{3} - 4 T^{2} - 8 T + 16$$
$29$ $$T^{3} + 10 T^{2} + 12 T - 40$$
$31$ $$T^{3} + 4 T^{2} - 48 T - 64$$
$37$ $$T^{3} - 20 T^{2} + 124 T - 244$$
$41$ $$T^{3} + 2 T^{2} - 36 T - 104$$
$43$ $$T^{3} - 4 T^{2} - 144 T + 592$$
$47$ $$T^{3} - 16T - 16$$
$53$ $$T^{3} - 16 T^{2} + 76 T - 92$$
$59$ $$T^{3} - 20 T^{2} + 112 T - 160$$
$61$ $$T^{3} + 2 T^{2} - 84 T + 232$$
$67$ $$T^{3} + 2 T^{2} - 76 T + 116$$
$71$ $$T^{3} - 4 T^{2} - 80 T + 64$$
$73$ $$T^{3} - 2 T^{2} - 20 T + 8$$
$79$ $$T^{3} - 192T + 160$$
$83$ $$T^{3} - 32 T^{2} + 328 T - 1072$$
$89$ $$T^{3} - 2 T^{2} - 132 T + 680$$
$97$ $$T^{3} - 20 T^{2} - 60 T + 1748$$