# Properties

 Label 1560.2.bg.f Level $1560$ Weight $2$ Character orbit 1560.bg Analytic conductor $12.457$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1560 = 2^{3} \cdot 3 \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1560.bg (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$12.4566627153$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{17})$$ Defining polynomial: $$x^{4} - x^{3} + 5x^{2} + 4x + 16$$ x^4 - x^3 + 5*x^2 + 4*x + 16 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{3} + q^{5} + (\beta_{3} + 2 \beta_{2} + \beta_1 - 2) q^{7} + (\beta_{2} - 1) q^{9}+O(q^{10})$$ q + b2 * q^3 + q^5 + (b3 + 2*b2 + b1 - 2) * q^7 + (b2 - 1) * q^9 $$q + \beta_{2} q^{3} + q^{5} + (\beta_{3} + 2 \beta_{2} + \beta_1 - 2) q^{7} + (\beta_{2} - 1) q^{9} + ( - 3 \beta_{2} + \beta_1) q^{11} + (2 \beta_{3} + \beta_1 + 1) q^{13} + \beta_{2} q^{15} + ( - 2 \beta_{2} + 2) q^{17} + (2 \beta_{3} - 2 \beta_{2} + 2 \beta_1 + 2) q^{19} + (\beta_{3} - 2) q^{21} + ( - 3 \beta_{2} + \beta_1) q^{23} + q^{25} - q^{27} + ( - \beta_{2} - 3 \beta_1) q^{29} + (2 \beta_{3} + 1) q^{31} + (\beta_{3} - 3 \beta_{2} + \beta_1 + 3) q^{33} + (\beta_{3} + 2 \beta_{2} + \beta_1 - 2) q^{35} + ( - 3 \beta_{2} + \beta_1) q^{37} + (\beta_{3} + \beta_{2} - \beta_1) q^{39} + 2 \beta_{2} q^{41} + (2 \beta_{3} - 3 \beta_{2} + 2 \beta_1 + 3) q^{43} + (\beta_{2} - 1) q^{45} + (\beta_{3} + 5) q^{47} + ( - \beta_{2} - 5 \beta_1) q^{49} + 2 q^{51} + (2 \beta_{3} - 2) q^{53} + ( - 3 \beta_{2} + \beta_1) q^{55} + (2 \beta_{3} + 2) q^{57} + (3 \beta_{3} + \beta_{2} + 3 \beta_1 - 1) q^{59} + ( - 3 \beta_{3} + 6 \beta_{2} - 3 \beta_1 - 6) q^{61} + ( - 2 \beta_{2} - \beta_1) q^{63} + (2 \beta_{3} + \beta_1 + 1) q^{65} + (4 \beta_{2} + \beta_1) q^{67} + (\beta_{3} - 3 \beta_{2} + \beta_1 + 3) q^{69} + ( - \beta_{3} + 6) q^{73} + \beta_{2} q^{75} + 2 q^{77} + (4 \beta_{3} - 7) q^{79} - \beta_{2} q^{81} + ( - 6 \beta_{3} - 2) q^{83} + ( - 2 \beta_{2} + 2) q^{85} + ( - 3 \beta_{3} - \beta_{2} - 3 \beta_1 + 1) q^{87} - 2 \beta_{2} q^{89} + ( - 2 \beta_{3} - 6 \beta_{2} - 5 \beta_1 + 2) q^{91} + (\beta_{2} - 2 \beta_1) q^{93} + (2 \beta_{3} - 2 \beta_{2} + 2 \beta_1 + 2) q^{95} + (\beta_{3} + 10 \beta_{2} + \beta_1 - 10) q^{97} + (\beta_{3} + 3) q^{99}+O(q^{100})$$ q + b2 * q^3 + q^5 + (b3 + 2*b2 + b1 - 2) * q^7 + (b2 - 1) * q^9 + (-3*b2 + b1) * q^11 + (2*b3 + b1 + 1) * q^13 + b2 * q^15 + (-2*b2 + 2) * q^17 + (2*b3 - 2*b2 + 2*b1 + 2) * q^19 + (b3 - 2) * q^21 + (-3*b2 + b1) * q^23 + q^25 - q^27 + (-b2 - 3*b1) * q^29 + (2*b3 + 1) * q^31 + (b3 - 3*b2 + b1 + 3) * q^33 + (b3 + 2*b2 + b1 - 2) * q^35 + (-3*b2 + b1) * q^37 + (b3 + b2 - b1) * q^39 + 2*b2 * q^41 + (2*b3 - 3*b2 + 2*b1 + 3) * q^43 + (b2 - 1) * q^45 + (b3 + 5) * q^47 + (-b2 - 5*b1) * q^49 + 2 * q^51 + (2*b3 - 2) * q^53 + (-3*b2 + b1) * q^55 + (2*b3 + 2) * q^57 + (3*b3 + b2 + 3*b1 - 1) * q^59 + (-3*b3 + 6*b2 - 3*b1 - 6) * q^61 + (-2*b2 - b1) * q^63 + (2*b3 + b1 + 1) * q^65 + (4*b2 + b1) * q^67 + (b3 - 3*b2 + b1 + 3) * q^69 + (-b3 + 6) * q^73 + b2 * q^75 + 2 * q^77 + (4*b3 - 7) * q^79 - b2 * q^81 + (-6*b3 - 2) * q^83 + (-2*b2 + 2) * q^85 + (-3*b3 - b2 - 3*b1 + 1) * q^87 - 2*b2 * q^89 + (-2*b3 - 6*b2 - 5*b1 + 2) * q^91 + (b2 - 2*b1) * q^93 + (2*b3 - 2*b2 + 2*b1 + 2) * q^95 + (b3 + 10*b2 + b1 - 10) * q^97 + (b3 + 3) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{3} + 4 q^{5} - 5 q^{7} - 2 q^{9}+O(q^{10})$$ 4 * q + 2 * q^3 + 4 * q^5 - 5 * q^7 - 2 * q^9 $$4 q + 2 q^{3} + 4 q^{5} - 5 q^{7} - 2 q^{9} - 5 q^{11} + q^{13} + 2 q^{15} + 4 q^{17} + 2 q^{19} - 10 q^{21} - 5 q^{23} + 4 q^{25} - 4 q^{27} - 5 q^{29} + 5 q^{33} - 5 q^{35} - 5 q^{37} - q^{39} + 4 q^{41} + 4 q^{43} - 2 q^{45} + 18 q^{47} - 7 q^{49} + 8 q^{51} - 12 q^{53} - 5 q^{55} + 4 q^{57} - 5 q^{59} - 9 q^{61} - 5 q^{63} + q^{65} + 9 q^{67} + 5 q^{69} + 26 q^{73} + 2 q^{75} + 8 q^{77} - 36 q^{79} - 2 q^{81} + 4 q^{83} + 4 q^{85} + 5 q^{87} - 4 q^{89} - 5 q^{91} + 2 q^{95} - 21 q^{97} + 10 q^{99}+O(q^{100})$$ 4 * q + 2 * q^3 + 4 * q^5 - 5 * q^7 - 2 * q^9 - 5 * q^11 + q^13 + 2 * q^15 + 4 * q^17 + 2 * q^19 - 10 * q^21 - 5 * q^23 + 4 * q^25 - 4 * q^27 - 5 * q^29 + 5 * q^33 - 5 * q^35 - 5 * q^37 - q^39 + 4 * q^41 + 4 * q^43 - 2 * q^45 + 18 * q^47 - 7 * q^49 + 8 * q^51 - 12 * q^53 - 5 * q^55 + 4 * q^57 - 5 * q^59 - 9 * q^61 - 5 * q^63 + q^65 + 9 * q^67 + 5 * q^69 + 26 * q^73 + 2 * q^75 + 8 * q^77 - 36 * q^79 - 2 * q^81 + 4 * q^83 + 4 * q^85 + 5 * q^87 - 4 * q^89 - 5 * q^91 + 2 * q^95 - 21 * q^97 + 10 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} + 5x^{2} + 4x + 16$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( -\nu^{3} + 5\nu^{2} - 5\nu + 16 ) / 20$$ (-v^3 + 5*v^2 - 5*v + 16) / 20 $$\beta_{3}$$ $$=$$ $$( \nu^{3} + 4 ) / 5$$ (v^3 + 4) / 5
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} + 4\beta_{2} + \beta _1 - 4$$ b3 + 4*b2 + b1 - 4 $$\nu^{3}$$ $$=$$ $$5\beta_{3} - 4$$ 5*b3 - 4

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1560\mathbb{Z}\right)^\times$$.

 $$n$$ $$391$$ $$521$$ $$781$$ $$937$$ $$1081$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$1$$ $$-1 + \beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
601.1
 1.28078 + 2.21837i −0.780776 − 1.35234i 1.28078 − 2.21837i −0.780776 + 1.35234i
0 0.500000 + 0.866025i 0 1.00000 0 −2.28078 + 3.95042i 0 −0.500000 + 0.866025i 0
601.2 0 0.500000 + 0.866025i 0 1.00000 0 −0.219224 + 0.379706i 0 −0.500000 + 0.866025i 0
841.1 0 0.500000 0.866025i 0 1.00000 0 −2.28078 3.95042i 0 −0.500000 0.866025i 0
841.2 0 0.500000 0.866025i 0 1.00000 0 −0.219224 0.379706i 0 −0.500000 0.866025i 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
13.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1560.2.bg.f 4
13.c even 3 1 inner 1560.2.bg.f 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1560.2.bg.f 4 1.a even 1 1 trivial
1560.2.bg.f 4 13.c even 3 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1560, [\chi])$$:

 $$T_{7}^{4} + 5T_{7}^{3} + 23T_{7}^{2} + 10T_{7} + 4$$ T7^4 + 5*T7^3 + 23*T7^2 + 10*T7 + 4 $$T_{11}^{4} + 5T_{11}^{3} + 23T_{11}^{2} + 10T_{11} + 4$$ T11^4 + 5*T11^3 + 23*T11^2 + 10*T11 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$(T^{2} - T + 1)^{2}$$
$5$ $$(T - 1)^{4}$$
$7$ $$T^{4} + 5 T^{3} + 23 T^{2} + 10 T + 4$$
$11$ $$T^{4} + 5 T^{3} + 23 T^{2} + 10 T + 4$$
$13$ $$T^{4} - T^{3} - 12 T^{2} - 13 T + 169$$
$17$ $$(T^{2} - 2 T + 4)^{2}$$
$19$ $$T^{4} - 2 T^{3} + 20 T^{2} + 32 T + 256$$
$23$ $$T^{4} + 5 T^{3} + 23 T^{2} + 10 T + 4$$
$29$ $$T^{4} + 5 T^{3} + 57 T^{2} + \cdots + 1024$$
$31$ $$(T^{2} - 17)^{2}$$
$37$ $$T^{4} + 5 T^{3} + 23 T^{2} + 10 T + 4$$
$41$ $$(T^{2} - 2 T + 4)^{2}$$
$43$ $$T^{4} - 4 T^{3} + 29 T^{2} + 52 T + 169$$
$47$ $$(T^{2} - 9 T + 16)^{2}$$
$53$ $$(T^{2} + 6 T - 8)^{2}$$
$59$ $$T^{4} + 5 T^{3} + 57 T^{2} + \cdots + 1024$$
$61$ $$T^{4} + 9 T^{3} + 99 T^{2} - 162 T + 324$$
$67$ $$T^{4} - 9 T^{3} + 65 T^{2} - 144 T + 256$$
$71$ $$T^{4}$$
$73$ $$(T^{2} - 13 T + 38)^{2}$$
$79$ $$(T^{2} + 18 T + 13)^{2}$$
$83$ $$(T^{2} - 2 T - 152)^{2}$$
$89$ $$(T^{2} + 2 T + 4)^{2}$$
$97$ $$T^{4} + 21 T^{3} + 335 T^{2} + \cdots + 11236$$