# Properties

 Label 1560.2.bg.e 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: $$\Q(\sqrt{-3}, \sqrt{10})$$ Defining polynomial: $$x^{4} + 10x^{2} + 100$$ x^4 + 10*x^2 + 100 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.

 $$f(q)$$ $$=$$ $$q - \beta_{2} q^{3} - q^{5} + ( - \beta_{2} + \beta_1 - 1) q^{7} + ( - \beta_{2} - 1) q^{9}+O(q^{10})$$ q - b2 * q^3 - q^5 + (-b2 + b1 - 1) * q^7 + (-b2 - 1) * q^9 $$q - \beta_{2} q^{3} - q^{5} + ( - \beta_{2} + \beta_1 - 1) q^{7} + ( - \beta_{2} - 1) q^{9} - 2 \beta_{2} q^{11} + ( - \beta_{2} - \beta_1 + 1) q^{13} + \beta_{2} q^{15} + (4 \beta_{2} + \beta_1 + 4) q^{17} + ( - 2 \beta_{2} - 2) q^{19} + ( - \beta_{3} - 1) q^{21} - 2 \beta_{2} q^{23} + q^{25} - q^{27} + (\beta_{3} - 4 \beta_{2} + \beta_1) q^{29} + ( - 2 \beta_{3} - 3) q^{31} + ( - 2 \beta_{2} - 2) q^{33} + (\beta_{2} - \beta_1 + 1) q^{35} + ( - 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1) q^{37} + (\beta_{3} - 2 \beta_{2} - 1) q^{39} + ( - \beta_{3} + 2 \beta_{2} - \beta_1) q^{41} + ( - 7 \beta_{2} + \beta_1 - 7) q^{43} + (\beta_{2} + 1) q^{45} + ( - 3 \beta_{3} + 2) q^{47} + ( - 2 \beta_{3} + 4 \beta_{2} - 2 \beta_1) q^{49} + ( - \beta_{3} + 4) q^{51} - 2 \beta_{3} q^{53} + 2 \beta_{2} q^{55} - 2 q^{57} + (4 \beta_{2} - \beta_1 + 4) q^{59} + ( - 13 \beta_{2} - 13) q^{61} + ( - \beta_{3} + \beta_{2} - \beta_1) q^{63} + (\beta_{2} + \beta_1 - 1) q^{65} + ( - \beta_{3} + 7 \beta_{2} - \beta_1) q^{67} + ( - 2 \beta_{2} - 2) q^{69} + (4 \beta_{2} + 3 \beta_1 + 4) q^{71} + ( - \beta_{3} - 3) q^{73} - \beta_{2} q^{75} + ( - 2 \beta_{3} - 2) q^{77} + 11 q^{79} + \beta_{2} q^{81} + ( - 4 \beta_{2} - \beta_1 - 4) q^{85} + ( - 4 \beta_{2} + \beta_1 - 4) q^{87} + (3 \beta_{3} - 6 \beta_{2} + 3 \beta_1) q^{89} + ( - 11 \beta_{2} + 2 \beta_1 - 2) q^{91} + ( - 2 \beta_{3} + 3 \beta_{2} - 2 \beta_1) q^{93} + (2 \beta_{2} + 2) q^{95} + ( - 3 \beta_{2} - 5 \beta_1 - 3) q^{97} - 2 q^{99}+O(q^{100})$$ q - b2 * q^3 - q^5 + (-b2 + b1 - 1) * q^7 + (-b2 - 1) * q^9 - 2*b2 * q^11 + (-b2 - b1 + 1) * q^13 + b2 * q^15 + (4*b2 + b1 + 4) * q^17 + (-2*b2 - 2) * q^19 + (-b3 - 1) * q^21 - 2*b2 * q^23 + q^25 - q^27 + (b3 - 4*b2 + b1) * q^29 + (-2*b3 - 3) * q^31 + (-2*b2 - 2) * q^33 + (b2 - b1 + 1) * q^35 + (-2*b3 + 2*b2 - 2*b1) * q^37 + (b3 - 2*b2 - 1) * q^39 + (-b3 + 2*b2 - b1) * q^41 + (-7*b2 + b1 - 7) * q^43 + (b2 + 1) * q^45 + (-3*b3 + 2) * q^47 + (-2*b3 + 4*b2 - 2*b1) * q^49 + (-b3 + 4) * q^51 - 2*b3 * q^53 + 2*b2 * q^55 - 2 * q^57 + (4*b2 - b1 + 4) * q^59 + (-13*b2 - 13) * q^61 + (-b3 + b2 - b1) * q^63 + (b2 + b1 - 1) * q^65 + (-b3 + 7*b2 - b1) * q^67 + (-2*b2 - 2) * q^69 + (4*b2 + 3*b1 + 4) * q^71 + (-b3 - 3) * q^73 - b2 * q^75 + (-2*b3 - 2) * q^77 + 11 * q^79 + b2 * q^81 + (-4*b2 - b1 - 4) * q^85 + (-4*b2 + b1 - 4) * q^87 + (3*b3 - 6*b2 + 3*b1) * q^89 + (-11*b2 + 2*b1 - 2) * q^91 + (-2*b3 + 3*b2 - 2*b1) * q^93 + (2*b2 + 2) * q^95 + (-3*b2 - 5*b1 - 3) * q^97 - 2 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{3} - 4 q^{5} - 2 q^{7} - 2 q^{9}+O(q^{10})$$ 4 * q + 2 * q^3 - 4 * q^5 - 2 * q^7 - 2 * q^9 $$4 q + 2 q^{3} - 4 q^{5} - 2 q^{7} - 2 q^{9} + 4 q^{11} + 6 q^{13} - 2 q^{15} + 8 q^{17} - 4 q^{19} - 4 q^{21} + 4 q^{23} + 4 q^{25} - 4 q^{27} + 8 q^{29} - 12 q^{31} - 4 q^{33} + 2 q^{35} - 4 q^{37} - 4 q^{41} - 14 q^{43} + 2 q^{45} + 8 q^{47} - 8 q^{49} + 16 q^{51} - 4 q^{55} - 8 q^{57} + 8 q^{59} - 26 q^{61} - 2 q^{63} - 6 q^{65} - 14 q^{67} - 4 q^{69} + 8 q^{71} - 12 q^{73} + 2 q^{75} - 8 q^{77} + 44 q^{79} - 2 q^{81} - 8 q^{85} - 8 q^{87} + 12 q^{89} + 14 q^{91} - 6 q^{93} + 4 q^{95} - 6 q^{97} - 8 q^{99}+O(q^{100})$$ 4 * q + 2 * q^3 - 4 * q^5 - 2 * q^7 - 2 * q^9 + 4 * q^11 + 6 * q^13 - 2 * q^15 + 8 * q^17 - 4 * q^19 - 4 * q^21 + 4 * q^23 + 4 * q^25 - 4 * q^27 + 8 * q^29 - 12 * q^31 - 4 * q^33 + 2 * q^35 - 4 * q^37 - 4 * q^41 - 14 * q^43 + 2 * q^45 + 8 * q^47 - 8 * q^49 + 16 * q^51 - 4 * q^55 - 8 * q^57 + 8 * q^59 - 26 * q^61 - 2 * q^63 - 6 * q^65 - 14 * q^67 - 4 * q^69 + 8 * q^71 - 12 * q^73 + 2 * q^75 - 8 * q^77 + 44 * q^79 - 2 * q^81 - 8 * q^85 - 8 * q^87 + 12 * q^89 + 14 * q^91 - 6 * q^93 + 4 * q^95 - 6 * q^97 - 8 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 10x^{2} + 100$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} ) / 10$$ (v^2) / 10 $$\beta_{3}$$ $$=$$ $$( \nu^{3} ) / 10$$ (v^3) / 10
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$10\beta_{2}$$ 10*b2 $$\nu^{3}$$ $$=$$ $$10\beta_{3}$$ 10*b3

## 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
 −1.58114 + 2.73861i 1.58114 − 2.73861i −1.58114 − 2.73861i 1.58114 + 2.73861i
0 0.500000 + 0.866025i 0 −1.00000 0 −2.08114 + 3.60464i 0 −0.500000 + 0.866025i 0
601.2 0 0.500000 + 0.866025i 0 −1.00000 0 1.08114 1.87259i 0 −0.500000 + 0.866025i 0
841.1 0 0.500000 0.866025i 0 −1.00000 0 −2.08114 3.60464i 0 −0.500000 0.866025i 0
841.2 0 0.500000 0.866025i 0 −1.00000 0 1.08114 + 1.87259i 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.e 4
13.c even 3 1 inner 1560.2.bg.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1560.2.bg.e 4 1.a even 1 1 trivial
1560.2.bg.e 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 $$S_{2}^{\mathrm{new}}(1560, [\chi])$$:

 $$T_{7}^{4} + 2T_{7}^{3} + 13T_{7}^{2} - 18T_{7} + 81$$ T7^4 + 2*T7^3 + 13*T7^2 - 18*T7 + 81 $$T_{11}^{2} - 2T_{11} + 4$$ T11^2 - 2*T11 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$(T^{2} - T + 1)^{2}$$
$5$ $$(T + 1)^{4}$$
$7$ $$T^{4} + 2 T^{3} + 13 T^{2} - 18 T + 81$$
$11$ $$(T^{2} - 2 T + 4)^{2}$$
$13$ $$T^{4} - 6 T^{3} + 25 T^{2} - 78 T + 169$$
$17$ $$T^{4} - 8 T^{3} + 58 T^{2} - 48 T + 36$$
$19$ $$(T^{2} + 2 T + 4)^{2}$$
$23$ $$(T^{2} - 2 T + 4)^{2}$$
$29$ $$T^{4} - 8 T^{3} + 58 T^{2} - 48 T + 36$$
$31$ $$(T^{2} + 6 T - 31)^{2}$$
$37$ $$T^{4} + 4 T^{3} + 52 T^{2} + \cdots + 1296$$
$41$ $$T^{4} + 4 T^{3} + 22 T^{2} - 24 T + 36$$
$43$ $$T^{4} + 14 T^{3} + 157 T^{2} + \cdots + 1521$$
$47$ $$(T^{2} - 4 T - 86)^{2}$$
$53$ $$(T^{2} - 40)^{2}$$
$59$ $$T^{4} - 8 T^{3} + 58 T^{2} - 48 T + 36$$
$61$ $$(T^{2} + 13 T + 169)^{2}$$
$67$ $$T^{4} + 14 T^{3} + 157 T^{2} + \cdots + 1521$$
$71$ $$T^{4} - 8 T^{3} + 138 T^{2} + \cdots + 5476$$
$73$ $$(T^{2} + 6 T - 1)^{2}$$
$79$ $$(T - 11)^{4}$$
$83$ $$T^{4}$$
$89$ $$T^{4} - 12 T^{3} + 198 T^{2} + \cdots + 2916$$
$97$ $$T^{4} + 6 T^{3} + 277 T^{2} + \cdots + 58081$$