# Properties

 Label 1560.2.g.h Level $1560$ Weight $2$ Character orbit 1560.g 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.g (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$12.4566627153$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{17})$$ Defining polynomial: $$x^{4} + 9x^{2} + 16$$ x^4 + 9*x^2 + 16 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + q^{3} - \beta_{2} q^{5} + ( - \beta_{2} + \beta_1) q^{7} + q^{9}+O(q^{10})$$ q + q^3 - b2 * q^5 + (-b2 + b1) * q^7 + q^9 $$q + q^{3} - \beta_{2} q^{5} + ( - \beta_{2} + \beta_1) q^{7} + q^{9} + (\beta_{2} + \beta_1) q^{11} + (\beta_{3} + \beta_{2} - \beta_1 + 1) q^{13} - \beta_{2} q^{15} + (\beta_{3} + 2) q^{17} + 2 \beta_1 q^{19} + ( - \beta_{2} + \beta_1) q^{21} - \beta_{3} q^{23} - q^{25} + q^{27} - 2 q^{29} - 2 \beta_{2} q^{31} + (\beta_{2} + \beta_1) q^{33} + (\beta_{3} - 2) q^{35} + ( - 3 \beta_{2} + \beta_1) q^{37} + (\beta_{3} + \beta_{2} - \beta_1 + 1) q^{39} + (\beta_{2} + 3 \beta_1) q^{41} + (2 \beta_{3} + 4) q^{43} - \beta_{2} q^{45} + (3 \beta_{3} - 1) q^{49} + (\beta_{3} + 2) q^{51} + (3 \beta_{3} - 2) q^{53} + \beta_{3} q^{55} + 2 \beta_1 q^{57} + ( - 4 \beta_{2} - 4 \beta_1) q^{59} + (3 \beta_{3} - 2) q^{61} + ( - \beta_{2} + \beta_1) q^{63} + ( - \beta_{3} - 2 \beta_{2} - \beta_1 + 2) q^{65} + 6 \beta_1 q^{67} - \beta_{3} q^{69} + (7 \beta_{2} - \beta_1) q^{71} + ( - 6 \beta_{2} - 2 \beta_1) q^{73} - q^{75} + (\beta_{3} - 4) q^{77} + (3 \beta_{3} - 4) q^{79} + q^{81} + (2 \beta_{2} + 2 \beta_1) q^{83} + ( - 3 \beta_{2} - \beta_1) q^{85} - 2 q^{87} + ( - 3 \beta_{2} - \beta_1) q^{89} + ( - 3 \beta_{3} + 2 \beta_{2} + 8) q^{91} - 2 \beta_{2} q^{93} + (2 \beta_{3} - 2) q^{95} + (\beta_{2} + \beta_1) q^{97} + (\beta_{2} + \beta_1) q^{99}+O(q^{100})$$ q + q^3 - b2 * q^5 + (-b2 + b1) * q^7 + q^9 + (b2 + b1) * q^11 + (b3 + b2 - b1 + 1) * q^13 - b2 * q^15 + (b3 + 2) * q^17 + 2*b1 * q^19 + (-b2 + b1) * q^21 - b3 * q^23 - q^25 + q^27 - 2 * q^29 - 2*b2 * q^31 + (b2 + b1) * q^33 + (b3 - 2) * q^35 + (-3*b2 + b1) * q^37 + (b3 + b2 - b1 + 1) * q^39 + (b2 + 3*b1) * q^41 + (2*b3 + 4) * q^43 - b2 * q^45 + (3*b3 - 1) * q^49 + (b3 + 2) * q^51 + (3*b3 - 2) * q^53 + b3 * q^55 + 2*b1 * q^57 + (-4*b2 - 4*b1) * q^59 + (3*b3 - 2) * q^61 + (-b2 + b1) * q^63 + (-b3 - 2*b2 - b1 + 2) * q^65 + 6*b1 * q^67 - b3 * q^69 + (7*b2 - b1) * q^71 + (-6*b2 - 2*b1) * q^73 - q^75 + (b3 - 4) * q^77 + (3*b3 - 4) * q^79 + q^81 + (2*b2 + 2*b1) * q^83 + (-3*b2 - b1) * q^85 - 2 * q^87 + (-3*b2 - b1) * q^89 + (-3*b3 + 2*b2 + 8) * q^91 - 2*b2 * q^93 + (2*b3 - 2) * q^95 + (b2 + b1) * q^97 + (b2 + b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{3} + 4 q^{9}+O(q^{10})$$ 4 * q + 4 * q^3 + 4 * q^9 $$4 q + 4 q^{3} + 4 q^{9} + 6 q^{13} + 10 q^{17} - 2 q^{23} - 4 q^{25} + 4 q^{27} - 8 q^{29} - 6 q^{35} + 6 q^{39} + 20 q^{43} + 2 q^{49} + 10 q^{51} - 2 q^{53} + 2 q^{55} - 2 q^{61} + 6 q^{65} - 2 q^{69} - 4 q^{75} - 14 q^{77} - 10 q^{79} + 4 q^{81} - 8 q^{87} + 26 q^{91} - 4 q^{95}+O(q^{100})$$ 4 * q + 4 * q^3 + 4 * q^9 + 6 * q^13 + 10 * q^17 - 2 * q^23 - 4 * q^25 + 4 * q^27 - 8 * q^29 - 6 * q^35 + 6 * q^39 + 20 * q^43 + 2 * q^49 + 10 * q^51 - 2 * q^53 + 2 * q^55 - 2 * q^61 + 6 * q^65 - 2 * q^69 - 4 * q^75 - 14 * q^77 - 10 * q^79 + 4 * q^81 - 8 * q^87 + 26 * q^91 - 4 * q^95

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 9x^{2} + 16$$ :

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

## Character values

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

 $$n$$ $$391$$ $$521$$ $$781$$ $$937$$ $$1081$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
961.1
 − 2.56155i 1.56155i − 1.56155i 2.56155i
0 1.00000 0 1.00000i 0 3.56155i 0 1.00000 0
961.2 0 1.00000 0 1.00000i 0 0.561553i 0 1.00000 0
961.3 0 1.00000 0 1.00000i 0 0.561553i 0 1.00000 0
961.4 0 1.00000 0 1.00000i 0 3.56155i 0 1.00000 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.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1560.2.g.h 4
3.b odd 2 1 4680.2.g.g 4
4.b odd 2 1 3120.2.g.p 4
13.b even 2 1 inner 1560.2.g.h 4
39.d odd 2 1 4680.2.g.g 4
52.b odd 2 1 3120.2.g.p 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1560.2.g.h 4 1.a even 1 1 trivial
1560.2.g.h 4 13.b even 2 1 inner
3120.2.g.p 4 4.b odd 2 1
3120.2.g.p 4 52.b odd 2 1
4680.2.g.g 4 3.b odd 2 1
4680.2.g.g 4 39.d odd 2 1

## Hecke kernels

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

 $$T_{7}^{4} + 13T_{7}^{2} + 4$$ T7^4 + 13*T7^2 + 4 $$T_{11}^{4} + 9T_{11}^{2} + 16$$ T11^4 + 9*T11^2 + 16

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$(T - 1)^{4}$$
$5$ $$(T^{2} + 1)^{2}$$
$7$ $$T^{4} + 13T^{2} + 4$$
$11$ $$T^{4} + 9T^{2} + 16$$
$13$ $$T^{4} - 6 T^{3} + 18 T^{2} - 78 T + 169$$
$17$ $$(T^{2} - 5 T + 2)^{2}$$
$19$ $$T^{4} + 36T^{2} + 256$$
$23$ $$(T^{2} + T - 4)^{2}$$
$29$ $$(T + 2)^{4}$$
$31$ $$(T^{2} + 4)^{2}$$
$37$ $$T^{4} + 33T^{2} + 64$$
$41$ $$T^{4} + 77T^{2} + 1444$$
$43$ $$(T^{2} - 10 T + 8)^{2}$$
$47$ $$T^{4}$$
$53$ $$(T^{2} + T - 38)^{2}$$
$59$ $$T^{4} + 144T^{2} + 4096$$
$61$ $$(T^{2} + T - 38)^{2}$$
$67$ $$T^{4} + 324 T^{2} + 20736$$
$71$ $$T^{4} + 121T^{2} + 2704$$
$73$ $$T^{4} + 84T^{2} + 64$$
$79$ $$(T^{2} + 5 T - 32)^{2}$$
$83$ $$T^{4} + 36T^{2} + 256$$
$89$ $$T^{4} + 21T^{2} + 4$$
$97$ $$T^{4} + 9T^{2} + 16$$