# Properties

 Label 4560.2.d.g Level $4560$ Weight $2$ Character orbit 4560.d Analytic conductor $36.412$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4560 = 2^{4} \cdot 3 \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4560.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$36.4117833217$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.9821011968.3 Defining polynomial: $$x^{6} + 20 x^{4} + 118 x^{2} + 192$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2$$ 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 $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{3} - q^{5} + \beta_{1} q^{7} + q^{9} +O(q^{10})$$ $$q + q^{3} - q^{5} + \beta_{1} q^{7} + q^{9} + ( \beta_{1} + \beta_{2} ) q^{11} -\beta_{1} q^{13} - q^{15} + ( -1 + \beta_{3} ) q^{17} + ( \beta_{2} - \beta_{3} ) q^{19} + \beta_{1} q^{21} + ( \beta_{1} + \beta_{2} + \beta_{4} ) q^{23} + q^{25} + q^{27} + ( -\beta_{1} - \beta_{2} ) q^{29} + ( 2 + \beta_{5} ) q^{31} + ( \beta_{1} + \beta_{2} ) q^{33} -\beta_{1} q^{35} -\beta_{4} q^{37} -\beta_{1} q^{39} + ( -\beta_{1} + \beta_{2} ) q^{41} + \beta_{4} q^{43} - q^{45} + ( -\beta_{1} + \beta_{2} - \beta_{4} ) q^{47} + \beta_{3} q^{49} + ( -1 + \beta_{3} ) q^{51} + ( \beta_{1} + \beta_{2} + \beta_{4} ) q^{53} + ( -\beta_{1} - \beta_{2} ) q^{55} + ( \beta_{2} - \beta_{3} ) q^{57} + ( -2 + 2 \beta_{3} ) q^{59} -\beta_{5} q^{61} + \beta_{1} q^{63} + \beta_{1} q^{65} + ( -1 + \beta_{3} - \beta_{5} ) q^{67} + ( \beta_{1} + \beta_{2} + \beta_{4} ) q^{69} + ( -2 + 2 \beta_{3} ) q^{71} + ( -1 + 3 \beta_{3} + \beta_{5} ) q^{73} + q^{75} + ( -6 + 2 \beta_{3} - \beta_{5} ) q^{77} + ( -7 + \beta_{3} + \beta_{5} ) q^{79} + q^{81} + ( \beta_{1} + \beta_{2} - \beta_{4} ) q^{83} + ( 1 - \beta_{3} ) q^{85} + ( -\beta_{1} - \beta_{2} ) q^{87} + ( -2 \beta_{1} + \beta_{2} - \beta_{4} ) q^{89} + ( 7 - \beta_{3} ) q^{91} + ( 2 + \beta_{5} ) q^{93} + ( -\beta_{2} + \beta_{3} ) q^{95} + ( \beta_{1} + 4 \beta_{2} + 2 \beta_{4} ) q^{97} + ( \beta_{1} + \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q + 6q^{3} - 6q^{5} + 6q^{9} + O(q^{10})$$ $$6q + 6q^{3} - 6q^{5} + 6q^{9} - 6q^{15} - 4q^{17} - 2q^{19} + 6q^{25} + 6q^{27} + 12q^{31} - 6q^{45} + 2q^{49} - 4q^{51} - 2q^{57} - 8q^{59} - 4q^{67} - 8q^{71} + 6q^{75} - 32q^{77} - 40q^{79} + 6q^{81} + 4q^{85} + 40q^{91} + 12q^{93} + 2q^{95} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} + 20 x^{4} + 118 x^{2} + 192$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{5} + 12 \nu^{3} + 22 \nu$$$$)/8$$ $$\beta_{3}$$ $$=$$ $$\nu^{2} + 7$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{5} - 16 \nu^{3} - 58 \nu$$$$)/4$$ $$\beta_{5}$$ $$=$$ $$\nu^{4} + 13 \nu^{2} + 32$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} - 7$$ $$\nu^{3}$$ $$=$$ $$-\beta_{4} - 2 \beta_{2} - 9 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$\beta_{5} - 13 \beta_{3} + 59$$ $$\nu^{5}$$ $$=$$ $$12 \beta_{4} + 32 \beta_{2} + 86 \beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/4560\mathbb{Z}\right)^\times$$.

 $$n$$ $$1141$$ $$1711$$ $$1921$$ $$2737$$ $$3041$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2431.1
 − 3.24237i − 2.60808i − 1.63858i 1.63858i 2.60808i 3.24237i
0 1.00000 0 −1.00000 0 3.24237i 0 1.00000 0
2431.2 0 1.00000 0 −1.00000 0 2.60808i 0 1.00000 0
2431.3 0 1.00000 0 −1.00000 0 1.63858i 0 1.00000 0
2431.4 0 1.00000 0 −1.00000 0 1.63858i 0 1.00000 0
2431.5 0 1.00000 0 −1.00000 0 2.60808i 0 1.00000 0
2431.6 0 1.00000 0 −1.00000 0 3.24237i 0 1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 2431.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
76.d even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4560.2.d.g yes 6
4.b odd 2 1 4560.2.d.e 6
19.b odd 2 1 4560.2.d.e 6
76.d even 2 1 inner 4560.2.d.g yes 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4560.2.d.e 6 4.b odd 2 1
4560.2.d.e 6 19.b odd 2 1
4560.2.d.g yes 6 1.a even 1 1 trivial
4560.2.d.g yes 6 76.d even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(4560, [\chi])$$:

 $$T_{7}^{6} + 20 T_{7}^{4} + 118 T_{7}^{2} + 192$$ $$T_{31}^{3} - 6 T_{31}^{2} - 66 T_{31} + 404$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$( -1 + T )^{6}$$
$5$ $$( 1 + T )^{6}$$
$7$ $$192 + 118 T^{2} + 20 T^{4} + T^{6}$$
$11$ $$108 + 142 T^{2} + 38 T^{4} + T^{6}$$
$13$ $$192 + 118 T^{2} + 20 T^{4} + T^{6}$$
$17$ $$( -12 - 14 T + 2 T^{2} + T^{3} )^{2}$$
$19$ $$6859 + 722 T - 57 T^{2} + 52 T^{3} - 3 T^{4} + 2 T^{5} + T^{6}$$
$23$ $$3072 + 2176 T^{2} + 98 T^{4} + T^{6}$$
$29$ $$108 + 142 T^{2} + 38 T^{4} + T^{6}$$
$31$ $$( 404 - 66 T - 6 T^{2} + T^{3} )^{2}$$
$37$ $$48 + 742 T^{2} + 92 T^{4} + T^{6}$$
$41$ $$108 + 270 T^{2} + 54 T^{4} + T^{6}$$
$43$ $$48 + 742 T^{2} + 92 T^{4} + T^{6}$$
$47$ $$768 + 4672 T^{2} + 146 T^{4} + T^{6}$$
$53$ $$3072 + 2176 T^{2} + 98 T^{4} + T^{6}$$
$59$ $$( -96 - 56 T + 4 T^{2} + T^{3} )^{2}$$
$61$ $$( -256 - 78 T + T^{3} )^{2}$$
$67$ $$( -96 - 96 T + 2 T^{2} + T^{3} )^{2}$$
$71$ $$( -96 - 56 T + 4 T^{2} + T^{3} )^{2}$$
$73$ $$( -976 - 204 T + T^{3} )^{2}$$
$79$ $$( -128 + 44 T + 20 T^{2} + T^{3} )^{2}$$
$83$ $$84672 + 6912 T^{2} + 162 T^{4} + T^{6}$$
$89$ $$57132 + 6750 T^{2} + 198 T^{4} + T^{6}$$
$97$ $$4853952 + 103734 T^{2} + 612 T^{4} + T^{6}$$