# Properties

 Label 4560.2.d.h 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: 6.0.792772608.2 Defining polynomial: $$x^{6} + 22 x^{4} + 62 x^{2} + 12$$ 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.

 $$f(q)$$ $$=$$ $$q + q^{3} + q^{5} + ( -\beta_{1} - \beta_{2} ) q^{7} + q^{9} +O(q^{10})$$ $$q + q^{3} + q^{5} + ( -\beta_{1} - \beta_{2} ) q^{7} + q^{9} + \beta_{1} q^{11} + ( -\beta_{1} - \beta_{2} ) q^{13} + q^{15} + ( -1 - \beta_{3} + \beta_{4} ) q^{17} + ( 1 + \beta_{2} - \beta_{3} ) q^{19} + ( -\beta_{1} - \beta_{2} ) q^{21} + ( -\beta_{2} + \beta_{5} ) q^{23} + q^{25} + q^{27} + ( \beta_{1} - 2 \beta_{2} + 2 \beta_{5} ) q^{29} + ( 2 + \beta_{3} ) q^{31} + \beta_{1} q^{33} + ( -\beta_{1} - \beta_{2} ) q^{35} + ( \beta_{1} + \beta_{2} - \beta_{5} ) q^{37} + ( -\beta_{1} - \beta_{2} ) q^{39} + ( \beta_{1} - 2 \beta_{5} ) q^{41} + ( \beta_{1} + \beta_{2} - \beta_{5} ) q^{43} + q^{45} + ( -\beta_{2} + 3 \beta_{5} ) q^{47} + ( -2 - 3 \beta_{3} - \beta_{4} ) q^{49} + ( -1 - \beta_{3} + \beta_{4} ) q^{51} + ( -\beta_{2} + \beta_{5} ) q^{53} + \beta_{1} q^{55} + ( 1 + \beta_{2} - \beta_{3} ) q^{57} + ( -2 - 2 \beta_{3} - 2 \beta_{4} ) q^{59} + ( -2 - 3 \beta_{3} - 2 \beta_{4} ) q^{61} + ( -\beta_{1} - \beta_{2} ) q^{63} + ( -\beta_{1} - \beta_{2} ) q^{65} + ( 3 + 2 \beta_{3} + \beta_{4} ) q^{67} + ( -\beta_{2} + \beta_{5} ) q^{69} + ( 2 + 2 \beta_{3} + 2 \beta_{4} ) q^{71} + ( -1 - 2 \beta_{3} - \beta_{4} ) q^{73} + q^{75} + ( 4 + \beta_{3} ) q^{77} + ( -1 - \beta_{4} ) q^{79} + q^{81} + ( -2 \beta_{1} - 3 \beta_{2} - \beta_{5} ) q^{83} + ( -1 - \beta_{3} + \beta_{4} ) q^{85} + ( \beta_{1} - 2 \beta_{2} + 2 \beta_{5} ) q^{87} + ( -3 \beta_{1} - 4 \beta_{2} - \beta_{5} ) q^{89} + ( -9 - 3 \beta_{3} - \beta_{4} ) q^{91} + ( 2 + \beta_{3} ) q^{93} + ( 1 + \beta_{2} - \beta_{3} ) q^{95} + ( -\beta_{1} - \beta_{2} - 2 \beta_{5} ) q^{97} + \beta_{1} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q + 6q^{3} + 6q^{5} + 6q^{9} + O(q^{10})$$ $$6q + 6q^{3} + 6q^{5} + 6q^{9} + 6q^{15} - 4q^{17} + 6q^{19} + 6q^{25} + 6q^{27} + 12q^{31} + 6q^{45} - 14q^{49} - 4q^{51} + 6q^{57} - 16q^{59} - 16q^{61} + 20q^{67} + 16q^{71} - 8q^{73} + 6q^{75} + 24q^{77} - 8q^{79} + 6q^{81} - 4q^{85} - 56q^{91} + 12q^{93} + 6q^{95} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} + 22 x^{4} + 62 x^{2} + 12$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{5} + 26 \nu^{3} + 114 \nu$$$$)/26$$ $$\beta_{3}$$ $$=$$ $$($$$$-2 \nu^{4} - 39 \nu^{2} - 46$$$$)/13$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{4} + 26 \nu^{2} + 75$$$$)/13$$ $$\beta_{5}$$ $$=$$ $$($$$$3 \nu^{5} + 65 \nu^{3} + 160 \nu$$$$)/13$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$2 \beta_{4} + \beta_{3} - 8$$ $$\nu^{3}$$ $$=$$ $$-\beta_{5} + 6 \beta_{2} - 14 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$-39 \beta_{4} - 26 \beta_{3} + 133$$ $$\nu^{5}$$ $$=$$ $$26 \beta_{5} - 130 \beta_{2} + 250 \beta_{1}$$

## 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
 1.75167i 0.457038i 4.32698i − 4.32698i − 0.457038i − 1.75167i
0 1.00000 0 1.00000 0 4.69162i 0 1.00000 0
2431.2 0 1.00000 0 1.00000 0 2.36627i 0 1.00000 0
2431.3 0 1.00000 0 1.00000 0 0.624069i 0 1.00000 0
2431.4 0 1.00000 0 1.00000 0 0.624069i 0 1.00000 0
2431.5 0 1.00000 0 1.00000 0 2.36627i 0 1.00000 0
2431.6 0 1.00000 0 1.00000 0 4.69162i 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.h yes 6
4.b odd 2 1 4560.2.d.f 6
19.b odd 2 1 4560.2.d.f 6
76.d even 2 1 inner 4560.2.d.h yes 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4560.2.d.f 6 4.b odd 2 1
4560.2.d.f 6 19.b odd 2 1
4560.2.d.h yes 6 1.a even 1 1 trivial
4560.2.d.h 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 $$S_{2}^{\mathrm{new}}(4560, [\chi])$$:

 $$T_{7}^{6} + 28 T_{7}^{4} + 134 T_{7}^{2} + 48$$ $$T_{31}^{3} - 6 T_{31}^{2} - 2 T_{31} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$( -1 + T )^{6}$$
$5$ $$( -1 + T )^{6}$$
$7$ $$48 + 134 T^{2} + 28 T^{4} + T^{6}$$
$11$ $$12 + 62 T^{2} + 22 T^{4} + T^{6}$$
$13$ $$48 + 134 T^{2} + 28 T^{4} + T^{6}$$
$17$ $$( -156 - 46 T + 2 T^{2} + T^{3} )^{2}$$
$19$ $$6859 - 2166 T + 247 T^{2} + 4 T^{3} + 13 T^{4} - 6 T^{5} + T^{6}$$
$23$ $$768 + 320 T^{2} + 34 T^{4} + T^{6}$$
$29$ $$164268 + 9086 T^{2} + 166 T^{4} + T^{6}$$
$31$ $$( 4 - 2 T - 6 T^{2} + T^{3} )^{2}$$
$37$ $$1728 + 566 T^{2} + 52 T^{4} + T^{6}$$
$41$ $$137388 + 9662 T^{2} + 182 T^{4} + T^{6}$$
$43$ $$1728 + 566 T^{2} + 52 T^{4} + T^{6}$$
$47$ $$27648 + 10880 T^{2} + 242 T^{4} + T^{6}$$
$53$ $$768 + 320 T^{2} + 34 T^{4} + T^{6}$$
$59$ $$( -768 - 104 T + 8 T^{2} + T^{3} )^{2}$$
$61$ $$( -688 - 158 T + 8 T^{2} + T^{3} )^{2}$$
$67$ $$( 128 - 32 T - 10 T^{2} + T^{3} )^{2}$$
$71$ $$( 768 - 104 T - 8 T^{2} + T^{3} )^{2}$$
$73$ $$( -32 - 60 T + 4 T^{2} + T^{3} )^{2}$$
$79$ $$( -32 - 20 T + 4 T^{2} + T^{3} )^{2}$$
$83$ $$292032 + 20864 T^{2} + 274 T^{4} + T^{6}$$
$89$ $$629292 + 53966 T^{2} + 454 T^{4} + T^{6}$$
$97$ $$3888 + 1862 T^{2} + 172 T^{4} + T^{6}$$