# Properties

 Label 560.2.q.l Level $560$ Weight $2$ Character orbit 560.q Analytic conductor $4.472$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$560 = 2^{4} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 560.q (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.47162251319$$ Analytic rank: $$0$$ Dimension: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{3})$$ Coefficient field: 6.0.11337408.1 Defining polynomial: $$x^{6} + 18 x^{4} + 81 x^{2} + 12$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 280) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( \beta_{3} + \beta_{4} - \beta_{5} ) q^{3} -\beta_{2} q^{5} + ( -1 - \beta_{1} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{7} + ( -\beta_{1} - 3 \beta_{2} + \beta_{4} ) q^{9} +O(q^{10})$$ $$q + ( \beta_{3} + \beta_{4} - \beta_{5} ) q^{3} -\beta_{2} q^{5} + ( -1 - \beta_{1} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{7} + ( -\beta_{1} - 3 \beta_{2} + \beta_{4} ) q^{9} + ( 1 - 2 \beta_{1} - \beta_{2} - \beta_{4} ) q^{11} + ( 1 + \beta_{1} - 2 \beta_{3} ) q^{13} + ( -\beta_{3} - \beta_{4} ) q^{15} + ( -2 + 2 \beta_{2} ) q^{17} + ( \beta_{1} + \beta_{2} - \beta_{4} + 2 \beta_{5} ) q^{19} + ( 2 - \beta_{1} - 4 \beta_{2} - 2 \beta_{3} - \beta_{4} ) q^{21} + ( -\beta_{1} + \beta_{2} + \beta_{4} - \beta_{5} ) q^{23} + ( -1 + \beta_{2} ) q^{25} + ( 6 - 3 \beta_{3} - 3 \beta_{4} ) q^{27} + ( 4 + \beta_{1} + 2 \beta_{4} ) q^{29} + ( 4 - 4 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{31} + ( 6 \beta_{2} - 4 \beta_{5} ) q^{33} + ( \beta_{1} + \beta_{2} - \beta_{3} ) q^{35} + ( \beta_{1} - 3 \beta_{2} - \beta_{4} + 2 \beta_{5} ) q^{37} + ( -6 + 4 \beta_{1} + 6 \beta_{2} - 2 \beta_{3} + 2 \beta_{5} ) q^{39} + ( 3 - 2 \beta_{1} + 2 \beta_{3} - 2 \beta_{4} ) q^{41} + ( -4 - \beta_{3} - \beta_{4} ) q^{43} + ( -3 + 2 \beta_{1} + 3 \beta_{2} + \beta_{4} ) q^{45} + ( -3 \beta_{1} - 5 \beta_{2} + 3 \beta_{4} - 2 \beta_{5} ) q^{47} + ( -1 - 2 \beta_{2} - 2 \beta_{3} - 3 \beta_{4} ) q^{49} + 2 \beta_{5} q^{51} + ( -3 - 2 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} - 3 \beta_{4} + 2 \beta_{5} ) q^{53} + ( -1 + \beta_{1} + 2 \beta_{4} ) q^{55} + ( 6 - 2 \beta_{1} + 4 \beta_{3} ) q^{57} + ( 8 - 8 \beta_{2} ) q^{59} + ( -\beta_{1} + 2 \beta_{2} + \beta_{4} + 2 \beta_{5} ) q^{61} + ( -6 + \beta_{1} + 9 \beta_{2} - 2 \beta_{3} - 3 \beta_{4} - 2 \beta_{5} ) q^{63} + ( \beta_{1} - \beta_{2} - \beta_{4} + 2 \beta_{5} ) q^{65} + ( 2 - 2 \beta_{2} - 3 \beta_{3} - 3 \beta_{4} + 3 \beta_{5} ) q^{67} + ( \beta_{1} - 2 \beta_{3} ) q^{69} + ( -4 \beta_{1} + 4 \beta_{3} + 2 \beta_{4} - 4 \beta_{5} ) q^{73} + \beta_{5} q^{75} + ( -7 + 2 \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{4} - 2 \beta_{5} ) q^{77} + ( 2 \beta_{1} - 6 \beta_{2} - 2 \beta_{4} + 4 \beta_{5} ) q^{79} + ( -9 + 9 \beta_{2} + 6 \beta_{3} + 6 \beta_{4} - 6 \beta_{5} ) q^{81} + ( 10 + \beta_{3} + \beta_{4} ) q^{83} + 2 q^{85} + ( 6 - 6 \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{87} + ( \beta_{1} - \beta_{4} + 2 \beta_{5} ) q^{89} + ( -9 - \beta_{1} + 10 \beta_{2} + 2 \beta_{5} ) q^{91} + ( -2 \beta_{1} - 12 \beta_{2} + 2 \beta_{4} - 4 \beta_{5} ) q^{93} + ( 1 - 2 \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} - 2 \beta_{5} ) q^{95} -2 q^{97} + ( -21 + \beta_{1} + 6 \beta_{3} + 8 \beta_{4} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q - 3q^{5} - 6q^{7} - 9q^{9} + O(q^{10})$$ $$6q - 3q^{5} - 6q^{7} - 9q^{9} + 3q^{11} + 6q^{13} - 6q^{17} + 3q^{19} + 3q^{23} - 3q^{25} + 36q^{27} + 24q^{29} + 12q^{31} + 18q^{33} + 3q^{35} - 9q^{37} - 18q^{39} + 18q^{41} - 24q^{43} - 9q^{45} - 15q^{47} - 12q^{49} - 9q^{53} - 6q^{55} + 36q^{57} + 24q^{59} + 6q^{61} - 9q^{63} - 3q^{65} + 6q^{67} - 39q^{77} - 18q^{79} - 27q^{81} + 60q^{83} + 12q^{85} + 18q^{87} - 24q^{91} - 36q^{93} + 3q^{95} - 12q^{97} - 126q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} + 18 x^{4} + 81 x^{2} + 12$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + 9 \nu + 2$$$$)/4$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{2} + \nu + 6$$$$)/2$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{4} + 9 \nu^{2} - 2 \nu$$$$)/4$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{5} + \nu^{4} + 15 \nu^{3} + 11 \nu^{2} + 48 \nu + 12$$$$)/8$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$2 \beta_{3} - \beta_{1} - 6$$ $$\nu^{3}$$ $$=$$ $$4 \beta_{2} - 9 \beta_{1} - 2$$ $$\nu^{4}$$ $$=$$ $$4 \beta_{4} - 18 \beta_{3} + 11 \beta_{1} + 54$$ $$\nu^{5}$$ $$=$$ $$8 \beta_{5} - 4 \beta_{4} - 4 \beta_{3} - 60 \beta_{2} + 87 \beta_{1} + 30$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/560\mathbb{Z}\right)^\times$$.

 $$n$$ $$241$$ $$337$$ $$351$$ $$421$$ $$\chi(n)$$ $$-\beta_{2}$$ $$1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
81.1
 2.78499i − 3.17656i 0.391571i − 2.78499i 3.17656i − 0.391571i
0 −1.64497 + 2.84918i 0 −0.500000 0.866025i 0 −2.64497 + 0.0641892i 0 −3.91187 6.77556i 0
81.2 0 0.352860 0.611171i 0 −0.500000 0.866025i 0 −0.647140 + 2.56539i 0 1.25098 + 2.16676i 0
81.3 0 1.29211 2.23800i 0 −0.500000 0.866025i 0 0.292113 2.62958i 0 −1.83911 3.18543i 0
401.1 0 −1.64497 2.84918i 0 −0.500000 + 0.866025i 0 −2.64497 0.0641892i 0 −3.91187 + 6.77556i 0
401.2 0 0.352860 + 0.611171i 0 −0.500000 + 0.866025i 0 −0.647140 2.56539i 0 1.25098 2.16676i 0
401.3 0 1.29211 + 2.23800i 0 −0.500000 + 0.866025i 0 0.292113 + 2.62958i 0 −1.83911 + 3.18543i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 401.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 560.2.q.l 6
4.b odd 2 1 280.2.q.e 6
7.c even 3 1 inner 560.2.q.l 6
7.c even 3 1 3920.2.a.cc 3
7.d odd 6 1 3920.2.a.cb 3
12.b even 2 1 2520.2.bi.q 6
20.d odd 2 1 1400.2.q.j 6
20.e even 4 2 1400.2.bh.i 12
28.d even 2 1 1960.2.q.w 6
28.f even 6 1 1960.2.a.v 3
28.f even 6 1 1960.2.q.w 6
28.g odd 6 1 280.2.q.e 6
28.g odd 6 1 1960.2.a.w 3
84.n even 6 1 2520.2.bi.q 6
140.p odd 6 1 1400.2.q.j 6
140.p odd 6 1 9800.2.a.ce 3
140.s even 6 1 9800.2.a.cf 3
140.w even 12 2 1400.2.bh.i 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.q.e 6 4.b odd 2 1
280.2.q.e 6 28.g odd 6 1
560.2.q.l 6 1.a even 1 1 trivial
560.2.q.l 6 7.c even 3 1 inner
1400.2.q.j 6 20.d odd 2 1
1400.2.q.j 6 140.p odd 6 1
1400.2.bh.i 12 20.e even 4 2
1400.2.bh.i 12 140.w even 12 2
1960.2.a.v 3 28.f even 6 1
1960.2.a.w 3 28.g odd 6 1
1960.2.q.w 6 28.d even 2 1
1960.2.q.w 6 28.f even 6 1
2520.2.bi.q 6 12.b even 2 1
2520.2.bi.q 6 84.n even 6 1
3920.2.a.cb 3 7.d odd 6 1
3920.2.a.cc 3 7.c even 3 1
9800.2.a.ce 3 140.p odd 6 1
9800.2.a.cf 3 140.s even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(560, [\chi])$$:

 $$T_{3}^{6} + 9 T_{3}^{4} - 12 T_{3}^{3} + 81 T_{3}^{2} - 54 T_{3} + 36$$ $$T_{11}^{6} - 3 T_{11}^{5} + 33 T_{11}^{4} - 16 T_{11}^{3} + 708 T_{11}^{2} - 1056 T_{11} + 1936$$ $$T_{13}^{3} - 3 T_{13}^{2} - 24 T_{13} + 68$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$36 - 54 T + 81 T^{2} - 12 T^{3} + 9 T^{4} + T^{6}$$
$5$ $$( 1 + T + T^{2} )^{3}$$
$7$ $$343 + 294 T + 168 T^{2} + 80 T^{3} + 24 T^{4} + 6 T^{5} + T^{6}$$
$11$ $$1936 - 1056 T + 708 T^{2} - 16 T^{3} + 33 T^{4} - 3 T^{5} + T^{6}$$
$13$ $$( 68 - 24 T - 3 T^{2} + T^{3} )^{2}$$
$17$ $$( 4 + 2 T + T^{2} )^{3}$$
$19$ $$256 + 384 T + 528 T^{2} + 104 T^{3} + 33 T^{4} - 3 T^{5} + T^{6}$$
$23$ $$49 + 105 T + 204 T^{2} + 59 T^{3} + 24 T^{4} - 3 T^{5} + T^{6}$$
$29$ $$( 26 + 21 T - 12 T^{2} + T^{3} )^{2}$$
$31$ $$16384 + 1536 T + 1680 T^{2} - 400 T^{3} + 132 T^{4} - 12 T^{5} + T^{6}$$
$37$ $$9216 + 864 T^{2} + 192 T^{3} + 81 T^{4} + 9 T^{5} + T^{6}$$
$41$ $$( 381 - 45 T - 9 T^{2} + T^{3} )^{2}$$
$43$ $$( 22 + 39 T + 12 T^{2} + T^{3} )^{2}$$
$47$ $$2521744 + 152448 T + 33036 T^{2} + 1736 T^{3} + 321 T^{4} + 15 T^{5} + T^{6}$$
$53$ $$389376 + 44928 T + 10800 T^{2} + 600 T^{3} + 153 T^{4} + 9 T^{5} + T^{6}$$
$59$ $$( 64 - 8 T + T^{2} )^{3}$$
$61$ $$295936 - 47328 T + 10833 T^{2} - 566 T^{3} + 123 T^{4} - 6 T^{5} + T^{6}$$
$67$ $$64 + 552 T + 4713 T^{2} + 430 T^{3} + 105 T^{4} - 6 T^{5} + T^{6}$$
$71$ $$T^{6}$$
$73$ $$112896 + 36288 T + 11664 T^{2} + 672 T^{3} + 108 T^{4} + T^{6}$$
$79$ $$589824 + 13824 T^{2} + 1536 T^{3} + 324 T^{4} + 18 T^{5} + T^{6}$$
$83$ $$( -904 + 291 T - 30 T^{2} + T^{3} )^{2}$$
$89$ $$1764 + 1134 T + 729 T^{2} + 84 T^{3} + 27 T^{4} + T^{6}$$
$97$ $$( 2 + T )^{6}$$