Properties

 Label 1560.2.l.d Level $1560$ Weight $2$ Character orbit 1560.l Analytic conductor $12.457$ Analytic rank $0$ Dimension $8$ 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.l (of order $$2$$, degree $$1$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$12.4566627153$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.57815240704.2 Defining polynomial: $$x^{8} - 2 x^{7} + 2 x^{6} + 89 x^{4} - 170 x^{3} + 162 x^{2} - 72 x + 16$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ 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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{2} q^{3} -\beta_{7} q^{5} + ( -\beta_{1} + 2 \beta_{2} ) q^{7} - q^{9} +O(q^{10})$$ $$q -\beta_{2} q^{3} -\beta_{7} q^{5} + ( -\beta_{1} + 2 \beta_{2} ) q^{7} - q^{9} + ( \beta_{3} + \beta_{4} + \beta_{5} ) q^{11} + \beta_{2} q^{13} -\beta_{5} q^{15} + ( -\beta_{1} + \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} ) q^{17} + ( 2 + \beta_{3} ) q^{21} + ( -\beta_{1} - 2 \beta_{2} + \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} ) q^{23} + ( -\beta_{2} - 2 \beta_{3} - \beta_{5} - \beta_{6} ) q^{25} + \beta_{2} q^{27} + ( -2 + \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} ) q^{29} + ( \beta_{1} + \beta_{6} - \beta_{7} ) q^{33} + ( -2 + \beta_{1} + 2 \beta_{4} + 2 \beta_{5} ) q^{35} + ( \beta_{1} + \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} ) q^{37} + q^{39} + ( 2 - \beta_{3} - \beta_{6} - \beta_{7} ) q^{41} + ( -2 \beta_{2} + \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{43} + \beta_{7} q^{45} + ( 2 \beta_{2} - \beta_{6} + \beta_{7} ) q^{47} + ( -3 - 3 \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} ) q^{49} + ( \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{51} + ( \beta_{1} + \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} ) q^{53} + ( 1 + 2 \beta_{1} - 3 \beta_{2} - \beta_{3} - \beta_{4} + 2 \beta_{6} - \beta_{7} ) q^{55} + ( -2 + 2 \beta_{4} + 2 \beta_{5} + \beta_{6} + \beta_{7} ) q^{59} + ( 2 + \beta_{3} + 2 \beta_{4} + 2 \beta_{5} ) q^{61} + ( \beta_{1} - 2 \beta_{2} ) q^{63} + \beta_{5} q^{65} + ( 4 \beta_{2} + 2 \beta_{6} - 2 \beta_{7} ) q^{67} + ( -2 + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{69} + ( 4 + 3 \beta_{3} + \beta_{6} + \beta_{7} ) q^{71} + ( -2 \beta_{1} + 2 \beta_{2} + \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} ) q^{73} + ( -1 - 2 \beta_{1} + \beta_{4} + \beta_{7} ) q^{75} + ( -\beta_{1} + 2 \beta_{2} + \beta_{4} - \beta_{5} - 3 \beta_{6} + 3 \beta_{7} ) q^{77} + ( -\beta_{3} - 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{79} + q^{81} + ( 4 \beta_{1} - 6 \beta_{2} + \beta_{6} - \beta_{7} ) q^{83} + ( 2 - \beta_{1} - 4 \beta_{2} + 2 \beta_{3} + 3 \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{85} + ( 2 \beta_{2} + \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{87} + ( -2 + 3 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - \beta_{6} - \beta_{7} ) q^{89} + ( -2 - \beta_{3} ) q^{91} + ( 5 \beta_{1} + 2 \beta_{6} - 2 \beta_{7} ) q^{97} + ( -\beta_{3} - \beta_{4} - \beta_{5} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 2 q^{5} - 8 q^{9} + O(q^{10})$$ $$8 q - 2 q^{5} - 8 q^{9} + 2 q^{11} - 2 q^{15} + 14 q^{21} - 16 q^{29} - 8 q^{35} + 8 q^{39} + 14 q^{41} + 2 q^{45} - 18 q^{49} + 6 q^{51} + 10 q^{55} - 4 q^{59} + 22 q^{61} + 2 q^{65} - 10 q^{69} + 30 q^{71} - 4 q^{75} + 2 q^{79} + 8 q^{81} + 24 q^{85} - 18 q^{89} - 14 q^{91} - 2 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 2 x^{7} + 2 x^{6} + 89 x^{4} - 170 x^{3} + 162 x^{2} - 72 x + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$72 \nu^{7} + 747 \nu^{6} + 76 \nu^{5} + 32 \nu^{4} + 6804 \nu^{3} + 63783 \nu^{2} + 9448 \nu - 2736$$$$)/18170$$ $$\beta_{2}$$ $$=$$ $$($$$$-1881 \nu^{7} + 2970 \nu^{6} - 2894 \nu^{5} - 836 \nu^{4} - 167761 \nu^{3} + 244926 \nu^{2} - 234110 \nu + 67844$$$$)/36340$$ $$\beta_{3}$$ $$=$$ $$($$$$-1611 \nu^{7} + 1683 \nu^{6} - 792 \nu^{5} - 716 \nu^{4} - 144063 \nu^{3} + 136611 \nu^{2} - 60588 \nu + 28512$$$$)/18170$$ $$\beta_{4}$$ $$=$$ $$($$$$358 \nu^{7} - 374 \nu^{6} + 176 \nu^{5} + 361 \nu^{4} + 32014 \nu^{3} - 30358 \nu^{2} + 11647 \nu + 2749$$$$)/1817$$ $$\beta_{5}$$ $$=$$ $$($$$$1988 \nu^{7} - 2087 \nu^{6} + 1089 \nu^{5} + 1893 \nu^{4} + 178781 \nu^{3} - 169443 \nu^{2} + 84217 \nu + 6221$$$$)/9085$$ $$\beta_{6}$$ $$=$$ $$($$$$1665 \nu^{7} - 3394 \nu^{6} + 2666 \nu^{5} + 740 \nu^{4} + 147349 \nu^{3} - 289098 \nu^{2} + 213034 \nu - 59636$$$$)/7268$$ $$\beta_{7}$$ $$=$$ $$($$$$-9117 \nu^{7} + 17838 \nu^{6} - 14166 \nu^{5} - 4052 \nu^{4} - 811589 \nu^{3} + 1516102 \nu^{2} - 1096418 \nu + 328276$$$$)/36340$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{7} + \beta_{6} + \beta_{5} - \beta_{4}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} - 10 \beta_{2} - 4 \beta_{1}$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$-9 \beta_{7} - 9 \beta_{6} + 9 \beta_{5} - 9 \beta_{4} + 2 \beta_{3} + 4 \beta_{2} - 2 \beta_{1} + 4$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$9 \beta_{7} + 9 \beta_{6} + 9 \beta_{5} + 9 \beta_{4} + 40 \beta_{3} - 90$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$-85 \beta_{7} - 85 \beta_{6} - 85 \beta_{5} + 85 \beta_{4} + 22 \beta_{3} - 36 \beta_{2} + 22 \beta_{1} + 36$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$($$$$-85 \beta_{7} + 85 \beta_{6} + 77 \beta_{5} - 77 \beta_{4} + 850 \beta_{2} + 384 \beta_{1}$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$801 \beta_{7} + 809 \beta_{6} - 809 \beta_{5} + 801 \beta_{4} - 230 \beta_{3} - 300 \beta_{2} + 230 \beta_{1} - 300$$$$)/2$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1249.1
 2.20793 − 2.20793i 0.594137 − 0.594137i 0.353624 − 0.353624i −2.15569 + 2.15569i 2.20793 + 2.20793i 0.594137 + 0.594137i 0.353624 + 0.353624i −2.15569 − 2.15569i
0 1.00000i 0 −2.20793 0.353624i 0 1.65573i 0 −1.00000 0
1249.2 0 1.00000i 0 −0.594137 + 2.15569i 0 4.92778i 0 −1.00000 0
1249.3 0 1.00000i 0 −0.353624 2.20793i 0 3.09417i 0 −1.00000 0
1249.4 0 1.00000i 0 2.15569 0.594137i 0 0.633776i 0 −1.00000 0
1249.5 0 1.00000i 0 −2.20793 + 0.353624i 0 1.65573i 0 −1.00000 0
1249.6 0 1.00000i 0 −0.594137 2.15569i 0 4.92778i 0 −1.00000 0
1249.7 0 1.00000i 0 −0.353624 + 2.20793i 0 3.09417i 0 −1.00000 0
1249.8 0 1.00000i 0 2.15569 + 0.594137i 0 0.633776i 0 −1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1249.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.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.l.d 8
3.b odd 2 1 4680.2.l.g 8
4.b odd 2 1 3120.2.l.n 8
5.b even 2 1 inner 1560.2.l.d 8
5.c odd 4 1 7800.2.a.bt 4
5.c odd 4 1 7800.2.a.by 4
15.d odd 2 1 4680.2.l.g 8
20.d odd 2 1 3120.2.l.n 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1560.2.l.d 8 1.a even 1 1 trivial
1560.2.l.d 8 5.b even 2 1 inner
3120.2.l.n 8 4.b odd 2 1
3120.2.l.n 8 20.d odd 2 1
4680.2.l.g 8 3.b odd 2 1
4680.2.l.g 8 15.d odd 2 1
7800.2.a.bt 4 5.c odd 4 1
7800.2.a.by 4 5.c odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{8} + 37 T_{7}^{6} + 340 T_{7}^{4} + 768 T_{7}^{2} + 256$$ acting on $$S_{2}^{\mathrm{new}}(1560, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$( 1 + T^{2} )^{4}$$
$5$ $$625 + 250 T + 50 T^{2} - 30 T^{3} - 46 T^{4} - 6 T^{5} + 2 T^{6} + 2 T^{7} + T^{8}$$
$7$ $$256 + 768 T^{2} + 340 T^{4} + 37 T^{6} + T^{8}$$
$11$ $$( -4 - 26 T - 20 T^{2} - T^{3} + T^{4} )^{2}$$
$13$ $$( 1 + T^{2} )^{4}$$
$17$ $$64 + 1712 T^{2} + 908 T^{4} + 61 T^{6} + T^{8}$$
$19$ $$T^{8}$$
$23$ $$1024 + 3904 T^{2} + 1056 T^{4} + 65 T^{6} + T^{8}$$
$29$ $$( 256 - 144 T - 20 T^{2} + 8 T^{3} + T^{4} )^{2}$$
$31$ $$T^{8}$$
$37$ $$141376 + 60208 T^{2} + 5068 T^{4} + 133 T^{6} + T^{8}$$
$41$ $$( -4 + 22 T - 8 T^{2} - 7 T^{3} + T^{4} )^{2}$$
$43$ $$65536 + 30976 T^{2} + 3216 T^{4} + 104 T^{6} + T^{8}$$
$47$ $$16384 + 7312 T^{2} + 1108 T^{4} + 64 T^{6} + T^{8}$$
$53$ $$141376 + 60208 T^{2} + 5068 T^{4} + 133 T^{6} + T^{8}$$
$59$ $$( 1072 - 36 T - 86 T^{2} + 2 T^{3} + T^{4} )^{2}$$
$61$ $$( 32 + 128 T - 22 T^{2} - 11 T^{3} + T^{4} )^{2}$$
$67$ $$65536 + 37888 T^{2} + 5440 T^{4} + 192 T^{6} + T^{8}$$
$71$ $$( 1256 + 530 T - 30 T^{2} - 15 T^{3} + T^{4} )^{2}$$
$73$ $$1024 + 28992 T^{2} + 3568 T^{4} + 124 T^{6} + T^{8}$$
$79$ $$( 7328 + 52 T - 176 T^{2} - T^{3} + T^{4} )^{2}$$
$83$ $$7311616 + 2652624 T^{2} + 65620 T^{4} + 464 T^{6} + T^{8}$$
$89$ $$( 268 - 378 T - 136 T^{2} + 9 T^{3} + T^{4} )^{2}$$
$97$ $$41783296 + 3611280 T^{2} + 86296 T^{4} + 545 T^{6} + T^{8}$$