Properties

 Label 1560.2.l.a Level $1560$ Weight $2$ Character orbit 1560.l Analytic conductor $12.457$ Analytic rank $0$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ 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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + i q^{3} + ( -1 + 2 i ) q^{5} + 5 i q^{7} - q^{9} +O(q^{10})$$ $$q + i q^{3} + ( -1 + 2 i ) q^{5} + 5 i q^{7} - q^{9} + 5 q^{11} + i q^{13} + ( -2 - i ) q^{15} + 3 i q^{17} + 4 q^{19} -5 q^{21} + 5 i q^{23} + ( -3 - 4 i ) q^{25} -i q^{27} + 4 q^{29} + 5 i q^{33} + ( -10 - 5 i ) q^{35} + 7 i q^{37} - q^{39} + 11 q^{41} -12 i q^{43} + ( 1 - 2 i ) q^{45} -6 i q^{47} -18 q^{49} -3 q^{51} + i q^{53} + ( -5 + 10 i ) q^{55} + 4 i q^{57} -12 q^{59} -7 q^{61} -5 i q^{63} + ( -2 - i ) q^{65} -4 i q^{67} -5 q^{69} -7 q^{71} -14 i q^{73} + ( 4 - 3 i ) q^{75} + 25 i q^{77} + 5 q^{79} + q^{81} -2 i q^{83} + ( -6 - 3 i ) q^{85} + 4 i q^{87} + 3 q^{89} -5 q^{91} + ( -4 + 8 i ) q^{95} + i q^{97} -5 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{5} - 2q^{9} + O(q^{10})$$ $$2q - 2q^{5} - 2q^{9} + 10q^{11} - 4q^{15} + 8q^{19} - 10q^{21} - 6q^{25} + 8q^{29} - 20q^{35} - 2q^{39} + 22q^{41} + 2q^{45} - 36q^{49} - 6q^{51} - 10q^{55} - 24q^{59} - 14q^{61} - 4q^{65} - 10q^{69} - 14q^{71} + 8q^{75} + 10q^{79} + 2q^{81} - 12q^{85} + 6q^{89} - 10q^{91} - 8q^{95} - 10q^{99} + O(q^{100})$$

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
 − 1.00000i 1.00000i
0 1.00000i 0 −1.00000 2.00000i 0 5.00000i 0 −1.00000 0
1249.2 0 1.00000i 0 −1.00000 + 2.00000i 0 5.00000i 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
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.a 2
3.b odd 2 1 4680.2.l.c 2
4.b odd 2 1 3120.2.l.b 2
5.b even 2 1 inner 1560.2.l.a 2
5.c odd 4 1 7800.2.a.l 1
5.c odd 4 1 7800.2.a.m 1
15.d odd 2 1 4680.2.l.c 2
20.d odd 2 1 3120.2.l.b 2

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{2} + 25$$ acting on $$S_{2}^{\mathrm{new}}(1560, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$1 + T^{2}$$
$5$ $$5 + 2 T + T^{2}$$
$7$ $$25 + T^{2}$$
$11$ $$( -5 + T )^{2}$$
$13$ $$1 + T^{2}$$
$17$ $$9 + T^{2}$$
$19$ $$( -4 + T )^{2}$$
$23$ $$25 + T^{2}$$
$29$ $$( -4 + T )^{2}$$
$31$ $$T^{2}$$
$37$ $$49 + T^{2}$$
$41$ $$( -11 + T )^{2}$$
$43$ $$144 + T^{2}$$
$47$ $$36 + T^{2}$$
$53$ $$1 + T^{2}$$
$59$ $$( 12 + T )^{2}$$
$61$ $$( 7 + T )^{2}$$
$67$ $$16 + T^{2}$$
$71$ $$( 7 + T )^{2}$$
$73$ $$196 + T^{2}$$
$79$ $$( -5 + T )^{2}$$
$83$ $$4 + T^{2}$$
$89$ $$( -3 + T )^{2}$$
$97$ $$1 + T^{2}$$