# Properties

 Label 1260.2.s.c Level $1260$ Weight $2$ Character orbit 1260.s Analytic conductor $10.061$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1260 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1260.s (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$10.0611506547$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 140) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\zeta_{6} q^{5} + ( 3 - \zeta_{6} ) q^{7} +O(q^{10})$$ $$q -\zeta_{6} q^{5} + ( 3 - \zeta_{6} ) q^{7} + ( 6 - 6 \zeta_{6} ) q^{11} + 2 q^{13} + ( -6 + 6 \zeta_{6} ) q^{17} -8 \zeta_{6} q^{19} + 3 \zeta_{6} q^{23} + ( -1 + \zeta_{6} ) q^{25} -3 q^{29} + ( -2 + 2 \zeta_{6} ) q^{31} + ( -1 - 2 \zeta_{6} ) q^{35} -8 \zeta_{6} q^{37} + 3 q^{41} + 5 q^{43} + ( 8 - 5 \zeta_{6} ) q^{49} + ( 12 - 12 \zeta_{6} ) q^{53} -6 q^{55} + \zeta_{6} q^{61} -2 \zeta_{6} q^{65} + ( 7 - 7 \zeta_{6} ) q^{67} + ( 10 - 10 \zeta_{6} ) q^{73} + ( 12 - 18 \zeta_{6} ) q^{77} + 4 \zeta_{6} q^{79} -3 q^{83} + 6 q^{85} -3 \zeta_{6} q^{89} + ( 6 - 2 \zeta_{6} ) q^{91} + ( -8 + 8 \zeta_{6} ) q^{95} -10 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{5} + 5q^{7} + O(q^{10})$$ $$2q - q^{5} + 5q^{7} + 6q^{11} + 4q^{13} - 6q^{17} - 8q^{19} + 3q^{23} - q^{25} - 6q^{29} - 2q^{31} - 4q^{35} - 8q^{37} + 6q^{41} + 10q^{43} + 11q^{49} + 12q^{53} - 12q^{55} + q^{61} - 2q^{65} + 7q^{67} + 10q^{73} + 6q^{77} + 4q^{79} - 6q^{83} + 12q^{85} - 3q^{89} + 10q^{91} - 8q^{95} - 20q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1260\mathbb{Z}\right)^\times$$.

 $$n$$ $$281$$ $$631$$ $$757$$ $$1081$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-\zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
361.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 0 0 −0.500000 0.866025i 0 2.50000 0.866025i 0 0 0
541.1 0 0 0 −0.500000 + 0.866025i 0 2.50000 + 0.866025i 0 0 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
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 1260.2.s.c 2
3.b odd 2 1 140.2.i.a 2
7.c even 3 1 inner 1260.2.s.c 2
7.c even 3 1 8820.2.a.p 1
7.d odd 6 1 8820.2.a.a 1
12.b even 2 1 560.2.q.f 2
15.d odd 2 1 700.2.i.b 2
15.e even 4 2 700.2.r.a 4
21.c even 2 1 980.2.i.f 2
21.g even 6 1 980.2.a.e 1
21.g even 6 1 980.2.i.f 2
21.h odd 6 1 140.2.i.a 2
21.h odd 6 1 980.2.a.g 1
84.j odd 6 1 3920.2.a.w 1
84.n even 6 1 560.2.q.f 2
84.n even 6 1 3920.2.a.k 1
105.o odd 6 1 700.2.i.b 2
105.o odd 6 1 4900.2.a.i 1
105.p even 6 1 4900.2.a.q 1
105.w odd 12 2 4900.2.e.n 2
105.x even 12 2 700.2.r.a 4
105.x even 12 2 4900.2.e.m 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.i.a 2 3.b odd 2 1
140.2.i.a 2 21.h odd 6 1
560.2.q.f 2 12.b even 2 1
560.2.q.f 2 84.n even 6 1
700.2.i.b 2 15.d odd 2 1
700.2.i.b 2 105.o odd 6 1
700.2.r.a 4 15.e even 4 2
700.2.r.a 4 105.x even 12 2
980.2.a.e 1 21.g even 6 1
980.2.a.g 1 21.h odd 6 1
980.2.i.f 2 21.c even 2 1
980.2.i.f 2 21.g even 6 1
1260.2.s.c 2 1.a even 1 1 trivial
1260.2.s.c 2 7.c even 3 1 inner
3920.2.a.k 1 84.n even 6 1
3920.2.a.w 1 84.j odd 6 1
4900.2.a.i 1 105.o odd 6 1
4900.2.a.q 1 105.p even 6 1
4900.2.e.m 2 105.x even 12 2
4900.2.e.n 2 105.w odd 12 2
8820.2.a.a 1 7.d odd 6 1
8820.2.a.p 1 7.c even 3 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1260, [\chi])$$:

 $$T_{11}^{2} - 6 T_{11} + 36$$ $$T_{13} - 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$1 + T + T^{2}$$
$7$ $$7 - 5 T + T^{2}$$
$11$ $$36 - 6 T + T^{2}$$
$13$ $$( -2 + T )^{2}$$
$17$ $$36 + 6 T + T^{2}$$
$19$ $$64 + 8 T + T^{2}$$
$23$ $$9 - 3 T + T^{2}$$
$29$ $$( 3 + T )^{2}$$
$31$ $$4 + 2 T + T^{2}$$
$37$ $$64 + 8 T + T^{2}$$
$41$ $$( -3 + T )^{2}$$
$43$ $$( -5 + T )^{2}$$
$47$ $$T^{2}$$
$53$ $$144 - 12 T + T^{2}$$
$59$ $$T^{2}$$
$61$ $$1 - T + T^{2}$$
$67$ $$49 - 7 T + T^{2}$$
$71$ $$T^{2}$$
$73$ $$100 - 10 T + T^{2}$$
$79$ $$16 - 4 T + T^{2}$$
$83$ $$( 3 + T )^{2}$$
$89$ $$9 + 3 T + T^{2}$$
$97$ $$( 10 + T )^{2}$$