# Properties

 Label 1260.2.s.b Level $1260$ Weight $2$ Character orbit 1260.s Analytic conductor $10.061$ Analytic rank $1$ 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: $$1$$ 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.

 $$f(q)$$ $$=$$ $$q -\zeta_{6} q^{5} + ( -1 + 3 \zeta_{6} ) q^{7} +O(q^{10})$$ $$q -\zeta_{6} q^{5} + ( -1 + 3 \zeta_{6} ) q^{7} + ( -2 + 2 \zeta_{6} ) q^{11} -6 q^{13} + ( 2 - 2 \zeta_{6} ) q^{17} -9 \zeta_{6} q^{23} + ( -1 + \zeta_{6} ) q^{25} -3 q^{29} + ( -2 + 2 \zeta_{6} ) q^{31} + ( 3 - 2 \zeta_{6} ) q^{35} -8 \zeta_{6} q^{37} -5 q^{41} + q^{43} + 8 \zeta_{6} q^{47} + ( -8 + 3 \zeta_{6} ) q^{49} + ( 4 - 4 \zeta_{6} ) q^{53} + 2 q^{55} + ( -8 + 8 \zeta_{6} ) q^{59} -7 \zeta_{6} q^{61} + 6 \zeta_{6} q^{65} + ( 3 - 3 \zeta_{6} ) q^{67} -8 q^{71} + ( -14 + 14 \zeta_{6} ) q^{73} + ( -4 - 2 \zeta_{6} ) q^{77} -4 \zeta_{6} q^{79} + q^{83} -2 q^{85} + 13 \zeta_{6} q^{89} + ( 6 - 18 \zeta_{6} ) q^{91} -10 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{5} + q^{7} + O(q^{10})$$ $$2q - q^{5} + q^{7} - 2q^{11} - 12q^{13} + 2q^{17} - 9q^{23} - q^{25} - 6q^{29} - 2q^{31} + 4q^{35} - 8q^{37} - 10q^{41} + 2q^{43} + 8q^{47} - 13q^{49} + 4q^{53} + 4q^{55} - 8q^{59} - 7q^{61} + 6q^{65} + 3q^{67} - 16q^{71} - 14q^{73} - 10q^{77} - 4q^{79} + 2q^{83} - 4q^{85} + 13q^{89} - 6q^{91} - 20q^{97} + O(q^{100})$$

## 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 0.500000 + 2.59808i 0 0 0
541.1 0 0 0 −0.500000 + 0.866025i 0 0.500000 2.59808i 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.b 2
3.b odd 2 1 140.2.i.b 2
7.c even 3 1 inner 1260.2.s.b 2
7.c even 3 1 8820.2.a.w 1
7.d odd 6 1 8820.2.a.k 1
12.b even 2 1 560.2.q.a 2
15.d odd 2 1 700.2.i.a 2
15.e even 4 2 700.2.r.c 4
21.c even 2 1 980.2.i.a 2
21.g even 6 1 980.2.a.i 1
21.g even 6 1 980.2.i.a 2
21.h odd 6 1 140.2.i.b 2
21.h odd 6 1 980.2.a.a 1
84.j odd 6 1 3920.2.a.d 1
84.n even 6 1 560.2.q.a 2
84.n even 6 1 3920.2.a.bi 1
105.o odd 6 1 700.2.i.a 2
105.o odd 6 1 4900.2.a.v 1
105.p even 6 1 4900.2.a.a 1
105.w odd 12 2 4900.2.e.b 2
105.x even 12 2 700.2.r.c 4
105.x even 12 2 4900.2.e.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.i.b 2 3.b odd 2 1
140.2.i.b 2 21.h odd 6 1
560.2.q.a 2 12.b even 2 1
560.2.q.a 2 84.n even 6 1
700.2.i.a 2 15.d odd 2 1
700.2.i.a 2 105.o odd 6 1
700.2.r.c 4 15.e even 4 2
700.2.r.c 4 105.x even 12 2
980.2.a.a 1 21.h odd 6 1
980.2.a.i 1 21.g even 6 1
980.2.i.a 2 21.c even 2 1
980.2.i.a 2 21.g even 6 1
1260.2.s.b 2 1.a even 1 1 trivial
1260.2.s.b 2 7.c even 3 1 inner
3920.2.a.d 1 84.j odd 6 1
3920.2.a.bi 1 84.n even 6 1
4900.2.a.a 1 105.p even 6 1
4900.2.a.v 1 105.o odd 6 1
4900.2.e.b 2 105.w odd 12 2
4900.2.e.c 2 105.x even 12 2
8820.2.a.k 1 7.d odd 6 1
8820.2.a.w 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 $$S_{2}^{\mathrm{new}}(1260, [\chi])$$:

 $$T_{11}^{2} + 2 T_{11} + 4$$ $$T_{13} + 6$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$1 + T + T^{2}$$
$7$ $$7 - T + T^{2}$$
$11$ $$4 + 2 T + T^{2}$$
$13$ $$( 6 + T )^{2}$$
$17$ $$4 - 2 T + T^{2}$$
$19$ $$T^{2}$$
$23$ $$81 + 9 T + T^{2}$$
$29$ $$( 3 + T )^{2}$$
$31$ $$4 + 2 T + T^{2}$$
$37$ $$64 + 8 T + T^{2}$$
$41$ $$( 5 + T )^{2}$$
$43$ $$( -1 + T )^{2}$$
$47$ $$64 - 8 T + T^{2}$$
$53$ $$16 - 4 T + T^{2}$$
$59$ $$64 + 8 T + T^{2}$$
$61$ $$49 + 7 T + T^{2}$$
$67$ $$9 - 3 T + T^{2}$$
$71$ $$( 8 + T )^{2}$$
$73$ $$196 + 14 T + T^{2}$$
$79$ $$16 + 4 T + T^{2}$$
$83$ $$( -1 + T )^{2}$$
$89$ $$169 - 13 T + T^{2}$$
$97$ $$( 10 + T )^{2}$$