# Properties

 Label 1260.2.s.a 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 420) 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.

 $$f(q)$$ $$=$$ $$q -\zeta_{6} q^{5} + ( -2 - \zeta_{6} ) q^{7} +O(q^{10})$$ $$q -\zeta_{6} q^{5} + ( -2 - \zeta_{6} ) q^{7} + ( -4 + 4 \zeta_{6} ) q^{11} + 7 q^{13} + ( -6 + 6 \zeta_{6} ) q^{17} -3 \zeta_{6} q^{19} -2 \zeta_{6} q^{23} + ( -1 + \zeta_{6} ) q^{25} + 2 q^{29} + ( -7 + 7 \zeta_{6} ) q^{31} + ( -1 + 3 \zeta_{6} ) q^{35} + 7 \zeta_{6} q^{37} + 8 q^{41} + 5 q^{43} + 10 \zeta_{6} q^{47} + ( 3 + 5 \zeta_{6} ) q^{49} + ( -8 + 8 \zeta_{6} ) q^{53} + 4 q^{55} + ( 10 - 10 \zeta_{6} ) q^{59} + 6 \zeta_{6} q^{61} -7 \zeta_{6} q^{65} + ( -3 + 3 \zeta_{6} ) q^{67} + ( -15 + 15 \zeta_{6} ) q^{73} + ( 12 - 8 \zeta_{6} ) q^{77} -\zeta_{6} q^{79} -8 q^{83} + 6 q^{85} + 2 \zeta_{6} q^{89} + ( -14 - 7 \zeta_{6} ) q^{91} + ( -3 + 3 \zeta_{6} ) q^{95} -10 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{5} - 5 q^{7} + O(q^{10})$$ $$2 q - q^{5} - 5 q^{7} - 4 q^{11} + 14 q^{13} - 6 q^{17} - 3 q^{19} - 2 q^{23} - q^{25} + 4 q^{29} - 7 q^{31} + q^{35} + 7 q^{37} + 16 q^{41} + 10 q^{43} + 10 q^{47} + 11 q^{49} - 8 q^{53} + 8 q^{55} + 10 q^{59} + 6 q^{61} - 7 q^{65} - 3 q^{67} - 15 q^{73} + 16 q^{77} - q^{79} - 16 q^{83} + 12 q^{85} + 2 q^{89} - 35 q^{91} - 3 q^{95} - 20 q^{97} + O(q^{100})$$

## 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.a 2
3.b odd 2 1 420.2.q.b 2
7.c even 3 1 inner 1260.2.s.a 2
7.c even 3 1 8820.2.a.bb 1
7.d odd 6 1 8820.2.a.l 1
12.b even 2 1 1680.2.bg.j 2
15.d odd 2 1 2100.2.q.d 2
15.e even 4 2 2100.2.bc.d 4
21.c even 2 1 2940.2.q.c 2
21.g even 6 1 2940.2.a.k 1
21.g even 6 1 2940.2.q.c 2
21.h odd 6 1 420.2.q.b 2
21.h odd 6 1 2940.2.a.b 1
84.n even 6 1 1680.2.bg.j 2
105.o odd 6 1 2100.2.q.d 2
105.x even 12 2 2100.2.bc.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
420.2.q.b 2 3.b odd 2 1
420.2.q.b 2 21.h odd 6 1
1260.2.s.a 2 1.a even 1 1 trivial
1260.2.s.a 2 7.c even 3 1 inner
1680.2.bg.j 2 12.b even 2 1
1680.2.bg.j 2 84.n even 6 1
2100.2.q.d 2 15.d odd 2 1
2100.2.q.d 2 105.o odd 6 1
2100.2.bc.d 4 15.e even 4 2
2100.2.bc.d 4 105.x even 12 2
2940.2.a.b 1 21.h odd 6 1
2940.2.a.k 1 21.g even 6 1
2940.2.q.c 2 21.c even 2 1
2940.2.q.c 2 21.g even 6 1
8820.2.a.l 1 7.d odd 6 1
8820.2.a.bb 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 $$S_{2}^{\mathrm{new}}(1260, [\chi])$$:

 $$T_{11}^{2} + 4 T_{11} + 16$$ $$T_{13} - 7$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$1 + T + T^{2}$$
$7$ $$7 + 5 T + T^{2}$$
$11$ $$16 + 4 T + T^{2}$$
$13$ $$( -7 + T )^{2}$$
$17$ $$36 + 6 T + T^{2}$$
$19$ $$9 + 3 T + T^{2}$$
$23$ $$4 + 2 T + T^{2}$$
$29$ $$( -2 + T )^{2}$$
$31$ $$49 + 7 T + T^{2}$$
$37$ $$49 - 7 T + T^{2}$$
$41$ $$( -8 + T )^{2}$$
$43$ $$( -5 + T )^{2}$$
$47$ $$100 - 10 T + T^{2}$$
$53$ $$64 + 8 T + T^{2}$$
$59$ $$100 - 10 T + T^{2}$$
$61$ $$36 - 6 T + T^{2}$$
$67$ $$9 + 3 T + T^{2}$$
$71$ $$T^{2}$$
$73$ $$225 + 15 T + T^{2}$$
$79$ $$1 + T + T^{2}$$
$83$ $$( 8 + T )^{2}$$
$89$ $$4 - 2 T + T^{2}$$
$97$ $$( 10 + T )^{2}$$