# Properties

 Label 1260.1.bt.b Level $1260$ Weight $1$ Character orbit 1260.bt Analytic conductor $0.629$ Analytic rank $0$ Dimension $2$ Projective image $D_{3}$ CM discriminant -35 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1260 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1260.bt (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.628821915918$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{6})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{3}$$ Projective field: Galois closure of 3.1.11340.2 Artin image: $C_3\times S_3$ Artin field: Galois closure of 6.0.55566000.2

## $q$-expansion

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q + \zeta_{6}^{2} q^{3} -\zeta_{6} q^{5} + \zeta_{6}^{2} q^{7} -\zeta_{6} q^{9} +O(q^{10})$$ $$q + \zeta_{6}^{2} q^{3} -\zeta_{6} q^{5} + \zeta_{6}^{2} q^{7} -\zeta_{6} q^{9} -\zeta_{6}^{2} q^{11} -2 \zeta_{6} q^{13} + q^{15} + 2 q^{17} -\zeta_{6} q^{21} + \zeta_{6}^{2} q^{25} + q^{27} -\zeta_{6}^{2} q^{29} + \zeta_{6} q^{33} + q^{35} + 2 q^{39} + \zeta_{6}^{2} q^{45} -\zeta_{6}^{2} q^{47} -\zeta_{6} q^{49} + 2 \zeta_{6}^{2} q^{51} - q^{55} + q^{63} + 2 \zeta_{6}^{2} q^{65} - q^{71} - q^{73} -\zeta_{6} q^{75} + \zeta_{6} q^{77} -\zeta_{6}^{2} q^{79} + \zeta_{6}^{2} q^{81} -\zeta_{6}^{2} q^{83} -2 \zeta_{6} q^{85} + \zeta_{6} q^{87} + 2 q^{91} -\zeta_{6}^{2} q^{97} - q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{3} - q^{5} - q^{7} - q^{9} + O(q^{10})$$ $$2q - q^{3} - q^{5} - q^{7} - q^{9} + q^{11} - 2q^{13} + 2q^{15} + 4q^{17} - q^{21} - q^{25} + 2q^{27} + q^{29} + q^{33} + 2q^{35} + 4q^{39} - q^{45} + q^{47} - q^{49} - 2q^{51} - 2q^{55} + 2q^{63} - 2q^{65} - 2q^{71} - 2q^{73} - q^{75} + q^{77} + q^{79} - q^{81} + q^{83} - 2q^{85} + q^{87} + 4q^{91} + q^{97} - 2q^{99} + O(q^{100})$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
349.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 −0.500000 + 0.866025i 0 −0.500000 0.866025i 0 −0.500000 + 0.866025i 0 −0.500000 0.866025i 0
769.1 0 −0.500000 0.866025i 0 −0.500000 + 0.866025i 0 −0.500000 0.866025i 0 −0.500000 + 0.866025i 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
35.c odd 2 1 CM by $$\Q(\sqrt{-35})$$
9.c even 3 1 inner
315.bg odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1260.1.bt.b 2
3.b odd 2 1 3780.1.bt.c 2
5.b even 2 1 1260.1.bt.d yes 2
7.b odd 2 1 1260.1.bt.d yes 2
9.c even 3 1 inner 1260.1.bt.b 2
9.d odd 6 1 3780.1.bt.c 2
15.d odd 2 1 3780.1.bt.a 2
21.c even 2 1 3780.1.bt.a 2
35.c odd 2 1 CM 1260.1.bt.b 2
45.h odd 6 1 3780.1.bt.a 2
45.j even 6 1 1260.1.bt.d yes 2
63.l odd 6 1 1260.1.bt.d yes 2
63.o even 6 1 3780.1.bt.a 2
105.g even 2 1 3780.1.bt.c 2
315.z even 6 1 3780.1.bt.c 2
315.bg odd 6 1 inner 1260.1.bt.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1260.1.bt.b 2 1.a even 1 1 trivial
1260.1.bt.b 2 9.c even 3 1 inner
1260.1.bt.b 2 35.c odd 2 1 CM
1260.1.bt.b 2 315.bg odd 6 1 inner
1260.1.bt.d yes 2 5.b even 2 1
1260.1.bt.d yes 2 7.b odd 2 1
1260.1.bt.d yes 2 45.j even 6 1
1260.1.bt.d yes 2 63.l odd 6 1
3780.1.bt.a 2 15.d odd 2 1
3780.1.bt.a 2 21.c even 2 1
3780.1.bt.a 2 45.h odd 6 1
3780.1.bt.a 2 63.o even 6 1
3780.1.bt.c 2 3.b odd 2 1
3780.1.bt.c 2 9.d odd 6 1
3780.1.bt.c 2 105.g even 2 1
3780.1.bt.c 2 315.z even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{1}^{\mathrm{new}}(1260, [\chi])$$:

 $$T_{11}^{2} - T_{11} + 1$$ $$T_{13}^{2} + 2 T_{13} + 4$$

## Hecke characteristic polynomials

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