# Properties

 Label 1368.1.bt.b Level $1368$ Weight $1$ Character orbit 1368.bt Analytic conductor $0.683$ Analytic rank $0$ Dimension $2$ Projective image $D_{6}$ CM discriminant -8 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1368 = 2^{3} \cdot 3^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1368.bt (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.682720937282$$ 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: $$D_{6}$$ Projective field: Galois closure of 6.2.3119171623488.7

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + q^{2} -\zeta_{6}^{2} q^{3} + q^{4} -\zeta_{6}^{2} q^{6} + q^{8} -\zeta_{6} q^{9} +O(q^{10})$$ $$q + q^{2} -\zeta_{6}^{2} q^{3} + q^{4} -\zeta_{6}^{2} q^{6} + q^{8} -\zeta_{6} q^{9} -\zeta_{6}^{2} q^{12} + q^{16} + ( -1 + \zeta_{6}^{2} ) q^{17} -\zeta_{6} q^{18} - q^{19} -\zeta_{6}^{2} q^{24} + \zeta_{6} q^{25} - q^{27} + q^{32} + ( -1 + \zeta_{6}^{2} ) q^{34} -\zeta_{6} q^{36} - q^{38} + \zeta_{6}^{2} q^{41} + q^{43} -\zeta_{6}^{2} q^{48} -\zeta_{6} q^{49} + \zeta_{6} q^{50} + ( \zeta_{6} + \zeta_{6}^{2} ) q^{51} - q^{54} + \zeta_{6}^{2} q^{57} -\zeta_{6}^{2} q^{59} + q^{64} + ( -\zeta_{6} - \zeta_{6}^{2} ) q^{67} + ( -1 + \zeta_{6}^{2} ) q^{68} -\zeta_{6} q^{72} + 2 \zeta_{6} q^{73} + q^{75} - q^{76} + \zeta_{6}^{2} q^{81} + \zeta_{6}^{2} q^{82} + ( -1 - \zeta_{6} ) q^{83} + q^{86} + 2 \zeta_{6}^{2} q^{89} -\zeta_{6}^{2} q^{96} -\zeta_{6} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + q^{3} + 2 q^{4} + q^{6} + 2 q^{8} - q^{9} + O(q^{10})$$ $$2 q + 2 q^{2} + q^{3} + 2 q^{4} + q^{6} + 2 q^{8} - q^{9} + q^{12} + 2 q^{16} - 3 q^{17} - q^{18} - 2 q^{19} + q^{24} + q^{25} - 2 q^{27} + 2 q^{32} - 3 q^{34} - q^{36} - 2 q^{38} - q^{41} + 2 q^{43} + q^{48} - q^{49} + q^{50} - 2 q^{54} - q^{57} + q^{59} + 2 q^{64} - 3 q^{68} - q^{72} + 2 q^{73} + 2 q^{75} - 2 q^{76} - q^{81} - q^{82} - 3 q^{83} + 2 q^{86} - 2 q^{89} + q^{96} - q^{98} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1368\mathbb{Z}\right)^\times$$.

 $$n$$ $$343$$ $$685$$ $$1009$$ $$1217$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-\zeta_{6}^{2}$$ $$\zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
635.1
 0.5 − 0.866025i 0.5 + 0.866025i
1.00000 0.500000 + 0.866025i 1.00000 0 0.500000 + 0.866025i 0 1.00000 −0.500000 + 0.866025i 0
1019.1 1.00000 0.500000 0.866025i 1.00000 0 0.500000 0.866025i 0 1.00000 −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
8.d odd 2 1 CM by $$\Q(\sqrt{-2})$$
171.t even 6 1 inner
1368.bt odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1368.1.bt.b 2
8.d odd 2 1 CM 1368.1.bt.b 2
9.d odd 6 1 1368.1.cz.b yes 2
19.d odd 6 1 1368.1.cz.b yes 2
72.l even 6 1 1368.1.cz.b yes 2
152.o even 6 1 1368.1.cz.b yes 2
171.t even 6 1 inner 1368.1.bt.b 2
1368.bt odd 6 1 inner 1368.1.bt.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1368.1.bt.b 2 1.a even 1 1 trivial
1368.1.bt.b 2 8.d odd 2 1 CM
1368.1.bt.b 2 171.t even 6 1 inner
1368.1.bt.b 2 1368.bt odd 6 1 inner
1368.1.cz.b yes 2 9.d odd 6 1
1368.1.cz.b yes 2 19.d odd 6 1
1368.1.cz.b yes 2 72.l even 6 1
1368.1.cz.b yes 2 152.o even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{11}$$ acting on $$S_{1}^{\mathrm{new}}(1368, [\chi])$$.

## Hecke characteristic polynomials

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