# Properties

 Label 1368.1.cj.c Level $1368$ Weight $1$ Character orbit 1368.cj Analytic conductor $0.683$ Analytic rank $0$ Dimension $4$ Projective image $A_{4}$ CM/RM no Inner twists $4$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$0.682720937282$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$A_{4}$$ Projective field: Galois closure of 4.0.1871424.3

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q -\zeta_{12}^{3} q^{2} - q^{3} - q^{4} -\zeta_{12}^{5} q^{5} + \zeta_{12}^{3} q^{6} + \zeta_{12}^{5} q^{7} + \zeta_{12}^{3} q^{8} + q^{9} +O(q^{10})$$ $$q -\zeta_{12}^{3} q^{2} - q^{3} - q^{4} -\zeta_{12}^{5} q^{5} + \zeta_{12}^{3} q^{6} + \zeta_{12}^{5} q^{7} + \zeta_{12}^{3} q^{8} + q^{9} -\zeta_{12}^{2} q^{10} + \zeta_{12}^{2} q^{11} + q^{12} + \zeta_{12}^{2} q^{14} + \zeta_{12}^{5} q^{15} + q^{16} + \zeta_{12}^{4} q^{17} -\zeta_{12}^{3} q^{18} + q^{19} + \zeta_{12}^{5} q^{20} -\zeta_{12}^{5} q^{21} -\zeta_{12}^{5} q^{22} -\zeta_{12}^{3} q^{24} - q^{27} -\zeta_{12}^{5} q^{28} + \zeta_{12} q^{29} + \zeta_{12}^{2} q^{30} + \zeta_{12} q^{31} -\zeta_{12}^{3} q^{32} -\zeta_{12}^{2} q^{33} + \zeta_{12} q^{34} + \zeta_{12}^{4} q^{35} - q^{36} -2 \zeta_{12}^{3} q^{37} -\zeta_{12}^{3} q^{38} + \zeta_{12}^{2} q^{40} + \zeta_{12}^{2} q^{41} -\zeta_{12}^{2} q^{42} -\zeta_{12}^{2} q^{44} -\zeta_{12}^{5} q^{45} -\zeta_{12} q^{47} - q^{48} -\zeta_{12}^{4} q^{51} -\zeta_{12}^{5} q^{53} + \zeta_{12}^{3} q^{54} + \zeta_{12} q^{55} -\zeta_{12}^{2} q^{56} - q^{57} -\zeta_{12}^{4} q^{58} -\zeta_{12}^{2} q^{59} -\zeta_{12}^{5} q^{60} + \zeta_{12} q^{61} -\zeta_{12}^{4} q^{62} + \zeta_{12}^{5} q^{63} - q^{64} + \zeta_{12}^{5} q^{66} + 2 q^{67} -\zeta_{12}^{4} q^{68} + \zeta_{12} q^{70} -\zeta_{12} q^{71} + \zeta_{12}^{3} q^{72} + \zeta_{12}^{4} q^{73} -2 q^{74} - q^{76} -\zeta_{12} q^{77} -\zeta_{12}^{5} q^{80} + q^{81} -\zeta_{12}^{5} q^{82} + \zeta_{12}^{2} q^{83} + \zeta_{12}^{5} q^{84} + \zeta_{12}^{3} q^{85} -\zeta_{12} q^{87} + \zeta_{12}^{5} q^{88} -\zeta_{12}^{2} q^{89} -\zeta_{12}^{2} q^{90} -\zeta_{12} q^{93} + \zeta_{12}^{4} q^{94} -\zeta_{12}^{5} q^{95} + \zeta_{12}^{3} q^{96} + \zeta_{12}^{2} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{3} - 4 q^{4} + 4 q^{9} + O(q^{10})$$ $$4 q - 4 q^{3} - 4 q^{4} + 4 q^{9} - 2 q^{10} + 2 q^{11} + 4 q^{12} + 2 q^{14} + 4 q^{16} - 2 q^{17} + 4 q^{19} - 4 q^{27} + 2 q^{30} - 2 q^{33} - 2 q^{35} - 4 q^{36} + 2 q^{40} + 2 q^{41} - 2 q^{42} - 2 q^{44} - 4 q^{48} + 2 q^{51} - 2 q^{56} - 4 q^{57} + 2 q^{58} - 2 q^{59} + 2 q^{62} - 4 q^{64} + 8 q^{67} + 2 q^{68} - 2 q^{73} - 8 q^{74} - 4 q^{76} + 4 q^{81} + 2 q^{83} - 2 q^{89} - 2 q^{90} - 2 q^{94} + 2 q^{99} + O(q^{100})$$

## Character values

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

 $$n$$ $$343$$ $$685$$ $$1009$$ $$1217$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-\zeta_{12}^{2}$$ $$\zeta_{12}^{4}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
691.1
 −0.866025 + 0.500000i 0.866025 − 0.500000i 0.866025 + 0.500000i −0.866025 − 0.500000i
1.00000i −1.00000 −1.00000 −0.866025 0.500000i 1.00000i 0.866025 + 0.500000i 1.00000i 1.00000 −0.500000 + 0.866025i
691.2 1.00000i −1.00000 −1.00000 0.866025 + 0.500000i 1.00000i −0.866025 0.500000i 1.00000i 1.00000 −0.500000 + 0.866025i
1075.1 1.00000i −1.00000 −1.00000 0.866025 0.500000i 1.00000i −0.866025 + 0.500000i 1.00000i 1.00000 −0.500000 0.866025i
1075.2 1.00000i −1.00000 −1.00000 −0.866025 + 0.500000i 1.00000i 0.866025 0.500000i 1.00000i 1.00000 −0.500000 0.866025i
 $$n$$: 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 inner
171.h even 3 1 inner
1368.cj 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.cj.c yes 4
8.d odd 2 1 inner 1368.1.cj.c yes 4
9.c even 3 1 1368.1.ba.c 4
19.c even 3 1 1368.1.ba.c 4
72.p odd 6 1 1368.1.ba.c 4
152.k odd 6 1 1368.1.ba.c 4
171.h even 3 1 inner 1368.1.cj.c yes 4
1368.cj odd 6 1 inner 1368.1.cj.c yes 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1368.1.ba.c 4 9.c even 3 1
1368.1.ba.c 4 19.c even 3 1
1368.1.ba.c 4 72.p odd 6 1
1368.1.ba.c 4 152.k odd 6 1
1368.1.cj.c yes 4 1.a even 1 1 trivial
1368.1.cj.c yes 4 8.d odd 2 1 inner
1368.1.cj.c yes 4 171.h even 3 1 inner
1368.1.cj.c yes 4 1368.cj odd 6 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{1}^{\mathrm{new}}(1368, [\chi])$$:

 $$T_{5}^{4} - T_{5}^{2} + 1$$ $$T_{11}^{2} - T_{11} + 1$$

## Hecke characteristic polynomials

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