# Properties

 Label 1386.2.k.w Level $1386$ Weight $2$ Character orbit 1386.k Analytic conductor $11.067$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1386.k (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$11.0672657201$$ Analytic rank: $$0$$ Dimension: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{3})$$ Coefficient field: 6.0.1156923.1 Defining polynomial: $$x^{6} - 3 x^{5} + 12 x^{4} - 19 x^{3} + 27 x^{2} - 18 x + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 462) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{2} q^{2} + ( -1 - \beta_{2} ) q^{4} + \beta_{4} q^{5} + ( \beta_{3} - \beta_{4} + \beta_{5} ) q^{7} - q^{8} +O(q^{10})$$ $$q -\beta_{2} q^{2} + ( -1 - \beta_{2} ) q^{4} + \beta_{4} q^{5} + ( \beta_{3} - \beta_{4} + \beta_{5} ) q^{7} - q^{8} + ( -\beta_{1} + \beta_{4} ) q^{10} + ( 1 + \beta_{2} ) q^{11} + 2 \beta_{1} q^{13} + ( -1 + \beta_{3} - \beta_{4} ) q^{14} + \beta_{2} q^{16} + ( 1 + \beta_{2} ) q^{17} + ( 1 + \beta_{1} + \beta_{3} + 2 \beta_{5} ) q^{19} -\beta_{1} q^{20} + q^{22} + ( -1 - \beta_{1} - 4 \beta_{2} - \beta_{3} + \beta_{4} - 2 \beta_{5} ) q^{23} + ( -1 + \beta_{1} + 2 \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{25} + 2 \beta_{4} q^{26} + ( -1 - \beta_{5} ) q^{28} + ( -4 + \beta_{3} - \beta_{5} ) q^{29} + ( -2 + 2 \beta_{1} + 4 \beta_{3} - 4 \beta_{4} + 2 \beta_{5} ) q^{31} + ( 1 + \beta_{2} ) q^{32} + q^{34} + ( -3 \beta_{1} + 3 \beta_{2} + 2 \beta_{4} - \beta_{5} ) q^{35} + ( 1 + \beta_{1} + 4 \beta_{2} + \beta_{3} + 2 \beta_{5} ) q^{37} + ( -1 - \beta_{1} + 2 \beta_{3} + \beta_{5} ) q^{38} -\beta_{4} q^{40} + ( -5 + \beta_{3} - \beta_{5} ) q^{41} + ( 2 - 2 \beta_{1} + \beta_{3} - \beta_{5} ) q^{43} -\beta_{2} q^{44} + ( -3 - 4 \beta_{2} - 2 \beta_{3} + \beta_{4} - \beta_{5} ) q^{46} + ( -1 - \beta_{1} + 4 \beta_{2} - \beta_{3} + \beta_{4} - 2 \beta_{5} ) q^{47} + ( -4 + \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{49} + ( -2 + \beta_{3} - \beta_{5} ) q^{50} + ( -2 \beta_{1} + 2 \beta_{4} ) q^{52} + ( 2 \beta_{1} - 2 \beta_{4} ) q^{53} + \beta_{1} q^{55} + ( -\beta_{3} + \beta_{4} - \beta_{5} ) q^{56} + ( -1 - \beta_{1} + 2 \beta_{2} - \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{58} + ( 3 - 3 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{59} + ( -2 - 2 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} + 5 \beta_{4} - 4 \beta_{5} ) q^{61} + ( -4 + 2 \beta_{3} - 2 \beta_{5} ) q^{62} + q^{64} + ( 2 + 2 \beta_{1} - 10 \beta_{2} + 2 \beta_{3} - 4 \beta_{4} + 4 \beta_{5} ) q^{65} + ( -2 + \beta_{1} - \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{67} -\beta_{2} q^{68} + ( 3 - 2 \beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{5} ) q^{70} + ( -2 + \beta_{3} - \beta_{5} ) q^{71} + ( -2 \beta_{2} - 4 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{73} + ( 3 - \beta_{1} + 4 \beta_{2} + 2 \beta_{3} + \beta_{5} ) q^{74} + ( -2 - 2 \beta_{1} + \beta_{3} - \beta_{5} ) q^{76} + ( 1 + \beta_{5} ) q^{77} + ( 4 \beta_{2} - 3 \beta_{4} ) q^{79} + ( \beta_{1} - \beta_{4} ) q^{80} + ( -1 - \beta_{1} + 3 \beta_{2} - \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{82} + ( 3 + 4 \beta_{1} - \beta_{3} + \beta_{5} ) q^{83} + \beta_{1} q^{85} + ( -1 - \beta_{1} - 4 \beta_{2} - \beta_{3} - 2 \beta_{5} ) q^{86} + ( -1 - \beta_{2} ) q^{88} + ( 12 \beta_{2} + 2 \beta_{4} ) q^{89} + ( -6 - 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + 4 \beta_{4} ) q^{91} + ( -2 + \beta_{1} - \beta_{3} + \beta_{5} ) q^{92} + ( 5 + 4 \beta_{2} - 2 \beta_{3} + \beta_{4} - \beta_{5} ) q^{94} + ( -8 - 2 \beta_{1} - 6 \beta_{2} + 4 \beta_{3} + 2 \beta_{5} ) q^{95} + ( 1 + 2 \beta_{3} - 2 \beta_{5} ) q^{97} + ( 3 \beta_{1} + 4 \beta_{2} - 2 \beta_{4} + \beta_{5} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q + 3q^{2} - 3q^{4} - 6q^{8} + O(q^{10})$$ $$6q + 3q^{2} - 3q^{4} - 6q^{8} + 3q^{11} - 3q^{14} - 3q^{16} + 3q^{17} + 3q^{19} + 6q^{22} + 9q^{23} - 3q^{25} - 3q^{28} - 18q^{29} - 6q^{31} + 3q^{32} + 6q^{34} - 6q^{35} - 9q^{37} - 3q^{38} - 24q^{41} + 18q^{43} + 3q^{44} - 9q^{46} - 15q^{47} - 24q^{49} - 6q^{50} - 9q^{58} + 9q^{59} + 6q^{61} - 12q^{62} + 6q^{64} + 36q^{65} - 6q^{67} + 3q^{68} + 12q^{70} - 6q^{71} + 9q^{74} - 6q^{76} + 3q^{77} - 12q^{79} - 12q^{82} + 12q^{83} + 9q^{86} - 3q^{88} - 36q^{89} - 36q^{91} - 18q^{92} + 15q^{94} - 24q^{95} + 18q^{97} - 15q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 3 x^{5} + 12 x^{4} - 19 x^{3} + 27 x^{2} - 18 x + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{2} - \nu + 3$$ $$\beta_{2}$$ $$=$$ $$($$$$2 \nu^{5} - 5 \nu^{4} + 22 \nu^{3} - 28 \nu^{2} + 43 \nu - 18$$$$)/2$$ $$\beta_{3}$$ $$=$$ $$-3 \nu^{5} + 7 \nu^{4} - 31 \nu^{3} + 37 \nu^{2} - 56 \nu + 22$$ $$\beta_{4}$$ $$=$$ $$($$$$-6 \nu^{5} + 15 \nu^{4} - 64 \nu^{3} + 82 \nu^{2} - 121 \nu + 50$$$$)/2$$ $$\beta_{5}$$ $$=$$ $$-3 \nu^{5} + 8 \nu^{4} - 33 \nu^{3} + 44 \nu^{2} - 62 \nu + 24$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{5} - 2 \beta_{4} + \beta_{3} + \beta_{1} + 1$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{5} - 2 \beta_{4} + \beta_{3} + 3 \beta_{1} - 5$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$-3 \beta_{5} + 8 \beta_{4} - 3 \beta_{3} + 6 \beta_{2} - \beta_{1} - 5$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$-5 \beta_{5} + 18 \beta_{4} - 9 \beta_{3} + 12 \beta_{2} - 17 \beta_{1} + 27$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$13 \beta_{5} - 28 \beta_{4} + 3 \beta_{3} - 34 \beta_{2} - 11 \beta_{1} + 49$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1386\mathbb{Z}\right)^\times$$.

 $$n$$ $$155$$ $$199$$ $$1135$$ $$\chi(n)$$ $$1$$ $$\beta_{2}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
793.1
 0.5 − 2.43956i 0.5 + 1.51496i 0.5 + 0.0585812i 0.5 + 2.43956i 0.5 − 1.51496i 0.5 − 0.0585812i
0.500000 0.866025i 0 −0.500000 0.866025i −1.60074 + 2.77256i 0 1.60074 2.10657i −1.00000 0 1.60074 + 2.77256i
793.2 0.500000 0.866025i 0 −0.500000 0.866025i 0.227452 0.393958i 0 −0.227452 + 2.63596i −1.00000 0 −0.227452 0.393958i
793.3 0.500000 0.866025i 0 −0.500000 0.866025i 1.37328 2.37860i 0 −1.37328 2.26144i −1.00000 0 −1.37328 2.37860i
991.1 0.500000 + 0.866025i 0 −0.500000 + 0.866025i −1.60074 2.77256i 0 1.60074 + 2.10657i −1.00000 0 1.60074 2.77256i
991.2 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0.227452 + 0.393958i 0 −0.227452 2.63596i −1.00000 0 −0.227452 + 0.393958i
991.3 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 1.37328 + 2.37860i 0 −1.37328 + 2.26144i −1.00000 0 −1.37328 + 2.37860i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 991.3 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 1386.2.k.w 6
3.b odd 2 1 462.2.i.f 6
7.c even 3 1 inner 1386.2.k.w 6
7.c even 3 1 9702.2.a.dt 3
7.d odd 6 1 9702.2.a.du 3
21.g even 6 1 3234.2.a.bg 3
21.h odd 6 1 462.2.i.f 6
21.h odd 6 1 3234.2.a.bi 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.i.f 6 3.b odd 2 1
462.2.i.f 6 21.h odd 6 1
1386.2.k.w 6 1.a even 1 1 trivial
1386.2.k.w 6 7.c even 3 1 inner
3234.2.a.bg 3 21.g even 6 1
3234.2.a.bi 3 21.h odd 6 1
9702.2.a.dt 3 7.c even 3 1
9702.2.a.du 3 7.d odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1386, [\chi])$$:

 $$T_{5}^{6} + 9 T_{5}^{4} - 8 T_{5}^{3} + 81 T_{5}^{2} - 36 T_{5} + 16$$ $$T_{13}^{3} - 36 T_{13} + 32$$ $$T_{17}^{2} - T_{17} + 1$$ $$T_{23}^{6} - 9 T_{23}^{5} + 78 T_{23}^{4} - 69 T_{23}^{3} + 198 T_{23}^{2} + 63 T_{23} + 441$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T + T^{2} )^{3}$$
$3$ $$T^{6}$$
$5$ $$16 - 36 T + 81 T^{2} - 8 T^{3} + 9 T^{4} + T^{6}$$
$7$ $$343 + 84 T^{2} - 4 T^{3} + 12 T^{4} + T^{6}$$
$11$ $$( 1 - T + T^{2} )^{3}$$
$13$ $$( 32 - 36 T + T^{3} )^{2}$$
$17$ $$( 1 - T + T^{2} )^{3}$$
$19$ $$1296 + 1296 T + 1188 T^{2} + 180 T^{3} + 45 T^{4} - 3 T^{5} + T^{6}$$
$23$ $$441 + 63 T + 198 T^{2} - 69 T^{3} + 78 T^{4} - 9 T^{5} + T^{6}$$
$29$ $$( -92 + 9 T^{2} + T^{3} )^{2}$$
$31$ $$262144 + 49152 T + 12288 T^{2} + 448 T^{3} + 132 T^{4} + 6 T^{5} + T^{6}$$
$37$ $$26896 + 1968 T + 1620 T^{2} + 220 T^{3} + 93 T^{4} + 9 T^{5} + T^{6}$$
$41$ $$( -82 + 21 T + 12 T^{2} + T^{3} )^{2}$$
$43$ $$( 164 - 12 T - 9 T^{2} + T^{3} )^{2}$$
$47$ $$361 - 969 T + 2886 T^{2} + 803 T^{3} + 174 T^{4} + 15 T^{5} + T^{6}$$
$53$ $$1024 - 1152 T + 1296 T^{2} - 64 T^{3} + 36 T^{4} + T^{6}$$
$59$ $$589824 - 73728 T + 16128 T^{2} - 672 T^{3} + 177 T^{4} - 9 T^{5} + T^{6}$$
$61$ $$51076 - 29154 T + 17997 T^{2} + 322 T^{3} + 165 T^{4} - 6 T^{5} + T^{6}$$
$67$ $$7056 + 1260 T + 729 T^{2} + 78 T^{3} + 51 T^{4} + 6 T^{5} + T^{6}$$
$71$ $$( -64 - 24 T + 3 T^{2} + T^{3} )^{2}$$
$73$ $$36864 - 18432 T + 9216 T^{2} - 384 T^{3} + 96 T^{4} + T^{6}$$
$79$ $$135424 + 12144 T + 5505 T^{2} + 340 T^{3} + 177 T^{4} + 12 T^{5} + T^{6}$$
$83$ $$( -188 - 111 T - 6 T^{2} + T^{3} )^{2}$$
$89$ $$1763584 + 525888 T + 109008 T^{2} + 11600 T^{3} + 900 T^{4} + 36 T^{5} + T^{6}$$
$97$ $$( -7 - 81 T - 9 T^{2} + T^{3} )^{2}$$