# Properties

 Label 1386.2.k.h Level $1386$ Weight $2$ Character orbit 1386.k Analytic conductor $11.067$ Analytic rank $0$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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.

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

## Character values

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

 $$n$$ $$155$$ $$199$$ $$1135$$ $$\chi(n)$$ $$1$$ $$-\zeta_{6}$$ $$1$$

## 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 − 0.866025i 0.5 + 0.866025i
−0.500000 + 0.866025i 0 −0.500000 0.866025i 1.00000 1.73205i 0 2.00000 1.73205i 1.00000 0 1.00000 + 1.73205i
991.1 −0.500000 0.866025i 0 −0.500000 + 0.866025i 1.00000 + 1.73205i 0 2.00000 + 1.73205i 1.00000 0 1.00000 1.73205i
 $$n$$: 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 1386.2.k.h 2
3.b odd 2 1 1386.2.k.l yes 2
7.c even 3 1 inner 1386.2.k.h 2
7.c even 3 1 9702.2.a.bi 1
7.d odd 6 1 9702.2.a.ca 1
21.g even 6 1 9702.2.a.f 1
21.h odd 6 1 1386.2.k.l yes 2
21.h odd 6 1 9702.2.a.s 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1386.2.k.h 2 1.a even 1 1 trivial
1386.2.k.h 2 7.c even 3 1 inner
1386.2.k.l yes 2 3.b odd 2 1
1386.2.k.l yes 2 21.h odd 6 1
9702.2.a.f 1 21.g even 6 1
9702.2.a.s 1 21.h odd 6 1
9702.2.a.bi 1 7.c even 3 1
9702.2.a.ca 1 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 $$S_{2}^{\mathrm{new}}(1386, [\chi])$$:

 $$T_{5}^{2} - 2 T_{5} + 4$$ $$T_{13} - 2$$ $$T_{17}^{2} + T_{17} + 1$$ $$T_{23}^{2} - T_{23} + 1$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T + T^{2}$$
$3$ $$T^{2}$$
$5$ $$4 - 2 T + T^{2}$$
$7$ $$7 - 4 T + T^{2}$$
$11$ $$1 + T + T^{2}$$
$13$ $$( -2 + T )^{2}$$
$17$ $$1 + T + T^{2}$$
$19$ $$9 - 3 T + T^{2}$$
$23$ $$1 - T + T^{2}$$
$29$ $$( 1 + T )^{2}$$
$31$ $$4 - 2 T + T^{2}$$
$37$ $$25 - 5 T + T^{2}$$
$41$ $$( 10 + T )^{2}$$
$43$ $$( -1 + T )^{2}$$
$47$ $$49 + 7 T + T^{2}$$
$53$ $$144 + 12 T + T^{2}$$
$59$ $$9 - 3 T + T^{2}$$
$61$ $$196 - 14 T + T^{2}$$
$67$ $$144 + 12 T + T^{2}$$
$71$ $$( -5 + T )^{2}$$
$73$ $$64 - 8 T + T^{2}$$
$79$ $$T^{2}$$
$83$ $$( 6 + T )^{2}$$
$89$ $$36 - 6 T + T^{2}$$
$97$ $$( -7 + T )^{2}$$