# Properties

 Label 1386.2.k.o Level $1386$ Weight $2$ Character orbit 1386.k Analytic conductor $11.067$ Analytic rank $1$ 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: $$1$$ 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: no (minimal twist has level 154) 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.

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

## 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 −2.50000 + 0.866025i −1.00000 0 −1.00000 1.73205i
991.1 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 1.00000 + 1.73205i 0 −2.50000 0.866025i −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.o 2
3.b odd 2 1 154.2.e.a 2
7.c even 3 1 inner 1386.2.k.o 2
7.c even 3 1 9702.2.a.i 1
7.d odd 6 1 9702.2.a.y 1
12.b even 2 1 1232.2.q.e 2
21.c even 2 1 1078.2.e.f 2
21.g even 6 1 1078.2.a.g 1
21.g even 6 1 1078.2.e.f 2
21.h odd 6 1 154.2.e.a 2
21.h odd 6 1 1078.2.a.m 1
84.j odd 6 1 8624.2.a.be 1
84.n even 6 1 1232.2.q.e 2
84.n even 6 1 8624.2.a.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.2.e.a 2 3.b odd 2 1
154.2.e.a 2 21.h odd 6 1
1078.2.a.g 1 21.g even 6 1
1078.2.a.m 1 21.h odd 6 1
1078.2.e.f 2 21.c even 2 1
1078.2.e.f 2 21.g even 6 1
1232.2.q.e 2 12.b even 2 1
1232.2.q.e 2 84.n even 6 1
1386.2.k.o 2 1.a even 1 1 trivial
1386.2.k.o 2 7.c even 3 1 inner
8624.2.a.b 1 84.n even 6 1
8624.2.a.be 1 84.j odd 6 1
9702.2.a.i 1 7.c even 3 1
9702.2.a.y 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 $$S_{2}^{\mathrm{new}}(1386, [\chi])$$:

 $$T_{5}^{2} - 2 T_{5} + 4$$ $$T_{13} + 7$$ $$T_{17}^{2} - 2 T_{17} + 4$$ $$T_{23}^{2} + 8 T_{23} + 64$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T + T^{2}$$
$3$ $$T^{2}$$
$5$ $$4 - 2 T + T^{2}$$
$7$ $$7 + 5 T + T^{2}$$
$11$ $$1 + T + T^{2}$$
$13$ $$( 7 + T )^{2}$$
$17$ $$4 - 2 T + T^{2}$$
$19$ $$T^{2}$$
$23$ $$64 + 8 T + T^{2}$$
$29$ $$( -5 + T )^{2}$$
$31$ $$16 + 4 T + T^{2}$$
$37$ $$16 + 4 T + T^{2}$$
$41$ $$( 4 + T )^{2}$$
$43$ $$( 8 + T )^{2}$$
$47$ $$4 - 2 T + T^{2}$$
$53$ $$36 + 6 T + T^{2}$$
$59$ $$9 - 3 T + T^{2}$$
$61$ $$1 + T + T^{2}$$
$67$ $$81 + 9 T + T^{2}$$
$71$ $$( -2 + T )^{2}$$
$73$ $$16 + 4 T + T^{2}$$
$79$ $$81 + 9 T + T^{2}$$
$83$ $$( 6 + T )^{2}$$
$89$ $$36 - 6 T + T^{2}$$
$97$ $$( -7 + T )^{2}$$