# Properties

 Label 1386.2.k.e 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.

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

## 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 0 0 −0.500000 2.59808i 1.00000 0 0
991.1 −0.500000 0.866025i 0 −0.500000 + 0.866025i 0 0 −0.500000 + 2.59808i 1.00000 0 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
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.e 2
3.b odd 2 1 154.2.e.c 2
7.c even 3 1 inner 1386.2.k.e 2
7.c even 3 1 9702.2.a.br 1
7.d odd 6 1 9702.2.a.bs 1
12.b even 2 1 1232.2.q.d 2
21.c even 2 1 1078.2.e.k 2
21.g even 6 1 1078.2.a.c 1
21.g even 6 1 1078.2.e.k 2
21.h odd 6 1 154.2.e.c 2
21.h odd 6 1 1078.2.a.e 1
84.j odd 6 1 8624.2.a.u 1
84.n even 6 1 1232.2.q.d 2
84.n even 6 1 8624.2.a.k 1

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

 $$T_{5}$$ $$T_{13} + 1$$ $$T_{17}^{2} + 6 T_{17} + 36$$ $$T_{23}^{2} + 6 T_{23} + 36$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T + T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$7 + T + T^{2}$$
$11$ $$1 + T + T^{2}$$
$13$ $$( 1 + T )^{2}$$
$17$ $$36 + 6 T + T^{2}$$
$19$ $$4 + 2 T + T^{2}$$
$23$ $$36 + 6 T + T^{2}$$
$29$ $$( 9 + T )^{2}$$
$31$ $$16 - 4 T + T^{2}$$
$37$ $$4 + 2 T + T^{2}$$
$41$ $$( -6 + T )^{2}$$
$43$ $$( 4 + T )^{2}$$
$47$ $$36 + 6 T + T^{2}$$
$53$ $$T^{2}$$
$59$ $$9 + 3 T + T^{2}$$
$61$ $$121 + 11 T + T^{2}$$
$67$ $$121 + 11 T + T^{2}$$
$71$ $$T^{2}$$
$73$ $$4 + 2 T + T^{2}$$
$79$ $$25 + 5 T + T^{2}$$
$83$ $$( -6 + T )^{2}$$
$89$ $$324 + 18 T + T^{2}$$
$97$ $$( 13 + T )^{2}$$