# Properties

 Label 1386.2.k.s Level $1386$ Weight $2$ Character orbit 1386.k Analytic conductor $11.067$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{7})$$ Defining polynomial: $$x^{4} + 7 x^{2} + 49$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ 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 basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 7 x^{2} + 49$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/7$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$$$/7$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$7 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$7 \beta_{3}$$

## Character values

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

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

## 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
 −1.32288 + 2.29129i 1.32288 − 2.29129i −1.32288 − 2.29129i 1.32288 + 2.29129i
−0.500000 + 0.866025i 0 −0.500000 0.866025i −0.822876 + 1.42526i 0 −1.32288 + 2.29129i 1.00000 0 −0.822876 1.42526i
793.2 −0.500000 + 0.866025i 0 −0.500000 0.866025i 1.82288 3.15731i 0 1.32288 2.29129i 1.00000 0 1.82288 + 3.15731i
991.1 −0.500000 0.866025i 0 −0.500000 + 0.866025i −0.822876 1.42526i 0 −1.32288 2.29129i 1.00000 0 −0.822876 + 1.42526i
991.2 −0.500000 0.866025i 0 −0.500000 + 0.866025i 1.82288 + 3.15731i 0 1.32288 + 2.29129i 1.00000 0 1.82288 3.15731i
 $$n$$: 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.s 4
3.b odd 2 1 154.2.e.f 4
7.c even 3 1 inner 1386.2.k.s 4
7.c even 3 1 9702.2.a.cz 2
7.d odd 6 1 9702.2.a.dr 2
12.b even 2 1 1232.2.q.g 4
21.c even 2 1 1078.2.e.v 4
21.g even 6 1 1078.2.a.n 2
21.g even 6 1 1078.2.e.v 4
21.h odd 6 1 154.2.e.f 4
21.h odd 6 1 1078.2.a.s 2
84.j odd 6 1 8624.2.a.bk 2
84.n even 6 1 1232.2.q.g 4
84.n even 6 1 8624.2.a.ca 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.2.e.f 4 3.b odd 2 1
154.2.e.f 4 21.h odd 6 1
1078.2.a.n 2 21.g even 6 1
1078.2.a.s 2 21.h odd 6 1
1078.2.e.v 4 21.c even 2 1
1078.2.e.v 4 21.g even 6 1
1232.2.q.g 4 12.b even 2 1
1232.2.q.g 4 84.n even 6 1
1386.2.k.s 4 1.a even 1 1 trivial
1386.2.k.s 4 7.c even 3 1 inner
8624.2.a.bk 2 84.j odd 6 1
8624.2.a.ca 2 84.n even 6 1
9702.2.a.cz 2 7.c even 3 1
9702.2.a.dr 2 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 $$S_{2}^{\mathrm{new}}(1386, [\chi])$$:

 $$T_{5}^{4} - 2 T_{5}^{3} + 10 T_{5}^{2} + 12 T_{5} + 36$$ $$T_{13} - 5$$ $$T_{17}^{2} - 6 T_{17} + 36$$ $$T_{23}^{4} + 2 T_{23}^{3} + 10 T_{23}^{2} - 12 T_{23} + 36$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T + T^{2} )^{2}$$
$3$ $$T^{4}$$
$5$ $$36 + 12 T + 10 T^{2} - 2 T^{3} + T^{4}$$
$7$ $$49 + 7 T^{2} + T^{4}$$
$11$ $$( 1 - T + T^{2} )^{2}$$
$13$ $$( -5 + T )^{4}$$
$17$ $$( 36 - 6 T + T^{2} )^{2}$$
$19$ $$4 - 12 T + 34 T^{2} - 6 T^{3} + T^{4}$$
$23$ $$36 - 12 T + 10 T^{2} + 2 T^{3} + T^{4}$$
$29$ $$( -27 + 2 T + T^{2} )^{2}$$
$31$ $$( 16 - 4 T + T^{2} )^{2}$$
$37$ $$36 - 12 T + 10 T^{2} + 2 T^{3} + T^{4}$$
$41$ $$( -54 + 6 T + T^{2} )^{2}$$
$43$ $$( 4 + T )^{4}$$
$47$ $$1296 - 576 T + 220 T^{2} - 16 T^{3} + T^{4}$$
$53$ $$36 - 12 T + 10 T^{2} + 2 T^{3} + T^{4}$$
$59$ $$9 + 12 T + 19 T^{2} - 4 T^{3} + T^{4}$$
$61$ $$2809 + 954 T + 271 T^{2} + 18 T^{3} + T^{4}$$
$67$ $$2209 + 376 T + 111 T^{2} - 8 T^{3} + T^{4}$$
$71$ $$( 42 + 14 T + T^{2} )^{2}$$
$73$ $$4 + 12 T + 34 T^{2} + 6 T^{3} + T^{4}$$
$79$ $$49 + 7 T^{2} + T^{4}$$
$83$ $$( 36 + 16 T + T^{2} )^{2}$$
$89$ $$9216 - 768 T + 160 T^{2} + 8 T^{3} + T^{4}$$
$97$ $$( 93 + 22 T + T^{2} )^{2}$$