# Properties

 Label 1386.2.k.r 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: $$x^{4} + 7x^{2} + 49$$ 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 462) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{2} - 1) q^{2} + \beta_{2} q^{4} + \beta_1 q^{5} - \beta_1 q^{7} + q^{8}+O(q^{10})$$ q + (-b2 - 1) * q^2 + b2 * q^4 + b1 * q^5 - b1 * q^7 + q^8 $$q + ( - \beta_{2} - 1) q^{2} + \beta_{2} q^{4} + \beta_1 q^{5} - \beta_1 q^{7} + q^{8} + ( - \beta_{3} - \beta_1) q^{10} - \beta_{2} q^{11} - 4 q^{13} + (\beta_{3} + \beta_1) q^{14} + ( - \beta_{2} - 1) q^{16} + 3 \beta_{2} q^{17} - 2 \beta_1 q^{19} + \beta_{3} q^{20} - q^{22} - \beta_1 q^{23} + 2 \beta_{2} q^{25} + (4 \beta_{2} + 4) q^{26} - \beta_{3} q^{28} - 2 q^{29} - 4 \beta_{2} q^{31} + \beta_{2} q^{32} + 3 q^{34} - 7 \beta_{2} q^{35} + (4 \beta_{2} - 2 \beta_1 + 4) q^{37} + (2 \beta_{3} + 2 \beta_1) q^{38} + \beta_1 q^{40} - 9 q^{41} + ( - 2 \beta_{3} + 4) q^{43} + (\beta_{2} + 1) q^{44} + (\beta_{3} + \beta_1) q^{46} + (4 \beta_{2} - 3 \beta_1 + 4) q^{47} + 7 \beta_{2} q^{49} + 2 q^{50} - 4 \beta_{2} q^{52} - 4 \beta_{2} q^{53} - \beta_{3} q^{55} - \beta_1 q^{56} + (2 \beta_{2} + 2) q^{58} + ( - 4 \beta_{3} + 4 \beta_{2} - 4 \beta_1) q^{59} + ( - 4 \beta_{2} - 3 \beta_1 - 4) q^{61} - 4 q^{62} + q^{64} - 4 \beta_1 q^{65} + (4 \beta_{3} - 3 \beta_{2} + 4 \beta_1) q^{67} + ( - 3 \beta_{2} - 3) q^{68} - 7 q^{70} + (2 \beta_{3} + 8) q^{71} + ( - 2 \beta_{3} - 10 \beta_{2} - 2 \beta_1) q^{73} + (2 \beta_{3} - 4 \beta_{2} + 2 \beta_1) q^{74} - 2 \beta_{3} q^{76} + \beta_{3} q^{77} + ( - 8 \beta_{2} - \beta_1 - 8) q^{79} + ( - \beta_{3} - \beta_1) q^{80} + (9 \beta_{2} + 9) q^{82} + (4 \beta_{3} - 5) q^{83} + 3 \beta_{3} q^{85} + ( - 4 \beta_{2} - 2 \beta_1 - 4) q^{86} - \beta_{2} q^{88} + (8 \beta_{2} - 2 \beta_1 + 8) q^{89} + 4 \beta_1 q^{91} - \beta_{3} q^{92} + (3 \beta_{3} - 4 \beta_{2} + 3 \beta_1) q^{94} - 14 \beta_{2} q^{95} + (4 \beta_{3} - 1) q^{97} + 7 q^{98}+O(q^{100})$$ q + (-b2 - 1) * q^2 + b2 * q^4 + b1 * q^5 - b1 * q^7 + q^8 + (-b3 - b1) * q^10 - b2 * q^11 - 4 * q^13 + (b3 + b1) * q^14 + (-b2 - 1) * q^16 + 3*b2 * q^17 - 2*b1 * q^19 + b3 * q^20 - q^22 - b1 * q^23 + 2*b2 * q^25 + (4*b2 + 4) * q^26 - b3 * q^28 - 2 * q^29 - 4*b2 * q^31 + b2 * q^32 + 3 * q^34 - 7*b2 * q^35 + (4*b2 - 2*b1 + 4) * q^37 + (2*b3 + 2*b1) * q^38 + b1 * q^40 - 9 * q^41 + (-2*b3 + 4) * q^43 + (b2 + 1) * q^44 + (b3 + b1) * q^46 + (4*b2 - 3*b1 + 4) * q^47 + 7*b2 * q^49 + 2 * q^50 - 4*b2 * q^52 - 4*b2 * q^53 - b3 * q^55 - b1 * q^56 + (2*b2 + 2) * q^58 + (-4*b3 + 4*b2 - 4*b1) * q^59 + (-4*b2 - 3*b1 - 4) * q^61 - 4 * q^62 + q^64 - 4*b1 * q^65 + (4*b3 - 3*b2 + 4*b1) * q^67 + (-3*b2 - 3) * q^68 - 7 * q^70 + (2*b3 + 8) * q^71 + (-2*b3 - 10*b2 - 2*b1) * q^73 + (2*b3 - 4*b2 + 2*b1) * q^74 - 2*b3 * q^76 + b3 * q^77 + (-8*b2 - b1 - 8) * q^79 + (-b3 - b1) * q^80 + (9*b2 + 9) * q^82 + (4*b3 - 5) * q^83 + 3*b3 * q^85 + (-4*b2 - 2*b1 - 4) * q^86 - b2 * q^88 + (8*b2 - 2*b1 + 8) * q^89 + 4*b1 * q^91 - b3 * q^92 + (3*b3 - 4*b2 + 3*b1) * q^94 - 14*b2 * q^95 + (4*b3 - 1) * q^97 + 7 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{2} - 2 q^{4} + 4 q^{8}+O(q^{10})$$ 4 * q - 2 * q^2 - 2 * q^4 + 4 * q^8 $$4 q - 2 q^{2} - 2 q^{4} + 4 q^{8} + 2 q^{11} - 16 q^{13} - 2 q^{16} - 6 q^{17} - 4 q^{22} - 4 q^{25} + 8 q^{26} - 8 q^{29} + 8 q^{31} - 2 q^{32} + 12 q^{34} + 14 q^{35} + 8 q^{37} - 36 q^{41} + 16 q^{43} + 2 q^{44} + 8 q^{47} - 14 q^{49} + 8 q^{50} + 8 q^{52} + 8 q^{53} + 4 q^{58} - 8 q^{59} - 8 q^{61} - 16 q^{62} + 4 q^{64} + 6 q^{67} - 6 q^{68} - 28 q^{70} + 32 q^{71} + 20 q^{73} + 8 q^{74} - 16 q^{79} + 18 q^{82} - 20 q^{83} - 8 q^{86} + 2 q^{88} + 16 q^{89} + 8 q^{94} + 28 q^{95} - 4 q^{97} + 28 q^{98}+O(q^{100})$$ 4 * q - 2 * q^2 - 2 * q^4 + 4 * q^8 + 2 * q^11 - 16 * q^13 - 2 * q^16 - 6 * q^17 - 4 * q^22 - 4 * q^25 + 8 * q^26 - 8 * q^29 + 8 * q^31 - 2 * q^32 + 12 * q^34 + 14 * q^35 + 8 * q^37 - 36 * q^41 + 16 * q^43 + 2 * q^44 + 8 * q^47 - 14 * q^49 + 8 * q^50 + 8 * q^52 + 8 * q^53 + 4 * q^58 - 8 * q^59 - 8 * q^61 - 16 * q^62 + 4 * q^64 + 6 * q^67 - 6 * q^68 - 28 * q^70 + 32 * q^71 + 20 * q^73 + 8 * q^74 - 16 * q^79 + 18 * q^82 - 20 * q^83 - 8 * q^86 + 2 * q^88 + 16 * q^89 + 8 * q^94 + 28 * q^95 - 4 * q^97 + 28 * q^98

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

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} ) / 7$$ (v^2) / 7 $$\beta_{3}$$ $$=$$ $$( \nu^{3} ) / 7$$ (v^3) / 7
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$7\beta_{2}$$ 7*b2 $$\nu^{3}$$ $$=$$ $$7\beta_{3}$$ 7*b3

## 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 −1.32288 + 2.29129i 0 1.32288 2.29129i 1.00000 0 −1.32288 2.29129i
793.2 −0.500000 + 0.866025i 0 −0.500000 0.866025i 1.32288 2.29129i 0 −1.32288 + 2.29129i 1.00000 0 1.32288 + 2.29129i
991.1 −0.500000 0.866025i 0 −0.500000 + 0.866025i −1.32288 2.29129i 0 1.32288 + 2.29129i 1.00000 0 −1.32288 + 2.29129i
991.2 −0.500000 0.866025i 0 −0.500000 + 0.866025i 1.32288 + 2.29129i 0 −1.32288 2.29129i 1.00000 0 1.32288 2.29129i
 $$n$$: 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.r 4
3.b odd 2 1 462.2.i.e 4
7.c even 3 1 inner 1386.2.k.r 4
7.c even 3 1 9702.2.a.db 2
7.d odd 6 1 9702.2.a.dm 2
21.g even 6 1 3234.2.a.ba 2
21.h odd 6 1 462.2.i.e 4
21.h odd 6 1 3234.2.a.w 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.i.e 4 3.b odd 2 1
462.2.i.e 4 21.h odd 6 1
1386.2.k.r 4 1.a even 1 1 trivial
1386.2.k.r 4 7.c even 3 1 inner
3234.2.a.w 2 21.h odd 6 1
3234.2.a.ba 2 21.g even 6 1
9702.2.a.db 2 7.c even 3 1
9702.2.a.dm 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 $$S_{2}^{\mathrm{new}}(1386, [\chi])$$:

 $$T_{5}^{4} + 7T_{5}^{2} + 49$$ T5^4 + 7*T5^2 + 49 $$T_{13} + 4$$ T13 + 4 $$T_{17}^{2} + 3T_{17} + 9$$ T17^2 + 3*T17 + 9 $$T_{23}^{4} + 7T_{23}^{2} + 49$$ T23^4 + 7*T23^2 + 49

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + T + 1)^{2}$$
$3$ $$T^{4}$$
$5$ $$T^{4} + 7T^{2} + 49$$
$7$ $$T^{4} + 7T^{2} + 49$$
$11$ $$(T^{2} - T + 1)^{2}$$
$13$ $$(T + 4)^{4}$$
$17$ $$(T^{2} + 3 T + 9)^{2}$$
$19$ $$T^{4} + 28T^{2} + 784$$
$23$ $$T^{4} + 7T^{2} + 49$$
$29$ $$(T + 2)^{4}$$
$31$ $$(T^{2} - 4 T + 16)^{2}$$
$37$ $$T^{4} - 8 T^{3} + 76 T^{2} + 96 T + 144$$
$41$ $$(T + 9)^{4}$$
$43$ $$(T^{2} - 8 T - 12)^{2}$$
$47$ $$T^{4} - 8 T^{3} + 111 T^{2} + \cdots + 2209$$
$53$ $$(T^{2} - 4 T + 16)^{2}$$
$59$ $$T^{4} + 8 T^{3} + 160 T^{2} + \cdots + 9216$$
$61$ $$T^{4} + 8 T^{3} + 111 T^{2} + \cdots + 2209$$
$67$ $$T^{4} - 6 T^{3} + 139 T^{2} + \cdots + 10609$$
$71$ $$(T^{2} - 16 T + 36)^{2}$$
$73$ $$T^{4} - 20 T^{3} + 328 T^{2} + \cdots + 5184$$
$79$ $$T^{4} + 16 T^{3} + 199 T^{2} + \cdots + 3249$$
$83$ $$(T^{2} + 10 T - 87)^{2}$$
$89$ $$T^{4} - 16 T^{3} + 220 T^{2} + \cdots + 1296$$
$97$ $$(T^{2} + 2 T - 111)^{2}$$