# Properties

 Label 1386.2.e.c Level $1386$ Weight $2$ Character orbit 1386.e Analytic conductor $11.067$ Analytic rank $0$ Dimension $8$ CM no Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1386.e (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$11.0672657201$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.157351936.1 Defining polynomial: $$x^{8} + x^{4} + 16$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{10}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{2} - q^{4} -\beta_{5} q^{5} + \beta_{1} q^{7} -\beta_{2} q^{8} +O(q^{10})$$ $$q + \beta_{2} q^{2} - q^{4} -\beta_{5} q^{5} + \beta_{1} q^{7} -\beta_{2} q^{8} -\beta_{3} q^{10} + ( 2 \beta_{2} + \beta_{4} ) q^{11} + 2 \beta_{3} q^{13} -\beta_{4} q^{14} + q^{16} + \beta_{7} q^{17} -\beta_{3} q^{19} + \beta_{5} q^{20} + ( -2 + \beta_{1} ) q^{22} + 2 \beta_{4} q^{23} -3 q^{25} -2 \beta_{5} q^{26} -\beta_{1} q^{28} + 6 \beta_{2} q^{29} -\beta_{6} q^{31} + \beta_{2} q^{32} + \beta_{6} q^{34} -\beta_{7} q^{35} -10 q^{37} + \beta_{5} q^{38} + \beta_{3} q^{40} -\beta_{7} q^{41} + 2 \beta_{1} q^{43} + ( -2 \beta_{2} - \beta_{4} ) q^{44} + 2 \beta_{1} q^{46} + \beta_{5} q^{47} -7 q^{49} -3 \beta_{2} q^{50} -2 \beta_{3} q^{52} + 4 \beta_{4} q^{53} + ( -2 \beta_{3} + \beta_{6} ) q^{55} + \beta_{4} q^{56} -6 q^{58} + 4 \beta_{5} q^{59} + \beta_{7} q^{62} - q^{64} + 16 \beta_{2} q^{65} -12 q^{67} -\beta_{7} q^{68} -\beta_{6} q^{70} -2 \beta_{4} q^{71} + 3 \beta_{3} q^{73} -10 \beta_{2} q^{74} + \beta_{3} q^{76} + ( 7 \beta_{2} - 2 \beta_{4} ) q^{77} + 2 \beta_{1} q^{79} -\beta_{5} q^{80} -\beta_{6} q^{82} + \beta_{7} q^{83} + 8 \beta_{1} q^{85} -2 \beta_{4} q^{86} + ( 2 - \beta_{1} ) q^{88} + 2 \beta_{6} q^{91} -2 \beta_{4} q^{92} + \beta_{3} q^{94} -8 \beta_{2} q^{95} -2 \beta_{6} q^{97} -7 \beta_{2} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 8q^{4} + O(q^{10})$$ $$8q - 8q^{4} + 8q^{16} - 16q^{22} - 24q^{25} - 80q^{37} - 56q^{49} - 48q^{58} - 8q^{64} - 96q^{67} + 16q^{88} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + x^{4} + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$2 \nu^{4} + 1$$$$)/3$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{6} + 5 \nu^{2}$$$$)/12$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{7} - 4 \nu^{5} + 7 \nu^{3} + 4 \nu$$$$)/12$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{6} + 3 \nu^{2}$$$$)/4$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{7} - 4 \nu^{5} - 7 \nu^{3} + 4 \nu$$$$)/12$$ $$\beta_{6}$$ $$=$$ $$($$$$5 \nu^{7} + 4 \nu^{5} + 13 \nu^{3} + 44 \nu$$$$)/12$$ $$\beta_{7}$$ $$=$$ $$($$$$-5 \nu^{7} + 4 \nu^{5} - 13 \nu^{3} + 44 \nu$$$$)/12$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{7} + \beta_{6} + \beta_{5} + \beta_{3}$$$$)/8$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{4} + 3 \beta_{2}$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$-\beta_{7} + \beta_{6} - 5 \beta_{5} + 5 \beta_{3}$$$$)/8$$ $$\nu^{4}$$ $$=$$ $$($$$$3 \beta_{1} - 1$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$\beta_{7} + \beta_{6} - 11 \beta_{5} - 11 \beta_{3}$$$$)/8$$ $$\nu^{6}$$ $$=$$ $$($$$$-5 \beta_{4} + 9 \beta_{2}$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$-7 \beta_{7} + 7 \beta_{6} + 13 \beta_{5} - 13 \beta_{3}$$$$)/8$$

## Character values

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

 $$n$$ $$155$$ $$199$$ $$1135$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
307.1
 1.28897 − 0.581861i −0.581861 + 1.28897i −1.28897 + 0.581861i 0.581861 − 1.28897i 0.581861 + 1.28897i −1.28897 − 0.581861i −0.581861 − 1.28897i 1.28897 + 0.581861i
1.00000i 0 −1.00000 2.82843i 0 2.64575i 1.00000i 0 −2.82843
307.2 1.00000i 0 −1.00000 2.82843i 0 2.64575i 1.00000i 0 −2.82843
307.3 1.00000i 0 −1.00000 2.82843i 0 2.64575i 1.00000i 0 2.82843
307.4 1.00000i 0 −1.00000 2.82843i 0 2.64575i 1.00000i 0 2.82843
307.5 1.00000i 0 −1.00000 2.82843i 0 2.64575i 1.00000i 0 2.82843
307.6 1.00000i 0 −1.00000 2.82843i 0 2.64575i 1.00000i 0 2.82843
307.7 1.00000i 0 −1.00000 2.82843i 0 2.64575i 1.00000i 0 −2.82843
307.8 1.00000i 0 −1.00000 2.82843i 0 2.64575i 1.00000i 0 −2.82843
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 307.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.b odd 2 1 inner
11.b odd 2 1 inner
21.c even 2 1 inner
33.d even 2 1 inner
77.b even 2 1 inner
231.h odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1386.2.e.c 8
3.b odd 2 1 inner 1386.2.e.c 8
7.b odd 2 1 inner 1386.2.e.c 8
11.b odd 2 1 inner 1386.2.e.c 8
21.c even 2 1 inner 1386.2.e.c 8
33.d even 2 1 inner 1386.2.e.c 8
77.b even 2 1 inner 1386.2.e.c 8
231.h odd 2 1 inner 1386.2.e.c 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1386.2.e.c 8 1.a even 1 1 trivial
1386.2.e.c 8 3.b odd 2 1 inner
1386.2.e.c 8 7.b odd 2 1 inner
1386.2.e.c 8 11.b odd 2 1 inner
1386.2.e.c 8 21.c even 2 1 inner
1386.2.e.c 8 33.d even 2 1 inner
1386.2.e.c 8 77.b even 2 1 inner
1386.2.e.c 8 231.h odd 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1386, [\chi])$$:

 $$T_{5}^{2} + 8$$ $$T_{13}^{2} - 32$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{2} )^{4}$$
$3$ $$T^{8}$$
$5$ $$( 8 + T^{2} )^{4}$$
$7$ $$( 7 + T^{2} )^{4}$$
$11$ $$( 121 - 6 T^{2} + T^{4} )^{2}$$
$13$ $$( -32 + T^{2} )^{4}$$
$17$ $$( -56 + T^{2} )^{4}$$
$19$ $$( -8 + T^{2} )^{4}$$
$23$ $$( -28 + T^{2} )^{4}$$
$29$ $$( 36 + T^{2} )^{4}$$
$31$ $$( 56 + T^{2} )^{4}$$
$37$ $$( 10 + T )^{8}$$
$41$ $$( -56 + T^{2} )^{4}$$
$43$ $$( 28 + T^{2} )^{4}$$
$47$ $$( 8 + T^{2} )^{4}$$
$53$ $$( -112 + T^{2} )^{4}$$
$59$ $$( 128 + T^{2} )^{4}$$
$61$ $$T^{8}$$
$67$ $$( 12 + T )^{8}$$
$71$ $$( -28 + T^{2} )^{4}$$
$73$ $$( -72 + T^{2} )^{4}$$
$79$ $$( 28 + T^{2} )^{4}$$
$83$ $$( -56 + T^{2} )^{4}$$
$89$ $$T^{8}$$
$97$ $$( 224 + T^{2} )^{4}$$