# Properties

 Label 1386.2.e.b Level $1386$ Weight $2$ Character orbit 1386.e Analytic conductor $11.067$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# 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: 8.0.12745506816.1 Defining polynomial: $$x^{8} + 23 x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: no (minimal twist has level 154) 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 23 x^{4} + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{6} + 24 \nu^{2}$$$$)/5$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{7} + 24 \nu^{3} + 5 \nu$$$$)/5$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{7} + 24 \nu^{3} - 5 \nu$$$$)/5$$ $$\beta_{4}$$ $$=$$ $$($$$$-3 \nu^{6} - \nu^{4} - 67 \nu^{2} - 9$$$$)/5$$ $$\beta_{5}$$ $$=$$ $$($$$$-3 \nu^{6} + \nu^{4} - 67 \nu^{2} + 9$$$$)/5$$ $$\beta_{6}$$ $$=$$ $$($$$$-5 \nu^{7} - \nu^{5} - 115 \nu^{3} - 24 \nu$$$$)/5$$ $$\beta_{7}$$ $$=$$ $$($$$$-5 \nu^{7} + \nu^{5} - 115 \nu^{3} + 24 \nu$$$$)/5$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{3} + \beta_{2}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{5} + \beta_{4} + 6 \beta_{1}$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{7} + \beta_{6} + 5 \beta_{3} + 5 \beta_{2}$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$5 \beta_{5} - 5 \beta_{4} - 18$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$5 \beta_{7} - 5 \beta_{6} + 24 \beta_{3} - 24 \beta_{2}$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$-12 \beta_{5} - 12 \beta_{4} - 67 \beta_{1}$$ $$\nu^{7}$$ $$=$$ $$($$$$-24 \beta_{7} - 24 \beta_{6} - 115 \beta_{3} - 115 \beta_{2}$$$$)/2$$

## 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
 1.54779 + 1.54779i 0.323042 + 0.323042i −0.323042 − 0.323042i −1.54779 − 1.54779i −1.54779 + 1.54779i −0.323042 + 0.323042i 0.323042 − 0.323042i 1.54779 − 1.54779i
1.00000i 0 −1.00000 3.09557i 0 −2.44949 1.00000i 1.00000i 0 −3.09557
307.2 1.00000i 0 −1.00000 0.646084i 0 2.44949 1.00000i 1.00000i 0 −0.646084
307.3 1.00000i 0 −1.00000 0.646084i 0 −2.44949 1.00000i 1.00000i 0 0.646084
307.4 1.00000i 0 −1.00000 3.09557i 0 2.44949 1.00000i 1.00000i 0 3.09557
307.5 1.00000i 0 −1.00000 3.09557i 0 2.44949 + 1.00000i 1.00000i 0 3.09557
307.6 1.00000i 0 −1.00000 0.646084i 0 −2.44949 + 1.00000i 1.00000i 0 0.646084
307.7 1.00000i 0 −1.00000 0.646084i 0 2.44949 + 1.00000i 1.00000i 0 −0.646084
307.8 1.00000i 0 −1.00000 3.09557i 0 −2.44949 + 1.00000i 1.00000i 0 −3.09557
 $$n$$: 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
7.b odd 2 1 inner
11.b odd 2 1 inner
77.b even 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.b 8
3.b odd 2 1 154.2.c.a 8
7.b odd 2 1 inner 1386.2.e.b 8
11.b odd 2 1 inner 1386.2.e.b 8
12.b even 2 1 1232.2.e.e 8
21.c even 2 1 154.2.c.a 8
21.g even 6 2 1078.2.i.b 16
21.h odd 6 2 1078.2.i.b 16
33.d even 2 1 154.2.c.a 8
77.b even 2 1 inner 1386.2.e.b 8
84.h odd 2 1 1232.2.e.e 8
132.d odd 2 1 1232.2.e.e 8
231.h odd 2 1 154.2.c.a 8
231.k odd 6 2 1078.2.i.b 16
231.l even 6 2 1078.2.i.b 16
924.n even 2 1 1232.2.e.e 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.2.c.a 8 3.b odd 2 1
154.2.c.a 8 21.c even 2 1
154.2.c.a 8 33.d even 2 1
154.2.c.a 8 231.h odd 2 1
1078.2.i.b 16 21.g even 6 2
1078.2.i.b 16 21.h odd 6 2
1078.2.i.b 16 231.k odd 6 2
1078.2.i.b 16 231.l even 6 2
1232.2.e.e 8 12.b even 2 1
1232.2.e.e 8 84.h odd 2 1
1232.2.e.e 8 132.d odd 2 1
1232.2.e.e 8 924.n even 2 1
1386.2.e.b 8 1.a even 1 1 trivial
1386.2.e.b 8 7.b odd 2 1 inner
1386.2.e.b 8 11.b odd 2 1 inner
1386.2.e.b 8 77.b even 2 1 inner

## Hecke kernels

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

 $$T_{5}^{4} + 10 T_{5}^{2} + 4$$ $$T_{13}^{4} - 10 T_{13}^{2} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{2} )^{4}$$
$3$ $$T^{8}$$
$5$ $$( 4 + 10 T^{2} + T^{4} )^{2}$$
$7$ $$( 49 - 10 T^{2} + T^{4} )^{2}$$
$11$ $$( 121 + 22 T + 2 T^{2} + 2 T^{3} + T^{4} )^{2}$$
$13$ $$( 4 - 10 T^{2} + T^{4} )^{2}$$
$17$ $$( -14 + T^{2} )^{4}$$
$19$ $$( 100 - 34 T^{2} + T^{4} )^{2}$$
$23$ $$( 4 + T )^{8}$$
$29$ $$( 144 + 60 T^{2} + T^{4} )^{2}$$
$31$ $$( 100 + 76 T^{2} + T^{4} )^{2}$$
$37$ $$( -20 + 2 T + T^{2} )^{4}$$
$41$ $$( 2500 - 124 T^{2} + T^{4} )^{2}$$
$43$ $$( 6400 + 176 T^{2} + T^{4} )^{2}$$
$47$ $$( 2500 + 124 T^{2} + T^{4} )^{2}$$
$53$ $$( 28 - 14 T + T^{2} )^{4}$$
$59$ $$( 4 + 10 T^{2} + T^{4} )^{2}$$
$61$ $$( 324 - 90 T^{2} + T^{4} )^{2}$$
$67$ $$( -12 + 6 T + T^{2} )^{4}$$
$71$ $$( 2 + T )^{8}$$
$73$ $$( 10404 - 300 T^{2} + T^{4} )^{2}$$
$79$ $$( 16 + T^{2} )^{4}$$
$83$ $$( 13924 - 250 T^{2} + T^{4} )^{2}$$
$89$ $$( 96 + T^{2} )^{4}$$
$97$ $$( 18496 + 328 T^{2} + T^{4} )^{2}$$