# Properties

 Label 1386.2.e.e Level $1386$ Weight $2$ Character orbit 1386.e Analytic conductor $11.067$ Analytic rank $0$ Dimension $8$ 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.e (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$11.0672657201$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.6679465984.1 Defining polynomial: $$x^{8} + 14 x^{6} + 61 x^{4} + 88 x^{2} + 36$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: no (minimal twist has level 462) 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 14 x^{6} + 61 x^{4} + 88 x^{2} + 36$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{7} + 14 \nu^{5} + 55 \nu^{3} + 46 \nu$$$$)/12$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{4} + 9 \nu^{2} + 12$$$$)/2$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{7} + 12 \nu^{5} + 41 \nu^{3} + 38 \nu$$$$)/4$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{7} + 3 \nu^{6} - 11 \nu^{5} + 33 \nu^{4} - 28 \nu^{3} + 96 \nu^{2} + 8 \nu + 60$$$$)/12$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{7} + 3 \nu^{6} + 11 \nu^{5} + 33 \nu^{4} + 28 \nu^{3} + 96 \nu^{2} - 8 \nu + 60$$$$)/12$$ $$\beta_{6}$$ $$=$$ $$($$$$-\nu^{7} + 3 \nu^{6} - 11 \nu^{5} + 39 \nu^{4} - 28 \nu^{3} + 138 \nu^{2} - 4 \nu + 96$$$$)/12$$ $$\beta_{7}$$ $$=$$ $$($$$$\nu^{7} + 3 \nu^{6} + 11 \nu^{5} + 39 \nu^{4} + 28 \nu^{3} + 138 \nu^{2} + 4 \nu + 96$$$$)/12$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{7} - \beta_{6} - \beta_{5} + \beta_{4}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} + 2 \beta_{2} - 6$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$-7 \beta_{7} + 7 \beta_{6} + 5 \beta_{5} - 5 \beta_{4} + 2 \beta_{3} - 2 \beta_{1}$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$9 \beta_{7} + 9 \beta_{6} - 9 \beta_{5} - 9 \beta_{4} - 14 \beta_{2} + 30$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$45 \beta_{7} - 45 \beta_{6} - 31 \beta_{5} + 31 \beta_{4} - 18 \beta_{3} + 26 \beta_{1}$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$($$$$-67 \beta_{7} - 67 \beta_{6} + 71 \beta_{5} + 71 \beta_{4} + 90 \beta_{2} - 178$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$-291 \beta_{7} + 291 \beta_{6} + 205 \beta_{5} - 205 \beta_{4} + 142 \beta_{3} - 230 \beta_{1}$$$$)/2$$

## 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
 − 2.20392i 2.62511i − 1.25051i 0.829319i − 0.829319i 1.25051i − 2.62511i 2.20392i
1.00000i 0 −1.00000 3.06118i 0 2.37330 + 1.16938i 1.00000i 0 −3.06118
307.2 1.00000i 0 −1.00000 0.266088i 0 −1.34926 + 2.27585i 1.00000i 0 −0.266088
307.3 1.00000i 0 −1.00000 1.18572i 0 −1.74138 1.99189i 1.00000i 0 1.18572
307.4 1.00000i 0 −1.00000 4.14155i 0 0.717333 + 2.54665i 1.00000i 0 4.14155
307.5 1.00000i 0 −1.00000 4.14155i 0 0.717333 2.54665i 1.00000i 0 4.14155
307.6 1.00000i 0 −1.00000 1.18572i 0 −1.74138 + 1.99189i 1.00000i 0 1.18572
307.7 1.00000i 0 −1.00000 0.266088i 0 −1.34926 2.27585i 1.00000i 0 −0.266088
307.8 1.00000i 0 −1.00000 3.06118i 0 2.37330 1.16938i 1.00000i 0 −3.06118
 $$n$$: 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
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.e 8
3.b odd 2 1 462.2.e.a 8
7.b odd 2 1 1386.2.e.a 8
11.b odd 2 1 1386.2.e.a 8
12.b even 2 1 3696.2.q.b 8
21.c even 2 1 462.2.e.b yes 8
33.d even 2 1 462.2.e.b yes 8
77.b even 2 1 inner 1386.2.e.e 8
84.h odd 2 1 3696.2.q.c 8
132.d odd 2 1 3696.2.q.c 8
231.h odd 2 1 462.2.e.a 8
924.n even 2 1 3696.2.q.b 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.e.a 8 3.b odd 2 1
462.2.e.a 8 231.h odd 2 1
462.2.e.b yes 8 21.c even 2 1
462.2.e.b yes 8 33.d even 2 1
1386.2.e.a 8 7.b odd 2 1
1386.2.e.a 8 11.b odd 2 1
1386.2.e.e 8 1.a even 1 1 trivial
1386.2.e.e 8 77.b even 2 1 inner
3696.2.q.b 8 12.b even 2 1
3696.2.q.b 8 924.n even 2 1
3696.2.q.c 8 84.h odd 2 1
3696.2.q.c 8 132.d odd 2 1

## Hecke kernels

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

 $$T_{5}^{8} + 28 T_{5}^{6} + 200 T_{5}^{4} + 240 T_{5}^{2} + 16$$ $$T_{13}^{4} - 26 T_{13}^{2} + 56 T_{13} - 16$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{2} )^{4}$$
$3$ $$T^{8}$$
$5$ $$16 + 240 T^{2} + 200 T^{4} + 28 T^{6} + T^{8}$$
$7$ $$2401 + 294 T^{2} - 112 T^{3} + 50 T^{4} - 16 T^{5} + 6 T^{6} + T^{8}$$
$11$ $$14641 - 10648 T + 4840 T^{2} - 1848 T^{3} + 606 T^{4} - 168 T^{5} + 40 T^{6} - 8 T^{7} + T^{8}$$
$13$ $$( -16 + 56 T - 26 T^{2} + T^{4} )^{2}$$
$17$ $$( -24 + 80 T - 10 T^{2} - 6 T^{3} + T^{4} )^{2}$$
$19$ $$( -40 + 208 T - 58 T^{2} - 2 T^{3} + T^{4} )^{2}$$
$23$ $$( -424 + 288 T - 46 T^{2} - 4 T^{3} + T^{4} )^{2}$$
$29$ $$9216 + 5632 T^{2} + 976 T^{4} + 56 T^{6} + T^{8}$$
$31$ $$6400 + 13504 T^{2} + 2436 T^{4} + 112 T^{6} + T^{8}$$
$37$ $$( 432 + 448 T - 64 T^{2} - 8 T^{3} + T^{4} )^{2}$$
$41$ $$( -656 + 384 T - 22 T^{2} - 10 T^{3} + T^{4} )^{2}$$
$43$ $$73984 + 43264 T^{2} + 4448 T^{4} + 144 T^{6} + T^{8}$$
$47$ $$1507984 + 423440 T^{2} + 16872 T^{4} + 228 T^{6} + T^{8}$$
$53$ $$( 512 + 64 T - 182 T^{2} + T^{4} )^{2}$$
$59$ $$11943936 + 993280 T^{2} + 25920 T^{4} + 272 T^{6} + T^{8}$$
$61$ $$( -3232 + 96 T + 206 T^{2} - 28 T^{3} + T^{4} )^{2}$$
$67$ $$( 2624 + 1248 T - 172 T^{2} - 8 T^{3} + T^{4} )^{2}$$
$71$ $$( -1416 - 1360 T - 222 T^{2} + 4 T^{3} + T^{4} )^{2}$$
$73$ $$( 64 - 22 T^{2} + 2 T^{3} + T^{4} )^{2}$$
$79$ $$20502784 + 1359104 T^{2} + 31524 T^{4} + 300 T^{6} + T^{8}$$
$83$ $$( 4296 - 176 T - 146 T^{2} + 2 T^{3} + T^{4} )^{2}$$
$89$ $$147456 + 77824 T^{2} + 10512 T^{4} + 200 T^{6} + T^{8}$$
$97$ $$1024 + 47616 T^{2} + 4688 T^{4} + 136 T^{6} + T^{8}$$