# Properties

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

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## Newspace parameters

 Level: $$N$$ $$=$$ $$1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1386.r (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$11.0672657201$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{24})$$ Defining polynomial: $$x^{8} - x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}\cdot 3^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{24}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\zeta_{24}^{2} q^{2} + \zeta_{24}^{4} q^{4} + ( -2 \zeta_{24} + \zeta_{24}^{3} + \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{5} + ( -1 - \zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{7} -\zeta_{24}^{6} q^{8} +O(q^{10})$$ $$q -\zeta_{24}^{2} q^{2} + \zeta_{24}^{4} q^{4} + ( -2 \zeta_{24} + \zeta_{24}^{3} + \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{5} + ( -1 - \zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{7} -\zeta_{24}^{6} q^{8} + ( \zeta_{24} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{10} + ( -\zeta_{24}^{2} + \zeta_{24}^{6} ) q^{11} + ( -\zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{13} + ( -2 \zeta_{24} + \zeta_{24}^{2} + \zeta_{24}^{3} + \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{14} + ( -1 + \zeta_{24}^{4} ) q^{16} + ( \zeta_{24}^{2} + \zeta_{24}^{6} ) q^{17} + ( -1 - \zeta_{24}^{4} ) q^{19} + ( -\zeta_{24} - \zeta_{24}^{3} - \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{20} + q^{22} + ( 3 \zeta_{24}^{2} + 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} - 3 \zeta_{24}^{7} ) q^{23} -\zeta_{24}^{4} q^{25} + ( -2 \zeta_{24} + \zeta_{24}^{3} + \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{26} + ( \zeta_{24} + 2 \zeta_{24}^{3} - \zeta_{24}^{4} - 2 \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{28} + ( 3 \zeta_{24} - 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} + 3 \zeta_{24}^{6} ) q^{29} + ( -4 + 2 \zeta_{24} + 4 \zeta_{24}^{3} + 2 \zeta_{24}^{4} - 4 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{31} + ( \zeta_{24}^{2} - \zeta_{24}^{6} ) q^{32} + ( 1 - 2 \zeta_{24}^{4} ) q^{34} + ( 2 \zeta_{24} + 6 \zeta_{24}^{2} - \zeta_{24}^{3} - \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{35} + ( -1 + 3 \zeta_{24}^{3} + \zeta_{24}^{4} - 3 \zeta_{24}^{5} - 3 \zeta_{24}^{7} ) q^{37} + ( \zeta_{24}^{2} + \zeta_{24}^{6} ) q^{38} + ( 2 \zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{40} + ( -2 \zeta_{24} + 4 \zeta_{24}^{2} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - 2 \zeta_{24}^{6} + 4 \zeta_{24}^{7} ) q^{41} + q^{43} -\zeta_{24}^{2} q^{44} + ( -3 \zeta_{24} - 3 \zeta_{24}^{4} - 3 \zeta_{24}^{7} ) q^{46} + ( -2 \zeta_{24} - 3 \zeta_{24}^{2} + \zeta_{24}^{3} + \zeta_{24}^{5} + 6 \zeta_{24}^{6} + \zeta_{24}^{7} ) q^{47} + ( -5 + 2 \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{49} + \zeta_{24}^{6} q^{50} + ( \zeta_{24} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{52} + ( -\zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{55} + ( -\zeta_{24} - \zeta_{24}^{3} - \zeta_{24}^{5} + \zeta_{24}^{6} + 2 \zeta_{24}^{7} ) q^{56} + ( 3 - 3 \zeta_{24}^{3} - 3 \zeta_{24}^{4} + 3 \zeta_{24}^{5} + 3 \zeta_{24}^{7} ) q^{58} + ( -4 \zeta_{24} + 3 \zeta_{24}^{2} + 8 \zeta_{24}^{3} + 8 \zeta_{24}^{5} + 3 \zeta_{24}^{6} - 4 \zeta_{24}^{7} ) q^{59} + ( 6 + 6 \zeta_{24}^{4} ) q^{61} + ( -2 \zeta_{24} + 4 \zeta_{24}^{2} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - 2 \zeta_{24}^{6} + 4 \zeta_{24}^{7} ) q^{62} - q^{64} + 6 \zeta_{24}^{2} q^{65} + ( 6 \zeta_{24} + 4 \zeta_{24}^{4} + 6 \zeta_{24}^{7} ) q^{67} + ( -\zeta_{24}^{2} + 2 \zeta_{24}^{6} ) q^{68} + ( -\zeta_{24} - 2 \zeta_{24}^{3} - 6 \zeta_{24}^{4} + 2 \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{70} + ( -3 \zeta_{24} + 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} + 3 \zeta_{24}^{6} ) q^{71} + ( 2 \zeta_{24} + 4 \zeta_{24}^{3} - 4 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{73} + ( -3 \zeta_{24} + \zeta_{24}^{2} - \zeta_{24}^{6} + 3 \zeta_{24}^{7} ) q^{74} + ( 1 - 2 \zeta_{24}^{4} ) q^{76} + ( -\zeta_{24} + \zeta_{24}^{2} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - \zeta_{24}^{6} - \zeta_{24}^{7} ) q^{77} + ( -8 + 8 \zeta_{24}^{4} ) q^{79} + ( \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{80} + ( -2 + 4 \zeta_{24} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{4} - 2 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{82} + ( -\zeta_{24} - 12 \zeta_{24}^{2} - \zeta_{24}^{3} - \zeta_{24}^{5} + 6 \zeta_{24}^{6} + 2 \zeta_{24}^{7} ) q^{83} + ( -3 \zeta_{24} - 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} ) q^{85} -\zeta_{24}^{2} q^{86} + \zeta_{24}^{4} q^{88} + ( -2 \zeta_{24} + 2 \zeta_{24}^{2} + \zeta_{24}^{3} + \zeta_{24}^{5} - 4 \zeta_{24}^{6} + \zeta_{24}^{7} ) q^{89} + ( -6 + \zeta_{24} - \zeta_{24}^{3} + \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{91} + ( -3 \zeta_{24} + 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} + 3 \zeta_{24}^{6} ) q^{92} + ( 6 + \zeta_{24} + 2 \zeta_{24}^{3} - 3 \zeta_{24}^{4} - 2 \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{94} + ( 3 \zeta_{24} - 3 \zeta_{24}^{7} ) q^{95} + ( -1 - 2 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{4} - 2 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{97} + ( 4 \zeta_{24} + 5 \zeta_{24}^{2} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 4 q^{4} - 8 q^{7} + O(q^{10})$$ $$8 q + 4 q^{4} - 8 q^{7} - 4 q^{16} - 12 q^{19} + 8 q^{22} - 4 q^{25} - 4 q^{28} - 24 q^{31} - 4 q^{37} + 8 q^{43} - 12 q^{46} - 40 q^{49} + 12 q^{58} + 72 q^{61} - 8 q^{64} + 16 q^{67} - 24 q^{70} - 32 q^{79} - 24 q^{82} + 4 q^{88} - 48 q^{91} + 36 q^{94} + O(q^{100})$$

## Character values

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

 $$n$$ $$155$$ $$199$$ $$1135$$ $$\chi(n)$$ $$-1$$ $$1 - \zeta_{24}^{2}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
89.1
 0.965926 + 0.258819i −0.965926 − 0.258819i 0.258819 − 0.965926i −0.258819 + 0.965926i 0.965926 − 0.258819i −0.965926 + 0.258819i 0.258819 + 0.965926i −0.258819 − 0.965926i
−0.866025 0.500000i 0 0.500000 + 0.866025i −1.22474 + 2.12132i 0 −1.00000 2.44949i 1.00000i 0 2.12132 1.22474i
89.2 −0.866025 0.500000i 0 0.500000 + 0.866025i 1.22474 2.12132i 0 −1.00000 + 2.44949i 1.00000i 0 −2.12132 + 1.22474i
89.3 0.866025 + 0.500000i 0 0.500000 + 0.866025i −1.22474 + 2.12132i 0 −1.00000 + 2.44949i 1.00000i 0 −2.12132 + 1.22474i
89.4 0.866025 + 0.500000i 0 0.500000 + 0.866025i 1.22474 2.12132i 0 −1.00000 2.44949i 1.00000i 0 2.12132 1.22474i
1277.1 −0.866025 + 0.500000i 0 0.500000 0.866025i −1.22474 2.12132i 0 −1.00000 + 2.44949i 1.00000i 0 2.12132 + 1.22474i
1277.2 −0.866025 + 0.500000i 0 0.500000 0.866025i 1.22474 + 2.12132i 0 −1.00000 2.44949i 1.00000i 0 −2.12132 1.22474i
1277.3 0.866025 0.500000i 0 0.500000 0.866025i −1.22474 2.12132i 0 −1.00000 2.44949i 1.00000i 0 −2.12132 1.22474i
1277.4 0.866025 0.500000i 0 0.500000 0.866025i 1.22474 + 2.12132i 0 −1.00000 + 2.44949i 1.00000i 0 2.12132 + 1.22474i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1277.4 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.d odd 6 1 inner
21.g even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1386.2.r.b 8
3.b odd 2 1 inner 1386.2.r.b 8
7.d odd 6 1 inner 1386.2.r.b 8
21.g even 6 1 inner 1386.2.r.b 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1386.2.r.b 8 1.a even 1 1 trivial
1386.2.r.b 8 3.b odd 2 1 inner
1386.2.r.b 8 7.d odd 6 1 inner
1386.2.r.b 8 21.g even 6 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} + 6 T_{5}^{2} + 36$$ acting on $$S_{2}^{\mathrm{new}}(1386, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T^{2} + T^{4} )^{2}$$
$3$ $$T^{8}$$
$5$ $$( 36 + 6 T^{2} + T^{4} )^{2}$$
$7$ $$( 7 + 2 T + T^{2} )^{4}$$
$11$ $$( 1 - T^{2} + T^{4} )^{2}$$
$13$ $$( 6 + T^{2} )^{4}$$
$17$ $$( 9 + 3 T^{2} + T^{4} )^{2}$$
$19$ $$( 3 + 3 T + T^{2} )^{4}$$
$23$ $$6561 - 4374 T^{2} + 2835 T^{4} - 54 T^{6} + T^{8}$$
$29$ $$( 81 + 54 T^{2} + T^{4} )^{2}$$
$31$ $$( 144 - 144 T + 36 T^{2} + 12 T^{3} + T^{4} )^{2}$$
$37$ $$( 289 - 34 T + 21 T^{2} + 2 T^{3} + T^{4} )^{2}$$
$41$ $$( 144 - 72 T^{2} + T^{4} )^{2}$$
$43$ $$( -1 + T )^{8}$$
$47$ $$194481 + 29106 T^{2} + 3915 T^{4} + 66 T^{6} + T^{8}$$
$53$ $$T^{8}$$
$59$ $$22667121 + 1171206 T^{2} + 55755 T^{4} + 246 T^{6} + T^{8}$$
$61$ $$( 108 - 18 T + T^{2} )^{4}$$
$67$ $$( 3136 + 448 T + 120 T^{2} - 8 T^{3} + T^{4} )^{2}$$
$71$ $$( 81 + 54 T^{2} + T^{4} )^{2}$$
$73$ $$( 576 - 24 T^{2} + T^{4} )^{2}$$
$79$ $$( 64 + 8 T + T^{2} )^{4}$$
$83$ $$( 10404 - 228 T^{2} + T^{4} )^{2}$$
$89$ $$1296 + 1296 T^{2} + 1260 T^{4} + 36 T^{6} + T^{8}$$
$97$ $$( 441 + 54 T^{2} + T^{4} )^{2}$$
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