Properties

 Label 1386.2.r.c Level $1386$ Weight $2$ Character orbit 1386.r 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.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: $$1$$ 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.

 $$f(q)$$ $$=$$ $$q -\zeta_{24}^{2} q^{2} + \zeta_{24}^{4} q^{4} + ( 2 + \zeta_{24} - 2 \zeta_{24}^{4} - \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{5} + ( 2 \zeta_{24} + \zeta_{24}^{5} ) q^{7} -\zeta_{24}^{6} q^{8} +O(q^{10})$$ $$q -\zeta_{24}^{2} q^{2} + \zeta_{24}^{4} q^{4} + ( 2 + \zeta_{24} - 2 \zeta_{24}^{4} - \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{5} + ( 2 \zeta_{24} + \zeta_{24}^{5} ) q^{7} -\zeta_{24}^{6} q^{8} + ( -\zeta_{24} - 2 \zeta_{24}^{2} - \zeta_{24}^{3} + \zeta_{24}^{5} + 2 \zeta_{24}^{6} + \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}^{3} - \zeta_{24}^{7} ) q^{14} + ( -1 + \zeta_{24}^{4} ) q^{16} + ( -2 \zeta_{24} - 2 \zeta_{24}^{2} - \zeta_{24}^{4} - 2 \zeta_{24}^{6} - 2 \zeta_{24}^{7} ) q^{17} + ( 2 + 2 \zeta_{24} + 2 \zeta_{24}^{2} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{4} - 4 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{19} + ( 2 + \zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{7} ) q^{20} + q^{22} + ( 2 + \zeta_{24} - 2 \zeta_{24}^{2} + \zeta_{24}^{3} + 2 \zeta_{24}^{4} ) q^{23} + ( -\zeta_{24}^{2} - 4 \zeta_{24}^{3} - \zeta_{24}^{4} - 4 \zeta_{24}^{5} - \zeta_{24}^{6} ) q^{25} + ( -2 \zeta_{24} + \zeta_{24}^{3} + \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{26} + ( -\zeta_{24} + 3 \zeta_{24}^{5} ) q^{28} + ( 3 + 2 \zeta_{24} - 2 \zeta_{24}^{3} - 6 \zeta_{24}^{4} - \zeta_{24}^{6} + 2 \zeta_{24}^{7} ) q^{29} + ( 2 - 3 \zeta_{24}^{2} - 2 \zeta_{24}^{3} - \zeta_{24}^{4} + 2 \zeta_{24}^{5} + 3 \zeta_{24}^{6} ) q^{31} + ( \zeta_{24}^{2} - \zeta_{24}^{6} ) q^{32} + ( -2 - 2 \zeta_{24} + 2 \zeta_{24}^{3} + 4 \zeta_{24}^{4} + 2 \zeta_{24}^{5} + \zeta_{24}^{6} ) q^{34} + ( 3 + 6 \zeta_{24} + 3 \zeta_{24}^{2} - 2 \zeta_{24}^{4} - 4 \zeta_{24}^{5} - 2 \zeta_{24}^{6} ) q^{35} + ( 3 + 4 \zeta_{24} + \zeta_{24}^{2} - 6 \zeta_{24}^{3} - 3 \zeta_{24}^{4} + 2 \zeta_{24}^{5} - 2 \zeta_{24}^{6} + 2 \zeta_{24}^{7} ) q^{37} + ( 4 \zeta_{24} - 2 \zeta_{24}^{2} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{4} - 2 \zeta_{24}^{5} - 2 \zeta_{24}^{6} + 4 \zeta_{24}^{7} ) q^{38} + ( -\zeta_{24} - 2 \zeta_{24}^{2} - \zeta_{24}^{3} ) q^{40} + ( 2 + 4 \zeta_{24} - 2 \zeta_{24}^{2} + 4 \zeta_{24}^{3} - 4 \zeta_{24}^{5} + \zeta_{24}^{6} ) q^{41} + ( -4 - 5 \zeta_{24} + 4 \zeta_{24}^{2} - 5 \zeta_{24}^{3} - \zeta_{24}^{5} - 2 \zeta_{24}^{6} + 6 \zeta_{24}^{7} ) q^{43} -\zeta_{24}^{2} q^{44} + ( -2 \zeta_{24}^{2} - \zeta_{24}^{3} + 2 \zeta_{24}^{4} - \zeta_{24}^{5} - 2 \zeta_{24}^{6} ) q^{46} + ( -3 \zeta_{24} - 2 \zeta_{24}^{3} + 5 \zeta_{24}^{5} + 5 \zeta_{24}^{7} ) q^{47} + ( 3 \zeta_{24}^{2} + 5 \zeta_{24}^{6} ) q^{49} + ( -1 + 2 \zeta_{24}^{4} + 4 \zeta_{24}^{5} + \zeta_{24}^{6} + 4 \zeta_{24}^{7} ) q^{50} + ( \zeta_{24} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{52} + ( 8 + \zeta_{24} + 4 \zeta_{24}^{2} + 2 \zeta_{24}^{3} - 4 \zeta_{24}^{4} - 2 \zeta_{24}^{5} - 4 \zeta_{24}^{6} - \zeta_{24}^{7} ) q^{53} + ( \zeta_{24}^{5} + 2 \zeta_{24}^{6} + \zeta_{24}^{7} ) q^{55} + ( \zeta_{24}^{3} - 3 \zeta_{24}^{7} ) q^{56} + ( -1 + 2 \zeta_{24} - 3 \zeta_{24}^{2} - 2 \zeta_{24}^{3} + \zeta_{24}^{4} + 6 \zeta_{24}^{6} ) q^{58} + ( \zeta_{24} - 2 \zeta_{24}^{2} + 4 \zeta_{24}^{4} - 2 \zeta_{24}^{6} + \zeta_{24}^{7} ) q^{59} + ( -\zeta_{24} - 2 \zeta_{24}^{2} + 6 \zeta_{24}^{3} + 7 \zeta_{24}^{5} - 7 \zeta_{24}^{7} ) q^{61} + ( 3 - 2 \zeta_{24}^{2} + 2 \zeta_{24}^{5} + \zeta_{24}^{6} - 2 \zeta_{24}^{7} ) q^{62} - q^{64} + ( -1 - 4 \zeta_{24} - 3 \zeta_{24}^{2} - 2 \zeta_{24}^{3} - \zeta_{24}^{4} + 2 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{65} + ( 3 \zeta_{24}^{2} - 2 \zeta_{24}^{3} - 6 \zeta_{24}^{4} - 2 \zeta_{24}^{5} + 3 \zeta_{24}^{6} ) q^{67} + ( 1 + 2 \zeta_{24}^{2} + 2 \zeta_{24}^{3} - \zeta_{24}^{4} - 2 \zeta_{24}^{5} - 4 \zeta_{24}^{6} - 2 \zeta_{24}^{7} ) q^{68} + ( -2 - 3 \zeta_{24}^{2} - 6 \zeta_{24}^{3} - \zeta_{24}^{4} + 2 \zeta_{24}^{6} + 4 \zeta_{24}^{7} ) q^{70} + ( -6 - 3 \zeta_{24} + 3 \zeta_{24}^{3} + 12 \zeta_{24}^{4} - \zeta_{24}^{5} + 2 \zeta_{24}^{6} - 4 \zeta_{24}^{7} ) q^{71} + ( -4 - 3 \zeta_{24} + 6 \zeta_{24}^{2} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{4} + 2 \zeta_{24}^{5} - 6 \zeta_{24}^{6} + 3 \zeta_{24}^{7} ) q^{73} + ( -2 + 2 \zeta_{24} - 3 \zeta_{24}^{2} - 4 \zeta_{24}^{3} + \zeta_{24}^{4} + 4 \zeta_{24}^{5} + 3 \zeta_{24}^{6} - 2 \zeta_{24}^{7} ) q^{74} + ( -2 + 4 \zeta_{24} - 4 \zeta_{24}^{3} + 4 \zeta_{24}^{4} - 2 \zeta_{24}^{5} + 2 \zeta_{24}^{6} + 2 \zeta_{24}^{7} ) q^{76} + ( -3 \zeta_{24}^{3} + 2 \zeta_{24}^{7} ) q^{77} + ( 10 + \zeta_{24} + 2 \zeta_{24}^{2} - 10 \zeta_{24}^{4} - \zeta_{24}^{5} - 4 \zeta_{24}^{6} - \zeta_{24}^{7} ) q^{79} + ( \zeta_{24}^{3} + 2 \zeta_{24}^{4} + \zeta_{24}^{5} ) q^{80} + ( 1 - 2 \zeta_{24}^{2} - 4 \zeta_{24}^{3} + \zeta_{24}^{4} - 4 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{82} + ( 9 - 6 \zeta_{24} - 4 \zeta_{24}^{2} - 6 \zeta_{24}^{3} + 2 \zeta_{24}^{6} + 6 \zeta_{24}^{7} ) q^{83} + ( -4 - 9 \zeta_{24} - 12 \zeta_{24}^{2} - 9 \zeta_{24}^{3} + 6 \zeta_{24}^{5} + 6 \zeta_{24}^{6} + 3 \zeta_{24}^{7} ) q^{85} + ( -2 + 6 \zeta_{24} + 4 \zeta_{24}^{2} + 5 \zeta_{24}^{3} - 2 \zeta_{24}^{4} - \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{86} + \zeta_{24}^{4} q^{88} + ( -4 - 8 \zeta_{24} + 2 \zeta_{24}^{2} + 3 \zeta_{24}^{3} + 4 \zeta_{24}^{4} + 5 \zeta_{24}^{5} - 4 \zeta_{24}^{6} + 5 \zeta_{24}^{7} ) q^{89} + ( 5 - \zeta_{24}^{2} - \zeta_{24}^{4} - 4 \zeta_{24}^{6} ) q^{91} + ( -2 + 4 \zeta_{24}^{4} + \zeta_{24}^{5} - 2 \zeta_{24}^{6} + \zeta_{24}^{7} ) q^{92} + ( 5 \zeta_{24} + 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} - 5 \zeta_{24}^{7} ) q^{94} + ( 4 + 2 \zeta_{24} + 4 \zeta_{24}^{2} + 8 \zeta_{24}^{3} - 2 \zeta_{24}^{4} - 8 \zeta_{24}^{5} - 4 \zeta_{24}^{6} - 2 \zeta_{24}^{7} ) q^{95} + ( -4 - 4 \zeta_{24} + 4 \zeta_{24}^{3} + 8 \zeta_{24}^{4} + \zeta_{24}^{6} - 4 \zeta_{24}^{7} ) q^{97} + ( 5 - 8 \zeta_{24}^{4} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 4q^{4} + 8q^{5} + O(q^{10})$$ $$8q + 4q^{4} + 8q^{5} - 4q^{16} - 4q^{17} + 24q^{19} + 16q^{20} + 8q^{22} + 24q^{23} - 4q^{25} + 12q^{31} + 16q^{35} + 12q^{37} - 8q^{38} + 16q^{41} - 32q^{43} + 8q^{46} + 48q^{53} - 4q^{58} + 16q^{59} + 24q^{62} - 8q^{64} - 12q^{65} - 24q^{67} + 4q^{68} - 20q^{70} - 24q^{73} - 12q^{74} + 40q^{79} + 8q^{80} + 12q^{82} + 72q^{83} - 32q^{85} - 24q^{86} + 4q^{88} - 16q^{89} + 36q^{91} + 24q^{95} + 8q^{98} + O(q^{100})$$

Character values

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

 $$n$$ $$155$$ $$199$$ $$1135$$ $$\chi(n)$$ $$-1$$ $$1 - \zeta_{24}^{4}$$ $$1$$

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 0.0340742 0.0590182i 0 −2.19067 1.48356i 1.00000i 0 −0.0590182 + 0.0340742i
89.2 −0.866025 0.500000i 0 0.500000 + 0.866025i 1.96593 3.40508i 0 2.19067 + 1.48356i 1.00000i 0 −3.40508 + 1.96593i
89.3 0.866025 + 0.500000i 0 0.500000 + 0.866025i 0.741181 1.28376i 0 −1.48356 + 2.19067i 1.00000i 0 1.28376 0.741181i
89.4 0.866025 + 0.500000i 0 0.500000 + 0.866025i 1.25882 2.18034i 0 1.48356 2.19067i 1.00000i 0 2.18034 1.25882i
1277.1 −0.866025 + 0.500000i 0 0.500000 0.866025i 0.0340742 + 0.0590182i 0 −2.19067 + 1.48356i 1.00000i 0 −0.0590182 0.0340742i
1277.2 −0.866025 + 0.500000i 0 0.500000 0.866025i 1.96593 + 3.40508i 0 2.19067 1.48356i 1.00000i 0 −3.40508 1.96593i
1277.3 0.866025 0.500000i 0 0.500000 0.866025i 0.741181 + 1.28376i 0 −1.48356 2.19067i 1.00000i 0 1.28376 + 0.741181i
1277.4 0.866025 0.500000i 0 0.500000 0.866025i 1.25882 + 2.18034i 0 1.48356 + 2.19067i 1.00000i 0 2.18034 + 1.25882i
 $$n$$: 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
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.c yes 8
3.b odd 2 1 1386.2.r.a 8
7.d odd 6 1 1386.2.r.a 8
21.g even 6 1 inner 1386.2.r.c yes 8

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{8} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(1386, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T^{2} + T^{4} )^{2}$$
$3$ $$T^{8}$$
$5$ $$1 - 16 T + 236 T^{2} - 304 T^{3} + 271 T^{4} - 128 T^{5} + 44 T^{6} - 8 T^{7} + T^{8}$$
$7$ $$2401 + 71 T^{4} + T^{8}$$
$11$ $$( 1 - T^{2} + T^{4} )^{2}$$
$13$ $$( 6 + T^{2} )^{4}$$
$17$ $$529 + 1748 T + 4994 T^{2} + 2768 T^{3} + 1483 T^{4} + 16 T^{5} + 50 T^{6} + 4 T^{7} + T^{8}$$
$19$ $$1290496 - 763392 T + 132352 T^{2} + 10752 T^{3} - 3984 T^{4} - 384 T^{5} + 208 T^{6} - 24 T^{7} + T^{8}$$
$23$ $$2209 + 5640 T + 1980 T^{2} - 7200 T^{3} + 4607 T^{4} - 1440 T^{5} + 252 T^{6} - 24 T^{7} + T^{8}$$
$29$ $$10000 + 91200 T^{2} + 6536 T^{4} + 144 T^{6} + T^{8}$$
$31$ $$16 + 96 T + 128 T^{2} - 384 T^{3} + 156 T^{4} + 192 T^{5} + 32 T^{6} - 12 T^{7} + T^{8}$$
$37$ $$5740816 - 2357664 T + 814912 T^{2} - 120480 T^{3} + 18300 T^{4} - 1200 T^{5} + 208 T^{6} - 12 T^{7} + T^{8}$$
$41$ $$( 577 + 248 T - 46 T^{2} - 8 T^{3} + T^{4} )^{2}$$
$43$ $$( -3644 - 1504 T - 52 T^{2} + 16 T^{3} + T^{4} )^{2}$$
$47$ $$14641 + 9196 T^{2} + 5655 T^{4} + 76 T^{6} + T^{8}$$
$53$ $$85264 - 364416 T + 590416 T^{2} - 304512 T^{3} + 79212 T^{4} - 11712 T^{5} + 1012 T^{6} - 48 T^{7} + T^{8}$$
$59$ $$8464 + 2944 T + 7280 T^{2} - 5120 T^{3} + 4204 T^{4} - 1024 T^{5} + 188 T^{6} - 16 T^{7} + T^{8}$$
$61$ $$38651089 - 1939704 T - 1086612 T^{2} + 56160 T^{3} + 26183 T^{4} - 180 T^{6} + T^{8}$$
$67$ $$3150625 + 213000 T + 273550 T^{2} + 67680 T^{3} + 25971 T^{4} + 3744 T^{5} + 430 T^{6} + 24 T^{7} + T^{8}$$
$71$ $$18696976 + 4721568 T^{2} + 94040 T^{4} + 552 T^{6} + T^{8}$$
$73$ $$3671056 + 2851008 T + 791696 T^{2} + 41664 T^{3} - 9204 T^{4} - 672 T^{5} + 164 T^{6} + 24 T^{7} + T^{8}$$
$79$ $$56806369 - 26108168 T + 7688132 T^{2} - 1378448 T^{3} + 181087 T^{4} - 15952 T^{5} + 1028 T^{6} - 40 T^{7} + T^{8}$$
$83$ $$( -15111 + 972 T + 318 T^{2} - 36 T^{3} + T^{4} )^{2}$$
$89$ $$3341584 - 3451264 T + 3791216 T^{2} + 175616 T^{3} + 43756 T^{4} + 1792 T^{5} + 380 T^{6} + 16 T^{7} + T^{8}$$
$97$ $$2070721 + 731844 T^{2} + 29510 T^{4} + 324 T^{6} + T^{8}$$