Properties

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

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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: \( 1 \)
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.

\(f(q)\) \(=\) \( q -\zeta_{24}^{2} q^{2} + \zeta_{24}^{4} q^{4} + ( -2 + \zeta_{24} - \zeta_{24}^{3} + 2 \zeta_{24}^{4} ) q^{5} + ( -\zeta_{24}^{3} + 3 \zeta_{24}^{7} ) q^{7} -\zeta_{24}^{6} q^{8} +O(q^{10})\) \( q -\zeta_{24}^{2} q^{2} + \zeta_{24}^{4} q^{4} + ( -2 + \zeta_{24} - \zeta_{24}^{3} + 2 \zeta_{24}^{4} ) q^{5} + ( -\zeta_{24}^{3} + 3 \zeta_{24}^{7} ) q^{7} -\zeta_{24}^{6} q^{8} + ( 2 \zeta_{24}^{2} - \zeta_{24}^{3} + \zeta_{24}^{5} - 2 \zeta_{24}^{6} ) q^{10} + ( -\zeta_{24}^{2} + \zeta_{24}^{6} ) q^{11} + ( -\zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{13} + ( 3 \zeta_{24} - 2 \zeta_{24}^{5} ) 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} + 4 \zeta_{24}^{3} + 2 \zeta_{24}^{4} + 2 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{19} + ( -2 + \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{20} + q^{22} + ( -2 - \zeta_{24} - 2 \zeta_{24}^{2} - 2 \zeta_{24}^{4} + \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{23} + ( -4 \zeta_{24} + \zeta_{24}^{2} + 4 \zeta_{24}^{3} - \zeta_{24}^{4} + 4 \zeta_{24}^{5} + \zeta_{24}^{6} - 4 \zeta_{24}^{7} ) q^{25} + ( -2 \zeta_{24} + \zeta_{24}^{3} + \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{26} + ( -3 \zeta_{24}^{3} + 2 \zeta_{24}^{7} ) q^{28} + ( -3 + 6 \zeta_{24}^{4} - 2 \zeta_{24}^{5} - \zeta_{24}^{6} - 2 \zeta_{24}^{7} ) q^{29} + ( 2 - 2 \zeta_{24} + 3 \zeta_{24}^{2} - 2 \zeta_{24}^{3} - \zeta_{24}^{4} + 2 \zeta_{24}^{5} - 3 \zeta_{24}^{6} + 2 \zeta_{24}^{7} ) 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 + 3 \zeta_{24}^{2} - 4 \zeta_{24}^{3} + 2 \zeta_{24}^{4} - 2 \zeta_{24}^{6} - 2 \zeta_{24}^{7} ) q^{35} + ( 3 - 4 \zeta_{24} - \zeta_{24}^{2} - 2 \zeta_{24}^{3} - 3 \zeta_{24}^{4} + 6 \zeta_{24}^{5} + 2 \zeta_{24}^{6} + 6 \zeta_{24}^{7} ) q^{37} + ( -2 \zeta_{24} - 2 \zeta_{24}^{2} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{4} - 2 \zeta_{24}^{5} - 2 \zeta_{24}^{6} - 2 \zeta_{24}^{7} ) q^{38} + ( -\zeta_{24} + 2 \zeta_{24}^{2} + \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{40} + ( -2 - 4 \zeta_{24} - 2 \zeta_{24}^{2} - 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} + \zeta_{24}^{6} ) q^{41} + ( -4 + \zeta_{24} - 4 \zeta_{24}^{2} + \zeta_{24}^{3} + 5 \zeta_{24}^{5} + 2 \zeta_{24}^{6} - 6 \zeta_{24}^{7} ) q^{43} -\zeta_{24}^{2} q^{44} + ( -\zeta_{24} + 2 \zeta_{24}^{2} + \zeta_{24}^{3} + 2 \zeta_{24}^{4} + \zeta_{24}^{5} + 2 \zeta_{24}^{6} - \zeta_{24}^{7} ) q^{46} + ( -3 \zeta_{24} + 5 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{47} + ( -3 \zeta_{24}^{2} - 5 \zeta_{24}^{6} ) q^{49} + ( 1 - 4 \zeta_{24} + 4 \zeta_{24}^{3} - 2 \zeta_{24}^{4} + \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} - \zeta_{24}^{3} - 2 \zeta_{24}^{6} + \zeta_{24}^{7} ) q^{55} + ( 2 \zeta_{24} + \zeta_{24}^{5} ) q^{56} + ( -1 - 2 \zeta_{24} + 3 \zeta_{24}^{2} + \zeta_{24}^{4} + 2 \zeta_{24}^{5} - 6 \zeta_{24}^{6} + 2 \zeta_{24}^{7} ) 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} - 7 \zeta_{24}^{3} - 6 \zeta_{24}^{5} + 6 \zeta_{24}^{7} ) q^{61} + ( -3 + 2 \zeta_{24} - 2 \zeta_{24}^{2} + 2 \zeta_{24}^{3} + \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} + ( -2 \zeta_{24} - 3 \zeta_{24}^{2} + 2 \zeta_{24}^{3} - 6 \zeta_{24}^{4} + 2 \zeta_{24}^{5} - 3 \zeta_{24}^{6} - 2 \zeta_{24}^{7} ) 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 - 2 \zeta_{24} + 3 \zeta_{24}^{2} - \zeta_{24}^{4} + 6 \zeta_{24}^{5} - 2 \zeta_{24}^{6} ) q^{70} + ( 6 + \zeta_{24} - \zeta_{24}^{3} - 12 \zeta_{24}^{4} + 3 \zeta_{24}^{5} + 2 \zeta_{24}^{6} + 4 \zeta_{24}^{7} ) q^{71} + ( -4 + \zeta_{24} - 6 \zeta_{24}^{2} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{4} + 2 \zeta_{24}^{5} + 6 \zeta_{24}^{6} - \zeta_{24}^{7} ) q^{73} + ( 2 + 6 \zeta_{24} - 3 \zeta_{24}^{2} + 4 \zeta_{24}^{3} - \zeta_{24}^{4} - 4 \zeta_{24}^{5} + 3 \zeta_{24}^{6} - 6 \zeta_{24}^{7} ) q^{74} + ( -2 - 2 \zeta_{24} + 2 \zeta_{24}^{3} + 4 \zeta_{24}^{4} + 4 \zeta_{24}^{5} - 2 \zeta_{24}^{6} + 2 \zeta_{24}^{7} ) q^{76} + ( \zeta_{24} - 3 \zeta_{24}^{5} ) q^{77} + ( 10 - \zeta_{24} - 2 \zeta_{24}^{2} + \zeta_{24}^{3} - 10 \zeta_{24}^{4} + 4 \zeta_{24}^{6} ) q^{79} + ( -\zeta_{24} + \zeta_{24}^{3} - 2 \zeta_{24}^{4} + \zeta_{24}^{5} - \zeta_{24}^{7} ) 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 - 4 \zeta_{24}^{2} - 6 \zeta_{24}^{5} + 2 \zeta_{24}^{6} + 6 \zeta_{24}^{7} ) q^{83} + ( -4 - 6 \zeta_{24} + 12 \zeta_{24}^{2} - 6 \zeta_{24}^{3} + 9 \zeta_{24}^{5} - 6 \zeta_{24}^{6} - 3 \zeta_{24}^{7} ) q^{85} + ( 2 - 6 \zeta_{24} + 4 \zeta_{24}^{2} - \zeta_{24}^{3} + 2 \zeta_{24}^{4} + 5 \zeta_{24}^{5} - 5 \zeta_{24}^{7} ) q^{86} + \zeta_{24}^{4} q^{88} + ( 4 - 8 \zeta_{24} + 2 \zeta_{24}^{2} + 5 \zeta_{24}^{3} - 4 \zeta_{24}^{4} + 3 \zeta_{24}^{5} - 4 \zeta_{24}^{6} + 3 \zeta_{24}^{7} ) q^{89} + ( 5 + \zeta_{24}^{2} - \zeta_{24}^{4} + 4 \zeta_{24}^{6} ) q^{91} + ( 2 - \zeta_{24} + \zeta_{24}^{3} - 4 \zeta_{24}^{4} - 2 \zeta_{24}^{6} - \zeta_{24}^{7} ) q^{92} + ( -2 \zeta_{24} + 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{94} + ( -4 - 6 \zeta_{24} + 4 \zeta_{24}^{2} - 8 \zeta_{24}^{3} + 2 \zeta_{24}^{4} + 8 \zeta_{24}^{5} - 4 \zeta_{24}^{6} + 6 \zeta_{24}^{7} ) q^{95} + ( -4 + 8 \zeta_{24}^{4} - 4 \zeta_{24}^{5} - \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 −1.25882 + 2.18034i 0 1.48356 2.19067i 1.00000i 0 2.18034 1.25882i
89.2 −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
89.3 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
89.4 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
1277.1 −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
1277.2 −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
1277.3 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
1277.4 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
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1277.4
Significant digits:
Format:

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

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

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