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})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 4q^{4} - 8q^{7} + O(q^{10}) \) \( 8q + 4q^{4} - 8q^{7} - 4q^{16} - 12q^{19} + 8q^{22} - 4q^{25} - 4q^{28} - 24q^{31} - 4q^{37} + 8q^{43} - 12q^{46} - 40q^{49} + 12q^{58} + 72q^{61} - 8q^{64} + 16q^{67} - 24q^{70} - 32q^{79} - 24q^{82} + 4q^{88} - 48q^{91} + 36q^{94} + O(q^{100}) \)

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:

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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