Properties

Label 1386.4.a.d
Level $1386$
Weight $4$
Character orbit 1386.a
Self dual yes
Analytic conductor $81.777$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1386.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(81.7766472680\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 462)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 2 q^{2} + 4 q^{4} - q^{5} - 7 q^{7} - 8 q^{8} + O(q^{10}) \) \( q - 2 q^{2} + 4 q^{4} - q^{5} - 7 q^{7} - 8 q^{8} + 2 q^{10} + 11 q^{11} - 43 q^{13} + 14 q^{14} + 16 q^{16} - 100 q^{17} - 87 q^{19} - 4 q^{20} - 22 q^{22} + 58 q^{23} - 124 q^{25} + 86 q^{26} - 28 q^{28} + 223 q^{29} + 88 q^{31} - 32 q^{32} + 200 q^{34} + 7 q^{35} + 37 q^{37} + 174 q^{38} + 8 q^{40} - 128 q^{41} - 458 q^{43} + 44 q^{44} - 116 q^{46} + 341 q^{47} + 49 q^{49} + 248 q^{50} - 172 q^{52} + 342 q^{53} - 11 q^{55} + 56 q^{56} - 446 q^{58} + 105 q^{59} + 190 q^{61} - 176 q^{62} + 64 q^{64} + 43 q^{65} - 579 q^{67} - 400 q^{68} - 14 q^{70} - 128 q^{71} - 161 q^{73} - 74 q^{74} - 348 q^{76} - 77 q^{77} - 396 q^{79} - 16 q^{80} + 256 q^{82} + 420 q^{83} + 100 q^{85} + 916 q^{86} - 88 q^{88} + 798 q^{89} + 301 q^{91} + 232 q^{92} - 682 q^{94} + 87 q^{95} + 1414 q^{97} - 98 q^{98} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
−2.00000 0 4.00000 −1.00000 0 −7.00000 −8.00000 0 2.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(7\) \(1\)
\(11\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1386.4.a.d 1
3.b odd 2 1 462.4.a.f 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.4.a.f 1 3.b odd 2 1
1386.4.a.d 1 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1386))\):

\( T_{5} + 1 \)
\( T_{13} + 43 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 2 + T \)
$3$ \( T \)
$5$ \( 1 + T \)
$7$ \( 7 + T \)
$11$ \( -11 + T \)
$13$ \( 43 + T \)
$17$ \( 100 + T \)
$19$ \( 87 + T \)
$23$ \( -58 + T \)
$29$ \( -223 + T \)
$31$ \( -88 + T \)
$37$ \( -37 + T \)
$41$ \( 128 + T \)
$43$ \( 458 + T \)
$47$ \( -341 + T \)
$53$ \( -342 + T \)
$59$ \( -105 + T \)
$61$ \( -190 + T \)
$67$ \( 579 + T \)
$71$ \( 128 + T \)
$73$ \( 161 + T \)
$79$ \( 396 + T \)
$83$ \( -420 + T \)
$89$ \( -798 + T \)
$97$ \( -1414 + T \)
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