Properties

Label 1386.4.a.m
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

\(f(q)\) \(=\) \( q + 2 q^{2} + 4 q^{4} + 14 q^{5} - 7 q^{7} + 8 q^{8} + O(q^{10}) \) \( q + 2 q^{2} + 4 q^{4} + 14 q^{5} - 7 q^{7} + 8 q^{8} + 28 q^{10} - 11 q^{11} + 38 q^{13} - 14 q^{14} + 16 q^{16} - 54 q^{17} + 40 q^{19} + 56 q^{20} - 22 q^{22} - 8 q^{23} + 71 q^{25} + 76 q^{26} - 28 q^{28} + 170 q^{29} + 92 q^{31} + 32 q^{32} - 108 q^{34} - 98 q^{35} + 294 q^{37} + 80 q^{38} + 112 q^{40} + 258 q^{41} - 52 q^{43} - 44 q^{44} - 16 q^{46} + 76 q^{47} + 49 q^{49} + 142 q^{50} + 152 q^{52} + 322 q^{53} - 154 q^{55} - 56 q^{56} + 340 q^{58} - 260 q^{59} + 22 q^{61} + 184 q^{62} + 64 q^{64} + 532 q^{65} - 436 q^{67} - 216 q^{68} - 196 q^{70} + 368 q^{71} - 2 q^{73} + 588 q^{74} + 160 q^{76} + 77 q^{77} - 200 q^{79} + 224 q^{80} + 516 q^{82} + 952 q^{83} - 756 q^{85} - 104 q^{86} - 88 q^{88} + 70 q^{89} - 266 q^{91} - 32 q^{92} + 152 q^{94} + 560 q^{95} - 1086 q^{97} + 98 q^{98} + O(q^{100}) \)

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 14.0000 0 −7.00000 8.00000 0 28.0000
\(n\): 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.m 1
3.b odd 2 1 462.4.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.4.a.b 1 3.b odd 2 1
1386.4.a.m 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1386))\):

\( T_{5} - 14 \)
\( T_{13} - 38 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -2 + T \)
$3$ \( T \)
$5$ \( -14 + T \)
$7$ \( 7 + T \)
$11$ \( 11 + T \)
$13$ \( -38 + T \)
$17$ \( 54 + T \)
$19$ \( -40 + T \)
$23$ \( 8 + T \)
$29$ \( -170 + T \)
$31$ \( -92 + T \)
$37$ \( -294 + T \)
$41$ \( -258 + T \)
$43$ \( 52 + T \)
$47$ \( -76 + T \)
$53$ \( -322 + T \)
$59$ \( 260 + T \)
$61$ \( -22 + T \)
$67$ \( 436 + T \)
$71$ \( -368 + T \)
$73$ \( 2 + T \)
$79$ \( 200 + T \)
$83$ \( -952 + T \)
$89$ \( -70 + T \)
$97$ \( 1086 + T \)
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