Properties

Label 1386.4.a.l
Level $1386$
Weight $4$
Character orbit 1386.a
Self dual yes
Analytic conductor $81.777$
Analytic rank $1$
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: \(1\)
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

\(f(q)\) \(=\) \( q + 2 q^{2} + 4 q^{4} + 7 q^{5} + 7 q^{7} + 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{2} + 4 q^{4} + 7 q^{5} + 7 q^{7} + 8 q^{8} + 14 q^{10} - 11 q^{11} - 67 q^{13} + 14 q^{14} + 16 q^{16} - 30 q^{17} - 7 q^{19} + 28 q^{20} - 22 q^{22} - 28 q^{23} - 76 q^{25} - 134 q^{26} + 28 q^{28} - 121 q^{29} - 310 q^{31} + 32 q^{32} - 60 q^{34} + 49 q^{35} - 71 q^{37} - 14 q^{38} + 56 q^{40} + 180 q^{41} - 108 q^{43} - 44 q^{44} - 56 q^{46} - 71 q^{47} + 49 q^{49} - 152 q^{50} - 268 q^{52} - 128 q^{53} - 77 q^{55} + 56 q^{56} - 242 q^{58} + 429 q^{59} + 22 q^{61} - 620 q^{62} + 64 q^{64} - 469 q^{65} - 803 q^{67} - 120 q^{68} + 98 q^{70} - 468 q^{71} - 117 q^{73} - 142 q^{74} - 28 q^{76} - 77 q^{77} - 96 q^{79} + 112 q^{80} + 360 q^{82} + 1122 q^{83} - 210 q^{85} - 216 q^{86} - 88 q^{88} + 1146 q^{89} - 469 q^{91} - 112 q^{92} - 142 q^{94} - 49 q^{95} - 92 q^{97} + 98 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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

\( T_{5} - 7 \) Copy content Toggle raw display
\( T_{13} + 67 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 2 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T - 7 \) Copy content Toggle raw display
$7$ \( T - 7 \) Copy content Toggle raw display
$11$ \( T + 11 \) Copy content Toggle raw display
$13$ \( T + 67 \) Copy content Toggle raw display
$17$ \( T + 30 \) Copy content Toggle raw display
$19$ \( T + 7 \) Copy content Toggle raw display
$23$ \( T + 28 \) Copy content Toggle raw display
$29$ \( T + 121 \) Copy content Toggle raw display
$31$ \( T + 310 \) Copy content Toggle raw display
$37$ \( T + 71 \) Copy content Toggle raw display
$41$ \( T - 180 \) Copy content Toggle raw display
$43$ \( T + 108 \) Copy content Toggle raw display
$47$ \( T + 71 \) Copy content Toggle raw display
$53$ \( T + 128 \) Copy content Toggle raw display
$59$ \( T - 429 \) Copy content Toggle raw display
$61$ \( T - 22 \) Copy content Toggle raw display
$67$ \( T + 803 \) Copy content Toggle raw display
$71$ \( T + 468 \) Copy content Toggle raw display
$73$ \( T + 117 \) Copy content Toggle raw display
$79$ \( T + 96 \) Copy content Toggle raw display
$83$ \( T - 1122 \) Copy content Toggle raw display
$89$ \( T - 1146 \) Copy content Toggle raw display
$97$ \( T + 92 \) Copy content Toggle raw display
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