Properties

Label 1386.4.a.a
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 154)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 2 q^{2} + 4 q^{4} - 18 q^{5} + 7 q^{7} - 8 q^{8} + O(q^{10}) \) \( q - 2 q^{2} + 4 q^{4} - 18 q^{5} + 7 q^{7} - 8 q^{8} + 36 q^{10} + 11 q^{11} + 56 q^{13} - 14 q^{14} + 16 q^{16} - 36 q^{17} - 28 q^{19} - 72 q^{20} - 22 q^{22} - 180 q^{23} + 199 q^{25} - 112 q^{26} + 28 q^{28} + 54 q^{29} - 334 q^{31} - 32 q^{32} + 72 q^{34} - 126 q^{35} + 386 q^{37} + 56 q^{38} + 144 q^{40} + 444 q^{41} - 316 q^{43} + 44 q^{44} + 360 q^{46} + 402 q^{47} + 49 q^{49} - 398 q^{50} + 224 q^{52} + 486 q^{53} - 198 q^{55} - 56 q^{56} - 108 q^{58} + 282 q^{59} + 380 q^{61} + 668 q^{62} + 64 q^{64} - 1008 q^{65} + 176 q^{67} - 144 q^{68} + 252 q^{70} + 324 q^{71} + 800 q^{73} - 772 q^{74} - 112 q^{76} + 77 q^{77} - 1144 q^{79} - 288 q^{80} - 888 q^{82} - 468 q^{83} + 648 q^{85} + 632 q^{86} - 88 q^{88} + 870 q^{89} + 392 q^{91} - 720 q^{92} - 804 q^{94} + 504 q^{95} - 1330 q^{97} - 98 q^{98} + O(q^{100}) \)

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 −18.0000 0 7.00000 −8.00000 0 36.0000
\(n\): 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.a 1
3.b odd 2 1 154.4.a.d 1
12.b even 2 1 1232.4.a.f 1
21.c even 2 1 1078.4.a.g 1
33.d even 2 1 1694.4.a.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.4.a.d 1 3.b odd 2 1
1078.4.a.g 1 21.c even 2 1
1232.4.a.f 1 12.b even 2 1
1386.4.a.a 1 1.a even 1 1 trivial
1694.4.a.c 1 33.d even 2 1

Hecke kernels

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

\( T_{5} + 18 \)
\( T_{13} - 56 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 2 + T \)
$3$ \( T \)
$5$ \( 18 + T \)
$7$ \( -7 + T \)
$11$ \( -11 + T \)
$13$ \( -56 + T \)
$17$ \( 36 + T \)
$19$ \( 28 + T \)
$23$ \( 180 + T \)
$29$ \( -54 + T \)
$31$ \( 334 + T \)
$37$ \( -386 + T \)
$41$ \( -444 + T \)
$43$ \( 316 + T \)
$47$ \( -402 + T \)
$53$ \( -486 + T \)
$59$ \( -282 + T \)
$61$ \( -380 + T \)
$67$ \( -176 + T \)
$71$ \( -324 + T \)
$73$ \( -800 + T \)
$79$ \( 1144 + T \)
$83$ \( 468 + T \)
$89$ \( -870 + T \)
$97$ \( 1330 + T \)
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