Properties

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

Related objects

Downloads

Learn more

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

\(f(q)\) \(=\) \( q - 2 q^{2} + 4 q^{4} + 17 q^{5} + 7 q^{7} - 8 q^{8} + O(q^{10}) \) \( q - 2 q^{2} + 4 q^{4} + 17 q^{5} + 7 q^{7} - 8 q^{8} - 34 q^{10} + 11 q^{11} - 21 q^{13} - 14 q^{14} + 16 q^{16} + 104 q^{17} - 161 q^{19} + 68 q^{20} - 22 q^{22} - 194 q^{23} + 164 q^{25} + 42 q^{26} + 28 q^{28} - 9 q^{29} - 180 q^{31} - 32 q^{32} - 208 q^{34} + 119 q^{35} - 363 q^{37} + 322 q^{38} - 136 q^{40} + 108 q^{41} - 386 q^{43} + 44 q^{44} + 388 q^{46} - 333 q^{47} + 49 q^{49} - 328 q^{50} - 84 q^{52} + 122 q^{53} + 187 q^{55} - 56 q^{56} + 18 q^{58} - 537 q^{59} - 950 q^{61} + 360 q^{62} + 64 q^{64} - 357 q^{65} - 83 q^{67} + 416 q^{68} - 238 q^{70} - 180 q^{71} + 177 q^{73} + 726 q^{74} - 644 q^{76} + 77 q^{77} - 220 q^{79} + 272 q^{80} - 216 q^{82} - 1112 q^{83} + 1768 q^{85} + 772 q^{86} - 88 q^{88} + 394 q^{89} - 147 q^{91} - 776 q^{92} + 666 q^{94} - 2737 q^{95} + 826 q^{97} - 98 q^{98} + O(q^{100}) \)

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 17.0000 0 7.00000 −8.00000 0 −34.0000
\(n\): 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.h 1
3.b odd 2 1 462.4.a.h 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.4.a.h 1 3.b odd 2 1
1386.4.a.h 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1386))\):

\( T_{5} - 17 \)
\( T_{13} + 21 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 2 + T \)
$3$ \( T \)
$5$ \( -17 + T \)
$7$ \( -7 + T \)
$11$ \( -11 + T \)
$13$ \( 21 + T \)
$17$ \( -104 + T \)
$19$ \( 161 + T \)
$23$ \( 194 + T \)
$29$ \( 9 + T \)
$31$ \( 180 + T \)
$37$ \( 363 + T \)
$41$ \( -108 + T \)
$43$ \( 386 + T \)
$47$ \( 333 + T \)
$53$ \( -122 + T \)
$59$ \( 537 + T \)
$61$ \( 950 + T \)
$67$ \( 83 + T \)
$71$ \( 180 + T \)
$73$ \( -177 + T \)
$79$ \( 220 + T \)
$83$ \( 1112 + T \)
$89$ \( -394 + T \)
$97$ \( -826 + T \)
show more
show less