Properties

Label 19.9.b.a
Level 19
Weight 9
Character orbit 19.b
Self dual yes
Analytic conductor 7.740
Analytic rank 0
Dimension 1
CM discriminant -19
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 19 \)
Weight: \( k \) = \( 9 \)
Character orbit: \([\chi]\) = 19.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(7.74019359116\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

\(f(q)\) \(=\) \( q + 256q^{4} - 289q^{5} + 527q^{7} + 6561q^{9} + O(q^{10}) \) \( q + 256q^{4} - 289q^{5} + 527q^{7} + 6561q^{9} + 25007q^{11} + 65536q^{16} - 42433q^{17} + 130321q^{19} - 73984q^{20} - 534718q^{23} - 307104q^{25} + 134912q^{28} - 152303q^{35} + 1679616q^{36} + 5602127q^{43} + 6401792q^{44} - 1896129q^{45} - 8302513q^{47} - 5487072q^{49} - 7227023q^{55} - 17661793q^{61} + 3457647q^{63} + 16777216q^{64} - 10862848q^{68} + 43864607q^{73} + 33362176q^{76} + 13178689q^{77} - 18939904q^{80} + 43046721q^{81} - 62676958q^{83} + 12263137q^{85} - 136887808q^{92} - 37662769q^{95} + 164070927q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/19\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
18.1
0
0 0 256.000 −289.000 0 527.000 0 6561.00 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.b odd 2 1 CM by \(\Q(\sqrt{-19}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 19.9.b.a 1
3.b odd 2 1 171.9.c.a 1
4.b odd 2 1 304.9.e.a 1
19.b odd 2 1 CM 19.9.b.a 1
57.d even 2 1 171.9.c.a 1
76.d even 2 1 304.9.e.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.9.b.a 1 1.a even 1 1 trivial
19.9.b.a 1 19.b odd 2 1 CM
171.9.c.a 1 3.b odd 2 1
171.9.c.a 1 57.d even 2 1
304.9.e.a 1 4.b odd 2 1
304.9.e.a 1 76.d even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{9}^{\mathrm{new}}(19, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - 16 T )( 1 + 16 T ) \)
$3$ \( ( 1 - 81 T )( 1 + 81 T ) \)
$5$ \( 1 + 289 T + 390625 T^{2} \)
$7$ \( 1 - 527 T + 5764801 T^{2} \)
$11$ \( 1 - 25007 T + 214358881 T^{2} \)
$13$ \( ( 1 - 28561 T )( 1 + 28561 T ) \)
$17$ \( 1 + 42433 T + 6975757441 T^{2} \)
$19$ \( 1 - 130321 T \)
$23$ \( 1 + 534718 T + 78310985281 T^{2} \)
$29$ \( ( 1 - 707281 T )( 1 + 707281 T ) \)
$31$ \( ( 1 - 923521 T )( 1 + 923521 T ) \)
$37$ \( ( 1 - 1874161 T )( 1 + 1874161 T ) \)
$41$ \( ( 1 - 2825761 T )( 1 + 2825761 T ) \)
$43$ \( 1 - 5602127 T + 11688200277601 T^{2} \)
$47$ \( 1 + 8302513 T + 23811286661761 T^{2} \)
$53$ \( ( 1 - 7890481 T )( 1 + 7890481 T ) \)
$59$ \( ( 1 - 12117361 T )( 1 + 12117361 T ) \)
$61$ \( 1 + 17661793 T + 191707312997281 T^{2} \)
$67$ \( ( 1 - 20151121 T )( 1 + 20151121 T ) \)
$71$ \( ( 1 - 25411681 T )( 1 + 25411681 T ) \)
$73$ \( 1 - 43864607 T + 806460091894081 T^{2} \)
$79$ \( ( 1 - 38950081 T )( 1 + 38950081 T ) \)
$83$ \( 1 + 62676958 T + 2252292232139041 T^{2} \)
$89$ \( ( 1 - 62742241 T )( 1 + 62742241 T ) \)
$97$ \( ( 1 - 88529281 T )( 1 + 88529281 T ) \)
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