Properties

Label 19.5.b.a
Level 19
Weight 5
Character orbit 19.b
Self dual yes
Analytic conductor 1.964
Analytic rank 0
Dimension 1
CM discriminant -19
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 19 \)
Weight: \( k \) = \( 5 \)
Character orbit: \([\chi]\) = 19.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(1.96402929859\)
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

\(f(q)\) \(=\) \( q + 16q^{4} + 31q^{5} - 73q^{7} + 81q^{9} + O(q^{10}) \) \( q + 16q^{4} + 31q^{5} - 73q^{7} + 81q^{9} - 233q^{11} + 256q^{16} - 353q^{17} + 361q^{19} + 496q^{20} - 158q^{23} + 336q^{25} - 1168q^{28} - 2263q^{35} + 1296q^{36} + 3527q^{43} - 3728q^{44} + 2511q^{45} + 1207q^{47} + 2928q^{49} - 7223q^{55} + 3167q^{61} - 5913q^{63} + 4096q^{64} - 5648q^{68} - 10033q^{73} + 5776q^{76} + 17009q^{77} + 7936q^{80} + 6561q^{81} - 5678q^{83} - 10943q^{85} - 2528q^{92} + 11191q^{95} - 18873q^{99} + O(q^{100}) \)

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 16.0000 31.0000 0 −73.0000 0 81.0000 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 19.5.b.a 1
3.b odd 2 1 171.5.c.a 1
4.b odd 2 1 304.5.e.a 1
19.b odd 2 1 CM 19.5.b.a 1
57.d even 2 1 171.5.c.a 1
76.d even 2 1 304.5.e.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.5.b.a 1 1.a even 1 1 trivial
19.5.b.a 1 19.b odd 2 1 CM
171.5.c.a 1 3.b odd 2 1
171.5.c.a 1 57.d even 2 1
304.5.e.a 1 4.b odd 2 1
304.5.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 \(S_{5}^{\mathrm{new}}(19, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - 4 T )( 1 + 4 T ) \)
$3$ \( ( 1 - 9 T )( 1 + 9 T ) \)
$5$ \( 1 - 31 T + 625 T^{2} \)
$7$ \( 1 + 73 T + 2401 T^{2} \)
$11$ \( 1 + 233 T + 14641 T^{2} \)
$13$ \( ( 1 - 169 T )( 1 + 169 T ) \)
$17$ \( 1 + 353 T + 83521 T^{2} \)
$19$ \( 1 - 361 T \)
$23$ \( 1 + 158 T + 279841 T^{2} \)
$29$ \( ( 1 - 841 T )( 1 + 841 T ) \)
$31$ \( ( 1 - 961 T )( 1 + 961 T ) \)
$37$ \( ( 1 - 1369 T )( 1 + 1369 T ) \)
$41$ \( ( 1 - 1681 T )( 1 + 1681 T ) \)
$43$ \( 1 - 3527 T + 3418801 T^{2} \)
$47$ \( 1 - 1207 T + 4879681 T^{2} \)
$53$ \( ( 1 - 2809 T )( 1 + 2809 T ) \)
$59$ \( ( 1 - 3481 T )( 1 + 3481 T ) \)
$61$ \( 1 - 3167 T + 13845841 T^{2} \)
$67$ \( ( 1 - 4489 T )( 1 + 4489 T ) \)
$71$ \( ( 1 - 5041 T )( 1 + 5041 T ) \)
$73$ \( 1 + 10033 T + 28398241 T^{2} \)
$79$ \( ( 1 - 6241 T )( 1 + 6241 T ) \)
$83$ \( 1 + 5678 T + 47458321 T^{2} \)
$89$ \( ( 1 - 7921 T )( 1 + 7921 T ) \)
$97$ \( ( 1 - 9409 T )( 1 + 9409 T ) \)
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