Properties

Label 19.3.b.a
Level 19
Weight 3
Character orbit 19.b
Self dual yes
Analytic conductor 0.518
Analytic rank 0
Dimension 1
CM discriminant -19
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 19 \)
Weight: \( k \) = \( 3 \)
Character orbit: \([\chi]\) = 19.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(0.517712502285\)
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

\(f(q)\) \(=\) \( q + 4q^{4} - 9q^{5} - 5q^{7} + 9q^{9} + O(q^{10}) \) \( q + 4q^{4} - 9q^{5} - 5q^{7} + 9q^{9} + 3q^{11} + 16q^{16} + 15q^{17} - 19q^{19} - 36q^{20} - 30q^{23} + 56q^{25} - 20q^{28} + 45q^{35} + 36q^{36} - 85q^{43} + 12q^{44} - 81q^{45} + 75q^{47} - 24q^{49} - 27q^{55} + 103q^{61} - 45q^{63} + 64q^{64} + 60q^{68} - 25q^{73} - 76q^{76} - 15q^{77} - 144q^{80} + 81q^{81} + 90q^{83} - 135q^{85} - 120q^{92} + 171q^{95} + 27q^{99} + O(q^{100}) \)

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 4.00000 −9.00000 0 −5.00000 0 9.00000 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 19.3.b.a 1
3.b odd 2 1 171.3.c.a 1
4.b odd 2 1 304.3.e.a 1
5.b even 2 1 475.3.c.a 1
5.c odd 4 2 475.3.d.a 2
8.b even 2 1 1216.3.e.a 1
8.d odd 2 1 1216.3.e.b 1
12.b even 2 1 2736.3.o.a 1
19.b odd 2 1 CM 19.3.b.a 1
19.c even 3 2 361.3.d.a 2
19.d odd 6 2 361.3.d.a 2
19.e even 9 6 361.3.f.a 6
19.f odd 18 6 361.3.f.a 6
57.d even 2 1 171.3.c.a 1
76.d even 2 1 304.3.e.a 1
95.d odd 2 1 475.3.c.a 1
95.g even 4 2 475.3.d.a 2
152.b even 2 1 1216.3.e.b 1
152.g odd 2 1 1216.3.e.a 1
228.b odd 2 1 2736.3.o.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.3.b.a 1 1.a even 1 1 trivial
19.3.b.a 1 19.b odd 2 1 CM
171.3.c.a 1 3.b odd 2 1
171.3.c.a 1 57.d even 2 1
304.3.e.a 1 4.b odd 2 1
304.3.e.a 1 76.d even 2 1
361.3.d.a 2 19.c even 3 2
361.3.d.a 2 19.d odd 6 2
361.3.f.a 6 19.e even 9 6
361.3.f.a 6 19.f odd 18 6
475.3.c.a 1 5.b even 2 1
475.3.c.a 1 95.d odd 2 1
475.3.d.a 2 5.c odd 4 2
475.3.d.a 2 95.g even 4 2
1216.3.e.a 1 8.b even 2 1
1216.3.e.a 1 152.g odd 2 1
1216.3.e.b 1 8.d odd 2 1
1216.3.e.b 1 152.b even 2 1
2736.3.o.a 1 12.b even 2 1
2736.3.o.a 1 228.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{3}^{\mathrm{new}}(19, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - 2 T )( 1 + 2 T ) \)
$3$ \( ( 1 - 3 T )( 1 + 3 T ) \)
$5$ \( 1 + 9 T + 25 T^{2} \)
$7$ \( 1 + 5 T + 49 T^{2} \)
$11$ \( 1 - 3 T + 121 T^{2} \)
$13$ \( ( 1 - 13 T )( 1 + 13 T ) \)
$17$ \( 1 - 15 T + 289 T^{2} \)
$19$ \( 1 + 19 T \)
$23$ \( 1 + 30 T + 529 T^{2} \)
$29$ \( ( 1 - 29 T )( 1 + 29 T ) \)
$31$ \( ( 1 - 31 T )( 1 + 31 T ) \)
$37$ \( ( 1 - 37 T )( 1 + 37 T ) \)
$41$ \( ( 1 - 41 T )( 1 + 41 T ) \)
$43$ \( 1 + 85 T + 1849 T^{2} \)
$47$ \( 1 - 75 T + 2209 T^{2} \)
$53$ \( ( 1 - 53 T )( 1 + 53 T ) \)
$59$ \( ( 1 - 59 T )( 1 + 59 T ) \)
$61$ \( 1 - 103 T + 3721 T^{2} \)
$67$ \( ( 1 - 67 T )( 1 + 67 T ) \)
$71$ \( ( 1 - 71 T )( 1 + 71 T ) \)
$73$ \( 1 + 25 T + 5329 T^{2} \)
$79$ \( ( 1 - 79 T )( 1 + 79 T ) \)
$83$ \( 1 - 90 T + 6889 T^{2} \)
$89$ \( ( 1 - 89 T )( 1 + 89 T ) \)
$97$ \( ( 1 - 97 T )( 1 + 97 T ) \)
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