Properties

Label 19.11.b.a
Level 19
Weight 11
Character orbit 19.b
Self dual yes
Analytic conductor 12.072
Analytic rank 0
Dimension 1
CM discriminant -19
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 19 \)
Weight: \( k \) = \( 11 \)
Character orbit: \([\chi]\) = 19.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(12.0717878008\)
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

\(f(q)\) \(=\) \( q + 1024q^{4} + 3951q^{5} - 32525q^{7} + 59049q^{9} + O(q^{10}) \) \( q + 1024q^{4} + 3951q^{5} - 32525q^{7} + 59049q^{9} + 203523q^{11} + 1048576q^{16} + 2146575q^{17} - 2476099q^{19} + 4045824q^{20} + 5138850q^{23} + 5844776q^{25} - 33305600q^{28} - 128506275q^{35} + 60466176q^{36} - 212457925q^{43} + 208407552q^{44} + 233302599q^{45} - 456682125q^{47} + 775400376q^{49} + 804119373q^{55} - 1606836977q^{61} - 1920568725q^{63} + 1073741824q^{64} + 2198092800q^{68} - 3143217625q^{73} - 2535525376q^{76} - 6619585575q^{77} + 4142923776q^{80} + 3486784401q^{81} + 2150739450q^{83} + 8481117825q^{85} + 5262182400q^{92} - 9783067149q^{95} + 12017829627q^{99} + O(q^{100}) \)

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 1024.00 3951.00 0 −32525.0 0 59049.0 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 19.11.b.a 1
3.b odd 2 1 171.11.c.a 1
4.b odd 2 1 304.11.e.a 1
19.b odd 2 1 CM 19.11.b.a 1
57.d even 2 1 171.11.c.a 1
76.d even 2 1 304.11.e.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.11.b.a 1 1.a even 1 1 trivial
19.11.b.a 1 19.b odd 2 1 CM
171.11.c.a 1 3.b odd 2 1
171.11.c.a 1 57.d even 2 1
304.11.e.a 1 4.b odd 2 1
304.11.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 \(S_{11}^{\mathrm{new}}(19, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - 32 T )( 1 + 32 T ) \)
$3$ \( ( 1 - 243 T )( 1 + 243 T ) \)
$5$ \( 1 - 3951 T + 9765625 T^{2} \)
$7$ \( 1 + 32525 T + 282475249 T^{2} \)
$11$ \( 1 - 203523 T + 25937424601 T^{2} \)
$13$ \( ( 1 - 371293 T )( 1 + 371293 T ) \)
$17$ \( 1 - 2146575 T + 2015993900449 T^{2} \)
$19$ \( 1 + 2476099 T \)
$23$ \( 1 - 5138850 T + 41426511213649 T^{2} \)
$29$ \( ( 1 - 20511149 T )( 1 + 20511149 T ) \)
$31$ \( ( 1 - 28629151 T )( 1 + 28629151 T ) \)
$37$ \( ( 1 - 69343957 T )( 1 + 69343957 T ) \)
$41$ \( ( 1 - 115856201 T )( 1 + 115856201 T ) \)
$43$ \( 1 + 212457925 T + 21611482313284249 T^{2} \)
$47$ \( 1 + 456682125 T + 52599132235830049 T^{2} \)
$53$ \( ( 1 - 418195493 T )( 1 + 418195493 T ) \)
$59$ \( ( 1 - 714924299 T )( 1 + 714924299 T ) \)
$61$ \( 1 + 1606836977 T + 713342911662882601 T^{2} \)
$67$ \( ( 1 - 1350125107 T )( 1 + 1350125107 T ) \)
$71$ \( ( 1 - 1804229351 T )( 1 + 1804229351 T ) \)
$73$ \( 1 + 3143217625 T + 4297625829703557649 T^{2} \)
$79$ \( ( 1 - 3077056399 T )( 1 + 3077056399 T ) \)
$83$ \( 1 - 2150739450 T + 15516041187205853449 T^{2} \)
$89$ \( ( 1 - 5584059449 T )( 1 + 5584059449 T ) \)
$97$ \( ( 1 - 8587340257 T )( 1 + 8587340257 T ) \)
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