Properties

Label 19.3.b.b
Level 19
Weight 3
Character orbit 19.b
Analytic conductor 0.518
Analytic rank 0
Dimension 2
CM no
Inner twists 2

Related objects

Downloads

Learn more about

Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(0.517712502285\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-13}) \)
Defining polynomial: \(x^{2} + 13\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{-13}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} -\beta q^{3} -9 q^{4} + 4 q^{5} + 13 q^{6} -5 q^{7} -5 \beta q^{8} -4 q^{9} +O(q^{10})\) \( q + \beta q^{2} -\beta q^{3} -9 q^{4} + 4 q^{5} + 13 q^{6} -5 q^{7} -5 \beta q^{8} -4 q^{9} + 4 \beta q^{10} -10 q^{11} + 9 \beta q^{12} + \beta q^{13} -5 \beta q^{14} -4 \beta q^{15} + 29 q^{16} + 15 q^{17} -4 \beta q^{18} + ( -6 + 5 \beta ) q^{19} -36 q^{20} + 5 \beta q^{21} -10 \beta q^{22} + 35 q^{23} -65 q^{24} -9 q^{25} -13 q^{26} -5 \beta q^{27} + 45 q^{28} + 5 \beta q^{29} + 52 q^{30} -10 \beta q^{31} + 9 \beta q^{32} + 10 \beta q^{33} + 15 \beta q^{34} -20 q^{35} + 36 q^{36} -6 \beta q^{37} + ( -65 - 6 \beta ) q^{38} + 13 q^{39} -20 \beta q^{40} + 10 \beta q^{41} -65 q^{42} -20 q^{43} + 90 q^{44} -16 q^{45} + 35 \beta q^{46} + 10 q^{47} -29 \beta q^{48} -24 q^{49} -9 \beta q^{50} -15 \beta q^{51} -9 \beta q^{52} -21 \beta q^{53} + 65 q^{54} -40 q^{55} + 25 \beta q^{56} + ( 65 + 6 \beta ) q^{57} -65 q^{58} + 5 \beta q^{59} + 36 \beta q^{60} -40 q^{61} + 130 q^{62} + 20 q^{63} - q^{64} + 4 \beta q^{65} -130 q^{66} + 11 \beta q^{67} -135 q^{68} -35 \beta q^{69} -20 \beta q^{70} + 30 \beta q^{71} + 20 \beta q^{72} + 105 q^{73} + 78 q^{74} + 9 \beta q^{75} + ( 54 - 45 \beta ) q^{76} + 50 q^{77} + 13 \beta q^{78} -10 \beta q^{79} + 116 q^{80} -101 q^{81} -130 q^{82} -40 q^{83} -45 \beta q^{84} + 60 q^{85} -20 \beta q^{86} + 65 q^{87} + 50 \beta q^{88} -16 \beta q^{90} -5 \beta q^{91} -315 q^{92} -130 q^{93} + 10 \beta q^{94} + ( -24 + 20 \beta ) q^{95} + 117 q^{96} + 34 \beta q^{97} -24 \beta q^{98} + 40 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 18q^{4} + 8q^{5} + 26q^{6} - 10q^{7} - 8q^{9} + O(q^{10}) \) \( 2q - 18q^{4} + 8q^{5} + 26q^{6} - 10q^{7} - 8q^{9} - 20q^{11} + 58q^{16} + 30q^{17} - 12q^{19} - 72q^{20} + 70q^{23} - 130q^{24} - 18q^{25} - 26q^{26} + 90q^{28} + 104q^{30} - 40q^{35} + 72q^{36} - 130q^{38} + 26q^{39} - 130q^{42} - 40q^{43} + 180q^{44} - 32q^{45} + 20q^{47} - 48q^{49} + 130q^{54} - 80q^{55} + 130q^{57} - 130q^{58} - 80q^{61} + 260q^{62} + 40q^{63} - 2q^{64} - 260q^{66} - 270q^{68} + 210q^{73} + 156q^{74} + 108q^{76} + 100q^{77} + 232q^{80} - 202q^{81} - 260q^{82} - 80q^{83} + 120q^{85} + 130q^{87} - 630q^{92} - 260q^{93} - 48q^{95} + 234q^{96} + 80q^{99} + O(q^{100}) \)

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
3.60555i
3.60555i
3.60555i 3.60555i −9.00000 4.00000 13.0000 −5.00000 18.0278i −4.00000 14.4222i
18.2 3.60555i 3.60555i −9.00000 4.00000 13.0000 −5.00000 18.0278i −4.00000 14.4222i
\(n\): 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 inner

Twists

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 13 \) acting on \(S_{3}^{\mathrm{new}}(19, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 5 T^{2} + 16 T^{4} \)
$3$ \( 1 - 5 T^{2} + 81 T^{4} \)
$5$ \( ( 1 - 4 T + 25 T^{2} )^{2} \)
$7$ \( ( 1 + 5 T + 49 T^{2} )^{2} \)
$11$ \( ( 1 + 10 T + 121 T^{2} )^{2} \)
$13$ \( 1 - 325 T^{2} + 28561 T^{4} \)
$17$ \( ( 1 - 15 T + 289 T^{2} )^{2} \)
$19$ \( 1 + 12 T + 361 T^{2} \)
$23$ \( ( 1 - 35 T + 529 T^{2} )^{2} \)
$29$ \( 1 - 1357 T^{2} + 707281 T^{4} \)
$31$ \( 1 - 622 T^{2} + 923521 T^{4} \)
$37$ \( 1 - 2270 T^{2} + 1874161 T^{4} \)
$41$ \( 1 - 2062 T^{2} + 2825761 T^{4} \)
$43$ \( ( 1 + 20 T + 1849 T^{2} )^{2} \)
$47$ \( ( 1 - 10 T + 2209 T^{2} )^{2} \)
$53$ \( 1 + 115 T^{2} + 7890481 T^{4} \)
$59$ \( 1 - 6637 T^{2} + 12117361 T^{4} \)
$61$ \( ( 1 + 40 T + 3721 T^{2} )^{2} \)
$67$ \( 1 - 7405 T^{2} + 20151121 T^{4} \)
$71$ \( ( 1 - 92 T + 5041 T^{2} )( 1 + 92 T + 5041 T^{2} ) \)
$73$ \( ( 1 - 105 T + 5329 T^{2} )^{2} \)
$79$ \( 1 - 11182 T^{2} + 38950081 T^{4} \)
$83$ \( ( 1 + 40 T + 6889 T^{2} )^{2} \)
$89$ \( ( 1 - 89 T )^{2}( 1 + 89 T )^{2} \)
$97$ \( 1 - 3790 T^{2} + 88529281 T^{4} \)
show more
show less