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

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Newspace parameters

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

Newform invariants

Self dual: No
Analytic conductor: \(0.517712502285\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-13}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
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.

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

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. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
19.b Odd 1 yes

Hecke kernels

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