Properties

Label 19.4.a.a
Level 19
Weight 4
Character orbit 19.a
Self dual Yes
Analytic conductor 1.121
Analytic rank 1
Dimension 1
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 19 \)
Weight: \( k \) = \( 4 \)
Character orbit: \([\chi]\) = 19.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(1.12103629011\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \(q \) \(\mathstrut -\mathstrut 3q^{2} \) \(\mathstrut -\mathstrut 5q^{3} \) \(\mathstrut +\mathstrut q^{4} \) \(\mathstrut -\mathstrut 12q^{5} \) \(\mathstrut +\mathstrut 15q^{6} \) \(\mathstrut +\mathstrut 11q^{7} \) \(\mathstrut +\mathstrut 21q^{8} \) \(\mathstrut -\mathstrut 2q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(q \) \(\mathstrut -\mathstrut 3q^{2} \) \(\mathstrut -\mathstrut 5q^{3} \) \(\mathstrut +\mathstrut q^{4} \) \(\mathstrut -\mathstrut 12q^{5} \) \(\mathstrut +\mathstrut 15q^{6} \) \(\mathstrut +\mathstrut 11q^{7} \) \(\mathstrut +\mathstrut 21q^{8} \) \(\mathstrut -\mathstrut 2q^{9} \) \(\mathstrut +\mathstrut 36q^{10} \) \(\mathstrut -\mathstrut 54q^{11} \) \(\mathstrut -\mathstrut 5q^{12} \) \(\mathstrut +\mathstrut 11q^{13} \) \(\mathstrut -\mathstrut 33q^{14} \) \(\mathstrut +\mathstrut 60q^{15} \) \(\mathstrut -\mathstrut 71q^{16} \) \(\mathstrut -\mathstrut 93q^{17} \) \(\mathstrut +\mathstrut 6q^{18} \) \(\mathstrut +\mathstrut 19q^{19} \) \(\mathstrut -\mathstrut 12q^{20} \) \(\mathstrut -\mathstrut 55q^{21} \) \(\mathstrut +\mathstrut 162q^{22} \) \(\mathstrut +\mathstrut 183q^{23} \) \(\mathstrut -\mathstrut 105q^{24} \) \(\mathstrut +\mathstrut 19q^{25} \) \(\mathstrut -\mathstrut 33q^{26} \) \(\mathstrut +\mathstrut 145q^{27} \) \(\mathstrut +\mathstrut 11q^{28} \) \(\mathstrut -\mathstrut 249q^{29} \) \(\mathstrut -\mathstrut 180q^{30} \) \(\mathstrut +\mathstrut 56q^{31} \) \(\mathstrut +\mathstrut 45q^{32} \) \(\mathstrut +\mathstrut 270q^{33} \) \(\mathstrut +\mathstrut 279q^{34} \) \(\mathstrut -\mathstrut 132q^{35} \) \(\mathstrut -\mathstrut 2q^{36} \) \(\mathstrut -\mathstrut 250q^{37} \) \(\mathstrut -\mathstrut 57q^{38} \) \(\mathstrut -\mathstrut 55q^{39} \) \(\mathstrut -\mathstrut 252q^{40} \) \(\mathstrut +\mathstrut 240q^{41} \) \(\mathstrut +\mathstrut 165q^{42} \) \(\mathstrut -\mathstrut 196q^{43} \) \(\mathstrut -\mathstrut 54q^{44} \) \(\mathstrut +\mathstrut 24q^{45} \) \(\mathstrut -\mathstrut 549q^{46} \) \(\mathstrut -\mathstrut 168q^{47} \) \(\mathstrut +\mathstrut 355q^{48} \) \(\mathstrut -\mathstrut 222q^{49} \) \(\mathstrut -\mathstrut 57q^{50} \) \(\mathstrut +\mathstrut 465q^{51} \) \(\mathstrut +\mathstrut 11q^{52} \) \(\mathstrut +\mathstrut 435q^{53} \) \(\mathstrut -\mathstrut 435q^{54} \) \(\mathstrut +\mathstrut 648q^{55} \) \(\mathstrut +\mathstrut 231q^{56} \) \(\mathstrut -\mathstrut 95q^{57} \) \(\mathstrut +\mathstrut 747q^{58} \) \(\mathstrut +\mathstrut 195q^{59} \) \(\mathstrut +\mathstrut 60q^{60} \) \(\mathstrut -\mathstrut 358q^{61} \) \(\mathstrut -\mathstrut 168q^{62} \) \(\mathstrut -\mathstrut 22q^{63} \) \(\mathstrut +\mathstrut 433q^{64} \) \(\mathstrut -\mathstrut 132q^{65} \) \(\mathstrut -\mathstrut 810q^{66} \) \(\mathstrut -\mathstrut 961q^{67} \) \(\mathstrut -\mathstrut 93q^{68} \) \(\mathstrut -\mathstrut 915q^{69} \) \(\mathstrut +\mathstrut 396q^{70} \) \(\mathstrut -\mathstrut 246q^{71} \) \(\mathstrut -\mathstrut 42q^{72} \) \(\mathstrut +\mathstrut 353q^{73} \) \(\mathstrut +\mathstrut 750q^{74} \) \(\mathstrut -\mathstrut 95q^{75} \) \(\mathstrut +\mathstrut 19q^{76} \) \(\mathstrut -\mathstrut 594q^{77} \) \(\mathstrut +\mathstrut 165q^{78} \) \(\mathstrut -\mathstrut 34q^{79} \) \(\mathstrut +\mathstrut 852q^{80} \) \(\mathstrut -\mathstrut 671q^{81} \) \(\mathstrut -\mathstrut 720q^{82} \) \(\mathstrut +\mathstrut 234q^{83} \) \(\mathstrut -\mathstrut 55q^{84} \) \(\mathstrut +\mathstrut 1116q^{85} \) \(\mathstrut +\mathstrut 588q^{86} \) \(\mathstrut +\mathstrut 1245q^{87} \) \(\mathstrut -\mathstrut 1134q^{88} \) \(\mathstrut -\mathstrut 168q^{89} \) \(\mathstrut -\mathstrut 72q^{90} \) \(\mathstrut +\mathstrut 121q^{91} \) \(\mathstrut +\mathstrut 183q^{92} \) \(\mathstrut -\mathstrut 280q^{93} \) \(\mathstrut +\mathstrut 504q^{94} \) \(\mathstrut -\mathstrut 228q^{95} \) \(\mathstrut -\mathstrut 225q^{96} \) \(\mathstrut +\mathstrut 758q^{97} \) \(\mathstrut +\mathstrut 666q^{98} \) \(\mathstrut +\mathstrut 108q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
−3.00000 −5.00000 1.00000 −12.0000 15.0000 11.0000 21.0000 −2.00000 36.0000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Atkin-Lehner signs

\( p \) Sign
\(19\) \(-1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2} \) \(\mathstrut +\mathstrut 3 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(19))\).