Properties

Label 4019.2.a.b
Level 4019
Weight 2
Character orbit 4019.a
Self dual yes
Analytic conductor 32.092
Analytic rank 0
Dimension 186
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4019 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4019.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(32.0918765724\)
Analytic rank: \(0\)
Dimension: \(186\)
Coefficient ring index: multiple of None
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 186q + 6q^{2} + 10q^{3} + 212q^{4} + 38q^{5} + 47q^{6} + 32q^{7} + 15q^{8} + 216q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 186q + 6q^{2} + 10q^{3} + 212q^{4} + 38q^{5} + 47q^{6} + 32q^{7} + 15q^{8} + 216q^{9} + 50q^{10} + 25q^{11} + 17q^{12} + 113q^{13} + 12q^{14} + 12q^{15} + 252q^{16} + 35q^{17} + 13q^{18} + 97q^{19} + 55q^{20} + 115q^{21} + 14q^{22} + 27q^{23} + 122q^{24} + 244q^{25} + 39q^{26} + 34q^{27} + 66q^{28} + 91q^{29} + 4q^{30} + 135q^{31} + 21q^{32} + 32q^{33} + 58q^{34} + 17q^{35} + 273q^{36} + 133q^{37} - 3q^{38} + 55q^{39} + 142q^{40} + 97q^{41} - 8q^{42} + 67q^{43} + 44q^{44} + 154q^{45} + 101q^{46} + 20q^{47} - 7q^{48} + 312q^{49} + 21q^{50} + 23q^{51} + 193q^{52} + 22q^{53} + 141q^{54} + 88q^{55} + 28q^{56} + 65q^{57} + 62q^{58} + 41q^{59} + q^{60} + 377q^{61} + 29q^{62} + 39q^{63} + 311q^{64} + 21q^{65} + 35q^{66} + 42q^{67} + 24q^{68} + 137q^{69} + 35q^{70} + 17q^{71} - 8q^{72} + 213q^{73} - 9q^{74} + 2q^{75} + 242q^{76} + 60q^{77} + 103q^{79} + 80q^{80} + 270q^{81} + 84q^{82} + 42q^{83} + 137q^{84} + 294q^{85} - 9q^{86} + 22q^{87} - 13q^{88} + 78q^{89} + 69q^{90} + 118q^{91} + 49q^{92} + 51q^{93} + 93q^{94} + 10q^{95} + 260q^{96} + 142q^{97} - 31q^{98} + 78q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −2.80607 −1.31751 5.87400 −0.399238 3.69703 −2.36747 −10.8707 −1.26416 1.12029
1.2 −2.79148 −3.39321 5.79236 3.31167 9.47208 −0.0588258 −10.5863 8.51387 −9.24445
1.3 −2.78390 2.40470 5.75012 −2.23828 −6.69445 −0.0567746 −10.4400 2.78257 6.23116
1.4 −2.77200 −0.494167 5.68400 0.905775 1.36983 1.77801 −10.2121 −2.75580 −2.51081
1.5 −2.72152 −2.92759 5.40665 −3.58721 7.96750 −4.23646 −9.27126 5.57081 9.76265
1.6 −2.71067 −2.48956 5.34775 −1.96358 6.74838 3.14635 −9.07467 3.19790 5.32263
1.7 −2.69011 0.353602 5.23670 −0.999886 −0.951229 3.25121 −8.70707 −2.87497 2.68981
1.8 −2.67918 1.79491 5.17799 3.40719 −4.80888 4.24264 −8.51439 0.221695 −9.12846
1.9 −2.62160 −1.74677 4.87277 −3.59409 4.57932 5.00470 −7.53124 0.0512031 9.42226
1.10 −2.60285 −0.581840 4.77483 3.23074 1.51444 −2.22687 −7.22248 −2.66146 −8.40914
1.11 −2.59341 1.69325 4.72575 −3.22826 −4.39129 −0.750732 −7.06898 −0.132895 8.37219
1.12 −2.56992 −0.512548 4.60446 −4.01178 1.31721 0.174790 −6.69325 −2.73729 10.3099
1.13 −2.56492 −2.35383 4.57881 1.72936 6.03739 −3.13382 −6.61443 2.54053 −4.43566
1.14 −2.55030 3.02512 4.50404 −1.05624 −7.71496 5.05580 −6.38607 6.15132 2.69374
1.15 −2.52937 0.978887 4.39770 −0.161932 −2.47596 −2.77531 −6.06468 −2.04178 0.409586
1.16 −2.43886 −0.628245 3.94805 3.87591 1.53220 −4.09962 −4.75103 −2.60531 −9.45281
1.17 −2.43453 2.49356 3.92696 3.69129 −6.07065 0.601351 −4.69125 3.21782 −8.98658
1.18 −2.40616 −0.556574 3.78961 1.68267 1.33921 2.28423 −4.30608 −2.69023 −4.04878
1.19 −2.39579 3.13150 3.73980 0.529965 −7.50242 −1.46029 −4.16820 6.80631 −1.26968
1.20 −2.34165 −0.437788 3.48332 0.358726 1.02515 3.39349 −3.47342 −2.80834 −0.840010
See next 80 embeddings (of 186 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.186
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4019.2.a.b 186
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4019.2.a.b 186 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(4019\) \(-1\)