Properties

Label 19.4.a.b
Level 19
Weight 4
Character orbit 19.a
Self dual Yes
Analytic conductor 1.121
Analytic rank 0
Dimension 3
CM No
Inner twists 1

Related objects

Downloads

Learn more about

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: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.3144.1
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + ( 1 + \beta_{1} - \beta_{2} ) q^{2} \) \( + ( -\beta_{1} + 2 \beta_{2} ) q^{3} \) \( + ( 8 - \beta_{1} - 2 \beta_{2} ) q^{4} \) \( + ( 5 - 3 \beta_{1} + 2 \beta_{2} ) q^{5} \) \( + ( -24 + 3 \beta_{1} + 4 \beta_{2} ) q^{6} \) \( + ( -13 + 4 \beta_{1} ) q^{7} \) \( + ( 12 - \beta_{1} - 8 \beta_{2} ) q^{8} \) \( + ( 19 - 3 \beta_{1} - 6 \beta_{2} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + ( 1 + \beta_{1} - \beta_{2} ) q^{2} \) \( + ( -\beta_{1} + 2 \beta_{2} ) q^{3} \) \( + ( 8 - \beta_{1} - 2 \beta_{2} ) q^{4} \) \( + ( 5 - 3 \beta_{1} + 2 \beta_{2} ) q^{5} \) \( + ( -24 + 3 \beta_{1} + 4 \beta_{2} ) q^{6} \) \( + ( -13 + 4 \beta_{1} ) q^{7} \) \( + ( 12 - \beta_{1} - 8 \beta_{2} ) q^{8} \) \( + ( 19 - 3 \beta_{1} - 6 \beta_{2} ) q^{9} \) \( + ( -31 + 10 \beta_{1} - 5 \beta_{2} ) q^{10} \) \( + ( 5 + 3 \beta_{1} - 2 \beta_{2} ) q^{11} \) \( + ( -42 - 15 \beta_{1} + 26 \beta_{2} ) q^{12} \) \( + ( 22 + 5 \beta_{1} - 6 \beta_{2} ) q^{13} \) \( + ( 11 - 17 \beta_{1} + 21 \beta_{2} ) q^{14} \) \( + ( 50 - 18 \beta_{1} + 8 \beta_{2} ) q^{15} \) \( + ( 14 + 13 \beta_{1} - 22 \beta_{2} ) q^{16} \) \( + ( 1 + 22 \beta_{1} + 4 \beta_{2} ) q^{17} \) \( + ( 55 + 16 \beta_{1} - 43 \beta_{2} ) q^{18} \) \( -19 q^{19} \) \( + ( 34 - 22 \beta_{1} + 20 \beta_{2} ) q^{20} \) \( + ( -8 + 33 \beta_{1} - 34 \beta_{2} ) q^{21} \) \( + ( 41 - 5 \beta_{2} ) q^{22} \) \( + ( -42 + 11 \beta_{1} + 14 \beta_{2} ) q^{23} \) \( + ( -174 - 25 \beta_{1} + 58 \beta_{2} ) q^{24} \) \( + ( -6 - 49 \beta_{1} + 30 \beta_{2} ) q^{25} \) \( + ( 106 + 11 \beta_{1} - 30 \beta_{2} ) q^{26} \) \( + ( -126 - 13 \beta_{1} + 14 \beta_{2} ) q^{27} \) \( + ( -176 + 17 \beta_{1} + 18 \beta_{2} ) q^{28} \) \( + ( 144 - 25 \beta_{1} - 30 \beta_{2} ) q^{29} \) \( + ( -130 + 76 \beta_{1} - 62 \beta_{2} ) q^{30} \) \( + ( -32 - 56 \beta_{1} + 12 \beta_{2} ) q^{31} \) \( + ( 194 - 13 \beta_{1} + 10 \beta_{2} ) q^{32} \) \( + ( -50 + 8 \beta_{1} + 12 \beta_{2} ) q^{33} \) \( + ( 97 - 17 \beta_{1} + 55 \beta_{2} ) q^{34} \) \( + ( -153 + 71 \beta_{1} - 50 \beta_{2} ) q^{35} \) \( + ( 386 + 20 \beta_{1} - 104 \beta_{2} ) q^{36} \) \( + ( -122 + 32 \beta_{1} + 44 \beta_{2} ) q^{37} \) \( + ( -19 - 19 \beta_{1} + 19 \beta_{2} ) q^{38} \) \( + ( -142 - 3 \beta_{1} + 58 \beta_{2} ) q^{39} \) \( + ( -30 - 4 \beta_{1} + 22 \beta_{2} ) q^{40} \) \( + ( 314 + 6 \beta_{1} + 8 \beta_{2} ) q^{41} \) \( + ( 496 - 75 \beta_{1} - 28 \beta_{2} ) q^{42} \) \( + ( -163 + 29 \beta_{1} - 110 \beta_{2} ) q^{43} \) \( + ( 46 + 12 \beta_{1} - 40 \beta_{2} ) q^{44} \) \( + ( 77 - 51 \beta_{1} + 50 \beta_{2} ) q^{45} \) \( + ( -102 - 39 \beta_{1} + 106 \beta_{2} ) q^{46} \) \( + ( 39 - 33 \beta_{1} - 18 \beta_{2} ) q^{47} \) \( + ( -510 + 29 \beta_{1} + 90 \beta_{2} ) q^{48} \) \( + ( -14 - 88 \beta_{1} + 32 \beta_{2} ) q^{49} \) \( + ( -570 + 73 \beta_{1} - 2 \beta_{2} ) q^{50} \) \( + ( 44 + 113 \beta_{1} - 58 \beta_{2} ) q^{51} \) \( + ( 266 + 25 \beta_{1} - 126 \beta_{2} ) q^{52} \) \( + ( 266 + 29 \beta_{1} - 10 \beta_{2} ) q^{53} \) \( + ( -330 - 99 \beta_{1} + 142 \beta_{2} ) q^{54} \) \( + ( -69 + 19 \beta_{1} - 10 \beta_{2} ) q^{55} \) \( + ( -324 - 39 \beta_{1} + 96 \beta_{2} ) q^{56} \) \( + ( 19 \beta_{1} - 38 \beta_{2} ) q^{57} \) \( + ( 264 + 139 \beta_{1} - 284 \beta_{2} ) q^{58} \) \( + ( 128 - 53 \beta_{1} - 66 \beta_{2} ) q^{59} \) \( + ( 484 - 124 \beta_{1} + 32 \beta_{2} ) q^{60} \) \( + ( 285 + 23 \beta_{1} + 110 \beta_{2} ) q^{61} \) \( + ( -476 + 36 \beta_{1} - 44 \beta_{2} ) q^{62} \) \( + ( -463 + 31 \beta_{1} + 54 \beta_{2} ) q^{63} \) \( + ( -86 + 113 \beta_{1} - 14 \beta_{2} ) q^{64} \) \( + ( -84 + 4 \beta_{1} + 8 \beta_{2} ) q^{65} \) \( + ( -110 - 46 \beta_{1} + 102 \beta_{2} ) q^{66} \) \( + ( -94 + 29 \beta_{1} + 46 \beta_{2} ) q^{67} \) \( + ( -508 - 7 \beta_{1} + 2 \beta_{2} ) q^{68} \) \( + ( 286 + 111 \beta_{1} - 162 \beta_{2} ) q^{69} \) \( + ( 723 - 274 \beta_{1} + 145 \beta_{2} ) q^{70} \) \( + ( 270 - 28 \beta_{1} + 64 \beta_{2} ) q^{71} \) \( + ( 1002 + 134 \beta_{1} - 314 \beta_{2} ) q^{72} \) \( + ( 265 - 176 \beta_{1} + 8 \beta_{2} ) q^{73} \) \( + ( -326 - 110 \beta_{1} + 318 \beta_{2} ) q^{74} \) \( + ( 758 - 209 \beta_{1} - 34 \beta_{2} ) q^{75} \) \( + ( -152 + 19 \beta_{1} + 38 \beta_{2} ) q^{76} \) \( + ( 23 - 31 \beta_{1} + 50 \beta_{2} ) q^{77} \) \( + ( -682 - 81 \beta_{1} + 310 \beta_{2} ) q^{78} \) \( + ( 56 + 206 \beta_{1} + 8 \beta_{2} ) q^{79} \) \( + ( -524 + 172 \beta_{1} - 72 \beta_{2} ) q^{80} \) \( + ( -179 + 156 \beta_{1} - 120 \beta_{2} ) q^{81} \) \( + ( 278 + 316 \beta_{1} - 278 \beta_{2} ) q^{82} \) \( + ( -232 - 130 \beta_{1} + 60 \beta_{2} ) q^{83} \) \( + ( 362 + 279 \beta_{1} - 458 \beta_{2} ) q^{84} \) \( + ( -423 + 153 \beta_{1} - 126 \beta_{2} ) q^{85} \) \( + ( 1001 - 302 \beta_{1} - 109 \beta_{2} ) q^{86} \) \( + ( -610 - 299 \beta_{1} + 458 \beta_{2} ) q^{87} \) \( + ( 150 - 6 \beta_{1} - 102 \beta_{2} ) q^{88} \) \( + ( -40 + 168 \beta_{1} - 220 \beta_{2} ) q^{89} \) \( + ( -679 + 178 \beta_{1} - 29 \beta_{2} ) q^{90} \) \( + ( -182 - 29 \beta_{1} + 118 \beta_{2} ) q^{91} \) \( + ( -954 - 45 \beta_{1} + 230 \beta_{2} ) q^{92} \) \( + ( 376 - 236 \beta_{1} ) q^{93} \) \( + ( 3 + 54 \beta_{1} - 159 \beta_{2} ) q^{94} \) \( + ( -95 + 57 \beta_{1} - 38 \beta_{2} ) q^{95} \) \( + ( 246 - 249 \beta_{1} + 374 \beta_{2} ) q^{96} \) \( + ( -852 - 90 \beta_{1} + 196 \beta_{2} ) q^{97} \) \( + ( -830 + 106 \beta_{1} - 66 \beta_{2} ) q^{98} \) \( + ( 113 + 21 \beta_{1} - 110 \beta_{2} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(3q \) \(\mathstrut +\mathstrut 3q^{2} \) \(\mathstrut +\mathstrut q^{3} \) \(\mathstrut +\mathstrut 21q^{4} \) \(\mathstrut +\mathstrut 14q^{5} \) \(\mathstrut -\mathstrut 65q^{6} \) \(\mathstrut -\mathstrut 35q^{7} \) \(\mathstrut +\mathstrut 27q^{8} \) \(\mathstrut +\mathstrut 48q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(3q \) \(\mathstrut +\mathstrut 3q^{2} \) \(\mathstrut +\mathstrut q^{3} \) \(\mathstrut +\mathstrut 21q^{4} \) \(\mathstrut +\mathstrut 14q^{5} \) \(\mathstrut -\mathstrut 65q^{6} \) \(\mathstrut -\mathstrut 35q^{7} \) \(\mathstrut +\mathstrut 27q^{8} \) \(\mathstrut +\mathstrut 48q^{9} \) \(\mathstrut -\mathstrut 88q^{10} \) \(\mathstrut +\mathstrut 16q^{11} \) \(\mathstrut -\mathstrut 115q^{12} \) \(\mathstrut +\mathstrut 65q^{13} \) \(\mathstrut +\mathstrut 37q^{14} \) \(\mathstrut +\mathstrut 140q^{15} \) \(\mathstrut +\mathstrut 33q^{16} \) \(\mathstrut +\mathstrut 29q^{17} \) \(\mathstrut +\mathstrut 138q^{18} \) \(\mathstrut -\mathstrut 57q^{19} \) \(\mathstrut +\mathstrut 100q^{20} \) \(\mathstrut -\mathstrut 25q^{21} \) \(\mathstrut +\mathstrut 118q^{22} \) \(\mathstrut -\mathstrut 101q^{23} \) \(\mathstrut -\mathstrut 489q^{24} \) \(\mathstrut -\mathstrut 37q^{25} \) \(\mathstrut +\mathstrut 299q^{26} \) \(\mathstrut -\mathstrut 377q^{27} \) \(\mathstrut -\mathstrut 493q^{28} \) \(\mathstrut +\mathstrut 377q^{29} \) \(\mathstrut -\mathstrut 376q^{30} \) \(\mathstrut -\mathstrut 140q^{31} \) \(\mathstrut +\mathstrut 579q^{32} \) \(\mathstrut -\mathstrut 130q^{33} \) \(\mathstrut +\mathstrut 329q^{34} \) \(\mathstrut -\mathstrut 438q^{35} \) \(\mathstrut +\mathstrut 1074q^{36} \) \(\mathstrut -\mathstrut 290q^{37} \) \(\mathstrut -\mathstrut 57q^{38} \) \(\mathstrut -\mathstrut 371q^{39} \) \(\mathstrut -\mathstrut 72q^{40} \) \(\mathstrut +\mathstrut 956q^{41} \) \(\mathstrut +\mathstrut 1385q^{42} \) \(\mathstrut -\mathstrut 570q^{43} \) \(\mathstrut +\mathstrut 110q^{44} \) \(\mathstrut +\mathstrut 230q^{45} \) \(\mathstrut -\mathstrut 239q^{46} \) \(\mathstrut +\mathstrut 66q^{47} \) \(\mathstrut -\mathstrut 1411q^{48} \) \(\mathstrut -\mathstrut 98q^{49} \) \(\mathstrut -\mathstrut 1639q^{50} \) \(\mathstrut +\mathstrut 187q^{51} \) \(\mathstrut +\mathstrut 697q^{52} \) \(\mathstrut +\mathstrut 817q^{53} \) \(\mathstrut -\mathstrut 947q^{54} \) \(\mathstrut -\mathstrut 198q^{55} \) \(\mathstrut -\mathstrut 915q^{56} \) \(\mathstrut -\mathstrut 19q^{57} \) \(\mathstrut +\mathstrut 647q^{58} \) \(\mathstrut +\mathstrut 265q^{59} \) \(\mathstrut +\mathstrut 1360q^{60} \) \(\mathstrut +\mathstrut 988q^{61} \) \(\mathstrut -\mathstrut 1436q^{62} \) \(\mathstrut -\mathstrut 1304q^{63} \) \(\mathstrut -\mathstrut 159q^{64} \) \(\mathstrut -\mathstrut 240q^{65} \) \(\mathstrut -\mathstrut 274q^{66} \) \(\mathstrut -\mathstrut 207q^{67} \) \(\mathstrut -\mathstrut 1529q^{68} \) \(\mathstrut +\mathstrut 807q^{69} \) \(\mathstrut +\mathstrut 2040q^{70} \) \(\mathstrut +\mathstrut 846q^{71} \) \(\mathstrut +\mathstrut 2826q^{72} \) \(\mathstrut +\mathstrut 627q^{73} \) \(\mathstrut -\mathstrut 770q^{74} \) \(\mathstrut +\mathstrut 2031q^{75} \) \(\mathstrut -\mathstrut 399q^{76} \) \(\mathstrut +\mathstrut 88q^{77} \) \(\mathstrut -\mathstrut 1817q^{78} \) \(\mathstrut +\mathstrut 382q^{79} \) \(\mathstrut -\mathstrut 1472q^{80} \) \(\mathstrut -\mathstrut 501q^{81} \) \(\mathstrut +\mathstrut 872q^{82} \) \(\mathstrut -\mathstrut 766q^{83} \) \(\mathstrut +\mathstrut 907q^{84} \) \(\mathstrut -\mathstrut 1242q^{85} \) \(\mathstrut +\mathstrut 2592q^{86} \) \(\mathstrut -\mathstrut 1671q^{87} \) \(\mathstrut +\mathstrut 342q^{88} \) \(\mathstrut -\mathstrut 172q^{89} \) \(\mathstrut -\mathstrut 1888q^{90} \) \(\mathstrut -\mathstrut 457q^{91} \) \(\mathstrut -\mathstrut 2677q^{92} \) \(\mathstrut +\mathstrut 892q^{93} \) \(\mathstrut -\mathstrut 96q^{94} \) \(\mathstrut -\mathstrut 266q^{95} \) \(\mathstrut +\mathstrut 863q^{96} \) \(\mathstrut -\mathstrut 2450q^{97} \) \(\mathstrut -\mathstrut 2450q^{98} \) \(\mathstrut +\mathstrut 250q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3}\mathstrut -\mathstrut \) \(x^{2}\mathstrut -\mathstrut \) \(16\) \(x\mathstrut -\mathstrut \) \(8\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{2} - \nu - 10 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(2\) \(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(10\)

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
−3.20905
4.73549
−0.526440
−3.96257 6.71610 7.70200 18.1342 −26.6130 −25.8362 1.18085 18.1060 −71.8581
1.2 1.89080 2.95388 −4.42486 −1.51710 5.58521 5.94196 −23.4930 −18.2746 −2.86853
1.3 5.07177 −8.66998 17.7229 −2.61710 −43.9722 −15.1058 49.3121 48.1686 −13.2733
\(n\): 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 \(T_{2}^{3} \) \(\mathstrut -\mathstrut 3 T_{2}^{2} \) \(\mathstrut -\mathstrut 18 T_{2} \) \(\mathstrut +\mathstrut 38 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(19))\).