Properties

Label 19.8.a.a
Level 19
Weight 8
Character orbit 19.a
Self dual Yes
Analytic conductor 5.935
Analytic rank 1
Dimension 4
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 19 \)
Weight: \( k \) = \( 8 \)
Character orbit: \([\chi]\) = 19.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(5.9353154842\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2\cdot 3 \)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + ( -2 + \beta_{1} ) q^{2} \) \( + ( -4 - \beta_{1} - \beta_{2} ) q^{3} \) \( + ( 8 - 10 \beta_{1} + 4 \beta_{2} - \beta_{3} ) q^{4} \) \( + ( -59 - 11 \beta_{1} + 3 \beta_{2} + 6 \beta_{3} ) q^{5} \) \( + ( -156 - 18 \beta_{1} - 6 \beta_{2} - 3 \beta_{3} ) q^{6} \) \( + ( -311 - 20 \beta_{1} - 4 \beta_{2} - 26 \beta_{3} ) q^{7} \) \( + ( -892 + 50 \beta_{1} - 36 \beta_{2} + 27 \beta_{3} ) q^{8} \) \( + ( -1201 + 77 \beta_{1} + 23 \beta_{2} + 6 \beta_{3} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + ( -2 + \beta_{1} ) q^{2} \) \( + ( -4 - \beta_{1} - \beta_{2} ) q^{3} \) \( + ( 8 - 10 \beta_{1} + 4 \beta_{2} - \beta_{3} ) q^{4} \) \( + ( -59 - 11 \beta_{1} + 3 \beta_{2} + 6 \beta_{3} ) q^{5} \) \( + ( -156 - 18 \beta_{1} - 6 \beta_{2} - 3 \beta_{3} ) q^{6} \) \( + ( -311 - 20 \beta_{1} - 4 \beta_{2} - 26 \beta_{3} ) q^{7} \) \( + ( -892 + 50 \beta_{1} - 36 \beta_{2} + 27 \beta_{3} ) q^{8} \) \( + ( -1201 + 77 \beta_{1} + 23 \beta_{2} + 6 \beta_{3} ) q^{9} \) \( + ( -1598 + 83 \beta_{1} - 14 \beta_{2} + 17 \beta_{3} ) q^{10} \) \( + ( -2121 + 177 \beta_{1} + 87 \beta_{2} + 30 \beta_{3} ) q^{11} \) \( + ( -1564 - 10 \beta_{1} + 32 \beta_{2} - 3 \beta_{3} ) q^{12} \) \( + ( -1054 - 99 \beta_{1} + 123 \beta_{2} - 240 \beta_{3} ) q^{13} \) \( + ( -586 - 187 \beta_{1} - 192 \beta_{2} + 30 \beta_{3} ) q^{14} \) \( + ( -690 + 60 \beta_{1} + 90 \beta_{2} + 150 \beta_{3} ) q^{15} \) \( + ( 4588 - 858 \beta_{1} - 276 \beta_{2} - 93 \beta_{3} ) q^{16} \) \( + ( -1175 + 778 \beta_{1} - 510 \beta_{2} + 528 \beta_{3} ) q^{17} \) \( + ( 12942 - 1323 \beta_{1} + 378 \beta_{2} + 9 \beta_{3} ) q^{18} \) \( + 6859 q^{19} \) \( + ( 20236 - 1196 \beta_{1} - 12 \beta_{2} - 924 \beta_{3} ) q^{20} \) \( + ( 9100 + 1183 \beta_{1} + 619 \beta_{2} - 474 \beta_{3} ) q^{21} \) \( + ( 28590 - 1683 \beta_{1} + 1002 \beta_{2} + 141 \beta_{3} ) q^{22} \) \( + ( -7042 + 4217 \beta_{1} - 429 \beta_{2} + 1428 \beta_{3} ) q^{23} \) \( + ( 22980 + 1530 \beta_{1} + 780 \beta_{2} + 525 \beta_{3} ) q^{24} \) \( + ( -5722 + 887 \beta_{1} - 1931 \beta_{2} - 2062 \beta_{3} ) q^{25} \) \( + ( 7376 + 2924 \beta_{1} - 1110 \beta_{2} + 831 \beta_{3} ) q^{26} \) \( + ( -17926 + 569 \beta_{1} + 2495 \beta_{2} - 174 \beta_{3} ) q^{27} \) \( + ( 8352 - 814 \beta_{1} - 500 \beta_{2} + 2717 \beta_{3} ) q^{28} \) \( + ( -63172 + 953 \beta_{1} - 2907 \beta_{2} - 3510 \beta_{3} ) q^{29} \) \( + ( 3180 + 510 \beta_{1} + 1020 \beta_{2} + 150 \beta_{3} ) q^{30} \) \( + ( -79472 - 7694 \beta_{1} - 2428 \beta_{2} + 4306 \beta_{3} ) q^{31} \) \( + ( -11508 - 834 \beta_{1} + 252 \beta_{2} - 3609 \beta_{3} ) q^{32} \) \( + ( -92226 - 6330 \beta_{1} - 372 \beta_{2} - 162 \beta_{3} ) q^{33} \) \( + ( 57046 - 19675 \beta_{1} + 4204 \beta_{2} - 3346 \beta_{3} ) q^{34} \) \( + ( -140419 - 2431 \beta_{1} + 6363 \beta_{2} + 1746 \beta_{3} ) q^{35} \) \( + ( -35236 + 21968 \beta_{1} - 7444 \beta_{2} + 2058 \beta_{3} ) q^{36} \) \( + ( -69986 - 18722 \beta_{1} + 9572 \beta_{2} + 334 \beta_{3} ) q^{37} \) \( + ( -13718 + 6859 \beta_{1} ) q^{38} \) \( + ( -66206 - 1085 \beta_{1} + 1453 \beta_{2} - 5112 \beta_{3} ) q^{39} \) \( + ( 61256 + 20764 \beta_{1} - 6712 \beta_{2} - 104 \beta_{3} ) q^{40} \) \( + ( -207854 + 17284 \beta_{1} - 6306 \beta_{2} + 13830 \beta_{3} ) q^{41} \) \( + ( 186204 + 14202 \beta_{1} + 4074 \beta_{2} + 1767 \beta_{3} ) q^{42} \) \( + ( 2927 - 15777 \beta_{1} - 12495 \beta_{2} - 2526 \beta_{3} ) q^{43} \) \( + ( 15756 + 41160 \beta_{1} - 15300 \beta_{2} + 1710 \beta_{3} ) q^{44} \) \( + ( 41613 + 17277 \beta_{1} - 7941 \beta_{2} - 10422 \beta_{3} ) q^{45} \) \( + ( 471320 - 53072 \beta_{1} + 21722 \beta_{2} - 7361 \beta_{3} ) q^{46} \) \( + ( 88385 - 30067 \beta_{1} + 33855 \beta_{2} - 14082 \beta_{3} ) q^{47} \) \( + ( 349652 + 28130 \beta_{1} + 5684 \beta_{2} + 1449 \beta_{3} ) q^{48} \) \( + ( -16918 + 29344 \beta_{1} - 15400 \beta_{2} + 14056 \beta_{3} ) q^{49} \) \( + ( 190456 - 51176 \beta_{1} - 8562 \beta_{2} - 6549 \beta_{3} ) q^{50} \) \( + ( 260712 + 7365 \beta_{1} - 1551 \beta_{2} + 10284 \beta_{3} ) q^{51} \) \( + ( 420748 - 29426 \beta_{1} - 2944 \beta_{2} + 22525 \beta_{3} ) q^{52} \) \( + ( 3542 - 61405 \beta_{1} + 6159 \beta_{2} + 7314 \beta_{3} ) q^{53} \) \( + ( 201240 + 32760 \beta_{1} + 6570 \beta_{2} + 9585 \beta_{3} ) q^{54} \) \( + ( 261861 + 33129 \beta_{1} - 23757 \beta_{2} - 29154 \beta_{3} ) q^{55} \) \( + ( -228164 + 22366 \beta_{1} + 31188 \beta_{2} - 7743 \beta_{3} ) q^{56} \) \( + ( -27436 - 6859 \beta_{1} - 6859 \beta_{2} ) q^{57} \) \( + ( 369716 - 127730 \beta_{1} - 16042 \beta_{2} - 9071 \beta_{3} ) q^{58} \) \( + ( -767012 + 122233 \beta_{1} + 10575 \beta_{2} + 9522 \beta_{3} ) q^{59} \) \( + ( 172920 + 13560 \beta_{1} - 6840 \beta_{2} - 15780 \beta_{3} ) q^{60} \) \( + ( 82493 - 30189 \beta_{1} - 13935 \beta_{2} - 27546 \beta_{3} ) q^{61} \) \( + ( -1192720 - 79948 \beta_{1} - 18408 \beta_{2} - 6324 \beta_{3} ) q^{62} \) \( + ( -24521 - 17087 \beta_{1} - 14729 \beta_{2} + 41502 \beta_{3} ) q^{63} \) \( + ( -449732 + 117750 \beta_{1} + 18060 \beta_{2} + 17355 \beta_{3} ) q^{64} \) \( + ( -1127764 - 49516 \beta_{1} + 48468 \beta_{2} - 2964 \beta_{3} ) q^{65} \) \( + ( -653292 - 49446 \beta_{1} - 26712 \beta_{2} + 5004 \beta_{3} ) q^{66} \) \( + ( -33310 + 100737 \beta_{1} + 74733 \beta_{2} + 38388 \beta_{3} ) q^{67} \) \( + ( -2225504 + 214042 \beta_{1} - 18396 \beta_{2} - 27747 \beta_{3} ) q^{68} \) \( + ( -391902 - 68841 \beta_{1} - 27687 \beta_{2} + 18624 \beta_{3} ) q^{69} \) \( + ( 58802 + 15523 \beta_{1} + 9986 \beta_{2} + 26137 \beta_{3} ) q^{70} \) \( + ( -542294 - 70856 \beta_{1} - 85524 \beta_{2} - 53328 \beta_{3} ) q^{71} \) \( + ( 951984 - 209520 \beta_{1} + 32832 \beta_{2} - 54954 \beta_{3} ) q^{72} \) \( + ( -37495 + 204564 \beta_{1} + 23496 \beta_{2} - 38004 \beta_{3} ) q^{73} \) \( + ( -2045068 + 289706 \beta_{1} - 54408 \beta_{2} + 56676 \beta_{3} ) q^{74} \) \( + ( 1541030 + 110945 \beta_{1} + 28115 \beta_{2} - 40170 \beta_{3} ) q^{75} \) \( + ( 54872 - 68590 \beta_{1} + 27436 \beta_{2} - 6859 \beta_{3} ) q^{76} \) \( + ( -893121 - 91371 \beta_{1} - 36297 \beta_{2} + 94302 \beta_{3} ) q^{77} \) \( + ( 342408 - 15336 \beta_{1} - 21882 \beta_{2} + 12009 \beta_{3} ) q^{78} \) \( + ( 4023872 - 24108 \beta_{1} + 39378 \beta_{2} - 54354 \beta_{3} ) q^{79} \) \( + ( -180416 - 99224 \beta_{1} + 70752 \beta_{2} + 70764 \beta_{3} ) q^{80} \) \( + ( 603053 - 292348 \beta_{1} - 63676 \beta_{2} - 25968 \beta_{3} ) q^{81} \) \( + ( 1665604 - 512518 \beta_{1} + 111844 \beta_{2} - 56338 \beta_{3} ) q^{82} \) \( + ( 2086224 - 173070 \beta_{1} + 40110 \beta_{2} + 46272 \beta_{3} ) q^{83} \) \( + ( 361804 + 7258 \beta_{1} - 7208 \beta_{2} + 60999 \beta_{3} ) q^{84} \) \( + ( 1093417 + 199973 \beta_{1} - 99089 \beta_{2} + 50402 \beta_{3} ) q^{85} \) \( + ( -2336698 - 140695 \beta_{1} - 98202 \beta_{2} - 31677 \beta_{3} ) q^{86} \) \( + ( 2621958 + 233283 \beta_{1} + 101565 \beta_{2} - 67848 \beta_{3} ) q^{87} \) \( + ( 1149888 - 438120 \beta_{1} + 12624 \beta_{2} - 122118 \beta_{3} ) q^{88} \) \( + ( 803576 + 204626 \beta_{1} + 1716 \beta_{2} - 108870 \beta_{3} ) q^{89} \) \( + ( 2568546 - 250461 \beta_{1} + 11538 \beta_{2} - 38619 \beta_{3} ) q^{90} \) \( + ( 5714162 + 75005 \beta_{1} - 230993 \beta_{2} + 176582 \beta_{3} ) q^{91} \) \( + ( -5910004 + 848726 \beta_{1} - 143376 \beta_{2} - 35463 \beta_{3} ) q^{92} \) \( + ( 3312760 + 318976 \beta_{1} + 150508 \beta_{2} + 120792 \beta_{3} ) q^{93} \) \( + ( -2217334 + 1101895 \beta_{1} - 108886 \beta_{2} + 179569 \beta_{3} ) q^{94} \) \( + ( -404681 - 75449 \beta_{1} + 20577 \beta_{2} + 41154 \beta_{3} ) q^{95} \) \( + ( 167364 + 50922 \beta_{1} + 29844 \beta_{2} - 74043 \beta_{3} ) q^{96} \) \( + ( -969756 + 482594 \beta_{1} + 216394 \beta_{2} - 188896 \beta_{3} ) q^{97} \) \( + ( 2571084 - 618582 \beta_{1} + 142800 \beta_{2} - 105000 \beta_{3} ) q^{98} \) \( + ( 6353907 - 146937 \beta_{1} - 42663 \beta_{2} - 48906 \beta_{3} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(4q \) \(\mathstrut -\mathstrut 9q^{2} \) \(\mathstrut -\mathstrut 14q^{3} \) \(\mathstrut +\mathstrut 37q^{4} \) \(\mathstrut -\mathstrut 222q^{5} \) \(\mathstrut -\mathstrut 603q^{6} \) \(\mathstrut -\mathstrut 1246q^{7} \) \(\mathstrut -\mathstrut 3555q^{8} \) \(\mathstrut -\mathstrut 4898q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(4q \) \(\mathstrut -\mathstrut 9q^{2} \) \(\mathstrut -\mathstrut 14q^{3} \) \(\mathstrut +\mathstrut 37q^{4} \) \(\mathstrut -\mathstrut 222q^{5} \) \(\mathstrut -\mathstrut 603q^{6} \) \(\mathstrut -\mathstrut 1246q^{7} \) \(\mathstrut -\mathstrut 3555q^{8} \) \(\mathstrut -\mathstrut 4898q^{9} \) \(\mathstrut -\mathstrut 6444q^{10} \) \(\mathstrut -\mathstrut 8718q^{11} \) \(\mathstrut -\mathstrut 6281q^{12} \) \(\mathstrut -\mathstrut 4480q^{13} \) \(\mathstrut -\mathstrut 1935q^{14} \) \(\mathstrut -\mathstrut 2760q^{15} \) \(\mathstrut +\mathstrut 19393q^{16} \) \(\mathstrut -\mathstrut 4440q^{17} \) \(\mathstrut +\mathstrut 52722q^{18} \) \(\mathstrut +\mathstrut 27436q^{19} \) \(\mathstrut +\mathstrut 81228q^{20} \) \(\mathstrut +\mathstrut 34124q^{21} \) \(\mathstrut +\mathstrut 115182q^{22} \) \(\mathstrut -\mathstrut 30528q^{23} \) \(\mathstrut +\mathstrut 90135q^{24} \) \(\mathstrut -\mathstrut 23906q^{25} \) \(\mathstrut +\mathstrut 28521q^{26} \) \(\mathstrut -\mathstrut 74942q^{27} \) \(\mathstrut +\mathstrut 37439q^{28} \) \(\mathstrut -\mathstrut 254244q^{29} \) \(\mathstrut +\mathstrut 11340q^{30} \) \(\mathstrut -\mathstrut 303460q^{31} \) \(\mathstrut -\mathstrut 49059q^{32} \) \(\mathstrut -\mathstrut 362364q^{33} \) \(\mathstrut +\mathstrut 240309q^{34} \) \(\mathstrut -\mathstrut 563862q^{35} \) \(\mathstrut -\mathstrut 153410q^{36} \) \(\mathstrut -\mathstrut 270460q^{37} \) \(\mathstrut -\mathstrut 61731q^{38} \) \(\mathstrut -\mathstrut 270304q^{39} \) \(\mathstrut +\mathstrut 230868q^{40} \) \(\mathstrut -\mathstrut 828564q^{41} \) \(\mathstrut +\mathstrut 728307q^{42} \) \(\mathstrut +\mathstrut 37454q^{43} \) \(\mathstrut +\mathstrut 38874q^{44} \) \(\mathstrut +\mathstrut 146694q^{45} \) \(\mathstrut +\mathstrut 1909269q^{46} \) \(\mathstrut +\mathstrut 335670q^{47} \) \(\mathstrut +\mathstrut 1366243q^{48} \) \(\mathstrut -\mathstrut 67560q^{49} \) \(\mathstrut +\mathstrut 815013q^{50} \) \(\mathstrut +\mathstrut 1047318q^{51} \) \(\mathstrut +\mathstrut 1737887q^{52} \) \(\mathstrut +\mathstrut 76728q^{53} \) \(\mathstrut +\mathstrut 775215q^{54} \) \(\mathstrut +\mathstrut 1008918q^{55} \) \(\mathstrut -\mathstrut 973953q^{56} \) \(\mathstrut -\mathstrut 96026q^{57} \) \(\mathstrut +\mathstrut 1613565q^{58} \) \(\mathstrut -\mathstrut 3191334q^{59} \) \(\mathstrut +\mathstrut 669180q^{60} \) \(\mathstrut +\mathstrut 346550q^{61} \) \(\mathstrut -\mathstrut 4678848q^{62} \) \(\mathstrut -\mathstrut 24766q^{63} \) \(\mathstrut -\mathstrut 1917383q^{64} \) \(\mathstrut -\mathstrut 4512972q^{65} \) \(\mathstrut -\mathstrut 2532006q^{66} \) \(\mathstrut -\mathstrut 270322q^{67} \) \(\mathstrut -\mathstrut 9125409q^{68} \) \(\mathstrut -\mathstrut 1452456q^{69} \) \(\mathstrut +\mathstrut 235836q^{70} \) \(\mathstrut -\mathstrut 2066124q^{71} \) \(\mathstrut +\mathstrut 3929670q^{72} \) \(\mathstrut -\mathstrut 416044q^{73} \) \(\mathstrut -\mathstrut 8358894q^{74} \) \(\mathstrut +\mathstrut 5984890q^{75} \) \(\mathstrut +\mathstrut 253783q^{76} \) \(\mathstrut -\mathstrut 3350514q^{77} \) \(\mathstrut +\mathstrut 1418859q^{78} \) \(\mathstrut +\mathstrut 16025864q^{79} \) \(\mathstrut -\mathstrut 622428q^{80} \) \(\mathstrut +\mathstrut 2742268q^{81} \) \(\mathstrut +\mathstrut 7006752q^{82} \) \(\mathstrut +\mathstrut 8524128q^{83} \) \(\mathstrut +\mathstrut 1508165q^{84} \) \(\mathstrut +\mathstrut 4323186q^{85} \) \(\mathstrut -\mathstrut 9139572q^{86} \) \(\mathstrut +\mathstrut 10085136q^{87} \) \(\mathstrut +\mathstrut 4902930q^{88} \) \(\mathstrut +\mathstrut 2899092q^{89} \) \(\mathstrut +\mathstrut 10474488q^{90} \) \(\mathstrut +\mathstrut 23189218q^{91} \) \(\mathstrut -\mathstrut 24380829q^{92} \) \(\mathstrut +\mathstrut 12902348q^{93} \) \(\mathstrut -\mathstrut 9682776q^{94} \) \(\mathstrut -\mathstrut 1522698q^{95} \) \(\mathstrut +\mathstrut 514647q^{96} \) \(\mathstrut -\mathstrut 4766908q^{97} \) \(\mathstrut +\mathstrut 10655118q^{98} \) \(\mathstrut +\mathstrut 25556322q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4}\mathstrut -\mathstrut \) \(x^{3}\mathstrut -\mathstrut \) \(255\) \(x^{2}\mathstrut +\mathstrut \) \(475\) \(x\mathstrut +\mathstrut \) \(500\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} + 4 \nu^{2} - 235 \nu - 300 \)\()/50\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{3} + \nu^{2} + 305 \nu - 375 \)\()/25\)
\(\beta_{3}\)\(=\)\((\)\( 3 \nu^{3} - 8 \nu^{2} - 685 \nu + 1700 \)\()/50\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3}\mathstrut +\mathstrut \) \(2\) \(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(2\)\()/6\)
\(\nu^{2}\)\(=\)\((\)\(-\)\(7\) \(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(23\) \(\beta_{1}\mathstrut +\mathstrut \) \(391\)\()/3\)
\(\nu^{3}\)\(=\)\((\)\(97\) \(\beta_{3}\mathstrut +\mathstrut \) \(154\) \(\beta_{2}\mathstrut +\mathstrut \) \(117\) \(\beta_{1}\mathstrut -\mathstrut \) \(286\)\()/2\)

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
2.64844
−0.751230
−16.3077
15.4104
−19.5150 −3.33349 252.835 170.463 65.0531 31.4807 −2436.16 −2175.89 −3326.58
1.2 −4.43255 22.5580 −108.352 160.438 −99.9896 −1314.43 1047.64 −1678.14 −711.151
1.3 3.18411 20.6576 −117.861 −477.630 65.7761 883.696 −782.850 −1760.26 −1520.83
1.4 11.7634 −53.8822 10.3786 −75.2709 −633.840 −846.751 −1383.63 716.286 −885.446
\(n\): 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 \(T_{2}^{4} \) \(\mathstrut +\mathstrut 9 T_{2}^{3} \) \(\mathstrut -\mathstrut 234 T_{2}^{2} \) \(\mathstrut -\mathstrut 396 T_{2} \) \(\mathstrut +\mathstrut 3240 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(19))\).