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})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 9q^{2} - 14q^{3} + 37q^{4} - 222q^{5} - 603q^{6} - 1246q^{7} - 3555q^{8} - 4898q^{9} + O(q^{10}) \) \( 4q - 9q^{2} - 14q^{3} + 37q^{4} - 222q^{5} - 603q^{6} - 1246q^{7} - 3555q^{8} - 4898q^{9} - 6444q^{10} - 8718q^{11} - 6281q^{12} - 4480q^{13} - 1935q^{14} - 2760q^{15} + 19393q^{16} - 4440q^{17} + 52722q^{18} + 27436q^{19} + 81228q^{20} + 34124q^{21} + 115182q^{22} - 30528q^{23} + 90135q^{24} - 23906q^{25} + 28521q^{26} - 74942q^{27} + 37439q^{28} - 254244q^{29} + 11340q^{30} - 303460q^{31} - 49059q^{32} - 362364q^{33} + 240309q^{34} - 563862q^{35} - 153410q^{36} - 270460q^{37} - 61731q^{38} - 270304q^{39} + 230868q^{40} - 828564q^{41} + 728307q^{42} + 37454q^{43} + 38874q^{44} + 146694q^{45} + 1909269q^{46} + 335670q^{47} + 1366243q^{48} - 67560q^{49} + 815013q^{50} + 1047318q^{51} + 1737887q^{52} + 76728q^{53} + 775215q^{54} + 1008918q^{55} - 973953q^{56} - 96026q^{57} + 1613565q^{58} - 3191334q^{59} + 669180q^{60} + 346550q^{61} - 4678848q^{62} - 24766q^{63} - 1917383q^{64} - 4512972q^{65} - 2532006q^{66} - 270322q^{67} - 9125409q^{68} - 1452456q^{69} + 235836q^{70} - 2066124q^{71} + 3929670q^{72} - 416044q^{73} - 8358894q^{74} + 5984890q^{75} + 253783q^{76} - 3350514q^{77} + 1418859q^{78} + 16025864q^{79} - 622428q^{80} + 2742268q^{81} + 7006752q^{82} + 8524128q^{83} + 1508165q^{84} + 4323186q^{85} - 9139572q^{86} + 10085136q^{87} + 4902930q^{88} + 2899092q^{89} + 10474488q^{90} + 23189218q^{91} - 24380829q^{92} + 12902348q^{93} - 9682776q^{94} - 1522698q^{95} + 514647q^{96} - 4766908q^{97} + 10655118q^{98} + 25556322q^{99} + O(q^{100}) \)

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

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

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 \( T_{2}^{4} + 9 T_{2}^{3} - 234 T_{2}^{2} - 396 T_{2} + 3240 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(19))\).