Properties

Label 8001.2.a.v
Level 8001
Weight 2
Character orbit 8001.a
Self dual Yes
Analytic conductor 63.888
Analytic rank 0
Dimension 19
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 8001 = 3^{2} \cdot 7 \cdot 127 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8001.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(63.8883066572\)
Analytic rank: \(0\)
Dimension: \(19\)
Coefficient field: \(\mathbb{Q}[x]/(x^{19} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{18}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{1} q^{2} + ( 1 + \beta_{2} ) q^{4} + \beta_{10} q^{5} + q^{7} + ( -\beta_{1} - \beta_{3} ) q^{8} +O(q^{10})\) \( q -\beta_{1} q^{2} + ( 1 + \beta_{2} ) q^{4} + \beta_{10} q^{5} + q^{7} + ( -\beta_{1} - \beta_{3} ) q^{8} + ( \beta_{1} - \beta_{6} - \beta_{12} + \beta_{17} ) q^{10} + \beta_{8} q^{11} + ( 1 + \beta_{12} ) q^{13} -\beta_{1} q^{14} + ( \beta_{2} + \beta_{8} + \beta_{9} ) q^{16} + ( -1 + \beta_{7} - \beta_{16} ) q^{17} + ( 2 - \beta_{2} - \beta_{4} + \beta_{6} + \beta_{7} + \beta_{11} - \beta_{16} ) q^{19} + ( -\beta_{5} + \beta_{10} + \beta_{12} ) q^{20} + ( -\beta_{1} - \beta_{3} - \beta_{7} + \beta_{15} ) q^{22} + ( 1 + \beta_{2} - \beta_{6} + \beta_{10} + \beta_{18} ) q^{23} + ( 3 + \beta_{2} - \beta_{9} - \beta_{15} - \beta_{16} + \beta_{18} ) q^{25} + ( -1 - \beta_{1} - \beta_{2} - \beta_{4} + \beta_{6} - 2 \beta_{10} - \beta_{12} - \beta_{13} - \beta_{16} - \beta_{18} ) q^{26} + ( 1 + \beta_{2} ) q^{28} + ( 1 - \beta_{1} - \beta_{7} + \beta_{10} + \beta_{11} + \beta_{18} ) q^{29} + ( 1 + \beta_{10} - \beta_{13} - \beta_{17} ) q^{31} + ( -1 - 2 \beta_{1} - \beta_{3} + \beta_{5} - \beta_{7} + \beta_{14} + \beta_{15} ) q^{32} + ( 1 + \beta_{1} - \beta_{5} - \beta_{10} - \beta_{12} - \beta_{13} + \beta_{15} - \beta_{18} ) q^{34} + \beta_{10} q^{35} + ( 3 - \beta_{6} + \beta_{10} + \beta_{14} + \beta_{18} ) q^{37} + ( -1 - \beta_{1} + \beta_{5} - 2 \beta_{10} - \beta_{12} - \beta_{13} + \beta_{15} - \beta_{18} ) q^{38} + ( -2 + \beta_{2} + \beta_{4} - \beta_{9} - 2 \beta_{10} - 2 \beta_{11} - \beta_{13} - \beta_{16} - \beta_{17} - \beta_{18} ) q^{40} + ( 1 + \beta_{1} + \beta_{11} - \beta_{12} - \beta_{13} - \beta_{14} ) q^{41} + ( 1 - \beta_{1} - \beta_{3} + \beta_{4} + \beta_{10} + \beta_{12} + \beta_{16} - \beta_{17} ) q^{43} + ( 1 + 3 \beta_{2} + \beta_{7} + \beta_{8} + \beta_{9} + \beta_{13} - 2 \beta_{15} - \beta_{16} + \beta_{17} ) q^{44} + ( 1 - 2 \beta_{1} - \beta_{6} + 2 \beta_{10} + \beta_{13} + \beta_{16} ) q^{46} + ( -1 + \beta_{1} - \beta_{2} - \beta_{6} + \beta_{10} + \beta_{13} + \beta_{16} + \beta_{17} + \beta_{18} ) q^{47} + q^{49} + ( -2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{5} - \beta_{7} - \beta_{11} + \beta_{12} + \beta_{13} - \beta_{14} + \beta_{15} + 2 \beta_{16} - \beta_{18} ) q^{50} + ( 1 + 2 \beta_{1} + \beta_{2} + \beta_{6} + \beta_{7} + \beta_{10} + \beta_{12} + \beta_{13} - \beta_{15} + \beta_{17} ) q^{52} + ( -2 + \beta_{1} + \beta_{6} + \beta_{7} + \beta_{9} - 2 \beta_{10} - \beta_{11} - \beta_{15} - \beta_{16} - \beta_{18} ) q^{53} + ( 2 + \beta_{1} - \beta_{2} - \beta_{5} + \beta_{7} + \beta_{10} + \beta_{11} - \beta_{12} + \beta_{13} - \beta_{14} + \beta_{17} + \beta_{18} ) q^{55} + ( -\beta_{1} - \beta_{3} ) q^{56} + ( \beta_{1} + \beta_{5} + \beta_{8} - \beta_{10} - \beta_{11} + \beta_{13} + \beta_{16} + \beta_{17} - \beta_{18} ) q^{58} + ( -1 - \beta_{1} + \beta_{3} - \beta_{6} + 2 \beta_{10} + 2 \beta_{12} + \beta_{13} + \beta_{14} + \beta_{16} + \beta_{18} ) q^{59} + ( 1 - \beta_{1} + \beta_{5} - \beta_{7} - \beta_{11} + \beta_{12} + \beta_{13} - \beta_{14} + \beta_{16} ) q^{61} + ( -1 + \beta_{1} - \beta_{2} - \beta_{6} + \beta_{7} - \beta_{8} + \beta_{13} + \beta_{16} + 2 \beta_{17} ) q^{62} + ( 4 - 2 \beta_{4} + 2 \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} + 2 \beta_{11} + \beta_{13} - \beta_{14} - 2 \beta_{15} - \beta_{16} + \beta_{17} ) q^{64} + ( -1 + \beta_{2} + \beta_{3} + \beta_{5} - \beta_{6} - \beta_{8} + 4 \beta_{10} + \beta_{11} + \beta_{13} + \beta_{14} + \beta_{16} - \beta_{17} + \beta_{18} ) q^{65} + ( 1 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{5} + \beta_{9} + \beta_{11} - \beta_{12} + \beta_{13} ) q^{67} + ( -\beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{4} - \beta_{5} - \beta_{6} - \beta_{9} + \beta_{10} - \beta_{11} + \beta_{12} - \beta_{15} + \beta_{16} + \beta_{17} ) q^{68} + ( \beta_{1} - \beta_{6} - \beta_{12} + \beta_{17} ) q^{70} + ( 2 + \beta_{2} + \beta_{3} - \beta_{8} - \beta_{9} - \beta_{10} - \beta_{12} - \beta_{13} - \beta_{16} - \beta_{18} ) q^{71} + ( 3 + \beta_{2} - \beta_{3} - \beta_{5} + \beta_{7} + \beta_{9} - \beta_{11} - \beta_{15} + \beta_{17} ) q^{73} + ( -2 \beta_{1} + \beta_{3} - \beta_{6} + \beta_{9} + 2 \beta_{10} + \beta_{13} - \beta_{14} + \beta_{16} ) q^{74} + ( 2 - 2 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{5} + 2 \beta_{6} + \beta_{9} + \beta_{10} + \beta_{11} + 2 \beta_{12} - \beta_{15} + \beta_{16} ) q^{76} + \beta_{8} q^{77} + ( 1 - \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{5} - \beta_{6} - \beta_{7} + \beta_{10} + \beta_{11} - \beta_{12} + \beta_{15} - \beta_{17} ) q^{79} + ( 2 + 2 \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{5} + \beta_{6} + \beta_{7} - 2 \beta_{10} + \beta_{13} - \beta_{14} - \beta_{15} + \beta_{17} ) q^{80} + ( -2 - 2 \beta_{2} - \beta_{3} + \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} + \beta_{12} + \beta_{14} + \beta_{15} + \beta_{16} + \beta_{17} - \beta_{18} ) q^{82} + ( \beta_{2} - \beta_{3} + \beta_{6} - \beta_{8} - 2 \beta_{10} - 2 \beta_{11} + \beta_{12} - \beta_{15} + \beta_{17} - \beta_{18} ) q^{83} + ( 2 - \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} - \beta_{7} + \beta_{8} - \beta_{9} - \beta_{11} + \beta_{12} + \beta_{16} - \beta_{17} ) q^{85} + ( 2 \beta_{2} + \beta_{9} - 3 \beta_{10} - 2 \beta_{12} - \beta_{13} + \beta_{15} ) q^{86} + ( -1 - 6 \beta_{1} + \beta_{2} - 3 \beta_{3} - 2 \beta_{7} - \beta_{8} - \beta_{10} - 2 \beta_{11} - 2 \beta_{13} + \beta_{14} + 2 \beta_{15} + \beta_{16} - \beta_{17} - \beta_{18} ) q^{88} + ( 1 + \beta_{1} - \beta_{2} - 2 \beta_{4} + \beta_{5} + \beta_{6} + 2 \beta_{7} - \beta_{10} + \beta_{11} - 2 \beta_{12} - \beta_{14} - \beta_{15} - 2 \beta_{16} + \beta_{17} - \beta_{18} ) q^{89} + ( 1 + \beta_{12} ) q^{91} + ( 4 - \beta_{1} + 3 \beta_{2} + \beta_{4} - \beta_{6} - \beta_{7} - \beta_{11} - \beta_{12} ) q^{92} + ( 2 \beta_{1} + 2 \beta_{3} + \beta_{5} - \beta_{6} - \beta_{7} + 3 \beta_{10} + \beta_{11} - \beta_{12} - 2 \beta_{17} + \beta_{18} ) q^{94} + ( 2 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} + 3 \beta_{10} + \beta_{16} - 2 \beta_{17} + \beta_{18} ) q^{95} + ( -2 - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} - 2 \beta_{6} + \beta_{10} - \beta_{11} + \beta_{12} + \beta_{13} + \beta_{15} + \beta_{16} - \beta_{17} ) q^{97} -\beta_{1} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 19q - 4q^{2} + 22q^{4} - 5q^{5} + 19q^{7} - 9q^{8} + O(q^{10}) \) \( 19q - 4q^{2} + 22q^{4} - 5q^{5} + 19q^{7} - 9q^{8} + 9q^{11} + 24q^{13} - 4q^{14} + 20q^{16} - 17q^{17} + 23q^{19} - 5q^{20} - 3q^{22} + 17q^{23} + 38q^{25} - 28q^{26} + 22q^{28} - 2q^{29} + 16q^{31} - 17q^{32} + 29q^{34} - 5q^{35} + 56q^{37} - 2q^{38} - 13q^{40} + 7q^{41} + 19q^{43} + 29q^{44} + 10q^{46} - 25q^{47} + 19q^{49} + 9q^{50} + 16q^{52} - 18q^{53} + 10q^{55} - 9q^{56} + 31q^{58} - 11q^{59} + 26q^{61} - 26q^{62} + 45q^{64} - 27q^{65} + 24q^{67} - 14q^{68} + 32q^{71} + 51q^{73} + 12q^{76} + 9q^{77} + 30q^{79} + 30q^{80} - 52q^{82} - q^{83} + 44q^{85} + 24q^{86} - 30q^{88} - 5q^{89} + 24q^{91} + 88q^{92} + 7q^{94} + 24q^{95} + 5q^{97} - 4q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{19} - 4 x^{18} - 22 x^{17} + 101 x^{16} + 178 x^{15} - 1035 x^{14} - 583 x^{13} + 5572 x^{12} + 21 x^{11} - 17032 x^{10} + 4985 x^{9} + 29792 x^{8} - 13249 x^{7} - 28600 x^{6} + 14000 x^{5} + 13725 x^{4} - 5723 x^{3} - 2913 x^{2} + 608 x + 210\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - 5 \nu \)
\(\beta_{4}\)\(=\)\((\)\(11967908 \nu^{18} + 10330919 \nu^{17} - 422890858 \nu^{16} - 185012986 \nu^{15} + 5929350010 \nu^{14} + 1055361561 \nu^{13} - 43092462154 \nu^{12} - 1159028944 \nu^{11} + 177057585547 \nu^{10} - 8859733933 \nu^{9} - 419706704682 \nu^{8} + 32463907935 \nu^{7} + 560761757568 \nu^{6} - 42523921020 \nu^{5} - 394138444469 \nu^{4} + 26352010178 \nu^{3} + 124055335687 \nu^{2} - 9280440934 \nu - 9272820218\)\()/ 579103703 \)
\(\beta_{5}\)\(=\)\((\)\(65114179 \nu^{18} - 132926940 \nu^{17} - 1671831523 \nu^{16} + 3285269092 \nu^{15} + 17368062950 \nu^{14} - 32561720490 \nu^{13} - 94107501027 \nu^{12} + 166375659285 \nu^{11} + 285664374904 \nu^{10} - 469100157707 \nu^{9} - 486402019943 \nu^{8} + 726626199106 \nu^{7} + 448072525313 \nu^{6} - 582478309828 \nu^{5} - 222447224597 \nu^{4} + 206901778527 \nu^{3} + 71510619076 \nu^{2} - 23752461453 \nu - 6128233325\)\()/ 2895518515 \)
\(\beta_{6}\)\(=\)\((\)\(138645363 \nu^{18} - 343698145 \nu^{17} - 3866824621 \nu^{16} + 9112645479 \nu^{15} + 44954282930 \nu^{14} - 98469596705 \nu^{13} - 284325273789 \nu^{12} + 559933954855 \nu^{11} + 1071360077368 \nu^{10} - 1803665054044 \nu^{9} - 2478737837371 \nu^{8} + 3295314382092 \nu^{7} + 3500770767616 \nu^{6} - 3225526895581 \nu^{5} - 2872131807524 \nu^{4} + 1473343414389 \nu^{3} + 1168583011302 \nu^{2} - 242700042466 \nu - 133199717500\)\()/ 5791037030 \)
\(\beta_{7}\)\(=\)\((\)\(318716503 \nu^{18} - 1140866135 \nu^{17} - 7501449651 \nu^{16} + 29542139284 \nu^{15} + 67595392300 \nu^{14} - 309929331465 \nu^{13} - 281749529274 \nu^{12} + 1696786234950 \nu^{11} + 451103298313 \nu^{10} - 5196857369249 \nu^{9} + 375217402964 \nu^{8} + 8851478759162 \nu^{7} - 2162801145399 \nu^{6} - 7808662348931 \nu^{5} + 2393312892281 \nu^{4} + 2961548762254 \nu^{3} - 839166774003 \nu^{2} - 274887004146 \nu + 54702171430\)\()/ 11582074060 \)
\(\beta_{8}\)\(=\)\((\)\(-93609108 \nu^{18} + 195011795 \nu^{17} + 2566480776 \nu^{16} - 4911478344 \nu^{15} - 29294218685 \nu^{14} + 49897035585 \nu^{13} + 181459391204 \nu^{12} - 263787283760 \nu^{11} - 664790856488 \nu^{10} + 781891378959 \nu^{9} + 1466329373136 \nu^{8} - 1310268250057 \nu^{7} - 1887181446716 \nu^{6} + 1200884819526 \nu^{5} + 1292110192984 \nu^{4} - 547670340074 \nu^{3} - 388056127817 \nu^{2} + 88500571266 \nu + 40044597930\)\()/ 2895518515 \)
\(\beta_{9}\)\(=\)\((\)\(93609108 \nu^{18} - 195011795 \nu^{17} - 2566480776 \nu^{16} + 4911478344 \nu^{15} + 29294218685 \nu^{14} - 49897035585 \nu^{13} - 181459391204 \nu^{12} + 263787283760 \nu^{11} + 664790856488 \nu^{10} - 781891378959 \nu^{9} - 1466329373136 \nu^{8} + 1310268250057 \nu^{7} + 1887181446716 \nu^{6} - 1200884819526 \nu^{5} - 1289214674469 \nu^{4} + 547670340074 \nu^{3} + 367787498212 \nu^{2} - 88500571266 \nu - 19775968325\)\()/ 2895518515 \)
\(\beta_{10}\)\(=\)\((\)\(-37953488 \nu^{18} + 92196111 \nu^{17} + 991696752 \nu^{16} - 2324375148 \nu^{15} - 10672704577 \nu^{14} + 23745817195 \nu^{13} + 61534319327 \nu^{12} - 126913768361 \nu^{11} - 206178478291 \nu^{10} + 381112300142 \nu^{9} + 404304603883 \nu^{8} - 638066586796 \nu^{7} - 439470286205 \nu^{6} + 548407227881 \nu^{5} + 230702575789 \nu^{4} - 194336671842 \nu^{3} - 44204269770 \nu^{2} + 15157612018 \nu + 2068831656\)\()/ 1158207406 \)
\(\beta_{11}\)\(=\)\((\)\(-240105167 \nu^{18} + 881338715 \nu^{17} + 5495659804 \nu^{16} - 21651992771 \nu^{15} - 49786456290 \nu^{14} + 215066182570 \nu^{13} + 231300589556 \nu^{12} - 1119608505200 \nu^{11} - 605499617407 \nu^{10} + 3307955295931 \nu^{9} + 961255585349 \nu^{8} - 5591713835683 \nu^{7} - 1060121848224 \nu^{6} + 5135218235504 \nu^{5} + 906752365551 \nu^{4} - 2201652718176 \nu^{3} - 455951257588 \nu^{2} + 287009377064 \nu + 53652831630\)\()/ 5791037030 \)
\(\beta_{12}\)\(=\)\((\)\(290773938 \nu^{18} - 662122355 \nu^{17} - 7697023186 \nu^{16} + 16815829469 \nu^{15} + 83341546495 \nu^{14} - 172281371545 \nu^{13} - 477184016914 \nu^{12} + 916160179920 \nu^{11} + 1554464292428 \nu^{10} - 2705691492119 \nu^{9} - 2868279746356 \nu^{8} + 4389554678512 \nu^{7} + 2773105119646 \nu^{6} - 3584941413081 \nu^{5} - 1121219155374 \nu^{4} + 1145884621839 \nu^{3} + 63474192337 \nu^{2} - 53779704136 \nu + 19309791560\)\()/ 5791037030 \)
\(\beta_{13}\)\(=\)\((\)\(-485961931 \nu^{18} + 1113420900 \nu^{17} + 13007283232 \nu^{16} - 28691667118 \nu^{15} - 142770006155 \nu^{14} + 298637498085 \nu^{13} + 833859292088 \nu^{12} - 1614947153800 \nu^{11} - 2816533502951 \nu^{10} + 4855571864208 \nu^{9} + 5615791269077 \nu^{8} - 8046688000049 \nu^{7} - 6493467022012 \nu^{6} + 6795149697847 \nu^{5} + 4085986343623 \nu^{4} - 2351933693773 \nu^{3} - 1176825672339 \nu^{2} + 161926194492 \nu + 74147034150\)\()/ 5791037030 \)
\(\beta_{14}\)\(=\)\((\)\(-244538816 \nu^{18} + 640007340 \nu^{17} + 6214873087 \nu^{16} - 15917066553 \nu^{15} - 64356454145 \nu^{14} + 159447001730 \nu^{13} + 351910167043 \nu^{12} - 829200724505 \nu^{11} - 1098123323401 \nu^{10} + 2402070934223 \nu^{9} + 1964936315422 \nu^{8} - 3854034717714 \nu^{7} - 1924408194587 \nu^{6} + 3182220496297 \nu^{5} + 959561891823 \nu^{4} - 1107518683308 \nu^{3} - 255693379414 \nu^{2} + 88860693382 \nu + 28681664520\)\()/ 2895518515 \)
\(\beta_{15}\)\(=\)\((\)\(1036415051 \nu^{18} - 3169187735 \nu^{17} - 25673615907 \nu^{16} + 80069329128 \nu^{15} + 255548957080 \nu^{14} - 817470456425 \nu^{13} - 1312960193338 \nu^{12} + 4348086495830 \nu^{11} + 3700939092301 \nu^{10} - 12928740475313 \nu^{9} - 5538919778952 \nu^{8} + 21361112833594 \nu^{7} + 3742541531697 \nu^{6} - 18219213168867 \nu^{5} - 555145776623 \nu^{4} + 6668255047918 \nu^{3} - 102435732651 \nu^{2} - 709051042762 \nu - 23929479290\)\()/ 11582074060 \)
\(\beta_{16}\)\(=\)\((\)\(-1177420721 \nu^{18} + 3022806135 \nu^{17} + 30214864317 \nu^{16} - 76041669698 \nu^{15} - 316167323730 \nu^{14} + 773094156925 \nu^{13} + 1747199273478 \nu^{12} - 4098812389930 \nu^{11} - 5505446366291 \nu^{10} + 12181293389813 \nu^{9} + 9922760972072 \nu^{8} - 20239239560654 \nu^{7} - 9698265109967 \nu^{6} + 17589452904587 \nu^{5} + 4616300955633 \nu^{4} - 6763494721768 \nu^{3} - 1017780451669 \nu^{2} + 804092581722 \nu + 101431035930\)\()/ 11582074060 \)
\(\beta_{17}\)\(=\)\((\)\(727508506 \nu^{18} - 1789420580 \nu^{17} - 19108483507 \nu^{16} + 45513393513 \nu^{15} + 205976043850 \nu^{14} - 467788147365 \nu^{13} - 1184324624578 \nu^{12} + 2503001409990 \nu^{11} + 3952381907166 \nu^{10} - 7476870253978 \nu^{9} - 7810236222227 \nu^{8} + 12396449304189 \nu^{7} + 8959188531857 \nu^{6} - 10620725347607 \nu^{5} - 5626225717688 \nu^{4} + 3926288444198 \nu^{3} + 1709061696269 \nu^{2} - 427993545432 \nu - 153741088340\)\()/ 5791037030 \)
\(\beta_{18}\)\(=\)\((\)\(3366745827 \nu^{18} - 9263333215 \nu^{17} - 85049828069 \nu^{16} + 231373969696 \nu^{15} + 877139208820 \nu^{14} - 2334094800655 \nu^{13} - 4806101920096 \nu^{12} + 12274426272960 \nu^{11} + 15248263236747 \nu^{10} - 36169011220831 \nu^{9} - 28613582641044 \nu^{8} + 59507326593548 \nu^{7} + 31177087450109 \nu^{6} - 50962697977249 \nu^{5} - 18711356928001 \nu^{4} + 18937119806716 \nu^{3} + 5559535596393 \nu^{2} - 1938290594534 \nu - 509625256950\)\()/ 11582074060 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{3} + 5 \beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{9} + \beta_{8} + 7 \beta_{2} + 14\)
\(\nu^{5}\)\(=\)\(-\beta_{15} - \beta_{14} + \beta_{7} - \beta_{5} + 9 \beta_{3} + 30 \beta_{1} + 1\)
\(\nu^{6}\)\(=\)\(\beta_{17} - \beta_{16} - 2 \beta_{15} - \beta_{14} + \beta_{13} + 2 \beta_{11} + 11 \beta_{9} + 11 \beta_{8} + \beta_{7} + 2 \beta_{6} - 2 \beta_{4} + 46 \beta_{2} + 80\)
\(\nu^{7}\)\(=\)\(\beta_{18} + \beta_{17} - \beta_{16} - 14 \beta_{15} - 12 \beta_{14} + 2 \beta_{13} + 2 \beta_{11} + 3 \beta_{10} + \beta_{9} + \beta_{8} + 14 \beta_{7} - 14 \beta_{5} + 70 \beta_{3} - \beta_{2} + 195 \beta_{1} + 14\)
\(\nu^{8}\)\(=\)\(2 \beta_{18} + 13 \beta_{17} - 14 \beta_{16} - 31 \beta_{15} - 13 \beta_{14} + 16 \beta_{13} + 4 \beta_{12} + 31 \beta_{11} + 4 \beta_{10} + 96 \beta_{9} + 95 \beta_{8} + 17 \beta_{7} + 30 \beta_{6} - \beta_{5} - 29 \beta_{4} + 2 \beta_{3} + 305 \beta_{2} - 2 \beta_{1} + 508\)
\(\nu^{9}\)\(=\)\(17 \beta_{18} + 17 \beta_{17} - 16 \beta_{16} - 145 \beta_{15} - 109 \beta_{14} + 34 \beta_{13} + 8 \beta_{12} + 32 \beta_{11} + 54 \beta_{10} + 16 \beta_{9} + 20 \beta_{8} + 142 \beta_{7} + 3 \beta_{6} - 143 \beta_{5} - \beta_{4} + 525 \beta_{3} - 12 \beta_{2} + 1321 \beta_{1} + 141\)
\(\nu^{10}\)\(=\)\(38 \beta_{18} + 123 \beta_{17} - 137 \beta_{16} - 338 \beta_{15} - 125 \beta_{14} + 181 \beta_{13} + 78 \beta_{12} + 334 \beta_{11} + 80 \beta_{10} + 777 \beta_{9} + 762 \beta_{8} + 199 \beta_{7} + 318 \beta_{6} - 21 \beta_{5} - 295 \beta_{4} + 39 \beta_{3} + 2061 \beta_{2} - 28 \beta_{1} + 3420\)
\(\nu^{11}\)\(=\)\(196 \beta_{18} + 201 \beta_{17} - 175 \beta_{16} - 1338 \beta_{15} - 902 \beta_{14} + 401 \beta_{13} + 165 \beta_{12} + 354 \beta_{11} + 651 \beta_{10} + 185 \beta_{9} + 258 \beta_{8} + 1281 \beta_{7} + 63 \beta_{6} - 1291 \beta_{5} - 22 \beta_{4} + 3896 \beta_{3} - 92 \beta_{2} + 9180 \beta_{1} + 1270\)
\(\nu^{12}\)\(=\)\(472 \beta_{18} + 1052 \beta_{17} - 1168 \beta_{16} - 3209 \beta_{15} - 1087 \beta_{14} + 1783 \beta_{13} + 1001 \beta_{12} + 3109 \beta_{11} + 1036 \beta_{10} + 6089 \beta_{9} + 5939 \beta_{8} + 1997 \beta_{7} + 2936 \beta_{6} - 291 \beta_{5} - 2617 \beta_{4} + 509 \beta_{3} + 14191 \beta_{2} - 222 \beta_{1} + 23845\)
\(\nu^{13}\)\(=\)\(1930 \beta_{18} + 2048 \beta_{17} - 1628 \beta_{16} - 11637 \beta_{15} - 7176 \beta_{14} + 4070 \beta_{13} + 2196 \beta_{12} + 3394 \beta_{11} + 6618 \beta_{10} + 1887 \beta_{9} + 2764 \beta_{8} + 10931 \beta_{7} + 856 \beta_{6} - 10946 \beta_{5} - 312 \beta_{4} + 28856 \beta_{3} - 508 \beta_{2} + 64948 \beta_{1} + 10927\)
\(\nu^{14}\)\(=\)\(4884 \beta_{18} + 8685 \beta_{17} - 9323 \beta_{16} - 28414 \beta_{15} - 9063 \beta_{14} + 16327 \beta_{13} + 10700 \beta_{12} + 26856 \beta_{11} + 11110 \beta_{10} + 46978 \beta_{9} + 45726 \beta_{8} + 18471 \beta_{7} + 25254 \beta_{6} - 3357 \beta_{5} - 21718 \beta_{4} + 5607 \beta_{3} + 99346 \beta_{2} - 954 \beta_{1} + 170076\)
\(\nu^{15}\)\(=\)\(17519 \beta_{18} + 19249 \beta_{17} - 13913 \beta_{16} - 97749 \beta_{15} - 56041 \beta_{14} + 38110 \beta_{13} + 24088 \beta_{12} + 30340 \beta_{11} + 61369 \beta_{10} + 18027 \beta_{9} + 26842 \beta_{8} + 90467 \beta_{7} + 9628 \beta_{6} - 89511 \beta_{5} - 3656 \beta_{4} + 214022 \beta_{3} - 1303 \beta_{2} + 465784 \beta_{1} + 92083\)
\(\nu^{16}\)\(=\)\(45824 \beta_{18} + 70814 \beta_{17} - 71931 \beta_{16} - 241742 \beta_{15} - 74068 \beta_{14} + 142969 \beta_{13} + 103396 \beta_{12} + 222271 \beta_{11} + 107566 \beta_{10} + 359574 \beta_{9} + 350215 \beta_{8} + 162701 \beta_{7} + 208480 \beta_{6} - 34917 \beta_{5} - 173795 \beta_{4} + 56387 \beta_{3} + 705414 \beta_{2} + 4037 \beta_{1} + 1232416\)
\(\nu^{17}\)\(=\)\(151648 \beta_{18} + 171997 \beta_{17} - 113356 \beta_{16} - 803134 \beta_{15} - 433642 \beta_{14} + 339563 \beta_{13} + 237420 \beta_{12} + 260928 \beta_{11} + 538331 \beta_{10} + 165372 \beta_{9} + 245930 \beta_{8} + 734926 \beta_{7} + 97683 \beta_{6} - 715742 \beta_{5} - 38593 \beta_{4} + 1591650 \beta_{3} + 15758 \beta_{2} + 3376342 \beta_{1} + 767873\)
\(\nu^{18}\)\(=\)\(405545 \beta_{18} + 574450 \beta_{17} - 545365 \beta_{16} - 2006325 \beta_{15} - 599014 \beta_{14} + 1215030 \beta_{13} + 938630 \beta_{12} + 1792211 \beta_{11} + 978901 \beta_{10} + 2741034 \beta_{9} + 2676791 \beta_{8} + 1388627 \beta_{7} + 1677626 \beta_{6} - 339940 \beta_{5} - 1361630 \beta_{4} + 536340 \beta_{3} + 5069187 \beta_{2} + 153210 \beta_{1} + 9034683\)

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
2.79341
2.49323
2.42782
2.26573
1.88777
1.49024
1.35771
1.20823
0.823573
0.395540
−0.249163
−0.407951
−0.782325
−1.30029
−1.48888
−1.91759
−1.94177
−2.35941
−2.69590
−2.79341 0 5.80316 1.63704 0 1.00000 −10.6238 0 −4.57293
1.2 −2.49323 0 4.21621 0.262973 0 1.00000 −5.52554 0 −0.655653
1.3 −2.42782 0 3.89430 −4.13136 0 1.00000 −4.59902 0 10.0302
1.4 −2.26573 0 3.13354 0.263451 0 1.00000 −2.56830 0 −0.596909
1.5 −1.88777 0 1.56369 3.95777 0 1.00000 0.823654 0 −7.47138
1.6 −1.49024 0 0.220813 −2.04744 0 1.00000 2.65141 0 3.05118
1.7 −1.35771 0 −0.156613 −4.06757 0 1.00000 2.92806 0 5.52260
1.8 −1.20823 0 −0.540168 −0.486834 0 1.00000 3.06912 0 0.588209
1.9 −0.823573 0 −1.32173 2.87784 0 1.00000 2.73569 0 −2.37011
1.10 −0.395540 0 −1.84355 −1.95228 0 1.00000 1.52028 0 0.772204
1.11 0.249163 0 −1.93792 −0.989651 0 1.00000 −0.981182 0 −0.246584
1.12 0.407951 0 −1.83358 1.08992 0 1.00000 −1.56391 0 0.444635
1.13 0.782325 0 −1.38797 2.85652 0 1.00000 −2.65049 0 2.23472
1.14 1.30029 0 −0.309249 −3.89453 0 1.00000 −3.00269 0 −5.06401
1.15 1.48888 0 0.216778 2.91159 0 1.00000 −2.65501 0 4.33502
1.16 1.91759 0 1.67713 −1.63006 0 1.00000 −0.619124 0 −3.12578
1.17 1.94177 0 1.77048 −3.66642 0 1.00000 −0.445683 0 −7.11935
1.18 2.35941 0 3.56679 3.48367 0 1.00000 3.69670 0 8.21939
1.19 2.69590 0 5.26787 −1.47463 0 1.00000 8.80983 0 −3.97544
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.19
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(7\) \(-1\)
\(127\) \(-1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8001))\):

\(T_{2}^{19} + \cdots\)
\(T_{5}^{19} + \cdots\)