Properties

Label 6018.2.a.bb
Level 6018
Weight 2
Character orbit 6018.a
Self dual yes
Analytic conductor 48.054
Analytic rank 0
Dimension 13
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 6018 = 2 \cdot 3 \cdot 17 \cdot 59 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6018.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(48.0539719364\)
Analytic rank: \(0\)
Dimension: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{12}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + q^{3} + q^{4} + \beta_{1} q^{5} + q^{6} + ( 1 - \beta_{3} ) q^{7} + q^{8} + q^{9} +O(q^{10})\) \( q + q^{2} + q^{3} + q^{4} + \beta_{1} q^{5} + q^{6} + ( 1 - \beta_{3} ) q^{7} + q^{8} + q^{9} + \beta_{1} q^{10} + ( -\beta_{4} - \beta_{5} ) q^{11} + q^{12} + ( -\beta_{5} + \beta_{8} ) q^{13} + ( 1 - \beta_{3} ) q^{14} + \beta_{1} q^{15} + q^{16} + q^{17} + q^{18} + ( -\beta_{4} - \beta_{5} + \beta_{7} - \beta_{10} ) q^{19} + \beta_{1} q^{20} + ( 1 - \beta_{3} ) q^{21} + ( -\beta_{4} - \beta_{5} ) q^{22} + ( \beta_{1} - \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} + \beta_{12} ) q^{23} + q^{24} + ( 3 + \beta_{3} + \beta_{5} + \beta_{6} + \beta_{9} + \beta_{10} ) q^{25} + ( -\beta_{5} + \beta_{8} ) q^{26} + q^{27} + ( 1 - \beta_{3} ) q^{28} + ( 2 - 2 \beta_{1} - \beta_{2} + \beta_{4} + \beta_{7} - \beta_{9} ) q^{29} + \beta_{1} q^{30} + ( -1 - \beta_{2} - \beta_{6} - \beta_{9} - \beta_{11} ) q^{31} + q^{32} + ( -\beta_{4} - \beta_{5} ) q^{33} + q^{34} + ( 2 + 2 \beta_{2} - \beta_{4} + 2 \beta_{6} - \beta_{10} - \beta_{12} ) q^{35} + q^{36} + ( 1 + \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} + 2 \beta_{10} + \beta_{12} ) q^{37} + ( -\beta_{4} - \beta_{5} + \beta_{7} - \beta_{10} ) q^{38} + ( -\beta_{5} + \beta_{8} ) q^{39} + \beta_{1} q^{40} + ( 3 - \beta_{2} + \beta_{3} + 2 \beta_{4} + \beta_{5} - \beta_{6} - \beta_{8} + \beta_{10} + \beta_{12} ) q^{41} + ( 1 - \beta_{3} ) q^{42} + ( 1 + \beta_{1} + \beta_{2} + 2 \beta_{5} - \beta_{7} - \beta_{8} + \beta_{9} + \beta_{11} - \beta_{12} ) q^{43} + ( -\beta_{4} - \beta_{5} ) q^{44} + \beta_{1} q^{45} + ( \beta_{1} - \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} + \beta_{12} ) q^{46} + ( 1 + \beta_{1} + \beta_{2} - \beta_{4} + \beta_{5} - \beta_{7} + \beta_{9} - 2 \beta_{12} ) q^{47} + q^{48} + ( 3 - 2 \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} + \beta_{11} ) q^{49} + ( 3 + \beta_{3} + \beta_{5} + \beta_{6} + \beta_{9} + \beta_{10} ) q^{50} + q^{51} + ( -\beta_{5} + \beta_{8} ) q^{52} + ( 1 - 2 \beta_{1} - \beta_{2} - 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{7} + \beta_{8} - \beta_{10} - \beta_{12} ) q^{53} + q^{54} + ( -1 + \beta_{2} - 2 \beta_{4} + \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} - 2 \beta_{10} - \beta_{12} ) q^{55} + ( 1 - \beta_{3} ) q^{56} + ( -\beta_{4} - \beta_{5} + \beta_{7} - \beta_{10} ) q^{57} + ( 2 - 2 \beta_{1} - \beta_{2} + \beta_{4} + \beta_{7} - \beta_{9} ) q^{58} - q^{59} + \beta_{1} q^{60} + ( 1 + 2 \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} - 2 \beta_{8} + 2 \beta_{9} + \beta_{10} ) q^{61} + ( -1 - \beta_{2} - \beta_{6} - \beta_{9} - \beta_{11} ) q^{62} + ( 1 - \beta_{3} ) q^{63} + q^{64} + ( 5 + 2 \beta_{1} + 2 \beta_{2} + \beta_{3} + 3 \beta_{4} + 3 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} - \beta_{8} + 3 \beta_{9} + 2 \beta_{10} + \beta_{11} + \beta_{12} ) q^{65} + ( -\beta_{4} - \beta_{5} ) q^{66} + ( 3 - \beta_{1} - \beta_{8} - 2 \beta_{10} - 2 \beta_{12} ) q^{67} + q^{68} + ( \beta_{1} - \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} + \beta_{12} ) q^{69} + ( 2 + 2 \beta_{2} - \beta_{4} + 2 \beta_{6} - \beta_{10} - \beta_{12} ) q^{70} + ( 1 - \beta_{1} - 2 \beta_{2} - \beta_{6} - \beta_{12} ) q^{71} + q^{72} + ( -1 - \beta_{1} - \beta_{2} + \beta_{3} - \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} - 2 \beta_{9} - \beta_{10} + \beta_{12} ) q^{73} + ( 1 + \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} + 2 \beta_{10} + \beta_{12} ) q^{74} + ( 3 + \beta_{3} + \beta_{5} + \beta_{6} + \beta_{9} + \beta_{10} ) q^{75} + ( -\beta_{4} - \beta_{5} + \beta_{7} - \beta_{10} ) q^{76} + ( 3 - \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{5} - \beta_{7} - \beta_{9} + 3 \beta_{10} + \beta_{12} ) q^{77} + ( -\beta_{5} + \beta_{8} ) q^{78} + ( -2 + \beta_{1} - 2 \beta_{2} + \beta_{4} + \beta_{5} - 3 \beta_{6} - \beta_{8} - 2 \beta_{9} + \beta_{12} ) q^{79} + \beta_{1} q^{80} + q^{81} + ( 3 - \beta_{2} + \beta_{3} + 2 \beta_{4} + \beta_{5} - \beta_{6} - \beta_{8} + \beta_{10} + \beta_{12} ) q^{82} + ( 1 - 2 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + 2 \beta_{7} + \beta_{8} - 2 \beta_{10} - \beta_{12} ) q^{83} + ( 1 - \beta_{3} ) q^{84} + \beta_{1} q^{85} + ( 1 + \beta_{1} + \beta_{2} + 2 \beta_{5} - \beta_{7} - \beta_{8} + \beta_{9} + \beta_{11} - \beta_{12} ) q^{86} + ( 2 - 2 \beta_{1} - \beta_{2} + \beta_{4} + \beta_{7} - \beta_{9} ) q^{87} + ( -\beta_{4} - \beta_{5} ) q^{88} + ( 1 + 2 \beta_{2} - \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} + 2 \beta_{9} + \beta_{10} - \beta_{11} ) q^{89} + \beta_{1} q^{90} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} + 2 \beta_{6} - 2 \beta_{7} + \beta_{8} + 2 \beta_{10} ) q^{91} + ( \beta_{1} - \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} + \beta_{12} ) q^{92} + ( -1 - \beta_{2} - \beta_{6} - \beta_{9} - \beta_{11} ) q^{93} + ( 1 + \beta_{1} + \beta_{2} - \beta_{4} + \beta_{5} - \beta_{7} + \beta_{9} - 2 \beta_{12} ) q^{94} + ( 1 - 2 \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} + \beta_{7} - 3 \beta_{9} - \beta_{10} - \beta_{11} ) q^{95} + q^{96} + ( \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} + \beta_{10} + \beta_{12} ) q^{97} + ( 3 - 2 \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} + \beta_{11} ) q^{98} + ( -\beta_{4} - \beta_{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13q + 13q^{2} + 13q^{3} + 13q^{4} + 4q^{5} + 13q^{6} + 11q^{7} + 13q^{8} + 13q^{9} + O(q^{10}) \) \( 13q + 13q^{2} + 13q^{3} + 13q^{4} + 4q^{5} + 13q^{6} + 11q^{7} + 13q^{8} + 13q^{9} + 4q^{10} + 7q^{11} + 13q^{12} + 6q^{13} + 11q^{14} + 4q^{15} + 13q^{16} + 13q^{17} + 13q^{18} + 9q^{19} + 4q^{20} + 11q^{21} + 7q^{22} + 2q^{23} + 13q^{24} + 33q^{25} + 6q^{26} + 13q^{27} + 11q^{28} + 14q^{29} + 4q^{30} - 5q^{31} + 13q^{32} + 7q^{33} + 13q^{34} + 24q^{35} + 13q^{36} + 4q^{37} + 9q^{38} + 6q^{39} + 4q^{40} + 28q^{41} + 11q^{42} + q^{43} + 7q^{44} + 4q^{45} + 2q^{46} + 12q^{47} + 13q^{48} + 32q^{49} + 33q^{50} + 13q^{51} + 6q^{52} + 22q^{53} + 13q^{54} - 7q^{55} + 11q^{56} + 9q^{57} + 14q^{58} - 13q^{59} + 4q^{60} - 9q^{61} - 5q^{62} + 11q^{63} + 13q^{64} + 34q^{65} + 7q^{66} + 26q^{67} + 13q^{68} + 2q^{69} + 24q^{70} + 8q^{71} + 13q^{72} + 4q^{73} + 4q^{74} + 33q^{75} + 9q^{76} + 38q^{77} + 6q^{78} - 17q^{79} + 4q^{80} + 13q^{81} + 28q^{82} + 14q^{83} + 11q^{84} + 4q^{85} + q^{86} + 14q^{87} + 7q^{88} + 19q^{89} + 4q^{90} - 5q^{91} + 2q^{92} - 5q^{93} + 12q^{94} + 25q^{95} + 13q^{96} - 5q^{97} + 32q^{98} + 7q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{13} - 4 x^{12} - 41 x^{11} + 179 x^{10} + 540 x^{9} - 2773 x^{8} - 2260 x^{7} + 17621 x^{6} - 838 x^{5} - 44478 x^{4} + 16472 x^{3} + 29944 x^{2} - 6856 x + 128\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(-1796388205553 \nu^{12} - 1489108919712 \nu^{11} + 81660319298773 \nu^{10} + 60295853640717 \nu^{9} - 1346976232371876 \nu^{8} - 922973188703411 \nu^{7} + 9899380051264956 \nu^{6} + 6716883469612343 \nu^{5} - 32386874868109258 \nu^{4} - 21507116682985566 \nu^{3} + 37358751314234452 \nu^{2} + 20860161005825920 \nu - 1768227501996184\)\()/ 689049844074136 \)
\(\beta_{3}\)\(=\)\((\)\(-10253913982431 \nu^{12} + 12143449436408 \nu^{11} + 459309915620523 \nu^{10} - 550253809975141 \nu^{9} - 7291775225900956 \nu^{8} + 8262582263045907 \nu^{7} + 49539905362356236 \nu^{6} - 46501901494687967 \nu^{5} - 141537061577422086 \nu^{4} + 86711893064448398 \nu^{3} + 122348683101464684 \nu^{2} - 15147039293522512 \nu - 2559670711426040\)\()/ 689049844074136 \)
\(\beta_{4}\)\(=\)\((\)\(2643124952245 \nu^{12} - 3844423118131 \nu^{11} - 118025134245614 \nu^{10} + 172279045954540 \nu^{9} + 1859746846457838 \nu^{8} - 2577999496689658 \nu^{7} - 12435310500957227 \nu^{6} + 14674795233528832 \nu^{5} + 34437813698810893 \nu^{4} - 28641901734087195 \nu^{3} - 27309473007286086 \nu^{2} + 7953201992223538 \nu + 191018910747970\)\()/ 172262461018534 \)
\(\beta_{5}\)\(=\)\((\)\(-2806630645981 \nu^{12} + 3962699821337 \nu^{11} + 124624713157090 \nu^{10} - 179172235132576 \nu^{9} - 1946700518875936 \nu^{8} + 2711523524200234 \nu^{7} + 12817031913020937 \nu^{6} - 15728312750603394 \nu^{5} - 34425778503897719 \nu^{4} + 31982336352673793 \nu^{3} + 25008787374956530 \nu^{2} - 10497946895775678 \nu + 558668664048170\)\()/ 172262461018534 \)
\(\beta_{6}\)\(=\)\((\)\(-11419786805749 \nu^{12} + 16698434980110 \nu^{11} + 504336575792465 \nu^{10} - 755378470856885 \nu^{9} - 7810944784378346 \nu^{8} + 11452358170544105 \nu^{7} + 50713678365610526 \nu^{6} - 66799597776713869 \nu^{5} - 133463695266617588 \nu^{4} + 138168015452056174 \nu^{3} + 94078149732007496 \nu^{2} - 48634450370623920 \nu + 1161596938398896\)\()/ 689049844074136 \)
\(\beta_{7}\)\(=\)\((\)\(5836486040157 \nu^{12} - 6831410905550 \nu^{11} - 260080147076221 \nu^{10} + 311196222902321 \nu^{9} + 4094961906303710 \nu^{8} - 4701200261840201 \nu^{7} - 27429895945466362 \nu^{6} + 26705214956593781 \nu^{5} + 76513736980135184 \nu^{4} - 50745708591403802 \nu^{3} - 63294691908648740 \nu^{2} + 9206122997578932 \nu + 990949223000480\)\()/ 344524922037068 \)
\(\beta_{8}\)\(=\)\((\)\(-3800074510933 \nu^{12} + 4801952864360 \nu^{11} + 168177451801958 \nu^{10} - 219681987395845 \nu^{9} - 2617125025528437 \nu^{8} + 3357195618452128 \nu^{7} + 17159458829299488 \nu^{6} - 19652933906266972 \nu^{5} - 45999555735978216 \nu^{4} + 40631636419194187 \nu^{3} + 34418056162936816 \nu^{2} - 14039929218321888 \nu + 228430125895582\)\()/ 172262461018534 \)
\(\beta_{9}\)\(=\)\((\)\(-15713129420919 \nu^{12} + 20121588852528 \nu^{11} + 698648560224535 \nu^{10} - 915337629159613 \nu^{9} - 10951008089524496 \nu^{8} + 13885279897829999 \nu^{7} + 72691678822176548 \nu^{6} - 80202504730163387 \nu^{5} - 199066645616036446 \nu^{4} + 160388343465178226 \nu^{3} + 155400440504302388 \nu^{2} - 46826258636980632 \nu + 927871747678256\)\()/ 689049844074136 \)
\(\beta_{10}\)\(=\)\((\)\(48613352793023 \nu^{12} - 64814272554394 \nu^{11} - 2160793904265883 \nu^{10} + 2937658850521943 \nu^{9} + 33840530175307542 \nu^{8} - 44446314428220947 \nu^{7} - 224213390202227058 \nu^{6} + 256417255003978799 \nu^{5} + 611770516475666996 \nu^{4} - 513197597392377970 \nu^{3} - 471173372993526552 \nu^{2} + 152599535884229776 \nu - 7276871383436880\)\()/ 689049844074136 \)
\(\beta_{11}\)\(=\)\((\)\(98658980574257 \nu^{12} - 121059374370454 \nu^{11} - 4393285532037869 \nu^{10} + 5495632108851737 \nu^{9} + 69064662601312658 \nu^{8} - 82857036982229621 \nu^{7} - 461116336742527902 \nu^{6} + 471078241980980913 \nu^{5} + 1277245628332544140 \nu^{4} - 904590081886303550 \nu^{3} - 1027333801147935896 \nu^{2} + 200744255133819480 \nu + 497584607825056\)\()/ 689049844074136 \)
\(\beta_{12}\)\(=\)\((\)\(-114490409117923 \nu^{12} + 149510059488566 \nu^{11} + 5090085024310823 \nu^{10} - 6771601944160655 \nu^{9} - 79766880804874594 \nu^{8} + 102183142140723143 \nu^{7} + 529363065829535614 \nu^{6} - 585411706046996443 \nu^{5} - 1450584604586481880 \nu^{4} + 1152338156086936082 \nu^{3} + 1135181228111546144 \nu^{2} - 313510477401767040 \nu + 7990135291149864\)\()/ 689049844074136 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{10} + \beta_{9} + \beta_{6} + \beta_{5} + \beta_{3} + 8\)
\(\nu^{3}\)\(=\)\(-\beta_{11} - 3 \beta_{9} + 3 \beta_{8} + 3 \beta_{7} - 3 \beta_{6} - 2 \beta_{5} - 3 \beta_{2} + 9 \beta_{1} - 4\)
\(\nu^{4}\)\(=\)\(3 \beta_{12} + 2 \beta_{11} + 19 \beta_{10} + 20 \beta_{9} - 5 \beta_{8} + 14 \beta_{6} + 24 \beta_{5} + 9 \beta_{4} + 20 \beta_{3} + \beta_{1} + 107\)
\(\nu^{5}\)\(=\)\(4 \beta_{12} - 20 \beta_{11} + 3 \beta_{10} - 71 \beta_{9} + 61 \beta_{8} + 53 \beta_{7} - 73 \beta_{6} - 34 \beta_{5} + 16 \beta_{4} + 4 \beta_{3} - 75 \beta_{2} + 108 \beta_{1} - 100\)
\(\nu^{6}\)\(=\)\(83 \beta_{12} + 57 \beta_{11} + 343 \beta_{10} + 370 \beta_{9} - 121 \beta_{8} - 20 \beta_{7} + 197 \beta_{6} + 449 \beta_{5} + 220 \beta_{4} + 354 \beta_{3} - 4 \beta_{2} + 26 \beta_{1} + 1611\)
\(\nu^{7}\)\(=\)\(120 \beta_{12} - 324 \beta_{11} + 63 \beta_{10} - 1398 \beta_{9} + 1079 \beta_{8} + 814 \beta_{7} - 1413 \beta_{6} - 497 \beta_{5} + 434 \beta_{4} + 123 \beta_{3} - 1409 \beta_{2} + 1527 \beta_{1} - 2028\)
\(\nu^{8}\)\(=\)\(1747 \beta_{12} + 1234 \beta_{11} + 6054 \beta_{10} + 6709 \beta_{9} - 2374 \beta_{8} - 717 \beta_{7} + 2841 \beta_{6} + 7974 \beta_{5} + 4195 \beta_{4} + 5929 \beta_{3} - 42 \beta_{2} + 615 \beta_{1} + 25577\)
\(\nu^{9}\)\(=\)\(2658 \beta_{12} - 5011 \beta_{11} + 949 \beta_{10} - 26224 \beta_{9} + 18554 \beta_{8} + 12170 \beta_{7} - 25538 \beta_{6} - 7021 \beta_{5} + 8879 \beta_{4} + 2710 \beta_{3} - 24450 \beta_{2} + 23401 \beta_{1} - 38176\)
\(\nu^{10}\)\(=\)\(33381 \beta_{12} + 24253 \beta_{11} + 105562 \beta_{10} + 121001 \beta_{9} - 43955 \beta_{8} - 17839 \beta_{7} + 42232 \beta_{6} + 138935 \beta_{5} + 73809 \beta_{4} + 97328 \beta_{3} + 891 \beta_{2} + 13849 \beta_{1} + 417089\)
\(\nu^{11}\)\(=\)\(52093 \beta_{12} - 77476 \beta_{11} + 10881 \beta_{10} - 483001 \beta_{9} + 317710 \beta_{8} + 182279 \beta_{7} - 449090 \beta_{6} - 99772 \beta_{5} + 165057 \beta_{4} + 53295 \beta_{3} - 414050 \beta_{2} + 371483 \beta_{1} - 695315\)
\(\nu^{12}\)\(=\)\(610129 \beta_{12} + 454952 \beta_{11} + 1829309 \beta_{10} + 2177976 \beta_{9} - 797491 \beta_{8} - 382102 \beta_{7} + 647576 \beta_{6} + 2398243 \beta_{5} + 1257836 \beta_{4} + 1588185 \beta_{3} + 51549 \beta_{2} + 294293 \beta_{1} + 6906799\)

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
−4.21367
−3.91861
−2.41505
−2.08866
−0.877543
0.0205303
0.197152
1.47504
2.66604
2.71857
2.75551
3.59366
4.08702
1.00000 1.00000 1.00000 −4.21367 1.00000 1.46512 1.00000 1.00000 −4.21367
1.2 1.00000 1.00000 1.00000 −3.91861 1.00000 −4.71846 1.00000 1.00000 −3.91861
1.3 1.00000 1.00000 1.00000 −2.41505 1.00000 1.11075 1.00000 1.00000 −2.41505
1.4 1.00000 1.00000 1.00000 −2.08866 1.00000 3.96496 1.00000 1.00000 −2.08866
1.5 1.00000 1.00000 1.00000 −0.877543 1.00000 −4.30430 1.00000 1.00000 −0.877543
1.6 1.00000 1.00000 1.00000 0.0205303 1.00000 5.09020 1.00000 1.00000 0.0205303
1.7 1.00000 1.00000 1.00000 0.197152 1.00000 1.50880 1.00000 1.00000 0.197152
1.8 1.00000 1.00000 1.00000 1.47504 1.00000 −0.770392 1.00000 1.00000 1.47504
1.9 1.00000 1.00000 1.00000 2.66604 1.00000 4.24276 1.00000 1.00000 2.66604
1.10 1.00000 1.00000 1.00000 2.71857 1.00000 1.56988 1.00000 1.00000 2.71857
1.11 1.00000 1.00000 1.00000 2.75551 1.00000 1.74830 1.00000 1.00000 2.75551
1.12 1.00000 1.00000 1.00000 3.59366 1.00000 2.37015 1.00000 1.00000 3.59366
1.13 1.00000 1.00000 1.00000 4.08702 1.00000 −2.27776 1.00000 1.00000 4.08702
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.13
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 6018.2.a.bb 13
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6018.2.a.bb 13 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(17\) \(-1\)
\(59\) \(1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6018))\):

\(T_{5}^{13} - \cdots\)
\(T_{7}^{13} - \cdots\)