# 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 $1$

# Learn more about

## 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: $$\mathbb{Q}[x]/(x^{13} - \cdots)$$ Defining polynomial: $$x^{13} - 4 x^{12} - 41 x^{11} + 179 x^{10} + 540 x^{9} - 2773 x^{8} - 2260 x^{7} + 17621 x^{6} - 838 x^{5} - 44478 x^{4} + 16472 x^{3} + 29944 x^{2} - 6856 x + 128$$ 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$-1$$
$$17$$ $$-1$$
$$59$$ $$1$$

## 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

## Hecke kernels

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

 $$T_{5}^{13} - \cdots$$ $$T_{7}^{13} - \cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T )^{13}$$
$3$ $$( 1 - T )^{13}$$
$5$ $$1 - 4 T + 24 T^{2} - 61 T^{3} + 235 T^{4} - 423 T^{5} + 1415 T^{6} - 1924 T^{7} + 7312 T^{8} - 9448 T^{9} + 44647 T^{10} - 68991 T^{11} + 286974 T^{12} - 425882 T^{13} + 1434870 T^{14} - 1724775 T^{15} + 5580875 T^{16} - 5905000 T^{17} + 22850000 T^{18} - 30062500 T^{19} + 110546875 T^{20} - 165234375 T^{21} + 458984375 T^{22} - 595703125 T^{23} + 1171875000 T^{24} - 976562500 T^{25} + 1220703125 T^{26}$$
$7$ $$1 - 11 T + 90 T^{2} - 504 T^{3} + 2365 T^{4} - 9084 T^{5} + 31374 T^{6} - 97541 T^{7} + 295191 T^{8} - 872287 T^{9} + 2616353 T^{10} - 7686385 T^{11} + 22070788 T^{12} - 59818944 T^{13} + 154495516 T^{14} - 376632865 T^{15} + 897409079 T^{16} - 2094361087 T^{17} + 4961275137 T^{18} - 11475601109 T^{19} + 25837838082 T^{20} - 52367452284 T^{21} + 95436280555 T^{22} - 142367525496 T^{23} + 177959406870 T^{24} - 152254159211 T^{25} + 96889010407 T^{26}$$
$11$ $$1 - 7 T + 79 T^{2} - 466 T^{3} + 3068 T^{4} - 16053 T^{5} + 81139 T^{6} - 381061 T^{7} + 1650510 T^{8} - 6918423 T^{9} + 26974560 T^{10} - 100898785 T^{11} + 360239839 T^{12} - 1215156762 T^{13} + 3962638229 T^{14} - 12208752985 T^{15} + 35903139360 T^{16} - 101292631143 T^{17} + 265816286010 T^{18} - 675072806221 T^{19} + 1581169567769 T^{20} - 3441103116693 T^{21} + 7234183515988 T^{22} - 12086839864066 T^{23} + 22539621978269 T^{24} - 21968998637047 T^{25} + 34522712143931 T^{26}$$
$13$ $$1 - 6 T + 86 T^{2} - 408 T^{3} + 3814 T^{4} - 15822 T^{5} + 117156 T^{6} - 436096 T^{7} + 2755554 T^{8} - 9332762 T^{9} + 52168947 T^{10} - 161742440 T^{11} + 814406662 T^{12} - 2309105956 T^{13} + 10587286606 T^{14} - 27334472360 T^{15} + 114615176559 T^{16} - 266553015482 T^{17} + 1023117911322 T^{18} - 2104952097664 T^{19} + 7351365257652 T^{20} - 12906491467662 T^{21} + 40445560608622 T^{22} - 56246264674392 T^{23} + 154125793887182 T^{24} - 139788510734886 T^{25} + 302875106592253 T^{26}$$
$17$ $$( 1 - T )^{13}$$
$19$ $$1 - 9 T + 178 T^{2} - 1317 T^{3} + 15226 T^{4} - 96635 T^{5} + 836403 T^{6} - 4666436 T^{7} + 33145555 T^{8} - 164719457 T^{9} + 1003778522 T^{10} - 4468587847 T^{11} + 23927051309 T^{12} - 95382954198 T^{13} + 454613974871 T^{14} - 1613160212767 T^{15} + 6884916882398 T^{16} - 21466404355697 T^{17} + 82071675589945 T^{18} - 219536592750116 T^{19} + 747637004114817 T^{20} - 1641206614467035 T^{21} + 4913242886383054 T^{22} - 8074614261523917 T^{23} + 20735266083882982 T^{24} - 19919834271595449 T^{25} + 42052983462257059 T^{26}$$
$23$ $$1 - 2 T + 123 T^{2} - 240 T^{3} + 7693 T^{4} - 17743 T^{5} + 333729 T^{6} - 920731 T^{7} + 11469350 T^{8} - 35670582 T^{9} + 338293479 T^{10} - 1089616869 T^{11} + 8795671823 T^{12} - 27344426722 T^{13} + 202300451929 T^{14} - 576407323701 T^{15} + 4116016758993 T^{16} - 9982091337462 T^{17} + 73820670587050 T^{18} - 136301232114859 T^{19} + 1136288991601863 T^{20} - 1389471811840783 T^{21} + 13856267424634859 T^{22} - 9942362691275760 T^{23} + 117195600223413021 T^{24} - 43829248864040642 T^{25} + 504036361936467383 T^{26}$$
$29$ $$1 - 14 T + 272 T^{2} - 3043 T^{3} + 35419 T^{4} - 319831 T^{5} + 2875159 T^{6} - 21817640 T^{7} + 164254990 T^{8} - 1088427838 T^{9} + 7158239785 T^{10} - 42462414021 T^{11} + 251106573838 T^{12} - 1352059496186 T^{13} + 7282090641302 T^{14} - 35710890191661 T^{15} + 174582310116365 T^{16} - 769824329688478 T^{17} + 3369058573883510 T^{18} - 12977641081182440 T^{19} + 49596137118708131 T^{20} - 159994310503729591 T^{21} + 513828603319304111 T^{22} - 1280212110932511643 T^{23} + 3318538656271985488 T^{24} - 4953406964876566574 T^{25} + 10260628712958602189 T^{26}$$
$31$ $$1 + 5 T + 247 T^{2} + 956 T^{3} + 29081 T^{4} + 90752 T^{5} + 2243853 T^{6} + 5915632 T^{7} + 129377689 T^{8} + 300080107 T^{9} + 5927039795 T^{10} + 12387878868 T^{11} + 222008919780 T^{12} + 421997430048 T^{13} + 6882276513180 T^{14} + 11904751592148 T^{15} + 176572442532845 T^{16} + 277130280496747 T^{17} + 3703973394412039 T^{18} + 5250145175441392 T^{19} + 61734261710809683 T^{20} + 77401567429845632 T^{21} + 768890652054473351 T^{22} + 783564642353645756 T^{23} + 6275893793411993257 T^{24} + 3938313918942748805 T^{25} + 24417546297445042591 T^{26}$$
$37$ $$1 - 4 T + 268 T^{2} - 1505 T^{3} + 37311 T^{4} - 238203 T^{5} + 3602425 T^{6} - 23178978 T^{7} + 263083522 T^{8} - 1608410860 T^{9} + 15077457257 T^{10} - 84978732605 T^{11} + 691733609360 T^{12} - 3528174773018 T^{13} + 25594143546320 T^{14} - 116335884936245 T^{15} + 763718442438821 T^{16} - 3014420905788460 T^{17} + 18243252436976554 T^{18} - 59470915988230002 T^{19} + 341984967480847525 T^{20} - 836683143362343963 T^{21} + 4849002473494117947 T^{22} - 7236919480488862745 T^{23} + 47681922636895390684 T^{24} - 26331808023360141124 T^{25} +$$$$24\!\cdots\!97$$$$T^{26}$$
$41$ $$1 - 28 T + 697 T^{2} - 11614 T^{3} + 174983 T^{4} - 2150799 T^{5} + 24426115 T^{6} - 242135695 T^{7} + 2254444482 T^{8} - 18937084194 T^{9} + 151269704149 T^{10} - 1110551182841 T^{11} + 7818480943245 T^{12} - 51045147899962 T^{13} + 320557718673045 T^{14} - 1866836538355721 T^{15} + 10425659279653229 T^{16} - 53511673969121634 T^{17} + 261191373049932882 T^{18} - 1150169791716982495 T^{19} + 4757090290558802315 T^{20} - 17173969197868217679 T^{21} + 57286273026058477663 T^{22} -$$$$15\!\cdots\!14$$$$T^{23} +$$$$38\!\cdots\!77$$$$T^{24} -$$$$63\!\cdots\!68$$$$T^{25} +$$$$92\!\cdots\!21$$$$T^{26}$$
$43$ $$1 - T + 245 T^{2} + 474 T^{3} + 28168 T^{4} + 117291 T^{5} + 2361755 T^{6} + 11329137 T^{7} + 165872558 T^{8} + 728083927 T^{9} + 9503533718 T^{10} + 39269716061 T^{11} + 454851744743 T^{12} + 1836059905822 T^{13} + 19558625023949 T^{14} + 72609704996789 T^{15} + 755597455317026 T^{16} + 2489174057711527 T^{17} + 24384666488007194 T^{18} + 71615588008858713 T^{19} + 641968963875012785 T^{20} + 1370920698760098891 T^{21} + 14157028693036993624 T^{22} + 10243842616496734026 T^{23} +$$$$22\!\cdots\!15$$$$T^{24} - 39959630797262576401 T^{25} +$$$$17\!\cdots\!43$$$$T^{26}$$
$47$ $$1 - 12 T + 269 T^{2} - 2553 T^{3} + 35227 T^{4} - 296262 T^{5} + 3255365 T^{6} - 25570409 T^{7} + 241265717 T^{8} - 1791933732 T^{9} + 15000154241 T^{10} - 105278719614 T^{11} + 803719754514 T^{12} - 5311053194900 T^{13} + 37774828462158 T^{14} - 232560691627326 T^{15} + 1557361013763343 T^{16} - 8744064985299492 T^{17} + 55333087554225019 T^{18} - 275628944661599561 T^{19} + 1649243174546033995 T^{20} - 7054379408986637382 T^{21} + 39423609175991173109 T^{22} -$$$$13\!\cdots\!97$$$$T^{23} +$$$$66\!\cdots\!07$$$$T^{24} -$$$$13\!\cdots\!92$$$$T^{25} +$$$$54\!\cdots\!27$$$$T^{26}$$
$53$ $$1 - 22 T + 630 T^{2} - 9498 T^{3} + 165545 T^{4} - 1991081 T^{5} + 26678715 T^{6} - 272042655 T^{7} + 3035565744 T^{8} - 27002679610 T^{9} + 260501235021 T^{10} - 2051310947231 T^{11} + 17441725444192 T^{12} - 122279337375630 T^{13} + 924411448542176 T^{14} - 5762132450771879 T^{15} + 38782642366221417 T^{16} - 213064130411792410 T^{17} + 1269459912845991792 T^{18} - 6029651647911957495 T^{19} + 31339783707036469455 T^{20} -$$$$12\!\cdots\!41$$$$T^{21} +$$$$54\!\cdots\!85$$$$T^{22} -$$$$16\!\cdots\!02$$$$T^{23} +$$$$58\!\cdots\!10$$$$T^{24} -$$$$10\!\cdots\!02$$$$T^{25} +$$$$26\!\cdots\!73$$$$T^{26}$$
$59$ $$( 1 + T )^{13}$$
$61$ $$1 + 9 T + 411 T^{2} + 2790 T^{3} + 82608 T^{4} + 438291 T^{5} + 11026399 T^{6} + 45697727 T^{7} + 1105499164 T^{8} + 3535757071 T^{9} + 89544485756 T^{10} + 226475559115 T^{11} + 6195513309525 T^{12} + 13712578106586 T^{13} + 377926311881025 T^{14} + 842715555466915 T^{15} + 20324896921382636 T^{16} + 48955530219691711 T^{17} + 933700504672992364 T^{18} + 2354364002486777447 T^{19} + 34653136464359118379 T^{20} + 84023589920891286771 T^{21} +$$$$96\!\cdots\!28$$$$T^{22} +$$$$19\!\cdots\!90$$$$T^{23} +$$$$17\!\cdots\!71$$$$T^{24} +$$$$23\!\cdots\!89$$$$T^{25} +$$$$16\!\cdots\!81$$$$T^{26}$$
$67$ $$1 - 26 T + 587 T^{2} - 7685 T^{3} + 98035 T^{4} - 859948 T^{5} + 8994736 T^{6} - 71815972 T^{7} + 844292078 T^{8} - 7159101516 T^{9} + 79343800464 T^{10} - 584508284135 T^{11} + 5678322537989 T^{12} - 38606203816252 T^{13} + 380447610045263 T^{14} - 2623857687482015 T^{15} + 23863679458954032 T^{16} - 144263920900199436 T^{17} + 1139899932149002346 T^{18} - 6496356641014203268 T^{19} + 54514500862016579728 T^{20} -$$$$34\!\cdots\!68$$$$T^{21} +$$$$26\!\cdots\!45$$$$T^{22} -$$$$14\!\cdots\!65$$$$T^{23} +$$$$71\!\cdots\!21$$$$T^{24} -$$$$21\!\cdots\!86$$$$T^{25} +$$$$54\!\cdots\!87$$$$T^{26}$$
$71$ $$1 - 8 T + 707 T^{2} - 4807 T^{3} + 234473 T^{4} - 1355460 T^{5} + 48766033 T^{6} - 239904951 T^{7} + 7180648081 T^{8} - 30240148472 T^{9} + 801213215407 T^{10} - 2931832489746 T^{11} + 70588396333664 T^{12} - 229575459404024 T^{13} + 5011776139690144 T^{14} - 14779367580809586 T^{15} + 286763023139534777 T^{16} - 768453006363101432 T^{17} + 12955536026942025431 T^{18} - 30731892337153592871 T^{19} +$$$$44\!\cdots\!03$$$$T^{20} -$$$$87\!\cdots\!60$$$$T^{21} +$$$$10\!\cdots\!63$$$$T^{22} -$$$$15\!\cdots\!07$$$$T^{23} +$$$$16\!\cdots\!97$$$$T^{24} -$$$$13\!\cdots\!28$$$$T^{25} +$$$$11\!\cdots\!11$$$$T^{26}$$
$73$ $$1 - 4 T + 611 T^{2} - 1682 T^{3} + 182525 T^{4} - 333829 T^{5} + 35779349 T^{6} - 42252171 T^{7} + 5168952286 T^{8} - 3954446354 T^{9} + 581980660823 T^{10} - 309268883687 T^{11} + 52471512642205 T^{12} - 22682225317618 T^{13} + 3830420422880965 T^{14} - 1648093881168023 T^{15} + 226400370731380991 T^{16} - 112299320582463314 T^{17} + 10715608149679011598 T^{18} - 6394199607315523419 T^{19} +$$$$39\!\cdots\!53$$$$T^{20} -$$$$26\!\cdots\!49$$$$T^{21} +$$$$10\!\cdots\!25$$$$T^{22} -$$$$72\!\cdots\!18$$$$T^{23} +$$$$19\!\cdots\!47$$$$T^{24} -$$$$91\!\cdots\!84$$$$T^{25} +$$$$16\!\cdots\!33$$$$T^{26}$$
$79$ $$1 + 17 T + 675 T^{2} + 10217 T^{3} + 230523 T^{4} + 3085967 T^{5} + 51902335 T^{6} + 617521660 T^{7} + 8516985261 T^{8} + 90620190381 T^{9} + 1071807738223 T^{10} + 10225800559523 T^{11} + 106256985825752 T^{12} + 907693023024310 T^{13} + 8394301880234408 T^{14} + 63819221291983043 T^{15} + 528443015445729697 T^{16} + 3529663755575370861 T^{17} + 26207243997548735139 T^{18} +$$$$15\!\cdots\!60$$$$T^{19} +$$$$99\!\cdots\!65$$$$T^{20} +$$$$46\!\cdots\!87$$$$T^{21} +$$$$27\!\cdots\!37$$$$T^{22} +$$$$96\!\cdots\!17$$$$T^{23} +$$$$50\!\cdots\!25$$$$T^{24} +$$$$10\!\cdots\!97$$$$T^{25} +$$$$46\!\cdots\!39$$$$T^{26}$$
$83$ $$1 - 14 T + 673 T^{2} - 8914 T^{3} + 218926 T^{4} - 2742155 T^{5} + 46577271 T^{6} - 545990071 T^{7} + 7358136112 T^{8} - 79494323920 T^{9} + 918676881756 T^{10} - 9035028131695 T^{11} + 93126797064565 T^{12} - 829717859738958 T^{13} + 7729524156358895 T^{14} - 62242308799246855 T^{15} + 525287498188617972 T^{16} - 3772667142273338320 T^{17} + 28983997201894000016 T^{18} -$$$$17\!\cdots\!99$$$$T^{19} +$$$$12\!\cdots\!17$$$$T^{20} -$$$$61\!\cdots\!55$$$$T^{21} +$$$$40\!\cdots\!78$$$$T^{22} -$$$$13\!\cdots\!86$$$$T^{23} +$$$$86\!\cdots\!91$$$$T^{24} -$$$$14\!\cdots\!54$$$$T^{25} +$$$$88\!\cdots\!63$$$$T^{26}$$
$89$ $$1 - 19 T + 649 T^{2} - 8445 T^{3} + 174410 T^{4} - 1832313 T^{5} + 30874796 T^{6} - 298103433 T^{7} + 4471307735 T^{8} - 40990309666 T^{9} + 543114645966 T^{10} - 4621933851892 T^{11} + 55130137152671 T^{12} - 438959927249304 T^{13} + 4906582206587719 T^{14} - 36610338040836532 T^{15} + 382878988852005054 T^{16} - 2571823887728801506 T^{17} + 24968048207013538015 T^{18} -$$$$14\!\cdots\!13$$$$T^{19} +$$$$13\!\cdots\!84$$$$T^{20} -$$$$72\!\cdots\!53$$$$T^{21} +$$$$61\!\cdots\!90$$$$T^{22} -$$$$26\!\cdots\!45$$$$T^{23} +$$$$18\!\cdots\!61$$$$T^{24} -$$$$46\!\cdots\!99$$$$T^{25} +$$$$21\!\cdots\!69$$$$T^{26}$$
$97$ $$1 + 5 T + 977 T^{2} + 3604 T^{3} + 446756 T^{4} + 1122012 T^{5} + 127803015 T^{6} + 191161537 T^{7} + 25826661398 T^{8} + 18132723153 T^{9} + 3947586637000 T^{10} + 801378421739 T^{11} + 475729449369125 T^{12} + 18087734525852 T^{13} + 46145756588805125 T^{14} + 7540169570142251 T^{15} + 3602855738750701000 T^{16} + 1605276943307142993 T^{17} +$$$$22\!\cdots\!86$$$$T^{18} +$$$$15\!\cdots\!73$$$$T^{19} +$$$$10\!\cdots\!95$$$$T^{20} +$$$$87\!\cdots\!32$$$$T^{21} +$$$$33\!\cdots\!52$$$$T^{22} +$$$$26\!\cdots\!96$$$$T^{23} +$$$$69\!\cdots\!81$$$$T^{24} +$$$$34\!\cdots\!05$$$$T^{25} +$$$$67\!\cdots\!77$$$$T^{26}$$
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