Properties

Label 1.40.a.a
Level 1
Weight 40
Character orbit 1.a
Self dual yes
Analytic conductor 9.634
Analytic rank 0
Dimension 3
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 1 \)
Weight: \( k \) \(=\) \( 40 \)
Character orbit: \([\chi]\) \(=\) 1.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(9.63395513897\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
Defining polynomial: \(x^{3} - 175630027 x - 142249227846\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{10}\cdot 3^{5}\cdot 5\cdot 13 \)
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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 182952 - \beta_{1} ) q^{2} + ( 369814284 - 729 \beta_{1} + \beta_{2} ) q^{3} + ( 90692993728 - 278496 \beta_{1} + 168 \beta_{2} ) q^{4} + ( 5808972166830 - 47874308 \beta_{1} - 12636 \beta_{2} ) q^{5} + ( 509909282622432 - 2200469004 \beta_{1} + 392832 \beta_{2} ) q^{6} + ( -5998765906211848 + 32827389462 \beta_{1} - 7335174 \beta_{2} ) q^{7} + ( 85014747935377920 + 136605670144 \beta_{1} + 92207808 \beta_{2} ) q^{8} + ( 2768626634026225797 - 3975102694872 \beta_{1} - 804651624 \beta_{2} ) q^{9} +O(q^{10})\) \( q +(182952 - \beta_{1}) q^{2} +(369814284 - 729 \beta_{1} + \beta_{2}) q^{3} +(90692993728 - 278496 \beta_{1} + 168 \beta_{2}) q^{4} +(5808972166830 - 47874308 \beta_{1} - 12636 \beta_{2}) q^{5} +(509909282622432 - 2200469004 \beta_{1} + 392832 \beta_{2}) q^{6} +(-5998765906211848 + 32827389462 \beta_{1} - 7335174 \beta_{2}) q^{7} +(85014747935377920 + 136605670144 \beta_{1} + 92207808 \beta_{2}) q^{8} +(2768626634026225797 - 3975102694872 \beta_{1} - 804651624 \beta_{2}) q^{9} +(30124360388455890480 + 11868960677202 \beta_{1} + 4626614784 \beta_{2}) q^{10} +(\)\(22\!\cdots\!92\)\( + 134509309731005 \beta_{1} - 11834988165 \beta_{2}) q^{11} +(\)\(12\!\cdots\!32\)\( - 1011157291880064 \beta_{1} - 73870961696 \beta_{2}) q^{12} +(\)\(10\!\cdots\!54\)\( + 160821112991868 \beta_{1} + 1115908456740 \beta_{2}) q^{13} +(-\)\(21\!\cdots\!24\)\( + 22052490480756232 \beta_{1} - 7498139072256 \beta_{2}) q^{14} +(-\)\(57\!\cdots\!40\)\( - 75414140480539566 \beta_{1} + 32630722986078 \beta_{2}) q^{15} +(-\)\(11\!\cdots\!24\)\( - 81236340431935488 \beta_{1} - 90379426346496 \beta_{2}) q^{16} +(\)\(29\!\cdots\!02\)\( + 1013853378421933608 \beta_{1} + 94395618447768 \beta_{2}) q^{17} +(\)\(29\!\cdots\!24\)\( - 1731429806216370309 \beta_{1} + 450271639673856 \beta_{2}) q^{18} +(\)\(30\!\cdots\!60\)\( - 433737587411141229 \beta_{1} - 2625847472007243 \beta_{2}) q^{19} +(-\)\(48\!\cdots\!60\)\( - 10818656431349393984 \beta_{1} + 6203580643521072 \beta_{2}) q^{20} +(-\)\(63\!\cdots\!48\)\( + 88357570761272298192 \beta_{1} - 3151366507609936 \beta_{2}) q^{21} +(-\)\(40\!\cdots\!16\)\( - \)\(18\!\cdots\!12\)\( \beta_{1} - 25797271435098240 \beta_{2}) q^{22} +(\)\(10\!\cdots\!24\)\( + 43986670167470841250 \beta_{1} + 80143699389496974 \beta_{2}) q^{23} +(\)\(55\!\cdots\!80\)\( + 17673851445949744128 \beta_{1} - 66059004049530624 \beta_{2}) q^{24} +(\)\(62\!\cdots\!75\)\( + \)\(16\!\cdots\!80\)\( \beta_{1} - 148686722850746640 \beta_{2}) q^{25} +(\)\(94\!\cdots\!32\)\( - \)\(30\!\cdots\!22\)\( \beta_{1} + 274679063381592576 \beta_{2}) q^{26} +(-\)\(38\!\cdots\!20\)\( - \)\(55\!\cdots\!34\)\( \beta_{1} + 618515661005911962 \beta_{2}) q^{27} +(-\)\(13\!\cdots\!04\)\( + \)\(18\!\cdots\!40\)\( \beta_{1} - 1699460727962082624 \beta_{2}) q^{28} +(-\)\(83\!\cdots\!10\)\( - \)\(16\!\cdots\!96\)\( \beta_{1} - 3905238531403525932 \beta_{2}) q^{29} +(\)\(35\!\cdots\!60\)\( - \)\(75\!\cdots\!96\)\( \beta_{1} + 21491617867246695168 \beta_{2}) q^{30} +(\)\(10\!\cdots\!72\)\( - \)\(11\!\cdots\!60\)\( \beta_{1} - 22561852423742474520 \beta_{2}) q^{31} +(-\)\(18\!\cdots\!88\)\( + \)\(19\!\cdots\!04\)\( \beta_{1} - 61479055048341934080 \beta_{2}) q^{32} +(-\)\(52\!\cdots\!72\)\( + \)\(14\!\cdots\!92\)\( \beta_{1} + \)\(21\!\cdots\!12\)\( \beta_{2}) q^{33} +(-\)\(56\!\cdots\!04\)\( - \)\(36\!\cdots\!42\)\( \beta_{1} - \)\(14\!\cdots\!64\)\( \beta_{2}) q^{34} +(-\)\(39\!\cdots\!20\)\( + \)\(14\!\cdots\!52\)\( \beta_{1} - \)\(38\!\cdots\!16\)\( \beta_{2}) q^{35} +(\)\(62\!\cdots\!16\)\( - \)\(16\!\cdots\!48\)\( \beta_{1} + \)\(85\!\cdots\!84\)\( \beta_{2}) q^{36} +(\)\(25\!\cdots\!02\)\( + \)\(35\!\cdots\!60\)\( \beta_{1} - \)\(25\!\cdots\!28\)\( \beta_{2}) q^{37} +(\)\(82\!\cdots\!80\)\( + \)\(15\!\cdots\!56\)\( \beta_{1} - \)\(63\!\cdots\!08\)\( \beta_{2}) q^{38} +(\)\(74\!\cdots\!64\)\( - \)\(28\!\cdots\!18\)\( \beta_{1} - \)\(46\!\cdots\!06\)\( \beta_{2}) q^{39} +(-\)\(10\!\cdots\!00\)\( - \)\(13\!\cdots\!80\)\( \beta_{1} + \)\(95\!\cdots\!40\)\( \beta_{2}) q^{40} +(\)\(79\!\cdots\!62\)\( - \)\(25\!\cdots\!60\)\( \beta_{1} + \)\(66\!\cdots\!80\)\( \beta_{2}) q^{41} +(-\)\(65\!\cdots\!76\)\( + \)\(77\!\cdots\!80\)\( \beta_{1} - \)\(15\!\cdots\!16\)\( \beta_{2}) q^{42} +(\)\(54\!\cdots\!64\)\( - \)\(77\!\cdots\!91\)\( \beta_{1} + \)\(14\!\cdots\!07\)\( \beta_{2}) q^{43} +(-\)\(15\!\cdots\!24\)\( - \)\(56\!\cdots\!92\)\( \beta_{1} + \)\(31\!\cdots\!36\)\( \beta_{2}) q^{44} +(\)\(19\!\cdots\!10\)\( + \)\(79\!\cdots\!04\)\( \beta_{1} - \)\(33\!\cdots\!32\)\( \beta_{2}) q^{45} +(-\)\(80\!\cdots\!68\)\( - \)\(23\!\cdots\!80\)\( \beta_{1} + \)\(14\!\cdots\!40\)\( \beta_{2}) q^{46} +(\)\(67\!\cdots\!52\)\( + \)\(63\!\cdots\!16\)\( \beta_{1} - \)\(44\!\cdots\!16\)\( \beta_{2}) q^{47} +(-\)\(58\!\cdots\!76\)\( + \)\(11\!\cdots\!40\)\( \beta_{1} + \)\(19\!\cdots\!04\)\( \beta_{2}) q^{48} +(\)\(12\!\cdots\!93\)\( - \)\(11\!\cdots\!20\)\( \beta_{1} + \)\(21\!\cdots\!60\)\( \beta_{2}) q^{49} +(-\)\(87\!\cdots\!00\)\( - \)\(20\!\cdots\!95\)\( \beta_{1} - \)\(31\!\cdots\!40\)\( \beta_{2}) q^{50} +(\)\(26\!\cdots\!92\)\( + \)\(16\!\cdots\!74\)\( \beta_{1} - \)\(41\!\cdots\!42\)\( \beta_{2}) q^{51} +(\)\(12\!\cdots\!92\)\( - \)\(95\!\cdots\!28\)\( \beta_{1} - \)\(34\!\cdots\!64\)\( \beta_{2}) q^{52} +(\)\(19\!\cdots\!34\)\( + \)\(17\!\cdots\!36\)\( \beta_{1} + \)\(61\!\cdots\!56\)\( \beta_{2}) q^{53} +(\)\(26\!\cdots\!60\)\( + \)\(22\!\cdots\!96\)\( \beta_{1} + \)\(10\!\cdots\!32\)\( \beta_{2}) q^{54} +(-\)\(16\!\cdots\!40\)\( - \)\(12\!\cdots\!86\)\( \beta_{1} - \)\(41\!\cdots\!62\)\( \beta_{2}) q^{55} +(-\)\(20\!\cdots\!60\)\( + \)\(65\!\cdots\!04\)\( \beta_{1} + \)\(59\!\cdots\!68\)\( \beta_{2}) q^{56} +(-\)\(15\!\cdots\!40\)\( + \)\(26\!\cdots\!12\)\( \beta_{1} + \)\(66\!\cdots\!84\)\( \beta_{2}) q^{57} +(\)\(85\!\cdots\!20\)\( + \)\(75\!\cdots\!94\)\( \beta_{1} - \)\(77\!\cdots\!92\)\( \beta_{2}) q^{58} +(-\)\(18\!\cdots\!20\)\( + \)\(13\!\cdots\!13\)\( \beta_{1} - \)\(83\!\cdots\!29\)\( \beta_{2}) q^{59} +(\)\(42\!\cdots\!80\)\( - \)\(32\!\cdots\!68\)\( \beta_{1} - \)\(10\!\cdots\!56\)\( \beta_{2}) q^{60} +(\)\(13\!\cdots\!42\)\( - \)\(26\!\cdots\!00\)\( \beta_{1} + \)\(28\!\cdots\!00\)\( \beta_{2}) q^{61} +(\)\(87\!\cdots\!44\)\( - \)\(73\!\cdots\!32\)\( \beta_{1} + \)\(12\!\cdots\!80\)\( \beta_{2}) q^{62} +(-\)\(58\!\cdots\!76\)\( + \)\(10\!\cdots\!42\)\( \beta_{1} - \)\(48\!\cdots\!42\)\( \beta_{2}) q^{63} +(-\)\(56\!\cdots\!92\)\( + \)\(19\!\cdots\!28\)\( \beta_{1} + \)\(55\!\cdots\!76\)\( \beta_{2}) q^{64} +(-\)\(88\!\cdots\!40\)\( - \)\(13\!\cdots\!96\)\( \beta_{1} + \)\(21\!\cdots\!68\)\( \beta_{2}) q^{65} +(-\)\(95\!\cdots\!56\)\( - \)\(30\!\cdots\!08\)\( \beta_{1} + \)\(32\!\cdots\!64\)\( \beta_{2}) q^{66} +(-\)\(46\!\cdots\!48\)\( + \)\(25\!\cdots\!63\)\( \beta_{1} - \)\(11\!\cdots\!47\)\( \beta_{2}) q^{67} +(-\)\(43\!\cdots\!04\)\( + \)\(22\!\cdots\!68\)\( \beta_{1} - \)\(29\!\cdots\!68\)\( \beta_{2}) q^{68} +(\)\(52\!\cdots\!64\)\( - \)\(15\!\cdots\!44\)\( \beta_{1} - \)\(13\!\cdots\!48\)\( \beta_{2}) q^{69} +(-\)\(16\!\cdots\!20\)\( + \)\(10\!\cdots\!12\)\( \beta_{1} - \)\(12\!\cdots\!96\)\( \beta_{2}) q^{70} +(\)\(12\!\cdots\!32\)\( - \)\(72\!\cdots\!50\)\( \beta_{1} + \)\(23\!\cdots\!50\)\( \beta_{2}) q^{71} +(-\)\(56\!\cdots\!60\)\( - \)\(77\!\cdots\!32\)\( \beta_{1} + \)\(26\!\cdots\!76\)\( \beta_{2}) q^{72} +(-\)\(60\!\cdots\!26\)\( + \)\(59\!\cdots\!20\)\( \beta_{1} - \)\(33\!\cdots\!36\)\( \beta_{2}) q^{73} +(-\)\(16\!\cdots\!64\)\( - \)\(17\!\cdots\!30\)\( \beta_{1} - \)\(66\!\cdots\!60\)\( \beta_{2}) q^{74} +(-\)\(14\!\cdots\!00\)\( + \)\(32\!\cdots\!85\)\( \beta_{1} + \)\(47\!\cdots\!95\)\( \beta_{2}) q^{75} +(-\)\(24\!\cdots\!20\)\( + \)\(67\!\cdots\!88\)\( \beta_{1} + \)\(10\!\cdots\!96\)\( \beta_{2}) q^{76} +(\)\(18\!\cdots\!84\)\( + \)\(43\!\cdots\!84\)\( \beta_{1} - \)\(85\!\cdots\!48\)\( \beta_{2}) q^{77} +(\)\(31\!\cdots\!48\)\( - \)\(68\!\cdots\!92\)\( \beta_{1} + \)\(35\!\cdots\!64\)\( \beta_{2}) q^{78} +(\)\(46\!\cdots\!40\)\( - \)\(12\!\cdots\!96\)\( \beta_{1} - \)\(69\!\cdots\!32\)\( \beta_{2}) q^{79} +(\)\(89\!\cdots\!80\)\( + \)\(13\!\cdots\!12\)\( \beta_{1} - \)\(86\!\cdots\!96\)\( \beta_{2}) q^{80} +(-\)\(62\!\cdots\!79\)\( + \)\(61\!\cdots\!16\)\( \beta_{1} - \)\(23\!\cdots\!28\)\( \beta_{2}) q^{81} +(\)\(16\!\cdots\!24\)\( - \)\(12\!\cdots\!22\)\( \beta_{1} + \)\(22\!\cdots\!80\)\( \beta_{2}) q^{82} +(-\)\(69\!\cdots\!56\)\( + \)\(35\!\cdots\!67\)\( \beta_{1} + \)\(95\!\cdots\!53\)\( \beta_{2}) q^{83} +(-\)\(24\!\cdots\!44\)\( + \)\(51\!\cdots\!04\)\( \beta_{1} - \)\(15\!\cdots\!32\)\( \beta_{2}) q^{84} +(-\)\(35\!\cdots\!20\)\( - \)\(41\!\cdots\!48\)\( \beta_{1} - \)\(77\!\cdots\!16\)\( \beta_{2}) q^{85} +(\)\(56\!\cdots\!32\)\( - \)\(64\!\cdots\!76\)\( \beta_{1} + \)\(13\!\cdots\!08\)\( \beta_{2}) q^{86} +(-\)\(24\!\cdots\!60\)\( + \)\(57\!\cdots\!38\)\( \beta_{1} + \)\(47\!\cdots\!66\)\( \beta_{2}) q^{87} +(\)\(23\!\cdots\!40\)\( + \)\(63\!\cdots\!48\)\( \beta_{1} + \)\(23\!\cdots\!36\)\( \beta_{2}) q^{88} +(\)\(68\!\cdots\!70\)\( + \)\(10\!\cdots\!32\)\( \beta_{1} - \)\(40\!\cdots\!56\)\( \beta_{2}) q^{89} +(\)\(31\!\cdots\!60\)\( - \)\(13\!\cdots\!26\)\( \beta_{1} - \)\(10\!\cdots\!92\)\( \beta_{2}) q^{90} +(-\)\(55\!\cdots\!48\)\( + \)\(10\!\cdots\!84\)\( \beta_{1} - \)\(22\!\cdots\!72\)\( \beta_{2}) q^{91} +(\)\(87\!\cdots\!52\)\( - \)\(64\!\cdots\!12\)\( \beta_{1} - \)\(97\!\cdots\!72\)\( \beta_{2}) q^{92} +(-\)\(55\!\cdots\!52\)\( - \)\(25\!\cdots\!08\)\( \beta_{1} + \)\(15\!\cdots\!32\)\( \beta_{2}) q^{93} +(-\)\(37\!\cdots\!44\)\( + \)\(71\!\cdots\!56\)\( \beta_{1} - \)\(11\!\cdots\!48\)\( \beta_{2}) q^{94} +(\)\(24\!\cdots\!00\)\( + \)\(52\!\cdots\!30\)\( \beta_{1} - \)\(12\!\cdots\!90\)\( \beta_{2}) q^{95} +(-\)\(48\!\cdots\!08\)\( + \)\(54\!\cdots\!96\)\( \beta_{1} + \)\(22\!\cdots\!32\)\( \beta_{2}) q^{96} +(\)\(57\!\cdots\!02\)\( + \)\(14\!\cdots\!16\)\( \beta_{1} + \)\(78\!\cdots\!84\)\( \beta_{2}) q^{97} +(\)\(74\!\cdots\!36\)\( - \)\(61\!\cdots\!13\)\( \beta_{1} + \)\(25\!\cdots\!60\)\( \beta_{2}) q^{98} +(\)\(35\!\cdots\!24\)\( - \)\(67\!\cdots\!39\)\( \beta_{1} - \)\(31\!\cdots\!13\)\( \beta_{2}) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 548856q^{2} + 1109442852q^{3} + 272078981184q^{4} + 17426916500490q^{5} + 1529727847867296q^{6} - 17996297718635544q^{7} + 255044243806133760q^{8} + 8305879902078677391q^{9} + O(q^{10}) \) \( 3q + 548856q^{2} + 1109442852q^{3} + 272078981184q^{4} + 17426916500490q^{5} + 1529727847867296q^{6} - 17996297718635544q^{7} + 255044243806133760q^{8} + 8305879902078677391q^{9} + 90373081165367671440q^{10} + \)\(66\!\cdots\!76\)\(q^{11} + \)\(36\!\cdots\!96\)\(q^{12} + \)\(31\!\cdots\!62\)\(q^{13} - \)\(63\!\cdots\!72\)\(q^{14} - \)\(17\!\cdots\!20\)\(q^{15} - \)\(35\!\cdots\!72\)\(q^{16} + \)\(89\!\cdots\!06\)\(q^{17} + \)\(87\!\cdots\!72\)\(q^{18} + \)\(92\!\cdots\!80\)\(q^{19} - \)\(14\!\cdots\!80\)\(q^{20} - \)\(19\!\cdots\!44\)\(q^{21} - \)\(12\!\cdots\!48\)\(q^{22} + \)\(30\!\cdots\!72\)\(q^{23} + \)\(16\!\cdots\!40\)\(q^{24} + \)\(18\!\cdots\!25\)\(q^{25} + \)\(28\!\cdots\!96\)\(q^{26} - \)\(11\!\cdots\!60\)\(q^{27} - \)\(41\!\cdots\!12\)\(q^{28} - \)\(24\!\cdots\!30\)\(q^{29} + \)\(10\!\cdots\!80\)\(q^{30} + \)\(30\!\cdots\!16\)\(q^{31} - \)\(56\!\cdots\!64\)\(q^{32} - \)\(15\!\cdots\!16\)\(q^{33} - \)\(16\!\cdots\!12\)\(q^{34} - \)\(11\!\cdots\!60\)\(q^{35} + \)\(18\!\cdots\!48\)\(q^{36} + \)\(75\!\cdots\!06\)\(q^{37} + \)\(24\!\cdots\!40\)\(q^{38} + \)\(22\!\cdots\!92\)\(q^{39} - \)\(32\!\cdots\!00\)\(q^{40} + \)\(23\!\cdots\!86\)\(q^{41} - \)\(19\!\cdots\!28\)\(q^{42} + \)\(16\!\cdots\!92\)\(q^{43} - \)\(45\!\cdots\!72\)\(q^{44} + \)\(58\!\cdots\!30\)\(q^{45} - \)\(24\!\cdots\!04\)\(q^{46} + \)\(20\!\cdots\!56\)\(q^{47} - \)\(17\!\cdots\!28\)\(q^{48} + \)\(36\!\cdots\!79\)\(q^{49} - \)\(26\!\cdots\!00\)\(q^{50} + \)\(78\!\cdots\!76\)\(q^{51} + \)\(37\!\cdots\!76\)\(q^{52} + \)\(58\!\cdots\!02\)\(q^{53} + \)\(79\!\cdots\!80\)\(q^{54} - \)\(49\!\cdots\!20\)\(q^{55} - \)\(62\!\cdots\!80\)\(q^{56} - \)\(46\!\cdots\!20\)\(q^{57} + \)\(25\!\cdots\!60\)\(q^{58} - \)\(56\!\cdots\!60\)\(q^{59} + \)\(12\!\cdots\!40\)\(q^{60} + \)\(39\!\cdots\!26\)\(q^{61} + \)\(26\!\cdots\!32\)\(q^{62} - \)\(17\!\cdots\!28\)\(q^{63} - \)\(16\!\cdots\!76\)\(q^{64} - \)\(26\!\cdots\!20\)\(q^{65} - \)\(28\!\cdots\!68\)\(q^{66} - \)\(14\!\cdots\!44\)\(q^{67} - \)\(13\!\cdots\!12\)\(q^{68} + \)\(15\!\cdots\!92\)\(q^{69} - \)\(48\!\cdots\!60\)\(q^{70} + \)\(38\!\cdots\!96\)\(q^{71} - \)\(16\!\cdots\!80\)\(q^{72} - \)\(18\!\cdots\!78\)\(q^{73} - \)\(50\!\cdots\!92\)\(q^{74} - \)\(43\!\cdots\!00\)\(q^{75} - \)\(73\!\cdots\!60\)\(q^{76} + \)\(56\!\cdots\!52\)\(q^{77} + \)\(93\!\cdots\!44\)\(q^{78} + \)\(13\!\cdots\!20\)\(q^{79} + \)\(26\!\cdots\!40\)\(q^{80} - \)\(18\!\cdots\!37\)\(q^{81} + \)\(50\!\cdots\!72\)\(q^{82} - \)\(20\!\cdots\!68\)\(q^{83} - \)\(72\!\cdots\!32\)\(q^{84} - \)\(10\!\cdots\!60\)\(q^{85} + \)\(17\!\cdots\!96\)\(q^{86} - \)\(73\!\cdots\!80\)\(q^{87} + \)\(69\!\cdots\!20\)\(q^{88} + \)\(20\!\cdots\!10\)\(q^{89} + \)\(93\!\cdots\!80\)\(q^{90} - \)\(16\!\cdots\!44\)\(q^{91} + \)\(26\!\cdots\!56\)\(q^{92} - \)\(16\!\cdots\!56\)\(q^{93} - \)\(11\!\cdots\!32\)\(q^{94} + \)\(72\!\cdots\!00\)\(q^{95} - \)\(14\!\cdots\!24\)\(q^{96} + \)\(17\!\cdots\!06\)\(q^{97} + \)\(22\!\cdots\!08\)\(q^{98} + \)\(10\!\cdots\!72\)\(q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - 175630027 x - 142249227846\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 72 \nu \)
\(\beta_{2}\)\(=\)\((\)\( 216 \nu^{2} - 262224 \nu - 25290723888 \)\()/7\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)\(/72\)
\(\nu^{2}\)\(=\)\((\)\(7 \beta_{2} + 3642 \beta_{1} + 25290723888\)\()/216\)

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
13640.3
−812.996
−12827.3
−799152. 1.27116e9 8.88877e10 −6.16448e13 −1.01585e15 1.43779e16 3.68304e17 −2.43670e18 4.92635e19
1.2 241488. −3.14962e9 −4.91439e11 5.36221e13 −7.60595e14 1.82084e16 −2.51436e17 5.86757e18 1.29491e19
1.3 1.10652e6 2.98790e9 6.74631e11 2.54496e13 3.30617e15 −5.05826e16 1.38177e17 4.87501e18 2.81604e19
\(n\): e.g. 2-40 or 990-1000
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 1.40.a.a 3
3.b odd 2 1 9.40.a.b 3
4.b odd 2 1 16.40.a.c 3
5.b even 2 1 25.40.a.a 3
5.c odd 4 2 25.40.b.a 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.40.a.a 3 1.a even 1 1 trivial
9.40.a.b 3 3.b odd 2 1
16.40.a.c 3 4.b odd 2 1
25.40.a.a 3 5.b even 2 1
25.40.b.a 6 5.c odd 4 2

Hecke kernels

This newform subspace is the entire newspace \(S_{40}^{\mathrm{new}}(\Gamma_0(1))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 548856 T + 839215684608 T^{2} - 389931393425080320 T^{3} + \)\(46\!\cdots\!04\)\( T^{4} - \)\(16\!\cdots\!64\)\( T^{5} + \)\(16\!\cdots\!72\)\( T^{6} \)
$3$ \( 1 - 1109442852 T + 2541324499416072657 T^{2} + \)\(29\!\cdots\!20\)\( T^{3} + \)\(10\!\cdots\!19\)\( T^{4} - \)\(18\!\cdots\!28\)\( T^{5} + \)\(66\!\cdots\!63\)\( T^{6} \)
$5$ \( 1 - 17426916500490 T + \)\(19\!\cdots\!75\)\( T^{2} + \)\(20\!\cdots\!00\)\( T^{3} + \)\(35\!\cdots\!75\)\( T^{4} - \)\(57\!\cdots\!50\)\( T^{5} + \)\(60\!\cdots\!25\)\( T^{6} \)
$7$ \( 1 + 17996297718635544 T + \)\(13\!\cdots\!93\)\( T^{2} + \)\(45\!\cdots\!00\)\( T^{3} + \)\(12\!\cdots\!99\)\( T^{4} + \)\(14\!\cdots\!56\)\( T^{5} + \)\(75\!\cdots\!07\)\( T^{6} \)
$11$ \( 1 - \)\(66\!\cdots\!76\)\( T + \)\(25\!\cdots\!65\)\( T^{2} - \)\(61\!\cdots\!20\)\( T^{3} + \)\(10\!\cdots\!15\)\( T^{4} - \)\(11\!\cdots\!56\)\( T^{5} + \)\(69\!\cdots\!71\)\( T^{6} \)
$13$ \( 1 - \)\(31\!\cdots\!62\)\( T + \)\(74\!\cdots\!07\)\( T^{2} - \)\(14\!\cdots\!60\)\( T^{3} + \)\(20\!\cdots\!39\)\( T^{4} - \)\(24\!\cdots\!98\)\( T^{5} + \)\(21\!\cdots\!33\)\( T^{6} \)
$17$ \( 1 - \)\(89\!\cdots\!06\)\( T + \)\(21\!\cdots\!63\)\( T^{2} - \)\(17\!\cdots\!60\)\( T^{3} + \)\(20\!\cdots\!39\)\( T^{4} - \)\(84\!\cdots\!54\)\( T^{5} + \)\(91\!\cdots\!77\)\( T^{6} \)
$19$ \( 1 - \)\(92\!\cdots\!80\)\( T + \)\(18\!\cdots\!37\)\( T^{2} - \)\(14\!\cdots\!40\)\( T^{3} + \)\(13\!\cdots\!23\)\( T^{4} - \)\(51\!\cdots\!80\)\( T^{5} + \)\(41\!\cdots\!39\)\( T^{6} \)
$23$ \( 1 - \)\(30\!\cdots\!72\)\( T + \)\(35\!\cdots\!57\)\( T^{2} - \)\(67\!\cdots\!40\)\( T^{3} + \)\(45\!\cdots\!59\)\( T^{4} - \)\(50\!\cdots\!68\)\( T^{5} + \)\(20\!\cdots\!03\)\( T^{6} \)
$29$ \( 1 + \)\(24\!\cdots\!30\)\( T + \)\(30\!\cdots\!07\)\( T^{2} + \)\(45\!\cdots\!40\)\( T^{3} + \)\(33\!\cdots\!83\)\( T^{4} + \)\(29\!\cdots\!30\)\( T^{5} + \)\(12\!\cdots\!09\)\( T^{6} \)
$31$ \( 1 - \)\(30\!\cdots\!16\)\( T + \)\(58\!\cdots\!65\)\( T^{2} - \)\(75\!\cdots\!20\)\( T^{3} + \)\(85\!\cdots\!15\)\( T^{4} - \)\(64\!\cdots\!56\)\( T^{5} + \)\(30\!\cdots\!11\)\( T^{6} \)
$37$ \( 1 - \)\(75\!\cdots\!06\)\( T + \)\(50\!\cdots\!03\)\( T^{2} - \)\(19\!\cdots\!80\)\( T^{3} + \)\(73\!\cdots\!19\)\( T^{4} - \)\(15\!\cdots\!74\)\( T^{5} + \)\(30\!\cdots\!17\)\( T^{6} \)
$41$ \( 1 - \)\(23\!\cdots\!86\)\( T + \)\(19\!\cdots\!15\)\( T^{2} - \)\(49\!\cdots\!20\)\( T^{3} + \)\(15\!\cdots\!15\)\( T^{4} - \)\(14\!\cdots\!06\)\( T^{5} + \)\(49\!\cdots\!81\)\( T^{6} \)
$43$ \( 1 - \)\(16\!\cdots\!92\)\( T + \)\(18\!\cdots\!57\)\( T^{2} - \)\(15\!\cdots\!00\)\( T^{3} + \)\(94\!\cdots\!99\)\( T^{4} - \)\(41\!\cdots\!08\)\( T^{5} + \)\(13\!\cdots\!43\)\( T^{6} \)
$47$ \( 1 - \)\(20\!\cdots\!56\)\( T + \)\(11\!\cdots\!73\)\( T^{2} + \)\(52\!\cdots\!60\)\( T^{3} + \)\(18\!\cdots\!59\)\( T^{4} - \)\(53\!\cdots\!84\)\( T^{5} + \)\(43\!\cdots\!87\)\( T^{6} \)
$53$ \( 1 - \)\(58\!\cdots\!02\)\( T + \)\(58\!\cdots\!07\)\( T^{2} - \)\(20\!\cdots\!80\)\( T^{3} + \)\(10\!\cdots\!19\)\( T^{4} - \)\(18\!\cdots\!78\)\( T^{5} + \)\(54\!\cdots\!13\)\( T^{6} \)
$59$ \( 1 + \)\(56\!\cdots\!60\)\( T + \)\(36\!\cdots\!17\)\( T^{2} + \)\(12\!\cdots\!80\)\( T^{3} + \)\(42\!\cdots\!63\)\( T^{4} + \)\(75\!\cdots\!60\)\( T^{5} + \)\(15\!\cdots\!19\)\( T^{6} \)
$61$ \( 1 - \)\(39\!\cdots\!26\)\( T + \)\(57\!\cdots\!15\)\( T^{2} + \)\(46\!\cdots\!80\)\( T^{3} + \)\(24\!\cdots\!15\)\( T^{4} - \)\(71\!\cdots\!06\)\( T^{5} + \)\(76\!\cdots\!21\)\( T^{6} \)
$67$ \( 1 + \)\(14\!\cdots\!44\)\( T + \)\(49\!\cdots\!13\)\( T^{2} + \)\(46\!\cdots\!40\)\( T^{3} + \)\(82\!\cdots\!39\)\( T^{4} + \)\(38\!\cdots\!96\)\( T^{5} + \)\(44\!\cdots\!27\)\( T^{6} \)
$71$ \( 1 - \)\(38\!\cdots\!96\)\( T + \)\(86\!\cdots\!65\)\( T^{2} - \)\(13\!\cdots\!20\)\( T^{3} + \)\(13\!\cdots\!15\)\( T^{4} - \)\(96\!\cdots\!56\)\( T^{5} + \)\(39\!\cdots\!91\)\( T^{6} \)
$73$ \( 1 + \)\(18\!\cdots\!78\)\( T + \)\(12\!\cdots\!07\)\( T^{2} + \)\(12\!\cdots\!60\)\( T^{3} + \)\(60\!\cdots\!59\)\( T^{4} + \)\(39\!\cdots\!82\)\( T^{5} + \)\(10\!\cdots\!53\)\( T^{6} \)
$79$ \( 1 - \)\(13\!\cdots\!20\)\( T + \)\(22\!\cdots\!57\)\( T^{2} - \)\(18\!\cdots\!60\)\( T^{3} + \)\(23\!\cdots\!83\)\( T^{4} - \)\(14\!\cdots\!20\)\( T^{5} + \)\(10\!\cdots\!59\)\( T^{6} \)
$83$ \( 1 + \)\(20\!\cdots\!68\)\( T + \)\(13\!\cdots\!57\)\( T^{2} + \)\(33\!\cdots\!80\)\( T^{3} + \)\(95\!\cdots\!79\)\( T^{4} + \)\(10\!\cdots\!12\)\( T^{5} + \)\(34\!\cdots\!23\)\( T^{6} \)
$89$ \( 1 - \)\(20\!\cdots\!10\)\( T + \)\(19\!\cdots\!27\)\( T^{2} - \)\(19\!\cdots\!80\)\( T^{3} + \)\(21\!\cdots\!43\)\( T^{4} - \)\(23\!\cdots\!10\)\( T^{5} + \)\(11\!\cdots\!29\)\( T^{6} \)
$97$ \( 1 - \)\(17\!\cdots\!06\)\( T + \)\(84\!\cdots\!23\)\( T^{2} - \)\(99\!\cdots\!40\)\( T^{3} + \)\(25\!\cdots\!59\)\( T^{4} - \)\(16\!\cdots\!34\)\( T^{5} + \)\(28\!\cdots\!37\)\( T^{6} \)
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