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: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{10}\cdot 3^{5}\cdot 5\cdot 13 \)
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.

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

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

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

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.

Hecke kernels

There are no other newforms in \(S_{40}^{\mathrm{new}}(\Gamma_0(1))\).