Properties

Label 1.48.a.a
Level 1
Weight 48
Character orbit 1.a
Self dual yes
Analytic conductor 13.991
Analytic rank 0
Dimension 4
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 1 \)
Weight: \( k \) \(=\) \( 48 \)
Character orbit: \([\chi]\) \(=\) 1.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(13.9907662655\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Defining polynomial: \(x^{4} - x^{3} - 832803191366 x^{2} + 3710135215485780 x + 13175318942671469337000\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{20}\cdot 3^{7}\cdot 5^{3}\cdot 7^{2} \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1446390 + \beta_{1} ) q^{2} + ( 9615373740 + 2162 \beta_{1} + \beta_{2} ) q^{3} + ( 101201874790288 + 2732288 \beta_{1} + 485 \beta_{2} + \beta_{3} ) q^{4} + ( -7778670060568050 + 214816712 \beta_{1} + 145572 \beta_{2} - 96 \beta_{3} ) q^{5} + ( 532521763270289352 + 55065393132 \beta_{1} + 5052720 \beta_{2} + 4464 \beta_{3} ) q^{6} + ( -9792304681472105800 + 3698310091796 \beta_{1} - 395027494 \beta_{2} - 133760 \beta_{3} ) q^{7} + ( 598179247018446084480 + 185572515537920 \beta_{1} + 8521403448 \beta_{2} + 2897880 \beta_{3} ) q^{8} + ( -4267993104339535550043 + 450437911104816 \beta_{1} - 84024248040 \beta_{2} - 48264768 \beta_{3} ) q^{9} +O(q^{10})\) \( q +(1446390 + \beta_{1}) q^{2} +(9615373740 + 2162 \beta_{1} + \beta_{2}) q^{3} +(101201874790288 + 2732288 \beta_{1} + 485 \beta_{2} + \beta_{3}) q^{4} +(-7778670060568050 + 214816712 \beta_{1} + 145572 \beta_{2} - 96 \beta_{3}) q^{5} +(532521763270289352 + 55065393132 \beta_{1} + 5052720 \beta_{2} + 4464 \beta_{3}) q^{6} +(-9792304681472105800 + 3698310091796 \beta_{1} - 395027494 \beta_{2} - 133760 \beta_{3}) q^{7} +(\)\(59\!\cdots\!80\)\( + 185572515537920 \beta_{1} + 8521403448 \beta_{2} + 2897880 \beta_{3}) q^{8} +(-\)\(42\!\cdots\!43\)\( + 450437911104816 \beta_{1} - 84024248040 \beta_{2} - 48264768 \beta_{3}) q^{9} +(\)\(40\!\cdots\!00\)\( - 20577272733644018 \beta_{1} + 182672051392 \beta_{2} + 641207744 \beta_{3}) q^{10} +(-\)\(47\!\cdots\!28\)\( - 80174485775730170 \beta_{1} + 5305339596315 \beta_{2} - 6954435840 \beta_{3}) q^{11} +(\)\(12\!\cdots\!80\)\( + 1411471507786689536 \beta_{1} - 70344036633956 \beta_{2} + 62452035180 \beta_{3}) q^{12} +(\)\(31\!\cdots\!30\)\( + 2187437718817535432 \beta_{1} + 397156124541860 \beta_{2} - 467536231520 \beta_{3}) q^{13} +(\)\(87\!\cdots\!76\)\( - 48764796509034861256 \beta_{1} - 490874886546720 \beta_{2} + 2916146241888 \beta_{3}) q^{14} +(\)\(31\!\cdots\!00\)\( - 45558691269230379876 \beta_{1} - 9078822530317506 \beta_{2} - 14993052561792 \beta_{3}) q^{15} +(\)\(31\!\cdots\!96\)\( + \)\(13\!\cdots\!24\)\( \beta_{1} + 71092096782308160 \beta_{2} + 61695767581248 \beta_{3}) q^{16} +(\)\(52\!\cdots\!70\)\( - \)\(16\!\cdots\!44\)\( \beta_{1} - 203408765056797672 \beta_{2} - 187599812159040 \beta_{3}) q^{17} +(\)\(10\!\cdots\!90\)\( - \)\(16\!\cdots\!83\)\( \beta_{1} - 371577547005704064 \beta_{2} + 302907998183040 \beta_{3}) q^{18} +(-\)\(26\!\cdots\!60\)\( + \)\(32\!\cdots\!02\)\( \beta_{1} + 5939652197374925685 \beta_{2} + 676931170946304 \beta_{3}) q^{19} +(-\)\(37\!\cdots\!00\)\( + \)\(12\!\cdots\!16\)\( \beta_{1} - 26366915018591276154 \beta_{2} - 7255673126427378 \beta_{3}) q^{20} +(-\)\(65\!\cdots\!28\)\( - \)\(74\!\cdots\!76\)\( \beta_{1} + 61298669387830063280 \beta_{2} + 28908305661771648 \beta_{3}) q^{21} +(-\)\(19\!\cdots\!20\)\( - \)\(17\!\cdots\!88\)\( \beta_{1} - 54181462559863310320 \beta_{2} - 61348773982345520 \beta_{3}) q^{22} +(\)\(34\!\cdots\!20\)\( + \)\(10\!\cdots\!72\)\( \beta_{1} - 66809044933520290386 \beta_{2} - 4835096198025600 \beta_{3}) q^{23} +(\)\(28\!\cdots\!20\)\( + \)\(16\!\cdots\!96\)\( \beta_{1} + 19925905205939828640 \beta_{2} + 561903121129250592 \beta_{3}) q^{24} +(\)\(25\!\cdots\!75\)\( - \)\(21\!\cdots\!00\)\( \beta_{1} + \)\(79\!\cdots\!00\)\( \beta_{2} - 2144521474142940800 \beta_{3}) q^{25} +(\)\(57\!\cdots\!72\)\( - \)\(42\!\cdots\!94\)\( \beta_{1} + \)\(19\!\cdots\!60\)\( \beta_{2} + 3546261034268171712 \beta_{3}) q^{26} +(-\)\(18\!\cdots\!80\)\( - \)\(11\!\cdots\!20\)\( \beta_{1} - \)\(14\!\cdots\!38\)\( \beta_{2} + 3095826871020721920 \beta_{3}) q^{27} +(-\)\(90\!\cdots\!20\)\( + \)\(85\!\cdots\!68\)\( \beta_{1} + \)\(45\!\cdots\!56\)\( \beta_{2} - 33842643361650715080 \beta_{3}) q^{28} +(-\)\(56\!\cdots\!90\)\( - \)\(72\!\cdots\!92\)\( \beta_{1} - \)\(30\!\cdots\!00\)\( \beta_{2} + 85281536978719677216 \beta_{3}) q^{29} +(-\)\(64\!\cdots\!00\)\( - \)\(34\!\cdots\!36\)\( \beta_{1} - \)\(13\!\cdots\!16\)\( \beta_{2} - 52201576900169625312 \beta_{3}) q^{30} +(\)\(18\!\cdots\!12\)\( - \)\(72\!\cdots\!60\)\( \beta_{1} + \)\(29\!\cdots\!20\)\( \beta_{2} - \)\(31\!\cdots\!20\)\( \beta_{3}) q^{31} +(\)\(29\!\cdots\!40\)\( + \)\(22\!\cdots\!96\)\( \beta_{1} + \)\(87\!\cdots\!60\)\( \beta_{2} + \)\(10\!\cdots\!60\)\( \beta_{3}) q^{32} +(\)\(65\!\cdots\!80\)\( - \)\(97\!\cdots\!56\)\( \beta_{1} - \)\(38\!\cdots\!68\)\( \beta_{2} - \)\(12\!\cdots\!40\)\( \beta_{3}) q^{33} +(-\)\(31\!\cdots\!04\)\( + \)\(41\!\cdots\!66\)\( \beta_{1} - \)\(25\!\cdots\!60\)\( \beta_{2} - \)\(19\!\cdots\!68\)\( \beta_{3}) q^{34} +(-\)\(32\!\cdots\!00\)\( - \)\(99\!\cdots\!28\)\( \beta_{1} + \)\(73\!\cdots\!32\)\( \beta_{2} + \)\(89\!\cdots\!24\)\( \beta_{3}) q^{35} +(-\)\(33\!\cdots\!84\)\( + \)\(61\!\cdots\!24\)\( \beta_{1} + \)\(36\!\cdots\!85\)\( \beta_{2} - \)\(11\!\cdots\!27\)\( \beta_{3}) q^{36} +(-\)\(28\!\cdots\!90\)\( + \)\(21\!\cdots\!96\)\( \beta_{1} - \)\(40\!\cdots\!08\)\( \beta_{2} - \)\(78\!\cdots\!60\)\( \beta_{3}) q^{37} +(\)\(74\!\cdots\!20\)\( + \)\(16\!\cdots\!00\)\( \beta_{1} + \)\(43\!\cdots\!92\)\( \beta_{2} + \)\(45\!\cdots\!60\)\( \beta_{3}) q^{38} +(\)\(98\!\cdots\!84\)\( - \)\(11\!\cdots\!36\)\( \beta_{1} + \)\(61\!\cdots\!70\)\( \beta_{2} - \)\(49\!\cdots\!72\)\( \beta_{3}) q^{39} +(\)\(17\!\cdots\!00\)\( - \)\(33\!\cdots\!20\)\( \beta_{1} - \)\(11\!\cdots\!20\)\( \beta_{2} - \)\(23\!\cdots\!40\)\( \beta_{3}) q^{40} +(\)\(32\!\cdots\!82\)\( + \)\(37\!\cdots\!40\)\( \beta_{1} - \)\(17\!\cdots\!80\)\( \beta_{2} + \)\(77\!\cdots\!80\)\( \beta_{3}) q^{41} +(-\)\(27\!\cdots\!80\)\( + \)\(18\!\cdots\!52\)\( \beta_{1} + \)\(36\!\cdots\!04\)\( \beta_{2} + \)\(39\!\cdots\!40\)\( \beta_{3}) q^{42} +(-\)\(11\!\cdots\!00\)\( + \)\(10\!\cdots\!82\)\( \beta_{1} + \)\(59\!\cdots\!47\)\( \beta_{2} - \)\(19\!\cdots\!00\)\( \beta_{3}) q^{43} +(-\)\(38\!\cdots\!64\)\( - \)\(25\!\cdots\!24\)\( \beta_{1} - \)\(21\!\cdots\!60\)\( \beta_{2} - \)\(83\!\cdots\!48\)\( \beta_{3}) q^{44} +(\)\(21\!\cdots\!50\)\( - \)\(17\!\cdots\!96\)\( \beta_{1} + \)\(19\!\cdots\!24\)\( \beta_{2} + \)\(17\!\cdots\!68\)\( \beta_{3}) q^{45} +(\)\(30\!\cdots\!92\)\( + \)\(31\!\cdots\!00\)\( \beta_{1} + \)\(22\!\cdots\!20\)\( \beta_{2} + \)\(92\!\cdots\!00\)\( \beta_{3}) q^{46} +(\)\(50\!\cdots\!80\)\( + \)\(56\!\cdots\!76\)\( \beta_{1} - \)\(58\!\cdots\!16\)\( \beta_{2} - \)\(73\!\cdots\!00\)\( \beta_{3}) q^{47} +(\)\(25\!\cdots\!20\)\( + \)\(21\!\cdots\!52\)\( \beta_{1} + \)\(20\!\cdots\!84\)\( \beta_{2} + \)\(69\!\cdots\!60\)\( \beta_{3}) q^{48} +(\)\(22\!\cdots\!93\)\( - \)\(32\!\cdots\!80\)\( \beta_{1} - \)\(28\!\cdots\!80\)\( \beta_{2} + \)\(89\!\cdots\!40\)\( \beta_{3}) q^{49} +(-\)\(48\!\cdots\!50\)\( - \)\(17\!\cdots\!25\)\( \beta_{1} - \)\(18\!\cdots\!00\)\( \beta_{2} - \)\(17\!\cdots\!00\)\( \beta_{3}) q^{50} +(-\)\(46\!\cdots\!88\)\( - \)\(20\!\cdots\!72\)\( \beta_{1} + \)\(92\!\cdots\!90\)\( \beta_{2} - \)\(10\!\cdots\!44\)\( \beta_{3}) q^{51} +(-\)\(13\!\cdots\!00\)\( + \)\(92\!\cdots\!36\)\( \beta_{1} - \)\(57\!\cdots\!54\)\( \beta_{2} + \)\(20\!\cdots\!50\)\( \beta_{3}) q^{52} +(\)\(72\!\cdots\!90\)\( + \)\(11\!\cdots\!52\)\( \beta_{1} - \)\(27\!\cdots\!24\)\( \beta_{2} + \)\(80\!\cdots\!80\)\( \beta_{3}) q^{53} +(-\)\(29\!\cdots\!60\)\( - \)\(19\!\cdots\!28\)\( \beta_{1} - \)\(96\!\cdots\!40\)\( \beta_{2} - \)\(14\!\cdots\!56\)\( \beta_{3}) q^{54} +(\)\(48\!\cdots\!00\)\( + \)\(46\!\cdots\!64\)\( \beta_{1} + \)\(12\!\cdots\!34\)\( \beta_{2} - \)\(15\!\cdots\!12\)\( \beta_{3}) q^{55} +(\)\(69\!\cdots\!60\)\( - \)\(59\!\cdots\!72\)\( \beta_{1} + \)\(48\!\cdots\!00\)\( \beta_{2} + \)\(58\!\cdots\!56\)\( \beta_{3}) q^{56} +(\)\(13\!\cdots\!40\)\( + \)\(36\!\cdots\!40\)\( \beta_{1} - \)\(83\!\cdots\!36\)\( \beta_{2} - \)\(14\!\cdots\!00\)\( \beta_{3}) q^{57} +(-\)\(18\!\cdots\!20\)\( + \)\(91\!\cdots\!10\)\( \beta_{1} - \)\(28\!\cdots\!32\)\( \beta_{2} - \)\(88\!\cdots\!40\)\( \beta_{3}) q^{58} +(\)\(11\!\cdots\!20\)\( + \)\(91\!\cdots\!66\)\( \beta_{1} + \)\(19\!\cdots\!95\)\( \beta_{2} + \)\(27\!\cdots\!32\)\( \beta_{3}) q^{59} +(-\)\(53\!\cdots\!00\)\( - \)\(16\!\cdots\!68\)\( \beta_{1} + \)\(28\!\cdots\!92\)\( \beta_{2} + \)\(14\!\cdots\!44\)\( \beta_{3}) q^{60} +(\)\(15\!\cdots\!22\)\( - \)\(41\!\cdots\!00\)\( \beta_{1} + \)\(29\!\cdots\!00\)\( \beta_{2} + \)\(82\!\cdots\!00\)\( \beta_{3}) q^{61} +(-\)\(17\!\cdots\!20\)\( - \)\(41\!\cdots\!68\)\( \beta_{1} - \)\(40\!\cdots\!60\)\( \beta_{2} - \)\(62\!\cdots\!60\)\( \beta_{3}) q^{62} +(\)\(14\!\cdots\!60\)\( - \)\(70\!\cdots\!48\)\( \beta_{1} - \)\(63\!\cdots\!62\)\( \beta_{2} + \)\(19\!\cdots\!60\)\( \beta_{3}) q^{63} +(\)\(13\!\cdots\!68\)\( + \)\(35\!\cdots\!96\)\( \beta_{1} + \)\(68\!\cdots\!60\)\( \beta_{2} + \)\(12\!\cdots\!92\)\( \beta_{3}) q^{64} +(\)\(32\!\cdots\!00\)\( - \)\(11\!\cdots\!56\)\( \beta_{1} + \)\(17\!\cdots\!64\)\( \beta_{2} - \)\(11\!\cdots\!52\)\( \beta_{3}) q^{65} +(-\)\(22\!\cdots\!56\)\( - \)\(20\!\cdots\!36\)\( \beta_{1} - \)\(12\!\cdots\!80\)\( \beta_{2} - \)\(94\!\cdots\!72\)\( \beta_{3}) q^{66} +(\)\(46\!\cdots\!20\)\( + \)\(71\!\cdots\!06\)\( \beta_{1} - \)\(31\!\cdots\!87\)\( \beta_{2} + \)\(34\!\cdots\!60\)\( \beta_{3}) q^{67} +(-\)\(68\!\cdots\!60\)\( - \)\(57\!\cdots\!32\)\( \beta_{1} + \)\(10\!\cdots\!02\)\( \beta_{2} + \)\(26\!\cdots\!70\)\( \beta_{3}) q^{68} +(-\)\(52\!\cdots\!76\)\( + \)\(10\!\cdots\!52\)\( \beta_{1} + \)\(45\!\cdots\!20\)\( \beta_{2} + \)\(82\!\cdots\!04\)\( \beta_{3}) q^{69} +(-\)\(24\!\cdots\!00\)\( + \)\(16\!\cdots\!92\)\( \beta_{1} + \)\(28\!\cdots\!52\)\( \beta_{2} - \)\(90\!\cdots\!36\)\( \beta_{3}) q^{70} +(\)\(55\!\cdots\!92\)\( + \)\(41\!\cdots\!00\)\( \beta_{1} - \)\(11\!\cdots\!50\)\( \beta_{2} + \)\(65\!\cdots\!00\)\( \beta_{3}) q^{71} +(-\)\(43\!\cdots\!40\)\( - \)\(29\!\cdots\!00\)\( \beta_{1} + \)\(37\!\cdots\!96\)\( \beta_{2} + \)\(38\!\cdots\!80\)\( \beta_{3}) q^{72} +(\)\(26\!\cdots\!70\)\( + \)\(30\!\cdots\!52\)\( \beta_{1} + \)\(13\!\cdots\!44\)\( \beta_{2} - \)\(55\!\cdots\!40\)\( \beta_{3}) q^{73} +(\)\(46\!\cdots\!36\)\( - \)\(58\!\cdots\!70\)\( \beta_{1} - \)\(10\!\cdots\!20\)\( \beta_{2} + \)\(12\!\cdots\!60\)\( \beta_{3}) q^{74} +(\)\(80\!\cdots\!00\)\( - \)\(22\!\cdots\!50\)\( \beta_{1} + \)\(37\!\cdots\!75\)\( \beta_{2} - \)\(31\!\cdots\!00\)\( \beta_{3}) q^{75} +(\)\(87\!\cdots\!20\)\( + \)\(14\!\cdots\!96\)\( \beta_{1} - \)\(34\!\cdots\!00\)\( \beta_{2} + \)\(12\!\cdots\!92\)\( \beta_{3}) q^{76} +(-\)\(65\!\cdots\!00\)\( - \)\(17\!\cdots\!48\)\( \beta_{1} + \)\(15\!\cdots\!12\)\( \beta_{2} + \)\(21\!\cdots\!60\)\( \beta_{3}) q^{77} +(-\)\(12\!\cdots\!00\)\( + \)\(22\!\cdots\!04\)\( \beta_{1} - \)\(70\!\cdots\!56\)\( \beta_{2} + \)\(77\!\cdots\!20\)\( \beta_{3}) q^{78} +(-\)\(33\!\cdots\!40\)\( - \)\(91\!\cdots\!52\)\( \beta_{1} - \)\(11\!\cdots\!20\)\( \beta_{2} + \)\(34\!\cdots\!96\)\( \beta_{3}) q^{79} +(-\)\(23\!\cdots\!00\)\( - \)\(13\!\cdots\!68\)\( \beta_{1} + \)\(15\!\cdots\!92\)\( \beta_{2} - \)\(25\!\cdots\!56\)\( \beta_{3}) q^{80} +(-\)\(26\!\cdots\!79\)\( - \)\(23\!\cdots\!68\)\( \beta_{1} + \)\(10\!\cdots\!40\)\( \beta_{2} + \)\(18\!\cdots\!64\)\( \beta_{3}) q^{81} +(\)\(95\!\cdots\!80\)\( + \)\(45\!\cdots\!02\)\( \beta_{1} + \)\(15\!\cdots\!40\)\( \beta_{2} + \)\(33\!\cdots\!40\)\( \beta_{3}) q^{82} +(-\)\(35\!\cdots\!40\)\( + \)\(22\!\cdots\!02\)\( \beta_{1} - \)\(65\!\cdots\!27\)\( \beta_{2} - \)\(34\!\cdots\!00\)\( \beta_{3}) q^{83} +(\)\(13\!\cdots\!36\)\( + \)\(88\!\cdots\!48\)\( \beta_{1} - \)\(60\!\cdots\!00\)\( \beta_{2} - \)\(14\!\cdots\!04\)\( \beta_{3}) q^{84} +(-\)\(25\!\cdots\!00\)\( + \)\(16\!\cdots\!52\)\( \beta_{1} + \)\(16\!\cdots\!12\)\( \beta_{2} - \)\(46\!\cdots\!16\)\( \beta_{3}) q^{85} +(\)\(23\!\cdots\!32\)\( - \)\(76\!\cdots\!12\)\( \beta_{1} + \)\(74\!\cdots\!20\)\( \beta_{2} + \)\(12\!\cdots\!76\)\( \beta_{3}) q^{86} +(-\)\(10\!\cdots\!40\)\( + \)\(16\!\cdots\!80\)\( \beta_{1} - \)\(16\!\cdots\!94\)\( \beta_{2} + \)\(52\!\cdots\!40\)\( \beta_{3}) q^{87} +(-\)\(38\!\cdots\!40\)\( - \)\(42\!\cdots\!60\)\( \beta_{1} - \)\(17\!\cdots\!44\)\( \beta_{2} - \)\(21\!\cdots\!40\)\( \beta_{3}) q^{88} +(-\)\(19\!\cdots\!70\)\( + \)\(45\!\cdots\!04\)\( \beta_{1} + \)\(18\!\cdots\!00\)\( \beta_{2} + \)\(79\!\cdots\!08\)\( \beta_{3}) q^{89} +(-\)\(42\!\cdots\!00\)\( + \)\(42\!\cdots\!94\)\( \beta_{1} + \)\(82\!\cdots\!64\)\( \beta_{2} - \)\(14\!\cdots\!52\)\( \beta_{3}) q^{90} +(\)\(13\!\cdots\!92\)\( - \)\(29\!\cdots\!92\)\( \beta_{1} - \)\(11\!\cdots\!00\)\( \beta_{2} + \)\(30\!\cdots\!16\)\( \beta_{3}) q^{91} +(\)\(32\!\cdots\!20\)\( + \)\(39\!\cdots\!96\)\( \beta_{1} + \)\(30\!\cdots\!08\)\( \beta_{2} + \)\(32\!\cdots\!40\)\( \beta_{3}) q^{92} +(\)\(26\!\cdots\!80\)\( - \)\(52\!\cdots\!16\)\( \beta_{1} + \)\(46\!\cdots\!92\)\( \beta_{2} - \)\(74\!\cdots\!20\)\( \beta_{3}) q^{93} +(\)\(86\!\cdots\!56\)\( - \)\(12\!\cdots\!28\)\( \beta_{1} - \)\(61\!\cdots\!40\)\( \beta_{2} - \)\(58\!\cdots\!56\)\( \beta_{3}) q^{94} +(\)\(18\!\cdots\!00\)\( - \)\(64\!\cdots\!80\)\( \beta_{1} - \)\(12\!\cdots\!30\)\( \beta_{2} - \)\(37\!\cdots\!60\)\( \beta_{3}) q^{95} +(\)\(16\!\cdots\!32\)\( + \)\(28\!\cdots\!32\)\( \beta_{1} + \)\(22\!\cdots\!80\)\( \beta_{2} + \)\(18\!\cdots\!64\)\( \beta_{3}) q^{96} +(\)\(23\!\cdots\!30\)\( - \)\(49\!\cdots\!64\)\( \beta_{1} + \)\(13\!\cdots\!64\)\( \beta_{2} + \)\(75\!\cdots\!20\)\( \beta_{3}) q^{97} +(-\)\(74\!\cdots\!30\)\( + \)\(24\!\cdots\!13\)\( \beta_{1} - \)\(22\!\cdots\!80\)\( \beta_{2} - \)\(39\!\cdots\!60\)\( \beta_{3}) q^{98} +(\)\(13\!\cdots\!04\)\( + \)\(56\!\cdots\!62\)\( \beta_{1} + \)\(88\!\cdots\!75\)\( \beta_{2} + \)\(68\!\cdots\!24\)\( \beta_{3}) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 5785560q^{2} + 38461494960q^{3} + 404807499161152q^{4} - 31114680242272200q^{5} + 2130087053081157408q^{6} - 39169218725888423200q^{7} + 2392716988073784337920q^{8} - 17071972417358142200172q^{9} + O(q^{10}) \) \( 4q + 5785560q^{2} + 38461494960q^{3} + 404807499161152q^{4} - 31114680242272200q^{5} + 2130087053081157408q^{6} - 39169218725888423200q^{7} + \)\(23\!\cdots\!20\)\(q^{8} - \)\(17\!\cdots\!72\)\(q^{9} + \)\(16\!\cdots\!00\)\(q^{10} - \)\(19\!\cdots\!12\)\(q^{11} + \)\(50\!\cdots\!20\)\(q^{12} + \)\(12\!\cdots\!20\)\(q^{13} + \)\(34\!\cdots\!04\)\(q^{14} + \)\(12\!\cdots\!00\)\(q^{15} + \)\(12\!\cdots\!84\)\(q^{16} + \)\(21\!\cdots\!80\)\(q^{17} + \)\(40\!\cdots\!60\)\(q^{18} - \)\(10\!\cdots\!40\)\(q^{19} - \)\(15\!\cdots\!00\)\(q^{20} - \)\(26\!\cdots\!12\)\(q^{21} - \)\(79\!\cdots\!80\)\(q^{22} + \)\(13\!\cdots\!80\)\(q^{23} + \)\(11\!\cdots\!80\)\(q^{24} + \)\(10\!\cdots\!00\)\(q^{25} + \)\(22\!\cdots\!88\)\(q^{26} - \)\(73\!\cdots\!20\)\(q^{27} - \)\(36\!\cdots\!80\)\(q^{28} - \)\(22\!\cdots\!60\)\(q^{29} - \)\(25\!\cdots\!00\)\(q^{30} + \)\(75\!\cdots\!48\)\(q^{31} + \)\(11\!\cdots\!60\)\(q^{32} + \)\(26\!\cdots\!20\)\(q^{33} - \)\(12\!\cdots\!16\)\(q^{34} - \)\(13\!\cdots\!00\)\(q^{35} - \)\(13\!\cdots\!36\)\(q^{36} - \)\(11\!\cdots\!60\)\(q^{37} + \)\(29\!\cdots\!80\)\(q^{38} + \)\(39\!\cdots\!36\)\(q^{39} + \)\(70\!\cdots\!00\)\(q^{40} + \)\(13\!\cdots\!28\)\(q^{41} - \)\(10\!\cdots\!20\)\(q^{42} - \)\(44\!\cdots\!00\)\(q^{43} - \)\(15\!\cdots\!56\)\(q^{44} + \)\(86\!\cdots\!00\)\(q^{45} + \)\(12\!\cdots\!68\)\(q^{46} + \)\(20\!\cdots\!20\)\(q^{47} + \)\(10\!\cdots\!80\)\(q^{48} + \)\(91\!\cdots\!72\)\(q^{49} - \)\(19\!\cdots\!00\)\(q^{50} - \)\(18\!\cdots\!52\)\(q^{51} - \)\(55\!\cdots\!00\)\(q^{52} + \)\(29\!\cdots\!60\)\(q^{53} - \)\(11\!\cdots\!40\)\(q^{54} + \)\(19\!\cdots\!00\)\(q^{55} + \)\(27\!\cdots\!40\)\(q^{56} + \)\(55\!\cdots\!60\)\(q^{57} - \)\(73\!\cdots\!80\)\(q^{58} + \)\(47\!\cdots\!80\)\(q^{59} - \)\(21\!\cdots\!00\)\(q^{60} + \)\(62\!\cdots\!88\)\(q^{61} - \)\(68\!\cdots\!80\)\(q^{62} + \)\(58\!\cdots\!40\)\(q^{63} + \)\(55\!\cdots\!72\)\(q^{64} + \)\(12\!\cdots\!00\)\(q^{65} - \)\(89\!\cdots\!24\)\(q^{66} + \)\(18\!\cdots\!80\)\(q^{67} - \)\(27\!\cdots\!40\)\(q^{68} - \)\(20\!\cdots\!04\)\(q^{69} - \)\(97\!\cdots\!00\)\(q^{70} + \)\(22\!\cdots\!68\)\(q^{71} - \)\(17\!\cdots\!60\)\(q^{72} + \)\(10\!\cdots\!80\)\(q^{73} + \)\(18\!\cdots\!44\)\(q^{74} + \)\(32\!\cdots\!00\)\(q^{75} + \)\(35\!\cdots\!80\)\(q^{76} - \)\(26\!\cdots\!00\)\(q^{77} - \)\(49\!\cdots\!00\)\(q^{78} - \)\(13\!\cdots\!60\)\(q^{79} - \)\(94\!\cdots\!00\)\(q^{80} - \)\(10\!\cdots\!16\)\(q^{81} + \)\(38\!\cdots\!20\)\(q^{82} - \)\(14\!\cdots\!60\)\(q^{83} + \)\(52\!\cdots\!44\)\(q^{84} - \)\(10\!\cdots\!00\)\(q^{85} + \)\(95\!\cdots\!28\)\(q^{86} - \)\(43\!\cdots\!60\)\(q^{87} - \)\(15\!\cdots\!60\)\(q^{88} - \)\(79\!\cdots\!80\)\(q^{89} - \)\(16\!\cdots\!00\)\(q^{90} + \)\(53\!\cdots\!68\)\(q^{91} + \)\(13\!\cdots\!80\)\(q^{92} + \)\(10\!\cdots\!20\)\(q^{93} + \)\(34\!\cdots\!24\)\(q^{94} + \)\(75\!\cdots\!00\)\(q^{95} + \)\(64\!\cdots\!28\)\(q^{96} + \)\(95\!\cdots\!20\)\(q^{97} - \)\(29\!\cdots\!20\)\(q^{98} + \)\(53\!\cdots\!16\)\(q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - 832803191366 x^{2} + 3710135215485780 x + 13175318942671469337000\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 24 \nu - 6 \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} + 60053 \nu^{2} - 800757750008 \nu - 22223988026907900 \)\()/514752\)
\(\beta_{3}\)\(=\)\((\)\( -485 \nu^{3} + 267371447 \nu^{2} + 390350086376920 \nu - 112683253510933193364 \)\()/514752\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1} + 6\)\()/24\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} + 485 \beta_{2} - 160480 \beta_{1} + 239847319113552\)\()/576\)
\(\nu^{3}\)\(=\)\((\)\(-60053 \beta_{3} + 267371447 \beta_{2} + 19227823305632 \beta_{1} - 1602418642111186704\)\()/576\)

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
−906006.
−124721.
129356.
901372.
−2.02978e7 −2.01642e10 2.71261e14 −3.11682e16 4.09288e17 −1.26592e20 −2.64934e21 −2.61822e22 6.32644e23
1.2 −1.54692e6 1.52034e11 −1.38345e14 4.23962e16 −2.35185e17 −3.90714e19 4.31719e20 −3.47448e21 −6.55837e22
1.3 4.55092e6 −2.21918e11 −1.20027e14 −3.08341e16 −1.00993e18 1.11073e20 −1.18672e21 2.26588e22 −1.40324e23
1.4 2.30793e7 1.28510e11 3.91917e14 −1.15086e16 2.96592e18 1.54211e19 5.79706e21 −1.00741e22 −2.65611e23
\(n\): 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 1.48.a.a 4
3.b odd 2 1 9.48.a.c 4
4.b odd 2 1 16.48.a.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.48.a.a 4 1.a even 1 1 trivial
9.48.a.c 4 3.b odd 2 1
16.48.a.d 4 4.b odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{48}^{\mathrm{new}}(\Gamma_0(1))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 5785560 T + 95807579386880 T^{2} - \)\(10\!\cdots\!80\)\( T^{3} - \)\(93\!\cdots\!32\)\( T^{4} - \)\(14\!\cdots\!40\)\( T^{5} + \)\(18\!\cdots\!20\)\( T^{6} - \)\(16\!\cdots\!20\)\( T^{7} + \)\(39\!\cdots\!56\)\( T^{8} \)
$3$ \( 1 - 38461494960 T + \)\(62\!\cdots\!60\)\( T^{2} + \)\(40\!\cdots\!80\)\( T^{3} + \)\(19\!\cdots\!38\)\( T^{4} + \)\(10\!\cdots\!60\)\( T^{5} + \)\(44\!\cdots\!40\)\( T^{6} - \)\(72\!\cdots\!80\)\( T^{7} + \)\(49\!\cdots\!61\)\( T^{8} \)
$5$ \( 1 + 31114680242272200 T + \)\(14\!\cdots\!00\)\( T^{2} + \)\(63\!\cdots\!00\)\( T^{3} + \)\(51\!\cdots\!50\)\( T^{4} + \)\(45\!\cdots\!00\)\( T^{5} + \)\(70\!\cdots\!00\)\( T^{6} + \)\(11\!\cdots\!00\)\( T^{7} + \)\(25\!\cdots\!25\)\( T^{8} \)
$7$ \( 1 + 39169218725888423200 T + \)\(66\!\cdots\!00\)\( T^{2} + \)\(27\!\cdots\!00\)\( T^{3} + \)\(23\!\cdots\!98\)\( T^{4} + \)\(14\!\cdots\!00\)\( T^{5} + \)\(18\!\cdots\!00\)\( T^{6} + \)\(56\!\cdots\!00\)\( T^{7} + \)\(75\!\cdots\!01\)\( T^{8} \)
$11$ \( 1 + \)\(19\!\cdots\!12\)\( T + \)\(27\!\cdots\!88\)\( T^{2} + \)\(38\!\cdots\!64\)\( T^{3} + \)\(33\!\cdots\!70\)\( T^{4} + \)\(34\!\cdots\!44\)\( T^{5} + \)\(21\!\cdots\!08\)\( T^{6} + \)\(13\!\cdots\!32\)\( T^{7} + \)\(60\!\cdots\!81\)\( T^{8} \)
$13$ \( 1 - \)\(12\!\cdots\!20\)\( T + \)\(66\!\cdots\!20\)\( T^{2} - \)\(82\!\cdots\!40\)\( T^{3} + \)\(19\!\cdots\!78\)\( T^{4} - \)\(18\!\cdots\!80\)\( T^{5} + \)\(34\!\cdots\!80\)\( T^{6} - \)\(14\!\cdots\!60\)\( T^{7} + \)\(26\!\cdots\!21\)\( T^{8} \)
$17$ \( 1 - \)\(21\!\cdots\!80\)\( T + \)\(37\!\cdots\!40\)\( T^{2} - \)\(40\!\cdots\!40\)\( T^{3} + \)\(39\!\cdots\!58\)\( T^{4} - \)\(27\!\cdots\!20\)\( T^{5} + \)\(17\!\cdots\!60\)\( T^{6} - \)\(65\!\cdots\!60\)\( T^{7} + \)\(21\!\cdots\!41\)\( T^{8} \)
$19$ \( 1 + \)\(10\!\cdots\!40\)\( T + \)\(34\!\cdots\!56\)\( T^{2} + \)\(31\!\cdots\!80\)\( T^{3} + \)\(58\!\cdots\!26\)\( T^{4} + \)\(40\!\cdots\!20\)\( T^{5} + \)\(54\!\cdots\!76\)\( T^{6} + \)\(21\!\cdots\!60\)\( T^{7} + \)\(25\!\cdots\!41\)\( T^{8} \)
$23$ \( 1 - \)\(13\!\cdots\!80\)\( T + \)\(46\!\cdots\!80\)\( T^{2} - \)\(42\!\cdots\!60\)\( T^{3} + \)\(73\!\cdots\!18\)\( T^{4} - \)\(42\!\cdots\!20\)\( T^{5} + \)\(46\!\cdots\!20\)\( T^{6} - \)\(13\!\cdots\!40\)\( T^{7} + \)\(10\!\cdots\!81\)\( T^{8} \)
$29$ \( 1 + \)\(22\!\cdots\!60\)\( T + \)\(13\!\cdots\!36\)\( T^{2} + \)\(20\!\cdots\!20\)\( T^{3} + \)\(81\!\cdots\!86\)\( T^{4} + \)\(11\!\cdots\!80\)\( T^{5} + \)\(39\!\cdots\!16\)\( T^{6} + \)\(35\!\cdots\!40\)\( T^{7} + \)\(85\!\cdots\!61\)\( T^{8} \)
$31$ \( 1 - \)\(75\!\cdots\!48\)\( T + \)\(12\!\cdots\!08\)\( T^{2} - \)\(40\!\cdots\!96\)\( T^{3} + \)\(15\!\cdots\!70\)\( T^{4} - \)\(50\!\cdots\!56\)\( T^{5} + \)\(19\!\cdots\!68\)\( T^{6} - \)\(14\!\cdots\!88\)\( T^{7} + \)\(23\!\cdots\!41\)\( T^{8} \)
$37$ \( 1 + \)\(11\!\cdots\!60\)\( T + \)\(15\!\cdots\!20\)\( T^{2} + \)\(93\!\cdots\!80\)\( T^{3} + \)\(88\!\cdots\!78\)\( T^{4} + \)\(47\!\cdots\!40\)\( T^{5} + \)\(39\!\cdots\!80\)\( T^{6} + \)\(14\!\cdots\!20\)\( T^{7} + \)\(66\!\cdots\!21\)\( T^{8} \)
$41$ \( 1 - \)\(13\!\cdots\!28\)\( T + \)\(22\!\cdots\!68\)\( T^{2} - \)\(20\!\cdots\!76\)\( T^{3} + \)\(21\!\cdots\!70\)\( T^{4} - \)\(13\!\cdots\!56\)\( T^{5} + \)\(91\!\cdots\!48\)\( T^{6} - \)\(33\!\cdots\!48\)\( T^{7} + \)\(15\!\cdots\!21\)\( T^{8} \)
$43$ \( 1 + \)\(44\!\cdots\!00\)\( T + \)\(24\!\cdots\!00\)\( T^{2} + \)\(66\!\cdots\!00\)\( T^{3} + \)\(21\!\cdots\!98\)\( T^{4} + \)\(39\!\cdots\!00\)\( T^{5} + \)\(85\!\cdots\!00\)\( T^{6} + \)\(92\!\cdots\!00\)\( T^{7} + \)\(12\!\cdots\!01\)\( T^{8} \)
$47$ \( 1 - \)\(20\!\cdots\!20\)\( T + \)\(10\!\cdots\!60\)\( T^{2} - \)\(18\!\cdots\!60\)\( T^{3} + \)\(59\!\cdots\!38\)\( T^{4} - \)\(70\!\cdots\!80\)\( T^{5} + \)\(15\!\cdots\!40\)\( T^{6} - \)\(11\!\cdots\!40\)\( T^{7} + \)\(22\!\cdots\!61\)\( T^{8} \)
$53$ \( 1 - \)\(29\!\cdots\!60\)\( T + \)\(33\!\cdots\!60\)\( T^{2} - \)\(87\!\cdots\!20\)\( T^{3} + \)\(50\!\cdots\!38\)\( T^{4} - \)\(96\!\cdots\!40\)\( T^{5} + \)\(40\!\cdots\!40\)\( T^{6} - \)\(38\!\cdots\!80\)\( T^{7} + \)\(14\!\cdots\!61\)\( T^{8} \)
$59$ \( 1 - \)\(47\!\cdots\!80\)\( T + \)\(71\!\cdots\!76\)\( T^{2} - \)\(24\!\cdots\!60\)\( T^{3} + \)\(18\!\cdots\!66\)\( T^{4} - \)\(41\!\cdots\!40\)\( T^{5} + \)\(20\!\cdots\!36\)\( T^{6} - \)\(23\!\cdots\!20\)\( T^{7} + \)\(83\!\cdots\!21\)\( T^{8} \)
$61$ \( 1 - \)\(62\!\cdots\!88\)\( T + \)\(29\!\cdots\!88\)\( T^{2} - \)\(12\!\cdots\!36\)\( T^{3} + \)\(34\!\cdots\!70\)\( T^{4} - \)\(10\!\cdots\!56\)\( T^{5} + \)\(19\!\cdots\!08\)\( T^{6} - \)\(33\!\cdots\!68\)\( T^{7} + \)\(43\!\cdots\!81\)\( T^{8} \)
$67$ \( 1 - \)\(18\!\cdots\!80\)\( T + \)\(35\!\cdots\!40\)\( T^{2} - \)\(38\!\cdots\!40\)\( T^{3} + \)\(38\!\cdots\!58\)\( T^{4} - \)\(25\!\cdots\!20\)\( T^{5} + \)\(15\!\cdots\!60\)\( T^{6} - \)\(56\!\cdots\!60\)\( T^{7} + \)\(20\!\cdots\!41\)\( T^{8} \)
$71$ \( 1 - \)\(22\!\cdots\!68\)\( T + \)\(32\!\cdots\!48\)\( T^{2} - \)\(42\!\cdots\!16\)\( T^{3} + \)\(43\!\cdots\!70\)\( T^{4} - \)\(43\!\cdots\!56\)\( T^{5} + \)\(33\!\cdots\!88\)\( T^{6} - \)\(23\!\cdots\!28\)\( T^{7} + \)\(10\!\cdots\!61\)\( T^{8} \)
$73$ \( 1 - \)\(10\!\cdots\!80\)\( T + \)\(14\!\cdots\!80\)\( T^{2} - \)\(90\!\cdots\!60\)\( T^{3} + \)\(74\!\cdots\!18\)\( T^{4} - \)\(34\!\cdots\!20\)\( T^{5} + \)\(20\!\cdots\!20\)\( T^{6} - \)\(56\!\cdots\!40\)\( T^{7} + \)\(20\!\cdots\!81\)\( T^{8} \)
$79$ \( 1 + \)\(13\!\cdots\!60\)\( T + \)\(11\!\cdots\!36\)\( T^{2} + \)\(68\!\cdots\!20\)\( T^{3} + \)\(31\!\cdots\!86\)\( T^{4} + \)\(10\!\cdots\!80\)\( T^{5} + \)\(27\!\cdots\!16\)\( T^{6} + \)\(48\!\cdots\!40\)\( T^{7} + \)\(56\!\cdots\!61\)\( T^{8} \)
$83$ \( 1 + \)\(14\!\cdots\!60\)\( T + \)\(39\!\cdots\!40\)\( T^{2} + \)\(29\!\cdots\!20\)\( T^{3} + \)\(63\!\cdots\!58\)\( T^{4} + \)\(45\!\cdots\!40\)\( T^{5} + \)\(96\!\cdots\!60\)\( T^{6} + \)\(55\!\cdots\!80\)\( T^{7} + \)\(61\!\cdots\!41\)\( T^{8} \)
$89$ \( 1 + \)\(79\!\cdots\!80\)\( T + \)\(72\!\cdots\!16\)\( T^{2} + \)\(58\!\cdots\!60\)\( T^{3} + \)\(76\!\cdots\!46\)\( T^{4} + \)\(24\!\cdots\!40\)\( T^{5} + \)\(12\!\cdots\!56\)\( T^{6} + \)\(58\!\cdots\!20\)\( T^{7} + \)\(30\!\cdots\!81\)\( T^{8} \)
$97$ \( 1 - \)\(95\!\cdots\!20\)\( T + \)\(11\!\cdots\!60\)\( T^{2} - \)\(65\!\cdots\!60\)\( T^{3} + \)\(43\!\cdots\!38\)\( T^{4} - \)\(15\!\cdots\!80\)\( T^{5} + \)\(65\!\cdots\!40\)\( T^{6} - \)\(12\!\cdots\!40\)\( T^{7} + \)\(32\!\cdots\!61\)\( T^{8} \)
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