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: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{20}\cdot 3^{7}\cdot 5^{3}\cdot 7^{2} \)
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.

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

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

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

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.

Hecke kernels

There are no other newforms in \(S_{48}^{\mathrm{new}}(\Gamma_0(1))\).