Properties

Label 1.72.a.a
Level 1
Weight 72
Character orbit 1.a
Self dual yes
Analytic conductor 31.925
Analytic rank 0
Dimension 6
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 1 \)
Weight: \( k \) \(=\) \( 72 \)
Character orbit: \([\chi]\) \(=\) 1.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(31.9246160561\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Defining polynomial: \(x^{6} - 3 x^{5} - 18795872470415627868 x^{4} - 3619866001951210071877967584 x^{3} + 86589594972295987044342709087705454976 x^{2} + 35564509575256206574563332087691180650482000128 x - 11608057594368249332625196663730471558406093241461869568\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{55}\cdot 3^{20}\cdot 5^{6}\cdot 7^{3} \)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q +(11026222740 - \beta_{1}) q^{2} +(14982825462961740 + 223315 \beta_{1} + \beta_{2}) q^{3} +(\)\(13\!\cdots\!28\)\( - 15119266233 \beta_{1} + 6859 \beta_{2} + \beta_{3}) q^{4} +(-\)\(71\!\cdots\!70\)\( + 9986066292712 \beta_{1} + 19552951 \beta_{2} - 40 \beta_{3} + \beta_{4}) q^{5} +(-\)\(64\!\cdots\!48\)\( - 36755474236375553 \beta_{1} + 23852632969 \beta_{2} - 37549 \beta_{3} + 569 \beta_{4} + \beta_{5}) q^{6} +(\)\(56\!\cdots\!00\)\( + 6085020597208821870 \beta_{1} + 704467142494 \beta_{2} + 68764912 \beta_{3} + 104364 \beta_{4} - 144 \beta_{5}) q^{7} +(\)\(43\!\cdots\!80\)\( - \)\(93\!\cdots\!76\)\( \beta_{1} + 373662647287184 \beta_{2} + 33223483504 \beta_{3} - 85912 \beta_{4} + 10152 \beta_{5}) q^{8} +(\)\(48\!\cdots\!57\)\( - \)\(30\!\cdots\!44\)\( \beta_{1} + 12565567900962762 \beta_{2} + 3101985626448 \beta_{3} - 549102138 \beta_{4} - 466752 \beta_{5}) q^{9} +O(q^{10})\) \( q +(11026222740 - \beta_{1}) q^{2} +(14982825462961740 + 223315 \beta_{1} + \beta_{2}) q^{3} +(\)\(13\!\cdots\!28\)\( - 15119266233 \beta_{1} + 6859 \beta_{2} + \beta_{3}) q^{4} +(-\)\(71\!\cdots\!70\)\( + 9986066292712 \beta_{1} + 19552951 \beta_{2} - 40 \beta_{3} + \beta_{4}) q^{5} +(-\)\(64\!\cdots\!48\)\( - 36755474236375553 \beta_{1} + 23852632969 \beta_{2} - 37549 \beta_{3} + 569 \beta_{4} + \beta_{5}) q^{6} +(\)\(56\!\cdots\!00\)\( + 6085020597208821870 \beta_{1} + 704467142494 \beta_{2} + 68764912 \beta_{3} + 104364 \beta_{4} - 144 \beta_{5}) q^{7} +(\)\(43\!\cdots\!80\)\( - \)\(93\!\cdots\!76\)\( \beta_{1} + 373662647287184 \beta_{2} + 33223483504 \beta_{3} - 85912 \beta_{4} + 10152 \beta_{5}) q^{8} +(\)\(48\!\cdots\!57\)\( - \)\(30\!\cdots\!44\)\( \beta_{1} + 12565567900962762 \beta_{2} + 3101985626448 \beta_{3} - 549102138 \beta_{4} - 466752 \beta_{5}) q^{9} +(-\)\(43\!\cdots\!20\)\( + \)\(37\!\cdots\!22\)\( \beta_{1} + 2030985545242499396 \beta_{2} - 28842510271060 \beta_{3} + 32895007236 \beta_{4} + 15727140 \beta_{5}) q^{10} +(-\)\(12\!\cdots\!88\)\( + \)\(64\!\cdots\!57\)\( \beta_{1} + 48683821575851961539 \beta_{2} - 2448158380737824 \beta_{3} - 1029826965928 \beta_{4} - 413754912 \beta_{5}) q^{11} +(\)\(90\!\cdots\!80\)\( - \)\(51\!\cdots\!80\)\( \beta_{1} + \)\(31\!\cdots\!60\)\( \beta_{2} + 52097436999376524 \beta_{3} + 21112675878528 \beta_{4} + 8840886912 \beta_{5}) q^{12} +(\)\(32\!\cdots\!30\)\( + \)\(21\!\cdots\!16\)\( \beta_{1} - \)\(13\!\cdots\!09\)\( \beta_{2} - 35883420918929576 \beta_{3} - 304264002590847 \beta_{4} - 157563209088 \beta_{5}) q^{13} +(-\)\(21\!\cdots\!44\)\( - \)\(16\!\cdots\!34\)\( \beta_{1} - \)\(39\!\cdots\!18\)\( \beta_{2} - 10523246779171378802 \beta_{3} + 3088202886856490 \beta_{4} + 2386586841210 \beta_{5}) q^{14} +(\)\(23\!\cdots\!60\)\( - \)\(59\!\cdots\!06\)\( \beta_{1} - \)\(20\!\cdots\!58\)\( \beta_{2} + \)\(14\!\cdots\!80\)\( \beta_{3} - 19837023582415628 \beta_{4} - 31143253089520 \beta_{5}) q^{15} +(\)\(61\!\cdots\!36\)\( - \)\(76\!\cdots\!00\)\( \beta_{1} + \)\(40\!\cdots\!00\)\( \beta_{2} - \)\(55\!\cdots\!40\)\( \beta_{3} + 28441592013392064 \beta_{4} + 353579854769856 \beta_{5}) q^{16} +(\)\(53\!\cdots\!70\)\( - \)\(66\!\cdots\!44\)\( \beta_{1} + \)\(11\!\cdots\!70\)\( \beta_{2} - \)\(57\!\cdots\!72\)\( \beta_{3} + 1061975335613560966 \beta_{4} - 3516764067625536 \beta_{5}) q^{17} +(\)\(16\!\cdots\!40\)\( - \)\(10\!\cdots\!41\)\( \beta_{1} - \)\(18\!\cdots\!16\)\( \beta_{2} + \)\(85\!\cdots\!52\)\( \beta_{3} - 14799504281961238056 \beta_{4} + 30776805316993176 \beta_{5}) q^{18} +(-\)\(35\!\cdots\!80\)\( - \)\(10\!\cdots\!41\)\( \beta_{1} - \)\(97\!\cdots\!07\)\( \beta_{2} - \)\(39\!\cdots\!08\)\( \beta_{3} + \)\(10\!\cdots\!36\)\( \beta_{4} - 237437426670073056 \beta_{5}) q^{19} +(-\)\(16\!\cdots\!60\)\( + \)\(25\!\cdots\!86\)\( \beta_{1} + \)\(11\!\cdots\!78\)\( \beta_{2} - \)\(73\!\cdots\!70\)\( \beta_{3} - \)\(45\!\cdots\!72\)\( \beta_{4} + 1613400289468070400 \beta_{5}) q^{20} +(\)\(14\!\cdots\!12\)\( + \)\(14\!\cdots\!36\)\( \beta_{1} + \)\(95\!\cdots\!72\)\( \beta_{2} + \)\(18\!\cdots\!08\)\( \beta_{3} - 75478716933227776980 \beta_{4} - 9612887800803073920 \beta_{5}) q^{21} +(-\)\(24\!\cdots\!20\)\( + \)\(45\!\cdots\!61\)\( \beta_{1} - \)\(26\!\cdots\!29\)\( \beta_{2} - \)\(84\!\cdots\!11\)\( \beta_{3} + \)\(17\!\cdots\!83\)\( \beta_{4} + 49716646820528064507 \beta_{5}) q^{22} +(\)\(16\!\cdots\!20\)\( + \)\(75\!\cdots\!46\)\( \beta_{1} - \)\(52\!\cdots\!86\)\( \beta_{2} - \)\(55\!\cdots\!80\)\( \beta_{3} - \)\(14\!\cdots\!60\)\( \beta_{4} - \)\(21\!\cdots\!40\)\( \beta_{5}) q^{23} +(\)\(43\!\cdots\!60\)\( - \)\(16\!\cdots\!40\)\( \beta_{1} + \)\(99\!\cdots\!20\)\( \beta_{2} + \)\(18\!\cdots\!20\)\( \beta_{3} + \)\(66\!\cdots\!36\)\( \beta_{4} + \)\(78\!\cdots\!44\)\( \beta_{5}) q^{24} +(-\)\(32\!\cdots\!25\)\( - \)\(33\!\cdots\!40\)\( \beta_{1} - \)\(15\!\cdots\!20\)\( \beta_{2} - \)\(76\!\cdots\!00\)\( \beta_{3} - \)\(12\!\cdots\!20\)\( \beta_{4} - \)\(20\!\cdots\!00\)\( \beta_{5}) q^{25} +(-\)\(42\!\cdots\!88\)\( - \)\(22\!\cdots\!54\)\( \beta_{1} - \)\(11\!\cdots\!08\)\( \beta_{2} - \)\(18\!\cdots\!32\)\( \beta_{3} - \)\(64\!\cdots\!88\)\( \beta_{4} + \)\(16\!\cdots\!48\)\( \beta_{5}) q^{26} +(\)\(85\!\cdots\!20\)\( + \)\(65\!\cdots\!86\)\( \beta_{1} + \)\(37\!\cdots\!46\)\( \beta_{2} + \)\(10\!\cdots\!96\)\( \beta_{3} + \)\(72\!\cdots\!12\)\( \beta_{4} + \)\(17\!\cdots\!48\)\( \beta_{5}) q^{27} +(\)\(22\!\cdots\!80\)\( + \)\(33\!\cdots\!56\)\( \beta_{1} + \)\(24\!\cdots\!60\)\( \beta_{2} - \)\(25\!\cdots\!84\)\( \beta_{3} - \)\(37\!\cdots\!48\)\( \beta_{4} - \)\(12\!\cdots\!92\)\( \beta_{5}) q^{28} +(-\)\(21\!\cdots\!70\)\( - \)\(21\!\cdots\!36\)\( \beta_{1} + \)\(22\!\cdots\!03\)\( \beta_{2} - \)\(97\!\cdots\!28\)\( \beta_{3} + \)\(11\!\cdots\!77\)\( \beta_{4} + \)\(49\!\cdots\!08\)\( \beta_{5}) q^{29} +(\)\(23\!\cdots\!60\)\( - \)\(45\!\cdots\!26\)\( \beta_{1} - \)\(14\!\cdots\!98\)\( \beta_{2} + \)\(76\!\cdots\!70\)\( \beta_{3} - \)\(20\!\cdots\!98\)\( \beta_{4} - \)\(11\!\cdots\!50\)\( \beta_{5}) q^{30} +(\)\(34\!\cdots\!32\)\( - \)\(42\!\cdots\!84\)\( \beta_{1} + \)\(91\!\cdots\!32\)\( \beta_{2} - \)\(17\!\cdots\!12\)\( \beta_{3} - \)\(18\!\cdots\!64\)\( \beta_{4} + \)\(34\!\cdots\!44\)\( \beta_{5}) q^{31} +(\)\(18\!\cdots\!40\)\( + \)\(12\!\cdots\!80\)\( \beta_{1} + \)\(20\!\cdots\!52\)\( \beta_{2} + \)\(45\!\cdots\!68\)\( \beta_{3} + \)\(24\!\cdots\!96\)\( \beta_{4} + \)\(10\!\cdots\!84\)\( \beta_{5}) q^{32} +(\)\(61\!\cdots\!80\)\( + \)\(10\!\cdots\!96\)\( \beta_{1} - \)\(43\!\cdots\!06\)\( \beta_{2} + \)\(31\!\cdots\!88\)\( \beta_{3} - \)\(80\!\cdots\!14\)\( \beta_{4} - \)\(52\!\cdots\!56\)\( \beta_{5}) q^{33} +(\)\(24\!\cdots\!16\)\( - \)\(72\!\cdots\!70\)\( \beta_{1} - \)\(46\!\cdots\!40\)\( \beta_{2} + \)\(48\!\cdots\!80\)\( \beta_{3} + \)\(12\!\cdots\!56\)\( \beta_{4} + \)\(13\!\cdots\!24\)\( \beta_{5}) q^{34} +(\)\(38\!\cdots\!80\)\( + \)\(48\!\cdots\!32\)\( \beta_{1} - \)\(30\!\cdots\!24\)\( \beta_{2} - \)\(25\!\cdots\!60\)\( \beta_{3} + \)\(24\!\cdots\!16\)\( \beta_{4} - \)\(95\!\cdots\!60\)\( \beta_{5}) q^{35} +(\)\(28\!\cdots\!96\)\( - \)\(23\!\cdots\!45\)\( \beta_{1} + \)\(96\!\cdots\!35\)\( \beta_{2} + \)\(53\!\cdots\!25\)\( \beta_{3} - \)\(54\!\cdots\!96\)\( \beta_{4} - \)\(76\!\cdots\!84\)\( \beta_{5}) q^{36} +(\)\(34\!\cdots\!10\)\( - \)\(31\!\cdots\!28\)\( \beta_{1} + \)\(20\!\cdots\!35\)\( \beta_{2} - \)\(22\!\cdots\!48\)\( \beta_{3} + \)\(94\!\cdots\!69\)\( \beta_{4} + \)\(38\!\cdots\!76\)\( \beta_{5}) q^{37} +(\)\(35\!\cdots\!20\)\( + \)\(12\!\cdots\!95\)\( \beta_{1} - \)\(92\!\cdots\!91\)\( \beta_{2} - \)\(54\!\cdots\!77\)\( \beta_{3} - \)\(43\!\cdots\!19\)\( \beta_{4} - \)\(83\!\cdots\!51\)\( \beta_{5}) q^{38} +(-\)\(99\!\cdots\!76\)\( + \)\(37\!\cdots\!58\)\( \beta_{1} + \)\(75\!\cdots\!66\)\( \beta_{2} - \)\(25\!\cdots\!96\)\( \beta_{3} + \)\(38\!\cdots\!12\)\( \beta_{4} + \)\(19\!\cdots\!48\)\( \beta_{5}) q^{39} +(\)\(11\!\cdots\!00\)\( - \)\(19\!\cdots\!40\)\( \beta_{1} + \)\(31\!\cdots\!80\)\( \beta_{2} + \)\(10\!\cdots\!00\)\( \beta_{3} - \)\(28\!\cdots\!20\)\( \beta_{4} + \)\(53\!\cdots\!00\)\( \beta_{5}) q^{40} +(-\)\(48\!\cdots\!58\)\( - \)\(75\!\cdots\!84\)\( \beta_{1} + \)\(49\!\cdots\!32\)\( \beta_{2} + \)\(15\!\cdots\!88\)\( \beta_{3} + \)\(70\!\cdots\!36\)\( \beta_{4} - \)\(19\!\cdots\!56\)\( \beta_{5}) q^{41} +(-\)\(51\!\cdots\!80\)\( - \)\(41\!\cdots\!40\)\( \beta_{1} - \)\(42\!\cdots\!08\)\( \beta_{2} - \)\(13\!\cdots\!68\)\( \beta_{3} + \)\(27\!\cdots\!04\)\( \beta_{4} + \)\(27\!\cdots\!16\)\( \beta_{5}) q^{42} +(-\)\(33\!\cdots\!00\)\( - \)\(59\!\cdots\!23\)\( \beta_{1} + \)\(38\!\cdots\!27\)\( \beta_{2} + \)\(24\!\cdots\!40\)\( \beta_{3} - \)\(59\!\cdots\!20\)\( \beta_{4} + \)\(33\!\cdots\!20\)\( \beta_{5}) q^{43} +(-\)\(16\!\cdots\!64\)\( + \)\(31\!\cdots\!40\)\( \beta_{1} - \)\(72\!\cdots\!20\)\( \beta_{2} + \)\(12\!\cdots\!60\)\( \beta_{3} + \)\(13\!\cdots\!56\)\( \beta_{4} - \)\(26\!\cdots\!76\)\( \beta_{5}) q^{44} +(-\)\(20\!\cdots\!90\)\( + \)\(33\!\cdots\!84\)\( \beta_{1} + \)\(44\!\cdots\!07\)\( \beta_{2} - \)\(24\!\cdots\!80\)\( \beta_{3} - \)\(25\!\cdots\!43\)\( \beta_{4} + \)\(58\!\cdots\!00\)\( \beta_{5}) q^{45} +(-\)\(25\!\cdots\!28\)\( - \)\(14\!\cdots\!74\)\( \beta_{1} - \)\(12\!\cdots\!98\)\( \beta_{2} - \)\(10\!\cdots\!82\)\( \beta_{3} - \)\(50\!\cdots\!94\)\( \beta_{4} - \)\(19\!\cdots\!26\)\( \beta_{5}) q^{46} +(\)\(13\!\cdots\!80\)\( - \)\(10\!\cdots\!08\)\( \beta_{1} + \)\(41\!\cdots\!04\)\( \beta_{2} + \)\(15\!\cdots\!80\)\( \beta_{3} + \)\(73\!\cdots\!60\)\( \beta_{4} - \)\(24\!\cdots\!60\)\( \beta_{5}) q^{47} +(\)\(43\!\cdots\!20\)\( - \)\(91\!\cdots\!68\)\( \beta_{1} + \)\(37\!\cdots\!56\)\( \beta_{2} + \)\(96\!\cdots\!08\)\( \beta_{3} + \)\(10\!\cdots\!76\)\( \beta_{4} + \)\(72\!\cdots\!04\)\( \beta_{5}) q^{48} +(\)\(46\!\cdots\!93\)\( + \)\(92\!\cdots\!96\)\( \beta_{1} + \)\(61\!\cdots\!92\)\( \beta_{2} - \)\(29\!\cdots\!72\)\( \beta_{3} - \)\(25\!\cdots\!04\)\( \beta_{4} - \)\(66\!\cdots\!16\)\( \beta_{5}) q^{49} +(\)\(11\!\cdots\!00\)\( + \)\(18\!\cdots\!85\)\( \beta_{1} - \)\(12\!\cdots\!20\)\( \beta_{2} + \)\(15\!\cdots\!00\)\( \beta_{3} - \)\(77\!\cdots\!20\)\( \beta_{4} - \)\(15\!\cdots\!00\)\( \beta_{5}) q^{50} +(\)\(86\!\cdots\!32\)\( + \)\(79\!\cdots\!86\)\( \beta_{1} + \)\(66\!\cdots\!22\)\( \beta_{2} + \)\(34\!\cdots\!68\)\( \beta_{3} + \)\(31\!\cdots\!84\)\( \beta_{4} + \)\(61\!\cdots\!36\)\( \beta_{5}) q^{51} +(-\)\(24\!\cdots\!00\)\( + \)\(24\!\cdots\!66\)\( \beta_{1} - \)\(28\!\cdots\!54\)\( \beta_{2} + \)\(27\!\cdots\!30\)\( \beta_{3} - \)\(73\!\cdots\!40\)\( \beta_{4} - \)\(76\!\cdots\!60\)\( \beta_{5}) q^{52} +(-\)\(42\!\cdots\!10\)\( - \)\(78\!\cdots\!92\)\( \beta_{1} + \)\(11\!\cdots\!27\)\( \beta_{2} - \)\(11\!\cdots\!96\)\( \beta_{3} - \)\(16\!\cdots\!87\)\( \beta_{4} - \)\(59\!\cdots\!48\)\( \beta_{5}) q^{53} +(-\)\(22\!\cdots\!80\)\( - \)\(32\!\cdots\!26\)\( \beta_{1} + \)\(24\!\cdots\!98\)\( \beta_{2} - \)\(53\!\cdots\!38\)\( \beta_{3} + \)\(32\!\cdots\!06\)\( \beta_{4} + \)\(35\!\cdots\!74\)\( \beta_{5}) q^{54} +(-\)\(19\!\cdots\!40\)\( - \)\(14\!\cdots\!06\)\( \beta_{1} - \)\(28\!\cdots\!38\)\( \beta_{2} + \)\(17\!\cdots\!20\)\( \beta_{3} + \)\(11\!\cdots\!12\)\( \beta_{4} - \)\(47\!\cdots\!00\)\( \beta_{5}) q^{55} +(-\)\(66\!\cdots\!20\)\( + \)\(68\!\cdots\!24\)\( \beta_{1} - \)\(34\!\cdots\!52\)\( \beta_{2} - \)\(70\!\cdots\!48\)\( \beta_{3} - \)\(12\!\cdots\!08\)\( \beta_{4} - \)\(10\!\cdots\!32\)\( \beta_{5}) q^{56} +(-\)\(13\!\cdots\!60\)\( + \)\(22\!\cdots\!08\)\( \beta_{1} - \)\(75\!\cdots\!86\)\( \beta_{2} - \)\(25\!\cdots\!60\)\( \beta_{3} + \)\(89\!\cdots\!30\)\( \beta_{4} + \)\(11\!\cdots\!20\)\( \beta_{5}) q^{57} +(\)\(74\!\cdots\!80\)\( + \)\(17\!\cdots\!62\)\( \beta_{1} + \)\(38\!\cdots\!40\)\( \beta_{2} - \)\(87\!\cdots\!52\)\( \beta_{3} + \)\(28\!\cdots\!56\)\( \beta_{4} - \)\(11\!\cdots\!76\)\( \beta_{5}) q^{58} +(\)\(42\!\cdots\!60\)\( - \)\(37\!\cdots\!91\)\( \beta_{1} - \)\(10\!\cdots\!57\)\( \beta_{2} + \)\(58\!\cdots\!52\)\( \beta_{3} - \)\(37\!\cdots\!20\)\( \beta_{4} + \)\(28\!\cdots\!20\)\( \beta_{5}) q^{59} +(\)\(13\!\cdots\!80\)\( - \)\(21\!\cdots\!68\)\( \beta_{1} - \)\(25\!\cdots\!24\)\( \beta_{2} + \)\(23\!\cdots\!40\)\( \beta_{3} - \)\(10\!\cdots\!84\)\( \beta_{4} - \)\(26\!\cdots\!60\)\( \beta_{5}) q^{60} +(\)\(79\!\cdots\!62\)\( + \)\(76\!\cdots\!00\)\( \beta_{1} - \)\(90\!\cdots\!25\)\( \beta_{2} - \)\(68\!\cdots\!00\)\( \beta_{3} + \)\(27\!\cdots\!25\)\( \beta_{4} + \)\(83\!\cdots\!00\)\( \beta_{5}) q^{61} +(\)\(19\!\cdots\!80\)\( - \)\(78\!\cdots\!08\)\( \beta_{1} + \)\(18\!\cdots\!48\)\( \beta_{2} + \)\(18\!\cdots\!32\)\( \beta_{3} + \)\(82\!\cdots\!04\)\( \beta_{4} + \)\(35\!\cdots\!16\)\( \beta_{5}) q^{62} +(\)\(21\!\cdots\!60\)\( + \)\(13\!\cdots\!50\)\( \beta_{1} + \)\(28\!\cdots\!50\)\( \beta_{2} + \)\(72\!\cdots\!68\)\( \beta_{3} - \)\(11\!\cdots\!04\)\( \beta_{4} - \)\(57\!\cdots\!16\)\( \beta_{5}) q^{63} +(-\)\(41\!\cdots\!32\)\( - \)\(47\!\cdots\!04\)\( \beta_{1} + \)\(24\!\cdots\!92\)\( \beta_{2} + \)\(17\!\cdots\!68\)\( \beta_{3} + \)\(15\!\cdots\!72\)\( \beta_{4} + \)\(95\!\cdots\!88\)\( \beta_{5}) q^{64} +(-\)\(10\!\cdots\!40\)\( + \)\(10\!\cdots\!64\)\( \beta_{1} + \)\(44\!\cdots\!52\)\( \beta_{2} + \)\(13\!\cdots\!80\)\( \beta_{3} + \)\(60\!\cdots\!32\)\( \beta_{4} + \)\(37\!\cdots\!80\)\( \beta_{5}) q^{65} +(-\)\(31\!\cdots\!76\)\( - \)\(47\!\cdots\!12\)\( \beta_{1} - \)\(50\!\cdots\!24\)\( \beta_{2} - \)\(10\!\cdots\!56\)\( \beta_{3} - \)\(39\!\cdots\!68\)\( \beta_{4} - \)\(33\!\cdots\!72\)\( \beta_{5}) q^{66} +(-\)\(24\!\cdots\!80\)\( - \)\(58\!\cdots\!09\)\( \beta_{1} + \)\(43\!\cdots\!85\)\( \beta_{2} + \)\(89\!\cdots\!48\)\( \beta_{3} + \)\(52\!\cdots\!56\)\( \beta_{4} + \)\(33\!\cdots\!24\)\( \beta_{5}) q^{67} +(\)\(17\!\cdots\!40\)\( - \)\(16\!\cdots\!22\)\( \beta_{1} + \)\(37\!\cdots\!86\)\( \beta_{2} + \)\(22\!\cdots\!06\)\( \beta_{3} - \)\(50\!\cdots\!68\)\( \beta_{4} + \)\(30\!\cdots\!28\)\( \beta_{5}) q^{68} +(-\)\(54\!\cdots\!56\)\( + \)\(40\!\cdots\!12\)\( \beta_{1} + \)\(42\!\cdots\!24\)\( \beta_{2} - \)\(20\!\cdots\!04\)\( \beta_{3} + \)\(34\!\cdots\!04\)\( \beta_{4} - \)\(58\!\cdots\!84\)\( \beta_{5}) q^{69} +(\)\(24\!\cdots\!80\)\( + \)\(56\!\cdots\!72\)\( \beta_{1} - \)\(60\!\cdots\!44\)\( \beta_{2} - \)\(50\!\cdots\!40\)\( \beta_{3} + \)\(11\!\cdots\!56\)\( \beta_{4} - \)\(76\!\cdots\!00\)\( \beta_{5}) q^{70} +(\)\(20\!\cdots\!72\)\( + \)\(50\!\cdots\!50\)\( \beta_{1} - \)\(41\!\cdots\!50\)\( \beta_{2} + \)\(31\!\cdots\!00\)\( \beta_{3} - \)\(42\!\cdots\!00\)\( \beta_{4} + \)\(95\!\cdots\!00\)\( \beta_{5}) q^{71} +(\)\(79\!\cdots\!60\)\( - \)\(15\!\cdots\!20\)\( \beta_{1} + \)\(51\!\cdots\!52\)\( \beta_{2} + \)\(13\!\cdots\!24\)\( \beta_{3} + \)\(60\!\cdots\!28\)\( \beta_{4} + \)\(53\!\cdots\!12\)\( \beta_{5}) q^{72} +(\)\(39\!\cdots\!70\)\( + \)\(77\!\cdots\!72\)\( \beta_{1} - \)\(38\!\cdots\!94\)\( \beta_{2} - \)\(19\!\cdots\!32\)\( \beta_{3} - \)\(46\!\cdots\!54\)\( \beta_{4} - \)\(11\!\cdots\!16\)\( \beta_{5}) q^{73} +(\)\(15\!\cdots\!36\)\( - \)\(31\!\cdots\!26\)\( \beta_{1} + \)\(17\!\cdots\!48\)\( \beta_{2} + \)\(34\!\cdots\!32\)\( \beta_{3} + \)\(11\!\cdots\!24\)\( \beta_{4} - \)\(57\!\cdots\!04\)\( \beta_{5}) q^{74} +(-\)\(22\!\cdots\!00\)\( + \)\(24\!\cdots\!45\)\( \beta_{1} - \)\(78\!\cdots\!65\)\( \beta_{2} - \)\(78\!\cdots\!00\)\( \beta_{3} + \)\(19\!\cdots\!60\)\( \beta_{4} + \)\(40\!\cdots\!00\)\( \beta_{5}) q^{75} +(-\)\(30\!\cdots\!40\)\( + \)\(25\!\cdots\!68\)\( \beta_{1} - \)\(41\!\cdots\!64\)\( \beta_{2} - \)\(73\!\cdots\!36\)\( \beta_{3} - \)\(73\!\cdots\!16\)\( \beta_{4} - \)\(27\!\cdots\!64\)\( \beta_{5}) q^{76} +(-\)\(22\!\cdots\!00\)\( + \)\(63\!\cdots\!48\)\( \beta_{1} + \)\(20\!\cdots\!44\)\( \beta_{2} + \)\(49\!\cdots\!88\)\( \beta_{3} + \)\(89\!\cdots\!36\)\( \beta_{4} - \)\(59\!\cdots\!56\)\( \beta_{5}) q^{77} +(-\)\(13\!\cdots\!00\)\( + \)\(52\!\cdots\!66\)\( \beta_{1} + \)\(40\!\cdots\!38\)\( \beta_{2} - \)\(41\!\cdots\!74\)\( \beta_{3} + \)\(60\!\cdots\!22\)\( \beta_{4} + \)\(62\!\cdots\!38\)\( \beta_{5}) q^{78} +(-\)\(10\!\cdots\!20\)\( - \)\(89\!\cdots\!72\)\( \beta_{1} + \)\(91\!\cdots\!56\)\( \beta_{2} - \)\(48\!\cdots\!76\)\( \beta_{3} - \)\(25\!\cdots\!24\)\( \beta_{4} + \)\(12\!\cdots\!04\)\( \beta_{5}) q^{79} +(\)\(74\!\cdots\!80\)\( - \)\(32\!\cdots\!68\)\( \beta_{1} - \)\(77\!\cdots\!64\)\( \beta_{2} + \)\(78\!\cdots\!60\)\( \beta_{3} + \)\(15\!\cdots\!36\)\( \beta_{4} - \)\(92\!\cdots\!00\)\( \beta_{5}) q^{80} +(\)\(15\!\cdots\!61\)\( - \)\(25\!\cdots\!32\)\( \beta_{1} - \)\(27\!\cdots\!14\)\( \beta_{2} - \)\(26\!\cdots\!96\)\( \beta_{3} - \)\(37\!\cdots\!30\)\( \beta_{4} - \)\(55\!\cdots\!20\)\( \beta_{5}) q^{81} +(\)\(21\!\cdots\!80\)\( + \)\(74\!\cdots\!82\)\( \beta_{1} - \)\(42\!\cdots\!52\)\( \beta_{2} + \)\(49\!\cdots\!32\)\( \beta_{3} + \)\(46\!\cdots\!04\)\( \beta_{4} + \)\(76\!\cdots\!16\)\( \beta_{5}) q^{82} +(\)\(79\!\cdots\!60\)\( - \)\(44\!\cdots\!21\)\( \beta_{1} + \)\(21\!\cdots\!33\)\( \beta_{2} + \)\(63\!\cdots\!00\)\( \beta_{3} - \)\(34\!\cdots\!00\)\( \beta_{4} + \)\(10\!\cdots\!00\)\( \beta_{5}) q^{83} +(\)\(59\!\cdots\!36\)\( + \)\(47\!\cdots\!84\)\( \beta_{1} - \)\(48\!\cdots\!32\)\( \beta_{2} - \)\(26\!\cdots\!68\)\( \beta_{3} - \)\(19\!\cdots\!68\)\( \beta_{4} - \)\(27\!\cdots\!72\)\( \beta_{5}) q^{84} +(\)\(35\!\cdots\!80\)\( - \)\(24\!\cdots\!48\)\( \beta_{1} + \)\(96\!\cdots\!86\)\( \beta_{2} - \)\(48\!\cdots\!60\)\( \beta_{3} + \)\(33\!\cdots\!26\)\( \beta_{4} - \)\(18\!\cdots\!60\)\( \beta_{5}) q^{85} +(\)\(17\!\cdots\!92\)\( - \)\(18\!\cdots\!95\)\( \beta_{1} + \)\(19\!\cdots\!35\)\( \beta_{2} + \)\(12\!\cdots\!85\)\( \beta_{3} - \)\(20\!\cdots\!57\)\( \beta_{4} + \)\(44\!\cdots\!47\)\( \beta_{5}) q^{86} +(\)\(62\!\cdots\!60\)\( - \)\(13\!\cdots\!94\)\( \beta_{1} - \)\(33\!\cdots\!66\)\( \beta_{2} + \)\(59\!\cdots\!52\)\( \beta_{3} - \)\(20\!\cdots\!56\)\( \beta_{4} + \)\(10\!\cdots\!76\)\( \beta_{5}) q^{87} +(-\)\(74\!\cdots\!40\)\( + \)\(67\!\cdots\!88\)\( \beta_{1} - \)\(38\!\cdots\!92\)\( \beta_{2} - \)\(21\!\cdots\!52\)\( \beta_{3} - \)\(23\!\cdots\!44\)\( \beta_{4} - \)\(95\!\cdots\!76\)\( \beta_{5}) q^{88} +(-\)\(79\!\cdots\!10\)\( - \)\(78\!\cdots\!48\)\( \beta_{1} - \)\(86\!\cdots\!46\)\( \beta_{2} - \)\(43\!\cdots\!04\)\( \beta_{3} - \)\(79\!\cdots\!34\)\( \beta_{4} - \)\(67\!\cdots\!36\)\( \beta_{5}) q^{89} +(-\)\(14\!\cdots\!40\)\( + \)\(16\!\cdots\!54\)\( \beta_{1} + \)\(30\!\cdots\!72\)\( \beta_{2} - \)\(14\!\cdots\!20\)\( \beta_{3} + \)\(22\!\cdots\!52\)\( \beta_{4} + \)\(35\!\cdots\!80\)\( \beta_{5}) q^{90} +(-\)\(13\!\cdots\!28\)\( + \)\(13\!\cdots\!04\)\( \beta_{1} + \)\(57\!\cdots\!08\)\( \beta_{2} + \)\(86\!\cdots\!92\)\( \beta_{3} + \)\(20\!\cdots\!12\)\( \beta_{4} - \)\(74\!\cdots\!52\)\( \beta_{5}) q^{91} +(-\)\(65\!\cdots\!80\)\( + \)\(54\!\cdots\!04\)\( \beta_{1} - \)\(35\!\cdots\!84\)\( \beta_{2} - \)\(33\!\cdots\!08\)\( \beta_{3} - \)\(41\!\cdots\!76\)\( \beta_{4} - \)\(71\!\cdots\!04\)\( \beta_{5}) q^{92} +(\)\(12\!\cdots\!80\)\( - \)\(84\!\cdots\!12\)\( \beta_{1} + \)\(13\!\cdots\!48\)\( \beta_{2} - \)\(45\!\cdots\!56\)\( \beta_{3} - \)\(70\!\cdots\!32\)\( \beta_{4} + \)\(51\!\cdots\!72\)\( \beta_{5}) q^{93} +(\)\(51\!\cdots\!96\)\( - \)\(17\!\cdots\!36\)\( \beta_{1} - \)\(60\!\cdots\!72\)\( \beta_{2} + \)\(61\!\cdots\!32\)\( \beta_{3} + \)\(45\!\cdots\!36\)\( \beta_{4} + \)\(75\!\cdots\!44\)\( \beta_{5}) q^{94} +(\)\(13\!\cdots\!00\)\( + \)\(62\!\cdots\!90\)\( \beta_{1} + \)\(19\!\cdots\!70\)\( \beta_{2} - \)\(50\!\cdots\!00\)\( \beta_{3} + \)\(13\!\cdots\!20\)\( \beta_{4} - \)\(23\!\cdots\!00\)\( \beta_{5}) q^{95} +(\)\(27\!\cdots\!92\)\( - \)\(25\!\cdots\!00\)\( \beta_{1} + \)\(13\!\cdots\!00\)\( \beta_{2} + \)\(81\!\cdots\!80\)\( \beta_{3} - \)\(18\!\cdots\!28\)\( \beta_{4} - \)\(21\!\cdots\!12\)\( \beta_{5}) q^{96} +(\)\(15\!\cdots\!30\)\( + \)\(24\!\cdots\!24\)\( \beta_{1} + \)\(64\!\cdots\!38\)\( \beta_{2} + \)\(46\!\cdots\!96\)\( \beta_{3} - \)\(31\!\cdots\!38\)\( \beta_{4} + \)\(31\!\cdots\!48\)\( \beta_{5}) q^{97} +(-\)\(28\!\cdots\!80\)\( - \)\(53\!\cdots\!89\)\( \beta_{1} - \)\(22\!\cdots\!32\)\( \beta_{2} - \)\(14\!\cdots\!08\)\( \beta_{3} + \)\(44\!\cdots\!24\)\( \beta_{4} + \)\(61\!\cdots\!96\)\( \beta_{5}) q^{98} +(-\)\(24\!\cdots\!16\)\( + \)\(13\!\cdots\!01\)\( \beta_{1} + \)\(25\!\cdots\!27\)\( \beta_{2} + \)\(45\!\cdots\!48\)\( \beta_{3} + \)\(51\!\cdots\!28\)\( \beta_{4} - \)\(56\!\cdots\!88\)\( \beta_{5}) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 66157336440q^{2} + 89896952777770440q^{3} + \)\(82\!\cdots\!68\)\(q^{4} - \)\(42\!\cdots\!20\)\(q^{5} - \)\(38\!\cdots\!88\)\(q^{6} + \)\(33\!\cdots\!00\)\(q^{7} + \)\(26\!\cdots\!80\)\(q^{8} + \)\(28\!\cdots\!42\)\(q^{9} + O(q^{10}) \) \( 6q + 66157336440q^{2} + 89896952777770440q^{3} + \)\(82\!\cdots\!68\)\(q^{4} - \)\(42\!\cdots\!20\)\(q^{5} - \)\(38\!\cdots\!88\)\(q^{6} + \)\(33\!\cdots\!00\)\(q^{7} + \)\(26\!\cdots\!80\)\(q^{8} + \)\(28\!\cdots\!42\)\(q^{9} - \)\(26\!\cdots\!20\)\(q^{10} - \)\(72\!\cdots\!28\)\(q^{11} + \)\(54\!\cdots\!80\)\(q^{12} + \)\(19\!\cdots\!80\)\(q^{13} - \)\(12\!\cdots\!64\)\(q^{14} + \)\(13\!\cdots\!60\)\(q^{15} + \)\(36\!\cdots\!16\)\(q^{16} + \)\(31\!\cdots\!20\)\(q^{17} + \)\(97\!\cdots\!40\)\(q^{18} - \)\(21\!\cdots\!80\)\(q^{19} - \)\(10\!\cdots\!60\)\(q^{20} + \)\(84\!\cdots\!72\)\(q^{21} - \)\(14\!\cdots\!20\)\(q^{22} + \)\(96\!\cdots\!20\)\(q^{23} + \)\(26\!\cdots\!60\)\(q^{24} - \)\(19\!\cdots\!50\)\(q^{25} - \)\(25\!\cdots\!28\)\(q^{26} + \)\(51\!\cdots\!20\)\(q^{27} + \)\(13\!\cdots\!80\)\(q^{28} - \)\(13\!\cdots\!20\)\(q^{29} + \)\(14\!\cdots\!60\)\(q^{30} + \)\(20\!\cdots\!92\)\(q^{31} + \)\(10\!\cdots\!40\)\(q^{32} + \)\(36\!\cdots\!80\)\(q^{33} + \)\(14\!\cdots\!96\)\(q^{34} + \)\(23\!\cdots\!80\)\(q^{35} + \)\(17\!\cdots\!76\)\(q^{36} + \)\(20\!\cdots\!60\)\(q^{37} + \)\(21\!\cdots\!20\)\(q^{38} - \)\(59\!\cdots\!56\)\(q^{39} + \)\(68\!\cdots\!00\)\(q^{40} - \)\(29\!\cdots\!48\)\(q^{41} - \)\(30\!\cdots\!80\)\(q^{42} - \)\(20\!\cdots\!00\)\(q^{43} - \)\(97\!\cdots\!84\)\(q^{44} - \)\(12\!\cdots\!40\)\(q^{45} - \)\(15\!\cdots\!68\)\(q^{46} + \)\(80\!\cdots\!80\)\(q^{47} + \)\(26\!\cdots\!20\)\(q^{48} + \)\(27\!\cdots\!58\)\(q^{49} + \)\(70\!\cdots\!00\)\(q^{50} + \)\(52\!\cdots\!92\)\(q^{51} - \)\(14\!\cdots\!00\)\(q^{52} - \)\(25\!\cdots\!60\)\(q^{53} - \)\(13\!\cdots\!80\)\(q^{54} - \)\(11\!\cdots\!40\)\(q^{55} - \)\(39\!\cdots\!20\)\(q^{56} - \)\(78\!\cdots\!60\)\(q^{57} + \)\(44\!\cdots\!80\)\(q^{58} + \)\(25\!\cdots\!60\)\(q^{59} + \)\(81\!\cdots\!80\)\(q^{60} + \)\(47\!\cdots\!72\)\(q^{61} + \)\(11\!\cdots\!80\)\(q^{62} + \)\(12\!\cdots\!60\)\(q^{63} - \)\(24\!\cdots\!92\)\(q^{64} - \)\(64\!\cdots\!40\)\(q^{65} - \)\(19\!\cdots\!56\)\(q^{66} - \)\(14\!\cdots\!80\)\(q^{67} + \)\(10\!\cdots\!40\)\(q^{68} - \)\(32\!\cdots\!36\)\(q^{69} + \)\(14\!\cdots\!80\)\(q^{70} + \)\(12\!\cdots\!32\)\(q^{71} + \)\(47\!\cdots\!60\)\(q^{72} + \)\(23\!\cdots\!20\)\(q^{73} + \)\(90\!\cdots\!16\)\(q^{74} - \)\(13\!\cdots\!00\)\(q^{75} - \)\(18\!\cdots\!40\)\(q^{76} - \)\(13\!\cdots\!00\)\(q^{77} - \)\(82\!\cdots\!00\)\(q^{78} - \)\(60\!\cdots\!20\)\(q^{79} + \)\(44\!\cdots\!80\)\(q^{80} + \)\(90\!\cdots\!66\)\(q^{81} + \)\(13\!\cdots\!80\)\(q^{82} + \)\(47\!\cdots\!60\)\(q^{83} + \)\(35\!\cdots\!16\)\(q^{84} + \)\(21\!\cdots\!80\)\(q^{85} + \)\(10\!\cdots\!52\)\(q^{86} + \)\(37\!\cdots\!60\)\(q^{87} - \)\(44\!\cdots\!40\)\(q^{88} - \)\(47\!\cdots\!60\)\(q^{89} - \)\(86\!\cdots\!40\)\(q^{90} - \)\(78\!\cdots\!68\)\(q^{91} - \)\(39\!\cdots\!80\)\(q^{92} + \)\(76\!\cdots\!80\)\(q^{93} + \)\(31\!\cdots\!76\)\(q^{94} + \)\(83\!\cdots\!00\)\(q^{95} + \)\(16\!\cdots\!52\)\(q^{96} + \)\(90\!\cdots\!80\)\(q^{97} - \)\(16\!\cdots\!80\)\(q^{98} - \)\(14\!\cdots\!96\)\(q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 3 x^{5} - 18795872470415627868 x^{4} - 3619866001951210071877967584 x^{3} + 86589594972295987044342709087705454976 x^{2} + 35564509575256206574563332087691180650482000128 x - 11608057594368249332625196663730471558406093241461869568\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 24 \nu - 12 \)
\(\beta_{2}\)\(=\)\((\)\(1725241141 \nu^{5} + 52370219295084693165 \nu^{4} - 49498807386833879951613619104 \nu^{3} - 528775848438262569600754436589454711264 \nu^{2} + 234519987467691230702307692269422720705140245248 \nu + 211778903923627520139425348255827122068172212718485498112\)\()/ \)\(63\!\cdots\!64\)\( \)
\(\beta_{3}\)\(=\)\((\)\(-622812051901 \nu^{5} - 18905649165525574232565 \nu^{4} + 17869069466647030662532516496544 \nu^{3} + 382695705267597042646168653945716402011360 \nu^{2} - 140071575584482192899887077227607649818311761885952 \nu - 1278182730691235506753379480697661790587354972675857065283840\)\()/ \)\(33\!\cdots\!56\)\( \)
\(\beta_{4}\)\(=\)\((\)\(-687580198211778593 \nu^{5} + 261149215098436825652325447 \nu^{4} + 11559216341078523099923912171229656352 \nu^{3} - 1831957080661742895586860354230197619463579040 \nu^{2} - 45068668662605440818298141845697038686098198714223785728 \nu - 6514727813269645208229386763944768843836092419206003637663237376\)\()/ \)\(20\!\cdots\!00\)\( \)
\(\beta_{5}\)\(=\)\((\)\(-1079038690906363417057 \nu^{5} - 18626326216530086775315818821497 \nu^{4} - 189888900813385683268026335502486713225952 \nu^{3} + 187306220558887538447517093208279813704122145555040 \nu^{2} + 1949929883211596391434605232574955365872129939317018400540928 \nu + 317594279206985688318277247102119634699065221984689566533231834053376\)\()/ \)\(15\!\cdots\!00\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1} + 12\)\()/24\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} + 6859 \beta_{2} + 6933179271 \beta_{1} + 3608807514319800551520\)\()/576\)
\(\nu^{3}\)\(=\)\((\)\(-1269 \beta_{5} + 10739 \beta_{4} - 18101906 \beta_{3} - 18347007714910 \beta_{2} + 689917724744602829892 \beta_{1} + 3127564274404746945419871434208\)\()/1728\)
\(\nu^{4}\)\(=\)\((\)\(-490484368455 \beta_{5} + 167868392758705 \beta_{4} + 30169391367716115638 \beta_{3} + 406073038002404360756170 \beta_{2} + 310989773189543403973035133392 \beta_{1} + 103740844032723052780031393952765939440928\)\()/1728\)
\(\nu^{5}\)\(=\)\((\)\(-7173337831791001913787 \beta_{5} - 1595861326796321577498403 \beta_{4} + 1053742672105498200047372126 \beta_{3} - 69677960216467034790725601629790 \beta_{2} + 2313948240110542399345406901819640590768 \beta_{1} + 15587534859598899444262181668865818402373281376928\)\()/576\)

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
3.39604e9
2.88052e9
2.15491e8
−6.56883e8
−2.55796e9
−3.27721e9
−7.04788e10 1.46397e17 2.60608e21 3.41813e23 −1.03179e28 1.73558e30 −1.72603e31 1.39227e34 −2.40906e34
1.2 −5.81063e10 −8.58335e16 1.01516e21 8.21398e23 4.98747e27 −4.94216e29 7.82125e31 −1.42084e32 −4.77284e34
1.3 5.85445e9 5.36564e16 −2.32691e21 −7.86170e24 3.14128e26 −1.67406e30 −2.74462e31 −4.63046e33 −4.60259e34
1.4 2.67914e10 −1.17507e16 −1.64340e21 8.36674e24 −3.14817e26 1.48756e30 −1.07289e32 −7.37139e33 2.24157e35
1.5 7.24174e10 −1.51457e17 2.88309e21 −9.47506e24 −1.09681e28 1.67109e28 3.77951e31 1.54297e34 −6.86158e35
1.6 8.96793e10 1.38884e17 5.68119e21 3.52866e24 1.24550e28 −7.32344e29 2.97736e32 1.17794e34 3.16448e35
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
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 1.72.a.a 6
3.b odd 2 1 9.72.a.b 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.72.a.a 6 1.a even 1 1 trivial
9.72.a.b 6 3.b odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{72}^{\mathrm{new}}(\Gamma_0(1))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 66157336440 T + \)\(51\!\cdots\!60\)\( T^{2} - \)\(28\!\cdots\!20\)\( T^{3} + \)\(17\!\cdots\!12\)\( T^{4} - \)\(98\!\cdots\!20\)\( T^{5} + \)\(57\!\cdots\!80\)\( T^{6} - \)\(23\!\cdots\!60\)\( T^{7} + \)\(10\!\cdots\!48\)\( T^{8} - \)\(36\!\cdots\!40\)\( T^{9} + \)\(16\!\cdots\!60\)\( T^{10} - \)\(48\!\cdots\!20\)\( T^{11} + \)\(17\!\cdots\!64\)\( T^{12} \)
$3$ \( 1 - 89896952777770440 T + \)\(12\!\cdots\!70\)\( T^{2} - \)\(78\!\cdots\!80\)\( T^{3} + \)\(94\!\cdots\!27\)\( T^{4} - \)\(42\!\cdots\!20\)\( T^{5} + \)\(73\!\cdots\!60\)\( T^{6} - \)\(31\!\cdots\!40\)\( T^{7} + \)\(53\!\cdots\!43\)\( T^{8} - \)\(33\!\cdots\!40\)\( T^{9} + \)\(38\!\cdots\!70\)\( T^{10} - \)\(21\!\cdots\!80\)\( T^{11} + \)\(17\!\cdots\!29\)\( T^{12} \)
$5$ \( 1 + \)\(42\!\cdots\!20\)\( T + \)\(14\!\cdots\!50\)\( T^{2} + \)\(65\!\cdots\!00\)\( T^{3} + \)\(11\!\cdots\!75\)\( T^{4} + \)\(41\!\cdots\!00\)\( T^{5} + \)\(57\!\cdots\!00\)\( T^{6} + \)\(17\!\cdots\!00\)\( T^{7} + \)\(20\!\cdots\!75\)\( T^{8} + \)\(49\!\cdots\!00\)\( T^{9} + \)\(46\!\cdots\!50\)\( T^{10} + \)\(58\!\cdots\!00\)\( T^{11} + \)\(57\!\cdots\!25\)\( T^{12} \)
$7$ \( 1 - \)\(33\!\cdots\!00\)\( T + \)\(16\!\cdots\!50\)\( T^{2} - \)\(13\!\cdots\!00\)\( T^{3} + \)\(19\!\cdots\!47\)\( T^{4} - \)\(77\!\cdots\!00\)\( T^{5} + \)\(25\!\cdots\!00\)\( T^{6} - \)\(78\!\cdots\!00\)\( T^{7} + \)\(19\!\cdots\!03\)\( T^{8} - \)\(13\!\cdots\!00\)\( T^{9} + \)\(17\!\cdots\!50\)\( T^{10} - \)\(34\!\cdots\!00\)\( T^{11} + \)\(10\!\cdots\!49\)\( T^{12} \)
$11$ \( 1 + \)\(72\!\cdots\!28\)\( T + \)\(17\!\cdots\!26\)\( T^{2} - \)\(23\!\cdots\!20\)\( T^{3} + \)\(12\!\cdots\!95\)\( T^{4} - \)\(37\!\cdots\!92\)\( T^{5} + \)\(16\!\cdots\!44\)\( T^{6} - \)\(32\!\cdots\!12\)\( T^{7} + \)\(97\!\cdots\!95\)\( T^{8} - \)\(15\!\cdots\!20\)\( T^{9} + \)\(97\!\cdots\!66\)\( T^{10} + \)\(36\!\cdots\!28\)\( T^{11} + \)\(42\!\cdots\!61\)\( T^{12} \)
$13$ \( 1 - \)\(19\!\cdots\!80\)\( T + \)\(62\!\cdots\!90\)\( T^{2} - \)\(10\!\cdots\!60\)\( T^{3} + \)\(17\!\cdots\!07\)\( T^{4} - \)\(24\!\cdots\!40\)\( T^{5} + \)\(28\!\cdots\!20\)\( T^{6} - \)\(30\!\cdots\!80\)\( T^{7} + \)\(26\!\cdots\!83\)\( T^{8} - \)\(19\!\cdots\!80\)\( T^{9} + \)\(14\!\cdots\!90\)\( T^{10} - \)\(55\!\cdots\!60\)\( T^{11} + \)\(34\!\cdots\!09\)\( T^{12} \)
$17$ \( 1 - \)\(31\!\cdots\!20\)\( T + \)\(65\!\cdots\!30\)\( T^{2} + \)\(16\!\cdots\!40\)\( T^{3} + \)\(12\!\cdots\!67\)\( T^{4} + \)\(79\!\cdots\!40\)\( T^{5} + \)\(12\!\cdots\!40\)\( T^{6} + \)\(18\!\cdots\!20\)\( T^{7} + \)\(65\!\cdots\!63\)\( T^{8} + \)\(20\!\cdots\!80\)\( T^{9} + \)\(18\!\cdots\!30\)\( T^{10} - \)\(20\!\cdots\!60\)\( T^{11} + \)\(14\!\cdots\!69\)\( T^{12} \)
$19$ \( 1 + \)\(21\!\cdots\!80\)\( T + \)\(22\!\cdots\!14\)\( T^{2} + \)\(27\!\cdots\!00\)\( T^{3} + \)\(24\!\cdots\!15\)\( T^{4} + \)\(22\!\cdots\!00\)\( T^{5} + \)\(17\!\cdots\!80\)\( T^{6} + \)\(13\!\cdots\!00\)\( T^{7} + \)\(92\!\cdots\!15\)\( T^{8} + \)\(64\!\cdots\!00\)\( T^{9} + \)\(33\!\cdots\!94\)\( T^{10} + \)\(19\!\cdots\!20\)\( T^{11} + \)\(56\!\cdots\!81\)\( T^{12} \)
$23$ \( 1 - \)\(96\!\cdots\!20\)\( T + \)\(19\!\cdots\!10\)\( T^{2} - \)\(15\!\cdots\!40\)\( T^{3} + \)\(17\!\cdots\!87\)\( T^{4} - \)\(11\!\cdots\!60\)\( T^{5} + \)\(10\!\cdots\!80\)\( T^{6} - \)\(53\!\cdots\!20\)\( T^{7} + \)\(41\!\cdots\!23\)\( T^{8} - \)\(17\!\cdots\!20\)\( T^{9} + \)\(10\!\cdots\!10\)\( T^{10} - \)\(24\!\cdots\!40\)\( T^{11} + \)\(12\!\cdots\!89\)\( T^{12} \)
$29$ \( 1 + \)\(13\!\cdots\!20\)\( T + \)\(20\!\cdots\!74\)\( T^{2} + \)\(17\!\cdots\!00\)\( T^{3} + \)\(24\!\cdots\!15\)\( T^{4} + \)\(73\!\cdots\!00\)\( T^{5} + \)\(19\!\cdots\!80\)\( T^{6} + \)\(49\!\cdots\!00\)\( T^{7} + \)\(11\!\cdots\!15\)\( T^{8} + \)\(55\!\cdots\!00\)\( T^{9} + \)\(43\!\cdots\!94\)\( T^{10} + \)\(18\!\cdots\!80\)\( T^{11} + \)\(95\!\cdots\!21\)\( T^{12} \)
$31$ \( 1 - \)\(20\!\cdots\!92\)\( T + \)\(55\!\cdots\!46\)\( T^{2} - \)\(77\!\cdots\!20\)\( T^{3} + \)\(11\!\cdots\!95\)\( T^{4} - \)\(11\!\cdots\!92\)\( T^{5} + \)\(12\!\cdots\!84\)\( T^{6} - \)\(89\!\cdots\!52\)\( T^{7} + \)\(68\!\cdots\!95\)\( T^{8} - \)\(35\!\cdots\!20\)\( T^{9} + \)\(19\!\cdots\!66\)\( T^{10} - \)\(56\!\cdots\!92\)\( T^{11} + \)\(20\!\cdots\!81\)\( T^{12} \)
$37$ \( 1 - \)\(20\!\cdots\!60\)\( T + \)\(27\!\cdots\!90\)\( T^{2} - \)\(25\!\cdots\!80\)\( T^{3} + \)\(19\!\cdots\!07\)\( T^{4} - \)\(11\!\cdots\!80\)\( T^{5} + \)\(59\!\cdots\!20\)\( T^{6} - \)\(25\!\cdots\!40\)\( T^{7} + \)\(92\!\cdots\!83\)\( T^{8} - \)\(27\!\cdots\!60\)\( T^{9} + \)\(64\!\cdots\!90\)\( T^{10} - \)\(10\!\cdots\!80\)\( T^{11} + \)\(11\!\cdots\!09\)\( T^{12} \)
$41$ \( 1 + \)\(29\!\cdots\!48\)\( T + \)\(15\!\cdots\!06\)\( T^{2} + \)\(39\!\cdots\!80\)\( T^{3} + \)\(11\!\cdots\!95\)\( T^{4} + \)\(22\!\cdots\!08\)\( T^{5} + \)\(49\!\cdots\!04\)\( T^{6} + \)\(72\!\cdots\!28\)\( T^{7} + \)\(11\!\cdots\!95\)\( T^{8} + \)\(13\!\cdots\!80\)\( T^{9} + \)\(16\!\cdots\!66\)\( T^{10} + \)\(10\!\cdots\!48\)\( T^{11} + \)\(11\!\cdots\!41\)\( T^{12} \)
$43$ \( 1 + \)\(20\!\cdots\!00\)\( T + \)\(48\!\cdots\!50\)\( T^{2} + \)\(67\!\cdots\!00\)\( T^{3} + \)\(90\!\cdots\!47\)\( T^{4} + \)\(98\!\cdots\!00\)\( T^{5} + \)\(10\!\cdots\!00\)\( T^{6} + \)\(93\!\cdots\!00\)\( T^{7} + \)\(80\!\cdots\!03\)\( T^{8} + \)\(56\!\cdots\!00\)\( T^{9} + \)\(39\!\cdots\!50\)\( T^{10} + \)\(15\!\cdots\!00\)\( T^{11} + \)\(72\!\cdots\!49\)\( T^{12} \)
$47$ \( 1 - \)\(80\!\cdots\!80\)\( T + \)\(47\!\cdots\!70\)\( T^{2} - \)\(19\!\cdots\!40\)\( T^{3} + \)\(68\!\cdots\!27\)\( T^{4} - \)\(19\!\cdots\!40\)\( T^{5} + \)\(48\!\cdots\!60\)\( T^{6} - \)\(10\!\cdots\!20\)\( T^{7} + \)\(18\!\cdots\!43\)\( T^{8} - \)\(27\!\cdots\!80\)\( T^{9} + \)\(35\!\cdots\!70\)\( T^{10} - \)\(31\!\cdots\!40\)\( T^{11} + \)\(20\!\cdots\!29\)\( T^{12} \)
$53$ \( 1 + \)\(25\!\cdots\!60\)\( T + \)\(14\!\cdots\!70\)\( T^{2} + \)\(29\!\cdots\!20\)\( T^{3} + \)\(91\!\cdots\!27\)\( T^{4} + \)\(14\!\cdots\!80\)\( T^{5} + \)\(31\!\cdots\!60\)\( T^{6} + \)\(38\!\cdots\!60\)\( T^{7} + \)\(64\!\cdots\!43\)\( T^{8} + \)\(54\!\cdots\!60\)\( T^{9} + \)\(72\!\cdots\!70\)\( T^{10} + \)\(33\!\cdots\!20\)\( T^{11} + \)\(34\!\cdots\!29\)\( T^{12} \)
$59$ \( 1 - \)\(25\!\cdots\!60\)\( T + \)\(55\!\cdots\!54\)\( T^{2} - \)\(78\!\cdots\!00\)\( T^{3} + \)\(95\!\cdots\!15\)\( T^{4} - \)\(89\!\cdots\!00\)\( T^{5} + \)\(73\!\cdots\!80\)\( T^{6} - \)\(48\!\cdots\!00\)\( T^{7} + \)\(27\!\cdots\!15\)\( T^{8} - \)\(12\!\cdots\!00\)\( T^{9} + \)\(46\!\cdots\!94\)\( T^{10} - \)\(11\!\cdots\!40\)\( T^{11} + \)\(24\!\cdots\!41\)\( T^{12} \)
$61$ \( 1 - \)\(47\!\cdots\!72\)\( T + \)\(13\!\cdots\!26\)\( T^{2} - \)\(25\!\cdots\!20\)\( T^{3} + \)\(10\!\cdots\!95\)\( T^{4} - \)\(28\!\cdots\!92\)\( T^{5} + \)\(65\!\cdots\!44\)\( T^{6} - \)\(16\!\cdots\!12\)\( T^{7} + \)\(35\!\cdots\!95\)\( T^{8} - \)\(48\!\cdots\!20\)\( T^{9} + \)\(14\!\cdots\!66\)\( T^{10} - \)\(29\!\cdots\!72\)\( T^{11} + \)\(35\!\cdots\!61\)\( T^{12} \)
$67$ \( 1 + \)\(14\!\cdots\!80\)\( T + \)\(23\!\cdots\!30\)\( T^{2} + \)\(20\!\cdots\!40\)\( T^{3} + \)\(18\!\cdots\!67\)\( T^{4} + \)\(13\!\cdots\!40\)\( T^{5} + \)\(95\!\cdots\!40\)\( T^{6} + \)\(59\!\cdots\!20\)\( T^{7} + \)\(38\!\cdots\!63\)\( T^{8} + \)\(18\!\cdots\!80\)\( T^{9} + \)\(92\!\cdots\!30\)\( T^{10} + \)\(26\!\cdots\!40\)\( T^{11} + \)\(80\!\cdots\!69\)\( T^{12} \)
$71$ \( 1 - \)\(12\!\cdots\!32\)\( T + \)\(11\!\cdots\!86\)\( T^{2} - \)\(85\!\cdots\!20\)\( T^{3} + \)\(53\!\cdots\!95\)\( T^{4} - \)\(31\!\cdots\!92\)\( T^{5} + \)\(17\!\cdots\!64\)\( T^{6} - \)\(87\!\cdots\!32\)\( T^{7} + \)\(40\!\cdots\!95\)\( T^{8} - \)\(17\!\cdots\!20\)\( T^{9} + \)\(67\!\cdots\!66\)\( T^{10} - \)\(19\!\cdots\!32\)\( T^{11} + \)\(43\!\cdots\!21\)\( T^{12} \)
$73$ \( 1 - \)\(23\!\cdots\!20\)\( T + \)\(10\!\cdots\!10\)\( T^{2} - \)\(14\!\cdots\!40\)\( T^{3} + \)\(39\!\cdots\!87\)\( T^{4} - \)\(40\!\cdots\!60\)\( T^{5} + \)\(89\!\cdots\!80\)\( T^{6} - \)\(80\!\cdots\!20\)\( T^{7} + \)\(15\!\cdots\!23\)\( T^{8} - \)\(11\!\cdots\!20\)\( T^{9} + \)\(15\!\cdots\!10\)\( T^{10} - \)\(71\!\cdots\!40\)\( T^{11} + \)\(59\!\cdots\!89\)\( T^{12} \)
$79$ \( 1 + \)\(60\!\cdots\!20\)\( T + \)\(30\!\cdots\!74\)\( T^{2} + \)\(83\!\cdots\!00\)\( T^{3} + \)\(22\!\cdots\!15\)\( T^{4} + \)\(35\!\cdots\!00\)\( T^{5} + \)\(91\!\cdots\!80\)\( T^{6} + \)\(19\!\cdots\!00\)\( T^{7} + \)\(64\!\cdots\!15\)\( T^{8} + \)\(13\!\cdots\!00\)\( T^{9} + \)\(25\!\cdots\!94\)\( T^{10} + \)\(27\!\cdots\!80\)\( T^{11} + \)\(24\!\cdots\!21\)\( T^{12} \)
$83$ \( 1 - \)\(47\!\cdots\!60\)\( T + \)\(18\!\cdots\!30\)\( T^{2} - \)\(46\!\cdots\!20\)\( T^{3} + \)\(10\!\cdots\!67\)\( T^{4} - \)\(17\!\cdots\!80\)\( T^{5} + \)\(26\!\cdots\!40\)\( T^{6} - \)\(31\!\cdots\!60\)\( T^{7} + \)\(33\!\cdots\!63\)\( T^{8} - \)\(27\!\cdots\!60\)\( T^{9} + \)\(19\!\cdots\!30\)\( T^{10} - \)\(89\!\cdots\!20\)\( T^{11} + \)\(33\!\cdots\!69\)\( T^{12} \)
$89$ \( 1 + \)\(47\!\cdots\!60\)\( T + \)\(20\!\cdots\!34\)\( T^{2} + \)\(52\!\cdots\!00\)\( T^{3} + \)\(12\!\cdots\!15\)\( T^{4} + \)\(22\!\cdots\!00\)\( T^{5} + \)\(40\!\cdots\!80\)\( T^{6} + \)\(57\!\cdots\!00\)\( T^{7} + \)\(81\!\cdots\!15\)\( T^{8} + \)\(86\!\cdots\!00\)\( T^{9} + \)\(85\!\cdots\!94\)\( T^{10} + \)\(51\!\cdots\!40\)\( T^{11} + \)\(27\!\cdots\!61\)\( T^{12} \)
$97$ \( 1 - \)\(90\!\cdots\!80\)\( T + \)\(81\!\cdots\!70\)\( T^{2} - \)\(43\!\cdots\!40\)\( T^{3} + \)\(22\!\cdots\!27\)\( T^{4} - \)\(85\!\cdots\!40\)\( T^{5} + \)\(32\!\cdots\!60\)\( T^{6} - \)\(98\!\cdots\!20\)\( T^{7} + \)\(29\!\cdots\!43\)\( T^{8} - \)\(65\!\cdots\!80\)\( T^{9} + \)\(14\!\cdots\!70\)\( T^{10} - \)\(18\!\cdots\!40\)\( T^{11} + \)\(23\!\cdots\!29\)\( T^{12} \)
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