Properties

Label 3.34.a.a
Level $3$
Weight $34$
Character orbit 3.a
Self dual yes
Analytic conductor $20.695$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3 \)
Weight: \( k \) \(=\) \( 34 \)
Character orbit: \([\chi]\) \(=\) 3.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(20.6948486643\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
Defining polynomial: \(x^{3} - x^{2} - 35900150 x + 10469144400\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{6}\cdot 3^{6}\cdot 5\cdot 7\cdot 11 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 13734 - \beta_{1} ) q^{2} + 43046721 q^{3} + ( -369155924 - 63216 \beta_{1} + \beta_{2} ) q^{4} + ( 17087274630 + 280984 \beta_{1} + 108 \beta_{2} ) q^{5} + ( 591203666214 - 43046721 \beta_{1} ) q^{6} + ( 25371244671752 - 323610696 \beta_{1} - 6404 \beta_{2} ) q^{7} + ( 384716449496664 + 3092877952 \beta_{1} + 41202 \beta_{2} ) q^{8} + 1853020188851841 q^{9} +O(q^{10})\) \( q +(13734 - \beta_{1}) q^{2} +43046721 q^{3} +(-369155924 - 63216 \beta_{1} + \beta_{2}) q^{4} +(17087274630 + 280984 \beta_{1} + 108 \beta_{2}) q^{5} +(591203666214 - 43046721 \beta_{1}) q^{6} +(25371244671752 - 323610696 \beta_{1} - 6404 \beta_{2}) q^{7} +(384716449496664 + 3092877952 \beta_{1} + 41202 \beta_{2}) q^{8} +1853020188851841 q^{9} +(-2022248971884060 - 298904737158 \beta_{1} - 2658496 \beta_{2}) q^{10} +(-34507916295151860 - 1124553276112 \beta_{1} + 13291992 \beta_{2}) q^{11} +(-15890952065925204 - 2721241514736 \beta_{1} + 43046721 \beta_{2}) q^{12} +(638470867119282830 - 20317957756848 \beta_{1} - 460386584 \beta_{2}) q^{13} +(2947741324795972656 - 23848982404616 \beta_{1} + 464588352 \beta_{2}) q^{14} +(735551143647988230 + 12095439853464 \beta_{1} + 4649045868 \beta_{2}) q^{15} +(-16387763951764585808 + 198529037952768 \beta_{1} - 12589833372 \beta_{2}) q^{16} +(-5650418010665548782 + 3009063412790256 \beta_{1} - 5037146568 \beta_{2}) q^{17} +(25449379273691184294 - 1853020188851841 \beta_{1}) q^{18} +(\)\(87\!\cdots\!88\)\( - 3612192985132560 \beta_{1} + 273191138616 \beta_{2}) q^{19} +(\)\(22\!\cdots\!60\)\( - 7902406121558432 \beta_{1} - 570284067834 \beta_{2}) q^{20} +(\)\(10\!\cdots\!92\)\( - 13930379343327816 \beta_{1} - 275671201284 \beta_{2}) q^{21} +(\)\(85\!\cdots\!76\)\( - 57532809152138508 \beta_{1} + 831943364224 \beta_{2}) q^{22} +(\)\(26\!\cdots\!56\)\( - 20101240887203216 \beta_{1} + 2300566232952 \beta_{2}) q^{23} +(\)\(16\!\cdots\!44\)\( + 133138254286795392 \beta_{1} + 1773610998642 \beta_{2}) q^{24} +(\)\(14\!\cdots\!75\)\( + 1314683245233833760 \beta_{1} - 25998439792880 \beta_{2}) q^{25} +(\)\(17\!\cdots\!52\)\( - 383232636676627598 \beta_{1} + 30452908017024 \beta_{2}) q^{26} +\)\(79\!\cdots\!61\)\( q^{27} +(\)\(14\!\cdots\!44\)\( - 2620158482959136640 \beta_{1} + 68631475550856 \beta_{2}) q^{28} +(\)\(39\!\cdots\!86\)\( - 9122515363106364664 \beta_{1} - 173787588879516 \beta_{2}) q^{29} +(-\)\(87\!\cdots\!60\)\( - 12866868826018758918 \beta_{1} - 114439535591616 \beta_{2}) q^{30} +(-\)\(57\!\cdots\!92\)\( + 32335952639255595288 \beta_{1} + 532179934052908 \beta_{2}) q^{31} +(-\)\(51\!\cdots\!28\)\( + 34116717155837996544 \beta_{1} - 275298931161144 \beta_{2}) q^{32} +(-\)\(14\!\cdots\!60\)\( - 48408331126429228752 \beta_{1} + 572176671158232 \beta_{2}) q^{33} +(-\)\(24\!\cdots\!48\)\( + \)\(16\!\cdots\!10\)\( \beta_{1} - 2898175668242304 \beta_{2}) q^{34} +(-\)\(15\!\cdots\!60\)\( - \)\(16\!\cdots\!08\)\( \beta_{1} + 3119819378072904 \beta_{2}) q^{35} +(-\)\(68\!\cdots\!84\)\( - \)\(11\!\cdots\!56\)\( \beta_{1} + 1853020188851841 \beta_{2}) q^{36} +(\)\(76\!\cdots\!62\)\( + \)\(69\!\cdots\!92\)\( \beta_{1} - 1782384058562208 \beta_{2}) q^{37} +(\)\(41\!\cdots\!68\)\( - \)\(18\!\cdots\!40\)\( \beta_{1} - 2401836740360064 \beta_{2}) q^{38} +(\)\(27\!\cdots\!30\)\( - \)\(87\!\cdots\!08\)\( \beta_{1} - 19818132833591064 \beta_{2}) q^{39} +(\)\(11\!\cdots\!00\)\( + \)\(15\!\cdots\!20\)\( \beta_{1} + 43292946343949740 \beta_{2}) q^{40} +(\)\(14\!\cdots\!38\)\( + \)\(37\!\cdots\!80\)\( \beta_{1} - 7030519097942808 \beta_{2}) q^{41} +(\)\(12\!\cdots\!76\)\( - \)\(10\!\cdots\!36\)\( \beta_{1} + 19999005168393792 \beta_{2}) q^{42} +(\)\(31\!\cdots\!48\)\( + \)\(59\!\cdots\!56\)\( \beta_{1} - 91355347912557080 \beta_{2}) q^{43} +(\)\(87\!\cdots\!84\)\( - \)\(40\!\cdots\!76\)\( \beta_{1} - 74958933945275892 \beta_{2}) q^{44} +(\)\(31\!\cdots\!30\)\( + \)\(52\!\cdots\!44\)\( \beta_{1} + 200126180395998828 \beta_{2}) q^{45} +(\)\(52\!\cdots\!28\)\( - \)\(33\!\cdots\!72\)\( \beta_{1} - 30543424165002112 \beta_{2}) q^{46} +(-\)\(29\!\cdots\!84\)\( + \)\(95\!\cdots\!60\)\( \beta_{1} + 411342274012752936 \beta_{2}) q^{47} +(-\)\(70\!\cdots\!68\)\( + \)\(85\!\cdots\!28\)\( \beta_{1} - 541951044600973212 \beta_{2}) q^{48} +(-\)\(53\!\cdots\!91\)\( - \)\(48\!\cdots\!36\)\( \beta_{1} - 184357709370325568 \beta_{2}) q^{49} +(-\)\(86\!\cdots\!50\)\( - \)\(47\!\cdots\!95\)\( \beta_{1} - 742353591633373440 \beta_{2}) q^{50} +(-\)\(24\!\cdots\!22\)\( + \)\(12\!\cdots\!76\)\( \beta_{1} - 216832642948803528 \beta_{2}) q^{51} +(-\)\(44\!\cdots\!32\)\( - \)\(99\!\cdots\!20\)\( \beta_{1} + 3667532963184174990 \beta_{2}) q^{52} +(-\)\(39\!\cdots\!34\)\( + \)\(21\!\cdots\!20\)\( \beta_{1} + 996010849755971604 \beta_{2}) q^{53} +(\)\(10\!\cdots\!74\)\( - \)\(79\!\cdots\!61\)\( \beta_{1}) q^{54} +(\)\(28\!\cdots\!60\)\( - \)\(19\!\cdots\!72\)\( \beta_{1} - 11954579525329437664 \beta_{2}) q^{55} +(-\)\(40\!\cdots\!80\)\( - \)\(12\!\cdots\!64\)\( \beta_{1} - 2881478375702479728 \beta_{2}) q^{56} +(\)\(37\!\cdots\!48\)\( - \)\(15\!\cdots\!60\)\( \beta_{1} + 11759982723675278136 \beta_{2}) q^{57} +(\)\(78\!\cdots\!36\)\( - \)\(36\!\cdots\!02\)\( \beta_{1} + 12948275344700029888 \beta_{2}) q^{58} +(\)\(43\!\cdots\!52\)\( - \)\(40\!\cdots\!64\)\( \beta_{1} + 10719599070768289056 \beta_{2}) q^{59} +(\)\(95\!\cdots\!60\)\( - \)\(34\!\cdots\!72\)\( \beta_{1} - 24548859158795272314 \beta_{2}) q^{60} +(\)\(12\!\cdots\!54\)\( + \)\(37\!\cdots\!04\)\( \beta_{1} - 4256599947344600048 \beta_{2}) q^{61} +(-\)\(26\!\cdots\!56\)\( + \)\(71\!\cdots\!92\)\( \beta_{1} - 44051361707496312000 \beta_{2}) q^{62} +(\)\(47\!\cdots\!32\)\( - \)\(59\!\cdots\!36\)\( \beta_{1} - 11866741289407189764 \beta_{2}) q^{63} +(-\)\(20\!\cdots\!48\)\( + \)\(58\!\cdots\!68\)\( \beta_{1} + 80089558704402231696 \beta_{2}) q^{64} +(-\)\(11\!\cdots\!80\)\( - \)\(11\!\cdots\!24\)\( \beta_{1} + \)\(10\!\cdots\!12\)\( \beta_{2}) q^{65} +(\)\(36\!\cdots\!96\)\( - \)\(24\!\cdots\!68\)\( \beta_{1} + 35812433887551909504 \beta_{2}) q^{66} +(-\)\(52\!\cdots\!32\)\( - \)\(11\!\cdots\!68\)\( \beta_{1} - \)\(28\!\cdots\!16\)\( \beta_{2}) q^{67} +(-\)\(16\!\cdots\!72\)\( + \)\(14\!\cdots\!24\)\( \beta_{1} - 61268200541361628398 \beta_{2}) q^{68} +(\)\(11\!\cdots\!76\)\( - \)\(86\!\cdots\!36\)\( \beta_{1} + 99031832771905750392 \beta_{2}) q^{69} +(\)\(11\!\cdots\!20\)\( - \)\(11\!\cdots\!04\)\( \beta_{1} + 94795686836474695552 \beta_{2}) q^{70} +(-\)\(22\!\cdots\!28\)\( + \)\(63\!\cdots\!84\)\( \beta_{1} - 45679595098071025512 \beta_{2}) q^{71} +(\)\(71\!\cdots\!24\)\( + \)\(57\!\cdots\!32\)\( \beta_{1} + 76348137821073552882 \beta_{2}) q^{72} +(\)\(58\!\cdots\!06\)\( - \)\(18\!\cdots\!12\)\( \beta_{1} + \)\(86\!\cdots\!60\)\( \beta_{2}) q^{73} +(-\)\(54\!\cdots\!24\)\( + \)\(31\!\cdots\!98\)\( \beta_{1} - \)\(65\!\cdots\!80\)\( \beta_{2}) q^{74} +(\)\(60\!\cdots\!75\)\( + \)\(56\!\cdots\!60\)\( \beta_{1} - \)\(11\!\cdots\!80\)\( \beta_{2}) q^{75} +(\)\(75\!\cdots\!72\)\( - \)\(92\!\cdots\!00\)\( \beta_{1} - \)\(49\!\cdots\!36\)\( \beta_{2}) q^{76} +(\)\(17\!\cdots\!64\)\( - \)\(30\!\cdots\!68\)\( \beta_{1} + \)\(12\!\cdots\!80\)\( \beta_{2}) q^{77} +(\)\(74\!\cdots\!92\)\( - \)\(16\!\cdots\!58\)\( \beta_{1} + \)\(13\!\cdots\!04\)\( \beta_{2}) q^{78} +(\)\(15\!\cdots\!80\)\( - \)\(38\!\cdots\!16\)\( \beta_{1} - \)\(15\!\cdots\!32\)\( \beta_{2}) q^{79} +(-\)\(29\!\cdots\!20\)\( - \)\(87\!\cdots\!96\)\( \beta_{1} + \)\(24\!\cdots\!48\)\( \beta_{2}) q^{80} +\)\(34\!\cdots\!81\)\( q^{81} +(-\)\(27\!\cdots\!96\)\( + \)\(60\!\cdots\!38\)\( \beta_{1} - \)\(35\!\cdots\!68\)\( \beta_{2}) q^{82} +(-\)\(36\!\cdots\!88\)\( + \)\(13\!\cdots\!72\)\( \beta_{1} - \)\(53\!\cdots\!40\)\( \beta_{2}) q^{83} +(\)\(60\!\cdots\!24\)\( - \)\(11\!\cdots\!40\)\( \beta_{1} + \)\(29\!\cdots\!76\)\( \beta_{2}) q^{84} +(-\)\(52\!\cdots\!60\)\( + \)\(85\!\cdots\!52\)\( \beta_{1} + \)\(13\!\cdots\!24\)\( \beta_{2}) q^{85} +(-\)\(43\!\cdots\!20\)\( + \)\(23\!\cdots\!04\)\( \beta_{1} - \)\(39\!\cdots\!36\)\( \beta_{2}) q^{86} +(\)\(16\!\cdots\!06\)\( - \)\(39\!\cdots\!44\)\( \beta_{1} - \)\(74\!\cdots\!36\)\( \beta_{2}) q^{87} +(-\)\(29\!\cdots\!96\)\( - \)\(37\!\cdots\!96\)\( \beta_{1} - \)\(14\!\cdots\!44\)\( \beta_{2}) q^{88} +(\)\(93\!\cdots\!98\)\( - \)\(84\!\cdots\!68\)\( \beta_{1} + \)\(37\!\cdots\!80\)\( \beta_{2}) q^{89} +(-\)\(37\!\cdots\!60\)\( - \)\(55\!\cdots\!78\)\( \beta_{1} - \)\(49\!\cdots\!36\)\( \beta_{2}) q^{90} +(\)\(13\!\cdots\!28\)\( + \)\(95\!\cdots\!32\)\( \beta_{1} - \)\(72\!\cdots\!64\)\( \beta_{2}) q^{91} +(\)\(51\!\cdots\!72\)\( - \)\(19\!\cdots\!36\)\( \beta_{1} + \)\(14\!\cdots\!56\)\( \beta_{2}) q^{92} +(-\)\(24\!\cdots\!32\)\( + \)\(13\!\cdots\!48\)\( \beta_{1} + \)\(22\!\cdots\!68\)\( \beta_{2}) q^{93} +(-\)\(11\!\cdots\!00\)\( + \)\(22\!\cdots\!32\)\( \beta_{1} - \)\(18\!\cdots\!64\)\( \beta_{2}) q^{94} +(\)\(65\!\cdots\!00\)\( + \)\(22\!\cdots\!80\)\( \beta_{1} + \)\(63\!\cdots\!60\)\( \beta_{2}) q^{95} +(-\)\(22\!\cdots\!88\)\( + \)\(14\!\cdots\!24\)\( \beta_{1} - \)\(11\!\cdots\!24\)\( \beta_{2}) q^{96} +(\)\(27\!\cdots\!18\)\( - \)\(63\!\cdots\!48\)\( \beta_{1} - \)\(75\!\cdots\!76\)\( \beta_{2}) q^{97} +(-\)\(34\!\cdots\!50\)\( + \)\(56\!\cdots\!75\)\( \beta_{1} + \)\(88\!\cdots\!88\)\( \beta_{2}) q^{98} +(-\)\(63\!\cdots\!60\)\( - \)\(20\!\cdots\!92\)\( \beta_{1} + \)\(24\!\cdots\!72\)\( \beta_{2}) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 41202q^{2} + 129140163q^{3} - 1107467772q^{4} + 51261823890q^{5} + 1773610998642q^{6} + 76113734015256q^{7} + 1154149348489992q^{8} + 5559060566555523q^{9} + O(q^{10}) \) \( 3q + 41202q^{2} + 129140163q^{3} - 1107467772q^{4} + 51261823890q^{5} + 1773610998642q^{6} + 76113734015256q^{7} + 1154149348489992q^{8} + 5559060566555523q^{9} - 6066746915652180q^{10} - 103523748885455580q^{11} - 47672856197775612q^{12} + 1915412601357848490q^{13} + 8843223974387917968q^{14} + 2206653430943964690q^{15} - 49163291855293757424q^{16} - 16951254031996646346q^{17} + 76348137821073552882q^{18} + \)\(26\!\cdots\!64\)\(q^{19} + \)\(66\!\cdots\!80\)\(q^{20} + \)\(32\!\cdots\!76\)\(q^{21} + \)\(25\!\cdots\!28\)\(q^{22} + \)\(79\!\cdots\!68\)\(q^{23} + \)\(49\!\cdots\!32\)\(q^{24} + \)\(42\!\cdots\!25\)\(q^{25} + \)\(51\!\cdots\!56\)\(q^{26} + \)\(23\!\cdots\!83\)\(q^{27} + \)\(42\!\cdots\!32\)\(q^{28} + \)\(11\!\cdots\!58\)\(q^{29} - \)\(26\!\cdots\!80\)\(q^{30} - \)\(17\!\cdots\!76\)\(q^{31} - \)\(15\!\cdots\!84\)\(q^{32} - \)\(44\!\cdots\!80\)\(q^{33} - \)\(72\!\cdots\!44\)\(q^{34} - \)\(46\!\cdots\!80\)\(q^{35} - \)\(20\!\cdots\!52\)\(q^{36} + \)\(23\!\cdots\!86\)\(q^{37} + \)\(12\!\cdots\!04\)\(q^{38} + \)\(82\!\cdots\!90\)\(q^{39} + \)\(33\!\cdots\!00\)\(q^{40} + \)\(42\!\cdots\!14\)\(q^{41} + \)\(38\!\cdots\!28\)\(q^{42} + \)\(93\!\cdots\!44\)\(q^{43} + \)\(26\!\cdots\!52\)\(q^{44} + \)\(94\!\cdots\!90\)\(q^{45} + \)\(15\!\cdots\!84\)\(q^{46} - \)\(87\!\cdots\!52\)\(q^{47} - \)\(21\!\cdots\!04\)\(q^{48} - \)\(16\!\cdots\!73\)\(q^{49} - \)\(25\!\cdots\!50\)\(q^{50} - \)\(72\!\cdots\!66\)\(q^{51} - \)\(13\!\cdots\!96\)\(q^{52} - \)\(11\!\cdots\!02\)\(q^{53} + \)\(32\!\cdots\!22\)\(q^{54} + \)\(85\!\cdots\!80\)\(q^{55} - \)\(12\!\cdots\!40\)\(q^{56} + \)\(11\!\cdots\!44\)\(q^{57} + \)\(23\!\cdots\!08\)\(q^{58} + \)\(12\!\cdots\!56\)\(q^{59} + \)\(28\!\cdots\!80\)\(q^{60} + \)\(36\!\cdots\!62\)\(q^{61} - \)\(80\!\cdots\!68\)\(q^{62} + \)\(14\!\cdots\!96\)\(q^{63} - \)\(61\!\cdots\!44\)\(q^{64} - \)\(33\!\cdots\!40\)\(q^{65} + \)\(11\!\cdots\!88\)\(q^{66} - \)\(15\!\cdots\!96\)\(q^{67} - \)\(49\!\cdots\!16\)\(q^{68} + \)\(34\!\cdots\!28\)\(q^{69} + \)\(33\!\cdots\!60\)\(q^{70} - \)\(66\!\cdots\!84\)\(q^{71} + \)\(21\!\cdots\!72\)\(q^{72} + \)\(17\!\cdots\!18\)\(q^{73} - \)\(16\!\cdots\!72\)\(q^{74} + \)\(18\!\cdots\!25\)\(q^{75} + \)\(22\!\cdots\!16\)\(q^{76} + \)\(52\!\cdots\!92\)\(q^{77} + \)\(22\!\cdots\!76\)\(q^{78} + \)\(47\!\cdots\!40\)\(q^{79} - \)\(89\!\cdots\!60\)\(q^{80} + \)\(10\!\cdots\!43\)\(q^{81} - \)\(83\!\cdots\!88\)\(q^{82} - \)\(10\!\cdots\!64\)\(q^{83} + \)\(18\!\cdots\!72\)\(q^{84} - \)\(15\!\cdots\!80\)\(q^{85} - \)\(13\!\cdots\!60\)\(q^{86} + \)\(50\!\cdots\!18\)\(q^{87} - \)\(87\!\cdots\!88\)\(q^{88} + \)\(28\!\cdots\!94\)\(q^{89} - \)\(11\!\cdots\!80\)\(q^{90} + \)\(40\!\cdots\!84\)\(q^{91} + \)\(15\!\cdots\!16\)\(q^{92} - \)\(74\!\cdots\!96\)\(q^{93} - \)\(34\!\cdots\!00\)\(q^{94} + \)\(19\!\cdots\!00\)\(q^{95} - \)\(66\!\cdots\!64\)\(q^{96} + \)\(81\!\cdots\!54\)\(q^{97} - \)\(10\!\cdots\!50\)\(q^{98} - \)\(19\!\cdots\!80\)\(q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 35900150 x + 10469144400\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{2} + 1429 \nu - 23933910 \)\()/195\)
\(\beta_{2}\)\(=\)\((\)\( -5046 \nu^{2} + 57654066 \nu + 120748888260 \)\()/65\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} + 15138 \beta_{1} + 332640\)\()/997920\)
\(\nu^{2}\)\(=\)\((\)\(-1429 \beta_{2} + 172962198 \beta_{1} + 23883652124640\)\()/997920\)

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
5840.68
−6131.99
292.312
−81270.9 4.30467e7 −1.98497e9 5.17904e11 −3.49845e12 −3.34870e13 8.59432e14 1.85302e15 −4.20905e16
1.2 −11418.8 4.30467e7 −8.45955e9 −6.77881e11 −4.91541e11 5.88597e13 1.94684e14 1.85302e15 7.74057e15
1.3 133892. 4.30467e7 9.33705e9 2.11239e11 5.76360e12 5.07411e13 1.00033e14 1.85302e15 2.82832e16
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)

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 3.34.a.a 3
3.b odd 2 1 9.34.a.d 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.34.a.a 3 1.a even 1 1 trivial
9.34.a.d 3 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{3} - 41202 T_{2}^{2} - 11482365600 T_{2} - \)124253441040384

'>\(12\!\cdots\!84\)\( \) acting on \(S_{34}^{\mathrm{new}}(\Gamma_0(3))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 41202 T + 14287438176 T^{2} - 832098411159552 T^{3} + \)\(12\!\cdots\!92\)\( T^{4} - \)\(30\!\cdots\!28\)\( T^{5} + \)\(63\!\cdots\!88\)\( T^{6} \)
$3$ \( ( 1 - 43046721 T )^{3} \)
$5$ \( 1 - 51261823890 T - \)\(35\!\cdots\!25\)\( T^{2} + \)\(62\!\cdots\!00\)\( T^{3} - \)\(41\!\cdots\!25\)\( T^{4} - \)\(69\!\cdots\!50\)\( T^{5} + \)\(15\!\cdots\!25\)\( T^{6} \)
$7$ \( 1 - 76113734015256 T + \)\(22\!\cdots\!65\)\( T^{2} - \)\(10\!\cdots\!64\)\( T^{3} + \)\(17\!\cdots\!55\)\( T^{4} - \)\(45\!\cdots\!44\)\( T^{5} + \)\(46\!\cdots\!43\)\( T^{6} \)
$11$ \( 1 + 103523748885455580 T + \)\(52\!\cdots\!61\)\( T^{2} + \)\(32\!\cdots\!96\)\( T^{3} + \)\(12\!\cdots\!91\)\( T^{4} + \)\(55\!\cdots\!80\)\( T^{5} + \)\(12\!\cdots\!91\)\( T^{6} \)
$13$ \( 1 - 1915412601357848490 T + \)\(65\!\cdots\!31\)\( T^{2} - \)\(25\!\cdots\!52\)\( T^{3} + \)\(37\!\cdots\!43\)\( T^{4} - \)\(63\!\cdots\!10\)\( T^{5} + \)\(19\!\cdots\!77\)\( T^{6} \)
$17$ \( 1 + 16951254031996646346 T + \)\(10\!\cdots\!83\)\( T^{2} + \)\(11\!\cdots\!72\)\( T^{3} + \)\(44\!\cdots\!71\)\( T^{4} + \)\(27\!\cdots\!74\)\( T^{5} + \)\(65\!\cdots\!53\)\( T^{6} \)
$19$ \( 1 - \)\(26\!\cdots\!64\)\( T + \)\(44\!\cdots\!37\)\( T^{2} - \)\(50\!\cdots\!52\)\( T^{3} + \)\(70\!\cdots\!83\)\( T^{4} - \)\(65\!\cdots\!84\)\( T^{5} + \)\(39\!\cdots\!79\)\( T^{6} \)
$23$ \( 1 - \)\(79\!\cdots\!68\)\( T + \)\(45\!\cdots\!85\)\( T^{2} - \)\(15\!\cdots\!08\)\( T^{3} + \)\(39\!\cdots\!55\)\( T^{4} - \)\(59\!\cdots\!52\)\( T^{5} + \)\(64\!\cdots\!87\)\( T^{6} \)
$29$ \( 1 - \)\(11\!\cdots\!58\)\( T + \)\(39\!\cdots\!87\)\( T^{2} - \)\(24\!\cdots\!24\)\( T^{3} + \)\(70\!\cdots\!43\)\( T^{4} - \)\(38\!\cdots\!18\)\( T^{5} + \)\(59\!\cdots\!69\)\( T^{6} \)
$31$ \( 1 + \)\(17\!\cdots\!76\)\( T + \)\(28\!\cdots\!17\)\( T^{2} + \)\(47\!\cdots\!32\)\( T^{3} + \)\(46\!\cdots\!47\)\( T^{4} + \)\(46\!\cdots\!56\)\( T^{5} + \)\(44\!\cdots\!71\)\( T^{6} \)
$37$ \( 1 - \)\(23\!\cdots\!86\)\( T + \)\(11\!\cdots\!95\)\( T^{2} - \)\(65\!\cdots\!24\)\( T^{3} + \)\(62\!\cdots\!15\)\( T^{4} - \)\(73\!\cdots\!74\)\( T^{5} + \)\(17\!\cdots\!73\)\( T^{6} \)
$41$ \( 1 - \)\(42\!\cdots\!14\)\( T + \)\(39\!\cdots\!27\)\( T^{2} - \)\(10\!\cdots\!68\)\( T^{3} + \)\(65\!\cdots\!67\)\( T^{4} - \)\(11\!\cdots\!74\)\( T^{5} + \)\(46\!\cdots\!61\)\( T^{6} \)
$43$ \( 1 - \)\(93\!\cdots\!44\)\( T + \)\(19\!\cdots\!53\)\( T^{2} - \)\(11\!\cdots\!40\)\( T^{3} + \)\(16\!\cdots\!79\)\( T^{4} - \)\(60\!\cdots\!56\)\( T^{5} + \)\(51\!\cdots\!07\)\( T^{6} \)
$47$ \( 1 + \)\(87\!\cdots\!52\)\( T + \)\(64\!\cdots\!57\)\( T^{2} + \)\(26\!\cdots\!60\)\( T^{3} + \)\(97\!\cdots\!39\)\( T^{4} + \)\(19\!\cdots\!08\)\( T^{5} + \)\(34\!\cdots\!83\)\( T^{6} \)
$53$ \( 1 + \)\(11\!\cdots\!02\)\( T + \)\(18\!\cdots\!15\)\( T^{2} + \)\(16\!\cdots\!12\)\( T^{3} + \)\(14\!\cdots\!95\)\( T^{4} + \)\(75\!\cdots\!58\)\( T^{5} + \)\(50\!\cdots\!17\)\( T^{6} \)
$59$ \( 1 - \)\(12\!\cdots\!56\)\( T + \)\(82\!\cdots\!37\)\( T^{2} - \)\(68\!\cdots\!48\)\( T^{3} + \)\(22\!\cdots\!23\)\( T^{4} - \)\(97\!\cdots\!96\)\( T^{5} + \)\(20\!\cdots\!39\)\( T^{6} \)
$61$ \( 1 - \)\(36\!\cdots\!62\)\( T + \)\(12\!\cdots\!59\)\( T^{2} - \)\(22\!\cdots\!36\)\( T^{3} + \)\(10\!\cdots\!79\)\( T^{4} - \)\(24\!\cdots\!82\)\( T^{5} + \)\(55\!\cdots\!41\)\( T^{6} \)
$67$ \( 1 + \)\(15\!\cdots\!96\)\( T + \)\(11\!\cdots\!33\)\( T^{2} + \)\(29\!\cdots\!72\)\( T^{3} + \)\(20\!\cdots\!71\)\( T^{4} + \)\(52\!\cdots\!24\)\( T^{5} + \)\(60\!\cdots\!03\)\( T^{6} \)
$71$ \( 1 + \)\(66\!\cdots\!84\)\( T + \)\(51\!\cdots\!85\)\( T^{2} + \)\(17\!\cdots\!00\)\( T^{3} + \)\(63\!\cdots\!35\)\( T^{4} + \)\(10\!\cdots\!64\)\( T^{5} + \)\(18\!\cdots\!31\)\( T^{6} \)
$73$ \( 1 - \)\(17\!\cdots\!18\)\( T + \)\(16\!\cdots\!55\)\( T^{2} - \)\(10\!\cdots\!88\)\( T^{3} + \)\(52\!\cdots\!15\)\( T^{4} - \)\(16\!\cdots\!02\)\( T^{5} + \)\(29\!\cdots\!37\)\( T^{6} \)
$79$ \( 1 - \)\(47\!\cdots\!40\)\( T + \)\(18\!\cdots\!17\)\( T^{2} - \)\(41\!\cdots\!20\)\( T^{3} + \)\(79\!\cdots\!63\)\( T^{4} - \)\(82\!\cdots\!40\)\( T^{5} + \)\(73\!\cdots\!19\)\( T^{6} \)
$83$ \( 1 + \)\(10\!\cdots\!64\)\( T + \)\(92\!\cdots\!69\)\( T^{2} + \)\(46\!\cdots\!56\)\( T^{3} + \)\(19\!\cdots\!47\)\( T^{4} + \)\(50\!\cdots\!16\)\( T^{5} + \)\(97\!\cdots\!47\)\( T^{6} \)
$89$ \( 1 - \)\(28\!\cdots\!94\)\( T + \)\(81\!\cdots\!67\)\( T^{2} - \)\(12\!\cdots\!72\)\( T^{3} + \)\(17\!\cdots\!23\)\( T^{4} - \)\(12\!\cdots\!34\)\( T^{5} + \)\(97\!\cdots\!09\)\( T^{6} \)
$97$ \( 1 - \)\(81\!\cdots\!54\)\( T + \)\(65\!\cdots\!03\)\( T^{2} - \)\(25\!\cdots\!48\)\( T^{3} + \)\(23\!\cdots\!31\)\( T^{4} - \)\(10\!\cdots\!66\)\( T^{5} + \)\(49\!\cdots\!33\)\( T^{6} \)
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