Properties

Label 3.36.a.a
Level 3
Weight 36
Character orbit 3.a
Self dual Yes
Analytic conductor 23.279
Analytic rank 1
Dimension 2
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 3 \)
Weight: \( k \) = \( 36 \)
Character orbit: \([\chi]\) = 3.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(23.2785391901\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2196841}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{4}\cdot 3\cdot 7 \)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 168\sqrt{2196841}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + ( -30456 - \beta ) q^{2} \) \( + 129140163 q^{3} \) \( + ( 28571469952 + 60912 \beta ) q^{4} \) \( + ( -666889748370 + 1210960 \beta ) q^{5} \) \( + ( -3933092804328 - 129140163 \beta ) q^{6} \) \( + ( -600756391476472 - 142131024 \beta ) q^{7} \) \( + ( -3600478240192512 + 3933132544 \beta ) q^{8} \) \( + 16677181699666569 q^{9} \) \(+O(q^{10})\) \( q\) \(+(-30456 - \beta) q^{2}\) \(+129140163 q^{3}\) \(+(28571469952 + 60912 \beta) q^{4}\) \(+(-666889748370 + 1210960 \beta) q^{5}\) \(+(-3933092804328 - 129140163 \beta) q^{6}\) \(+(-600756391476472 - 142131024 \beta) q^{7}\) \(+(-3600478240192512 + 3933132544 \beta) q^{8}\) \(+16677181699666569 q^{9}\) \(+(-54773134183051920 + 630008750610 \beta) q^{10}\) \(+(-737221926110660316 + 3560156783456 \beta) q^{11}\) \(+(3689724286750882176 + 7866185608656 \beta) q^{12}\) \(+(15002606829102839414 + 161668794040416 \beta) q^{13}\) \(+(27109277558313104448 + 605085133943416 \beta) q^{14}\) \(+(-86122250807530784310 + 156383571786480 \beta) q^{15}\) \(+(-\)\(11\!\cdots\!60\)\( + 1387770371960832 \beta) q^{16}\) \(+(\)\(30\!\cdots\!66\)\( - 5140788294161760 \beta) q^{17}\) \(+(-\)\(50\!\cdots\!64\)\( - 16677181699666569 \beta) q^{18}\) \(+(-\)\(80\!\cdots\!32\)\( - 101300863898712096 \beta) q^{19}\) \(+(-\)\(14\!\cdots\!60\)\( - 6022681099639520 \beta) q^{20}\) \(+(-\)\(77\!\cdots\!36\)\( - 18354823606716912 \beta) q^{21}\) \(+(-\)\(19\!\cdots\!08\)\( + 628793791113724380 \beta) q^{22}\) \(+(\)\(24\!\cdots\!36\)\( + 3053203101283215712 \beta) q^{23}\) \(+(-\)\(46\!\cdots\!56\)\( + 507925377832764672 \beta) q^{24}\) \(+(-\)\(23\!\cdots\!25\)\( - 1615153619372270400 \beta) q^{25}\) \(+(-\)\(10\!\cdots\!28\)\( - 19926391620397749110 \beta) q^{26}\) \(+\)\(21\!\cdots\!47\)\( q^{27}\) \(+(-\)\(17\!\cdots\!36\)\( - 40654165599077853312 \beta) q^{28}\) \(+(-\)\(39\!\cdots\!74\)\( - 22123136384176647952 \beta) q^{29}\) \(+(-\)\(70\!\cdots\!60\)\( + 81359432745201749430 \beta) q^{30}\) \(+(-\)\(60\!\cdots\!24\)\( + \)\(15\!\cdots\!92\)\( \beta) q^{31}\) \(+(\)\(71\!\cdots\!88\)\( + \)\(93\!\cdots\!76\)\( \beta) q^{32}\) \(+(-\)\(95\!\cdots\!08\)\( + \)\(45\!\cdots\!28\)\( \beta) q^{33}\) \(+(\)\(22\!\cdots\!44\)\( - \)\(29\!\cdots\!06\)\( \beta) q^{34}\) \(+(\)\(38\!\cdots\!80\)\( - \)\(63\!\cdots\!40\)\( \beta) q^{35}\) \(+(\)\(47\!\cdots\!88\)\( + \)\(10\!\cdots\!28\)\( \beta) q^{36}\) \(+(\)\(83\!\cdots\!58\)\( - \)\(15\!\cdots\!04\)\( \beta) q^{37}\) \(+(\)\(65\!\cdots\!56\)\( + \)\(11\!\cdots\!08\)\( \beta) q^{38}\) \(+(\)\(19\!\cdots\!82\)\( + \)\(20\!\cdots\!08\)\( \beta) q^{39}\) \(+(\)\(26\!\cdots\!00\)\( - \)\(69\!\cdots\!00\)\( \beta) q^{40}\) \(+(-\)\(16\!\cdots\!94\)\( + \)\(49\!\cdots\!32\)\( \beta) q^{41}\) \(+(\)\(35\!\cdots\!24\)\( + \)\(78\!\cdots\!08\)\( \beta) q^{42}\) \(+(\)\(50\!\cdots\!60\)\( - \)\(25\!\cdots\!56\)\( \beta) q^{43}\) \(+(-\)\(76\!\cdots\!84\)\( + \)\(56\!\cdots\!20\)\( \beta) q^{44}\) \(+(-\)\(11\!\cdots\!30\)\( + \)\(20\!\cdots\!40\)\( \beta) q^{45}\) \(+(-\)\(19\!\cdots\!24\)\( - \)\(33\!\cdots\!08\)\( \beta) q^{46}\) \(+(-\)\(26\!\cdots\!60\)\( - \)\(10\!\cdots\!88\)\( \beta) q^{47}\) \(+(-\)\(14\!\cdots\!80\)\( + \)\(17\!\cdots\!16\)\( \beta) q^{48}\) \(+(-\)\(16\!\cdots\!75\)\( + \)\(17\!\cdots\!56\)\( \beta) q^{49}\) \(+(\)\(17\!\cdots\!00\)\( + \)\(24\!\cdots\!25\)\( \beta) q^{50}\) \(+(\)\(39\!\cdots\!58\)\( - \)\(66\!\cdots\!80\)\( \beta) q^{51}\) \(+(\)\(10\!\cdots\!56\)\( + \)\(55\!\cdots\!00\)\( \beta) q^{52}\) \(+(-\)\(15\!\cdots\!34\)\( - \)\(99\!\cdots\!88\)\( \beta) q^{53}\) \(+(-\)\(65\!\cdots\!32\)\( - \)\(21\!\cdots\!47\)\( \beta) q^{54}\) \(+(\)\(75\!\cdots\!60\)\( - \)\(32\!\cdots\!80\)\( \beta) q^{55}\) \(+(\)\(21\!\cdots\!60\)\( - \)\(18\!\cdots\!80\)\( \beta) q^{56}\) \(+(-\)\(10\!\cdots\!16\)\( - \)\(13\!\cdots\!48\)\( \beta) q^{57}\) \(+(\)\(25\!\cdots\!12\)\( + \)\(40\!\cdots\!86\)\( \beta) q^{58}\) \(+(\)\(41\!\cdots\!32\)\( - \)\(66\!\cdots\!64\)\( \beta) q^{59}\) \(+(-\)\(18\!\cdots\!80\)\( - \)\(77\!\cdots\!60\)\( \beta) q^{60}\) \(+(-\)\(20\!\cdots\!70\)\( + \)\(26\!\cdots\!24\)\( \beta) q^{61}\) \(+(-\)\(76\!\cdots\!84\)\( + \)\(55\!\cdots\!72\)\( \beta) q^{62}\) \(+(-\)\(10\!\cdots\!68\)\( - \)\(23\!\cdots\!56\)\( \beta) q^{63}\) \(+(-\)\(22\!\cdots\!32\)\( - \)\(14\!\cdots\!20\)\( \beta) q^{64}\) \(+(\)\(21\!\cdots\!60\)\( - \)\(89\!\cdots\!80\)\( \beta) q^{65}\) \(+(-\)\(25\!\cdots\!04\)\( + \)\(81\!\cdots\!40\)\( \beta) q^{66}\) \(+(\)\(48\!\cdots\!56\)\( - \)\(28\!\cdots\!40\)\( \beta) q^{67}\) \(+(\)\(68\!\cdots\!52\)\( + \)\(41\!\cdots\!72\)\( \beta) q^{68}\) \(+(\)\(31\!\cdots\!68\)\( + \)\(39\!\cdots\!56\)\( \beta) q^{69}\) \(+(\)\(27\!\cdots\!80\)\( - \)\(37\!\cdots\!40\)\( \beta) q^{70}\) \(+(\)\(22\!\cdots\!12\)\( + \)\(10\!\cdots\!80\)\( \beta) q^{71}\) \(+(-\)\(60\!\cdots\!28\)\( + \)\(65\!\cdots\!36\)\( \beta) q^{72}\) \(+(-\)\(68\!\cdots\!54\)\( + \)\(82\!\cdots\!92\)\( \beta) q^{73}\) \(+(\)\(90\!\cdots\!88\)\( - \)\(37\!\cdots\!34\)\( \beta) q^{74}\) \(+(-\)\(30\!\cdots\!75\)\( - \)\(20\!\cdots\!00\)\( \beta) q^{75}\) \(+(-\)\(61\!\cdots\!32\)\( - \)\(33\!\cdots\!76\)\( \beta) q^{76}\) \(+(\)\(41\!\cdots\!56\)\( - \)\(20\!\cdots\!48\)\( \beta) q^{77}\) \(+(-\)\(13\!\cdots\!64\)\( - \)\(25\!\cdots\!30\)\( \beta) q^{78}\) \(+(-\)\(53\!\cdots\!40\)\( + \)\(64\!\cdots\!20\)\( \beta) q^{79}\) \(+(\)\(84\!\cdots\!80\)\( - \)\(22\!\cdots\!40\)\( \beta) q^{80}\) \(+\)\(27\!\cdots\!61\)\( q^{81}\) \(+(-\)\(25\!\cdots\!24\)\( + \)\(14\!\cdots\!02\)\( \beta) q^{82}\) \(+(-\)\(27\!\cdots\!72\)\( - \)\(21\!\cdots\!32\)\( \beta) q^{83}\) \(+(-\)\(22\!\cdots\!68\)\( - \)\(52\!\cdots\!56\)\( \beta) q^{84}\) \(+(-\)\(24\!\cdots\!20\)\( + \)\(71\!\cdots\!60\)\( \beta) q^{85}\) \(+(\)\(15\!\cdots\!44\)\( + \)\(27\!\cdots\!76\)\( \beta) q^{86}\) \(+(-\)\(50\!\cdots\!62\)\( - \)\(28\!\cdots\!76\)\( \beta) q^{87}\) \(+(\)\(35\!\cdots\!68\)\( - \)\(15\!\cdots\!76\)\( \beta) q^{88}\) \(+(\)\(12\!\cdots\!18\)\( - \)\(35\!\cdots\!36\)\( \beta) q^{89}\) \(+(-\)\(91\!\cdots\!80\)\( + \)\(10\!\cdots\!90\)\( \beta) q^{90}\) \(+(-\)\(10\!\cdots\!64\)\( - \)\(99\!\cdots\!88\)\( \beta) q^{91}\) \(+(\)\(18\!\cdots\!68\)\( + \)\(10\!\cdots\!56\)\( \beta) q^{92}\) \(+(-\)\(78\!\cdots\!12\)\( + \)\(19\!\cdots\!96\)\( \beta) q^{93}\) \(+(\)\(72\!\cdots\!52\)\( + \)\(29\!\cdots\!88\)\( \beta) q^{94}\) \(+(-\)\(22\!\cdots\!00\)\( + \)\(57\!\cdots\!00\)\( \beta) q^{95}\) \(+(\)\(92\!\cdots\!44\)\( + \)\(12\!\cdots\!88\)\( \beta) q^{96}\) \(+(\)\(41\!\cdots\!26\)\( - \)\(60\!\cdots\!00\)\( \beta) q^{97}\) \(+(-\)\(10\!\cdots\!04\)\( + \)\(11\!\cdots\!39\)\( \beta) q^{98}\) \(+(-\)\(12\!\cdots\!04\)\( + \)\(59\!\cdots\!64\)\( \beta) q^{99}\) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut -\mathstrut 60912q^{2} \) \(\mathstrut +\mathstrut 258280326q^{3} \) \(\mathstrut +\mathstrut 57142939904q^{4} \) \(\mathstrut -\mathstrut 1333779496740q^{5} \) \(\mathstrut -\mathstrut 7866185608656q^{6} \) \(\mathstrut -\mathstrut 1201512782952944q^{7} \) \(\mathstrut -\mathstrut 7200956480385024q^{8} \) \(\mathstrut +\mathstrut 33354363399333138q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut -\mathstrut 60912q^{2} \) \(\mathstrut +\mathstrut 258280326q^{3} \) \(\mathstrut +\mathstrut 57142939904q^{4} \) \(\mathstrut -\mathstrut 1333779496740q^{5} \) \(\mathstrut -\mathstrut 7866185608656q^{6} \) \(\mathstrut -\mathstrut 1201512782952944q^{7} \) \(\mathstrut -\mathstrut 7200956480385024q^{8} \) \(\mathstrut +\mathstrut 33354363399333138q^{9} \) \(\mathstrut -\mathstrut 109546268366103840q^{10} \) \(\mathstrut -\mathstrut 1474443852221320632q^{11} \) \(\mathstrut +\mathstrut 7379448573501764352q^{12} \) \(\mathstrut +\mathstrut 30005213658205678828q^{13} \) \(\mathstrut +\mathstrut 54218555116626208896q^{14} \) \(\mathstrut -\mathstrut \)\(17\!\cdots\!20\)\(q^{15} \) \(\mathstrut -\mathstrut \)\(22\!\cdots\!20\)\(q^{16} \) \(\mathstrut +\mathstrut \)\(61\!\cdots\!32\)\(q^{17} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!28\)\(q^{18} \) \(\mathstrut -\mathstrut \)\(16\!\cdots\!64\)\(q^{19} \) \(\mathstrut -\mathstrut \)\(28\!\cdots\!20\)\(q^{20} \) \(\mathstrut -\mathstrut \)\(15\!\cdots\!72\)\(q^{21} \) \(\mathstrut -\mathstrut \)\(39\!\cdots\!16\)\(q^{22} \) \(\mathstrut +\mathstrut \)\(49\!\cdots\!72\)\(q^{23} \) \(\mathstrut -\mathstrut \)\(92\!\cdots\!12\)\(q^{24} \) \(\mathstrut -\mathstrut \)\(47\!\cdots\!50\)\(q^{25} \) \(\mathstrut -\mathstrut \)\(20\!\cdots\!56\)\(q^{26} \) \(\mathstrut +\mathstrut \)\(43\!\cdots\!94\)\(q^{27} \) \(\mathstrut -\mathstrut \)\(35\!\cdots\!72\)\(q^{28} \) \(\mathstrut -\mathstrut \)\(78\!\cdots\!48\)\(q^{29} \) \(\mathstrut -\mathstrut \)\(14\!\cdots\!20\)\(q^{30} \) \(\mathstrut -\mathstrut \)\(12\!\cdots\!48\)\(q^{31} \) \(\mathstrut +\mathstrut \)\(14\!\cdots\!76\)\(q^{32} \) \(\mathstrut -\mathstrut \)\(19\!\cdots\!16\)\(q^{33} \) \(\mathstrut +\mathstrut \)\(44\!\cdots\!88\)\(q^{34} \) \(\mathstrut +\mathstrut \)\(77\!\cdots\!60\)\(q^{35} \) \(\mathstrut +\mathstrut \)\(95\!\cdots\!76\)\(q^{36} \) \(\mathstrut +\mathstrut \)\(16\!\cdots\!16\)\(q^{37} \) \(\mathstrut +\mathstrut \)\(13\!\cdots\!12\)\(q^{38} \) \(\mathstrut +\mathstrut \)\(38\!\cdots\!64\)\(q^{39} \) \(\mathstrut +\mathstrut \)\(53\!\cdots\!00\)\(q^{40} \) \(\mathstrut -\mathstrut \)\(32\!\cdots\!88\)\(q^{41} \) \(\mathstrut +\mathstrut \)\(70\!\cdots\!48\)\(q^{42} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!20\)\(q^{43} \) \(\mathstrut -\mathstrut \)\(15\!\cdots\!68\)\(q^{44} \) \(\mathstrut -\mathstrut \)\(22\!\cdots\!60\)\(q^{45} \) \(\mathstrut -\mathstrut \)\(39\!\cdots\!48\)\(q^{46} \) \(\mathstrut -\mathstrut \)\(52\!\cdots\!20\)\(q^{47} \) \(\mathstrut -\mathstrut \)\(28\!\cdots\!60\)\(q^{48} \) \(\mathstrut -\mathstrut \)\(33\!\cdots\!50\)\(q^{49} \) \(\mathstrut +\mathstrut \)\(34\!\cdots\!00\)\(q^{50} \) \(\mathstrut +\mathstrut \)\(79\!\cdots\!16\)\(q^{51} \) \(\mathstrut +\mathstrut \)\(20\!\cdots\!12\)\(q^{52} \) \(\mathstrut -\mathstrut \)\(31\!\cdots\!68\)\(q^{53} \) \(\mathstrut -\mathstrut \)\(13\!\cdots\!64\)\(q^{54} \) \(\mathstrut +\mathstrut \)\(15\!\cdots\!20\)\(q^{55} \) \(\mathstrut +\mathstrut \)\(42\!\cdots\!20\)\(q^{56} \) \(\mathstrut -\mathstrut \)\(20\!\cdots\!32\)\(q^{57} \) \(\mathstrut +\mathstrut \)\(51\!\cdots\!24\)\(q^{58} \) \(\mathstrut +\mathstrut \)\(82\!\cdots\!64\)\(q^{59} \) \(\mathstrut -\mathstrut \)\(37\!\cdots\!60\)\(q^{60} \) \(\mathstrut -\mathstrut \)\(40\!\cdots\!40\)\(q^{61} \) \(\mathstrut -\mathstrut \)\(15\!\cdots\!68\)\(q^{62} \) \(\mathstrut -\mathstrut \)\(20\!\cdots\!36\)\(q^{63} \) \(\mathstrut -\mathstrut \)\(44\!\cdots\!64\)\(q^{64} \) \(\mathstrut +\mathstrut \)\(42\!\cdots\!20\)\(q^{65} \) \(\mathstrut -\mathstrut \)\(51\!\cdots\!08\)\(q^{66} \) \(\mathstrut +\mathstrut \)\(96\!\cdots\!12\)\(q^{67} \) \(\mathstrut +\mathstrut \)\(13\!\cdots\!04\)\(q^{68} \) \(\mathstrut +\mathstrut \)\(63\!\cdots\!36\)\(q^{69} \) \(\mathstrut +\mathstrut \)\(54\!\cdots\!60\)\(q^{70} \) \(\mathstrut +\mathstrut \)\(44\!\cdots\!24\)\(q^{71} \) \(\mathstrut -\mathstrut \)\(12\!\cdots\!56\)\(q^{72} \) \(\mathstrut -\mathstrut \)\(13\!\cdots\!08\)\(q^{73} \) \(\mathstrut +\mathstrut \)\(18\!\cdots\!76\)\(q^{74} \) \(\mathstrut -\mathstrut \)\(61\!\cdots\!50\)\(q^{75} \) \(\mathstrut -\mathstrut \)\(12\!\cdots\!64\)\(q^{76} \) \(\mathstrut +\mathstrut \)\(82\!\cdots\!12\)\(q^{77} \) \(\mathstrut -\mathstrut \)\(27\!\cdots\!28\)\(q^{78} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!80\)\(q^{79} \) \(\mathstrut +\mathstrut \)\(16\!\cdots\!60\)\(q^{80} \) \(\mathstrut +\mathstrut \)\(55\!\cdots\!22\)\(q^{81} \) \(\mathstrut -\mathstrut \)\(51\!\cdots\!48\)\(q^{82} \) \(\mathstrut -\mathstrut \)\(55\!\cdots\!44\)\(q^{83} \) \(\mathstrut -\mathstrut \)\(45\!\cdots\!36\)\(q^{84} \) \(\mathstrut -\mathstrut \)\(48\!\cdots\!40\)\(q^{85} \) \(\mathstrut +\mathstrut \)\(31\!\cdots\!88\)\(q^{86} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!24\)\(q^{87} \) \(\mathstrut +\mathstrut \)\(70\!\cdots\!36\)\(q^{88} \) \(\mathstrut +\mathstrut \)\(24\!\cdots\!36\)\(q^{89} \) \(\mathstrut -\mathstrut \)\(18\!\cdots\!60\)\(q^{90} \) \(\mathstrut -\mathstrut \)\(20\!\cdots\!28\)\(q^{91} \) \(\mathstrut +\mathstrut \)\(37\!\cdots\!36\)\(q^{92} \) \(\mathstrut -\mathstrut \)\(15\!\cdots\!24\)\(q^{93} \) \(\mathstrut +\mathstrut \)\(14\!\cdots\!04\)\(q^{94} \) \(\mathstrut -\mathstrut \)\(45\!\cdots\!00\)\(q^{95} \) \(\mathstrut +\mathstrut \)\(18\!\cdots\!88\)\(q^{96} \) \(\mathstrut +\mathstrut \)\(83\!\cdots\!52\)\(q^{97} \) \(\mathstrut -\mathstrut \)\(20\!\cdots\!08\)\(q^{98} \) \(\mathstrut -\mathstrut \)\(24\!\cdots\!08\)\(q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
741.587
−740.587
−279461. 1.29140e8 4.37389e10 −3.65354e11 −3.60897e13 −6.36148e14 −2.62111e15 1.66772e16 1.02102e17
1.2 218549. 1.29140e8 1.34041e10 −9.68425e11 2.82235e13 −5.65365e14 −4.57985e15 1.66772e16 −2.11649e17
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Atkin-Lehner signs

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

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2}^{2} \) \(\mathstrut +\mathstrut 60912 T_{2} \) \(\mathstrut -\mathstrut 61076072448 \) acting on \(S_{36}^{\mathrm{new}}(\Gamma_0(3))\).