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.

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

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 \( T_{2}^{2} + 60912 T_{2} - 61076072448 \) acting on \(S_{36}^{\mathrm{new}}(\Gamma_0(3))\).