Properties

Label 2.44.a.b
Level 2
Weight 44
Character orbit 2.a
Self dual Yes
Analytic conductor 23.422
Analytic rank 1
Dimension 2
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 2 \)
Weight: \( k \) = \( 44 \)
Character orbit: \([\chi]\) = 2.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(23.4220790691\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\mathbb{Q}[x]/(x^{2} - \cdots)\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{8}\cdot 3^{3}\cdot 5 \)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 17280\sqrt{1589985537001}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + 2097152 q^{2} \) \( + ( -11170817028 - \beta ) q^{3} \) \( + 4398046511104 q^{4} \) \( + ( -23660329199010 + 19308 \beta ) q^{5} \) \( + ( -23426901271904256 - 2097152 \beta ) q^{6} \) \( + ( -111230065673519464 + 87838422 \beta ) q^{7} \) \( + 9223372036854775808 q^{8} \) \( + ( 271297323050157073557 + 22341634056 \beta ) q^{9} \) \(+O(q^{10})\) \( q\) \(+2097152 q^{2}\) \(+(-11170817028 - \beta) q^{3}\) \(+4398046511104 q^{4}\) \(+(-23660329199010 + 19308 \beta) q^{5}\) \(+(-23426901271904256 - 2097152 \beta) q^{6}\) \(+(-111230065673519464 + 87838422 \beta) q^{7}\) \(+9223372036854775808 q^{8}\) \(+(\)\(27\!\cdots\!57\)\( + 22341634056 \beta) q^{9}\) \(+(-49619306700362219520 + 40491810816 \beta) q^{10}\) \(+(-\)\(20\!\cdots\!08\)\( - 567604269771 \beta) q^{11}\) \(+(-\)\(49\!\cdots\!12\)\( - 4398046511104 \beta) q^{12}\) \(+(-\)\(75\!\cdots\!98\)\( - 1877690293332 \beta) q^{13}\) \(+(-\)\(23\!\cdots\!28\)\( + 184210522374144 \beta) q^{14}\) \(+(-\)\(89\!\cdots\!20\)\( - 192025805977614 \beta) q^{15}\) \(+\)\(19\!\cdots\!16\)\( q^{16}\) \(+(-\)\(33\!\cdots\!74\)\( - 13871340963470712 \beta) q^{17}\) \(+(\)\(56\!\cdots\!64\)\( + 46853802543808512 \beta) q^{18}\) \(+(\)\(61\!\cdots\!20\)\( + 60601238932971 \beta) q^{19}\) \(+(-\)\(10\!\cdots\!40\)\( + 84917482036396032 \beta) q^{20}\) \(+(-\)\(40\!\cdots\!08\)\( - 869996874516730352 \beta) q^{21}\) \(+(-\)\(43\!\cdots\!16\)\( - 1190352429558792192 \beta) q^{22}\) \(+(\)\(19\!\cdots\!12\)\( + 10312189640309144994 \beta) q^{23}\) \(+(-\)\(10\!\cdots\!24\)\( - 9223372036854775808 \beta) q^{24}\) \(+(-\)\(95\!\cdots\!25\)\( - 913667272348970160 \beta) q^{25}\) \(+(-\)\(15\!\cdots\!96\)\( - 3937801954041790464 \beta) q^{26}\) \(+(-\)\(99\!\cdots\!40\)\( - \)\(19\!\cdots\!98\)\( \beta) q^{27}\) \(+(-\)\(48\!\cdots\!56\)\( + \)\(38\!\cdots\!88\)\( \beta) q^{28}\) \(+(-\)\(36\!\cdots\!50\)\( + \)\(73\!\cdots\!32\)\( \beta) q^{29}\) \(+(-\)\(18\!\cdots\!40\)\( - \)\(40\!\cdots\!28\)\( \beta) q^{30}\) \(+(\)\(20\!\cdots\!12\)\( - \)\(55\!\cdots\!72\)\( \beta) q^{31}\) \(+\)\(40\!\cdots\!32\)\( q^{32}\) \(+(\)\(50\!\cdots\!24\)\( + \)\(27\!\cdots\!96\)\( \beta) q^{33}\) \(+(-\)\(70\!\cdots\!48\)\( - \)\(29\!\cdots\!24\)\( \beta) q^{34}\) \(+(\)\(80\!\cdots\!40\)\( - \)\(42\!\cdots\!32\)\( \beta) q^{35}\) \(+(\)\(11\!\cdots\!28\)\( + \)\(98\!\cdots\!24\)\( \beta) q^{36}\) \(+(\)\(16\!\cdots\!26\)\( - \)\(30\!\cdots\!52\)\( \beta) q^{37}\) \(+(\)\(12\!\cdots\!40\)\( + \)\(12\!\cdots\!92\)\( \beta) q^{38}\) \(+(\)\(93\!\cdots\!44\)\( + \)\(77\!\cdots\!94\)\( \beta) q^{39}\) \(+(-\)\(21\!\cdots\!80\)\( + \)\(17\!\cdots\!64\)\( \beta) q^{40}\) \(+(-\)\(29\!\cdots\!98\)\( - \)\(10\!\cdots\!68\)\( \beta) q^{41}\) \(+(-\)\(84\!\cdots\!16\)\( - \)\(18\!\cdots\!04\)\( \beta) q^{42}\) \(+(-\)\(14\!\cdots\!28\)\( + \)\(19\!\cdots\!17\)\( \beta) q^{43}\) \(+(-\)\(92\!\cdots\!32\)\( - \)\(24\!\cdots\!84\)\( \beta) q^{44}\) \(+(\)\(19\!\cdots\!30\)\( + \)\(47\!\cdots\!96\)\( \beta) q^{45}\) \(+(\)\(40\!\cdots\!24\)\( + \)\(21\!\cdots\!88\)\( \beta) q^{46}\) \(+(\)\(91\!\cdots\!36\)\( - \)\(29\!\cdots\!68\)\( \beta) q^{47}\) \(+(-\)\(21\!\cdots\!48\)\( - \)\(19\!\cdots\!16\)\( \beta) q^{48}\) \(+(\)\(14\!\cdots\!53\)\( - \)\(19\!\cdots\!16\)\( \beta) q^{49}\) \(+(-\)\(20\!\cdots\!00\)\( - \)\(19\!\cdots\!20\)\( \beta) q^{50}\) \(+(\)\(69\!\cdots\!72\)\( + \)\(18\!\cdots\!10\)\( \beta) q^{51}\) \(+(-\)\(33\!\cdots\!92\)\( - \)\(82\!\cdots\!28\)\( \beta) q^{52}\) \(+(\)\(67\!\cdots\!82\)\( - \)\(35\!\cdots\!92\)\( \beta) q^{53}\) \(+(-\)\(20\!\cdots\!80\)\( - \)\(40\!\cdots\!96\)\( \beta) q^{54}\) \(+(-\)\(47\!\cdots\!20\)\( - \)\(39\!\cdots\!54\)\( \beta) q^{55}\) \(+(-\)\(10\!\cdots\!12\)\( + \)\(81\!\cdots\!76\)\( \beta) q^{56}\) \(+(-\)\(69\!\cdots\!60\)\( - \)\(61\!\cdots\!08\)\( \beta) q^{57}\) \(+(-\)\(76\!\cdots\!00\)\( + \)\(15\!\cdots\!64\)\( \beta) q^{58}\) \(+(\)\(32\!\cdots\!00\)\( + \)\(18\!\cdots\!69\)\( \beta) q^{59}\) \(+(-\)\(39\!\cdots\!80\)\( - \)\(84\!\cdots\!56\)\( \beta) q^{60}\) \(+(\)\(42\!\cdots\!22\)\( - \)\(57\!\cdots\!64\)\( \beta) q^{61}\) \(+(\)\(42\!\cdots\!24\)\( - \)\(11\!\cdots\!44\)\( \beta) q^{62}\) \(+(\)\(90\!\cdots\!52\)\( + \)\(21\!\cdots\!70\)\( \beta) q^{63}\) \(+\)\(85\!\cdots\!64\)\( q^{64}\) \(+(\)\(66\!\cdots\!80\)\( - \)\(14\!\cdots\!64\)\( \beta) q^{65}\) \(+(\)\(10\!\cdots\!48\)\( + \)\(57\!\cdots\!92\)\( \beta) q^{66}\) \(+(-\)\(19\!\cdots\!04\)\( - \)\(15\!\cdots\!09\)\( \beta) q^{67}\) \(+(-\)\(14\!\cdots\!96\)\( - \)\(61\!\cdots\!48\)\( \beta) q^{68}\) \(+(-\)\(51\!\cdots\!36\)\( - \)\(13\!\cdots\!44\)\( \beta) q^{69}\) \(+(\)\(16\!\cdots\!80\)\( - \)\(88\!\cdots\!64\)\( \beta) q^{70}\) \(+(-\)\(22\!\cdots\!48\)\( + \)\(40\!\cdots\!06\)\( \beta) q^{71}\) \(+(\)\(25\!\cdots\!56\)\( + \)\(20\!\cdots\!48\)\( \beta) q^{72}\) \(+(-\)\(51\!\cdots\!78\)\( - \)\(43\!\cdots\!52\)\( \beta) q^{73}\) \(+(\)\(35\!\cdots\!52\)\( - \)\(63\!\cdots\!04\)\( \beta) q^{74}\) \(+(\)\(11\!\cdots\!00\)\( + \)\(96\!\cdots\!05\)\( \beta) q^{75}\) \(+(\)\(27\!\cdots\!80\)\( + \)\(26\!\cdots\!84\)\( \beta) q^{76}\) \(+(-\)\(21\!\cdots\!88\)\( - \)\(17\!\cdots\!32\)\( \beta) q^{77}\) \(+(\)\(19\!\cdots\!88\)\( + \)\(16\!\cdots\!88\)\( \beta) q^{78}\) \(+(\)\(53\!\cdots\!60\)\( - \)\(42\!\cdots\!44\)\( \beta) q^{79}\) \(+(-\)\(45\!\cdots\!60\)\( + \)\(37\!\cdots\!28\)\( \beta) q^{80}\) \(+(\)\(11\!\cdots\!81\)\( + \)\(47\!\cdots\!72\)\( \beta) q^{81}\) \(+(-\)\(61\!\cdots\!96\)\( - \)\(21\!\cdots\!36\)\( \beta) q^{82}\) \(+(-\)\(49\!\cdots\!28\)\( - \)\(69\!\cdots\!45\)\( \beta) q^{83}\) \(+(-\)\(17\!\cdots\!32\)\( - \)\(38\!\cdots\!08\)\( \beta) q^{84}\) \(+(-\)\(12\!\cdots\!60\)\( - \)\(32\!\cdots\!72\)\( \beta) q^{85}\) \(+(-\)\(29\!\cdots\!56\)\( + \)\(41\!\cdots\!84\)\( \beta) q^{86}\) \(+(\)\(56\!\cdots\!00\)\( + \)\(28\!\cdots\!54\)\( \beta) q^{87}\) \(+(-\)\(19\!\cdots\!64\)\( - \)\(52\!\cdots\!68\)\( \beta) q^{88}\) \(+(\)\(87\!\cdots\!10\)\( - \)\(20\!\cdots\!20\)\( \beta) q^{89}\) \(+(\)\(41\!\cdots\!60\)\( + \)\(98\!\cdots\!92\)\( \beta) q^{90}\) \(+(\)\(57\!\cdots\!72\)\( - \)\(66\!\cdots\!08\)\( \beta) q^{91}\) \(+(\)\(83\!\cdots\!48\)\( + \)\(45\!\cdots\!76\)\( \beta) q^{92}\) \(+(\)\(24\!\cdots\!64\)\( + \)\(42\!\cdots\!04\)\( \beta) q^{93}\) \(+(\)\(19\!\cdots\!72\)\( - \)\(61\!\cdots\!36\)\( \beta) q^{94}\) \(+(-\)\(14\!\cdots\!00\)\( + \)\(11\!\cdots\!50\)\( \beta) q^{95}\) \(+(-\)\(45\!\cdots\!96\)\( - \)\(40\!\cdots\!32\)\( \beta) q^{96}\) \(+(-\)\(18\!\cdots\!94\)\( + \)\(37\!\cdots\!40\)\( \beta) q^{97}\) \(+(\)\(31\!\cdots\!56\)\( - \)\(40\!\cdots\!32\)\( \beta) q^{98}\) \(+(-\)\(11\!\cdots\!56\)\( - \)\(62\!\cdots\!95\)\( \beta) q^{99}\) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut +\mathstrut 4194304q^{2} \) \(\mathstrut -\mathstrut 22341634056q^{3} \) \(\mathstrut +\mathstrut 8796093022208q^{4} \) \(\mathstrut -\mathstrut 47320658398020q^{5} \) \(\mathstrut -\mathstrut 46853802543808512q^{6} \) \(\mathstrut -\mathstrut 222460131347038928q^{7} \) \(\mathstrut +\mathstrut 18446744073709551616q^{8} \) \(\mathstrut +\mathstrut 542594646100314147114q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut +\mathstrut 4194304q^{2} \) \(\mathstrut -\mathstrut 22341634056q^{3} \) \(\mathstrut +\mathstrut 8796093022208q^{4} \) \(\mathstrut -\mathstrut 47320658398020q^{5} \) \(\mathstrut -\mathstrut 46853802543808512q^{6} \) \(\mathstrut -\mathstrut 222460131347038928q^{7} \) \(\mathstrut +\mathstrut 18446744073709551616q^{8} \) \(\mathstrut +\mathstrut \)\(54\!\cdots\!14\)\(q^{9} \) \(\mathstrut -\mathstrut 99238613400724439040q^{10} \) \(\mathstrut -\mathstrut \)\(41\!\cdots\!16\)\(q^{11} \) \(\mathstrut -\mathstrut \)\(98\!\cdots\!24\)\(q^{12} \) \(\mathstrut -\mathstrut \)\(15\!\cdots\!96\)\(q^{13} \) \(\mathstrut -\mathstrut \)\(46\!\cdots\!56\)\(q^{14} \) \(\mathstrut -\mathstrut \)\(17\!\cdots\!40\)\(q^{15} \) \(\mathstrut +\mathstrut \)\(38\!\cdots\!32\)\(q^{16} \) \(\mathstrut -\mathstrut \)\(67\!\cdots\!48\)\(q^{17} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!28\)\(q^{18} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!40\)\(q^{19} \) \(\mathstrut -\mathstrut \)\(20\!\cdots\!80\)\(q^{20} \) \(\mathstrut -\mathstrut \)\(80\!\cdots\!16\)\(q^{21} \) \(\mathstrut -\mathstrut \)\(87\!\cdots\!32\)\(q^{22} \) \(\mathstrut +\mathstrut \)\(38\!\cdots\!24\)\(q^{23} \) \(\mathstrut -\mathstrut \)\(20\!\cdots\!48\)\(q^{24} \) \(\mathstrut -\mathstrut \)\(19\!\cdots\!50\)\(q^{25} \) \(\mathstrut -\mathstrut \)\(31\!\cdots\!92\)\(q^{26} \) \(\mathstrut -\mathstrut \)\(19\!\cdots\!80\)\(q^{27} \) \(\mathstrut -\mathstrut \)\(97\!\cdots\!12\)\(q^{28} \) \(\mathstrut -\mathstrut \)\(72\!\cdots\!00\)\(q^{29} \) \(\mathstrut -\mathstrut \)\(37\!\cdots\!80\)\(q^{30} \) \(\mathstrut +\mathstrut \)\(40\!\cdots\!24\)\(q^{31} \) \(\mathstrut +\mathstrut \)\(81\!\cdots\!64\)\(q^{32} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!48\)\(q^{33} \) \(\mathstrut -\mathstrut \)\(14\!\cdots\!96\)\(q^{34} \) \(\mathstrut +\mathstrut \)\(16\!\cdots\!80\)\(q^{35} \) \(\mathstrut +\mathstrut \)\(23\!\cdots\!56\)\(q^{36} \) \(\mathstrut +\mathstrut \)\(33\!\cdots\!52\)\(q^{37} \) \(\mathstrut +\mathstrut \)\(25\!\cdots\!80\)\(q^{38} \) \(\mathstrut +\mathstrut \)\(18\!\cdots\!88\)\(q^{39} \) \(\mathstrut -\mathstrut \)\(43\!\cdots\!60\)\(q^{40} \) \(\mathstrut -\mathstrut \)\(58\!\cdots\!96\)\(q^{41} \) \(\mathstrut -\mathstrut \)\(16\!\cdots\!32\)\(q^{42} \) \(\mathstrut -\mathstrut \)\(28\!\cdots\!56\)\(q^{43} \) \(\mathstrut -\mathstrut \)\(18\!\cdots\!64\)\(q^{44} \) \(\mathstrut +\mathstrut \)\(39\!\cdots\!60\)\(q^{45} \) \(\mathstrut +\mathstrut \)\(80\!\cdots\!48\)\(q^{46} \) \(\mathstrut +\mathstrut \)\(18\!\cdots\!72\)\(q^{47} \) \(\mathstrut -\mathstrut \)\(43\!\cdots\!96\)\(q^{48} \) \(\mathstrut +\mathstrut \)\(29\!\cdots\!06\)\(q^{49} \) \(\mathstrut -\mathstrut \)\(40\!\cdots\!00\)\(q^{50} \) \(\mathstrut +\mathstrut \)\(13\!\cdots\!44\)\(q^{51} \) \(\mathstrut -\mathstrut \)\(66\!\cdots\!84\)\(q^{52} \) \(\mathstrut +\mathstrut \)\(13\!\cdots\!64\)\(q^{53} \) \(\mathstrut -\mathstrut \)\(41\!\cdots\!60\)\(q^{54} \) \(\mathstrut -\mathstrut \)\(94\!\cdots\!40\)\(q^{55} \) \(\mathstrut -\mathstrut \)\(20\!\cdots\!24\)\(q^{56} \) \(\mathstrut -\mathstrut \)\(13\!\cdots\!20\)\(q^{57} \) \(\mathstrut -\mathstrut \)\(15\!\cdots\!00\)\(q^{58} \) \(\mathstrut +\mathstrut \)\(65\!\cdots\!00\)\(q^{59} \) \(\mathstrut -\mathstrut \)\(78\!\cdots\!60\)\(q^{60} \) \(\mathstrut +\mathstrut \)\(84\!\cdots\!44\)\(q^{61} \) \(\mathstrut +\mathstrut \)\(84\!\cdots\!48\)\(q^{62} \) \(\mathstrut +\mathstrut \)\(18\!\cdots\!04\)\(q^{63} \) \(\mathstrut +\mathstrut \)\(17\!\cdots\!28\)\(q^{64} \) \(\mathstrut +\mathstrut \)\(13\!\cdots\!60\)\(q^{65} \) \(\mathstrut +\mathstrut \)\(21\!\cdots\!96\)\(q^{66} \) \(\mathstrut -\mathstrut \)\(38\!\cdots\!08\)\(q^{67} \) \(\mathstrut -\mathstrut \)\(29\!\cdots\!92\)\(q^{68} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!72\)\(q^{69} \) \(\mathstrut +\mathstrut \)\(33\!\cdots\!60\)\(q^{70} \) \(\mathstrut -\mathstrut \)\(45\!\cdots\!96\)\(q^{71} \) \(\mathstrut +\mathstrut \)\(50\!\cdots\!12\)\(q^{72} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!56\)\(q^{73} \) \(\mathstrut +\mathstrut \)\(70\!\cdots\!04\)\(q^{74} \) \(\mathstrut +\mathstrut \)\(22\!\cdots\!00\)\(q^{75} \) \(\mathstrut +\mathstrut \)\(54\!\cdots\!60\)\(q^{76} \) \(\mathstrut -\mathstrut \)\(42\!\cdots\!76\)\(q^{77} \) \(\mathstrut +\mathstrut \)\(39\!\cdots\!76\)\(q^{78} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!20\)\(q^{79} \) \(\mathstrut -\mathstrut \)\(91\!\cdots\!20\)\(q^{80} \) \(\mathstrut +\mathstrut \)\(22\!\cdots\!62\)\(q^{81} \) \(\mathstrut -\mathstrut \)\(12\!\cdots\!92\)\(q^{82} \) \(\mathstrut -\mathstrut \)\(99\!\cdots\!56\)\(q^{83} \) \(\mathstrut -\mathstrut \)\(35\!\cdots\!64\)\(q^{84} \) \(\mathstrut -\mathstrut \)\(25\!\cdots\!20\)\(q^{85} \) \(\mathstrut -\mathstrut \)\(59\!\cdots\!12\)\(q^{86} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!00\)\(q^{87} \) \(\mathstrut -\mathstrut \)\(38\!\cdots\!28\)\(q^{88} \) \(\mathstrut +\mathstrut \)\(17\!\cdots\!20\)\(q^{89} \) \(\mathstrut +\mathstrut \)\(83\!\cdots\!20\)\(q^{90} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!44\)\(q^{91} \) \(\mathstrut +\mathstrut \)\(16\!\cdots\!96\)\(q^{92} \) \(\mathstrut +\mathstrut \)\(48\!\cdots\!28\)\(q^{93} \) \(\mathstrut +\mathstrut \)\(38\!\cdots\!44\)\(q^{94} \) \(\mathstrut -\mathstrut \)\(28\!\cdots\!00\)\(q^{95} \) \(\mathstrut -\mathstrut \)\(90\!\cdots\!92\)\(q^{96} \) \(\mathstrut -\mathstrut \)\(36\!\cdots\!88\)\(q^{97} \) \(\mathstrut +\mathstrut \)\(62\!\cdots\!12\)\(q^{98} \) \(\mathstrut -\mathstrut \)\(23\!\cdots\!12\)\(q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
630474.
−630473.
2.09715e6 −3.29600e10 4.39805e12 3.97045e14 −6.91221e16 1.80269e18 9.22337e18 7.58103e20 8.32663e20
1.2 2.09715e6 1.06183e10 4.39805e12 −4.44365e14 2.22683e16 −2.02515e18 9.22337e18 −2.15508e20 −9.31902e20
\(n\): 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
\(2\) \(-1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{3}^{2} \) \(\mathstrut +\mathstrut 22341634056 T_{3} \) \(\mathstrut -\mathstrut \)\(34\!\cdots\!16\)\( \) acting on \(S_{44}^{\mathrm{new}}(\Gamma_0(2))\).