Properties

Label 2.44.a.a
Level 2
Weight 44
Character orbit 2.a
Self dual Yes
Analytic conductor 23.422
Analytic rank 0
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: \(0\)
Dimension: \(2\)
Coefficient field: \(\mathbb{Q}[x]/(x^{2} - \cdots)\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{8}\cdot 3\cdot 5\cdot 11 \)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 21120\sqrt{97578078049}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( -2097152 q^{2} \) \( + ( -6490815492 - \beta ) q^{3} \) \( + 4398046511104 q^{4} \) \( + ( -199331355641250 + 283500 \beta ) q^{5} \) \( + ( 13612226690678784 + 2097152 \beta ) q^{6} \) \( + ( 587435016771620504 - 190730538 \beta ) q^{7} \) \( -9223372036854775808 q^{8} \) \( + ( -242601150586050009963 + 12981630984 \beta ) q^{9} \) \(+O(q^{10})\) \( q\) \(-2097152 q^{2}\) \(+(-6490815492 - \beta) q^{3}\) \(+4398046511104 q^{4}\) \(+(-199331355641250 + 283500 \beta) q^{5}\) \(+(13612226690678784 + 2097152 \beta) q^{6}\) \(+(587435016771620504 - 190730538 \beta) q^{7}\) \(-9223372036854775808 q^{8}\) \(+(-\)\(24\!\cdots\!63\)\( + 12981630984 \beta) q^{9}\) \(+(\)\(41\!\cdots\!00\)\( - 594542592000 \beta) q^{10}\) \(+(-\)\(58\!\cdots\!88\)\( + 4407651659061 \beta) q^{11}\) \(+(-\)\(28\!\cdots\!68\)\( - 4398046511104 \beta) q^{12}\) \(+(-\)\(82\!\cdots\!42\)\( - 59681996510292 \beta) q^{13}\) \(+(-\)\(12\!\cdots\!08\)\( + 399990929227776 \beta) q^{14}\) \(+(-\)\(11\!\cdots\!00\)\( - 1640814836340750 \beta) q^{15}\) \(+\)\(19\!\cdots\!16\)\( q^{16}\) \(+(\)\(17\!\cdots\!34\)\( + 26766690090947208 \beta) q^{17}\) \(+(\)\(50\!\cdots\!76\)\( - 27224453381357568 \beta) q^{18}\) \(+(-\)\(26\!\cdots\!00\)\( - 403392382384796181 \beta) q^{19}\) \(+(-\)\(87\!\cdots\!00\)\( + 1246846185897984000 \beta) q^{20}\) \(+(\)\(44\!\cdots\!32\)\( + 650561714076274192 \beta) q^{21}\) \(+(\)\(12\!\cdots\!76\)\( - 9243515492103094272 \beta) q^{22}\) \(+(\)\(11\!\cdots\!68\)\( + 16797585459534273954 \beta) q^{23}\) \(+(\)\(59\!\cdots\!36\)\( + 9223372036854775808 \beta) q^{24}\) \(+(\)\(24\!\cdots\!75\)\( - \)\(11\!\cdots\!00\)\( \beta) q^{25}\) \(+(\)\(17\!\cdots\!84\)\( + \)\(12\!\cdots\!84\)\( \beta) q^{26}\) \(+(\)\(31\!\cdots\!80\)\( + \)\(48\!\cdots\!62\)\( \beta) q^{27}\) \(+(\)\(25\!\cdots\!16\)\( - \)\(83\!\cdots\!52\)\( \beta) q^{28}\) \(+(\)\(17\!\cdots\!10\)\( - \)\(19\!\cdots\!92\)\( \beta) q^{29}\) \(+(\)\(23\!\cdots\!00\)\( + \)\(34\!\cdots\!00\)\( \beta) q^{30}\) \(+(\)\(81\!\cdots\!52\)\( + \)\(49\!\cdots\!12\)\( \beta) q^{31}\) \(-\)\(40\!\cdots\!32\)\( q^{32}\) \(+(-\)\(15\!\cdots\!04\)\( - \)\(22\!\cdots\!24\)\( \beta) q^{33}\) \(+(-\)\(35\!\cdots\!68\)\( - \)\(56\!\cdots\!16\)\( \beta) q^{34}\) \(+(-\)\(24\!\cdots\!00\)\( + \)\(20\!\cdots\!00\)\( \beta) q^{35}\) \(+(-\)\(10\!\cdots\!52\)\( + \)\(57\!\cdots\!36\)\( \beta) q^{36}\) \(+(-\)\(21\!\cdots\!66\)\( - \)\(83\!\cdots\!32\)\( \beta) q^{37}\) \(+(\)\(55\!\cdots\!00\)\( + \)\(84\!\cdots\!12\)\( \beta) q^{38}\) \(+(\)\(79\!\cdots\!64\)\( + \)\(12\!\cdots\!06\)\( \beta) q^{39}\) \(+(\)\(18\!\cdots\!00\)\( - \)\(26\!\cdots\!00\)\( \beta) q^{40}\) \(+(\)\(36\!\cdots\!62\)\( - \)\(78\!\cdots\!52\)\( \beta) q^{41}\) \(+(-\)\(94\!\cdots\!64\)\( - \)\(13\!\cdots\!84\)\( \beta) q^{42}\) \(+(\)\(11\!\cdots\!68\)\( + \)\(12\!\cdots\!77\)\( \beta) q^{43}\) \(+(-\)\(25\!\cdots\!52\)\( + \)\(19\!\cdots\!44\)\( \beta) q^{44}\) \(+(\)\(20\!\cdots\!50\)\( - \)\(71\!\cdots\!00\)\( \beta) q^{45}\) \(+(-\)\(24\!\cdots\!36\)\( - \)\(35\!\cdots\!08\)\( \beta) q^{46}\) \(+(-\)\(19\!\cdots\!96\)\( + \)\(19\!\cdots\!12\)\( \beta) q^{47}\) \(+(-\)\(12\!\cdots\!72\)\( - \)\(19\!\cdots\!16\)\( \beta) q^{48}\) \(+(-\)\(25\!\cdots\!27\)\( - \)\(22\!\cdots\!04\)\( \beta) q^{49}\) \(+(-\)\(50\!\cdots\!00\)\( + \)\(23\!\cdots\!00\)\( \beta) q^{50}\) \(+(-\)\(22\!\cdots\!28\)\( - \)\(34\!\cdots\!70\)\( \beta) q^{51}\) \(+(-\)\(36\!\cdots\!68\)\( - \)\(26\!\cdots\!68\)\( \beta) q^{52}\) \(+(\)\(10\!\cdots\!78\)\( + \)\(18\!\cdots\!88\)\( \beta) q^{53}\) \(+(-\)\(65\!\cdots\!60\)\( - \)\(10\!\cdots\!24\)\( \beta) q^{54}\) \(+(\)\(55\!\cdots\!00\)\( - \)\(25\!\cdots\!50\)\( \beta) q^{55}\) \(+(-\)\(54\!\cdots\!32\)\( + \)\(17\!\cdots\!04\)\( \beta) q^{56}\) \(+(\)\(19\!\cdots\!00\)\( + \)\(28\!\cdots\!52\)\( \beta) q^{57}\) \(+(-\)\(35\!\cdots\!20\)\( + \)\(41\!\cdots\!84\)\( \beta) q^{58}\) \(+(\)\(33\!\cdots\!20\)\( - \)\(14\!\cdots\!39\)\( \beta) q^{59}\) \(+(-\)\(48\!\cdots\!00\)\( - \)\(72\!\cdots\!00\)\( \beta) q^{60}\) \(+(\)\(52\!\cdots\!02\)\( + \)\(23\!\cdots\!44\)\( \beta) q^{61}\) \(+(-\)\(17\!\cdots\!04\)\( - \)\(10\!\cdots\!24\)\( \beta) q^{62}\) \(+(-\)\(25\!\cdots\!52\)\( + \)\(53\!\cdots\!30\)\( \beta) q^{63}\) \(+\)\(85\!\cdots\!64\)\( q^{64}\) \(+(-\)\(57\!\cdots\!00\)\( - \)\(22\!\cdots\!00\)\( \beta) q^{65}\) \(+(\)\(32\!\cdots\!08\)\( + \)\(47\!\cdots\!48\)\( \beta) q^{66}\) \(+(-\)\(64\!\cdots\!76\)\( + \)\(27\!\cdots\!71\)\( \beta) q^{67}\) \(+(\)\(75\!\cdots\!36\)\( + \)\(11\!\cdots\!32\)\( \beta) q^{68}\) \(+(-\)\(14\!\cdots\!56\)\( - \)\(22\!\cdots\!36\)\( \beta) q^{69}\) \(+(\)\(51\!\cdots\!00\)\( - \)\(42\!\cdots\!00\)\( \beta) q^{70}\) \(+(\)\(52\!\cdots\!32\)\( + \)\(34\!\cdots\!34\)\( \beta) q^{71}\) \(+(\)\(22\!\cdots\!04\)\( - \)\(11\!\cdots\!72\)\( \beta) q^{72}\) \(+(-\)\(54\!\cdots\!62\)\( - \)\(13\!\cdots\!72\)\( \beta) q^{73}\) \(+(\)\(46\!\cdots\!32\)\( + \)\(17\!\cdots\!64\)\( \beta) q^{74}\) \(+(-\)\(10\!\cdots\!00\)\( - \)\(16\!\cdots\!75\)\( \beta) q^{75}\) \(+(-\)\(11\!\cdots\!00\)\( - \)\(17\!\cdots\!24\)\( \beta) q^{76}\) \(+(-\)\(40\!\cdots\!52\)\( + \)\(36\!\cdots\!88\)\( \beta) q^{77}\) \(+(-\)\(16\!\cdots\!28\)\( - \)\(25\!\cdots\!12\)\( \beta) q^{78}\) \(+(-\)\(33\!\cdots\!60\)\( + \)\(88\!\cdots\!24\)\( \beta) q^{79}\) \(+(-\)\(38\!\cdots\!00\)\( + \)\(54\!\cdots\!00\)\( \beta) q^{80}\) \(+(\)\(38\!\cdots\!41\)\( - \)\(10\!\cdots\!52\)\( \beta) q^{81}\) \(+(-\)\(76\!\cdots\!24\)\( + \)\(16\!\cdots\!04\)\( \beta) q^{82}\) \(+(\)\(27\!\cdots\!28\)\( - \)\(49\!\cdots\!65\)\( \beta) q^{83}\) \(+(\)\(19\!\cdots\!28\)\( + \)\(28\!\cdots\!68\)\( \beta) q^{84}\) \(+(\)\(29\!\cdots\!00\)\( + \)\(43\!\cdots\!00\)\( \beta) q^{85}\) \(+(-\)\(24\!\cdots\!36\)\( - \)\(25\!\cdots\!04\)\( \beta) q^{86}\) \(+(-\)\(24\!\cdots\!20\)\( - \)\(42\!\cdots\!46\)\( \beta) q^{87}\) \(+(\)\(53\!\cdots\!04\)\( - \)\(40\!\cdots\!88\)\( \beta) q^{88}\) \(+(\)\(14\!\cdots\!10\)\( - \)\(10\!\cdots\!80\)\( \beta) q^{89}\) \(+(-\)\(43\!\cdots\!00\)\( + \)\(14\!\cdots\!00\)\( \beta) q^{90}\) \(+(\)\(82\!\cdots\!32\)\( + \)\(12\!\cdots\!28\)\( \beta) q^{91}\) \(+(\)\(50\!\cdots\!72\)\( + \)\(73\!\cdots\!16\)\( \beta) q^{92}\) \(+(-\)\(74\!\cdots\!84\)\( - \)\(11\!\cdots\!56\)\( \beta) q^{93}\) \(+(\)\(41\!\cdots\!92\)\( - \)\(40\!\cdots\!24\)\( \beta) q^{94}\) \(+(-\)\(49\!\cdots\!00\)\( + \)\(57\!\cdots\!50\)\( \beta) q^{95}\) \(+(\)\(26\!\cdots\!44\)\( + \)\(40\!\cdots\!32\)\( \beta) q^{96}\) \(+(-\)\(11\!\cdots\!06\)\( + \)\(69\!\cdots\!20\)\( \beta) q^{97}\) \(+(\)\(53\!\cdots\!04\)\( + \)\(46\!\cdots\!08\)\( \beta) q^{98}\) \(+(\)\(38\!\cdots\!44\)\( - \)\(11\!\cdots\!35\)\( \beta) q^{99}\) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut -\mathstrut 4194304q^{2} \) \(\mathstrut -\mathstrut 12981630984q^{3} \) \(\mathstrut +\mathstrut 8796093022208q^{4} \) \(\mathstrut -\mathstrut 398662711282500q^{5} \) \(\mathstrut +\mathstrut 27224453381357568q^{6} \) \(\mathstrut +\mathstrut 1174870033543241008q^{7} \) \(\mathstrut -\mathstrut 18446744073709551616q^{8} \) \(\mathstrut -\mathstrut 485202301172100019926q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut -\mathstrut 4194304q^{2} \) \(\mathstrut -\mathstrut 12981630984q^{3} \) \(\mathstrut +\mathstrut 8796093022208q^{4} \) \(\mathstrut -\mathstrut 398662711282500q^{5} \) \(\mathstrut +\mathstrut 27224453381357568q^{6} \) \(\mathstrut +\mathstrut 1174870033543241008q^{7} \) \(\mathstrut -\mathstrut 18446744073709551616q^{8} \) \(\mathstrut -\mathstrut \)\(48\!\cdots\!26\)\(q^{9} \) \(\mathstrut +\mathstrut \)\(83\!\cdots\!00\)\(q^{10} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!76\)\(q^{11} \) \(\mathstrut -\mathstrut \)\(57\!\cdots\!36\)\(q^{12} \) \(\mathstrut -\mathstrut \)\(16\!\cdots\!84\)\(q^{13} \) \(\mathstrut -\mathstrut \)\(24\!\cdots\!16\)\(q^{14} \) \(\mathstrut -\mathstrut \)\(22\!\cdots\!00\)\(q^{15} \) \(\mathstrut +\mathstrut \)\(38\!\cdots\!32\)\(q^{16} \) \(\mathstrut +\mathstrut \)\(34\!\cdots\!68\)\(q^{17} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!52\)\(q^{18} \) \(\mathstrut -\mathstrut \)\(52\!\cdots\!00\)\(q^{19} \) \(\mathstrut -\mathstrut \)\(17\!\cdots\!00\)\(q^{20} \) \(\mathstrut +\mathstrut \)\(89\!\cdots\!64\)\(q^{21} \) \(\mathstrut +\mathstrut \)\(24\!\cdots\!52\)\(q^{22} \) \(\mathstrut +\mathstrut \)\(22\!\cdots\!36\)\(q^{23} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!72\)\(q^{24} \) \(\mathstrut +\mathstrut \)\(48\!\cdots\!50\)\(q^{25} \) \(\mathstrut +\mathstrut \)\(34\!\cdots\!68\)\(q^{26} \) \(\mathstrut +\mathstrut \)\(62\!\cdots\!60\)\(q^{27} \) \(\mathstrut +\mathstrut \)\(51\!\cdots\!32\)\(q^{28} \) \(\mathstrut +\mathstrut \)\(34\!\cdots\!20\)\(q^{29} \) \(\mathstrut +\mathstrut \)\(46\!\cdots\!00\)\(q^{30} \) \(\mathstrut +\mathstrut \)\(16\!\cdots\!04\)\(q^{31} \) \(\mathstrut -\mathstrut \)\(81\!\cdots\!64\)\(q^{32} \) \(\mathstrut -\mathstrut \)\(30\!\cdots\!08\)\(q^{33} \) \(\mathstrut -\mathstrut \)\(71\!\cdots\!36\)\(q^{34} \) \(\mathstrut -\mathstrut \)\(49\!\cdots\!00\)\(q^{35} \) \(\mathstrut -\mathstrut \)\(21\!\cdots\!04\)\(q^{36} \) \(\mathstrut -\mathstrut \)\(43\!\cdots\!32\)\(q^{37} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!00\)\(q^{38} \) \(\mathstrut +\mathstrut \)\(15\!\cdots\!28\)\(q^{39} \) \(\mathstrut +\mathstrut \)\(36\!\cdots\!00\)\(q^{40} \) \(\mathstrut +\mathstrut \)\(73\!\cdots\!24\)\(q^{41} \) \(\mathstrut -\mathstrut \)\(18\!\cdots\!28\)\(q^{42} \) \(\mathstrut +\mathstrut \)\(23\!\cdots\!36\)\(q^{43} \) \(\mathstrut -\mathstrut \)\(51\!\cdots\!04\)\(q^{44} \) \(\mathstrut +\mathstrut \)\(41\!\cdots\!00\)\(q^{45} \) \(\mathstrut -\mathstrut \)\(48\!\cdots\!72\)\(q^{46} \) \(\mathstrut -\mathstrut \)\(39\!\cdots\!92\)\(q^{47} \) \(\mathstrut -\mathstrut \)\(25\!\cdots\!44\)\(q^{48} \) \(\mathstrut -\mathstrut \)\(51\!\cdots\!54\)\(q^{49} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!00\)\(q^{50} \) \(\mathstrut -\mathstrut \)\(45\!\cdots\!56\)\(q^{51} \) \(\mathstrut -\mathstrut \)\(72\!\cdots\!36\)\(q^{52} \) \(\mathstrut +\mathstrut \)\(21\!\cdots\!56\)\(q^{53} \) \(\mathstrut -\mathstrut \)\(13\!\cdots\!20\)\(q^{54} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!00\)\(q^{55} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!64\)\(q^{56} \) \(\mathstrut +\mathstrut \)\(38\!\cdots\!00\)\(q^{57} \) \(\mathstrut -\mathstrut \)\(71\!\cdots\!40\)\(q^{58} \) \(\mathstrut +\mathstrut \)\(67\!\cdots\!40\)\(q^{59} \) \(\mathstrut -\mathstrut \)\(97\!\cdots\!00\)\(q^{60} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!04\)\(q^{61} \) \(\mathstrut -\mathstrut \)\(34\!\cdots\!08\)\(q^{62} \) \(\mathstrut -\mathstrut \)\(50\!\cdots\!04\)\(q^{63} \) \(\mathstrut +\mathstrut \)\(17\!\cdots\!28\)\(q^{64} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!00\)\(q^{65} \) \(\mathstrut +\mathstrut \)\(64\!\cdots\!16\)\(q^{66} \) \(\mathstrut -\mathstrut \)\(12\!\cdots\!52\)\(q^{67} \) \(\mathstrut +\mathstrut \)\(15\!\cdots\!72\)\(q^{68} \) \(\mathstrut -\mathstrut \)\(29\!\cdots\!12\)\(q^{69} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!00\)\(q^{70} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!64\)\(q^{71} \) \(\mathstrut +\mathstrut \)\(44\!\cdots\!08\)\(q^{72} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!24\)\(q^{73} \) \(\mathstrut +\mathstrut \)\(92\!\cdots\!64\)\(q^{74} \) \(\mathstrut -\mathstrut \)\(21\!\cdots\!00\)\(q^{75} \) \(\mathstrut -\mathstrut \)\(23\!\cdots\!00\)\(q^{76} \) \(\mathstrut -\mathstrut \)\(80\!\cdots\!04\)\(q^{77} \) \(\mathstrut -\mathstrut \)\(33\!\cdots\!56\)\(q^{78} \) \(\mathstrut -\mathstrut \)\(66\!\cdots\!20\)\(q^{79} \) \(\mathstrut -\mathstrut \)\(77\!\cdots\!00\)\(q^{80} \) \(\mathstrut +\mathstrut \)\(76\!\cdots\!82\)\(q^{81} \) \(\mathstrut -\mathstrut \)\(15\!\cdots\!48\)\(q^{82} \) \(\mathstrut +\mathstrut \)\(54\!\cdots\!56\)\(q^{83} \) \(\mathstrut +\mathstrut \)\(39\!\cdots\!56\)\(q^{84} \) \(\mathstrut +\mathstrut \)\(59\!\cdots\!00\)\(q^{85} \) \(\mathstrut -\mathstrut \)\(49\!\cdots\!72\)\(q^{86} \) \(\mathstrut -\mathstrut \)\(49\!\cdots\!40\)\(q^{87} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!08\)\(q^{88} \) \(\mathstrut +\mathstrut \)\(29\!\cdots\!20\)\(q^{89} \) \(\mathstrut -\mathstrut \)\(87\!\cdots\!00\)\(q^{90} \) \(\mathstrut +\mathstrut \)\(16\!\cdots\!64\)\(q^{91} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!44\)\(q^{92} \) \(\mathstrut -\mathstrut \)\(14\!\cdots\!68\)\(q^{93} \) \(\mathstrut +\mathstrut \)\(82\!\cdots\!84\)\(q^{94} \) \(\mathstrut -\mathstrut \)\(98\!\cdots\!00\)\(q^{95} \) \(\mathstrut +\mathstrut \)\(52\!\cdots\!88\)\(q^{96} \) \(\mathstrut -\mathstrut \)\(22\!\cdots\!12\)\(q^{97} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!08\)\(q^{98} \) \(\mathstrut +\mathstrut \)\(77\!\cdots\!88\)\(q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
156188.
−156187.
−2.09715e6 −1.30882e10 4.39805e12 1.67102e15 2.74479e16 −6.70883e17 −9.22337e18 −1.56957e20 −3.50438e21
1.2 −2.09715e6 1.06542e8 4.39805e12 −2.06968e15 −2.23436e14 1.84575e18 −9.22337e18 −3.28246e20 4.34044e21
\(n\): 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 \(T_{3}^{2} \) \(\mathstrut +\mathstrut 12981630984 T_{3} \) \(\mathstrut -\mathstrut \)\(13\!\cdots\!36\)\( \) acting on \(S_{44}^{\mathrm{new}}(\Gamma_0(2))\).