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.

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

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
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 \( T_{3}^{2} + 12981630984 T_{3} - \)\(13\!\cdots\!36\)\( \) acting on \(S_{44}^{\mathrm{new}}(\Gamma_0(2))\).