Properties

Label 2.42.a.b
Level 2
Weight 42
Character orbit 2.a
Self dual Yes
Analytic conductor 21.294
Analytic rank 0
Dimension 2
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 2 \)
Weight: \( k \) = \( 42 \)
Character orbit: \([\chi]\) = 2.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(21.2943340913\)
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: \( 2^{7}\cdot 3^{2}\cdot 5 \)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2880\sqrt{4559670239569}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + 1048576 q^{2} \) \( + ( 4431673764 - \beta ) q^{3} \) \( + 1099511627776 q^{4} \) \( + ( 48799592162790 + 49284 \beta ) q^{5} \) \( + ( 4646946748760064 - 1048576 \beta ) q^{6} \) \( + ( 108553401186544328 - 30482802 \beta ) q^{7} \) \( + 1152921504606846976 q^{8} \) \( + ( 20986464808436254893 - 8863347528 \beta ) q^{9} \) \(+O(q^{10})\) \( q\) \(+1048576 q^{2}\) \(+(4431673764 - \beta) q^{3}\) \(+1099511627776 q^{4}\) \(+(48799592162790 + 49284 \beta) q^{5}\) \(+(4646946748760064 - 1048576 \beta) q^{6}\) \(+(108553401186544328 - 30482802 \beta) q^{7}\) \(+1152921504606846976 q^{8}\) \(+(20986464808436254893 - 8863347528 \beta) q^{9}\) \(+(51170081151689687040 + 51678019584 \beta) q^{10}\) \(+(55297240742469782892 + 187241620797 \beta) q^{11}\) \(+(\)\(48\!\cdots\!64\)\( - 1099511627776 \beta) q^{12}\) \(+(\)\(86\!\cdots\!34\)\( + 929010448548 \beta) q^{13}\) \(+(\)\(11\!\cdots\!28\)\( - 31963534589952 \beta) q^{14}\) \(+(-\)\(16\!\cdots\!40\)\( + 169611017622186 \beta) q^{15}\) \(+\)\(12\!\cdots\!76\)\( q^{16}\) \(+(-\)\(51\!\cdots\!02\)\( + 30456449636472 \beta) q^{17}\) \(+(\)\(22\!\cdots\!68\)\( - 9293893497520128 \beta) q^{18}\) \(+(\)\(10\!\cdots\!00\)\( + 22472953406377947 \beta) q^{19}\) \(+(\)\(53\!\cdots\!40\)\( + 54188331063312384 \beta) q^{20}\) \(+(\)\(16\!\cdots\!92\)\( - 243643235063151056 \beta) q^{21}\) \(+(\)\(57\!\cdots\!92\)\( + 196337069768835072 \beta) q^{22}\) \(+(\)\(74\!\cdots\!84\)\( + 125759442615183594 \beta) q^{23}\) \(+(\)\(51\!\cdots\!64\)\( - 1152921504606846976 \beta) q^{24}\) \(+(\)\(48\!\cdots\!75\)\( + 4810078200301884720 \beta) q^{25}\) \(+(\)\(90\!\cdots\!84\)\( + 974138060096667648 \beta) q^{26}\) \(+(\)\(26\!\cdots\!60\)\( - 23792933132317323882 \beta) q^{27}\) \(+(\)\(11\!\cdots\!28\)\( - 33516195246193508352 \beta) q^{28}\) \(+(-\)\(68\!\cdots\!10\)\( + \)\(15\!\cdots\!44\)\( \beta) q^{29}\) \(+(-\)\(17\!\cdots\!40\)\( + \)\(17\!\cdots\!36\)\( \beta) q^{30}\) \(+(-\)\(18\!\cdots\!08\)\( - \)\(74\!\cdots\!56\)\( \beta) q^{31}\) \(+\)\(12\!\cdots\!76\)\( q^{32}\) \(+(-\)\(68\!\cdots\!12\)\( + \)\(77\!\cdots\!16\)\( \beta) q^{33}\) \(+(-\)\(53\!\cdots\!52\)\( + 31935902134013263872 \beta) q^{34}\) \(+(-\)\(51\!\cdots\!80\)\( + \)\(38\!\cdots\!72\)\( \beta) q^{35}\) \(+(\)\(23\!\cdots\!68\)\( - \)\(97\!\cdots\!28\)\( \beta) q^{36}\) \(+(\)\(60\!\cdots\!18\)\( + \)\(22\!\cdots\!92\)\( \beta) q^{37}\) \(+(\)\(10\!\cdots\!00\)\( + \)\(23\!\cdots\!72\)\( \beta) q^{38}\) \(+(\)\(34\!\cdots\!76\)\( - \)\(82\!\cdots\!62\)\( \beta) q^{39}\) \(+(\)\(56\!\cdots\!40\)\( + \)\(56\!\cdots\!84\)\( \beta) q^{40}\) \(+(\)\(63\!\cdots\!82\)\( + \)\(24\!\cdots\!16\)\( \beta) q^{41}\) \(+(\)\(17\!\cdots\!92\)\( - \)\(25\!\cdots\!56\)\( \beta) q^{42}\) \(+(-\)\(30\!\cdots\!76\)\( - \)\(43\!\cdots\!83\)\( \beta) q^{43}\) \(+(\)\(60\!\cdots\!92\)\( + \)\(20\!\cdots\!72\)\( \beta) q^{44}\) \(+(-\)\(15\!\cdots\!30\)\( + \)\(60\!\cdots\!92\)\( \beta) q^{45}\) \(+(\)\(78\!\cdots\!84\)\( + \)\(13\!\cdots\!44\)\( \beta) q^{46}\) \(+(-\)\(12\!\cdots\!32\)\( + \)\(10\!\cdots\!88\)\( \beta) q^{47}\) \(+(\)\(53\!\cdots\!64\)\( - \)\(12\!\cdots\!76\)\( \beta) q^{48}\) \(+(\)\(23\!\cdots\!77\)\( - \)\(66\!\cdots\!12\)\( \beta) q^{49}\) \(+(\)\(51\!\cdots\!00\)\( + \)\(50\!\cdots\!20\)\( \beta) q^{50}\) \(+(-\)\(23\!\cdots\!28\)\( + \)\(52\!\cdots\!10\)\( \beta) q^{51}\) \(+(\)\(95\!\cdots\!84\)\( + \)\(10\!\cdots\!48\)\( \beta) q^{52}\) \(+(-\)\(15\!\cdots\!26\)\( + \)\(82\!\cdots\!88\)\( \beta) q^{53}\) \(+(\)\(27\!\cdots\!60\)\( - \)\(24\!\cdots\!32\)\( \beta) q^{54}\) \(+(\)\(35\!\cdots\!80\)\( + \)\(11\!\cdots\!58\)\( \beta) q^{55}\) \(+(\)\(12\!\cdots\!28\)\( - \)\(35\!\cdots\!52\)\( \beta) q^{56}\) \(+(-\)\(39\!\cdots\!00\)\( - \)\(35\!\cdots\!92\)\( \beta) q^{57}\) \(+(-\)\(72\!\cdots\!60\)\( + \)\(16\!\cdots\!44\)\( \beta) q^{58}\) \(+(-\)\(16\!\cdots\!20\)\( - \)\(27\!\cdots\!87\)\( \beta) q^{59}\) \(+(-\)\(18\!\cdots\!40\)\( + \)\(18\!\cdots\!36\)\( \beta) q^{60}\) \(+(-\)\(22\!\cdots\!18\)\( + \)\(70\!\cdots\!88\)\( \beta) q^{61}\) \(+(-\)\(19\!\cdots\!08\)\( - \)\(78\!\cdots\!56\)\( \beta) q^{62}\) \(+(\)\(12\!\cdots\!04\)\( - \)\(16\!\cdots\!70\)\( \beta) q^{63}\) \(+\)\(13\!\cdots\!76\)\( q^{64}\) \(+(\)\(59\!\cdots\!60\)\( + \)\(43\!\cdots\!76\)\( \beta) q^{65}\) \(+(-\)\(71\!\cdots\!12\)\( + \)\(81\!\cdots\!16\)\( \beta) q^{66}\) \(+(\)\(27\!\cdots\!28\)\( - \)\(41\!\cdots\!61\)\( \beta) q^{67}\) \(+(-\)\(56\!\cdots\!52\)\( + \)\(33\!\cdots\!72\)\( \beta) q^{68}\) \(+(\)\(28\!\cdots\!76\)\( - \)\(69\!\cdots\!68\)\( \beta) q^{69}\) \(+(-\)\(54\!\cdots\!80\)\( + \)\(40\!\cdots\!72\)\( \beta) q^{70}\) \(+(-\)\(24\!\cdots\!08\)\( + \)\(13\!\cdots\!38\)\( \beta) q^{71}\) \(+(\)\(24\!\cdots\!68\)\( - \)\(10\!\cdots\!28\)\( \beta) q^{72}\) \(+(-\)\(24\!\cdots\!06\)\( + \)\(66\!\cdots\!08\)\( \beta) q^{73}\) \(+(\)\(63\!\cdots\!68\)\( + \)\(23\!\cdots\!92\)\( \beta) q^{74}\) \(+(\)\(34\!\cdots\!00\)\( - \)\(27\!\cdots\!95\)\( \beta) q^{75}\) \(+(\)\(11\!\cdots\!00\)\( + \)\(24\!\cdots\!72\)\( \beta) q^{76}\) \(+(-\)\(20\!\cdots\!24\)\( + \)\(18\!\cdots\!32\)\( \beta) q^{77}\) \(+(\)\(36\!\cdots\!76\)\( - \)\(86\!\cdots\!12\)\( \beta) q^{78}\) \(+(-\)\(45\!\cdots\!40\)\( + \)\(33\!\cdots\!32\)\( \beta) q^{79}\) \(+(\)\(58\!\cdots\!40\)\( + \)\(59\!\cdots\!84\)\( \beta) q^{80}\) \(+(\)\(13\!\cdots\!61\)\( - \)\(48\!\cdots\!24\)\( \beta) q^{81}\) \(+(\)\(66\!\cdots\!32\)\( + \)\(25\!\cdots\!16\)\( \beta) q^{82}\) \(+(\)\(63\!\cdots\!84\)\( - \)\(10\!\cdots\!65\)\( \beta) q^{83}\) \(+(\)\(17\!\cdots\!92\)\( - \)\(26\!\cdots\!56\)\( \beta) q^{84}\) \(+(-\)\(19\!\cdots\!80\)\( - \)\(25\!\cdots\!88\)\( \beta) q^{85}\) \(+(-\)\(31\!\cdots\!76\)\( - \)\(45\!\cdots\!08\)\( \beta) q^{86}\) \(+(-\)\(88\!\cdots\!40\)\( + \)\(13\!\cdots\!26\)\( \beta) q^{87}\) \(+(\)\(63\!\cdots\!92\)\( + \)\(21\!\cdots\!72\)\( \beta) q^{88}\) \(+(\)\(13\!\cdots\!90\)\( - \)\(13\!\cdots\!40\)\( \beta) q^{89}\) \(+(-\)\(16\!\cdots\!80\)\( + \)\(63\!\cdots\!92\)\( \beta) q^{90}\) \(+(\)\(83\!\cdots\!52\)\( - \)\(25\!\cdots\!24\)\( \beta) q^{91}\) \(+(\)\(82\!\cdots\!84\)\( + \)\(13\!\cdots\!44\)\( \beta) q^{92}\) \(+(\)\(20\!\cdots\!88\)\( - \)\(14\!\cdots\!76\)\( \beta) q^{93}\) \(+(-\)\(13\!\cdots\!32\)\( + \)\(10\!\cdots\!88\)\( \beta) q^{94}\) \(+(\)\(46\!\cdots\!00\)\( + \)\(61\!\cdots\!30\)\( \beta) q^{95}\) \(+(\)\(56\!\cdots\!64\)\( - \)\(12\!\cdots\!76\)\( \beta) q^{96}\) \(+(-\)\(15\!\cdots\!02\)\( - \)\(12\!\cdots\!20\)\( \beta) q^{97}\) \(+(\)\(24\!\cdots\!52\)\( - \)\(69\!\cdots\!12\)\( \beta) q^{98}\) \(+(-\)\(61\!\cdots\!44\)\( + \)\(34\!\cdots\!45\)\( \beta) q^{99}\) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut +\mathstrut 2097152q^{2} \) \(\mathstrut +\mathstrut 8863347528q^{3} \) \(\mathstrut +\mathstrut 2199023255552q^{4} \) \(\mathstrut +\mathstrut 97599184325580q^{5} \) \(\mathstrut +\mathstrut 9293893497520128q^{6} \) \(\mathstrut +\mathstrut 217106802373088656q^{7} \) \(\mathstrut +\mathstrut 2305843009213693952q^{8} \) \(\mathstrut +\mathstrut 41972929616872509786q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut +\mathstrut 2097152q^{2} \) \(\mathstrut +\mathstrut 8863347528q^{3} \) \(\mathstrut +\mathstrut 2199023255552q^{4} \) \(\mathstrut +\mathstrut 97599184325580q^{5} \) \(\mathstrut +\mathstrut 9293893497520128q^{6} \) \(\mathstrut +\mathstrut 217106802373088656q^{7} \) \(\mathstrut +\mathstrut 2305843009213693952q^{8} \) \(\mathstrut +\mathstrut 41972929616872509786q^{9} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!80\)\(q^{10} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!84\)\(q^{11} \) \(\mathstrut +\mathstrut \)\(97\!\cdots\!28\)\(q^{12} \) \(\mathstrut +\mathstrut \)\(17\!\cdots\!68\)\(q^{13} \) \(\mathstrut +\mathstrut \)\(22\!\cdots\!56\)\(q^{14} \) \(\mathstrut -\mathstrut \)\(32\!\cdots\!80\)\(q^{15} \) \(\mathstrut +\mathstrut \)\(24\!\cdots\!52\)\(q^{16} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!04\)\(q^{17} \) \(\mathstrut +\mathstrut \)\(44\!\cdots\!36\)\(q^{18} \) \(\mathstrut +\mathstrut \)\(20\!\cdots\!00\)\(q^{19} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!80\)\(q^{20} \) \(\mathstrut +\mathstrut \)\(32\!\cdots\!84\)\(q^{21} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!84\)\(q^{22} \) \(\mathstrut +\mathstrut \)\(14\!\cdots\!68\)\(q^{23} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!28\)\(q^{24} \) \(\mathstrut +\mathstrut \)\(97\!\cdots\!50\)\(q^{25} \) \(\mathstrut +\mathstrut \)\(18\!\cdots\!68\)\(q^{26} \) \(\mathstrut +\mathstrut \)\(53\!\cdots\!20\)\(q^{27} \) \(\mathstrut +\mathstrut \)\(23\!\cdots\!56\)\(q^{28} \) \(\mathstrut -\mathstrut \)\(13\!\cdots\!20\)\(q^{29} \) \(\mathstrut -\mathstrut \)\(34\!\cdots\!80\)\(q^{30} \) \(\mathstrut -\mathstrut \)\(36\!\cdots\!16\)\(q^{31} \) \(\mathstrut +\mathstrut \)\(25\!\cdots\!52\)\(q^{32} \) \(\mathstrut -\mathstrut \)\(13\!\cdots\!24\)\(q^{33} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!04\)\(q^{34} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!60\)\(q^{35} \) \(\mathstrut +\mathstrut \)\(46\!\cdots\!36\)\(q^{36} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!36\)\(q^{37} \) \(\mathstrut +\mathstrut \)\(21\!\cdots\!00\)\(q^{38} \) \(\mathstrut +\mathstrut \)\(69\!\cdots\!52\)\(q^{39} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!80\)\(q^{40} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!64\)\(q^{41} \) \(\mathstrut +\mathstrut \)\(34\!\cdots\!84\)\(q^{42} \) \(\mathstrut -\mathstrut \)\(60\!\cdots\!52\)\(q^{43} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!84\)\(q^{44} \) \(\mathstrut -\mathstrut \)\(30\!\cdots\!60\)\(q^{45} \) \(\mathstrut +\mathstrut \)\(15\!\cdots\!68\)\(q^{46} \) \(\mathstrut -\mathstrut \)\(25\!\cdots\!64\)\(q^{47} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!28\)\(q^{48} \) \(\mathstrut +\mathstrut \)\(47\!\cdots\!54\)\(q^{49} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!00\)\(q^{50} \) \(\mathstrut -\mathstrut \)\(47\!\cdots\!56\)\(q^{51} \) \(\mathstrut +\mathstrut \)\(19\!\cdots\!68\)\(q^{52} \) \(\mathstrut -\mathstrut \)\(30\!\cdots\!52\)\(q^{53} \) \(\mathstrut +\mathstrut \)\(55\!\cdots\!20\)\(q^{54} \) \(\mathstrut +\mathstrut \)\(70\!\cdots\!60\)\(q^{55} \) \(\mathstrut +\mathstrut \)\(25\!\cdots\!56\)\(q^{56} \) \(\mathstrut -\mathstrut \)\(78\!\cdots\!00\)\(q^{57} \) \(\mathstrut -\mathstrut \)\(14\!\cdots\!20\)\(q^{58} \) \(\mathstrut -\mathstrut \)\(32\!\cdots\!40\)\(q^{59} \) \(\mathstrut -\mathstrut \)\(36\!\cdots\!80\)\(q^{60} \) \(\mathstrut -\mathstrut \)\(44\!\cdots\!36\)\(q^{61} \) \(\mathstrut -\mathstrut \)\(38\!\cdots\!16\)\(q^{62} \) \(\mathstrut +\mathstrut \)\(24\!\cdots\!08\)\(q^{63} \) \(\mathstrut +\mathstrut \)\(26\!\cdots\!52\)\(q^{64} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!20\)\(q^{65} \) \(\mathstrut -\mathstrut \)\(14\!\cdots\!24\)\(q^{66} \) \(\mathstrut +\mathstrut \)\(55\!\cdots\!56\)\(q^{67} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!04\)\(q^{68} \) \(\mathstrut +\mathstrut \)\(56\!\cdots\!52\)\(q^{69} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!60\)\(q^{70} \) \(\mathstrut -\mathstrut \)\(48\!\cdots\!16\)\(q^{71} \) \(\mathstrut +\mathstrut \)\(48\!\cdots\!36\)\(q^{72} \) \(\mathstrut -\mathstrut \)\(48\!\cdots\!12\)\(q^{73} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!36\)\(q^{74} \) \(\mathstrut +\mathstrut \)\(68\!\cdots\!00\)\(q^{75} \) \(\mathstrut +\mathstrut \)\(22\!\cdots\!00\)\(q^{76} \) \(\mathstrut -\mathstrut \)\(41\!\cdots\!48\)\(q^{77} \) \(\mathstrut +\mathstrut \)\(73\!\cdots\!52\)\(q^{78} \) \(\mathstrut -\mathstrut \)\(90\!\cdots\!80\)\(q^{79} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!80\)\(q^{80} \) \(\mathstrut +\mathstrut \)\(26\!\cdots\!22\)\(q^{81} \) \(\mathstrut +\mathstrut \)\(13\!\cdots\!64\)\(q^{82} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!68\)\(q^{83} \) \(\mathstrut +\mathstrut \)\(35\!\cdots\!84\)\(q^{84} \) \(\mathstrut -\mathstrut \)\(38\!\cdots\!60\)\(q^{85} \) \(\mathstrut -\mathstrut \)\(63\!\cdots\!52\)\(q^{86} \) \(\mathstrut -\mathstrut \)\(17\!\cdots\!80\)\(q^{87} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!84\)\(q^{88} \) \(\mathstrut +\mathstrut \)\(26\!\cdots\!80\)\(q^{89} \) \(\mathstrut -\mathstrut \)\(32\!\cdots\!60\)\(q^{90} \) \(\mathstrut +\mathstrut \)\(16\!\cdots\!04\)\(q^{91} \) \(\mathstrut +\mathstrut \)\(16\!\cdots\!68\)\(q^{92} \) \(\mathstrut +\mathstrut \)\(40\!\cdots\!76\)\(q^{93} \) \(\mathstrut -\mathstrut \)\(26\!\cdots\!64\)\(q^{94} \) \(\mathstrut +\mathstrut \)\(93\!\cdots\!00\)\(q^{95} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!28\)\(q^{96} \) \(\mathstrut -\mathstrut \)\(31\!\cdots\!04\)\(q^{97} \) \(\mathstrut +\mathstrut \)\(49\!\cdots\!04\)\(q^{98} \) \(\mathstrut -\mathstrut \)\(12\!\cdots\!88\)\(q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
1.06767e6
−1.06767e6
1.04858e6 −1.71810e9 1.09951e12 3.51885e14 −1.80156e15 −7.89090e16 1.15292e18 −3.35211e19 3.68978e20
1.2 1.04858e6 1.05814e10 1.09951e12 −2.54286e14 1.10955e16 2.96016e17 1.15292e18 7.54941e19 −2.66638e20
\(n\): 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 \(T_{3}^{2} \) \(\mathstrut -\mathstrut 8863347528 T_{3} \) \(\mathstrut -\mathstrut \)\(18\!\cdots\!04\)\( \) acting on \(S_{42}^{\mathrm{new}}(\Gamma_0(2))\).