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.

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

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
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 \( T_{3}^{2} - 8863347528 T_{3} - \)\(18\!\cdots\!04\)\( \) acting on \(S_{42}^{\mathrm{new}}(\Gamma_0(2))\).