Properties

Label 2.40.a.b
Level 2
Weight 40
Character orbit 2.a
Self dual Yes
Analytic conductor 19.268
Analytic rank 0
Dimension 2
CM No
Inner twists 1

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 2 \)
Weight: \( k \) = \( 40 \)
Character orbit: \([\chi]\) = 2.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(19.2679102779\)
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: \( 2^{7}\cdot 3\cdot 5 \)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 960\sqrt{4202094647521}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( -524288 q^{2} \) \( + ( 143709132 - \beta ) q^{3} \) \( + 274877906944 q^{4} \) \( + ( 26811369083310 - 21924 \beta ) q^{5} \) \( + ( -75344973398016 + 524288 \beta ) q^{6} \) \( + ( 3748995812923256 - 26490618 \beta ) q^{7} \) \( -144115188075855872 q^{8} \) \( + ( -159252411243429243 - 287418264 \beta ) q^{9} \) \(+O(q^{10})\) \( q\) \(-524288 q^{2}\) \(+(143709132 - \beta) q^{3}\) \(+274877906944 q^{4}\) \(+(26811369083310 - 21924 \beta) q^{5}\) \(+(-75344973398016 + 524288 \beta) q^{6}\) \(+(3748995812923256 - 26490618 \beta) q^{7}\) \(-144115188075855872 q^{8}\) \(+(-159252411243429243 - 287418264 \beta) q^{9}\) \(+(-14056879073950433280 + 11494490112 \beta) q^{10}\) \(+(-\)\(21\!\cdots\!08\)\( - 1640149371 \beta) q^{11}\) \(+(39502465412899012608 - 274877906944 \beta) q^{12}\) \(+(-\)\(55\!\cdots\!78\)\( - 1090692827172 \beta) q^{13}\) \(+(-\)\(19\!\cdots\!28\)\( + 13888713129984 \beta) q^{14}\) \(+(\)\(88\!\cdots\!20\)\( - 29962048093278 \beta) q^{15}\) \(+\)\(75\!\cdots\!36\)\( q^{16}\) \(+(-\)\(18\!\cdots\!54\)\( - 81667210689432 \beta) q^{17}\) \(+(\)\(83\!\cdots\!84\)\( + 150689946796032 \beta) q^{18}\) \(+(\)\(57\!\cdots\!80\)\( + 2573198844112971 \beta) q^{19}\) \(+(\)\(73\!\cdots\!40\)\( - 6026423231840256 \beta) q^{20}\) \(+(\)\(10\!\cdots\!92\)\( - 7555939531846832 \beta) q^{21}\) \(+(\)\(11\!\cdots\!04\)\( + 859910633422848 \beta) q^{22}\) \(+(\)\(36\!\cdots\!12\)\( + 98470758272206194 \beta) q^{23}\) \(+(-\)\(20\!\cdots\!04\)\( + 144115188075855872 \beta) q^{24}\) \(+(\)\(76\!\cdots\!75\)\( - 1175624911564976880 \beta) q^{25}\) \(+(\)\(29\!\cdots\!64\)\( + 571837160972353536 \beta) q^{26}\) \(+(\)\(50\!\cdots\!80\)\( + 4170502935022018662 \beta) q^{27}\) \(+(\)\(10\!\cdots\!64\)\( - 7281685629493051392 \beta) q^{28}\) \(+(-\)\(33\!\cdots\!30\)\( + 3076733487202628652 \beta) q^{29}\) \(+(-\)\(46\!\cdots\!60\)\( + 15708742270728536064 \beta) q^{30}\) \(+(-\)\(93\!\cdots\!28\)\( - 57411334880600430312 \beta) q^{31}\) \(-\)\(39\!\cdots\!68\)\( q^{32}\) \(+(-\)\(24\!\cdots\!56\)\( + \)\(21\!\cdots\!36\)\( \beta) q^{33}\) \(+(\)\(94\!\cdots\!52\)\( + 42817138557940924416 \beta) q^{34}\) \(+(\)\(23\!\cdots\!60\)\( - \)\(79\!\cdots\!24\)\( \beta) q^{35}\) \(+(-\)\(43\!\cdots\!92\)\( - 79004930825798025216 \beta) q^{36}\) \(+(\)\(14\!\cdots\!26\)\( + \)\(20\!\cdots\!68\)\( \beta) q^{37}\) \(+(-\)\(30\!\cdots\!40\)\( - \)\(13\!\cdots\!48\)\( \beta) q^{38}\) \(+(\)\(41\!\cdots\!04\)\( + \)\(39\!\cdots\!74\)\( \beta) q^{39}\) \(+(-\)\(38\!\cdots\!20\)\( + \)\(31\!\cdots\!28\)\( \beta) q^{40}\) \(+(\)\(83\!\cdots\!62\)\( - \)\(13\!\cdots\!08\)\( \beta) q^{41}\) \(+(-\)\(54\!\cdots\!96\)\( + \)\(39\!\cdots\!16\)\( \beta) q^{42}\) \(+(-\)\(49\!\cdots\!08\)\( + \)\(21\!\cdots\!17\)\( \beta) q^{43}\) \(+(-\)\(59\!\cdots\!52\)\( - \)\(45\!\cdots\!24\)\( \beta) q^{44}\) \(+(\)\(20\!\cdots\!70\)\( - \)\(42\!\cdots\!08\)\( \beta) q^{45}\) \(+(-\)\(19\!\cdots\!56\)\( - \)\(51\!\cdots\!72\)\( \beta) q^{46}\) \(+(\)\(33\!\cdots\!16\)\( + \)\(63\!\cdots\!92\)\( \beta) q^{47}\) \(+(\)\(10\!\cdots\!52\)\( - \)\(75\!\cdots\!36\)\( \beta) q^{48}\) \(+(\)\(18\!\cdots\!93\)\( - \)\(19\!\cdots\!16\)\( \beta) q^{49}\) \(+(-\)\(39\!\cdots\!00\)\( + \)\(61\!\cdots\!40\)\( \beta) q^{50}\) \(+(\)\(29\!\cdots\!72\)\( + \)\(16\!\cdots\!30\)\( \beta) q^{51}\) \(+(-\)\(15\!\cdots\!32\)\( - \)\(29\!\cdots\!68\)\( \beta) q^{52}\) \(+(-\)\(43\!\cdots\!18\)\( - \)\(11\!\cdots\!32\)\( \beta) q^{53}\) \(+(-\)\(26\!\cdots\!40\)\( - \)\(21\!\cdots\!56\)\( \beta) q^{54}\) \(+(-\)\(56\!\cdots\!80\)\( + \)\(47\!\cdots\!82\)\( \beta) q^{55}\) \(+(-\)\(54\!\cdots\!32\)\( + \)\(38\!\cdots\!96\)\( \beta) q^{56}\) \(+(-\)\(91\!\cdots\!40\)\( - \)\(53\!\cdots\!08\)\( \beta) q^{57}\) \(+(\)\(17\!\cdots\!40\)\( - \)\(16\!\cdots\!76\)\( \beta) q^{58}\) \(+(-\)\(64\!\cdots\!60\)\( + \)\(12\!\cdots\!49\)\( \beta) q^{59}\) \(+(\)\(24\!\cdots\!80\)\( - \)\(82\!\cdots\!32\)\( \beta) q^{60}\) \(+(-\)\(10\!\cdots\!58\)\( - \)\(45\!\cdots\!04\)\( \beta) q^{61}\) \(+(\)\(48\!\cdots\!64\)\( + \)\(30\!\cdots\!56\)\( \beta) q^{62}\) \(+(\)\(28\!\cdots\!92\)\( + \)\(31\!\cdots\!90\)\( \beta) q^{63}\) \(+\)\(20\!\cdots\!84\)\( q^{64}\) \(+(\)\(77\!\cdots\!20\)\( - \)\(17\!\cdots\!48\)\( \beta) q^{65}\) \(+(\)\(12\!\cdots\!28\)\( - \)\(11\!\cdots\!68\)\( \beta) q^{66}\) \(+(-\)\(21\!\cdots\!04\)\( + \)\(18\!\cdots\!31\)\( \beta) q^{67}\) \(+(-\)\(49\!\cdots\!76\)\( - \)\(22\!\cdots\!08\)\( \beta) q^{68}\) \(+(-\)\(32\!\cdots\!16\)\( - \)\(35\!\cdots\!04\)\( \beta) q^{69}\) \(+(-\)\(12\!\cdots\!80\)\( + \)\(41\!\cdots\!12\)\( \beta) q^{70}\) \(+(-\)\(25\!\cdots\!68\)\( + \)\(24\!\cdots\!26\)\( \beta) q^{71}\) \(+(\)\(22\!\cdots\!96\)\( + \)\(41\!\cdots\!08\)\( \beta) q^{72}\) \(+(\)\(26\!\cdots\!62\)\( - \)\(26\!\cdots\!32\)\( \beta) q^{73}\) \(+(-\)\(78\!\cdots\!88\)\( - \)\(10\!\cdots\!84\)\( \beta) q^{74}\) \(+(\)\(46\!\cdots\!00\)\( - \)\(93\!\cdots\!35\)\( \beta) q^{75}\) \(+(\)\(15\!\cdots\!20\)\( + \)\(70\!\cdots\!24\)\( \beta) q^{76}\) \(+(-\)\(64\!\cdots\!48\)\( + \)\(57\!\cdots\!68\)\( \beta) q^{77}\) \(+(-\)\(21\!\cdots\!52\)\( - \)\(20\!\cdots\!12\)\( \beta) q^{78}\) \(+(\)\(25\!\cdots\!20\)\( - \)\(74\!\cdots\!04\)\( \beta) q^{79}\) \(+(\)\(20\!\cdots\!60\)\( - \)\(16\!\cdots\!64\)\( \beta) q^{80}\) \(+(-\)\(15\!\cdots\!59\)\( + \)\(12\!\cdots\!92\)\( \beta) q^{81}\) \(+(-\)\(43\!\cdots\!56\)\( + \)\(73\!\cdots\!04\)\( \beta) q^{82}\) \(+(-\)\(32\!\cdots\!48\)\( - \)\(21\!\cdots\!85\)\( \beta) q^{83}\) \(+(\)\(28\!\cdots\!48\)\( - \)\(20\!\cdots\!08\)\( \beta) q^{84}\) \(+(\)\(20\!\cdots\!60\)\( + \)\(17\!\cdots\!76\)\( \beta) q^{85}\) \(+(\)\(25\!\cdots\!04\)\( - \)\(11\!\cdots\!96\)\( \beta) q^{86}\) \(+(-\)\(16\!\cdots\!60\)\( + \)\(33\!\cdots\!94\)\( \beta) q^{87}\) \(+(\)\(31\!\cdots\!76\)\( + \)\(23\!\cdots\!12\)\( \beta) q^{88}\) \(+(-\)\(17\!\cdots\!90\)\( - \)\(61\!\cdots\!20\)\( \beta) q^{89}\) \(+(-\)\(10\!\cdots\!60\)\( + \)\(22\!\cdots\!04\)\( \beta) q^{90}\) \(+(\)\(10\!\cdots\!32\)\( + \)\(10\!\cdots\!72\)\( \beta) q^{91}\) \(+(\)\(10\!\cdots\!28\)\( + \)\(27\!\cdots\!36\)\( \beta) q^{92}\) \(+(\)\(20\!\cdots\!04\)\( + \)\(85\!\cdots\!44\)\( \beta) q^{93}\) \(+(-\)\(17\!\cdots\!08\)\( - \)\(33\!\cdots\!96\)\( \beta) q^{94}\) \(+(-\)\(63\!\cdots\!00\)\( - \)\(57\!\cdots\!10\)\( \beta) q^{95}\) \(+(-\)\(56\!\cdots\!76\)\( + \)\(39\!\cdots\!68\)\( \beta) q^{96}\) \(+(-\)\(72\!\cdots\!34\)\( - \)\(19\!\cdots\!00\)\( \beta) q^{97}\) \(+(-\)\(95\!\cdots\!84\)\( + \)\(10\!\cdots\!08\)\( \beta) q^{98}\) \(+(\)\(36\!\cdots\!44\)\( + \)\(62\!\cdots\!65\)\( \beta) q^{99}\) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut -\mathstrut 1048576q^{2} \) \(\mathstrut +\mathstrut 287418264q^{3} \) \(\mathstrut +\mathstrut 549755813888q^{4} \) \(\mathstrut +\mathstrut 53622738166620q^{5} \) \(\mathstrut -\mathstrut 150689946796032q^{6} \) \(\mathstrut +\mathstrut 7497991625846512q^{7} \) \(\mathstrut -\mathstrut 288230376151711744q^{8} \) \(\mathstrut -\mathstrut 318504822486858486q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut -\mathstrut 1048576q^{2} \) \(\mathstrut +\mathstrut 287418264q^{3} \) \(\mathstrut +\mathstrut 549755813888q^{4} \) \(\mathstrut +\mathstrut 53622738166620q^{5} \) \(\mathstrut -\mathstrut 150689946796032q^{6} \) \(\mathstrut +\mathstrut 7497991625846512q^{7} \) \(\mathstrut -\mathstrut 288230376151711744q^{8} \) \(\mathstrut -\mathstrut 318504822486858486q^{9} \) \(\mathstrut -\mathstrut 28113758147900866560q^{10} \) \(\mathstrut -\mathstrut \)\(43\!\cdots\!16\)\(q^{11} \) \(\mathstrut +\mathstrut 79004930825798025216q^{12} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!56\)\(q^{13} \) \(\mathstrut -\mathstrut \)\(39\!\cdots\!56\)\(q^{14} \) \(\mathstrut +\mathstrut \)\(17\!\cdots\!40\)\(q^{15} \) \(\mathstrut +\mathstrut \)\(15\!\cdots\!72\)\(q^{16} \) \(\mathstrut -\mathstrut \)\(36\!\cdots\!08\)\(q^{17} \) \(\mathstrut +\mathstrut \)\(16\!\cdots\!68\)\(q^{18} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!60\)\(q^{19} \) \(\mathstrut +\mathstrut \)\(14\!\cdots\!80\)\(q^{20} \) \(\mathstrut +\mathstrut \)\(20\!\cdots\!84\)\(q^{21} \) \(\mathstrut +\mathstrut \)\(22\!\cdots\!08\)\(q^{22} \) \(\mathstrut +\mathstrut \)\(73\!\cdots\!24\)\(q^{23} \) \(\mathstrut -\mathstrut \)\(41\!\cdots\!08\)\(q^{24} \) \(\mathstrut +\mathstrut \)\(15\!\cdots\!50\)\(q^{25} \) \(\mathstrut +\mathstrut \)\(58\!\cdots\!28\)\(q^{26} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!60\)\(q^{27} \) \(\mathstrut +\mathstrut \)\(20\!\cdots\!28\)\(q^{28} \) \(\mathstrut -\mathstrut \)\(66\!\cdots\!60\)\(q^{29} \) \(\mathstrut -\mathstrut \)\(93\!\cdots\!20\)\(q^{30} \) \(\mathstrut -\mathstrut \)\(18\!\cdots\!56\)\(q^{31} \) \(\mathstrut -\mathstrut \)\(79\!\cdots\!36\)\(q^{32} \) \(\mathstrut -\mathstrut \)\(49\!\cdots\!12\)\(q^{33} \) \(\mathstrut +\mathstrut \)\(18\!\cdots\!04\)\(q^{34} \) \(\mathstrut +\mathstrut \)\(46\!\cdots\!20\)\(q^{35} \) \(\mathstrut -\mathstrut \)\(87\!\cdots\!84\)\(q^{36} \) \(\mathstrut +\mathstrut \)\(29\!\cdots\!52\)\(q^{37} \) \(\mathstrut -\mathstrut \)\(60\!\cdots\!80\)\(q^{38} \) \(\mathstrut +\mathstrut \)\(82\!\cdots\!08\)\(q^{39} \) \(\mathstrut -\mathstrut \)\(77\!\cdots\!40\)\(q^{40} \) \(\mathstrut +\mathstrut \)\(16\!\cdots\!24\)\(q^{41} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!92\)\(q^{42} \) \(\mathstrut -\mathstrut \)\(98\!\cdots\!16\)\(q^{43} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!04\)\(q^{44} \) \(\mathstrut +\mathstrut \)\(40\!\cdots\!40\)\(q^{45} \) \(\mathstrut -\mathstrut \)\(38\!\cdots\!12\)\(q^{46} \) \(\mathstrut +\mathstrut \)\(67\!\cdots\!32\)\(q^{47} \) \(\mathstrut +\mathstrut \)\(21\!\cdots\!04\)\(q^{48} \) \(\mathstrut +\mathstrut \)\(36\!\cdots\!86\)\(q^{49} \) \(\mathstrut -\mathstrut \)\(79\!\cdots\!00\)\(q^{50} \) \(\mathstrut +\mathstrut \)\(58\!\cdots\!44\)\(q^{51} \) \(\mathstrut -\mathstrut \)\(30\!\cdots\!64\)\(q^{52} \) \(\mathstrut -\mathstrut \)\(87\!\cdots\!36\)\(q^{53} \) \(\mathstrut -\mathstrut \)\(53\!\cdots\!80\)\(q^{54} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!60\)\(q^{55} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!64\)\(q^{56} \) \(\mathstrut -\mathstrut \)\(18\!\cdots\!80\)\(q^{57} \) \(\mathstrut +\mathstrut \)\(34\!\cdots\!80\)\(q^{58} \) \(\mathstrut -\mathstrut \)\(12\!\cdots\!20\)\(q^{59} \) \(\mathstrut +\mathstrut \)\(48\!\cdots\!60\)\(q^{60} \) \(\mathstrut -\mathstrut \)\(20\!\cdots\!16\)\(q^{61} \) \(\mathstrut +\mathstrut \)\(97\!\cdots\!28\)\(q^{62} \) \(\mathstrut +\mathstrut \)\(57\!\cdots\!84\)\(q^{63} \) \(\mathstrut +\mathstrut \)\(41\!\cdots\!68\)\(q^{64} \) \(\mathstrut +\mathstrut \)\(15\!\cdots\!40\)\(q^{65} \) \(\mathstrut +\mathstrut \)\(25\!\cdots\!56\)\(q^{66} \) \(\mathstrut -\mathstrut \)\(43\!\cdots\!08\)\(q^{67} \) \(\mathstrut -\mathstrut \)\(99\!\cdots\!52\)\(q^{68} \) \(\mathstrut -\mathstrut \)\(65\!\cdots\!32\)\(q^{69} \) \(\mathstrut -\mathstrut \)\(24\!\cdots\!60\)\(q^{70} \) \(\mathstrut -\mathstrut \)\(50\!\cdots\!36\)\(q^{71} \) \(\mathstrut +\mathstrut \)\(45\!\cdots\!92\)\(q^{72} \) \(\mathstrut +\mathstrut \)\(52\!\cdots\!24\)\(q^{73} \) \(\mathstrut -\mathstrut \)\(15\!\cdots\!76\)\(q^{74} \) \(\mathstrut +\mathstrut \)\(93\!\cdots\!00\)\(q^{75} \) \(\mathstrut +\mathstrut \)\(31\!\cdots\!40\)\(q^{76} \) \(\mathstrut -\mathstrut \)\(12\!\cdots\!96\)\(q^{77} \) \(\mathstrut -\mathstrut \)\(43\!\cdots\!04\)\(q^{78} \) \(\mathstrut +\mathstrut \)\(51\!\cdots\!40\)\(q^{79} \) \(\mathstrut +\mathstrut \)\(40\!\cdots\!20\)\(q^{80} \) \(\mathstrut -\mathstrut \)\(30\!\cdots\!18\)\(q^{81} \) \(\mathstrut -\mathstrut \)\(87\!\cdots\!12\)\(q^{82} \) \(\mathstrut -\mathstrut \)\(64\!\cdots\!96\)\(q^{83} \) \(\mathstrut +\mathstrut \)\(56\!\cdots\!96\)\(q^{84} \) \(\mathstrut +\mathstrut \)\(41\!\cdots\!20\)\(q^{85} \) \(\mathstrut +\mathstrut \)\(51\!\cdots\!08\)\(q^{86} \) \(\mathstrut -\mathstrut \)\(33\!\cdots\!20\)\(q^{87} \) \(\mathstrut +\mathstrut \)\(62\!\cdots\!52\)\(q^{88} \) \(\mathstrut -\mathstrut \)\(35\!\cdots\!80\)\(q^{89} \) \(\mathstrut -\mathstrut \)\(21\!\cdots\!20\)\(q^{90} \) \(\mathstrut +\mathstrut \)\(21\!\cdots\!64\)\(q^{91} \) \(\mathstrut +\mathstrut \)\(20\!\cdots\!56\)\(q^{92} \) \(\mathstrut +\mathstrut \)\(41\!\cdots\!08\)\(q^{93} \) \(\mathstrut -\mathstrut \)\(35\!\cdots\!16\)\(q^{94} \) \(\mathstrut -\mathstrut \)\(12\!\cdots\!00\)\(q^{95} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!52\)\(q^{96} \) \(\mathstrut -\mathstrut \)\(14\!\cdots\!68\)\(q^{97} \) \(\mathstrut -\mathstrut \)\(19\!\cdots\!68\)\(q^{98} \) \(\mathstrut +\mathstrut \)\(72\!\cdots\!88\)\(q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
1.02495e6
−1.02495e6
−524288. −1.82420e9 2.74878e11 −1.63330e13 9.56404e14 −4.83820e16 −1.44115e17 −7.24864e17 8.56319e18
1.2 −524288. 2.11161e9 2.74878e11 6.99557e13 −1.10709e15 5.58800e16 −1.44115e17 4.06359e17 −3.66769e19
\(n\): 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 \(T_{3}^{2} \) \(\mathstrut -\mathstrut 287418264 T_{3} \) \(\mathstrut -\mathstrut \)\(38\!\cdots\!76\)\( \) acting on \(S_{40}^{\mathrm{new}}(\Gamma_0(2))\).