Properties

Label 1.38.a.a
Level 1
Weight 38
Character orbit 1.a
Self dual Yes
Analytic conductor 8.671
Analytic rank 1
Dimension 2
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 1 \)
Weight: \( k \) = \( 38 \)
Character orbit: \([\chi]\) = 1.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(8.67140381246\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\mathbb{Q}[x]/(x^{2} - \cdots)\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{5}\cdot 3 \)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 48\sqrt{63737521}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + ( -97200 - \beta ) q^{2} \) \( + ( 6995700 + 72 \beta ) q^{3} \) \( + ( 18860134912 + 194400 \beta ) q^{4} \) \( + ( 2764792192950 + 28866400 \beta ) q^{5} \) \( + ( -11253271923648 - 13994100 \beta ) q^{6} \) \( + ( -1724221976743000 - 9650004336 \beta ) q^{7} \) \( + ( -17022021521817600 + 99683138560 \beta ) q^{8} \) \( + ( -449473689200884707 + 1007380800 \beta ) q^{9} \) \(+O(q^{10})\) \( q\) \(+(-97200 - \beta) q^{2}\) \(+(6995700 + 72 \beta) q^{3}\) \(+(18860134912 + 194400 \beta) q^{4}\) \(+(2764792192950 + 28866400 \beta) q^{5}\) \(+(-11253271923648 - 13994100 \beta) q^{6}\) \(+(-1724221976743000 - 9650004336 \beta) q^{7}\) \(+(-17022021521817600 + 99683138560 \beta) q^{8}\) \(+(-449473689200884707 + 1007380800 \beta) q^{9}\) \(+(-4507804677506637600 - 5570606272950 \beta) q^{10}\) \(+(-13367018177424269028 + 32186306471000 \beta) q^{11}\) \(+(2187387399185049600 + 2717893793664 \beta) q^{12}\) \(+(\)\(26\!\cdots\!50\)\( - 724443400725408 \beta) q^{13}\) \(+(\)\(15\!\cdots\!24\)\( + 2662202398202200 \beta) q^{14}\) \(+(\)\(32\!\cdots\!00\)\( + 401005712372400 \beta) q^{15}\) \(+(-\)\(15\!\cdots\!04\)\( - 19385312101171200 \beta) q^{16}\) \(+(-\)\(44\!\cdots\!50\)\( - 24633489938571456 \beta) q^{17}\) \(+(\)\(43\!\cdots\!00\)\( + 449375771787124707 \beta) q^{18}\) \(+(\)\(18\!\cdots\!60\)\( - 1292976182287157400 \beta) q^{19}\) \(+(\)\(87\!\cdots\!00\)\( + 1081899800733236800 \beta) q^{20}\) \(+(-\)\(11\!\cdots\!28\)\( - 191652517658851200 \beta) q^{21}\) \(+(-\)\(34\!\cdots\!00\)\( + 10238509188443069028 \beta) q^{22}\) \(+(-\)\(13\!\cdots\!00\)\( - 34024622863285905488 \beta) q^{23}\) \(+(\)\(93\!\cdots\!80\)\( - 528232217146675200 \beta) q^{24}\) \(+(\)\(57\!\cdots\!75\)\( + \)\(15\!\cdots\!00\)\( \beta) q^{25}\) \(+(\)\(80\!\cdots\!72\)\( - \)\(19\!\cdots\!50\)\( \beta) q^{26}\) \(+(-\)\(62\!\cdots\!00\)\( - 64775499512752949040 \beta) q^{27}\) \(+(-\)\(30\!\cdots\!00\)\( - \)\(51\!\cdots\!32\)\( \beta) q^{28}\) \(+(-\)\(63\!\cdots\!10\)\( + \)\(23\!\cdots\!00\)\( \beta) q^{29}\) \(+(-\)\(90\!\cdots\!00\)\( - \)\(36\!\cdots\!00\)\( \beta) q^{30}\) \(+(\)\(13\!\cdots\!12\)\( - \)\(64\!\cdots\!00\)\( \beta) q^{31}\) \(+(\)\(67\!\cdots\!00\)\( + \)\(37\!\cdots\!84\)\( \beta) q^{32}\) \(+(\)\(24\!\cdots\!00\)\( - \)\(73\!\cdots\!16\)\( \beta) q^{33}\) \(+(\)\(79\!\cdots\!04\)\( + \)\(47\!\cdots\!50\)\( \beta) q^{34}\) \(+(-\)\(45\!\cdots\!00\)\( - \)\(76\!\cdots\!00\)\( \beta) q^{35}\) \(+(-\)\(84\!\cdots\!84\)\( - \)\(87\!\cdots\!00\)\( \beta) q^{36}\) \(+(-\)\(34\!\cdots\!50\)\( + \)\(25\!\cdots\!04\)\( \beta) q^{37}\) \(+(\)\(17\!\cdots\!00\)\( - \)\(60\!\cdots\!60\)\( \beta) q^{38}\) \(+(-\)\(58\!\cdots\!84\)\( + \)\(14\!\cdots\!00\)\( \beta) q^{39}\) \(+(\)\(37\!\cdots\!00\)\( - \)\(21\!\cdots\!00\)\( \beta) q^{40}\) \(+(-\)\(63\!\cdots\!18\)\( - \)\(58\!\cdots\!00\)\( \beta) q^{41}\) \(+(\)\(39\!\cdots\!00\)\( + \)\(13\!\cdots\!28\)\( \beta) q^{42}\) \(+(-\)\(12\!\cdots\!00\)\( + \)\(42\!\cdots\!52\)\( \beta) q^{43}\) \(+(\)\(66\!\cdots\!64\)\( - \)\(19\!\cdots\!00\)\( \beta) q^{44}\) \(+(-\)\(12\!\cdots\!50\)\( - \)\(12\!\cdots\!00\)\( \beta) q^{45}\) \(+(\)\(62\!\cdots\!92\)\( + \)\(16\!\cdots\!00\)\( \beta) q^{46}\) \(+(\)\(21\!\cdots\!00\)\( - \)\(38\!\cdots\!16\)\( \beta) q^{47}\) \(+(-\)\(31\!\cdots\!00\)\( - \)\(12\!\cdots\!88\)\( \beta) q^{48}\) \(+(-\)\(19\!\cdots\!43\)\( + \)\(33\!\cdots\!00\)\( \beta) q^{49}\) \(+(-\)\(29\!\cdots\!00\)\( - \)\(72\!\cdots\!75\)\( \beta) q^{50}\) \(+(-\)\(57\!\cdots\!88\)\( - \)\(33\!\cdots\!00\)\( \beta) q^{51}\) \(+(-\)\(15\!\cdots\!00\)\( + \)\(37\!\cdots\!04\)\( \beta) q^{52}\) \(+(\)\(79\!\cdots\!50\)\( + \)\(11\!\cdots\!72\)\( \beta) q^{53}\) \(+(\)\(10\!\cdots\!60\)\( + \)\(12\!\cdots\!00\)\( \beta) q^{54}\) \(+(\)\(99\!\cdots\!00\)\( - \)\(29\!\cdots\!00\)\( \beta) q^{55}\) \(+(-\)\(11\!\cdots\!40\)\( - \)\(76\!\cdots\!00\)\( \beta) q^{56}\) \(+(-\)\(12\!\cdots\!00\)\( + \)\(43\!\cdots\!20\)\( \beta) q^{57}\) \(+(-\)\(27\!\cdots\!00\)\( + \)\(41\!\cdots\!10\)\( \beta) q^{58}\) \(+(-\)\(11\!\cdots\!20\)\( + \)\(48\!\cdots\!00\)\( \beta) q^{59}\) \(+(\)\(17\!\cdots\!00\)\( + \)\(70\!\cdots\!00\)\( \beta) q^{60}\) \(+(\)\(50\!\cdots\!22\)\( - \)\(28\!\cdots\!00\)\( \beta) q^{61}\) \(+(\)\(93\!\cdots\!00\)\( + \)\(49\!\cdots\!88\)\( \beta) q^{62}\) \(+(\)\(77\!\cdots\!00\)\( + \)\(43\!\cdots\!52\)\( \beta) q^{63}\) \(+(\)\(93\!\cdots\!32\)\( - \)\(44\!\cdots\!00\)\( \beta) q^{64}\) \(+(-\)\(23\!\cdots\!00\)\( + \)\(56\!\cdots\!00\)\( \beta) q^{65}\) \(+(\)\(84\!\cdots\!44\)\( - \)\(17\!\cdots\!00\)\( \beta) q^{66}\) \(+(-\)\(54\!\cdots\!00\)\( - \)\(78\!\cdots\!56\)\( \beta) q^{67}\) \(+(-\)\(15\!\cdots\!00\)\( - \)\(91\!\cdots\!72\)\( \beta) q^{68}\) \(+(-\)\(45\!\cdots\!24\)\( - \)\(11\!\cdots\!00\)\( \beta) q^{69}\) \(+(\)\(15\!\cdots\!00\)\( + \)\(53\!\cdots\!00\)\( \beta) q^{70}\) \(+(-\)\(37\!\cdots\!08\)\( - \)\(17\!\cdots\!00\)\( \beta) q^{71}\) \(+(\)\(76\!\cdots\!00\)\( - \)\(44\!\cdots\!20\)\( \beta) q^{72}\) \(+(\)\(98\!\cdots\!50\)\( - \)\(11\!\cdots\!88\)\( \beta) q^{73}\) \(+(-\)\(33\!\cdots\!36\)\( + \)\(96\!\cdots\!50\)\( \beta) q^{74}\) \(+(\)\(20\!\cdots\!00\)\( + \)\(52\!\cdots\!00\)\( \beta) q^{75}\) \(+(-\)\(33\!\cdots\!80\)\( + \)\(11\!\cdots\!00\)\( \beta) q^{76}\) \(+(-\)\(22\!\cdots\!00\)\( + \)\(73\!\cdots\!08\)\( \beta) q^{77}\) \(+(-\)\(14\!\cdots\!00\)\( + \)\(44\!\cdots\!84\)\( \beta) q^{78}\) \(+(\)\(13\!\cdots\!40\)\( + \)\(40\!\cdots\!00\)\( \beta) q^{79}\) \(+(-\)\(12\!\cdots\!00\)\( - \)\(50\!\cdots\!00\)\( \beta) q^{80}\) \(+(\)\(20\!\cdots\!21\)\( - \)\(13\!\cdots\!00\)\( \beta) q^{81}\) \(+(\)\(14\!\cdots\!00\)\( + \)\(68\!\cdots\!18\)\( \beta) q^{82}\) \(+(-\)\(23\!\cdots\!00\)\( + \)\(23\!\cdots\!32\)\( \beta) q^{83}\) \(+(-\)\(76\!\cdots\!36\)\( - \)\(25\!\cdots\!00\)\( \beta) q^{84}\) \(+(-\)\(22\!\cdots\!00\)\( - \)\(13\!\cdots\!00\)\( \beta) q^{85}\) \(+(-\)\(50\!\cdots\!68\)\( + \)\(86\!\cdots\!00\)\( \beta) q^{86}\) \(+(\)\(19\!\cdots\!00\)\( - \)\(29\!\cdots\!20\)\( \beta) q^{87}\) \(+(\)\(69\!\cdots\!00\)\( - \)\(18\!\cdots\!80\)\( \beta) q^{88}\) \(+(-\)\(66\!\cdots\!30\)\( + \)\(23\!\cdots\!00\)\( \beta) q^{89}\) \(+(\)\(20\!\cdots\!00\)\( + \)\(24\!\cdots\!50\)\( \beta) q^{90}\) \(+(\)\(56\!\cdots\!92\)\( - \)\(13\!\cdots\!00\)\( \beta) q^{91}\) \(+(-\)\(12\!\cdots\!00\)\( - \)\(31\!\cdots\!56\)\( \beta) q^{92}\) \(+(-\)\(67\!\cdots\!00\)\( - \)\(35\!\cdots\!36\)\( \beta) q^{93}\) \(+(\)\(35\!\cdots\!44\)\( - \)\(17\!\cdots\!00\)\( \beta) q^{94}\) \(+(-\)\(49\!\cdots\!00\)\( + \)\(18\!\cdots\!00\)\( \beta) q^{95}\) \(+(\)\(86\!\cdots\!32\)\( + \)\(50\!\cdots\!00\)\( \beta) q^{96}\) \(+(\)\(30\!\cdots\!50\)\( + \)\(10\!\cdots\!84\)\( \beta) q^{97}\) \(+(-\)\(47\!\cdots\!00\)\( - \)\(13\!\cdots\!57\)\( \beta) q^{98}\) \(+(\)\(60\!\cdots\!96\)\( - \)\(14\!\cdots\!00\)\( \beta) q^{99}\) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut -\mathstrut 194400q^{2} \) \(\mathstrut +\mathstrut 13991400q^{3} \) \(\mathstrut +\mathstrut 37720269824q^{4} \) \(\mathstrut +\mathstrut 5529584385900q^{5} \) \(\mathstrut -\mathstrut 22506543847296q^{6} \) \(\mathstrut -\mathstrut 3448443953486000q^{7} \) \(\mathstrut -\mathstrut 34044043043635200q^{8} \) \(\mathstrut -\mathstrut 898947378401769414q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut -\mathstrut 194400q^{2} \) \(\mathstrut +\mathstrut 13991400q^{3} \) \(\mathstrut +\mathstrut 37720269824q^{4} \) \(\mathstrut +\mathstrut 5529584385900q^{5} \) \(\mathstrut -\mathstrut 22506543847296q^{6} \) \(\mathstrut -\mathstrut 3448443953486000q^{7} \) \(\mathstrut -\mathstrut 34044043043635200q^{8} \) \(\mathstrut -\mathstrut 898947378401769414q^{9} \) \(\mathstrut -\mathstrut 9015609355013275200q^{10} \) \(\mathstrut -\mathstrut 26734036354848538056q^{11} \) \(\mathstrut +\mathstrut 4374774798370099200q^{12} \) \(\mathstrut +\mathstrut \)\(53\!\cdots\!00\)\(q^{13} \) \(\mathstrut +\mathstrut \)\(31\!\cdots\!48\)\(q^{14} \) \(\mathstrut +\mathstrut \)\(64\!\cdots\!00\)\(q^{15} \) \(\mathstrut -\mathstrut \)\(31\!\cdots\!08\)\(q^{16} \) \(\mathstrut -\mathstrut \)\(89\!\cdots\!00\)\(q^{17} \) \(\mathstrut +\mathstrut \)\(87\!\cdots\!00\)\(q^{18} \) \(\mathstrut +\mathstrut \)\(37\!\cdots\!20\)\(q^{19} \) \(\mathstrut +\mathstrut \)\(17\!\cdots\!00\)\(q^{20} \) \(\mathstrut -\mathstrut \)\(22\!\cdots\!56\)\(q^{21} \) \(\mathstrut -\mathstrut \)\(68\!\cdots\!00\)\(q^{22} \) \(\mathstrut -\mathstrut \)\(26\!\cdots\!00\)\(q^{23} \) \(\mathstrut +\mathstrut \)\(18\!\cdots\!60\)\(q^{24} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!50\)\(q^{25} \) \(\mathstrut +\mathstrut \)\(16\!\cdots\!44\)\(q^{26} \) \(\mathstrut -\mathstrut \)\(12\!\cdots\!00\)\(q^{27} \) \(\mathstrut -\mathstrut \)\(61\!\cdots\!00\)\(q^{28} \) \(\mathstrut -\mathstrut \)\(12\!\cdots\!20\)\(q^{29} \) \(\mathstrut -\mathstrut \)\(18\!\cdots\!00\)\(q^{30} \) \(\mathstrut +\mathstrut \)\(26\!\cdots\!24\)\(q^{31} \) \(\mathstrut +\mathstrut \)\(13\!\cdots\!00\)\(q^{32} \) \(\mathstrut +\mathstrut \)\(49\!\cdots\!00\)\(q^{33} \) \(\mathstrut +\mathstrut \)\(15\!\cdots\!08\)\(q^{34} \) \(\mathstrut -\mathstrut \)\(91\!\cdots\!00\)\(q^{35} \) \(\mathstrut -\mathstrut \)\(16\!\cdots\!68\)\(q^{36} \) \(\mathstrut -\mathstrut \)\(68\!\cdots\!00\)\(q^{37} \) \(\mathstrut +\mathstrut \)\(34\!\cdots\!00\)\(q^{38} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!68\)\(q^{39} \) \(\mathstrut +\mathstrut \)\(75\!\cdots\!00\)\(q^{40} \) \(\mathstrut -\mathstrut \)\(12\!\cdots\!36\)\(q^{41} \) \(\mathstrut +\mathstrut \)\(78\!\cdots\!00\)\(q^{42} \) \(\mathstrut -\mathstrut \)\(25\!\cdots\!00\)\(q^{43} \) \(\mathstrut +\mathstrut \)\(13\!\cdots\!28\)\(q^{44} \) \(\mathstrut -\mathstrut \)\(24\!\cdots\!00\)\(q^{45} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!84\)\(q^{46} \) \(\mathstrut +\mathstrut \)\(42\!\cdots\!00\)\(q^{47} \) \(\mathstrut -\mathstrut \)\(62\!\cdots\!00\)\(q^{48} \) \(\mathstrut -\mathstrut \)\(38\!\cdots\!86\)\(q^{49} \) \(\mathstrut -\mathstrut \)\(58\!\cdots\!00\)\(q^{50} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!76\)\(q^{51} \) \(\mathstrut -\mathstrut \)\(31\!\cdots\!00\)\(q^{52} \) \(\mathstrut +\mathstrut \)\(15\!\cdots\!00\)\(q^{53} \) \(\mathstrut +\mathstrut \)\(20\!\cdots\!20\)\(q^{54} \) \(\mathstrut +\mathstrut \)\(19\!\cdots\!00\)\(q^{55} \) \(\mathstrut -\mathstrut \)\(22\!\cdots\!80\)\(q^{56} \) \(\mathstrut -\mathstrut \)\(24\!\cdots\!00\)\(q^{57} \) \(\mathstrut -\mathstrut \)\(55\!\cdots\!00\)\(q^{58} \) \(\mathstrut -\mathstrut \)\(23\!\cdots\!40\)\(q^{59} \) \(\mathstrut +\mathstrut \)\(35\!\cdots\!00\)\(q^{60} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!44\)\(q^{61} \) \(\mathstrut +\mathstrut \)\(18\!\cdots\!00\)\(q^{62} \) \(\mathstrut +\mathstrut \)\(15\!\cdots\!00\)\(q^{63} \) \(\mathstrut +\mathstrut \)\(18\!\cdots\!64\)\(q^{64} \) \(\mathstrut -\mathstrut \)\(46\!\cdots\!00\)\(q^{65} \) \(\mathstrut +\mathstrut \)\(16\!\cdots\!88\)\(q^{66} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!00\)\(q^{67} \) \(\mathstrut -\mathstrut \)\(30\!\cdots\!00\)\(q^{68} \) \(\mathstrut -\mathstrut \)\(90\!\cdots\!48\)\(q^{69} \) \(\mathstrut +\mathstrut \)\(31\!\cdots\!00\)\(q^{70} \) \(\mathstrut -\mathstrut \)\(74\!\cdots\!16\)\(q^{71} \) \(\mathstrut +\mathstrut \)\(15\!\cdots\!00\)\(q^{72} \) \(\mathstrut +\mathstrut \)\(19\!\cdots\!00\)\(q^{73} \) \(\mathstrut -\mathstrut \)\(67\!\cdots\!72\)\(q^{74} \) \(\mathstrut +\mathstrut \)\(41\!\cdots\!00\)\(q^{75} \) \(\mathstrut -\mathstrut \)\(66\!\cdots\!60\)\(q^{76} \) \(\mathstrut -\mathstrut \)\(45\!\cdots\!00\)\(q^{77} \) \(\mathstrut -\mathstrut \)\(29\!\cdots\!00\)\(q^{78} \) \(\mathstrut +\mathstrut \)\(27\!\cdots\!80\)\(q^{79} \) \(\mathstrut -\mathstrut \)\(25\!\cdots\!00\)\(q^{80} \) \(\mathstrut +\mathstrut \)\(40\!\cdots\!42\)\(q^{81} \) \(\mathstrut +\mathstrut \)\(29\!\cdots\!00\)\(q^{82} \) \(\mathstrut -\mathstrut \)\(47\!\cdots\!00\)\(q^{83} \) \(\mathstrut -\mathstrut \)\(15\!\cdots\!72\)\(q^{84} \) \(\mathstrut -\mathstrut \)\(45\!\cdots\!00\)\(q^{85} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!36\)\(q^{86} \) \(\mathstrut +\mathstrut \)\(39\!\cdots\!00\)\(q^{87} \) \(\mathstrut +\mathstrut \)\(13\!\cdots\!00\)\(q^{88} \) \(\mathstrut -\mathstrut \)\(13\!\cdots\!60\)\(q^{89} \) \(\mathstrut +\mathstrut \)\(40\!\cdots\!00\)\(q^{90} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!84\)\(q^{91} \) \(\mathstrut -\mathstrut \)\(24\!\cdots\!00\)\(q^{92} \) \(\mathstrut -\mathstrut \)\(13\!\cdots\!00\)\(q^{93} \) \(\mathstrut +\mathstrut \)\(70\!\cdots\!88\)\(q^{94} \) \(\mathstrut -\mathstrut \)\(99\!\cdots\!00\)\(q^{95} \) \(\mathstrut +\mathstrut \)\(17\!\cdots\!64\)\(q^{96} \) \(\mathstrut +\mathstrut \)\(60\!\cdots\!00\)\(q^{97} \) \(\mathstrut -\mathstrut \)\(94\!\cdots\!00\)\(q^{98} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!92\)\(q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
3992.29
−3991.29
−480412. 3.45869e7 9.33565e10 1.38267e13 −1.66160e13 −5.42222e15 2.11777e16 −4.49088e17 −6.64253e18
1.2 286012. −2.05955e7 −5.56362e10 −8.29715e12 −5.89057e12 1.97377e15 −5.52218e16 −4.49860e17 −2.37308e18
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Hecke kernels

There are no other newforms in \(S_{38}^{\mathrm{new}}(\Gamma_0(1))\).