Properties

Label 1.32.a.a
Level 1
Weight 32
Character orbit 1.a
Self dual Yes
Analytic conductor 6.088
Analytic rank 0
Dimension 2
CM No
Inner twists 1

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 1 \)
Weight: \( k \) = \( 32 \)
Character orbit: \([\chi]\) = 1.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(6.0877132819\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\mathbb{Q}[x]/(x^{2} - \cdots)\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{3}\cdot 3 \)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 12\sqrt{18295489}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + ( 19980 - \beta ) q^{2} \) \( + ( 8681580 - 432 \beta ) q^{3} \) \( + ( 886267168 - 39960 \beta ) q^{4} \) \( + ( -9695609010 + 1418560 \beta ) q^{5} \) \( + ( 1311583748112 - 17312940 \beta ) q^{6} \) \( + ( 15128763788600 + 71928864 \beta ) q^{7} \) \( + ( 80077529352960 + 462815680 \beta ) q^{8} \) \( + ( -50633228151963 - 7500885120 \beta ) q^{9} \) \(+O(q^{10})\) \( q\) \(+(19980 - \beta) q^{2}\) \(+(8681580 - 432 \beta) q^{3}\) \(+(886267168 - 39960 \beta) q^{4}\) \(+(-9695609010 + 1418560 \beta) q^{5}\) \(+(1311583748112 - 17312940 \beta) q^{6}\) \(+(15128763788600 + 71928864 \beta) q^{7}\) \(+(80077529352960 + 462815680 \beta) q^{8}\) \(+(-50633228151963 - 7500885120 \beta) q^{9}\) \(+(-3930986106140760 + 38038437810 \beta) q^{10}\) \(+(-3891176872559388 - 29000909200 \beta) q^{11}\) \(+(53173705477656960 - 729783353376 \beta) q^{12}\) \(+(37354476525130310 + 4661016429888 \beta) q^{13}\) \(+(112772481922620576 - 13691625085880 \beta) q^{14}\) \(+(-1698672911337290520 + 16503845217120 \beta) q^{15}\) \(+(-1522606456842450944 + 14982974507520 \beta) q^{16}\) \(+(8612303914493544690 - 45085348093056 \beta) q^{17}\) \(+(18749808114787989180 - 99234456545637 \beta) q^{18}\) \(+(-6185281664011082020 + 157805792764560 \beta) q^{19}\) \(+(-\)\(15\!\cdots\!80\)\( + 1644659689877680 \beta) q^{20}\) \(+(49477478708035580832 - 5911169769550080 \beta) q^{21}\) \(+(-1341356516498345040 + 3311738706743388 \beta) q^{22}\) \(+(\)\(94\!\cdots\!40\)\( + 18557618179251808 \beta) q^{23}\) \(+(\)\(16\!\cdots\!40\)\( - 30575521329304320 \beta) q^{24}\) \(+(\)\(73\!\cdots\!75\)\( - 27507606234451200 \beta) q^{25}\) \(+(-\)\(11\!\cdots\!08\)\( + 55772631744031930 \beta) q^{26}\) \(+(\)\(27\!\cdots\!40\)\( + 223588927516223520 \beta) q^{27}\) \(+(\)\(58\!\cdots\!60\)\( - 540797210397718848 \beta) q^{28}\) \(+(\)\(64\!\cdots\!70\)\( - 234192357384448960 \beta) q^{29}\) \(+(-\)\(77\!\cdots\!20\)\( + 2028419738775348120 \beta) q^{30}\) \(+(\)\(62\!\cdots\!32\)\( - 2412621171020361600 \beta) q^{31}\) \(+(-\)\(24\!\cdots\!20\)\( + 828077182664699904 \beta) q^{32}\) \(+(-\)\(77\!\cdots\!40\)\( + 1429214695653119616 \beta) q^{33}\) \(+(\)\(29\!\cdots\!96\)\( - 9513109169392803570 \beta) q^{34}\) \(+(\)\(12\!\cdots\!40\)\( + 20763665018078951360 \beta) q^{35}\) \(+(\)\(74\!\cdots\!16\)\( - 4624484415843298680 \beta) q^{36}\) \(+(-\)\(41\!\cdots\!30\)\( - 32224244113578511296 \beta) q^{37}\) \(+(-\)\(53\!\cdots\!60\)\( + 9338241403446990820 \beta) q^{38}\) \(+(-\)\(49\!\cdots\!56\)\( + 24327853158530769120 \beta) q^{39}\) \(+(\)\(95\!\cdots\!00\)\( + \)\(10\!\cdots\!00\)\( \beta) q^{40}\) \(+(\)\(43\!\cdots\!42\)\( - \)\(18\!\cdots\!00\)\( \beta) q^{41}\) \(+(\)\(16\!\cdots\!40\)\( - \)\(16\!\cdots\!32\)\( \beta) q^{42}\) \(+(-\)\(91\!\cdots\!00\)\( + \)\(34\!\cdots\!48\)\( \beta) q^{43}\) \(+(-\)\(39\!\cdots\!84\)\( + \)\(12\!\cdots\!80\)\( \beta) q^{44}\) \(+(-\)\(27\!\cdots\!70\)\( + 899377225198297920 \beta) q^{45}\) \(+(-\)\(29\!\cdots\!28\)\( - \)\(57\!\cdots\!00\)\( \beta) q^{46}\) \(+(\)\(47\!\cdots\!60\)\( - \)\(51\!\cdots\!16\)\( \beta) q^{47}\) \(+(-\)\(30\!\cdots\!60\)\( + \)\(78\!\cdots\!08\)\( \beta) q^{48}\) \(+(\)\(84\!\cdots\!93\)\( + \)\(21\!\cdots\!00\)\( \beta) q^{49}\) \(+(\)\(87\!\cdots\!00\)\( - \)\(12\!\cdots\!75\)\( \beta) q^{50}\) \(+(\)\(12\!\cdots\!72\)\( - \)\(41\!\cdots\!60\)\( \beta) q^{51}\) \(+(-\)\(45\!\cdots\!00\)\( + \)\(26\!\cdots\!84\)\( \beta) q^{52}\) \(+(\)\(97\!\cdots\!30\)\( + \)\(12\!\cdots\!68\)\( \beta) q^{53}\) \(+(-\)\(53\!\cdots\!20\)\( + \)\(17\!\cdots\!60\)\( \beta) q^{54}\) \(+(-\)\(70\!\cdots\!20\)\( - \)\(52\!\cdots\!80\)\( \beta) q^{55}\) \(+(\)\(12\!\cdots\!20\)\( + \)\(12\!\cdots\!40\)\( \beta) q^{56}\) \(+(-\)\(23\!\cdots\!20\)\( + \)\(40\!\cdots\!40\)\( \beta) q^{57}\) \(+(\)\(19\!\cdots\!60\)\( - \)\(68\!\cdots\!70\)\( \beta) q^{58}\) \(+(-\)\(99\!\cdots\!60\)\( + \)\(44\!\cdots\!80\)\( \beta) q^{59}\) \(+(-\)\(32\!\cdots\!60\)\( + \)\(82\!\cdots\!60\)\( \beta) q^{60}\) \(+(-\)\(60\!\cdots\!38\)\( - \)\(11\!\cdots\!00\)\( \beta) q^{61}\) \(+(\)\(76\!\cdots\!60\)\( - \)\(11\!\cdots\!32\)\( \beta) q^{62}\) \(+(-\)\(21\!\cdots\!80\)\( - \)\(11\!\cdots\!32\)\( \beta) q^{63}\) \(+(-\)\(37\!\cdots\!52\)\( + \)\(22\!\cdots\!80\)\( \beta) q^{64}\) \(+(\)\(17\!\cdots\!80\)\( + \)\(77\!\cdots\!20\)\( \beta) q^{65}\) \(+(-\)\(37\!\cdots\!56\)\( + \)\(29\!\cdots\!20\)\( \beta) q^{66}\) \(+(-\)\(48\!\cdots\!60\)\( + \)\(88\!\cdots\!44\)\( \beta) q^{67}\) \(+(\)\(12\!\cdots\!80\)\( - \)\(38\!\cdots\!08\)\( \beta) q^{68}\) \(+(-\)\(12\!\cdots\!96\)\( - \)\(24\!\cdots\!40\)\( \beta) q^{69}\) \(+(-\)\(52\!\cdots\!60\)\( + \)\(29\!\cdots\!60\)\( \beta) q^{70}\) \(+(\)\(27\!\cdots\!72\)\( + \)\(95\!\cdots\!00\)\( \beta) q^{71}\) \(+(-\)\(13\!\cdots\!80\)\( - \)\(62\!\cdots\!40\)\( \beta) q^{72}\) \(+(\)\(31\!\cdots\!90\)\( - \)\(15\!\cdots\!92\)\( \beta) q^{73}\) \(+(\)\(76\!\cdots\!36\)\( - \)\(22\!\cdots\!50\)\( \beta) q^{74}\) \(+(\)\(37\!\cdots\!00\)\( - \)\(55\!\cdots\!00\)\( \beta) q^{75}\) \(+(-\)\(22\!\cdots\!60\)\( + \)\(38\!\cdots\!80\)\( \beta) q^{76}\) \(+(-\)\(64\!\cdots\!00\)\( - \)\(71\!\cdots\!32\)\( \beta) q^{77}\) \(+(-\)\(16\!\cdots\!00\)\( + \)\(54\!\cdots\!56\)\( \beta) q^{78}\) \(+(-\)\(59\!\cdots\!80\)\( - \)\(46\!\cdots\!60\)\( \beta) q^{79}\) \(+(\)\(70\!\cdots\!40\)\( - \)\(23\!\cdots\!40\)\( \beta) q^{80}\) \(+(-\)\(19\!\cdots\!79\)\( + \)\(53\!\cdots\!60\)\( \beta) q^{81}\) \(+(\)\(57\!\cdots\!60\)\( - \)\(80\!\cdots\!42\)\( \beta) q^{82}\) \(+(-\)\(13\!\cdots\!80\)\( + \)\(62\!\cdots\!28\)\( \beta) q^{83}\) \(+(\)\(66\!\cdots\!76\)\( - \)\(72\!\cdots\!60\)\( \beta) q^{84}\) \(+(-\)\(25\!\cdots\!60\)\( + \)\(12\!\cdots\!60\)\( \beta) q^{85}\) \(+(-\)\(10\!\cdots\!68\)\( + \)\(16\!\cdots\!40\)\( \beta) q^{86}\) \(+(\)\(82\!\cdots\!20\)\( - \)\(29\!\cdots\!40\)\( \beta) q^{87}\) \(+(-\)\(34\!\cdots\!80\)\( - \)\(41\!\cdots\!40\)\( \beta) q^{88}\) \(+(-\)\(10\!\cdots\!90\)\( - \)\(26\!\cdots\!80\)\( \beta) q^{89}\) \(+(-\)\(55\!\cdots\!20\)\( + \)\(27\!\cdots\!70\)\( \beta) q^{90}\) \(+(\)\(14\!\cdots\!12\)\( + \)\(73\!\cdots\!40\)\( \beta) q^{91}\) \(+(-\)\(11\!\cdots\!60\)\( - \)\(21\!\cdots\!56\)\( \beta) q^{92}\) \(+(\)\(32\!\cdots\!60\)\( - \)\(48\!\cdots\!24\)\( \beta) q^{93}\) \(+(\)\(23\!\cdots\!56\)\( - \)\(57\!\cdots\!40\)\( \beta) q^{94}\) \(+(\)\(64\!\cdots\!00\)\( - \)\(10\!\cdots\!00\)\( \beta) q^{95}\) \(+(-\)\(30\!\cdots\!48\)\( + \)\(11\!\cdots\!60\)\( \beta) q^{96}\) \(+(-\)\(45\!\cdots\!90\)\( + \)\(16\!\cdots\!84\)\( \beta) q^{97}\) \(+(-\)\(40\!\cdots\!60\)\( - \)\(41\!\cdots\!93\)\( \beta) q^{98}\) \(+(\)\(77\!\cdots\!44\)\( + \)\(30\!\cdots\!60\)\( \beta) q^{99}\) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut +\mathstrut 39960q^{2} \) \(\mathstrut +\mathstrut 17363160q^{3} \) \(\mathstrut +\mathstrut 1772534336q^{4} \) \(\mathstrut -\mathstrut 19391218020q^{5} \) \(\mathstrut +\mathstrut 2623167496224q^{6} \) \(\mathstrut +\mathstrut 30257527577200q^{7} \) \(\mathstrut +\mathstrut 160155058705920q^{8} \) \(\mathstrut -\mathstrut 101266456303926q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut +\mathstrut 39960q^{2} \) \(\mathstrut +\mathstrut 17363160q^{3} \) \(\mathstrut +\mathstrut 1772534336q^{4} \) \(\mathstrut -\mathstrut 19391218020q^{5} \) \(\mathstrut +\mathstrut 2623167496224q^{6} \) \(\mathstrut +\mathstrut 30257527577200q^{7} \) \(\mathstrut +\mathstrut 160155058705920q^{8} \) \(\mathstrut -\mathstrut 101266456303926q^{9} \) \(\mathstrut -\mathstrut 7861972212281520q^{10} \) \(\mathstrut -\mathstrut 7782353745118776q^{11} \) \(\mathstrut +\mathstrut 106347410955313920q^{12} \) \(\mathstrut +\mathstrut 74708953050260620q^{13} \) \(\mathstrut +\mathstrut 225544963845241152q^{14} \) \(\mathstrut -\mathstrut 3397345822674581040q^{15} \) \(\mathstrut -\mathstrut 3045212913684901888q^{16} \) \(\mathstrut +\mathstrut 17224607828987089380q^{17} \) \(\mathstrut +\mathstrut 37499616229575978360q^{18} \) \(\mathstrut -\mathstrut 12370563328022164040q^{19} \) \(\mathstrut -\mathstrut \)\(31\!\cdots\!60\)\(q^{20} \) \(\mathstrut +\mathstrut 98954957416071161664q^{21} \) \(\mathstrut -\mathstrut 2682713032996690080q^{22} \) \(\mathstrut +\mathstrut \)\(18\!\cdots\!80\)\(q^{23} \) \(\mathstrut +\mathstrut \)\(33\!\cdots\!80\)\(q^{24} \) \(\mathstrut +\mathstrut \)\(14\!\cdots\!50\)\(q^{25} \) \(\mathstrut -\mathstrut \)\(23\!\cdots\!16\)\(q^{26} \) \(\mathstrut +\mathstrut \)\(54\!\cdots\!80\)\(q^{27} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!20\)\(q^{28} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!40\)\(q^{29} \) \(\mathstrut -\mathstrut \)\(15\!\cdots\!40\)\(q^{30} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!64\)\(q^{31} \) \(\mathstrut -\mathstrut \)\(48\!\cdots\!40\)\(q^{32} \) \(\mathstrut -\mathstrut \)\(15\!\cdots\!80\)\(q^{33} \) \(\mathstrut +\mathstrut \)\(58\!\cdots\!92\)\(q^{34} \) \(\mathstrut +\mathstrut \)\(24\!\cdots\!80\)\(q^{35} \) \(\mathstrut +\mathstrut \)\(14\!\cdots\!32\)\(q^{36} \) \(\mathstrut -\mathstrut \)\(83\!\cdots\!60\)\(q^{37} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!20\)\(q^{38} \) \(\mathstrut -\mathstrut \)\(99\!\cdots\!12\)\(q^{39} \) \(\mathstrut +\mathstrut \)\(19\!\cdots\!00\)\(q^{40} \) \(\mathstrut +\mathstrut \)\(87\!\cdots\!84\)\(q^{41} \) \(\mathstrut +\mathstrut \)\(33\!\cdots\!80\)\(q^{42} \) \(\mathstrut -\mathstrut \)\(18\!\cdots\!00\)\(q^{43} \) \(\mathstrut -\mathstrut \)\(79\!\cdots\!68\)\(q^{44} \) \(\mathstrut -\mathstrut \)\(55\!\cdots\!40\)\(q^{45} \) \(\mathstrut -\mathstrut \)\(59\!\cdots\!56\)\(q^{46} \) \(\mathstrut +\mathstrut \)\(95\!\cdots\!20\)\(q^{47} \) \(\mathstrut -\mathstrut \)\(60\!\cdots\!20\)\(q^{48} \) \(\mathstrut +\mathstrut \)\(16\!\cdots\!86\)\(q^{49} \) \(\mathstrut +\mathstrut \)\(17\!\cdots\!00\)\(q^{50} \) \(\mathstrut +\mathstrut \)\(25\!\cdots\!44\)\(q^{51} \) \(\mathstrut -\mathstrut \)\(91\!\cdots\!00\)\(q^{52} \) \(\mathstrut +\mathstrut \)\(19\!\cdots\!60\)\(q^{53} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!40\)\(q^{54} \) \(\mathstrut -\mathstrut \)\(14\!\cdots\!40\)\(q^{55} \) \(\mathstrut +\mathstrut \)\(25\!\cdots\!40\)\(q^{56} \) \(\mathstrut -\mathstrut \)\(46\!\cdots\!40\)\(q^{57} \) \(\mathstrut +\mathstrut \)\(38\!\cdots\!20\)\(q^{58} \) \(\mathstrut -\mathstrut \)\(19\!\cdots\!20\)\(q^{59} \) \(\mathstrut -\mathstrut \)\(64\!\cdots\!20\)\(q^{60} \) \(\mathstrut -\mathstrut \)\(12\!\cdots\!76\)\(q^{61} \) \(\mathstrut +\mathstrut \)\(15\!\cdots\!20\)\(q^{62} \) \(\mathstrut -\mathstrut \)\(43\!\cdots\!60\)\(q^{63} \) \(\mathstrut -\mathstrut \)\(74\!\cdots\!04\)\(q^{64} \) \(\mathstrut +\mathstrut \)\(34\!\cdots\!60\)\(q^{65} \) \(\mathstrut -\mathstrut \)\(75\!\cdots\!12\)\(q^{66} \) \(\mathstrut -\mathstrut \)\(96\!\cdots\!20\)\(q^{67} \) \(\mathstrut +\mathstrut \)\(24\!\cdots\!60\)\(q^{68} \) \(\mathstrut -\mathstrut \)\(25\!\cdots\!92\)\(q^{69} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!20\)\(q^{70} \) \(\mathstrut +\mathstrut \)\(55\!\cdots\!44\)\(q^{71} \) \(\mathstrut -\mathstrut \)\(26\!\cdots\!60\)\(q^{72} \) \(\mathstrut +\mathstrut \)\(62\!\cdots\!80\)\(q^{73} \) \(\mathstrut +\mathstrut \)\(15\!\cdots\!72\)\(q^{74} \) \(\mathstrut +\mathstrut \)\(75\!\cdots\!00\)\(q^{75} \) \(\mathstrut -\mathstrut \)\(44\!\cdots\!20\)\(q^{76} \) \(\mathstrut -\mathstrut \)\(12\!\cdots\!00\)\(q^{77} \) \(\mathstrut -\mathstrut \)\(32\!\cdots\!00\)\(q^{78} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!60\)\(q^{79} \) \(\mathstrut +\mathstrut \)\(14\!\cdots\!80\)\(q^{80} \) \(\mathstrut -\mathstrut \)\(39\!\cdots\!58\)\(q^{81} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!20\)\(q^{82} \) \(\mathstrut -\mathstrut \)\(26\!\cdots\!60\)\(q^{83} \) \(\mathstrut +\mathstrut \)\(13\!\cdots\!52\)\(q^{84} \) \(\mathstrut -\mathstrut \)\(50\!\cdots\!20\)\(q^{85} \) \(\mathstrut -\mathstrut \)\(21\!\cdots\!36\)\(q^{86} \) \(\mathstrut +\mathstrut \)\(16\!\cdots\!40\)\(q^{87} \) \(\mathstrut -\mathstrut \)\(69\!\cdots\!60\)\(q^{88} \) \(\mathstrut -\mathstrut \)\(21\!\cdots\!80\)\(q^{89} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!40\)\(q^{90} \) \(\mathstrut +\mathstrut \)\(28\!\cdots\!24\)\(q^{91} \) \(\mathstrut -\mathstrut \)\(22\!\cdots\!20\)\(q^{92} \) \(\mathstrut +\mathstrut \)\(65\!\cdots\!20\)\(q^{93} \) \(\mathstrut +\mathstrut \)\(46\!\cdots\!12\)\(q^{94} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!00\)\(q^{95} \) \(\mathstrut -\mathstrut \)\(60\!\cdots\!96\)\(q^{96} \) \(\mathstrut -\mathstrut \)\(90\!\cdots\!80\)\(q^{97} \) \(\mathstrut -\mathstrut \)\(80\!\cdots\!20\)\(q^{98} \) \(\mathstrut +\mathstrut \)\(15\!\cdots\!88\)\(q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
2139.16
−2138.16
−31347.9 −1.34921e7 −1.16479e9 6.31161e10 4.22947e11 1.88207e13 1.03833e14 −4.35638e14 −1.97855e15
1.2 71307.9 3.08552e7 2.93733e9 −8.25073e10 2.20022e12 1.14368e13 5.63222e13 3.34371e14 −5.88342e15
\(n\): 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 \(S_{32}^{\mathrm{new}}(\Gamma_0(1))\).