Properties

Label 2.32.a.b
Level 2
Weight 32
Character orbit 2.a
Self dual Yes
Analytic conductor 12.175
Analytic rank 0
Dimension 2
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 2 \)
Weight: \( k \) = \( 32 \)
Character orbit: \([\chi]\) = 2.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(12.1754265638\)
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 \(\beta = 960\sqrt{987507049}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( -32768 q^{2} \) \( + ( 8358252 - \beta ) q^{3} \) \( + 1073741824 q^{4} \) \( + ( -13381532850 - 3780 \beta ) q^{5} \) \( + ( -273883201536 + 32768 \beta ) q^{6} \) \( + ( -11751851055304 + 38502 \beta ) q^{7} \) \( -35184372088832 q^{8} \) \( + ( 362273476569957 - 16716504 \beta ) q^{9} \) \(+O(q^{10})\) \( q\) \(-32768 q^{2}\) \(+(8358252 - \beta) q^{3}\) \(+1073741824 q^{4}\) \(+(-13381532850 - 3780 \beta) q^{5}\) \(+(-273883201536 + 32768 \beta) q^{6}\) \(+(-11751851055304 + 38502 \beta) q^{7}\) \(-35184372088832 q^{8}\) \(+(362273476569957 - 16716504 \beta) q^{9}\) \(+(438486068428800 + 123863040 \beta) q^{10}\) \(+(12575568739256292 - 209041371 \beta) q^{11}\) \(+(8974604747931648 - 1073741824 \beta) q^{12}\) \(+(94662172279854662 + 4745042748 \beta) q^{13}\) \(+(385084655380201472 - 1261633536 \beta) q^{14}\) \(+(3328280732528173800 - 18212659710 \beta) q^{15}\) \(+1152921504606846976 q^{16}\) \(+(460408414945337586 + 5927641128 \beta) q^{17}\) \(+(-11870977280244350976 + 547766403072 \beta) q^{18}\) \(+(-50043450866248766500 - 917688823029 \beta) q^{19}\) \(+(-14368311490274918400 - 4058744094720 \beta) q^{20}\) \(+(-\)\(13\!\cdots\!08\)\( + 12073660473808 \beta) q^{21}\) \(+(-\)\(41\!\cdots\!56\)\( + 6849867644928 \beta) q^{22}\) \(+(\)\(25\!\cdots\!12\)\( - 57371978548206 \beta) q^{23}\) \(+(-\)\(29\!\cdots\!64\)\( + 35184372088832 \beta) q^{24}\) \(+(\)\(85\!\cdots\!75\)\( + 101164388346000 \beta) q^{25}\) \(+(-\)\(31\!\cdots\!16\)\( - 155485560766464 \beta) q^{26}\) \(+(\)\(13\!\cdots\!20\)\( + 115679166722982 \beta) q^{27}\) \(+(-\)\(12\!\cdots\!96\)\( + 41341207707648 \beta) q^{28}\) \(+(-\)\(19\!\cdots\!90\)\( - 912080882507508 \beta) q^{29}\) \(+(-\)\(10\!\cdots\!00\)\( + 596792433377280 \beta) q^{30}\) \(+(\)\(24\!\cdots\!92\)\( + 3254688883162008 \beta) q^{31}\) \(-\)\(37\!\cdots\!68\)\( q^{32}\) \(+(\)\(29\!\cdots\!84\)\( - 14322789196499784 \beta) q^{33}\) \(+(-\)\(15\!\cdots\!48\)\( - 194236944482304 \beta) q^{34}\) \(+(\)\(24\!\cdots\!00\)\( + 43906781211258420 \beta) q^{35}\) \(+(\)\(38\!\cdots\!68\)\( - 17949209495863296 \beta) q^{36}\) \(+(\)\(94\!\cdots\!26\)\( - 51326225439746292 \beta) q^{37}\) \(+(\)\(16\!\cdots\!00\)\( + 30070827353014272 \beta) q^{38}\) \(+(-\)\(35\!\cdots\!76\)\( - 55001909241298166 \beta) q^{39}\) \(+(\)\(47\!\cdots\!00\)\( + 132996926495784960 \beta) q^{40}\) \(+(-\)\(14\!\cdots\!18\)\( + 104709028836970512 \beta) q^{41}\) \(+(\)\(43\!\cdots\!44\)\( - 395629706405740544 \beta) q^{42}\) \(+(-\)\(14\!\cdots\!68\)\( + 414868077394813917 \beta) q^{43}\) \(+(\)\(13\!\cdots\!08\)\( - 224456462989000704 \beta) q^{44}\) \(+(\)\(52\!\cdots\!50\)\( - 1145701294021281060 \beta) q^{45}\) \(+(-\)\(83\!\cdots\!16\)\( + 1879964993067614208 \beta) q^{46}\) \(+(\)\(66\!\cdots\!76\)\( + 900502500303526212 \beta) q^{47}\) \(+(\)\(96\!\cdots\!52\)\( - 1152921504606846976 \beta) q^{48}\) \(+(-\)\(18\!\cdots\!27\)\( - 904939538662629216 \beta) q^{49}\) \(+(-\)\(27\!\cdots\!00\)\( - 3314954677321728000 \beta) q^{50}\) \(+(-\)\(15\!\cdots\!28\)\( - 410863696631949330 \beta) q^{51}\) \(+(\)\(10\!\cdots\!88\)\( + 5094950855195492352 \beta) q^{52}\) \(+(-\)\(22\!\cdots\!18\)\( + 16855972608155554188 \beta) q^{53}\) \(+(-\)\(42\!\cdots\!60\)\( - 3790574935178674176 \beta) q^{54}\) \(+(\)\(55\!\cdots\!00\)\( - 44738355861343246410 \beta) q^{55}\) \(+(\)\(41\!\cdots\!28\)\( - 1354668694164209664 \beta) q^{56}\) \(+(\)\(41\!\cdots\!00\)\( + 42373176425788981192 \beta) q^{57}\) \(+(\)\(64\!\cdots\!20\)\( + 29887066358006022144 \beta) q^{58}\) \(+(-\)\(12\!\cdots\!80\)\( - 15588326499987499191 \beta) q^{59}\) \(+(\)\(35\!\cdots\!00\)\( - 19555694456906711040 \beta) q^{60}\) \(+(\)\(36\!\cdots\!82\)\( - 22159967545724485284 \beta) q^{61}\) \(+(-\)\(81\!\cdots\!56\)\( - \)\(10\!\cdots\!44\)\( \beta) q^{62}\) \(+(-\)\(48\!\cdots\!28\)\( + \)\(21\!\cdots\!30\)\( \beta) q^{63}\) \(+\)\(12\!\cdots\!24\)\( q^{64}\) \(+(-\)\(17\!\cdots\!00\)\( - \)\(42\!\cdots\!60\)\( \beta) q^{65}\) \(+(-\)\(96\!\cdots\!12\)\( + \)\(46\!\cdots\!12\)\( \beta) q^{66}\) \(+(\)\(10\!\cdots\!96\)\( + \)\(30\!\cdots\!11\)\( \beta) q^{67}\) \(+(\)\(49\!\cdots\!64\)\( + 6364756196796137472 \beta) q^{68}\) \(+(\)\(54\!\cdots\!24\)\( - \)\(73\!\cdots\!24\)\( \beta) q^{69}\) \(+(-\)\(81\!\cdots\!00\)\( - \)\(14\!\cdots\!60\)\( \beta) q^{70}\) \(+(\)\(18\!\cdots\!92\)\( + \)\(14\!\cdots\!66\)\( \beta) q^{71}\) \(+(-\)\(12\!\cdots\!24\)\( + \)\(58\!\cdots\!28\)\( \beta) q^{72}\) \(+(\)\(82\!\cdots\!42\)\( + \)\(19\!\cdots\!08\)\( \beta) q^{73}\) \(+(-\)\(30\!\cdots\!68\)\( + \)\(16\!\cdots\!56\)\( \beta) q^{74}\) \(+(-\)\(20\!\cdots\!00\)\( - \)\(76\!\cdots\!75\)\( \beta) q^{75}\) \(+(-\)\(53\!\cdots\!00\)\( - \)\(98\!\cdots\!96\)\( \beta) q^{76}\) \(+(-\)\(15\!\cdots\!68\)\( + \)\(29\!\cdots\!68\)\( \beta) q^{77}\) \(+(\)\(11\!\cdots\!68\)\( + \)\(18\!\cdots\!88\)\( \beta) q^{78}\) \(+(-\)\(21\!\cdots\!60\)\( + \)\(70\!\cdots\!76\)\( \beta) q^{79}\) \(+(-\)\(15\!\cdots\!00\)\( - \)\(43\!\cdots\!80\)\( \beta) q^{80}\) \(+(-\)\(21\!\cdots\!39\)\( - \)\(17\!\cdots\!68\)\( \beta) q^{81}\) \(+(\)\(47\!\cdots\!24\)\( - \)\(34\!\cdots\!16\)\( \beta) q^{82}\) \(+(\)\(20\!\cdots\!12\)\( - \)\(69\!\cdots\!65\)\( \beta) q^{83}\) \(+(-\)\(14\!\cdots\!92\)\( + \)\(12\!\cdots\!92\)\( \beta) q^{84}\) \(+(-\)\(26\!\cdots\!00\)\( - \)\(18\!\cdots\!80\)\( \beta) q^{85}\) \(+(\)\(47\!\cdots\!24\)\( - \)\(13\!\cdots\!56\)\( \beta) q^{86}\) \(+(\)\(66\!\cdots\!20\)\( + \)\(12\!\cdots\!74\)\( \beta) q^{87}\) \(+(-\)\(44\!\cdots\!44\)\( + \)\(73\!\cdots\!72\)\( \beta) q^{88}\) \(+(\)\(26\!\cdots\!10\)\( + \)\(42\!\cdots\!80\)\( \beta) q^{89}\) \(+(-\)\(17\!\cdots\!00\)\( + \)\(37\!\cdots\!80\)\( \beta) q^{90}\) \(+(-\)\(94\!\cdots\!48\)\( - \)\(52\!\cdots\!68\)\( \beta) q^{91}\) \(+(\)\(27\!\cdots\!88\)\( - \)\(61\!\cdots\!44\)\( \beta) q^{92}\) \(+(-\)\(27\!\cdots\!16\)\( + \)\(23\!\cdots\!24\)\( \beta) q^{93}\) \(+(-\)\(21\!\cdots\!68\)\( - \)\(29\!\cdots\!16\)\( \beta) q^{94}\) \(+(\)\(38\!\cdots\!00\)\( + \)\(20\!\cdots\!50\)\( \beta) q^{95}\) \(+(-\)\(31\!\cdots\!36\)\( + \)\(37\!\cdots\!68\)\( \beta) q^{96}\) \(+(\)\(46\!\cdots\!86\)\( - \)\(87\!\cdots\!80\)\( \beta) q^{97}\) \(+(\)\(60\!\cdots\!36\)\( + \)\(29\!\cdots\!88\)\( \beta) q^{98}\) \(+(\)\(77\!\cdots\!44\)\( - \)\(28\!\cdots\!15\)\( \beta) q^{99}\) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut -\mathstrut 65536q^{2} \) \(\mathstrut +\mathstrut 16716504q^{3} \) \(\mathstrut +\mathstrut 2147483648q^{4} \) \(\mathstrut -\mathstrut 26763065700q^{5} \) \(\mathstrut -\mathstrut 547766403072q^{6} \) \(\mathstrut -\mathstrut 23503702110608q^{7} \) \(\mathstrut -\mathstrut 70368744177664q^{8} \) \(\mathstrut +\mathstrut 724546953139914q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut -\mathstrut 65536q^{2} \) \(\mathstrut +\mathstrut 16716504q^{3} \) \(\mathstrut +\mathstrut 2147483648q^{4} \) \(\mathstrut -\mathstrut 26763065700q^{5} \) \(\mathstrut -\mathstrut 547766403072q^{6} \) \(\mathstrut -\mathstrut 23503702110608q^{7} \) \(\mathstrut -\mathstrut 70368744177664q^{8} \) \(\mathstrut +\mathstrut 724546953139914q^{9} \) \(\mathstrut +\mathstrut 876972136857600q^{10} \) \(\mathstrut +\mathstrut 25151137478512584q^{11} \) \(\mathstrut +\mathstrut 17949209495863296q^{12} \) \(\mathstrut +\mathstrut 189324344559709324q^{13} \) \(\mathstrut +\mathstrut 770169310760402944q^{14} \) \(\mathstrut +\mathstrut 6656561465056347600q^{15} \) \(\mathstrut +\mathstrut 2305843009213693952q^{16} \) \(\mathstrut +\mathstrut 920816829890675172q^{17} \) \(\mathstrut -\mathstrut 23741954560488701952q^{18} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!00\)\(q^{19} \) \(\mathstrut -\mathstrut 28736622980549836800q^{20} \) \(\mathstrut -\mathstrut \)\(26\!\cdots\!16\)\(q^{21} \) \(\mathstrut -\mathstrut \)\(82\!\cdots\!12\)\(q^{22} \) \(\mathstrut +\mathstrut \)\(51\!\cdots\!24\)\(q^{23} \) \(\mathstrut -\mathstrut \)\(58\!\cdots\!28\)\(q^{24} \) \(\mathstrut +\mathstrut \)\(17\!\cdots\!50\)\(q^{25} \) \(\mathstrut -\mathstrut \)\(62\!\cdots\!32\)\(q^{26} \) \(\mathstrut +\mathstrut \)\(26\!\cdots\!40\)\(q^{27} \) \(\mathstrut -\mathstrut \)\(25\!\cdots\!92\)\(q^{28} \) \(\mathstrut -\mathstrut \)\(39\!\cdots\!80\)\(q^{29} \) \(\mathstrut -\mathstrut \)\(21\!\cdots\!00\)\(q^{30} \) \(\mathstrut +\mathstrut \)\(49\!\cdots\!84\)\(q^{31} \) \(\mathstrut -\mathstrut \)\(75\!\cdots\!36\)\(q^{32} \) \(\mathstrut +\mathstrut \)\(59\!\cdots\!68\)\(q^{33} \) \(\mathstrut -\mathstrut \)\(30\!\cdots\!96\)\(q^{34} \) \(\mathstrut +\mathstrut \)\(49\!\cdots\!00\)\(q^{35} \) \(\mathstrut +\mathstrut \)\(77\!\cdots\!36\)\(q^{36} \) \(\mathstrut +\mathstrut \)\(18\!\cdots\!52\)\(q^{37} \) \(\mathstrut +\mathstrut \)\(32\!\cdots\!00\)\(q^{38} \) \(\mathstrut -\mathstrut \)\(70\!\cdots\!52\)\(q^{39} \) \(\mathstrut +\mathstrut \)\(94\!\cdots\!00\)\(q^{40} \) \(\mathstrut -\mathstrut \)\(29\!\cdots\!36\)\(q^{41} \) \(\mathstrut +\mathstrut \)\(87\!\cdots\!88\)\(q^{42} \) \(\mathstrut -\mathstrut \)\(29\!\cdots\!36\)\(q^{43} \) \(\mathstrut +\mathstrut \)\(27\!\cdots\!16\)\(q^{44} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!00\)\(q^{45} \) \(\mathstrut -\mathstrut \)\(16\!\cdots\!32\)\(q^{46} \) \(\mathstrut +\mathstrut \)\(13\!\cdots\!52\)\(q^{47} \) \(\mathstrut +\mathstrut \)\(19\!\cdots\!04\)\(q^{48} \) \(\mathstrut -\mathstrut \)\(36\!\cdots\!54\)\(q^{49} \) \(\mathstrut -\mathstrut \)\(55\!\cdots\!00\)\(q^{50} \) \(\mathstrut -\mathstrut \)\(30\!\cdots\!56\)\(q^{51} \) \(\mathstrut +\mathstrut \)\(20\!\cdots\!76\)\(q^{52} \) \(\mathstrut -\mathstrut \)\(45\!\cdots\!36\)\(q^{53} \) \(\mathstrut -\mathstrut \)\(85\!\cdots\!20\)\(q^{54} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!00\)\(q^{55} \) \(\mathstrut +\mathstrut \)\(82\!\cdots\!56\)\(q^{56} \) \(\mathstrut +\mathstrut \)\(83\!\cdots\!00\)\(q^{57} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!40\)\(q^{58} \) \(\mathstrut -\mathstrut \)\(25\!\cdots\!60\)\(q^{59} \) \(\mathstrut +\mathstrut \)\(71\!\cdots\!00\)\(q^{60} \) \(\mathstrut +\mathstrut \)\(73\!\cdots\!64\)\(q^{61} \) \(\mathstrut -\mathstrut \)\(16\!\cdots\!12\)\(q^{62} \) \(\mathstrut -\mathstrut \)\(96\!\cdots\!56\)\(q^{63} \) \(\mathstrut +\mathstrut \)\(24\!\cdots\!48\)\(q^{64} \) \(\mathstrut -\mathstrut \)\(35\!\cdots\!00\)\(q^{65} \) \(\mathstrut -\mathstrut \)\(19\!\cdots\!24\)\(q^{66} \) \(\mathstrut +\mathstrut \)\(21\!\cdots\!92\)\(q^{67} \) \(\mathstrut +\mathstrut \)\(98\!\cdots\!28\)\(q^{68} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!48\)\(q^{69} \) \(\mathstrut -\mathstrut \)\(16\!\cdots\!00\)\(q^{70} \) \(\mathstrut +\mathstrut \)\(36\!\cdots\!84\)\(q^{71} \) \(\mathstrut -\mathstrut \)\(25\!\cdots\!48\)\(q^{72} \) \(\mathstrut +\mathstrut \)\(16\!\cdots\!84\)\(q^{73} \) \(\mathstrut -\mathstrut \)\(61\!\cdots\!36\)\(q^{74} \) \(\mathstrut -\mathstrut \)\(41\!\cdots\!00\)\(q^{75} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!00\)\(q^{76} \) \(\mathstrut -\mathstrut \)\(31\!\cdots\!36\)\(q^{77} \) \(\mathstrut +\mathstrut \)\(23\!\cdots\!36\)\(q^{78} \) \(\mathstrut -\mathstrut \)\(43\!\cdots\!20\)\(q^{79} \) \(\mathstrut -\mathstrut \)\(30\!\cdots\!00\)\(q^{80} \) \(\mathstrut -\mathstrut \)\(43\!\cdots\!78\)\(q^{81} \) \(\mathstrut +\mathstrut \)\(95\!\cdots\!48\)\(q^{82} \) \(\mathstrut +\mathstrut \)\(41\!\cdots\!24\)\(q^{83} \) \(\mathstrut -\mathstrut \)\(28\!\cdots\!84\)\(q^{84} \) \(\mathstrut -\mathstrut \)\(53\!\cdots\!00\)\(q^{85} \) \(\mathstrut +\mathstrut \)\(95\!\cdots\!48\)\(q^{86} \) \(\mathstrut +\mathstrut \)\(13\!\cdots\!40\)\(q^{87} \) \(\mathstrut -\mathstrut \)\(88\!\cdots\!88\)\(q^{88} \) \(\mathstrut +\mathstrut \)\(52\!\cdots\!20\)\(q^{89} \) \(\mathstrut -\mathstrut \)\(34\!\cdots\!00\)\(q^{90} \) \(\mathstrut -\mathstrut \)\(18\!\cdots\!96\)\(q^{91} \) \(\mathstrut +\mathstrut \)\(54\!\cdots\!76\)\(q^{92} \) \(\mathstrut -\mathstrut \)\(55\!\cdots\!32\)\(q^{93} \) \(\mathstrut -\mathstrut \)\(43\!\cdots\!36\)\(q^{94} \) \(\mathstrut +\mathstrut \)\(76\!\cdots\!00\)\(q^{95} \) \(\mathstrut -\mathstrut \)\(63\!\cdots\!72\)\(q^{96} \) \(\mathstrut +\mathstrut \)\(93\!\cdots\!72\)\(q^{97} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!72\)\(q^{98} \) \(\mathstrut +\mathstrut \)\(15\!\cdots\!88\)\(q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
15712.8
−15711.8
−32768.0 −2.18094e7 1.07374e9 −1.27415e11 7.14650e11 −1.05903e13 −3.51844e13 −1.42024e14 4.17514e15
1.2 −32768.0 3.85259e7 1.07374e9 1.00652e11 −1.26242e12 −1.29134e13 −3.51844e13 8.66571e14 −3.29817e15
\(n\): 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 \(T_{3}^{2} \) \(\mathstrut -\mathstrut 16716504 T_{3} \) \(\mathstrut -\mathstrut \)\(84\!\cdots\!96\)\( \) acting on \(S_{32}^{\mathrm{new}}(\Gamma_0(2))\).