Properties

Label 3.26.a.a
Level 3
Weight 26
Character orbit 3.a
Self dual Yes
Analytic conductor 11.880
Analytic rank 1
Dimension 2
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 3 \)
Weight: \( k \) = \( 26 \)
Character orbit: \([\chi]\) = 3.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(11.8799033986\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{1287001}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 6\sqrt{1287001}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -162 - \beta ) q^{2} -531441 q^{3} + ( 12803848 + 324 \beta ) q^{4} + ( 285430878 + 22912 \beta ) q^{5} + ( 86093442 + 531441 \beta ) q^{6} + ( -14843692864 + 5152896 \beta ) q^{7} + ( -11649985056 + 20698096 \beta ) q^{8} + 282429536481 q^{9} +O(q^{10})\) \( q +(-162 - \beta) q^{2} -531441 q^{3} +(12803848 + 324 \beta) q^{4} +(285430878 + 22912 \beta) q^{5} +(86093442 + 531441 \beta) q^{6} +(-14843692864 + 5152896 \beta) q^{7} +(-11649985056 + 20698096 \beta) q^{8} +282429536481 q^{9} +(-1107799411068 - 289142622 \beta) q^{10} +(-9093892809276 - 983057152 \beta) q^{11} +(-6804489784968 - 172186884 \beta) q^{12} +(-52571818339618 + 8456327424 \beta) q^{13} +(-236339484732288 + 14008923712 \beta) q^{14} +(-151689671235198 - 12176376192 \beta) q^{15} +(-1386723478478720 - 2574742464 \beta) q^{16} +(-228493682174718 - 310973856000 \beta) q^{17} +(-45753584909922 - 282429536481 \beta) q^{18} +(6948719247673532 + 1780077627648 \beta) q^{19} +(3998558889680112 + 385841369848 \beta) q^{20} +(7888546979337024 - 2738460203136 \beta) q^{21} +(47020249991624184 + 9253148067900 \beta) q^{22} +(69671149450176888 - 20198378016512 \beta) q^{23} +(6191279708145696 - 10999816836336 \beta) q^{24} +(-192229984003543457 + 13079584553472 \beta) q^{25} +(-383282232065537148 + 51201893296930 \beta) q^{26} -150094635296999121 q^{27} +(-112703278385033728 + 61167540655872 \beta) q^{28} +(-1251016745130354906 - 126217146058624 \beta) q^{29} +(588730026817388988 + 153662244178302 \beta) q^{30} +(537736051246227656 - 1052676722058624 \beta) q^{31} +(734850895408897536 + 692627731996416 \beta) q^{32} +(4832867488454446716 + 522436875916032 \beta) q^{33} +(14445067867763120316 + 278871446846718 \beta) q^{34} +(1233257975178122880 + 1130696938622720 \beta) q^{35} +(3616184855813178888 + 91507169819844 \beta) q^{36} +(-985735474619156314 + 295504419926016 \beta) q^{37} +(-83600313245104843512 - 7237091823352508 \beta) q^{38} +(27938819710224929538 - 4494039102537984 \beta) q^{39} +(18646997230106224704 + 5640951256605216 \beta) q^{40} +(-\)\(20\!\cdots\!34\)\( + 5221410776755456 \beta) q^{41} +(\)\(12\!\cdots\!08\)\( - 7444916426428992 \beta) q^{42} +(19465197949129913540 + 45176076103631616 \beta) q^{43} +(-\)\(13\!\cdots\!76\)\( - 15533335619726320 \beta) q^{44} +(80614110570904860318 + 6471025539852672 \beta) q^{45} +(\)\(92\!\cdots\!76\)\( - 66399012211501944 \beta) q^{46} +(-\)\(36\!\cdots\!20\)\( - 9934991015089408 \beta) q^{47} +(\)\(73\!\cdots\!20\)\( + 1368323709810624 \beta) q^{48} +(\)\(10\!\cdots\!65\)\( - 152976011168268288 \beta) q^{49} +(-\)\(57\!\cdots\!58\)\( + 190111091305880993 \beta) q^{50} +(\)\(12\!\cdots\!38\)\( + 165264257006496000 \beta) q^{51} +(-\)\(54\!\cdots\!28\)\( + 91240261833091320 \beta) q^{52} +(-\)\(42\!\cdots\!82\)\( + 15631770768985728 \beta) q^{53} +(24315330918113857602 + 150094635296999121 \beta) q^{54} +(-\)\(36\!\cdots\!92\)\( - 488954138065671168 \beta) q^{55} +(\)\(51\!\cdots\!60\)\( - 367267341288709120 \beta) q^{56} +(-\)\(36\!\cdots\!12\)\( - 946006234514880768 \beta) q^{57} +(\)\(60\!\cdots\!36\)\( + 1271463922791851994 \beta) q^{58} +(\)\(42\!\cdots\!48\)\( - 397484438615856128 \beta) q^{59} +(-\)\(21\!\cdots\!92\)\( - 205051923433390968 \beta) q^{60} +(-\)\(12\!\cdots\!30\)\( + 2464520538925757952 \beta) q^{61} +(\)\(48\!\cdots\!92\)\( - 367202422272730568 \beta) q^{62} +(-\)\(41\!\cdots\!84\)\( + 1455330028814798976 \beta) q^{63} +(\)\(14\!\cdots\!32\)\( - 760662567066516480 \beta) q^{64} +(-\)\(60\!\cdots\!36\)\( + 1209171459490470656 \beta) q^{65} +(-\)\(24\!\cdots\!44\)\( - 4917502262352843900 \beta) q^{66} +(-\)\(67\!\cdots\!28\)\( - 7885549958339957760 \beta) q^{67} +(-\)\(75\!\cdots\!64\)\( - 4055693937222496632 \beta) q^{68} +(-\)\(37\!\cdots\!08\)\( + 10734246211473153792 \beta) q^{69} +(-\)\(52\!\cdots\!80\)\( - 1416430879235003520 \beta) q^{70} +(\)\(60\!\cdots\!92\)\( + 18643323271852842240 \beta) q^{71} +(-\)\(32\!\cdots\!36\)\( + 5845753659319240176 \beta) q^{72} +(\)\(42\!\cdots\!38\)\( - 13385360579839137792 \beta) q^{73} +(-\)\(13\!\cdots\!08\)\( + 937863758591141722 \beta) q^{74} +(\)\(10\!\cdots\!37\)\( - 6951027494681713152 \beta) q^{75} +(\)\(11\!\cdots\!08\)\( + 25043228408851813872 \beta) q^{76} +(-\)\(99\!\cdots\!48\)\( - 32267685449300499968 \beta) q^{77} +(\)\(20\!\cdots\!68\)\( - 27210785375613776130 \beta) q^{78} +(\)\(63\!\cdots\!40\)\( - 27385251957433054080 \beta) q^{79} +(-\)\(39\!\cdots\!08\)\( - 32507519341027836032 \beta) q^{80} +\)\(79\!\cdots\!61\)\( q^{81} +(-\)\(20\!\cdots\!08\)\( + \)\(20\!\cdots\!62\)\( \beta) q^{82} +(-\)\(43\!\cdots\!56\)\( + 42027911871325874432 \beta) q^{83} +(\)\(59\!\cdots\!48\)\( - 32506938973697271552 \beta) q^{84} +(-\)\(39\!\cdots\!04\)\( - 93996787999112706816 \beta) q^{85} +(-\)\(20\!\cdots\!56\)\( - 26783722277918235332 \beta) q^{86} +(\)\(66\!\cdots\!46\)\( + 67076966318541197184 \beta) q^{87} +(-\)\(83\!\cdots\!56\)\( - \)\(17\!\cdots\!84\)\( \beta) q^{88} +(\)\(10\!\cdots\!22\)\( + \)\(13\!\cdots\!68\)\( \beta) q^{89} +(-\)\(31\!\cdots\!08\)\( - 81662416708360993182 \beta) q^{90} +(\)\(27\!\cdots\!96\)\( - \)\(39\!\cdots\!64\)\( \beta) q^{91} +(\)\(58\!\cdots\!56\)\( - \)\(23\!\cdots\!64\)\( \beta) q^{92} +(-\)\(28\!\cdots\!96\)\( + \)\(55\!\cdots\!84\)\( \beta) q^{93} +(\)\(52\!\cdots\!28\)\( + \)\(37\!\cdots\!16\)\( \beta) q^{94} +(\)\(38\!\cdots\!32\)\( + \)\(66\!\cdots\!28\)\( \beta) q^{95} +(-\)\(39\!\cdots\!76\)\( - \)\(36\!\cdots\!56\)\( \beta) q^{96} +(-\)\(27\!\cdots\!58\)\( - 71593944273965145600 \beta) q^{97} +(\)\(70\!\cdots\!38\)\( - 84708326791237796409 \beta) q^{98} +(-\)\(25\!\cdots\!56\)\( - \)\(27\!\cdots\!12\)\( \beta) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 324q^{2} - 1062882q^{3} + 25607696q^{4} + 570861756q^{5} + 172186884q^{6} - 29687385728q^{7} - 23299970112q^{8} + 564859072962q^{9} + O(q^{10}) \) \( 2q - 324q^{2} - 1062882q^{3} + 25607696q^{4} + 570861756q^{5} + 172186884q^{6} - 29687385728q^{7} - 23299970112q^{8} + 564859072962q^{9} - 2215598822136q^{10} - 18187785618552q^{11} - 13608979569936q^{12} - 105143636679236q^{13} - 472678969464576q^{14} - 303379342470396q^{15} - 2773446956957440q^{16} - 456987364349436q^{17} - 91507169819844q^{18} + 13897438495347064q^{19} + 7997117779360224q^{20} + 15777093958674048q^{21} + 94040499983248368q^{22} + 139342298900353776q^{23} + 12382559416291392q^{24} - 384459968007086914q^{25} - 766564464131074296q^{26} - 300189270593998242q^{27} - 225406556770067456q^{28} - 2502033490260709812q^{29} + 1177460053634777976q^{30} + 1075472102492455312q^{31} + 1469701790817795072q^{32} + 9665734976908893432q^{33} + 28890135735526240632q^{34} + 2466515950356245760q^{35} + 7232369711626357776q^{36} - 1971470949238312628q^{37} - \)\(16\!\cdots\!24\)\(q^{38} + 55877639420449859076q^{39} + 37293994460212449408q^{40} - \)\(41\!\cdots\!68\)\(q^{41} + \)\(25\!\cdots\!16\)\(q^{42} + 38930395898259827080q^{43} - \)\(26\!\cdots\!52\)\(q^{44} + \)\(16\!\cdots\!36\)\(q^{45} + \)\(18\!\cdots\!52\)\(q^{46} - \)\(73\!\cdots\!40\)\(q^{47} + \)\(14\!\cdots\!40\)\(q^{48} + \)\(21\!\cdots\!30\)\(q^{49} - \)\(11\!\cdots\!16\)\(q^{50} + \)\(24\!\cdots\!76\)\(q^{51} - \)\(10\!\cdots\!56\)\(q^{52} - \)\(84\!\cdots\!64\)\(q^{53} + 48630661836227715204q^{54} - \)\(72\!\cdots\!84\)\(q^{55} + \)\(10\!\cdots\!20\)\(q^{56} - \)\(73\!\cdots\!24\)\(q^{57} + \)\(12\!\cdots\!72\)\(q^{58} + \)\(84\!\cdots\!96\)\(q^{59} - \)\(42\!\cdots\!84\)\(q^{60} - \)\(24\!\cdots\!60\)\(q^{61} + \)\(97\!\cdots\!84\)\(q^{62} - \)\(83\!\cdots\!68\)\(q^{63} + \)\(28\!\cdots\!64\)\(q^{64} - \)\(12\!\cdots\!72\)\(q^{65} - \)\(49\!\cdots\!88\)\(q^{66} - \)\(13\!\cdots\!56\)\(q^{67} - \)\(15\!\cdots\!28\)\(q^{68} - \)\(74\!\cdots\!16\)\(q^{69} - \)\(10\!\cdots\!60\)\(q^{70} + \)\(12\!\cdots\!84\)\(q^{71} - \)\(65\!\cdots\!72\)\(q^{72} + \)\(84\!\cdots\!76\)\(q^{73} - \)\(27\!\cdots\!16\)\(q^{74} + \)\(20\!\cdots\!74\)\(q^{75} + \)\(23\!\cdots\!16\)\(q^{76} - \)\(19\!\cdots\!96\)\(q^{77} + \)\(40\!\cdots\!36\)\(q^{78} + \)\(12\!\cdots\!80\)\(q^{79} - \)\(79\!\cdots\!16\)\(q^{80} + \)\(15\!\cdots\!22\)\(q^{81} - \)\(41\!\cdots\!16\)\(q^{82} - \)\(87\!\cdots\!12\)\(q^{83} + \)\(11\!\cdots\!96\)\(q^{84} - \)\(79\!\cdots\!08\)\(q^{85} - \)\(41\!\cdots\!12\)\(q^{86} + \)\(13\!\cdots\!92\)\(q^{87} - \)\(16\!\cdots\!12\)\(q^{88} + \)\(21\!\cdots\!44\)\(q^{89} - \)\(62\!\cdots\!16\)\(q^{90} + \)\(55\!\cdots\!92\)\(q^{91} + \)\(11\!\cdots\!12\)\(q^{92} - \)\(57\!\cdots\!92\)\(q^{93} + \)\(10\!\cdots\!56\)\(q^{94} + \)\(77\!\cdots\!64\)\(q^{95} - \)\(78\!\cdots\!52\)\(q^{96} - \)\(55\!\cdots\!16\)\(q^{97} + \)\(14\!\cdots\!76\)\(q^{98} - \)\(51\!\cdots\!12\)\(q^{99} + O(q^{100}) \)

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
567.730
−566.730
−6968.76 −531441. 1.50092e7 4.41387e8 3.70349e9 2.02309e10 1.29237e11 2.82430e11 −3.07592e12
1.2 6644.76 −531441. 1.05985e7 1.29474e8 −3.53130e9 −4.99182e10 −1.52537e11 2.82430e11 8.60326e11
\(n\): 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
\(3\) \(1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \( T_{2}^{2} + 324 T_{2} - 46305792 \) acting on \(S_{26}^{\mathrm{new}}(\Gamma_0(3))\).