Properties

Label 3.14.a.b
Level 3
Weight 14
Character orbit 3.a
Self dual Yes
Analytic conductor 3.217
Analytic rank 0
Dimension 2
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 3 \)
Weight: \( k \) = \( 14 \)
Character orbit: \([\chi]\) = 3.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(3.21692786856\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{1969}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2\cdot 3 \)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 3\sqrt{1969}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + ( -27 - \beta ) q^{2} \) \( + 729 q^{3} \) \( + ( 10258 + 54 \beta ) q^{4} \) \( + ( 20358 - 128 \beta ) q^{5} \) \( + ( -19683 - 729 \beta ) q^{6} \) \( + ( -10504 + 3456 \beta ) q^{7} \) \( + ( -1012716 - 3524 \beta ) q^{8} \) \( + 531441 q^{9} \) \(+O(q^{10})\) \( q\) \( + ( -27 - \beta ) q^{2} \) \( + 729 q^{3} \) \( + ( 10258 + 54 \beta ) q^{4} \) \( + ( 20358 - 128 \beta ) q^{5} \) \( + ( -19683 - 729 \beta ) q^{6} \) \( + ( -10504 + 3456 \beta ) q^{7} \) \( + ( -1012716 - 3524 \beta ) q^{8} \) \( + 531441 q^{9} \) \( + ( 1718622 - 16902 \beta ) q^{10} \) \( + ( 336204 + 36608 \beta ) q^{11} \) \( + ( 7478082 + 39366 \beta ) q^{12} \) \( + ( 8766302 - 89856 \beta ) q^{13} \) \( + ( -60960168 - 82808 \beta ) q^{14} \) \( + ( 14840982 - 93312 \beta ) q^{15} \) \( + ( 5758600 + 665496 \beta ) q^{16} \) \( + ( 41919282 - 72960 \beta ) q^{17} \) \( + ( -14348907 - 531441 \beta ) q^{18} \) \( + ( 128146772 - 767232 \beta ) q^{19} \) \( + ( 86344812 - 213692 \beta ) q^{20} \) \( + ( -7657416 + 2519424 \beta ) q^{21} \) \( + ( -657807876 - 1324620 \beta ) q^{22} \) \( + ( 429790968 + 3697408 \beta ) q^{23} \) \( + ( -738269964 - 2568996 \beta ) q^{24} \) \( + ( -515914097 - 5211648 \beta ) q^{25} \) \( + ( 1355648022 - 6340190 \beta ) q^{26} \) \( + 387420489 q^{27} \) \( + ( 3199413872 + 34884432 \beta ) q^{28} \) \( + ( -2364237666 - 6108544 \beta ) q^{29} \) \( + ( 1252875438 - 12321558 \beta ) q^{30} \) \( + ( -2991275824 - 35040384 \beta ) q^{31} \) \( + ( -3652567344 + 5141616 \beta ) q^{32} \) \( + ( 245092716 + 26687232 \beta ) q^{33} \) \( + ( 161103546 - 39949362 \beta ) q^{34} \) \( + ( -8053043760 + 71701760 \beta ) q^{35} \) \( + ( 5451521778 + 28697814 \beta ) q^{36} \) \( + ( 13705597046 + 17335296 \beta ) q^{37} \) \( + ( 10136155428 - 107431508 \beta ) q^{38} \) \( + ( 6390634158 - 65505024 \beta ) q^{39} \) \( + ( -12623425416 + 57886056 \beta ) q^{40} \) \( + ( 7629487146 - 98057984 \beta ) q^{41} \) \( + ( -44439962472 - 60367032 \beta ) q^{42} \) \( + ( -5657249620 + 311613696 \beta ) q^{43} \) \( + ( 38480220504 + 393679880 \beta ) q^{44} \) \( + ( 10819075878 - 68024448 \beta ) q^{45} \) \( + ( -77126123304 - 529620984 \beta ) q^{46} \) \( + ( -34517571120 - 690673408 \beta ) q^{47} \) \( + ( 4198019400 + 485146584 \beta ) q^{48} \) \( + ( 114879813465 - 72603648 \beta ) q^{49} \) \( + ( 106285294827 + 656628593 \beta ) q^{50} \) \( + ( 30559156578 - 53187840 \beta ) q^{51} \) \( + ( 3938464412 - 448362540 \beta ) q^{52} \) \( + ( -113168447082 + 1307299968 \beta ) q^{53} \) \( + ( -10460353203 - 387420489 \beta ) q^{54} \) \( + ( -76193046072 + 702231552 \beta ) q^{55} \) \( + ( -205185497760 - 3462930400 \beta ) q^{56} \) \( + ( 93418996788 - 559312128 \beta ) q^{57} \) \( + ( 172083925206 + 2529168354 \beta ) q^{58} \) \( + ( -463910412132 + 397652992 \beta ) q^{59} \) \( + ( 62945367948 - 155781468 \beta ) q^{60} \) \( + ( 89697730670 - 1553776128 \beta ) q^{61} \) \( + ( 701715092112 + 3937366192 \beta ) q^{62} \) \( + ( -5582256264 + 1836660096 \beta ) q^{63} \) \( + ( -39669710048 - 1937999520 \beta ) q^{64} \) \( + ( 382283662644 - 2951375104 \beta ) q^{65} \) \( + ( -479541941604 - 965647980 \beta ) q^{66} \) \( + ( -349157530588 - 6092098560 \beta ) q^{67} \) \( + ( 360190090116 + 1515217548 \beta ) q^{68} \) \( + ( 313317615672 + 2695410432 \beta ) q^{69} \) \( + ( -1053194707440 + 6117096240 \beta ) q^{70} \) \( + ( -392229274968 + 4890812160 \beta ) q^{71} \) \( + ( -538198803756 - 1872798084 \beta ) q^{72} \) \( + ( 928700122538 + 7102660608 \beta ) q^{73} \) \( + ( -677249900658 - 14173650038 \beta ) q^{74} \) \( + ( -376101376713 - 3799291392 \beta ) q^{75} \) \( + ( 580339200488 - 950340168 \beta ) q^{76} \) \( + ( 2238480664992 + 777390592 \beta ) q^{77} \) \( + ( 988267408038 - 4621998510 \beta ) q^{78} \) \( + ( -357012735040 + 1007856000 \beta ) q^{79} \) \( + ( -1392303012048 + 12811066768 \beta ) q^{80} \) \( + 282429536481 q^{81} \) \( + ( 1531689381522 - 4981921578 \beta ) q^{82} \) \( + ( -2287146958956 + 1485492992 \beta ) q^{83} \) \( + ( 2332372712688 + 25430750928 \beta ) q^{84} \) \( + ( 1018887035436 - 6850987776 \beta ) q^{85} \) \( + ( -5369360567076 - 2756320172 \beta ) q^{86} \) \( + ( -1723529258514 - 4453128576 \beta ) q^{87} \) \( + ( -2626604986896 - 38258290224 \beta ) q^{88} \) \( + ( 1635089350842 - 22362004992 \beta ) q^{89} \) \( + ( 913346194302 - 8982415782 \beta ) q^{90} \) \( + ( -5595201972464 + 31240187136 \beta ) q^{91} \) \( + ( 7946971176816 + 61136723536 \beta ) q^{92} \) \( + ( -2180640075696 - 25544439936 \beta ) q^{93} \) \( + ( 13171397883408 + 53165753136 \beta ) q^{94} \) \( + ( 4349115123192 - 32022095872 \beta ) q^{95} \) \( + ( -2662721593776 + 3748238064 \beta ) q^{96} \) \( + ( -4937463078238 + 37066775040 \beta ) q^{97} \) \( + ( -1815145717347 - 112919514969 \beta ) q^{98} \) \( + ( 178672589964 + 19454992128 \beta ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut -\mathstrut 54q^{2} \) \(\mathstrut +\mathstrut 1458q^{3} \) \(\mathstrut +\mathstrut 20516q^{4} \) \(\mathstrut +\mathstrut 40716q^{5} \) \(\mathstrut -\mathstrut 39366q^{6} \) \(\mathstrut -\mathstrut 21008q^{7} \) \(\mathstrut -\mathstrut 2025432q^{8} \) \(\mathstrut +\mathstrut 1062882q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut -\mathstrut 54q^{2} \) \(\mathstrut +\mathstrut 1458q^{3} \) \(\mathstrut +\mathstrut 20516q^{4} \) \(\mathstrut +\mathstrut 40716q^{5} \) \(\mathstrut -\mathstrut 39366q^{6} \) \(\mathstrut -\mathstrut 21008q^{7} \) \(\mathstrut -\mathstrut 2025432q^{8} \) \(\mathstrut +\mathstrut 1062882q^{9} \) \(\mathstrut +\mathstrut 3437244q^{10} \) \(\mathstrut +\mathstrut 672408q^{11} \) \(\mathstrut +\mathstrut 14956164q^{12} \) \(\mathstrut +\mathstrut 17532604q^{13} \) \(\mathstrut -\mathstrut 121920336q^{14} \) \(\mathstrut +\mathstrut 29681964q^{15} \) \(\mathstrut +\mathstrut 11517200q^{16} \) \(\mathstrut +\mathstrut 83838564q^{17} \) \(\mathstrut -\mathstrut 28697814q^{18} \) \(\mathstrut +\mathstrut 256293544q^{19} \) \(\mathstrut +\mathstrut 172689624q^{20} \) \(\mathstrut -\mathstrut 15314832q^{21} \) \(\mathstrut -\mathstrut 1315615752q^{22} \) \(\mathstrut +\mathstrut 859581936q^{23} \) \(\mathstrut -\mathstrut 1476539928q^{24} \) \(\mathstrut -\mathstrut 1031828194q^{25} \) \(\mathstrut +\mathstrut 2711296044q^{26} \) \(\mathstrut +\mathstrut 774840978q^{27} \) \(\mathstrut +\mathstrut 6398827744q^{28} \) \(\mathstrut -\mathstrut 4728475332q^{29} \) \(\mathstrut +\mathstrut 2505750876q^{30} \) \(\mathstrut -\mathstrut 5982551648q^{31} \) \(\mathstrut -\mathstrut 7305134688q^{32} \) \(\mathstrut +\mathstrut 490185432q^{33} \) \(\mathstrut +\mathstrut 322207092q^{34} \) \(\mathstrut -\mathstrut 16106087520q^{35} \) \(\mathstrut +\mathstrut 10903043556q^{36} \) \(\mathstrut +\mathstrut 27411194092q^{37} \) \(\mathstrut +\mathstrut 20272310856q^{38} \) \(\mathstrut +\mathstrut 12781268316q^{39} \) \(\mathstrut -\mathstrut 25246850832q^{40} \) \(\mathstrut +\mathstrut 15258974292q^{41} \) \(\mathstrut -\mathstrut 88879924944q^{42} \) \(\mathstrut -\mathstrut 11314499240q^{43} \) \(\mathstrut +\mathstrut 76960441008q^{44} \) \(\mathstrut +\mathstrut 21638151756q^{45} \) \(\mathstrut -\mathstrut 154252246608q^{46} \) \(\mathstrut -\mathstrut 69035142240q^{47} \) \(\mathstrut +\mathstrut 8396038800q^{48} \) \(\mathstrut +\mathstrut 229759626930q^{49} \) \(\mathstrut +\mathstrut 212570589654q^{50} \) \(\mathstrut +\mathstrut 61118313156q^{51} \) \(\mathstrut +\mathstrut 7876928824q^{52} \) \(\mathstrut -\mathstrut 226336894164q^{53} \) \(\mathstrut -\mathstrut 20920706406q^{54} \) \(\mathstrut -\mathstrut 152386092144q^{55} \) \(\mathstrut -\mathstrut 410370995520q^{56} \) \(\mathstrut +\mathstrut 186837993576q^{57} \) \(\mathstrut +\mathstrut 344167850412q^{58} \) \(\mathstrut -\mathstrut 927820824264q^{59} \) \(\mathstrut +\mathstrut 125890735896q^{60} \) \(\mathstrut +\mathstrut 179395461340q^{61} \) \(\mathstrut +\mathstrut 1403430184224q^{62} \) \(\mathstrut -\mathstrut 11164512528q^{63} \) \(\mathstrut -\mathstrut 79339420096q^{64} \) \(\mathstrut +\mathstrut 764567325288q^{65} \) \(\mathstrut -\mathstrut 959083883208q^{66} \) \(\mathstrut -\mathstrut 698315061176q^{67} \) \(\mathstrut +\mathstrut 720380180232q^{68} \) \(\mathstrut +\mathstrut 626635231344q^{69} \) \(\mathstrut -\mathstrut 2106389414880q^{70} \) \(\mathstrut -\mathstrut 784458549936q^{71} \) \(\mathstrut -\mathstrut 1076397607512q^{72} \) \(\mathstrut +\mathstrut 1857400245076q^{73} \) \(\mathstrut -\mathstrut 1354499801316q^{74} \) \(\mathstrut -\mathstrut 752202753426q^{75} \) \(\mathstrut +\mathstrut 1160678400976q^{76} \) \(\mathstrut +\mathstrut 4476961329984q^{77} \) \(\mathstrut +\mathstrut 1976534816076q^{78} \) \(\mathstrut -\mathstrut 714025470080q^{79} \) \(\mathstrut -\mathstrut 2784606024096q^{80} \) \(\mathstrut +\mathstrut 564859072962q^{81} \) \(\mathstrut +\mathstrut 3063378763044q^{82} \) \(\mathstrut -\mathstrut 4574293917912q^{83} \) \(\mathstrut +\mathstrut 4664745425376q^{84} \) \(\mathstrut +\mathstrut 2037774070872q^{85} \) \(\mathstrut -\mathstrut 10738721134152q^{86} \) \(\mathstrut -\mathstrut 3447058517028q^{87} \) \(\mathstrut -\mathstrut 5253209973792q^{88} \) \(\mathstrut +\mathstrut 3270178701684q^{89} \) \(\mathstrut +\mathstrut 1826692388604q^{90} \) \(\mathstrut -\mathstrut 11190403944928q^{91} \) \(\mathstrut +\mathstrut 15893942353632q^{92} \) \(\mathstrut -\mathstrut 4361280151392q^{93} \) \(\mathstrut +\mathstrut 26342795766816q^{94} \) \(\mathstrut +\mathstrut 8698230246384q^{95} \) \(\mathstrut -\mathstrut 5325443187552q^{96} \) \(\mathstrut -\mathstrut 9874926156476q^{97} \) \(\mathstrut -\mathstrut 3630291434694q^{98} \) \(\mathstrut +\mathstrut 357345179928q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
22.6867
−21.6867
−160.120 729.000 17446.5 3318.61 −116728. 449560. −1.48183e6 531441. −531376.
1.2 106.120 729.000 3069.51 37397.4 77361.7 −470568. −543600. 531441. 3.96862e6
\(n\): 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 \(T_{2}^{2} \) \(\mathstrut +\mathstrut 54 T_{2} \) \(\mathstrut -\mathstrut 16992 \) acting on \(S_{14}^{\mathrm{new}}(\Gamma_0(3))\).