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 $1$

Related objects

Downloads

Learn more

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: \(\Q(\sqrt{1969}) \)
Defining polynomial: \( x^{2} - x - 492 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2\cdot 3 \)
Twist minimal: yes
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.

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

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:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3.14.a.b 2
3.b odd 2 1 9.14.a.c 2
4.b odd 2 1 48.14.a.g 2
5.b even 2 1 75.14.a.e 2
5.c odd 4 2 75.14.b.c 4
7.b odd 2 1 147.14.a.b 2
8.b even 2 1 192.14.a.k 2
8.d odd 2 1 192.14.a.o 2
12.b even 2 1 144.14.a.m 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.14.a.b 2 1.a even 1 1 trivial
9.14.a.c 2 3.b odd 2 1
48.14.a.g 2 4.b odd 2 1
75.14.a.e 2 5.b even 2 1
75.14.b.c 4 5.c odd 4 2
144.14.a.m 2 12.b even 2 1
147.14.a.b 2 7.b odd 2 1
192.14.a.k 2 8.b even 2 1
192.14.a.o 2 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 54T_{2} - 16992 \) acting on \(S_{14}^{\mathrm{new}}(\Gamma_0(3))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 54T - 16992 \) Copy content Toggle raw display
$3$ \( (T - 729)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 40716 T + 124107300 \) Copy content Toggle raw display
$7$ \( T^{2} + 21008 T - 211548155840 \) Copy content Toggle raw display
$11$ \( T^{2} - 672408 T - 23635688182128 \) Copy content Toggle raw display
$13$ \( T^{2} - 17532604 T - 66233088387452 \) Copy content Toggle raw display
$17$ \( T^{2} - 83838564 T + 16\!\cdots\!24 \) Copy content Toggle raw display
$19$ \( T^{2} - 256293544 T + 59\!\cdots\!80 \) Copy content Toggle raw display
$23$ \( T^{2} - 859581936 T - 57\!\cdots\!20 \) Copy content Toggle raw display
$29$ \( T^{2} + 4728475332 T + 49\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{2} + 5982551648 T - 12\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{2} - 27411194092 T + 18\!\cdots\!80 \) Copy content Toggle raw display
$41$ \( T^{2} - 15258974292 T - 11\!\cdots\!60 \) Copy content Toggle raw display
$43$ \( T^{2} + 11314499240 T - 16\!\cdots\!36 \) Copy content Toggle raw display
$47$ \( T^{2} + 69035142240 T - 72\!\cdots\!44 \) Copy content Toggle raw display
$53$ \( T^{2} + 226336894164 T - 17\!\cdots\!80 \) Copy content Toggle raw display
$59$ \( T^{2} + 927820824264 T + 21\!\cdots\!80 \) Copy content Toggle raw display
$61$ \( T^{2} - 179395461340 T - 34\!\cdots\!64 \) Copy content Toggle raw display
$67$ \( T^{2} + 698315061176 T - 53\!\cdots\!56 \) Copy content Toggle raw display
$71$ \( T^{2} + 784458549936 T - 27\!\cdots\!76 \) Copy content Toggle raw display
$73$ \( T^{2} - 1857400245076 T - 31\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{2} + 714025470080 T + 10\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{2} + 4574293917912 T + 51\!\cdots\!92 \) Copy content Toggle raw display
$89$ \( T^{2} - 3270178701684 T - 61\!\cdots\!80 \) Copy content Toggle raw display
$97$ \( T^{2} + 9874926156476 T + 30\!\cdots\!44 \) Copy content Toggle raw display
show more
show less