Properties

Label 15.12.a.d
Level 15
Weight 12
Character orbit 15.a
Self dual Yes
Analytic conductor 11.525
Analytic rank 0
Dimension 3
CM No
Inner twists 1

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 15 = 3 \cdot 5 \)
Weight: \( k \) = \( 12 \)
Character orbit: \([\chi]\) = 15.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(11.5251477084\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2\cdot 3\cdot 5 \)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( -\beta_{1} q^{2} \) \( + 243 q^{3} \) \( + ( 1585 + 3 \beta_{1} + \beta_{2} ) q^{4} \) \( + 3125 q^{5} \) \( -243 \beta_{1} q^{6} \) \( + ( -4708 - 496 \beta_{1} - 12 \beta_{2} ) q^{7} \) \( + ( -10881 - 1357 \beta_{1} - \beta_{2} ) q^{8} \) \( + 59049 q^{9} \) \(+O(q^{10})\) \( q\) \( -\beta_{1} q^{2} \) \( + 243 q^{3} \) \( + ( 1585 + 3 \beta_{1} + \beta_{2} ) q^{4} \) \( + 3125 q^{5} \) \( -243 \beta_{1} q^{6} \) \( + ( -4708 - 496 \beta_{1} - 12 \beta_{2} ) q^{7} \) \( + ( -10881 - 1357 \beta_{1} - \beta_{2} ) q^{8} \) \( + 59049 q^{9} \) \( -3125 \beta_{1} q^{10} \) \( + ( 180588 - 1056 \beta_{1} + 88 \beta_{2} ) q^{11} \) \( + ( 385155 + 729 \beta_{1} + 243 \beta_{2} ) q^{12} \) \( + ( 278498 + 4848 \beta_{1} - 628 \beta_{2} ) q^{13} \) \( + ( 1801752 + 27928 \beta_{1} + 472 \beta_{2} ) q^{14} \) \( + 759375 q^{15} \) \( + ( 1683883 + 10619 \beta_{1} - 693 \beta_{2} ) q^{16} \) \( + ( 5016966 + 112656 \beta_{1} - 1484 \beta_{2} ) q^{17} \) \( -59049 \beta_{1} q^{18} \) \( + ( 5936504 - 61200 \beta_{1} + 4556 \beta_{2} ) q^{19} \) \( + ( 4953125 + 9375 \beta_{1} + 3125 \beta_{2} ) q^{20} \) \( + ( -1144044 - 120528 \beta_{1} - 2916 \beta_{2} ) q^{21} \) \( + ( 3838032 - 336788 \beta_{1} + 1232 \beta_{2} ) q^{22} \) \( + ( -9388236 + 14896 \beta_{1} - 8996 \beta_{2} ) q^{23} \) \( + ( -2644083 - 329751 \beta_{1} - 243 \beta_{2} ) q^{24} \) \( + 9765625 q^{25} \) \( + ( -17624088 + 844266 \beta_{1} - 6104 \beta_{2} ) q^{26} \) \( + 14348907 q^{27} \) \( + ( -91811944 - 1724520 \beta_{1} - 2408 \beta_{2} ) q^{28} \) \( + ( -23186442 + 2213312 \beta_{1} + 36784 \beta_{2} ) q^{29} \) \( -759375 \beta_{1} q^{30} \) \( + ( -70054156 + 3260944 \beta_{1} + 17972 \beta_{2} ) q^{31} \) \( + ( -16307013 + 2318419 \beta_{1} - 9957 \beta_{2} ) q^{32} \) \( + ( 43882884 - 256608 \beta_{1} + 21384 \beta_{2} ) q^{33} \) \( + ( -409305960 - 2667410 \beta_{1} - 115624 \beta_{2} ) q^{34} \) \( + ( -14712500 - 1550000 \beta_{1} - 37500 \beta_{2} ) q^{35} \) \( + ( 93592665 + 177147 \beta_{1} + 59049 \beta_{2} ) q^{36} \) \( + ( -107248198 + 3445552 \beta_{1} + 38236 \beta_{2} ) q^{37} \) \( + ( 222421608 - 14003820 \beta_{1} + 70312 \beta_{2} ) q^{38} \) \( + ( 67675014 + 1178064 \beta_{1} - 152604 \beta_{2} ) q^{39} \) \( + ( -34003125 - 4240625 \beta_{1} - 3125 \beta_{2} ) q^{40} \) \( + ( 702449634 + 2708000 \beta_{1} - 28952 \beta_{2} ) q^{41} \) \( + ( 437825736 + 6786504 \beta_{1} + 114696 \beta_{2} ) q^{42} \) \( + ( 143001716 - 10532416 \beta_{1} + 213040 \beta_{2} ) q^{43} \) \( + ( 853728756 - 2896132 \beta_{1} + 159028 \beta_{2} ) q^{44} \) \( + 184528125 q^{45} \) \( + ( -54279096 + 25635304 \beta_{1} - 32888 \beta_{2} ) q^{46} \) \( + ( -530606652 - 8380080 \beta_{1} - 531452 \beta_{2} ) q^{47} \) \( + ( 409183569 + 2580417 \beta_{1} - 168399 \beta_{2} ) q^{48} \) \( + ( -114158391 + 26442368 \beta_{1} + 74272 \beta_{2} ) q^{49} \) \( -9765625 \beta_{1} q^{50} \) \( + ( 1219122738 + 27375408 \beta_{1} - 360612 \beta_{2} ) q^{51} \) \( + ( -3637692154 + 16216930 \beta_{1} + 429670 \beta_{2} ) q^{52} \) \( + ( 1506839574 - 31569760 \beta_{1} - 193272 \beta_{2} ) q^{53} \) \( -14348907 \beta_{1} q^{54} \) \( + ( 564337500 - 3300000 \beta_{1} + 275000 \beta_{2} ) q^{55} \) \( + ( 2575149720 + 44149848 \beta_{1} + 753048 \beta_{2} ) q^{56} \) \( + ( 1442570472 - 14871600 \beta_{1} + 1107108 \beta_{2} ) q^{57} \) \( + ( -8040300384 - 50069318 \beta_{1} - 2139744 \beta_{2} ) q^{58} \) \( + ( 3711359436 - 29339488 \beta_{1} + 2571336 \beta_{2} ) q^{59} \) \( + ( 1203609375 + 2278125 \beta_{1} + 759375 \beta_{2} ) q^{60} \) \( + ( -1144911154 - 134332768 \beta_{1} - 945272 \beta_{2} ) q^{61} \) \( + ( -11846686056 + 27724032 \beta_{1} - 3225000 \beta_{2} ) q^{62} \) \( + ( -278002692 - 29288304 \beta_{1} - 708588 \beta_{2} ) q^{63} \) \( + ( -11871587837 + 5636171 \beta_{1} - 919069 \beta_{2} ) q^{64} \) \( + ( 870306250 + 15150000 \beta_{1} - 1962500 \beta_{2} ) q^{65} \) \( + ( 932641776 - 81839484 \beta_{1} + 299376 \beta_{2} ) q^{66} \) \( + ( 741905516 + 6924032 \beta_{1} + 2697792 \beta_{2} ) q^{67} \) \( + ( -586127070 + 395983766 \beta_{1} + 5475394 \beta_{2} ) q^{68} \) \( + ( -2281341348 + 3619728 \beta_{1} - 2186028 \beta_{2} ) q^{69} \) \( + ( 5630475000 + 87275000 \beta_{1} + 1475000 \beta_{2} ) q^{70} \) \( + ( 16634962392 - 64285568 \beta_{1} - 2165088 \beta_{2} ) q^{71} \) \( + ( -642512169 - 80129493 \beta_{1} - 59049 \beta_{2} ) q^{72} \) \( + ( 13524622514 + 181311712 \beta_{1} + 2959320 \beta_{2} ) q^{73} \) \( + ( -12517002168 + 27666146 \beta_{1} - 3369080 \beta_{2} ) q^{74} \) \( + 2373046875 q^{75} \) \( + ( 38719183484 - 182407580 \beta_{1} + 4813756 \beta_{2} ) q^{76} \) \( + ( -5894674896 - 135282368 \beta_{1} - 82160 \beta_{2} ) q^{77} \) \( + ( -4282653384 + 205156638 \beta_{1} - 1483272 \beta_{2} ) q^{78} \) \( + ( -20364580 + 169248368 \beta_{1} - 5005332 \beta_{2} ) q^{79} \) \( + ( 5262134375 + 33184375 \beta_{1} - 2165625 \beta_{2} ) q^{80} \) \( + 3486784401 q^{81} \) \( + ( -9838685136 - 658141562 \beta_{1} - 2765904 \beta_{2} ) q^{82} \) \( + ( 2074466220 + 32952768 \beta_{1} - 3308880 \beta_{2} ) q^{83} \) \( + ( -22310302392 - 419058360 \beta_{1} - 585144 \beta_{2} ) q^{84} \) \( + ( 15678018750 + 352050000 \beta_{1} - 4637500 \beta_{2} ) q^{85} \) \( + ( 38268102048 - 497219908 \beta_{1} + 10958496 \beta_{2} ) q^{86} \) \( + ( -5634305406 + 537834816 \beta_{1} + 8938512 \beta_{2} ) q^{87} \) \( + ( 2664220524 - 443298244 \beta_{1} + 691052 \beta_{2} ) q^{88} \) \( + ( -19666758366 - 959666976 \beta_{1} + 12459480 \beta_{2} ) q^{89} \) \( -184528125 \beta_{1} q^{90} \) \( + ( 39531088744 + 263143776 \beta_{1} - 16951816 \beta_{2} ) q^{91} \) \( + ( -73906544088 + 6426344 \beta_{1} - 7277272 \beta_{2} ) q^{92} \) \( + ( -17023159908 + 792409392 \beta_{1} + 4367196 \beta_{2} ) q^{93} \) \( + ( 30435264504 + 1518206464 \beta_{1} + 7317176 \beta_{2} ) q^{94} \) \( + ( 18551575000 - 191250000 \beta_{1} + 14237500 \beta_{2} ) q^{95} \) \( + ( -3962604159 + 563375817 \beta_{1} - 2419551 \beta_{2} ) q^{96} \) \( + ( -69658780174 + 1148939072 \beta_{1} + 3884432 \beta_{2} ) q^{97} \) \( + ( -96063786048 - 99675305 \beta_{1} - 26293824 \beta_{2} ) q^{98} \) \( + ( 10663540812 - 62355744 \beta_{1} + 5196312 \beta_{2} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(3q \) \(\mathstrut -\mathstrut q^{2} \) \(\mathstrut +\mathstrut 729q^{3} \) \(\mathstrut +\mathstrut 4757q^{4} \) \(\mathstrut +\mathstrut 9375q^{5} \) \(\mathstrut -\mathstrut 243q^{6} \) \(\mathstrut -\mathstrut 14608q^{7} \) \(\mathstrut -\mathstrut 33999q^{8} \) \(\mathstrut +\mathstrut 177147q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(3q \) \(\mathstrut -\mathstrut q^{2} \) \(\mathstrut +\mathstrut 729q^{3} \) \(\mathstrut +\mathstrut 4757q^{4} \) \(\mathstrut +\mathstrut 9375q^{5} \) \(\mathstrut -\mathstrut 243q^{6} \) \(\mathstrut -\mathstrut 14608q^{7} \) \(\mathstrut -\mathstrut 33999q^{8} \) \(\mathstrut +\mathstrut 177147q^{9} \) \(\mathstrut -\mathstrut 3125q^{10} \) \(\mathstrut +\mathstrut 540620q^{11} \) \(\mathstrut +\mathstrut 1155951q^{12} \) \(\mathstrut +\mathstrut 840970q^{13} \) \(\mathstrut +\mathstrut 5432712q^{14} \) \(\mathstrut +\mathstrut 2278125q^{15} \) \(\mathstrut +\mathstrut 5062961q^{16} \) \(\mathstrut +\mathstrut 15165038q^{17} \) \(\mathstrut -\mathstrut 59049q^{18} \) \(\mathstrut +\mathstrut 17743756q^{19} \) \(\mathstrut +\mathstrut 14865625q^{20} \) \(\mathstrut -\mathstrut 3549744q^{21} \) \(\mathstrut +\mathstrut 11176076q^{22} \) \(\mathstrut -\mathstrut 28140816q^{23} \) \(\mathstrut -\mathstrut 8261757q^{24} \) \(\mathstrut +\mathstrut 29296875q^{25} \) \(\mathstrut -\mathstrut 52021894q^{26} \) \(\mathstrut +\mathstrut 43046721q^{27} \) \(\mathstrut -\mathstrut 277157944q^{28} \) \(\mathstrut -\mathstrut 67382798q^{29} \) \(\mathstrut -\mathstrut 759375q^{30} \) \(\mathstrut -\mathstrut 206919496q^{31} \) \(\mathstrut -\mathstrut 46592663q^{32} \) \(\mathstrut +\mathstrut 131370660q^{33} \) \(\mathstrut -\mathstrut 1230469666q^{34} \) \(\mathstrut -\mathstrut 45650000q^{35} \) \(\mathstrut +\mathstrut 280896093q^{36} \) \(\mathstrut -\mathstrut 318337278q^{37} \) \(\mathstrut +\mathstrut 653190692q^{38} \) \(\mathstrut +\mathstrut 204355710q^{39} \) \(\mathstrut -\mathstrut 106246875q^{40} \) \(\mathstrut +\mathstrut 2110085854q^{41} \) \(\mathstrut +\mathstrut 1320149016q^{42} \) \(\mathstrut +\mathstrut 418259692q^{43} \) \(\mathstrut +\mathstrut 2558131108q^{44} \) \(\mathstrut +\mathstrut 553584375q^{45} \) \(\mathstrut -\mathstrut 137169096q^{46} \) \(\mathstrut -\mathstrut 1599668584q^{47} \) \(\mathstrut +\mathstrut 1230299523q^{48} \) \(\mathstrut -\mathstrut 316107077q^{49} \) \(\mathstrut -\mathstrut 9765625q^{50} \) \(\mathstrut +\mathstrut 3685104234q^{51} \) \(\mathstrut -\mathstrut 10897289202q^{52} \) \(\mathstrut +\mathstrut 4489142234q^{53} \) \(\mathstrut -\mathstrut 14348907q^{54} \) \(\mathstrut +\mathstrut 1689437500q^{55} \) \(\mathstrut +\mathstrut 7768845960q^{56} \) \(\mathstrut +\mathstrut 4311732708q^{57} \) \(\mathstrut -\mathstrut 24168830726q^{58} \) \(\mathstrut +\mathstrut 11102167484q^{59} \) \(\mathstrut +\mathstrut 3612346875q^{60} \) \(\mathstrut -\mathstrut 3568120958q^{61} \) \(\mathstrut -\mathstrut 35509109136q^{62} \) \(\mathstrut -\mathstrut 862587792q^{63} \) \(\mathstrut -\mathstrut 35608208271q^{64} \) \(\mathstrut +\mathstrut 2628031250q^{65} \) \(\mathstrut +\mathstrut 2715786468q^{66} \) \(\mathstrut +\mathstrut 2229942788q^{67} \) \(\mathstrut -\mathstrut 1367872838q^{68} \) \(\mathstrut -\mathstrut 6838218288q^{69} \) \(\mathstrut +\mathstrut 16977225000q^{70} \) \(\mathstrut +\mathstrut 49842766696q^{71} \) \(\mathstrut -\mathstrut 2007606951q^{72} \) \(\mathstrut +\mathstrut 40752219934q^{73} \) \(\mathstrut -\mathstrut 37519971278q^{74} \) \(\mathstrut +\mathstrut 7119140625q^{75} \) \(\mathstrut +\mathstrut 115970329116q^{76} \) \(\mathstrut -\mathstrut 17819224896q^{77} \) \(\mathstrut -\mathstrut 12641320242q^{78} \) \(\mathstrut +\mathstrut 113159960q^{79} \) \(\mathstrut +\mathstrut 15821753125q^{80} \) \(\mathstrut +\mathstrut 10460353203q^{81} \) \(\mathstrut -\mathstrut 30171431066q^{82} \) \(\mathstrut +\mathstrut 6259660308q^{83} \) \(\mathstrut -\mathstrut 67349380392q^{84} \) \(\mathstrut +\mathstrut 47390743750q^{85} \) \(\mathstrut +\mathstrut 114296127740q^{86} \) \(\mathstrut -\mathstrut 16374019914q^{87} \) \(\mathstrut +\mathstrut 7548672276q^{88} \) \(\mathstrut -\mathstrut 59972401554q^{89} \) \(\mathstrut -\mathstrut 184528125q^{90} \) \(\mathstrut +\mathstrut 118873361824q^{91} \) \(\mathstrut -\mathstrut 221705928648q^{92} \) \(\mathstrut -\mathstrut 50281437528q^{93} \) \(\mathstrut +\mathstrut 92816682800q^{94} \) \(\mathstrut +\mathstrut 55449237500q^{95} \) \(\mathstrut -\mathstrut 11322017109q^{96} \) \(\mathstrut -\mathstrut 207831285882q^{97} \) \(\mathstrut -\mathstrut 288264739625q^{98} \) \(\mathstrut +\mathstrut 31923070380q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3}\mathstrut -\mathstrut \) \(x^{2}\mathstrut -\mathstrut \) \(5450\) \(x\mathstrut -\mathstrut \) \(7248\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \nu - 3633 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(3\) \(\beta_{1}\mathstrut +\mathstrut \) \(3633\)

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
74.9776
−1.33067
−72.6470
−74.9776 243.000 3573.65 3125.00 −18219.6 −63061.5 −114389. 59049.0 −234305.
1.2 1.33067 243.000 −2046.23 3125.00 323.352 39478.9 −5448.05 59049.0 4158.33
1.3 72.6470 243.000 3229.58 3125.00 17653.2 8974.61 85838.4 59049.0 227022.
\(n\): 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
\(3\) \(-1\)
\(5\) \(-1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2}^{3} \) \(\mathstrut +\mathstrut T_{2}^{2} \) \(\mathstrut -\mathstrut 5450 T_{2} \) \(\mathstrut +\mathstrut 7248 \) acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(15))\).