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

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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})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - q^{2} + 729q^{3} + 4757q^{4} + 9375q^{5} - 243q^{6} - 14608q^{7} - 33999q^{8} + 177147q^{9} + O(q^{10}) \) \( 3q - q^{2} + 729q^{3} + 4757q^{4} + 9375q^{5} - 243q^{6} - 14608q^{7} - 33999q^{8} + 177147q^{9} - 3125q^{10} + 540620q^{11} + 1155951q^{12} + 840970q^{13} + 5432712q^{14} + 2278125q^{15} + 5062961q^{16} + 15165038q^{17} - 59049q^{18} + 17743756q^{19} + 14865625q^{20} - 3549744q^{21} + 11176076q^{22} - 28140816q^{23} - 8261757q^{24} + 29296875q^{25} - 52021894q^{26} + 43046721q^{27} - 277157944q^{28} - 67382798q^{29} - 759375q^{30} - 206919496q^{31} - 46592663q^{32} + 131370660q^{33} - 1230469666q^{34} - 45650000q^{35} + 280896093q^{36} - 318337278q^{37} + 653190692q^{38} + 204355710q^{39} - 106246875q^{40} + 2110085854q^{41} + 1320149016q^{42} + 418259692q^{43} + 2558131108q^{44} + 553584375q^{45} - 137169096q^{46} - 1599668584q^{47} + 1230299523q^{48} - 316107077q^{49} - 9765625q^{50} + 3685104234q^{51} - 10897289202q^{52} + 4489142234q^{53} - 14348907q^{54} + 1689437500q^{55} + 7768845960q^{56} + 4311732708q^{57} - 24168830726q^{58} + 11102167484q^{59} + 3612346875q^{60} - 3568120958q^{61} - 35509109136q^{62} - 862587792q^{63} - 35608208271q^{64} + 2628031250q^{65} + 2715786468q^{66} + 2229942788q^{67} - 1367872838q^{68} - 6838218288q^{69} + 16977225000q^{70} + 49842766696q^{71} - 2007606951q^{72} + 40752219934q^{73} - 37519971278q^{74} + 7119140625q^{75} + 115970329116q^{76} - 17819224896q^{77} - 12641320242q^{78} + 113159960q^{79} + 15821753125q^{80} + 10460353203q^{81} - 30171431066q^{82} + 6259660308q^{83} - 67349380392q^{84} + 47390743750q^{85} + 114296127740q^{86} - 16374019914q^{87} + 7548672276q^{88} - 59972401554q^{89} - 184528125q^{90} + 118873361824q^{91} - 221705928648q^{92} - 50281437528q^{93} + 92816682800q^{94} + 55449237500q^{95} - 11322017109q^{96} - 207831285882q^{97} - 288264739625q^{98} + 31923070380q^{99} + O(q^{100}) \)

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

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

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 \( T_{2}^{3} + T_{2}^{2} - 5450 T_{2} + 7248 \) acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(15))\).