Properties

Label 15.12.a.c
Level 15
Weight 12
Character orbit 15.a
Self dual Yes
Analytic conductor 11.525
Analytic rank 0
Dimension 2
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: \(2\)
Coefficient field: \(\Q(\sqrt{1801}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{1801})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + ( -6 - \beta ) q^{2} \) \( -243 q^{3} \) \( + ( -1562 + 13 \beta ) q^{4} \) \( -3125 q^{5} \) \( + ( 1458 + 243 \beta ) q^{6} \) \( + ( 5684 - 3584 \beta ) q^{7} \) \( + ( 15810 + 3519 \beta ) q^{8} \) \( + 59049 q^{9} \) \(+O(q^{10})\) \( q\) \( + ( -6 - \beta ) q^{2} \) \( -243 q^{3} \) \( + ( -1562 + 13 \beta ) q^{4} \) \( -3125 q^{5} \) \( + ( 1458 + 243 \beta ) q^{6} \) \( + ( 5684 - 3584 \beta ) q^{7} \) \( + ( 15810 + 3519 \beta ) q^{8} \) \( + 59049 q^{9} \) \( + ( 18750 + 3125 \beta ) q^{10} \) \( + ( 163272 - 30976 \beta ) q^{11} \) \( + ( 379566 - 3159 \beta ) q^{12} \) \( + ( 292778 + 71936 \beta ) q^{13} \) \( + ( 1578696 + 19404 \beta ) q^{14} \) \( + 759375 q^{15} \) \( + ( 1520566 - 67067 \beta ) q^{16} \) \( + ( 4250934 + 78080 \beta ) q^{17} \) \( + ( -354294 - 59049 \beta ) q^{18} \) \( + ( 8879780 - 131584 \beta ) q^{19} \) \( + ( 4881250 - 40625 \beta ) q^{20} \) \( + ( -1381212 + 870912 \beta ) q^{21} \) \( + ( 12959568 + 53560 \beta ) q^{22} \) \( + ( -14134872 - 1571328 \beta ) q^{23} \) \( + ( -3841830 - 855117 \beta ) q^{24} \) \( + 9765625 q^{25} \) \( + ( -34127868 - 796330 \beta ) q^{26} \) \( -14348907 q^{27} \) \( + ( -29844808 + 5625508 \beta ) q^{28} \) \( + ( -103051950 + 4221952 \beta ) q^{29} \) \( + ( -4556250 - 759375 \beta ) q^{30} \) \( + ( -32308408 - 6440192 \beta ) q^{31} \) \( + ( -11322126 - 8258009 \beta ) q^{32} \) \( + ( -39675096 + 7527168 \beta ) q^{33} \) \( + ( -60641604 - 4797494 \beta ) q^{34} \) \( + ( -17762500 + 11200000 \beta ) q^{35} \) \( + ( -92234538 + 767637 \beta ) q^{36} \) \( + ( 346513754 + 12830976 \beta ) q^{37} \) \( + ( 5934120 - 7958692 \beta ) q^{38} \) \( + ( -71145054 - 17480448 \beta ) q^{39} \) \( + ( -49406250 - 10996875 \beta ) q^{40} \) \( + ( -154634358 - 18386432 \beta ) q^{41} \) \( + ( -383623128 - 4715172 \beta ) q^{42} \) \( + ( 1598048348 - 3544576 \beta ) q^{43} \) \( + ( -436240464 + 50104360 \beta ) q^{44} \) \( -184528125 q^{45} \) \( + ( 791906832 + 25134168 \beta ) q^{46} \) \( + ( 1022573064 + 7918592 \beta ) q^{47} \) \( + ( -369497538 + 16297281 \beta ) q^{48} \) \( + ( 3835256313 - 27897856 \beta ) q^{49} \) \( + ( -58593750 - 9765625 \beta ) q^{50} \) \( + ( -1032976962 - 18973440 \beta ) q^{51} \) \( + ( -36493636 - 107622750 \beta ) q^{52} \) \( + ( -1208121822 + 111944192 \beta ) q^{53} \) \( + ( 86093442 + 14348907 \beta ) q^{54} \) \( + ( -510225000 + 96800000 \beta ) q^{55} \) \( + ( -5585579160 - 49273140 \beta ) q^{56} \) \( + ( -2157786540 + 31974912 \beta ) q^{57} \) \( + ( -1281566700 + 73498286 \beta ) q^{58} \) \( + ( -619281480 - 240207616 \beta ) q^{59} \) \( + ( -1186143750 + 9871875 \beta ) q^{60} \) \( + ( -4074042658 - 116806144 \beta ) q^{61} \) \( + ( 3091936848 + 77389752 \beta ) q^{62} \) \( + ( 335634516 - 211631616 \beta ) q^{63} \) \( + ( 669917638 + 206481405 \beta ) q^{64} \) \( + ( -914931250 - 224800000 \beta ) q^{65} \) \( + ( -3149175024 - 13015080 \beta ) q^{66} \) \( + ( 11999581244 + 213015040 \beta ) q^{67} \) \( + ( -6183190908 - 65683778 \beta ) q^{68} \) \( + ( 3434773896 + 381832704 \beta ) q^{69} \) \( + ( -4933425000 - 60637500 \beta ) q^{70} \) \( + ( -10521135648 + 823383040 \beta ) q^{71} \) \( + ( 933564690 + 207793431 \beta ) q^{72} \) \( + ( 13394501138 - 909419008 \beta ) q^{73} \) \( + ( -7853021724 - 436330586 \beta ) q^{74} \) \( -2373046875 q^{75} \) \( + ( -14639982760 + 319260756 \beta ) q^{76} \) \( + ( 50886130848 - 650216448 \beta ) q^{77} \) \( + ( 8293071924 + 193508190 \beta ) q^{78} \) \( + ( 10842173240 + 640648960 \beta ) q^{79} \) \( + ( -4751768750 + 209584375 \beta ) q^{80} \) \( + 3486784401 q^{81} \) \( + ( 9201700548 + 283339382 \beta ) q^{82} \) \( + ( 23726037468 + 562434048 \beta ) q^{83} \) \( + ( 7252288344 - 1366998444 \beta ) q^{84} \) \( + ( -13284168750 - 244000000 \beta ) q^{85} \) \( + ( -7993230888 - 1573236316 \beta ) q^{86} \) \( + ( 25041623850 - 1025934336 \beta ) q^{87} \) \( + ( -46470714480 - 24180936 \beta ) q^{88} \) \( + ( 38776245570 + 1657191936 \beta ) q^{89} \) \( + ( 1107168750 + 184528125 \beta ) q^{90} \) \( + ( -114354230648 - 898250752 \beta ) q^{91} \) \( + ( 12886401264 + 2250233736 \beta ) q^{92} \) \( + ( 7850943144 + 1564966656 \beta ) q^{93} \) \( + ( -9698804784 - 1078003208 \beta ) q^{94} \) \( + ( -27749312500 + 411200000 \beta ) q^{95} \) \( + ( 2751276618 + 2006696187 \beta ) q^{96} \) \( + ( -18582564766 + 89902080 \beta ) q^{97} \) \( + ( -10457502678 - 3639971321 \beta ) q^{98} \) \( + ( 9641048328 - 1829101824 \beta ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut -\mathstrut 13q^{2} \) \(\mathstrut -\mathstrut 486q^{3} \) \(\mathstrut -\mathstrut 3111q^{4} \) \(\mathstrut -\mathstrut 6250q^{5} \) \(\mathstrut +\mathstrut 3159q^{6} \) \(\mathstrut +\mathstrut 7784q^{7} \) \(\mathstrut +\mathstrut 35139q^{8} \) \(\mathstrut +\mathstrut 118098q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut -\mathstrut 13q^{2} \) \(\mathstrut -\mathstrut 486q^{3} \) \(\mathstrut -\mathstrut 3111q^{4} \) \(\mathstrut -\mathstrut 6250q^{5} \) \(\mathstrut +\mathstrut 3159q^{6} \) \(\mathstrut +\mathstrut 7784q^{7} \) \(\mathstrut +\mathstrut 35139q^{8} \) \(\mathstrut +\mathstrut 118098q^{9} \) \(\mathstrut +\mathstrut 40625q^{10} \) \(\mathstrut +\mathstrut 295568q^{11} \) \(\mathstrut +\mathstrut 755973q^{12} \) \(\mathstrut +\mathstrut 657492q^{13} \) \(\mathstrut +\mathstrut 3176796q^{14} \) \(\mathstrut +\mathstrut 1518750q^{15} \) \(\mathstrut +\mathstrut 2974065q^{16} \) \(\mathstrut +\mathstrut 8579948q^{17} \) \(\mathstrut -\mathstrut 767637q^{18} \) \(\mathstrut +\mathstrut 17627976q^{19} \) \(\mathstrut +\mathstrut 9721875q^{20} \) \(\mathstrut -\mathstrut 1891512q^{21} \) \(\mathstrut +\mathstrut 25972696q^{22} \) \(\mathstrut -\mathstrut 29841072q^{23} \) \(\mathstrut -\mathstrut 8538777q^{24} \) \(\mathstrut +\mathstrut 19531250q^{25} \) \(\mathstrut -\mathstrut 69052066q^{26} \) \(\mathstrut -\mathstrut 28697814q^{27} \) \(\mathstrut -\mathstrut 54064108q^{28} \) \(\mathstrut -\mathstrut 201881948q^{29} \) \(\mathstrut -\mathstrut 9871875q^{30} \) \(\mathstrut -\mathstrut 71057008q^{31} \) \(\mathstrut -\mathstrut 30902261q^{32} \) \(\mathstrut -\mathstrut 71823024q^{33} \) \(\mathstrut -\mathstrut 126080702q^{34} \) \(\mathstrut -\mathstrut 24325000q^{35} \) \(\mathstrut -\mathstrut 183701439q^{36} \) \(\mathstrut +\mathstrut 705858484q^{37} \) \(\mathstrut +\mathstrut 3909548q^{38} \) \(\mathstrut -\mathstrut 159770556q^{39} \) \(\mathstrut -\mathstrut 109809375q^{40} \) \(\mathstrut -\mathstrut 327655148q^{41} \) \(\mathstrut -\mathstrut 771961428q^{42} \) \(\mathstrut +\mathstrut 3192552120q^{43} \) \(\mathstrut -\mathstrut 822376568q^{44} \) \(\mathstrut -\mathstrut 369056250q^{45} \) \(\mathstrut +\mathstrut 1608947832q^{46} \) \(\mathstrut +\mathstrut 2053064720q^{47} \) \(\mathstrut -\mathstrut 722697795q^{48} \) \(\mathstrut +\mathstrut 7642614770q^{49} \) \(\mathstrut -\mathstrut 126953125q^{50} \) \(\mathstrut -\mathstrut 2084927364q^{51} \) \(\mathstrut -\mathstrut 180610022q^{52} \) \(\mathstrut -\mathstrut 2304299452q^{53} \) \(\mathstrut +\mathstrut 186535791q^{54} \) \(\mathstrut -\mathstrut 923650000q^{55} \) \(\mathstrut -\mathstrut 11220431460q^{56} \) \(\mathstrut -\mathstrut 4283598168q^{57} \) \(\mathstrut -\mathstrut 2489635114q^{58} \) \(\mathstrut -\mathstrut 1478770576q^{59} \) \(\mathstrut -\mathstrut 2362415625q^{60} \) \(\mathstrut -\mathstrut 8264891460q^{61} \) \(\mathstrut +\mathstrut 6261263448q^{62} \) \(\mathstrut +\mathstrut 459637416q^{63} \) \(\mathstrut +\mathstrut 1546316681q^{64} \) \(\mathstrut -\mathstrut 2054662500q^{65} \) \(\mathstrut -\mathstrut 6311365128q^{66} \) \(\mathstrut +\mathstrut 24212177528q^{67} \) \(\mathstrut -\mathstrut 12432065594q^{68} \) \(\mathstrut +\mathstrut 7251380496q^{69} \) \(\mathstrut -\mathstrut 9927487500q^{70} \) \(\mathstrut -\mathstrut 20218888256q^{71} \) \(\mathstrut +\mathstrut 2074922811q^{72} \) \(\mathstrut +\mathstrut 25879583268q^{73} \) \(\mathstrut -\mathstrut 16142374034q^{74} \) \(\mathstrut -\mathstrut 4746093750q^{75} \) \(\mathstrut -\mathstrut 28960704764q^{76} \) \(\mathstrut +\mathstrut 101122045248q^{77} \) \(\mathstrut +\mathstrut 16779652038q^{78} \) \(\mathstrut +\mathstrut 22324995440q^{79} \) \(\mathstrut -\mathstrut 9293953125q^{80} \) \(\mathstrut +\mathstrut 6973568802q^{81} \) \(\mathstrut +\mathstrut 18686740478q^{82} \) \(\mathstrut +\mathstrut 48014508984q^{83} \) \(\mathstrut +\mathstrut 13137578244q^{84} \) \(\mathstrut -\mathstrut 26812337500q^{85} \) \(\mathstrut -\mathstrut 17559698092q^{86} \) \(\mathstrut +\mathstrut 49057313364q^{87} \) \(\mathstrut -\mathstrut 92965609896q^{88} \) \(\mathstrut +\mathstrut 79209683076q^{89} \) \(\mathstrut +\mathstrut 2398865625q^{90} \) \(\mathstrut -\mathstrut 229606712048q^{91} \) \(\mathstrut +\mathstrut 28023036264q^{92} \) \(\mathstrut +\mathstrut 17266852944q^{93} \) \(\mathstrut -\mathstrut 20475612776q^{94} \) \(\mathstrut -\mathstrut 55087425000q^{95} \) \(\mathstrut +\mathstrut 7509249423q^{96} \) \(\mathstrut -\mathstrut 37075227452q^{97} \) \(\mathstrut -\mathstrut 24554976677q^{98} \) \(\mathstrut +\mathstrut 17452994832q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
21.7191
−20.7191
−27.7191 −243.000 −1279.65 −3125.00 6735.74 −72157.2 92239.5 59049.0 86622.2
1.2 14.7191 −243.000 −1831.35 −3125.00 −3576.74 79941.2 −57100.5 59049.0 −45997.2
\(n\): 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 \(T_{2}^{2} \) \(\mathstrut +\mathstrut 13 T_{2} \) \(\mathstrut -\mathstrut 408 \) acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(15))\).