Properties

Label 15.12.a.b
Level 15
Weight 12
Character orbit 15.a
Self dual Yes
Analytic conductor 11.525
Analytic rank 1
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{1609}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{1609}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + ( -11 - \beta ) q^{2} \) \( + 243 q^{3} \) \( + ( -318 + 22 \beta ) q^{4} \) \( -3125 q^{5} \) \( + ( -2673 - 243 \beta ) q^{6} \) \( + ( -5432 + 1276 \beta ) q^{7} \) \( + ( -9372 + 2124 \beta ) q^{8} \) \( + 59049 q^{9} \) \(+O(q^{10})\) \( q\) \( + ( -11 - \beta ) q^{2} \) \( + 243 q^{3} \) \( + ( -318 + 22 \beta ) q^{4} \) \( -3125 q^{5} \) \( + ( -2673 - 243 \beta ) q^{6} \) \( + ( -5432 + 1276 \beta ) q^{7} \) \( + ( -9372 + 2124 \beta ) q^{8} \) \( + 59049 q^{9} \) \( + ( 34375 + 3125 \beta ) q^{10} \) \( + ( -180896 - 10384 \beta ) q^{11} \) \( + ( -77274 + 5346 \beta ) q^{12} \) \( + ( -1066866 + 1916 \beta ) q^{13} \) \( + ( -1993332 - 8604 \beta ) q^{14} \) \( -759375 q^{15} \) \( + ( -2663160 - 59048 \beta ) q^{16} \) \( + ( -3902294 - 67540 \beta ) q^{17} \) \( + ( -649539 - 59049 \beta ) q^{18} \) \( + ( -7781112 + 208364 \beta ) q^{19} \) \( + ( 993750 - 68750 \beta ) q^{20} \) \( + ( -1319976 + 310068 \beta ) q^{21} \) \( + ( 18697712 + 295120 \beta ) q^{22} \) \( + ( -18725124 - 692028 \beta ) q^{23} \) \( + ( -2277396 + 516132 \beta ) q^{24} \) \( + 9765625 q^{25} \) \( + ( 8652682 + 1045790 \beta ) q^{26} \) \( + 14348907 q^{27} \) \( + ( 46895224 - 525272 \beta ) q^{28} \) \( + ( -35160334 - 3089792 \beta ) q^{29} \) \( + ( 8353125 + 759375 \beta ) q^{30} \) \( + ( 149292436 - 1112588 \beta ) q^{31} \) \( + ( 143496848 - 1037264 \beta ) q^{32} \) \( + ( -43957728 - 2523312 \beta ) q^{33} \) \( + ( 151597094 + 4645234 \beta ) q^{34} \) \( + ( 16975000 - 3987500 \beta ) q^{35} \) \( + ( -18777582 + 1299078 \beta ) q^{36} \) \( + ( 118000478 + 9027036 \beta ) q^{37} \) \( + ( -249665444 + 5489108 \beta ) q^{38} \) \( + ( -259248438 + 465588 \beta ) q^{39} \) \( + ( 29287500 - 6637500 \beta ) q^{40} \) \( + ( -232471294 + 22327912 \beta ) q^{41} \) \( + ( -484379676 - 2090772 \beta ) q^{42} \) \( + ( -121104300 - 41066656 \beta ) q^{43} \) \( + ( -310047904 - 677600 \beta ) q^{44} \) \( -184528125 q^{45} \) \( + ( 1319449416 + 26337432 \beta ) q^{46} \) \( + ( -2187898460 + 19806332 \beta ) q^{47} \) \( + ( -647147880 - 14348664 \beta ) q^{48} \) \( + ( 671915065 - 13862464 \beta ) q^{49} \) \( + ( -107421875 - 9765625 \beta ) q^{50} \) \( + ( -948257442 - 16412220 \beta ) q^{51} \) \( + ( 407085956 - 24080340 \beta ) q^{52} \) \( + ( -1094770694 - 29631928 \beta ) q^{53} \) \( + ( -157837977 - 14348907 \beta ) q^{54} \) \( + ( 565300000 + 32450000 \beta ) q^{55} \) \( + ( 4411659120 - 23496240 \beta ) q^{56} \) \( + ( -1890810216 + 50632452 \beta ) q^{57} \) \( + ( 5358239002 + 69148046 \beta ) q^{58} \) \( + ( -2740192928 + 146298896 \beta ) q^{59} \) \( + ( 241481250 - 16706250 \beta ) q^{60} \) \( + ( 7278951990 - 14026936 \beta ) q^{61} \) \( + ( 147937296 - 137053968 \beta ) q^{62} \) \( + ( -320754168 + 75346524 \beta ) q^{63} \) \( + ( 5544644128 - 11156640 \beta ) q^{64} \) \( + ( 3333956250 - 5987500 \beta ) q^{65} \) \( + ( 4543544016 + 71714160 \beta ) q^{66} \) \( + ( -7959194444 + 50570080 \beta ) q^{67} \) \( + ( -1149851428 - 64372748 \beta ) q^{68} \) \( + ( -4550205132 - 168162804 \beta ) q^{69} \) \( + ( 6229162500 + 26887500 \beta ) q^{70} \) \( + ( 560280512 - 295730240 \beta ) q^{71} \) \( + ( -553407228 + 125420076 \beta ) q^{72} \) \( + ( -12260787174 + 71528792 \beta ) q^{73} \) \( + ( -15822506182 - 217297874 \beta ) q^{74} \) \( + 2373046875 q^{75} \) \( + ( 9850062488 - 237444216 \beta ) q^{76} \) \( + ( -20336597184 - 174417408 \beta ) q^{77} \) \( + ( 2102601726 + 254126970 \beta ) q^{78} \) \( + ( -39621527780 + 90478060 \beta ) q^{79} \) \( + ( 8322375000 + 184525000 \beta ) q^{80} \) \( + 3486784401 q^{81} \) \( + ( -33368426174 - 13135738 \beta ) q^{82} \) \( + ( 4622613348 + 1383234048 \beta ) q^{83} \) \( + ( 11395539432 - 127641096 \beta ) q^{84} \) \( + ( 12194668750 + 211062500 \beta ) q^{85} \) \( + ( 67408396804 + 572837516 \beta ) q^{86} \) \( + ( -8543961162 - 750819456 \beta ) q^{87} \) \( + ( -33792128832 - 286904256 \beta ) q^{88} \) \( + ( 11058660618 - 312314856 \beta ) q^{89} \) \( + ( 2029809375 + 184528125 \beta ) q^{90} \) \( + ( 9728925056 - 1371728728 \beta ) q^{91} \) \( + ( -18541817712 - 191887824 \beta ) q^{92} \) \( + ( 36278061948 - 270358884 \beta ) q^{93} \) \( + ( -7801505128 + 1970028808 \beta ) q^{94} \) \( + ( 24315975000 - 651137500 \beta ) q^{95} \) \( + ( 34869734064 - 252055152 \beta ) q^{96} \) \( + ( -80181836734 + 127385280 \beta ) q^{97} \) \( + ( 14913638861 - 519427961 \beta ) q^{98} \) \( + ( -10681727904 - 613164816 \beta ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut -\mathstrut 22q^{2} \) \(\mathstrut +\mathstrut 486q^{3} \) \(\mathstrut -\mathstrut 636q^{4} \) \(\mathstrut -\mathstrut 6250q^{5} \) \(\mathstrut -\mathstrut 5346q^{6} \) \(\mathstrut -\mathstrut 10864q^{7} \) \(\mathstrut -\mathstrut 18744q^{8} \) \(\mathstrut +\mathstrut 118098q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut -\mathstrut 22q^{2} \) \(\mathstrut +\mathstrut 486q^{3} \) \(\mathstrut -\mathstrut 636q^{4} \) \(\mathstrut -\mathstrut 6250q^{5} \) \(\mathstrut -\mathstrut 5346q^{6} \) \(\mathstrut -\mathstrut 10864q^{7} \) \(\mathstrut -\mathstrut 18744q^{8} \) \(\mathstrut +\mathstrut 118098q^{9} \) \(\mathstrut +\mathstrut 68750q^{10} \) \(\mathstrut -\mathstrut 361792q^{11} \) \(\mathstrut -\mathstrut 154548q^{12} \) \(\mathstrut -\mathstrut 2133732q^{13} \) \(\mathstrut -\mathstrut 3986664q^{14} \) \(\mathstrut -\mathstrut 1518750q^{15} \) \(\mathstrut -\mathstrut 5326320q^{16} \) \(\mathstrut -\mathstrut 7804588q^{17} \) \(\mathstrut -\mathstrut 1299078q^{18} \) \(\mathstrut -\mathstrut 15562224q^{19} \) \(\mathstrut +\mathstrut 1987500q^{20} \) \(\mathstrut -\mathstrut 2639952q^{21} \) \(\mathstrut +\mathstrut 37395424q^{22} \) \(\mathstrut -\mathstrut 37450248q^{23} \) \(\mathstrut -\mathstrut 4554792q^{24} \) \(\mathstrut +\mathstrut 19531250q^{25} \) \(\mathstrut +\mathstrut 17305364q^{26} \) \(\mathstrut +\mathstrut 28697814q^{27} \) \(\mathstrut +\mathstrut 93790448q^{28} \) \(\mathstrut -\mathstrut 70320668q^{29} \) \(\mathstrut +\mathstrut 16706250q^{30} \) \(\mathstrut +\mathstrut 298584872q^{31} \) \(\mathstrut +\mathstrut 286993696q^{32} \) \(\mathstrut -\mathstrut 87915456q^{33} \) \(\mathstrut +\mathstrut 303194188q^{34} \) \(\mathstrut +\mathstrut 33950000q^{35} \) \(\mathstrut -\mathstrut 37555164q^{36} \) \(\mathstrut +\mathstrut 236000956q^{37} \) \(\mathstrut -\mathstrut 499330888q^{38} \) \(\mathstrut -\mathstrut 518496876q^{39} \) \(\mathstrut +\mathstrut 58575000q^{40} \) \(\mathstrut -\mathstrut 464942588q^{41} \) \(\mathstrut -\mathstrut 968759352q^{42} \) \(\mathstrut -\mathstrut 242208600q^{43} \) \(\mathstrut -\mathstrut 620095808q^{44} \) \(\mathstrut -\mathstrut 369056250q^{45} \) \(\mathstrut +\mathstrut 2638898832q^{46} \) \(\mathstrut -\mathstrut 4375796920q^{47} \) \(\mathstrut -\mathstrut 1294295760q^{48} \) \(\mathstrut +\mathstrut 1343830130q^{49} \) \(\mathstrut -\mathstrut 214843750q^{50} \) \(\mathstrut -\mathstrut 1896514884q^{51} \) \(\mathstrut +\mathstrut 814171912q^{52} \) \(\mathstrut -\mathstrut 2189541388q^{53} \) \(\mathstrut -\mathstrut 315675954q^{54} \) \(\mathstrut +\mathstrut 1130600000q^{55} \) \(\mathstrut +\mathstrut 8823318240q^{56} \) \(\mathstrut -\mathstrut 3781620432q^{57} \) \(\mathstrut +\mathstrut 10716478004q^{58} \) \(\mathstrut -\mathstrut 5480385856q^{59} \) \(\mathstrut +\mathstrut 482962500q^{60} \) \(\mathstrut +\mathstrut 14557903980q^{61} \) \(\mathstrut +\mathstrut 295874592q^{62} \) \(\mathstrut -\mathstrut 641508336q^{63} \) \(\mathstrut +\mathstrut 11089288256q^{64} \) \(\mathstrut +\mathstrut 6667912500q^{65} \) \(\mathstrut +\mathstrut 9087088032q^{66} \) \(\mathstrut -\mathstrut 15918388888q^{67} \) \(\mathstrut -\mathstrut 2299702856q^{68} \) \(\mathstrut -\mathstrut 9100410264q^{69} \) \(\mathstrut +\mathstrut 12458325000q^{70} \) \(\mathstrut +\mathstrut 1120561024q^{71} \) \(\mathstrut -\mathstrut 1106814456q^{72} \) \(\mathstrut -\mathstrut 24521574348q^{73} \) \(\mathstrut -\mathstrut 31645012364q^{74} \) \(\mathstrut +\mathstrut 4746093750q^{75} \) \(\mathstrut +\mathstrut 19700124976q^{76} \) \(\mathstrut -\mathstrut 40673194368q^{77} \) \(\mathstrut +\mathstrut 4205203452q^{78} \) \(\mathstrut -\mathstrut 79243055560q^{79} \) \(\mathstrut +\mathstrut 16644750000q^{80} \) \(\mathstrut +\mathstrut 6973568802q^{81} \) \(\mathstrut -\mathstrut 66736852348q^{82} \) \(\mathstrut +\mathstrut 9245226696q^{83} \) \(\mathstrut +\mathstrut 22791078864q^{84} \) \(\mathstrut +\mathstrut 24389337500q^{85} \) \(\mathstrut +\mathstrut 134816793608q^{86} \) \(\mathstrut -\mathstrut 17087922324q^{87} \) \(\mathstrut -\mathstrut 67584257664q^{88} \) \(\mathstrut +\mathstrut 22117321236q^{89} \) \(\mathstrut +\mathstrut 4059618750q^{90} \) \(\mathstrut +\mathstrut 19457850112q^{91} \) \(\mathstrut -\mathstrut 37083635424q^{92} \) \(\mathstrut +\mathstrut 72556123896q^{93} \) \(\mathstrut -\mathstrut 15603010256q^{94} \) \(\mathstrut +\mathstrut 48631950000q^{95} \) \(\mathstrut +\mathstrut 69739468128q^{96} \) \(\mathstrut -\mathstrut 160363673468q^{97} \) \(\mathstrut +\mathstrut 29827277722q^{98} \) \(\mathstrut -\mathstrut 21363455808q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
20.5562
−19.5562
−51.1123 243.000 564.472 −3125.00 −12420.3 45751.3 75826.6 59049.0 159726.
1.2 29.1123 243.000 −1200.47 −3125.00 7074.30 −56615.3 −94570.6 59049.0 −90976.1
\(n\): 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 \(T_{2}^{2} \) \(\mathstrut +\mathstrut 22 T_{2} \) \(\mathstrut -\mathstrut 1488 \) acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(15))\).