# 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

# Related objects

## 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})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 22q^{2} + 486q^{3} - 636q^{4} - 6250q^{5} - 5346q^{6} - 10864q^{7} - 18744q^{8} + 118098q^{9} + O(q^{10})$$ $$2q - 22q^{2} + 486q^{3} - 636q^{4} - 6250q^{5} - 5346q^{6} - 10864q^{7} - 18744q^{8} + 118098q^{9} + 68750q^{10} - 361792q^{11} - 154548q^{12} - 2133732q^{13} - 3986664q^{14} - 1518750q^{15} - 5326320q^{16} - 7804588q^{17} - 1299078q^{18} - 15562224q^{19} + 1987500q^{20} - 2639952q^{21} + 37395424q^{22} - 37450248q^{23} - 4554792q^{24} + 19531250q^{25} + 17305364q^{26} + 28697814q^{27} + 93790448q^{28} - 70320668q^{29} + 16706250q^{30} + 298584872q^{31} + 286993696q^{32} - 87915456q^{33} + 303194188q^{34} + 33950000q^{35} - 37555164q^{36} + 236000956q^{37} - 499330888q^{38} - 518496876q^{39} + 58575000q^{40} - 464942588q^{41} - 968759352q^{42} - 242208600q^{43} - 620095808q^{44} - 369056250q^{45} + 2638898832q^{46} - 4375796920q^{47} - 1294295760q^{48} + 1343830130q^{49} - 214843750q^{50} - 1896514884q^{51} + 814171912q^{52} - 2189541388q^{53} - 315675954q^{54} + 1130600000q^{55} + 8823318240q^{56} - 3781620432q^{57} + 10716478004q^{58} - 5480385856q^{59} + 482962500q^{60} + 14557903980q^{61} + 295874592q^{62} - 641508336q^{63} + 11089288256q^{64} + 6667912500q^{65} + 9087088032q^{66} - 15918388888q^{67} - 2299702856q^{68} - 9100410264q^{69} + 12458325000q^{70} + 1120561024q^{71} - 1106814456q^{72} - 24521574348q^{73} - 31645012364q^{74} + 4746093750q^{75} + 19700124976q^{76} - 40673194368q^{77} + 4205203452q^{78} - 79243055560q^{79} + 16644750000q^{80} + 6973568802q^{81} - 66736852348q^{82} + 9245226696q^{83} + 22791078864q^{84} + 24389337500q^{85} + 134816793608q^{86} - 17087922324q^{87} - 67584257664q^{88} + 22117321236q^{89} + 4059618750q^{90} + 19457850112q^{91} - 37083635424q^{92} + 72556123896q^{93} - 15603010256q^{94} + 48631950000q^{95} + 69739468128q^{96} - 160363673468q^{97} + 29827277722q^{98} - 21363455808q^{99} + O(q^{100})$$

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 $$T_{2}^{2} + 22 T_{2} - 1488$$ acting on $$S_{12}^{\mathrm{new}}(\Gamma_0(15))$$.