# Properties

 Label 15.10.a.d Level 15 Weight 10 Character orbit 15.a Self dual yes Analytic conductor 7.726 Analytic rank 0 Dimension 2 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$15 = 3 \cdot 5$$ Weight: $$k$$ $$=$$ $$10$$ Character orbit: $$[\chi]$$ $$=$$ 15.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$7.72553754246$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{241})$$ Defining polynomial: $$x^{2} - x - 60$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$3$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(-1 + 3\sqrt{241})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 15 - \beta ) q^{2} -81 q^{3} + ( 255 - 31 \beta ) q^{4} + 625 q^{5} + ( -1215 + 81 \beta ) q^{6} + ( 6944 - 224 \beta ) q^{7} + ( 12947 - 239 \beta ) q^{8} + 6561 q^{9} +O(q^{10})$$ $$q + ( 15 - \beta ) q^{2} -81 q^{3} + ( 255 - 31 \beta ) q^{4} + 625 q^{5} + ( -1215 + 81 \beta ) q^{6} + ( 6944 - 224 \beta ) q^{7} + ( 12947 - 239 \beta ) q^{8} + 6561 q^{9} + ( 9375 - 625 \beta ) q^{10} + ( -9572 + 2368 \beta ) q^{11} + ( -20655 + 2511 \beta ) q^{12} + ( 14814 + 5344 \beta ) q^{13} + ( 225568 - 10528 \beta ) q^{14} -50625 q^{15} + ( 193183 - 899 \beta ) q^{16} + ( -82238 - 7520 \beta ) q^{17} + ( 98415 - 6561 \beta ) q^{18} + ( -45084 + 5728 \beta ) q^{19} + ( 159375 - 19375 \beta ) q^{20} + ( -562464 + 18144 \beta ) q^{21} + ( -1427036 + 47460 \beta ) q^{22} + ( -380768 - 26272 \beta ) q^{23} + ( -1048707 + 19359 \beta ) q^{24} + 390625 q^{25} + ( -2674238 + 70690 \beta ) q^{26} -531441 q^{27} + ( 5534368 - 279328 \beta ) q^{28} + ( -1254818 + 168576 \beta ) q^{29} + ( -759375 + 50625 \beta ) q^{30} + ( 5467584 + 152736 \beta ) q^{31} + ( -3243861 - 85199 \beta ) q^{32} + ( 775332 - 191808 \beta ) q^{33} + ( 2842270 - 38082 \beta ) q^{34} + ( 4340000 - 140000 \beta ) q^{35} + ( 1673055 - 203391 \beta ) q^{36} + ( 11083414 + 198496 \beta ) q^{37} + ( -3780836 + 136732 \beta ) q^{38} + ( -1199934 - 432864 \beta ) q^{39} + ( 8091875 - 149375 \beta ) q^{40} + ( 13276234 + 492096 \beta ) q^{41} + ( -18271008 + 852768 \beta ) q^{42} + ( -3244412 + 702336 \beta ) q^{43} + ( -42227996 + 973980 \beta ) q^{44} + 4100625 q^{45} + ( 8527904 - 39584 \beta ) q^{46} + ( -16775384 - 1970528 \beta ) q^{47} + ( -15647823 + 72819 \beta ) q^{48} + ( 35060921 - 3161088 \beta ) q^{49} + ( 5859375 - 390625 \beta ) q^{50} + ( 6661278 + 609120 \beta ) q^{51} + ( -86012318 + 1069150 \beta ) q^{52} + ( 1654422 + 177728 \beta ) q^{53} + ( -7971615 + 531441 \beta ) q^{54} + ( -5982500 + 1480000 \beta ) q^{55} + ( 118920480 - 4613280 \beta ) q^{56} + ( 3651804 - 463968 \beta ) q^{57} + ( -110190462 + 3952034 \beta ) q^{58} + ( -15573156 + 4348352 \beta ) q^{59} + ( -12909375 + 1569375 \beta ) q^{60} + ( 170273566 - 950208 \beta ) q^{61} + ( -769152 - 3023808 \beta ) q^{62} + ( 45559584 - 1469664 \beta ) q^{63} + ( -101389753 + 2340965 \beta ) q^{64} + ( 9258750 + 3340000 \beta ) q^{65} + ( 115589916 - 3844260 \beta ) q^{66} + ( -144611188 - 1026560 \beta ) q^{67} + ( 105380350 + 398658 \beta ) q^{68} + ( 30842208 + 2128032 \beta ) q^{69} + ( 140980000 - 6580000 \beta ) q^{70} + ( 107415672 + 4545280 \beta ) q^{71} + ( 84945267 - 1568079 \beta ) q^{72} + ( -116915638 - 1168192 \beta ) q^{73} + ( 58666378 - 7907478 \beta ) q^{74} -31640625 q^{75} + ( -107738276 + 3035812 \beta ) q^{76} + ( -353962112 + 19117952 \beta ) q^{77} + ( 216613278 - 5725890 \beta ) q^{78} + ( -21902080 - 19049120 \beta ) q^{79} + ( 120739375 - 561875 \beta ) q^{80} + 43046721 q^{81} + ( -67572522 - 5402698 \beta ) q^{82} + ( -189671220 - 7260288 \beta ) q^{83} + ( -448283808 + 22625568 \beta ) q^{84} + ( -51398750 - 4700000 \beta ) q^{85} + ( -429332292 + 14481788 \beta ) q^{86} + ( 101640258 - 13654656 \beta ) q^{87} + ( -430674668 + 33512156 \beta ) q^{88} + ( -218344134 - 9049152 \beta ) q^{89} + ( 61509375 - 4100625 \beta ) q^{90} + ( -545935936 + 34987456 \beta ) q^{91} + ( 344326304 + 4290016 \beta ) q^{92} + ( -442874304 - 12371616 \beta ) q^{93} + ( 816395416 - 14753064 \beta ) q^{94} + ( -28177500 + 3580000 \beta ) q^{95} + ( 262752741 + 6901119 \beta ) q^{96} + ( 892554882 + 13450880 \beta ) q^{97} + ( 2239223511 - 85638329 \beta ) q^{98} + ( -62801892 + 15536448 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 31q^{2} - 162q^{3} + 541q^{4} + 1250q^{5} - 2511q^{6} + 14112q^{7} + 26133q^{8} + 13122q^{9} + O(q^{10})$$ $$2q + 31q^{2} - 162q^{3} + 541q^{4} + 1250q^{5} - 2511q^{6} + 14112q^{7} + 26133q^{8} + 13122q^{9} + 19375q^{10} - 21512q^{11} - 43821q^{12} + 24284q^{13} + 461664q^{14} - 101250q^{15} + 387265q^{16} - 156956q^{17} + 203391q^{18} - 95896q^{19} + 338125q^{20} - 1143072q^{21} - 2901532q^{22} - 735264q^{23} - 2116773q^{24} + 781250q^{25} - 5419166q^{26} - 1062882q^{27} + 11348064q^{28} - 2678212q^{29} - 1569375q^{30} + 10782432q^{31} - 6402523q^{32} + 1742472q^{33} + 5722622q^{34} + 8820000q^{35} + 3549501q^{36} + 21968332q^{37} - 7698404q^{38} - 1967004q^{39} + 16333125q^{40} + 26060372q^{41} - 37394784q^{42} - 7191160q^{43} - 85429972q^{44} + 8201250q^{45} + 17095392q^{46} - 31580240q^{47} - 31368465q^{48} + 73282930q^{49} + 12109375q^{50} + 12713436q^{51} - 173093786q^{52} + 3131116q^{53} - 16474671q^{54} - 13445000q^{55} + 242454240q^{56} + 7767576q^{57} - 224332958q^{58} - 35494664q^{59} - 27388125q^{60} + 341497340q^{61} + 1485504q^{62} + 92588832q^{63} - 205120471q^{64} + 15177500q^{65} + 235024092q^{66} - 288195816q^{67} + 210362042q^{68} + 59556384q^{69} + 288540000q^{70} + 210286064q^{71} + 171458613q^{72} - 232663084q^{73} + 125240234q^{74} - 63281250q^{75} - 218512364q^{76} - 727042176q^{77} + 438952446q^{78} - 24755040q^{79} + 242040625q^{80} + 86093442q^{81} - 129742346q^{82} - 372082152q^{83} - 919193184q^{84} - 98097500q^{85} - 873146372q^{86} + 216935172q^{87} - 894861492q^{88} - 427639116q^{89} + 127119375q^{90} - 1126859328q^{91} + 684362592q^{92} - 873376992q^{93} + 1647543896q^{94} - 59935000q^{95} + 518604363q^{96} + 1771658884q^{97} + 4564085351q^{98} - 141140232q^{99} + O(q^{100})$$

## 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
 8.26209 −7.26209
−7.78626 −81.0000 −451.374 625.000 630.687 1839.88 7501.08 6561.00 −4866.41
1.2 38.7863 −81.0000 992.374 625.000 −3141.69 12272.1 18631.9 6561.00 24241.4
 $$n$$: 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.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 15.10.a.d 2
3.b odd 2 1 45.10.a.d 2
4.b odd 2 1 240.10.a.r 2
5.b even 2 1 75.10.a.f 2
5.c odd 4 2 75.10.b.f 4
15.d odd 2 1 225.10.a.k 2
15.e even 4 2 225.10.b.i 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.10.a.d 2 1.a even 1 1 trivial
45.10.a.d 2 3.b odd 2 1
75.10.a.f 2 5.b even 2 1
75.10.b.f 4 5.c odd 4 2
225.10.a.k 2 15.d odd 2 1
225.10.b.i 4 15.e even 4 2
240.10.a.r 2 4.b odd 2 1

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$1$$
$$5$$ $$-1$$

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} - 31 T_{2} - 302$$ acting on $$S_{10}^{\mathrm{new}}(\Gamma_0(15))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 31 T + 722 T^{2} - 15872 T^{3} + 262144 T^{4}$$
$3$ $$( 1 + 81 T )^{2}$$
$5$ $$( 1 - 625 T )^{2}$$
$7$ $$1 - 14112 T + 103286414 T^{2} - 569470101984 T^{3} + 1628413597910449 T^{4}$$
$11$ $$1 + 21512 T + 1790961254 T^{2} + 50724170728792 T^{3} + 5559917313492231481 T^{4}$$
$13$ $$1 - 24284 T + 5870669214 T^{2} - 257519662773932 T^{3} +$$$$11\!\cdots\!29$$$$T^{4}$$
$17$ $$1 + 156956 T + 212670095078 T^{2} + 18613078743463132 T^{3} +$$$$14\!\cdots\!09$$$$T^{4}$$
$19$ $$1 + 95896 T + 629883192438 T^{2} + 30944459466214984 T^{3} +$$$$10\!\cdots\!41$$$$T^{4}$$
$23$ $$1 + 735264 T + 3363187908526 T^{2} + 1324322710477931232 T^{3} +$$$$32\!\cdots\!69$$$$T^{4}$$
$29$ $$1 + 2678212 T + 15397908029438 T^{2} + 38853212438324066228 T^{3} +$$$$21\!\cdots\!61$$$$T^{4}$$
$31$ $$1 - 10782432 T + 69294691361342 T^{2} -$$$$28\!\cdots\!72$$$$T^{3} +$$$$69\!\cdots\!41$$$$T^{4}$$
$37$ $$1 - 21968332 T + 359210373327534 T^{2} -$$$$28\!\cdots\!64$$$$T^{3} +$$$$16\!\cdots\!29$$$$T^{4}$$
$41$ $$1 - 26060372 T + 693239183881142 T^{2} -$$$$85\!\cdots\!92$$$$T^{3} +$$$$10\!\cdots\!21$$$$T^{4}$$
$43$ $$1 + 7191160 T + 750634586008230 T^{2} +$$$$36\!\cdots\!80$$$$T^{3} +$$$$25\!\cdots\!49$$$$T^{4}$$
$47$ $$1 + 31580240 T + 382042606129310 T^{2} +$$$$35\!\cdots\!80$$$$T^{3} +$$$$12\!\cdots\!89$$$$T^{4}$$
$53$ $$1 - 3131116 T + 6584849973489806 T^{2} -$$$$10\!\cdots\!28$$$$T^{3} +$$$$10\!\cdots\!89$$$$T^{4}$$
$59$ $$1 + 35494664 T + 7388006896329158 T^{2} +$$$$30\!\cdots\!96$$$$T^{3} +$$$$75\!\cdots\!21$$$$T^{4}$$
$61$ $$1 - 341497340 T + 52053805546777278 T^{2} -$$$$39\!\cdots\!40$$$$T^{3} +$$$$13\!\cdots\!81$$$$T^{4}$$
$67$ $$1 + 288195816 T + 74605839041196758 T^{2} +$$$$78\!\cdots\!52$$$$T^{3} +$$$$74\!\cdots\!09$$$$T^{4}$$
$71$ $$1 - 210286064 T + 91549406631588686 T^{2} -$$$$96\!\cdots\!84$$$$T^{3} +$$$$21\!\cdots\!61$$$$T^{4}$$
$73$ $$1 + 232663084 T + 130536207391012086 T^{2} +$$$$13\!\cdots\!92$$$$T^{3} +$$$$34\!\cdots\!69$$$$T^{4}$$
$79$ $$1 + 24755040 T + 43090694479668638 T^{2} +$$$$29\!\cdots\!60$$$$T^{3} +$$$$14\!\cdots\!61$$$$T^{4}$$
$83$ $$1 + 372082152 T + 379908828789982198 T^{2} +$$$$69\!\cdots\!56$$$$T^{3} +$$$$34\!\cdots\!09$$$$T^{4}$$
$89$ $$1 + 427639116 T + 702028302670151638 T^{2} +$$$$14\!\cdots\!44$$$$T^{3} +$$$$12\!\cdots\!81$$$$T^{4}$$
$97$ $$1 - 1771658884 T + 2207048700436243398 T^{2} -$$$$13\!\cdots\!28$$$$T^{3} +$$$$57\!\cdots\!89$$$$T^{4}$$