# Properties

 Label 15.10.a.c 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: $$\Q(\sqrt{4729})$$ Defining polynomial: $$x^{2} - x - 1182$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{4729})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 10 - \beta ) q^{2} + 81 q^{3} + ( 770 - 19 \beta ) q^{4} -625 q^{5} + ( 810 - 81 \beta ) q^{6} + ( -5964 + 56 \beta ) q^{7} + ( 25038 - 429 \beta ) q^{8} + 6561 q^{9} +O(q^{10})$$ $$q + ( 10 - \beta ) q^{2} + 81 q^{3} + ( 770 - 19 \beta ) q^{4} -625 q^{5} + ( 810 - 81 \beta ) q^{6} + ( -5964 + 56 \beta ) q^{7} + ( 25038 - 429 \beta ) q^{8} + 6561 q^{9} + ( -6250 + 625 \beta ) q^{10} + ( 16768 + 1952 \beta ) q^{11} + ( 62370 - 1539 \beta ) q^{12} + ( 71146 + 1384 \beta ) q^{13} + ( -125832 + 6468 \beta ) q^{14} -50625 q^{15} + ( 363218 - 19171 \beta ) q^{16} + ( 193678 - 2200 \beta ) q^{17} + ( 65610 - 6561 \beta ) q^{18} + ( -201164 - 968 \beta ) q^{19} + ( -481250 + 11875 \beta ) q^{20} + ( -483084 + 4536 \beta ) q^{21} + ( -2139584 + 800 \beta ) q^{22} + ( 79368 + 64968 \beta ) q^{23} + ( 2028078 - 34749 \beta ) q^{24} + 390625 q^{25} + ( -924428 - 58690 \beta ) q^{26} + 531441 q^{27} + ( -5849928 + 155372 \beta ) q^{28} + ( -31078 - 12416 \beta ) q^{29} + ( -506250 + 50625 \beta ) q^{30} + ( -2475696 - 75736 \beta ) q^{31} + ( 13472846 - 316109 \beta ) q^{32} + ( 1358208 + 158112 \beta ) q^{33} + ( 4537180 - 213478 \beta ) q^{34} + ( 3727500 - 35000 \beta ) q^{35} + ( 5051970 - 124659 \beta ) q^{36} + ( 2599466 + 174696 \beta ) q^{37} + ( -867464 + 192452 \beta ) q^{38} + ( 5762826 + 112104 \beta ) q^{39} + ( -15648750 + 268125 \beta ) q^{40} + ( 7340714 - 470096 \beta ) q^{41} + ( -10192392 + 523908 \beta ) q^{42} + ( 13950652 - 152384 \beta ) q^{43} + ( -30926656 + 1147360 \beta ) q^{44} -4100625 q^{45} + ( -75998496 + 505344 \beta ) q^{46} + ( 48198904 - 431368 \beta ) q^{47} + ( 29420658 - 1552851 \beta ) q^{48} + ( -1077559 - 664832 \beta ) q^{49} + ( 3906250 - 390625 \beta ) q^{50} + ( 15687918 - 178200 \beta ) q^{51} + ( 23700548 - 312390 \beta ) q^{52} + ( -31687862 - 929872 \beta ) q^{53} + ( 5314410 - 531441 \beta ) q^{54} + ( -10480000 - 1220000 \beta ) q^{55} + ( -177723000 + 3936660 \beta ) q^{56} + ( -16294284 - 78408 \beta ) q^{57} + ( 14364932 - 80666 \beta ) q^{58} + ( 93124864 + 1613408 \beta ) q^{59} + ( -38981250 + 961875 \beta ) q^{60} + ( 75911686 + 2256688 \beta ) q^{61} + ( 64762992 + 1794072 \beta ) q^{62} + ( -39129804 + 367416 \beta ) q^{63} + ( 322401682 - 6502275 \beta ) q^{64} + ( -44466250 - 865000 \beta ) q^{65} + ( -173306304 + 64800 \beta ) q^{66} + ( 13074108 + 7444160 \beta ) q^{67} + ( 198539660 - 5332082 \beta ) q^{68} + ( 6428808 + 5262408 \beta ) q^{69} + ( 78645000 - 4042500 \beta ) q^{70} + ( -110604928 - 7061120 \beta ) q^{71} + ( 164274318 - 2814669 \beta ) q^{72} + ( -19801262 + 6480208 \beta ) q^{73} + ( -180496012 - 1027202 \beta ) q^{74} + 31640625 q^{75} + ( -133156936 + 3095148 \beta ) q^{76} + ( 29202432 - 10593408 \beta ) q^{77} + ( -74878668 - 4753890 \beta ) q^{78} + ( -467102400 + 1798040 \beta ) q^{79} + ( -227011250 + 11981875 \beta ) q^{80} + 43046721 q^{81} + ( 629060612 - 11571578 \beta ) q^{82} + ( 105100620 - 3161088 \beta ) q^{83} + ( -473844168 + 12585132 \beta ) q^{84} + ( -121048750 + 1375000 \beta ) q^{85} + ( 319624408 - 15322108 \beta ) q^{86} + ( -2517318 - 1005696 \beta ) q^{87} + ( -569979072 + 40843296 \beta ) q^{88} + ( 107605986 + 9306192 \beta ) q^{89} + ( -41006250 + 4100625 \beta ) q^{90} + ( -332705016 - 4192496 \beta ) q^{91} + ( -1397937984 + 47282976 \beta ) q^{92} + ( -200531376 - 6134616 \beta ) q^{93} + ( 991866016 - 52081216 \beta ) q^{94} + ( 125727500 + 605000 \beta ) q^{95} + ( 1091300526 - 25604829 \beta ) q^{96} + ( 215586818 - 44039040 \beta ) q^{97} + ( 775055834 - 4905929 \beta ) q^{98} + ( 110014848 + 12807072 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 19q^{2} + 162q^{3} + 1521q^{4} - 1250q^{5} + 1539q^{6} - 11872q^{7} + 49647q^{8} + 13122q^{9} + O(q^{10})$$ $$2q + 19q^{2} + 162q^{3} + 1521q^{4} - 1250q^{5} + 1539q^{6} - 11872q^{7} + 49647q^{8} + 13122q^{9} - 11875q^{10} + 35488q^{11} + 123201q^{12} + 143676q^{13} - 245196q^{14} - 101250q^{15} + 707265q^{16} + 385156q^{17} + 124659q^{18} - 403296q^{19} - 950625q^{20} - 961632q^{21} - 4278368q^{22} + 223704q^{23} + 4021407q^{24} + 781250q^{25} - 1907546q^{26} + 1062882q^{27} - 11544484q^{28} - 74572q^{29} - 961875q^{30} - 5027128q^{31} + 26629583q^{32} + 2874528q^{33} + 8860882q^{34} + 7420000q^{35} + 9979281q^{36} + 5373628q^{37} - 1542476q^{38} + 11637756q^{39} - 31029375q^{40} + 14211332q^{41} - 19860876q^{42} + 27748920q^{43} - 60705952q^{44} - 8201250q^{45} - 151491648q^{46} + 95966440q^{47} + 57288465q^{48} - 2819950q^{49} + 7421875q^{50} + 31197636q^{51} + 47088706q^{52} - 64305596q^{53} + 10097379q^{54} - 22180000q^{55} - 351509340q^{56} - 32666976q^{57} + 28649198q^{58} + 187863136q^{59} - 77000625q^{60} + 154080060q^{61} + 131320056q^{62} - 77892192q^{63} + 638301089q^{64} - 89797500q^{65} - 346547808q^{66} + 33592376q^{67} + 391747238q^{68} + 18120024q^{69} + 153247500q^{70} - 228270976q^{71} + 325733967q^{72} - 33122316q^{73} - 362019226q^{74} + 63281250q^{75} - 263218724q^{76} + 47811456q^{77} - 154511226q^{78} - 932406760q^{79} - 442040625q^{80} + 86093442q^{81} + 1246549646q^{82} + 207040152q^{83} - 935103204q^{84} - 240722500q^{85} + 623926708q^{86} - 6040332q^{87} - 1099114848q^{88} + 224518164q^{89} - 77911875q^{90} - 669602528q^{91} - 2748592992q^{92} - 407197368q^{93} + 1931650816q^{94} + 252060000q^{95} + 2156996223q^{96} + 387134596q^{97} + 1545205739q^{98} + 232836768q^{99} + O(q^{100})$$

## 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
 34.8839 −33.8839
−24.8839 81.0000 107.207 −625.000 −2015.59 −4010.50 10072.8 6561.00 15552.4
1.2 43.8839 81.0000 1413.79 −625.000 3554.59 −7861.50 39574.2 6561.00 −27427.4
 $$n$$: 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.c 2
3.b odd 2 1 45.10.a.e 2
4.b odd 2 1 240.10.a.m 2
5.b even 2 1 75.10.a.g 2
5.c odd 4 2 75.10.b.e 4
15.d odd 2 1 225.10.a.j 2
15.e even 4 2 225.10.b.g 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.10.a.c 2 1.a even 1 1 trivial
45.10.a.e 2 3.b odd 2 1
75.10.a.g 2 5.b even 2 1
75.10.b.e 4 5.c odd 4 2
225.10.a.j 2 15.d odd 2 1
225.10.b.g 4 15.e even 4 2
240.10.a.m 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 $$T_{2}^{2} - 19 T_{2} - 1092$$ acting on $$S_{10}^{\mathrm{new}}(\Gamma_0(15))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 19 T - 68 T^{2} - 9728 T^{3} + 262144 T^{4}$$
$3$ $$( 1 - 81 T )^{2}$$
$5$ $$( 1 + 625 T )^{2}$$
$7$ $$1 + 11872 T + 112235774 T^{2} + 479078022304 T^{3} + 1628413597910449 T^{4}$$
$11$ $$1 - 35488 T + 526013014 T^{2} - 83678847658208 T^{3} + 5559917313492231481 T^{4}$$
$13$ $$1 - 143676 T + 24105149134 T^{2} - 1523612051915148 T^{3} +$$$$11\!\cdots\!29$$$$T^{4}$$
$17$ $$1 - 385156 T + 268539949078 T^{2} - 45674832160078532 T^{3} +$$$$14\!\cdots\!09$$$$T^{4}$$
$19$ $$1 + 403296 T + 684929514838 T^{2} + 130138657763479584 T^{3} +$$$$10\!\cdots\!41$$$$T^{4}$$
$23$ $$1 - 223704 T - 1375273107794 T^{2} - 402925054979918952 T^{3} +$$$$32\!\cdots\!69$$$$T^{4}$$
$29$ $$1 + 74572 T + 28833430018078 T^{2} + 1081826889712503068 T^{3} +$$$$21\!\cdots\!61$$$$T^{4}$$
$31$ $$1 + 5027128 T + 52415931233342 T^{2} +$$$$13\!\cdots\!88$$$$T^{3} +$$$$69\!\cdots\!41$$$$T^{4}$$
$37$ $$1 - 5373628 T + 231061724951934 T^{2} -$$$$69\!\cdots\!56$$$$T^{3} +$$$$16\!\cdots\!29$$$$T^{4}$$
$41$ $$1 - 14211332 T + 443988635955862 T^{2} -$$$$46\!\cdots\!52$$$$T^{3} +$$$$10\!\cdots\!21$$$$T^{4}$$
$43$ $$1 - 27748920 T + 1170232974699430 T^{2} -$$$$13\!\cdots\!60$$$$T^{3} +$$$$25\!\cdots\!49$$$$T^{4}$$
$47$ $$1 - 95966440 T + 4320659216802910 T^{2} -$$$$10\!\cdots\!80$$$$T^{3} +$$$$12\!\cdots\!89$$$$T^{4}$$
$53$ $$1 + 64305596 T + 6611083028543086 T^{2} +$$$$21\!\cdots\!68$$$$T^{3} +$$$$10\!\cdots\!89$$$$T^{4}$$
$59$ $$1 - 187863136 T + 23071633420288438 T^{2} -$$$$16\!\cdots\!04$$$$T^{3} +$$$$75\!\cdots\!21$$$$T^{4}$$
$61$ $$1 - 154080060 T + 23302683905802238 T^{2} -$$$$18\!\cdots\!60$$$$T^{3} +$$$$13\!\cdots\!81$$$$T^{4}$$
$67$ $$1 - 33592376 T - 10819815556424362 T^{2} -$$$$91\!\cdots\!72$$$$T^{3} +$$$$74\!\cdots\!09$$$$T^{4}$$
$71$ $$1 + 228270976 T + 45777616900481806 T^{2} +$$$$10\!\cdots\!56$$$$T^{3} +$$$$21\!\cdots\!61$$$$T^{4}$$
$73$ $$1 + 33122316 T + 68371107952007926 T^{2} +$$$$19\!\cdots\!08$$$$T^{3} +$$$$34\!\cdots\!69$$$$T^{4}$$
$79$ $$1 + 932406760 T + 453226630902929438 T^{2} +$$$$11\!\cdots\!40$$$$T^{3} +$$$$14\!\cdots\!61$$$$T^{4}$$
$83$ $$1 - 207040152 T + 372783310330485238 T^{2} -$$$$38\!\cdots\!56$$$$T^{3} +$$$$34\!\cdots\!09$$$$T^{4}$$
$89$ $$1 - 224518164 T + 610925899926766678 T^{2} -$$$$78\!\cdots\!76$$$$T^{3} +$$$$12\!\cdots\!81$$$$T^{4}$$
$97$ $$1 - 387134596 T - 734969029248610362 T^{2} -$$$$29\!\cdots\!32$$$$T^{3} +$$$$57\!\cdots\!89$$$$T^{4}$$