# Properties

 Label 16.15.c.c Level 16 Weight 15 Character orbit 16.c Analytic conductor 19.893 Analytic rank 0 Dimension 4 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$16 = 2^{4}$$ Weight: $$k$$ $$=$$ $$15$$ Character orbit: $$[\chi]$$ $$=$$ 16.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$19.8926349043$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\mathbb{Q}[x]/(x^{4} - \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{25}\cdot 3$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{3} + ( -21750 + \beta_{3} ) q^{5} + ( 245 \beta_{1} - \beta_{2} ) q^{7} + ( -5000199 + 226 \beta_{3} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{3} + ( -21750 + \beta_{3} ) q^{5} + ( 245 \beta_{1} - \beta_{2} ) q^{7} + ( -5000199 + 226 \beta_{3} ) q^{9} + ( 3057 \beta_{1} - 270 \beta_{2} ) q^{11} + ( 16716154 - 1131 \beta_{3} ) q^{13} + ( -52977 \beta_{1} - 9531 \beta_{2} ) q^{15} + ( 3279138 - 7030 \beta_{3} ) q^{17} + ( -80045 \beta_{1} - 112826 \beta_{2} ) q^{19} + ( -2395550976 + 55084 \beta_{3} ) q^{21} + ( 454707 \beta_{1} - 651807 \beta_{2} ) q^{23} + ( -4222801285 - 43500 \beta_{3} ) q^{25} + ( -7274532 \beta_{1} - 2154006 \beta_{2} ) q^{27} + ( -4375446438 - 407767 \beta_{3} ) q^{29} + ( 4798512 \beta_{1} - 4065288 \beta_{2} ) q^{31} + ( -29549344896 + 613662 \beta_{3} ) q^{33} + ( -12933984 \beta_{1} - 2344572 \beta_{2} ) q^{35} + ( -34996433590 + 1942029 \beta_{3} ) q^{37} + ( 52033891 \beta_{1} + 10779561 \beta_{2} ) q^{39} + ( 165522102642 - 2823372 \beta_{3} ) q^{41} + ( -62935245 \beta_{1} + 39873372 \beta_{2} ) q^{43} + ( 426883644090 - 9915699 \beta_{3} ) q^{45} + ( 107341674 \beta_{1} + 72228726 \beta_{2} ) q^{47} + ( 91599855697 + 13424712 \beta_{3} ) q^{49} + ( 222804948 \beta_{1} + 67002930 \beta_{2} ) q^{51} + ( -743793406038 + 33160533 \beta_{3} ) q^{53} + ( -149697819 \beta_{1} - 31695057 \beta_{2} ) q^{55} + ( 932608892544 - 50358406 \beta_{3} ) q^{57} + ( 339074283 \beta_{1} - 249381720 \beta_{2} ) q^{59} + ( -1656242712934 - 32995419 \beta_{3} ) q^{61} + ( -2943831639 \beta_{1} - 529788573 \beta_{2} ) q^{63} + ( -1955630580540 + 41315404 \beta_{3} ) q^{65} + ( 1229905543 \beta_{1} - 679535978 \beta_{2} ) q^{67} + ( -3584710764288 - 83653020 \beta_{3} ) q^{69} + ( 2286480849 \beta_{1} - 411388821 \beta_{2} ) q^{71} + ( -3008283147278 + 355525386 \beta_{3} ) q^{73} + ( -2864426785 \beta_{1} + 414598500 \beta_{2} ) q^{75} + ( -7245770902272 + 149257428 \beta_{3} ) q^{77} + ( 3951839050 \beta_{1} + 1805219230 \beta_{2} ) q^{79} + ( 50106626153649 - 1179138954 \beta_{3} ) q^{81} + ( -5754348195 \beta_{1} + 3410578656 \beta_{2} ) q^{83} + ( -9967113686700 + 156181638 \beta_{3} ) q^{85} + ( 8357893671 \beta_{1} + 3886427277 \beta_{2} ) q^{87} + ( 26576753624274 + 1035147034 \beta_{3} ) q^{89} + ( 12696977384 \beta_{1} + 2659594028 \beta_{2} ) q^{91} + ( -41557394433024 - 78208656 \beta_{3} ) q^{93} + ( 9360700671 \beta_{1} - 306343107 \beta_{2} ) q^{95} + ( 23948291484034 + 1255201890 \beta_{3} ) q^{97} + ( -34090631937 \beta_{1} - 7140214152 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 87000q^{5} - 20000796q^{9} + O(q^{10})$$ $$4q - 87000q^{5} - 20000796q^{9} + 66864616q^{13} + 13116552q^{17} - 9582203904q^{21} - 16891205140q^{25} - 17501785752q^{29} - 118197379584q^{33} - 139985734360q^{37} + 662088410568q^{41} + 1707534576360q^{45} + 366399422788q^{49} - 2975173624152q^{53} + 3730435570176q^{57} - 6624970851736q^{61} - 7822522322160q^{65} - 14338843057152q^{69} - 12033132589112q^{73} - 28983083609088q^{77} + 200426504614596q^{81} - 39868454746800q^{85} + 106307014497096q^{89} - 166229577732096q^{93} + 95793165936136q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} + 9547 x^{2} + 9546 x + 91126116$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-25760 \nu^{3} + 2902288 \nu^{2} - 488959152 \nu + 13606715664$$$$)/15189277$$ $$\beta_{2}$$ $$=$$ $$($$$$73184 \nu^{3} + 30397648 \nu^{2} + 1427772944 \nu + 145786588368$$$$)/45567831$$ $$\beta_{3}$$ $$=$$ $$($$$$-384 \nu^{3} - 5498688$$$$)/9547$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$32 \beta_{3} + 171 \beta_{2} - 597 \beta_{1} + 6144$$$$)/24576$$ $$\nu^{2}$$ $$=$$ $$($$$$-32 \beta_{3} + 28809 \beta_{2} + 28041 \beta_{1} - 117307392$$$$)/24576$$ $$\nu^{3}$$ $$=$$ $$($$$$-9547 \beta_{3} - 5498688$$$$)/384$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/16\mathbb{Z}\right)^\times$$.

 $$n$$ $$5$$ $$15$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
15.1
 −48.6025 + 84.1819i 49.1025 + 85.0479i 49.1025 − 85.0479i −48.6025 − 84.1819i
0 4273.45i 0 −59268.7 0 1.04417e6i 0 −1.34794e7 0
15.2 0 1141.90i 0 15768.7 0 288003.i 0 3.47902e6 0
15.3 0 1141.90i 0 15768.7 0 288003.i 0 3.47902e6 0
15.4 0 4273.45i 0 −59268.7 0 1.04417e6i 0 −1.34794e7 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 16.15.c.c 4
3.b odd 2 1 144.15.g.g 4
4.b odd 2 1 inner 16.15.c.c 4
8.b even 2 1 64.15.c.c 4
8.d odd 2 1 64.15.c.c 4
12.b even 2 1 144.15.g.g 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
16.15.c.c 4 1.a even 1 1 trivial
16.15.c.c 4 4.b odd 2 1 inner
64.15.c.c 4 8.b even 2 1
64.15.c.c 4 8.d odd 2 1
144.15.g.g 4 3.b odd 2 1
144.15.g.g 4 12.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} + 19566336 T_{3}^{2} +$$$$23\!\cdots\!84$$ acting on $$S_{15}^{\mathrm{new}}(16, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 + 434460 T^{2} - 26096451597018 T^{4} + 9939051249982356060 T^{6} +$$$$52\!\cdots\!21$$$$T^{8}$$
$5$ $$( 1 + 43500 T + 11272441910 T^{2} + 265502929687500 T^{3} + 37252902984619140625 T^{4} )^{2}$$
$7$ $$1 - 1539645857092 T^{2} +$$$$12\!\cdots\!58$$$$T^{4} -$$$$70\!\cdots\!92$$$$T^{6} +$$$$21\!\cdots\!01$$$$T^{8}$$
$11$ $$1 - 1334996682152740 T^{2} +$$$$73\!\cdots\!22$$$$T^{4} -$$$$19\!\cdots\!40$$$$T^{6} +$$$$20\!\cdots\!61$$$$T^{8}$$
$13$ $$( 1 - 33432308 T + 6353569240644054 T^{2} -$$$$13\!\cdots\!12$$$$T^{3} +$$$$15\!\cdots\!21$$$$T^{4} )^{2}$$
$17$ $$( 1 - 6558276 T + 267198985045368902 T^{2} -$$$$11\!\cdots\!04$$$$T^{3} +$$$$28\!\cdots\!41$$$$T^{4} )^{2}$$
$19$ $$1 - 2057925827207927140 T^{2} +$$$$23\!\cdots\!22$$$$T^{4} -$$$$13\!\cdots\!40$$$$T^{6} +$$$$40\!\cdots\!81$$$$T^{8}$$
$23$ $$1 - 11694673362806264260 T^{2} +$$$$27\!\cdots\!22$$$$T^{4} -$$$$15\!\cdots\!60$$$$T^{6} +$$$$18\!\cdots\!61$$$$T^{8}$$
$29$ $$( 1 + 8750892876 T +$$$$38\!\cdots\!46$$$$T^{2} +$$$$26\!\cdots\!56$$$$T^{3} +$$$$88\!\cdots\!61$$$$T^{4} )^{2}$$
$31$ $$1 -$$$$14\!\cdots\!60$$$$T^{2} +$$$$11\!\cdots\!82$$$$T^{4} -$$$$81\!\cdots\!60$$$$T^{6} +$$$$32\!\cdots\!81$$$$T^{8}$$
$37$ $$( 1 + 69992867180 T +$$$$13\!\cdots\!38$$$$T^{2} +$$$$63\!\cdots\!20$$$$T^{3} +$$$$81\!\cdots\!21$$$$T^{4} )^{2}$$
$41$ $$( 1 - 331044205284 T +$$$$92\!\cdots\!26$$$$T^{2} -$$$$12\!\cdots\!24$$$$T^{3} +$$$$14\!\cdots\!21$$$$T^{4} )^{2}$$
$43$ $$1 -$$$$11\!\cdots\!20$$$$T^{2} +$$$$93\!\cdots\!42$$$$T^{4} -$$$$60\!\cdots\!20$$$$T^{6} +$$$$29\!\cdots\!01$$$$T^{8}$$
$47$ $$1 -$$$$36\!\cdots\!52$$$$T^{2} +$$$$15\!\cdots\!58$$$$T^{4} -$$$$24\!\cdots\!72$$$$T^{6} +$$$$43\!\cdots\!21$$$$T^{8}$$
$53$ $$( 1 + 1487586812076 T +$$$$17\!\cdots\!22$$$$T^{2} +$$$$20\!\cdots\!44$$$$T^{3} +$$$$19\!\cdots\!61$$$$T^{4} )^{2}$$
$59$ $$1 -$$$$18\!\cdots\!00$$$$T^{2} +$$$$15\!\cdots\!02$$$$T^{4} -$$$$70\!\cdots\!00$$$$T^{6} +$$$$14\!\cdots\!41$$$$T^{8}$$
$61$ $$( 1 + 3312485425868 T +$$$$20\!\cdots\!98$$$$T^{2} +$$$$32\!\cdots\!88$$$$T^{3} +$$$$97\!\cdots\!81$$$$T^{4} )^{2}$$
$67$ $$1 -$$$$86\!\cdots\!80$$$$T^{2} +$$$$42\!\cdots\!42$$$$T^{4} -$$$$11\!\cdots\!80$$$$T^{6} +$$$$18\!\cdots\!81$$$$T^{8}$$
$71$ $$1 -$$$$22\!\cdots\!48$$$$T^{2} +$$$$25\!\cdots\!98$$$$T^{4} -$$$$15\!\cdots\!28$$$$T^{6} +$$$$46\!\cdots\!21$$$$T^{8}$$
$73$ $$( 1 + 6016566294556 T +$$$$75\!\cdots\!62$$$$T^{2} +$$$$73\!\cdots\!04$$$$T^{3} +$$$$14\!\cdots\!81$$$$T^{4} )^{2}$$
$79$ $$1 -$$$$88\!\cdots\!24$$$$T^{2} +$$$$43\!\cdots\!66$$$$T^{4} -$$$$12\!\cdots\!64$$$$T^{6} +$$$$18\!\cdots\!21$$$$T^{8}$$
$83$ $$1 -$$$$15\!\cdots\!72$$$$T^{2} +$$$$14\!\cdots\!18$$$$T^{4} -$$$$82\!\cdots\!52$$$$T^{6} +$$$$29\!\cdots\!81$$$$T^{8}$$
$89$ $$( 1 - 53153507248548 T +$$$$31\!\cdots\!18$$$$T^{2} -$$$$10\!\cdots\!68$$$$T^{3} +$$$$38\!\cdots\!81$$$$T^{4} )^{2}$$
$97$ $$( 1 - 47896582968068 T +$$$$11\!\cdots\!94$$$$T^{2} -$$$$31\!\cdots\!92$$$$T^{3} +$$$$42\!\cdots\!61$$$$T^{4} )^{2}$$