# Properties

 Label 16.44.a.c Level $16$ Weight $44$ Character orbit 16.a Self dual yes Analytic conductor $187.377$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$16 = 2^{4}$$ Weight: $$k$$ $$=$$ $$44$$ Character orbit: $$[\chi]$$ $$=$$ 16.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$187.376632553$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: $$\mathbb{Q}[x]/(x^{3} - \cdots)$$ Defining polynomial: $$x^{3} - 11258260111 x - 264759545317170$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{25}\cdot 3^{4}\cdot 5\cdot 7$$ Twist minimal: no (minimal twist has level 1) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + ( -8133812604 + \beta_{1} ) q^{3} + ( 178401793591390 - 24511 \beta_{1} + 5333 \beta_{2} ) q^{5} + ( -100657141888492952 + 27736590 \beta_{1} - 33060636 \beta_{2} ) q^{7} + ( 159160708390355581077 - 13504373514 \beta_{1} - 5599537938 \beta_{2} ) q^{9} +O(q^{10})$$ $$q +(-8133812604 + \beta_{1}) q^{3} +(178401793591390 - 24511 \beta_{1} + 5333 \beta_{2}) q^{5} +(-100657141888492952 + 27736590 \beta_{1} - 33060636 \beta_{2}) q^{7} +($$$$15\!\cdots\!77$$$$- 13504373514 \beta_{1} - 5599537938 \beta_{2}) q^{9} +(-$$$$88\!\cdots\!32$$$$- 400454830765 \beta_{1} - 263213129080 \beta_{2}) q^{11} +($$$$89\!\cdots\!94$$$$+ 20246528902353 \beta_{1} - 2561391288027 \beta_{2}) q^{13} +(-$$$$11\!\cdots\!80$$$$+ 116708291400522 \beta_{1} + 79136230497084 \beta_{2}) q^{15} +(-$$$$13\!\cdots\!98$$$$- 13097493598399338 \beta_{1} + 2285660650779150 \beta_{2}) q^{17} +(-$$$$52\!\cdots\!00$$$$+ 145387562524099533 \beta_{1} + 38079513958814136 \beta_{2}) q^{19} +($$$$13\!\cdots\!12$$$$+ 948892297661770924 \beta_{1} + 204951788249022108 \beta_{2}) q^{21} +($$$$40\!\cdots\!16$$$$+ 1718851991611118938 \beta_{1} + 3594965504577966844 \beta_{2}) q^{23} +(-$$$$77\!\cdots\!25$$$$+ 1975197243032212380 \beta_{1} + 292136831915638860 \beta_{2}) q^{25} +(-$$$$42\!\cdots\!60$$$$+$$$$10\!\cdots\!98$$$$\beta_{1} +$$$$13\!\cdots\!56$$$$\beta_{2}) q^{27} +($$$$19\!\cdots\!50$$$$-$$$$99\!\cdots\!07$$$$\beta_{1} + 23728961726491800481 \beta_{2}) q^{29} +($$$$85\!\cdots\!08$$$$-$$$$22\!\cdots\!20$$$$\beta_{1} +$$$$11\!\cdots\!60$$$$\beta_{2}) q^{31} +(-$$$$91\!\cdots\!72$$$$+$$$$28\!\cdots\!98$$$$\beta_{1} +$$$$51\!\cdots\!10$$$$\beta_{2}) q^{33} +(-$$$$79\!\cdots\!40$$$$-$$$$26\!\cdots\!84$$$$\beta_{1} -$$$$11\!\cdots\!48$$$$\beta_{2}) q^{35} +(-$$$$77\!\cdots\!98$$$$+$$$$67\!\cdots\!21$$$$\beta_{1} +$$$$20\!\cdots\!13$$$$\beta_{2}) q^{37} +($$$$13\!\cdots\!76$$$$+$$$$87\!\cdots\!46$$$$\beta_{1} -$$$$85\!\cdots\!68$$$$\beta_{2}) q^{39} +($$$$85\!\cdots\!22$$$$+$$$$14\!\cdots\!20$$$$\beta_{1} -$$$$83\!\cdots\!60$$$$\beta_{2}) q^{41} +(-$$$$80\!\cdots\!64$$$$-$$$$23\!\cdots\!65$$$$\beta_{1} -$$$$52\!\cdots\!92$$$$\beta_{2}) q^{43} +($$$$85\!\cdots\!30$$$$-$$$$73\!\cdots\!47$$$$\beta_{1} -$$$$32\!\cdots\!59$$$$\beta_{2}) q^{45} +(-$$$$10\!\cdots\!52$$$$+$$$$23\!\cdots\!72$$$$\beta_{1} -$$$$13\!\cdots\!12$$$$\beta_{2}) q^{47} +($$$$11\!\cdots\!93$$$$+$$$$47\!\cdots\!00$$$$\beta_{1} +$$$$47\!\cdots\!00$$$$\beta_{2}) q^{49} +(-$$$$44\!\cdots\!52$$$$-$$$$14\!\cdots\!18$$$$\beta_{1} +$$$$48\!\cdots\!44$$$$\beta_{2}) q^{51} +($$$$50\!\cdots\!54$$$$-$$$$10\!\cdots\!91$$$$\beta_{1} -$$$$17\!\cdots\!55$$$$\beta_{2}) q^{53} +(-$$$$13\!\cdots\!80$$$$+$$$$37\!\cdots\!02$$$$\beta_{1} -$$$$21\!\cdots\!56$$$$\beta_{2}) q^{55} +($$$$64\!\cdots\!20$$$$-$$$$26\!\cdots\!06$$$$\beta_{1} -$$$$12\!\cdots\!82$$$$\beta_{2}) q^{57} +(-$$$$75\!\cdots\!00$$$$-$$$$36\!\cdots\!01$$$$\beta_{1} +$$$$10\!\cdots\!08$$$$\beta_{2}) q^{59} +(-$$$$31\!\cdots\!18$$$$+$$$$15\!\cdots\!25$$$$\beta_{1} +$$$$46\!\cdots\!25$$$$\beta_{2}) q^{61} +($$$$32\!\cdots\!16$$$$-$$$$85\!\cdots\!86$$$$\beta_{1} +$$$$33\!\cdots\!76$$$$\beta_{2}) q^{63} +(-$$$$90\!\cdots\!20$$$$-$$$$25\!\cdots\!32$$$$\beta_{1} +$$$$61\!\cdots\!96$$$$\beta_{2}) q^{65} +($$$$24\!\cdots\!48$$$$-$$$$55\!\cdots\!47$$$$\beta_{1} -$$$$81\!\cdots\!20$$$$\beta_{2}) q^{67} +($$$$65\!\cdots\!44$$$$-$$$$13\!\cdots\!68$$$$\beta_{1} -$$$$48\!\cdots\!56$$$$\beta_{2}) q^{69} +($$$$61\!\cdots\!88$$$$-$$$$11\!\cdots\!50$$$$\beta_{1} -$$$$16\!\cdots\!00$$$$\beta_{2}) q^{71} +(-$$$$67\!\cdots\!66$$$$-$$$$22\!\cdots\!38$$$$\beta_{1} -$$$$95\!\cdots\!94$$$$\beta_{2}) q^{73} +($$$$70\!\cdots\!00$$$$-$$$$79\!\cdots\!85$$$$\beta_{1} -$$$$14\!\cdots\!20$$$$\beta_{2}) q^{75} +($$$$19\!\cdots\!64$$$$-$$$$57\!\cdots\!40$$$$\beta_{1} +$$$$68\!\cdots\!32$$$$\beta_{2}) q^{77} +(-$$$$50\!\cdots\!00$$$$-$$$$88\!\cdots\!28$$$$\beta_{1} +$$$$31\!\cdots\!24$$$$\beta_{2}) q^{79} +($$$$24\!\cdots\!21$$$$-$$$$53\!\cdots\!58$$$$\beta_{1} -$$$$24\!\cdots\!86$$$$\beta_{2}) q^{81} +($$$$29\!\cdots\!76$$$$-$$$$71\!\cdots\!91$$$$\beta_{1} -$$$$52\!\cdots\!04$$$$\beta_{2}) q^{83} +($$$$14\!\cdots\!40$$$$+$$$$62\!\cdots\!14$$$$\beta_{1} -$$$$13\!\cdots\!42$$$$\beta_{2}) q^{85} +(-$$$$57\!\cdots\!80$$$$+$$$$23\!\cdots\!74$$$$\beta_{1} +$$$$52\!\cdots\!28$$$$\beta_{2}) q^{87} +($$$$67\!\cdots\!50$$$$+$$$$27\!\cdots\!14$$$$\beta_{1} -$$$$37\!\cdots\!62$$$$\beta_{2}) q^{89} +($$$$38\!\cdots\!28$$$$+$$$$49\!\cdots\!52$$$$\beta_{1} -$$$$27\!\cdots\!16$$$$\beta_{2}) q^{91} +(-$$$$16\!\cdots\!32$$$$+$$$$55\!\cdots\!48$$$$\beta_{1} +$$$$40\!\cdots\!80$$$$\beta_{2}) q^{93} +(-$$$$10\!\cdots\!00$$$$+$$$$31\!\cdots\!50$$$$\beta_{1} +$$$$31\!\cdots\!00$$$$\beta_{2}) q^{95} +(-$$$$12\!\cdots\!98$$$$+$$$$88\!\cdots\!98$$$$\beta_{1} +$$$$67\!\cdots\!02$$$$\beta_{2}) q^{97} +($$$$47\!\cdots\!36$$$$-$$$$16\!\cdots\!57$$$$\beta_{1} +$$$$15\!\cdots\!56$$$$\beta_{2}) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q - 24401437812q^{3} + 535205380774170q^{5} - 301971425665478856q^{7} + 477482125171066743231q^{9} + O(q^{10})$$ $$3q - 24401437812q^{3} + 535205380774170q^{5} - 301971425665478856q^{7} +$$$$47\!\cdots\!31$$$$q^{9} -$$$$26\!\cdots\!96$$$$q^{11} +$$$$26\!\cdots\!82$$$$q^{13} -$$$$35\!\cdots\!40$$$$q^{15} -$$$$40\!\cdots\!94$$$$q^{17} -$$$$15\!\cdots\!00$$$$q^{19} +$$$$39\!\cdots\!36$$$$q^{21} +$$$$12\!\cdots\!48$$$$q^{23} -$$$$23\!\cdots\!75$$$$q^{25} -$$$$12\!\cdots\!80$$$$q^{27} +$$$$57\!\cdots\!50$$$$q^{29} +$$$$25\!\cdots\!24$$$$q^{31} -$$$$27\!\cdots\!16$$$$q^{33} -$$$$23\!\cdots\!20$$$$q^{35} -$$$$23\!\cdots\!94$$$$q^{37} +$$$$39\!\cdots\!28$$$$q^{39} +$$$$25\!\cdots\!66$$$$q^{41} -$$$$24\!\cdots\!92$$$$q^{43} +$$$$25\!\cdots\!90$$$$q^{45} -$$$$30\!\cdots\!56$$$$q^{47} +$$$$34\!\cdots\!79$$$$q^{49} -$$$$13\!\cdots\!56$$$$q^{51} +$$$$15\!\cdots\!62$$$$q^{53} -$$$$39\!\cdots\!40$$$$q^{55} +$$$$19\!\cdots\!60$$$$q^{57} -$$$$22\!\cdots\!00$$$$q^{59} -$$$$93\!\cdots\!54$$$$q^{61} +$$$$96\!\cdots\!48$$$$q^{63} -$$$$27\!\cdots\!60$$$$q^{65} +$$$$73\!\cdots\!44$$$$q^{67} +$$$$19\!\cdots\!32$$$$q^{69} +$$$$18\!\cdots\!64$$$$q^{71} -$$$$20\!\cdots\!98$$$$q^{73} +$$$$21\!\cdots\!00$$$$q^{75} +$$$$59\!\cdots\!92$$$$q^{77} -$$$$15\!\cdots\!00$$$$q^{79} +$$$$73\!\cdots\!63$$$$q^{81} +$$$$89\!\cdots\!28$$$$q^{83} +$$$$43\!\cdots\!20$$$$q^{85} -$$$$17\!\cdots\!40$$$$q^{87} +$$$$20\!\cdots\!50$$$$q^{89} +$$$$11\!\cdots\!84$$$$q^{91} -$$$$49\!\cdots\!96$$$$q^{93} -$$$$31\!\cdots\!00$$$$q^{95} -$$$$38\!\cdots\!94$$$$q^{97} +$$$$14\!\cdots\!08$$$$q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - 11258260111 x - 264759545317170$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$27648 \nu^{2} + 443743488 \nu - 207512250365952$$$$)/8369$$ $$\beta_{2}$$ $$=$$ $$($$$$70656 \nu^{2} - 6064667904 \nu - 530309084268544$$$$)/8369$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-9 \beta_{2} + 23 \beta_{1}$$$$)/7741440$$ $$\nu^{2}$$ $$=$$ $$($$$$577791 \beta_{2} + 7896703 \beta_{1} + 232413720409866240$$$$)/30965760$$

## 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
 −24885.9 −91450.2 116336.
0 −3.22027e10 0 5.54482e14 0 5.57604e17 0 7.08758e20 0
1.2 0 −1.01494e10 0 6.19839e14 0 −2.58688e18 0 −2.25246e20 0
1.3 0 1.79507e10 0 −6.39116e14 0 1.72730e18 0 −6.02939e18 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$

## 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 16.44.a.c 3
4.b odd 2 1 1.44.a.a 3
12.b even 2 1 9.44.a.b 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.44.a.a 3 4.b odd 2 1
9.44.a.b 3 12.b even 2 1
16.44.a.c 3 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{3} + 24401437812 T_{3}^{2} -$$$$43\!\cdots\!84$$$$T_{3} -$$$$58\!\cdots\!08$$ acting on $$S_{44}^{\mathrm{new}}(\Gamma_0(16))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$-$$$$58\!\cdots\!08$$$$-$$$$43\!\cdots\!84$$$$T + 24401437812 T^{2} + T^{3}$$
$5$ $$21\!\cdots\!00$$$$-$$$$40\!\cdots\!00$$$$T - 535205380774170 T^{2} + T^{3}$$
$7$ $$24\!\cdots\!84$$$$-$$$$49\!\cdots\!36$$$$T + 301971425665478856 T^{2} + T^{3}$$
$11$ $$-$$$$32\!\cdots\!32$$$$-$$$$14\!\cdots\!28$$$$T +$$$$26\!\cdots\!96$$$$T^{2} + T^{3}$$
$13$ $$-$$$$50\!\cdots\!72$$$$+$$$$20\!\cdots\!96$$$$T -$$$$26\!\cdots\!82$$$$T^{2} + T^{3}$$
$17$ $$28\!\cdots\!76$$$$-$$$$75\!\cdots\!96$$$$T +$$$$40\!\cdots\!94$$$$T^{2} + T^{3}$$
$19$ $$21\!\cdots\!00$$$$-$$$$18\!\cdots\!00$$$$T +$$$$15\!\cdots\!00$$$$T^{2} + T^{3}$$
$23$ $$-$$$$36\!\cdots\!48$$$$-$$$$54\!\cdots\!24$$$$T -$$$$12\!\cdots\!48$$$$T^{2} + T^{3}$$
$29$ $$73\!\cdots\!00$$$$+$$$$46\!\cdots\!00$$$$T -$$$$57\!\cdots\!50$$$$T^{2} + T^{3}$$
$31$ $$17\!\cdots\!88$$$$+$$$$13\!\cdots\!92$$$$T -$$$$25\!\cdots\!24$$$$T^{2} + T^{3}$$
$37$ $$56\!\cdots\!96$$$$-$$$$26\!\cdots\!16$$$$T +$$$$23\!\cdots\!94$$$$T^{2} + T^{3}$$
$41$ $$27\!\cdots\!52$$$$-$$$$40\!\cdots\!48$$$$T -$$$$25\!\cdots\!66$$$$T^{2} + T^{3}$$
$43$ $$49\!\cdots\!12$$$$+$$$$14\!\cdots\!36$$$$T +$$$$24\!\cdots\!92$$$$T^{2} + T^{3}$$
$47$ $$-$$$$86\!\cdots\!56$$$$-$$$$33\!\cdots\!76$$$$T +$$$$30\!\cdots\!56$$$$T^{2} + T^{3}$$
$53$ $$-$$$$90\!\cdots\!92$$$$+$$$$66\!\cdots\!16$$$$T -$$$$15\!\cdots\!62$$$$T^{2} + T^{3}$$
$59$ $$-$$$$78\!\cdots\!00$$$$+$$$$87\!\cdots\!00$$$$T +$$$$22\!\cdots\!00$$$$T^{2} + T^{3}$$
$61$ $$-$$$$87\!\cdots\!68$$$$-$$$$87\!\cdots\!28$$$$T +$$$$93\!\cdots\!54$$$$T^{2} + T^{3}$$
$67$ $$-$$$$42\!\cdots\!76$$$$-$$$$19\!\cdots\!96$$$$T -$$$$73\!\cdots\!44$$$$T^{2} + T^{3}$$
$71$ $$-$$$$18\!\cdots\!72$$$$+$$$$10\!\cdots\!32$$$$T -$$$$18\!\cdots\!64$$$$T^{2} + T^{3}$$
$73$ $$-$$$$31\!\cdots\!52$$$$+$$$$69\!\cdots\!76$$$$T +$$$$20\!\cdots\!98$$$$T^{2} + T^{3}$$
$79$ $$26\!\cdots\!00$$$$-$$$$83\!\cdots\!00$$$$T +$$$$15\!\cdots\!00$$$$T^{2} + T^{3}$$
$83$ $$17\!\cdots\!32$$$$-$$$$30\!\cdots\!44$$$$T -$$$$89\!\cdots\!28$$$$T^{2} + T^{3}$$
$89$ $$-$$$$10\!\cdots\!00$$$$-$$$$52\!\cdots\!00$$$$T -$$$$20\!\cdots\!50$$$$T^{2} + T^{3}$$
$97$ $$-$$$$26\!\cdots\!44$$$$-$$$$17\!\cdots\!76$$$$T +$$$$38\!\cdots\!94$$$$T^{2} + T^{3}$$