# Properties

 Label 16.34.a.b Level 16 Weight 34 Character orbit 16.a Self dual yes Analytic conductor 110.373 Analytic rank 0 Dimension 2 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$16 = 2^{4}$$ Weight: $$k$$ $$=$$ $$34$$ Character orbit: $$[\chi]$$ $$=$$ 16.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$110.372526210$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\mathbb{Q}[x]/(x^{2} - \cdots)$$ Defining polynomial: $$x^{2} - x - 589050$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{7}\cdot 3^{2}$$ 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 $$\beta = 576\sqrt{2356201}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -18959940 - 39 \beta ) q^{3} + ( -90530768250 - 125500 \beta ) q^{5} + ( 33576540033400 - 100199358 \beta ) q^{7} + ( -4010568477485427 + 1478875320 \beta ) q^{9} +O(q^{10})$$ $$q +(-18959940 - 39 \beta) q^{3} +(-90530768250 - 125500 \beta) q^{5} +(33576540033400 - 100199358 \beta) q^{7} +(-4010568477485427 + 1478875320 \beta) q^{9} +(-66935907720957132 + 168326472875 \beta) q^{11} +(-1490805239129721970 - 467324482716 \beta) q^{13} +(5542640034569937000 + 5910172431750 \beta) q^{15} +(-39680574630587762670 - 332761335850632 \beta) q^{17} +($$$$68\!\cdots\!00$$$$- 610587893846115 \beta) q^{19} +($$$$24\!\cdots\!12$$$$+ 590288754415920 \beta) q^{21} +(-$$$$13\!\cdots\!20$$$$- 20578649420360954 \beta) q^{23} +(-$$$$95\!\cdots\!25$$$$+ 22723222830750000 \beta) q^{25} +($$$$13\!\cdots\!20$$$$+ 345176145382916250 \beta) q^{27} +(-$$$$83\!\cdots\!50$$$$- 760643170023021740 \beta) q^{29} +($$$$31\!\cdots\!08$$$$- 4114562133058947000 \beta) q^{31} +(-$$$$38\!\cdots\!20$$$$- 580959425004299352 \beta) q^{33} +($$$$67\!\cdots\!00$$$$+ 4857269083705083500 \beta) q^{35} +(-$$$$52\!\cdots\!10$$$$+ 25166008023558787188 \beta) q^{37} +($$$$42\!\cdots\!24$$$$+ 67001848478885553870 \beta) q^{39} +($$$$13\!\cdots\!22$$$$-$$$$40\!\cdots\!00$$$$\beta) q^{41} +(-$$$$78\!\cdots\!00$$$$-$$$$11\!\cdots\!69$$$$\beta) q^{43} +($$$$21\!\cdots\!50$$$$+$$$$36\!\cdots\!00$$$$\beta) q^{45} +(-$$$$27\!\cdots\!20$$$$-$$$$61\!\cdots\!48$$$$\beta) q^{47} +($$$$12\!\cdots\!57$$$$-$$$$67\!\cdots\!00$$$$\beta) q^{49} +($$$$10\!\cdots\!48$$$$+$$$$78\!\cdots\!10$$$$\beta) q^{51} +(-$$$$13\!\cdots\!10$$$$+$$$$25\!\cdots\!64$$$$\beta) q^{53} +(-$$$$10\!\cdots\!00$$$$-$$$$68\!\cdots\!50$$$$\beta) q^{55} +($$$$57\!\cdots\!60$$$$-$$$$14\!\cdots\!00$$$$\beta) q^{57} +($$$$15\!\cdots\!00$$$$-$$$$39\!\cdots\!45$$$$\beta) q^{59} +(-$$$$28\!\cdots\!18$$$$-$$$$79\!\cdots\!00$$$$\beta) q^{61} +(-$$$$25\!\cdots\!60$$$$+$$$$45\!\cdots\!66$$$$\beta) q^{63} +($$$$18\!\cdots\!00$$$$+$$$$22\!\cdots\!00$$$$\beta) q^{65} +(-$$$$79\!\cdots\!80$$$$-$$$$35\!\cdots\!43$$$$\beta) q^{67} +($$$$87\!\cdots\!56$$$$+$$$$90\!\cdots\!40$$$$\beta) q^{69} +($$$$13\!\cdots\!88$$$$-$$$$23\!\cdots\!50$$$$\beta) q^{71} +($$$$47\!\cdots\!70$$$$-$$$$73\!\cdots\!96$$$$\beta) q^{73} +($$$$11\!\cdots\!00$$$$+$$$$33\!\cdots\!75$$$$\beta) q^{75} +(-$$$$15\!\cdots\!00$$$$+$$$$12\!\cdots\!56$$$$\beta) q^{77} +($$$$42\!\cdots\!00$$$$+$$$$14\!\cdots\!40$$$$\beta) q^{79} +($$$$91\!\cdots\!21$$$$-$$$$20\!\cdots\!40$$$$\beta) q^{81} +(-$$$$14\!\cdots\!60$$$$-$$$$30\!\cdots\!99$$$$\beta) q^{83} +($$$$36\!\cdots\!00$$$$+$$$$35\!\cdots\!00$$$$\beta) q^{85} +($$$$39\!\cdots\!60$$$$+$$$$47\!\cdots\!50$$$$\beta) q^{87} +($$$$66\!\cdots\!50$$$$-$$$$47\!\cdots\!20$$$$\beta) q^{89} +(-$$$$13\!\cdots\!72$$$$+$$$$13\!\cdots\!60$$$$\beta) q^{91} +($$$$66\!\cdots\!80$$$$-$$$$43\!\cdots\!12$$$$\beta) q^{93} +(-$$$$16\!\cdots\!00$$$$-$$$$30\!\cdots\!50$$$$\beta) q^{95} +(-$$$$18\!\cdots\!30$$$$+$$$$66\!\cdots\!48$$$$\beta) q^{97} +($$$$46\!\cdots\!64$$$$-$$$$77\!\cdots\!65$$$$\beta) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 37919880q^{3} - 181061536500q^{5} + 67153080066800q^{7} - 8021136954970854q^{9} + O(q^{10})$$ $$2q - 37919880q^{3} - 181061536500q^{5} + 67153080066800q^{7} - 8021136954970854q^{9} - 133871815441914264q^{11} - 2981610478259443940q^{13} + 11085280069139874000q^{15} - 79361149261175525340q^{17} +$$$$13\!\cdots\!00$$$$q^{19} +$$$$48\!\cdots\!24$$$$q^{21} -$$$$26\!\cdots\!40$$$$q^{23} -$$$$19\!\cdots\!50$$$$q^{25} +$$$$27\!\cdots\!40$$$$q^{27} -$$$$16\!\cdots\!00$$$$q^{29} +$$$$62\!\cdots\!16$$$$q^{31} -$$$$77\!\cdots\!40$$$$q^{33} +$$$$13\!\cdots\!00$$$$q^{35} -$$$$10\!\cdots\!20$$$$q^{37} +$$$$85\!\cdots\!48$$$$q^{39} +$$$$27\!\cdots\!44$$$$q^{41} -$$$$15\!\cdots\!00$$$$q^{43} +$$$$43\!\cdots\!00$$$$q^{45} -$$$$54\!\cdots\!40$$$$q^{47} +$$$$24\!\cdots\!14$$$$q^{49} +$$$$21\!\cdots\!96$$$$q^{51} -$$$$26\!\cdots\!20$$$$q^{53} -$$$$20\!\cdots\!00$$$$q^{55} +$$$$11\!\cdots\!20$$$$q^{57} +$$$$30\!\cdots\!00$$$$q^{59} -$$$$57\!\cdots\!36$$$$q^{61} -$$$$50\!\cdots\!20$$$$q^{63} +$$$$36\!\cdots\!00$$$$q^{65} -$$$$15\!\cdots\!60$$$$q^{67} +$$$$17\!\cdots\!12$$$$q^{69} +$$$$26\!\cdots\!76$$$$q^{71} +$$$$94\!\cdots\!40$$$$q^{73} +$$$$22\!\cdots\!00$$$$q^{75} -$$$$30\!\cdots\!00$$$$q^{77} +$$$$85\!\cdots\!00$$$$q^{79} +$$$$18\!\cdots\!42$$$$q^{81} -$$$$29\!\cdots\!20$$$$q^{83} +$$$$72\!\cdots\!00$$$$q^{85} +$$$$78\!\cdots\!20$$$$q^{87} +$$$$13\!\cdots\!00$$$$q^{89} -$$$$26\!\cdots\!44$$$$q^{91} +$$$$13\!\cdots\!60$$$$q^{93} -$$$$33\!\cdots\!00$$$$q^{95} -$$$$36\!\cdots\!60$$$$q^{97} +$$$$92\!\cdots\!28$$$$q^{99} + O(q^{100})$$

## 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
 767.996 −766.996
0 −5.34420e7 0 −2.01492e11 0 −5.50153e13 0 −2.70301e15 0
1.2 0 1.55221e7 0 2.04307e10 0 1.22168e14 0 −5.31812e15 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 16.34.a.b 2
4.b odd 2 1 1.34.a.a 2
12.b even 2 1 9.34.a.b 2
20.d odd 2 1 25.34.a.a 2
20.e even 4 2 25.34.b.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.34.a.a 2 4.b odd 2 1
9.34.a.b 2 12.b even 2 1
16.34.a.b 2 1.a even 1 1 trivial
25.34.a.a 2 20.d odd 2 1
25.34.b.a 4 20.e even 4 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + 37919880 T_{3} -$$$$82\!\cdots\!96$$ acting on $$S_{34}^{\mathrm{new}}(\Gamma_0(16))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 + 37919880 T + 10288587693648150 T^{2} +$$$$21\!\cdots\!40$$$$T^{3} +$$$$30\!\cdots\!29$$$$T^{4}$$
$5$ $$1 + 181061536500 T +$$$$22\!\cdots\!50$$$$T^{2} +$$$$21\!\cdots\!00$$$$T^{3} +$$$$13\!\cdots\!25$$$$T^{4}$$
$7$ $$1 - 67153080066800 T +$$$$87\!\cdots\!50$$$$T^{2} -$$$$51\!\cdots\!00$$$$T^{3} +$$$$59\!\cdots\!49$$$$T^{4}$$
$11$ $$1 + 133871815441914264 T +$$$$28\!\cdots\!86$$$$T^{2} +$$$$31\!\cdots\!84$$$$T^{3} +$$$$53\!\cdots\!61$$$$T^{4}$$
$13$ $$1 + 2981610478259443940 T +$$$$13\!\cdots\!50$$$$T^{2} +$$$$17\!\cdots\!20$$$$T^{3} +$$$$33\!\cdots\!09$$$$T^{4}$$
$17$ $$1 + 79361149261175525340 T -$$$$44\!\cdots\!50$$$$T^{2} +$$$$31\!\cdots\!80$$$$T^{3} +$$$$16\!\cdots\!69$$$$T^{4}$$
$19$ $$1 -$$$$13\!\cdots\!00$$$$T +$$$$33\!\cdots\!18$$$$T^{2} -$$$$21\!\cdots\!00$$$$T^{3} +$$$$24\!\cdots\!81$$$$T^{4}$$
$23$ $$1 +$$$$26\!\cdots\!40$$$$T +$$$$15\!\cdots\!50$$$$T^{2} +$$$$22\!\cdots\!20$$$$T^{3} +$$$$74\!\cdots\!89$$$$T^{4}$$
$29$ $$1 +$$$$16\!\cdots\!00$$$$T +$$$$38\!\cdots\!78$$$$T^{2} +$$$$30\!\cdots\!00$$$$T^{3} +$$$$32\!\cdots\!21$$$$T^{4}$$
$31$ $$1 -$$$$62\!\cdots\!16$$$$T +$$$$29\!\cdots\!46$$$$T^{2} -$$$$10\!\cdots\!56$$$$T^{3} +$$$$26\!\cdots\!81$$$$T^{4}$$
$37$ $$1 +$$$$10\!\cdots\!20$$$$T +$$$$13\!\cdots\!50$$$$T^{2} +$$$$59\!\cdots\!40$$$$T^{3} +$$$$31\!\cdots\!09$$$$T^{4}$$
$41$ $$1 -$$$$27\!\cdots\!44$$$$T +$$$$22\!\cdots\!26$$$$T^{2} -$$$$46\!\cdots\!24$$$$T^{3} +$$$$27\!\cdots\!41$$$$T^{4}$$
$43$ $$1 +$$$$15\!\cdots\!00$$$$T +$$$$12\!\cdots\!50$$$$T^{2} +$$$$12\!\cdots\!00$$$$T^{3} +$$$$64\!\cdots\!49$$$$T^{4}$$
$47$ $$1 +$$$$54\!\cdots\!40$$$$T +$$$$37\!\cdots\!50$$$$T^{2} +$$$$81\!\cdots\!80$$$$T^{3} +$$$$22\!\cdots\!29$$$$T^{4}$$
$53$ $$1 +$$$$26\!\cdots\!20$$$$T +$$$$12\!\cdots\!50$$$$T^{2} +$$$$21\!\cdots\!60$$$$T^{3} +$$$$63\!\cdots\!29$$$$T^{4}$$
$59$ $$1 -$$$$30\!\cdots\!00$$$$T +$$$$77\!\cdots\!58$$$$T^{2} -$$$$83\!\cdots\!00$$$$T^{3} +$$$$75\!\cdots\!41$$$$T^{4}$$
$61$ $$1 +$$$$57\!\cdots\!36$$$$T +$$$$16\!\cdots\!86$$$$T^{2} +$$$$47\!\cdots\!16$$$$T^{3} +$$$$67\!\cdots\!61$$$$T^{4}$$
$67$ $$1 +$$$$15\!\cdots\!60$$$$T +$$$$41\!\cdots\!50$$$$T^{2} +$$$$29\!\cdots\!20$$$$T^{3} +$$$$33\!\cdots\!69$$$$T^{4}$$
$71$ $$1 -$$$$26\!\cdots\!76$$$$T +$$$$22\!\cdots\!66$$$$T^{2} -$$$$32\!\cdots\!36$$$$T^{3} +$$$$15\!\cdots\!21$$$$T^{4}$$
$73$ $$1 -$$$$94\!\cdots\!40$$$$T +$$$$19\!\cdots\!50$$$$T^{2} -$$$$29\!\cdots\!20$$$$T^{3} +$$$$95\!\cdots\!89$$$$T^{4}$$
$79$ $$1 -$$$$85\!\cdots\!00$$$$T +$$$$85\!\cdots\!78$$$$T^{2} -$$$$35\!\cdots\!00$$$$T^{3} +$$$$17\!\cdots\!21$$$$T^{4}$$
$83$ $$1 +$$$$29\!\cdots\!20$$$$T +$$$$37\!\cdots\!50$$$$T^{2} +$$$$62\!\cdots\!60$$$$T^{3} +$$$$45\!\cdots\!69$$$$T^{4}$$
$89$ $$1 -$$$$13\!\cdots\!00$$$$T +$$$$47\!\cdots\!38$$$$T^{2} -$$$$28\!\cdots\!00$$$$T^{3} +$$$$45\!\cdots\!61$$$$T^{4}$$
$97$ $$1 +$$$$36\!\cdots\!60$$$$T +$$$$42\!\cdots\!50$$$$T^{2} +$$$$13\!\cdots\!20$$$$T^{3} +$$$$13\!\cdots\!29$$$$T^{4}$$