# Properties

 Label 9.44.a.b Level $9$ Weight $44$ Character orbit 9.a Self dual yes Analytic conductor $105.399$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$9 = 3^{2}$$ Weight: $$k$$ $$=$$ $$44$$ Character orbit: $$[\chi]$$ $$=$$ 9.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$105.399355811$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: $$\mathbb{Q}[x]/(x^{3} - \cdots)$$ Defining polynomial: $$x^{3} - 11258260111 x - 264759545317170$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{11}\cdot 3^{5}\cdot 5\cdot 7$$ 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.

 $$f(q)$$ $$=$$ $$q + ( 736648 + \beta_{1} ) q^{2} + ( 3274206140608 + 810680 \beta_{1} - 136 \beta_{2} ) q^{4} + ( -178401793591390 + 23668880 \beta_{1} + 9948 \beta_{2} ) q^{5} + ( 100657141888492952 - 411883648456 \beta_{1} - 20265742 \beta_{2} ) q^{7} + ( 5276954637929116160 - 137106139840 \beta_{1} - 300552384 \beta_{2} ) q^{8} +O(q^{10})$$ $$q +(736648 + \beta_{1}) q^{2} +(3274206140608 + 810680 \beta_{1} - 136 \beta_{2}) q^{4} +(-178401793591390 + 23668880 \beta_{1} + 9948 \beta_{2}) q^{5} +(100657141888492952 - 411883648456 \beta_{1} - 20265742 \beta_{2}) q^{7} +(5276954637929116160 - 137106139840 \beta_{1} - 300552384 \beta_{2}) q^{8} +($$$$14\!\cdots\!40$$$$- 566140399707230 \beta_{1} + 10700910592 \beta_{2}) q^{10} +(-$$$$88\!\cdots\!32$$$$+ 4605449192710300 \beta_{1} - 45747233895 \beta_{2}) q^{11} +($$$$89\!\cdots\!94$$$$- 6721472103363472 \beta_{1} + 7602640063460 \beta_{2}) q^{13} +(-$$$$46\!\cdots\!76$$$$+ 863622683114558360 \beta_{1} + 27659052976128 \beta_{2}) q^{14} +(-$$$$26\!\cdots\!64$$$$+ 9903419018930019840 \beta_{1} + 794362954993152 \beta_{2}) q^{16} +($$$$13\!\cdots\!98$$$$+ 4610357498452581216 \beta_{1} + 5127718083059496 \beta_{2}) q^{17} +($$$$52\!\cdots\!00$$$$+$$$$85\!\cdots\!20$$$$\beta_{1} - 35769349521761799 \beta_{2}) q^{19} +(-$$$$48\!\cdots\!20$$$$-$$$$52\!\cdots\!60$$$$\beta_{1} + 4464959933862384 \beta_{2}) q^{20} +($$$$46\!\cdots\!64$$$$-$$$$67\!\cdots\!32$$$$\beta_{1} - 690353547697454080 \beta_{2}) q^{22} +($$$$40\!\cdots\!16$$$$-$$$$54\!\cdots\!88$$$$\beta_{1} - 625371170988949302 \beta_{2}) q^{23} +(-$$$$77\!\cdots\!25$$$$-$$$$83\!\cdots\!00$$$$\beta_{1} + 561020137038857840 \beta_{2}) q^{25} +($$$$57\!\cdots\!48$$$$+$$$$59\!\cdots\!90$$$$\beta_{1} + 11552220751814725632 \beta_{2}) q^{26} +($$$$56\!\cdots\!56$$$$-$$$$20\!\cdots\!08$$$$\beta_{1} + 99508983996076431168 \beta_{2}) q^{28} +(-$$$$19\!\cdots\!50$$$$-$$$$17\!\cdots\!80$$$$\beta_{1} +$$$$33\!\cdots\!96$$$$\beta_{2}) q^{29} +(-$$$$85\!\cdots\!08$$$$+$$$$11\!\cdots\!00$$$$\beta_{1} +$$$$11\!\cdots\!60$$$$\beta_{2}) q^{31} +($$$$48\!\cdots\!28$$$$-$$$$55\!\cdots\!64$$$$\beta_{1} +$$$$24\!\cdots\!60$$$$\beta_{2}) q^{32} +($$$$15\!\cdots\!96$$$$-$$$$64\!\cdots\!90$$$$\beta_{1} +$$$$65\!\cdots\!68$$$$\beta_{2}) q^{34} +(-$$$$79\!\cdots\!40$$$$+$$$$22\!\cdots\!80$$$$\beta_{1} -$$$$49\!\cdots\!12$$$$\beta_{2}) q^{35} +(-$$$$77\!\cdots\!98$$$$-$$$$43\!\cdots\!36$$$$\beta_{1} +$$$$15\!\cdots\!36$$$$\beta_{2}) q^{37} +($$$$10\!\cdots\!40$$$$+$$$$19\!\cdots\!40$$$$\beta_{1} -$$$$16\!\cdots\!56$$$$\beta_{2}) q^{38} +(-$$$$10\!\cdots\!00$$$$-$$$$86\!\cdots\!00$$$$\beta_{1} -$$$$16\!\cdots\!00$$$$\beta_{2}) q^{40} +(-$$$$85\!\cdots\!22$$$$-$$$$87\!\cdots\!00$$$$\beta_{1} -$$$$76\!\cdots\!60$$$$\beta_{2}) q^{41} +($$$$80\!\cdots\!64$$$$-$$$$12\!\cdots\!92$$$$\beta_{1} +$$$$59\!\cdots\!91$$$$\beta_{2}) q^{43} +($$$$34\!\cdots\!44$$$$+$$$$32\!\cdots\!40$$$$\beta_{1} +$$$$35\!\cdots\!92$$$$\beta_{2}) q^{44} +(-$$$$63\!\cdots\!88$$$$+$$$$20\!\cdots\!00$$$$\beta_{1} +$$$$65\!\cdots\!40$$$$\beta_{2}) q^{46} +(-$$$$10\!\cdots\!52$$$$-$$$$31\!\cdots\!04$$$$\beta_{1} +$$$$83\!\cdots\!28$$$$\beta_{2}) q^{47} +($$$$11\!\cdots\!93$$$$-$$$$77\!\cdots\!00$$$$\beta_{1} -$$$$10\!\cdots\!00$$$$\beta_{2}) q^{49} +(-$$$$66\!\cdots\!00$$$$-$$$$79\!\cdots\!25$$$$\beta_{1} +$$$$19\!\cdots\!60$$$$\beta_{2}) q^{50} +(-$$$$57\!\cdots\!68$$$$+$$$$23\!\cdots\!04$$$$\beta_{1} -$$$$13\!\cdots\!72$$$$\beta_{2}) q^{52} +(-$$$$50\!\cdots\!54$$$$-$$$$47\!\cdots\!68$$$$\beta_{1} +$$$$30\!\cdots\!12$$$$\beta_{2}) q^{53} +($$$$13\!\cdots\!80$$$$-$$$$29\!\cdots\!60$$$$\beta_{1} -$$$$83\!\cdots\!86$$$$\beta_{2}) q^{55} +($$$$21\!\cdots\!00$$$$-$$$$60\!\cdots\!80$$$$\beta_{1} +$$$$17\!\cdots\!16$$$$\beta_{2}) q^{56} +(-$$$$34\!\cdots\!60$$$$-$$$$32\!\cdots\!10$$$$\beta_{1} +$$$$71\!\cdots\!24$$$$\beta_{2}) q^{58} +(-$$$$75\!\cdots\!00$$$$+$$$$62\!\cdots\!40$$$$\beta_{1} -$$$$12\!\cdots\!03$$$$\beta_{2}) q^{59} +(-$$$$31\!\cdots\!18$$$$-$$$$68\!\cdots\!00$$$$\beta_{1} -$$$$10\!\cdots\!00$$$$\beta_{2}) q^{61} +($$$$69\!\cdots\!16$$$$-$$$$12\!\cdots\!08$$$$\beta_{1} +$$$$68\!\cdots\!40$$$$\beta_{2}) q^{62} +(-$$$$37\!\cdots\!12$$$$-$$$$13\!\cdots\!40$$$$\beta_{1} +$$$$39\!\cdots\!28$$$$\beta_{2}) q^{64} +($$$$90\!\cdots\!20$$$$+$$$$32\!\cdots\!60$$$$\beta_{1} +$$$$10\!\cdots\!76$$$$\beta_{2}) q^{65} +(-$$$$24\!\cdots\!48$$$$-$$$$23\!\cdots\!16$$$$\beta_{1} +$$$$15\!\cdots\!09$$$$\beta_{2}) q^{67} +(-$$$$18\!\cdots\!56$$$$-$$$$14\!\cdots\!12$$$$\beta_{1} -$$$$27\!\cdots\!76$$$$\beta_{2}) q^{68} +($$$$19\!\cdots\!40$$$$-$$$$58\!\cdots\!80$$$$\beta_{1} -$$$$37\!\cdots\!48$$$$\beta_{2}) q^{70} +($$$$61\!\cdots\!88$$$$+$$$$47\!\cdots\!00$$$$\beta_{1} -$$$$32\!\cdots\!50$$$$\beta_{2}) q^{71} +(-$$$$67\!\cdots\!66$$$$+$$$$18\!\cdots\!88$$$$\beta_{1} -$$$$43\!\cdots\!48$$$$\beta_{2}) q^{73} +(-$$$$55\!\cdots\!36$$$$-$$$$14\!\cdots\!50$$$$\beta_{1} +$$$$80\!\cdots\!00$$$$\beta_{2}) q^{74} +($$$$25\!\cdots\!00$$$$+$$$$93\!\cdots\!60$$$$\beta_{1} -$$$$18\!\cdots\!32$$$$\beta_{2}) q^{76} +(-$$$$19\!\cdots\!64$$$$+$$$$84\!\cdots\!92$$$$\beta_{1} +$$$$42\!\cdots\!24$$$$\beta_{2}) q^{77} +($$$$50\!\cdots\!00$$$$+$$$$26\!\cdots\!80$$$$\beta_{1} +$$$$39\!\cdots\!84$$$$\beta_{2}) q^{79} +($$$$33\!\cdots\!60$$$$-$$$$56\!\cdots\!20$$$$\beta_{1} -$$$$50\!\cdots\!72$$$$\beta_{2}) q^{80} +(-$$$$10\!\cdots\!56$$$$+$$$$20\!\cdots\!78$$$$\beta_{1} +$$$$10\!\cdots\!60$$$$\beta_{2}) q^{82} +($$$$29\!\cdots\!76$$$$+$$$$22\!\cdots\!52$$$$\beta_{1} -$$$$22\!\cdots\!29$$$$\beta_{2}) q^{83} +($$$$14\!\cdots\!40$$$$+$$$$55\!\cdots\!20$$$$\beta_{1} +$$$$25\!\cdots\!52$$$$\beta_{2}) q^{85} +(-$$$$83\!\cdots\!32$$$$+$$$$55\!\cdots\!20$$$$\beta_{1} +$$$$25\!\cdots\!36$$$$\beta_{2}) q^{86} +(-$$$$85\!\cdots\!20$$$$+$$$$82\!\cdots\!80$$$$\beta_{1} +$$$$21\!\cdots\!88$$$$\beta_{2}) q^{88} +(-$$$$67\!\cdots\!50$$$$+$$$$45\!\cdots\!60$$$$\beta_{1} -$$$$10\!\cdots\!92$$$$\beta_{2}) q^{89} +(-$$$$38\!\cdots\!28$$$$-$$$$28\!\cdots\!20$$$$\beta_{1} -$$$$25\!\cdots\!56$$$$\beta_{2}) q^{91} +(-$$$$22\!\cdots\!52$$$$-$$$$40\!\cdots\!84$$$$\beta_{1} +$$$$11\!\cdots\!76$$$$\beta_{2}) q^{92} +(-$$$$44\!\cdots\!44$$$$-$$$$43\!\cdots\!80$$$$\beta_{1} +$$$$16\!\cdots\!36$$$$\beta_{2}) q^{94} +(-$$$$10\!\cdots\!00$$$$-$$$$52\!\cdots\!00$$$$\beta_{1} -$$$$42\!\cdots\!50$$$$\beta_{2}) q^{95} +(-$$$$12\!\cdots\!98$$$$-$$$$11\!\cdots\!96$$$$\beta_{1} +$$$$70\!\cdots\!32$$$$\beta_{2}) q^{97} +(-$$$$80\!\cdots\!36$$$$+$$$$10\!\cdots\!93$$$$\beta_{1} +$$$$10\!\cdots\!00$$$$\beta_{2}) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q + 2209944q^{2} + 9822618421824q^{4} - 535205380774170q^{5} + 301971425665478856q^{7} + 15830863913787348480q^{8} + O(q^{10})$$ $$3q + 2209944q^{2} + 9822618421824q^{4} - 535205380774170q^{5} + 301971425665478856q^{7} + 15830863913787348480q^{8} +$$$$42\!\cdots\!20$$$$q^{10} -$$$$26\!\cdots\!96$$$$q^{11} +$$$$26\!\cdots\!82$$$$q^{13} -$$$$14\!\cdots\!28$$$$q^{14} -$$$$79\!\cdots\!92$$$$q^{16} +$$$$40\!\cdots\!94$$$$q^{17} +$$$$15\!\cdots\!00$$$$q^{19} -$$$$14\!\cdots\!60$$$$q^{20} +$$$$13\!\cdots\!92$$$$q^{22} +$$$$12\!\cdots\!48$$$$q^{23} -$$$$23\!\cdots\!75$$$$q^{25} +$$$$17\!\cdots\!44$$$$q^{26} +$$$$16\!\cdots\!68$$$$q^{28} -$$$$57\!\cdots\!50$$$$q^{29} -$$$$25\!\cdots\!24$$$$q^{31} +$$$$14\!\cdots\!84$$$$q^{32} +$$$$46\!\cdots\!88$$$$q^{34} -$$$$23\!\cdots\!20$$$$q^{35} -$$$$23\!\cdots\!94$$$$q^{37} +$$$$30\!\cdots\!20$$$$q^{38} -$$$$32\!\cdots\!00$$$$q^{40} -$$$$25\!\cdots\!66$$$$q^{41} +$$$$24\!\cdots\!92$$$$q^{43} +$$$$10\!\cdots\!32$$$$q^{44} -$$$$18\!\cdots\!64$$$$q^{46} -$$$$30\!\cdots\!56$$$$q^{47} +$$$$34\!\cdots\!79$$$$q^{49} -$$$$19\!\cdots\!00$$$$q^{50} -$$$$17\!\cdots\!04$$$$q^{52} -$$$$15\!\cdots\!62$$$$q^{53} +$$$$39\!\cdots\!40$$$$q^{55} +$$$$64\!\cdots\!00$$$$q^{56} -$$$$10\!\cdots\!80$$$$q^{58} -$$$$22\!\cdots\!00$$$$q^{59} -$$$$93\!\cdots\!54$$$$q^{61} +$$$$20\!\cdots\!48$$$$q^{62} -$$$$11\!\cdots\!36$$$$q^{64} +$$$$27\!\cdots\!60$$$$q^{65} -$$$$73\!\cdots\!44$$$$q^{67} -$$$$54\!\cdots\!68$$$$q^{68} +$$$$59\!\cdots\!20$$$$q^{70} +$$$$18\!\cdots\!64$$$$q^{71} -$$$$20\!\cdots\!98$$$$q^{73} -$$$$16\!\cdots\!08$$$$q^{74} +$$$$77\!\cdots\!00$$$$q^{76} -$$$$59\!\cdots\!92$$$$q^{77} +$$$$15\!\cdots\!00$$$$q^{79} +$$$$10\!\cdots\!80$$$$q^{80} -$$$$32\!\cdots\!68$$$$q^{82} +$$$$89\!\cdots\!28$$$$q^{83} +$$$$43\!\cdots\!20$$$$q^{85} -$$$$25\!\cdots\!96$$$$q^{86} -$$$$25\!\cdots\!60$$$$q^{88} -$$$$20\!\cdots\!50$$$$q^{89} -$$$$11\!\cdots\!84$$$$q^{91} -$$$$68\!\cdots\!56$$$$q^{92} -$$$$13\!\cdots\!32$$$$q^{94} -$$$$31\!\cdots\!00$$$$q^{95} -$$$$38\!\cdots\!94$$$$q^{97} -$$$$24\!\cdots\!08$$$$q^{98} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-6 \nu^{2} + 343074 \nu + 45033040444$$$$)/8369$$ $$\beta_{2}$$ $$=$$ $$($$$$44520 \nu^{2} + 3528276360 \nu - 334145160094480$$$$)/8369$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} + 7420 \beta_{1}$$$$)/725760$$ $$\nu^{2}$$ $$=$$ $$($$$$57179 \beta_{2} - 588046060 \beta_{1} + 5447196572106240$$$$)/725760$$

## 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
 −91450.2 116336. −24885.9
−3.62707e6 0 4.35955e12 −6.19839e14 0 2.58688e18 1.60917e19 0 2.24820e21
1.2 1.18359e6 0 −7.39521e12 6.39116e14 0 −1.72730e18 −1.91639e19 0 7.56451e20
1.3 4.65343e6 0 1.28583e13 −5.54482e14 0 −5.57604e17 1.89031e19 0 −2.58024e21
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

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

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

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{3} - 2209944 T_{2}^{2} -$$$$15\!\cdots\!56$$$$T_{2} +$$$$19\!\cdots\!64$$ acting on $$S_{44}^{\mathrm{new}}(\Gamma_0(9))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$19976984434430705664 - 15663522502656 T - 2209944 T^{2} + T^{3}$$
$3$ $$T^{3}$$
$5$ $$-$$$$21\!\cdots\!00$$$$-$$$$40\!\cdots\!00$$$$T + 535205380774170 T^{2} + T^{3}$$
$7$ $$-$$$$24\!\cdots\!84$$$$-$$$$49\!\cdots\!36$$$$T - 301971425665478856 T^{2} + T^{3}$$
$11$ $$-$$$$32\!\cdots\!32$$$$-$$$$14\!\cdots\!28$$$$T +$$$$26\!\cdots\!96$$$$T^{2} + T^{3}$$
$13$ $$-$$$$50\!\cdots\!72$$$$+$$$$20\!\cdots\!96$$$$T -$$$$26\!\cdots\!82$$$$T^{2} + T^{3}$$
$17$ $$-$$$$28\!\cdots\!76$$$$-$$$$75\!\cdots\!96$$$$T -$$$$40\!\cdots\!94$$$$T^{2} + T^{3}$$
$19$ $$-$$$$21\!\cdots\!00$$$$-$$$$18\!\cdots\!00$$$$T -$$$$15\!\cdots\!00$$$$T^{2} + T^{3}$$
$23$ $$-$$$$36\!\cdots\!48$$$$-$$$$54\!\cdots\!24$$$$T -$$$$12\!\cdots\!48$$$$T^{2} + T^{3}$$
$29$ $$-$$$$73\!\cdots\!00$$$$+$$$$46\!\cdots\!00$$$$T +$$$$57\!\cdots\!50$$$$T^{2} + T^{3}$$
$31$ $$-$$$$17\!\cdots\!88$$$$+$$$$13\!\cdots\!92$$$$T +$$$$25\!\cdots\!24$$$$T^{2} + T^{3}$$
$37$ $$56\!\cdots\!96$$$$-$$$$26\!\cdots\!16$$$$T +$$$$23\!\cdots\!94$$$$T^{2} + T^{3}$$
$41$ $$-$$$$27\!\cdots\!52$$$$-$$$$40\!\cdots\!48$$$$T +$$$$25\!\cdots\!66$$$$T^{2} + T^{3}$$
$43$ $$-$$$$49\!\cdots\!12$$$$+$$$$14\!\cdots\!36$$$$T -$$$$24\!\cdots\!92$$$$T^{2} + T^{3}$$
$47$ $$-$$$$86\!\cdots\!56$$$$-$$$$33\!\cdots\!76$$$$T +$$$$30\!\cdots\!56$$$$T^{2} + T^{3}$$
$53$ $$90\!\cdots\!92$$$$+$$$$66\!\cdots\!16$$$$T +$$$$15\!\cdots\!62$$$$T^{2} + T^{3}$$
$59$ $$-$$$$78\!\cdots\!00$$$$+$$$$87\!\cdots\!00$$$$T +$$$$22\!\cdots\!00$$$$T^{2} + T^{3}$$
$61$ $$-$$$$87\!\cdots\!68$$$$-$$$$87\!\cdots\!28$$$$T +$$$$93\!\cdots\!54$$$$T^{2} + T^{3}$$
$67$ $$42\!\cdots\!76$$$$-$$$$19\!\cdots\!96$$$$T +$$$$73\!\cdots\!44$$$$T^{2} + T^{3}$$
$71$ $$-$$$$18\!\cdots\!72$$$$+$$$$10\!\cdots\!32$$$$T -$$$$18\!\cdots\!64$$$$T^{2} + T^{3}$$
$73$ $$-$$$$31\!\cdots\!52$$$$+$$$$69\!\cdots\!76$$$$T +$$$$20\!\cdots\!98$$$$T^{2} + T^{3}$$
$79$ $$-$$$$26\!\cdots\!00$$$$-$$$$83\!\cdots\!00$$$$T -$$$$15\!\cdots\!00$$$$T^{2} + T^{3}$$
$83$ $$17\!\cdots\!32$$$$-$$$$30\!\cdots\!44$$$$T -$$$$89\!\cdots\!28$$$$T^{2} + T^{3}$$
$89$ $$10\!\cdots\!00$$$$-$$$$52\!\cdots\!00$$$$T +$$$$20\!\cdots\!50$$$$T^{2} + T^{3}$$
$97$ $$-$$$$26\!\cdots\!44$$$$-$$$$17\!\cdots\!76$$$$T +$$$$38\!\cdots\!94$$$$T^{2} + T^{3}$$