# Properties

 Label 9.36.a.a Level $9$ Weight $36$ Character orbit 9.a Self dual yes Analytic conductor $69.836$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$9 = 3^{2}$$ Weight: $$k$$ $$=$$ $$36$$ Character orbit: $$[\chi]$$ $$=$$ 9.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$69.8356175703$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2196841})$$ Defining polynomial: $$x^{2} - x - 549210$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2^{4}\cdot 3\cdot 7$$ Twist minimal: no (minimal twist has level 3) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 168\sqrt{2196841}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 30456 - \beta ) q^{2} + ( 28571469952 - 60912 \beta ) q^{4} + ( 666889748370 + 1210960 \beta ) q^{5} + ( -600756391476472 + 142131024 \beta ) q^{7} + ( 3600478240192512 + 3933132544 \beta ) q^{8} +O(q^{10})$$ $$q +(30456 - \beta) q^{2} +(28571469952 - 60912 \beta) q^{4} +(666889748370 + 1210960 \beta) q^{5} +(-600756391476472 + 142131024 \beta) q^{7} +(3600478240192512 + 3933132544 \beta) q^{8} +(-54773134183051920 - 630008750610 \beta) q^{10} +(737221926110660316 + 3560156783456 \beta) q^{11} +(15002606829102839414 - 161668794040416 \beta) q^{13} +(-27109277558313104448 + 605085133943416 \beta) q^{14} +(-$$$$11\!\cdots\!60$$$$- 1387770371960832 \beta) q^{16} +(-$$$$30\!\cdots\!66$$$$- 5140788294161760 \beta) q^{17} +(-$$$$80\!\cdots\!32$$$$+ 101300863898712096 \beta) q^{19} +($$$$14\!\cdots\!60$$$$- 6022681099639520 \beta) q^{20} +(-$$$$19\!\cdots\!08$$$$- 628793791113724380 \beta) q^{22} +(-$$$$24\!\cdots\!36$$$$+ 3053203101283215712 \beta) q^{23} +(-$$$$23\!\cdots\!25$$$$+ 1615153619372270400 \beta) q^{25} +($$$$10\!\cdots\!28$$$$- 19926391620397749110 \beta) q^{26} +(-$$$$17\!\cdots\!36$$$$+ 40654165599077853312 \beta) q^{28} +($$$$39\!\cdots\!74$$$$- 22123136384176647952 \beta) q^{29} +(-$$$$60\!\cdots\!24$$$$-$$$$15\!\cdots\!92$$$$\beta) q^{31} +(-$$$$71\!\cdots\!88$$$$+$$$$93\!\cdots\!76$$$$\beta) q^{32} +($$$$22\!\cdots\!44$$$$+$$$$29\!\cdots\!06$$$$\beta) q^{34} +(-$$$$38\!\cdots\!80$$$$-$$$$63\!\cdots\!40$$$$\beta) q^{35} +($$$$83\!\cdots\!58$$$$+$$$$15\!\cdots\!04$$$$\beta) q^{37} +(-$$$$65\!\cdots\!56$$$$+$$$$11\!\cdots\!08$$$$\beta) q^{38} +($$$$26\!\cdots\!00$$$$+$$$$69\!\cdots\!00$$$$\beta) q^{40} +($$$$16\!\cdots\!94$$$$+$$$$49\!\cdots\!32$$$$\beta) q^{41} +($$$$50\!\cdots\!60$$$$+$$$$25\!\cdots\!56$$$$\beta) q^{43} +($$$$76\!\cdots\!84$$$$+$$$$56\!\cdots\!20$$$$\beta) q^{44} +(-$$$$19\!\cdots\!24$$$$+$$$$33\!\cdots\!08$$$$\beta) q^{46} +($$$$26\!\cdots\!60$$$$-$$$$10\!\cdots\!88$$$$\beta) q^{47} +(-$$$$16\!\cdots\!75$$$$-$$$$17\!\cdots\!56$$$$\beta) q^{49} +(-$$$$17\!\cdots\!00$$$$+$$$$24\!\cdots\!25$$$$\beta) q^{50} +($$$$10\!\cdots\!56$$$$-$$$$55\!\cdots\!00$$$$\beta) q^{52} +($$$$15\!\cdots\!34$$$$-$$$$99\!\cdots\!88$$$$\beta) q^{53} +($$$$75\!\cdots\!60$$$$+$$$$32\!\cdots\!80$$$$\beta) q^{55} +(-$$$$21\!\cdots\!60$$$$-$$$$18\!\cdots\!80$$$$\beta) q^{56} +($$$$25\!\cdots\!12$$$$-$$$$40\!\cdots\!86$$$$\beta) q^{58} +(-$$$$41\!\cdots\!32$$$$-$$$$66\!\cdots\!64$$$$\beta) q^{59} +(-$$$$20\!\cdots\!70$$$$-$$$$26\!\cdots\!24$$$$\beta) q^{61} +($$$$76\!\cdots\!84$$$$+$$$$55\!\cdots\!72$$$$\beta) q^{62} +(-$$$$22\!\cdots\!32$$$$+$$$$14\!\cdots\!20$$$$\beta) q^{64} +(-$$$$21\!\cdots\!60$$$$-$$$$89\!\cdots\!80$$$$\beta) q^{65} +($$$$48\!\cdots\!56$$$$+$$$$28\!\cdots\!40$$$$\beta) q^{67} +(-$$$$68\!\cdots\!52$$$$+$$$$41\!\cdots\!72$$$$\beta) q^{68} +($$$$27\!\cdots\!80$$$$+$$$$37\!\cdots\!40$$$$\beta) q^{70} +(-$$$$22\!\cdots\!12$$$$+$$$$10\!\cdots\!80$$$$\beta) q^{71} +(-$$$$68\!\cdots\!54$$$$-$$$$82\!\cdots\!92$$$$\beta) q^{73} +(-$$$$90\!\cdots\!88$$$$-$$$$37\!\cdots\!34$$$$\beta) q^{74} +(-$$$$61\!\cdots\!32$$$$+$$$$33\!\cdots\!76$$$$\beta) q^{76} +(-$$$$41\!\cdots\!56$$$$-$$$$20\!\cdots\!48$$$$\beta) q^{77} +(-$$$$53\!\cdots\!40$$$$-$$$$64\!\cdots\!20$$$$\beta) q^{79} +(-$$$$84\!\cdots\!80$$$$-$$$$22\!\cdots\!40$$$$\beta) q^{80} +(-$$$$25\!\cdots\!24$$$$-$$$$14\!\cdots\!02$$$$\beta) q^{82} +($$$$27\!\cdots\!72$$$$-$$$$21\!\cdots\!32$$$$\beta) q^{83} +(-$$$$24\!\cdots\!20$$$$-$$$$71\!\cdots\!60$$$$\beta) q^{85} +(-$$$$15\!\cdots\!44$$$$+$$$$27\!\cdots\!76$$$$\beta) q^{86} +($$$$35\!\cdots\!68$$$$+$$$$15\!\cdots\!76$$$$\beta) q^{88} +(-$$$$12\!\cdots\!18$$$$-$$$$35\!\cdots\!36$$$$\beta) q^{89} +(-$$$$10\!\cdots\!64$$$$+$$$$99\!\cdots\!88$$$$\beta) q^{91} +(-$$$$18\!\cdots\!68$$$$+$$$$10\!\cdots\!56$$$$\beta) q^{92} +($$$$72\!\cdots\!52$$$$-$$$$29\!\cdots\!88$$$$\beta) q^{94} +($$$$22\!\cdots\!00$$$$+$$$$57\!\cdots\!00$$$$\beta) q^{95} +($$$$41\!\cdots\!26$$$$+$$$$60\!\cdots\!00$$$$\beta) q^{97} +($$$$10\!\cdots\!04$$$$+$$$$11\!\cdots\!39$$$$\beta) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 60912q^{2} + 57142939904q^{4} + 1333779496740q^{5} - 1201512782952944q^{7} + 7200956480385024q^{8} + O(q^{10})$$ $$2q + 60912q^{2} + 57142939904q^{4} + 1333779496740q^{5} - 1201512782952944q^{7} + 7200956480385024q^{8} - 109546268366103840q^{10} + 1474443852221320632q^{11} + 30005213658205678828q^{13} - 54218555116626208896q^{14} -$$$$22\!\cdots\!20$$$$q^{16} -$$$$61\!\cdots\!32$$$$q^{17} -$$$$16\!\cdots\!64$$$$q^{19} +$$$$28\!\cdots\!20$$$$q^{20} -$$$$39\!\cdots\!16$$$$q^{22} -$$$$49\!\cdots\!72$$$$q^{23} -$$$$47\!\cdots\!50$$$$q^{25} +$$$$20\!\cdots\!56$$$$q^{26} -$$$$35\!\cdots\!72$$$$q^{28} +$$$$78\!\cdots\!48$$$$q^{29} -$$$$12\!\cdots\!48$$$$q^{31} -$$$$14\!\cdots\!76$$$$q^{32} +$$$$44\!\cdots\!88$$$$q^{34} -$$$$77\!\cdots\!60$$$$q^{35} +$$$$16\!\cdots\!16$$$$q^{37} -$$$$13\!\cdots\!12$$$$q^{38} +$$$$53\!\cdots\!00$$$$q^{40} +$$$$32\!\cdots\!88$$$$q^{41} +$$$$10\!\cdots\!20$$$$q^{43} +$$$$15\!\cdots\!68$$$$q^{44} -$$$$39\!\cdots\!48$$$$q^{46} +$$$$52\!\cdots\!20$$$$q^{47} -$$$$33\!\cdots\!50$$$$q^{49} -$$$$34\!\cdots\!00$$$$q^{50} +$$$$20\!\cdots\!12$$$$q^{52} +$$$$31\!\cdots\!68$$$$q^{53} +$$$$15\!\cdots\!20$$$$q^{55} -$$$$42\!\cdots\!20$$$$q^{56} +$$$$51\!\cdots\!24$$$$q^{58} -$$$$82\!\cdots\!64$$$$q^{59} -$$$$40\!\cdots\!40$$$$q^{61} +$$$$15\!\cdots\!68$$$$q^{62} -$$$$44\!\cdots\!64$$$$q^{64} -$$$$42\!\cdots\!20$$$$q^{65} +$$$$96\!\cdots\!12$$$$q^{67} -$$$$13\!\cdots\!04$$$$q^{68} +$$$$54\!\cdots\!60$$$$q^{70} -$$$$44\!\cdots\!24$$$$q^{71} -$$$$13\!\cdots\!08$$$$q^{73} -$$$$18\!\cdots\!76$$$$q^{74} -$$$$12\!\cdots\!64$$$$q^{76} -$$$$82\!\cdots\!12$$$$q^{77} -$$$$10\!\cdots\!80$$$$q^{79} -$$$$16\!\cdots\!60$$$$q^{80} -$$$$51\!\cdots\!48$$$$q^{82} +$$$$55\!\cdots\!44$$$$q^{83} -$$$$48\!\cdots\!40$$$$q^{85} -$$$$31\!\cdots\!88$$$$q^{86} +$$$$70\!\cdots\!36$$$$q^{88} -$$$$24\!\cdots\!36$$$$q^{89} -$$$$20\!\cdots\!28$$$$q^{91} -$$$$37\!\cdots\!36$$$$q^{92} +$$$$14\!\cdots\!04$$$$q^{94} +$$$$45\!\cdots\!00$$$$q^{95} +$$$$83\!\cdots\!52$$$$q^{97} +$$$$20\!\cdots\!08$$$$q^{98} + O(q^{100})$$

## 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
 741.587 −740.587
−218549. 0 1.34041e10 9.68425e11 0 −5.65365e14 4.57985e15 0 −2.11649e17
1.2 279461. 0 4.37389e10 3.65354e11 0 −6.36148e14 2.62111e15 0 1.02102e17
 $$n$$: 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 9.36.a.a 2
3.b odd 2 1 3.36.a.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.36.a.a 2 3.b odd 2 1
9.36.a.a 2 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} - 60912 T_{2} - 61076072448$$ acting on $$S_{36}^{\mathrm{new}}(\Gamma_0(9))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 60912 T + 7643404288 T^{2} - 2092920383471616 T^{3} +$$$$11\!\cdots\!24$$$$T^{4}$$
$3$ 1
$5$ $$1 - 1333779496740 T +$$$$61\!\cdots\!50$$$$T^{2} -$$$$38\!\cdots\!00$$$$T^{3} +$$$$84\!\cdots\!25$$$$T^{4}$$
$7$ $$1 + 1201512782952944 T +$$$$11\!\cdots\!86$$$$T^{2} +$$$$45\!\cdots\!92$$$$T^{3} +$$$$14\!\cdots\!49$$$$T^{4}$$
$11$ $$1 - 1474443852221320632 T +$$$$53\!\cdots\!34$$$$T^{2} -$$$$41\!\cdots\!32$$$$T^{3} +$$$$78\!\cdots\!01$$$$T^{4}$$
$13$ $$1 - 30005213658205678828 T +$$$$55\!\cdots\!06$$$$T^{2} -$$$$29\!\cdots\!96$$$$T^{3} +$$$$94\!\cdots\!49$$$$T^{4}$$
$17$ $$1 +$$$$61\!\cdots\!32$$$$T +$$$$31\!\cdots\!42$$$$T^{2} +$$$$71\!\cdots\!76$$$$T^{3} +$$$$13\!\cdots\!49$$$$T^{4}$$
$19$ $$1 +$$$$16\!\cdots\!64$$$$T +$$$$56\!\cdots\!78$$$$T^{2} +$$$$91\!\cdots\!36$$$$T^{3} +$$$$32\!\cdots\!01$$$$T^{4}$$
$23$ $$1 +$$$$49\!\cdots\!72$$$$T +$$$$39\!\cdots\!14$$$$T^{2} +$$$$22\!\cdots\!04$$$$T^{3} +$$$$20\!\cdots\!49$$$$T^{4}$$
$29$ $$1 -$$$$78\!\cdots\!48$$$$T +$$$$45\!\cdots\!38$$$$T^{2} -$$$$12\!\cdots\!52$$$$T^{3} +$$$$23\!\cdots\!01$$$$T^{4}$$
$31$ $$1 +$$$$12\!\cdots\!48$$$$T +$$$$33\!\cdots\!02$$$$T^{2} +$$$$19\!\cdots\!48$$$$T^{3} +$$$$24\!\cdots\!01$$$$T^{4}$$
$37$ $$1 -$$$$16\!\cdots\!16$$$$T +$$$$20\!\cdots\!06$$$$T^{2} -$$$$12\!\cdots\!88$$$$T^{3} +$$$$59\!\cdots\!49$$$$T^{4}$$
$41$ $$1 -$$$$32\!\cdots\!88$$$$T +$$$$68\!\cdots\!22$$$$T^{2} -$$$$92\!\cdots\!88$$$$T^{3} +$$$$78\!\cdots\!01$$$$T^{4}$$
$43$ $$1 -$$$$10\!\cdots\!20$$$$T -$$$$10\!\cdots\!10$$$$T^{2} -$$$$15\!\cdots\!40$$$$T^{3} +$$$$22\!\cdots\!49$$$$T^{4}$$
$47$ $$1 -$$$$52\!\cdots\!20$$$$T +$$$$66\!\cdots\!90$$$$T^{2} -$$$$17\!\cdots\!60$$$$T^{3} +$$$$11\!\cdots\!49$$$$T^{4}$$
$53$ $$1 -$$$$31\!\cdots\!68$$$$T -$$$$16\!\cdots\!26$$$$T^{2} -$$$$69\!\cdots\!76$$$$T^{3} +$$$$50\!\cdots\!49$$$$T^{4}$$
$59$ $$1 +$$$$82\!\cdots\!64$$$$T +$$$$20\!\cdots\!58$$$$T^{2} +$$$$78\!\cdots\!36$$$$T^{3} +$$$$91\!\cdots\!01$$$$T^{4}$$
$61$ $$1 +$$$$40\!\cdots\!40$$$$T +$$$$98\!\cdots\!18$$$$T^{2} +$$$$12\!\cdots\!40$$$$T^{3} +$$$$93\!\cdots\!01$$$$T^{4}$$
$67$ $$1 -$$$$96\!\cdots\!12$$$$T +$$$$11\!\cdots\!22$$$$T^{2} -$$$$78\!\cdots\!16$$$$T^{3} +$$$$66\!\cdots\!49$$$$T^{4}$$
$71$ $$1 +$$$$44\!\cdots\!24$$$$T +$$$$99\!\cdots\!46$$$$T^{2} +$$$$27\!\cdots\!24$$$$T^{3} +$$$$38\!\cdots\!01$$$$T^{4}$$
$73$ $$1 +$$$$13\!\cdots\!08$$$$T +$$$$29\!\cdots\!54$$$$T^{2} +$$$$22\!\cdots\!56$$$$T^{3} +$$$$27\!\cdots\!49$$$$T^{4}$$
$79$ $$1 +$$$$10\!\cdots\!80$$$$T +$$$$29\!\cdots\!98$$$$T^{2} +$$$$27\!\cdots\!20$$$$T^{3} +$$$$68\!\cdots\!01$$$$T^{4}$$
$83$ $$1 -$$$$55\!\cdots\!44$$$$T +$$$$36\!\cdots\!82$$$$T^{2} -$$$$80\!\cdots\!08$$$$T^{3} +$$$$21\!\cdots\!49$$$$T^{4}$$
$89$ $$1 +$$$$24\!\cdots\!36$$$$T +$$$$40\!\cdots\!58$$$$T^{2} +$$$$40\!\cdots\!64$$$$T^{3} +$$$$28\!\cdots\!01$$$$T^{4}$$
$97$ $$1 -$$$$83\!\cdots\!52$$$$T +$$$$66\!\cdots\!62$$$$T^{2} -$$$$28\!\cdots\!36$$$$T^{3} +$$$$11\!\cdots\!49$$$$T^{4}$$