# Properties

 Label 9.40.a.c Level $9$ Weight $40$ Character orbit 9.a Self dual yes Analytic conductor $86.706$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$9 = 3^{2}$$ Weight: $$k$$ $$=$$ $$40$$ Character orbit: $$[\chi]$$ $$=$$ 9.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$86.7055962508$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: $$\mathbb{Q}[x]/(x^{3} - \cdots)$$ Defining polynomial: $$x^{3} - 3876249523 x - 18467420411022$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2^{8}\cdot 3^{6}\cdot 5\cdot 7\cdot 13$$ 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 a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -177858 + \beta_{1} ) q^{2} + ( 319147551244 - 227081 \beta_{1} + \beta_{2} ) q^{4} + ( 17793991314810 - 12092638 \beta_{1} - 54 \beta_{2} ) q^{5} + ( -525770892074176 - 12596523938 \beta_{1} + 56758 \beta_{2} ) q^{7} + ( -149112778455452232 + 182657330678 \beta_{1} - 533574 \beta_{2} ) q^{8} +O(q^{10})$$ $$q +(-177858 + \beta_{1}) q^{2} +(319147551244 - 227081 \beta_{1} + \beta_{2}) q^{4} +(17793991314810 - 12092638 \beta_{1} - 54 \beta_{2}) q^{5} +(-525770892074176 - 12596523938 \beta_{1} + 56758 \beta_{2}) q^{7} +(-149112778455452232 + 182657330678 \beta_{1} - 533574 \beta_{2}) q^{8} +(-13289593742193286260 - 3323523968902 \beta_{1} + 4457984 \beta_{2}) q^{10} +($$$$17\!\cdots\!80$$$$- 2617390166972 \beta_{1} + 42093972 \beta_{2}) q^{11} +($$$$18\!\cdots\!30$$$$- 3296990601565580 \beta_{1} + 2302171012 \beta_{2}) q^{13} +(-$$$$10\!\cdots\!84$$$$+ 22915977673056320 \beta_{1} - 29992453632 \beta_{2}) q^{14} +($$$$40\!\cdots\!12$$$$- 247808314916500164 \beta_{1} - 203561787228 \beta_{2}) q^{16} +(-$$$$24\!\cdots\!86$$$$+ 1432579830459091908 \beta_{1} - 73008839916 \beta_{2}) q^{17} +(-$$$$35\!\cdots\!08$$$$- 3709293048070314324 \beta_{1} + 14019280007196 \beta_{2}) q^{19} +(-$$$$10\!\cdots\!20$$$$- 4685500071252208714 \beta_{1} + 24996949090938 \beta_{2}) q^{20} +(-$$$$33\!\cdots\!08$$$$+$$$$19\!\cdots\!84$$$$\beta_{1} - 15518897927168 \beta_{2}) q^{22} +(-$$$$13\!\cdots\!88$$$$- 71070379474158455740 \beta_{1} - 864912659360364 \beta_{2}) q^{23} +(-$$$$39\!\cdots\!25$$$$- 37236871224450578840 \beta_{1} - 3116005674041720 \beta_{2}) q^{25} +(-$$$$30\!\cdots\!92$$$$+$$$$29\!\cdots\!78$$$$\beta_{1} - 4002589901546496 \beta_{2}) q^{26} +($$$$21\!\cdots\!52$$$$-$$$$16\!\cdots\!88$$$$\beta_{1} + 905414279433792 \beta_{2}) q^{28} +(-$$$$24\!\cdots\!54$$$$+$$$$60\!\cdots\!54$$$$\beta_{1} - 10883586973384674 \beta_{2}) q^{29} +(-$$$$12\!\cdots\!52$$$$+$$$$13\!\cdots\!26$$$$\beta_{1} - 51025473408711938 \beta_{2}) q^{31} +(-$$$$12\!\cdots\!04$$$$-$$$$16\!\cdots\!32$$$$\beta_{1} + 107917356575846952 \beta_{2}) q^{32} +($$$$12\!\cdots\!48$$$$-$$$$34\!\cdots\!14$$$$\beta_{1} + 1454956528831466496 \beta_{2}) q^{34} +(-$$$$91\!\cdots\!40$$$$-$$$$42\!\cdots\!08$$$$\beta_{1} + 2423830555879147836 \beta_{2}) q^{35} +($$$$96\!\cdots\!74$$$$-$$$$33\!\cdots\!92$$$$\beta_{1} - 3950416425949998144 \beta_{2}) q^{37} +(-$$$$24\!\cdots\!64$$$$+$$$$22\!\cdots\!08$$$$\beta_{1} - 8006104235315837952 \beta_{2}) q^{38} +($$$$51\!\cdots\!00$$$$+$$$$19\!\cdots\!20$$$$\beta_{1} - 14797692611200750940 \beta_{2}) q^{40} +($$$$46\!\cdots\!42$$$$-$$$$18\!\cdots\!28$$$$\beta_{1} - 41049028078685916228 \beta_{2}) q^{41} +(-$$$$80\!\cdots\!76$$$$-$$$$20\!\cdots\!56$$$$\beta_{1} + 42956960985072135700 \beta_{2}) q^{43} +($$$$71\!\cdots\!04$$$$-$$$$48\!\cdots\!16$$$$\beta_{1} +$$$$17\!\cdots\!72$$$$\beta_{2}) q^{44} +(-$$$$34\!\cdots\!52$$$$-$$$$48\!\cdots\!44$$$$\beta_{1} +$$$$19\!\cdots\!12$$$$\beta_{2}) q^{46} +($$$$36\!\cdots\!68$$$$+$$$$27\!\cdots\!32$$$$\beta_{1} -$$$$15\!\cdots\!28$$$$\beta_{2}) q^{47} +($$$$30\!\cdots\!81$$$$-$$$$70\!\cdots\!80$$$$\beta_{1} -$$$$10\!\cdots\!68$$$$\beta_{2}) q^{49} +($$$$39\!\cdots\!50$$$$-$$$$16\!\cdots\!85$$$$\beta_{1} +$$$$91\!\cdots\!20$$$$\beta_{2}) q^{50} +($$$$19\!\cdots\!96$$$$-$$$$30\!\cdots\!10$$$$\beta_{1} +$$$$28\!\cdots\!50$$$$\beta_{2}) q^{52} +(-$$$$20\!\cdots\!78$$$$-$$$$33\!\cdots\!18$$$$\beta_{1} -$$$$29\!\cdots\!98$$$$\beta_{2}) q^{53} +($$$$24\!\cdots\!80$$$$-$$$$22\!\cdots\!24$$$$\beta_{1} -$$$$78\!\cdots\!92$$$$\beta_{2}) q^{55} +(-$$$$12\!\cdots\!00$$$$+$$$$99\!\cdots\!44$$$$\beta_{1} -$$$$50\!\cdots\!28$$$$\beta_{2}) q^{56} +($$$$93\!\cdots\!28$$$$-$$$$28\!\cdots\!62$$$$\beta_{1} +$$$$93\!\cdots\!36$$$$\beta_{2}) q^{58} +($$$$25\!\cdots\!52$$$$-$$$$10\!\cdots\!28$$$$\beta_{1} -$$$$27\!\cdots\!56$$$$\beta_{2}) q^{59} +(-$$$$23\!\cdots\!66$$$$+$$$$74\!\cdots\!00$$$$\beta_{1} +$$$$14\!\cdots\!48$$$$\beta_{2}) q^{61} +($$$$11\!\cdots\!72$$$$-$$$$28\!\cdots\!92$$$$\beta_{1} +$$$$14\!\cdots\!60$$$$\beta_{2}) q^{62} +(-$$$$11\!\cdots\!52$$$$+$$$$61\!\cdots\!32$$$$\beta_{1} -$$$$87\!\cdots\!04$$$$\beta_{2}) q^{64} +($$$$23\!\cdots\!20$$$$-$$$$87\!\cdots\!96$$$$\beta_{1} +$$$$12\!\cdots\!32$$$$\beta_{2}) q^{65} +(-$$$$17\!\cdots\!24$$$$+$$$$39\!\cdots\!84$$$$\beta_{1} +$$$$35\!\cdots\!32$$$$\beta_{2}) q^{67} +(-$$$$37\!\cdots\!64$$$$+$$$$10\!\cdots\!30$$$$\beta_{1} -$$$$74\!\cdots\!34$$$$\beta_{2}) q^{68} +($$$$12\!\cdots\!40$$$$+$$$$63\!\cdots\!68$$$$\beta_{1} -$$$$78\!\cdots\!56$$$$\beta_{2}) q^{70} +(-$$$$28\!\cdots\!92$$$$-$$$$73\!\cdots\!04$$$$\beta_{1} +$$$$74\!\cdots\!88$$$$\beta_{2}) q^{71} +($$$$21\!\cdots\!38$$$$+$$$$25\!\cdots\!92$$$$\beta_{1} +$$$$69\!\cdots\!00$$$$\beta_{2}) q^{73} +(-$$$$29\!\cdots\!04$$$$-$$$$45\!\cdots\!06$$$$\beta_{1} -$$$$21\!\cdots\!00$$$$\beta_{2}) q^{74} +($$$$42\!\cdots\!12$$$$-$$$$37\!\cdots\!04$$$$\beta_{1} -$$$$29\!\cdots\!04$$$$\beta_{2}) q^{76} +($$$$73\!\cdots\!12$$$$-$$$$26\!\cdots\!32$$$$\beta_{1} +$$$$91\!\cdots\!40$$$$\beta_{2}) q^{77} +(-$$$$54\!\cdots\!00$$$$-$$$$11\!\cdots\!74$$$$\beta_{1} +$$$$11\!\cdots\!38$$$$\beta_{2}) q^{79} +($$$$62\!\cdots\!60$$$$+$$$$17\!\cdots\!12$$$$\beta_{1} -$$$$72\!\cdots\!04$$$$\beta_{2}) q^{80} +(-$$$$16\!\cdots\!12$$$$-$$$$10\!\cdots\!66$$$$\beta_{1} -$$$$59\!\cdots\!24$$$$\beta_{2}) q^{82} +(-$$$$19\!\cdots\!16$$$$-$$$$12\!\cdots\!88$$$$\beta_{1} +$$$$40\!\cdots\!00$$$$\beta_{2}) q^{83} +(-$$$$17\!\cdots\!60$$$$-$$$$47\!\cdots\!32$$$$\beta_{1} +$$$$26\!\cdots\!44$$$$\beta_{2}) q^{85} +(-$$$$28\!\cdots\!00$$$$-$$$$62\!\cdots\!88$$$$\beta_{1} -$$$$33\!\cdots\!56$$$$\beta_{2}) q^{86} +(-$$$$34\!\cdots\!88$$$$+$$$$37\!\cdots\!28$$$$\beta_{1} -$$$$93\!\cdots\!28$$$$\beta_{2}) q^{88} +(-$$$$60\!\cdots\!62$$$$+$$$$73\!\cdots\!36$$$$\beta_{1} -$$$$13\!\cdots\!40$$$$\beta_{2}) q^{89} +($$$$77\!\cdots\!08$$$$-$$$$10\!\cdots\!60$$$$\beta_{1} +$$$$11\!\cdots\!64$$$$\beta_{2}) q^{91} +(-$$$$32\!\cdots\!44$$$$+$$$$10\!\cdots\!88$$$$\beta_{1} -$$$$67\!\cdots\!28$$$$\beta_{2}) q^{92} +(-$$$$41\!\cdots\!40$$$$+$$$$29\!\cdots\!80$$$$\beta_{1} +$$$$75\!\cdots\!36$$$$\beta_{2}) q^{94} +(-$$$$28\!\cdots\!00$$$$+$$$$34\!\cdots\!60$$$$\beta_{1} +$$$$78\!\cdots\!80$$$$\beta_{2}) q^{95} +(-$$$$33\!\cdots\!14$$$$+$$$$51\!\cdots\!24$$$$\beta_{1} +$$$$22\!\cdots\!12$$$$\beta_{2}) q^{97} +(-$$$$64\!\cdots\!70$$$$-$$$$64\!\cdots\!91$$$$\beta_{1} -$$$$39\!\cdots\!56$$$$\beta_{2}) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q - 533574q^{2} + 957442653732q^{4} + 53381973944430q^{5} - 1577312676222528q^{7} - 447338335366356696q^{8} + O(q^{10})$$ $$3q - 533574q^{2} + 957442653732q^{4} + 53381973944430q^{5} - 1577312676222528q^{7} - 447338335366356696q^{8} - 39868781226579858780q^{10} +$$$$53\!\cdots\!40$$$$q^{11} +$$$$54\!\cdots\!90$$$$q^{13} -$$$$31\!\cdots\!52$$$$q^{14} +$$$$12\!\cdots\!36$$$$q^{16} -$$$$72\!\cdots\!58$$$$q^{17} -$$$$10\!\cdots\!24$$$$q^{19} -$$$$30\!\cdots\!60$$$$q^{20} -$$$$10\!\cdots\!24$$$$q^{22} -$$$$41\!\cdots\!64$$$$q^{23} -$$$$11\!\cdots\!75$$$$q^{25} -$$$$92\!\cdots\!76$$$$q^{26} +$$$$64\!\cdots\!56$$$$q^{28} -$$$$72\!\cdots\!62$$$$q^{29} -$$$$38\!\cdots\!56$$$$q^{31} -$$$$37\!\cdots\!12$$$$q^{32} +$$$$37\!\cdots\!44$$$$q^{34} -$$$$27\!\cdots\!20$$$$q^{35} +$$$$29\!\cdots\!22$$$$q^{37} -$$$$74\!\cdots\!92$$$$q^{38} +$$$$15\!\cdots\!00$$$$q^{40} +$$$$13\!\cdots\!26$$$$q^{41} -$$$$24\!\cdots\!28$$$$q^{43} +$$$$21\!\cdots\!12$$$$q^{44} -$$$$10\!\cdots\!56$$$$q^{46} +$$$$10\!\cdots\!04$$$$q^{47} +$$$$92\!\cdots\!43$$$$q^{49} +$$$$11\!\cdots\!50$$$$q^{50} +$$$$59\!\cdots\!88$$$$q^{52} -$$$$62\!\cdots\!34$$$$q^{53} +$$$$72\!\cdots\!40$$$$q^{55} -$$$$36\!\cdots\!00$$$$q^{56} +$$$$28\!\cdots\!84$$$$q^{58} +$$$$75\!\cdots\!56$$$$q^{59} -$$$$71\!\cdots\!98$$$$q^{61} +$$$$33\!\cdots\!16$$$$q^{62} -$$$$35\!\cdots\!56$$$$q^{64} +$$$$71\!\cdots\!60$$$$q^{65} -$$$$51\!\cdots\!72$$$$q^{67} -$$$$11\!\cdots\!92$$$$q^{68} +$$$$38\!\cdots\!20$$$$q^{70} -$$$$84\!\cdots\!76$$$$q^{71} +$$$$63\!\cdots\!14$$$$q^{73} -$$$$89\!\cdots\!12$$$$q^{74} +$$$$12\!\cdots\!36$$$$q^{76} +$$$$22\!\cdots\!36$$$$q^{77} -$$$$16\!\cdots\!00$$$$q^{79} +$$$$18\!\cdots\!80$$$$q^{80} -$$$$48\!\cdots\!36$$$$q^{82} -$$$$59\!\cdots\!48$$$$q^{83} -$$$$52\!\cdots\!80$$$$q^{85} -$$$$85\!\cdots\!00$$$$q^{86} -$$$$10\!\cdots\!64$$$$q^{88} -$$$$18\!\cdots\!86$$$$q^{89} +$$$$23\!\cdots\!24$$$$q^{91} -$$$$96\!\cdots\!32$$$$q^{92} -$$$$12\!\cdots\!20$$$$q^{94} -$$$$84\!\cdots\!00$$$$q^{95} -$$$$99\!\cdots\!42$$$$q^{97} -$$$$19\!\cdots\!10$$$$q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - 3876249523 x - 18467420411022$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$18 \nu$$ $$\beta_{2}$$ $$=$$ $$324 \nu^{2} - 2315430 \nu - 837269896968$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$$$/18$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{2} + 128635 \beta_{1} + 837269896968$$$$)/324$$

## 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
 −59724.7 −4792.65 64517.4
−1.25290e6 0 1.02001e12 6.13018e12 0 3.89397e16 −5.89182e17 0 −7.68052e18
1.2 −264126. 0 −4.79993e11 6.30487e13 0 −4.59086e16 2.71983e17 0 −1.66528e19
1.3 983454. 0 4.17427e11 −1.57969e13 0 5.39162e15 −1.30140e17 0 −1.55355e19
 $$n$$: 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 9.40.a.c 3
3.b odd 2 1 3.40.a.b 3

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{3} + 533574 T_{2}^{2} -$$$$11\!\cdots\!60$$$$T_{2} -$$$$32\!\cdots\!08$$ acting on $$S_{40}^{\mathrm{new}}(\Gamma_0(9))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-325448454458769408 - 1161004440960 T + 533574 T^{2} + T^{3}$$
$3$ $$T^{3}$$
$5$ $$61\!\cdots\!00$$$$-$$$$70\!\cdots\!00$$$$T - 53381973944430 T^{2} + T^{3}$$
$7$ $$96\!\cdots\!40$$$$-$$$$18\!\cdots\!44$$$$T + 1577312676222528 T^{2} + T^{3}$$
$11$ $$-$$$$53\!\cdots\!36$$$$+$$$$93\!\cdots\!48$$$$T -$$$$53\!\cdots\!40$$$$T^{2} + T^{3}$$
$13$ $$16\!\cdots\!36$$$$-$$$$63\!\cdots\!92$$$$T -$$$$54\!\cdots\!90$$$$T^{2} + T^{3}$$
$17$ $$-$$$$77\!\cdots\!44$$$$-$$$$24\!\cdots\!12$$$$T +$$$$72\!\cdots\!58$$$$T^{2} + T^{3}$$
$19$ $$-$$$$28\!\cdots\!00$$$$-$$$$78\!\cdots\!40$$$$T +$$$$10\!\cdots\!24$$$$T^{2} + T^{3}$$
$23$ $$-$$$$14\!\cdots\!40$$$$-$$$$32\!\cdots\!96$$$$T +$$$$41\!\cdots\!64$$$$T^{2} + T^{3}$$
$29$ $$12\!\cdots\!00$$$$+$$$$16\!\cdots\!80$$$$T +$$$$72\!\cdots\!62$$$$T^{2} + T^{3}$$
$31$ $$65\!\cdots\!00$$$$-$$$$23\!\cdots\!16$$$$T +$$$$38\!\cdots\!56$$$$T^{2} + T^{3}$$
$37$ $$53\!\cdots\!80$$$$-$$$$19\!\cdots\!84$$$$T -$$$$29\!\cdots\!22$$$$T^{2} + T^{3}$$
$41$ $$72\!\cdots\!00$$$$-$$$$12\!\cdots\!56$$$$T -$$$$13\!\cdots\!26$$$$T^{2} + T^{3}$$
$43$ $$39\!\cdots\!68$$$$+$$$$17\!\cdots\!56$$$$T +$$$$24\!\cdots\!28$$$$T^{2} + T^{3}$$
$47$ $$-$$$$42\!\cdots\!16$$$$+$$$$38\!\cdots\!04$$$$T -$$$$10\!\cdots\!04$$$$T^{2} + T^{3}$$
$53$ $$61\!\cdots\!80$$$$-$$$$51\!\cdots\!96$$$$T +$$$$62\!\cdots\!34$$$$T^{2} + T^{3}$$
$59$ $$56\!\cdots\!00$$$$-$$$$50\!\cdots\!80$$$$T -$$$$75\!\cdots\!56$$$$T^{2} + T^{3}$$
$61$ $$-$$$$27\!\cdots\!52$$$$-$$$$53\!\cdots\!24$$$$T +$$$$71\!\cdots\!98$$$$T^{2} + T^{3}$$
$67$ $$-$$$$88\!\cdots\!76$$$$-$$$$16\!\cdots\!72$$$$T +$$$$51\!\cdots\!72$$$$T^{2} + T^{3}$$
$71$ $$-$$$$61\!\cdots\!12$$$$-$$$$72\!\cdots\!08$$$$T +$$$$84\!\cdots\!76$$$$T^{2} + T^{3}$$
$73$ $$-$$$$42\!\cdots\!60$$$$-$$$$81\!\cdots\!96$$$$T -$$$$63\!\cdots\!14$$$$T^{2} + T^{3}$$
$79$ $$-$$$$23\!\cdots\!00$$$$-$$$$13\!\cdots\!00$$$$T +$$$$16\!\cdots\!00$$$$T^{2} + T^{3}$$
$83$ $$76\!\cdots\!76$$$$+$$$$11\!\cdots\!80$$$$T +$$$$59\!\cdots\!48$$$$T^{2} + T^{3}$$
$89$ $$-$$$$15\!\cdots\!00$$$$+$$$$41\!\cdots\!20$$$$T +$$$$18\!\cdots\!86$$$$T^{2} + T^{3}$$
$97$ $$-$$$$15\!\cdots\!56$$$$-$$$$29\!\cdots\!12$$$$T +$$$$99\!\cdots\!42$$$$T^{2} + T^{3}$$