# Properties

 Label 9.34.a.b Level $9$ Weight $34$ Character orbit 9.a Self dual yes Analytic conductor $62.085$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$9 = 3^{2}$$ Weight: $$k$$ $$=$$ $$34$$ Character orbit: $$[\chi]$$ $$=$$ 9.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$62.0845459929$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\mathbb{Q}[x]/(x^{2} - \cdots)$$ Defining polynomial: $$x^{2} - x - 589050$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2^{4}\cdot 3^{2}$$ 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 $$\beta = 72\sqrt{2356201}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 60840 - \beta ) q^{2} + ( 7326116992 - 121680 \beta ) q^{4} + ( 90530768250 - 1004000 \beta ) q^{5} + ( -33576540033400 - 801594864 \beta ) q^{7} + ( 1409375292549120 - 6139193600 \beta ) q^{8} +O(q^{10})$$ $$q +(60840 - \beta) q^{2} +(7326116992 - 121680 \beta) q^{4} +(90530768250 - 1004000 \beta) q^{5} +(-33576540033400 - 801594864 \beta) q^{7} +(1409375292549120 - 6139193600 \beta) q^{8} +(17771296108266000 - 151614128250 \beta) q^{10} +(-66935907720957132 - 1346611783000 \beta) q^{11} +(-1490805239129721970 + 3738595861728 \beta) q^{13} +(7748320631234170176 - 15192491492360 \beta) q^{14} +(97802989555947175936 - 737660590018560 \beta) q^{16} +(39680574630587762670 - 2662090686805056 \beta) q^{17} +(-$$$$68\!\cdots\!00$$$$- 4884703150768920 \beta) q^{19} +($$$$21\!\cdots\!00$$$$- 18371205340628000 \beta) q^{20} +($$$$12\!\cdots\!20$$$$- 14991953156762868 \beta) q^{22} +(-$$$$13\!\cdots\!20$$$$+ 164629195362887632 \beta) q^{23} +(-$$$$95\!\cdots\!25$$$$- 181785782646000000 \beta) q^{25} +(-$$$$13\!\cdots\!52$$$$+ 1718261411357253490 \beta) q^{26} +($$$$94\!\cdots\!80$$$$- 1786984362586217088 \beta) q^{28} +($$$$83\!\cdots\!50$$$$- 6085145360184173920 \beta) q^{29} +(-$$$$31\!\cdots\!08$$$$- 32916497064471576000 \beta) q^{31} +($$$$28\!\cdots\!40$$$$- 89946988381051355136 \beta) q^{32} +($$$$34\!\cdots\!04$$$$-$$$$20\!\cdots\!10$$$$\beta) q^{34} +($$$$67\!\cdots\!00$$$$- 38858152669640668000 \beta) q^{35} +(-$$$$52\!\cdots\!10$$$$-$$$$20\!\cdots\!04$$$$\beta) q^{37} +($$$$18\!\cdots\!80$$$$+$$$$38\!\cdots\!00$$$$\beta) q^{38} +($$$$20\!\cdots\!00$$$$-$$$$19\!\cdots\!00$$$$\beta) q^{40} +(-$$$$13\!\cdots\!22$$$$-$$$$32\!\cdots\!00$$$$\beta) q^{41} +($$$$78\!\cdots\!00$$$$-$$$$88\!\cdots\!52$$$$\beta) q^{43} +($$$$15\!\cdots\!56$$$$-$$$$17\!\cdots\!40$$$$\beta) q^{44} +(-$$$$28\!\cdots\!88$$$$+$$$$23\!\cdots\!00$$$$\beta) q^{46} +(-$$$$27\!\cdots\!20$$$$+$$$$49\!\cdots\!84$$$$\beta) q^{47} +($$$$12\!\cdots\!57$$$$+$$$$53\!\cdots\!00$$$$\beta) q^{49} +(-$$$$36\!\cdots\!00$$$$+$$$$84\!\cdots\!25$$$$\beta) q^{50} +(-$$$$16\!\cdots\!00$$$$+$$$$20\!\cdots\!76$$$$\beta) q^{52} +($$$$13\!\cdots\!10$$$$+$$$$20\!\cdots\!12$$$$\beta) q^{53} +($$$$10\!\cdots\!00$$$$-$$$$54\!\cdots\!00$$$$\beta) q^{55} +($$$$12\!\cdots\!00$$$$-$$$$92\!\cdots\!80$$$$\beta) q^{56} +($$$$12\!\cdots\!80$$$$-$$$$12\!\cdots\!50$$$$\beta) q^{58} +($$$$15\!\cdots\!00$$$$+$$$$31\!\cdots\!60$$$$\beta) q^{59} +(-$$$$28\!\cdots\!18$$$$+$$$$63\!\cdots\!00$$$$\beta) q^{61} +($$$$21\!\cdots\!80$$$$+$$$$11\!\cdots\!08$$$$\beta) q^{62} +($$$$43\!\cdots\!12$$$$-$$$$19\!\cdots\!60$$$$\beta) q^{64} +(-$$$$18\!\cdots\!00$$$$+$$$$18\!\cdots\!00$$$$\beta) q^{65} +($$$$79\!\cdots\!80$$$$-$$$$28\!\cdots\!44$$$$\beta) q^{67} +($$$$42\!\cdots\!60$$$$-$$$$24\!\cdots\!52$$$$\beta) q^{68} +($$$$88\!\cdots\!00$$$$-$$$$91\!\cdots\!00$$$$\beta) q^{70} +($$$$13\!\cdots\!88$$$$+$$$$18\!\cdots\!00$$$$\beta) q^{71} +($$$$47\!\cdots\!70$$$$+$$$$59\!\cdots\!68$$$$\beta) q^{73} +(-$$$$72\!\cdots\!64$$$$+$$$$40\!\cdots\!50$$$$\beta) q^{74} +($$$$22\!\cdots\!00$$$$+$$$$46\!\cdots\!60$$$$\beta) q^{76} +($$$$15\!\cdots\!00$$$$+$$$$98\!\cdots\!48$$$$\beta) q^{77} +(-$$$$42\!\cdots\!00$$$$+$$$$11\!\cdots\!20$$$$\beta) q^{79} +($$$$17\!\cdots\!00$$$$-$$$$16\!\cdots\!00$$$$\beta) q^{80} +($$$$31\!\cdots\!20$$$$-$$$$58\!\cdots\!78$$$$\beta) q^{82} +(-$$$$14\!\cdots\!60$$$$+$$$$24\!\cdots\!92$$$$\beta) q^{83} +($$$$36\!\cdots\!00$$$$-$$$$28\!\cdots\!00$$$$\beta) q^{85} +($$$$15\!\cdots\!68$$$$-$$$$13\!\cdots\!80$$$$\beta) q^{86} +($$$$66\!\cdots\!60$$$$-$$$$14\!\cdots\!00$$$$\beta) q^{88} +(-$$$$66\!\cdots\!50$$$$-$$$$38\!\cdots\!60$$$$\beta) q^{89} +($$$$13\!\cdots\!72$$$$+$$$$10\!\cdots\!80$$$$\beta) q^{91} +(-$$$$34\!\cdots\!80$$$$+$$$$27\!\cdots\!44$$$$\beta) q^{92} +(-$$$$22\!\cdots\!56$$$$+$$$$30\!\cdots\!80$$$$\beta) q^{94} +(-$$$$16\!\cdots\!00$$$$+$$$$24\!\cdots\!00$$$$\beta) q^{95} +(-$$$$18\!\cdots\!30$$$$-$$$$53\!\cdots\!84$$$$\beta) q^{97} +(-$$$$58\!\cdots\!20$$$$+$$$$20\!\cdots\!43$$$$\beta) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 121680q^{2} + 14652233984q^{4} + 181061536500q^{5} - 67153080066800q^{7} + 2818750585098240q^{8} + O(q^{10})$$ $$2q + 121680q^{2} + 14652233984q^{4} + 181061536500q^{5} - 67153080066800q^{7} + 2818750585098240q^{8} + 35542592216532000q^{10} - 133871815441914264q^{11} - 2981610478259443940q^{13} + 15496641262468340352q^{14} +$$$$19\!\cdots\!72$$$$q^{16} + 79361149261175525340q^{17} -$$$$13\!\cdots\!00$$$$q^{19} +$$$$43\!\cdots\!00$$$$q^{20} +$$$$24\!\cdots\!40$$$$q^{22} -$$$$26\!\cdots\!40$$$$q^{23} -$$$$19\!\cdots\!50$$$$q^{25} -$$$$27\!\cdots\!04$$$$q^{26} +$$$$18\!\cdots\!60$$$$q^{28} +$$$$16\!\cdots\!00$$$$q^{29} -$$$$62\!\cdots\!16$$$$q^{31} +$$$$57\!\cdots\!80$$$$q^{32} +$$$$69\!\cdots\!08$$$$q^{34} +$$$$13\!\cdots\!00$$$$q^{35} -$$$$10\!\cdots\!20$$$$q^{37} +$$$$36\!\cdots\!60$$$$q^{38} +$$$$40\!\cdots\!00$$$$q^{40} -$$$$27\!\cdots\!44$$$$q^{41} +$$$$15\!\cdots\!00$$$$q^{43} +$$$$30\!\cdots\!12$$$$q^{44} -$$$$56\!\cdots\!76$$$$q^{46} -$$$$54\!\cdots\!40$$$$q^{47} +$$$$24\!\cdots\!14$$$$q^{49} -$$$$72\!\cdots\!00$$$$q^{50} -$$$$32\!\cdots\!00$$$$q^{52} +$$$$26\!\cdots\!20$$$$q^{53} +$$$$20\!\cdots\!00$$$$q^{55} +$$$$25\!\cdots\!00$$$$q^{56} +$$$$25\!\cdots\!60$$$$q^{58} +$$$$30\!\cdots\!00$$$$q^{59} -$$$$57\!\cdots\!36$$$$q^{61} +$$$$42\!\cdots\!60$$$$q^{62} +$$$$86\!\cdots\!24$$$$q^{64} -$$$$36\!\cdots\!00$$$$q^{65} +$$$$15\!\cdots\!60$$$$q^{67} +$$$$84\!\cdots\!20$$$$q^{68} +$$$$17\!\cdots\!00$$$$q^{70} +$$$$26\!\cdots\!76$$$$q^{71} +$$$$94\!\cdots\!40$$$$q^{73} -$$$$14\!\cdots\!28$$$$q^{74} +$$$$45\!\cdots\!00$$$$q^{76} +$$$$30\!\cdots\!00$$$$q^{77} -$$$$85\!\cdots\!00$$$$q^{79} +$$$$35\!\cdots\!00$$$$q^{80} +$$$$62\!\cdots\!40$$$$q^{82} -$$$$29\!\cdots\!20$$$$q^{83} +$$$$72\!\cdots\!00$$$$q^{85} +$$$$31\!\cdots\!36$$$$q^{86} +$$$$13\!\cdots\!20$$$$q^{88} -$$$$13\!\cdots\!00$$$$q^{89} +$$$$26\!\cdots\!44$$$$q^{91} -$$$$68\!\cdots\!60$$$$q^{92} -$$$$45\!\cdots\!12$$$$q^{94} -$$$$33\!\cdots\!00$$$$q^{95} -$$$$36\!\cdots\!60$$$$q^{97} -$$$$11\!\cdots\!40$$$$q^{98} + O(q^{100})$$

## 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
 767.996 −766.996
−49679.4 0 −6.12189e9 −2.04307e10 0 −1.22168e14 7.30875e14 0 1.01499e15
1.2 171359. 0 2.07741e10 2.01492e11 0 5.50153e13 2.08788e15 0 3.45276e16
 $$n$$: 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 9.34.a.b 2
3.b odd 2 1 1.34.a.a 2
12.b even 2 1 16.34.a.b 2
15.d odd 2 1 25.34.a.a 2
15.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 3.b odd 2 1
9.34.a.b 2 1.a even 1 1 trivial
16.34.a.b 2 12.b even 2 1
25.34.a.a 2 15.d odd 2 1
25.34.b.a 4 15.e even 4 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} - 121680 T_{2} - 8513040384$$ acting on $$S_{34}^{\mathrm{new}}(\Gamma_0(9))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-8513040384 - 121680 T + T^{2}$$
$3$ $$T^{2}$$
$5$ $$-$$$$41\!\cdots\!00$$$$- 181061536500 T + T^{2}$$
$7$ $$-$$$$67\!\cdots\!64$$$$+ 67153080066800 T + T^{2}$$
$11$ $$-$$$$17\!\cdots\!76$$$$+ 133871815441914264 T + T^{2}$$
$13$ $$20\!\cdots\!44$$$$+ 2981610478259443940 T + T^{2}$$
$17$ $$-$$$$84\!\cdots\!24$$$$- 79361149261175525340 T + T^{2}$$
$19$ $$17\!\cdots\!00$$$$+$$$$13\!\cdots\!00$$$$T + T^{2}$$
$23$ $$-$$$$15\!\cdots\!16$$$$+$$$$26\!\cdots\!40$$$$T + T^{2}$$
$29$ $$24\!\cdots\!00$$$$-$$$$16\!\cdots\!00$$$$T + T^{2}$$
$31$ $$-$$$$35\!\cdots\!36$$$$+$$$$62\!\cdots\!16$$$$T + T^{2}$$
$37$ $$22\!\cdots\!56$$$$+$$$$10\!\cdots\!20$$$$T + T^{2}$$
$41$ $$-$$$$10\!\cdots\!16$$$$+$$$$27\!\cdots\!44$$$$T + T^{2}$$
$43$ $$-$$$$35\!\cdots\!36$$$$-$$$$15\!\cdots\!00$$$$T + T^{2}$$
$47$ $$70\!\cdots\!96$$$$+$$$$54\!\cdots\!40$$$$T + T^{2}$$
$53$ $$-$$$$33\!\cdots\!96$$$$-$$$$26\!\cdots\!20$$$$T + T^{2}$$
$59$ $$22\!\cdots\!00$$$$-$$$$30\!\cdots\!00$$$$T + T^{2}$$
$61$ $$-$$$$41\!\cdots\!76$$$$+$$$$57\!\cdots\!36$$$$T + T^{2}$$
$67$ $$54\!\cdots\!76$$$$-$$$$15\!\cdots\!60$$$$T + T^{2}$$
$71$ $$-$$$$26\!\cdots\!56$$$$-$$$$26\!\cdots\!76$$$$T + T^{2}$$
$73$ $$-$$$$42\!\cdots\!16$$$$-$$$$94\!\cdots\!40$$$$T + T^{2}$$
$79$ $$16\!\cdots\!00$$$$+$$$$85\!\cdots\!00$$$$T + T^{2}$$
$83$ $$-$$$$50\!\cdots\!76$$$$+$$$$29\!\cdots\!20$$$$T + T^{2}$$
$89$ $$43\!\cdots\!00$$$$+$$$$13\!\cdots\!00$$$$T + T^{2}$$
$97$ $$-$$$$31\!\cdots\!04$$$$+$$$$36\!\cdots\!60$$$$T + T^{2}$$