# Properties

 Label 9.32.a.a Level $9$ Weight $32$ Character orbit 9.a Self dual yes Analytic conductor $54.789$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$9 = 3^{2}$$ Weight: $$k$$ $$=$$ $$32$$ Character orbit: $$[\chi]$$ $$=$$ 9.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$54.7894195371$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\mathbb{Q}[x]/(x^{2} - \cdots)$$ Defining polynomial: $$x^{2} - x - 4573872$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2^{3}\cdot 3$$ 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 $$\beta = 12\sqrt{18295489}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -19980 - \beta ) q^{2} + ( 886267168 + 39960 \beta ) q^{4} + ( 9695609010 + 1418560 \beta ) q^{5} + ( 15128763788600 - 71928864 \beta ) q^{7} + ( -80077529352960 + 462815680 \beta ) q^{8} +O(q^{10})$$ $$q +(-19980 - \beta) q^{2} +(886267168 + 39960 \beta) q^{4} +(9695609010 + 1418560 \beta) q^{5} +(15128763788600 - 71928864 \beta) q^{7} +(-80077529352960 + 462815680 \beta) q^{8} +(-3930986106140760 - 38038437810 \beta) q^{10} +(3891176872559388 - 29000909200 \beta) q^{11} +(37354476525130310 - 4661016429888 \beta) q^{13} +(-112772481922620576 - 13691625085880 \beta) q^{14} +(-1522606456842450944 - 14982974507520 \beta) q^{16} +(-8612303914493544690 - 45085348093056 \beta) q^{17} +(-6185281664011082020 - 157805792764560 \beta) q^{19} +($$$$15\!\cdots\!80$$$$+ 1644659689877680 \beta) q^{20} +(-1341356516498345040 - 3311738706743388 \beta) q^{22} +(-$$$$94\!\cdots\!40$$$$+ 18557618179251808 \beta) q^{23} +($$$$73\!\cdots\!75$$$$+ 27507606234451200 \beta) q^{25} +($$$$11\!\cdots\!08$$$$+ 55772631744031930 \beta) q^{26} +($$$$58\!\cdots\!60$$$$+ 540797210397718848 \beta) q^{28} +(-$$$$64\!\cdots\!70$$$$- 234192357384448960 \beta) q^{29} +($$$$62\!\cdots\!32$$$$+ 2412621171020361600 \beta) q^{31} +($$$$24\!\cdots\!20$$$$+ 828077182664699904 \beta) q^{32} +($$$$29\!\cdots\!96$$$$+ 9513109169392803570 \beta) q^{34} +(-$$$$12\!\cdots\!40$$$$+ 20763665018078951360 \beta) q^{35} +(-$$$$41\!\cdots\!30$$$$+ 32224244113578511296 \beta) q^{37} +($$$$53\!\cdots\!60$$$$+ 9338241403446990820 \beta) q^{38} +($$$$95\!\cdots\!00$$$$-$$$$10\!\cdots\!00$$$$\beta) q^{40} +(-$$$$43\!\cdots\!42$$$$-$$$$18\!\cdots\!00$$$$\beta) q^{41} +(-$$$$91\!\cdots\!00$$$$-$$$$34\!\cdots\!48$$$$\beta) q^{43} +($$$$39\!\cdots\!84$$$$+$$$$12\!\cdots\!80$$$$\beta) q^{44} +(-$$$$29\!\cdots\!28$$$$+$$$$57\!\cdots\!00$$$$\beta) q^{46} +(-$$$$47\!\cdots\!60$$$$-$$$$51\!\cdots\!16$$$$\beta) q^{47} +($$$$84\!\cdots\!93$$$$-$$$$21\!\cdots\!00$$$$\beta) q^{49} +(-$$$$87\!\cdots\!00$$$$-$$$$12\!\cdots\!75$$$$\beta) q^{50} +(-$$$$45\!\cdots\!00$$$$-$$$$26\!\cdots\!84$$$$\beta) q^{52} +(-$$$$97\!\cdots\!30$$$$+$$$$12\!\cdots\!68$$$$\beta) q^{53} +(-$$$$70\!\cdots\!20$$$$+$$$$52\!\cdots\!80$$$$\beta) q^{55} +(-$$$$12\!\cdots\!20$$$$+$$$$12\!\cdots\!40$$$$\beta) q^{56} +($$$$19\!\cdots\!60$$$$+$$$$68\!\cdots\!70$$$$\beta) q^{58} +($$$$99\!\cdots\!60$$$$+$$$$44\!\cdots\!80$$$$\beta) q^{59} +(-$$$$60\!\cdots\!38$$$$+$$$$11\!\cdots\!00$$$$\beta) q^{61} +(-$$$$76\!\cdots\!60$$$$-$$$$11\!\cdots\!32$$$$\beta) q^{62} +(-$$$$37\!\cdots\!52$$$$-$$$$22\!\cdots\!80$$$$\beta) q^{64} +(-$$$$17\!\cdots\!80$$$$+$$$$77\!\cdots\!20$$$$\beta) q^{65} +(-$$$$48\!\cdots\!60$$$$-$$$$88\!\cdots\!44$$$$\beta) q^{67} +(-$$$$12\!\cdots\!80$$$$-$$$$38\!\cdots\!08$$$$\beta) q^{68} +(-$$$$52\!\cdots\!60$$$$-$$$$29\!\cdots\!60$$$$\beta) q^{70} +(-$$$$27\!\cdots\!72$$$$+$$$$95\!\cdots\!00$$$$\beta) q^{71} +($$$$31\!\cdots\!90$$$$+$$$$15\!\cdots\!92$$$$\beta) q^{73} +(-$$$$76\!\cdots\!36$$$$-$$$$22\!\cdots\!50$$$$\beta) q^{74} +(-$$$$22\!\cdots\!60$$$$-$$$$38\!\cdots\!80$$$$\beta) q^{76} +($$$$64\!\cdots\!00$$$$-$$$$71\!\cdots\!32$$$$\beta) q^{77} +(-$$$$59\!\cdots\!80$$$$+$$$$46\!\cdots\!60$$$$\beta) q^{79} +(-$$$$70\!\cdots\!40$$$$-$$$$23\!\cdots\!40$$$$\beta) q^{80} +($$$$57\!\cdots\!60$$$$+$$$$80\!\cdots\!42$$$$\beta) q^{82} +($$$$13\!\cdots\!80$$$$+$$$$62\!\cdots\!28$$$$\beta) q^{83} +(-$$$$25\!\cdots\!60$$$$-$$$$12\!\cdots\!60$$$$\beta) q^{85} +($$$$10\!\cdots\!68$$$$+$$$$16\!\cdots\!40$$$$\beta) q^{86} +(-$$$$34\!\cdots\!80$$$$+$$$$41\!\cdots\!40$$$$\beta) q^{88} +($$$$10\!\cdots\!90$$$$-$$$$26\!\cdots\!80$$$$\beta) q^{89} +($$$$14\!\cdots\!12$$$$-$$$$73\!\cdots\!40$$$$\beta) q^{91} +($$$$11\!\cdots\!60$$$$-$$$$21\!\cdots\!56$$$$\beta) q^{92} +($$$$23\!\cdots\!56$$$$+$$$$57\!\cdots\!40$$$$\beta) q^{94} +(-$$$$64\!\cdots\!00$$$$-$$$$10\!\cdots\!00$$$$\beta) q^{95} +(-$$$$45\!\cdots\!90$$$$-$$$$16\!\cdots\!84$$$$\beta) q^{97} +($$$$40\!\cdots\!60$$$$-$$$$41\!\cdots\!93$$$$\beta) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 39960q^{2} + 1772534336q^{4} + 19391218020q^{5} + 30257527577200q^{7} - 160155058705920q^{8} + O(q^{10})$$ $$2q - 39960q^{2} + 1772534336q^{4} + 19391218020q^{5} + 30257527577200q^{7} - 160155058705920q^{8} - 7861972212281520q^{10} + 7782353745118776q^{11} + 74708953050260620q^{13} - 225544963845241152q^{14} - 3045212913684901888q^{16} - 17224607828987089380q^{17} - 12370563328022164040q^{19} +$$$$31\!\cdots\!60$$$$q^{20} - 2682713032996690080q^{22} -$$$$18\!\cdots\!80$$$$q^{23} +$$$$14\!\cdots\!50$$$$q^{25} +$$$$23\!\cdots\!16$$$$q^{26} +$$$$11\!\cdots\!20$$$$q^{28} -$$$$12\!\cdots\!40$$$$q^{29} +$$$$12\!\cdots\!64$$$$q^{31} +$$$$48\!\cdots\!40$$$$q^{32} +$$$$58\!\cdots\!92$$$$q^{34} -$$$$24\!\cdots\!80$$$$q^{35} -$$$$83\!\cdots\!60$$$$q^{37} +$$$$10\!\cdots\!20$$$$q^{38} +$$$$19\!\cdots\!00$$$$q^{40} -$$$$87\!\cdots\!84$$$$q^{41} -$$$$18\!\cdots\!00$$$$q^{43} +$$$$79\!\cdots\!68$$$$q^{44} -$$$$59\!\cdots\!56$$$$q^{46} -$$$$95\!\cdots\!20$$$$q^{47} +$$$$16\!\cdots\!86$$$$q^{49} -$$$$17\!\cdots\!00$$$$q^{50} -$$$$91\!\cdots\!00$$$$q^{52} -$$$$19\!\cdots\!60$$$$q^{53} -$$$$14\!\cdots\!40$$$$q^{55} -$$$$25\!\cdots\!40$$$$q^{56} +$$$$38\!\cdots\!20$$$$q^{58} +$$$$19\!\cdots\!20$$$$q^{59} -$$$$12\!\cdots\!76$$$$q^{61} -$$$$15\!\cdots\!20$$$$q^{62} -$$$$74\!\cdots\!04$$$$q^{64} -$$$$34\!\cdots\!60$$$$q^{65} -$$$$96\!\cdots\!20$$$$q^{67} -$$$$24\!\cdots\!60$$$$q^{68} -$$$$10\!\cdots\!20$$$$q^{70} -$$$$55\!\cdots\!44$$$$q^{71} +$$$$62\!\cdots\!80$$$$q^{73} -$$$$15\!\cdots\!72$$$$q^{74} -$$$$44\!\cdots\!20$$$$q^{76} +$$$$12\!\cdots\!00$$$$q^{77} -$$$$11\!\cdots\!60$$$$q^{79} -$$$$14\!\cdots\!80$$$$q^{80} +$$$$11\!\cdots\!20$$$$q^{82} +$$$$26\!\cdots\!60$$$$q^{83} -$$$$50\!\cdots\!20$$$$q^{85} +$$$$21\!\cdots\!36$$$$q^{86} -$$$$69\!\cdots\!60$$$$q^{88} +$$$$21\!\cdots\!80$$$$q^{89} +$$$$28\!\cdots\!24$$$$q^{91} +$$$$22\!\cdots\!20$$$$q^{92} +$$$$46\!\cdots\!12$$$$q^{94} -$$$$12\!\cdots\!00$$$$q^{95} -$$$$90\!\cdots\!80$$$$q^{97} +$$$$80\!\cdots\!20$$$$q^{98} + O(q^{100})$$

## 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
 2139.16 −2138.16
−71307.9 0 2.93733e9 8.25073e10 0 1.14368e13 −5.63222e13 0 −5.88342e15
1.2 31347.9 0 −1.16479e9 −6.31161e10 0 1.88207e13 −1.03833e14 0 −1.97855e15
 $$n$$: 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 9.32.a.a 2
3.b odd 2 1 1.32.a.a 2
12.b even 2 1 16.32.a.b 2
15.d odd 2 1 25.32.a.a 2
15.e even 4 2 25.32.b.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.32.a.a 2 3.b odd 2 1
9.32.a.a 2 1.a even 1 1 trivial
16.32.a.b 2 12.b even 2 1
25.32.a.a 2 15.d odd 2 1
25.32.b.a 4 15.e even 4 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 39960 T_{2} - 2235350016$$ acting on $$S_{32}^{\mathrm{new}}(\Gamma_0(9))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-2235350016 + 39960 T + T^{2}$$
$3$ $$T^{2}$$
$5$ $$-$$$$52\!\cdots\!00$$$$- 19391218020 T + T^{2}$$
$7$ $$21\!\cdots\!64$$$$- 30257527577200 T + T^{2}$$
$11$ $$12\!\cdots\!44$$$$- 7782353745118776 T + T^{2}$$
$13$ $$-$$$$55\!\cdots\!04$$$$- 74708953050260620 T + T^{2}$$
$17$ $$68\!\cdots\!24$$$$+ 17224607828987089380 T + T^{2}$$
$19$ $$-$$$$27\!\cdots\!00$$$$+ 12370563328022164040 T + T^{2}$$
$23$ $$-$$$$73\!\cdots\!24$$$$+$$$$18\!\cdots\!80$$$$T + T^{2}$$
$29$ $$39\!\cdots\!00$$$$+$$$$12\!\cdots\!40$$$$T + T^{2}$$
$31$ $$-$$$$11\!\cdots\!76$$$$-$$$$12\!\cdots\!64$$$$T + T^{2}$$
$37$ $$-$$$$25\!\cdots\!56$$$$+$$$$83\!\cdots\!60$$$$T + T^{2}$$
$41$ $$-$$$$70\!\cdots\!36$$$$+$$$$87\!\cdots\!84$$$$T + T^{2}$$
$43$ $$-$$$$22\!\cdots\!64$$$$+$$$$18\!\cdots\!00$$$$T + T^{2}$$
$47$ $$15\!\cdots\!04$$$$+$$$$95\!\cdots\!20$$$$T + T^{2}$$
$53$ $$56\!\cdots\!16$$$$+$$$$19\!\cdots\!60$$$$T + T^{2}$$
$59$ $$-$$$$51\!\cdots\!00$$$$-$$$$19\!\cdots\!20$$$$T + T^{2}$$
$61$ $$36\!\cdots\!44$$$$+$$$$12\!\cdots\!76$$$$T + T^{2}$$
$67$ $$26\!\cdots\!24$$$$+$$$$96\!\cdots\!20$$$$T + T^{2}$$
$71$ $$-$$$$16\!\cdots\!16$$$$+$$$$55\!\cdots\!44$$$$T + T^{2}$$
$73$ $$90\!\cdots\!76$$$$-$$$$62\!\cdots\!80$$$$T + T^{2}$$
$79$ $$-$$$$53\!\cdots\!00$$$$+$$$$11\!\cdots\!60$$$$T + T^{2}$$
$83$ $$-$$$$86\!\cdots\!44$$$$-$$$$26\!\cdots\!60$$$$T + T^{2}$$
$89$ $$-$$$$62\!\cdots\!00$$$$-$$$$21\!\cdots\!80$$$$T + T^{2}$$
$97$ $$19\!\cdots\!04$$$$+$$$$90\!\cdots\!80$$$$T + T^{2}$$