# Properties

 Label 9.62.a.a Level 9 Weight 62 Character orbit 9.a Self dual yes Analytic conductor 212.091 Analytic rank 0 Dimension 4 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$9 = 3^{2}$$ Weight: $$k$$ $$=$$ $$62$$ Character orbit: $$[\chi]$$ $$=$$ 9.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$212.090564938$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\mathbb{Q}[x]/(x^{4} - \cdots)$$ Defining polynomial: $$x^{4} - x^{3} - 180363795469121 x^{2} + 166129321978984507920 x + 2785609847439483545242446300$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{28}\cdot 3^{10}\cdot 5^{2}\cdot 7$$ 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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -286578000 + \beta_{1} ) q^{2} + ( 1100749418957151232 - 838426626 \beta_{1} + 38 \beta_{2} + \beta_{3} ) q^{4} + ( 131031499005073809450 - 100252987760 \beta_{1} + 5380 \beta_{2} + 120 \beta_{3} ) q^{5} + ( -15834642365846289693295000 + 14513887843120512 \beta_{1} - 5219268334 \beta_{2} - 19628000 \beta_{3} ) q^{7} + ( -2441967096789168519757824000 + 1373003764151712384 \beta_{1} + 833101117568 \beta_{2} - 1190856000 \beta_{3} ) q^{8} +O(q^{10})$$ $$q +(-286578000 + \beta_{1}) q^{2} +(1100749418957151232 - 838426626 \beta_{1} + 38 \beta_{2} + \beta_{3}) q^{4} +($$$$13\!\cdots\!50$$$$- 100252987760 \beta_{1} + 5380 \beta_{2} + 120 \beta_{3}) q^{5} +(-$$$$15\!\cdots\!00$$$$+ 14513887843120512 \beta_{1} - 5219268334 \beta_{2} - 19628000 \beta_{3}) q^{7} +(-$$$$24\!\cdots\!00$$$$+ 1373003764151712384 \beta_{1} + 833101117568 \beta_{2} - 1190856000 \beta_{3}) q^{8} +(-$$$$37\!\cdots\!00$$$$+$$$$44\!\cdots\!70$$$$\beta_{1} + 100061893877440 \beta_{2} - 141501833440 \beta_{3}) q^{10} +($$$$10\!\cdots\!88$$$$-$$$$12\!\cdots\!20$$$$\beta_{1} - 14517391394747515 \beta_{2} - 11035029161280 \beta_{3}) q^{11} +($$$$26\!\cdots\!50$$$$+$$$$39\!\cdots\!92$$$$\beta_{1} - 1387217511377341540 \beta_{2} + 2435574015901000 \beta_{3}) q^{13} +($$$$52\!\cdots\!76$$$$-$$$$65\!\cdots\!72$$$$\beta_{1} - 16841348233782444064 \beta_{2} + 15743169581051472 \beta_{3}) q^{14} +($$$$27\!\cdots\!56$$$$-$$$$36\!\cdots\!12$$$$\beta_{1} -$$$$98\!\cdots\!44$$$$\beta_{2} + 603732751100608512 \beta_{3}) q^{16} +($$$$10\!\cdots\!50$$$$+$$$$38\!\cdots\!72$$$$\beta_{1} -$$$$37\!\cdots\!28$$$$\beta_{2} + 7263830520990282000 \beta_{3}) q^{17} +(-$$$$89\!\cdots\!80$$$$+$$$$48\!\cdots\!36$$$$\beta_{1} -$$$$18\!\cdots\!43$$$$\beta_{2} -$$$$10\!\cdots\!36$$$$\beta_{3}) q^{19} +($$$$12\!\cdots\!00$$$$-$$$$66\!\cdots\!20$$$$\beta_{1} -$$$$10\!\cdots\!40$$$$\beta_{2} +$$$$34\!\cdots\!90$$$$\beta_{3}) q^{20} +(-$$$$43\!\cdots\!00$$$$-$$$$79\!\cdots\!32$$$$\beta_{1} -$$$$11\!\cdots\!40$$$$\beta_{2} -$$$$26\!\cdots\!00$$$$\beta_{3}) q^{22} +($$$$10\!\cdots\!00$$$$-$$$$24\!\cdots\!20$$$$\beta_{1} +$$$$27\!\cdots\!74$$$$\beta_{2} -$$$$75\!\cdots\!00$$$$\beta_{3}) q^{23} +(-$$$$41\!\cdots\!25$$$$-$$$$79\!\cdots\!00$$$$\beta_{1} -$$$$13\!\cdots\!00$$$$\beta_{2} +$$$$41\!\cdots\!00$$$$\beta_{3}) q^{25} +($$$$12\!\cdots\!08$$$$+$$$$54\!\cdots\!58$$$$\beta_{1} +$$$$21\!\cdots\!96$$$$\beta_{2} +$$$$11\!\cdots\!92$$$$\beta_{3}) q^{26} +(-$$$$19\!\cdots\!00$$$$+$$$$86\!\cdots\!40$$$$\beta_{1} +$$$$21\!\cdots\!64$$$$\beta_{2} -$$$$48\!\cdots\!00$$$$\beta_{3}) q^{28} +(-$$$$16\!\cdots\!30$$$$-$$$$32\!\cdots\!24$$$$\beta_{1} +$$$$60\!\cdots\!12$$$$\beta_{2} -$$$$93\!\cdots\!76$$$$\beta_{3}) q^{29} +($$$$81\!\cdots\!32$$$$-$$$$32\!\cdots\!40$$$$\beta_{1} +$$$$53\!\cdots\!20$$$$\beta_{2} -$$$$92\!\cdots\!60$$$$\beta_{3}) q^{31} +(-$$$$73\!\cdots\!00$$$$+$$$$27\!\cdots\!36$$$$\beta_{1} -$$$$16\!\cdots\!20$$$$\beta_{2} -$$$$24\!\cdots\!00$$$$\beta_{3}) q^{32} +($$$$99\!\cdots\!04$$$$+$$$$22\!\cdots\!58$$$$\beta_{1} +$$$$60\!\cdots\!96$$$$\beta_{2} -$$$$38\!\cdots\!08$$$$\beta_{3}) q^{34} +(-$$$$23\!\cdots\!00$$$$+$$$$10\!\cdots\!80$$$$\beta_{1} +$$$$26\!\cdots\!60$$$$\beta_{2} -$$$$57\!\cdots\!60$$$$\beta_{3}) q^{35} +(-$$$$17\!\cdots\!50$$$$-$$$$22\!\cdots\!00$$$$\beta_{1} -$$$$18\!\cdots\!68$$$$\beta_{2} -$$$$11\!\cdots\!00$$$$\beta_{3}) q^{37} +($$$$16\!\cdots\!00$$$$-$$$$61\!\cdots\!24$$$$\beta_{1} -$$$$92\!\cdots\!48$$$$\beta_{2} +$$$$29\!\cdots\!00$$$$\beta_{3}) q^{38} +(-$$$$17\!\cdots\!00$$$$+$$$$13\!\cdots\!00$$$$\beta_{1} +$$$$33\!\cdots\!00$$$$\beta_{2} -$$$$62\!\cdots\!00$$$$\beta_{3}) q^{40} +(-$$$$40\!\cdots\!42$$$$+$$$$27\!\cdots\!40$$$$\beta_{1} +$$$$11\!\cdots\!80$$$$\beta_{2} +$$$$34\!\cdots\!60$$$$\beta_{3}) q^{41} +($$$$18\!\cdots\!00$$$$+$$$$70\!\cdots\!16$$$$\beta_{1} +$$$$16\!\cdots\!63$$$$\beta_{2} -$$$$33\!\cdots\!00$$$$\beta_{3}) q^{43} +(-$$$$38\!\cdots\!84$$$$-$$$$67\!\cdots\!28$$$$\beta_{1} +$$$$96\!\cdots\!64$$$$\beta_{2} +$$$$13\!\cdots\!28$$$$\beta_{3}) q^{44} +(-$$$$37\!\cdots\!28$$$$-$$$$52\!\cdots\!20$$$$\beta_{1} -$$$$62\!\cdots\!40$$$$\beta_{2} +$$$$62\!\cdots\!20$$$$\beta_{3}) q^{46} +($$$$54\!\cdots\!00$$$$+$$$$53\!\cdots\!44$$$$\beta_{1} -$$$$10\!\cdots\!24$$$$\beta_{2} -$$$$28\!\cdots\!00$$$$\beta_{3}) q^{47} +($$$$38\!\cdots\!57$$$$-$$$$12\!\cdots\!20$$$$\beta_{1} -$$$$56\!\cdots\!40$$$$\beta_{2} +$$$$74\!\cdots\!20$$$$\beta_{3}) q^{49} +($$$$93\!\cdots\!00$$$$-$$$$40\!\cdots\!25$$$$\beta_{1} +$$$$31\!\cdots\!00$$$$\beta_{2} -$$$$11\!\cdots\!00$$$$\beta_{3}) q^{50} +($$$$85\!\cdots\!00$$$$+$$$$29\!\cdots\!92$$$$\beta_{1} +$$$$46\!\cdots\!36$$$$\beta_{2} +$$$$20\!\cdots\!50$$$$\beta_{3}) q^{52} +(-$$$$21\!\cdots\!50$$$$+$$$$18\!\cdots\!16$$$$\beta_{1} +$$$$10\!\cdots\!16$$$$\beta_{2} +$$$$81\!\cdots\!00$$$$\beta_{3}) q^{53} +(-$$$$47\!\cdots\!00$$$$-$$$$80\!\cdots\!80$$$$\beta_{1} +$$$$12\!\cdots\!90$$$$\beta_{2} +$$$$16\!\cdots\!60$$$$\beta_{3}) q^{55} +($$$$22\!\cdots\!80$$$$-$$$$19\!\cdots\!36$$$$\beta_{1} +$$$$24\!\cdots\!68$$$$\beta_{2} +$$$$97\!\cdots\!36$$$$\beta_{3}) q^{56} +(-$$$$10\!\cdots\!00$$$$-$$$$17\!\cdots\!34$$$$\beta_{1} -$$$$76\!\cdots\!68$$$$\beta_{2} +$$$$82\!\cdots\!00$$$$\beta_{3}) q^{58} +($$$$53\!\cdots\!40$$$$+$$$$31\!\cdots\!12$$$$\beta_{1} +$$$$74\!\cdots\!69$$$$\beta_{2} +$$$$38\!\cdots\!88$$$$\beta_{3}) q^{59} +($$$$10\!\cdots\!62$$$$+$$$$63\!\cdots\!00$$$$\beta_{1} +$$$$27\!\cdots\!00$$$$\beta_{2} +$$$$50\!\cdots\!00$$$$\beta_{3}) q^{61} +(-$$$$12\!\cdots\!00$$$$-$$$$94\!\cdots\!08$$$$\beta_{1} -$$$$80\!\cdots\!80$$$$\beta_{2} +$$$$11\!\cdots\!00$$$$\beta_{3}) q^{62} +($$$$49\!\cdots\!52$$$$-$$$$57\!\cdots\!12$$$$\beta_{1} +$$$$12\!\cdots\!56$$$$\beta_{2} +$$$$25\!\cdots\!12$$$$\beta_{3}) q^{64} +($$$$10\!\cdots\!00$$$$+$$$$36\!\cdots\!40$$$$\beta_{1} +$$$$55\!\cdots\!80$$$$\beta_{2} +$$$$25\!\cdots\!20$$$$\beta_{3}) q^{65} +(-$$$$37\!\cdots\!00$$$$+$$$$19\!\cdots\!08$$$$\beta_{1} +$$$$65\!\cdots\!13$$$$\beta_{2} +$$$$26\!\cdots\!00$$$$\beta_{3}) q^{67} +($$$$49\!\cdots\!00$$$$-$$$$18\!\cdots\!72$$$$\beta_{1} +$$$$68\!\cdots\!12$$$$\beta_{2} +$$$$15\!\cdots\!50$$$$\beta_{3}) q^{68} +($$$$41\!\cdots\!00$$$$-$$$$41\!\cdots\!60$$$$\beta_{1} -$$$$43\!\cdots\!20$$$$\beta_{2} +$$$$15\!\cdots\!20$$$$\beta_{3}) q^{70} +(-$$$$66\!\cdots\!72$$$$+$$$$92\!\cdots\!00$$$$\beta_{1} -$$$$79\!\cdots\!50$$$$\beta_{2} -$$$$40\!\cdots\!00$$$$\beta_{3}) q^{71} +($$$$10\!\cdots\!50$$$$+$$$$79\!\cdots\!00$$$$\beta_{1} +$$$$19\!\cdots\!36$$$$\beta_{2} -$$$$19\!\cdots\!00$$$$\beta_{3}) q^{73} +(-$$$$68\!\cdots\!64$$$$-$$$$10\!\cdots\!70$$$$\beta_{1} -$$$$35\!\cdots\!40$$$$\beta_{2} -$$$$45\!\cdots\!80$$$$\beta_{3}) q^{74} +(-$$$$22\!\cdots\!60$$$$+$$$$14\!\cdots\!32$$$$\beta_{1} +$$$$64\!\cdots\!84$$$$\beta_{2} -$$$$59\!\cdots\!32$$$$\beta_{3}) q^{76} +($$$$15\!\cdots\!00$$$$+$$$$14\!\cdots\!76$$$$\beta_{1} -$$$$43\!\cdots\!52$$$$\beta_{2} -$$$$34\!\cdots\!00$$$$\beta_{3}) q^{77} +(-$$$$54\!\cdots\!20$$$$-$$$$32\!\cdots\!76$$$$\beta_{1} -$$$$16\!\cdots\!12$$$$\beta_{2} +$$$$16\!\cdots\!76$$$$\beta_{3}) q^{79} +($$$$20\!\cdots\!00$$$$-$$$$22\!\cdots\!60$$$$\beta_{1} -$$$$23\!\cdots\!20$$$$\beta_{2} +$$$$83\!\cdots\!20$$$$\beta_{3}) q^{80} +($$$$10\!\cdots\!00$$$$+$$$$18\!\cdots\!98$$$$\beta_{1} +$$$$32\!\cdots\!80$$$$\beta_{2} +$$$$29\!\cdots\!00$$$$\beta_{3}) q^{82} +($$$$48\!\cdots\!00$$$$-$$$$14\!\cdots\!08$$$$\beta_{1} -$$$$72\!\cdots\!37$$$$\beta_{2} +$$$$66\!\cdots\!00$$$$\beta_{3}) q^{83} +($$$$59\!\cdots\!00$$$$-$$$$22\!\cdots\!80$$$$\beta_{1} +$$$$82\!\cdots\!40$$$$\beta_{2} +$$$$18\!\cdots\!60$$$$\beta_{3}) q^{85} +($$$$17\!\cdots\!68$$$$+$$$$99\!\cdots\!04$$$$\beta_{1} -$$$$10\!\cdots\!52$$$$\beta_{2} +$$$$29\!\cdots\!96$$$$\beta_{3}) q^{86} +(-$$$$11\!\cdots\!00$$$$+$$$$47\!\cdots\!92$$$$\beta_{1} +$$$$36\!\cdots\!84$$$$\beta_{2} -$$$$24\!\cdots\!00$$$$\beta_{3}) q^{88} +($$$$15\!\cdots\!10$$$$-$$$$39\!\cdots\!72$$$$\beta_{1} +$$$$22\!\cdots\!36$$$$\beta_{2} -$$$$35\!\cdots\!28$$$$\beta_{3}) q^{89} +(-$$$$11\!\cdots\!88$$$$-$$$$72\!\cdots\!84$$$$\beta_{1} -$$$$10\!\cdots\!08$$$$\beta_{2} -$$$$80\!\cdots\!16$$$$\beta_{3}) q^{91} +(-$$$$40\!\cdots\!00$$$$+$$$$12\!\cdots\!92$$$$\beta_{1} -$$$$16\!\cdots\!88$$$$\beta_{2} +$$$$16\!\cdots\!00$$$$\beta_{3}) q^{92} +($$$$16\!\cdots\!44$$$$+$$$$17\!\cdots\!96$$$$\beta_{1} -$$$$13\!\cdots\!48$$$$\beta_{2} +$$$$41\!\cdots\!04$$$$\beta_{3}) q^{94} +(-$$$$27\!\cdots\!00$$$$+$$$$17\!\cdots\!00$$$$\beta_{1} +$$$$78\!\cdots\!50$$$$\beta_{2} -$$$$70\!\cdots\!00$$$$\beta_{3}) q^{95} +(-$$$$20\!\cdots\!50$$$$+$$$$47\!\cdots\!56$$$$\beta_{1} -$$$$33\!\cdots\!76$$$$\beta_{2} -$$$$42\!\cdots\!00$$$$\beta_{3}) q^{97} +(-$$$$42\!\cdots\!00$$$$+$$$$25\!\cdots\!37$$$$\beta_{1} +$$$$54\!\cdots\!60$$$$\beta_{2} -$$$$22\!\cdots\!00$$$$\beta_{3}) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 1146312000q^{2} + 4402997675828604928q^{4} + 524125996020295237800q^{5} - 63338569463385158773180000q^{7} - 9767868387156674079031296000q^{8} + O(q^{10})$$ $$4q - 1146312000q^{2} + 4402997675828604928q^{4} +$$$$52\!\cdots\!00$$$$q^{5} -$$$$63\!\cdots\!00$$$$q^{7} -$$$$97\!\cdots\!00$$$$q^{8} -$$$$14\!\cdots\!00$$$$q^{10} +$$$$41\!\cdots\!52$$$$q^{11} +$$$$10\!\cdots\!00$$$$q^{13} +$$$$21\!\cdots\!04$$$$q^{14} +$$$$10\!\cdots\!24$$$$q^{16} +$$$$40\!\cdots\!00$$$$q^{17} -$$$$35\!\cdots\!20$$$$q^{19} +$$$$50\!\cdots\!00$$$$q^{20} -$$$$17\!\cdots\!00$$$$q^{22} +$$$$41\!\cdots\!00$$$$q^{23} -$$$$16\!\cdots\!00$$$$q^{25} +$$$$49\!\cdots\!32$$$$q^{26} -$$$$79\!\cdots\!00$$$$q^{28} -$$$$66\!\cdots\!20$$$$q^{29} +$$$$32\!\cdots\!28$$$$q^{31} -$$$$29\!\cdots\!00$$$$q^{32} +$$$$39\!\cdots\!16$$$$q^{34} -$$$$94\!\cdots\!00$$$$q^{35} -$$$$71\!\cdots\!00$$$$q^{37} +$$$$65\!\cdots\!00$$$$q^{38} -$$$$69\!\cdots\!00$$$$q^{40} -$$$$16\!\cdots\!68$$$$q^{41} +$$$$75\!\cdots\!00$$$$q^{43} -$$$$15\!\cdots\!36$$$$q^{44} -$$$$15\!\cdots\!12$$$$q^{46} +$$$$21\!\cdots\!00$$$$q^{47} +$$$$15\!\cdots\!28$$$$q^{49} +$$$$37\!\cdots\!00$$$$q^{50} +$$$$34\!\cdots\!00$$$$q^{52} -$$$$84\!\cdots\!00$$$$q^{53} -$$$$18\!\cdots\!00$$$$q^{55} +$$$$89\!\cdots\!20$$$$q^{56} -$$$$42\!\cdots\!00$$$$q^{58} +$$$$21\!\cdots\!60$$$$q^{59} +$$$$42\!\cdots\!48$$$$q^{61} -$$$$51\!\cdots\!00$$$$q^{62} +$$$$19\!\cdots\!08$$$$q^{64} +$$$$40\!\cdots\!00$$$$q^{65} -$$$$15\!\cdots\!00$$$$q^{67} +$$$$19\!\cdots\!00$$$$q^{68} +$$$$16\!\cdots\!00$$$$q^{70} -$$$$26\!\cdots\!88$$$$q^{71} +$$$$43\!\cdots\!00$$$$q^{73} -$$$$27\!\cdots\!56$$$$q^{74} -$$$$91\!\cdots\!40$$$$q^{76} +$$$$62\!\cdots\!00$$$$q^{77} -$$$$21\!\cdots\!80$$$$q^{79} +$$$$81\!\cdots\!00$$$$q^{80} +$$$$41\!\cdots\!00$$$$q^{82} +$$$$19\!\cdots\!00$$$$q^{83} +$$$$23\!\cdots\!00$$$$q^{85} +$$$$71\!\cdots\!72$$$$q^{86} -$$$$45\!\cdots\!00$$$$q^{88} +$$$$60\!\cdots\!40$$$$q^{89} -$$$$45\!\cdots\!52$$$$q^{91} -$$$$16\!\cdots\!00$$$$q^{92} +$$$$64\!\cdots\!76$$$$q^{94} -$$$$11\!\cdots\!00$$$$q^{95} -$$$$80\!\cdots\!00$$$$q^{97} -$$$$17\!\cdots\!00$$$$q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 180363795469121 x^{2} + 166129321978984507920 x + 2785609847439483545242446300$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$192 \nu - 48$$ $$\beta_{2}$$ $$=$$ $$($$$$108 \nu^{3} + 186256044 \nu^{2} - 15885966350153172 \nu - 3340459070129947691688$$$$)/13402613$$ $$\beta_{3}$$ $$=$$ $$($$$$-4104 \nu^{3} + 486996195960 \nu^{2} + 1286287826053639416 \nu - 44429586960648335446356720$$$$)/13402613$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{1} + 48$$$$)/192$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} + 38 \beta_{2} - 265270530 \beta_{1} + 3324465478086847488$$$$)/36864$$ $$\nu^{3}$$ $$=$$ $$($$$$-5173779 \beta_{3} + 13527672110 \beta_{2} + 86097604964916454 \beta_{1} - 13779415522130239634116608$$$$)/110592$$

## 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
 −1.33086e7 −3.61406e6 4.76281e6 1.21599e7
−2.84183e9 0 5.77017e18 6.89620e20 0 −9.80127e25 −9.84504e27 0 −1.95979e30
1.2 −9.80478e8 0 −1.34451e18 −1.59503e20 0 1.63507e25 3.57909e27 0 1.56389e29
1.3 6.27881e8 0 −1.91161e18 −2.33985e20 0 6.25792e25 −2.64806e27 0 −1.46915e29
1.4 2.04812e9 0 1.88894e18 2.27994e20 0 −4.42559e25 −8.53860e26 0 4.66958e29
 $$n$$: 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 9.62.a.a 4
3.b odd 2 1 1.62.a.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.62.a.a 4 3.b odd 2 1
9.62.a.a 4 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} + 1146312000 T_{2}^{3} -$$$$61\!\cdots\!68$$$$T_{2}^{2} -$$$$25\!\cdots\!00$$$$T_{2} +$$$$35\!\cdots\!56$$ acting on $$S_{62}^{\mathrm{new}}(\Gamma_0(9))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 1146312000 T + 3067202781185085440 T^{2} +$$$$53\!\cdots\!00$$$$T^{3} +$$$$70\!\cdots\!08$$$$T^{4} +$$$$12\!\cdots\!00$$$$T^{5} +$$$$16\!\cdots\!60$$$$T^{6} +$$$$14\!\cdots\!00$$$$T^{7} +$$$$28\!\cdots\!16$$$$T^{8}$$
$3$ 1
$5$ $$1 -$$$$52\!\cdots\!00$$$$T +$$$$17\!\cdots\!00$$$$T^{2} -$$$$67\!\cdots\!00$$$$T^{3} +$$$$11\!\cdots\!50$$$$T^{4} -$$$$29\!\cdots\!00$$$$T^{5} +$$$$32\!\cdots\!00$$$$T^{6} -$$$$42\!\cdots\!00$$$$T^{7} +$$$$35\!\cdots\!25$$$$T^{8}$$
$7$ $$1 +$$$$63\!\cdots\!00$$$$T +$$$$83\!\cdots\!00$$$$T^{2} +$$$$47\!\cdots\!00$$$$T^{3} +$$$$38\!\cdots\!98$$$$T^{4} +$$$$17\!\cdots\!00$$$$T^{5} +$$$$10\!\cdots\!00$$$$T^{6} +$$$$28\!\cdots\!00$$$$T^{7} +$$$$15\!\cdots\!01$$$$T^{8}$$
$11$ $$1 -$$$$41\!\cdots\!52$$$$T +$$$$70\!\cdots\!08$$$$T^{2} -$$$$24\!\cdots\!04$$$$T^{3} +$$$$24\!\cdots\!70$$$$T^{4} -$$$$81\!\cdots\!44$$$$T^{5} +$$$$78\!\cdots\!68$$$$T^{6} -$$$$15\!\cdots\!12$$$$T^{7} +$$$$12\!\cdots\!41$$$$T^{8}$$
$13$ $$1 -$$$$10\!\cdots\!00$$$$T +$$$$17\!\cdots\!60$$$$T^{2} -$$$$12\!\cdots\!00$$$$T^{3} +$$$$14\!\cdots\!38$$$$T^{4} -$$$$10\!\cdots\!00$$$$T^{5} +$$$$13\!\cdots\!40$$$$T^{6} -$$$$76\!\cdots\!00$$$$T^{7} +$$$$63\!\cdots\!61$$$$T^{8}$$
$17$ $$1 -$$$$40\!\cdots\!00$$$$T +$$$$40\!\cdots\!20$$$$T^{2} -$$$$10\!\cdots\!00$$$$T^{3} +$$$$63\!\cdots\!78$$$$T^{4} -$$$$11\!\cdots\!00$$$$T^{5} +$$$$52\!\cdots\!80$$$$T^{6} -$$$$60\!\cdots\!00$$$$T^{7} +$$$$16\!\cdots\!21$$$$T^{8}$$
$19$ $$1 +$$$$35\!\cdots\!20$$$$T +$$$$16\!\cdots\!76$$$$T^{2} +$$$$29\!\cdots\!40$$$$T^{3} +$$$$19\!\cdots\!66$$$$T^{4} +$$$$29\!\cdots\!60$$$$T^{5} +$$$$16\!\cdots\!36$$$$T^{6} +$$$$36\!\cdots\!80$$$$T^{7} +$$$$10\!\cdots\!21$$$$T^{8}$$
$23$ $$1 -$$$$41\!\cdots\!00$$$$T +$$$$43\!\cdots\!40$$$$T^{2} -$$$$13\!\cdots\!00$$$$T^{3} +$$$$73\!\cdots\!58$$$$T^{4} -$$$$15\!\cdots\!00$$$$T^{5} +$$$$58\!\cdots\!60$$$$T^{6} -$$$$64\!\cdots\!00$$$$T^{7} +$$$$18\!\cdots\!41$$$$T^{8}$$
$29$ $$1 +$$$$66\!\cdots\!20$$$$T +$$$$43\!\cdots\!16$$$$T^{2} -$$$$46\!\cdots\!60$$$$T^{3} +$$$$85\!\cdots\!46$$$$T^{4} -$$$$74\!\cdots\!40$$$$T^{5} +$$$$11\!\cdots\!56$$$$T^{6} +$$$$27\!\cdots\!80$$$$T^{7} +$$$$66\!\cdots\!81$$$$T^{8}$$
$31$ $$1 -$$$$32\!\cdots\!28$$$$T +$$$$28\!\cdots\!68$$$$T^{2} -$$$$75\!\cdots\!76$$$$T^{3} +$$$$38\!\cdots\!70$$$$T^{4} -$$$$70\!\cdots\!56$$$$T^{5} +$$$$25\!\cdots\!48$$$$T^{6} -$$$$27\!\cdots\!48$$$$T^{7} +$$$$78\!\cdots\!21$$$$T^{8}$$
$37$ $$1 +$$$$71\!\cdots\!00$$$$T +$$$$98\!\cdots\!60$$$$T^{2} +$$$$68\!\cdots\!00$$$$T^{3} +$$$$69\!\cdots\!38$$$$T^{4} +$$$$31\!\cdots\!00$$$$T^{5} +$$$$20\!\cdots\!40$$$$T^{6} +$$$$67\!\cdots\!00$$$$T^{7} +$$$$43\!\cdots\!61$$$$T^{8}$$
$41$ $$1 +$$$$16\!\cdots\!68$$$$T +$$$$80\!\cdots\!48$$$$T^{2} +$$$$85\!\cdots\!16$$$$T^{3} +$$$$26\!\cdots\!70$$$$T^{4} +$$$$20\!\cdots\!56$$$$T^{5} +$$$$46\!\cdots\!88$$$$T^{6} +$$$$22\!\cdots\!28$$$$T^{7} +$$$$33\!\cdots\!61$$$$T^{8}$$
$43$ $$1 -$$$$75\!\cdots\!00$$$$T +$$$$13\!\cdots\!00$$$$T^{2} -$$$$79\!\cdots\!00$$$$T^{3} +$$$$84\!\cdots\!98$$$$T^{4} -$$$$34\!\cdots\!00$$$$T^{5} +$$$$25\!\cdots\!00$$$$T^{6} -$$$$63\!\cdots\!00$$$$T^{7} +$$$$36\!\cdots\!01$$$$T^{8}$$
$47$ $$1 -$$$$21\!\cdots\!00$$$$T +$$$$36\!\cdots\!80$$$$T^{2} -$$$$46\!\cdots\!00$$$$T^{3} +$$$$54\!\cdots\!18$$$$T^{4} -$$$$45\!\cdots\!00$$$$T^{5} +$$$$36\!\cdots\!20$$$$T^{6} -$$$$21\!\cdots\!00$$$$T^{7} +$$$$98\!\cdots\!81$$$$T^{8}$$
$53$ $$1 +$$$$84\!\cdots\!00$$$$T +$$$$55\!\cdots\!80$$$$T^{2} +$$$$20\!\cdots\!00$$$$T^{3} +$$$$89\!\cdots\!18$$$$T^{4} +$$$$31\!\cdots\!00$$$$T^{5} +$$$$12\!\cdots\!20$$$$T^{6} +$$$$29\!\cdots\!00$$$$T^{7} +$$$$52\!\cdots\!81$$$$T^{8}$$
$59$ $$1 -$$$$21\!\cdots\!60$$$$T +$$$$37\!\cdots\!36$$$$T^{2} -$$$$50\!\cdots\!20$$$$T^{3} +$$$$62\!\cdots\!86$$$$T^{4} -$$$$53\!\cdots\!80$$$$T^{5} +$$$$41\!\cdots\!16$$$$T^{6} -$$$$24\!\cdots\!40$$$$T^{7} +$$$$12\!\cdots\!61$$$$T^{8}$$
$61$ $$1 -$$$$42\!\cdots\!48$$$$T +$$$$31\!\cdots\!08$$$$T^{2} -$$$$10\!\cdots\!96$$$$T^{3} +$$$$37\!\cdots\!70$$$$T^{4} -$$$$81\!\cdots\!56$$$$T^{5} +$$$$20\!\cdots\!68$$$$T^{6} -$$$$22\!\cdots\!88$$$$T^{7} +$$$$41\!\cdots\!41$$$$T^{8}$$
$67$ $$1 +$$$$15\!\cdots\!00$$$$T +$$$$14\!\cdots\!20$$$$T^{2} +$$$$10\!\cdots\!00$$$$T^{3} +$$$$56\!\cdots\!78$$$$T^{4} +$$$$24\!\cdots\!00$$$$T^{5} +$$$$89\!\cdots\!80$$$$T^{6} +$$$$22\!\cdots\!00$$$$T^{7} +$$$$36\!\cdots\!21$$$$T^{8}$$
$71$ $$1 +$$$$26\!\cdots\!88$$$$T +$$$$15\!\cdots\!88$$$$T^{2} -$$$$22\!\cdots\!64$$$$T^{3} +$$$$64\!\cdots\!70$$$$T^{4} -$$$$18\!\cdots\!44$$$$T^{5} +$$$$10\!\cdots\!08$$$$T^{6} +$$$$16\!\cdots\!68$$$$T^{7} +$$$$50\!\cdots\!81$$$$T^{8}$$
$73$ $$1 -$$$$43\!\cdots\!00$$$$T +$$$$13\!\cdots\!40$$$$T^{2} -$$$$45\!\cdots\!00$$$$T^{3} +$$$$84\!\cdots\!58$$$$T^{4} -$$$$20\!\cdots\!00$$$$T^{5} +$$$$28\!\cdots\!60$$$$T^{6} -$$$$42\!\cdots\!00$$$$T^{7} +$$$$44\!\cdots\!41$$$$T^{8}$$
$79$ $$1 +$$$$21\!\cdots\!80$$$$T +$$$$11\!\cdots\!16$$$$T^{2} -$$$$36\!\cdots\!40$$$$T^{3} +$$$$56\!\cdots\!46$$$$T^{4} -$$$$20\!\cdots\!60$$$$T^{5} +$$$$38\!\cdots\!56$$$$T^{6} +$$$$40\!\cdots\!20$$$$T^{7} +$$$$10\!\cdots\!81$$$$T^{8}$$
$83$ $$1 -$$$$19\!\cdots\!00$$$$T +$$$$16\!\cdots\!20$$$$T^{2} -$$$$86\!\cdots\!00$$$$T^{3} +$$$$18\!\cdots\!78$$$$T^{4} -$$$$10\!\cdots\!00$$$$T^{5} +$$$$22\!\cdots\!80$$$$T^{6} -$$$$29\!\cdots\!00$$$$T^{7} +$$$$17\!\cdots\!21$$$$T^{8}$$
$89$ $$1 -$$$$60\!\cdots\!40$$$$T +$$$$42\!\cdots\!56$$$$T^{2} -$$$$15\!\cdots\!80$$$$T^{3} +$$$$56\!\cdots\!26$$$$T^{4} -$$$$12\!\cdots\!20$$$$T^{5} +$$$$28\!\cdots\!76$$$$T^{6} -$$$$33\!\cdots\!60$$$$T^{7} +$$$$44\!\cdots\!41$$$$T^{8}$$
$97$ $$1 +$$$$80\!\cdots\!00$$$$T +$$$$80\!\cdots\!80$$$$T^{2} +$$$$38\!\cdots\!00$$$$T^{3} +$$$$20\!\cdots\!18$$$$T^{4} +$$$$59\!\cdots\!00$$$$T^{5} +$$$$19\!\cdots\!20$$$$T^{6} +$$$$30\!\cdots\!00$$$$T^{7} +$$$$59\!\cdots\!81$$$$T^{8}$$