# Properties

 Label 9.38.a.a Level $9$ Weight $38$ Character orbit 9.a Self dual yes Analytic conductor $78.043$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$9 = 3^{2}$$ Weight: $$k$$ $$=$$ $$38$$ Character orbit: $$[\chi]$$ $$=$$ 9.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$78.0426343121$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\mathbb{Q}[x]/(x^{2} - \cdots)$$ Defining polynomial: $$x^{2} - x - 15934380$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2^{5}\cdot 3$$ 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 $$\beta = 48\sqrt{63737521}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 97200 - \beta ) q^{2} + ( 18860134912 - 194400 \beta ) q^{4} + ( -2764792192950 + 28866400 \beta ) q^{5} + ( -1724221976743000 + 9650004336 \beta ) q^{7} + ( 17022021521817600 + 99683138560 \beta ) q^{8} +O(q^{10})$$ $$q +(97200 - \beta) q^{2} +(18860134912 - 194400 \beta) q^{4} +(-2764792192950 + 28866400 \beta) q^{5} +(-1724221976743000 + 9650004336 \beta) q^{7} +(17022021521817600 + 99683138560 \beta) q^{8} +(-4507804677506637600 + 5570606272950 \beta) q^{10} +(13367018177424269028 + 32186306471000 \beta) q^{11} +($$$$26\!\cdots\!50$$$$+ 724443400725408 \beta) q^{13} +(-$$$$15\!\cdots\!24$$$$+ 2662202398202200 \beta) q^{14} +(-$$$$15\!\cdots\!04$$$$+ 19385312101171200 \beta) q^{16} +($$$$44\!\cdots\!50$$$$- 24633489938571456 \beta) q^{17} +($$$$18\!\cdots\!60$$$$+ 1292976182287157400 \beta) q^{19} +(-$$$$87\!\cdots\!00$$$$+ 1081899800733236800 \beta) q^{20} +(-$$$$34\!\cdots\!00$$$$- 10238509188443069028 \beta) q^{22} +($$$$13\!\cdots\!00$$$$- 34024622863285905488 \beta) q^{23} +($$$$57\!\cdots\!75$$$$-$$$$15\!\cdots\!00$$$$\beta) q^{25} +(-$$$$80\!\cdots\!72$$$$-$$$$19\!\cdots\!50$$$$\beta) q^{26} +(-$$$$30\!\cdots\!00$$$$+$$$$51\!\cdots\!32$$$$\beta) q^{28} +($$$$63\!\cdots\!10$$$$+$$$$23\!\cdots\!00$$$$\beta) q^{29} +($$$$13\!\cdots\!12$$$$+$$$$64\!\cdots\!00$$$$\beta) q^{31} +(-$$$$67\!\cdots\!00$$$$+$$$$37\!\cdots\!84$$$$\beta) q^{32} +($$$$79\!\cdots\!04$$$$-$$$$47\!\cdots\!50$$$$\beta) q^{34} +($$$$45\!\cdots\!00$$$$-$$$$76\!\cdots\!00$$$$\beta) q^{35} +(-$$$$34\!\cdots\!50$$$$-$$$$25\!\cdots\!04$$$$\beta) q^{37} +(-$$$$17\!\cdots\!00$$$$-$$$$60\!\cdots\!60$$$$\beta) q^{38} +($$$$37\!\cdots\!00$$$$+$$$$21\!\cdots\!00$$$$\beta) q^{40} +($$$$63\!\cdots\!18$$$$-$$$$58\!\cdots\!00$$$$\beta) q^{41} +(-$$$$12\!\cdots\!00$$$$-$$$$42\!\cdots\!52$$$$\beta) q^{43} +(-$$$$66\!\cdots\!64$$$$-$$$$19\!\cdots\!00$$$$\beta) q^{44} +($$$$62\!\cdots\!92$$$$-$$$$16\!\cdots\!00$$$$\beta) q^{46} +(-$$$$21\!\cdots\!00$$$$-$$$$38\!\cdots\!16$$$$\beta) q^{47} +(-$$$$19\!\cdots\!43$$$$-$$$$33\!\cdots\!00$$$$\beta) q^{49} +($$$$29\!\cdots\!00$$$$-$$$$72\!\cdots\!75$$$$\beta) q^{50} +(-$$$$15\!\cdots\!00$$$$-$$$$37\!\cdots\!04$$$$\beta) q^{52} +(-$$$$79\!\cdots\!50$$$$+$$$$11\!\cdots\!72$$$$\beta) q^{53} +($$$$99\!\cdots\!00$$$$+$$$$29\!\cdots\!00$$$$\beta) q^{55} +($$$$11\!\cdots\!40$$$$-$$$$76\!\cdots\!00$$$$\beta) q^{56} +(-$$$$27\!\cdots\!00$$$$-$$$$41\!\cdots\!10$$$$\beta) q^{58} +($$$$11\!\cdots\!20$$$$+$$$$48\!\cdots\!00$$$$\beta) q^{59} +($$$$50\!\cdots\!22$$$$+$$$$28\!\cdots\!00$$$$\beta) q^{61} +(-$$$$93\!\cdots\!00$$$$+$$$$49\!\cdots\!88$$$$\beta) q^{62} +($$$$93\!\cdots\!32$$$$+$$$$44\!\cdots\!00$$$$\beta) q^{64} +($$$$23\!\cdots\!00$$$$+$$$$56\!\cdots\!00$$$$\beta) q^{65} +(-$$$$54\!\cdots\!00$$$$+$$$$78\!\cdots\!56$$$$\beta) q^{67} +($$$$15\!\cdots\!00$$$$-$$$$91\!\cdots\!72$$$$\beta) q^{68} +($$$$15\!\cdots\!00$$$$-$$$$53\!\cdots\!00$$$$\beta) q^{70} +($$$$37\!\cdots\!08$$$$-$$$$17\!\cdots\!00$$$$\beta) q^{71} +($$$$98\!\cdots\!50$$$$+$$$$11\!\cdots\!88$$$$\beta) q^{73} +($$$$33\!\cdots\!36$$$$+$$$$96\!\cdots\!50$$$$\beta) q^{74} +(-$$$$33\!\cdots\!80$$$$-$$$$11\!\cdots\!00$$$$\beta) q^{76} +($$$$22\!\cdots\!00$$$$+$$$$73\!\cdots\!08$$$$\beta) q^{77} +($$$$13\!\cdots\!40$$$$-$$$$40\!\cdots\!00$$$$\beta) q^{79} +($$$$12\!\cdots\!00$$$$-$$$$50\!\cdots\!00$$$$\beta) q^{80} +($$$$14\!\cdots\!00$$$$-$$$$68\!\cdots\!18$$$$\beta) q^{82} +($$$$23\!\cdots\!00$$$$+$$$$23\!\cdots\!32$$$$\beta) q^{83} +(-$$$$22\!\cdots\!00$$$$+$$$$13\!\cdots\!00$$$$\beta) q^{85} +($$$$50\!\cdots\!68$$$$+$$$$86\!\cdots\!00$$$$\beta) q^{86} +($$$$69\!\cdots\!00$$$$+$$$$18\!\cdots\!80$$$$\beta) q^{88} +($$$$66\!\cdots\!30$$$$+$$$$23\!\cdots\!00$$$$\beta) q^{89} +($$$$56\!\cdots\!92$$$$+$$$$13\!\cdots\!00$$$$\beta) q^{91} +($$$$12\!\cdots\!00$$$$-$$$$31\!\cdots\!56$$$$\beta) q^{92} +($$$$35\!\cdots\!44$$$$+$$$$17\!\cdots\!00$$$$\beta) q^{94} +($$$$49\!\cdots\!00$$$$+$$$$18\!\cdots\!00$$$$\beta) q^{95} +($$$$30\!\cdots\!50$$$$-$$$$10\!\cdots\!84$$$$\beta) q^{97} +($$$$47\!\cdots\!00$$$$-$$$$13\!\cdots\!57$$$$\beta) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 194400q^{2} + 37720269824q^{4} - 5529584385900q^{5} - 3448443953486000q^{7} + 34044043043635200q^{8} + O(q^{10})$$ $$2q + 194400q^{2} + 37720269824q^{4} - 5529584385900q^{5} - 3448443953486000q^{7} + 34044043043635200q^{8} - 9015609355013275200q^{10} + 26734036354848538056q^{11} +$$$$53\!\cdots\!00$$$$q^{13} -$$$$31\!\cdots\!48$$$$q^{14} -$$$$31\!\cdots\!08$$$$q^{16} +$$$$89\!\cdots\!00$$$$q^{17} +$$$$37\!\cdots\!20$$$$q^{19} -$$$$17\!\cdots\!00$$$$q^{20} -$$$$68\!\cdots\!00$$$$q^{22} +$$$$26\!\cdots\!00$$$$q^{23} +$$$$11\!\cdots\!50$$$$q^{25} -$$$$16\!\cdots\!44$$$$q^{26} -$$$$61\!\cdots\!00$$$$q^{28} +$$$$12\!\cdots\!20$$$$q^{29} +$$$$26\!\cdots\!24$$$$q^{31} -$$$$13\!\cdots\!00$$$$q^{32} +$$$$15\!\cdots\!08$$$$q^{34} +$$$$91\!\cdots\!00$$$$q^{35} -$$$$68\!\cdots\!00$$$$q^{37} -$$$$34\!\cdots\!00$$$$q^{38} +$$$$75\!\cdots\!00$$$$q^{40} +$$$$12\!\cdots\!36$$$$q^{41} -$$$$25\!\cdots\!00$$$$q^{43} -$$$$13\!\cdots\!28$$$$q^{44} +$$$$12\!\cdots\!84$$$$q^{46} -$$$$42\!\cdots\!00$$$$q^{47} -$$$$38\!\cdots\!86$$$$q^{49} +$$$$58\!\cdots\!00$$$$q^{50} -$$$$31\!\cdots\!00$$$$q^{52} -$$$$15\!\cdots\!00$$$$q^{53} +$$$$19\!\cdots\!00$$$$q^{55} +$$$$22\!\cdots\!80$$$$q^{56} -$$$$55\!\cdots\!00$$$$q^{58} +$$$$23\!\cdots\!40$$$$q^{59} +$$$$10\!\cdots\!44$$$$q^{61} -$$$$18\!\cdots\!00$$$$q^{62} +$$$$18\!\cdots\!64$$$$q^{64} +$$$$46\!\cdots\!00$$$$q^{65} -$$$$10\!\cdots\!00$$$$q^{67} +$$$$30\!\cdots\!00$$$$q^{68} +$$$$31\!\cdots\!00$$$$q^{70} +$$$$74\!\cdots\!16$$$$q^{71} +$$$$19\!\cdots\!00$$$$q^{73} +$$$$67\!\cdots\!72$$$$q^{74} -$$$$66\!\cdots\!60$$$$q^{76} +$$$$45\!\cdots\!00$$$$q^{77} +$$$$27\!\cdots\!80$$$$q^{79} +$$$$25\!\cdots\!00$$$$q^{80} +$$$$29\!\cdots\!00$$$$q^{82} +$$$$47\!\cdots\!00$$$$q^{83} -$$$$45\!\cdots\!00$$$$q^{85} +$$$$10\!\cdots\!36$$$$q^{86} +$$$$13\!\cdots\!00$$$$q^{88} +$$$$13\!\cdots\!60$$$$q^{89} +$$$$11\!\cdots\!84$$$$q^{91} +$$$$24\!\cdots\!00$$$$q^{92} +$$$$70\!\cdots\!88$$$$q^{94} +$$$$99\!\cdots\!00$$$$q^{95} +$$$$60\!\cdots\!00$$$$q^{97} +$$$$94\!\cdots\!00$$$$q^{98} + O(q^{100})$$

## 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
 3992.29 −3991.29
−286012. 0 −5.56362e10 8.29715e12 0 1.97377e15 5.52218e16 0 −2.37308e18
1.2 480412. 0 9.33565e10 −1.38267e13 0 −5.42222e15 −2.11777e16 0 −6.64253e18
 $$n$$: 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 9.38.a.a 2
3.b odd 2 1 1.38.a.a 2
12.b even 2 1 16.38.a.b 2
15.d odd 2 1 25.38.a.a 2
15.e even 4 2 25.38.b.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.38.a.a 2 3.b odd 2 1
9.38.a.a 2 1.a even 1 1 trivial
16.38.a.b 2 12.b even 2 1
25.38.a.a 2 15.d odd 2 1
25.38.b.a 4 15.e even 4 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} - 194400 T_{2} - 137403408384$$ acting on $$S_{38}^{\mathrm{new}}(\Gamma_0(9))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 194400 T + 137474498560 T^{2} - 26718132554956800 T^{3} +$$$$18\!\cdots\!84$$$$T^{4}$$
$3$ 1
$5$ $$1 + 5529584385900 T +$$$$30\!\cdots\!50$$$$T^{2} +$$$$40\!\cdots\!00$$$$T^{3} +$$$$52\!\cdots\!25$$$$T^{4}$$
$7$ $$1 + 3448443953486000 T +$$$$26\!\cdots\!50$$$$T^{2} +$$$$64\!\cdots\!00$$$$T^{3} +$$$$34\!\cdots\!49$$$$T^{4}$$
$11$ $$1 - 26734036354848538056 T +$$$$70\!\cdots\!26$$$$T^{2} -$$$$90\!\cdots\!76$$$$T^{3} +$$$$11\!\cdots\!41$$$$T^{4}$$
$13$ $$1 -$$$$53\!\cdots\!00$$$$T +$$$$32\!\cdots\!90$$$$T^{2} -$$$$87\!\cdots\!00$$$$T^{3} +$$$$27\!\cdots\!89$$$$T^{4}$$
$17$ $$1 -$$$$89\!\cdots\!00$$$$T +$$$$86\!\cdots\!30$$$$T^{2} -$$$$30\!\cdots\!00$$$$T^{3} +$$$$11\!\cdots\!29$$$$T^{4}$$
$19$ $$1 -$$$$37\!\cdots\!20$$$$T +$$$$20\!\cdots\!78$$$$T^{2} -$$$$76\!\cdots\!80$$$$T^{3} +$$$$42\!\cdots\!21$$$$T^{4}$$
$23$ $$1 -$$$$26\!\cdots\!00$$$$T +$$$$48\!\cdots\!10$$$$T^{2} -$$$$63\!\cdots\!00$$$$T^{3} +$$$$58\!\cdots\!09$$$$T^{4}$$
$29$ $$1 -$$$$12\!\cdots\!20$$$$T +$$$$21\!\cdots\!18$$$$T^{2} -$$$$16\!\cdots\!80$$$$T^{3} +$$$$16\!\cdots\!81$$$$T^{4}$$
$31$ $$1 -$$$$26\!\cdots\!24$$$$T +$$$$24\!\cdots\!66$$$$T^{2} -$$$$39\!\cdots\!64$$$$T^{3} +$$$$22\!\cdots\!21$$$$T^{4}$$
$37$ $$1 +$$$$68\!\cdots\!00$$$$T +$$$$13\!\cdots\!90$$$$T^{2} +$$$$71\!\cdots\!00$$$$T^{3} +$$$$11\!\cdots\!89$$$$T^{4}$$
$41$ $$1 -$$$$12\!\cdots\!36$$$$T +$$$$12\!\cdots\!86$$$$T^{2} -$$$$59\!\cdots\!16$$$$T^{3} +$$$$22\!\cdots\!61$$$$T^{4}$$
$43$ $$1 +$$$$25\!\cdots\!00$$$$T +$$$$44\!\cdots\!50$$$$T^{2} +$$$$70\!\cdots\!00$$$$T^{3} +$$$$75\!\cdots\!49$$$$T^{4}$$
$47$ $$1 +$$$$42\!\cdots\!00$$$$T +$$$$14\!\cdots\!70$$$$T^{2} +$$$$31\!\cdots\!00$$$$T^{3} +$$$$54\!\cdots\!69$$$$T^{4}$$
$53$ $$1 +$$$$15\!\cdots\!00$$$$T +$$$$16\!\cdots\!70$$$$T^{2} +$$$$10\!\cdots\!00$$$$T^{3} +$$$$39\!\cdots\!69$$$$T^{4}$$
$59$ $$1 -$$$$23\!\cdots\!40$$$$T +$$$$64\!\cdots\!38$$$$T^{2} -$$$$79\!\cdots\!60$$$$T^{3} +$$$$11\!\cdots\!61$$$$T^{4}$$
$61$ $$1 -$$$$10\!\cdots\!44$$$$T +$$$$10\!\cdots\!26$$$$T^{2} -$$$$11\!\cdots\!24$$$$T^{3} +$$$$13\!\cdots\!41$$$$T^{4}$$
$67$ $$1 +$$$$10\!\cdots\!00$$$$T +$$$$94\!\cdots\!30$$$$T^{2} +$$$$39\!\cdots\!00$$$$T^{3} +$$$$13\!\cdots\!29$$$$T^{4}$$
$71$ $$1 -$$$$74\!\cdots\!16$$$$T +$$$$58\!\cdots\!46$$$$T^{2} -$$$$23\!\cdots\!56$$$$T^{3} +$$$$98\!\cdots\!81$$$$T^{4}$$
$73$ $$1 -$$$$19\!\cdots\!00$$$$T +$$$$18\!\cdots\!10$$$$T^{2} -$$$$17\!\cdots\!00$$$$T^{3} +$$$$76\!\cdots\!09$$$$T^{4}$$
$79$ $$1 -$$$$27\!\cdots\!80$$$$T +$$$$50\!\cdots\!18$$$$T^{2} -$$$$44\!\cdots\!20$$$$T^{3} +$$$$26\!\cdots\!81$$$$T^{4}$$
$83$ $$1 -$$$$47\!\cdots\!00$$$$T +$$$$24\!\cdots\!30$$$$T^{2} -$$$$47\!\cdots\!00$$$$T^{3} +$$$$10\!\cdots\!29$$$$T^{4}$$
$89$ $$1 -$$$$13\!\cdots\!60$$$$T +$$$$23\!\cdots\!58$$$$T^{2} -$$$$17\!\cdots\!40$$$$T^{3} +$$$$17\!\cdots\!41$$$$T^{4}$$
$97$ $$1 -$$$$60\!\cdots\!00$$$$T +$$$$57\!\cdots\!70$$$$T^{2} -$$$$19\!\cdots\!00$$$$T^{3} +$$$$10\!\cdots\!69$$$$T^{4}$$