# Properties

 Label 3.34.a.a Level $3$ Weight $34$ Character orbit 3.a Self dual yes Analytic conductor $20.695$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: yes Analytic conductor: $$20.6948486643$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: $$\mathbb{Q}[x]/(x^{3} - \cdots)$$ Defining polynomial: $$x^{3} - x^{2} - 35900150 x + 10469144400$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2^{6}\cdot 3^{6}\cdot 5\cdot 7\cdot 11$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + ( 13734 - \beta_{1} ) q^{2} + 43046721 q^{3} + ( -369155924 - 63216 \beta_{1} + \beta_{2} ) q^{4} + ( 17087274630 + 280984 \beta_{1} + 108 \beta_{2} ) q^{5} + ( 591203666214 - 43046721 \beta_{1} ) q^{6} + ( 25371244671752 - 323610696 \beta_{1} - 6404 \beta_{2} ) q^{7} + ( 384716449496664 + 3092877952 \beta_{1} + 41202 \beta_{2} ) q^{8} + 1853020188851841 q^{9} +O(q^{10})$$ $$q +(13734 - \beta_{1}) q^{2} +43046721 q^{3} +(-369155924 - 63216 \beta_{1} + \beta_{2}) q^{4} +(17087274630 + 280984 \beta_{1} + 108 \beta_{2}) q^{5} +(591203666214 - 43046721 \beta_{1}) q^{6} +(25371244671752 - 323610696 \beta_{1} - 6404 \beta_{2}) q^{7} +(384716449496664 + 3092877952 \beta_{1} + 41202 \beta_{2}) q^{8} +1853020188851841 q^{9} +(-2022248971884060 - 298904737158 \beta_{1} - 2658496 \beta_{2}) q^{10} +(-34507916295151860 - 1124553276112 \beta_{1} + 13291992 \beta_{2}) q^{11} +(-15890952065925204 - 2721241514736 \beta_{1} + 43046721 \beta_{2}) q^{12} +(638470867119282830 - 20317957756848 \beta_{1} - 460386584 \beta_{2}) q^{13} +(2947741324795972656 - 23848982404616 \beta_{1} + 464588352 \beta_{2}) q^{14} +(735551143647988230 + 12095439853464 \beta_{1} + 4649045868 \beta_{2}) q^{15} +(-16387763951764585808 + 198529037952768 \beta_{1} - 12589833372 \beta_{2}) q^{16} +(-5650418010665548782 + 3009063412790256 \beta_{1} - 5037146568 \beta_{2}) q^{17} +(25449379273691184294 - 1853020188851841 \beta_{1}) q^{18} +($$$$87\!\cdots\!88$$$$- 3612192985132560 \beta_{1} + 273191138616 \beta_{2}) q^{19} +($$$$22\!\cdots\!60$$$$- 7902406121558432 \beta_{1} - 570284067834 \beta_{2}) q^{20} +($$$$10\!\cdots\!92$$$$- 13930379343327816 \beta_{1} - 275671201284 \beta_{2}) q^{21} +($$$$85\!\cdots\!76$$$$- 57532809152138508 \beta_{1} + 831943364224 \beta_{2}) q^{22} +($$$$26\!\cdots\!56$$$$- 20101240887203216 \beta_{1} + 2300566232952 \beta_{2}) q^{23} +($$$$16\!\cdots\!44$$$$+ 133138254286795392 \beta_{1} + 1773610998642 \beta_{2}) q^{24} +($$$$14\!\cdots\!75$$$$+ 1314683245233833760 \beta_{1} - 25998439792880 \beta_{2}) q^{25} +($$$$17\!\cdots\!52$$$$- 383232636676627598 \beta_{1} + 30452908017024 \beta_{2}) q^{26} +$$$$79\!\cdots\!61$$$$q^{27} +($$$$14\!\cdots\!44$$$$- 2620158482959136640 \beta_{1} + 68631475550856 \beta_{2}) q^{28} +($$$$39\!\cdots\!86$$$$- 9122515363106364664 \beta_{1} - 173787588879516 \beta_{2}) q^{29} +(-$$$$87\!\cdots\!60$$$$- 12866868826018758918 \beta_{1} - 114439535591616 \beta_{2}) q^{30} +(-$$$$57\!\cdots\!92$$$$+ 32335952639255595288 \beta_{1} + 532179934052908 \beta_{2}) q^{31} +(-$$$$51\!\cdots\!28$$$$+ 34116717155837996544 \beta_{1} - 275298931161144 \beta_{2}) q^{32} +(-$$$$14\!\cdots\!60$$$$- 48408331126429228752 \beta_{1} + 572176671158232 \beta_{2}) q^{33} +(-$$$$24\!\cdots\!48$$$$+$$$$16\!\cdots\!10$$$$\beta_{1} - 2898175668242304 \beta_{2}) q^{34} +(-$$$$15\!\cdots\!60$$$$-$$$$16\!\cdots\!08$$$$\beta_{1} + 3119819378072904 \beta_{2}) q^{35} +(-$$$$68\!\cdots\!84$$$$-$$$$11\!\cdots\!56$$$$\beta_{1} + 1853020188851841 \beta_{2}) q^{36} +($$$$76\!\cdots\!62$$$$+$$$$69\!\cdots\!92$$$$\beta_{1} - 1782384058562208 \beta_{2}) q^{37} +($$$$41\!\cdots\!68$$$$-$$$$18\!\cdots\!40$$$$\beta_{1} - 2401836740360064 \beta_{2}) q^{38} +($$$$27\!\cdots\!30$$$$-$$$$87\!\cdots\!08$$$$\beta_{1} - 19818132833591064 \beta_{2}) q^{39} +($$$$11\!\cdots\!00$$$$+$$$$15\!\cdots\!20$$$$\beta_{1} + 43292946343949740 \beta_{2}) q^{40} +($$$$14\!\cdots\!38$$$$+$$$$37\!\cdots\!80$$$$\beta_{1} - 7030519097942808 \beta_{2}) q^{41} +($$$$12\!\cdots\!76$$$$-$$$$10\!\cdots\!36$$$$\beta_{1} + 19999005168393792 \beta_{2}) q^{42} +($$$$31\!\cdots\!48$$$$+$$$$59\!\cdots\!56$$$$\beta_{1} - 91355347912557080 \beta_{2}) q^{43} +($$$$87\!\cdots\!84$$$$-$$$$40\!\cdots\!76$$$$\beta_{1} - 74958933945275892 \beta_{2}) q^{44} +($$$$31\!\cdots\!30$$$$+$$$$52\!\cdots\!44$$$$\beta_{1} + 200126180395998828 \beta_{2}) q^{45} +($$$$52\!\cdots\!28$$$$-$$$$33\!\cdots\!72$$$$\beta_{1} - 30543424165002112 \beta_{2}) q^{46} +(-$$$$29\!\cdots\!84$$$$+$$$$95\!\cdots\!60$$$$\beta_{1} + 411342274012752936 \beta_{2}) q^{47} +(-$$$$70\!\cdots\!68$$$$+$$$$85\!\cdots\!28$$$$\beta_{1} - 541951044600973212 \beta_{2}) q^{48} +(-$$$$53\!\cdots\!91$$$$-$$$$48\!\cdots\!36$$$$\beta_{1} - 184357709370325568 \beta_{2}) q^{49} +(-$$$$86\!\cdots\!50$$$$-$$$$47\!\cdots\!95$$$$\beta_{1} - 742353591633373440 \beta_{2}) q^{50} +(-$$$$24\!\cdots\!22$$$$+$$$$12\!\cdots\!76$$$$\beta_{1} - 216832642948803528 \beta_{2}) q^{51} +(-$$$$44\!\cdots\!32$$$$-$$$$99\!\cdots\!20$$$$\beta_{1} + 3667532963184174990 \beta_{2}) q^{52} +(-$$$$39\!\cdots\!34$$$$+$$$$21\!\cdots\!20$$$$\beta_{1} + 996010849755971604 \beta_{2}) q^{53} +($$$$10\!\cdots\!74$$$$-$$$$79\!\cdots\!61$$$$\beta_{1}) q^{54} +($$$$28\!\cdots\!60$$$$-$$$$19\!\cdots\!72$$$$\beta_{1} - 11954579525329437664 \beta_{2}) q^{55} +(-$$$$40\!\cdots\!80$$$$-$$$$12\!\cdots\!64$$$$\beta_{1} - 2881478375702479728 \beta_{2}) q^{56} +($$$$37\!\cdots\!48$$$$-$$$$15\!\cdots\!60$$$$\beta_{1} + 11759982723675278136 \beta_{2}) q^{57} +($$$$78\!\cdots\!36$$$$-$$$$36\!\cdots\!02$$$$\beta_{1} + 12948275344700029888 \beta_{2}) q^{58} +($$$$43\!\cdots\!52$$$$-$$$$40\!\cdots\!64$$$$\beta_{1} + 10719599070768289056 \beta_{2}) q^{59} +($$$$95\!\cdots\!60$$$$-$$$$34\!\cdots\!72$$$$\beta_{1} - 24548859158795272314 \beta_{2}) q^{60} +($$$$12\!\cdots\!54$$$$+$$$$37\!\cdots\!04$$$$\beta_{1} - 4256599947344600048 \beta_{2}) q^{61} +(-$$$$26\!\cdots\!56$$$$+$$$$71\!\cdots\!92$$$$\beta_{1} - 44051361707496312000 \beta_{2}) q^{62} +($$$$47\!\cdots\!32$$$$-$$$$59\!\cdots\!36$$$$\beta_{1} - 11866741289407189764 \beta_{2}) q^{63} +(-$$$$20\!\cdots\!48$$$$+$$$$58\!\cdots\!68$$$$\beta_{1} + 80089558704402231696 \beta_{2}) q^{64} +(-$$$$11\!\cdots\!80$$$$-$$$$11\!\cdots\!24$$$$\beta_{1} +$$$$10\!\cdots\!12$$$$\beta_{2}) q^{65} +($$$$36\!\cdots\!96$$$$-$$$$24\!\cdots\!68$$$$\beta_{1} + 35812433887551909504 \beta_{2}) q^{66} +(-$$$$52\!\cdots\!32$$$$-$$$$11\!\cdots\!68$$$$\beta_{1} -$$$$28\!\cdots\!16$$$$\beta_{2}) q^{67} +(-$$$$16\!\cdots\!72$$$$+$$$$14\!\cdots\!24$$$$\beta_{1} - 61268200541361628398 \beta_{2}) q^{68} +($$$$11\!\cdots\!76$$$$-$$$$86\!\cdots\!36$$$$\beta_{1} + 99031832771905750392 \beta_{2}) q^{69} +($$$$11\!\cdots\!20$$$$-$$$$11\!\cdots\!04$$$$\beta_{1} + 94795686836474695552 \beta_{2}) q^{70} +(-$$$$22\!\cdots\!28$$$$+$$$$63\!\cdots\!84$$$$\beta_{1} - 45679595098071025512 \beta_{2}) q^{71} +($$$$71\!\cdots\!24$$$$+$$$$57\!\cdots\!32$$$$\beta_{1} + 76348137821073552882 \beta_{2}) q^{72} +($$$$58\!\cdots\!06$$$$-$$$$18\!\cdots\!12$$$$\beta_{1} +$$$$86\!\cdots\!60$$$$\beta_{2}) q^{73} +(-$$$$54\!\cdots\!24$$$$+$$$$31\!\cdots\!98$$$$\beta_{1} -$$$$65\!\cdots\!80$$$$\beta_{2}) q^{74} +($$$$60\!\cdots\!75$$$$+$$$$56\!\cdots\!60$$$$\beta_{1} -$$$$11\!\cdots\!80$$$$\beta_{2}) q^{75} +($$$$75\!\cdots\!72$$$$-$$$$92\!\cdots\!00$$$$\beta_{1} -$$$$49\!\cdots\!36$$$$\beta_{2}) q^{76} +($$$$17\!\cdots\!64$$$$-$$$$30\!\cdots\!68$$$$\beta_{1} +$$$$12\!\cdots\!80$$$$\beta_{2}) q^{77} +($$$$74\!\cdots\!92$$$$-$$$$16\!\cdots\!58$$$$\beta_{1} +$$$$13\!\cdots\!04$$$$\beta_{2}) q^{78} +($$$$15\!\cdots\!80$$$$-$$$$38\!\cdots\!16$$$$\beta_{1} -$$$$15\!\cdots\!32$$$$\beta_{2}) q^{79} +(-$$$$29\!\cdots\!20$$$$-$$$$87\!\cdots\!96$$$$\beta_{1} +$$$$24\!\cdots\!48$$$$\beta_{2}) q^{80} +$$$$34\!\cdots\!81$$$$q^{81} +(-$$$$27\!\cdots\!96$$$$+$$$$60\!\cdots\!38$$$$\beta_{1} -$$$$35\!\cdots\!68$$$$\beta_{2}) q^{82} +(-$$$$36\!\cdots\!88$$$$+$$$$13\!\cdots\!72$$$$\beta_{1} -$$$$53\!\cdots\!40$$$$\beta_{2}) q^{83} +($$$$60\!\cdots\!24$$$$-$$$$11\!\cdots\!40$$$$\beta_{1} +$$$$29\!\cdots\!76$$$$\beta_{2}) q^{84} +(-$$$$52\!\cdots\!60$$$$+$$$$85\!\cdots\!52$$$$\beta_{1} +$$$$13\!\cdots\!24$$$$\beta_{2}) q^{85} +(-$$$$43\!\cdots\!20$$$$+$$$$23\!\cdots\!04$$$$\beta_{1} -$$$$39\!\cdots\!36$$$$\beta_{2}) q^{86} +($$$$16\!\cdots\!06$$$$-$$$$39\!\cdots\!44$$$$\beta_{1} -$$$$74\!\cdots\!36$$$$\beta_{2}) q^{87} +(-$$$$29\!\cdots\!96$$$$-$$$$37\!\cdots\!96$$$$\beta_{1} -$$$$14\!\cdots\!44$$$$\beta_{2}) q^{88} +($$$$93\!\cdots\!98$$$$-$$$$84\!\cdots\!68$$$$\beta_{1} +$$$$37\!\cdots\!80$$$$\beta_{2}) q^{89} +(-$$$$37\!\cdots\!60$$$$-$$$$55\!\cdots\!78$$$$\beta_{1} -$$$$49\!\cdots\!36$$$$\beta_{2}) q^{90} +($$$$13\!\cdots\!28$$$$+$$$$95\!\cdots\!32$$$$\beta_{1} -$$$$72\!\cdots\!64$$$$\beta_{2}) q^{91} +($$$$51\!\cdots\!72$$$$-$$$$19\!\cdots\!36$$$$\beta_{1} +$$$$14\!\cdots\!56$$$$\beta_{2}) q^{92} +(-$$$$24\!\cdots\!32$$$$+$$$$13\!\cdots\!48$$$$\beta_{1} +$$$$22\!\cdots\!68$$$$\beta_{2}) q^{93} +(-$$$$11\!\cdots\!00$$$$+$$$$22\!\cdots\!32$$$$\beta_{1} -$$$$18\!\cdots\!64$$$$\beta_{2}) q^{94} +($$$$65\!\cdots\!00$$$$+$$$$22\!\cdots\!80$$$$\beta_{1} +$$$$63\!\cdots\!60$$$$\beta_{2}) q^{95} +(-$$$$22\!\cdots\!88$$$$+$$$$14\!\cdots\!24$$$$\beta_{1} -$$$$11\!\cdots\!24$$$$\beta_{2}) q^{96} +($$$$27\!\cdots\!18$$$$-$$$$63\!\cdots\!48$$$$\beta_{1} -$$$$75\!\cdots\!76$$$$\beta_{2}) q^{97} +(-$$$$34\!\cdots\!50$$$$+$$$$56\!\cdots\!75$$$$\beta_{1} +$$$$88\!\cdots\!88$$$$\beta_{2}) q^{98} +(-$$$$63\!\cdots\!60$$$$-$$$$20\!\cdots\!92$$$$\beta_{1} +$$$$24\!\cdots\!72$$$$\beta_{2}) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q + 41202q^{2} + 129140163q^{3} - 1107467772q^{4} + 51261823890q^{5} + 1773610998642q^{6} + 76113734015256q^{7} + 1154149348489992q^{8} + 5559060566555523q^{9} + O(q^{10})$$ $$3q + 41202q^{2} + 129140163q^{3} - 1107467772q^{4} + 51261823890q^{5} + 1773610998642q^{6} + 76113734015256q^{7} + 1154149348489992q^{8} + 5559060566555523q^{9} - 6066746915652180q^{10} - 103523748885455580q^{11} - 47672856197775612q^{12} + 1915412601357848490q^{13} + 8843223974387917968q^{14} + 2206653430943964690q^{15} - 49163291855293757424q^{16} - 16951254031996646346q^{17} + 76348137821073552882q^{18} +$$$$26\!\cdots\!64$$$$q^{19} +$$$$66\!\cdots\!80$$$$q^{20} +$$$$32\!\cdots\!76$$$$q^{21} +$$$$25\!\cdots\!28$$$$q^{22} +$$$$79\!\cdots\!68$$$$q^{23} +$$$$49\!\cdots\!32$$$$q^{24} +$$$$42\!\cdots\!25$$$$q^{25} +$$$$51\!\cdots\!56$$$$q^{26} +$$$$23\!\cdots\!83$$$$q^{27} +$$$$42\!\cdots\!32$$$$q^{28} +$$$$11\!\cdots\!58$$$$q^{29} -$$$$26\!\cdots\!80$$$$q^{30} -$$$$17\!\cdots\!76$$$$q^{31} -$$$$15\!\cdots\!84$$$$q^{32} -$$$$44\!\cdots\!80$$$$q^{33} -$$$$72\!\cdots\!44$$$$q^{34} -$$$$46\!\cdots\!80$$$$q^{35} -$$$$20\!\cdots\!52$$$$q^{36} +$$$$23\!\cdots\!86$$$$q^{37} +$$$$12\!\cdots\!04$$$$q^{38} +$$$$82\!\cdots\!90$$$$q^{39} +$$$$33\!\cdots\!00$$$$q^{40} +$$$$42\!\cdots\!14$$$$q^{41} +$$$$38\!\cdots\!28$$$$q^{42} +$$$$93\!\cdots\!44$$$$q^{43} +$$$$26\!\cdots\!52$$$$q^{44} +$$$$94\!\cdots\!90$$$$q^{45} +$$$$15\!\cdots\!84$$$$q^{46} -$$$$87\!\cdots\!52$$$$q^{47} -$$$$21\!\cdots\!04$$$$q^{48} -$$$$16\!\cdots\!73$$$$q^{49} -$$$$25\!\cdots\!50$$$$q^{50} -$$$$72\!\cdots\!66$$$$q^{51} -$$$$13\!\cdots\!96$$$$q^{52} -$$$$11\!\cdots\!02$$$$q^{53} +$$$$32\!\cdots\!22$$$$q^{54} +$$$$85\!\cdots\!80$$$$q^{55} -$$$$12\!\cdots\!40$$$$q^{56} +$$$$11\!\cdots\!44$$$$q^{57} +$$$$23\!\cdots\!08$$$$q^{58} +$$$$12\!\cdots\!56$$$$q^{59} +$$$$28\!\cdots\!80$$$$q^{60} +$$$$36\!\cdots\!62$$$$q^{61} -$$$$80\!\cdots\!68$$$$q^{62} +$$$$14\!\cdots\!96$$$$q^{63} -$$$$61\!\cdots\!44$$$$q^{64} -$$$$33\!\cdots\!40$$$$q^{65} +$$$$11\!\cdots\!88$$$$q^{66} -$$$$15\!\cdots\!96$$$$q^{67} -$$$$49\!\cdots\!16$$$$q^{68} +$$$$34\!\cdots\!28$$$$q^{69} +$$$$33\!\cdots\!60$$$$q^{70} -$$$$66\!\cdots\!84$$$$q^{71} +$$$$21\!\cdots\!72$$$$q^{72} +$$$$17\!\cdots\!18$$$$q^{73} -$$$$16\!\cdots\!72$$$$q^{74} +$$$$18\!\cdots\!25$$$$q^{75} +$$$$22\!\cdots\!16$$$$q^{76} +$$$$52\!\cdots\!92$$$$q^{77} +$$$$22\!\cdots\!76$$$$q^{78} +$$$$47\!\cdots\!40$$$$q^{79} -$$$$89\!\cdots\!60$$$$q^{80} +$$$$10\!\cdots\!43$$$$q^{81} -$$$$83\!\cdots\!88$$$$q^{82} -$$$$10\!\cdots\!64$$$$q^{83} +$$$$18\!\cdots\!72$$$$q^{84} -$$$$15\!\cdots\!80$$$$q^{85} -$$$$13\!\cdots\!60$$$$q^{86} +$$$$50\!\cdots\!18$$$$q^{87} -$$$$87\!\cdots\!88$$$$q^{88} +$$$$28\!\cdots\!94$$$$q^{89} -$$$$11\!\cdots\!80$$$$q^{90} +$$$$40\!\cdots\!84$$$$q^{91} +$$$$15\!\cdots\!16$$$$q^{92} -$$$$74\!\cdots\!96$$$$q^{93} -$$$$34\!\cdots\!00$$$$q^{94} +$$$$19\!\cdots\!00$$$$q^{95} -$$$$66\!\cdots\!64$$$$q^{96} +$$$$81\!\cdots\!54$$$$q^{97} -$$$$10\!\cdots\!50$$$$q^{98} -$$$$19\!\cdots\!80$$$$q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 35900150 x + 10469144400$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{2} + 1429 \nu - 23933910$$$$)/195$$ $$\beta_{2}$$ $$=$$ $$($$$$-5046 \nu^{2} + 57654066 \nu + 120748888260$$$$)/65$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} + 15138 \beta_{1} + 332640$$$$)/997920$$ $$\nu^{2}$$ $$=$$ $$($$$$-1429 \beta_{2} + 172962198 \beta_{1} + 23883652124640$$$$)/997920$$

## 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
 5840.68 −6131.99 292.312
−81270.9 4.30467e7 −1.98497e9 5.17904e11 −3.49845e12 −3.34870e13 8.59432e14 1.85302e15 −4.20905e16
1.2 −11418.8 4.30467e7 −8.45955e9 −6.77881e11 −4.91541e11 5.88597e13 1.94684e14 1.85302e15 7.74057e15
1.3 133892. 4.30467e7 9.33705e9 2.11239e11 5.76360e12 5.07411e13 1.00033e14 1.85302e15 2.82832e16
 $$n$$: 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 3.34.a.a 3
3.b odd 2 1 9.34.a.d 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.34.a.a 3 1.a even 1 1 trivial
9.34.a.d 3 3.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{3} - 41202 T_{2}^{2} - 11482365600 T_{2} -$$124253441040384

'>$$12\!\cdots\!84$$ acting on $$S_{34}^{\mathrm{new}}(\Gamma_0(3))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 41202 T + 14287438176 T^{2} - 832098411159552 T^{3} +$$$$12\!\cdots\!92$$$$T^{4} -$$$$30\!\cdots\!28$$$$T^{5} +$$$$63\!\cdots\!88$$$$T^{6}$$
$3$ $$( 1 - 43046721 T )^{3}$$
$5$ $$1 - 51261823890 T -$$$$35\!\cdots\!25$$$$T^{2} +$$$$62\!\cdots\!00$$$$T^{3} -$$$$41\!\cdots\!25$$$$T^{4} -$$$$69\!\cdots\!50$$$$T^{5} +$$$$15\!\cdots\!25$$$$T^{6}$$
$7$ $$1 - 76113734015256 T +$$$$22\!\cdots\!65$$$$T^{2} -$$$$10\!\cdots\!64$$$$T^{3} +$$$$17\!\cdots\!55$$$$T^{4} -$$$$45\!\cdots\!44$$$$T^{5} +$$$$46\!\cdots\!43$$$$T^{6}$$
$11$ $$1 + 103523748885455580 T +$$$$52\!\cdots\!61$$$$T^{2} +$$$$32\!\cdots\!96$$$$T^{3} +$$$$12\!\cdots\!91$$$$T^{4} +$$$$55\!\cdots\!80$$$$T^{5} +$$$$12\!\cdots\!91$$$$T^{6}$$
$13$ $$1 - 1915412601357848490 T +$$$$65\!\cdots\!31$$$$T^{2} -$$$$25\!\cdots\!52$$$$T^{3} +$$$$37\!\cdots\!43$$$$T^{4} -$$$$63\!\cdots\!10$$$$T^{5} +$$$$19\!\cdots\!77$$$$T^{6}$$
$17$ $$1 + 16951254031996646346 T +$$$$10\!\cdots\!83$$$$T^{2} +$$$$11\!\cdots\!72$$$$T^{3} +$$$$44\!\cdots\!71$$$$T^{4} +$$$$27\!\cdots\!74$$$$T^{5} +$$$$65\!\cdots\!53$$$$T^{6}$$
$19$ $$1 -$$$$26\!\cdots\!64$$$$T +$$$$44\!\cdots\!37$$$$T^{2} -$$$$50\!\cdots\!52$$$$T^{3} +$$$$70\!\cdots\!83$$$$T^{4} -$$$$65\!\cdots\!84$$$$T^{5} +$$$$39\!\cdots\!79$$$$T^{6}$$
$23$ $$1 -$$$$79\!\cdots\!68$$$$T +$$$$45\!\cdots\!85$$$$T^{2} -$$$$15\!\cdots\!08$$$$T^{3} +$$$$39\!\cdots\!55$$$$T^{4} -$$$$59\!\cdots\!52$$$$T^{5} +$$$$64\!\cdots\!87$$$$T^{6}$$
$29$ $$1 -$$$$11\!\cdots\!58$$$$T +$$$$39\!\cdots\!87$$$$T^{2} -$$$$24\!\cdots\!24$$$$T^{3} +$$$$70\!\cdots\!43$$$$T^{4} -$$$$38\!\cdots\!18$$$$T^{5} +$$$$59\!\cdots\!69$$$$T^{6}$$
$31$ $$1 +$$$$17\!\cdots\!76$$$$T +$$$$28\!\cdots\!17$$$$T^{2} +$$$$47\!\cdots\!32$$$$T^{3} +$$$$46\!\cdots\!47$$$$T^{4} +$$$$46\!\cdots\!56$$$$T^{5} +$$$$44\!\cdots\!71$$$$T^{6}$$
$37$ $$1 -$$$$23\!\cdots\!86$$$$T +$$$$11\!\cdots\!95$$$$T^{2} -$$$$65\!\cdots\!24$$$$T^{3} +$$$$62\!\cdots\!15$$$$T^{4} -$$$$73\!\cdots\!74$$$$T^{5} +$$$$17\!\cdots\!73$$$$T^{6}$$
$41$ $$1 -$$$$42\!\cdots\!14$$$$T +$$$$39\!\cdots\!27$$$$T^{2} -$$$$10\!\cdots\!68$$$$T^{3} +$$$$65\!\cdots\!67$$$$T^{4} -$$$$11\!\cdots\!74$$$$T^{5} +$$$$46\!\cdots\!61$$$$T^{6}$$
$43$ $$1 -$$$$93\!\cdots\!44$$$$T +$$$$19\!\cdots\!53$$$$T^{2} -$$$$11\!\cdots\!40$$$$T^{3} +$$$$16\!\cdots\!79$$$$T^{4} -$$$$60\!\cdots\!56$$$$T^{5} +$$$$51\!\cdots\!07$$$$T^{6}$$
$47$ $$1 +$$$$87\!\cdots\!52$$$$T +$$$$64\!\cdots\!57$$$$T^{2} +$$$$26\!\cdots\!60$$$$T^{3} +$$$$97\!\cdots\!39$$$$T^{4} +$$$$19\!\cdots\!08$$$$T^{5} +$$$$34\!\cdots\!83$$$$T^{6}$$
$53$ $$1 +$$$$11\!\cdots\!02$$$$T +$$$$18\!\cdots\!15$$$$T^{2} +$$$$16\!\cdots\!12$$$$T^{3} +$$$$14\!\cdots\!95$$$$T^{4} +$$$$75\!\cdots\!58$$$$T^{5} +$$$$50\!\cdots\!17$$$$T^{6}$$
$59$ $$1 -$$$$12\!\cdots\!56$$$$T +$$$$82\!\cdots\!37$$$$T^{2} -$$$$68\!\cdots\!48$$$$T^{3} +$$$$22\!\cdots\!23$$$$T^{4} -$$$$97\!\cdots\!96$$$$T^{5} +$$$$20\!\cdots\!39$$$$T^{6}$$
$61$ $$1 -$$$$36\!\cdots\!62$$$$T +$$$$12\!\cdots\!59$$$$T^{2} -$$$$22\!\cdots\!36$$$$T^{3} +$$$$10\!\cdots\!79$$$$T^{4} -$$$$24\!\cdots\!82$$$$T^{5} +$$$$55\!\cdots\!41$$$$T^{6}$$
$67$ $$1 +$$$$15\!\cdots\!96$$$$T +$$$$11\!\cdots\!33$$$$T^{2} +$$$$29\!\cdots\!72$$$$T^{3} +$$$$20\!\cdots\!71$$$$T^{4} +$$$$52\!\cdots\!24$$$$T^{5} +$$$$60\!\cdots\!03$$$$T^{6}$$
$71$ $$1 +$$$$66\!\cdots\!84$$$$T +$$$$51\!\cdots\!85$$$$T^{2} +$$$$17\!\cdots\!00$$$$T^{3} +$$$$63\!\cdots\!35$$$$T^{4} +$$$$10\!\cdots\!64$$$$T^{5} +$$$$18\!\cdots\!31$$$$T^{6}$$
$73$ $$1 -$$$$17\!\cdots\!18$$$$T +$$$$16\!\cdots\!55$$$$T^{2} -$$$$10\!\cdots\!88$$$$T^{3} +$$$$52\!\cdots\!15$$$$T^{4} -$$$$16\!\cdots\!02$$$$T^{5} +$$$$29\!\cdots\!37$$$$T^{6}$$
$79$ $$1 -$$$$47\!\cdots\!40$$$$T +$$$$18\!\cdots\!17$$$$T^{2} -$$$$41\!\cdots\!20$$$$T^{3} +$$$$79\!\cdots\!63$$$$T^{4} -$$$$82\!\cdots\!40$$$$T^{5} +$$$$73\!\cdots\!19$$$$T^{6}$$
$83$ $$1 +$$$$10\!\cdots\!64$$$$T +$$$$92\!\cdots\!69$$$$T^{2} +$$$$46\!\cdots\!56$$$$T^{3} +$$$$19\!\cdots\!47$$$$T^{4} +$$$$50\!\cdots\!16$$$$T^{5} +$$$$97\!\cdots\!47$$$$T^{6}$$
$89$ $$1 -$$$$28\!\cdots\!94$$$$T +$$$$81\!\cdots\!67$$$$T^{2} -$$$$12\!\cdots\!72$$$$T^{3} +$$$$17\!\cdots\!23$$$$T^{4} -$$$$12\!\cdots\!34$$$$T^{5} +$$$$97\!\cdots\!09$$$$T^{6}$$
$97$ $$1 -$$$$81\!\cdots\!54$$$$T +$$$$65\!\cdots\!03$$$$T^{2} -$$$$25\!\cdots\!48$$$$T^{3} +$$$$23\!\cdots\!31$$$$T^{4} -$$$$10\!\cdots\!66$$$$T^{5} +$$$$49\!\cdots\!33$$$$T^{6}$$