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Properties

Label 9.40.a
Level $9$
Weight $40$
Character orbit 9.a
Rep. character $\chi_{9}(1,\cdot)$
Character field $\Q$
Dimension $16$
Newform subspaces $5$
Sturm bound $40$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 40 \)
Character orbit: \([\chi]\) \(=\) 9.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 5 \)
Sturm bound: \(40\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{40}(\Gamma_0(9))\).

Total New Old
Modular forms 41 17 24
Cusp forms 37 16 21
Eisenstein series 4 1 3

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(3\)Dim
\(+\)\(7\)
\(-\)\(9\)

Trace form

\( 16 q + 24570 q^{2} + 4877804040820 q^{4} + 26598008014650 q^{5} + 49198557759573452 q^{7} + 824174401178644776 q^{8} + O(q^{10}) \) \( 16 q + 24570 q^{2} + 4877804040820 q^{4} + 26598008014650 q^{5} + 49198557759573452 q^{7} + 824174401178644776 q^{8} + 23912709817263761700 q^{10} - 96603933748882378140 q^{11} + 1528348738613991910988 q^{13} - 32438878052385141058176 q^{14} + 1335324948041928667109776 q^{16} - 1444044991745780321113746 q^{17} - 11880953827345624954920340 q^{19} - 49248998873986355606479320 q^{20} + 440242109844382705374495432 q^{22} - 464578411237568409552907464 q^{23} + 8754682317482564686812313780 q^{25} - 15959435943545492584110641028 q^{26} + 32115238539674546270510390336 q^{28} - 24181416174488518208428470846 q^{29} - 191468079258876939700698356092 q^{31} + 796354124163191957429007604128 q^{32} - 3210000619864340353396250723508 q^{34} + 1235052190041803622592390355040 q^{35} + 2100658469057721679432326221552 q^{37} - 5819252554454923343927500395768 q^{38} + 68340014467144907709719187617040 q^{40} - 36231609161791301069619931249242 q^{41} - 57142624678424645945471669699644 q^{43} - 124698482394740326748251332134064 q^{44} - 362070080062210563991089387056208 q^{46} + 896955127891754406022395555136992 q^{47} + 281959269125121535847612419348608 q^{49} + 713033921754647756598731002672950 q^{50} - 8293880217875696786715435811599304 q^{52} - 1121050001690284327919151100226694 q^{53} + 41845359535731710008202931270192360 q^{55} - 36174568312893481766951354549061120 q^{56} + 6311657182783595744819658738260916 q^{58} + 28007565422217664828166586333889668 q^{59} - 225845678706512725935658230660016240 q^{61} + 169717652916455991720289743380082096 q^{62} + 59908272136397569470528640098691648 q^{64} + 75961436480195577416143965009961020 q^{65} + 299180472218614819537711422535442516 q^{67} - 2255364005750439516720984034487726280 q^{68} + 4659440951721943848204923763958536960 q^{70} - 1557657649520306473052535018282331128 q^{71} - 3999499724994177536445922957476832396 q^{73} - 1306092906589109018281726348776846996 q^{74} + 2455330270919108923846158673597009232 q^{76} - 2479255534577150778884115955503466944 q^{77} + 6253693257839274334974146785633674068 q^{79} - 53763494042926121281306584921891853920 q^{80} + 90760744278726815817629878318622226684 q^{82} - 108064196286510845819514580635075002820 q^{83} + 122440237518085641686977859885794320780 q^{85} - 338331433073940367870589872937592813000 q^{86} + 436172177610118818798606440778582095904 q^{88} - 523175087046324189427370857977433141338 q^{89} + 457556607041527833495639442716243948472 q^{91} - 1092562149093814140538547221127018590752 q^{92} + 1081926518817581292662758464813182580480 q^{94} - 2029295360966506659313386917171050122840 q^{95} + 971139899401954844432905721575923777716 q^{97} - 5919824420838707612925908288440376552502 q^{98} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{40}^{\mathrm{new}}(\Gamma_0(9))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 3
9.40.a.a 9.a 1.a $1$ $86.706$ \(\Q\) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(54\!\cdots\!40\) $+$ $N(\mathrm{U}(1))$ \(q-2^{39}q^{4}+54595696320612740q^{7}+\cdots\)
9.40.a.b 9.a 1.a $3$ $86.706$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(-548856\) \(0\) \(-17\!\cdots\!90\) \(-17\!\cdots\!44\) $-$ $\mathrm{SU}(2)$ \(q+(-182952+\beta _{1})q^{2}+(90692993728+\cdots)q^{4}+\cdots\)
9.40.a.c 9.a 1.a $3$ $86.706$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(-533574\) \(0\) \(53\!\cdots\!30\) \(-15\!\cdots\!28\) $-$ $\mathrm{SU}(2)$ \(q+(-177858+\beta _{1})q^{2}+(319147551244+\cdots)q^{4}+\cdots\)
9.40.a.d 9.a 1.a $3$ $86.706$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(1107000\) \(0\) \(-93\!\cdots\!90\) \(13\!\cdots\!04\) $-$ $\mathrm{SU}(2)$ \(q+(369000+\beta _{1})q^{2}+(335300075200+\cdots)q^{4}+\cdots\)
9.40.a.e 9.a 1.a $6$ $86.706$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(0\) \(0\) \(0\) \(33\!\cdots\!80\) $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(532022999032+\beta _{3})q^{4}+\cdots\)

Decomposition of \(S_{40}^{\mathrm{old}}(\Gamma_0(9))\) into lower level spaces

\( S_{40}^{\mathrm{old}}(\Gamma_0(9)) \cong \) \(S_{40}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 3}\)\(\oplus\)\(S_{40}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 2}\)