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Properties

Label 9.36.a
Level $9$
Weight $36$
Character orbit 9.a
Rep. character $\chi_{9}(1,\cdot)$
Character field $\Q$
Dimension $14$
Newform subspaces $4$
Sturm bound $36$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 36 \)
Character orbit: \([\chi]\) \(=\) 9.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 4 \)
Sturm bound: \(36\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{36}(\Gamma_0(9))\).

Total New Old
Modular forms 37 15 22
Cusp forms 33 14 19
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
\(+\)\(6\)
\(-\)\(8\)

Trace form

\( 14 q + 8586 q^{2} + 248788772660 q^{4} - 2327548792680 q^{5} - 325661273688224 q^{7} + 3548093233178376 q^{8} + O(q^{10}) \) \( 14 q + 8586 q^{2} + 248788772660 q^{4} - 2327548792680 q^{5} - 325661273688224 q^{7} + 3548093233178376 q^{8} - 418890575786577660 q^{10} - 793966894352917128 q^{11} + 10936590773150199676 q^{13} + 27752464588746402720 q^{14} + 3608825719437173307536 q^{16} - 4982453747585548937280 q^{17} + 67740411387471149318824 q^{19} - 206773699131822907111320 q^{20} + 765705713148407319395400 q^{22} - 1785977265605479802544672 q^{23} + 1302534037661664059533610 q^{25} + 4215230269758474922110204 q^{26} - 13930647854137953771865472 q^{28} + 38429489471848618782187944 q^{29} + 53185388604358686732317488 q^{31} - 346213089487332946777146336 q^{32} + 590010735549622573662571116 q^{34} - 1216667287674384464115123120 q^{35} - 98226231076051533607947164 q^{37} + 8216979949201899339485858952 q^{38} - 20716376486383241012565514800 q^{40} + 29666567872561517581465815840 q^{41} - 48398212566080617797031358216 q^{43} - 98159335855306491081458146608 q^{44} + 425261172693809451870104788080 q^{46} - 404675717564364283871234752752 q^{47} + 578936434886989318347288915678 q^{49} - 17483239402153894828787456250 q^{50} - 2519537449012819482233937574280 q^{52} + 2091023533860069030883237028808 q^{53} - 3878582676958227152392107519600 q^{55} - 579987986995181474369880312960 q^{56} + 16697724917975781609557464130196 q^{58} - 23897521555024737849387514625592 q^{59} + 61075635599311472421948750902836 q^{61} - 115473118704338972918189194894512 q^{62} + 55176988368140110112680904608832 q^{64} - 72320045364248936227439592289680 q^{65} + 88984539627491889764985070463080 q^{67} - 441298800092359075120020492863112 q^{68} + 354430267976306591768729164691520 q^{70} - 761982219265256943809686960337856 q^{71} + 1423540922517751256780113313866612 q^{73} - 4056881255737235337606300466650900 q^{74} + 5979211973086947156218347728898000 q^{76} - 6821260280434697638855000036028832 q^{77} + 2638446736741780802110021891610224 q^{79} - 15630400274938532108950689231089760 q^{80} + 14571306529709028984474126515129052 q^{82} - 15945119597137239474209504756315928 q^{83} + 16261006521954825223552088566919280 q^{85} - 50224491501996729597245189161393608 q^{86} + 61407173299338599148163548124892064 q^{88} - 66034474709518254578423626138361328 q^{89} + 29355479797463648185097828474930720 q^{91} - 101601030565166522836460997307160736 q^{92} + 17784772125161894085242315023285824 q^{94} - 70553289549211174447030246624432560 q^{95} + 64637925215971752484048022684301700 q^{97} + 157835675028834944356308316152117786 q^{98} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{36}^{\mathrm{new}}(\Gamma_0(9))\) 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
9.36.a.a 9.a 1.a $2$ $69.836$ \(\Q(\sqrt{2196841}) \) None \(60912\) \(0\) \(13\!\cdots\!40\) \(-12\!\cdots\!44\) $-$ $\mathrm{SU}(2)$ \(q+(30456-\beta )q^{2}+(28571469952+\cdots)q^{4}+\cdots\)
9.36.a.b 9.a 1.a $3$ $69.836$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(-139656\) \(0\) \(-892652054010\) \(87\!\cdots\!56\) $-$ $\mathrm{SU}(2)$ \(q+(-46552-\beta _{1})q^{2}+(11613754048+\cdots)q^{4}+\cdots\)
9.36.a.c 9.a 1.a $3$ $69.836$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(87330\) \(0\) \(-27\!\cdots\!10\) \(48\!\cdots\!64\) $-$ $\mathrm{SU}(2)$ \(q+(29110-\beta _{1})q^{2}+(10829584300+\cdots)q^{4}+\cdots\)
9.36.a.d 9.a 1.a $6$ $69.836$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(0\) \(0\) \(0\) \(-49\!\cdots\!00\) $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(20719302952+\beta _{3})q^{4}+\cdots\)

Decomposition of \(S_{36}^{\mathrm{old}}(\Gamma_0(9))\) into lower level spaces

\( S_{36}^{\mathrm{old}}(\Gamma_0(9)) \cong \) \(S_{36}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 3}\)\(\oplus\)\(S_{36}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 2}\)