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Properties

Label 9.32.a
Level $9$
Weight $32$
Character orbit 9.a
Rep. character $\chi_{9}(1,\cdot)$
Character field $\Q$
Dimension $13$
Newform subspaces $4$
Sturm bound $32$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 32 \)
Character orbit: \([\chi]\) \(=\) 9.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 4 \)
Sturm bound: \(32\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{32}(\Gamma_0(9))\).

Total New Old
Modular forms 33 14 19
Cusp forms 29 13 16
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
\(+\)\(6\)
\(-\)\(7\)

Trace form

\( 13 q + 7194 q^{2} + 15638503924 q^{4} - 23300436810 q^{5} - 442528069864 q^{7} + 42179398927080 q^{8} + O(q^{10}) \) \( 13 q + 7194 q^{2} + 15638503924 q^{4} - 23300436810 q^{5} - 442528069864 q^{7} + 42179398927080 q^{8} + 2505892109770980 q^{10} + 365632217559948 q^{11} + 449121957114156662 q^{13} - 1173697142965308480 q^{14} + 25929524369835536272 q^{16} - 28653181893416591406 q^{17} + 84528754912184057468 q^{19} - 71719822857383704920 q^{20} - 1445193502437105820728 q^{22} + 1666449014512952991048 q^{23} - 3553155848691893080685 q^{25} - 7833506226096826079364 q^{26} + 70197489765192790547648 q^{28} - 34560617905498851829554 q^{29} - 19268582536644992024992 q^{31} + 182554002407928540813984 q^{32} - 2002242581761801685847156 q^{34} + 789820933036675788405120 q^{35} + 3672562482938456459944406 q^{37} - 6151070867585606655039480 q^{38} + 20775091956866389991010960 q^{40} - 14807896691555278093869030 q^{41} - 8483790800530896422175868 q^{43} + 13420512533556909968654928 q^{44} - 69306951432505566054206160 q^{46} - 41111485607702432070992256 q^{47} + 245702088261422495200051221 q^{49} - 899020322490285117652151850 q^{50} + 2731510955615547134535381176 q^{52} - 2132017445969214318141044202 q^{53} + 1453927083534440632148464440 q^{55} - 6291326455243885846824288000 q^{56} + 6482147672955867365804817780 q^{58} - 9092403397705915054654365108 q^{59} + 9019834763673147014784368318 q^{61} - 43011291474908108117888988432 q^{62} + 94130382160565120995374282304 q^{64} - 75615214268603195164049994780 q^{65} + 77005573660007845255087368716 q^{67} - 188151787295209920781897768008 q^{68} + 176791794608719264107729617280 q^{70} - 156007314603783709040726911944 q^{71} - 22751032062053487469566793798 q^{73} - 208898686771560468619550151060 q^{74} + 371046835242057820803791338064 q^{76} - 373295852489331115468973755392 q^{77} + 523974823460867647554701351504 q^{79} + 185026613671501092082737348000 q^{80} - 1280534441908657281556162074948 q^{82} + 1149178443895902958554267123348 q^{83} - 1859644725824848164262687144620 q^{85} + 5564909531166519405252599099448 q^{86} - 9578960845114247472222362792160 q^{88} + 6412669309584508339254305972538 q^{89} - 2409745259502551201309721140240 q^{91} + 12966333160767916831006224670944 q^{92} - 9871566742987333999510209681024 q^{94} + 9765549548731579210636673174520 q^{95} - 1038720369807690991086659189854 q^{97} + 23395069058419468947647874826602 q^{98} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{32}^{\mathrm{new}}(\Gamma_0(9))\) 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
9.32.a.a 9.a 1.a $2$ $54.789$ \(\mathbb{Q}[x]/(x^{2} - \cdots)\) None \(-39960\) \(0\) \(19391218020\) \(30\!\cdots\!00\) $-$ $\mathrm{SU}(2)$ \(q+(-19980-\beta )q^{2}+(886267168+\cdots)q^{4}+\cdots\)
9.32.a.b 9.a 1.a $2$ $54.789$ \(\mathbb{Q}[x]/(x^{2} - \cdots)\) None \(39528\) \(0\) \(7930517220\) \(-10\!\cdots\!76\) $-$ $\mathrm{SU}(2)$ \(q+(19764-\beta )q^{2}+(-100683488+\cdots)q^{4}+\cdots\)
9.32.a.c 9.a 1.a $3$ $54.789$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(7626\) \(0\) \(-50622172050\) \(-25\!\cdots\!08\) $-$ $\mathrm{SU}(2)$ \(q+(2542-\beta _{1})q^{2}+(1406276044-15591\beta _{1}+\cdots)q^{4}+\cdots\)
9.32.a.d 9.a 1.a $6$ $54.789$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(0\) \(0\) \(0\) \(55\!\cdots\!20\) $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(1641418072+\beta _{3})q^{4}+\cdots\)

Decomposition of \(S_{32}^{\mathrm{old}}(\Gamma_0(9))\) into lower level spaces

\( S_{32}^{\mathrm{old}}(\Gamma_0(9)) \cong \) \(S_{32}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 3}\)\(\oplus\)\(S_{32}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 2}\)