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Properties

Label 9.64.a
Level $9$
Weight $64$
Character orbit 9.a
Rep. character $\chi_{9}(1,\cdot)$
Character field $\Q$
Dimension $26$
Newform subspaces $5$
Sturm bound $64$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 64 \)
Character orbit: \([\chi]\) \(=\) 9.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 5 \)
Sturm bound: \(64\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{64}(\Gamma_0(9))\).

Total New Old
Modular forms 65 27 38
Cusp forms 61 26 35
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
\(+\)\(11\)
\(-\)\(15\)

Trace form

\( 26 q - 1640168550 q^{2} + 122247942644929885172 q^{4} - 1056910554666109360170 q^{5} + 574491297497385755241115612 q^{7} + 33082259935738270458913704936 q^{8} + O(q^{10}) \) \( 26 q - 1640168550 q^{2} + 122247942644929885172 q^{4} - 1056910554666109360170 q^{5} + 574491297497385755241115612 q^{7} + 33082259935738270458913704936 q^{8} + 24149851229855045120544921829860 q^{10} - 622002176146132136348440304988900 q^{11} + 106310709050446714068698142111953848 q^{13} - 2775842094457414154056814705904932160 q^{14} + 652321205282000774920916238579480631184 q^{16} - 816869417055564184709219922877131054366 q^{17} - 22434191985227392988325419818557815812604 q^{19} + 440985322331750993982044534978373411751080 q^{20} - 117798309348856435643147489048150841505848 q^{22} - 3021678646207587783809381601922600851872184 q^{23} + 353551617034229385638213534689832006861703050 q^{25} - 74435276649779189912772244862508094072808580 q^{26} + 2620602002570473925986422702158486457676593856 q^{28} - 10190736265498108022754486512359077890945810610 q^{29} - 254134157021654520889586864900778135474826008092 q^{31} + 118274295400131962120348342575953668989536532128 q^{32} + 6369087067376028583232452254671298948322434695820 q^{34} - 1094777174657071574170001443953788895663335157600 q^{35} + 19901633005060651539832873665423142482603022767292 q^{37} + 143704000500202279908446084579887771651123913565192 q^{38} + 221221850772173423283993135361647800649592353217680 q^{40} - 731436304040680770138574523837894993815143986080470 q^{41} - 6017969392415174883379178383705661499402804124974164 q^{43} + 6718864113137196867250960117506892330890166510036560 q^{44} - 51244885032953752831956343236174014588402101351043280 q^{46} - 25277997955864975492257049627651260575443442222278368 q^{47} + 466099879019441822654080603361746967558153551796626842 q^{49} - 731737948337520826621551915847973447178416584671662250 q^{50} + 2700251515809688114601558638813803457761575140066893496 q^{52} - 1650683711989795663029765629097224388540430489690087914 q^{53} + 11380558525352755520042178933475811271235568544104340120 q^{55} - 54923568677782341738821258854787810933562707806341446400 q^{56} + 31571968363166029655673959928845176560011886305514657396 q^{58} - 258987945354998194231192369595305112281014522133928604420 q^{59} + 205046464886071714613976700344273771280638002496612724012 q^{61} - 1279469460936283878952868337231915865782933934827273763344 q^{62} + 5054224546217783855358469103746467736863703695147761780288 q^{64} - 7510312448510330565090693497792873219813878426884815265180 q^{65} + 5779473067993996345122884572705594361896061967284661482956 q^{67} - 12020001786279936126666919328644983417828921045545297407560 q^{68} - 29228732413510703270156715436656532167198002911417863043200 q^{70} - 37298197833099900897850233444935408084315476196018599103880 q^{71} + 79879438942691279756172487838556756497796720940362381566824 q^{73} + 377768207526805425214866872771917148488484841019596769208940 q^{74} - 727272489799281017149332194802014585742378820409864495446448 q^{76} + 1007285305512367427230255951493651672663898629174175758540736 q^{77} - 1005570220178738715688080284726303320324848713881904773607276 q^{79} + 3535472164128357347906851293009897041004641591744645164002720 q^{80} - 2751236341971023737421452338574060166843722552799661201399876 q^{82} + 61474564476706975369043361632054345120571900051318199944900 q^{83} + 5329140306694440586105094247658908490880753061978768307159380 q^{85} - 11553838499237540942145906900651964269139833423337832030099400 q^{86} - 3284704885064093208980047439099537524275480975139135846730976 q^{88} - 68015872104948535839993673102409476916106090318875009048921430 q^{89} + 168537401535355956137899630255943412566039181711951136563096920 q^{91} - 249138849240756923758063308669323250143936512218967825095289632 q^{92} + 292912070616202390429255226636608895119196203931291243866800000 q^{94} - 141269359505142723609016920167160971090658303882613757162535720 q^{95} - 1325350744490696926007330396286866287575323520777392867332521224 q^{97} + 1843004971008039663634302603135942234961136107498335651141149418 q^{98} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{64}^{\mathrm{new}}(\Gamma_0(9))\) 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
9.64.a.a 9.a 1.a $1$ $226.225$ \(\Q\) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(-46\!\cdots\!20\) $+$ $N(\mathrm{U}(1))$ \(q-2^{63}q^{4}-463417932763745775997075420q^{7}+\cdots\)
9.64.a.b 9.a 1.a $5$ $226.225$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(-1604947032\) \(0\) \(14\!\cdots\!30\) \(-61\!\cdots\!92\) $-$ $\mathrm{SU}(2)$ \(q+(-320989406-\beta _{1})q^{2}+\cdots\)
9.64.a.c 9.a 1.a $5$ $226.225$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(-507315096\) \(0\) \(50\!\cdots\!30\) \(37\!\cdots\!56\) $-$ $\mathrm{SU}(2)$ \(q+(-101463019+\beta _{1})q^{2}+\cdots\)
9.64.a.d 9.a 1.a $5$ $226.225$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(472093578\) \(0\) \(-15\!\cdots\!30\) \(27\!\cdots\!88\) $-$ $\mathrm{SU}(2)$ \(q+(94418716+\beta _{1})q^{2}+(3972280023974348032+\cdots)q^{4}+\cdots\)
9.64.a.e 9.a 1.a $10$ $226.225$ \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None \(0\) \(0\) \(0\) \(12\!\cdots\!80\) $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(5993700990147690472+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{64}^{\mathrm{old}}(\Gamma_0(9))\) into lower level spaces

\( S_{64}^{\mathrm{old}}(\Gamma_0(9)) \cong \) \(S_{64}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 3}\)\(\oplus\)\(S_{64}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 2}\)