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Properties

Label 9.48.a
Level $9$
Weight $48$
Character orbit 9.a
Rep. character $\chi_{9}(1,\cdot)$
Character field $\Q$
Dimension $19$
Newform subspaces $4$
Sturm bound $48$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 48 \)
Character orbit: \([\chi]\) \(=\) 9.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 4 \)
Sturm bound: \(48\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{48}(\Gamma_0(9))\).

Total New Old
Modular forms 49 20 29
Cusp forms 45 19 26
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
\(+\)\(8\)
\(-\)\(11\)

Trace form

\( 19 q - 2603046 q^{2} + 1095469028793076 q^{4} - 22808265611820090 q^{5} - 21646563725382979864 q^{7} - 1123559395851858086040 q^{8} + O(q^{10}) \) \( 19 q - 2603046 q^{2} + 1095469028793076 q^{4} - 22808265611820090 q^{5} - 21646563725382979864 q^{7} - 1123559395851858086040 q^{8} - 622746595069512256679580 q^{10} + 2555337957125736891641172 q^{11} - 289536941457002779404204718 q^{13} - 1064942250015251610138869952 q^{14} + 30683657610205970147058639760 q^{16} - 42074010384113876870980325286 q^{17} + 1757135030793877899925887552212 q^{19} + 557675694594991170286428947880 q^{20} - 3796475352783823438192606838328 q^{22} - 186706652173700663912196373712952 q^{23} + 2535501763995039689583274503296245 q^{25} - 3269482147044981899873779058265732 q^{26} - 6885013842452387927143599897596992 q^{28} + 10821918188909686593873904659653262 q^{29} + 51355596214563054927534779834606048 q^{31} - 713693857240511598248209126522894176 q^{32} + 462874713599034626587707548329421580 q^{34} - 8407721780045704429359779911746163440 q^{35} + 6921258981480546587844866967245235626 q^{37} - 67205060684026176276843949677471001080 q^{38} + 50799665169881170287935292632493893520 q^{40} - 36635354676543389370200046799144117854 q^{41} - 304393798282118946111148257103288600948 q^{43} + 1422986513520592609301855758172252591184 q^{44} + 2093188960239250207499977212104461103664 q^{46} + 544273691620330958131222305612867435984 q^{47} + 6839667497211206135890529185809760631595 q^{49} + 1118532443327880339539624380725742848150 q^{50} - 37637562422486909847110317633387817162824 q^{52} - 66222136717605163198740404411171581442922 q^{53} + 191106857214325798266445374235904066114280 q^{55} - 248956502905565336064928252319448948030720 q^{56} - 318315440542638250220293289807551129302540 q^{58} - 300425301105242260919213709381292567055196 q^{59} + 534523722102668258881122936273153843900002 q^{61} + 4946722114367130319616172301775097282369648 q^{62} - 4102309233812007976285481975481541519360448 q^{64} + 9024792227160033705356964614022412933277220 q^{65} - 9981804290661615202276502613953267470689724 q^{67} - 260406302263030837817737563235797516600648 q^{68} + 42309220292050577134619253708422481071629440 q^{70} - 97632340809776490413007827387632888395585192 q^{71} + 127465758935191568242678815674015744081671022 q^{73} - 230469949698779055414176589773546518300246932 q^{74} + 51530925140628868367591464129839154027315280 q^{76} + 332277669364633831649038617842162666935500768 q^{77} - 1170506464081711097493067090760821283352651568 q^{79} + 1297433239429975397864909869103112750641542560 q^{80} + 1736516804954327794893263992779513721041875772 q^{82} + 2288694904120648851239569143686088906678776748 q^{83} - 578081158671149469976394436036942132735573420 q^{85} - 7472469656887834544887667751561150628787078088 q^{86} - 2954063081539170575156560264019239009562615520 q^{88} + 334491228376705326854649705486422031274565106 q^{89} + 3834062938848909443793680824843175642738961904 q^{91} - 35347240653200516790269905048943105367808263456 q^{92} - 1805108100425056790653401950646638451703982976 q^{94} + 3547744902279047537765433681127246851907281000 q^{95} + 23244881619292186663037380031398600474887473126 q^{97} + 44708619595647804990166201593526337517638085162 q^{98} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{48}^{\mathrm{new}}(\Gamma_0(9))\) 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
9.48.a.a 9.a 1.a $3$ $125.917$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(15384840\) \(0\) \(-15\!\cdots\!50\) \(59\!\cdots\!84\) $-$ $\mathrm{SU}(2)$ \(q+(5128280+\beta _{1})q^{2}+(55047382225600+\cdots)q^{4}+\cdots\)
9.48.a.b 9.a 1.a $4$ $125.917$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(-12202326\) \(0\) \(-38\!\cdots\!40\) \(-39\!\cdots\!08\) $-$ $\mathrm{SU}(2)$ \(q+(-3050581-\beta _{1})q^{2}+(48210021465679+\cdots)q^{4}+\cdots\)
9.48.a.c 9.a 1.a $4$ $125.917$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(-5785560\) \(0\) \(31\!\cdots\!00\) \(-39\!\cdots\!00\) $-$ $\mathrm{SU}(2)$ \(q+(-1446390-\beta _{1})q^{2}+(101201874790288+\cdots)q^{4}+\cdots\)
9.48.a.d 9.a 1.a $8$ $125.917$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(0\) \(0\) \(0\) \(-28\!\cdots\!40\) $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(41584911618802+\beta _{4}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{48}^{\mathrm{old}}(\Gamma_0(9))\) into lower level spaces

\( S_{48}^{\mathrm{old}}(\Gamma_0(9)) \cong \) \(S_{48}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 3}\)\(\oplus\)\(S_{48}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 2}\)