Properties

Label 9.26.a
Level $9$
Weight $26$
Character orbit 9.a
Rep. character $\chi_{9}(1,\cdot)$
Character field $\Q$
Dimension $10$
Newform subspaces $4$
Sturm bound $26$
Trace bound $1$

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Defining parameters

Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 26 \)
Character orbit: \([\chi]\) \(=\) 9.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 4 \)
Sturm bound: \(26\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{26}(\Gamma_0(9))\).

Total New Old
Modular forms 27 11 16
Cusp forms 23 10 13
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.
\(+\)\(4\)
\(-\)\(6\)

Trace form

\( 10 q + 4050 q^{2} + 220267348 q^{4} + 334280844 q^{5} - 40918831120 q^{7} + 288091685880 q^{8} + O(q^{10}) \) \( 10 q + 4050 q^{2} + 220267348 q^{4} + 334280844 q^{5} - 40918831120 q^{7} + 288091685880 q^{8} - 11209064898636 q^{10} + 15715268450280 q^{11} - 74761346748100 q^{13} - 279806803135056 q^{14} + 5569970029723408 q^{16} - 3663997807947660 q^{17} - 21217736873658616 q^{19} + 52107134347831176 q^{20} - 70457763987482040 q^{22} + 10211937347530800 q^{23} + 1133420517164978086 q^{25} - 151760101572249732 q^{26} - 3245902148646774880 q^{28} + 2773786800594119004 q^{29} - 8033045133786377440 q^{31} + 22116218905429083360 q^{32} - 27215628268059889668 q^{34} + 39426602932449584640 q^{35} + 18268446607706969660 q^{37} + 318540211201856069640 q^{38} - 657835176129802083792 q^{40} + 781145503713308249892 q^{41} + 36975571414944336440 q^{43} + 2254010900914313367696 q^{44} - 1941282469414653512112 q^{46} + 2241742133351605489920 q^{47} - 2196918340425253819014 q^{49} + 8760281440407049589166 q^{50} - 10596633413935934040040 q^{52} + 428284381587667438860 q^{53} + 13949658121609170948816 q^{55} - 17467291510967593247040 q^{56} + 39443922956707978162500 q^{58} - 45253971693438198786072 q^{59} - 7509152988113527282708 q^{61} - 75283372091267901036000 q^{62} + 108502130134040005518400 q^{64} - 137816045072347030379928 q^{65} - 37704903622586737043800 q^{67} - 388192003918481236625640 q^{68} + 339494380072244591171040 q^{70} - 190967914817177968333872 q^{71} + 465679359436374271589780 q^{73} + 526199316467462788703724 q^{74} - 2035038633209335713564400 q^{76} + 1256909847221159799955200 q^{77} - 1431283368219208577770816 q^{79} + 3172673989155704153574816 q^{80} + 2655719126416945746477900 q^{82} + 1867990788574425981440280 q^{83} - 3527790254390856980458008 q^{85} + 2149286748163902122519160 q^{86} - 2939391296691021839417760 q^{88} + 2992722485060481035826852 q^{89} + 7744187915268002880698848 q^{91} - 2459380751814926996678880 q^{92} - 19150855182255804939678240 q^{94} - 6852384385171181748854064 q^{95} + 432380286540481372534340 q^{97} - 38189763006479864037209790 q^{98} + O(q^{100}) \)

Decomposition of \(S_{26}^{\mathrm{new}}(\Gamma_0(9))\) 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
9.26.a.a 9.a 1.a $1$ $35.640$ \(\Q\) None \(48\) \(0\) \(741989850\) \(39080597192\) $-$ $\mathrm{SU}(2)$ \(q+48q^{2}-33552128q^{4}+741989850q^{5}+\cdots\)
9.26.a.b 9.a 1.a $2$ $35.640$ \(\Q(\sqrt{1287001}) \) None \(324\) \(0\) \(-570861756\) \(-29687385728\) $-$ $\mathrm{SU}(2)$ \(q+(162-\beta )q^{2}+(12803848-18^{2}\beta )q^{4}+\cdots\)
9.26.a.c 9.a 1.a $3$ $35.640$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(3678\) \(0\) \(163152750\) \(-9622572744\) $-$ $\mathrm{SU}(2)$ \(q+(1226-\beta _{1})q^{2}+(30249196-1499\beta _{1}+\cdots)q^{4}+\cdots\)
9.26.a.d 9.a 1.a $4$ $35.640$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(0\) \(0\) \(0\) \(-40689469840\) $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(34366048+\beta _{3})q^{4}+(-64017\beta _{1}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{26}^{\mathrm{old}}(\Gamma_0(9))\) into lower level spaces

\( S_{26}^{\mathrm{old}}(\Gamma_0(9)) \cong \) \(S_{26}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 3}\)\(\oplus\)\(S_{26}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 2}\)