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Properties

Label 36.9.d
Level $36$
Weight $9$
Character orbit 36.d
Rep. character $\chi_{36}(19,\cdot)$
Character field $\Q$
Dimension $19$
Newform subspaces $4$
Sturm bound $54$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 36 = 2^{2} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 36.d (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 4 \)
Character field: \(\Q\)
Newform subspaces: \( 4 \)
Sturm bound: \(54\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{9}(36, [\chi])\).

Total New Old
Modular forms 52 21 31
Cusp forms 44 19 25
Eisenstein series 8 2 6

Trace form

\( 19 q - 2 q^{2} - 332 q^{4} + 170 q^{5} + 1504 q^{8} + O(q^{10}) \) \( 19 q - 2 q^{2} - 332 q^{4} + 170 q^{5} + 1504 q^{8} + 12908 q^{10} + 39974 q^{13} - 17784 q^{14} + 127120 q^{16} + 110378 q^{17} + 2552 q^{20} + 421656 q^{22} + 1295865 q^{25} - 123316 q^{26} + 248016 q^{28} - 399190 q^{29} + 2067808 q^{32} + 787988 q^{34} - 518458 q^{37} - 4146120 q^{38} - 6733120 q^{40} + 993770 q^{41} + 6839280 q^{44} + 5290176 q^{46} - 18765101 q^{49} - 26476998 q^{50} - 19633432 q^{52} - 734614 q^{53} + 50597568 q^{56} + 19674332 q^{58} + 17764742 q^{61} - 82481256 q^{62} - 60826112 q^{64} - 1123820 q^{65} + 141693992 q^{68} + 158548368 q^{70} + 73820102 q^{73} - 203803780 q^{74} - 95408208 q^{76} + 5254272 q^{77} + 305234912 q^{80} + 129868244 q^{82} - 8406740 q^{85} - 388412568 q^{86} - 255189312 q^{88} + 7922090 q^{89} + 506168064 q^{92} + 454893648 q^{94} - 182185210 q^{97} - 707159906 q^{98} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{9}^{\mathrm{new}}(36, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
36.9.d.a 36.d 4.b $1$ $14.666$ \(\Q\) \(\Q(\sqrt{-1}) \) \(-16\) \(0\) \(1054\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q-2^{4}q^{2}+2^{8}q^{4}+1054q^{5}-2^{12}q^{8}+\cdots\)
36.9.d.b 36.d 4.b $2$ $14.666$ \(\Q(\sqrt{-39}) \) None \(20\) \(0\) \(-1220\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(10-\beta )q^{2}+(-56-20\beta )q^{4}-610q^{5}+\cdots\)
36.9.d.c 36.d 4.b $8$ $14.666$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(-6\) \(0\) \(336\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(-1-\beta _{1})q^{2}+(-6+\beta _{1}+\beta _{3})q^{4}+\cdots\)
36.9.d.d 36.d 4.b $8$ $14.666$ \(\mathbb{Q}[x]/(x^{8} + \cdots)\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{2}+(-53-\beta _{5})q^{4}+(3\beta _{1}+\beta _{3}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{9}^{\mathrm{old}}(36, [\chi])\) into lower level spaces

\( S_{9}^{\mathrm{old}}(36, [\chi]) \cong \) \(S_{9}^{\mathrm{new}}(4, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(12, [\chi])\)\(^{\oplus 2}\)