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Properties

Label 126.9.n
Level $126$
Weight $9$
Character orbit 126.n
Rep. character $\chi_{126}(19,\cdot)$
Character field $\Q(\zeta_{6})$
Dimension $52$
Newform subspaces $4$
Sturm bound $216$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 126 = 2 \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 126.n (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 7 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 4 \)
Sturm bound: \(216\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{9}(126, [\chi])\).

Total New Old
Modular forms 400 52 348
Cusp forms 368 52 316
Eisenstein series 32 0 32

Trace form

\( 52 q - 3328 q^{4} + 1674 q^{5} - 1696 q^{7} + O(q^{10}) \) \( 52 q - 3328 q^{4} + 1674 q^{5} - 1696 q^{7} - 17664 q^{10} - 12474 q^{11} - 69120 q^{14} - 425984 q^{16} + 46278 q^{17} - 302586 q^{19} + 594432 q^{22} + 113346 q^{23} + 1559752 q^{25} - 566784 q^{26} - 392960 q^{28} + 728568 q^{29} - 1563810 q^{31} + 2949642 q^{35} - 2519822 q^{37} + 1278720 q^{38} + 2260992 q^{40} + 7186288 q^{43} - 1596672 q^{44} + 1257984 q^{46} + 5180490 q^{47} - 13816664 q^{49} - 11805696 q^{50} + 781824 q^{52} + 38805642 q^{53} + 12386304 q^{56} + 9048576 q^{58} - 65346858 q^{59} + 651690 q^{61} + 109051904 q^{64} + 24715692 q^{65} - 50986702 q^{67} - 5923584 q^{68} + 17505024 q^{70} + 1807056 q^{71} + 209922270 q^{73} + 52538112 q^{74} + 241045146 q^{77} + 89321666 q^{79} - 27426816 q^{80} + 39143424 q^{82} + 190526436 q^{85} + 124180992 q^{86} - 38043648 q^{88} - 209473290 q^{89} - 132232452 q^{91} - 29016576 q^{92} + 172609536 q^{94} + 165996594 q^{95} - 39204864 q^{98} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{9}^{\mathrm{new}}(126, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
126.9.n.a 126.n 7.d $8$ $51.330$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(0\) \(0\) \(2226\) \(-140\) $\mathrm{SU}(2)[C_{6}]$ \(q+(-\beta _{2}+\beta _{3})q^{2}+2^{7}\beta _{1}q^{4}+(185+\cdots)q^{5}+\cdots\)
126.9.n.b 126.n 7.d $12$ $51.330$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(0\) \(0\) \(-1674\) \(-1308\) $\mathrm{SU}(2)[C_{6}]$ \(q+\beta _{3}q^{2}+2^{7}\beta _{1}q^{4}+(-93+93\beta _{1}+\cdots)q^{5}+\cdots\)
126.9.n.c 126.n 7.d $12$ $51.330$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(0\) \(0\) \(1122\) \(-4434\) $\mathrm{SU}(2)[C_{6}]$ \(q+(\beta _{2}-\beta _{3})q^{2}+(-2^{7}-2^{7}\beta _{1})q^{4}+\cdots\)
126.9.n.d 126.n 7.d $20$ $51.330$ \(\mathbb{Q}[x]/(x^{20} - \cdots)\) None \(0\) \(0\) \(0\) \(4186\) $\mathrm{SU}(2)[C_{6}]$ \(q+\beta _{2}q^{2}+(-2^{7}+2^{7}\beta _{1})q^{4}+(-5\beta _{2}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{9}^{\mathrm{old}}(126, [\chi])\) into lower level spaces

\( S_{9}^{\mathrm{old}}(126, [\chi]) \cong \) \(S_{9}^{\mathrm{new}}(7, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(14, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(42, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(63, [\chi])\)\(^{\oplus 2}\)