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Properties

Label 49.9.d
Level $49$
Weight $9$
Character orbit 49.d
Rep. character $\chi_{49}(19,\cdot)$
Character field $\Q(\zeta_{6})$
Dimension $50$
Newform subspaces $4$
Sturm bound $42$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 49 = 7^{2} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 49.d (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 7 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 4 \)
Sturm bound: \(42\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{9}(49, [\chi])\).

Total New Old
Modular forms 82 58 24
Cusp forms 66 50 16
Eisenstein series 16 8 8

Trace form

\( 50 q + 7 q^{2} + 84 q^{3} - 3257 q^{4} + 840 q^{5} - 3862 q^{8} + 54223 q^{9} + O(q^{10}) \) \( 50 q + 7 q^{2} + 84 q^{3} - 3257 q^{4} + 840 q^{5} - 3862 q^{8} + 54223 q^{9} - 5796 q^{10} - 5978 q^{11} + 40908 q^{12} - 268336 q^{15} - 273309 q^{16} + 141456 q^{17} + 11315 q^{18} + 257544 q^{19} - 2420332 q^{22} + 877282 q^{23} + 895104 q^{24} + 1728715 q^{25} + 2913120 q^{26} - 7395428 q^{29} + 2436912 q^{30} + 2376696 q^{31} - 2376357 q^{32} + 5719140 q^{33} - 29363270 q^{36} - 2804774 q^{37} + 7088088 q^{38} + 8300740 q^{39} + 7601832 q^{40} - 7023980 q^{43} + 9539010 q^{44} - 3328164 q^{45} + 1549334 q^{46} - 2704128 q^{47} + 80096746 q^{50} - 6710352 q^{51} - 11135208 q^{52} - 21445214 q^{53} - 24553368 q^{54} - 9733288 q^{57} - 34247682 q^{58} - 25291140 q^{59} - 10762332 q^{60} - 59368764 q^{61} + 189569978 q^{64} + 19258596 q^{65} - 463428 q^{66} + 41755266 q^{67} - 44316972 q^{68} + 132970420 q^{71} - 40171117 q^{72} - 116758404 q^{73} - 190749766 q^{74} - 79832424 q^{75} + 167051360 q^{78} + 68451082 q^{79} + 93591624 q^{80} + 144278979 q^{81} + 91061712 q^{82} - 30370408 q^{85} - 76084498 q^{86} - 26702676 q^{87} + 214027166 q^{88} + 2322516 q^{89} - 1105126356 q^{92} - 106501988 q^{93} + 345566088 q^{94} + 143228460 q^{95} + 416455200 q^{96} - 438485892 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{9}^{\mathrm{new}}(49, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
49.9.d.a 49.d 7.d $2$ $19.962$ \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-7}) \) 7.9.b.a \(31\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{6}]$ \(q+31\zeta_{6}q^{2}+(-705+705\zeta_{6})q^{4}+\cdots\)
49.9.d.b 49.d 7.d $8$ $19.962$ \(\mathbb{Q}[x]/(x^{8} + \cdots)\) None 7.9.b.b \(-32\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+(8\beta _{2}+\beta _{4})q^{2}-\beta _{3}q^{3}+(8+8\beta _{2}+\cdots)q^{4}+\cdots\)
49.9.d.c 49.d 7.d $8$ $19.962$ \(\mathbb{Q}[x]/(x^{8} + \cdots)\) None 7.9.d.a \(-4\) \(84\) \(840\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+(-\beta _{1}+\beta _{2}+\beta _{3})q^{2}+(7-7\beta _{2}+\beta _{5}+\cdots)q^{3}+\cdots\)
49.9.d.d 49.d 7.d $32$ $19.962$ None 49.9.b.b \(12\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$

Decomposition of \(S_{9}^{\mathrm{old}}(49, [\chi])\) into lower level spaces

\( S_{9}^{\mathrm{old}}(49, [\chi]) \simeq \) \(S_{9}^{\mathrm{new}}(7, [\chi])\)\(^{\oplus 2}\)