Properties

Label 49.10.c
Level $49$
Weight $10$
Character orbit 49.c
Rep. character $\chi_{49}(18,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $56$
Newform subspaces $8$
Sturm bound $46$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 49 = 7^{2} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 49.c (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 7 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 8 \)
Sturm bound: \(46\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(2\), \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{10}(49, [\chi])\).

Total New Old
Modular forms 92 64 28
Cusp forms 76 56 20
Eisenstein series 16 8 8

Trace form

\( 56 q + 35 q^{2} - 161 q^{3} - 6825 q^{4} - 1533 q^{5} + 8708 q^{6} - 68250 q^{8} - 142261 q^{9} + O(q^{10}) \) \( 56 q + 35 q^{2} - 161 q^{3} - 6825 q^{4} - 1533 q^{5} + 8708 q^{6} - 68250 q^{8} - 142261 q^{9} - 4298 q^{10} - 59731 q^{11} - 135604 q^{12} + 319676 q^{13} - 181118 q^{15} - 2079973 q^{16} - 324681 q^{17} + 1711801 q^{18} + 16121 q^{19} + 350616 q^{20} + 1217244 q^{22} - 3510185 q^{23} - 8449728 q^{24} - 4701949 q^{25} - 4179252 q^{26} + 18331558 q^{27} - 21539672 q^{29} - 30056082 q^{30} - 19179237 q^{31} + 19278805 q^{32} - 1689359 q^{33} + 62909700 q^{34} - 31321990 q^{36} - 29326969 q^{37} - 67365270 q^{38} + 61073642 q^{39} - 5721744 q^{40} + 53436852 q^{41} - 3069192 q^{43} - 137957372 q^{44} - 85098230 q^{45} + 58191182 q^{46} - 32509659 q^{47} + 185141600 q^{48} - 97915678 q^{50} - 62181665 q^{51} - 103893272 q^{52} - 137763941 q^{53} - 51200926 q^{54} + 144695222 q^{55} - 245916930 q^{57} - 255256834 q^{58} - 46776513 q^{59} - 239485708 q^{60} + 113075039 q^{61} - 467465628 q^{62} + 552046362 q^{64} + 538560918 q^{65} + 836682602 q^{66} + 522906853 q^{67} - 32262636 q^{68} - 1323616182 q^{69} + 1645887824 q^{71} + 1290363501 q^{72} + 859257651 q^{73} - 426747972 q^{74} + 169061732 q^{75} - 1101475592 q^{76} - 1277033016 q^{78} + 1199023735 q^{79} + 1257352656 q^{80} - 1384909288 q^{81} + 1341703076 q^{82} + 144863208 q^{83} - 2245136390 q^{85} + 3411709392 q^{86} + 340781350 q^{87} - 1006995528 q^{88} - 1661554797 q^{89} - 1967758744 q^{90} + 7904319752 q^{92} + 640526215 q^{93} + 272580882 q^{94} + 1644723129 q^{95} + 1441922272 q^{96} - 869770188 q^{97} - 3174578428 q^{99} + O(q^{100}) \)

Decomposition of \(S_{10}^{\mathrm{new}}(49, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
49.10.c.a 49.c 7.c $2$ $25.237$ \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-7}) \) \(5\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{3}]$ \(q+5\zeta_{6}q^{2}+(487-487\zeta_{6})q^{4}+4995q^{8}+\cdots\)
49.10.c.b 49.c 7.c $4$ $25.237$ \(\Q(\sqrt{-3}, \sqrt{193})\) None \(6\) \(-86\) \(-2238\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(3\beta _{1}+\beta _{2})q^{2}+(-43+43\beta _{1}-11\beta _{2}+\cdots)q^{3}+\cdots\)
49.10.c.c 49.c 7.c $4$ $25.237$ \(\Q(\sqrt{-3}, \sqrt{193})\) None \(6\) \(86\) \(2238\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(3\beta _{1}+\beta _{2})q^{2}+(43-43\beta _{1}+11\beta _{2}+\cdots)q^{3}+\cdots\)
49.10.c.d 49.c 7.c $6$ $25.237$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(-21\) \(-84\) \(-1554\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(7\beta _{3}-\beta _{5})q^{2}+(-28+\beta _{1}+\beta _{2}+\cdots)q^{3}+\cdots\)
49.10.c.e 49.c 7.c $6$ $25.237$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(-21\) \(84\) \(1554\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(7\beta _{3}-\beta _{5})q^{2}+(28-\beta _{1}-\beta _{2}+28\beta _{3}+\cdots)q^{3}+\cdots\)
49.10.c.f 49.c 7.c $8$ $25.237$ \(\mathbb{Q}[x]/(x^{8} + \cdots)\) None \(12\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(3-3\beta _{1}-\beta _{3}+\beta _{5})q^{2}+\beta _{2}q^{3}+\cdots\)
49.10.c.g 49.c 7.c $10$ $25.237$ \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None \(-18\) \(-161\) \(-1533\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-\beta _{1}+\beta _{2}-4\beta _{3})q^{2}+(-33+\beta _{1}+\cdots)q^{3}+\cdots\)
49.10.c.h 49.c 7.c $16$ $25.237$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(66\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-8\beta _{1}+\beta _{2}-\beta _{6})q^{2}+(-\beta _{4}-\beta _{8}+\cdots)q^{3}+\cdots\)

Decomposition of \(S_{10}^{\mathrm{old}}(49, [\chi])\) into lower level spaces

\( S_{10}^{\mathrm{old}}(49, [\chi]) \cong \) \(S_{10}^{\mathrm{new}}(7, [\chi])\)\(^{\oplus 2}\)