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Properties

Label 49.4.c
Level $49$
Weight $4$
Character orbit 49.c
Rep. character $\chi_{49}(18,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $16$
Newform subspaces $5$
Sturm bound $18$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 49 = 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 49.c (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 7 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 5 \)
Sturm bound: \(18\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(2\), \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(49, [\chi])\).

Total New Old
Modular forms 36 24 12
Cusp forms 20 16 4
Eisenstein series 16 8 8

Trace form

\( 16 q + 3 q^{2} + 7 q^{3} - 33 q^{4} + 7 q^{5} - 28 q^{6} + 54 q^{8} + 11 q^{9} + O(q^{10}) \) \( 16 q + 3 q^{2} + 7 q^{3} - 33 q^{4} + 7 q^{5} - 28 q^{6} + 54 q^{8} + 11 q^{9} + 14 q^{10} - 11 q^{11} - 28 q^{12} + 28 q^{13} - 158 q^{15} + 19 q^{16} - 21 q^{17} + 145 q^{18} + 49 q^{19} + 56 q^{20} + 12 q^{22} - 249 q^{23} - 168 q^{24} + 67 q^{25} - 28 q^{26} + 70 q^{27} - 136 q^{29} + 390 q^{30} + 147 q^{31} + 277 q^{32} - 35 q^{33} + 84 q^{34} - 550 q^{36} - 389 q^{37} + 98 q^{38} - 742 q^{39} - 168 q^{40} - 700 q^{41} + 984 q^{43} + 564 q^{44} + 154 q^{45} - 482 q^{46} + 525 q^{47} + 224 q^{48} + 3154 q^{50} - 65 q^{51} + 56 q^{52} - 585 q^{53} - 70 q^{54} + 70 q^{55} - 2202 q^{57} - 386 q^{58} - 105 q^{59} + 812 q^{60} - 413 q^{61} - 588 q^{62} - 3702 q^{64} - 42 q^{65} - 70 q^{66} + 1781 q^{67} + 84 q^{68} + 2226 q^{69} + 1936 q^{71} - 267 q^{72} - 1113 q^{73} + 1604 q^{74} - 532 q^{75} + 392 q^{76} - 2856 q^{78} + 1655 q^{79} - 112 q^{80} + 524 q^{81} + 700 q^{82} - 2184 q^{83} - 3406 q^{85} - 5216 q^{86} + 406 q^{87} - 4200 q^{88} - 329 q^{89} - 616 q^{90} + 5592 q^{92} + 4387 q^{93} + 1050 q^{94} + 985 q^{95} + 1120 q^{96} + 1764 q^{97} + 10868 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{4}^{\mathrm{new}}(49, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
49.4.c.a 49.c 7.c $2$ $2.891$ \(\Q(\sqrt{-3}) \) None 7.4.c.a \(-2\) \(7\) \(7\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q-2\zeta_{6}q^{2}+(7-7\zeta_{6})q^{3}+(4-4\zeta_{6})q^{4}+\cdots\)
49.4.c.b 49.c 7.c $2$ $2.891$ \(\Q(\sqrt{-3}) \) None 7.4.a.a \(1\) \(-2\) \(16\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+\zeta_{6}q^{2}+(-2+2\zeta_{6})q^{3}+(7-7\zeta_{6})q^{4}+\cdots\)
49.4.c.c 49.c 7.c $2$ $2.891$ \(\Q(\sqrt{-3}) \) None 7.4.a.a \(1\) \(2\) \(-16\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+\zeta_{6}q^{2}+(2-2\zeta_{6})q^{3}+(7-7\zeta_{6})q^{4}+\cdots\)
49.4.c.d 49.c 7.c $2$ $2.891$ \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-7}) \) 49.4.a.a \(5\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{3}]$ \(q+5\zeta_{6}q^{2}+(-17+17\zeta_{6})q^{4}-45q^{8}+\cdots\)
49.4.c.e 49.c 7.c $8$ $2.891$ 8.0.\(\cdots\).19 None 49.4.a.e \(-2\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1+\beta _{2}-\beta _{6})q^{2}+\beta _{3}q^{3}+(-8+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(49, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(49, [\chi]) \simeq \) \(S_{4}^{\mathrm{new}}(7, [\chi])\)\(^{\oplus 2}\)