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Properties

Label 49.2.e
Level $49$
Weight $2$
Character orbit 49.e
Rep. character $\chi_{49}(8,\cdot)$
Character field $\Q(\zeta_{7})$
Dimension $18$
Newform subspaces $2$
Sturm bound $9$
Trace bound $1$

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Defining parameters

Level: \( N \) \(=\) \( 49 = 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 49.e (of order \(7\) and degree \(6\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 49 \)
Character field: \(\Q(\zeta_{7})\)
Newform subspaces: \( 2 \)
Sturm bound: \(9\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(49, [\chi])\).

Total New Old
Modular forms 30 30 0
Cusp forms 18 18 0
Eisenstein series 12 12 0

Trace form

\( 18 q - 5 q^{2} - 3 q^{3} - 5 q^{4} - q^{5} - 9 q^{6} + 5 q^{8} - 14 q^{9} + O(q^{10}) \) \( 18 q - 5 q^{2} - 3 q^{3} - 5 q^{4} - q^{5} - 9 q^{6} + 5 q^{8} - 14 q^{9} + 11 q^{10} - 6 q^{11} - 14 q^{12} + 7 q^{13} - 21 q^{14} - 3 q^{15} + 13 q^{16} + 4 q^{17} + 16 q^{18} - 22 q^{19} + 7 q^{20} + 21 q^{21} + 5 q^{22} - 12 q^{23} + 11 q^{24} + 14 q^{25} + 21 q^{26} + 12 q^{27} - 27 q^{29} + 40 q^{30} - 24 q^{31} - 3 q^{32} + 41 q^{33} - 37 q^{34} + 21 q^{35} + q^{36} - 26 q^{37} + 25 q^{38} - 7 q^{39} - 15 q^{40} + 21 q^{41} - 49 q^{42} + 29 q^{43} - 13 q^{44} - 55 q^{45} + 3 q^{46} - 36 q^{47} + 12 q^{48} + 14 q^{49} - 76 q^{50} + 19 q^{51} - 7 q^{52} - 20 q^{53} + 50 q^{54} - 47 q^{55} - 14 q^{56} + 11 q^{57} - 31 q^{58} + 25 q^{59} - 35 q^{60} - 15 q^{61} + 61 q^{62} + 7 q^{63} + 27 q^{64} + 49 q^{65} - 31 q^{66} + 36 q^{67} + 28 q^{68} + 5 q^{69} + 35 q^{70} - 34 q^{71} + 4 q^{72} + 46 q^{73} + 59 q^{74} - 9 q^{75} + 56 q^{77} - 7 q^{78} + 44 q^{79} + 32 q^{80} - 48 q^{81} + 14 q^{82} - 21 q^{83} + 28 q^{84} + 11 q^{85} - 3 q^{86} + 43 q^{87} + 5 q^{88} - q^{89} - 4 q^{90} - 91 q^{91} + 44 q^{92} - 65 q^{93} - 80 q^{94} - 8 q^{95} - 98 q^{96} - 28 q^{97} - 7 q^{98} - 20 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(49, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
49.2.e.a 49.e 49.e $6$ $0.391$ \(\Q(\zeta_{14})\) None \(-3\) \(-3\) \(6\) \(7\) $\mathrm{SU}(2)[C_{7}]$ \(q+(-1+\zeta_{14}-\zeta_{14}^{4}+\zeta_{14}^{5})q^{2}+\cdots\)
49.2.e.b 49.e 49.e $12$ $0.391$ \(\Q(\zeta_{21})\) None \(-2\) \(0\) \(-7\) \(-7\) $\mathrm{SU}(2)[C_{7}]$ \(q+(\zeta_{21}^{2}+\zeta_{21}^{4}-\zeta_{21}^{5}-\zeta_{21}^{9}+\zeta_{21}^{11})q^{2}+\cdots\)