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Properties

Label 7.7.d
Level $7$
Weight $7$
Character orbit 7.d
Rep. character $\chi_{7}(3,\cdot)$
Character field $\Q(\zeta_{6})$
Dimension $6$
Newform subspaces $2$
Sturm bound $4$
Trace bound $1$

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Defining parameters

Level: \( N \) \(=\) \( 7 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 7.d (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 7 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 2 \)
Sturm bound: \(4\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{7}(7, [\chi])\).

Total New Old
Modular forms 10 10 0
Cusp forms 6 6 0
Eisenstein series 4 4 0

Trace form

\( 6 q + 4 q^{2} - 3 q^{3} - 20 q^{4} + 165 q^{5} - 406 q^{7} - 1312 q^{8} + 42 q^{9} + O(q^{10}) \) \( 6 q + 4 q^{2} - 3 q^{3} - 20 q^{4} + 165 q^{5} - 406 q^{7} - 1312 q^{8} + 42 q^{9} + 2580 q^{10} + 403 q^{11} + 5964 q^{12} - 5936 q^{14} - 20910 q^{15} + 3064 q^{16} + 8229 q^{17} + 6672 q^{18} + 29985 q^{19} - 8967 q^{21} - 63584 q^{22} + 8383 q^{23} + 28872 q^{24} + 19950 q^{25} + 33768 q^{26} - 18004 q^{28} - 26588 q^{29} - 10560 q^{30} - 39399 q^{31} - 39072 q^{32} - 36711 q^{33} + 46305 q^{35} + 175488 q^{36} + 17707 q^{37} + 60960 q^{38} + 16548 q^{39} - 137280 q^{40} - 22344 q^{42} + 285388 q^{43} - 122652 q^{44} - 370530 q^{45} - 184592 q^{46} - 407823 q^{47} + 375438 q^{49} + 578800 q^{50} + 64809 q^{51} + 128856 q^{52} + 165043 q^{53} - 44496 q^{54} - 214480 q^{56} - 141534 q^{57} - 11696 q^{58} - 436299 q^{59} - 207900 q^{60} + 141201 q^{61} + 380898 q^{63} - 778720 q^{64} + 10500 q^{65} + 1025532 q^{66} + 701271 q^{67} + 799596 q^{68} - 1634640 q^{70} - 1277684 q^{71} + 141072 q^{72} - 421599 q^{73} - 688900 q^{74} + 826050 q^{75} + 1149967 q^{77} - 1268736 q^{78} + 573759 q^{79} + 2654040 q^{80} + 872505 q^{81} + 2585016 q^{82} - 1982148 q^{84} - 1677390 q^{85} - 947680 q^{86} - 1909746 q^{87} - 556648 q^{88} - 2990823 q^{89} + 711480 q^{91} + 1178376 q^{92} + 551337 q^{93} + 223584 q^{94} + 958485 q^{95} + 972720 q^{96} + 1050364 q^{98} + 1879428 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{7}^{\mathrm{new}}(7, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
7.7.d.a 7.d 7.d $2$ $1.610$ \(\Q(\sqrt{-3}) \) None \(12\) \(-21\) \(315\) \(-686\) $\mathrm{SU}(2)[C_{6}]$ \(q+12\zeta_{6}q^{2}+(-7-7\zeta_{6})q^{3}+(-80+\cdots)q^{4}+\cdots\)
7.7.d.b 7.d 7.d $4$ $1.610$ \(\Q(\sqrt{2}, \sqrt{-3})\) None \(-8\) \(18\) \(-150\) \(280\) $\mathrm{SU}(2)[C_{6}]$ \(q+(-4-4\beta _{1}+\beta _{2}+\beta _{3})q^{2}+(6+3\beta _{1}+\cdots)q^{3}+\cdots\)