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Properties

Label 58.8.d
Level $58$
Weight $8$
Character orbit 58.d
Rep. character $\chi_{58}(7,\cdot)$
Character field $\Q(\zeta_{7})$
Dimension $102$
Newform subspaces $2$
Sturm bound $60$
Trace bound $1$

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Defining parameters

Level: \( N \) \(=\) \( 58 = 2 \cdot 29 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 58.d (of order \(7\) and degree \(6\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 29 \)
Character field: \(\Q(\zeta_{7})\)
Newform subspaces: \( 2 \)
Sturm bound: \(60\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(58, [\chi])\).

Total New Old
Modular forms 330 102 228
Cusp forms 306 102 204
Eisenstein series 24 0 24

Trace form

\( 102 q + 8 q^{2} - 1088 q^{4} + 140 q^{5} + 2752 q^{6} - 1348 q^{7} + 512 q^{8} - 7131 q^{9} + O(q^{10}) \) \( 102 q + 8 q^{2} - 1088 q^{4} + 140 q^{5} + 2752 q^{6} - 1348 q^{7} + 512 q^{8} - 7131 q^{9} - 5360 q^{10} - 11816 q^{11} - 24684 q^{13} + 14272 q^{14} - 48434 q^{15} - 69632 q^{16} - 82834 q^{17} + 29160 q^{18} - 51004 q^{19} + 33600 q^{20} - 360658 q^{21} + 54048 q^{22} + 421564 q^{23} + 176128 q^{24} + 152777 q^{25} + 105416 q^{26} - 1069584 q^{27} - 86272 q^{28} - 516233 q^{29} - 978496 q^{30} + 390636 q^{31} + 32768 q^{32} + 1581930 q^{33} - 99240 q^{34} + 964756 q^{35} - 794176 q^{36} - 1322274 q^{37} - 1147952 q^{38} - 1739604 q^{39} + 950784 q^{40} - 827598 q^{41} - 1227936 q^{42} - 2706556 q^{43} + 1890560 q^{44} + 5199159 q^{45} - 88288 q^{46} - 4452510 q^{47} + 2685737 q^{49} + 1967768 q^{50} + 3313372 q^{51} + 654848 q^{52} + 4685193 q^{53} - 6012640 q^{54} - 4821732 q^{55} + 913408 q^{56} + 6506420 q^{57} - 5440680 q^{58} - 6482016 q^{59} - 3099776 q^{60} + 7612694 q^{61} - 657872 q^{62} + 10408786 q^{63} - 4456448 q^{64} + 18236047 q^{65} + 3073408 q^{66} - 448252 q^{67} - 671744 q^{68} - 13144440 q^{69} + 20562208 q^{70} - 19333270 q^{71} + 1866240 q^{72} + 3221749 q^{73} + 7769888 q^{74} + 2121560 q^{75} + 3081216 q^{76} + 13680514 q^{77} + 27519840 q^{78} + 27525172 q^{79} + 573440 q^{80} - 6725933 q^{81} - 15044560 q^{82} - 15150052 q^{83} + 9612032 q^{84} - 10736100 q^{85} - 45648160 q^{86} - 34574300 q^{87} - 4282368 q^{88} + 480316 q^{89} - 45196096 q^{90} + 72650156 q^{91} + 26980096 q^{92} + 55651402 q^{93} + 6776352 q^{94} + 45238928 q^{95} - 2031616 q^{96} + 54852907 q^{97} + 14313160 q^{98} - 205933400 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{8}^{\mathrm{new}}(58, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
58.8.d.a 58.d 29.d $48$ $18.118$ None \(-64\) \(-31\) \(-356\) \(-3695\) $\mathrm{SU}(2)[C_{7}]$
58.8.d.b 58.d 29.d $54$ $18.118$ None \(72\) \(31\) \(496\) \(2347\) $\mathrm{SU}(2)[C_{7}]$

Decomposition of \(S_{8}^{\mathrm{old}}(58, [\chi])\) into lower level spaces

\( S_{8}^{\mathrm{old}}(58, [\chi]) \simeq \) \(S_{8}^{\mathrm{new}}(29, [\chi])\)\(^{\oplus 2}\)