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Properties

Label 98.13.d
Level $98$
Weight $13$
Character orbit 98.d
Rep. character $\chi_{98}(19,\cdot)$
Character field $\Q(\zeta_{6})$
Dimension $80$
Newform subspaces $4$
Sturm bound $182$
Trace bound $9$

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Defining parameters

Level: \( N \) \(=\) \( 98 = 2 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 13 \)
Character orbit: \([\chi]\) \(=\) 98.d (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 7 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 4 \)
Sturm bound: \(182\)
Trace bound: \(9\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{13}(98, [\chi])\).

Total New Old
Modular forms 352 80 272
Cusp forms 320 80 240
Eisenstein series 32 0 32

Trace form

\( 80 q - 81920 q^{4} - 18144 q^{5} + 5490184 q^{9} + O(q^{10}) \) \( 80 q - 81920 q^{4} - 18144 q^{5} + 5490184 q^{9} + 2290176 q^{10} - 3524760 q^{11} + 1351344 q^{15} - 167772160 q^{16} + 101561040 q^{17} + 18503680 q^{18} - 174931848 q^{19} + 42931200 q^{22} + 9526320 q^{23} - 242221056 q^{24} + 1929444320 q^{25} + 1434682368 q^{26} - 1093170672 q^{29} + 2852006912 q^{30} - 4583818344 q^{31} - 6054957720 q^{33} - 22487793664 q^{36} + 2499474680 q^{37} + 149506560 q^{38} - 17994033592 q^{39} - 4690280448 q^{40} + 19699404320 q^{43} - 7218708480 q^{44} - 57253352184 q^{45} - 8783818752 q^{46} - 18116171640 q^{47} + 23994040320 q^{50} + 11843483016 q^{51} - 8269578240 q^{52} - 22515039720 q^{53} - 105152205312 q^{54} + 174450989680 q^{57} + 69854361600 q^{58} - 201845459088 q^{59} - 1383776256 q^{60} - 336780254328 q^{61} + 687194767360 q^{64} - 421612669800 q^{65} - 268884080640 q^{66} + 250150859200 q^{67} - 207997009920 q^{68} + 1534392952704 q^{71} + 37895536640 q^{72} + 738414283320 q^{73} - 460128766464 q^{74} - 1537028640000 q^{75} + 1477482045440 q^{78} + 25780804624 q^{79} + 76101451776 q^{80} - 797681726976 q^{81} + 302578053120 q^{82} + 1383910944144 q^{85} - 74188892160 q^{86} - 1957017683880 q^{87} - 43961548800 q^{88} + 2485007442792 q^{89} - 39019806720 q^{92} - 2273978150760 q^{93} + 2021298693120 q^{94} - 40374222192 q^{95} + 496068722688 q^{96} - 15293282778064 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{13}^{\mathrm{new}}(98, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
98.13.d.a 98.d 7.d $16$ $89.571$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(0\) \(0\) \(-18144\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+\beta _{3}q^{2}+(-2\beta _{2}+5\beta _{3}-\beta _{5})q^{3}+\cdots\)
98.13.d.b 98.d 7.d $16$ $89.571$ \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+\beta _{2}q^{2}+(\beta _{9}-\beta _{10})q^{3}+2^{11}\beta _{1}q^{4}+\cdots\)
98.13.d.c 98.d 7.d $24$ $89.571$ None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$
98.13.d.d 98.d 7.d $24$ $89.571$ None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$

Decomposition of \(S_{13}^{\mathrm{old}}(98, [\chi])\) into lower level spaces

\( S_{13}^{\mathrm{old}}(98, [\chi]) \cong \) \(S_{13}^{\mathrm{new}}(7, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{13}^{\mathrm{new}}(14, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{13}^{\mathrm{new}}(49, [\chi])\)\(^{\oplus 2}\)