Properties

Label 98.2.c
Level 98
Weight 2
Character orbit c
Rep. character \(\chi_{98}(67,\cdot)\)
Character field \(\Q(\zeta_{3})\)
Dimension 8
Newform subspaces 3
Sturm bound 28
Trace bound 3

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Defining parameters

Level: \( N \) = \( 98 = 2 \cdot 7^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 98.c (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 7 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 3 \)
Sturm bound: \(28\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(98, [\chi])\).

Total New Old
Modular forms 44 8 36
Cusp forms 12 8 4
Eisenstein series 32 0 32

Trace form

\( 8q - 4q^{4} + O(q^{10}) \) \( 8q - 4q^{4} + 4q^{11} - 16q^{15} - 4q^{16} + 4q^{18} - 8q^{22} + 8q^{23} + 4q^{25} - 16q^{29} + 8q^{30} - 24q^{37} - 16q^{39} + 40q^{43} + 4q^{44} + 8q^{46} + 32q^{50} + 20q^{51} - 8q^{53} + 24q^{57} - 16q^{58} + 8q^{60} + 8q^{64} - 16q^{67} - 48q^{71} + 4q^{72} - 16q^{74} - 32q^{78} - 8q^{79} + 32q^{81} - 16q^{85} + 12q^{86} + 4q^{88} - 16q^{92} + 8q^{93} + 40q^{95} + 8q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(98, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
98.2.c.a \(2\) \(0.783\) \(\Q(\sqrt{-3}) \) None \(1\) \(-2\) \(0\) \(0\) \(q+\zeta_{6}q^{2}+(-2+2\zeta_{6})q^{3}+(-1+\zeta_{6})q^{4}+\cdots\)
98.2.c.b \(2\) \(0.783\) \(\Q(\sqrt{-3}) \) None \(1\) \(2\) \(0\) \(0\) \(q+\zeta_{6}q^{2}+(2-2\zeta_{6})q^{3}+(-1+\zeta_{6})q^{4}+\cdots\)
98.2.c.c \(4\) \(0.783\) \(\Q(\sqrt{2}, \sqrt{-3})\) None \(-2\) \(0\) \(0\) \(0\) \(q+\beta _{2}q^{2}+\beta _{1}q^{3}+(-1-\beta _{2})q^{4}+(2\beta _{1}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(98, [\chi])\) into lower level spaces

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

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 - T + T^{2} \))(\( 1 - T + T^{2} \))(\( ( 1 + T + T^{2} )^{2} \))
$3$ (\( 1 + 2 T + T^{2} + 6 T^{3} + 9 T^{4} \))(\( 1 - 2 T + T^{2} - 6 T^{3} + 9 T^{4} \))(\( 1 - 4 T^{2} + 7 T^{4} - 36 T^{6} + 81 T^{8} \))
$5$ (\( 1 - 5 T^{2} + 25 T^{4} \))(\( 1 - 5 T^{2} + 25 T^{4} \))(\( ( 1 - 6 T + 17 T^{2} - 30 T^{3} + 25 T^{4} )( 1 + 6 T + 17 T^{2} + 30 T^{3} + 25 T^{4} ) \))
$7$ 1
$11$ (\( 1 - 11 T^{2} + 121 T^{4} \))(\( 1 - 11 T^{2} + 121 T^{4} \))(\( ( 1 - 2 T - 7 T^{2} - 22 T^{3} + 121 T^{4} )^{2} \))
$13$ (\( ( 1 - 4 T + 13 T^{2} )^{2} \))(\( ( 1 + 4 T + 13 T^{2} )^{2} \))(\( ( 1 + 13 T^{2} )^{4} \))
$17$ (\( 1 - 6 T + 19 T^{2} - 102 T^{3} + 289 T^{4} \))(\( 1 + 6 T + 19 T^{2} + 102 T^{3} + 289 T^{4} \))(\( 1 - 32 T^{2} + 735 T^{4} - 9248 T^{6} + 83521 T^{8} \))
$19$ (\( 1 - 2 T - 15 T^{2} - 38 T^{3} + 361 T^{4} \))(\( 1 + 2 T - 15 T^{2} + 38 T^{3} + 361 T^{4} \))(\( 1 + 12 T^{2} - 217 T^{4} + 4332 T^{6} + 130321 T^{8} \))
$23$ (\( 1 - 23 T^{2} + 529 T^{4} \))(\( 1 - 23 T^{2} + 529 T^{4} \))(\( ( 1 - 4 T - 7 T^{2} - 92 T^{3} + 529 T^{4} )^{2} \))
$29$ (\( ( 1 + 6 T + 29 T^{2} )^{2} \))(\( ( 1 + 6 T + 29 T^{2} )^{2} \))(\( ( 1 - 2 T + 29 T^{2} )^{4} \))
$31$ (\( ( 1 - 7 T + 31 T^{2} )( 1 + 11 T + 31 T^{2} ) \))(\( ( 1 - 11 T + 31 T^{2} )( 1 + 7 T + 31 T^{2} ) \))(\( 1 + 10 T^{2} - 861 T^{4} + 9610 T^{6} + 923521 T^{8} \))
$37$ (\( 1 + 2 T - 33 T^{2} + 74 T^{3} + 1369 T^{4} \))(\( 1 + 2 T - 33 T^{2} + 74 T^{3} + 1369 T^{4} \))(\( ( 1 - T + 37 T^{2} )^{2}( 1 + 11 T + 37 T^{2} )^{2} \))
$41$ (\( ( 1 + 6 T + 41 T^{2} )^{2} \))(\( ( 1 - 6 T + 41 T^{2} )^{2} \))(\( ( 1 - 16 T^{2} + 1681 T^{4} )^{2} \))
$43$ (\( ( 1 - 8 T + 43 T^{2} )^{2} \))(\( ( 1 - 8 T + 43 T^{2} )^{2} \))(\( ( 1 - 2 T + 43 T^{2} )^{4} \))
$47$ (\( 1 + 12 T + 97 T^{2} + 564 T^{3} + 2209 T^{4} \))(\( 1 - 12 T + 97 T^{2} - 564 T^{3} + 2209 T^{4} \))(\( 1 - 86 T^{2} + 5187 T^{4} - 189974 T^{6} + 4879681 T^{8} \))
$53$ (\( 1 + 6 T - 17 T^{2} + 318 T^{3} + 2809 T^{4} \))(\( 1 + 6 T - 17 T^{2} + 318 T^{3} + 2809 T^{4} \))(\( ( 1 - 2 T - 49 T^{2} - 106 T^{3} + 2809 T^{4} )^{2} \))
$59$ (\( 1 + 6 T - 23 T^{2} + 354 T^{3} + 3481 T^{4} \))(\( 1 - 6 T - 23 T^{2} - 354 T^{3} + 3481 T^{4} \))(\( 1 - 116 T^{2} + 9975 T^{4} - 403796 T^{6} + 12117361 T^{8} \))
$61$ (\( 1 - 8 T + 3 T^{2} - 488 T^{3} + 3721 T^{4} \))(\( 1 + 8 T + 3 T^{2} + 488 T^{3} + 3721 T^{4} \))(\( 1 - 114 T^{2} + 9275 T^{4} - 424194 T^{6} + 13845841 T^{8} \))
$67$ (\( 1 - 4 T - 51 T^{2} - 268 T^{3} + 4489 T^{4} \))(\( 1 - 4 T - 51 T^{2} - 268 T^{3} + 4489 T^{4} \))(\( ( 1 + 12 T + 77 T^{2} + 804 T^{3} + 4489 T^{4} )^{2} \))
$71$ (\( ( 1 + 71 T^{2} )^{2} \))(\( ( 1 + 71 T^{2} )^{2} \))(\( ( 1 + 12 T + 71 T^{2} )^{4} \))
$73$ (\( 1 - 2 T - 69 T^{2} - 146 T^{3} + 5329 T^{4} \))(\( 1 + 2 T - 69 T^{2} + 146 T^{3} + 5329 T^{4} \))(\( 1 - 144 T^{2} + 15407 T^{4} - 767376 T^{6} + 28398241 T^{8} \))
$79$ (\( 1 + 8 T - 15 T^{2} + 632 T^{3} + 6241 T^{4} \))(\( 1 + 8 T - 15 T^{2} + 632 T^{3} + 6241 T^{4} \))(\( ( 1 - 17 T + 79 T^{2} )^{2}( 1 + 13 T + 79 T^{2} )^{2} \))
$83$ (\( ( 1 - 6 T + 83 T^{2} )^{2} \))(\( ( 1 + 6 T + 83 T^{2} )^{2} \))(\( ( 1 + 68 T^{2} + 6889 T^{4} )^{2} \))
$89$ (\( 1 + 6 T - 53 T^{2} + 534 T^{3} + 7921 T^{4} \))(\( 1 - 6 T - 53 T^{2} - 534 T^{3} + 7921 T^{4} \))(\( 1 - 128 T^{2} + 8463 T^{4} - 1013888 T^{6} + 62742241 T^{8} \))
$97$ (\( ( 1 - 10 T + 97 T^{2} )^{2} \))(\( ( 1 + 10 T + 97 T^{2} )^{2} \))(\( ( 1 + 96 T^{2} + 9409 T^{4} )^{2} \))
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