Properties

Label 90.2.e
Level 90
Weight 2
Character orbit e
Rep. character \(\chi_{90}(31,\cdot)\)
Character field \(\Q(\zeta_{3})\)
Dimension 8
Newform subspaces 3
Sturm bound 36
Trace bound 2

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Defining parameters

Level: \( N \) = \( 90 = 2 \cdot 3^{2} \cdot 5 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 90.e (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 9 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 3 \)
Sturm bound: \(36\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(90, [\chi])\).

Total New Old
Modular forms 44 8 36
Cusp forms 28 8 20
Eisenstein series 16 0 16

Trace form

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

Decomposition of \(S_{2}^{\mathrm{new}}(90, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
90.2.e.a \(2\) \(0.719\) \(\Q(\sqrt{-3}) \) None \(-1\) \(3\) \(1\) \(1\) \(q+(-1+\zeta_{6})q^{2}+(2-\zeta_{6})q^{3}-\zeta_{6}q^{4}+\cdots\)
90.2.e.b \(2\) \(0.719\) \(\Q(\sqrt{-3}) \) None \(1\) \(-3\) \(-1\) \(4\) \(q+(1-\zeta_{6})q^{2}+(-1-\zeta_{6})q^{3}-\zeta_{6}q^{4}+\cdots\)
90.2.e.c \(4\) \(0.719\) \(\Q(\sqrt{-3}, \sqrt{-11})\) None \(2\) \(2\) \(2\) \(-1\) \(q+\beta _{2}q^{2}+(1-\beta _{1}+\beta _{3})q^{3}+(-1+\beta _{2}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(90, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(90, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(18, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(45, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + T + T^{2} \))(\( 1 - T + T^{2} \))(\( ( 1 - T + T^{2} )^{2} \))
$3$ (\( 1 - 3 T + 3 T^{2} \))(\( 1 + 3 T + 3 T^{2} \))(\( ( 1 - T + 3 T^{2} )^{2} \))
$5$ (\( 1 - T + T^{2} \))(\( 1 + T + T^{2} \))(\( ( 1 - T + T^{2} )^{2} \))
$7$ (\( ( 1 - 5 T + 7 T^{2} )( 1 + 4 T + 7 T^{2} ) \))(\( ( 1 - 5 T + 7 T^{2} )( 1 + T + 7 T^{2} ) \))(\( 1 + T - 5 T^{2} - 8 T^{3} - 20 T^{4} - 56 T^{5} - 245 T^{6} + 343 T^{7} + 2401 T^{8} \))
$11$ (\( 1 + 6 T + 25 T^{2} + 66 T^{3} + 121 T^{4} \))(\( 1 + 3 T - 2 T^{2} + 33 T^{3} + 121 T^{4} \))(\( 1 - 3 T - 7 T^{2} + 18 T^{3} + 36 T^{4} + 198 T^{5} - 847 T^{6} - 3993 T^{7} + 14641 T^{8} \))
$13$ (\( ( 1 - 5 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} ) \))(\( 1 - 4 T + 3 T^{2} - 52 T^{3} + 169 T^{4} \))(\( 1 - 2 T + 10 T^{2} + 64 T^{3} - 185 T^{4} + 832 T^{5} + 1690 T^{6} - 4394 T^{7} + 28561 T^{8} \))
$17$ (\( ( 1 + 17 T^{2} )^{2} \))(\( ( 1 - 3 T + 17 T^{2} )^{2} \))(\( ( 1 + 9 T + 46 T^{2} + 153 T^{3} + 289 T^{4} )^{2} \))
$19$ (\( ( 1 + 4 T + 19 T^{2} )^{2} \))(\( ( 1 - 5 T + 19 T^{2} )^{2} \))(\( ( 1 - T + 30 T^{2} - 19 T^{3} + 361 T^{4} )^{2} \))
$23$ (\( 1 + 9 T + 58 T^{2} + 207 T^{3} + 529 T^{4} \))(\( 1 + 6 T + 13 T^{2} + 138 T^{3} + 529 T^{4} \))(\( 1 - 3 T - 31 T^{2} + 18 T^{3} + 864 T^{4} + 414 T^{5} - 16399 T^{6} - 36501 T^{7} + 279841 T^{8} \))
$29$ (\( 1 + 3 T - 20 T^{2} + 87 T^{3} + 841 T^{4} \))(\( 1 + 6 T + 7 T^{2} + 174 T^{3} + 841 T^{4} \))(\( 1 - 3 T - 43 T^{2} + 18 T^{3} + 1602 T^{4} + 522 T^{5} - 36163 T^{6} - 73167 T^{7} + 707281 T^{8} \))
$31$ (\( ( 1 - 11 T + 31 T^{2} )( 1 + 7 T + 31 T^{2} ) \))(\( 1 + 2 T - 27 T^{2} + 62 T^{3} + 961 T^{4} \))(\( 1 - 2 T - 26 T^{2} + 64 T^{3} - 185 T^{4} + 1984 T^{5} - 24986 T^{6} - 59582 T^{7} + 923521 T^{8} \))
$37$ (\( ( 1 - 8 T + 37 T^{2} )^{2} \))(\( ( 1 + 4 T + 37 T^{2} )^{2} \))(\( ( 1 + 4 T + 37 T^{2} )^{4} \))
$41$ (\( 1 - 3 T - 32 T^{2} - 123 T^{3} + 1681 T^{4} \))(\( 1 - 3 T - 32 T^{2} - 123 T^{3} + 1681 T^{4} \))(\( ( 1 - 3 T - 32 T^{2} - 123 T^{3} + 1681 T^{4} )^{2} \))
$43$ (\( ( 1 - 5 T + 43 T^{2} )( 1 + 13 T + 43 T^{2} ) \))(\( 1 + 11 T + 78 T^{2} + 473 T^{3} + 1849 T^{4} \))(\( 1 - 17 T + 139 T^{2} - 1088 T^{3} + 8224 T^{4} - 46784 T^{5} + 257011 T^{6} - 1351619 T^{7} + 3418801 T^{8} \))
$47$ (\( 1 - 3 T - 38 T^{2} - 141 T^{3} + 2209 T^{4} \))(\( 1 - 47 T^{2} + 2209 T^{4} \))(\( 1 - 9 T - 25 T^{2} - 108 T^{3} + 5220 T^{4} - 5076 T^{5} - 55225 T^{6} - 934407 T^{7} + 4879681 T^{8} \))
$53$ (\( ( 1 - 6 T + 53 T^{2} )^{2} \))(\( ( 1 - 6 T + 53 T^{2} )^{2} \))(\( ( 1 - 26 T^{2} + 2809 T^{4} )^{2} \))
$59$ (\( 1 + 6 T - 23 T^{2} + 354 T^{3} + 3481 T^{4} \))(\( 1 - 3 T - 50 T^{2} - 177 T^{3} + 3481 T^{4} \))(\( 1 + 3 T - 103 T^{2} - 18 T^{3} + 8532 T^{4} - 1062 T^{5} - 358543 T^{6} + 616137 T^{7} + 12117361 T^{8} \))
$61$ (\( ( 1 - 14 T + 61 T^{2} )( 1 + T + 61 T^{2} ) \))(\( 1 - 10 T + 39 T^{2} - 610 T^{3} + 3721 T^{4} \))(\( 1 + T - 47 T^{2} - 74 T^{3} - 1478 T^{4} - 4514 T^{5} - 174887 T^{6} + 226981 T^{7} + 13845841 T^{8} \))
$67$ (\( 1 - 13 T + 102 T^{2} - 871 T^{3} + 4489 T^{4} \))(\( ( 1 - 11 T + 67 T^{2} )( 1 + 16 T + 67 T^{2} ) \))(\( ( 1 - 7 T - 18 T^{2} - 469 T^{3} + 4489 T^{4} )^{2} \))
$71$ (\( ( 1 + 6 T + 71 T^{2} )^{2} \))(\( ( 1 - 6 T + 71 T^{2} )^{2} \))(\( ( 1 + 6 T + 71 T^{2} )^{4} \))
$73$ (\( ( 1 + 4 T + 73 T^{2} )^{2} \))(\( ( 1 + 7 T + 73 T^{2} )^{2} \))(\( ( 1 + 11 T + 102 T^{2} + 803 T^{3} + 5329 T^{4} )^{2} \))
$79$ (\( 1 - 10 T + 21 T^{2} - 790 T^{3} + 6241 T^{4} \))(\( 1 + 14 T + 117 T^{2} + 1106 T^{3} + 6241 T^{4} \))(\( ( 1 + 2 T - 75 T^{2} + 158 T^{3} + 6241 T^{4} )^{2} \))
$83$ (\( 1 - 9 T - 2 T^{2} - 747 T^{3} + 6889 T^{4} \))(\( 1 + 12 T + 61 T^{2} + 996 T^{3} + 6889 T^{4} \))(\( 1 + 9 T - 97 T^{2} + 108 T^{3} + 18072 T^{4} + 8964 T^{5} - 668233 T^{6} + 5146083 T^{7} + 47458321 T^{8} \))
$89$ (\( ( 1 - 9 T + 89 T^{2} )^{2} \))(\( ( 1 - 6 T + 89 T^{2} )^{2} \))(\( ( 1 - 15 T + 160 T^{2} - 1335 T^{3} + 7921 T^{4} )^{2} \))
$97$ (\( 1 + 2 T - 93 T^{2} + 194 T^{3} + 9409 T^{4} \))(\( 1 + 11 T + 24 T^{2} + 1067 T^{3} + 9409 T^{4} \))(\( 1 - 11 T - 95 T^{2} - 242 T^{3} + 25510 T^{4} - 23474 T^{5} - 893855 T^{6} - 10039403 T^{7} + 88529281 T^{8} \))
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