Properties

Label 90.2.i
Level 90
Weight 2
Character orbit i
Rep. character \(\chi_{90}(49,\cdot)\)
Character field \(\Q(\zeta_{6})\)
Dimension 12
Newform subspaces 2
Sturm bound 36
Trace bound 1

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Defining parameters

Level: \( N \) = \( 90 = 2 \cdot 3^{2} \cdot 5 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 90.i (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 45 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 2 \)
Sturm bound: \(36\)
Trace bound: \(1\)
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 12 32
Cusp forms 28 12 16
Eisenstein series 16 0 16

Trace form

\( 12q + 6q^{4} - 2q^{5} + 4q^{6} - 10q^{9} + O(q^{10}) \) \( 12q + 6q^{4} - 2q^{5} + 4q^{6} - 10q^{9} + 8q^{11} - 10q^{14} - 20q^{15} - 6q^{16} + 2q^{20} - 32q^{21} + 2q^{24} + 6q^{25} - 24q^{26} - 6q^{29} + 22q^{30} - 12q^{31} + 32q^{35} - 8q^{36} + 36q^{39} + 26q^{41} + 16q^{44} + 8q^{45} - 12q^{46} + 4q^{50} + 64q^{51} + 38q^{54} - 24q^{55} + 10q^{56} - 4q^{59} - 16q^{60} + 6q^{61} - 12q^{64} - 36q^{65} - 4q^{66} - 22q^{69} + 6q^{70} - 24q^{71} - 28q^{74} - 4q^{75} - 24q^{79} + 4q^{80} + 10q^{81} - 22q^{84} - 12q^{85} + 28q^{86} - 68q^{89} - 32q^{90} - 24q^{91} - 6q^{94} - 36q^{95} - 2q^{96} + 4q^{99} + O(q^{100}) \)

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.i.a \(4\) \(0.719\) \(\Q(\zeta_{12})\) None \(0\) \(0\) \(2\) \(0\) \(q+\zeta_{12}q^{2}+(-\zeta_{12}-\zeta_{12}^{3})q^{3}+\zeta_{12}^{2}q^{4}+\cdots\)
90.2.i.b \(8\) \(0.719\) \(\Q(\zeta_{24})\) None \(0\) \(0\) \(-4\) \(0\) \(q+\zeta_{24}^{2}q^{2}+(\zeta_{24}^{2}+\zeta_{24}^{3}-\zeta_{24}^{5}+\cdots)q^{3}+\cdots\)

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

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

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 - T^{2} + T^{4} \))(\( ( 1 - T^{2} + T^{4} )^{2} \))
$3$ (\( 1 + 3 T^{2} + 9 T^{4} \))(\( 1 + 2 T^{2} - 5 T^{4} + 18 T^{6} + 81 T^{8} \))
$5$ (\( 1 - 2 T - T^{2} - 10 T^{3} + 25 T^{4} \))(\( 1 + 4 T + 8 T^{2} - 8 T^{3} - 41 T^{4} - 40 T^{5} + 200 T^{6} + 500 T^{7} + 625 T^{8} \))
$7$ (\( ( 1 + 2 T^{2} + 49 T^{4} )( 1 + 11 T^{2} + 49 T^{4} ) \))(\( 1 + 8 T^{2} + 46 T^{4} - 640 T^{6} - 5213 T^{8} - 31360 T^{10} + 110446 T^{12} + 941192 T^{14} + 5764801 T^{16} \))
$11$ (\( ( 1 - 2 T - 7 T^{2} - 22 T^{3} + 121 T^{4} )^{2} \))(\( ( 1 - 2 T - 13 T^{2} + 10 T^{3} + 124 T^{4} + 110 T^{5} - 1573 T^{6} - 2662 T^{7} + 14641 T^{8} )^{2} \))
$13$ (\( ( 1 - 4 T + 3 T^{2} - 52 T^{3} + 169 T^{4} )( 1 + 4 T + 3 T^{2} + 52 T^{3} + 169 T^{4} ) \))(\( ( 1 + 20 T^{2} + 231 T^{4} + 3380 T^{6} + 28561 T^{8} )^{2} \))
$17$ (\( ( 1 - 8 T + 17 T^{2} )^{2}( 1 + 8 T + 17 T^{2} )^{2} \))(\( ( 1 - 18 T^{2} + 563 T^{4} - 5202 T^{6} + 83521 T^{8} )^{2} \))
$19$ (\( ( 1 + 6 T + 19 T^{2} )^{4} \))(\( ( 1 - 6 T + 41 T^{2} - 114 T^{3} + 361 T^{4} )^{4} \))
$23$ (\( 1 + 45 T^{2} + 1496 T^{4} + 23805 T^{6} + 279841 T^{8} \))(\( 1 + 36 T^{2} + 298 T^{4} - 2160 T^{6} + 30579 T^{8} - 1142640 T^{10} + 83392618 T^{12} + 5329292004 T^{14} + 78310985281 T^{16} \))
$29$ (\( ( 1 - 9 T + 52 T^{2} - 261 T^{3} + 841 T^{4} )^{2} \))(\( ( 1 + 6 T + 7 T^{2} + 174 T^{3} + 841 T^{4} )^{4} \))
$31$ (\( ( 1 - 2 T - 27 T^{2} - 62 T^{3} + 961 T^{4} )^{2} \))(\( ( 1 + 8 T - 8 T^{2} + 80 T^{3} + 2239 T^{4} + 2480 T^{5} - 7688 T^{6} + 238328 T^{7} + 923521 T^{8} )^{2} \))
$37$ (\( ( 1 - 12 T + 37 T^{2} )^{2}( 1 + 12 T + 37 T^{2} )^{2} \))(\( ( 1 - 10 T^{2} + 1369 T^{4} )^{4} \))
$41$ (\( ( 1 - 11 T + 80 T^{2} - 451 T^{3} + 1681 T^{4} )^{2} \))(\( ( 1 - T - 40 T^{2} - 41 T^{3} + 1681 T^{4} )^{4} \))
$43$ (\( 1 + 70 T^{2} + 3051 T^{4} + 129430 T^{6} + 3418801 T^{8} \))(\( 1 + 110 T^{2} + 5977 T^{4} + 266750 T^{6} + 11699428 T^{8} + 493220750 T^{10} + 20434173577 T^{12} + 695349935390 T^{14} + 11688200277601 T^{16} \))
$47$ (\( 1 + 45 T^{2} - 184 T^{4} + 99405 T^{6} + 4879681 T^{8} \))(\( 1 + 168 T^{2} + 16846 T^{4} + 1169280 T^{6} + 62232387 T^{8} + 2582939520 T^{10} + 82203106126 T^{12} + 1810908175272 T^{14} + 23811286661761 T^{16} \))
$53$ (\( ( 1 - 53 T^{2} )^{4} \))(\( ( 1 - 128 T^{2} + 8850 T^{4} - 359552 T^{6} + 7890481 T^{8} )^{2} \))
$59$ (\( ( 1 + 4 T - 43 T^{2} + 236 T^{3} + 3481 T^{4} )^{2} \))(\( ( 1 - 2 T + 35 T^{2} + 298 T^{3} - 2756 T^{4} + 17582 T^{5} + 121835 T^{6} - 410758 T^{7} + 12117361 T^{8} )^{2} \))
$61$ (\( ( 1 - 7 T - 12 T^{2} - 427 T^{3} + 3721 T^{4} )^{2} \))(\( ( 1 + 4 T - 104 T^{2} - 8 T^{3} + 9703 T^{4} - 488 T^{5} - 386984 T^{6} + 907924 T^{7} + 13845841 T^{8} )^{2} \))
$67$ (\( ( 1 - 109 T^{2} + 4489 T^{4} )( 1 + 122 T^{2} + 4489 T^{4} ) \))(\( 1 + 158 T^{2} + 10921 T^{4} + 800270 T^{6} + 64991332 T^{8} + 3592412030 T^{10} + 220070392441 T^{12} + 14292424382702 T^{14} + 406067677556641 T^{16} \))
$71$ (\( ( 1 + 6 T + 71 T^{2} )^{4} \))(\( ( 1 + 136 T^{2} + 5041 T^{4} )^{4} \))
$73$ (\( ( 1 - 130 T^{2} + 5329 T^{4} )^{2} \))(\( ( 1 - 50 T^{2} + 1683 T^{4} - 266450 T^{6} + 28398241 T^{8} )^{2} \))
$79$ (\( ( 1 + 12 T + 65 T^{2} + 948 T^{3} + 6241 T^{4} )^{2} \))(\( ( 1 - 104 T^{2} + 4575 T^{4} - 649064 T^{6} + 38950081 T^{8} )^{2} \))
$83$ (\( 1 + 45 T^{2} - 4864 T^{4} + 310005 T^{6} + 47458321 T^{8} \))(\( ( 1 + 150 T^{2} + 15611 T^{4} + 1033350 T^{6} + 47458321 T^{8} )^{2} \))
$89$ (\( ( 1 + T + 89 T^{2} )^{4} \))(\( ( 1 + 16 T + 218 T^{2} + 1424 T^{3} + 7921 T^{4} )^{4} \))
$97$ (\( ( 1 - 18 T + 227 T^{2} - 1746 T^{3} + 9409 T^{4} )( 1 + 18 T + 227 T^{2} + 1746 T^{3} + 9409 T^{4} ) \))(\( ( 1 + 25 T^{2} - 8784 T^{4} + 235225 T^{6} + 88529281 T^{8} )^{2} \))
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