Properties

Label 90.6.c
Level $90$
Weight $6$
Character orbit 90.c
Rep. character $\chi_{90}(19,\cdot)$
Character field $\Q$
Dimension $12$
Newform subspaces $4$
Sturm bound $108$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 90 = 2 \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 90.c (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 5 \)
Character field: \(\Q\)
Newform subspaces: \( 4 \)
Sturm bound: \(108\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(90, [\chi])\).

Total New Old
Modular forms 98 12 86
Cusp forms 82 12 70
Eisenstein series 16 0 16

Trace form

\( 12q - 192q^{4} - 6q^{5} + O(q^{10}) \) \( 12q - 192q^{4} - 6q^{5} + 408q^{10} + 948q^{11} - 384q^{14} + 3072q^{16} + 7464q^{19} + 96q^{20} + 4416q^{25} + 2640q^{26} - 10692q^{29} + 23472q^{31} + 10800q^{34} + 34116q^{35} - 6528q^{40} - 7200q^{41} - 15168q^{44} + 3984q^{46} + 12612q^{49} + 29952q^{50} - 85728q^{55} + 6144q^{56} - 183540q^{59} - 86400q^{61} - 49152q^{64} + 132492q^{65} + 4272q^{70} + 88248q^{71} - 164016q^{74} - 119424q^{76} - 136800q^{79} - 1536q^{80} - 38676q^{85} + 230928q^{86} - 245904q^{89} - 6456q^{91} + 268272q^{94} + 241800q^{95} + O(q^{100}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(90, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
90.6.c.a \(2\) \(14.435\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(-110\) \(0\) \(q+2iq^{2}-2^{4}q^{4}+(-55+5i)q^{5}+\cdots\)
90.6.c.b \(2\) \(14.435\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(110\) \(0\) \(q+2iq^{2}-2^{4}q^{4}+(55+5i)q^{5}+2iq^{7}+\cdots\)
90.6.c.c \(4\) \(14.435\) \(\Q(i, \sqrt{1249})\) None \(0\) \(0\) \(-6\) \(0\) \(q+\beta _{1}q^{2}-2^{4}q^{4}+(-1-\beta _{2})q^{5}+(-1+\cdots)q^{7}+\cdots\)
90.6.c.d \(4\) \(14.435\) \(\Q(i, \sqrt{19})\) None \(0\) \(0\) \(0\) \(0\) \(q+2\beta _{2}q^{2}-2^{4}q^{4}+(5\beta _{1}-15\beta _{2})q^{5}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(90, [\chi])\) into lower level spaces

\( S_{6}^{\mathrm{old}}(90, [\chi]) \cong \) \(S_{6}^{\mathrm{new}}(5, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(10, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(15, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(30, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(45, [\chi])\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 16 T^{2} \))(\( 1 + 16 T^{2} \))(\( ( 1 + 16 T^{2} )^{2} \))(\( ( 1 + 16 T^{2} )^{2} \))
$3$ 1
$5$ (\( 1 + 110 T + 3125 T^{2} \))(\( 1 - 110 T + 3125 T^{2} \))(\( 1 + 6 T + 5010 T^{2} + 18750 T^{3} + 9765625 T^{4} \))(\( 1 - 1350 T^{2} + 9765625 T^{4} \))
$7$ (\( 1 - 8650 T^{2} + 282475249 T^{4} \))(\( 1 - 33598 T^{2} + 282475249 T^{4} \))(\( 1 + 1720 T^{2} + 159889998 T^{4} + 485857428280 T^{6} + 79792266297612001 T^{8} \))(\( ( 1 - 33310 T^{2} + 282475249 T^{4} )^{2} \))
$11$ (\( ( 1 - 148 T + 161051 T^{2} )^{2} \))(\( ( 1 - 500 T + 161051 T^{2} )^{2} \))(\( ( 1 + 174 T + 298446 T^{2} + 28022874 T^{3} + 25937424601 T^{4} )^{2} \))(\( ( 1 - 94074 T^{2} + 25937424601 T^{4} )^{2} \))
$13$ (\( 1 - 274730 T^{2} + 137858491849 T^{4} \))(\( 1 - 659642 T^{2} + 137858491849 T^{4} \))(\( 1 - 717040 T^{2} + 288425185998 T^{4} - 98850052995406960 T^{6} + \)\(19\!\cdots\!01\)\( T^{8} \))(\( ( 1 + 463990 T^{2} + 137858491849 T^{4} )^{2} \))
$17$ (\( 1 + 1354590 T^{2} + 2015993900449 T^{4} \))(\( 1 - 541458 T^{2} + 2015993900449 T^{4} \))(\( 1 - 3971280 T^{2} + 7330589629598 T^{4} - 8006076256975104720 T^{6} + \)\(40\!\cdots\!01\)\( T^{8} \))(\( ( 1 - 1480158 T^{2} + 2015993900449 T^{4} )^{2} \))
$19$ (\( ( 1 + 2220 T + 2476099 T^{2} )^{2} \))(\( ( 1 - 1344 T + 2476099 T^{2} )^{2} \))(\( ( 1 - 120 T + 459398 T^{2} - 297131880 T^{3} + 6131066257801 T^{4} )^{2} \))(\( ( 1 - 2244 T + 2476099 T^{2} )^{4} \))
$23$ (\( 1 - 11320170 T^{2} + 41426511213649 T^{4} \))(\( 1 + 3937314 T^{2} + 41426511213649 T^{4} \))(\( 1 - 7282620 T^{2} + 10950776597798 T^{4} - \)\(30\!\cdots\!80\)\( T^{6} + \)\(17\!\cdots\!01\)\( T^{8} \))(\( ( 1 - 12622686 T^{2} + 41426511213649 T^{4} )^{2} \))
$29$ (\( ( 1 + 270 T + 20511149 T^{2} )^{2} \))(\( ( 1 + 2646 T + 20511149 T^{2} )^{2} \))(\( ( 1 + 2430 T + 19735498 T^{2} + 49842092070 T^{3} + 420707233300201 T^{4} )^{2} \))(\( ( 1 + 40800682 T^{2} + 420707233300201 T^{4} )^{2} \))
$31$ (\( ( 1 + 2048 T + 28629151 T^{2} )^{2} \))(\( ( 1 + 5612 T + 28629151 T^{2} )^{2} \))(\( ( 1 - 11684 T + 90263166 T^{2} - 334503000284 T^{3} + 819628286980801 T^{4} )^{2} \))(\( ( 1 - 3856 T + 28629151 T^{2} )^{4} \))
$37$ (\( 1 - 119573530 T^{2} + 4808584372417849 T^{4} \))(\( 1 - 85572970 T^{2} + 4808584372417849 T^{4} \))(\( 1 - 100197680 T^{2} + 8640930841305198 T^{4} - \)\(48\!\cdots\!20\)\( T^{6} + \)\(23\!\cdots\!01\)\( T^{8} \))(\( ( 1 - 89804410 T^{2} + 4808584372417849 T^{4} )^{2} \))
$41$ (\( ( 1 - 2398 T + 115856201 T^{2} )^{2} \))(\( ( 1 - 18986 T + 115856201 T^{2} )^{2} \))(\( ( 1 + 24984 T + 351666366 T^{2} + 2894551325784 T^{3} + 13422659310152401 T^{4} )^{2} \))(\( ( 1 + 139752402 T^{2} + 13422659310152401 T^{4} )^{2} \))
$43$ (\( 1 - 288754450 T^{2} + 21611482313284249 T^{4} \))(\( 1 - 288237670 T^{2} + 21611482313284249 T^{4} \))(\( 1 - 100144460 T^{2} - 11465052277897002 T^{4} - \)\(21\!\cdots\!40\)\( T^{6} + \)\(46\!\cdots\!01\)\( T^{8} \))(\( ( 1 - 265922422 T^{2} + 21611482313284249 T^{4} )^{2} \))
$47$ (\( 1 - 344584890 T^{2} + 52599132235830049 T^{4} \))(\( 1 - 379480014 T^{2} + 52599132235830049 T^{4} \))(\( 1 - 826473900 T^{2} + 275763720563232998 T^{4} - \)\(43\!\cdots\!00\)\( T^{6} + \)\(27\!\cdots\!01\)\( T^{8} \))(\( ( 1 - 62680014 T^{2} + 52599132235830049 T^{4} )^{2} \))
$53$ (\( 1 - 827605690 T^{2} + 174887470365513049 T^{4} \))(\( 1 + 747967430 T^{2} + 174887470365513049 T^{4} \))(\( 1 - 809517920 T^{2} + 441417135624894798 T^{4} - \)\(14\!\cdots\!80\)\( T^{6} + \)\(30\!\cdots\!01\)\( T^{8} \))(\( ( 1 - 835077670 T^{2} + 174887470365513049 T^{4} )^{2} \))
$59$ (\( ( 1 + 39740 T + 714924299 T^{2} )^{2} \))(\( ( 1 + 28300 T + 714924299 T^{2} )^{2} \))(\( ( 1 + 23730 T + 501951198 T^{2} + 16965153615270 T^{3} + 511116753300641401 T^{4} )^{2} \))(\( ( 1 + 33744102 T^{2} + 511116753300641401 T^{4} )^{2} \))
$61$ (\( ( 1 + 42298 T + 844596301 T^{2} )^{2} \))(\( ( 1 - 18290 T + 844596301 T^{2} )^{2} \))(\( ( 1 - 57124 T + 2217210846 T^{2} - 48246719098324 T^{3} + 713342911662882601 T^{4} )^{2} \))(\( ( 1 + 38158 T + 844596301 T^{2} )^{4} \))
$67$ (\( 1 - 1669968610 T^{2} + 1822837804551761449 T^{4} \))(\( 1 + 1649943722 T^{2} + 1822837804551761449 T^{4} \))(\( 1 - 4204562300 T^{2} + 7726214820152619798 T^{4} - \)\(76\!\cdots\!00\)\( T^{6} + \)\(33\!\cdots\!01\)\( T^{8} \))(\( ( 1 - 1387552678 T^{2} + 1822837804551761449 T^{4} )^{2} \))
$71$ (\( ( 1 - 4248 T + 1804229351 T^{2} )^{2} \))(\( ( 1 - 28800 T + 1804229351 T^{2} )^{2} \))(\( ( 1 - 11076 T + 940164046 T^{2} - 19983644291676 T^{3} + 3255243551009881201 T^{4} )^{2} \))(\( ( 1 + 3331783438 T^{2} + 3255243551009881201 T^{4} )^{2} \))
$73$ (\( 1 - 3239892370 T^{2} + 4297625829703557649 T^{4} \))(\( 1 - 3197010322 T^{2} + 4297625829703557649 T^{4} \))(\( 1 - 3772142420 T^{2} + 9801194247527769798 T^{4} - \)\(16\!\cdots\!80\)\( T^{6} + \)\(18\!\cdots\!01\)\( T^{8} \))(\( ( 1 + 669338414 T^{2} + 4297625829703557649 T^{4} )^{2} \))
$79$ (\( ( 1 + 35280 T + 3077056399 T^{2} )^{2} \))(\( ( 1 + 60228 T + 3077056399 T^{2} )^{2} \))(\( ( 1 + 14220 T + 4980519998 T^{2} + 43755741993780 T^{3} + 9468276082626847201 T^{4} )^{2} \))(\( ( 1 - 20664 T + 3077056399 T^{2} )^{4} \))
$83$ (\( 1 - 7103795010 T^{2} + 15516041187205853449 T^{4} \))(\( 1 - 7871990262 T^{2} + 15516041187205853449 T^{4} \))(\( 1 - 2811580140 T^{2} + 17974811933757230198 T^{4} - \)\(43\!\cdots\!60\)\( T^{6} + \)\(24\!\cdots\!01\)\( T^{8} \))(\( ( 1 + 1524711738 T^{2} + 15516041187205853449 T^{4} )^{2} \))
$89$ (\( ( 1 + 85210 T + 5584059449 T^{2} )^{2} \))(\( ( 1 - 22678 T + 5584059449 T^{2} )^{2} \))(\( ( 1 + 60420 T + 10831762998 T^{2} + 337388871908580 T^{3} + 31181719929966183601 T^{4} )^{2} \))(\( ( 1 + 7604090994 T^{2} + 31181719929966183601 T^{4} )^{2} \))
$97$ (\( 1 - 7720618690 T^{2} + 73742412689492826049 T^{4} \))(\( 1 - 15808047490 T^{2} + 73742412689492826049 T^{4} \))(\( 1 - 19675914500 T^{2} + \)\(24\!\cdots\!98\)\( T^{4} - \)\(14\!\cdots\!00\)\( T^{6} + \)\(54\!\cdots\!01\)\( T^{8} \))(\( ( 1 - 17140767490 T^{2} + 73742412689492826049 T^{4} )^{2} \))
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