Properties

Label 90.6.a
Level 90
Weight 6
Character orbit a
Rep. character \(\chi_{90}(1,\cdot)\)
Character field \(\Q\)
Dimension 7
Newform subspaces 7
Sturm bound 108
Trace bound 7

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Defining parameters

Level: \( N \) = \( 90 = 2 \cdot 3^{2} \cdot 5 \)
Weight: \( k \) = \( 6 \)
Character orbit: \([\chi]\) = 90.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 7 \)
Sturm bound: \(108\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(90))\).

Total New Old
Modular forms 98 7 91
Cusp forms 82 7 75
Eisenstein series 16 0 16

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)\(3\)\(5\)FrickeDim.
\(+\)\(+\)\(-\)\(-\)\(1\)
\(+\)\(-\)\(+\)\(-\)\(1\)
\(+\)\(-\)\(-\)\(+\)\(1\)
\(-\)\(+\)\(+\)\(-\)\(1\)
\(-\)\(-\)\(+\)\(+\)\(1\)
\(-\)\(-\)\(-\)\(-\)\(2\)
Plus space\(+\)\(2\)
Minus space\(-\)\(5\)

Trace form

\( 7q + 4q^{2} + 112q^{4} + 25q^{5} + 80q^{7} + 64q^{8} + O(q^{10}) \) \( 7q + 4q^{2} + 112q^{4} + 25q^{5} + 80q^{7} + 64q^{8} - 100q^{10} - 288q^{11} + 1466q^{13} + 224q^{14} + 1792q^{16} + 2226q^{17} + 1748q^{19} + 400q^{20} + 3312q^{22} + 2880q^{23} + 4375q^{25} - 4984q^{26} + 1280q^{28} + 4686q^{29} + 17888q^{31} + 1024q^{32} + 12888q^{34} + 4100q^{35} - 718q^{37} + 23024q^{38} - 1600q^{40} - 15402q^{41} + 4388q^{43} - 4608q^{44} - 18384q^{46} - 56880q^{47} - 26529q^{49} + 2500q^{50} + 23456q^{52} + 46374q^{53} - 17700q^{55} + 3584q^{56} - 43128q^{58} + 5040q^{59} - 47158q^{61} + 3392q^{62} + 28672q^{64} + 71450q^{65} - 57508q^{67} + 35616q^{68} + 26800q^{70} - 174528q^{71} - 82762q^{73} - 134200q^{74} + 27968q^{76} - 90000q^{77} - 217120q^{79} + 6400q^{80} - 139176q^{82} + 174012q^{83} - 34050q^{85} - 12160q^{86} + 52992q^{88} + 78822q^{89} + 221944q^{91} + 46080q^{92} + 22992q^{94} - 18100q^{95} - 6466q^{97} + 100836q^{98} + O(q^{100}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(90))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 3 5
90.6.a.a \(1\) \(14.435\) \(\Q\) None \(-4\) \(0\) \(-25\) \(32\) \(+\) \(-\) \(+\) \(q-4q^{2}+2^{4}q^{4}-5^{2}q^{5}+2^{5}q^{7}-2^{6}q^{8}+\cdots\)
90.6.a.b \(1\) \(14.435\) \(\Q\) None \(-4\) \(0\) \(25\) \(-118\) \(+\) \(-\) \(-\) \(q-4q^{2}+2^{4}q^{4}+5^{2}q^{5}-118q^{7}+\cdots\)
90.6.a.c \(1\) \(14.435\) \(\Q\) None \(-4\) \(0\) \(25\) \(98\) \(+\) \(+\) \(-\) \(q-4q^{2}+2^{4}q^{4}+5^{2}q^{5}+98q^{7}-2^{6}q^{8}+\cdots\)
90.6.a.d \(1\) \(14.435\) \(\Q\) None \(4\) \(0\) \(-25\) \(-172\) \(-\) \(-\) \(+\) \(q+4q^{2}+2^{4}q^{4}-5^{2}q^{5}-172q^{7}+\cdots\)
90.6.a.e \(1\) \(14.435\) \(\Q\) None \(4\) \(0\) \(-25\) \(98\) \(-\) \(+\) \(+\) \(q+4q^{2}+2^{4}q^{4}-5^{2}q^{5}+98q^{7}+2^{6}q^{8}+\cdots\)
90.6.a.f \(1\) \(14.435\) \(\Q\) None \(4\) \(0\) \(25\) \(-22\) \(-\) \(-\) \(-\) \(q+4q^{2}+2^{4}q^{4}+5^{2}q^{5}-22q^{7}+2^{6}q^{8}+\cdots\)
90.6.a.g \(1\) \(14.435\) \(\Q\) None \(4\) \(0\) \(25\) \(164\) \(-\) \(-\) \(-\) \(q+4q^{2}+2^{4}q^{4}+5^{2}q^{5}+164q^{7}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(90))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(90)) \cong \) \(S_{6}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(18))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 4 T \))(\( 1 + 4 T \))(\( 1 + 4 T \))(\( 1 - 4 T \))(\( 1 - 4 T \))(\( 1 - 4 T \))(\( 1 - 4 T \))
$3$ 1
$5$ (\( 1 + 25 T \))(\( 1 - 25 T \))(\( 1 - 25 T \))(\( 1 + 25 T \))(\( 1 + 25 T \))(\( 1 - 25 T \))(\( 1 - 25 T \))
$7$ (\( 1 - 32 T + 16807 T^{2} \))(\( 1 + 118 T + 16807 T^{2} \))(\( 1 - 98 T + 16807 T^{2} \))(\( 1 + 172 T + 16807 T^{2} \))(\( 1 - 98 T + 16807 T^{2} \))(\( 1 + 22 T + 16807 T^{2} \))(\( 1 - 164 T + 16807 T^{2} \))
$11$ (\( 1 + 12 T + 161051 T^{2} \))(\( 1 + 192 T + 161051 T^{2} \))(\( 1 + 354 T + 161051 T^{2} \))(\( 1 + 132 T + 161051 T^{2} \))(\( 1 - 354 T + 161051 T^{2} \))(\( 1 - 768 T + 161051 T^{2} \))(\( 1 + 720 T + 161051 T^{2} \))
$13$ (\( 1 + 154 T + 371293 T^{2} \))(\( 1 - 1106 T + 371293 T^{2} \))(\( 1 - 404 T + 371293 T^{2} \))(\( 1 + 946 T + 371293 T^{2} \))(\( 1 - 404 T + 371293 T^{2} \))(\( 1 + 46 T + 371293 T^{2} \))(\( 1 - 698 T + 371293 T^{2} \))
$17$ (\( 1 - 918 T + 1419857 T^{2} \))(\( 1 + 762 T + 1419857 T^{2} \))(\( 1 + 654 T + 1419857 T^{2} \))(\( 1 - 222 T + 1419857 T^{2} \))(\( 1 - 654 T + 1419857 T^{2} \))(\( 1 + 378 T + 1419857 T^{2} \))(\( 1 - 2226 T + 1419857 T^{2} \))
$19$ (\( 1 + 1060 T + 2476099 T^{2} \))(\( 1 + 2740 T + 2476099 T^{2} \))(\( 1 - 1796 T + 2476099 T^{2} \))(\( 1 - 500 T + 2476099 T^{2} \))(\( 1 - 1796 T + 2476099 T^{2} \))(\( 1 - 1100 T + 2476099 T^{2} \))(\( 1 - 356 T + 2476099 T^{2} \))
$23$ (\( 1 - 4224 T + 6436343 T^{2} \))(\( 1 + 1566 T + 6436343 T^{2} \))(\( 1 - 1080 T + 6436343 T^{2} \))(\( 1 + 3564 T + 6436343 T^{2} \))(\( 1 + 1080 T + 6436343 T^{2} \))(\( 1 - 1986 T + 6436343 T^{2} \))(\( 1 - 1800 T + 6436343 T^{2} \))
$29$ (\( 1 - 7890 T + 20511149 T^{2} \))(\( 1 + 5910 T + 20511149 T^{2} \))(\( 1 - 5754 T + 20511149 T^{2} \))(\( 1 + 2190 T + 20511149 T^{2} \))(\( 1 + 5754 T + 20511149 T^{2} \))(\( 1 - 5610 T + 20511149 T^{2} \))(\( 1 + 714 T + 20511149 T^{2} \))
$31$ (\( 1 - 5192 T + 28629151 T^{2} \))(\( 1 + 6868 T + 28629151 T^{2} \))(\( 1 - 10196 T + 28629151 T^{2} \))(\( 1 - 2312 T + 28629151 T^{2} \))(\( 1 - 10196 T + 28629151 T^{2} \))(\( 1 + 3988 T + 28629151 T^{2} \))(\( 1 - 848 T + 28629151 T^{2} \))
$37$ (\( 1 - 16382 T + 69343957 T^{2} \))(\( 1 + 5518 T + 69343957 T^{2} \))(\( 1 - 5552 T + 69343957 T^{2} \))(\( 1 + 11242 T + 69343957 T^{2} \))(\( 1 - 5552 T + 69343957 T^{2} \))(\( 1 + 142 T + 69343957 T^{2} \))(\( 1 + 11302 T + 69343957 T^{2} \))
$41$ (\( 1 + 3642 T + 115856201 T^{2} \))(\( 1 - 378 T + 115856201 T^{2} \))(\( 1 - 12960 T + 115856201 T^{2} \))(\( 1 + 1242 T + 115856201 T^{2} \))(\( 1 + 12960 T + 115856201 T^{2} \))(\( 1 + 1542 T + 115856201 T^{2} \))(\( 1 + 9354 T + 115856201 T^{2} \))
$43$ (\( 1 - 15116 T + 147008443 T^{2} \))(\( 1 + 2434 T + 147008443 T^{2} \))(\( 1 + 8968 T + 147008443 T^{2} \))(\( 1 - 20624 T + 147008443 T^{2} \))(\( 1 + 8968 T + 147008443 T^{2} \))(\( 1 + 5026 T + 147008443 T^{2} \))(\( 1 + 5956 T + 147008443 T^{2} \))
$47$ (\( 1 + 23592 T + 229345007 T^{2} \))(\( 1 + 13122 T + 229345007 T^{2} \))(\( 1 - 5400 T + 229345007 T^{2} \))(\( 1 + 6588 T + 229345007 T^{2} \))(\( 1 + 5400 T + 229345007 T^{2} \))(\( 1 + 24738 T + 229345007 T^{2} \))(\( 1 - 11160 T + 229345007 T^{2} \))
$53$ (\( 1 - 16074 T + 418195493 T^{2} \))(\( 1 - 9174 T + 418195493 T^{2} \))(\( 1 + 8214 T + 418195493 T^{2} \))(\( 1 - 21066 T + 418195493 T^{2} \))(\( 1 - 8214 T + 418195493 T^{2} \))(\( 1 - 14166 T + 418195493 T^{2} \))(\( 1 + 14106 T + 418195493 T^{2} \))
$59$ (\( 1 - 14340 T + 714924299 T^{2} \))(\( 1 - 34980 T + 714924299 T^{2} \))(\( 1 + 3954 T + 714924299 T^{2} \))(\( 1 + 7980 T + 714924299 T^{2} \))(\( 1 - 3954 T + 714924299 T^{2} \))(\( 1 + 28380 T + 714924299 T^{2} \))(\( 1 + 7920 T + 714924299 T^{2} \))
$61$ (\( 1 + 47938 T + 844596301 T^{2} \))(\( 1 + 9838 T + 844596301 T^{2} \))(\( 1 - 962 T + 844596301 T^{2} \))(\( 1 - 16622 T + 844596301 T^{2} \))(\( 1 - 962 T + 844596301 T^{2} \))(\( 1 - 5522 T + 844596301 T^{2} \))(\( 1 + 13450 T + 844596301 T^{2} \))
$67$ (\( 1 - 33092 T + 1350125107 T^{2} \))(\( 1 - 33722 T + 1350125107 T^{2} \))(\( 1 + 17956 T + 1350125107 T^{2} \))(\( 1 - 1808 T + 1350125107 T^{2} \))(\( 1 + 17956 T + 1350125107 T^{2} \))(\( 1 + 24742 T + 1350125107 T^{2} \))(\( 1 + 65476 T + 1350125107 T^{2} \))
$71$ (\( 1 + 51912 T + 1804229351 T^{2} \))(\( 1 + 70212 T + 1804229351 T^{2} \))(\( 1 - 56148 T + 1804229351 T^{2} \))(\( 1 - 24528 T + 1804229351 T^{2} \))(\( 1 + 56148 T + 1804229351 T^{2} \))(\( 1 + 42372 T + 1804229351 T^{2} \))(\( 1 + 34560 T + 1804229351 T^{2} \))
$73$ (\( 1 - 12026 T + 2073071593 T^{2} \))(\( 1 - 21986 T + 2073071593 T^{2} \))(\( 1 + 85690 T + 2073071593 T^{2} \))(\( 1 - 20474 T + 2073071593 T^{2} \))(\( 1 + 85690 T + 2073071593 T^{2} \))(\( 1 + 52126 T + 2073071593 T^{2} \))(\( 1 - 86258 T + 2073071593 T^{2} \))
$79$ (\( 1 - 25160 T + 3077056399 T^{2} \))(\( 1 - 4520 T + 3077056399 T^{2} \))(\( 1 + 26044 T + 3077056399 T^{2} \))(\( 1 + 46240 T + 3077056399 T^{2} \))(\( 1 + 26044 T + 3077056399 T^{2} \))(\( 1 + 39640 T + 3077056399 T^{2} \))(\( 1 + 108832 T + 3077056399 T^{2} \))
$83$ (\( 1 + 35796 T + 3939040643 T^{2} \))(\( 1 - 109074 T + 3939040643 T^{2} \))(\( 1 - 93468 T + 3939040643 T^{2} \))(\( 1 - 51576 T + 3939040643 T^{2} \))(\( 1 + 93468 T + 3939040643 T^{2} \))(\( 1 - 59826 T + 3939040643 T^{2} \))(\( 1 + 10668 T + 3939040643 T^{2} \))
$89$ (\( 1 - 75510 T + 5584059449 T^{2} \))(\( 1 + 38490 T + 5584059449 T^{2} \))(\( 1 + 73428 T + 5584059449 T^{2} \))(\( 1 - 110310 T + 5584059449 T^{2} \))(\( 1 - 73428 T + 5584059449 T^{2} \))(\( 1 + 57690 T + 5584059449 T^{2} \))(\( 1 + 10818 T + 5584059449 T^{2} \))
$97$ (\( 1 + 44158 T + 8587340257 T^{2} \))(\( 1 + 1918 T + 8587340257 T^{2} \))(\( 1 - 128978 T + 8587340257 T^{2} \))(\( 1 + 78382 T + 8587340257 T^{2} \))(\( 1 - 128978 T + 8587340257 T^{2} \))(\( 1 + 144382 T + 8587340257 T^{2} \))(\( 1 - 4418 T + 8587340257 T^{2} \))
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