Properties

Label 1440.2.d
Level 1440
Weight 2
Character orbit d
Rep. character \(\chi_{1440}(1009,\cdot)\)
Character field \(\Q\)
Dimension 28
Newform subspaces 6
Sturm bound 576
Trace bound 31

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Defining parameters

Level: \( N \) \(=\) \( 1440 = 2^{5} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1440.d (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 40 \)
Character field: \(\Q\)
Newform subspaces: \( 6 \)
Sturm bound: \(576\)
Trace bound: \(31\)
Distinguishing \(T_p\): \(7\), \(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(1440, [\chi])\).

Total New Old
Modular forms 320 32 288
Cusp forms 256 28 228
Eisenstein series 64 4 60

Trace form

\( 28q + O(q^{10}) \) \( 28q - 4q^{25} + 8q^{41} - 20q^{49} + 8q^{55} + 24q^{65} - 16q^{71} + 32q^{79} + 16q^{89} + 40q^{95} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(1440, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
1440.2.d.a \(4\) \(11.498\) \(\Q(\sqrt{-2}, \sqrt{-5})\) \(\Q(\sqrt{-30}) \) \(0\) \(0\) \(0\) \(0\) \(q+\beta _{2}q^{5}-2\beta _{2}q^{11}-\beta _{3}q^{13}+\beta _{1}q^{17}+\cdots\)
1440.2.d.b \(4\) \(11.498\) \(\Q(\sqrt{2}, \sqrt{-3})\) \(\Q(\sqrt{-6}) \) \(0\) \(0\) \(0\) \(0\) \(q+\beta _{1}q^{5}-\beta _{3}q^{7}+(\beta _{1}+\beta _{2})q^{11}+(-1+\cdots)q^{25}+\cdots\)
1440.2.d.c \(4\) \(11.498\) \(\Q(\sqrt{2}, \sqrt{-3})\) None \(0\) \(0\) \(0\) \(0\) \(q+(\beta _{1}-\beta _{2})q^{5}-\beta _{3}q^{7}-2\beta _{2}q^{11}+\cdots\)
1440.2.d.d \(4\) \(11.498\) \(\Q(\sqrt{-3}, \sqrt{5})\) \(\Q(\sqrt{-15}) \) \(0\) \(0\) \(0\) \(0\) \(q+\beta _{1}q^{5}+2\beta _{2}q^{17}-\beta _{3}q^{19}-\beta _{2}q^{23}+\cdots\)
1440.2.d.e \(6\) \(11.498\) 6.0.839056.1 None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{2}q^{5}-\beta _{1}q^{7}+(-\beta _{1}-\beta _{5})q^{11}+\cdots\)
1440.2.d.f \(6\) \(11.498\) 6.0.839056.1 None \(0\) \(0\) \(0\) \(0\) \(q-\beta _{2}q^{5}-\beta _{1}q^{7}+(\beta _{1}+\beta _{5})q^{11}+(1+\cdots)q^{13}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(1440, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(1440, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(40, [\chi])\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(120, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(160, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(360, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(480, [\chi])\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ (\( ( 1 + 5 T^{2} )^{2} \))(\( 1 + 2 T^{2} + 25 T^{4} \))(\( 1 + 2 T^{2} + 25 T^{4} \))(\( ( 1 - 5 T^{2} )^{2} \))(\( 1 - T^{2} + 8 T^{3} - 5 T^{4} + 125 T^{6} \))(\( 1 - T^{2} - 8 T^{3} - 5 T^{4} + 125 T^{6} \))
$7$ (\( ( 1 - 7 T^{2} )^{4} \))(\( ( 1 - 2 T + 7 T^{2} )^{2}( 1 + 2 T + 7 T^{2} )^{2} \))(\( ( 1 - 8 T^{2} + 49 T^{4} )^{2} \))(\( ( 1 - 7 T^{2} )^{4} \))(\( 1 - 18 T^{2} + 191 T^{4} - 1532 T^{6} + 9359 T^{8} - 43218 T^{10} + 117649 T^{12} \))(\( 1 - 18 T^{2} + 191 T^{4} - 1532 T^{6} + 9359 T^{8} - 43218 T^{10} + 117649 T^{12} \))
$11$ (\( ( 1 - 2 T^{2} + 121 T^{4} )^{2} \))(\( ( 1 - 10 T^{2} + 121 T^{4} )^{2} \))(\( ( 1 - 10 T^{2} + 121 T^{4} )^{2} \))(\( ( 1 - 11 T^{2} )^{4} \))(\( 1 - 34 T^{2} + 503 T^{4} - 5436 T^{6} + 60863 T^{8} - 497794 T^{10} + 1771561 T^{12} \))(\( 1 - 34 T^{2} + 503 T^{4} - 5436 T^{6} + 60863 T^{8} - 497794 T^{10} + 1771561 T^{12} \))
$13$ (\( ( 1 - 14 T^{2} + 169 T^{4} )^{2} \))(\( ( 1 + 13 T^{2} )^{4} \))(\( ( 1 + 13 T^{2} )^{4} \))(\( ( 1 + 13 T^{2} )^{4} \))(\( ( 1 + 4 T + 23 T^{2} + 48 T^{3} + 299 T^{4} + 676 T^{5} + 2197 T^{6} )^{2} \))(\( ( 1 - 4 T + 23 T^{2} - 48 T^{3} + 299 T^{4} - 676 T^{5} + 2197 T^{6} )^{2} \))
$17$ (\( ( 1 - 26 T^{2} + 289 T^{4} )^{2} \))(\( ( 1 - 17 T^{2} )^{4} \))(\( ( 1 - 10 T^{2} + 289 T^{4} )^{2} \))(\( ( 1 + 14 T^{2} + 289 T^{4} )^{2} \))(\( 1 - 66 T^{2} + 2255 T^{4} - 47324 T^{6} + 651695 T^{8} - 5512386 T^{10} + 24137569 T^{12} \))(\( 1 - 66 T^{2} + 2255 T^{4} - 47324 T^{6} + 651695 T^{8} - 5512386 T^{10} + 24137569 T^{12} \))
$19$ (\( ( 1 - 19 T^{2} )^{4} \))(\( ( 1 - 19 T^{2} )^{4} \))(\( ( 1 - 8 T + 19 T^{2} )^{2}( 1 + 8 T + 19 T^{2} )^{2} \))(\( ( 1 - 4 T + 19 T^{2} )^{2}( 1 + 4 T + 19 T^{2} )^{2} \))(\( 1 - 54 T^{2} + 1367 T^{4} - 25652 T^{6} + 493487 T^{8} - 7037334 T^{10} + 47045881 T^{12} \))(\( 1 - 54 T^{2} + 1367 T^{4} - 25652 T^{6} + 493487 T^{8} - 7037334 T^{10} + 47045881 T^{12} \))
$23$ (\( ( 1 - 14 T^{2} + 529 T^{4} )^{2} \))(\( ( 1 - 23 T^{2} )^{4} \))(\( ( 1 - 40 T^{2} + 529 T^{4} )^{2} \))(\( ( 1 - 34 T^{2} + 529 T^{4} )^{2} \))(\( 1 - 46 T^{2} + 1775 T^{4} - 40932 T^{6} + 938975 T^{8} - 12872686 T^{10} + 148035889 T^{12} \))(\( 1 - 46 T^{2} + 1775 T^{4} - 40932 T^{6} + 938975 T^{8} - 12872686 T^{10} + 148035889 T^{12} \))
$29$ (\( ( 1 - 38 T^{2} + 841 T^{4} )^{2} \))(\( ( 1 + 50 T^{2} + 841 T^{4} )^{2} \))(\( ( 1 - 29 T^{2} )^{4} \))(\( ( 1 - 29 T^{2} )^{4} \))(\( 1 - 66 T^{2} + 3207 T^{4} - 111228 T^{6} + 2697087 T^{8} - 46680546 T^{10} + 594823321 T^{12} \))(\( 1 - 66 T^{2} + 3207 T^{4} - 111228 T^{6} + 2697087 T^{8} - 46680546 T^{10} + 594823321 T^{12} \))
$31$ (\( ( 1 + 2 T + 31 T^{2} )^{4} \))(\( ( 1 + 10 T + 31 T^{2} )^{4} \))(\( ( 1 + 4 T + 31 T^{2} )^{4} \))(\( ( 1 - 8 T + 31 T^{2} )^{4} \))(\( ( 1 - 8 T + 89 T^{2} - 432 T^{3} + 2759 T^{4} - 7688 T^{5} + 29791 T^{6} )^{2} \))(\( ( 1 - 8 T + 89 T^{2} - 432 T^{3} + 2759 T^{4} - 7688 T^{5} + 29791 T^{6} )^{2} \))
$37$ (\( ( 1 + 34 T^{2} + 1369 T^{4} )^{2} \))(\( ( 1 + 37 T^{2} )^{4} \))(\( ( 1 + 2 T^{2} + 1369 T^{4} )^{2} \))(\( ( 1 + 37 T^{2} )^{4} \))(\( ( 1 + 8 T + 111 T^{2} + 584 T^{3} + 4107 T^{4} + 10952 T^{5} + 50653 T^{6} )^{2} \))(\( ( 1 - 8 T + 111 T^{2} - 584 T^{3} + 4107 T^{4} - 10952 T^{5} + 50653 T^{6} )^{2} \))
$41$ (\( ( 1 + 41 T^{2} )^{4} \))(\( ( 1 + 41 T^{2} )^{4} \))(\( ( 1 + 41 T^{2} )^{4} \))(\( ( 1 + 41 T^{2} )^{4} \))(\( ( 1 - 2 T + 23 T^{2} - 220 T^{3} + 943 T^{4} - 3362 T^{5} + 68921 T^{6} )^{2} \))(\( ( 1 - 2 T + 23 T^{2} - 220 T^{3} + 943 T^{4} - 3362 T^{5} + 68921 T^{6} )^{2} \))
$43$ (\( ( 1 - 74 T^{2} + 1849 T^{4} )^{2} \))(\( ( 1 + 43 T^{2} )^{4} \))(\( ( 1 + 68 T^{2} + 1849 T^{4} )^{2} \))(\( ( 1 + 43 T^{2} )^{4} \))(\( ( 1 + 65 T^{2} - 64 T^{3} + 2795 T^{4} + 79507 T^{6} )^{2} \))(\( ( 1 + 65 T^{2} + 64 T^{3} + 2795 T^{4} + 79507 T^{6} )^{2} \))
$47$ (\( ( 1 + 34 T^{2} + 2209 T^{4} )^{2} \))(\( ( 1 - 47 T^{2} )^{4} \))(\( ( 1 - 40 T^{2} + 2209 T^{4} )^{2} \))(\( ( 1 + 14 T^{2} + 2209 T^{4} )^{2} \))(\( 1 - 222 T^{2} + 22367 T^{4} - 1328324 T^{6} + 49408703 T^{8} - 1083289182 T^{10} + 10779215329 T^{12} \))(\( 1 - 222 T^{2} + 22367 T^{4} - 1328324 T^{6} + 49408703 T^{8} - 1083289182 T^{10} + 10779215329 T^{12} \))
$53$ (\( ( 1 + 53 T^{2} )^{4} \))(\( ( 1 - 94 T^{2} + 2809 T^{4} )^{2} \))(\( ( 1 + 74 T^{2} + 2809 T^{4} )^{2} \))(\( ( 1 + 86 T^{2} + 2809 T^{4} )^{2} \))(\( ( 1 + 12 T + 191 T^{2} + 1264 T^{3} + 10123 T^{4} + 33708 T^{5} + 148877 T^{6} )^{2} \))(\( ( 1 - 12 T + 191 T^{2} - 1264 T^{3} + 10123 T^{4} - 33708 T^{5} + 148877 T^{6} )^{2} \))
$59$ (\( ( 1 - 98 T^{2} + 3481 T^{4} )^{2} \))(\( ( 1 - 10 T^{2} + 3481 T^{4} )^{2} \))(\( ( 1 - 10 T^{2} + 3481 T^{4} )^{2} \))(\( ( 1 - 59 T^{2} )^{4} \))(\( 1 - 178 T^{2} + 20567 T^{4} - 1418652 T^{6} + 71593727 T^{8} - 2156890258 T^{10} + 42180533641 T^{12} \))(\( 1 - 178 T^{2} + 20567 T^{4} - 1418652 T^{6} + 71593727 T^{8} - 2156890258 T^{10} + 42180533641 T^{12} \))
$61$ (\( ( 1 - 61 T^{2} )^{4} \))(\( ( 1 - 61 T^{2} )^{4} \))(\( ( 1 - 110 T^{2} + 3721 T^{4} )^{2} \))(\( ( 1 - 2 T + 61 T^{2} )^{2}( 1 + 2 T + 61 T^{2} )^{2} \))(\( 1 - 190 T^{2} + 20039 T^{4} - 1419204 T^{6} + 74565119 T^{8} - 2630709790 T^{10} + 51520374361 T^{12} \))(\( 1 - 190 T^{2} + 20039 T^{4} - 1419204 T^{6} + 74565119 T^{8} - 2630709790 T^{10} + 51520374361 T^{12} \))
$67$ (\( ( 1 - 26 T^{2} + 4489 T^{4} )^{2} \))(\( ( 1 + 67 T^{2} )^{4} \))(\( ( 1 + 116 T^{2} + 4489 T^{4} )^{2} \))(\( ( 1 + 67 T^{2} )^{4} \))(\( ( 1 + 137 T^{2} + 64 T^{3} + 9179 T^{4} + 300763 T^{6} )^{2} \))(\( ( 1 + 137 T^{2} - 64 T^{3} + 9179 T^{4} + 300763 T^{6} )^{2} \))
$71$ (\( ( 1 + 71 T^{2} )^{4} \))(\( ( 1 + 71 T^{2} )^{4} \))(\( ( 1 + 12 T + 71 T^{2} )^{4} \))(\( ( 1 + 71 T^{2} )^{4} \))(\( ( 1 - 8 T + 133 T^{2} - 1008 T^{3} + 9443 T^{4} - 40328 T^{5} + 357911 T^{6} )^{2} \))(\( ( 1 - 8 T + 133 T^{2} - 1008 T^{3} + 9443 T^{4} - 40328 T^{5} + 357911 T^{6} )^{2} \))
$73$ (\( ( 1 - 73 T^{2} )^{4} \))(\( ( 1 - 14 T + 73 T^{2} )^{2}( 1 + 14 T + 73 T^{2} )^{2} \))(\( ( 1 - 122 T^{2} + 5329 T^{4} )^{2} \))(\( ( 1 - 73 T^{2} )^{4} \))(\( 1 - 54 T^{2} + 2367 T^{4} - 531700 T^{6} + 12613743 T^{8} - 1533505014 T^{10} + 151334226289 T^{12} \))(\( 1 - 54 T^{2} + 2367 T^{4} - 531700 T^{6} + 12613743 T^{8} - 1533505014 T^{10} + 151334226289 T^{12} \))
$79$ (\( ( 1 + 14 T + 79 T^{2} )^{4} \))(\( ( 1 - 10 T + 79 T^{2} )^{4} \))(\( ( 1 - 4 T + 79 T^{2} )^{4} \))(\( ( 1 - 16 T + 79 T^{2} )^{4} \))(\( ( 1 + 8 T + 233 T^{2} + 1200 T^{3} + 18407 T^{4} + 49928 T^{5} + 493039 T^{6} )^{2} \))(\( ( 1 + 8 T + 233 T^{2} + 1200 T^{3} + 18407 T^{4} + 49928 T^{5} + 493039 T^{6} )^{2} \))
$83$ (\( ( 1 + 83 T^{2} )^{4} \))(\( ( 1 + 134 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 + 68 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 - 154 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 + 8 T + 185 T^{2} + 880 T^{3} + 15355 T^{4} + 55112 T^{5} + 571787 T^{6} )^{2} \))(\( ( 1 - 8 T + 185 T^{2} - 880 T^{3} + 15355 T^{4} - 55112 T^{5} + 571787 T^{6} )^{2} \))
$89$ (\( ( 1 + 89 T^{2} )^{4} \))(\( ( 1 + 89 T^{2} )^{4} \))(\( ( 1 + 6 T + 89 T^{2} )^{4} \))(\( ( 1 + 89 T^{2} )^{4} \))(\( ( 1 - 10 T + 103 T^{2} - 396 T^{3} + 9167 T^{4} - 79210 T^{5} + 704969 T^{6} )^{2} \))(\( ( 1 - 10 T + 103 T^{2} - 396 T^{3} + 9167 T^{4} - 79210 T^{5} + 704969 T^{6} )^{2} \))
$97$ (\( ( 1 - 97 T^{2} )^{4} \))(\( ( 1 - 2 T + 97 T^{2} )^{2}( 1 + 2 T + 97 T^{2} )^{2} \))(\( ( 1 - 170 T^{2} + 9409 T^{4} )^{2} \))(\( ( 1 - 97 T^{2} )^{4} \))(\( 1 - 246 T^{2} + 39183 T^{4} - 4535476 T^{6} + 368672847 T^{8} - 21778203126 T^{10} + 832972004929 T^{12} \))(\( 1 - 246 T^{2} + 39183 T^{4} - 4535476 T^{6} + 368672847 T^{8} - 21778203126 T^{10} + 832972004929 T^{12} \))
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