Properties

Label 560.2.g
Level 560
Weight 2
Character orbit g
Rep. character \(\chi_{560}(449,\cdot)\)
Character field \(\Q\)
Dimension 18
Newform subspaces 6
Sturm bound 192
Trace bound 9

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Defining parameters

Level: \( N \) = \( 560 = 2^{4} \cdot 5 \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 560.g (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 5 \)
Character field: \(\Q\)
Newform subspaces: \( 6 \)
Sturm bound: \(192\)
Trace bound: \(9\)
Distinguishing \(T_p\): \(3\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(560, [\chi])\).

Total New Old
Modular forms 108 18 90
Cusp forms 84 18 66
Eisenstein series 24 0 24

Trace form

\( 18q + 2q^{5} - 18q^{9} + O(q^{10}) \) \( 18q + 2q^{5} - 18q^{9} - 12q^{11} + 12q^{15} + 8q^{19} + 2q^{25} - 4q^{29} - 8q^{31} - 28q^{39} - 4q^{41} - 10q^{45} - 18q^{49} - 20q^{51} + 40q^{55} + 36q^{61} - 16q^{65} - 16q^{69} + 40q^{71} - 12q^{79} + 34q^{81} + 20q^{89} - 12q^{91} + 32q^{95} + 56q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(560, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
560.2.g.a \(2\) \(4.472\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(-4\) \(0\) \(q+3iq^{3}+(-2-i)q^{5}-iq^{7}-6q^{9}+\cdots\)
560.2.g.b \(2\) \(4.472\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(-4\) \(0\) \(q+iq^{3}+(-2-i)q^{5}+iq^{7}+2q^{9}+\cdots\)
560.2.g.c \(2\) \(4.472\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(2\) \(0\) \(q+(1+2i)q^{5}-iq^{7}+3q^{9}-4iq^{13}+\cdots\)
560.2.g.d \(2\) \(4.472\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(4\) \(0\) \(q+iq^{3}+(2+i)q^{5}+iq^{7}+2q^{9}+q^{11}+\cdots\)
560.2.g.e \(4\) \(4.472\) \(\Q(i, \sqrt{6})\) None \(0\) \(0\) \(4\) \(0\) \(q+(\beta _{1}+\beta _{3})q^{3}+(1+\beta _{1}-\beta _{2})q^{5}-\beta _{2}q^{7}+\cdots\)
560.2.g.f \(6\) \(4.472\) 6.0.5161984.1 None \(0\) \(0\) \(0\) \(0\) \(q-\beta _{5}q^{3}+(\beta _{2}+\beta _{5})q^{5}+\beta _{4}q^{7}+(-1+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(560, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(560, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(35, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(40, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(70, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(80, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(140, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(280, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ (\( 1 + 3 T^{2} + 9 T^{4} \))(\( 1 - 5 T^{2} + 9 T^{4} \))(\( ( 1 - 3 T^{2} )^{2} \))(\( 1 - 5 T^{2} + 9 T^{4} \))(\( ( 1 + 9 T^{4} )^{2} \))(\( 1 - 5 T^{2} + 11 T^{4} - 26 T^{6} + 99 T^{8} - 405 T^{10} + 729 T^{12} \))
$5$ (\( 1 + 4 T + 5 T^{2} \))(\( 1 + 4 T + 5 T^{2} \))(\( 1 - 2 T + 5 T^{2} \))(\( 1 - 4 T + 5 T^{2} \))(\( 1 - 4 T + 8 T^{2} - 20 T^{3} + 25 T^{4} \))(\( 1 + 5 T^{2} + 8 T^{3} + 25 T^{4} + 125 T^{6} \))
$7$ (\( 1 + T^{2} \))(\( 1 + T^{2} \))(\( 1 + T^{2} \))(\( 1 + T^{2} \))(\( ( 1 + T^{2} )^{2} \))(\( ( 1 + T^{2} )^{3} \))
$11$ (\( ( 1 + 3 T + 11 T^{2} )^{2} \))(\( ( 1 - 3 T + 11 T^{2} )^{2} \))(\( ( 1 + 11 T^{2} )^{2} \))(\( ( 1 - T + 11 T^{2} )^{2} \))(\( ( 1 - 2 T^{2} + 121 T^{4} )^{2} \))(\( ( 1 + 7 T + 41 T^{2} + 146 T^{3} + 451 T^{4} + 847 T^{5} + 1331 T^{6} )^{2} \))
$13$ (\( 1 - 25 T^{2} + 169 T^{4} \))(\( 1 - 25 T^{2} + 169 T^{4} \))(\( ( 1 - 6 T + 13 T^{2} )( 1 + 6 T + 13 T^{2} ) \))(\( 1 - 25 T^{2} + 169 T^{4} \))(\( 1 - 32 T^{2} + 498 T^{4} - 5408 T^{6} + 28561 T^{8} \))(\( 1 - 9 T^{2} + 491 T^{4} - 2882 T^{6} + 82979 T^{8} - 257049 T^{10} + 4826809 T^{12} \))
$17$ (\( 1 - 9 T^{2} + 289 T^{4} \))(\( 1 + 15 T^{2} + 289 T^{4} \))(\( 1 - 18 T^{2} + 289 T^{4} \))(\( 1 - 25 T^{2} + 289 T^{4} \))(\( ( 1 - 8 T + 17 T^{2} )^{2}( 1 + 8 T + 17 T^{2} )^{2} \))(\( 1 - 53 T^{2} + 1539 T^{4} - 31118 T^{6} + 444771 T^{8} - 4426613 T^{10} + 24137569 T^{12} \))
$19$ (\( ( 1 + 8 T + 19 T^{2} )^{2} \))(\( ( 1 + 19 T^{2} )^{2} \))(\( ( 1 - 4 T + 19 T^{2} )^{2} \))(\( ( 1 + 4 T + 19 T^{2} )^{2} \))(\( ( 1 - 8 T + 48 T^{2} - 152 T^{3} + 361 T^{4} )^{2} \))(\( ( 1 - 4 T + 43 T^{2} - 160 T^{3} + 817 T^{4} - 1444 T^{5} + 6859 T^{6} )^{2} \))
$23$ (\( 1 - 42 T^{2} + 529 T^{4} \))(\( 1 - 10 T^{2} + 529 T^{4} \))(\( 1 + 18 T^{2} + 529 T^{4} \))(\( 1 - 42 T^{2} + 529 T^{4} \))(\( 1 - 36 T^{2} + 998 T^{4} - 19044 T^{6} + 279841 T^{8} \))(\( 1 - 46 T^{2} + 1887 T^{4} - 43972 T^{6} + 998223 T^{8} - 12872686 T^{10} + 148035889 T^{12} \))
$29$ (\( ( 1 - T + 29 T^{2} )^{2} \))(\( ( 1 - 5 T + 29 T^{2} )^{2} \))(\( ( 1 + 2 T + 29 T^{2} )^{2} \))(\( ( 1 - T + 29 T^{2} )^{2} \))(\( ( 1 + 4 T + 38 T^{2} + 116 T^{3} + 841 T^{4} )^{2} \))(\( ( 1 + 3 T + 15 T^{2} + 66 T^{3} + 435 T^{4} + 2523 T^{5} + 24389 T^{6} )^{2} \))
$31$ (\( ( 1 - 2 T + 31 T^{2} )^{2} \))(\( ( 1 + 2 T + 31 T^{2} )^{2} \))(\( ( 1 - 8 T + 31 T^{2} )^{2} \))(\( ( 1 - 6 T + 31 T^{2} )^{2} \))(\( ( 1 + 8 T + 54 T^{2} + 248 T^{3} + 961 T^{4} )^{2} \))(\( ( 1 + 10 T + 101 T^{2} + 540 T^{3} + 3131 T^{4} + 9610 T^{5} + 29791 T^{6} )^{2} \))
$37$ (\( 1 + 26 T^{2} + 1369 T^{4} \))(\( ( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} ) \))(\( 1 - 10 T^{2} + 1369 T^{4} \))(\( ( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} ) \))(\( ( 1 - 12 T + 37 T^{2} )^{2}( 1 + 12 T + 37 T^{2} )^{2} \))(\( ( 1 - 38 T^{2} + 1369 T^{4} )^{3} \))
$41$ (\( ( 1 + 6 T + 41 T^{2} )^{2} \))(\( ( 1 - 2 T + 41 T^{2} )^{2} \))(\( ( 1 - 6 T + 41 T^{2} )^{2} \))(\( ( 1 + 10 T + 41 T^{2} )^{2} \))(\( ( 1 + 12 T + 94 T^{2} + 492 T^{3} + 1681 T^{4} )^{2} \))(\( ( 1 - 18 T + 191 T^{2} - 1388 T^{3} + 7831 T^{4} - 30258 T^{5} + 68921 T^{6} )^{2} \))
$43$ (\( 1 - 70 T^{2} + 1849 T^{4} \))(\( 1 - 70 T^{2} + 1849 T^{4} \))(\( 1 - 22 T^{2} + 1849 T^{4} \))(\( ( 1 - 43 T^{2} )^{2} \))(\( 1 - 92 T^{2} + 4278 T^{4} - 170108 T^{6} + 3418801 T^{8} \))(\( 1 - 178 T^{2} + 15575 T^{4} - 836124 T^{6} + 28798175 T^{8} - 608546578 T^{10} + 6321363049 T^{12} \))
$47$ (\( 1 + 27 T^{2} + 2209 T^{4} \))(\( 1 - 85 T^{2} + 2209 T^{4} \))(\( 1 - 30 T^{2} + 2209 T^{4} \))(\( 1 - 13 T^{2} + 2209 T^{4} \))(\( 1 - 108 T^{2} + 5798 T^{4} - 238572 T^{6} + 4879681 T^{8} \))(\( 1 - 105 T^{2} + 6339 T^{4} - 285798 T^{6} + 14002851 T^{8} - 512366505 T^{10} + 10779215329 T^{12} \))
$53$ (\( 1 - 70 T^{2} + 2809 T^{4} \))(\( 1 - 70 T^{2} + 2809 T^{4} \))(\( ( 1 - 53 T^{2} )^{2} \))(\( ( 1 - 4 T + 53 T^{2} )( 1 + 4 T + 53 T^{2} ) \))(\( 1 - 92 T^{2} + 4278 T^{4} - 258428 T^{6} + 7890481 T^{8} \))(\( 1 - 130 T^{2} + 13655 T^{4} - 792060 T^{6} + 38356895 T^{8} - 1025762530 T^{10} + 22164361129 T^{12} \))
$59$ (\( ( 1 + 10 T + 59 T^{2} )^{2} \))(\( ( 1 - 10 T + 59 T^{2} )^{2} \))(\( ( 1 + 4 T + 59 T^{2} )^{2} \))(\( ( 1 - 6 T + 59 T^{2} )^{2} \))(\( ( 1 + 8 T + 128 T^{2} + 472 T^{3} + 3481 T^{4} )^{2} \))(\( ( 1 - 6 T + 99 T^{2} - 752 T^{3} + 5841 T^{4} - 20886 T^{5} + 205379 T^{6} )^{2} \))
$61$ (\( ( 1 + 61 T^{2} )^{2} \))(\( ( 1 + 8 T + 61 T^{2} )^{2} \))(\( ( 1 + 6 T + 61 T^{2} )^{2} \))(\( ( 1 + 4 T + 61 T^{2} )^{2} \))(\( ( 1 - 12 T + 152 T^{2} - 732 T^{3} + 3721 T^{4} )^{2} \))(\( ( 1 - 24 T + 365 T^{2} - 3368 T^{3} + 22265 T^{4} - 89304 T^{5} + 226981 T^{6} )^{2} \))
$67$ (\( 1 - 34 T^{2} + 4489 T^{4} \))(\( 1 - 130 T^{2} + 4489 T^{4} \))(\( 1 - 70 T^{2} + 4489 T^{4} \))(\( 1 - 34 T^{2} + 4489 T^{4} \))(\( ( 1 - 70 T^{2} + 4489 T^{4} )^{2} \))(\( 1 - 174 T^{2} + 20567 T^{4} - 1533188 T^{6} + 92325263 T^{8} - 3506295054 T^{10} + 90458382169 T^{12} \))
$71$ (\( ( 1 + 71 T^{2} )^{2} \))(\( ( 1 - 8 T + 71 T^{2} )^{2} \))(\( ( 1 + 12 T + 71 T^{2} )^{2} \))(\( ( 1 - 16 T + 71 T^{2} )^{2} \))(\( ( 1 - 12 T + 154 T^{2} - 852 T^{3} + 5041 T^{4} )^{2} \))(\( ( 1 + 4 T + 193 T^{2} + 504 T^{3} + 13703 T^{4} + 20164 T^{5} + 357911 T^{6} )^{2} \))
$73$ (\( 1 - 46 T^{2} + 5329 T^{4} \))(\( ( 1 - 16 T + 73 T^{2} )( 1 + 16 T + 73 T^{2} ) \))(\( 1 - 130 T^{2} + 5329 T^{4} \))(\( 1 - 46 T^{2} + 5329 T^{4} \))(\( 1 - 236 T^{2} + 24198 T^{4} - 1257644 T^{6} + 28398241 T^{8} \))(\( 1 - 346 T^{2} + 55487 T^{4} - 5172972 T^{6} + 295690223 T^{8} - 9825791386 T^{10} + 151334226289 T^{12} \))
$79$ (\( ( 1 + 7 T + 79 T^{2} )^{2} \))(\( ( 1 + 5 T + 79 T^{2} )^{2} \))(\( ( 1 + 4 T + 79 T^{2} )^{2} \))(\( ( 1 + 11 T + 79 T^{2} )^{2} \))(\( ( 1 - 4 T + 138 T^{2} - 316 T^{3} + 6241 T^{4} )^{2} \))(\( ( 1 - 17 T + 205 T^{2} - 2138 T^{3} + 16195 T^{4} - 106097 T^{5} + 493039 T^{6} )^{2} \))
$83$ (\( 1 - 22 T^{2} + 6889 T^{4} \))(\( 1 - 150 T^{2} + 6889 T^{4} \))(\( ( 1 - 83 T^{2} )^{2} \))(\( 1 - 150 T^{2} + 6889 T^{4} \))(\( ( 1 - 160 T^{2} + 6889 T^{4} )^{2} \))(\( 1 - 70 T^{2} + 1179 T^{4} + 304148 T^{6} + 8122131 T^{8} - 3322082470 T^{10} + 326940373369 T^{12} \))
$89$ (\( ( 1 + 8 T + 89 T^{2} )^{2} \))(\( ( 1 + 89 T^{2} )^{2} \))(\( ( 1 - 10 T + 89 T^{2} )^{2} \))(\( ( 1 + 12 T + 89 T^{2} )^{2} \))(\( ( 1 - 10 T + 89 T^{2} )^{4} \))(\( ( 1 + 95 T^{2} + 464 T^{3} + 8455 T^{4} + 704969 T^{6} )^{2} \))
$97$ (\( 1 - 185 T^{2} + 9409 T^{4} \))(\( 1 - 145 T^{2} + 9409 T^{4} \))(\( 1 - 50 T^{2} + 9409 T^{4} \))(\( 1 + 167 T^{2} + 9409 T^{4} \))(\( 1 - 124 T^{2} + 8838 T^{4} - 1166716 T^{6} + 88529281 T^{8} \))(\( 1 - 469 T^{2} + 98339 T^{4} - 12075534 T^{6} + 925271651 T^{8} - 41520232789 T^{10} + 832972004929 T^{12} \))
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