Properties

Label 572.2.f
Level 572
Weight 2
Character orbit f
Rep. character \(\chi_{572}(441,\cdot)\)
Character field \(\Q\)
Dimension 14
Newform subspaces 3
Sturm bound 168
Trace bound 1

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Defining parameters

Level: \( N \) = \( 572 = 2^{2} \cdot 11 \cdot 13 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 572.f (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 13 \)
Character field: \(\Q\)
Newform subspaces: \( 3 \)
Sturm bound: \(168\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(572, [\chi])\).

Total New Old
Modular forms 90 14 76
Cusp forms 78 14 64
Eisenstein series 12 0 12

Trace form

\( 14q + 22q^{9} + O(q^{10}) \) \( 14q + 22q^{9} - 6q^{13} + 8q^{23} - 22q^{25} - 12q^{27} - 4q^{29} - 8q^{35} + 20q^{39} + 12q^{43} - 18q^{49} - 8q^{51} + 32q^{53} - 16q^{61} + 36q^{65} - 28q^{69} + 20q^{75} - 8q^{77} - 8q^{79} + 30q^{81} + 20q^{87} - 12q^{91} - 12q^{95} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(572, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
572.2.f.a \(2\) \(4.567\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) \(q+4iq^{7}-3q^{9}-iq^{11}+(-3+2i)q^{13}+\cdots\)
572.2.f.b \(4\) \(4.567\) \(\Q(i, \sqrt{21})\) None \(0\) \(2\) \(0\) \(0\) \(q+(1-\beta _{3})q^{3}+(\beta _{1}-\beta _{2})q^{7}+(3-\beta _{3})q^{9}+\cdots\)
572.2.f.c \(8\) \(4.567\) \(\mathbb{Q}[x]/(x^{8} + \cdots)\) None \(0\) \(-2\) \(0\) \(0\) \(q-\beta _{2}q^{3}+(\beta _{1}+\beta _{5}+\beta _{6}-\beta _{7})q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(572, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(572, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(26, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(143, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(286, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( \))(\( \))(\( \))
$3$ (\( ( 1 + 3 T^{2} )^{2} \))(\( ( 1 - T + T^{2} - 3 T^{3} + 9 T^{4} )^{2} \))(\( ( 1 + T + 2 T^{2} + 6 T^{3} + 9 T^{4} + 18 T^{5} + 18 T^{6} + 27 T^{7} + 81 T^{8} )^{2} \))
$5$ (\( ( 1 - 5 T^{2} )^{2} \))(\( ( 1 - 5 T^{2} )^{4} \))(\( 1 + 6 T^{2} + 49 T^{4} + 306 T^{6} + 1740 T^{8} + 7650 T^{10} + 30625 T^{12} + 93750 T^{14} + 390625 T^{16} \))
$7$ (\( 1 + 2 T^{2} + 49 T^{4} \))(\( 1 - 13 T^{2} + 93 T^{4} - 637 T^{6} + 2401 T^{8} \))(\( 1 - 29 T^{2} + 407 T^{4} - 3831 T^{6} + 28856 T^{8} - 187719 T^{10} + 977207 T^{12} - 3411821 T^{14} + 5764801 T^{16} \))
$11$ (\( 1 + T^{2} \))(\( ( 1 + T^{2} )^{2} \))(\( ( 1 + T^{2} )^{4} \))
$13$ (\( 1 + 6 T + 13 T^{2} \))(\( ( 1 - 4 T + 13 T^{2} )^{2} \))(\( 1 + 8 T + 18 T^{2} + 8 T^{3} + 2 T^{4} + 104 T^{5} + 3042 T^{6} + 17576 T^{7} + 28561 T^{8} \))
$17$ (\( ( 1 + 2 T + 17 T^{2} )^{2} \))(\( ( 1 + 6 T + 22 T^{2} + 102 T^{3} + 289 T^{4} )^{2} \))(\( ( 1 - 8 T + 54 T^{2} - 192 T^{3} + 906 T^{4} - 3264 T^{5} + 15606 T^{6} - 39304 T^{7} + 83521 T^{8} )^{2} \))
$19$ (\( 1 - 22 T^{2} + 361 T^{4} \))(\( 1 - 25 T^{2} + 453 T^{4} - 9025 T^{6} + 130321 T^{8} \))(\( 1 - 45 T^{2} + 1551 T^{4} - 38759 T^{6} + 815964 T^{8} - 13991999 T^{10} + 202127871 T^{12} - 2117064645 T^{14} + 16983563041 T^{16} \))
$23$ (\( ( 1 + 4 T + 23 T^{2} )^{2} \))(\( ( 1 - 3 T + 43 T^{2} - 69 T^{3} + 529 T^{4} )^{2} \))(\( ( 1 - 5 T + 64 T^{2} - 222 T^{3} + 1775 T^{4} - 5106 T^{5} + 33856 T^{6} - 60835 T^{7} + 279841 T^{8} )^{2} \))
$29$ (\( ( 1 + 6 T + 29 T^{2} )^{2} \))(\( ( 1 - 6 T + 46 T^{2} - 174 T^{3} + 841 T^{4} )^{2} \))(\( ( 1 + 2 T + 46 T^{2} - 166 T^{3} + 642 T^{4} - 4814 T^{5} + 38686 T^{6} + 48778 T^{7} + 707281 T^{8} )^{2} \))
$31$ (\( 1 - 46 T^{2} + 961 T^{4} \))(\( 1 - 64 T^{2} + 2190 T^{4} - 61504 T^{6} + 923521 T^{8} \))(\( 1 - 58 T^{2} + 1945 T^{4} - 82822 T^{6} + 3288412 T^{8} - 79591942 T^{10} + 1796248345 T^{12} - 51475213498 T^{14} + 852891037441 T^{16} \))
$37$ (\( 1 - 10 T^{2} + 1369 T^{4} \))(\( ( 1 - 8 T + 37 T^{2} )^{2}( 1 + 8 T + 37 T^{2} )^{2} \))(\( 1 - 174 T^{2} + 16065 T^{4} - 976430 T^{6} + 42457908 T^{8} - 1336732670 T^{10} + 30108396465 T^{12} - 446436395166 T^{14} + 3512479453921 T^{16} \))
$41$ (\( 1 + 62 T^{2} + 1681 T^{4} \))(\( 1 - 65 T^{2} + 3993 T^{4} - 109265 T^{6} + 2825761 T^{8} \))(\( 1 - 53 T^{2} + 4715 T^{4} - 193263 T^{6} + 10976756 T^{8} - 324875103 T^{10} + 13323463115 T^{12} - 251755524773 T^{14} + 7984925229121 T^{16} \))
$43$ (\( ( 1 - 12 T + 43 T^{2} )^{2} \))(\( ( 1 - 2 T + 43 T^{2} )^{4} \))(\( ( 1 + 10 T + 138 T^{2} + 726 T^{3} + 6898 T^{4} + 31218 T^{5} + 255162 T^{6} + 795070 T^{7} + 3418801 T^{8} )^{2} \))
$47$ (\( 1 - 78 T^{2} + 2209 T^{4} \))(\( ( 1 - 58 T^{2} + 2209 T^{4} )^{2} \))(\( 1 - 244 T^{2} + 29108 T^{4} - 2252748 T^{6} + 124096406 T^{8} - 4976320332 T^{10} + 142037754548 T^{12} - 2630128540276 T^{14} + 23811286661761 T^{16} \))
$53$ (\( ( 1 - 6 T + 53 T^{2} )^{2} \))(\( ( 1 + 9 T + 121 T^{2} + 477 T^{3} + 2809 T^{4} )^{2} \))(\( ( 1 - 19 T + 271 T^{2} - 2733 T^{3} + 23324 T^{4} - 144849 T^{5} + 761239 T^{6} - 2828663 T^{7} + 7890481 T^{8} )^{2} \))
$59$ (\( 1 - 102 T^{2} + 3481 T^{4} \))(\( ( 1 - 82 T^{2} + 3481 T^{4} )^{2} \))(\( 1 - 206 T^{2} + 19241 T^{4} - 1196838 T^{6} + 67809284 T^{8} - 4166193078 T^{10} + 233150143001 T^{12} - 8689189930046 T^{14} + 146830437604321 T^{16} \))
$61$ (\( ( 1 + 10 T + 61 T^{2} )^{2} \))(\( ( 1 - 10 T + 61 T^{2} )^{4} \))(\( ( 1 + 18 T + 294 T^{2} + 2746 T^{3} + 25986 T^{4} + 167506 T^{5} + 1093974 T^{6} + 4085658 T^{7} + 13845841 T^{8} )^{2} \))
$67$ (\( 1 - 118 T^{2} + 4489 T^{4} \))(\( 1 - 28 T^{2} - 2922 T^{4} - 125692 T^{6} + 20151121 T^{8} \))(\( 1 - 314 T^{2} + 49337 T^{4} - 5171766 T^{6} + 398651756 T^{8} - 23216057574 T^{10} + 994195856777 T^{12} - 28403932001066 T^{14} + 406067677556641 T^{16} \))
$71$ (\( 1 - 126 T^{2} + 5041 T^{4} \))(\( 1 + 112 T^{2} + 6414 T^{4} + 564592 T^{6} + 25411681 T^{8} \))(\( 1 - 250 T^{2} + 27065 T^{4} - 1845414 T^{6} + 117329020 T^{8} - 9302731974 T^{10} + 687767146265 T^{12} - 32025070980250 T^{14} + 645753531245761 T^{16} \))
$73$ (\( 1 - 130 T^{2} + 5329 T^{4} \))(\( 1 - 25 T^{2} + 9633 T^{4} - 133225 T^{6} + 28398241 T^{8} \))(\( 1 - 265 T^{2} + 44059 T^{4} - 4880443 T^{6} + 414776080 T^{8} - 26007880747 T^{10} + 1251198100219 T^{12} - 40103569966585 T^{14} + 806460091894081 T^{16} \))
$79$ (\( ( 1 + 8 T + 79 T^{2} )^{2} \))(\( ( 1 + 8 T + 79 T^{2} )^{4} \))(\( ( 1 - 20 T + 314 T^{2} - 3732 T^{3} + 39586 T^{4} - 294828 T^{5} + 1959674 T^{6} - 9860780 T^{7} + 38950081 T^{8} )^{2} \))
$83$ (\( 1 - 22 T^{2} + 6889 T^{4} \))(\( 1 - 197 T^{2} + 19653 T^{4} - 1357133 T^{6} + 47458321 T^{8} \))(\( 1 - 457 T^{2} + 101207 T^{4} - 14199027 T^{6} + 1391424476 T^{8} - 97817097003 T^{10} + 4803114293447 T^{12} - 149411750629633 T^{14} + 2252292232139041 T^{16} \))
$89$ (\( ( 1 - 10 T + 89 T^{2} )( 1 + 10 T + 89 T^{2} ) \))(\( ( 1 - 142 T^{2} + 7921 T^{4} )^{2} \))(\( 1 - 266 T^{2} + 29505 T^{4} - 1036110 T^{6} - 9499940 T^{8} - 8207027310 T^{10} + 1851209820705 T^{12} - 132197023395626 T^{14} + 3936588805702081 T^{16} \))
$97$ (\( ( 1 - 18 T + 97 T^{2} )( 1 + 18 T + 97 T^{2} ) \))(\( 1 - 328 T^{2} + 44958 T^{4} - 3086152 T^{6} + 88529281 T^{8} \))(\( 1 - 474 T^{2} + 115649 T^{4} - 18494574 T^{6} + 2107503900 T^{8} - 174015446766 T^{10} + 10238322818369 T^{12} - 394828730336346 T^{14} + 7837433594376961 T^{16} \))
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