# 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

# Related objects

## 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

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