# Properties

 Label 560.2.bj Level 560 Weight 2 Character orbit bj Rep. character $$\chi_{560}(97,\cdot)$$ Character field $$\Q(\zeta_{4})$$ Dimension 44 Newform subspaces 4 Sturm bound 192 Trace bound 7

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$560 = 2^{4} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 560.bj (of order $$4$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$35$$ Character field: $$\Q(i)$$ Newform subspaces: $$4$$ Sturm bound: $$192$$ Trace bound: $$7$$ Distinguishing $$T_p$$: $$3$$

## Dimensions

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

Total New Old
Modular forms 216 52 164
Cusp forms 168 44 124
Eisenstein series 48 8 40

## Trace form

 $$44q + 2q^{7} + O(q^{10})$$ $$44q + 2q^{7} + 8q^{11} + 4q^{15} + 4q^{21} + 28q^{23} - 4q^{25} - 10q^{35} - 4q^{37} + 4q^{43} + 32q^{51} - 4q^{53} - 32q^{57} - 54q^{63} - 20q^{65} - 20q^{67} + 64q^{71} - 16q^{77} - 4q^{81} - 4q^{85} + 36q^{91} + 24q^{93} + 24q^{95} + O(q^{100})$$

## 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.bj.a $$4$$ $$4.472$$ $$\Q(i, \sqrt{10})$$ None $$0$$ $$0$$ $$0$$ $$-4$$ $$q+\beta _{1}q^{3}+\beta _{1}q^{5}+(-1-\beta _{2}+\beta _{3})q^{7}+\cdots$$
560.2.bj.b $$8$$ $$4.472$$ 8.0.$$\cdots$$.3 None $$0$$ $$0$$ $$0$$ $$2$$ $$q+\beta _{1}q^{3}+(\beta _{2}-\beta _{6})q^{5}+(-\beta _{4}+\beta _{6}+\cdots)q^{7}+\cdots$$
560.2.bj.c $$8$$ $$4.472$$ $$\Q(\zeta_{16})$$ None $$0$$ $$0$$ $$0$$ $$8$$ $$q+(\zeta_{16}+\zeta_{16}^{3})q^{3}+(-\zeta_{16}-2\zeta_{16}^{5}+\cdots)q^{5}+\cdots$$
560.2.bj.d $$24$$ $$4.472$$ None $$0$$ $$0$$ $$0$$ $$-4$$

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ ($$1 - 17 T^{4} + 81 T^{8}$$)($$1 - 3 T^{4} - 92 T^{8} - 243 T^{12} + 6561 T^{16}$$)($$( 1 - 8 T^{2} + 32 T^{4} - 72 T^{6} + 81 T^{8} )( 1 + 8 T^{2} + 32 T^{4} + 72 T^{6} + 81 T^{8} )$$)
$5$ ($$1 + 25 T^{4}$$)($$1 + 6 T^{2} + 18 T^{4} + 150 T^{6} + 625 T^{8}$$)($$1 - 48 T^{4} + 625 T^{8}$$)
$7$ ($$1 + 4 T + 8 T^{2} + 28 T^{3} + 49 T^{4}$$)($$1 - 2 T + 2 T^{2} + 26 T^{3} - 62 T^{4} + 182 T^{5} + 98 T^{6} - 686 T^{7} + 2401 T^{8}$$)($$1 - 8 T + 32 T^{2} - 88 T^{3} + 226 T^{4} - 616 T^{5} + 1568 T^{6} - 2744 T^{7} + 2401 T^{8}$$)
$11$ ($$( 1 - T + 11 T^{2} )^{4}$$)($$( 1 - 3 T + 14 T^{2} - 33 T^{3} + 121 T^{4} )^{4}$$)($$( 1 + 14 T^{2} + 121 T^{4} )^{4}$$)
$13$ ($$1 + 103 T^{4} + 28561 T^{8}$$)($$1 + 357 T^{4} + 68228 T^{8} + 10196277 T^{12} + 815730721 T^{16}$$)($$( 1 - 240 T^{4} + 28561 T^{8} )^{2}$$)
$17$ ($$1 + 263 T^{4} + 83521 T^{8}$$)($$1 + 69 T^{4} - 53260 T^{8} + 5762949 T^{12} + 6975757441 T^{16}$$)($$1 - 252 T^{4} + 17030 T^{8} - 21047292 T^{12} + 6975757441 T^{16}$$)
$19$ ($$( 1 + 28 T^{2} + 361 T^{4} )^{2}$$)($$( 1 + 18 T^{2} + 762 T^{4} + 6498 T^{6} + 130321 T^{8} )^{2}$$)($$( 1 + 24 T^{2} + 288 T^{4} + 8664 T^{6} + 130321 T^{8} )^{2}$$)
$23$ ($$( 1 + 4 T + 8 T^{2} + 92 T^{3} + 529 T^{4} )^{2}$$)($$( 1 + 2 T + 2 T^{2} + 6 T^{3} - 382 T^{4} + 138 T^{5} + 1058 T^{6} + 24334 T^{7} + 279841 T^{8} )^{2}$$)($$( 1 - 4 T + 8 T^{2} - 84 T^{3} + 878 T^{4} - 1932 T^{5} + 4232 T^{6} - 48668 T^{7} + 279841 T^{8} )^{2}$$)
$29$ ($$( 1 - 49 T^{2} + 841 T^{4} )^{2}$$)($$( 1 - 83 T^{2} + 3148 T^{4} - 69803 T^{6} + 707281 T^{8} )^{2}$$)($$( 1 - 92 T^{2} + 3670 T^{4} - 77372 T^{6} + 707281 T^{8} )^{2}$$)
$31$ ($$( 1 - 52 T^{2} + 961 T^{4} )^{2}$$)($$( 1 + 18 T^{2} + 1962 T^{4} + 17298 T^{6} + 923521 T^{8} )^{2}$$)($$( 1 - 108 T^{2} + 4806 T^{4} - 103788 T^{6} + 923521 T^{8} )^{2}$$)
$37$ ($$( 1 + 12 T + 72 T^{2} + 444 T^{3} + 1369 T^{4} )^{2}$$)($$( 1 + 2 T + 2 T^{2} + 34 T^{3} + 178 T^{4} + 1258 T^{5} + 2738 T^{6} + 101306 T^{7} + 1874161 T^{8} )^{2}$$)($$( 1 - 16 T + 128 T^{2} - 1040 T^{3} + 7666 T^{4} - 38480 T^{5} + 175232 T^{6} - 810448 T^{7} + 1874161 T^{8} )^{2}$$)
$41$ ($$( 1 + 8 T^{2} + 1681 T^{4} )^{2}$$)($$( 1 - 106 T^{2} + 6130 T^{4} - 178186 T^{6} + 2825761 T^{8} )^{2}$$)($$( 1 - 148 T^{2} + 8806 T^{4} - 248788 T^{6} + 2825761 T^{8} )^{2}$$)
$43$ ($$( 1 - 6 T + 18 T^{2} - 258 T^{3} + 1849 T^{4} )^{2}$$)($$( 1 - 10 T + 50 T^{2} - 430 T^{3} + 1849 T^{4} )^{4}$$)($$( 1 + 8 T + 32 T^{2} + 344 T^{3} + 1849 T^{4} )^{4}$$)
$47$ ($$1 - 2017 T^{4} + 4879681 T^{8}$$)($$1 - 2227 T^{4} + 6943924 T^{8} - 10867049587 T^{12} + 23811286661761 T^{16}$$)($$1 + 6020 T^{4} + 17872774 T^{8} + 29375679620 T^{12} + 23811286661761 T^{16}$$)
$53$ ($$( 1 - 2 T + 2 T^{2} - 106 T^{3} + 2809 T^{4} )^{2}$$)($$( 1 - 14 T + 53 T^{2} )^{4}( 1 + 4 T + 53 T^{2} )^{4}$$)($$( 1 + 16 T + 128 T^{2} + 784 T^{3} + 4786 T^{4} + 41552 T^{5} + 359552 T^{6} + 2382032 T^{7} + 7890481 T^{8} )^{2}$$)
$59$ ($$( 1 + 28 T^{2} + 3481 T^{4} )^{2}$$)($$( 1 + 178 T^{2} + 14842 T^{4} + 619618 T^{6} + 12117361 T^{8} )^{2}$$)($$( 1 + 136 T^{2} + 10336 T^{4} + 473416 T^{6} + 12117361 T^{8} )^{2}$$)
$61$ ($$( 1 - 82 T^{2} + 3721 T^{4} )^{2}$$)($$( 1 - 12 T^{2} + 6822 T^{4} - 44652 T^{6} + 13845841 T^{8} )^{2}$$)($$( 1 - 96 T^{2} + 8688 T^{4} - 357216 T^{6} + 13845841 T^{8} )^{2}$$)
$67$ ($$( 1 - 2 T + 2 T^{2} - 134 T^{3} + 4489 T^{4} )^{2}$$)($$( 1 + 10 T + 50 T^{2} + 670 T^{3} + 4489 T^{4} )^{4}$$)($$( 1 + 8 T + 32 T^{2} - 552 T^{3} - 8974 T^{4} - 36984 T^{5} + 143648 T^{6} + 2406104 T^{7} + 20151121 T^{8} )^{2}$$)
$71$ ($$( 1 - 6 T + 71 T^{2} )^{4}$$)($$( 1 + 2 T + 102 T^{2} + 142 T^{3} + 5041 T^{4} )^{4}$$)($$( 1 - 4 T + 144 T^{2} - 284 T^{3} + 5041 T^{4} )^{4}$$)
$73$ ($$( 1 + 5329 T^{4} )^{2}$$)($$1 + 14144 T^{4} + 103908670 T^{8} + 401664720704 T^{12} + 806460091894081 T^{16}$$)($$1 + 1220 T^{4} + 55730374 T^{8} + 34645854020 T^{12} + 806460091894081 T^{16}$$)
$79$ ($$( 1 + 11 T^{2} + 6241 T^{4} )^{2}$$)($$( 1 - 271 T^{2} + 30340 T^{4} - 1691311 T^{6} + 38950081 T^{8} )^{2}$$)($$( 1 - 208 T^{2} + 22498 T^{4} - 1298128 T^{6} + 38950081 T^{8} )^{2}$$)
$83$ ($$1 + 7538 T^{4} + 47458321 T^{8}$$)($$1 - 8400 T^{4} + 51732158 T^{8} - 398649896400 T^{12} + 2252292232139041 T^{16}$$)($$1 - 4224 T^{4} + 99283874 T^{8} - 200463947904 T^{12} + 2252292232139041 T^{16}$$)
$89$ ($$( 1 + 138 T^{2} + 7921 T^{4} )^{2}$$)($$( 1 + 168 T^{2} + 14862 T^{4} + 1330728 T^{6} + 62742241 T^{8} )^{2}$$)($$( 1 - 60 T^{2} + 10470 T^{4} - 475260 T^{6} + 62742241 T^{8} )^{2}$$)
$97$ ($$1 + 16903 T^{4} + 88529281 T^{8}$$)($$1 + 16837 T^{4} + 160234548 T^{8} + 1490567504197 T^{12} + 7837433594376961 T^{16}$$)($$1 - 11900 T^{4} + 108781062 T^{8} - 1053498443900 T^{12} + 7837433594376961 T^{16}$$)