# Properties

 Label 210.2.n Level 210 Weight 2 Character orbit n Rep. character $$\chi_{210}(79,\cdot)$$ Character field $$\Q(\zeta_{6})$$ Dimension 16 Newform subspaces 2 Sturm bound 96 Trace bound 1

# Related objects

## Defining parameters

 Level: $$N$$ = $$210 = 2 \cdot 3 \cdot 5 \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 210.n (of order $$6$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ = $$35$$ Character field: $$\Q(\zeta_{6})$$ Newform subspaces: $$2$$ Sturm bound: $$96$$ Trace bound: $$1$$ Distinguishing $$T_p$$: $$11$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(210, [\chi])$$.

Total New Old
Modular forms 112 16 96
Cusp forms 80 16 64
Eisenstein series 32 0 32

## Trace form

 $$16q + 8q^{4} - 4q^{5} + 8q^{6} + 8q^{9} + O(q^{10})$$ $$16q + 8q^{4} - 4q^{5} + 8q^{6} + 8q^{9} - 2q^{10} + 4q^{11} - 4q^{14} + 4q^{15} - 8q^{16} + 20q^{19} - 8q^{20} + 8q^{21} + 4q^{24} - 6q^{25} - 4q^{26} - 48q^{29} + 4q^{30} + 12q^{31} - 32q^{34} - 20q^{35} + 16q^{36} + 8q^{39} + 2q^{40} - 72q^{41} - 4q^{44} + 4q^{45} - 12q^{46} + 20q^{49} + 16q^{50} - 8q^{51} + 4q^{54} + 20q^{55} - 8q^{56} - 16q^{59} + 2q^{60} - 8q^{61} - 16q^{64} - 32q^{65} - 16q^{66} - 48q^{69} - 26q^{70} + 64q^{71} + 28q^{74} - 8q^{75} + 40q^{76} + 20q^{79} - 4q^{80} - 8q^{81} + 16q^{84} + 8q^{85} + 32q^{86} + 8q^{89} - 4q^{90} + 8q^{91} + 20q^{94} + 28q^{95} - 4q^{96} + 8q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(210, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
210.2.n.a $$4$$ $$1.677$$ $$\Q(\zeta_{12})$$ None $$0$$ $$0$$ $$-4$$ $$0$$ $$q+\zeta_{12}q^{2}+(-\zeta_{12}+\zeta_{12}^{3})q^{3}+\zeta_{12}^{2}q^{4}+\cdots$$
210.2.n.b $$12$$ $$1.677$$ 12.0.$$\cdots$$.1 None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{2}q^{2}+(\beta _{2}-\beta _{8})q^{3}+(1+\beta _{10})q^{4}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(210, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(210, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(35, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(70, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(105, [\chi])$$$$^{\oplus 2}$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ ($$1 - T^{2} + T^{4}$$)($$( 1 - T^{2} + T^{4} )^{3}$$)
$3$ ($$1 - T^{2} + T^{4}$$)($$( 1 - T^{2} + T^{4} )^{3}$$)
$5$ ($$1 + 4 T + 11 T^{2} + 20 T^{3} + 25 T^{4}$$)($$( 1 - 20 T^{3} + 125 T^{6} )^{2}$$)
$7$ ($$1 + 2 T^{2} + 49 T^{4}$$)($$1 - 12 T^{2} + 120 T^{4} - 790 T^{6} + 5880 T^{8} - 28812 T^{10} + 117649 T^{12}$$)
$11$ ($$( 1 - 5 T + 14 T^{2} - 55 T^{3} + 121 T^{4} )^{2}$$)($$( 1 + 3 T + 3 T^{2} - 16 T^{3} - 81 T^{4} + 69 T^{5} + 1466 T^{6} + 759 T^{7} - 9801 T^{8} - 21296 T^{9} + 43923 T^{10} + 483153 T^{11} + 1771561 T^{12} )^{2}$$)
$13$ ($$( 1 - 25 T^{2} + 169 T^{4} )^{2}$$)($$( 1 - 45 T^{2} + 1107 T^{4} - 17390 T^{6} + 187083 T^{8} - 1285245 T^{10} + 4826809 T^{12} )^{2}$$)
$17$ ($$( 1 - 8 T + 47 T^{2} - 136 T^{3} + 289 T^{4} )( 1 + 8 T + 47 T^{2} + 136 T^{3} + 289 T^{4} )$$)($$1 - 30 T^{2} - 27 T^{4} + 2790 T^{6} + 132426 T^{8} - 614550 T^{10} - 41149843 T^{12} - 177604950 T^{14} + 11060351946 T^{16} + 67343817510 T^{18} - 188345450907 T^{20} - 60479817013470 T^{22} + 582622237229761 T^{24}$$)
$19$ ($$( 1 - 8 T + 19 T^{2} )^{2}( 1 + T + 19 T^{2} )^{2}$$)($$( 1 - 3 T - 36 T^{2} + 45 T^{3} + 900 T^{4} - 3 T^{5} - 19906 T^{6} - 57 T^{7} + 324900 T^{8} + 308655 T^{9} - 4691556 T^{10} - 7428297 T^{11} + 47045881 T^{12} )^{2}$$)
$23$ ($$1 + 37 T^{2} + 840 T^{4} + 19573 T^{6} + 279841 T^{8}$$)($$1 + 81 T^{2} + 3582 T^{4} + 90087 T^{6} + 1174464 T^{8} - 7396947 T^{10} - 518277076 T^{12} - 3912984963 T^{14} + 328663180224 T^{16} + 13336109132343 T^{18} + 280509949276542 T^{20} + 3355547408305569 T^{22} + 21914624432020321 T^{24}$$)
$29$ ($$( 1 + 29 T^{2} )^{4}$$)($$( 1 + 12 T + 120 T^{2} + 720 T^{3} + 3480 T^{4} + 10092 T^{5} + 24389 T^{6} )^{4}$$)
$31$ ($$( 1 - 6 T + 5 T^{2} - 186 T^{3} + 961 T^{4} )^{2}$$)($$( 1 - 78 T^{2} - 20 T^{3} + 3666 T^{4} + 780 T^{5} - 128922 T^{6} + 24180 T^{7} + 3523026 T^{8} - 595820 T^{9} - 72034638 T^{10} + 887503681 T^{12} )^{2}$$)
$37$ ($$1 + 49 T^{2} + 1032 T^{4} + 67081 T^{6} + 1874161 T^{8}$$)($$1 + 165 T^{2} + 14838 T^{4} + 911635 T^{6} + 43071816 T^{8} + 1725566145 T^{10} + 64531645932 T^{12} + 2362300052505 T^{14} + 80723517746376 T^{16} + 2339005994868715 T^{18} + 52118170137279798 T^{20} + 793416421448945085 T^{22} + 6582952005840035281 T^{24}$$)
$41$ ($$( 1 + 9 T + 41 T^{2} )^{4}$$)($$( 1 + 9 T + 75 T^{2} + 610 T^{3} + 3075 T^{4} + 15129 T^{5} + 68921 T^{6} )^{4}$$)
$43$ ($$( 1 + 14 T^{2} + 1849 T^{4} )^{2}$$)($$( 1 - 82 T^{2} + 1849 T^{4} )^{6}$$)
$47$ ($$1 - 75 T^{2} + 3416 T^{4} - 165675 T^{6} + 4879681 T^{8}$$)($$1 + 9 T^{2} - 1578 T^{4} - 306297 T^{6} - 2357976 T^{8} + 244410597 T^{10} + 40880244764 T^{12} + 539903008773 T^{14} - 11506170685656 T^{16} - 3301641317626713 T^{18} - 37574210352258858 T^{20} + 473392190122470441 T^{22} +$$$$11\!\cdots\!41$$$$T^{24}$$)
$53$ ($$1 + 105 T^{2} + 8216 T^{4} + 294945 T^{6} + 7890481 T^{8}$$)($$1 + 51 T^{2} - 6213 T^{4} - 119588 T^{6} + 38415969 T^{8} + 385077393 T^{10} - 113446795386 T^{12} + 1081682396937 T^{14} + 303120473491089 T^{16} - 2650591618694852 T^{18} - 386819456525785893 T^{20} + 8919260988641165499 T^{22} +$$$$49\!\cdots\!41$$$$T^{24}$$)
$59$ ($$( 1 - 4 T - 43 T^{2} - 236 T^{3} + 3481 T^{4} )^{2}$$)($$( 1 + 12 T - 6 T^{2} - 160 T^{3} + 1890 T^{4} - 23628 T^{5} - 457606 T^{6} - 1394052 T^{7} + 6579090 T^{8} - 32860640 T^{9} - 72704166 T^{10} + 8579091588 T^{11} + 42180533641 T^{12} )^{2}$$)
$61$ ($$( 1 - 2 T - 57 T^{2} - 122 T^{3} + 3721 T^{4} )^{2}$$)($$( 1 + 6 T - 39 T^{2} + 410 T^{3} + 930 T^{4} - 31074 T^{5} - 22719 T^{6} - 1895514 T^{7} + 3460530 T^{8} + 93062210 T^{9} - 539987799 T^{10} + 5067577806 T^{11} + 51520374361 T^{12} )^{2}$$)
$67$ ($$1 + 98 T^{2} + 5115 T^{4} + 439922 T^{6} + 20151121 T^{8}$$)($$1 + 150 T^{2} + 6333 T^{4} - 110030 T^{6} - 1332774 T^{8} + 1742888430 T^{10} + 160875783717 T^{12} + 7823826162270 T^{14} - 26856890139654 T^{16} - 9953135790055070 T^{18} + 2571626601966207453 T^{20} +$$$$27\!\cdots\!50$$$$T^{22} +$$$$81\!\cdots\!61$$$$T^{24}$$)
$71$ ($$( 1 + 2 T + 71 T^{2} )^{4}$$)($$( 1 - 6 T + 71 T^{2} )^{12}$$)
$73$ ($$1 + 130 T^{2} + 11571 T^{4} + 692770 T^{6} + 28398241 T^{8}$$)($$( 1 + 130 T^{2} + 11571 T^{4} + 692770 T^{6} + 28398241 T^{8} )^{3}$$)
$79$ ($$( 1 + 14 T + 117 T^{2} + 1106 T^{3} + 6241 T^{4} )^{2}$$)($$( 1 - 24 T + 162 T^{2} - 1548 T^{3} + 38034 T^{4} - 316212 T^{5} + 1440614 T^{6} - 24980748 T^{7} + 237370194 T^{8} - 763224372 T^{9} + 6309913122 T^{10} - 73849353576 T^{11} + 243087455521 T^{12} )^{2}$$)
$83$ ($$( 1 - 66 T^{2} + 6889 T^{4} )^{2}$$)($$( 1 - 300 T^{2} + 40392 T^{4} - 3707850 T^{6} + 278260488 T^{8} - 14237496300 T^{10} + 326940373369 T^{12} )^{2}$$)
$89$ ($$( 1 - 10 T + 11 T^{2} - 890 T^{3} + 7921 T^{4} )^{2}$$)($$( 1 + 6 T - 123 T^{2} - 398 T^{3} + 7074 T^{4} - 14802 T^{5} - 668251 T^{6} - 1317378 T^{7} + 56033154 T^{8} - 280577662 T^{9} - 7717295643 T^{10} + 33504356694 T^{11} + 496981290961 T^{12} )^{2}$$)
$97$ ($$( 1 - 18 T + 97 T^{2} )^{2}( 1 + 18 T + 97 T^{2} )^{2}$$)($$( 1 - 240 T^{2} + 16752 T^{4} - 546370 T^{6} + 157619568 T^{8} - 21247027440 T^{10} + 832972004929 T^{12} )^{2}$$)