# Properties

 Label 21.4.a Level 21 Weight 4 Character orbit a Rep. character $$\chi_{21}(1,\cdot)$$ Character field $$\Q$$ Dimension 4 Newform subspaces 3 Sturm bound 10 Trace bound 2

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$21 = 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 21.a (trivial) Character field: $$\Q$$ Newform subspaces: $$3$$ Sturm bound: $$10$$ Trace bound: $$2$$ Distinguishing $$T_p$$: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_0(21))$$.

Total New Old
Modular forms 10 4 6
Cusp forms 6 4 2
Eisenstein series 4 0 4

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

$$3$$$$7$$FrickeDim.
$$+$$$$+$$$$+$$$$1$$
$$+$$$$-$$$$-$$$$1$$
$$-$$$$-$$$$+$$$$2$$
Plus space$$+$$$$3$$
Minus space$$-$$$$1$$

## Trace form

 $$4q - 2q^{2} + 26q^{4} - 16q^{5} - 12q^{6} + 14q^{7} - 66q^{8} + 36q^{9} + O(q^{10})$$ $$4q - 2q^{2} + 26q^{4} - 16q^{5} - 12q^{6} + 14q^{7} - 66q^{8} + 36q^{9} - 28q^{10} + 20q^{11} + 24q^{12} - 80q^{13} - 70q^{14} + 84q^{15} + 2q^{16} + 120q^{17} - 18q^{18} + 40q^{19} + 172q^{20} + 42q^{21} + 80q^{22} - 36q^{23} - 324q^{24} - 28q^{25} + 172q^{26} + 70q^{28} - 160q^{29} - 312q^{30} - 168q^{31} - 490q^{32} - 96q^{33} + 276q^{34} - 56q^{35} + 234q^{36} - 96q^{37} + 1360q^{38} + 336q^{39} - 924q^{40} - 1016q^{41} + 84q^{42} + 176q^{43} + 1264q^{44} - 144q^{45} + 792q^{46} + 552q^{47} + 816q^{48} + 196q^{49} - 1198q^{50} - 396q^{51} - 1420q^{52} - 1344q^{53} - 108q^{54} + 952q^{55} - 462q^{56} + 264q^{57} - 652q^{58} + 792q^{59} + 816q^{60} + 592q^{61} - 1824q^{62} + 126q^{63} + 1210q^{64} + 224q^{65} - 1896q^{66} + 1144q^{67} + 492q^{68} + 144q^{69} + 28q^{70} - 444q^{71} - 594q^{72} - 72q^{73} - 780q^{74} - 624q^{75} - 832q^{76} - 728q^{77} + 1392q^{78} - 856q^{79} + 3484q^{80} + 324q^{81} + 1492q^{82} + 2208q^{83} + 504q^{84} - 1224q^{85} + 1880q^{86} - 1032q^{87} - 1920q^{88} + 920q^{89} - 252q^{90} + 308q^{91} - 3192q^{92} + 744q^{93} - 1344q^{94} + 656q^{95} - 204q^{96} - 744q^{97} - 98q^{98} + 180q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_0(21))$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces A-L signs $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$ 3 7
21.4.a.a $$1$$ $$1.239$$ $$\Q$$ None $$-3$$ $$-3$$ $$-18$$ $$7$$ $$+$$ $$-$$ $$q-3q^{2}-3q^{3}+q^{4}-18q^{5}+9q^{6}+\cdots$$
21.4.a.b $$1$$ $$1.239$$ $$\Q$$ None $$4$$ $$-3$$ $$-4$$ $$-7$$ $$+$$ $$+$$ $$q+4q^{2}-3q^{3}+8q^{4}-4q^{5}-12q^{6}+\cdots$$
21.4.a.c $$2$$ $$1.239$$ $$\Q(\sqrt{57})$$ None $$-3$$ $$6$$ $$6$$ $$14$$ $$-$$ $$-$$ $$q+(-1-\beta )q^{2}+3q^{3}+(7+3\beta )q^{4}+\cdots$$

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_0(21))$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(\Gamma_0(21)) \cong$$ $$S_{4}^{\mathrm{new}}(\Gamma_0(7))$$$$^{\oplus 2}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ ($$1 + 3 T + 8 T^{2}$$)($$1 - 4 T + 8 T^{2}$$)($$1 + 3 T + 4 T^{2} + 24 T^{3} + 64 T^{4}$$)
$3$ ($$1 + 3 T$$)($$1 + 3 T$$)($$( 1 - 3 T )^{2}$$)
$5$ ($$1 + 18 T + 125 T^{2}$$)($$1 + 4 T + 125 T^{2}$$)($$1 - 6 T + 202 T^{2} - 750 T^{3} + 15625 T^{4}$$)
$7$ ($$1 - 7 T$$)($$1 + 7 T$$)($$( 1 - 7 T )^{2}$$)
$11$ ($$1 + 36 T + 1331 T^{2}$$)($$1 - 62 T + 1331 T^{2}$$)($$1 + 6 T + 1246 T^{2} + 7986 T^{3} + 1771561 T^{4}$$)
$13$ ($$1 + 34 T + 2197 T^{2}$$)($$1 + 62 T + 2197 T^{2}$$)($$1 - 16 T + 2406 T^{2} - 35152 T^{3} + 4826809 T^{4}$$)
$17$ ($$1 - 42 T + 4913 T^{2}$$)($$1 - 84 T + 4913 T^{2}$$)($$1 + 6 T + 9778 T^{2} + 29478 T^{3} + 24137569 T^{4}$$)
$19$ ($$1 + 124 T + 6859 T^{2}$$)($$1 - 100 T + 6859 T^{2}$$)($$1 - 64 T + 6534 T^{2} - 438976 T^{3} + 47045881 T^{4}$$)
$23$ ($$1 + 12167 T^{2}$$)($$1 + 42 T + 12167 T^{2}$$)($$1 - 6 T + 7870 T^{2} - 73002 T^{3} + 148035889 T^{4}$$)
$29$ ($$1 - 102 T + 24389 T^{2}$$)($$1 + 10 T + 24389 T^{2}$$)($$1 + 252 T + 56446 T^{2} + 6146028 T^{3} + 594823321 T^{4}$$)
$31$ ($$1 + 160 T + 29791 T^{2}$$)($$1 + 48 T + 29791 T^{2}$$)($$1 - 40 T - 13890 T^{2} - 1191640 T^{3} + 887503681 T^{4}$$)
$37$ ($$1 - 398 T + 50653 T^{2}$$)($$1 + 246 T + 50653 T^{2}$$)($$1 + 248 T + 98214 T^{2} + 12561944 T^{3} + 2565726409 T^{4}$$)
$41$ ($$1 + 318 T + 68921 T^{2}$$)($$1 + 248 T + 68921 T^{2}$$)($$1 + 450 T + 175642 T^{2} + 31014450 T^{3} + 4750104241 T^{4}$$)
$43$ ($$1 + 268 T + 79507 T^{2}$$)($$1 - 68 T + 79507 T^{2}$$)($$1 - 376 T + 161526 T^{2} - 29894632 T^{3} + 6321363049 T^{4}$$)
$47$ ($$1 - 240 T + 103823 T^{2}$$)($$1 - 324 T + 103823 T^{2}$$)($$1 + 12 T + 141790 T^{2} + 1245876 T^{3} + 10779215329 T^{4}$$)
$53$ ($$1 + 498 T + 148877 T^{2}$$)($$1 - 258 T + 148877 T^{2}$$)($$1 + 1104 T + 602230 T^{2} + 164360208 T^{3} + 22164361129 T^{4}$$)
$59$ ($$1 + 132 T + 205379 T^{2}$$)($$1 - 120 T + 205379 T^{2}$$)($$1 - 804 T + 380614 T^{2} - 165124716 T^{3} + 42180533641 T^{4}$$)
$61$ ($$1 - 398 T + 226981 T^{2}$$)($$1 - 622 T + 226981 T^{2}$$)($$1 + 428 T + 425886 T^{2} + 97147868 T^{3} + 51520374361 T^{4}$$)
$67$ ($$1 - 92 T + 300763 T^{2}$$)($$1 - 904 T + 300763 T^{2}$$)($$1 - 148 T + 440790 T^{2} - 44512924 T^{3} + 90458382169 T^{4}$$)
$71$ ($$1 + 720 T + 357911 T^{2}$$)($$1 + 678 T + 357911 T^{2}$$)($$1 - 954 T + 930526 T^{2} - 341447094 T^{3} + 128100283921 T^{4}$$)
$73$ ($$1 + 502 T + 389017 T^{2}$$)($$1 + 642 T + 389017 T^{2}$$)($$1 - 1072 T + 1063278 T^{2} - 417026224 T^{3} + 151334226289 T^{4}$$)
$79$ ($$1 + 1024 T + 493039 T^{2}$$)($$1 - 740 T + 493039 T^{2}$$)($$1 + 572 T + 901662 T^{2} + 282018308 T^{3} + 243087455521 T^{4}$$)
$83$ ($$1 + 204 T + 571787 T^{2}$$)($$1 - 468 T + 571787 T^{2}$$)($$1 - 1944 T + 1957030 T^{2} - 1111553928 T^{3} + 326940373369 T^{4}$$)
$89$ ($$1 - 354 T + 704969 T^{2}$$)($$1 - 200 T + 704969 T^{2}$$)($$1 - 366 T + 1156090 T^{2} - 258018654 T^{3} + 496981290961 T^{4}$$)
$97$ ($$1 + 286 T + 912673 T^{2}$$)($$1 + 1266 T + 912673 T^{2}$$)($$1 - 808 T + 903054 T^{2} - 737439784 T^{3} + 832972004929 T^{4}$$)