# Properties

 Label 2005.2 Level 2005 Weight 2 Dimension 146199 Nonzero newspaces 30 Sturm bound 643200 Trace bound 22

## Defining parameters

 Level: $$N$$ = $$2005 = 5 \cdot 401$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$30$$ Sturm bound: $$643200$$ Trace bound: $$22$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(2005))$$.

Total New Old
Modular forms 162400 148595 13805
Cusp forms 159201 146199 13002
Eisenstein series 3199 2396 803

## Trace form

 $$146199q - 403q^{2} - 404q^{3} - 407q^{4} - 601q^{5} - 1212q^{6} - 408q^{7} - 415q^{8} - 413q^{9} + O(q^{10})$$ $$146199q - 403q^{2} - 404q^{3} - 407q^{4} - 601q^{5} - 1212q^{6} - 408q^{7} - 415q^{8} - 413q^{9} - 603q^{10} - 1212q^{11} - 428q^{12} - 414q^{13} - 424q^{14} - 604q^{15} - 1231q^{16} - 418q^{17} - 439q^{18} - 420q^{19} - 607q^{20} - 1232q^{21} - 436q^{22} - 424q^{23} - 460q^{24} - 601q^{25} - 1242q^{26} - 440q^{27} - 456q^{28} - 430q^{29} - 612q^{30} - 1232q^{31} - 463q^{32} - 448q^{33} - 454q^{34} - 608q^{35} - 1291q^{36} - 438q^{37} - 460q^{38} - 456q^{39} - 615q^{40} - 1242q^{41} - 496q^{42} - 444q^{43} - 484q^{44} - 613q^{45} - 1272q^{46} - 448q^{47} - 524q^{48} - 457q^{49} - 603q^{50} - 1272q^{51} - 498q^{52} - 454q^{53} - 520q^{54} - 612q^{55} - 1320q^{56} - 480q^{57} - 490q^{58} - 460q^{59} - 628q^{60} - 1262q^{61} - 496q^{62} - 504q^{63} - 527q^{64} - 614q^{65} - 1344q^{66} - 468q^{67} - 526q^{68} - 496q^{69} - 624q^{70} - 1272q^{71} - 595q^{72} - 474q^{73} - 514q^{74} - 604q^{75} - 1340q^{76} - 496q^{77} - 568q^{78} - 480q^{79} - 631q^{80} - 1321q^{81} - 526q^{82} - 484q^{83} - 624q^{84} - 618q^{85} - 1332q^{86} - 520q^{87} - 580q^{88} - 490q^{89} - 639q^{90} - 1312q^{91} - 568q^{92} - 528q^{93} - 544q^{94} - 620q^{95} - 1452q^{96} - 498q^{97} - 571q^{98} - 556q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(2005))$$

We only show spaces with even parity, since no modular forms exist when this condition is not satisfied. Within each space $$S_k^{\mathrm{new}}(N, \chi)$$ we list the newforms together with their dimension.

Label $$\chi$$ Newforms Dimension $$\chi$$ degree
2005.2.a $$\chi_{2005}(1, \cdot)$$ 2005.2.a.a 1 1
2005.2.a.b 1
2005.2.a.c 3
2005.2.a.d 25
2005.2.a.e 29
2005.2.a.f 37
2005.2.a.g 37
2005.2.b $$\chi_{2005}(1204, \cdot)$$ n/a 200 1
2005.2.c $$\chi_{2005}(2004, \cdot)$$ n/a 200 1
2005.2.d $$\chi_{2005}(801, \cdot)$$ n/a 134 1
2005.2.e $$\chi_{2005}(381, \cdot)$$ n/a 268 2
2005.2.j $$\chi_{2005}(1584, \cdot)$$ n/a 400 2
2005.2.k $$\chi_{2005}(841, \cdot)$$ n/a 536 4
2005.2.m $$\chi_{2005}(356, \cdot)$$ n/a 536 4
2005.2.n $$\chi_{2005}(499, \cdot)$$ n/a 792 4
2005.2.p $$\chi_{2005}(831, \cdot)$$ n/a 536 4
2005.2.q $$\chi_{2005}(29, \cdot)$$ n/a 800 4
2005.2.r $$\chi_{2005}(39, \cdot)$$ n/a 800 4
2005.2.u $$\chi_{2005}(147, \cdot)$$ n/a 1592 8
2005.2.v $$\chi_{2005}(133, \cdot)$$ n/a 1592 8
2005.2.w $$\chi_{2005}(164, \cdot)$$ n/a 1600 8
2005.2.bb $$\chi_{2005}(56, \cdot)$$ n/a 1072 8
2005.2.bc $$\chi_{2005}(51, \cdot)$$ n/a 2680 20
2005.2.be $$\chi_{2005}(114, \cdot)$$ n/a 3168 16
2005.2.bf $$\chi_{2005}(126, \cdot)$$ n/a 2144 16
2005.2.bh $$\chi_{2005}(16, \cdot)$$ n/a 2680 20
2005.2.bi $$\chi_{2005}(224, \cdot)$$ n/a 4000 20
2005.2.bj $$\chi_{2005}(14, \cdot)$$ n/a 4000 20
2005.2.bk $$\chi_{2005}(48, \cdot)$$ n/a 6368 32
2005.2.bl $$\chi_{2005}(33, \cdot)$$ n/a 6368 32
2005.2.bo $$\chi_{2005}(81, \cdot)$$ n/a 5360 40
2005.2.bt $$\chi_{2005}(4, \cdot)$$ n/a 8000 40
2005.2.bu $$\chi_{2005}(9, \cdot)$$ n/a 15840 80
2005.2.bx $$\chi_{2005}(11, \cdot)$$ n/a 10720 80
2005.2.by $$\chi_{2005}(12, \cdot)$$ n/a 31840 160
2005.2.cb $$\chi_{2005}(3, \cdot)$$ n/a 31840 160

"n/a" means that newforms for that character have not been added to the database yet

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(2005))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(2005)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(401))$$$$^{\oplus 2}$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ ($$1 + T + 2 T^{2}$$)($$1 + T + 2 T^{2}$$)($$1 + 3 T + 6 T^{2} + 9 T^{3} + 12 T^{4} + 12 T^{5} + 8 T^{6}$$)
$3$ ($$1 + 3 T^{2}$$)($$1 + 3 T^{2}$$)($$1 + 3 T + 3 T^{2} + T^{3} + 9 T^{4} + 27 T^{5} + 27 T^{6}$$)
$5$ ($$1 - T$$)($$1 - T$$)($$( 1 - T )^{3}$$)
$7$ ($$1 + 7 T^{2}$$)($$1 + 7 T^{2}$$)($$1 + 3 T + 3 T^{2} - 15 T^{3} + 21 T^{4} + 147 T^{5} + 343 T^{6}$$)
$11$ ($$1 + 4 T + 11 T^{2}$$)($$1 - 4 T + 11 T^{2}$$)($$1 + 24 T^{2} + 9 T^{3} + 264 T^{4} + 1331 T^{6}$$)
$13$ ($$1 - 4 T + 13 T^{2}$$)($$1 + 2 T + 13 T^{2}$$)($$1 + 3 T + 21 T^{2} + 95 T^{3} + 273 T^{4} + 507 T^{5} + 2197 T^{6}$$)
$17$ ($$1 - 4 T + 17 T^{2}$$)($$1 + 6 T + 17 T^{2}$$)($$1 + 6 T + 42 T^{2} + 133 T^{3} + 714 T^{4} + 1734 T^{5} + 4913 T^{6}$$)
$19$ ($$1 + 6 T + 19 T^{2}$$)($$1 + 19 T^{2}$$)($$( 1 - 3 T + 19 T^{2} )^{3}$$)
$23$ ($$1 - 8 T + 23 T^{2}$$)($$1 - 4 T + 23 T^{2}$$)($$1 + 9 T + 87 T^{2} + 423 T^{3} + 2001 T^{4} + 4761 T^{5} + 12167 T^{6}$$)
$29$ ($$1 + 6 T + 29 T^{2}$$)($$1 + 2 T + 29 T^{2}$$)($$1 + 48 T^{2} - 19 T^{3} + 1392 T^{4} + 24389 T^{6}$$)
$31$ ($$1 + 2 T + 31 T^{2}$$)($$1 - 4 T + 31 T^{2}$$)($$1 - 6 T + 84 T^{2} - 301 T^{3} + 2604 T^{4} - 5766 T^{5} + 29791 T^{6}$$)
$37$ ($$1 - 4 T + 37 T^{2}$$)($$1 + 10 T + 37 T^{2}$$)($$1 - 15 T + 159 T^{2} - 1073 T^{3} + 5883 T^{4} - 20535 T^{5} + 50653 T^{6}$$)
$41$ ($$1 - 2 T + 41 T^{2}$$)($$1 + 6 T + 41 T^{2}$$)($$1 + 18 T + 210 T^{2} + 1603 T^{3} + 8610 T^{4} + 30258 T^{5} + 68921 T^{6}$$)
$43$ ($$1 + 4 T + 43 T^{2}$$)($$1 - 4 T + 43 T^{2}$$)($$1 + 6 T + 114 T^{2} + 497 T^{3} + 4902 T^{4} + 11094 T^{5} + 79507 T^{6}$$)
$47$ ($$1 + 47 T^{2}$$)($$1 - 8 T + 47 T^{2}$$)($$1 + 84 T^{2} - 107 T^{3} + 3948 T^{4} + 103823 T^{6}$$)
$53$ ($$1 + 4 T + 53 T^{2}$$)($$1 - 6 T + 53 T^{2}$$)($$1 + 15 T + 231 T^{2} + 1699 T^{3} + 12243 T^{4} + 42135 T^{5} + 148877 T^{6}$$)
$59$ ($$1 + 6 T + 59 T^{2}$$)($$1 - 8 T + 59 T^{2}$$)($$1 + 6 T + 96 T^{2} + 511 T^{3} + 5664 T^{4} + 20886 T^{5} + 205379 T^{6}$$)
$61$ ($$1 - 14 T + 61 T^{2}$$)($$1 + 10 T + 61 T^{2}$$)($$1 + 3 T + 123 T^{2} + 313 T^{3} + 7503 T^{4} + 11163 T^{5} + 226981 T^{6}$$)
$67$ ($$1 + 12 T + 67 T^{2}$$)($$1 + 67 T^{2}$$)($$1 + 27 T + 408 T^{2} + 3951 T^{3} + 27336 T^{4} + 121203 T^{5} + 300763 T^{6}$$)
$71$ ($$1 + 10 T + 71 T^{2}$$)($$1 - 4 T + 71 T^{2}$$)($$1 + 96 T^{2} - 153 T^{3} + 6816 T^{4} + 357911 T^{6}$$)
$73$ ($$1 + 6 T + 73 T^{2}$$)($$1 + 6 T + 73 T^{2}$$)($$1 - 3 T + 183 T^{2} - 489 T^{3} + 13359 T^{4} - 15987 T^{5} + 389017 T^{6}$$)
$79$ ($$1 - 6 T + 79 T^{2}$$)($$1 + 12 T + 79 T^{2}$$)($$1 - 3 T + 12 T^{2} + 609 T^{3} + 948 T^{4} - 18723 T^{5} + 493039 T^{6}$$)
$83$ ($$1 - 4 T + 83 T^{2}$$)($$1 + 12 T + 83 T^{2}$$)($$1 + 39 T + 747 T^{2} + 8545 T^{3} + 62001 T^{4} + 268671 T^{5} + 571787 T^{6}$$)
$89$ ($$1 - 6 T + 89 T^{2}$$)($$1 - 10 T + 89 T^{2}$$)($$1 + 3 T + 153 T^{2} + 265 T^{3} + 13617 T^{4} + 23763 T^{5} + 704969 T^{6}$$)
$97$ ($$1 + 97 T^{2}$$)($$1 + 6 T + 97 T^{2}$$)($$1 + 12 T + 210 T^{2} + 2325 T^{3} + 20370 T^{4} + 112908 T^{5} + 912673 T^{6}$$)