# Properties

 Label 34.2 Level 34 Weight 2 Dimension 11 Nonzero newspaces 4 Newform subspaces 5 Sturm bound 144 Trace bound 3

## Defining parameters

 Level: $$N$$ = $$34 = 2 \cdot 17$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$4$$ Newform subspaces: $$5$$ Sturm bound: $$144$$ Trace bound: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(34))$$.

Total New Old
Modular forms 52 11 41
Cusp forms 21 11 10
Eisenstein series 31 0 31

## Trace form

 $$11q - q^{2} - 4q^{3} - q^{4} - 6q^{5} - 4q^{6} - 8q^{7} - q^{8} - 13q^{9} + O(q^{10})$$ $$11q - q^{2} - 4q^{3} - q^{4} - 6q^{5} - 4q^{6} - 8q^{7} - q^{8} - 13q^{9} - 2q^{10} + 4q^{11} + 4q^{12} + 2q^{13} + 8q^{14} + 24q^{15} + 3q^{16} - q^{17} + 19q^{18} - 4q^{19} - 2q^{20} + 16q^{21} + 4q^{22} - 8q^{23} + 4q^{24} + 5q^{25} - 10q^{26} - 16q^{27} - 8q^{28} - 10q^{29} - 24q^{30} - q^{32} - 16q^{33} - 17q^{34} - 16q^{35} - 13q^{36} - 6q^{37} - 12q^{38} + 8q^{39} - 6q^{40} + 26q^{41} + 16q^{42} + 12q^{43} + 20q^{44} + 6q^{45} + 8q^{46} + 32q^{47} - 4q^{48} + 7q^{49} + 25q^{50} - 4q^{51} + 18q^{52} + 14q^{53} + 16q^{54} + 8q^{55} - 8q^{56} + 8q^{57} + 2q^{58} + 20q^{59} + 8q^{60} - 14q^{61} + 16q^{62} - 8q^{63} - q^{64} - 48q^{65} - 40q^{66} - 52q^{67} - 13q^{68} - 32q^{69} - 16q^{70} - 24q^{71} + 7q^{72} + 10q^{73} - 2q^{74} - 12q^{75} - 20q^{76} - 16q^{77} + 8q^{78} + 10q^{80} - 17q^{81} + 26q^{82} - 12q^{83} + 30q^{85} - 12q^{86} + 24q^{87} + 20q^{88} + 6q^{89} - 10q^{90} + 48q^{91} - 8q^{92} + 16q^{94} + 24q^{95} - 4q^{96} - 2q^{97} - 21q^{98} + 28q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(34))$$

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
34.2.a $$\chi_{34}(1, \cdot)$$ 34.2.a.a 1 1
34.2.b $$\chi_{34}(33, \cdot)$$ 34.2.b.a 2 1
34.2.c $$\chi_{34}(13, \cdot)$$ 34.2.c.a 2 2
34.2.c.b 2
34.2.d $$\chi_{34}(9, \cdot)$$ 34.2.d.a 4 4

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(34))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(34)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(17))$$$$^{\oplus 2}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ ($$1 - T$$)($$( 1 + T )^{2}$$)($$1 + T^{2}$$)($$1 + T^{2}$$)($$1 + T^{4}$$)
$3$ ($$1 + 2 T + 3 T^{2}$$)($$( 1 - 2 T + 3 T^{2} )( 1 + 2 T + 3 T^{2} )$$)($$1 + 2 T + 2 T^{2} + 6 T^{3} + 9 T^{4}$$)($$1 + 9 T^{4}$$)($$( 1 - 2 T + 3 T^{2} )^{2}( 1 + 4 T + 8 T^{2} + 12 T^{3} + 9 T^{4} )$$)
$5$ ($$1 + 5 T^{2}$$)($$1 - 2 T^{2} + 25 T^{4}$$)($$1 - 4 T + 8 T^{2} - 20 T^{3} + 25 T^{4}$$)($$( 1 - 2 T + 5 T^{2} )( 1 + 4 T + 5 T^{2} )$$)($$1 + 8 T + 24 T^{2} + 32 T^{3} + 32 T^{4} + 160 T^{5} + 600 T^{6} + 1000 T^{7} + 625 T^{8}$$)
$7$ ($$1 + 4 T + 7 T^{2}$$)($$( 1 - 7 T^{2} )^{2}$$)($$1 + 4 T + 8 T^{2} + 28 T^{3} + 49 T^{4}$$)($$1 + 49 T^{4}$$)($$1 + 8 T^{2} + 24 T^{3} + 32 T^{4} + 168 T^{5} + 392 T^{6} + 2401 T^{8}$$)
$11$ ($$1 - 6 T + 11 T^{2}$$)($$( 1 - 6 T + 11 T^{2} )( 1 + 6 T + 11 T^{2} )$$)($$1 - 2 T + 2 T^{2} - 22 T^{3} + 121 T^{4}$$)($$1 + 8 T + 32 T^{2} + 88 T^{3} + 121 T^{4}$$)($$( 1 - 4 T + 8 T^{2} - 44 T^{3} + 121 T^{4} )( 1 + 14 T^{2} + 121 T^{4} )$$)
$13$ ($$1 - 2 T + 13 T^{2}$$)($$( 1 - 2 T + 13 T^{2} )^{2}$$)($$( 1 + 6 T + 13 T^{2} )^{2}$$)($$( 1 - 4 T + 13 T^{2} )^{2}$$)($$1 - 28 T^{2} + 406 T^{4} - 4732 T^{6} + 28561 T^{8}$$)
$17$ ($$1 + T$$)($$1 + 6 T + 17 T^{2}$$)($$1 - 8 T + 17 T^{2}$$)($$1 + 2 T + 17 T^{2}$$)($$1 + 16 T^{2} + 289 T^{4}$$)
$19$ ($$1 + 4 T + 19 T^{2}$$)($$( 1 + 4 T + 19 T^{2} )^{2}$$)($$1 - 22 T^{2} + 361 T^{4}$$)($$1 - 22 T^{2} + 361 T^{4}$$)($$1 - 8 T + 32 T^{2} - 184 T^{3} + 1042 T^{4} - 3496 T^{5} + 11552 T^{6} - 54872 T^{7} + 130321 T^{8}$$)
$23$ ($$1 + 23 T^{2}$$)($$1 - 14 T^{2} + 529 T^{4}$$)($$1 + 529 T^{4}$$)($$1 - 8 T + 32 T^{2} - 184 T^{3} + 529 T^{4}$$)($$1 + 16 T + 96 T^{2} + 256 T^{3} + 512 T^{4} + 5888 T^{5} + 50784 T^{6} + 194672 T^{7} + 279841 T^{8}$$)
$29$ ($$1 + 29 T^{2}$$)($$1 - 50 T^{2} + 841 T^{4}$$)($$1 + 4 T + 8 T^{2} + 116 T^{3} + 841 T^{4}$$)($$( 1 - 4 T + 29 T^{2} )( 1 + 10 T + 29 T^{2} )$$)($$1 + 8 T^{2} - 200 T^{3} + 32 T^{4} - 5800 T^{5} + 6728 T^{6} + 707281 T^{8}$$)
$31$ ($$1 + 4 T + 31 T^{2}$$)($$( 1 - 31 T^{2} )^{2}$$)($$1 - 12 T + 72 T^{2} - 372 T^{3} + 961 T^{4}$$)($$1 + 8 T + 32 T^{2} + 248 T^{3} + 961 T^{4}$$)($$1 + 8 T^{2} + 216 T^{3} + 32 T^{4} + 6696 T^{5} + 7688 T^{6} + 923521 T^{8}$$)
$37$ ($$1 + 4 T + 37 T^{2}$$)($$1 - 2 T^{2} + 1369 T^{4}$$)($$1 + 1369 T^{4}$$)($$1 - 6 T + 18 T^{2} - 222 T^{3} + 1369 T^{4}$$)($$1 + 8 T + 16 T^{2} - 296 T^{3} - 2240 T^{4} - 10952 T^{5} + 21904 T^{6} + 405224 T^{7} + 1874161 T^{8}$$)
$41$ ($$1 - 6 T + 41 T^{2}$$)($$1 - 50 T^{2} + 1681 T^{4}$$)($$( 1 - 10 T + 41 T^{2} )( 1 + 8 T + 41 T^{2} )$$)($$( 1 - 10 T + 41 T^{2} )( 1 + 8 T + 41 T^{2} )$$)($$( 1 - 8 T + 41 T^{2} )^{2}( 1 - 80 T^{2} + 1681 T^{4} )$$)
$43$ ($$1 - 8 T + 43 T^{2}$$)($$( 1 + 4 T + 43 T^{2} )^{2}$$)($$1 - 50 T^{2} + 1849 T^{4}$$)($$1 - 70 T^{2} + 1849 T^{4}$$)($$1 - 12 T + 72 T^{2} - 300 T^{3} + 926 T^{4} - 12900 T^{5} + 133128 T^{6} - 954084 T^{7} + 3418801 T^{8}$$)
$47$ ($$1 + 47 T^{2}$$)($$( 1 + 47 T^{2} )^{2}$$)($$( 1 - 8 T + 47 T^{2} )^{2}$$)($$( 1 - 8 T + 47 T^{2} )^{2}$$)($$1 - 92 T^{2} + 4486 T^{4} - 203228 T^{6} + 4879681 T^{8}$$)
$53$ ($$1 + 6 T + 53 T^{2}$$)($$( 1 - 6 T + 53 T^{2} )^{2}$$)($$1 - 70 T^{2} + 2809 T^{4}$$)($$( 1 - 14 T + 53 T^{2} )( 1 + 14 T + 53 T^{2} )$$)($$1 - 8 T + 32 T^{2} - 456 T^{3} + 6482 T^{4} - 24168 T^{5} + 89888 T^{6} - 1191016 T^{7} + 7890481 T^{8}$$)
$59$ ($$1 + 59 T^{2}$$)($$( 1 - 12 T + 59 T^{2} )^{2}$$)($$1 - 82 T^{2} + 3481 T^{4}$$)($$1 - 102 T^{2} + 3481 T^{4}$$)($$1 + 4 T + 8 T^{2} + 228 T^{3} + 6494 T^{4} + 13452 T^{5} + 27848 T^{6} + 821516 T^{7} + 12117361 T^{8}$$)
$61$ ($$1 + 4 T + 61 T^{2}$$)($$1 - 50 T^{2} + 3721 T^{4}$$)($$1 + 8 T + 32 T^{2} + 488 T^{3} + 3721 T^{4}$$)($$1 + 18 T + 162 T^{2} + 1098 T^{3} + 3721 T^{4}$$)($$1 - 16 T + 96 T^{2} - 256 T^{3} + 512 T^{4} - 15616 T^{5} + 357216 T^{6} - 3631696 T^{7} + 13845841 T^{8}$$)
$67$ ($$1 - 8 T + 67 T^{2}$$)($$( 1 + 4 T + 67 T^{2} )^{2}$$)($$( 1 + 2 T + 67 T^{2} )^{2}$$)($$( 1 + 12 T + 67 T^{2} )^{2}$$)($$( 1 + 12 T + 168 T^{2} + 804 T^{3} + 4489 T^{4} )^{2}$$)
$71$ ($$1 + 71 T^{2}$$)($$1 - 110 T^{2} + 5041 T^{4}$$)($$1 + 8 T + 32 T^{2} + 568 T^{3} + 5041 T^{4}$$)($$1 + 8 T + 32 T^{2} + 568 T^{3} + 5041 T^{4}$$)($$1 + 8 T + 16 T^{2} - 568 T^{3} - 4416 T^{4} - 40328 T^{5} + 80656 T^{6} + 2863288 T^{7} + 25411681 T^{8}$$)
$73$ ($$1 - 2 T + 73 T^{2}$$)($$( 1 - 73 T^{2} )^{2}$$)($$1 + 2 T + 2 T^{2} + 146 T^{3} + 5329 T^{4}$$)($$( 1 - 16 T + 73 T^{2} )( 1 + 6 T + 73 T^{2} )$$)($$1 + 98 T^{2} - 672 T^{3} + 4802 T^{4} - 49056 T^{5} + 522242 T^{6} + 28398241 T^{8}$$)
$79$ ($$1 - 8 T + 79 T^{2}$$)($$1 + 130 T^{2} + 6241 T^{4}$$)($$1 + 16 T + 128 T^{2} + 1264 T^{3} + 6241 T^{4}$$)($$1 - 16 T + 128 T^{2} - 1264 T^{3} + 6241 T^{4}$$)($$1 + 8 T + 48 T^{2} + 760 T^{3} + 5184 T^{4} + 60040 T^{5} + 299568 T^{6} + 3944312 T^{7} + 38950081 T^{8}$$)
$83$ ($$1 + 83 T^{2}$$)($$( 1 + 12 T + 83 T^{2} )^{2}$$)($$1 + 30 T^{2} + 6889 T^{4}$$)($$1 - 150 T^{2} + 6889 T^{4}$$)($$1 - 12 T + 72 T^{2} - 1164 T^{3} + 18622 T^{4} - 96612 T^{5} + 496008 T^{6} - 6861444 T^{7} + 47458321 T^{8}$$)
$89$ ($$1 + 6 T + 89 T^{2}$$)($$( 1 - 6 T + 89 T^{2} )^{2}$$)($$( 1 + 89 T^{2} )^{2}$$)($$( 1 + 89 T^{2} )^{2}$$)($$1 - 128 T^{2} + 7138 T^{4} - 1013888 T^{6} + 62742241 T^{8}$$)
$97$ ($$1 - 14 T + 97 T^{2}$$)($$( 1 - 10 T + 97 T^{2} )( 1 + 10 T + 97 T^{2} )$$)($$( 1 - 8 T + 97 T^{2} )( 1 + 18 T + 97 T^{2} )$$)($$1 - 6 T + 18 T^{2} - 582 T^{3} + 9409 T^{4}$$)($$1 + 12 T + 54 T^{2} + 108 T^{3} + 162 T^{4} + 10476 T^{5} + 508086 T^{6} + 10952076 T^{7} + 88529281 T^{8}$$)