Properties

Label 334.2
Level 334
Weight 2
Dimension 1161
Nonzero newspaces 2
Newform subspaces 8
Sturm bound 13944
Trace bound 1

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Defining parameters

Level: \( N \) = \( 334\( 334 = 2 \cdot 167 \) \)
Weight: \( k \) = \( 2 \)
Nonzero newspaces: \( 2 \)
Newform subspaces: \( 8 \)
Sturm bound: \(13944\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(334))\).

Total New Old
Modular forms 3652 1161 2491
Cusp forms 3321 1161 2160
Eisenstein series 331 0 331

Trace form

\( 1161q - q^{2} - 4q^{3} - q^{4} - 6q^{5} - 4q^{6} - 8q^{7} - q^{8} - 13q^{9} + O(q^{10}) \) \( 1161q - q^{2} - 4q^{3} - q^{4} - 6q^{5} - 4q^{6} - 8q^{7} - q^{8} - 13q^{9} - 6q^{10} - 12q^{11} - 4q^{12} - 14q^{13} - 8q^{14} - 24q^{15} - q^{16} - 18q^{17} - 13q^{18} - 20q^{19} - 6q^{20} - 32q^{21} - 12q^{22} - 24q^{23} - 4q^{24} - 31q^{25} - 14q^{26} - 40q^{27} - 8q^{28} - 30q^{29} - 24q^{30} - 32q^{31} - q^{32} - 48q^{33} - 18q^{34} - 48q^{35} - 13q^{36} - 38q^{37} - 20q^{38} - 56q^{39} - 6q^{40} - 42q^{41} - 32q^{42} - 44q^{43} - 12q^{44} - 78q^{45} - 24q^{46} - 48q^{47} - 4q^{48} - 57q^{49} - 31q^{50} - 72q^{51} - 14q^{52} - 54q^{53} - 40q^{54} - 72q^{55} - 8q^{56} - 80q^{57} - 30q^{58} - 60q^{59} - 24q^{60} - 62q^{61} - 32q^{62} - 104q^{63} - q^{64} - 84q^{65} - 48q^{66} - 68q^{67} - 18q^{68} - 96q^{69} - 48q^{70} - 72q^{71} - 13q^{72} - 74q^{73} - 38q^{74} - 124q^{75} - 20q^{76} - 96q^{77} - 56q^{78} - 80q^{79} - 6q^{80} - 121q^{81} - 42q^{82} - 84q^{83} - 32q^{84} - 108q^{85} - 44q^{86} - 120q^{87} - 12q^{88} - 90q^{89} - 78q^{90} - 112q^{91} - 24q^{92} - 128q^{93} - 48q^{94} - 120q^{95} - 4q^{96} - 98q^{97} - 57q^{98} - 156q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(334))\)

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
334.2.a \(\chi_{334}(1, \cdot)\) 334.2.a.a 1 1
334.2.a.b 2
334.2.a.c 2
334.2.a.d 2
334.2.a.e 3
334.2.a.f 3
334.2.c \(\chi_{334}(3, \cdot)\) 334.2.c.a 574 82
334.2.c.b 574

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_1(334))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_1(334)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_1(167))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 - T \))(\( ( 1 + T )^{2} \))(\( ( 1 + T )^{2} \))(\( ( 1 - T )^{2} \))(\( ( 1 + T )^{3} \))(\( ( 1 - T )^{3} \))
$3$ (\( 1 + 3 T^{2} \))(\( 1 + T + 5 T^{2} + 3 T^{3} + 9 T^{4} \))(\( 1 - 2 T^{2} + 9 T^{4} \))(\( 1 + 3 T + 7 T^{2} + 9 T^{3} + 9 T^{4} \))(\( 1 + T + 4 T^{2} + 2 T^{3} + 12 T^{4} + 9 T^{5} + 27 T^{6} \))(\( 1 - T + 2 T^{2} + 2 T^{3} + 6 T^{4} - 9 T^{5} + 27 T^{6} \))
$5$ (\( 1 - 3 T + 5 T^{2} \))(\( ( 1 + T + 5 T^{2} )^{2} \))(\( 1 - 2 T + 9 T^{2} - 10 T^{3} + 25 T^{4} \))(\( 1 + 4 T + 9 T^{2} + 20 T^{3} + 25 T^{4} \))(\( 1 + 2 T^{2} + 16 T^{3} + 10 T^{4} + 125 T^{6} \))(\( ( 1 + T + 5 T^{2} )^{3} \))
$7$ (\( 1 - T + 7 T^{2} \))(\( 1 + 9 T^{2} + 49 T^{4} \))(\( ( 1 + 3 T + 7 T^{2} )^{2} \))(\( ( 1 + 3 T + 7 T^{2} )^{2} \))(\( 1 + 8 T^{2} + 16 T^{3} + 56 T^{4} + 343 T^{6} \))(\( ( 1 - T + 7 T^{2} )^{3} \))
$11$ (\( 1 + 11 T^{2} \))(\( 1 + 5 T + 27 T^{2} + 55 T^{3} + 121 T^{4} \))(\( 1 + 14 T^{2} + 121 T^{4} \))(\( 1 + 9 T + 41 T^{2} + 99 T^{3} + 121 T^{4} \))(\( 1 - 11 T + 66 T^{2} - 258 T^{3} + 726 T^{4} - 1331 T^{5} + 1331 T^{6} \))(\( 1 - 7 T + 24 T^{2} - 66 T^{3} + 264 T^{4} - 847 T^{5} + 1331 T^{6} \))
$13$ (\( 1 + 2 T + 13 T^{2} \))(\( 1 + 21 T^{2} + 169 T^{4} \))(\( 1 - 8 T + 34 T^{2} - 104 T^{3} + 169 T^{4} \))(\( 1 - 2 T + 7 T^{2} - 26 T^{3} + 169 T^{4} \))(\( 1 + 10 T + 60 T^{2} + 256 T^{3} + 780 T^{4} + 1690 T^{5} + 2197 T^{6} \))(\( 1 - 2 T + 30 T^{2} - 38 T^{3} + 390 T^{4} - 338 T^{5} + 2197 T^{6} \))
$17$ (\( 1 + 2 T + 17 T^{2} \))(\( 1 + 5 T + 29 T^{2} + 85 T^{3} + 289 T^{4} \))(\( 1 - 4 T + 30 T^{2} - 68 T^{3} + 289 T^{4} \))(\( 1 - T + 3 T^{2} - 17 T^{3} + 289 T^{4} \))(\( 1 - 9 T + 50 T^{2} - 200 T^{3} + 850 T^{4} - 2601 T^{5} + 4913 T^{6} \))(\( 1 - 3 T + 44 T^{2} - 104 T^{3} + 748 T^{4} - 867 T^{5} + 4913 T^{6} \))
$19$ (\( 1 - 2 T + 19 T^{2} \))(\( 1 + 33 T^{2} + 361 T^{4} \))(\( 1 - 4 T + 10 T^{2} - 76 T^{3} + 361 T^{4} \))(\( 1 + 2 T + 19 T^{2} + 38 T^{3} + 361 T^{4} \))(\( 1 + 4 T + 8 T^{2} - 72 T^{3} + 152 T^{4} + 1444 T^{5} + 6859 T^{6} \))(\( 1 + 4 T + 38 T^{2} + 102 T^{3} + 722 T^{4} + 1444 T^{5} + 6859 T^{6} \))
$23$ (\( 1 - 2 T + 23 T^{2} \))(\( 1 + 4 T + 45 T^{2} + 92 T^{3} + 529 T^{4} \))(\( 1 - 12 T + 74 T^{2} - 276 T^{3} + 529 T^{4} \))(\( 1 + 41 T^{2} + 529 T^{4} \))(\( 1 + 10 T + 90 T^{2} + 456 T^{3} + 2070 T^{4} + 5290 T^{5} + 12167 T^{6} \))(\( 1 - 4 T - 2 T^{2} + 150 T^{3} - 46 T^{4} - 2116 T^{5} + 12167 T^{6} \))
$29$ (\( 1 + 4 T + 29 T^{2} \))(\( 1 + 12 T + 89 T^{2} + 348 T^{3} + 841 T^{4} \))(\( 1 - 8 T + 66 T^{2} - 232 T^{3} + 841 T^{4} \))(\( 1 + 16 T + 117 T^{2} + 464 T^{3} + 841 T^{4} \))(\( 1 - 18 T + 182 T^{2} - 1166 T^{3} + 5278 T^{4} - 15138 T^{5} + 24389 T^{6} \))(\( 1 - 4 T + 82 T^{2} - 228 T^{3} + 2378 T^{4} - 3364 T^{5} + 24389 T^{6} \))
$31$ (\( 1 - T + 31 T^{2} \))(\( 1 + 4 T + 21 T^{2} + 124 T^{3} + 961 T^{4} \))(\( 1 + 10 T + 79 T^{2} + 310 T^{3} + 961 T^{4} \))(\( 1 - 2 T + 43 T^{2} - 62 T^{3} + 961 T^{4} \))(\( 1 - 4 T + 28 T^{2} - 376 T^{3} + 868 T^{4} - 3844 T^{5} + 29791 T^{6} \))(\( 1 + 5 T + 60 T^{2} + 305 T^{3} + 1860 T^{4} + 4805 T^{5} + 29791 T^{6} \))
$37$ (\( 1 + 3 T + 37 T^{2} \))(\( 1 - 3 T - 25 T^{2} - 111 T^{3} + 1369 T^{4} \))(\( 1 + 6 T + 65 T^{2} + 222 T^{3} + 1369 T^{4} \))(\( 1 + 3 T + 65 T^{2} + 111 T^{3} + 1369 T^{4} \))(\( 1 - T + 106 T^{2} - 70 T^{3} + 3922 T^{4} - 1369 T^{5} + 50653 T^{6} \))(\( 1 + 4 T + 91 T^{2} + 223 T^{3} + 3367 T^{4} + 5476 T^{5} + 50653 T^{6} \))
$41$ (\( 1 - 2 T + 41 T^{2} \))(\( 1 + 8 T + 93 T^{2} + 328 T^{3} + 1681 T^{4} \))(\( 1 - 12 T + 86 T^{2} - 492 T^{3} + 1681 T^{4} \))(\( 1 - 6 T + 71 T^{2} - 246 T^{3} + 1681 T^{4} \))(\( 1 - 16 T + 196 T^{2} - 1398 T^{3} + 8036 T^{4} - 26896 T^{5} + 68921 T^{6} \))(\( 1 + 2 T + 114 T^{2} + 150 T^{3} + 4674 T^{4} + 3362 T^{5} + 68921 T^{6} \))
$43$ (\( 1 - 4 T + 43 T^{2} \))(\( 1 - 9 T + 75 T^{2} - 387 T^{3} + 1849 T^{4} \))(\( ( 1 - 2 T + 43 T^{2} )^{2} \))(\( 1 + 5 T - 9 T^{2} + 215 T^{3} + 1849 T^{4} \))(\( 1 + 11 T + 94 T^{2} + 572 T^{3} + 4042 T^{4} + 20339 T^{5} + 79507 T^{6} \))(\( 1 + 7 T + 84 T^{2} + 502 T^{3} + 3612 T^{4} + 12943 T^{5} + 79507 T^{6} \))
$47$ (\( 1 + T + 47 T^{2} \))(\( 1 + 6 T + 58 T^{2} + 282 T^{3} + 2209 T^{4} \))(\( 1 + 6 T + 31 T^{2} + 282 T^{3} + 2209 T^{4} \))(\( 1 - 6 T + 58 T^{2} - 282 T^{3} + 2209 T^{4} \))(\( 1 + 2 T + 113 T^{2} + 220 T^{3} + 5311 T^{4} + 4418 T^{5} + 103823 T^{6} \))(\( 1 - T + 23 T^{2} - 266 T^{3} + 1081 T^{4} - 2209 T^{5} + 103823 T^{6} \))
$53$ (\( 1 + 11 T + 53 T^{2} \))(\( 1 - 5 T + 111 T^{2} - 265 T^{3} + 2809 T^{4} \))(\( 1 + 22 T + 225 T^{2} + 1166 T^{3} + 2809 T^{4} \))(\( 1 + 7 T + 117 T^{2} + 371 T^{3} + 2809 T^{4} \))(\( 1 - 21 T + 278 T^{2} - 2422 T^{3} + 14734 T^{4} - 58989 T^{5} + 148877 T^{6} \))(\( 1 - 26 T + 377 T^{2} - 3349 T^{3} + 19981 T^{4} - 73034 T^{5} + 148877 T^{6} \))
$59$ (\( 1 + 3 T + 59 T^{2} \))(\( 1 + 2 T + 99 T^{2} + 118 T^{3} + 3481 T^{4} \))(\( 1 - 10 T + 141 T^{2} - 590 T^{3} + 3481 T^{4} \))(\( 1 - 10 T + 123 T^{2} - 590 T^{3} + 3481 T^{4} \))(\( 1 + 6 T + 176 T^{2} + 706 T^{3} + 10384 T^{4} + 20886 T^{5} + 205379 T^{6} \))(\( 1 - 9 T + 164 T^{2} - 881 T^{3} + 9676 T^{4} - 31329 T^{5} + 205379 T^{6} \))
$61$ (\( 1 + 4 T + 61 T^{2} \))(\( 1 + 5 T + 127 T^{2} + 305 T^{3} + 3721 T^{4} \))(\( 1 + 50 T^{2} + 3721 T^{4} \))(\( 1 - 9 T + 131 T^{2} - 549 T^{3} + 3721 T^{4} \))(\( 1 + 11 T + 62 T^{2} + 480 T^{3} + 3782 T^{4} + 40931 T^{5} + 226981 T^{6} \))(\( 1 - T + 44 T^{2} + 546 T^{3} + 2684 T^{4} - 3721 T^{5} + 226981 T^{6} \))
$67$ (\( 1 + 3 T + 67 T^{2} \))(\( 1 - 11 T + 153 T^{2} - 737 T^{3} + 4489 T^{4} \))(\( 1 + 2 T + 133 T^{2} + 134 T^{3} + 4489 T^{4} \))(\( 1 - T + 73 T^{2} - 67 T^{3} + 4489 T^{4} \))(\( 1 + 29 T + 476 T^{2} + 4740 T^{3} + 31892 T^{4} + 130181 T^{5} + 300763 T^{6} \))(\( 1 - 8 T + 61 T^{2} - 605 T^{3} + 4087 T^{4} - 35912 T^{5} + 300763 T^{6} \))
$71$ (\( 1 - 6 T + 71 T^{2} \))(\( 1 - 4 T - 34 T^{2} - 284 T^{3} + 5041 T^{4} \))(\( 1 - 12 T + 106 T^{2} - 852 T^{3} + 5041 T^{4} \))(\( ( 1 + 12 T + 71 T^{2} )^{2} \))(\( ( 1 + 4 T + 71 T^{2} )^{3} \))(\( 1 - 10 T + 205 T^{2} - 1260 T^{3} + 14555 T^{4} - 50410 T^{5} + 357911 T^{6} \))
$73$ (\( 1 + 4 T + 73 T^{2} \))(\( 1 - T + 145 T^{2} - 73 T^{3} + 5329 T^{4} \))(\( 1 + 16 T + 178 T^{2} + 1168 T^{3} + 5329 T^{4} \))(\( 1 - 5 T + 151 T^{2} - 365 T^{3} + 5329 T^{4} \))(\( 1 + 7 T + 66 T^{2} + 100 T^{3} + 4818 T^{4} + 37303 T^{5} + 389017 T^{6} \))(\( 1 + 17 T + 254 T^{2} + 2286 T^{3} + 18542 T^{4} + 90593 T^{5} + 389017 T^{6} \))
$79$ (\( 1 + 12 T + 79 T^{2} \))(\( 1 - 12 T + 189 T^{2} - 948 T^{3} + 6241 T^{4} \))(\( 1 + 86 T^{2} + 6241 T^{4} \))(\( 1 + 12 T + 149 T^{2} + 948 T^{3} + 6241 T^{4} \))(\( 1 + 2 T + 166 T^{2} + 308 T^{3} + 13114 T^{4} + 12482 T^{5} + 493039 T^{6} \))(\( 1 - 14 T + 226 T^{2} - 2192 T^{3} + 17854 T^{4} - 87374 T^{5} + 493039 T^{6} \))
$83$ (\( 1 - 7 T + 83 T^{2} \))(\( 1 - 23 T + 287 T^{2} - 1909 T^{3} + 6889 T^{4} \))(\( 1 - 10 T + 93 T^{2} - 830 T^{3} + 6889 T^{4} \))(\( 1 - T + 15 T^{2} - 83 T^{3} + 6889 T^{4} \))(\( 1 + 19 T + 362 T^{2} + 3368 T^{3} + 30046 T^{4} + 130891 T^{5} + 571787 T^{6} \))(\( 1 + 12 T + 231 T^{2} + 1943 T^{3} + 19173 T^{4} + 82668 T^{5} + 571787 T^{6} \))
$89$ (\( 1 - 3 T + 89 T^{2} \))(\( 1 + 2 T + 54 T^{2} + 178 T^{3} + 7921 T^{4} \))(\( 1 - 6 T + 155 T^{2} - 534 T^{3} + 7921 T^{4} \))(\( 1 - 6 T + 182 T^{2} - 534 T^{3} + 7921 T^{4} \))(\( 1 - 12 T + 179 T^{2} - 2144 T^{3} + 15931 T^{4} - 95052 T^{5} + 704969 T^{6} \))(\( 1 + 3 T + 173 T^{2} + 74 T^{3} + 15397 T^{4} + 23763 T^{5} + 704969 T^{6} \))
$97$ (\( 1 - 7 T + 97 T^{2} \))(\( 1 + 6 T + 158 T^{2} + 582 T^{3} + 9409 T^{4} \))(\( 1 - 2 T + 187 T^{2} - 194 T^{3} + 9409 T^{4} \))(\( 1 - 2 T + 70 T^{2} - 194 T^{3} + 9409 T^{4} \))(\( 1 - 12 T + 75 T^{2} + 48 T^{3} + 7275 T^{4} - 112908 T^{5} + 912673 T^{6} \))(\( 1 + 27 T + 441 T^{2} + 4930 T^{3} + 42777 T^{4} + 254043 T^{5} + 912673 T^{6} \))
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