Properties

Label 6005.2.a
Level 6005
Weight 2
Character orbit a
Rep. character \(\chi_{6005}(1,\cdot)\)
Character field \(\Q\)
Dimension 401
Newform subspaces 7
Sturm bound 1202
Trace bound 2

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Defining parameters

Level: \( N \) \(=\) \( 6005 = 5 \cdot 1201 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6005.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 7 \)
Sturm bound: \(1202\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(6005))\).

Total New Old
Modular forms 602 401 201
Cusp forms 599 401 198
Eisenstein series 3 0 3

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

\(5\)\(1201\)FrickeDim.
\(+\)\(+\)\(+\)\(87\)
\(+\)\(-\)\(-\)\(113\)
\(-\)\(+\)\(-\)\(113\)
\(-\)\(-\)\(+\)\(88\)
Plus space\(+\)\(175\)
Minus space\(-\)\(226\)

Trace form

\( 401q + 3q^{2} + 4q^{3} + 407q^{4} + q^{5} + 8q^{7} + 3q^{8} + 413q^{9} + O(q^{10}) \) \( 401q + 3q^{2} + 4q^{3} + 407q^{4} + q^{5} + 8q^{7} + 3q^{8} + 413q^{9} + 3q^{10} + 12q^{11} - 4q^{12} + 10q^{13} - 12q^{14} + 4q^{15} + 423q^{16} + 2q^{17} - 9q^{18} + 20q^{19} - q^{20} + 4q^{21} - 8q^{23} - 8q^{24} + 401q^{25} - 2q^{26} - 8q^{27} + 32q^{28} + 6q^{29} + 12q^{30} + 12q^{31} - q^{32} - 8q^{33} - 14q^{34} + 4q^{35} + 479q^{36} + 10q^{37} + 16q^{38} + 12q^{39} + 15q^{40} + 10q^{41} + 20q^{42} + 36q^{43} + 56q^{44} + 5q^{45} + 64q^{46} + 8q^{47} + 48q^{48} + 421q^{49} + 3q^{50} - 20q^{51} + 6q^{52} - 22q^{53} + 40q^{54} - 4q^{55} + 32q^{56} + 8q^{57} + 26q^{58} - 24q^{59} + 28q^{60} + 18q^{61} - 24q^{62} - 16q^{63} + 403q^{64} + 2q^{65} + 56q^{66} + 60q^{67} + 34q^{68} - 56q^{69} - 8q^{70} - 20q^{71} - 9q^{72} + 38q^{73} - 26q^{74} + 4q^{75} + 76q^{76} - 60q^{77} - 60q^{78} + 52q^{79} + 15q^{80} + 417q^{81} - 78q^{82} - 44q^{83} - 20q^{84} + 10q^{85} + 52q^{86} - 68q^{87} - 12q^{88} + 6q^{89} + 23q^{90} + 4q^{91} - 8q^{92} - 52q^{93} - 44q^{94} + 4q^{95} - 4q^{96} + 34q^{97} + 7q^{98} + 76q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(6005))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 5 1201
6005.2.a.a \(1\) \(47.950\) \(\Q\) None \(-2\) \(-2\) \(1\) \(-2\) \(-\) \(+\) \(q-2q^{2}-2q^{3}+2q^{4}+q^{5}+4q^{6}+\cdots\)
6005.2.a.b \(1\) \(47.950\) \(\Q\) None \(-1\) \(0\) \(1\) \(4\) \(-\) \(+\) \(q-q^{2}-q^{4}+q^{5}+4q^{7}+3q^{8}-3q^{9}+\cdots\)
6005.2.a.c \(4\) \(47.950\) 4.4.2777.1 None \(2\) \(-2\) \(-4\) \(-7\) \(+\) \(+\) \(q+(1-\beta _{3})q^{2}+(-1-\beta _{2}+\beta _{3})q^{3}+\cdots\)
6005.2.a.d \(83\) \(47.950\) None \(1\) \(-4\) \(-83\) \(2\) \(+\) \(+\)
6005.2.a.e \(88\) \(47.950\) None \(-14\) \(-34\) \(88\) \(-35\) \(-\) \(-\)
6005.2.a.f \(111\) \(47.950\) None \(20\) \(40\) \(111\) \(39\) \(-\) \(+\)
6005.2.a.g \(113\) \(47.950\) None \(-3\) \(6\) \(-113\) \(7\) \(+\) \(-\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(6005))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(6005)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(1201))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 2 T + 2 T^{2} \))(\( 1 + T + 2 T^{2} \))(\( 1 - 2 T + 5 T^{2} - 9 T^{3} + 14 T^{4} - 18 T^{5} + 20 T^{6} - 16 T^{7} + 16 T^{8} \))
$3$ (\( 1 + 2 T + 3 T^{2} \))(\( 1 + 3 T^{2} \))(\( 1 + 2 T + 7 T^{2} + 11 T^{3} + 22 T^{4} + 33 T^{5} + 63 T^{6} + 54 T^{7} + 81 T^{8} \))
$5$ (\( 1 - T \))(\( 1 - T \))(\( ( 1 + T )^{4} \))
$7$ (\( 1 + 2 T + 7 T^{2} \))(\( 1 - 4 T + 7 T^{2} \))(\( 1 + 7 T + 42 T^{2} + 152 T^{3} + 486 T^{4} + 1064 T^{5} + 2058 T^{6} + 2401 T^{7} + 2401 T^{8} \))
$11$ (\( 1 + 2 T + 11 T^{2} \))(\( 1 + 4 T + 11 T^{2} \))(\( 1 + 4 T + 24 T^{2} + 76 T^{3} + 254 T^{4} + 836 T^{5} + 2904 T^{6} + 5324 T^{7} + 14641 T^{8} \))
$13$ (\( 1 + 3 T + 13 T^{2} \))(\( 1 - 4 T + 13 T^{2} \))(\( 1 - 2 T + 49 T^{2} - 73 T^{3} + 937 T^{4} - 949 T^{5} + 8281 T^{6} - 4394 T^{7} + 28561 T^{8} \))
$17$ (\( 1 - 3 T + 17 T^{2} \))(\( 1 + 4 T + 17 T^{2} \))(\( 1 + T + 45 T^{2} + 21 T^{3} + 1019 T^{4} + 357 T^{5} + 13005 T^{6} + 4913 T^{7} + 83521 T^{8} \))
$19$ (\( 1 + 5 T + 19 T^{2} \))(\( 1 + 4 T + 19 T^{2} \))(\( 1 - 11 T + 112 T^{2} - 663 T^{3} + 3542 T^{4} - 12597 T^{5} + 40432 T^{6} - 75449 T^{7} + 130321 T^{8} \))
$23$ (\( 1 + 4 T + 23 T^{2} \))(\( 1 + 6 T + 23 T^{2} \))(\( 1 + 9 T + 118 T^{2} + 648 T^{3} + 4378 T^{4} + 14904 T^{5} + 62422 T^{6} + 109503 T^{7} + 279841 T^{8} \))
$29$ (\( 1 + 10 T + 29 T^{2} \))(\( 1 - 2 T + 29 T^{2} \))(\( 1 + 8 T + 99 T^{2} + 429 T^{3} + 3524 T^{4} + 12441 T^{5} + 83259 T^{6} + 195112 T^{7} + 707281 T^{8} \))
$31$ (\( 1 + 2 T + 31 T^{2} \))(\( 1 + 4 T + 31 T^{2} \))(\( 1 + 5 T + 110 T^{2} + 418 T^{3} + 4990 T^{4} + 12958 T^{5} + 105710 T^{6} + 148955 T^{7} + 923521 T^{8} \))
$37$ (\( 1 + 7 T + 37 T^{2} \))(\( 1 - 4 T + 37 T^{2} \))(\( 1 - 4 T + 47 T^{2} - 285 T^{3} + 1683 T^{4} - 10545 T^{5} + 64343 T^{6} - 202612 T^{7} + 1874161 T^{8} \))
$41$ (\( 1 + 6 T + 41 T^{2} \))(\( 1 + 2 T + 41 T^{2} \))(\( 1 - 3 T + 126 T^{2} - 304 T^{3} + 7346 T^{4} - 12464 T^{5} + 211806 T^{6} - 206763 T^{7} + 2825761 T^{8} \))
$43$ (\( 1 + 8 T + 43 T^{2} \))(\( 1 + 43 T^{2} \))(\( 1 + 19 T + 266 T^{2} + 2482 T^{3} + 19134 T^{4} + 106726 T^{5} + 491834 T^{6} + 1510633 T^{7} + 3418801 T^{8} \))
$47$ (\( 1 - 9 T + 47 T^{2} \))(\( 1 - 6 T + 47 T^{2} \))(\( 1 - 4 T + 105 T^{2} - 129 T^{3} + 5429 T^{4} - 6063 T^{5} + 231945 T^{6} - 415292 T^{7} + 4879681 T^{8} \))
$53$ (\( 1 + 10 T + 53 T^{2} \))(\( 1 + 8 T + 53 T^{2} \))(\( 1 - 28 T + 489 T^{2} - 5571 T^{3} + 47702 T^{4} - 295263 T^{5} + 1373601 T^{6} - 4168556 T^{7} + 7890481 T^{8} \))
$59$ (\( 1 - T + 59 T^{2} \))(\( 1 - 12 T + 59 T^{2} \))(\( 1 + 9 T + 135 T^{2} + 819 T^{3} + 9981 T^{4} + 48321 T^{5} + 469935 T^{6} + 1848411 T^{7} + 12117361 T^{8} \))
$61$ (\( 1 - 5 T + 61 T^{2} \))(\( 1 - 14 T + 61 T^{2} \))(\( 1 + 23 T + 389 T^{2} + 4365 T^{3} + 39975 T^{4} + 266265 T^{5} + 1447469 T^{6} + 5220563 T^{7} + 13845841 T^{8} \))
$67$ (\( 1 + 6 T + 67 T^{2} \))(\( 1 + 4 T + 67 T^{2} \))(\( 1 + 11 T + 294 T^{2} + 2230 T^{3} + 30420 T^{4} + 149410 T^{5} + 1319766 T^{6} + 3308393 T^{7} + 20151121 T^{8} \))
$71$ (\( 1 + 71 T^{2} \))(\( 1 + 71 T^{2} \))(\( 1 - 27 T + 514 T^{2} - 6390 T^{3} + 63378 T^{4} - 453690 T^{5} + 2591074 T^{6} - 9663597 T^{7} + 25411681 T^{8} \))
$73$ (\( 1 + 6 T + 73 T^{2} \))(\( 1 + 73 T^{2} \))(\( 1 - 18 T + 268 T^{2} - 2374 T^{3} + 22454 T^{4} - 173302 T^{5} + 1428172 T^{6} - 7002306 T^{7} + 28398241 T^{8} \))
$79$ (\( 1 + 8 T + 79 T^{2} \))(\( 1 - 16 T + 79 T^{2} \))(\( 1 - 11 T + 338 T^{2} - 2558 T^{3} + 40914 T^{4} - 202082 T^{5} + 2109458 T^{6} - 5423429 T^{7} + 38950081 T^{8} \))
$83$ (\( 1 - 12 T + 83 T^{2} \))(\( 1 - 6 T + 83 T^{2} \))(\( 1 + 2 T + 132 T^{2} - 310 T^{3} + 8006 T^{4} - 25730 T^{5} + 909348 T^{6} + 1143574 T^{7} + 47458321 T^{8} \))
$89$ (\( 1 - T + 89 T^{2} \))(\( 1 - 6 T + 89 T^{2} \))(\( 1 - T + 112 T^{2} - 143 T^{3} + 18742 T^{4} - 12727 T^{5} + 887152 T^{6} - 704969 T^{7} + 62742241 T^{8} \))
$97$ (\( 1 + 9 T + 97 T^{2} \))(\( 1 - 12 T + 97 T^{2} \))(\( 1 - 8 T + 394 T^{2} - 2296 T^{3} + 57619 T^{4} - 222712 T^{5} + 3707146 T^{6} - 7301384 T^{7} + 88529281 T^{8} \))
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