Properties

Label 67.2.a
Level 67
Weight 2
Character orbit a
Rep. character \(\chi_{67}(1,\cdot)\)
Character field \(\Q\)
Dimension 5
Newform subspaces 3
Sturm bound 11
Trace bound 2

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Defining parameters

Level: \( N \) = \( 67 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 67.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 3 \)
Sturm bound: \(11\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(67))\).

Total New Old
Modular forms 6 6 0
Cusp forms 5 5 0
Eisenstein series 1 1 0

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

\(67\)Dim.
\(+\)\(2\)
\(-\)\(3\)

Trace form

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

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(67))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 67
67.2.a.a \(1\) \(0.535\) \(\Q\) None \(2\) \(-2\) \(2\) \(-2\) \(-\) \(q+2q^{2}-2q^{3}+2q^{4}+2q^{5}-4q^{6}+\cdots\)
67.2.a.b \(2\) \(0.535\) \(\Q(\sqrt{5}) \) None \(-3\) \(-3\) \(-6\) \(-1\) \(+\) \(q+(-1-\beta )q^{2}+(-2+\beta )q^{3}+3\beta q^{4}+\cdots\)
67.2.a.c \(2\) \(0.535\) \(\Q(\sqrt{5}) \) None \(-1\) \(1\) \(4\) \(1\) \(-\) \(q-\beta q^{2}+(1-\beta )q^{3}+(-1+\beta )q^{4}+(1+\cdots)q^{5}+\cdots\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 - 2 T + 2 T^{2} \))(\( 1 + 3 T + 5 T^{2} + 6 T^{3} + 4 T^{4} \))(\( 1 + T + 3 T^{2} + 2 T^{3} + 4 T^{4} \))
$3$ (\( 1 + 2 T + 3 T^{2} \))(\( 1 + 3 T + 7 T^{2} + 9 T^{3} + 9 T^{4} \))(\( 1 - T + 5 T^{2} - 3 T^{3} + 9 T^{4} \))
$5$ (\( 1 - 2 T + 5 T^{2} \))(\( ( 1 + 3 T + 5 T^{2} )^{2} \))(\( 1 - 4 T + 9 T^{2} - 20 T^{3} + 25 T^{4} \))
$7$ (\( 1 + 2 T + 7 T^{2} \))(\( 1 + T + 3 T^{2} + 7 T^{3} + 49 T^{4} \))(\( 1 - T + 13 T^{2} - 7 T^{3} + 49 T^{4} \))
$11$ (\( 1 + 4 T + 11 T^{2} \))(\( 1 + 17 T^{2} + 121 T^{4} \))(\( ( 1 - T + 11 T^{2} )^{2} \))
$13$ (\( 1 - 2 T + 13 T^{2} \))(\( 1 + 7 T + 27 T^{2} + 91 T^{3} + 169 T^{4} \))(\( 1 + T + 25 T^{2} + 13 T^{3} + 169 T^{4} \))
$17$ (\( 1 - 3 T + 17 T^{2} \))(\( 1 + 6 T + 38 T^{2} + 102 T^{3} + 289 T^{4} \))(\( 1 - 6 T + 38 T^{2} - 102 T^{3} + 289 T^{4} \))
$19$ (\( 1 - 7 T + 19 T^{2} \))(\( 1 - T + 27 T^{2} - 19 T^{3} + 361 T^{4} \))(\( 1 + 11 T + 67 T^{2} + 209 T^{3} + 361 T^{4} \))
$23$ (\( 1 - 9 T + 23 T^{2} \))(\( 1 - 6 T + 35 T^{2} - 138 T^{3} + 529 T^{4} \))(\( 1 + 2 T + 27 T^{2} + 46 T^{3} + 529 T^{4} \))
$29$ (\( 1 + 5 T + 29 T^{2} \))(\( 1 + 6 T + 47 T^{2} + 174 T^{3} + 841 T^{4} \))(\( 1 - 10 T + 63 T^{2} - 290 T^{3} + 841 T^{4} \))
$31$ (\( 1 + 10 T + 31 T^{2} \))(\( ( 1 + T + 31 T^{2} )^{2} \))(\( 1 + 17 T^{2} + 961 T^{4} \))
$37$ (\( 1 + T + 37 T^{2} \))(\( 1 + T + 63 T^{2} + 37 T^{3} + 1369 T^{4} \))(\( 1 - 3 T + 75 T^{2} - 111 T^{3} + 1369 T^{4} \))
$41$ (\( 1 + 41 T^{2} \))(\( 1 + 3 T + 83 T^{2} + 123 T^{3} + 1681 T^{4} \))(\( 1 - 5 T + 57 T^{2} - 205 T^{3} + 1681 T^{4} \))
$43$ (\( 1 + 2 T + 43 T^{2} \))(\( 1 - 3 T + 77 T^{2} - 129 T^{3} + 1849 T^{4} \))(\( 1 + 9 T + 75 T^{2} + 387 T^{3} + 1849 T^{4} \))
$47$ (\( 1 + T + 47 T^{2} \))(\( 1 + 15 T + 149 T^{2} + 705 T^{3} + 2209 T^{4} \))(\( 1 + 7 T + 105 T^{2} + 329 T^{3} + 2209 T^{4} \))
$53$ (\( 1 - 10 T + 53 T^{2} \))(\( ( 1 + 9 T + 53 T^{2} )^{2} \))(\( 1 + 61 T^{2} + 2809 T^{4} \))
$59$ (\( 1 - 9 T + 59 T^{2} \))(\( ( 1 - 6 T + 59 T^{2} )^{2} \))(\( ( 1 + 6 T + 59 T^{2} )^{2} \))
$61$ (\( 1 + 2 T + 61 T^{2} \))(\( 1 + 7 T + 33 T^{2} + 427 T^{3} + 3721 T^{4} \))(\( 1 + 9 T + 131 T^{2} + 549 T^{3} + 3721 T^{4} \))
$67$ (\( 1 - T \))(\( ( 1 + T )^{2} \))(\( ( 1 - T )^{2} \))
$71$ (\( 1 + 71 T^{2} \))(\( 1 - 12 T + 173 T^{2} - 852 T^{3} + 5041 T^{4} \))(\( 1 - 103 T^{2} + 5041 T^{4} \))
$73$ (\( 1 + 7 T + 73 T^{2} \))(\( ( 1 + 4 T + 73 T^{2} )^{2} \))(\( ( 1 - 8 T + 73 T^{2} )^{2} \))
$79$ (\( 1 + 8 T + 79 T^{2} \))(\( 1 + 7 T + 69 T^{2} + 553 T^{3} + 6241 T^{4} \))(\( 1 + 11 T + 127 T^{2} + 869 T^{3} + 6241 T^{4} \))
$83$ (\( 1 - 4 T + 83 T^{2} \))(\( 1 + 15 T + 161 T^{2} + 1245 T^{3} + 6889 T^{4} \))(\( 1 - 13 T + 197 T^{2} - 1079 T^{3} + 6889 T^{4} \))
$89$ (\( 1 - 7 T + 89 T^{2} \))(\( 1 + 173 T^{2} + 7921 T^{4} \))(\( 1 + 16 T + 197 T^{2} + 1424 T^{3} + 7921 T^{4} \))
$97$ (\( 1 + 97 T^{2} \))(\( 1 - 2 T + 15 T^{2} - 194 T^{3} + 9409 T^{4} \))(\( 1 + 149 T^{2} + 9409 T^{4} \))
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