Properties

Label 268.2.a
Level 268
Weight 2
Character orbit a
Rep. character \(\chi_{268}(1,\cdot)\)
Character field \(\Q\)
Dimension 5
Newform subspaces 3
Sturm bound 68
Trace bound 3

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Defining parameters

Level: \( N \) = \( 268 = 2^{2} \cdot 67 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 268.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 3 \)
Sturm bound: \(68\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(268))\).

Total New Old
Modular forms 37 5 32
Cusp forms 32 5 27
Eisenstein series 5 0 5

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

\(2\)\(67\)FrickeDim.
\(-\)\(+\)\(-\)\(3\)
\(-\)\(-\)\(+\)\(2\)
Plus space\(+\)\(2\)
Minus space\(-\)\(3\)

Trace form

\( 5q - 2q^{7} + 7q^{9} + O(q^{10}) \) \( 5q - 2q^{7} + 7q^{9} + 2q^{11} - 6q^{13} - 2q^{15} + 3q^{17} - 5q^{19} + 4q^{21} - 5q^{23} - 9q^{25} + 6q^{27} + 3q^{29} - 4q^{31} - 2q^{33} - 2q^{35} - 19q^{37} + 12q^{39} + 2q^{41} + 20q^{43} + 12q^{45} + 7q^{47} - 5q^{49} + 2q^{51} - 8q^{55} + 6q^{57} + 15q^{59} - 2q^{61} - 16q^{63} - 30q^{65} - q^{67} - 8q^{69} - 16q^{71} + 17q^{73} - 6q^{75} + 2q^{77} + 14q^{79} - 19q^{81} + 14q^{83} - 10q^{85} - 12q^{87} - 17q^{89} - 6q^{91} - 34q^{93} - 12q^{95} - 22q^{97} + 34q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(268))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 67
268.2.a.a \(1\) \(2.140\) \(\Q\) None \(0\) \(2\) \(2\) \(2\) \(-\) \(+\) \(q+2q^{3}+2q^{5}+2q^{7}+q^{9}-4q^{11}+\cdots\)
268.2.a.b \(2\) \(2.140\) \(\Q(\sqrt{5}) \) None \(0\) \(-3\) \(0\) \(-5\) \(-\) \(-\) \(q+(-1-\beta )q^{3}+(-1+2\beta )q^{5}+(-2+\cdots)q^{7}+\cdots\)
268.2.a.c \(2\) \(2.140\) \(\Q(\sqrt{21}) \) None \(0\) \(1\) \(-2\) \(1\) \(-\) \(+\) \(q+\beta q^{3}-q^{5}+(1-\beta )q^{7}+(2+\beta )q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(268))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(268)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(67))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(134))\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ (\( 1 - 2 T + 3 T^{2} \))(\( 1 + 3 T + 7 T^{2} + 9 T^{3} + 9 T^{4} \))(\( 1 - T + T^{2} - 3 T^{3} + 9 T^{4} \))
$5$ (\( 1 - 2 T + 5 T^{2} \))(\( 1 + 5 T^{2} + 25 T^{4} \))(\( ( 1 + T + 5 T^{2} )^{2} \))
$7$ (\( 1 - 2 T + 7 T^{2} \))(\( 1 + 5 T + 19 T^{2} + 35 T^{3} + 49 T^{4} \))(\( 1 - T + 9 T^{2} - 7 T^{3} + 49 T^{4} \))
$11$ (\( 1 + 4 T + 11 T^{2} \))(\( 1 + 4 T + 21 T^{2} + 44 T^{3} + 121 T^{4} \))(\( ( 1 - 5 T + 11 T^{2} )^{2} \))
$13$ (\( 1 + 6 T + 13 T^{2} \))(\( 1 + 3 T + 17 T^{2} + 39 T^{3} + 169 T^{4} \))(\( 1 - 3 T + 23 T^{2} - 39 T^{3} + 169 T^{4} \))
$17$ (\( 1 - 3 T + 17 T^{2} \))(\( 1 + 6 T + 38 T^{2} + 102 T^{3} + 289 T^{4} \))(\( 1 - 6 T + 22 T^{2} - 102 T^{3} + 289 T^{4} \))
$19$ (\( 1 - T + 19 T^{2} \))(\( 1 + 5 T + 33 T^{2} + 95 T^{3} + 361 T^{4} \))(\( 1 + T + 33 T^{2} + 19 T^{3} + 361 T^{4} \))
$23$ (\( 1 - 3 T + 23 T^{2} \))(\( 1 + 8 T + 57 T^{2} + 184 T^{3} + 529 T^{4} \))(\( 1 + 25 T^{2} + 529 T^{4} \))
$29$ (\( 1 + T + 29 T^{2} \))(\( ( 1 - 3 T + 29 T^{2} )^{2} \))(\( ( 1 + T + 29 T^{2} )^{2} \))
$31$ (\( 1 - 2 T + 31 T^{2} \))(\( 1 - 2 T + 43 T^{2} - 62 T^{3} + 961 T^{4} \))(\( 1 + 8 T + 57 T^{2} + 248 T^{3} + 961 T^{4} \))
$37$ (\( 1 + 5 T + 37 T^{2} \))(\( 1 + T + 43 T^{2} + 37 T^{3} + 1369 T^{4} \))(\( 1 + 13 T + 111 T^{2} + 481 T^{3} + 1369 T^{4} \))
$41$ (\( 1 - 8 T + 41 T^{2} \))(\( 1 + T + 21 T^{2} + 41 T^{3} + 1681 T^{4} \))(\( 1 + 5 T + 83 T^{2} + 205 T^{3} + 1681 T^{4} \))
$43$ (\( 1 - 10 T + 43 T^{2} \))(\( 1 - 11 T + 85 T^{2} - 473 T^{3} + 1849 T^{4} \))(\( 1 + T + 39 T^{2} + 43 T^{3} + 1849 T^{4} \))
$47$ (\( 1 + 3 T + 47 T^{2} \))(\( 1 + 11 T + 123 T^{2} + 517 T^{3} + 2209 T^{4} \))(\( 1 - 21 T + 199 T^{2} - 987 T^{3} + 2209 T^{4} \))
$53$ (\( 1 + 6 T + 53 T^{2} \))(\( 1 - 12 T + 137 T^{2} - 636 T^{3} + 2809 T^{4} \))(\( ( 1 + 3 T + 53 T^{2} )^{2} \))
$59$ (\( 1 - 7 T + 59 T^{2} \))(\( 1 - 62 T^{2} + 3481 T^{4} \))(\( 1 - 8 T + 50 T^{2} - 472 T^{3} + 3481 T^{4} \))
$61$ (\( 1 + 10 T + 61 T^{2} \))(\( 1 - 13 T + 103 T^{2} - 793 T^{3} + 3721 T^{4} \))(\( 1 + 5 T + 81 T^{2} + 305 T^{3} + 3721 T^{4} \))
$67$ (\( 1 + T \))(\( ( 1 - T )^{2} \))(\( ( 1 + T )^{2} \))
$71$ (\( 1 + 8 T + 71 T^{2} \))(\( 1 + 10 T + 87 T^{2} + 710 T^{3} + 5041 T^{4} \))(\( 1 - 2 T + 59 T^{2} - 142 T^{3} + 5041 T^{4} \))
$73$ (\( 1 + 15 T + 73 T^{2} \))(\( 1 - 8 T + 82 T^{2} - 584 T^{3} + 5329 T^{4} \))(\( ( 1 - 12 T + 73 T^{2} )^{2} \))
$79$ (\( 1 - 16 T + 79 T^{2} \))(\( 1 - 5 T + 133 T^{2} - 395 T^{3} + 6241 T^{4} \))(\( 1 + 7 T + 123 T^{2} + 553 T^{3} + 6241 T^{4} \))
$83$ (\( 1 - 12 T + 83 T^{2} \))(\( 1 + 13 T + 147 T^{2} + 1079 T^{3} + 6889 T^{4} \))(\( 1 - 15 T + 175 T^{2} - 1245 T^{3} + 6889 T^{4} \))
$89$ (\( 1 - 15 T + 89 T^{2} \))(\( 1 + 20 T + 273 T^{2} + 1780 T^{3} + 7921 T^{4} \))(\( 1 + 12 T + 193 T^{2} + 1068 T^{3} + 7921 T^{4} \))
$97$ (\( 1 + 8 T + 97 T^{2} \))(\( 1 + 16 T + 213 T^{2} + 1552 T^{3} + 9409 T^{4} \))(\( 1 - 2 T - 141 T^{2} - 194 T^{3} + 9409 T^{4} \))
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