Properties

Label 135.2.a
Level 135
Weight 2
Character orbit a
Rep. character \(\chi_{135}(1,\cdot)\)
Character field \(\Q\)
Dimension 6
Newform subspaces 4
Sturm bound 36
Trace bound 2

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Defining parameters

Level: \( N \) = \( 135 = 3^{3} \cdot 5 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 135.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 4 \)
Sturm bound: \(36\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(135))\).

Total New Old
Modular forms 24 6 18
Cusp forms 13 6 7
Eisenstein series 11 0 11

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

\(3\)\(5\)FrickeDim.
\(+\)\(+\)\(+\)\(1\)
\(+\)\(-\)\(-\)\(3\)
\(-\)\(+\)\(-\)\(2\)
Plus space\(+\)\(1\)
Minus space\(-\)\(5\)

Trace form

\( 6q + 10q^{4} - 2q^{7} + O(q^{10}) \) \( 6q + 10q^{4} - 2q^{7} + 2q^{10} + 2q^{13} - 14q^{16} + 2q^{19} - 20q^{22} + 6q^{25} - 32q^{28} - 8q^{31} + 10q^{34} + 18q^{37} - 12q^{40} - 4q^{43} - 18q^{46} + 32q^{49} + 24q^{52} + 8q^{55} + 8q^{58} + 26q^{61} - 24q^{64} - 50q^{67} + 12q^{70} + 26q^{73} + 30q^{76} - 38q^{79} + 68q^{82} + 8q^{85} - 12q^{88} - 10q^{91} + 40q^{94} + 6q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(135))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 3 5
135.2.a.a \(1\) \(1.078\) \(\Q\) None \(-2\) \(0\) \(-1\) \(-3\) \(+\) \(+\) \(q-2q^{2}+2q^{4}-q^{5}-3q^{7}+2q^{10}+\cdots\)
135.2.a.b \(1\) \(1.078\) \(\Q\) None \(2\) \(0\) \(1\) \(-3\) \(+\) \(-\) \(q+2q^{2}+2q^{4}+q^{5}-3q^{7}+2q^{10}+\cdots\)
135.2.a.c \(2\) \(1.078\) \(\Q(\sqrt{13}) \) None \(-1\) \(0\) \(2\) \(2\) \(+\) \(-\) \(q-\beta q^{2}+(1+\beta )q^{4}+q^{5}+(2-2\beta )q^{7}+\cdots\)
135.2.a.d \(2\) \(1.078\) \(\Q(\sqrt{13}) \) None \(1\) \(0\) \(-2\) \(2\) \(-\) \(+\) \(q+\beta q^{2}+(1+\beta )q^{4}-q^{5}+(2-2\beta )q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(135))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(135)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(27))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 2 T + 2 T^{2} \))(\( 1 - 2 T + 2 T^{2} \))(\( 1 + T + T^{2} + 2 T^{3} + 4 T^{4} \))(\( 1 - T + T^{2} - 2 T^{3} + 4 T^{4} \))
$3$ 1
$5$ (\( 1 + T \))(\( 1 - T \))(\( ( 1 - T )^{2} \))(\( ( 1 + T )^{2} \))
$7$ (\( 1 + 3 T + 7 T^{2} \))(\( 1 + 3 T + 7 T^{2} \))(\( 1 - 2 T + 2 T^{2} - 14 T^{3} + 49 T^{4} \))(\( 1 - 2 T + 2 T^{2} - 14 T^{3} + 49 T^{4} \))
$11$ (\( 1 + 2 T + 11 T^{2} \))(\( 1 - 2 T + 11 T^{2} \))(\( 1 - 2 T + 10 T^{2} - 22 T^{3} + 121 T^{4} \))(\( 1 + 2 T + 10 T^{2} + 22 T^{3} + 121 T^{4} \))
$13$ (\( 1 + 5 T + 13 T^{2} \))(\( 1 + 5 T + 13 T^{2} \))(\( 1 - 6 T + 22 T^{2} - 78 T^{3} + 169 T^{4} \))(\( 1 - 6 T + 22 T^{2} - 78 T^{3} + 169 T^{4} \))
$17$ (\( 1 + 8 T + 17 T^{2} \))(\( 1 - 8 T + 17 T^{2} \))(\( 1 + 4 T + 25 T^{2} + 68 T^{3} + 289 T^{4} \))(\( 1 - 4 T + 25 T^{2} - 68 T^{3} + 289 T^{4} \))
$19$ (\( 1 - T + 19 T^{2} \))(\( 1 - T + 19 T^{2} \))(\( 1 + 25 T^{2} + 361 T^{4} \))(\( 1 + 25 T^{2} + 361 T^{4} \))
$23$ (\( 1 - 6 T + 23 T^{2} \))(\( 1 + 6 T + 23 T^{2} \))(\( ( 1 + 3 T + 23 T^{2} )^{2} \))(\( ( 1 - 3 T + 23 T^{2} )^{2} \))
$29$ (\( 1 - 2 T + 29 T^{2} \))(\( 1 + 2 T + 29 T^{2} \))(\( 1 - 10 T + 70 T^{2} - 290 T^{3} + 841 T^{4} \))(\( 1 + 10 T + 70 T^{2} + 290 T^{3} + 841 T^{4} \))
$31$ (\( 1 + 31 T^{2} \))(\( 1 + 31 T^{2} \))(\( 1 + 4 T + 53 T^{2} + 124 T^{3} + 961 T^{4} \))(\( 1 + 4 T + 53 T^{2} + 124 T^{3} + 961 T^{4} \))
$37$ (\( 1 - 5 T + 37 T^{2} \))(\( 1 - 5 T + 37 T^{2} \))(\( ( 1 - 2 T + 37 T^{2} )^{2} \))(\( ( 1 - 2 T + 37 T^{2} )^{2} \))
$41$ (\( 1 + 10 T + 41 T^{2} \))(\( 1 - 10 T + 41 T^{2} \))(\( 1 + 2 T + 70 T^{2} + 82 T^{3} + 1681 T^{4} \))(\( 1 - 2 T + 70 T^{2} - 82 T^{3} + 1681 T^{4} \))
$43$ (\( 1 - 4 T + 43 T^{2} \))(\( 1 - 4 T + 43 T^{2} \))(\( 1 + 6 T + 82 T^{2} + 258 T^{3} + 1849 T^{4} \))(\( 1 + 6 T + 82 T^{2} + 258 T^{3} + 1849 T^{4} \))
$47$ (\( 1 - 4 T + 47 T^{2} \))(\( 1 + 4 T + 47 T^{2} \))(\( 1 + 4 T + 46 T^{2} + 188 T^{3} + 2209 T^{4} \))(\( 1 - 4 T + 46 T^{2} - 188 T^{3} + 2209 T^{4} \))
$53$ (\( 1 + 2 T + 53 T^{2} \))(\( 1 - 2 T + 53 T^{2} \))(\( 1 + 4 T + 97 T^{2} + 212 T^{3} + 2809 T^{4} \))(\( 1 - 4 T + 97 T^{2} - 212 T^{3} + 2809 T^{4} \))
$59$ (\( 1 + 8 T + 59 T^{2} \))(\( 1 - 8 T + 59 T^{2} \))(\( 1 + 10 T + 130 T^{2} + 590 T^{3} + 3481 T^{4} \))(\( 1 - 10 T + 130 T^{2} - 590 T^{3} + 3481 T^{4} \))
$61$ (\( 1 - 7 T + 61 T^{2} \))(\( 1 - 7 T + 61 T^{2} \))(\( 1 - 6 T + 79 T^{2} - 366 T^{3} + 3721 T^{4} \))(\( 1 - 6 T + 79 T^{2} - 366 T^{3} + 3721 T^{4} \))
$67$ (\( 1 + 9 T + 67 T^{2} \))(\( 1 + 9 T + 67 T^{2} \))(\( 1 + 16 T + 146 T^{2} + 1072 T^{3} + 4489 T^{4} \))(\( 1 + 16 T + 146 T^{2} + 1072 T^{3} + 4489 T^{4} \))
$71$ (\( 1 - 2 T + 71 T^{2} \))(\( 1 + 2 T + 71 T^{2} \))(\( 1 - 22 T + 250 T^{2} - 1562 T^{3} + 5041 T^{4} \))(\( 1 + 22 T + 250 T^{2} + 1562 T^{3} + 5041 T^{4} \))
$73$ (\( 1 + 5 T + 73 T^{2} \))(\( 1 + 5 T + 73 T^{2} \))(\( 1 - 18 T + 214 T^{2} - 1314 T^{3} + 5329 T^{4} \))(\( 1 - 18 T + 214 T^{2} - 1314 T^{3} + 5329 T^{4} \))
$79$ (\( 1 + 3 T + 79 T^{2} \))(\( 1 + 3 T + 79 T^{2} \))(\( 1 + 16 T + 209 T^{2} + 1264 T^{3} + 6241 T^{4} \))(\( 1 + 16 T + 209 T^{2} + 1264 T^{3} + 6241 T^{4} \))
$83$ (\( 1 - 6 T + 83 T^{2} \))(\( 1 + 6 T + 83 T^{2} \))(\( ( 1 - 3 T + 83 T^{2} )^{2} \))(\( ( 1 + 3 T + 83 T^{2} )^{2} \))
$89$ (\( 1 + 12 T + 89 T^{2} \))(\( 1 - 12 T + 89 T^{2} \))(\( 1 + 6 T + 70 T^{2} + 534 T^{3} + 7921 T^{4} \))(\( 1 - 6 T + 70 T^{2} - 534 T^{3} + 7921 T^{4} \))
$97$ (\( 1 + 13 T + 97 T^{2} \))(\( 1 + 13 T + 97 T^{2} \))(\( ( 1 - 8 T + 97 T^{2} )^{2} \))(\( ( 1 - 8 T + 97 T^{2} )^{2} \))
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