Properties

Label 147.2.a
Level 147
Weight 2
Character orbit a
Rep. character \(\chi_{147}(1,\cdot)\)
Character field \(\Q\)
Dimension 7
Newform subspaces 5
Sturm bound 37
Trace bound 3

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Defining parameters

Level: \( N \) \(=\) \( 147 = 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 147.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 5 \)
Sturm bound: \(37\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(2\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(147))\).

Total New Old
Modular forms 26 7 19
Cusp forms 11 7 4
Eisenstein series 15 0 15

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

\(3\)\(7\)FrickeDim.
\(+\)\(+\)\(+\)\(2\)
\(+\)\(-\)\(-\)\(2\)
\(-\)\(+\)\(-\)\(3\)
Plus space\(+\)\(2\)
Minus space\(-\)\(5\)

Trace form

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

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(147))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 3 7
147.2.a.a \(1\) \(1.174\) \(\Q\) None \(-1\) \(-1\) \(2\) \(0\) \(+\) \(-\) \(q-q^{2}-q^{3}-q^{4}+2q^{5}+q^{6}+3q^{8}+\cdots\)
147.2.a.b \(1\) \(1.174\) \(\Q\) None \(2\) \(-1\) \(2\) \(0\) \(+\) \(-\) \(q+2q^{2}-q^{3}+2q^{4}+2q^{5}-2q^{6}+\cdots\)
147.2.a.c \(1\) \(1.174\) \(\Q\) None \(2\) \(1\) \(-2\) \(0\) \(-\) \(+\) \(q+2q^{2}+q^{3}+2q^{4}-2q^{5}+2q^{6}+\cdots\)
147.2.a.d \(2\) \(1.174\) \(\Q(\sqrt{2}) \) None \(-2\) \(-2\) \(-4\) \(0\) \(+\) \(+\) \(q+(-1+\beta )q^{2}-q^{3}+(1-2\beta )q^{4}+(-2+\cdots)q^{5}+\cdots\)
147.2.a.e \(2\) \(1.174\) \(\Q(\sqrt{2}) \) None \(-2\) \(2\) \(4\) \(0\) \(-\) \(+\) \(q+(-1+\beta )q^{2}+q^{3}+(1-2\beta )q^{4}+(2+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(147))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(147)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + T + 2 T^{2} \))(\( 1 - 2 T + 2 T^{2} \))(\( 1 - 2 T + 2 T^{2} \))(\( 1 + 2 T + 3 T^{2} + 4 T^{3} + 4 T^{4} \))(\( 1 + 2 T + 3 T^{2} + 4 T^{3} + 4 T^{4} \))
$3$ (\( 1 + T \))(\( 1 + T \))(\( 1 - T \))(\( ( 1 + T )^{2} \))(\( ( 1 - T )^{2} \))
$5$ (\( 1 - 2 T + 5 T^{2} \))(\( 1 - 2 T + 5 T^{2} \))(\( 1 + 2 T + 5 T^{2} \))(\( 1 + 4 T + 12 T^{2} + 20 T^{3} + 25 T^{4} \))(\( 1 - 4 T + 12 T^{2} - 20 T^{3} + 25 T^{4} \))
$7$ 1
$11$ (\( 1 - 4 T + 11 T^{2} \))(\( 1 + 2 T + 11 T^{2} \))(\( 1 + 2 T + 11 T^{2} \))(\( ( 1 + 2 T + 11 T^{2} )^{2} \))(\( ( 1 + 2 T + 11 T^{2} )^{2} \))
$13$ (\( 1 - 2 T + 13 T^{2} \))(\( 1 + T + 13 T^{2} \))(\( 1 - T + 13 T^{2} \))(\( 1 + 8 T + 40 T^{2} + 104 T^{3} + 169 T^{4} \))(\( 1 - 8 T + 40 T^{2} - 104 T^{3} + 169 T^{4} \))
$17$ (\( 1 - 6 T + 17 T^{2} \))(\( 1 + 17 T^{2} \))(\( 1 + 17 T^{2} \))(\( 1 + 4 T + 20 T^{2} + 68 T^{3} + 289 T^{4} \))(\( 1 - 4 T + 20 T^{2} - 68 T^{3} + 289 T^{4} \))
$19$ (\( 1 + 4 T + 19 T^{2} \))(\( 1 + T + 19 T^{2} \))(\( 1 - T + 19 T^{2} \))(\( 1 + 30 T^{2} + 361 T^{4} \))(\( 1 + 30 T^{2} + 361 T^{4} \))
$23$ (\( 1 + 23 T^{2} \))(\( 1 + 23 T^{2} \))(\( 1 + 23 T^{2} \))(\( 1 + 4 T + 18 T^{2} + 92 T^{3} + 529 T^{4} \))(\( 1 + 4 T + 18 T^{2} + 92 T^{3} + 529 T^{4} \))
$29$ (\( 1 + 2 T + 29 T^{2} \))(\( 1 - 4 T + 29 T^{2} \))(\( 1 - 4 T + 29 T^{2} \))(\( 1 + 8 T + 66 T^{2} + 232 T^{3} + 841 T^{4} \))(\( 1 + 8 T + 66 T^{2} + 232 T^{3} + 841 T^{4} \))
$31$ (\( 1 + 31 T^{2} \))(\( 1 + 9 T + 31 T^{2} \))(\( 1 - 9 T + 31 T^{2} \))(\( 1 - 8 T + 70 T^{2} - 248 T^{3} + 961 T^{4} \))(\( 1 + 8 T + 70 T^{2} + 248 T^{3} + 961 T^{4} \))
$37$ (\( 1 - 6 T + 37 T^{2} \))(\( 1 - 3 T + 37 T^{2} \))(\( 1 - 3 T + 37 T^{2} \))(\( ( 1 + 4 T + 37 T^{2} )^{2} \))(\( ( 1 + 4 T + 37 T^{2} )^{2} \))
$41$ (\( 1 + 2 T + 41 T^{2} \))(\( 1 - 10 T + 41 T^{2} \))(\( 1 + 10 T + 41 T^{2} \))(\( 1 + 4 T + 68 T^{2} + 164 T^{3} + 1681 T^{4} \))(\( 1 - 4 T + 68 T^{2} - 164 T^{3} + 1681 T^{4} \))
$43$ (\( 1 + 4 T + 43 T^{2} \))(\( 1 - 5 T + 43 T^{2} \))(\( 1 - 5 T + 43 T^{2} \))(\( 1 + 54 T^{2} + 1849 T^{4} \))(\( 1 + 54 T^{2} + 1849 T^{4} \))
$47$ (\( 1 + 47 T^{2} \))(\( 1 - 6 T + 47 T^{2} \))(\( 1 + 6 T + 47 T^{2} \))(\( 1 + 86 T^{2} + 2209 T^{4} \))(\( 1 + 86 T^{2} + 2209 T^{4} \))
$53$ (\( 1 - 6 T + 53 T^{2} \))(\( 1 - 12 T + 53 T^{2} \))(\( 1 - 12 T + 53 T^{2} \))(\( ( 1 + 2 T + 53 T^{2} )^{2} \))(\( ( 1 + 2 T + 53 T^{2} )^{2} \))
$59$ (\( 1 + 12 T + 59 T^{2} \))(\( 1 - 12 T + 59 T^{2} \))(\( 1 + 12 T + 59 T^{2} \))(\( 1 - 8 T + 126 T^{2} - 472 T^{3} + 3481 T^{4} \))(\( 1 + 8 T + 126 T^{2} + 472 T^{3} + 3481 T^{4} \))
$61$ (\( 1 - 2 T + 61 T^{2} \))(\( 1 + 10 T + 61 T^{2} \))(\( 1 - 10 T + 61 T^{2} \))(\( 1 + 16 T + 168 T^{2} + 976 T^{3} + 3721 T^{4} \))(\( 1 - 16 T + 168 T^{2} - 976 T^{3} + 3721 T^{4} \))
$67$ (\( 1 - 4 T + 67 T^{2} \))(\( 1 + 5 T + 67 T^{2} \))(\( 1 + 5 T + 67 T^{2} \))(\( 1 + 102 T^{2} + 4489 T^{4} \))(\( 1 + 102 T^{2} + 4489 T^{4} \))
$71$ (\( 1 + 71 T^{2} \))(\( 1 + 6 T + 71 T^{2} \))(\( 1 + 6 T + 71 T^{2} \))(\( 1 + 4 T + 18 T^{2} + 284 T^{3} + 5041 T^{4} \))(\( 1 + 4 T + 18 T^{2} + 284 T^{3} + 5041 T^{4} \))
$73$ (\( 1 - 6 T + 73 T^{2} \))(\( 1 - 3 T + 73 T^{2} \))(\( 1 + 3 T + 73 T^{2} \))(\( 1 + 8 T + 64 T^{2} + 584 T^{3} + 5329 T^{4} \))(\( 1 - 8 T + 64 T^{2} - 584 T^{3} + 5329 T^{4} \))
$79$ (\( 1 + 16 T + 79 T^{2} \))(\( 1 + T + 79 T^{2} \))(\( 1 + T + 79 T^{2} \))(\( 1 - 16 T + 190 T^{2} - 1264 T^{3} + 6241 T^{4} \))(\( 1 - 16 T + 190 T^{2} - 1264 T^{3} + 6241 T^{4} \))
$83$ (\( 1 - 12 T + 83 T^{2} \))(\( 1 + 6 T + 83 T^{2} \))(\( 1 - 6 T + 83 T^{2} \))(\( 1 - 8 T + 54 T^{2} - 664 T^{3} + 6889 T^{4} \))(\( 1 + 8 T + 54 T^{2} + 664 T^{3} + 6889 T^{4} \))
$89$ (\( 1 - 14 T + 89 T^{2} \))(\( 1 + 16 T + 89 T^{2} \))(\( 1 - 16 T + 89 T^{2} \))(\( 1 - 20 T + 260 T^{2} - 1780 T^{3} + 7921 T^{4} \))(\( 1 + 20 T + 260 T^{2} + 1780 T^{3} + 7921 T^{4} \))
$97$ (\( 1 + 18 T + 97 T^{2} \))(\( 1 - 6 T + 97 T^{2} \))(\( 1 + 6 T + 97 T^{2} \))(\( 1 + 8 T + 208 T^{2} + 776 T^{3} + 9409 T^{4} \))(\( 1 - 8 T + 208 T^{2} - 776 T^{3} + 9409 T^{4} \))
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