Properties

Label 189.2.a
Level 189
Weight 2
Character orbit a
Rep. character \(\chi_{189}(1,\cdot)\)
Character field \(\Q\)
Dimension 8
Newform subspaces 6
Sturm bound 48
Trace bound 5

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Defining parameters

Level: \( N \) = \( 189 = 3^{3} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 189.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 6 \)
Sturm bound: \(48\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(2\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(189))\).

Total New Old
Modular forms 30 8 22
Cusp forms 19 8 11
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.

\(3\)\(7\)FrickeDim.
\(+\)\(+\)\(+\)\(1\)
\(+\)\(-\)\(-\)\(3\)
\(-\)\(+\)\(-\)\(3\)
\(-\)\(-\)\(+\)\(1\)
Plus space\(+\)\(2\)
Minus space\(-\)\(6\)

Trace form

\( 8q + 12q^{4} + O(q^{10}) \) \( 8q + 12q^{4} - 4q^{10} - 12q^{13} + 12q^{16} + 12q^{19} - 4q^{22} - 16q^{28} + 20q^{31} - 36q^{34} - 40q^{37} - 48q^{40} + 28q^{43} - 12q^{46} + 8q^{49} - 8q^{52} + 52q^{55} + 8q^{58} - 12q^{61} - 4q^{64} + 16q^{70} + 8q^{73} + 40q^{76} + 4q^{79} + 28q^{82} - 12q^{85} - 36q^{88} + 4q^{91} - 12q^{94} - 28q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(189))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 3 7
189.2.a.a \(1\) \(1.509\) \(\Q\) None \(-2\) \(0\) \(-1\) \(-1\) \(+\) \(+\) \(q-2q^{2}+2q^{4}-q^{5}-q^{7}+2q^{10}+\cdots\)
189.2.a.b \(1\) \(1.509\) \(\Q\) None \(0\) \(0\) \(-3\) \(1\) \(-\) \(-\) \(q-2q^{4}-3q^{5}+q^{7}-6q^{11}-4q^{13}+\cdots\)
189.2.a.c \(1\) \(1.509\) \(\Q\) None \(0\) \(0\) \(3\) \(1\) \(+\) \(-\) \(q-2q^{4}+3q^{5}+q^{7}+6q^{11}-4q^{13}+\cdots\)
189.2.a.d \(1\) \(1.509\) \(\Q\) None \(2\) \(0\) \(1\) \(-1\) \(-\) \(+\) \(q+2q^{2}+2q^{4}+q^{5}-q^{7}+2q^{10}+\cdots\)
189.2.a.e \(2\) \(1.509\) \(\Q(\sqrt{3}) \) None \(0\) \(0\) \(0\) \(2\) \(+\) \(-\) \(q+\beta q^{2}+q^{4}+\beta q^{5}+q^{7}-\beta q^{8}+3q^{10}+\cdots\)
189.2.a.f \(2\) \(1.509\) \(\Q(\sqrt{7}) \) None \(0\) \(0\) \(0\) \(-2\) \(-\) \(+\) \(q+\beta q^{2}+5q^{4}-\beta q^{5}-q^{7}+3\beta q^{8}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(189))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(189)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(27))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(63))\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 2 T + 2 T^{2} \))(\( 1 + 2 T^{2} \))(\( 1 + 2 T^{2} \))(\( 1 - 2 T + 2 T^{2} \))(\( 1 + T^{2} + 4 T^{4} \))(\( 1 - 3 T^{2} + 4 T^{4} \))
$3$ 1
$5$ (\( 1 + T + 5 T^{2} \))(\( 1 + 3 T + 5 T^{2} \))(\( 1 - 3 T + 5 T^{2} \))(\( 1 - T + 5 T^{2} \))(\( 1 + 7 T^{2} + 25 T^{4} \))(\( 1 + 3 T^{2} + 25 T^{4} \))
$7$ (\( 1 + T \))(\( 1 - T \))(\( 1 - T \))(\( 1 + T \))(\( ( 1 - T )^{2} \))(\( ( 1 + T )^{2} \))
$11$ (\( 1 + 4 T + 11 T^{2} \))(\( 1 + 6 T + 11 T^{2} \))(\( 1 - 6 T + 11 T^{2} \))(\( 1 - 4 T + 11 T^{2} \))(\( 1 + 19 T^{2} + 121 T^{4} \))(\( 1 + 15 T^{2} + 121 T^{4} \))
$13$ (\( 1 + 2 T + 13 T^{2} \))(\( 1 + 4 T + 13 T^{2} \))(\( 1 + 4 T + 13 T^{2} \))(\( 1 + 2 T + 13 T^{2} \))(\( ( 1 - 2 T + 13 T^{2} )^{2} \))(\( ( 1 + 2 T + 13 T^{2} )^{2} \))
$17$ (\( 1 - 3 T + 17 T^{2} \))(\( 1 + 3 T + 17 T^{2} \))(\( 1 - 3 T + 17 T^{2} \))(\( 1 + 3 T + 17 T^{2} \))(\( 1 - 14 T^{2} + 289 T^{4} \))(\( ( 1 + 17 T^{2} )^{2} \))
$19$ (\( 1 + 8 T + 19 T^{2} \))(\( 1 - 2 T + 19 T^{2} \))(\( 1 - 2 T + 19 T^{2} \))(\( 1 + 8 T + 19 T^{2} \))(\( ( 1 - 5 T + 19 T^{2} )^{2} \))(\( ( 1 - 7 T + 19 T^{2} )^{2} \))
$23$ (\( 1 + 6 T + 23 T^{2} \))(\( 1 - 6 T + 23 T^{2} \))(\( 1 + 6 T + 23 T^{2} \))(\( 1 - 6 T + 23 T^{2} \))(\( 1 + 43 T^{2} + 529 T^{4} \))(\( 1 - 17 T^{2} + 529 T^{4} \))
$29$ (\( 1 + 4 T + 29 T^{2} \))(\( 1 - 6 T + 29 T^{2} \))(\( 1 + 6 T + 29 T^{2} \))(\( 1 - 4 T + 29 T^{2} \))(\( 1 - 50 T^{2} + 841 T^{4} \))(\( 1 + 30 T^{2} + 841 T^{4} \))
$31$ (\( 1 - 6 T + 31 T^{2} \))(\( 1 + 4 T + 31 T^{2} \))(\( 1 + 4 T + 31 T^{2} \))(\( 1 - 6 T + 31 T^{2} \))(\( ( 1 - 5 T + 31 T^{2} )^{2} \))(\( ( 1 - 3 T + 31 T^{2} )^{2} \))
$37$ (\( 1 + 3 T + 37 T^{2} \))(\( 1 + 7 T + 37 T^{2} \))(\( 1 + 7 T + 37 T^{2} \))(\( 1 + 3 T + 37 T^{2} \))(\( ( 1 + 7 T + 37 T^{2} )^{2} \))(\( ( 1 + 3 T + 37 T^{2} )^{2} \))
$41$ (\( 1 - T + 41 T^{2} \))(\( 1 - 3 T + 41 T^{2} \))(\( 1 + 3 T + 41 T^{2} \))(\( 1 + T + 41 T^{2} \))(\( 1 + 55 T^{2} + 1681 T^{4} \))(\( 1 + 75 T^{2} + 1681 T^{4} \))
$43$ (\( 1 - 11 T + 43 T^{2} \))(\( 1 + T + 43 T^{2} \))(\( 1 + T + 43 T^{2} \))(\( 1 - 11 T + 43 T^{2} \))(\( ( 1 + 4 T + 43 T^{2} )^{2} \))(\( ( 1 - 8 T + 43 T^{2} )^{2} \))
$47$ (\( 1 - 9 T + 47 T^{2} \))(\( 1 + 9 T + 47 T^{2} \))(\( 1 - 9 T + 47 T^{2} \))(\( 1 + 9 T + 47 T^{2} \))(\( 1 + 46 T^{2} + 2209 T^{4} \))(\( ( 1 + 47 T^{2} )^{2} \))
$53$ (\( 1 - 6 T + 53 T^{2} \))(\( 1 - 6 T + 53 T^{2} \))(\( 1 + 6 T + 53 T^{2} \))(\( 1 + 6 T + 53 T^{2} \))(\( 1 - 86 T^{2} + 2809 T^{4} \))(\( ( 1 + 53 T^{2} )^{2} \))
$59$ (\( 1 + 15 T + 59 T^{2} \))(\( 1 + 9 T + 59 T^{2} \))(\( 1 - 9 T + 59 T^{2} \))(\( 1 - 15 T + 59 T^{2} \))(\( 1 + 70 T^{2} + 3481 T^{4} \))(\( ( 1 + 59 T^{2} )^{2} \))
$61$ (\( 1 - 4 T + 61 T^{2} \))(\( 1 + 10 T + 61 T^{2} \))(\( 1 + 10 T + 61 T^{2} \))(\( 1 - 4 T + 61 T^{2} \))(\( ( 1 - 8 T + 61 T^{2} )^{2} \))(\( ( 1 + 8 T + 61 T^{2} )^{2} \))
$67$ (\( 1 + 8 T + 67 T^{2} \))(\( 1 + 4 T + 67 T^{2} \))(\( 1 + 4 T + 67 T^{2} \))(\( 1 + 8 T + 67 T^{2} \))(\( ( 1 - 14 T + 67 T^{2} )^{2} \))(\( ( 1 + 2 T + 67 T^{2} )^{2} \))
$71$ (\( 1 + 12 T + 71 T^{2} \))(\( 1 + 71 T^{2} \))(\( 1 + 71 T^{2} \))(\( 1 - 12 T + 71 T^{2} \))(\( 1 + 115 T^{2} + 5041 T^{4} \))(\( 1 + 79 T^{2} + 5041 T^{4} \))
$73$ (\( 1 - 6 T + 73 T^{2} \))(\( 1 - 2 T + 73 T^{2} \))(\( 1 - 2 T + 73 T^{2} \))(\( 1 - 6 T + 73 T^{2} \))(\( ( 1 + 4 T + 73 T^{2} )^{2} \))(\( ( 1 + 73 T^{2} )^{2} \))
$79$ (\( 1 + T + 79 T^{2} \))(\( 1 + T + 79 T^{2} \))(\( 1 + T + 79 T^{2} \))(\( 1 + T + 79 T^{2} \))(\( ( 1 - 8 T + 79 T^{2} )^{2} \))(\( ( 1 + 4 T + 79 T^{2} )^{2} \))
$83$ (\( 1 + 9 T + 83 T^{2} \))(\( 1 + 3 T + 83 T^{2} \))(\( 1 - 3 T + 83 T^{2} \))(\( 1 - 9 T + 83 T^{2} \))(\( 1 + 58 T^{2} + 6889 T^{4} \))(\( 1 - 86 T^{2} + 6889 T^{4} \))
$89$ (\( 1 - 2 T + 89 T^{2} \))(\( 1 + 6 T + 89 T^{2} \))(\( 1 - 6 T + 89 T^{2} \))(\( 1 + 2 T + 89 T^{2} \))(\( 1 + 103 T^{2} + 7921 T^{4} \))(\( 1 - 165 T^{2} + 7921 T^{4} \))
$97$ (\( 1 - 12 T + 97 T^{2} \))(\( 1 + 10 T + 97 T^{2} \))(\( 1 + 10 T + 97 T^{2} \))(\( 1 - 12 T + 97 T^{2} \))(\( ( 1 + 4 T + 97 T^{2} )^{2} \))(\( ( 1 + 12 T + 97 T^{2} )^{2} \))
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