Properties

Label 756.2.a
Level 756
Weight 2
Character orbit a
Rep. character \(\chi_{756}(1,\cdot)\)
Character field \(\Q\)
Dimension 8
Newform subspaces 7
Sturm bound 288
Trace bound 11

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Defining parameters

Level: \( N \) = \( 756 = 2^{2} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 756.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 7 \)
Sturm bound: \(288\)
Trace bound: \(11\)
Distinguishing \(T_p\): \(5\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(756))\).

Total New Old
Modular forms 162 8 154
Cusp forms 127 8 119
Eisenstein series 35 0 35

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

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

Trace form

\( 8q + O(q^{10}) \) \( 8q + 20q^{13} + 4q^{19} + 24q^{25} - 4q^{31} + 16q^{37} + 4q^{43} + 8q^{49} - 4q^{55} + 20q^{61} + 24q^{73} - 4q^{79} + 44q^{85} - 4q^{91} - 12q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(756))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 3 7
756.2.a.a \(1\) \(6.037\) \(\Q\) None \(0\) \(0\) \(-3\) \(1\) \(-\) \(-\) \(-\) \(q-3q^{5}+q^{7}-3q^{11}+2q^{13}+6q^{17}+\cdots\)
756.2.a.b \(1\) \(6.037\) \(\Q\) None \(0\) \(0\) \(-3\) \(1\) \(-\) \(+\) \(-\) \(q-3q^{5}+q^{7}+2q^{13}-3q^{17}-4q^{19}+\cdots\)
756.2.a.c \(1\) \(6.037\) \(\Q\) None \(0\) \(0\) \(-1\) \(-1\) \(-\) \(-\) \(+\) \(q-q^{5}-q^{7}-2q^{11}-5q^{17}+2q^{19}+\cdots\)
756.2.a.d \(1\) \(6.037\) \(\Q\) None \(0\) \(0\) \(1\) \(-1\) \(-\) \(+\) \(+\) \(q+q^{5}-q^{7}+2q^{11}+5q^{17}+2q^{19}+\cdots\)
756.2.a.e \(1\) \(6.037\) \(\Q\) None \(0\) \(0\) \(3\) \(1\) \(-\) \(-\) \(-\) \(q+3q^{5}+q^{7}+2q^{13}+3q^{17}-4q^{19}+\cdots\)
756.2.a.f \(1\) \(6.037\) \(\Q\) None \(0\) \(0\) \(3\) \(1\) \(-\) \(-\) \(-\) \(q+3q^{5}+q^{7}+3q^{11}+2q^{13}-6q^{17}+\cdots\)
756.2.a.g \(2\) \(6.037\) \(\Q(\sqrt{13}) \) None \(0\) \(0\) \(0\) \(-2\) \(-\) \(+\) \(+\) \(q-\beta q^{5}-q^{7}+\beta q^{11}+6q^{13}-2\beta q^{17}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(756))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(756)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(27))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(42))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(54))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(63))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(84))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(108))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(126))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(189))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(252))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(378))\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ (\( 1 + 3 T + 5 T^{2} \))(\( 1 + 3 T + 5 T^{2} \))(\( 1 + T + 5 T^{2} \))(\( 1 - T + 5 T^{2} \))(\( 1 - 3 T + 5 T^{2} \))(\( 1 - 3 T + 5 T^{2} \))(\( 1 - 3 T^{2} + 25 T^{4} \))
$7$ (\( 1 - T \))(\( 1 - T \))(\( 1 + T \))(\( 1 + T \))(\( 1 - T \))(\( 1 - T \))(\( ( 1 + T )^{2} \))
$11$ (\( 1 + 3 T + 11 T^{2} \))(\( 1 + 11 T^{2} \))(\( 1 + 2 T + 11 T^{2} \))(\( 1 - 2 T + 11 T^{2} \))(\( 1 + 11 T^{2} \))(\( 1 - 3 T + 11 T^{2} \))(\( 1 + 9 T^{2} + 121 T^{4} \))
$13$ (\( 1 - 2 T + 13 T^{2} \))(\( 1 - 2 T + 13 T^{2} \))(\( 1 + 13 T^{2} \))(\( 1 + 13 T^{2} \))(\( 1 - 2 T + 13 T^{2} \))(\( 1 - 2 T + 13 T^{2} \))(\( ( 1 - 6 T + 13 T^{2} )^{2} \))
$17$ (\( 1 - 6 T + 17 T^{2} \))(\( 1 + 3 T + 17 T^{2} \))(\( 1 + 5 T + 17 T^{2} \))(\( 1 - 5 T + 17 T^{2} \))(\( 1 - 3 T + 17 T^{2} \))(\( 1 + 6 T + 17 T^{2} \))(\( 1 - 18 T^{2} + 289 T^{4} \))
$19$ (\( 1 - 5 T + 19 T^{2} \))(\( 1 + 4 T + 19 T^{2} \))(\( 1 - 2 T + 19 T^{2} \))(\( 1 - 2 T + 19 T^{2} \))(\( 1 + 4 T + 19 T^{2} \))(\( 1 - 5 T + 19 T^{2} \))(\( ( 1 + T + 19 T^{2} )^{2} \))
$23$ (\( 1 - 9 T + 23 T^{2} \))(\( 1 + 6 T + 23 T^{2} \))(\( 1 + 2 T + 23 T^{2} \))(\( 1 - 2 T + 23 T^{2} \))(\( 1 - 6 T + 23 T^{2} \))(\( 1 + 9 T + 23 T^{2} \))(\( 1 + 33 T^{2} + 529 T^{4} \))
$29$ (\( 1 + 6 T + 29 T^{2} \))(\( 1 + 29 T^{2} \))(\( 1 + 10 T + 29 T^{2} \))(\( 1 - 10 T + 29 T^{2} \))(\( 1 + 29 T^{2} \))(\( 1 - 6 T + 29 T^{2} \))(\( 1 + 6 T^{2} + 841 T^{4} \))
$31$ (\( 1 + T + 31 T^{2} \))(\( 1 + 10 T + 31 T^{2} \))(\( 1 + 31 T^{2} \))(\( 1 + 31 T^{2} \))(\( 1 + 10 T + 31 T^{2} \))(\( 1 + T + 31 T^{2} \))(\( ( 1 - 9 T + 31 T^{2} )^{2} \))
$37$ (\( 1 - 11 T + 37 T^{2} \))(\( 1 + 7 T + 37 T^{2} \))(\( 1 - 5 T + 37 T^{2} \))(\( 1 - 5 T + 37 T^{2} \))(\( 1 + 7 T + 37 T^{2} \))(\( 1 - 11 T + 37 T^{2} \))(\( ( 1 + T + 37 T^{2} )^{2} \))
$41$ (\( 1 + 3 T + 41 T^{2} \))(\( 1 + 9 T + 41 T^{2} \))(\( 1 + 3 T + 41 T^{2} \))(\( 1 - 3 T + 41 T^{2} \))(\( 1 - 9 T + 41 T^{2} \))(\( 1 - 3 T + 41 T^{2} \))(\( 1 - 35 T^{2} + 1681 T^{4} \))
$43$ (\( 1 + 4 T + 43 T^{2} \))(\( 1 - 5 T + 43 T^{2} \))(\( 1 + 7 T + 43 T^{2} \))(\( 1 + 7 T + 43 T^{2} \))(\( 1 - 5 T + 43 T^{2} \))(\( 1 + 4 T + 43 T^{2} \))(\( ( 1 - 8 T + 43 T^{2} )^{2} \))
$47$ (\( 1 - 12 T + 47 T^{2} \))(\( 1 + 3 T + 47 T^{2} \))(\( 1 - 3 T + 47 T^{2} \))(\( 1 + 3 T + 47 T^{2} \))(\( 1 - 3 T + 47 T^{2} \))(\( 1 + 12 T + 47 T^{2} \))(\( ( 1 + 47 T^{2} )^{2} \))
$53$ (\( 1 + 53 T^{2} \))(\( 1 + 6 T + 53 T^{2} \))(\( 1 + 6 T + 53 T^{2} \))(\( 1 - 6 T + 53 T^{2} \))(\( 1 - 6 T + 53 T^{2} \))(\( 1 + 53 T^{2} \))(\( ( 1 + 53 T^{2} )^{2} \))
$59$ (\( 1 + 59 T^{2} \))(\( 1 - 9 T + 59 T^{2} \))(\( 1 + T + 59 T^{2} \))(\( 1 - T + 59 T^{2} \))(\( 1 + 9 T + 59 T^{2} \))(\( 1 + 59 T^{2} \))(\( 1 - 90 T^{2} + 3481 T^{4} \))
$61$ (\( 1 - 8 T + 61 T^{2} \))(\( 1 - 8 T + 61 T^{2} \))(\( 1 + 6 T + 61 T^{2} \))(\( 1 + 6 T + 61 T^{2} \))(\( 1 - 8 T + 61 T^{2} \))(\( 1 - 8 T + 61 T^{2} \))(\( ( 1 + 61 T^{2} )^{2} \))
$67$ (\( 1 + 10 T + 67 T^{2} \))(\( 1 - 8 T + 67 T^{2} \))(\( 1 - 4 T + 67 T^{2} \))(\( 1 - 4 T + 67 T^{2} \))(\( 1 - 8 T + 67 T^{2} \))(\( 1 + 10 T + 67 T^{2} \))(\( ( 1 + 2 T + 67 T^{2} )^{2} \))
$71$ (\( 1 + 3 T + 71 T^{2} \))(\( 1 - 12 T + 71 T^{2} \))(\( 1 + 8 T + 71 T^{2} \))(\( 1 - 8 T + 71 T^{2} \))(\( 1 + 12 T + 71 T^{2} \))(\( 1 - 3 T + 71 T^{2} \))(\( 1 + 129 T^{2} + 5041 T^{4} \))
$73$ (\( 1 - 8 T + 73 T^{2} \))(\( 1 + 10 T + 73 T^{2} \))(\( 1 - 10 T + 73 T^{2} \))(\( 1 - 10 T + 73 T^{2} \))(\( 1 + 10 T + 73 T^{2} \))(\( 1 - 8 T + 73 T^{2} \))(\( ( 1 - 4 T + 73 T^{2} )^{2} \))
$79$ (\( 1 + 4 T + 79 T^{2} \))(\( 1 - 5 T + 79 T^{2} \))(\( 1 + 3 T + 79 T^{2} \))(\( 1 + 3 T + 79 T^{2} \))(\( 1 - 5 T + 79 T^{2} \))(\( 1 + 4 T + 79 T^{2} \))(\( ( 1 + 79 T^{2} )^{2} \))
$83$ (\( 1 + 6 T + 83 T^{2} \))(\( 1 + 9 T + 83 T^{2} \))(\( 1 - 13 T + 83 T^{2} \))(\( 1 + 13 T + 83 T^{2} \))(\( 1 - 9 T + 83 T^{2} \))(\( 1 - 6 T + 83 T^{2} \))(\( 1 + 114 T^{2} + 6889 T^{4} \))
$89$ (\( 1 + 3 T + 89 T^{2} \))(\( 1 + 18 T + 89 T^{2} \))(\( 1 - 6 T + 89 T^{2} \))(\( 1 + 6 T + 89 T^{2} \))(\( 1 - 18 T + 89 T^{2} \))(\( 1 - 3 T + 89 T^{2} \))(\( 1 + 61 T^{2} + 7921 T^{4} \))
$97$ (\( 1 - 8 T + 97 T^{2} \))(\( 1 - 8 T + 97 T^{2} \))(\( 1 + 14 T + 97 T^{2} \))(\( 1 + 14 T + 97 T^{2} \))(\( 1 - 8 T + 97 T^{2} \))(\( 1 - 8 T + 97 T^{2} \))(\( ( 1 + 8 T + 97 T^{2} )^{2} \))
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