Properties

Label 768.2.d
Level 768
Weight 2
Character orbit d
Rep. character \(\chi_{768}(385,\cdot)\)
Character field \(\Q\)
Dimension 16
Newform subspaces 8
Sturm bound 256
Trace bound 15

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Defining parameters

Level: \( N \) \(=\) \( 768 = 2^{8} \cdot 3 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 768.d (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 8 \)
Character field: \(\Q\)
Newform subspaces: \( 8 \)
Sturm bound: \(256\)
Trace bound: \(15\)
Distinguishing \(T_p\): \(5\), \(7\), \(23\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(768, [\chi])\).

Total New Old
Modular forms 152 16 136
Cusp forms 104 16 88
Eisenstein series 48 0 48

Trace form

\( 16q - 16q^{9} + O(q^{10}) \) \( 16q - 16q^{9} - 16q^{25} - 16q^{49} + 32q^{57} - 32q^{65} + 64q^{73} + 16q^{81} + 32q^{89} - 32q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(768, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
768.2.d.a \(2\) \(6.133\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(-8\) \(q+iq^{3}+2iq^{5}-4q^{7}-q^{9}-4iq^{11}+\cdots\)
768.2.d.b \(2\) \(6.133\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(-4\) \(q+iq^{3}-2q^{7}-q^{9}-4iq^{11}-6iq^{13}+\cdots\)
768.2.d.c \(2\) \(6.133\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(-4\) \(q-iq^{3}+4iq^{5}-2q^{7}-q^{9}-4iq^{11}+\cdots\)
768.2.d.d \(2\) \(6.133\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) \(q+iq^{3}+2iq^{5}-q^{9}+4iq^{11}-2iq^{13}+\cdots\)
768.2.d.e \(2\) \(6.133\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) \(q-iq^{3}+2iq^{5}-q^{9}-4iq^{11}-2iq^{13}+\cdots\)
768.2.d.f \(2\) \(6.133\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(4\) \(q+iq^{3}+4iq^{5}+2q^{7}-q^{9}+4iq^{11}+\cdots\)
768.2.d.g \(2\) \(6.133\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(4\) \(q-iq^{3}+2q^{7}-q^{9}+4iq^{11}-6iq^{13}+\cdots\)
768.2.d.h \(2\) \(6.133\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(8\) \(q-iq^{3}+2iq^{5}+4q^{7}-q^{9}+4iq^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(768, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(768, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(24, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(64, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(96, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(128, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(192, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(256, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(384, [\chi])\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ (\( 1 + T^{2} \))(\( 1 + T^{2} \))(\( 1 + T^{2} \))(\( 1 + T^{2} \))(\( 1 + T^{2} \))(\( 1 + T^{2} \))(\( 1 + T^{2} \))(\( 1 + T^{2} \))
$5$ (\( ( 1 - 4 T + 5 T^{2} )( 1 + 4 T + 5 T^{2} ) \))(\( ( 1 - 5 T^{2} )^{2} \))(\( ( 1 - 2 T + 5 T^{2} )( 1 + 2 T + 5 T^{2} ) \))(\( ( 1 - 4 T + 5 T^{2} )( 1 + 4 T + 5 T^{2} ) \))(\( ( 1 - 4 T + 5 T^{2} )( 1 + 4 T + 5 T^{2} ) \))(\( ( 1 - 2 T + 5 T^{2} )( 1 + 2 T + 5 T^{2} ) \))(\( ( 1 - 5 T^{2} )^{2} \))(\( ( 1 - 4 T + 5 T^{2} )( 1 + 4 T + 5 T^{2} ) \))
$7$ (\( ( 1 + 4 T + 7 T^{2} )^{2} \))(\( ( 1 + 2 T + 7 T^{2} )^{2} \))(\( ( 1 + 2 T + 7 T^{2} )^{2} \))(\( ( 1 + 7 T^{2} )^{2} \))(\( ( 1 + 7 T^{2} )^{2} \))(\( ( 1 - 2 T + 7 T^{2} )^{2} \))(\( ( 1 - 2 T + 7 T^{2} )^{2} \))(\( ( 1 - 4 T + 7 T^{2} )^{2} \))
$11$ (\( 1 - 6 T^{2} + 121 T^{4} \))(\( 1 - 6 T^{2} + 121 T^{4} \))(\( 1 - 6 T^{2} + 121 T^{4} \))(\( 1 - 6 T^{2} + 121 T^{4} \))(\( 1 - 6 T^{2} + 121 T^{4} \))(\( 1 - 6 T^{2} + 121 T^{4} \))(\( 1 - 6 T^{2} + 121 T^{4} \))(\( 1 - 6 T^{2} + 121 T^{4} \))
$13$ (\( 1 - 22 T^{2} + 169 T^{4} \))(\( ( 1 - 4 T + 13 T^{2} )( 1 + 4 T + 13 T^{2} ) \))(\( 1 - 22 T^{2} + 169 T^{4} \))(\( 1 - 22 T^{2} + 169 T^{4} \))(\( 1 - 22 T^{2} + 169 T^{4} \))(\( 1 - 22 T^{2} + 169 T^{4} \))(\( ( 1 - 4 T + 13 T^{2} )( 1 + 4 T + 13 T^{2} ) \))(\( 1 - 22 T^{2} + 169 T^{4} \))
$17$ (\( ( 1 + 6 T + 17 T^{2} )^{2} \))(\( ( 1 - 6 T + 17 T^{2} )^{2} \))(\( ( 1 + 2 T + 17 T^{2} )^{2} \))(\( ( 1 - 2 T + 17 T^{2} )^{2} \))(\( ( 1 - 2 T + 17 T^{2} )^{2} \))(\( ( 1 + 2 T + 17 T^{2} )^{2} \))(\( ( 1 - 6 T + 17 T^{2} )^{2} \))(\( ( 1 + 6 T + 17 T^{2} )^{2} \))
$19$ (\( 1 - 22 T^{2} + 361 T^{4} \))(\( ( 1 - 19 T^{2} )^{2} \))(\( 1 + 26 T^{2} + 361 T^{4} \))(\( 1 - 22 T^{2} + 361 T^{4} \))(\( 1 - 22 T^{2} + 361 T^{4} \))(\( 1 + 26 T^{2} + 361 T^{4} \))(\( ( 1 - 19 T^{2} )^{2} \))(\( 1 - 22 T^{2} + 361 T^{4} \))
$23$ (\( ( 1 + 23 T^{2} )^{2} \))(\( ( 1 + 4 T + 23 T^{2} )^{2} \))(\( ( 1 + 4 T + 23 T^{2} )^{2} \))(\( ( 1 + 8 T + 23 T^{2} )^{2} \))(\( ( 1 - 8 T + 23 T^{2} )^{2} \))(\( ( 1 - 4 T + 23 T^{2} )^{2} \))(\( ( 1 - 4 T + 23 T^{2} )^{2} \))(\( ( 1 + 23 T^{2} )^{2} \))
$29$ (\( 1 - 54 T^{2} + 841 T^{4} \))(\( ( 1 - 10 T + 29 T^{2} )( 1 + 10 T + 29 T^{2} ) \))(\( ( 1 - 29 T^{2} )^{2} \))(\( 1 - 22 T^{2} + 841 T^{4} \))(\( 1 - 22 T^{2} + 841 T^{4} \))(\( ( 1 - 29 T^{2} )^{2} \))(\( ( 1 - 10 T + 29 T^{2} )( 1 + 10 T + 29 T^{2} ) \))(\( 1 - 54 T^{2} + 841 T^{4} \))
$31$ (\( ( 1 + 4 T + 31 T^{2} )^{2} \))(\( ( 1 - 10 T + 31 T^{2} )^{2} \))(\( ( 1 + 6 T + 31 T^{2} )^{2} \))(\( ( 1 + 8 T + 31 T^{2} )^{2} \))(\( ( 1 - 8 T + 31 T^{2} )^{2} \))(\( ( 1 - 6 T + 31 T^{2} )^{2} \))(\( ( 1 + 10 T + 31 T^{2} )^{2} \))(\( ( 1 - 4 T + 31 T^{2} )^{2} \))
$37$ (\( ( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} ) \))(\( ( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} ) \))(\( ( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} ) \))(\( 1 - 38 T^{2} + 1369 T^{4} \))(\( 1 - 38 T^{2} + 1369 T^{4} \))(\( ( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} ) \))(\( ( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} ) \))(\( ( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} ) \))
$41$ (\( ( 1 + 2 T + 41 T^{2} )^{2} \))(\( ( 1 - 2 T + 41 T^{2} )^{2} \))(\( ( 1 + 6 T + 41 T^{2} )^{2} \))(\( ( 1 - 6 T + 41 T^{2} )^{2} \))(\( ( 1 - 6 T + 41 T^{2} )^{2} \))(\( ( 1 + 6 T + 41 T^{2} )^{2} \))(\( ( 1 - 2 T + 41 T^{2} )^{2} \))(\( ( 1 + 2 T + 41 T^{2} )^{2} \))
$43$ (\( 1 - 70 T^{2} + 1849 T^{4} \))(\( 1 - 22 T^{2} + 1849 T^{4} \))(\( ( 1 - 43 T^{2} )^{2} \))(\( 1 - 70 T^{2} + 1849 T^{4} \))(\( 1 - 70 T^{2} + 1849 T^{4} \))(\( ( 1 - 43 T^{2} )^{2} \))(\( 1 - 22 T^{2} + 1849 T^{4} \))(\( 1 - 70 T^{2} + 1849 T^{4} \))
$47$ (\( ( 1 + 8 T + 47 T^{2} )^{2} \))(\( ( 1 + 12 T + 47 T^{2} )^{2} \))(\( ( 1 - 4 T + 47 T^{2} )^{2} \))(\( ( 1 + 47 T^{2} )^{2} \))(\( ( 1 + 47 T^{2} )^{2} \))(\( ( 1 + 4 T + 47 T^{2} )^{2} \))(\( ( 1 - 12 T + 47 T^{2} )^{2} \))(\( ( 1 - 8 T + 47 T^{2} )^{2} \))
$53$ (\( 1 - 6 T^{2} + 2809 T^{4} \))(\( 1 + 38 T^{2} + 2809 T^{4} \))(\( ( 1 - 53 T^{2} )^{2} \))(\( 1 - 102 T^{2} + 2809 T^{4} \))(\( 1 - 102 T^{2} + 2809 T^{4} \))(\( ( 1 - 53 T^{2} )^{2} \))(\( 1 + 38 T^{2} + 2809 T^{4} \))(\( 1 - 6 T^{2} + 2809 T^{4} \))
$59$ (\( 1 - 102 T^{2} + 3481 T^{4} \))(\( 1 - 102 T^{2} + 3481 T^{4} \))(\( 1 - 102 T^{2} + 3481 T^{4} \))(\( 1 - 102 T^{2} + 3481 T^{4} \))(\( 1 - 102 T^{2} + 3481 T^{4} \))(\( 1 - 102 T^{2} + 3481 T^{4} \))(\( 1 - 102 T^{2} + 3481 T^{4} \))(\( 1 - 102 T^{2} + 3481 T^{4} \))
$61$ (\( 1 - 86 T^{2} + 3721 T^{4} \))(\( 1 - 118 T^{2} + 3721 T^{4} \))(\( 1 + 74 T^{2} + 3721 T^{4} \))(\( 1 - 118 T^{2} + 3721 T^{4} \))(\( 1 - 118 T^{2} + 3721 T^{4} \))(\( 1 + 74 T^{2} + 3721 T^{4} \))(\( 1 - 118 T^{2} + 3721 T^{4} \))(\( 1 - 86 T^{2} + 3721 T^{4} \))
$67$ (\( 1 - 118 T^{2} + 4489 T^{4} \))(\( 1 - 118 T^{2} + 4489 T^{4} \))(\( 1 - 118 T^{2} + 4489 T^{4} \))(\( 1 - 118 T^{2} + 4489 T^{4} \))(\( 1 - 118 T^{2} + 4489 T^{4} \))(\( 1 - 118 T^{2} + 4489 T^{4} \))(\( 1 - 118 T^{2} + 4489 T^{4} \))(\( 1 - 118 T^{2} + 4489 T^{4} \))
$71$ (\( ( 1 + 16 T + 71 T^{2} )^{2} \))(\( ( 1 - 4 T + 71 T^{2} )^{2} \))(\( ( 1 + 12 T + 71 T^{2} )^{2} \))(\( ( 1 - 8 T + 71 T^{2} )^{2} \))(\( ( 1 + 8 T + 71 T^{2} )^{2} \))(\( ( 1 - 12 T + 71 T^{2} )^{2} \))(\( ( 1 + 4 T + 71 T^{2} )^{2} \))(\( ( 1 - 16 T + 71 T^{2} )^{2} \))
$73$ (\( ( 1 - 6 T + 73 T^{2} )^{2} \))(\( ( 1 - 10 T + 73 T^{2} )^{2} \))(\( ( 1 - 10 T + 73 T^{2} )^{2} \))(\( ( 1 + 10 T + 73 T^{2} )^{2} \))(\( ( 1 + 10 T + 73 T^{2} )^{2} \))(\( ( 1 - 10 T + 73 T^{2} )^{2} \))(\( ( 1 - 10 T + 73 T^{2} )^{2} \))(\( ( 1 - 6 T + 73 T^{2} )^{2} \))
$79$ (\( ( 1 + 4 T + 79 T^{2} )^{2} \))(\( ( 1 + 6 T + 79 T^{2} )^{2} \))(\( ( 1 - 10 T + 79 T^{2} )^{2} \))(\( ( 1 - 8 T + 79 T^{2} )^{2} \))(\( ( 1 + 8 T + 79 T^{2} )^{2} \))(\( ( 1 + 10 T + 79 T^{2} )^{2} \))(\( ( 1 - 6 T + 79 T^{2} )^{2} \))(\( ( 1 - 4 T + 79 T^{2} )^{2} \))
$83$ (\( 1 - 22 T^{2} + 6889 T^{4} \))(\( 1 - 22 T^{2} + 6889 T^{4} \))(\( 1 - 22 T^{2} + 6889 T^{4} \))(\( 1 - 150 T^{2} + 6889 T^{4} \))(\( 1 - 150 T^{2} + 6889 T^{4} \))(\( 1 - 22 T^{2} + 6889 T^{4} \))(\( 1 - 22 T^{2} + 6889 T^{4} \))(\( 1 - 22 T^{2} + 6889 T^{4} \))
$89$ (\( ( 1 + 10 T + 89 T^{2} )^{2} \))(\( ( 1 + 2 T + 89 T^{2} )^{2} \))(\( ( 1 - 14 T + 89 T^{2} )^{2} \))(\( ( 1 - 6 T + 89 T^{2} )^{2} \))(\( ( 1 - 6 T + 89 T^{2} )^{2} \))(\( ( 1 - 14 T + 89 T^{2} )^{2} \))(\( ( 1 + 2 T + 89 T^{2} )^{2} \))(\( ( 1 + 10 T + 89 T^{2} )^{2} \))
$97$ (\( ( 1 + 14 T + 97 T^{2} )^{2} \))(\( ( 1 + 6 T + 97 T^{2} )^{2} \))(\( ( 1 - 10 T + 97 T^{2} )^{2} \))(\( ( 1 - 2 T + 97 T^{2} )^{2} \))(\( ( 1 - 2 T + 97 T^{2} )^{2} \))(\( ( 1 - 10 T + 97 T^{2} )^{2} \))(\( ( 1 + 6 T + 97 T^{2} )^{2} \))(\( ( 1 + 14 T + 97 T^{2} )^{2} \))
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