Properties

Label 68.10.a
Level $68$
Weight $10$
Character orbit 68.a
Rep. character $\chi_{68}(1,\cdot)$
Character field $\Q$
Dimension $12$
Newform subspaces $2$
Sturm bound $90$
Trace bound $1$

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Defining parameters

Level: \( N \) \(=\) \( 68 = 2^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 68.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(90\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{10}(\Gamma_0(68))\).

Total New Old
Modular forms 84 12 72
Cusp forms 78 12 66
Eisenstein series 6 0 6

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

\(2\)\(17\)FrickeDim
\(-\)\(+\)$-$\(7\)
\(-\)\(-\)$+$\(5\)
Plus space\(+\)\(5\)
Minus space\(-\)\(7\)

Trace form

\( 12 q - 310 q^{3} + 342 q^{5} + 8844 q^{7} + 24808 q^{9} + O(q^{10}) \) \( 12 q - 310 q^{3} + 342 q^{5} + 8844 q^{7} + 24808 q^{9} + 27642 q^{11} + 201876 q^{13} + 17288 q^{15} - 167042 q^{17} - 149992 q^{19} + 1809240 q^{21} - 1846704 q^{23} + 5861128 q^{25} - 4916788 q^{27} - 10974018 q^{29} + 16613684 q^{31} + 15237932 q^{33} + 12566976 q^{35} + 24634 q^{37} - 274636 q^{39} - 22875204 q^{41} - 13207588 q^{43} + 42912126 q^{45} + 55817376 q^{47} + 97234156 q^{49} - 13530402 q^{51} + 172083300 q^{53} + 69914488 q^{55} + 116467624 q^{57} + 23558244 q^{59} + 24557874 q^{61} + 404601164 q^{63} - 119961276 q^{65} - 62438060 q^{67} - 107280048 q^{69} + 471868128 q^{71} - 166396976 q^{73} - 58560946 q^{75} - 108564600 q^{77} - 584115040 q^{79} - 970357472 q^{81} + 450874476 q^{83} - 218657978 q^{85} - 898561656 q^{87} + 350755848 q^{89} + 983401064 q^{91} + 1411756296 q^{93} + 374773944 q^{95} - 2188078980 q^{97} - 44890870 q^{99} + O(q^{100}) \)

Decomposition of \(S_{10}^{\mathrm{new}}(\Gamma_0(68))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 17
68.10.a.a 68.a 1.a $5$ $35.022$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(0\) \(-236\) \(-1138\) \(3712\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-47+\beta _{1})q^{3}+(-227+\beta _{1}+\beta _{2}+\cdots)q^{5}+\cdots\)
68.10.a.b 68.a 1.a $7$ $35.022$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(0\) \(-74\) \(1480\) \(5132\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(-11+\beta _{1})q^{3}+(212-\beta _{1}-\beta _{4}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{10}^{\mathrm{old}}(\Gamma_0(68))\) into lower level spaces

\( S_{10}^{\mathrm{old}}(\Gamma_0(68)) \cong \) \(S_{10}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 4}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(4))\)\(^{\oplus 2}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(17))\)\(^{\oplus 3}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(34))\)\(^{\oplus 2}\)