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Properties

Label 52.7.g
Level $52$
Weight $7$
Character orbit 52.g
Rep. character $\chi_{52}(5,\cdot)$
Character field $\Q(\zeta_{4})$
Dimension $14$
Newform subspaces $1$
Sturm bound $49$
Trace bound $0$

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Defining parameters

Level: \( N \) \(=\) \( 52 = 2^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 52.g (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 13 \)
Character field: \(\Q(i)\)
Newform subspaces: \( 1 \)
Sturm bound: \(49\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{7}(52, [\chi])\).

Total New Old
Modular forms 90 14 76
Cusp forms 78 14 64
Eisenstein series 12 0 12

Trace form

\( 14 q - 66 q^{5} - 320 q^{7} + 3402 q^{9} + O(q^{10}) \) \( 14 q - 66 q^{5} - 320 q^{7} + 3402 q^{9} - 1560 q^{11} - 1164 q^{13} + 624 q^{15} + 8824 q^{19} + 9288 q^{21} + 67464 q^{27} + 35256 q^{29} - 103480 q^{31} - 46416 q^{33} + 96840 q^{35} - 41266 q^{37} - 53856 q^{39} - 49518 q^{41} - 73566 q^{45} + 36288 q^{47} - 364572 q^{53} - 31968 q^{55} + 36984 q^{57} + 89832 q^{59} + 71700 q^{61} + 285912 q^{63} + 107778 q^{65} + 629392 q^{67} - 766152 q^{71} - 1140314 q^{73} + 891984 q^{79} + 47934 q^{81} + 1429368 q^{83} + 1737252 q^{85} + 2198544 q^{87} - 1939326 q^{89} - 1479824 q^{91} + 20616 q^{93} - 1585666 q^{97} - 579600 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{7}^{\mathrm{new}}(52, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
52.7.g.a 52.g 13.d $14$ $11.963$ \(\mathbb{Q}[x]/(x^{14} + \cdots)\) None 52.7.g.a \(0\) \(0\) \(-66\) \(-320\) $\mathrm{SU}(2)[C_{4}]$ \(q+\beta _{2}q^{3}+(-5-5\beta _{5}-\beta _{6})q^{5}+(-23+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{7}^{\mathrm{old}}(52, [\chi])\) into lower level spaces

\( S_{7}^{\mathrm{old}}(52, [\chi]) \simeq \) \(S_{7}^{\mathrm{new}}(13, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(26, [\chi])\)\(^{\oplus 2}\)