Properties

Label 169.4.b
Level $169$
Weight $4$
Character orbit 169.b
Rep. character $\chi_{169}(168,\cdot)$
Character field $\Q$
Dimension $34$
Newform subspaces $7$
Sturm bound $60$
Trace bound $4$

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Defining parameters

Level: \( N \) \(=\) \( 169 = 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 169.b (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 13 \)
Character field: \(\Q\)
Newform subspaces: \( 7 \)
Sturm bound: \(60\)
Trace bound: \(4\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(169, [\chi])\).

Total New Old
Modular forms 52 44 8
Cusp forms 38 34 4
Eisenstein series 14 10 4

Trace form

\( 34 q + 2 q^{3} - 118 q^{4} + 268 q^{9} + O(q^{10}) \) \( 34 q + 2 q^{3} - 118 q^{4} + 268 q^{9} - 18 q^{10} + 10 q^{12} - 122 q^{14} + 466 q^{16} + 92 q^{17} + 424 q^{22} - 280 q^{23} - 356 q^{25} - 256 q^{27} + 350 q^{29} + 30 q^{30} + 304 q^{35} + 82 q^{36} + 202 q^{38} - 354 q^{40} + 52 q^{42} + 334 q^{43} - 552 q^{48} - 234 q^{49} - 216 q^{51} + 658 q^{53} - 912 q^{55} - 544 q^{56} - 950 q^{61} + 1190 q^{62} + 1292 q^{64} + 390 q^{66} - 1140 q^{68} + 700 q^{69} - 2272 q^{74} - 598 q^{75} - 1392 q^{77} + 1808 q^{79} - 2854 q^{81} - 1614 q^{82} - 1120 q^{87} + 2208 q^{88} + 3124 q^{90} + 1112 q^{92} - 1430 q^{94} + 248 q^{95} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(169, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
169.4.b.a $2$ $9.971$ \(\Q(\sqrt{-1}) \) None \(0\) \(-14\) \(0\) \(0\) \(q+5iq^{2}-7q^{3}-17q^{4}+7iq^{5}-35iq^{6}+\cdots\)
169.4.b.b $2$ $9.971$ \(\Q(\sqrt{-3}) \) None \(0\) \(-14\) \(0\) \(0\) \(q-2\zeta_{6}q^{2}-7q^{3}-4q^{4}-8\zeta_{6}q^{5}+\cdots\)
169.4.b.c $2$ $9.971$ \(\Q(\sqrt{-1}) \) None \(0\) \(4\) \(0\) \(0\) \(q+4iq^{2}+2q^{3}-8q^{4}+17iq^{5}+8iq^{6}+\cdots\)
169.4.b.d $2$ $9.971$ \(\Q(\sqrt{-3}) \) None \(0\) \(4\) \(0\) \(0\) \(q-\zeta_{6}q^{2}+2q^{3}+5q^{4}-\zeta_{6}q^{5}-2\zeta_{6}q^{6}+\cdots\)
169.4.b.e $4$ $9.971$ \(\Q(i, \sqrt{17})\) None \(0\) \(10\) \(0\) \(0\) \(q+(\beta _{1}+3\beta _{2})q^{2}+(1+3\beta _{3})q^{3}-5\beta _{3}q^{4}+\cdots\)
169.4.b.f $4$ $9.971$ \(\Q(i, \sqrt{17})\) None \(0\) \(10\) \(0\) \(0\) \(q+\beta _{1}q^{2}+(1+3\beta _{3})q^{3}+(3+\beta _{3})q^{4}+\cdots\)
169.4.b.g $18$ $9.971$ \(\mathbb{Q}[x]/(x^{18} + \cdots)\) None \(0\) \(2\) \(0\) \(0\) \(q+\beta _{9}q^{2}-\beta _{2}q^{3}+(-4+\beta _{1})q^{4}+(-\beta _{9}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(169, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(169, [\chi]) \cong \) \(S_{4}^{\mathrm{new}}(13, [\chi])\)\(^{\oplus 2}\)