Properties

Label 169.4.c
Level $169$
Weight $4$
Character orbit 169.c
Rep. character $\chi_{169}(22,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $66$
Newform subspaces $12$
Sturm bound $60$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 169 = 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 169.c (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 13 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 12 \)
Sturm bound: \(60\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(2\)

Dimensions

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

Total New Old
Modular forms 106 86 20
Cusp forms 78 66 12
Eisenstein series 28 20 8

Trace form

\( 66 q - q^{2} + 7 q^{3} - 107 q^{4} - 4 q^{5} - 30 q^{6} + 35 q^{7} + 30 q^{8} - 204 q^{9} + O(q^{10}) \) \( 66 q - q^{2} + 7 q^{3} - 107 q^{4} - 4 q^{5} - 30 q^{6} + 35 q^{7} + 30 q^{8} - 204 q^{9} + 99 q^{10} - 15 q^{11} - 336 q^{12} - 132 q^{14} + 124 q^{15} - 331 q^{16} + 59 q^{17} + 614 q^{18} - 111 q^{19} + 311 q^{20} - 206 q^{21} - 434 q^{22} + 267 q^{23} - 216 q^{24} + 58 q^{25} - 770 q^{27} + 240 q^{28} + 169 q^{29} - 28 q^{30} + 428 q^{31} - 151 q^{32} + 361 q^{33} + 182 q^{34} + 236 q^{35} + 675 q^{36} - 417 q^{37} - 1192 q^{38} + 418 q^{40} + 373 q^{41} - 172 q^{42} + 439 q^{43} - 848 q^{44} - 16 q^{45} + 230 q^{46} + 204 q^{47} + 42 q^{48} - 506 q^{49} + 1106 q^{50} + 770 q^{51} - 1616 q^{53} - 1314 q^{54} + 14 q^{55} - 86 q^{56} + 330 q^{57} + 193 q^{58} - 1673 q^{59} - 144 q^{60} + 845 q^{61} - 1190 q^{62} - 70 q^{63} + 2714 q^{64} + 5064 q^{66} + 387 q^{67} + 441 q^{68} + 53 q^{69} - 2560 q^{70} + 781 q^{71} - 1155 q^{72} - 1600 q^{73} + 395 q^{74} - 2083 q^{75} + 100 q^{76} + 674 q^{77} - 32 q^{79} - 2153 q^{80} + 2099 q^{81} - 2637 q^{82} - 1776 q^{83} + 1540 q^{84} - 1426 q^{85} + 5172 q^{86} + 2841 q^{87} - 828 q^{88} + 655 q^{89} + 1138 q^{90} + 2408 q^{92} + 2052 q^{93} + 1156 q^{94} + 2136 q^{95} - 2816 q^{96} - 177 q^{97} + 1513 q^{98} - 5892 q^{99} + O(q^{100}) \)

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

Label Char Prim Dim $A$ Field CM Minimal twist Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
169.4.c.a 169.c 13.c $2$ $9.971$ \(\Q(\sqrt{-3}) \) None 13.4.a.a \(-5\) \(7\) \(14\) \(-13\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-5+5\zeta_{6})q^{2}+(7-7\zeta_{6})q^{3}-17\zeta_{6}q^{4}+\cdots\)
169.4.c.b 169.c 13.c $2$ $9.971$ \(\Q(\sqrt{-3}) \) None 13.4.b.a \(-3\) \(1\) \(-18\) \(-15\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-3+3\zeta_{6})q^{2}+(1-\zeta_{6})q^{3}-\zeta_{6}q^{4}+\cdots\)
169.4.c.c 169.c 13.c $2$ $9.971$ \(\Q(\sqrt{-3}) \) None 13.4.b.a \(3\) \(1\) \(18\) \(15\) $\mathrm{SU}(2)[C_{3}]$ \(q+(3-3\zeta_{6})q^{2}+(1-\zeta_{6})q^{3}-\zeta_{6}q^{4}+\cdots\)
169.4.c.d 169.c 13.c $2$ $9.971$ \(\Q(\sqrt{-3}) \) None 13.4.c.a \(4\) \(-2\) \(-34\) \(20\) $\mathrm{SU}(2)[C_{3}]$ \(q+(4-4\zeta_{6})q^{2}+(-2+2\zeta_{6})q^{3}-8\zeta_{6}q^{4}+\cdots\)
169.4.c.e 169.c 13.c $2$ $9.971$ \(\Q(\sqrt{-3}) \) None 13.4.a.a \(5\) \(7\) \(-14\) \(13\) $\mathrm{SU}(2)[C_{3}]$ \(q+(5-5\zeta_{6})q^{2}+(7-7\zeta_{6})q^{3}-17\zeta_{6}q^{4}+\cdots\)
169.4.c.f 169.c 13.c $4$ $9.971$ \(\Q(\sqrt{-3}, \sqrt{17})\) None 13.4.c.b \(-5\) \(-5\) \(30\) \(15\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-2+\beta _{1}+2\beta _{2}+\beta _{3})q^{2}+(-1+\cdots)q^{3}+\cdots\)
169.4.c.g 169.c 13.c $4$ $9.971$ \(\Q(\sqrt{-3}, \sqrt{17})\) None 13.4.a.b \(-1\) \(-5\) \(-6\) \(9\) $\mathrm{SU}(2)[C_{3}]$ \(q-\beta _{1}q^{2}+(3\beta _{1}-4\beta _{2})q^{3}+(4+\beta _{1}+\cdots)q^{4}+\cdots\)
169.4.c.h 169.c 13.c $4$ $9.971$ \(\Q(\zeta_{12})\) None 13.4.e.b \(0\) \(-4\) \(0\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q-\zeta_{12}^{2}q^{2}-2\zeta_{12}q^{3}+(5-5\zeta_{12}+\cdots)q^{4}+\cdots\)
169.4.c.i 169.c 13.c $4$ $9.971$ \(\Q(\zeta_{12})\) None 13.4.e.a \(0\) \(14\) \(0\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q-2\zeta_{12}^{2}q^{2}+7\zeta_{12}q^{3}+(-4+4\zeta_{12}+\cdots)q^{4}+\cdots\)
169.4.c.j 169.c 13.c $4$ $9.971$ \(\Q(\sqrt{-3}, \sqrt{17})\) None 13.4.a.b \(1\) \(-5\) \(6\) \(-9\) $\mathrm{SU}(2)[C_{3}]$ \(q+\beta _{1}q^{2}+(3\beta _{1}-4\beta _{2})q^{3}+(4+\beta _{1}+\cdots)q^{4}+\cdots\)
169.4.c.k 169.c 13.c $18$ $9.971$ \(\mathbb{Q}[x]/(x^{18} - \cdots)\) None 169.4.a.k \(-5\) \(-1\) \(60\) \(-38\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-\beta _{1}+\beta _{2}-\beta _{3})q^{2}+(\beta _{11}-\beta _{14}+\cdots)q^{3}+\cdots\)
169.4.c.l 169.c 13.c $18$ $9.971$ \(\mathbb{Q}[x]/(x^{18} - \cdots)\) None 169.4.a.k \(5\) \(-1\) \(-60\) \(38\) $\mathrm{SU}(2)[C_{3}]$ \(q+(\beta _{1}-\beta _{2}+\beta _{3})q^{2}+(\beta _{11}-\beta _{14})q^{3}+\cdots\)

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

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