Properties

Label 208.4.i
Level $208$
Weight $4$
Character orbit 208.i
Rep. character $\chi_{208}(81,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $40$
Newform subspaces $8$
Sturm bound $112$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 208 = 2^{4} \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 208.i (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 13 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 8 \)
Sturm bound: \(112\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(208, [\chi])\).

Total New Old
Modular forms 180 44 136
Cusp forms 156 40 116
Eisenstein series 24 4 20

Trace form

\( 40 q - 5 q^{3} - 2 q^{5} - 17 q^{7} - 163 q^{9} + O(q^{10}) \) \( 40 q - 5 q^{3} - 2 q^{5} - 17 q^{7} - 163 q^{9} + q^{11} - 25 q^{13} + 28 q^{15} + 12 q^{17} + 181 q^{19} + 50 q^{21} + 139 q^{23} + 818 q^{25} + 514 q^{27} + 70 q^{29} - 236 q^{31} + 53 q^{33} - 294 q^{35} - 166 q^{37} - 421 q^{39} + 236 q^{41} - 209 q^{43} - 117 q^{45} + 748 q^{47} - 559 q^{49} + 1486 q^{51} - 682 q^{53} + 962 q^{55} + 1234 q^{57} - 1393 q^{59} + 294 q^{61} - 1306 q^{63} - 727 q^{65} - 341 q^{67} - 467 q^{69} - 763 q^{71} - 158 q^{73} - 1449 q^{75} + 1522 q^{77} - 2560 q^{79} - 848 q^{81} + 1344 q^{83} - 997 q^{85} + 1321 q^{87} + 197 q^{89} + 3483 q^{91} - 2108 q^{93} + 388 q^{95} - 187 q^{97} - 3052 q^{99} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(208, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
208.4.i.a 208.i 13.c $2$ $12.272$ \(\Q(\sqrt{-3}) \) None 26.4.c.a \(0\) \(-3\) \(4\) \(-5\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-3+3\zeta_{6})q^{3}+2q^{5}-5\zeta_{6}q^{7}+\cdots\)
208.4.i.b 208.i 13.c $2$ $12.272$ \(\Q(\sqrt{-3}) \) None 13.4.c.a \(0\) \(2\) \(34\) \(20\) $\mathrm{SU}(2)[C_{3}]$ \(q+(2-2\zeta_{6})q^{3}+17q^{5}+20\zeta_{6}q^{7}+\cdots\)
208.4.i.c 208.i 13.c $2$ $12.272$ \(\Q(\sqrt{-3}) \) None 104.4.i.a \(0\) \(8\) \(-18\) \(-4\) $\mathrm{SU}(2)[C_{3}]$ \(q+(8-8\zeta_{6})q^{3}-9q^{5}-4\zeta_{6}q^{7}-37\zeta_{6}q^{9}+\cdots\)
208.4.i.d 208.i 13.c $4$ $12.272$ \(\Q(\sqrt{-3}, \sqrt{217})\) None 26.4.c.b \(0\) \(-3\) \(-14\) \(-45\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-2+\beta _{1}+\beta _{2}+\beta _{3})q^{3}+(-3-\beta _{3})q^{5}+\cdots\)
208.4.i.e 208.i 13.c $4$ $12.272$ \(\Q(\sqrt{-3}, \sqrt{17})\) None 13.4.c.b \(0\) \(5\) \(-30\) \(15\) $\mathrm{SU}(2)[C_{3}]$ \(q+(3\beta _{1}+\beta _{2})q^{3}+(-10-5\beta _{3})q^{5}+\cdots\)
208.4.i.f 208.i 13.c $6$ $12.272$ 6.0.6622206867.2 None 52.4.e.a \(0\) \(0\) \(-10\) \(16\) $\mathrm{SU}(2)[C_{3}]$ \(q+(\beta _{1}+\beta _{5})q^{3}+(-2-\beta _{1}+\beta _{2}-\beta _{4}+\cdots)q^{5}+\cdots\)
208.4.i.g 208.i 13.c $8$ $12.272$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None 104.4.i.b \(0\) \(-11\) \(14\) \(-15\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-3+\beta _{1}+3\beta _{2})q^{3}+(2-\beta _{4})q^{5}+\cdots\)
208.4.i.h 208.i 13.c $12$ $12.272$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None 104.4.i.c \(0\) \(-3\) \(18\) \(1\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1+\beta _{1}-\beta _{2})q^{3}+(2+\beta _{3}-\beta _{4}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(208, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(208, [\chi]) \simeq \) \(S_{4}^{\mathrm{new}}(13, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(26, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(52, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(104, [\chi])\)\(^{\oplus 2}\)