Properties

Label 210.8.i
Level $210$
Weight $8$
Character orbit 210.i
Rep. character $\chi_{210}(121,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $72$
Newform subspaces $10$
Sturm bound $384$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 210 = 2 \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 210.i (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 7 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 10 \)
Sturm bound: \(384\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(210, [\chi])\).

Total New Old
Modular forms 688 72 616
Cusp forms 656 72 584
Eisenstein series 32 0 32

Trace form

\( 72 q - 2304 q^{4} - 560 q^{7} - 26244 q^{9} + O(q^{10}) \) \( 72 q - 2304 q^{4} - 560 q^{7} - 26244 q^{9} + 4000 q^{10} + 3028 q^{11} + 15808 q^{13} - 16224 q^{14} - 147456 q^{16} + 41336 q^{17} + 8404 q^{19} - 81648 q^{21} + 32128 q^{22} - 562500 q^{25} - 113312 q^{26} - 143360 q^{28} + 843328 q^{29} + 349560 q^{31} + 287496 q^{33} + 227456 q^{34} - 145500 q^{35} + 3359232 q^{36} + 998304 q^{37} - 707776 q^{38} + 27648 q^{39} + 256000 q^{40} + 1877928 q^{41} + 1302912 q^{42} + 1020464 q^{43} + 193792 q^{44} - 617760 q^{46} - 1452848 q^{47} + 2549184 q^{49} - 1259712 q^{51} - 505856 q^{52} + 1274472 q^{53} - 2098000 q^{55} + 688128 q^{56} - 1552176 q^{57} - 1981440 q^{58} - 9762848 q^{59} - 2834056 q^{61} - 3110656 q^{62} + 2041200 q^{63} + 18874368 q^{64} + 41500 q^{65} + 8482264 q^{67} + 2645504 q^{68} + 2075328 q^{69} + 20803152 q^{71} + 5534544 q^{73} - 2574048 q^{74} - 1075712 q^{76} - 28803288 q^{77} - 5104512 q^{78} + 6140024 q^{79} - 19131876 q^{81} + 29322368 q^{82} + 57943888 q^{83} - 2612736 q^{84} + 15766000 q^{85} - 6151360 q^{86} - 1028096 q^{88} - 31663640 q^{89} - 5832000 q^{90} - 38415968 q^{91} + 18983160 q^{93} + 7766624 q^{94} - 3019000 q^{95} - 75375168 q^{97} + 28815360 q^{98} - 4414824 q^{99} + O(q^{100}) \)

Decomposition of \(S_{8}^{\mathrm{new}}(210, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
210.8.i.a 210.i 7.c $2$ $65.601$ \(\Q(\sqrt{-3}) \) None \(8\) \(27\) \(125\) \(980\) $\mathrm{SU}(2)[C_{3}]$ \(q+8\zeta_{6}q^{2}+(3^{3}-3^{3}\zeta_{6})q^{3}+(-2^{6}+\cdots)q^{4}+\cdots\)
210.8.i.b 210.i 7.c $4$ $65.601$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(-16\) \(-54\) \(250\) \(-77\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-8+8\beta _{1})q^{2}-3^{3}\beta _{1}q^{3}-2^{6}\beta _{1}q^{4}+\cdots\)
210.8.i.c 210.i 7.c $6$ $65.601$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(-24\) \(-81\) \(375\) \(3202\) $\mathrm{SU}(2)[C_{3}]$ \(q-8\beta _{2}q^{2}+(-3^{3}+3^{3}\beta _{2})q^{3}+(-2^{6}+\cdots)q^{4}+\cdots\)
210.8.i.d 210.i 7.c $6$ $65.601$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(24\) \(81\) \(375\) \(-1057\) $\mathrm{SU}(2)[C_{3}]$ \(q-8\beta _{1}q^{2}+(3^{3}+3^{3}\beta _{1})q^{3}+(-2^{6}+\cdots)q^{4}+\cdots\)
210.8.i.e 210.i 7.c $8$ $65.601$ \(\mathbb{Q}[x]/(x^{8} + \cdots)\) None \(-32\) \(-108\) \(-500\) \(-287\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-8-8\beta _{1})q^{2}+3^{3}\beta _{1}q^{3}+2^{6}\beta _{1}q^{4}+\cdots\)
210.8.i.f 210.i 7.c $8$ $65.601$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(-32\) \(108\) \(-500\) \(-1197\) $\mathrm{SU}(2)[C_{3}]$ \(q+8\beta _{1}q^{2}+(3^{3}+3^{3}\beta _{1})q^{3}+(-2^{6}+\cdots)q^{4}+\cdots\)
210.8.i.g 210.i 7.c $8$ $65.601$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(32\) \(-108\) \(500\) \(1281\) $\mathrm{SU}(2)[C_{3}]$ \(q+(8+8\beta _{1})q^{2}+3^{3}\beta _{1}q^{3}+2^{6}\beta _{1}q^{4}+\cdots\)
210.8.i.h 210.i 7.c $10$ $65.601$ \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None \(-40\) \(135\) \(625\) \(-1249\) $\mathrm{SU}(2)[C_{3}]$ \(q+8\beta _{1}q^{2}+(3^{3}+3^{3}\beta _{1})q^{3}+(-2^{6}+\cdots)q^{4}+\cdots\)
210.8.i.i 210.i 7.c $10$ $65.601$ \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None \(40\) \(-135\) \(-625\) \(-2131\) $\mathrm{SU}(2)[C_{3}]$ \(q+(8+8\beta _{1})q^{2}+3^{3}\beta _{1}q^{3}+2^{6}\beta _{1}q^{4}+\cdots\)
210.8.i.j 210.i 7.c $10$ $65.601$ \(\mathbb{Q}[x]/(x^{10} + \cdots)\) None \(40\) \(135\) \(-625\) \(-25\) $\mathrm{SU}(2)[C_{3}]$ \(q+(8-8\beta _{1})q^{2}+3^{3}\beta _{1}q^{3}-2^{6}\beta _{1}q^{4}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(210, [\chi])\) into lower level spaces

\( S_{8}^{\mathrm{old}}(210, [\chi]) \cong \) \(S_{8}^{\mathrm{new}}(7, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(14, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(35, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(42, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(70, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(105, [\chi])\)\(^{\oplus 2}\)