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Properties

Label 1734.2.f
Level $1734$
Weight $2$
Character orbit 1734.f
Rep. character $\chi_{1734}(829,\cdot)$
Character field $\Q(\zeta_{4})$
Dimension $92$
Newform subspaces $15$
Sturm bound $612$
Trace bound $21$

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Defining parameters

Level: \( N \) \(=\) \( 1734 = 2 \cdot 3 \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1734.f (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 17 \)
Character field: \(\Q(i)\)
Newform subspaces: \( 15 \)
Sturm bound: \(612\)
Trace bound: \(21\)
Distinguishing \(T_p\): \(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(1734, [\chi])\).

Total New Old
Modular forms 684 92 592
Cusp forms 540 92 448
Eisenstein series 144 0 144

Trace form

\( 92 q - 92 q^{4} + 4 q^{5} - 8 q^{7} + O(q^{10}) \) \( 92 q - 92 q^{4} + 4 q^{5} - 8 q^{7} + 4 q^{10} - 8 q^{13} + 8 q^{14} + 92 q^{16} - 4 q^{18} - 4 q^{20} - 8 q^{21} - 8 q^{23} + 8 q^{28} - 4 q^{29} - 8 q^{30} - 8 q^{31} + 24 q^{33} + 16 q^{35} + 12 q^{37} - 32 q^{38} + 8 q^{39} - 4 q^{40} + 20 q^{41} - 4 q^{45} + 8 q^{46} - 32 q^{47} + 12 q^{50} + 8 q^{52} + 16 q^{55} - 8 q^{56} + 8 q^{57} - 4 q^{58} - 20 q^{61} - 8 q^{62} - 8 q^{63} - 92 q^{64} + 16 q^{65} + 8 q^{67} + 32 q^{69} - 24 q^{71} + 4 q^{72} - 4 q^{73} + 12 q^{74} - 16 q^{75} + 8 q^{78} + 24 q^{79} + 4 q^{80} - 92 q^{81} - 20 q^{82} + 8 q^{84} + 32 q^{86} - 32 q^{89} + 4 q^{90} + 16 q^{91} + 8 q^{92} + 4 q^{97} + 4 q^{98} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(1734, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
1734.2.f.a 1734.f 17.c $4$ $13.846$ \(\Q(\zeta_{8})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+\zeta_{8}^{2}q^{2}-\zeta_{8}q^{3}-q^{4}+\zeta_{8}q^{5}-\zeta_{8}^{3}q^{6}+\cdots\)
1734.2.f.b 1734.f 17.c $4$ $13.846$ \(\Q(\zeta_{8})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q-\zeta_{8}^{2}q^{2}+\zeta_{8}^{3}q^{3}-q^{4}+\zeta_{8}q^{6}+\cdots\)
1734.2.f.c 1734.f 17.c $4$ $13.846$ \(\Q(\zeta_{8})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q-\zeta_{8}^{2}q^{2}+\zeta_{8}q^{3}-q^{4}+3\zeta_{8}q^{5}+\cdots\)
1734.2.f.d 1734.f 17.c $4$ $13.846$ \(\Q(\zeta_{8})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q-\zeta_{8}^{2}q^{2}-\zeta_{8}^{3}q^{3}-q^{4}-\zeta_{8}q^{6}+\cdots\)
1734.2.f.e 1734.f 17.c $4$ $13.846$ \(\Q(\zeta_{8})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q-\zeta_{8}^{2}q^{2}+\zeta_{8}q^{3}-q^{4}+2\zeta_{8}q^{5}+\cdots\)
1734.2.f.f 1734.f 17.c $4$ $13.846$ \(\Q(\zeta_{8})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q-\zeta_{8}^{2}q^{2}-\zeta_{8}q^{3}-q^{4}+4\zeta_{8}q^{5}+\cdots\)
1734.2.f.g 1734.f 17.c $4$ $13.846$ \(\Q(\zeta_{8})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+\zeta_{8}^{2}q^{2}-\zeta_{8}q^{3}-q^{4}+4\zeta_{8}q^{5}+\cdots\)
1734.2.f.h 1734.f 17.c $4$ $13.846$ \(\Q(\zeta_{8})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+\zeta_{8}^{2}q^{2}+\zeta_{8}q^{3}-q^{4}+2\zeta_{8}q^{5}+\cdots\)
1734.2.f.i 1734.f 17.c $4$ $13.846$ \(\Q(\zeta_{8})\) None \(0\) \(0\) \(4\) \(-8\) $\mathrm{SU}(2)[C_{4}]$ \(q-\zeta_{8}^{2}q^{2}+\zeta_{8}q^{3}-q^{4}+(1+2\zeta_{8}+\cdots)q^{5}+\cdots\)
1734.2.f.j 1734.f 17.c $8$ $13.846$ \(\Q(\zeta_{16})\) None \(0\) \(0\) \(-8\) \(8\) $\mathrm{SU}(2)[C_{4}]$ \(q+\zeta_{16}^{3}q^{2}+\zeta_{16}^{5}q^{3}-q^{4}+(-1+\cdots)q^{5}+\cdots\)
1734.2.f.k 1734.f 17.c $8$ $13.846$ \(\Q(\zeta_{16})\) None \(0\) \(0\) \(-8\) \(8\) $\mathrm{SU}(2)[C_{4}]$ \(q+\zeta_{16}^{3}q^{2}+\zeta_{16}q^{3}-q^{4}+(-1+2\zeta_{16}+\cdots)q^{5}+\cdots\)
1734.2.f.l 1734.f 17.c $8$ $13.846$ \(\Q(\zeta_{16})\) None \(0\) \(0\) \(8\) \(-8\) $\mathrm{SU}(2)[C_{4}]$ \(q+\zeta_{16}^{3}q^{2}+\zeta_{16}q^{3}-q^{4}+(1+2\zeta_{16}+\cdots)q^{5}+\cdots\)
1734.2.f.m 1734.f 17.c $8$ $13.846$ \(\Q(\zeta_{16})\) None \(0\) \(0\) \(8\) \(-8\) $\mathrm{SU}(2)[C_{4}]$ \(q-\zeta_{16}^{3}q^{2}-\zeta_{16}q^{3}-q^{4}+(1+\zeta_{16}^{3}+\cdots)q^{5}+\cdots\)
1734.2.f.n 1734.f 17.c $12$ $13.846$ 12.0.\(\cdots\).1 None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+\beta _{7}q^{2}-\beta _{10}q^{3}-q^{4}+(2\beta _{10}-\beta _{11})q^{5}+\cdots\)
1734.2.f.o 1734.f 17.c $12$ $13.846$ 12.0.\(\cdots\).1 None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q-\beta _{7}q^{2}-\beta _{10}q^{3}-q^{4}+(2\beta _{10}-\beta _{11})q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(1734, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(1734, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(34, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(51, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(102, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(289, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(578, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(867, [\chi])\)\(^{\oplus 2}\)