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Properties

Label 1734.2.a
Level $1734$
Weight $2$
Character orbit 1734.a
Rep. character $\chi_{1734}(1,\cdot)$
Character field $\Q$
Dimension $45$
Newform subspaces $23$
Sturm bound $612$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 1734 = 2 \cdot 3 \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1734.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 23 \)
Sturm bound: \(612\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1734))\).

Total New Old
Modular forms 342 45 297
Cusp forms 271 45 226
Eisenstein series 71 0 71

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)\(3\)\(17\)FrickeDim
\(+\)\(+\)\(+\)$+$\(3\)
\(+\)\(+\)\(-\)$-$\(8\)
\(+\)\(-\)\(+\)$-$\(6\)
\(+\)\(-\)\(-\)$+$\(5\)
\(-\)\(+\)\(+\)$-$\(8\)
\(-\)\(+\)\(-\)$+$\(4\)
\(-\)\(-\)\(+\)$+$\(2\)
\(-\)\(-\)\(-\)$-$\(9\)
Plus space\(+\)\(14\)
Minus space\(-\)\(31\)

Trace form

\( 45 q + q^{2} - q^{3} + 45 q^{4} + 6 q^{5} - q^{6} + q^{8} + 45 q^{9} + O(q^{10}) \) \( 45 q + q^{2} - q^{3} + 45 q^{4} + 6 q^{5} - q^{6} + q^{8} + 45 q^{9} - 2 q^{10} + 4 q^{11} - q^{12} + 10 q^{13} + 2 q^{15} + 45 q^{16} + q^{18} + 6 q^{20} - 4 q^{21} + 4 q^{22} - q^{24} + 47 q^{25} - 2 q^{26} - q^{27} + 14 q^{29} + 6 q^{30} + 8 q^{31} + q^{32} + 8 q^{33} + 45 q^{36} - 2 q^{37} + 4 q^{38} - 6 q^{39} - 2 q^{40} - 6 q^{41} + 4 q^{42} + 8 q^{43} + 4 q^{44} + 6 q^{45} - 8 q^{47} - q^{48} + 69 q^{49} + 15 q^{50} + 10 q^{52} - 2 q^{53} - q^{54} + 4 q^{57} + 6 q^{58} + 4 q^{59} + 2 q^{60} + 6 q^{61} - 24 q^{62} + 45 q^{64} - 28 q^{65} + 4 q^{66} + 40 q^{67} + 16 q^{69} + 16 q^{70} + q^{72} - 14 q^{73} + 6 q^{74} + 17 q^{75} + 8 q^{77} + 10 q^{78} + 8 q^{79} + 6 q^{80} + 45 q^{81} - 14 q^{82} + 4 q^{83} - 4 q^{84} - 12 q^{86} + 10 q^{87} + 4 q^{88} + 42 q^{89} - 2 q^{90} - 16 q^{91} - 4 q^{93} + 24 q^{94} + 24 q^{95} - q^{96} - 6 q^{97} + 9 q^{98} + 4 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1734))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 3 17
1734.2.a.a 1734.a 1.a $1$ $13.846$ \(\Q\) None \(-1\) \(-1\) \(-1\) \(4\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}-q^{5}+q^{6}+4q^{7}+\cdots\)
1734.2.a.b 1734.a 1.a $1$ $13.846$ \(\Q\) None \(-1\) \(-1\) \(0\) \(-2\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+q^{6}-2q^{7}-q^{8}+\cdots\)
1734.2.a.c 1734.a 1.a $1$ $13.846$ \(\Q\) None \(-1\) \(-1\) \(0\) \(1\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+q^{6}+q^{7}-q^{8}+\cdots\)
1734.2.a.d 1734.a 1.a $1$ $13.846$ \(\Q\) None \(-1\) \(-1\) \(2\) \(2\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+2q^{5}+q^{6}+2q^{7}+\cdots\)
1734.2.a.e 1734.a 1.a $1$ $13.846$ \(\Q\) None \(-1\) \(1\) \(-2\) \(-2\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}-2q^{5}-q^{6}-2q^{7}+\cdots\)
1734.2.a.f 1734.a 1.a $1$ $13.846$ \(\Q\) None \(-1\) \(1\) \(0\) \(-1\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}-q^{6}-q^{7}-q^{8}+\cdots\)
1734.2.a.g 1734.a 1.a $1$ $13.846$ \(\Q\) None \(-1\) \(1\) \(1\) \(-4\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}+q^{5}-q^{6}-4q^{7}+\cdots\)
1734.2.a.h 1734.a 1.a $1$ $13.846$ \(\Q\) None \(-1\) \(1\) \(4\) \(2\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}+4q^{5}-q^{6}+2q^{7}+\cdots\)
1734.2.a.i 1734.a 1.a $1$ $13.846$ \(\Q\) None \(1\) \(-1\) \(-4\) \(-3\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}-4q^{5}-q^{6}-3q^{7}+\cdots\)
1734.2.a.j 1734.a 1.a $1$ $13.846$ \(\Q\) None \(1\) \(-1\) \(2\) \(0\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}+2q^{5}-q^{6}+q^{8}+\cdots\)
1734.2.a.k 1734.a 1.a $1$ $13.846$ \(\Q\) None \(1\) \(-1\) \(3\) \(4\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}+3q^{5}-q^{6}+4q^{7}+\cdots\)
1734.2.a.l 1734.a 1.a $1$ $13.846$ \(\Q\) None \(1\) \(1\) \(-3\) \(-4\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}-3q^{5}+q^{6}-4q^{7}+\cdots\)
1734.2.a.m 1734.a 1.a $1$ $13.846$ \(\Q\) None \(1\) \(1\) \(4\) \(3\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}+4q^{5}+q^{6}+3q^{7}+\cdots\)
1734.2.a.n 1734.a 1.a $2$ $13.846$ \(\Q(\sqrt{2}) \) None \(2\) \(-2\) \(4\) \(4\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}+(2+\beta )q^{5}-q^{6}+\cdots\)
1734.2.a.o 1734.a 1.a $2$ $13.846$ \(\Q(\sqrt{2}) \) None \(2\) \(2\) \(-4\) \(-4\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}+(-2+\beta )q^{5}+q^{6}+\cdots\)
1734.2.a.p 1734.a 1.a $3$ $13.846$ \(\Q(\zeta_{18})^+\) None \(-3\) \(-3\) \(-6\) \(-3\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+(-2+\beta _{1})q^{5}+q^{6}+\cdots\)
1734.2.a.q 1734.a 1.a $3$ $13.846$ \(\Q(\zeta_{18})^+\) None \(-3\) \(3\) \(6\) \(3\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}+(2-\beta _{1})q^{5}-q^{6}+\cdots\)
1734.2.a.r 1734.a 1.a $3$ $13.846$ \(\Q(\zeta_{18})^+\) None \(3\) \(-3\) \(-6\) \(3\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}+(-2+\beta _{1})q^{5}-q^{6}+\cdots\)
1734.2.a.s 1734.a 1.a $3$ $13.846$ \(\Q(\zeta_{18})^+\) None \(3\) \(3\) \(6\) \(-3\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}+(2-\beta _{1})q^{5}+q^{6}+\cdots\)
1734.2.a.t 1734.a 1.a $4$ $13.846$ \(\Q(\zeta_{16})^+\) None \(-4\) \(-4\) \(8\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+(2+\beta _{2}+\beta _{3})q^{5}+\cdots\)
1734.2.a.u 1734.a 1.a $4$ $13.846$ \(\Q(\zeta_{16})^+\) None \(-4\) \(4\) \(-8\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}+(-2-\beta _{2}+\beta _{3})q^{5}+\cdots\)
1734.2.a.v 1734.a 1.a $4$ $13.846$ \(\Q(\zeta_{16})^+\) None \(4\) \(-4\) \(0\) \(-8\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}+(-\beta _{2}+\beta _{3})q^{5}+\cdots\)
1734.2.a.w 1734.a 1.a $4$ $13.846$ \(\Q(\zeta_{16})^+\) None \(4\) \(4\) \(0\) \(8\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}+(\beta _{2}+\beta _{3})q^{5}+q^{6}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1734))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1734)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(17))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(34))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(51))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(102))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(289))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(578))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(867))\)\(^{\oplus 2}\)