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Properties

Label 1734.4.a
Level $1734$
Weight $4$
Character orbit 1734.a
Rep. character $\chi_{1734}(1,\cdot)$
Character field $\Q$
Dimension $135$
Newform subspaces $37$
Sturm bound $1224$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 1734 = 2 \cdot 3 \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1734.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 37 \)
Sturm bound: \(1224\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(1734))\).

Total New Old
Modular forms 954 135 819
Cusp forms 882 135 747
Eisenstein series 72 0 72

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
\(+\)\(+\)\(+\)$+$\(19\)
\(+\)\(+\)\(-\)$-$\(15\)
\(+\)\(-\)\(+\)$-$\(16\)
\(+\)\(-\)\(-\)$+$\(17\)
\(-\)\(+\)\(+\)$-$\(15\)
\(-\)\(+\)\(-\)$+$\(19\)
\(-\)\(-\)\(+\)$+$\(21\)
\(-\)\(-\)\(-\)$-$\(13\)
Plus space\(+\)\(76\)
Minus space\(-\)\(59\)

Trace form

\( 135 q + 2 q^{2} - 3 q^{3} + 540 q^{4} - 6 q^{5} + 6 q^{6} + 8 q^{8} + 1215 q^{9} + O(q^{10}) \) \( 135 q + 2 q^{2} - 3 q^{3} + 540 q^{4} - 6 q^{5} + 6 q^{6} + 8 q^{8} + 1215 q^{9} + 28 q^{10} + 44 q^{11} - 12 q^{12} - 90 q^{13} - 48 q^{14} - 66 q^{15} + 2160 q^{16} + 18 q^{18} - 88 q^{19} - 24 q^{20} - 36 q^{21} - 56 q^{22} + 336 q^{23} + 24 q^{24} + 3829 q^{25} - 148 q^{26} - 27 q^{27} + 34 q^{29} - 36 q^{30} - 496 q^{31} + 32 q^{32} + 24 q^{33} + 168 q^{35} + 4860 q^{36} + 706 q^{37} + 104 q^{38} - 138 q^{39} + 112 q^{40} + 1158 q^{41} - 312 q^{42} - 312 q^{43} + 176 q^{44} - 54 q^{45} + 488 q^{47} - 48 q^{48} + 5879 q^{49} + 366 q^{50} - 360 q^{52} - 94 q^{53} + 54 q^{54} + 352 q^{55} - 192 q^{56} - 132 q^{57} + 204 q^{58} + 1748 q^{59} - 264 q^{60} + 1194 q^{61} + 1248 q^{62} + 8640 q^{64} - 1316 q^{65} + 168 q^{66} - 808 q^{67} + 984 q^{69} - 96 q^{70} - 624 q^{71} + 72 q^{72} + 814 q^{73} + 252 q^{74} - 1317 q^{75} - 352 q^{76} + 520 q^{77} + 84 q^{78} - 1624 q^{79} - 96 q^{80} + 10935 q^{81} + 868 q^{82} + 836 q^{83} - 144 q^{84} - 1560 q^{86} - 402 q^{87} - 224 q^{88} - 1098 q^{89} + 252 q^{90} - 2584 q^{91} + 1344 q^{92} - 756 q^{93} + 272 q^{94} - 3096 q^{95} + 96 q^{96} + 1014 q^{97} + 1842 q^{98} + 396 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(1734))\) 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
1734.4.a.a 1734.a 1.a $1$ $102.309$ \(\Q\) None \(-2\) \(-3\) \(5\) \(32\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}-3q^{3}+4q^{4}+5q^{5}+6q^{6}+\cdots\)
1734.4.a.b 1734.a 1.a $1$ $102.309$ \(\Q\) None \(-2\) \(-3\) \(12\) \(11\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}-3q^{3}+4q^{4}+12q^{5}+6q^{6}+\cdots\)
1734.4.a.c 1734.a 1.a $1$ $102.309$ \(\Q\) None \(-2\) \(3\) \(-12\) \(-11\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+3q^{3}+4q^{4}-12q^{5}-6q^{6}+\cdots\)
1734.4.a.d 1734.a 1.a $1$ $102.309$ \(\Q\) None \(-2\) \(3\) \(-6\) \(16\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+3q^{3}+4q^{4}-6q^{5}-6q^{6}+\cdots\)
1734.4.a.e 1734.a 1.a $1$ $102.309$ \(\Q\) None \(-2\) \(3\) \(3\) \(-20\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+3q^{3}+4q^{4}+3q^{5}-6q^{6}+\cdots\)
1734.4.a.f 1734.a 1.a $1$ $102.309$ \(\Q\) None \(2\) \(3\) \(-5\) \(-12\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+3q^{3}+4q^{4}-5q^{5}+6q^{6}+\cdots\)
1734.4.a.g 1734.a 1.a $1$ $102.309$ \(\Q\) None \(2\) \(3\) \(12\) \(22\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+3q^{3}+4q^{4}+12q^{5}+6q^{6}+\cdots\)
1734.4.a.h 1734.a 1.a $2$ $102.309$ \(\Q(\sqrt{15}) \) None \(-4\) \(-6\) \(-12\) \(-16\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}-3q^{3}+4q^{4}+(-6+\beta )q^{5}+\cdots\)
1734.4.a.i 1734.a 1.a $2$ $102.309$ \(\Q(\sqrt{2}) \) None \(-4\) \(-6\) \(-12\) \(-12\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}-3q^{3}+4q^{4}+(-6+\beta )q^{5}+\cdots\)
1734.4.a.j 1734.a 1.a $2$ $102.309$ \(\Q(\sqrt{2}) \) None \(-4\) \(-6\) \(18\) \(12\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}-3q^{3}+4q^{4}+(9+8\beta )q^{5}+\cdots\)
1734.4.a.k 1734.a 1.a $2$ $102.309$ \(\Q(\sqrt{2}) \) None \(-4\) \(6\) \(-18\) \(-12\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+3q^{3}+4q^{4}+(-9+8\beta )q^{5}+\cdots\)
1734.4.a.l 1734.a 1.a $2$ $102.309$ \(\Q(\sqrt{2}) \) None \(-4\) \(6\) \(12\) \(12\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+3q^{3}+4q^{4}+(6+\beta )q^{5}-6q^{6}+\cdots\)
1734.4.a.m 1734.a 1.a $2$ $102.309$ \(\Q(\sqrt{569}) \) None \(4\) \(-6\) \(-9\) \(-12\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}-3q^{3}+4q^{4}+(-4-\beta )q^{5}+\cdots\)
1734.4.a.n 1734.a 1.a $2$ $102.309$ \(\Q(\sqrt{393}) \) None \(4\) \(-6\) \(-3\) \(-22\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}-3q^{3}+4q^{4}+(-1-\beta )q^{5}+\cdots\)
1734.4.a.o 1734.a 1.a $2$ $102.309$ \(\Q(\sqrt{577}) \) None \(4\) \(-6\) \(-1\) \(-25\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}-3q^{3}+4q^{4}-\beta q^{5}-6q^{6}+\cdots\)
1734.4.a.p 1734.a 1.a $2$ $102.309$ \(\Q(\sqrt{33}) \) None \(4\) \(-6\) \(12\) \(-10\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}-3q^{3}+4q^{4}+(6+2\beta )q^{5}+\cdots\)
1734.4.a.q 1734.a 1.a $2$ $102.309$ \(\Q(\sqrt{33}) \) None \(4\) \(6\) \(-12\) \(10\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+3q^{3}+4q^{4}+(-6-2\beta )q^{5}+\cdots\)
1734.4.a.r 1734.a 1.a $2$ $102.309$ \(\Q(\sqrt{577}) \) None \(4\) \(6\) \(1\) \(25\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+3q^{3}+4q^{4}+\beta q^{5}+6q^{6}+\cdots\)
1734.4.a.s 1734.a 1.a $2$ $102.309$ \(\Q(\sqrt{569}) \) None \(4\) \(6\) \(9\) \(12\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+3q^{3}+4q^{4}+(5-\beta )q^{5}+6q^{6}+\cdots\)
1734.4.a.t 1734.a 1.a $3$ $102.309$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(-6\) \(-9\) \(-1\) \(22\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}-3q^{3}+4q^{4}-\beta _{1}q^{5}+6q^{6}+\cdots\)
1734.4.a.u 1734.a 1.a $3$ $102.309$ 3.3.956972.1 None \(-6\) \(-9\) \(5\) \(-2\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}-3q^{3}+4q^{4}+(2+\beta _{2})q^{5}+\cdots\)
1734.4.a.v 1734.a 1.a $3$ $102.309$ 3.3.956972.1 None \(-6\) \(9\) \(-5\) \(2\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+3q^{3}+4q^{4}+(-2-\beta _{2})q^{5}+\cdots\)
1734.4.a.w 1734.a 1.a $3$ $102.309$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(-6\) \(9\) \(1\) \(-22\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+3q^{3}+4q^{4}+\beta _{1}q^{5}-6q^{6}+\cdots\)
1734.4.a.x 1734.a 1.a $3$ $102.309$ \(\Q(\zeta_{18})^+\) None \(6\) \(-9\) \(21\) \(21\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}-3q^{3}+4q^{4}+(7+5\beta _{1})q^{5}+\cdots\)
1734.4.a.y 1734.a 1.a $3$ $102.309$ \(\Q(\zeta_{18})^+\) None \(6\) \(9\) \(-21\) \(-21\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+3q^{3}+4q^{4}+(-7-5\beta _{1}+\cdots)q^{5}+\cdots\)
1734.4.a.z 1734.a 1.a $6$ $102.309$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(-12\) \(-18\) \(-9\) \(-30\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}-3q^{3}+4q^{4}+(-1+\beta _{2})q^{5}+\cdots\)
1734.4.a.ba 1734.a 1.a $6$ $102.309$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(-12\) \(-18\) \(21\) \(-21\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}-3q^{3}+4q^{4}+(4+\beta _{1}-\beta _{3}+\cdots)q^{5}+\cdots\)
1734.4.a.bb 1734.a 1.a $6$ $102.309$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(-12\) \(18\) \(-21\) \(21\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+3q^{3}+4q^{4}+(-3-\beta _{2}+\beta _{3}+\cdots)q^{5}+\cdots\)
1734.4.a.bc 1734.a 1.a $6$ $102.309$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(-12\) \(18\) \(9\) \(30\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+3q^{3}+4q^{4}+(1-\beta _{2})q^{5}+\cdots\)
1734.4.a.bd 1734.a 1.a $6$ $102.309$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(12\) \(-18\) \(-14\) \(-28\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}-3q^{3}+4q^{4}+(-2-\beta _{1}-\beta _{2}+\cdots)q^{5}+\cdots\)
1734.4.a.be 1734.a 1.a $6$ $102.309$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(12\) \(18\) \(14\) \(28\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+3q^{3}+4q^{4}+(2+\beta _{1}+\beta _{2}+\cdots)q^{5}+\cdots\)
1734.4.a.bf 1734.a 1.a $8$ $102.309$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(-16\) \(-24\) \(-28\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}-3q^{3}+4q^{4}+(-4+\beta _{2}-\beta _{3}+\cdots)q^{5}+\cdots\)
1734.4.a.bg 1734.a 1.a $8$ $102.309$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(-16\) \(24\) \(28\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+3q^{3}+4q^{4}+(4+\beta _{1}-\beta _{5}+\cdots)q^{5}+\cdots\)
1734.4.a.bh 1734.a 1.a $8$ $102.309$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(16\) \(-24\) \(12\) \(56\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}-3q^{3}+4q^{4}+(1-\beta _{4})q^{5}+\cdots\)
1734.4.a.bi 1734.a 1.a $8$ $102.309$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(16\) \(24\) \(-12\) \(-56\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+3q^{3}+4q^{4}+(-1+\beta _{4})q^{5}+\cdots\)
1734.4.a.bj 1734.a 1.a $9$ $102.309$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None \(18\) \(-27\) \(-9\) \(30\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}-3q^{3}+4q^{4}+(-1+\beta _{2})q^{5}+\cdots\)
1734.4.a.bk 1734.a 1.a $9$ $102.309$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None \(18\) \(27\) \(9\) \(-30\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+3q^{3}+4q^{4}+(1-\beta _{2})q^{5}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(1734))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(1734)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(17))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(34))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(51))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(102))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(289))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(578))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(867))\)\(^{\oplus 2}\)