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Properties

Label 693.4.a
Level $693$
Weight $4$
Character orbit 693.a
Rep. character $\chi_{693}(1,\cdot)$
Character field $\Q$
Dimension $76$
Newform subspaces $21$
Sturm bound $384$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 693 = 3^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 693.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 21 \)
Sturm bound: \(384\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(2\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(693))\).

Total New Old
Modular forms 296 76 220
Cusp forms 280 76 204
Eisenstein series 16 0 16

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

\(3\)\(7\)\(11\)FrickeDim
\(+\)\(+\)\(+\)$+$\(8\)
\(+\)\(+\)\(-\)$-$\(8\)
\(+\)\(-\)\(+\)$-$\(8\)
\(+\)\(-\)\(-\)$+$\(8\)
\(-\)\(+\)\(+\)$-$\(11\)
\(-\)\(+\)\(-\)$+$\(11\)
\(-\)\(-\)\(+\)$+$\(13\)
\(-\)\(-\)\(-\)$-$\(9\)
Plus space\(+\)\(40\)
Minus space\(-\)\(36\)

Trace form

\( 76 q - 4 q^{2} + 288 q^{4} - 8 q^{5} - 36 q^{8} + O(q^{10}) \) \( 76 q - 4 q^{2} + 288 q^{4} - 8 q^{5} - 36 q^{8} + 80 q^{10} - 44 q^{11} - 144 q^{13} - 56 q^{14} + 1384 q^{16} + 152 q^{17} - 8 q^{19} - 388 q^{20} + 44 q^{22} + 100 q^{23} + 1976 q^{25} + 772 q^{26} - 360 q^{29} + 544 q^{31} - 812 q^{32} + 1072 q^{34} - 56 q^{35} - 116 q^{37} + 1372 q^{38} + 720 q^{40} - 1208 q^{41} - 328 q^{43} - 440 q^{44} + 776 q^{46} - 404 q^{47} + 3724 q^{49} - 1460 q^{50} + 476 q^{52} - 2456 q^{53} - 396 q^{55} - 672 q^{56} + 208 q^{58} + 3188 q^{59} - 2552 q^{61} + 2860 q^{62} + 6816 q^{64} - 856 q^{65} - 92 q^{67} + 3164 q^{68} + 28 q^{70} - 1636 q^{71} - 1968 q^{73} + 4616 q^{74} + 2616 q^{76} - 308 q^{77} + 272 q^{79} + 172 q^{80} + 568 q^{82} - 376 q^{83} + 392 q^{85} + 1680 q^{86} + 924 q^{88} - 28 q^{89} + 1568 q^{91} + 2944 q^{92} - 6932 q^{94} + 3680 q^{95} - 3300 q^{97} - 196 q^{98} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(693))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 3 7 11
693.4.a.a 693.a 1.a $1$ $40.888$ \(\Q\) None \(-5\) \(0\) \(6\) \(7\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-5q^{2}+17q^{4}+6q^{5}+7q^{7}-45q^{8}+\cdots\)
693.4.a.b 693.a 1.a $1$ $40.888$ \(\Q\) None \(-3\) \(0\) \(-12\) \(7\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-3q^{2}+q^{4}-12q^{5}+7q^{7}+21q^{8}+\cdots\)
693.4.a.c 693.a 1.a $1$ $40.888$ \(\Q\) None \(-3\) \(0\) \(14\) \(-7\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-3q^{2}+q^{4}+14q^{5}-7q^{7}+21q^{8}+\cdots\)
693.4.a.d 693.a 1.a $1$ $40.888$ \(\Q\) None \(-2\) \(0\) \(-11\) \(-7\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}-4q^{4}-11q^{5}-7q^{7}+24q^{8}+\cdots\)
693.4.a.e 693.a 1.a $1$ $40.888$ \(\Q\) None \(2\) \(0\) \(-1\) \(-7\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}-4q^{4}-q^{5}-7q^{7}-24q^{8}+\cdots\)
693.4.a.f 693.a 1.a $1$ $40.888$ \(\Q\) None \(3\) \(0\) \(4\) \(-7\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+3q^{2}+q^{4}+4q^{5}-7q^{7}-21q^{8}+\cdots\)
693.4.a.g 693.a 1.a $2$ $40.888$ \(\Q(\sqrt{17}) \) None \(-3\) \(0\) \(-1\) \(14\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(-1-\beta )q^{2}+(-3+3\beta )q^{4}+(-2+\cdots)q^{5}+\cdots\)
693.4.a.h 693.a 1.a $2$ $40.888$ \(\Q(\sqrt{37}) \) None \(-3\) \(0\) \(-2\) \(14\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(-1-\beta )q^{2}+(2+3\beta )q^{4}+(1-4\beta )q^{5}+\cdots\)
693.4.a.i 693.a 1.a $2$ $40.888$ \(\Q(\sqrt{2}) \) None \(2\) \(0\) \(4\) \(14\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{2}+(1+2\beta )q^{4}+(2-3\beta )q^{5}+\cdots\)
693.4.a.j 693.a 1.a $2$ $40.888$ \(\Q(\sqrt{17}) \) None \(3\) \(0\) \(-25\) \(14\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{2}+(-3+3\beta )q^{4}+(-14+\cdots)q^{5}+\cdots\)
693.4.a.k 693.a 1.a $2$ $40.888$ \(\Q(\sqrt{17}) \) None \(3\) \(0\) \(19\) \(14\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{2}+(-3+3\beta )q^{4}+(8+3\beta )q^{5}+\cdots\)
693.4.a.l 693.a 1.a $4$ $40.888$ 4.4.522072.1 None \(2\) \(0\) \(-10\) \(-28\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{2}q^{2}+(6-\beta _{2}-2\beta _{3})q^{4}+(-3+\cdots)q^{5}+\cdots\)
693.4.a.m 693.a 1.a $4$ $40.888$ 4.4.509800.1 None \(4\) \(0\) \(18\) \(-28\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(1-\beta _{2})q^{2}+(6-\beta _{1}-2\beta _{2}-\beta _{3})q^{4}+\cdots\)
693.4.a.n 693.a 1.a $5$ $40.888$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(-5\) \(0\) \(-7\) \(-35\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{2}+(4-\beta _{1}+\beta _{2})q^{4}+\cdots\)
693.4.a.o 693.a 1.a $5$ $40.888$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(-1\) \(0\) \(24\) \(35\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+(9+\beta _{1}+\beta _{2})q^{4}+(4-\beta _{3}+\cdots)q^{5}+\cdots\)
693.4.a.p 693.a 1.a $5$ $40.888$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(1\) \(0\) \(-21\) \(35\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(4+\beta _{1}+\beta _{2})q^{4}+(-4+\beta _{4})q^{5}+\cdots\)
693.4.a.q 693.a 1.a $5$ $40.888$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(1\) \(0\) \(-7\) \(-35\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(4+\beta _{2})q^{4}+(-2+2\beta _{1}+\cdots)q^{5}+\cdots\)
693.4.a.r 693.a 1.a $8$ $40.888$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(-6\) \(0\) \(-10\) \(56\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{2}+(4-\beta _{1}+\beta _{2})q^{4}+\cdots\)
693.4.a.s 693.a 1.a $8$ $40.888$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(-2\) \(0\) \(-10\) \(-56\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+(4+\beta _{2})q^{4}+(-1-\beta _{5})q^{5}+\cdots\)
693.4.a.t 693.a 1.a $8$ $40.888$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(2\) \(0\) \(10\) \(-56\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(4+\beta _{2})q^{4}+(1+\beta _{5})q^{5}+\cdots\)
693.4.a.u 693.a 1.a $8$ $40.888$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(6\) \(0\) \(10\) \(56\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}+(4-\beta _{1}+\beta _{2})q^{4}+(1+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(693))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(693)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(33))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(63))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(77))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(99))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(231))\)\(^{\oplus 2}\)