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Properties

Label 1694.2.a
Level $1694$
Weight $2$
Character orbit 1694.a
Rep. character $\chi_{1694}(1,\cdot)$
Character field $\Q$
Dimension $54$
Newform subspaces $27$
Sturm bound $528$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 1694 = 2 \cdot 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1694.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 27 \)
Sturm bound: \(528\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(3\), \(5\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1694))\).

Total New Old
Modular forms 288 54 234
Cusp forms 241 54 187
Eisenstein series 47 0 47

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

\(2\)\(7\)\(11\)FrickeDim
\(+\)\(+\)\(+\)$+$\(4\)
\(+\)\(+\)\(-\)$-$\(9\)
\(+\)\(-\)\(+\)$-$\(10\)
\(+\)\(-\)\(-\)$+$\(4\)
\(-\)\(+\)\(+\)$-$\(8\)
\(-\)\(+\)\(-\)$+$\(6\)
\(-\)\(-\)\(+\)$+$\(2\)
\(-\)\(-\)\(-\)$-$\(11\)
Plus space\(+\)\(16\)
Minus space\(-\)\(38\)

Trace form

\( 54 q + 6 q^{3} + 54 q^{4} + 2 q^{5} + 2 q^{6} + 62 q^{9} + O(q^{10}) \) \( 54 q + 6 q^{3} + 54 q^{4} + 2 q^{5} + 2 q^{6} + 62 q^{9} - 6 q^{10} + 6 q^{12} + 6 q^{13} - 2 q^{14} + 16 q^{15} + 54 q^{16} - 4 q^{18} + 10 q^{19} + 2 q^{20} + 6 q^{21} + 2 q^{24} + 66 q^{25} - 2 q^{26} + 36 q^{27} + 8 q^{29} + 16 q^{30} + 36 q^{31} + 4 q^{34} - 2 q^{35} + 62 q^{36} + 12 q^{37} + 14 q^{38} + 8 q^{39} - 6 q^{40} - 24 q^{41} - 2 q^{42} + 12 q^{43} - 14 q^{45} + 8 q^{46} + 4 q^{47} + 6 q^{48} + 54 q^{49} + 4 q^{50} - 12 q^{51} + 6 q^{52} - 8 q^{53} + 20 q^{54} - 2 q^{56} - 4 q^{57} + 2 q^{59} + 16 q^{60} + 10 q^{61} - 12 q^{62} - 12 q^{63} + 54 q^{64} + 4 q^{65} + 36 q^{67} + 24 q^{69} + 2 q^{70} + 16 q^{71} - 4 q^{72} - 4 q^{73} + 12 q^{74} + 42 q^{75} + 10 q^{76} + 16 q^{78} - 24 q^{79} + 2 q^{80} + 82 q^{81} - 4 q^{82} + 10 q^{83} + 6 q^{84} + 4 q^{85} + 16 q^{86} - 36 q^{87} + 12 q^{89} - 6 q^{90} + 6 q^{91} + 32 q^{93} - 4 q^{94} - 32 q^{95} + 2 q^{96} + 8 q^{97} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1694))\) 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 7 11
1694.2.a.a 1694.a 1.a $1$ $13.527$ \(\Q\) None \(-1\) \(-1\) \(2\) \(-1\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+2q^{5}+q^{6}-q^{7}+\cdots\)
1694.2.a.b 1694.a 1.a $1$ $13.527$ \(\Q\) None \(-1\) \(0\) \(-1\) \(1\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}-q^{5}+q^{7}-q^{8}-3q^{9}+\cdots\)
1694.2.a.c 1694.a 1.a $1$ $13.527$ \(\Q\) None \(-1\) \(0\) \(2\) \(1\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}+2q^{5}+q^{7}-q^{8}-3q^{9}+\cdots\)
1694.2.a.d 1694.a 1.a $1$ $13.527$ \(\Q\) None \(-1\) \(3\) \(2\) \(-1\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+3q^{3}+q^{4}+2q^{5}-3q^{6}-q^{7}+\cdots\)
1694.2.a.e 1694.a 1.a $1$ $13.527$ \(\Q\) None \(1\) \(-2\) \(0\) \(-1\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-2q^{3}+q^{4}-2q^{6}-q^{7}+q^{8}+\cdots\)
1694.2.a.f 1694.a 1.a $1$ $13.527$ \(\Q\) None \(1\) \(-1\) \(2\) \(1\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}+2q^{5}-q^{6}+q^{7}+\cdots\)
1694.2.a.g 1694.a 1.a $1$ $13.527$ \(\Q\) None \(1\) \(0\) \(-4\) \(1\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}-4q^{5}+q^{7}+q^{8}-3q^{9}+\cdots\)
1694.2.a.h 1694.a 1.a $1$ $13.527$ \(\Q\) None \(1\) \(0\) \(-1\) \(-1\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}-q^{5}-q^{7}+q^{8}-3q^{9}+\cdots\)
1694.2.a.i 1694.a 1.a $1$ $13.527$ \(\Q\) None \(1\) \(2\) \(2\) \(1\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+2q^{3}+q^{4}+2q^{5}+2q^{6}+q^{7}+\cdots\)
1694.2.a.j 1694.a 1.a $1$ $13.527$ \(\Q\) None \(1\) \(3\) \(2\) \(1\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+3q^{3}+q^{4}+2q^{5}+3q^{6}+q^{7}+\cdots\)
1694.2.a.k 1694.a 1.a $2$ $13.527$ \(\Q(\sqrt{5}) \) None \(-2\) \(-2\) \(-3\) \(-2\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+(-1-\beta )q^{5}+q^{6}+\cdots\)
1694.2.a.l 1694.a 1.a $2$ $13.527$ \(\Q(\sqrt{5}) \) None \(-2\) \(-2\) \(2\) \(-2\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+(-1-\beta )q^{3}+q^{4}+(1+\beta )q^{5}+\cdots\)
1694.2.a.m 1694.a 1.a $2$ $13.527$ \(\Q(\sqrt{5}) \) None \(-2\) \(-1\) \(4\) \(-2\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-\beta q^{3}+q^{4}+2q^{5}+\beta q^{6}-q^{7}+\cdots\)
1694.2.a.n 1694.a 1.a $2$ $13.527$ \(\Q(\sqrt{5}) \) None \(-2\) \(0\) \(-3\) \(-2\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+(1-2\beta )q^{3}+q^{4}+(-1-\beta )q^{5}+\cdots\)
1694.2.a.o 1694.a 1.a $2$ $13.527$ \(\Q(\sqrt{5}) \) None \(-2\) \(0\) \(-1\) \(2\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+(1-2\beta )q^{3}+q^{4}+(-1+\beta )q^{5}+\cdots\)
1694.2.a.p 1694.a 1.a $2$ $13.527$ \(\Q(\sqrt{3}) \) None \(-2\) \(2\) \(2\) \(2\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+(1+\beta )q^{3}+q^{4}+(1+\beta )q^{5}+\cdots\)
1694.2.a.q 1694.a 1.a $2$ $13.527$ \(\Q(\sqrt{5}) \) None \(2\) \(-2\) \(-3\) \(2\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}+(-1-\beta )q^{5}-q^{6}+\cdots\)
1694.2.a.r 1694.a 1.a $2$ $13.527$ \(\Q(\sqrt{5}) \) None \(2\) \(-1\) \(4\) \(2\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-\beta q^{3}+q^{4}+2q^{5}-\beta q^{6}+q^{7}+\cdots\)
1694.2.a.s 1694.a 1.a $2$ $13.527$ \(\Q(\sqrt{5}) \) None \(2\) \(0\) \(-3\) \(2\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+(1-2\beta )q^{3}+q^{4}+(-1-\beta )q^{5}+\cdots\)
1694.2.a.t 1694.a 1.a $2$ $13.527$ \(\Q(\sqrt{5}) \) None \(2\) \(0\) \(-1\) \(-2\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+(1-2\beta )q^{3}+q^{4}+(-1+\beta )q^{5}+\cdots\)
1694.2.a.u 1694.a 1.a $2$ $13.527$ \(\Q(\sqrt{3}) \) None \(2\) \(2\) \(2\) \(-2\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+(1+\beta )q^{3}+q^{4}+(1+\beta )q^{5}+\cdots\)
1694.2.a.v 1694.a 1.a $3$ $13.527$ 3.3.2024.1 None \(-3\) \(2\) \(-1\) \(-3\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+(1-\beta _{1})q^{3}+q^{4}-\beta _{2}q^{5}+(-1+\cdots)q^{6}+\cdots\)
1694.2.a.w 1694.a 1.a $3$ $13.527$ 3.3.2024.1 None \(3\) \(2\) \(-1\) \(3\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+(1-\beta _{1})q^{3}+q^{4}-\beta _{2}q^{5}+(1+\cdots)q^{6}+\cdots\)
1694.2.a.x 1694.a 1.a $4$ $13.527$ 4.4.7625.1 None \(-4\) \(-3\) \(-5\) \(4\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+(-1+\beta _{1})q^{3}+q^{4}+(-1-\beta _{1}+\cdots)q^{5}+\cdots\)
1694.2.a.y 1694.a 1.a $4$ $13.527$ 4.4.13824.1 None \(-4\) \(4\) \(4\) \(4\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+(1+\beta _{1})q^{3}+q^{4}+(1+\beta _{2}-\beta _{3})q^{5}+\cdots\)
1694.2.a.z 1694.a 1.a $4$ $13.527$ 4.4.7625.1 None \(4\) \(-3\) \(-5\) \(-4\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+(-1+\beta _{1})q^{3}+q^{4}+(-1-\beta _{1}+\cdots)q^{5}+\cdots\)
1694.2.a.ba 1694.a 1.a $4$ $13.527$ 4.4.13824.1 None \(4\) \(4\) \(4\) \(-4\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+(1+\beta _{1})q^{3}+q^{4}+(1+\beta _{2}-\beta _{3})q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1694))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1694)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(22))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(77))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(121))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(154))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(242))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(847))\)\(^{\oplus 2}\)