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Properties

Label 1682.2.a
Level $1682$
Weight $2$
Character orbit 1682.a
Rep. character $\chi_{1682}(1,\cdot)$
Character field $\Q$
Dimension $68$
Newform subspaces $22$
Sturm bound $435$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 1682 = 2 \cdot 29^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1682.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 22 \)
Sturm bound: \(435\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(3\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1682))\).

Total New Old
Modular forms 247 68 179
Cusp forms 188 68 120
Eisenstein series 59 0 59

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

\(2\)\(29\)FrickeDim
\(+\)\(+\)\(+\)\(15\)
\(+\)\(-\)\(-\)\(19\)
\(-\)\(+\)\(-\)\(22\)
\(-\)\(-\)\(+\)\(12\)
Plus space\(+\)\(27\)
Minus space\(-\)\(41\)

Trace form

\( 68 q + 4 q^{3} + 68 q^{4} + 2 q^{5} - 2 q^{6} + 4 q^{7} + 66 q^{9} + O(q^{10}) \) \( 68 q + 4 q^{3} + 68 q^{4} + 2 q^{5} - 2 q^{6} + 4 q^{7} + 66 q^{9} - 4 q^{10} + 4 q^{11} + 4 q^{12} - 2 q^{13} - 8 q^{15} + 68 q^{16} - 4 q^{17} + 8 q^{18} + 8 q^{19} + 2 q^{20} - 8 q^{21} + 2 q^{22} - 4 q^{23} - 2 q^{24} + 70 q^{25} + 4 q^{26} + 4 q^{27} + 4 q^{28} + 4 q^{30} - 12 q^{33} - 12 q^{34} - 4 q^{35} + 66 q^{36} - 14 q^{38} + 8 q^{39} - 4 q^{40} + 4 q^{42} + 4 q^{43} + 4 q^{44} + 14 q^{45} - 4 q^{46} - 24 q^{47} + 4 q^{48} + 76 q^{49} + 8 q^{50} - 12 q^{51} - 2 q^{52} + 2 q^{53} - 20 q^{54} - 32 q^{57} + 2 q^{59} - 8 q^{60} + 4 q^{61} + 4 q^{62} + 68 q^{64} + 4 q^{65} + 14 q^{67} - 4 q^{68} + 4 q^{69} + 8 q^{70} - 8 q^{71} + 8 q^{72} + 8 q^{73} - 18 q^{74} + 8 q^{75} + 8 q^{76} - 8 q^{77} - 16 q^{78} - 8 q^{79} + 2 q^{80} + 52 q^{81} - 4 q^{82} - 6 q^{83} - 8 q^{84} - 20 q^{85} + 18 q^{86} + 2 q^{88} + 16 q^{89} - 16 q^{90} + 4 q^{91} - 4 q^{92} - 8 q^{93} - 4 q^{94} - 24 q^{95} - 2 q^{96} + 8 q^{97} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1682))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 29
1682.2.a.a 1682.a 1.a $1$ $13.431$ \(\Q\) None 1682.2.a.a \(-1\) \(-2\) \(-2\) \(-5\) $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-2q^{3}+q^{4}-2q^{5}+2q^{6}-5q^{7}+\cdots\)
1682.2.a.b 1682.a 1.a $1$ $13.431$ \(\Q\) None 1682.2.a.b \(-1\) \(-2\) \(0\) \(-1\) $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-2q^{3}+q^{4}+2q^{6}-q^{7}-q^{8}+\cdots\)
1682.2.a.c 1682.a 1.a $1$ $13.431$ \(\Q\) None 58.2.b.a \(-1\) \(-1\) \(-1\) \(-2\) $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}-q^{5}+q^{6}-2q^{7}+\cdots\)
1682.2.a.d 1682.a 1.a $1$ $13.431$ \(\Q\) None 58.2.a.b \(-1\) \(1\) \(1\) \(-2\) $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}+q^{5}-q^{6}-2q^{7}+\cdots\)
1682.2.a.e 1682.a 1.a $1$ $13.431$ \(\Q\) None 1682.2.a.e \(-1\) \(3\) \(0\) \(4\) $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+3q^{3}+q^{4}-3q^{6}+4q^{7}-q^{8}+\cdots\)
1682.2.a.f 1682.a 1.a $1$ $13.431$ \(\Q\) None 1682.2.a.e \(1\) \(-3\) \(0\) \(4\) $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-3q^{3}+q^{4}-3q^{6}+4q^{7}+q^{8}+\cdots\)
1682.2.a.g 1682.a 1.a $1$ $13.431$ \(\Q\) None 58.2.b.a \(1\) \(1\) \(-1\) \(-2\) $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}-q^{5}+q^{6}-2q^{7}+\cdots\)
1682.2.a.h 1682.a 1.a $1$ $13.431$ \(\Q\) None 1682.2.a.a \(1\) \(2\) \(-2\) \(-5\) $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+2q^{3}+q^{4}-2q^{5}+2q^{6}-5q^{7}+\cdots\)
1682.2.a.i 1682.a 1.a $1$ $13.431$ \(\Q\) None 1682.2.a.b \(1\) \(2\) \(0\) \(-1\) $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+2q^{3}+q^{4}+2q^{6}-q^{7}+q^{8}+\cdots\)
1682.2.a.j 1682.a 1.a $1$ $13.431$ \(\Q\) None 58.2.a.a \(1\) \(3\) \(-3\) \(-2\) $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+3q^{3}+q^{4}-3q^{5}+3q^{6}-2q^{7}+\cdots\)
1682.2.a.k 1682.a 1.a $2$ $13.431$ \(\Q(\sqrt{5}) \) None 1682.2.a.k \(-2\) \(-1\) \(4\) \(6\) $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-\beta q^{3}+q^{4}+2q^{5}+\beta q^{6}+(4+\cdots)q^{7}+\cdots\)
1682.2.a.l 1682.a 1.a $2$ $13.431$ \(\Q(\sqrt{5}) \) None 1682.2.a.k \(2\) \(1\) \(4\) \(6\) $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+\beta q^{3}+q^{4}+2q^{5}+\beta q^{6}+(4+\cdots)q^{7}+\cdots\)
1682.2.a.m 1682.a 1.a $3$ $13.431$ \(\Q(\zeta_{14})^+\) None 58.2.d.a \(-3\) \(-2\) \(5\) \(1\) $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+(-1-\beta _{2})q^{3}+q^{4}+(1+\beta _{1}+\cdots)q^{5}+\cdots\)
1682.2.a.n 1682.a 1.a $3$ $13.431$ \(\Q(\zeta_{14})^+\) None 58.2.d.a \(3\) \(2\) \(5\) \(1\) $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+(1-\beta _{1})q^{3}+q^{4}+(2+\beta _{2})q^{5}+\cdots\)
1682.2.a.o 1682.a 1.a $4$ $13.431$ \(\Q(\zeta_{15})^+\) None 1682.2.a.o \(-4\) \(2\) \(-3\) \(-5\) $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+(1-\beta _{1}-\beta _{2})q^{3}+q^{4}+(-1+\cdots)q^{5}+\cdots\)
1682.2.a.p 1682.a 1.a $4$ $13.431$ \(\Q(\zeta_{15})^+\) None 1682.2.a.o \(4\) \(-2\) \(-3\) \(-5\) $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+(-1+\beta _{1}+\beta _{2})q^{3}+q^{4}+(-1+\cdots)q^{5}+\cdots\)
1682.2.a.q 1682.a 1.a $6$ $13.431$ 6.6.13716913.1 None 58.2.d.b \(-6\) \(-2\) \(0\) \(-3\) $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+(-1+\beta _{1}+\beta _{2})q^{3}+q^{4}+(-\beta _{1}+\cdots)q^{5}+\cdots\)
1682.2.a.r 1682.a 1.a $6$ $13.431$ \(\Q(\zeta_{28})^+\) None 58.2.e.a \(-6\) \(8\) \(-6\) \(2\) $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+(1+\beta _{4})q^{3}+q^{4}+(-1-\beta _{1}+\cdots)q^{5}+\cdots\)
1682.2.a.s 1682.a 1.a $6$ $13.431$ \(\Q(\zeta_{28})^+\) None 58.2.e.a \(6\) \(-8\) \(-6\) \(2\) $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+(-2+\beta _{2}+\beta _{4})q^{3}+q^{4}+(-1+\cdots)q^{5}+\cdots\)
1682.2.a.t 1682.a 1.a $6$ $13.431$ 6.6.13716913.1 None 58.2.d.b \(6\) \(2\) \(0\) \(-3\) $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+\beta _{1}q^{3}+q^{4}+(-1+\beta _{1}+\beta _{2}+\cdots)q^{5}+\cdots\)
1682.2.a.u 1682.a 1.a $8$ $13.431$ 8.8.\(\cdots\).1 None 1682.2.a.u \(-8\) \(-1\) \(5\) \(7\) $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-\beta _{1}q^{3}+q^{4}+(1-\beta _{1}-\beta _{4}+\cdots)q^{5}+\cdots\)
1682.2.a.v 1682.a 1.a $8$ $13.431$ 8.8.\(\cdots\).1 None 1682.2.a.u \(8\) \(1\) \(5\) \(7\) $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+\beta _{1}q^{3}+q^{4}+(1-\beta _{1}-\beta _{4}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1682))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1682)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(29))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(58))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(841))\)\(^{\oplus 2}\)