Properties

Label 322.2.a
Level $322$
Weight $2$
Character orbit 322.a
Rep. character $\chi_{322}(1,\cdot)$
Character field $\Q$
Dimension $11$
Newform subspaces $7$
Sturm bound $96$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 322 = 2 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 322.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 7 \)
Sturm bound: \(96\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(3\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(322))\).

Total New Old
Modular forms 52 11 41
Cusp forms 45 11 34
Eisenstein series 7 0 7

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

\(2\)\(7\)\(23\)FrickeDim.
\(+\)\(+\)\(+\)\(+\)\(2\)
\(+\)\(-\)\(+\)\(-\)\(1\)
\(+\)\(-\)\(-\)\(+\)\(1\)
\(-\)\(+\)\(+\)\(-\)\(3\)
\(-\)\(+\)\(-\)\(+\)\(1\)
\(-\)\(-\)\(-\)\(-\)\(3\)
Plus space\(+\)\(4\)
Minus space\(-\)\(7\)

Trace form

\( 11q + 3q^{2} + 11q^{4} - 2q^{5} - q^{7} + 3q^{8} + 15q^{9} + O(q^{10}) \) \( 11q + 3q^{2} + 11q^{4} - 2q^{5} - q^{7} + 3q^{8} + 15q^{9} + 6q^{10} - 2q^{13} - q^{14} - 16q^{15} + 11q^{16} - 10q^{17} + 7q^{18} - 8q^{19} - 2q^{20} + 4q^{21} - q^{23} - 3q^{25} - 2q^{26} - 24q^{27} - q^{28} + 2q^{29} - 16q^{31} + 3q^{32} + 14q^{34} - 2q^{35} + 15q^{36} - 10q^{37} + 8q^{38} - 40q^{39} + 6q^{40} - 18q^{41} - 4q^{42} - 8q^{43} + 6q^{45} + 3q^{46} + 8q^{47} + 11q^{49} + 5q^{50} + 32q^{51} - 2q^{52} - 18q^{53} + 24q^{54} - 8q^{55} - q^{56} - 14q^{58} - 16q^{60} + 14q^{61} - 16q^{62} - 13q^{63} + 11q^{64} + 36q^{65} - 16q^{66} - 10q^{68} - 4q^{69} - 2q^{70} - 24q^{71} + 7q^{72} - 10q^{73} - 10q^{74} - 24q^{75} - 8q^{76} + 12q^{77} - 8q^{78} - 32q^{79} - 2q^{80} + 3q^{81} + 30q^{82} + 16q^{83} + 4q^{84} + 44q^{85} + 24q^{86} + 24q^{87} + 6q^{89} - 34q^{90} + 10q^{91} - q^{92} + 24q^{93} + 8q^{94} - 8q^{95} + 6q^{97} + 3q^{98} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(322))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 7 23
322.2.a.a \(1\) \(2.571\) \(\Q\) None \(-1\) \(0\) \(-2\) \(1\) \(+\) \(-\) \(-\) \(q-q^{2}+q^{4}-2q^{5}+q^{7}-q^{8}-3q^{9}+\cdots\)
322.2.a.b \(1\) \(2.571\) \(\Q\) None \(-1\) \(2\) \(0\) \(1\) \(+\) \(-\) \(+\) \(q-q^{2}+2q^{3}+q^{4}-2q^{6}+q^{7}-q^{8}+\cdots\)
322.2.a.c \(1\) \(2.571\) \(\Q\) None \(1\) \(-2\) \(-2\) \(-1\) \(-\) \(+\) \(-\) \(q+q^{2}-2q^{3}+q^{4}-2q^{5}-2q^{6}-q^{7}+\cdots\)
322.2.a.d \(1\) \(2.571\) \(\Q\) None \(1\) \(2\) \(-2\) \(1\) \(-\) \(-\) \(-\) \(q+q^{2}+2q^{3}+q^{4}-2q^{5}+2q^{6}+q^{7}+\cdots\)
322.2.a.e \(2\) \(2.571\) \(\Q(\sqrt{5}) \) None \(-2\) \(-2\) \(-2\) \(-2\) \(+\) \(+\) \(+\) \(q-q^{2}+(-1-\beta )q^{3}+q^{4}+(-1+\beta )q^{5}+\cdots\)
322.2.a.f \(2\) \(2.571\) \(\Q(\sqrt{3}) \) None \(2\) \(-2\) \(2\) \(2\) \(-\) \(-\) \(-\) \(q+q^{2}+(-1+\beta )q^{3}+q^{4}+(1+\beta )q^{5}+\cdots\)
322.2.a.g \(3\) \(2.571\) 3.3.316.1 None \(3\) \(2\) \(4\) \(-3\) \(-\) \(+\) \(+\) \(q+q^{2}+(1+\beta _{2})q^{3}+q^{4}+(1-\beta _{2})q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(322))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(322)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(23))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(46))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(161))\)\(^{\oplus 2}\)