Properties

Label 648.4.a
Level $648$
Weight $4$
Character orbit 648.a
Rep. character $\chi_{648}(1,\cdot)$
Character field $\Q$
Dimension $36$
Newform subspaces $12$
Sturm bound $432$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 648 = 2^{3} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 648.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 12 \)
Sturm bound: \(432\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(648))\).

Total New Old
Modular forms 348 36 312
Cusp forms 300 36 264
Eisenstein series 48 0 48

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

\(2\)\(3\)FrickeDim.
\(+\)\(+\)\(+\)\(10\)
\(+\)\(-\)\(-\)\(8\)
\(-\)\(+\)\(-\)\(9\)
\(-\)\(-\)\(+\)\(9\)
Plus space\(+\)\(19\)
Minus space\(-\)\(17\)

Trace form

\( 36q + O(q^{10}) \) \( 36q - 18q^{13} - 90q^{19} + 666q^{25} + 180q^{31} - 594q^{37} + 90q^{43} + 1368q^{49} - 612q^{55} + 738q^{61} + 774q^{67} + 144q^{73} + 252q^{79} + 2826q^{85} + 108q^{91} + 342q^{97} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(648))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 3
648.4.a.a \(1\) \(38.233\) \(\Q\) None \(0\) \(0\) \(-5\) \(36\) \(+\) \(+\) \(q-5q^{5}+6^{2}q^{7}+2^{6}q^{11}-65q^{13}+\cdots\)
648.4.a.b \(1\) \(38.233\) \(\Q\) None \(0\) \(0\) \(5\) \(36\) \(-\) \(+\) \(q+5q^{5}+6^{2}q^{7}-2^{6}q^{11}-65q^{13}+\cdots\)
648.4.a.c \(2\) \(38.233\) \(\Q(\sqrt{201}) \) None \(0\) \(0\) \(-8\) \(-30\) \(-\) \(+\) \(q+(-4-\beta )q^{5}+(-15+\beta )q^{7}+(23+\cdots)q^{11}+\cdots\)
648.4.a.d \(2\) \(38.233\) \(\Q(\sqrt{129}) \) None \(0\) \(0\) \(-4\) \(-6\) \(+\) \(+\) \(q+(-2-\beta )q^{5}+(-3-\beta )q^{7}+(-5+\cdots)q^{11}+\cdots\)
648.4.a.e \(2\) \(38.233\) \(\Q(\sqrt{129}) \) None \(0\) \(0\) \(4\) \(-6\) \(-\) \(+\) \(q+(2+\beta )q^{5}+(-3-\beta )q^{7}+(5+3\beta )q^{11}+\cdots\)
648.4.a.f \(2\) \(38.233\) \(\Q(\sqrt{201}) \) None \(0\) \(0\) \(8\) \(-30\) \(+\) \(+\) \(q+(4+\beta )q^{5}+(-15+\beta )q^{7}+(-23+\cdots)q^{11}+\cdots\)
648.4.a.g \(4\) \(38.233\) 4.4.29952.1 None \(0\) \(0\) \(-8\) \(0\) \(+\) \(-\) \(q+(-2-\beta _{1})q^{5}+(\beta _{1}-\beta _{2})q^{7}+(-2+\cdots)q^{11}+\cdots\)
648.4.a.h \(4\) \(38.233\) 4.4.72153.1 None \(0\) \(0\) \(-5\) \(-3\) \(+\) \(-\) \(q+(-1-\beta _{2})q^{5}+(-1-\beta _{3})q^{7}+(4+\cdots)q^{11}+\cdots\)
648.4.a.i \(4\) \(38.233\) 4.4.72153.1 None \(0\) \(0\) \(5\) \(-3\) \(-\) \(+\) \(q+(1+\beta _{2})q^{5}+(-1-\beta _{3})q^{7}+(-4+\cdots)q^{11}+\cdots\)
648.4.a.j \(4\) \(38.233\) 4.4.29952.1 None \(0\) \(0\) \(8\) \(0\) \(-\) \(-\) \(q+(2-\beta _{1})q^{5}+(-\beta _{1}+\beta _{2})q^{7}+(2-\beta _{1}+\cdots)q^{11}+\cdots\)
648.4.a.k \(5\) \(38.233\) \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(0\) \(0\) \(-5\) \(3\) \(-\) \(-\) \(q+(-1-\beta _{2})q^{5}+(1+\beta _{1})q^{7}+(5-2\beta _{2}+\cdots)q^{11}+\cdots\)
648.4.a.l \(5\) \(38.233\) \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(0\) \(0\) \(5\) \(3\) \(+\) \(+\) \(q+(1+\beta _{2})q^{5}+(1+\beta _{1})q^{7}+(-5+2\beta _{2}+\cdots)q^{11}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(648))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(648)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 5}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(18))\)\(^{\oplus 9}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(27))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(54))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(72))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(81))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(108))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(162))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(216))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(324))\)\(^{\oplus 2}\)