Properties

Label 588.4.a
Level $588$
Weight $4$
Character orbit 588.a
Rep. character $\chi_{588}(1,\cdot)$
Character field $\Q$
Dimension $21$
Newform subspaces $11$
Sturm bound $448$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 588.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 11 \)
Sturm bound: \(448\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(588))\).

Total New Old
Modular forms 360 21 339
Cusp forms 312 21 291
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.

\(2\)\(3\)\(7\)FrickeDim.
\(-\)\(+\)\(+\)\(-\)\(6\)
\(-\)\(+\)\(-\)\(+\)\(5\)
\(-\)\(-\)\(+\)\(+\)\(6\)
\(-\)\(-\)\(-\)\(-\)\(4\)
Plus space\(+\)\(11\)
Minus space\(-\)\(10\)

Trace form

\( 21q - 3q^{3} - 2q^{5} + 189q^{9} + O(q^{10}) \) \( 21q - 3q^{3} - 2q^{5} + 189q^{9} - 76q^{11} - 106q^{13} + 30q^{15} - 118q^{17} + 84q^{19} + 192q^{23} + 447q^{25} - 27q^{27} + 302q^{29} + 272q^{31} - 12q^{33} - 150q^{37} - 366q^{39} - 78q^{41} - 332q^{43} - 18q^{45} + 288q^{47} + 270q^{51} + 590q^{53} + 376q^{55} + 480q^{57} + 652q^{59} - 1354q^{61} - 1172q^{65} + 1188q^{67} + 168q^{69} - 640q^{71} - 494q^{73} - 1077q^{75} - 644q^{79} + 1701q^{81} + 1844q^{83} - 2132q^{85} + 1182q^{87} + 1074q^{89} + 1764q^{93} - 4208q^{95} + 1546q^{97} - 684q^{99} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(588))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 3 7
588.4.a.a \(1\) \(34.693\) \(\Q\) None \(0\) \(-3\) \(-14\) \(0\) \(-\) \(+\) \(-\) \(q-3q^{3}-14q^{5}+9q^{9}+4q^{11}-54q^{13}+\cdots\)
588.4.a.b \(1\) \(34.693\) \(\Q\) None \(0\) \(-3\) \(4\) \(0\) \(-\) \(+\) \(-\) \(q-3q^{3}+4q^{5}+9q^{9}-20q^{11}-4q^{13}+\cdots\)
588.4.a.c \(1\) \(34.693\) \(\Q\) None \(0\) \(-3\) \(18\) \(0\) \(-\) \(+\) \(-\) \(q-3q^{3}+18q^{5}+9q^{9}+6^{2}q^{11}+10q^{13}+\cdots\)
588.4.a.d \(1\) \(34.693\) \(\Q\) None \(0\) \(3\) \(-6\) \(0\) \(-\) \(-\) \(-\) \(q+3q^{3}-6q^{5}+9q^{9}+6^{2}q^{11}-62q^{13}+\cdots\)
588.4.a.e \(1\) \(34.693\) \(\Q\) None \(0\) \(3\) \(-4\) \(0\) \(-\) \(-\) \(-\) \(q+3q^{3}-4q^{5}+9q^{9}-20q^{11}+4q^{13}+\cdots\)
588.4.a.f \(2\) \(34.693\) \(\Q(\sqrt{193}) \) None \(0\) \(-6\) \(-11\) \(0\) \(-\) \(+\) \(-\) \(q-3q^{3}+(-5-\beta )q^{5}+9q^{9}+(1-7\beta )q^{11}+\cdots\)
588.4.a.g \(2\) \(34.693\) \(\Q(\sqrt{57}) \) None \(0\) \(-6\) \(-3\) \(0\) \(-\) \(+\) \(+\) \(q-3q^{3}+(-2-\beta )q^{5}+9q^{9}+(-26+\cdots)q^{11}+\cdots\)
588.4.a.h \(2\) \(34.693\) \(\Q(\sqrt{57}) \) None \(0\) \(6\) \(3\) \(0\) \(-\) \(-\) \(-\) \(q+3q^{3}+(2+\beta )q^{5}+9q^{9}+(-26-\beta )q^{11}+\cdots\)
588.4.a.i \(2\) \(34.693\) \(\Q(\sqrt{193}) \) None \(0\) \(6\) \(11\) \(0\) \(-\) \(-\) \(+\) \(q+3q^{3}+(6-\beta )q^{5}+9q^{9}+(-6+7\beta )q^{11}+\cdots\)
588.4.a.j \(4\) \(34.693\) 4.4.136768.1 None \(0\) \(-12\) \(0\) \(0\) \(-\) \(+\) \(+\) \(q-3q^{3}-\beta _{1}q^{5}+9q^{9}+(3\beta _{1}-\beta _{3})q^{11}+\cdots\)
588.4.a.k \(4\) \(34.693\) 4.4.136768.1 None \(0\) \(12\) \(0\) \(0\) \(-\) \(-\) \(+\) \(q+3q^{3}+\beta _{1}q^{5}+9q^{9}+(3\beta _{1}-\beta _{3})q^{11}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(588))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(588)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(28))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(42))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(84))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(98))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(147))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(196))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(294))\)\(^{\oplus 2}\)