Properties

Label 1764.2.a
Level $1764$
Weight $2$
Character orbit 1764.a
Rep. character $\chi_{1764}(1,\cdot)$
Character field $\Q$
Dimension $17$
Newform subspaces $13$
Sturm bound $672$
Trace bound $13$

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Defining parameters

Level: \( N \) \(=\) \( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1764.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 13 \)
Sturm bound: \(672\)
Trace bound: \(13\)
Distinguishing \(T_p\): \(5\), \(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1764))\).

Total New Old
Modular forms 384 17 367
Cusp forms 289 17 272
Eisenstein series 95 0 95

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.
\(-\)\(+\)\(+\)\(-\)\(5\)
\(-\)\(+\)\(-\)\(+\)\(2\)
\(-\)\(-\)\(+\)\(+\)\(4\)
\(-\)\(-\)\(-\)\(-\)\(6\)
Plus space\(+\)\(6\)
Minus space\(-\)\(11\)

Trace form

\( 17q + 4q^{5} + O(q^{10}) \) \( 17q + 4q^{5} - 6q^{11} + 2q^{13} - 4q^{17} + 2q^{23} + 25q^{25} + 4q^{29} - 4q^{31} + 24q^{37} + 12q^{41} + 20q^{43} + 24q^{47} + 6q^{53} - 8q^{55} - 8q^{59} - 10q^{61} + 4q^{65} + 38q^{67} + 20q^{71} + 22q^{73} + 14q^{79} - 16q^{83} + 54q^{85} + 12q^{89} - 22q^{95} - 2q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1764))\) 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
1764.2.a.a \(1\) \(14.086\) \(\Q\) None \(0\) \(0\) \(-3\) \(0\) \(-\) \(-\) \(+\) \(q-3q^{5}+3q^{11}+2q^{13}-3q^{17}-q^{19}+\cdots\)
1764.2.a.b \(1\) \(14.086\) \(\Q\) None \(0\) \(0\) \(-2\) \(0\) \(-\) \(-\) \(-\) \(q-2q^{5}-2q^{11}-4q^{13}-6q^{17}+8q^{19}+\cdots\)
1764.2.a.c \(1\) \(14.086\) \(\Q\) None \(0\) \(0\) \(-2\) \(0\) \(-\) \(-\) \(-\) \(q-2q^{5}-2q^{11}+3q^{13}+8q^{17}+q^{19}+\cdots\)
1764.2.a.d \(1\) \(14.086\) \(\Q\) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(0\) \(-\) \(+\) \(-\) \(q-5q^{13}+q^{19}-5q^{25}-11q^{31}+\cdots\)
1764.2.a.e \(1\) \(14.086\) \(\Q\) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(0\) \(-\) \(+\) \(-\) \(q-2q^{13}-8q^{19}-5q^{25}+4q^{31}+\cdots\)
1764.2.a.f \(1\) \(14.086\) \(\Q\) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(0\) \(-\) \(+\) \(+\) \(q+5q^{13}-q^{19}-5q^{25}+11q^{31}+\cdots\)
1764.2.a.g \(1\) \(14.086\) \(\Q\) None \(0\) \(0\) \(0\) \(0\) \(-\) \(-\) \(-\) \(q+6q^{11}-2q^{13}+4q^{19}+6q^{23}+\cdots\)
1764.2.a.h \(1\) \(14.086\) \(\Q\) None \(0\) \(0\) \(2\) \(0\) \(-\) \(-\) \(+\) \(q+2q^{5}-2q^{11}-3q^{13}-8q^{17}-q^{19}+\cdots\)
1764.2.a.i \(1\) \(14.086\) \(\Q\) None \(0\) \(0\) \(2\) \(0\) \(-\) \(-\) \(-\) \(q+2q^{5}-2q^{11}+4q^{13}+6q^{17}-8q^{19}+\cdots\)
1764.2.a.j \(1\) \(14.086\) \(\Q\) None \(0\) \(0\) \(3\) \(0\) \(-\) \(-\) \(-\) \(q+3q^{5}+3q^{11}-2q^{13}+3q^{17}+q^{19}+\cdots\)
1764.2.a.k \(1\) \(14.086\) \(\Q\) None \(0\) \(0\) \(4\) \(0\) \(-\) \(-\) \(-\) \(q+4q^{5}-2q^{11}+6q^{13}-4q^{17}+4q^{19}+\cdots\)
1764.2.a.l \(2\) \(14.086\) \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(0\) \(0\) \(-\) \(-\) \(+\) \(q+\beta q^{5}-4q^{11}-3\beta q^{13}+\beta q^{17}+\cdots\)
1764.2.a.m \(4\) \(14.086\) \(\Q(\sqrt{2}, \sqrt{7})\) None \(0\) \(0\) \(0\) \(0\) \(-\) \(+\) \(+\) \(q+\beta _{2}q^{5}-\beta _{3}q^{11}-3\beta _{1}q^{13}+\beta _{2}q^{17}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1764))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1764)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(42))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(63))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(84))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(98))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(126))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(147))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(196))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(252))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(294))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(441))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(588))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(882))\)\(^{\oplus 2}\)