Properties

Label 1323.2.a
Level $1323$
Weight $2$
Character orbit 1323.a
Rep. character $\chi_{1323}(1,\cdot)$
Character field $\Q$
Dimension $55$
Newform subspaces $31$
Sturm bound $336$
Trace bound $13$

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Defining parameters

Level: \( N \) \(=\) \( 1323 = 3^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1323.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 31 \)
Sturm bound: \(336\)
Trace bound: \(13\)
Distinguishing \(T_p\): \(2\), \(5\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1323))\).

Total New Old
Modular forms 192 55 137
Cusp forms 145 55 90
Eisenstein series 47 0 47

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

\(3\)\(7\)FrickeDim.
\(+\)\(+\)\(+\)\(12\)
\(+\)\(-\)\(-\)\(16\)
\(-\)\(+\)\(-\)\(15\)
\(-\)\(-\)\(+\)\(12\)
Plus space\(+\)\(24\)
Minus space\(-\)\(31\)

Trace form

\( 55q + 54q^{4} + O(q^{10}) \) \( 55q + 54q^{4} + 4q^{10} + 7q^{13} + 32q^{16} - 5q^{19} - 8q^{22} + 73q^{25} - 16q^{31} + 36q^{34} + 19q^{37} + 48q^{40} - 46q^{43} + 4q^{46} + 18q^{52} - 52q^{55} - 8q^{58} + 13q^{61} + 56q^{64} - 47q^{67} - q^{73} - 54q^{76} - 67q^{79} - 28q^{82} + 28q^{85} + 12q^{88} + 12q^{94} + 47q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1323))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 3 7
1323.2.a.a \(1\) \(10.564\) \(\Q\) None \(-2\) \(0\) \(-3\) \(0\) \(-\) \(-\) \(q-2q^{2}+2q^{4}-3q^{5}+6q^{10}-2q^{11}+\cdots\)
1323.2.a.b \(1\) \(10.564\) \(\Q\) None \(-2\) \(0\) \(1\) \(0\) \(-\) \(-\) \(q-2q^{2}+2q^{4}+q^{5}-2q^{10}-4q^{11}+\cdots\)
1323.2.a.c \(1\) \(10.564\) \(\Q\) None \(-2\) \(0\) \(3\) \(0\) \(+\) \(-\) \(q-2q^{2}+2q^{4}+3q^{5}-6q^{10}-2q^{11}+\cdots\)
1323.2.a.d \(1\) \(10.564\) \(\Q\) None \(-1\) \(0\) \(-4\) \(0\) \(+\) \(-\) \(q-q^{2}-q^{4}-4q^{5}+3q^{8}+4q^{10}+\cdots\)
1323.2.a.e \(1\) \(10.564\) \(\Q\) None \(-1\) \(0\) \(-3\) \(0\) \(-\) \(-\) \(q-q^{2}-q^{4}-3q^{5}+3q^{8}+3q^{10}+\cdots\)
1323.2.a.f \(1\) \(10.564\) \(\Q\) None \(-1\) \(0\) \(3\) \(0\) \(+\) \(-\) \(q-q^{2}-q^{4}+3q^{5}+3q^{8}-3q^{10}+\cdots\)
1323.2.a.g \(1\) \(10.564\) \(\Q\) None \(-1\) \(0\) \(4\) \(0\) \(-\) \(+\) \(q-q^{2}-q^{4}+4q^{5}+3q^{8}-4q^{10}+\cdots\)
1323.2.a.h \(1\) \(10.564\) \(\Q\) None \(0\) \(0\) \(-3\) \(0\) \(-\) \(-\) \(q-2q^{4}-3q^{5}+6q^{11}+4q^{13}+4q^{16}+\cdots\)
1323.2.a.i \(1\) \(10.564\) \(\Q\) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(0\) \(+\) \(-\) \(q-2q^{4}-5q^{13}+4q^{16}+7q^{19}-5q^{25}+\cdots\)
1323.2.a.j \(1\) \(10.564\) \(\Q\) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(0\) \(-\) \(-\) \(q-2q^{4}-2q^{13}+4q^{16}+7q^{19}-5q^{25}+\cdots\)
1323.2.a.k \(1\) \(10.564\) \(\Q\) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(0\) \(+\) \(+\) \(q-2q^{4}+2q^{13}+4q^{16}-7q^{19}-5q^{25}+\cdots\)
1323.2.a.l \(1\) \(10.564\) \(\Q\) None \(0\) \(0\) \(3\) \(0\) \(+\) \(-\) \(q-2q^{4}+3q^{5}-6q^{11}+4q^{13}+4q^{16}+\cdots\)
1323.2.a.m \(1\) \(10.564\) \(\Q\) None \(1\) \(0\) \(-4\) \(0\) \(-\) \(+\) \(q+q^{2}-q^{4}-4q^{5}-3q^{8}-4q^{10}+\cdots\)
1323.2.a.n \(1\) \(10.564\) \(\Q\) None \(1\) \(0\) \(-3\) \(0\) \(+\) \(-\) \(q+q^{2}-q^{4}-3q^{5}-3q^{8}-3q^{10}+\cdots\)
1323.2.a.o \(1\) \(10.564\) \(\Q\) None \(1\) \(0\) \(3\) \(0\) \(-\) \(-\) \(q+q^{2}-q^{4}+3q^{5}-3q^{8}+3q^{10}+\cdots\)
1323.2.a.p \(1\) \(10.564\) \(\Q\) None \(1\) \(0\) \(4\) \(0\) \(+\) \(-\) \(q+q^{2}-q^{4}+4q^{5}-3q^{8}+4q^{10}+\cdots\)
1323.2.a.q \(1\) \(10.564\) \(\Q\) None \(2\) \(0\) \(-3\) \(0\) \(-\) \(-\) \(q+2q^{2}+2q^{4}-3q^{5}-6q^{10}+2q^{11}+\cdots\)
1323.2.a.r \(1\) \(10.564\) \(\Q\) None \(2\) \(0\) \(-1\) \(0\) \(+\) \(-\) \(q+2q^{2}+2q^{4}-q^{5}-2q^{10}+4q^{11}+\cdots\)
1323.2.a.s \(1\) \(10.564\) \(\Q\) None \(2\) \(0\) \(3\) \(0\) \(+\) \(-\) \(q+2q^{2}+2q^{4}+3q^{5}+6q^{10}+2q^{11}+\cdots\)
1323.2.a.t \(2\) \(10.564\) \(\Q(\sqrt{3}) \) None \(0\) \(0\) \(0\) \(0\) \(-\) \(-\) \(q+\beta q^{2}+q^{4}-\beta q^{5}-\beta q^{8}-3q^{10}+\cdots\)
1323.2.a.u \(2\) \(10.564\) \(\Q(\sqrt{6}) \) None \(0\) \(0\) \(0\) \(0\) \(+\) \(-\) \(q+\beta q^{2}+4q^{4}-\beta q^{5}+2\beta q^{8}-6q^{10}+\cdots\)
1323.2.a.v \(2\) \(10.564\) \(\Q(\sqrt{6}) \) None \(0\) \(0\) \(0\) \(0\) \(-\) \(+\) \(q+\beta q^{2}+4q^{4}+\beta q^{5}+2\beta q^{8}+6q^{10}+\cdots\)
1323.2.a.w \(2\) \(10.564\) \(\Q(\sqrt{7}) \) None \(0\) \(0\) \(0\) \(0\) \(+\) \(-\) \(q+\beta q^{2}+5q^{4}+\beta q^{5}+3\beta q^{8}+7q^{10}+\cdots\)
1323.2.a.x \(3\) \(10.564\) 3.3.321.1 None \(-2\) \(0\) \(-1\) \(0\) \(+\) \(+\) \(q+(-1+\beta _{1})q^{2}+(2-\beta _{1}+\beta _{2})q^{4}+\cdots\)
1323.2.a.y \(3\) \(10.564\) 3.3.321.1 None \(-2\) \(0\) \(1\) \(0\) \(-\) \(-\) \(q+(-1+\beta _{1})q^{2}+(2-\beta _{1}+\beta _{2})q^{4}+\cdots\)
1323.2.a.z \(3\) \(10.564\) 3.3.321.1 None \(2\) \(0\) \(-1\) \(0\) \(+\) \(-\) \(q+(1-\beta _{1})q^{2}+(2-\beta _{1}+\beta _{2})q^{4}+\beta _{2}q^{5}+\cdots\)
1323.2.a.ba \(3\) \(10.564\) 3.3.321.1 None \(2\) \(0\) \(1\) \(0\) \(-\) \(+\) \(q+(1-\beta _{1})q^{2}+(2-\beta _{1}+\beta _{2})q^{4}-\beta _{2}q^{5}+\cdots\)
1323.2.a.bb \(4\) \(10.564\) \(\Q(\sqrt{2}, \sqrt{5})\) None \(0\) \(0\) \(-8\) \(0\) \(+\) \(+\) \(q+\beta _{1}q^{2}+(1+\beta _{3})q^{4}+(-2-\beta _{3})q^{5}+\cdots\)
1323.2.a.bc \(4\) \(10.564\) 4.4.7168.1 None \(0\) \(0\) \(0\) \(0\) \(+\) \(+\) \(q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}-\beta _{1}q^{5}+\beta _{3}q^{8}+\cdots\)
1323.2.a.bd \(4\) \(10.564\) 4.4.7168.1 None \(0\) \(0\) \(0\) \(0\) \(-\) \(+\) \(q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}+\beta _{1}q^{5}+\beta _{3}q^{8}+\cdots\)
1323.2.a.be \(4\) \(10.564\) \(\Q(\sqrt{2}, \sqrt{5})\) None \(0\) \(0\) \(8\) \(0\) \(-\) \(+\) \(q+\beta _{1}q^{2}+(1+\beta _{3})q^{4}+(2+\beta _{3})q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1323))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1323)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(27))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(63))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(147))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(189))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(441))\)\(^{\oplus 2}\)