Properties

Label 1323.4.a
Level $1323$
Weight $4$
Character orbit 1323.a
Rep. character $\chi_{1323}(1,\cdot)$
Character field $\Q$
Dimension $164$
Newform subspaces $43$
Sturm bound $672$
Trace bound $13$

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Defining parameters

Level: \( N \) \(=\) \( 1323 = 3^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1323.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 43 \)
Sturm bound: \(672\)
Trace bound: \(13\)
Distinguishing \(T_p\): \(2\), \(5\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(1323))\).

Total New Old
Modular forms 528 164 364
Cusp forms 480 164 316
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.

\(3\)\(7\)FrickeDim.
\(+\)\(+\)\(+\)\(41\)
\(+\)\(-\)\(-\)\(40\)
\(-\)\(+\)\(-\)\(39\)
\(-\)\(-\)\(+\)\(44\)
Plus space\(+\)\(85\)
Minus space\(-\)\(79\)

Trace form

\( 164q + 662q^{4} + O(q^{10}) \) \( 164q + 662q^{4} - 30q^{10} - 2q^{13} + 2810q^{16} - 62q^{19} - 270q^{22} + 4202q^{25} + 406q^{31} - 204q^{34} - 338q^{37} - 1266q^{40} + 592q^{43} - 1224q^{46} + 220q^{52} + 1122q^{55} + 300q^{58} + 2410q^{61} + 12302q^{64} + 1222q^{67} + 1540q^{73} + 2128q^{76} - 2486q^{79} + 1104q^{82} - 720q^{85} - 8730q^{88} + 1704q^{94} - 3056q^{97} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(1323))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 3 7
1323.4.a.a \(1\) \(78.060\) \(\Q\) None \(-3\) \(0\) \(-12\) \(0\) \(+\) \(-\) \(q-3q^{2}+q^{4}-12q^{5}+21q^{8}+6^{2}q^{10}+\cdots\)
1323.4.a.b \(1\) \(78.060\) \(\Q\) None \(-3\) \(0\) \(-6\) \(0\) \(+\) \(-\) \(q-3q^{2}+q^{4}-6q^{5}+21q^{8}+18q^{10}+\cdots\)
1323.4.a.c \(1\) \(78.060\) \(\Q\) None \(-3\) \(0\) \(6\) \(0\) \(-\) \(+\) \(q-3q^{2}+q^{4}+6q^{5}+21q^{8}-18q^{10}+\cdots\)
1323.4.a.d \(1\) \(78.060\) \(\Q\) None \(-3\) \(0\) \(15\) \(0\) \(+\) \(-\) \(q-3q^{2}+q^{4}+15q^{5}+21q^{8}-45q^{10}+\cdots\)
1323.4.a.e \(1\) \(78.060\) \(\Q\) None \(0\) \(0\) \(-21\) \(0\) \(+\) \(-\) \(q-8q^{4}-21q^{5}-21q^{11}-2q^{13}+\cdots\)
1323.4.a.f \(1\) \(78.060\) \(\Q\) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(0\) \(-\) \(-\) \(q-8q^{4}-89q^{13}+2^{6}q^{16}-56q^{19}+\cdots\)
1323.4.a.g \(1\) \(78.060\) \(\Q\) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(0\) \(-\) \(+\) \(q-8q^{4}-19q^{13}+2^{6}q^{16}+56q^{19}+\cdots\)
1323.4.a.h \(1\) \(78.060\) \(\Q\) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(0\) \(+\) \(-\) \(q-8q^{4}+19q^{13}+2^{6}q^{16}-56q^{19}+\cdots\)
1323.4.a.i \(1\) \(78.060\) \(\Q\) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(0\) \(+\) \(+\) \(q-8q^{4}+89q^{13}+2^{6}q^{16}+56q^{19}+\cdots\)
1323.4.a.j \(1\) \(78.060\) \(\Q\) None \(0\) \(0\) \(21\) \(0\) \(+\) \(-\) \(q-8q^{4}+21q^{5}+21q^{11}-2q^{13}+\cdots\)
1323.4.a.k \(1\) \(78.060\) \(\Q\) None \(3\) \(0\) \(-15\) \(0\) \(-\) \(-\) \(q+3q^{2}+q^{4}-15q^{5}-21q^{8}-45q^{10}+\cdots\)
1323.4.a.l \(1\) \(78.060\) \(\Q\) None \(3\) \(0\) \(-6\) \(0\) \(+\) \(+\) \(q+3q^{2}+q^{4}-6q^{5}-21q^{8}-18q^{10}+\cdots\)
1323.4.a.m \(1\) \(78.060\) \(\Q\) None \(3\) \(0\) \(6\) \(0\) \(-\) \(-\) \(q+3q^{2}+q^{4}+6q^{5}-21q^{8}+18q^{10}+\cdots\)
1323.4.a.n \(1\) \(78.060\) \(\Q\) None \(3\) \(0\) \(12\) \(0\) \(-\) \(-\) \(q+3q^{2}+q^{4}+12q^{5}-21q^{8}+6^{2}q^{10}+\cdots\)
1323.4.a.o \(2\) \(78.060\) \(\Q(\sqrt{3}) \) None \(-6\) \(0\) \(6\) \(0\) \(-\) \(-\) \(q+(-3+\beta )q^{2}+(4-6\beta )q^{4}+(3-4\beta )q^{5}+\cdots\)
1323.4.a.p \(2\) \(78.060\) \(\Q(\sqrt{2}) \) None \(-2\) \(0\) \(-22\) \(0\) \(-\) \(+\) \(q+(-1+2\beta )q^{2}+(1-4\beta )q^{4}+(-11+\cdots)q^{5}+\cdots\)
1323.4.a.q \(2\) \(78.060\) \(\Q(\sqrt{2}) \) None \(-2\) \(0\) \(22\) \(0\) \(+\) \(+\) \(q+(-1+2\beta )q^{2}+(1-4\beta )q^{4}+(11-\beta )q^{5}+\cdots\)
1323.4.a.r \(2\) \(78.060\) \(\Q(\sqrt{3}) \) None \(0\) \(0\) \(0\) \(0\) \(-\) \(-\) \(q+\beta q^{2}-5q^{4}-\beta q^{5}-13\beta q^{8}-3q^{10}+\cdots\)
1323.4.a.s \(2\) \(78.060\) \(\Q(\sqrt{7}) \) None \(0\) \(0\) \(0\) \(0\) \(+\) \(-\) \(q+\beta q^{2}-q^{4}+\beta q^{5}-9\beta q^{8}+7q^{10}+\cdots\)
1323.4.a.t \(2\) \(78.060\) \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(0\) \(0\) \(-\) \(-\) \(q+\beta q^{2}+10q^{4}+4\beta q^{5}+2\beta q^{8}+72q^{10}+\cdots\)
1323.4.a.u \(2\) \(78.060\) \(\Q(\sqrt{21}) \) None \(0\) \(0\) \(0\) \(0\) \(+\) \(-\) \(q-\beta q^{2}+13q^{4}+\beta q^{5}-5\beta q^{8}-21q^{10}+\cdots\)
1323.4.a.v \(2\) \(78.060\) \(\Q(\sqrt{2}) \) None \(2\) \(0\) \(-22\) \(0\) \(+\) \(+\) \(q+(1+2\beta )q^{2}+(1+4\beta )q^{4}+(-11-\beta )q^{5}+\cdots\)
1323.4.a.w \(2\) \(78.060\) \(\Q(\sqrt{2}) \) None \(2\) \(0\) \(22\) \(0\) \(-\) \(+\) \(q+(1+2\beta )q^{2}+(1+4\beta )q^{4}+(11+\beta )q^{5}+\cdots\)
1323.4.a.x \(2\) \(78.060\) \(\Q(\sqrt{3}) \) None \(6\) \(0\) \(-6\) \(0\) \(+\) \(-\) \(q+(3+\beta )q^{2}+(4+6\beta )q^{4}+(-3-4\beta )q^{5}+\cdots\)
1323.4.a.y \(3\) \(78.060\) 3.3.3576.1 None \(-1\) \(0\) \(2\) \(0\) \(+\) \(-\) \(q+\beta _{1}q^{2}+(6-2\beta _{1}+\beta _{2})q^{4}+(1+2\beta _{1}+\cdots)q^{5}+\cdots\)
1323.4.a.z \(3\) \(78.060\) 3.3.3576.1 None \(1\) \(0\) \(-2\) \(0\) \(-\) \(-\) \(q-\beta _{1}q^{2}+(6-2\beta _{1}+\beta _{2})q^{4}+(-1+\cdots)q^{5}+\cdots\)
1323.4.a.ba \(4\) \(78.060\) \(\Q(\sqrt{5}, \sqrt{13})\) None \(0\) \(0\) \(0\) \(0\) \(-\) \(-\) \(q+(-\beta _{1}-\beta _{2})q^{2}+(6-\beta _{3})q^{4}+(4\beta _{1}+\cdots)q^{5}+\cdots\)
1323.4.a.bb \(6\) \(78.060\) 6.6.346909504.1 None \(0\) \(0\) \(-24\) \(0\) \(+\) \(-\) \(q+\beta _{1}q^{2}+(3+\beta _{2})q^{4}+(-4+\beta _{4})q^{5}+\cdots\)
1323.4.a.bc \(6\) \(78.060\) 6.6.346909504.1 None \(0\) \(0\) \(24\) \(0\) \(-\) \(-\) \(q+\beta _{1}q^{2}+(3+\beta _{2})q^{4}+(4-\beta _{4})q^{5}+\cdots\)
1323.4.a.bd \(6\) \(78.060\) \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) \(-\) \(+\) \(q+\beta _{1}q^{2}+(5+\beta _{3})q^{4}+\beta _{4}q^{5}+(\beta _{1}+\cdots)q^{8}+\cdots\)
1323.4.a.be \(6\) \(78.060\) \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) \(+\) \(-\) \(q+\beta _{1}q^{2}+(5+\beta _{3})q^{4}-\beta _{4}q^{5}+(\beta _{1}+\cdots)q^{8}+\cdots\)
1323.4.a.bf \(6\) \(78.060\) \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) \(+\) \(-\) \(q+\beta _{1}q^{2}+(6+\beta _{2})q^{4}+(-2\beta _{1}-\beta _{3}+\cdots)q^{5}+\cdots\)
1323.4.a.bg \(6\) \(78.060\) \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) \(-\) \(-\) \(q+\beta _{1}q^{2}+(6+\beta _{2})q^{4}+(2\beta _{1}+\beta _{3})q^{5}+\cdots\)
1323.4.a.bh \(7\) \(78.060\) \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(-1\) \(0\) \(-1\) \(0\) \(+\) \(-\) \(q-\beta _{1}q^{2}+(4+\beta _{1}+\beta _{2})q^{4}-\beta _{4}q^{5}+\cdots\)
1323.4.a.bi \(7\) \(78.060\) \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(-1\) \(0\) \(1\) \(0\) \(-\) \(+\) \(q-\beta _{1}q^{2}+(4+\beta _{1}+\beta _{2})q^{4}+\beta _{4}q^{5}+\cdots\)
1323.4.a.bj \(7\) \(78.060\) \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(1\) \(0\) \(-1\) \(0\) \(+\) \(+\) \(q+\beta _{1}q^{2}+(4+\beta _{1}+\beta _{2})q^{4}-\beta _{4}q^{5}+\cdots\)
1323.4.a.bk \(7\) \(78.060\) \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(1\) \(0\) \(1\) \(0\) \(-\) \(-\) \(q+\beta _{1}q^{2}+(4+\beta _{1}+\beta _{2})q^{4}+\beta _{4}q^{5}+\cdots\)
1323.4.a.bl \(8\) \(78.060\) \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) \(-\) \(+\) \(q+\beta _{1}q^{2}+(6-\beta _{2}+\beta _{4})q^{4}+(-\beta _{1}+\cdots)q^{5}+\cdots\)
1323.4.a.bm \(8\) \(78.060\) \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) \(+\) \(+\) \(q+\beta _{1}q^{2}+(6-\beta _{2}+\beta _{4})q^{4}+(\beta _{1}+\beta _{3}+\cdots)q^{5}+\cdots\)
1323.4.a.bn \(8\) \(78.060\) \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) \(-\) \(-\) \(q-\beta _{2}q^{2}+(6+\beta _{1})q^{4}-\beta _{3}q^{5}+(-5\beta _{2}+\cdots)q^{8}+\cdots\)
1323.4.a.bo \(8\) \(78.060\) \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) \(+\) \(+\) \(q-\beta _{2}q^{2}+(6+\beta _{1})q^{4}+\beta _{3}q^{5}+(-5\beta _{2}+\cdots)q^{8}+\cdots\)
1323.4.a.bp \(12\) \(78.060\) \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(0\) \(0\) \(-40\) \(0\) \(-\) \(+\) \(q+\beta _{1}q^{2}+(4+\beta _{2})q^{4}+(-3-\beta _{5})q^{5}+\cdots\)
1323.4.a.bq \(12\) \(78.060\) \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(0\) \(0\) \(40\) \(0\) \(+\) \(+\) \(q+\beta _{1}q^{2}+(4+\beta _{2})q^{4}+(3+\beta _{5})q^{5}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(1323))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(1323)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(27))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(63))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(147))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(189))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(441))\)\(^{\oplus 2}\)