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

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

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

Label Char Prim Dim $A$ Field CM Minimal twist Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 3 7
1323.4.a.a 1323.a 1.a $1$ $78.060$ \(\Q\) None 189.4.a.a \(-3\) \(0\) \(-12\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q-3q^{2}+q^{4}-12q^{5}+21q^{8}+6^{2}q^{10}+\cdots\)
1323.4.a.b 1323.a 1.a $1$ $78.060$ \(\Q\) None 189.4.e.a \(-3\) \(0\) \(-6\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q-3q^{2}+q^{4}-6q^{5}+21q^{8}+18q^{10}+\cdots\)
1323.4.a.c 1323.a 1.a $1$ $78.060$ \(\Q\) None 189.4.e.a \(-3\) \(0\) \(6\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q-3q^{2}+q^{4}+6q^{5}+21q^{8}-18q^{10}+\cdots\)
1323.4.a.d 1323.a 1.a $1$ $78.060$ \(\Q\) None 27.4.a.a \(-3\) \(0\) \(15\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q-3q^{2}+q^{4}+15q^{5}+21q^{8}-45q^{10}+\cdots\)
1323.4.a.e 1323.a 1.a $1$ $78.060$ \(\Q\) None 189.4.a.b \(0\) \(0\) \(-21\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q-8q^{4}-21q^{5}-21q^{11}-2q^{13}+\cdots\)
1323.4.a.f 1323.a 1.a $1$ $78.060$ \(\Q\) \(\Q(\sqrt{-3}) \) 189.4.e.c \(0\) \(0\) \(0\) \(0\) $-$ $-$ $N(\mathrm{U}(1))$ \(q-8q^{4}-89q^{13}+2^{6}q^{16}-56q^{19}+\cdots\)
1323.4.a.g 1323.a 1.a $1$ $78.060$ \(\Q\) \(\Q(\sqrt{-3}) \) 189.4.e.b \(0\) \(0\) \(0\) \(0\) $-$ $+$ $N(\mathrm{U}(1))$ \(q-8q^{4}-19q^{13}+2^{6}q^{16}+56q^{19}+\cdots\)
1323.4.a.h 1323.a 1.a $1$ $78.060$ \(\Q\) \(\Q(\sqrt{-3}) \) 189.4.e.b \(0\) \(0\) \(0\) \(0\) $+$ $-$ $N(\mathrm{U}(1))$ \(q-8q^{4}+19q^{13}+2^{6}q^{16}-56q^{19}+\cdots\)
1323.4.a.i 1323.a 1.a $1$ $78.060$ \(\Q\) \(\Q(\sqrt{-3}) \) 189.4.e.c \(0\) \(0\) \(0\) \(0\) $+$ $+$ $N(\mathrm{U}(1))$ \(q-8q^{4}+89q^{13}+2^{6}q^{16}+56q^{19}+\cdots\)
1323.4.a.j 1323.a 1.a $1$ $78.060$ \(\Q\) None 189.4.a.b \(0\) \(0\) \(21\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q-8q^{4}+21q^{5}+21q^{11}-2q^{13}+\cdots\)
1323.4.a.k 1323.a 1.a $1$ $78.060$ \(\Q\) None 27.4.a.a \(3\) \(0\) \(-15\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+3q^{2}+q^{4}-15q^{5}-21q^{8}-45q^{10}+\cdots\)
1323.4.a.l 1323.a 1.a $1$ $78.060$ \(\Q\) None 189.4.e.a \(3\) \(0\) \(-6\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q+3q^{2}+q^{4}-6q^{5}-21q^{8}-18q^{10}+\cdots\)
1323.4.a.m 1323.a 1.a $1$ $78.060$ \(\Q\) None 189.4.e.a \(3\) \(0\) \(6\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+3q^{2}+q^{4}+6q^{5}-21q^{8}+18q^{10}+\cdots\)
1323.4.a.n 1323.a 1.a $1$ $78.060$ \(\Q\) None 189.4.a.a \(3\) \(0\) \(12\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+3q^{2}+q^{4}+12q^{5}-21q^{8}+6^{2}q^{10}+\cdots\)
1323.4.a.o 1323.a 1.a $2$ $78.060$ \(\Q(\sqrt{3}) \) None 189.4.a.e \(-6\) \(0\) \(6\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-3+\beta )q^{2}+(4-6\beta )q^{4}+(3-4\beta )q^{5}+\cdots\)
1323.4.a.p 1323.a 1.a $2$ $78.060$ \(\Q(\sqrt{2}) \) None 1323.4.a.p \(-2\) \(0\) \(-22\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(-1+2\beta )q^{2}+(1-4\beta )q^{4}+(-11+\cdots)q^{5}+\cdots\)
1323.4.a.q 1323.a 1.a $2$ $78.060$ \(\Q(\sqrt{2}) \) None 1323.4.a.p \(-2\) \(0\) \(22\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(-1+2\beta )q^{2}+(1-4\beta )q^{4}+(11-\beta )q^{5}+\cdots\)
1323.4.a.r 1323.a 1.a $2$ $78.060$ \(\Q(\sqrt{3}) \) None 189.4.a.f \(0\) \(0\) \(0\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{2}-5q^{4}-\beta q^{5}-13\beta q^{8}-3q^{10}+\cdots\)
1323.4.a.s 1323.a 1.a $2$ $78.060$ \(\Q(\sqrt{7}) \) None 189.4.a.g \(0\) \(0\) \(0\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{2}-q^{4}+\beta q^{5}-9\beta q^{8}+7q^{10}+\cdots\)
1323.4.a.t 1323.a 1.a $2$ $78.060$ \(\Q(\sqrt{2}) \) None 27.4.a.c \(0\) \(0\) \(0\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+10q^{4}+4\beta q^{5}+2\beta q^{8}+72q^{10}+\cdots\)
1323.4.a.u 1323.a 1.a $2$ $78.060$ \(\Q(\sqrt{21}) \) None 189.4.a.h \(0\) \(0\) \(0\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta q^{2}+13q^{4}+\beta q^{5}-5\beta q^{8}-21q^{10}+\cdots\)
1323.4.a.v 1323.a 1.a $2$ $78.060$ \(\Q(\sqrt{2}) \) None 1323.4.a.p \(2\) \(0\) \(-22\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(1+2\beta )q^{2}+(1+4\beta )q^{4}+(-11-\beta )q^{5}+\cdots\)
1323.4.a.w 1323.a 1.a $2$ $78.060$ \(\Q(\sqrt{2}) \) None 1323.4.a.p \(2\) \(0\) \(22\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(1+2\beta )q^{2}+(1+4\beta )q^{4}+(11+\beta )q^{5}+\cdots\)
1323.4.a.x 1323.a 1.a $2$ $78.060$ \(\Q(\sqrt{3}) \) None 189.4.a.e \(6\) \(0\) \(-6\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(3+\beta )q^{2}+(4+6\beta )q^{4}+(-3-4\beta )q^{5}+\cdots\)
1323.4.a.y 1323.a 1.a $3$ $78.060$ 3.3.3576.1 None 189.4.a.j \(-1\) \(0\) \(2\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(6-2\beta _{1}+\beta _{2})q^{4}+(1+2\beta _{1}+\cdots)q^{5}+\cdots\)
1323.4.a.z 1323.a 1.a $3$ $78.060$ 3.3.3576.1 None 189.4.a.j \(1\) \(0\) \(-2\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+(6-2\beta _{1}+\beta _{2})q^{4}+(-1+\cdots)q^{5}+\cdots\)
1323.4.a.ba 1323.a 1.a $4$ $78.060$ \(\Q(\sqrt{5}, \sqrt{13})\) None 189.4.a.l \(0\) \(0\) \(0\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-\beta _{1}-\beta _{2})q^{2}+(6-\beta _{3})q^{4}+(4\beta _{1}+\cdots)q^{5}+\cdots\)
1323.4.a.bb 1323.a 1.a $6$ $78.060$ 6.6.346909504.1 None 1323.4.a.bb \(0\) \(0\) \(-24\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(3+\beta _{2})q^{4}+(-4+\beta _{4})q^{5}+\cdots\)
1323.4.a.bc 1323.a 1.a $6$ $78.060$ 6.6.346909504.1 None 1323.4.a.bb \(0\) \(0\) \(24\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(3+\beta _{2})q^{4}+(4-\beta _{4})q^{5}+\cdots\)
1323.4.a.bd 1323.a 1.a $6$ $78.060$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None 189.4.e.e \(0\) \(0\) \(0\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(5+\beta _{3})q^{4}+\beta _{4}q^{5}+(\beta _{1}+\cdots)q^{8}+\cdots\)
1323.4.a.be 1323.a 1.a $6$ $78.060$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None 189.4.e.e \(0\) \(0\) \(0\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(5+\beta _{3})q^{4}-\beta _{4}q^{5}+(\beta _{1}+\cdots)q^{8}+\cdots\)
1323.4.a.bf 1323.a 1.a $6$ $78.060$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None 1323.4.a.bf \(0\) \(0\) \(0\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(6+\beta _{2})q^{4}+(-2\beta _{1}-\beta _{3}+\cdots)q^{5}+\cdots\)
1323.4.a.bg 1323.a 1.a $6$ $78.060$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None 1323.4.a.bf \(0\) \(0\) \(0\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(6+\beta _{2})q^{4}+(2\beta _{1}+\beta _{3})q^{5}+\cdots\)
1323.4.a.bh 1323.a 1.a $7$ $78.060$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None 189.4.e.f \(-1\) \(0\) \(-1\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+(4+\beta _{1}+\beta _{2})q^{4}-\beta _{4}q^{5}+\cdots\)
1323.4.a.bi 1323.a 1.a $7$ $78.060$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None 189.4.e.f \(-1\) \(0\) \(1\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+(4+\beta _{1}+\beta _{2})q^{4}+\beta _{4}q^{5}+\cdots\)
1323.4.a.bj 1323.a 1.a $7$ $78.060$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None 189.4.e.f \(1\) \(0\) \(-1\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(4+\beta _{1}+\beta _{2})q^{4}-\beta _{4}q^{5}+\cdots\)
1323.4.a.bk 1323.a 1.a $7$ $78.060$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None 189.4.e.f \(1\) \(0\) \(1\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(4+\beta _{1}+\beta _{2})q^{4}+\beta _{4}q^{5}+\cdots\)
1323.4.a.bl 1323.a 1.a $8$ $78.060$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None 1323.4.a.bl \(0\) \(0\) \(0\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(6-\beta _{2}+\beta _{4})q^{4}+(-\beta _{1}+\cdots)q^{5}+\cdots\)
1323.4.a.bm 1323.a 1.a $8$ $78.060$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None 1323.4.a.bl \(0\) \(0\) \(0\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(6-\beta _{2}+\beta _{4})q^{4}+(\beta _{1}+\beta _{3}+\cdots)q^{5}+\cdots\)
1323.4.a.bn 1323.a 1.a $8$ $78.060$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None 189.4.e.h \(0\) \(0\) \(0\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{2}q^{2}+(6+\beta _{1})q^{4}-\beta _{3}q^{5}+(-5\beta _{2}+\cdots)q^{8}+\cdots\)
1323.4.a.bo 1323.a 1.a $8$ $78.060$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None 189.4.e.h \(0\) \(0\) \(0\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{2}q^{2}+(6+\beta _{1})q^{4}+\beta _{3}q^{5}+(-5\beta _{2}+\cdots)q^{8}+\cdots\)
1323.4.a.bp 1323.a 1.a $12$ $78.060$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None 1323.4.a.bp \(0\) \(0\) \(-40\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(4+\beta _{2})q^{4}+(-3-\beta _{5})q^{5}+\cdots\)
1323.4.a.bq 1323.a 1.a $12$ $78.060$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None 1323.4.a.bp \(0\) \(0\) \(40\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(4+\beta _{2})q^{4}+(3+\beta _{5})q^{5}+\cdots\)

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

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