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

\( 55 q + 54 q^{4} + 4 q^{10} + 7 q^{13} + 32 q^{16} - 5 q^{19} - 8 q^{22} + 73 q^{25} - 16 q^{31} + 36 q^{34} + 19 q^{37} + 48 q^{40} - 46 q^{43} + 4 q^{46} + 18 q^{52} - 52 q^{55} - 8 q^{58} + 13 q^{61}+ \cdots + 47 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1323))\) 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
1323.2.a.a 1323.a 1.a $1$ $10.564$ \(\Q\) None 1323.2.a.a \(-2\) \(0\) \(-3\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+2q^{4}-3q^{5}+6q^{10}-2q^{11}+\cdots\)
1323.2.a.b 1323.a 1.a $1$ $10.564$ \(\Q\) None 189.2.a.a \(-2\) \(0\) \(1\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+2q^{4}+q^{5}-2q^{10}-4q^{11}+\cdots\)
1323.2.a.c 1323.a 1.a $1$ $10.564$ \(\Q\) None 1323.2.a.a \(-2\) \(0\) \(3\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+2q^{4}+3q^{5}-6q^{10}-2q^{11}+\cdots\)
1323.2.a.d 1323.a 1.a $1$ $10.564$ \(\Q\) None 189.2.e.a \(-1\) \(0\) \(-4\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{4}-4q^{5}+3q^{8}+4q^{10}+\cdots\)
1323.2.a.e 1323.a 1.a $1$ $10.564$ \(\Q\) None 1323.2.a.e \(-1\) \(0\) \(-3\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{4}-3q^{5}+3q^{8}+3q^{10}+\cdots\)
1323.2.a.f 1323.a 1.a $1$ $10.564$ \(\Q\) None 1323.2.a.e \(-1\) \(0\) \(3\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{4}+3q^{5}+3q^{8}-3q^{10}+\cdots\)
1323.2.a.g 1323.a 1.a $1$ $10.564$ \(\Q\) None 189.2.e.a \(-1\) \(0\) \(4\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{4}+4q^{5}+3q^{8}-4q^{10}+\cdots\)
1323.2.a.h 1323.a 1.a $1$ $10.564$ \(\Q\) None 189.2.a.b \(0\) \(0\) \(-3\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{4}-3q^{5}+6q^{11}+4q^{13}+4q^{16}+\cdots\)
1323.2.a.i 1323.a 1.a $1$ $10.564$ \(\Q\) \(\Q(\sqrt{-3}) \) 27.2.a.a \(0\) \(0\) \(0\) \(0\) $+$ $-$ $N(\mathrm{U}(1))$ \(q-2q^{4}-5q^{13}+4q^{16}+7q^{19}-5q^{25}+\cdots\)
1323.2.a.j 1323.a 1.a $1$ $10.564$ \(\Q\) \(\Q(\sqrt{-3}) \) 189.2.e.b \(0\) \(0\) \(0\) \(0\) $-$ $-$ $N(\mathrm{U}(1))$ \(q-2q^{4}-2q^{13}+4q^{16}+7q^{19}-5q^{25}+\cdots\)
1323.2.a.k 1323.a 1.a $1$ $10.564$ \(\Q\) \(\Q(\sqrt{-3}) \) 189.2.e.b \(0\) \(0\) \(0\) \(0\) $+$ $+$ $N(\mathrm{U}(1))$ \(q-2q^{4}+2q^{13}+4q^{16}-7q^{19}-5q^{25}+\cdots\)
1323.2.a.l 1323.a 1.a $1$ $10.564$ \(\Q\) None 189.2.a.b \(0\) \(0\) \(3\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{4}+3q^{5}-6q^{11}+4q^{13}+4q^{16}+\cdots\)
1323.2.a.m 1323.a 1.a $1$ $10.564$ \(\Q\) None 189.2.e.a \(1\) \(0\) \(-4\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{4}-4q^{5}-3q^{8}-4q^{10}+\cdots\)
1323.2.a.n 1323.a 1.a $1$ $10.564$ \(\Q\) None 1323.2.a.e \(1\) \(0\) \(-3\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{4}-3q^{5}-3q^{8}-3q^{10}+\cdots\)
1323.2.a.o 1323.a 1.a $1$ $10.564$ \(\Q\) None 1323.2.a.e \(1\) \(0\) \(3\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{4}+3q^{5}-3q^{8}+3q^{10}+\cdots\)
1323.2.a.p 1323.a 1.a $1$ $10.564$ \(\Q\) None 189.2.e.a \(1\) \(0\) \(4\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{4}+4q^{5}-3q^{8}+4q^{10}+\cdots\)
1323.2.a.q 1323.a 1.a $1$ $10.564$ \(\Q\) None 1323.2.a.a \(2\) \(0\) \(-3\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+2q^{4}-3q^{5}-6q^{10}+2q^{11}+\cdots\)
1323.2.a.r 1323.a 1.a $1$ $10.564$ \(\Q\) None 189.2.a.a \(2\) \(0\) \(-1\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+2q^{4}-q^{5}-2q^{10}+4q^{11}+\cdots\)
1323.2.a.s 1323.a 1.a $1$ $10.564$ \(\Q\) None 1323.2.a.a \(2\) \(0\) \(3\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+2q^{4}+3q^{5}+6q^{10}+2q^{11}+\cdots\)
1323.2.a.t 1323.a 1.a $2$ $10.564$ \(\Q(\sqrt{3}) \) None 189.2.a.e \(0\) \(0\) \(0\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+q^{4}-\beta q^{5}-\beta q^{8}-3q^{10}+\cdots\)
1323.2.a.u 1323.a 1.a $2$ $10.564$ \(\Q(\sqrt{6}) \) None 189.2.e.d \(0\) \(0\) \(0\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+4q^{4}-\beta q^{5}+2\beta q^{8}-6q^{10}+\cdots\)
1323.2.a.v 1323.a 1.a $2$ $10.564$ \(\Q(\sqrt{6}) \) None 189.2.e.d \(0\) \(0\) \(0\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+4q^{4}+\beta q^{5}+2\beta q^{8}+6q^{10}+\cdots\)
1323.2.a.w 1323.a 1.a $2$ $10.564$ \(\Q(\sqrt{7}) \) None 189.2.a.f \(0\) \(0\) \(0\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+5q^{4}+\beta q^{5}+3\beta q^{8}+7q^{10}+\cdots\)
1323.2.a.x 1323.a 1.a $3$ $10.564$ 3.3.321.1 None 189.2.e.e \(-2\) \(0\) \(-1\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{2}+(2-\beta _{1}+\beta _{2})q^{4}+\cdots\)
1323.2.a.y 1323.a 1.a $3$ $10.564$ 3.3.321.1 None 189.2.e.e \(-2\) \(0\) \(1\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{2}+(2-\beta _{1}+\beta _{2})q^{4}+\cdots\)
1323.2.a.z 1323.a 1.a $3$ $10.564$ 3.3.321.1 None 189.2.e.e \(2\) \(0\) \(-1\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}+(2-\beta _{1}+\beta _{2})q^{4}+\beta _{2}q^{5}+\cdots\)
1323.2.a.ba 1323.a 1.a $3$ $10.564$ 3.3.321.1 None 189.2.e.e \(2\) \(0\) \(1\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}+(2-\beta _{1}+\beta _{2})q^{4}-\beta _{2}q^{5}+\cdots\)
1323.2.a.bb 1323.a 1.a $4$ $10.564$ \(\Q(\sqrt{2}, \sqrt{5})\) None 1323.2.a.bb \(0\) \(0\) \(-8\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(1+\beta _{3})q^{4}+(-2-\beta _{3})q^{5}+\cdots\)
1323.2.a.bc 1323.a 1.a $4$ $10.564$ 4.4.7168.1 None 1323.2.a.bc \(0\) \(0\) \(0\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}-\beta _{1}q^{5}+\beta _{3}q^{8}+\cdots\)
1323.2.a.bd 1323.a 1.a $4$ $10.564$ 4.4.7168.1 None 1323.2.a.bc \(0\) \(0\) \(0\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}+\beta _{1}q^{5}+\beta _{3}q^{8}+\cdots\)
1323.2.a.be 1323.a 1.a $4$ $10.564$ \(\Q(\sqrt{2}, \sqrt{5})\) None 1323.2.a.bb \(0\) \(0\) \(8\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(1+\beta _{3})q^{4}+(2+\beta _{3})q^{5}+\cdots\)

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

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