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Properties

Label 1320.2.a
Level $1320$
Weight $2$
Character orbit 1320.a
Rep. character $\chi_{1320}(1,\cdot)$
Character field $\Q$
Dimension $20$
Newform subspaces $17$
Sturm bound $576$
Trace bound $13$

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Defining parameters

Level: \( N \) \(=\) \( 1320 = 2^{3} \cdot 3 \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1320.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 17 \)
Sturm bound: \(576\)
Trace bound: \(13\)
Distinguishing \(T_p\): \(7\), \(13\), \(17\), \(19\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1320))\).

Total New Old
Modular forms 304 20 284
Cusp forms 273 20 253
Eisenstein series 31 0 31

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

\(2\)\(3\)\(5\)\(11\)FrickeDim
\(+\)\(+\)\(+\)\(+\)$+$\(1\)
\(+\)\(+\)\(+\)\(-\)$-$\(2\)
\(+\)\(+\)\(-\)\(+\)$-$\(2\)
\(+\)\(+\)\(-\)\(-\)$+$\(1\)
\(+\)\(-\)\(+\)\(+\)$-$\(1\)
\(+\)\(-\)\(+\)\(-\)$+$\(1\)
\(+\)\(-\)\(-\)\(-\)$-$\(2\)
\(-\)\(+\)\(+\)\(+\)$-$\(2\)
\(-\)\(+\)\(+\)\(-\)$+$\(1\)
\(-\)\(+\)\(-\)\(+\)$+$\(2\)
\(-\)\(+\)\(-\)\(-\)$-$\(1\)
\(-\)\(-\)\(+\)\(+\)$+$\(1\)
\(-\)\(-\)\(+\)\(-\)$-$\(1\)
\(-\)\(-\)\(-\)\(+\)$-$\(1\)
\(-\)\(-\)\(-\)\(-\)$+$\(1\)
Plus space\(+\)\(8\)
Minus space\(-\)\(12\)

Trace form

\( 20 q - 4 q^{3} - 8 q^{7} + 20 q^{9} + O(q^{10}) \) \( 20 q - 4 q^{3} - 8 q^{7} + 20 q^{9} - 8 q^{13} - 8 q^{19} - 8 q^{21} + 8 q^{23} + 20 q^{25} - 4 q^{27} + 4 q^{33} + 8 q^{37} - 8 q^{39} + 16 q^{41} + 8 q^{43} + 8 q^{47} + 20 q^{49} + 16 q^{51} + 16 q^{53} - 8 q^{57} + 16 q^{59} - 8 q^{61} - 8 q^{63} + 8 q^{69} - 24 q^{71} + 8 q^{73} - 4 q^{75} + 24 q^{79} + 20 q^{81} + 16 q^{87} - 24 q^{89} + 32 q^{91} + 16 q^{95} + 8 q^{97} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1320))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 3 5 11
1320.2.a.a 1320.a 1.a $1$ $10.540$ \(\Q\) None \(0\) \(-1\) \(-1\) \(-2\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}-q^{5}-2q^{7}+q^{9}-q^{11}-4q^{13}+\cdots\)
1320.2.a.b 1320.a 1.a $1$ $10.540$ \(\Q\) None \(0\) \(-1\) \(-1\) \(-2\) $-$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}-q^{5}-2q^{7}+q^{9}+q^{11}+4q^{13}+\cdots\)
1320.2.a.c 1320.a 1.a $1$ $10.540$ \(\Q\) None \(0\) \(-1\) \(-1\) \(2\) $+$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}-q^{5}+2q^{7}+q^{9}-q^{11}-4q^{13}+\cdots\)
1320.2.a.d 1320.a 1.a $1$ $10.540$ \(\Q\) None \(0\) \(-1\) \(-1\) \(4\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}-q^{5}+4q^{7}+q^{9}-q^{11}+2q^{13}+\cdots\)
1320.2.a.e 1320.a 1.a $1$ $10.540$ \(\Q\) None \(0\) \(-1\) \(1\) \(-4\) $-$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+q^{5}-4q^{7}+q^{9}-q^{11}+2q^{13}+\cdots\)
1320.2.a.f 1320.a 1.a $1$ $10.540$ \(\Q\) None \(0\) \(-1\) \(1\) \(0\) $-$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+q^{5}+q^{9}-q^{11}-2q^{13}-q^{15}+\cdots\)
1320.2.a.g 1320.a 1.a $1$ $10.540$ \(\Q\) None \(0\) \(-1\) \(1\) \(0\) $+$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}+q^{5}+q^{9}+q^{11}-6q^{13}-q^{15}+\cdots\)
1320.2.a.h 1320.a 1.a $1$ $10.540$ \(\Q\) None \(0\) \(-1\) \(1\) \(4\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}+q^{5}+4q^{7}+q^{9}+q^{11}+2q^{13}+\cdots\)
1320.2.a.i 1320.a 1.a $1$ $10.540$ \(\Q\) None \(0\) \(1\) \(-1\) \(-2\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}-q^{5}-2q^{7}+q^{9}-q^{11}-4q^{13}+\cdots\)
1320.2.a.j 1320.a 1.a $1$ $10.540$ \(\Q\) None \(0\) \(1\) \(-1\) \(-2\) $+$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}-q^{5}-2q^{7}+q^{9}-q^{11}-q^{15}+\cdots\)
1320.2.a.k 1320.a 1.a $1$ $10.540$ \(\Q\) None \(0\) \(1\) \(-1\) \(-2\) $+$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}-q^{5}-2q^{7}+q^{9}+q^{11}-q^{15}+\cdots\)
1320.2.a.l 1320.a 1.a $1$ $10.540$ \(\Q\) None \(0\) \(1\) \(-1\) \(2\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}-q^{5}+2q^{7}+q^{9}+q^{11}-q^{15}+\cdots\)
1320.2.a.m 1320.a 1.a $1$ $10.540$ \(\Q\) None \(0\) \(1\) \(1\) \(-4\) $-$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+q^{5}-4q^{7}+q^{9}+q^{11}-6q^{13}+\cdots\)
1320.2.a.n 1320.a 1.a $1$ $10.540$ \(\Q\) None \(0\) \(1\) \(1\) \(0\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}+q^{5}+q^{9}-q^{11}-2q^{13}+q^{15}+\cdots\)
1320.2.a.o 1320.a 1.a $2$ $10.540$ \(\Q(\sqrt{17}) \) None \(0\) \(-2\) \(-2\) \(-2\) $+$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}-q^{5}+(-1-\beta )q^{7}+q^{9}+q^{11}+\cdots\)
1320.2.a.p 1320.a 1.a $2$ $10.540$ \(\Q(\sqrt{2}) \) None \(0\) \(-2\) \(2\) \(0\) $+$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+q^{5}+\beta q^{7}+q^{9}-q^{11}+(2+\cdots)q^{13}+\cdots\)
1320.2.a.q 1320.a 1.a $2$ $10.540$ \(\Q(\sqrt{2}) \) None \(0\) \(2\) \(2\) \(0\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+q^{5}+\beta q^{7}+q^{9}+q^{11}+(2+\cdots)q^{13}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1320))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1320)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(22))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(33))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(44))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(55))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(60))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(66))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(88))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(110))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(120))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(132))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(165))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(220))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(264))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(330))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(440))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(660))\)\(^{\oplus 2}\)