Properties

Label 1380.2.a
Level $1380$
Weight $2$
Character orbit 1380.a
Rep. character $\chi_{1380}(1,\cdot)$
Character field $\Q$
Dimension $16$
Newform subspaces $10$
Sturm bound $576$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 1380 = 2^{2} \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1380.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 10 \)
Sturm bound: \(576\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1380))\).

Total New Old
Modular forms 300 16 284
Cusp forms 277 16 261
Eisenstein series 23 0 23

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

\(2\)\(3\)\(5\)\(23\)FrickeDim.
\(-\)\(+\)\(+\)\(+\)\(-\)\(2\)
\(-\)\(+\)\(+\)\(-\)\(+\)\(2\)
\(-\)\(+\)\(-\)\(+\)\(+\)\(2\)
\(-\)\(+\)\(-\)\(-\)\(-\)\(2\)
\(-\)\(-\)\(+\)\(+\)\(+\)\(1\)
\(-\)\(-\)\(+\)\(-\)\(-\)\(3\)
\(-\)\(-\)\(-\)\(+\)\(-\)\(3\)
\(-\)\(-\)\(-\)\(-\)\(+\)\(1\)
Plus space\(+\)\(6\)
Minus space\(-\)\(10\)

Trace form

\( 16 q + 16 q^{9} + O(q^{10}) \) \( 16 q + 16 q^{9} - 8 q^{13} - 16 q^{17} + 16 q^{25} - 12 q^{29} + 12 q^{31} + 8 q^{33} + 4 q^{35} + 16 q^{37} + 8 q^{39} + 12 q^{41} + 24 q^{43} + 24 q^{47} + 28 q^{49} + 8 q^{51} + 8 q^{53} + 8 q^{57} + 4 q^{59} + 8 q^{67} - 4 q^{71} - 8 q^{73} - 24 q^{77} + 16 q^{81} - 16 q^{83} - 4 q^{85} - 8 q^{87} - 8 q^{89} - 16 q^{91} + 16 q^{93} - 16 q^{95} + 32 q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1380))\) 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 23
1380.2.a.a 1380.a 1.a $1$ $11.019$ \(\Q\) None \(0\) \(-1\) \(-1\) \(-5\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}-q^{5}-5q^{7}+q^{9}+4q^{13}+q^{15}+\cdots\)
1380.2.a.b 1380.a 1.a $1$ $11.019$ \(\Q\) None \(0\) \(-1\) \(-1\) \(0\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}-q^{5}+q^{9}-6q^{13}+q^{15}+2q^{17}+\cdots\)
1380.2.a.c 1380.a 1.a $1$ $11.019$ \(\Q\) None \(0\) \(1\) \(-1\) \(-1\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}-q^{5}-q^{7}+q^{9}-4q^{13}-q^{15}+\cdots\)
1380.2.a.d 1380.a 1.a $1$ $11.019$ \(\Q\) None \(0\) \(1\) \(1\) \(-4\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}+q^{5}-4q^{7}+q^{9}+2q^{13}+q^{15}+\cdots\)
1380.2.a.e 1380.a 1.a $1$ $11.019$ \(\Q\) None \(0\) \(1\) \(1\) \(-3\) $-$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+q^{5}-3q^{7}+q^{9}-2q^{11}-2q^{13}+\cdots\)
1380.2.a.f 1380.a 1.a $2$ $11.019$ \(\Q(\sqrt{6}) \) None \(0\) \(-2\) \(-2\) \(2\) $-$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}-q^{5}+q^{7}+q^{9}+(-2+\beta )q^{11}+\cdots\)
1380.2.a.g 1380.a 1.a $2$ $11.019$ \(\Q(\sqrt{17}) \) None \(0\) \(-2\) \(2\) \(1\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}+q^{5}+\beta q^{7}+q^{9}+2\beta q^{11}+\cdots\)
1380.2.a.h 1380.a 1.a $2$ $11.019$ \(\Q(\sqrt{3}) \) None \(0\) \(-2\) \(2\) \(2\) $-$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+q^{5}+(1+2\beta )q^{7}+q^{9}+(-1+\cdots)q^{11}+\cdots\)
1380.2.a.i 1380.a 1.a $2$ $11.019$ \(\Q(\sqrt{15}) \) None \(0\) \(2\) \(2\) \(6\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}+q^{5}+3q^{7}+q^{9}+(1+\beta )q^{11}+\cdots\)
1380.2.a.j 1380.a 1.a $3$ $11.019$ 3.3.3144.1 None \(0\) \(3\) \(-3\) \(2\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}-q^{5}+(1-\beta _{1})q^{7}+q^{9}+(1+\beta _{2})q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1380))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1380)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(23))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(46))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(69))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(92))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(115))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(138))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(230))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(276))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(345))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(460))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(690))\)\(^{\oplus 2}\)