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Properties

Label 1380.4.a
Level $1380$
Weight $4$
Character orbit 1380.a
Rep. character $\chi_{1380}(1,\cdot)$
Character field $\Q$
Dimension $44$
Newform subspaces $10$
Sturm bound $1152$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 1380 = 2^{2} \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1380.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 10 \)
Sturm bound: \(1152\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(1380))\).

Total New Old
Modular forms 876 44 832
Cusp forms 852 44 808
Eisenstein series 24 0 24

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.
\(-\)\(+\)\(+\)\(+\)\(-\)\(5\)
\(-\)\(+\)\(+\)\(-\)\(+\)\(5\)
\(-\)\(+\)\(-\)\(+\)\(+\)\(5\)
\(-\)\(+\)\(-\)\(-\)\(-\)\(5\)
\(-\)\(-\)\(+\)\(+\)\(+\)\(7\)
\(-\)\(-\)\(+\)\(-\)\(-\)\(5\)
\(-\)\(-\)\(-\)\(+\)\(-\)\(5\)
\(-\)\(-\)\(-\)\(-\)\(+\)\(7\)
Plus space\(+\)\(24\)
Minus space\(-\)\(20\)

Trace form

\( 44 q + 12 q^{3} - 8 q^{7} + 396 q^{9} + O(q^{10}) \) \( 44 q + 12 q^{3} - 8 q^{7} + 396 q^{9} + 192 q^{13} + 16 q^{17} - 104 q^{19} + 24 q^{21} + 1100 q^{25} + 108 q^{27} - 672 q^{29} - 144 q^{31} - 192 q^{33} - 40 q^{35} - 488 q^{37} - 504 q^{39} - 288 q^{41} - 280 q^{43} - 912 q^{47} + 500 q^{49} + 240 q^{51} + 304 q^{53} + 504 q^{57} + 968 q^{59} + 744 q^{61} - 72 q^{63} + 664 q^{67} + 1864 q^{71} + 2120 q^{73} + 300 q^{75} - 2400 q^{77} + 1496 q^{79} + 3564 q^{81} + 1552 q^{83} + 1240 q^{85} + 1920 q^{87} + 2288 q^{89} + 496 q^{91} - 96 q^{93} - 80 q^{95} + 776 q^{97} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(1380))\) 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
1380.4.a.a 1380.a 1.a $1$ $81.423$ \(\Q\) None \(0\) \(-3\) \(-5\) \(-16\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-3q^{3}-5q^{5}-2^{4}q^{7}+9q^{9}+30q^{11}+\cdots\)
1380.4.a.b 1380.a 1.a $2$ $81.423$ \(\Q(\sqrt{561}) \) None \(0\) \(-6\) \(10\) \(-17\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-3q^{3}+5q^{5}+(-8-\beta )q^{7}+9q^{9}+\cdots\)
1380.4.a.c 1380.a 1.a $3$ $81.423$ 3.3.72332.1 None \(0\) \(-9\) \(15\) \(22\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-3q^{3}+5q^{5}+(8-2\beta _{1}-\beta _{2})q^{7}+\cdots\)
1380.4.a.d 1380.a 1.a $4$ $81.423$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(0\) \(-12\) \(-20\) \(35\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-3q^{3}-5q^{5}+(9-\beta _{3})q^{7}+9q^{9}+\cdots\)
1380.4.a.e 1380.a 1.a $5$ $81.423$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(0\) \(-15\) \(-25\) \(-7\) $-$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-3q^{3}-5q^{5}+(-1-\beta _{2})q^{7}+9q^{9}+\cdots\)
1380.4.a.f 1380.a 1.a $5$ $81.423$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(0\) \(-15\) \(25\) \(-25\) $-$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-3q^{3}+5q^{5}+(-5-\beta _{1})q^{7}+9q^{9}+\cdots\)
1380.4.a.g 1380.a 1.a $5$ $81.423$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(0\) \(15\) \(-25\) \(-5\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+3q^{3}-5q^{5}+(-1-\beta _{1})q^{7}+9q^{9}+\cdots\)
1380.4.a.h 1380.a 1.a $5$ $81.423$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(0\) \(15\) \(25\) \(-23\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+3q^{3}+5q^{5}+(-5-\beta _{1}-\beta _{3}-\beta _{4})q^{7}+\cdots\)
1380.4.a.i 1380.a 1.a $7$ $81.423$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(0\) \(21\) \(-35\) \(-7\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+3q^{3}-5q^{5}+(-1+\beta _{1})q^{7}+9q^{9}+\cdots\)
1380.4.a.j 1380.a 1.a $7$ $81.423$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(0\) \(21\) \(35\) \(35\) $-$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+3q^{3}+5q^{5}+(5-\beta _{1})q^{7}+9q^{9}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(1380))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(1380)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(23))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(46))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(60))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(69))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(92))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(115))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(138))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(230))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(276))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(345))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(460))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(690))\)\(^{\oplus 2}\)