Properties

Label 387.4.a
Level $387$
Weight $4$
Character orbit 387.a
Rep. character $\chi_{387}(1,\cdot)$
Character field $\Q$
Dimension $52$
Newform subspaces $11$
Sturm bound $176$
Trace bound $4$

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Defining parameters

Level: \( N \) \(=\) \( 387 = 3^{2} \cdot 43 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 387.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 11 \)
Sturm bound: \(176\)
Trace bound: \(4\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(387))\).

Total New Old
Modular forms 136 52 84
Cusp forms 128 52 76
Eisenstein series 8 0 8

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

\(3\)\(43\)FrickeDim
\(+\)\(+\)$+$\(12\)
\(+\)\(-\)$-$\(8\)
\(-\)\(+\)$-$\(15\)
\(-\)\(-\)$+$\(17\)
Plus space\(+\)\(29\)
Minus space\(-\)\(23\)

Trace form

\( 52 q + 2 q^{2} + 216 q^{4} + 12 q^{7} - 24 q^{8} + O(q^{10}) \) \( 52 q + 2 q^{2} + 216 q^{4} + 12 q^{7} - 24 q^{8} - 90 q^{10} + 14 q^{11} + 42 q^{13} + 140 q^{14} + 748 q^{16} + 44 q^{17} + 72 q^{19} - 40 q^{20} + 154 q^{22} + 32 q^{23} + 1546 q^{25} - 282 q^{26} + 740 q^{28} + 240 q^{29} - 712 q^{31} - 860 q^{32} - 822 q^{34} - 168 q^{35} - 216 q^{37} + 1534 q^{38} - 670 q^{40} + 156 q^{41} - 86 q^{43} + 300 q^{44} + 562 q^{46} - 322 q^{47} + 2188 q^{49} + 878 q^{50} - 720 q^{52} - 206 q^{53} + 252 q^{55} + 2024 q^{56} - 170 q^{58} - 116 q^{59} + 1888 q^{61} + 2518 q^{62} + 4932 q^{64} + 1268 q^{65} + 2310 q^{67} - 1098 q^{68} - 2692 q^{70} - 2780 q^{71} - 1664 q^{73} + 1206 q^{74} + 1692 q^{76} + 840 q^{77} - 2750 q^{79} + 504 q^{80} + 1834 q^{82} - 1878 q^{83} + 988 q^{85} + 430 q^{86} + 488 q^{88} + 1480 q^{89} + 1992 q^{91} + 650 q^{92} - 956 q^{94} - 278 q^{95} + 3712 q^{97} - 3862 q^{98} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(387))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 3 43
387.4.a.a 387.a 1.a $1$ $22.834$ \(\Q\) None \(-4\) \(0\) \(-11\) \(9\) $-$ $+$ $\mathrm{SU}(2)$ \(q-4q^{2}+8q^{4}-11q^{5}+9q^{7}+44q^{10}+\cdots\)
387.4.a.b 387.a 1.a $1$ $22.834$ \(\Q\) None \(1\) \(0\) \(2\) \(6\) $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-7q^{4}+2q^{5}+6q^{7}-15q^{8}+\cdots\)
387.4.a.c 387.a 1.a $2$ $22.834$ \(\Q(\sqrt{7}) \) None \(0\) \(0\) \(0\) \(-58\) $+$ $+$ $\mathrm{SU}(2)$ \(q-8q^{4}-\beta q^{5}-29q^{7}-\beta q^{11}-21q^{13}+\cdots\)
387.4.a.d 387.a 1.a $2$ $22.834$ \(\Q(\sqrt{10}) \) None \(2\) \(0\) \(14\) \(-38\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{2}+(3+2\beta )q^{4}+(7-3\beta )q^{5}+\cdots\)
387.4.a.e 387.a 1.a $4$ $22.834$ 4.4.45868.1 None \(4\) \(0\) \(27\) \(-20\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(1-\beta _{3})q^{2}+(1+\beta _{1}-3\beta _{2}-4\beta _{3})q^{4}+\cdots\)
387.4.a.f 387.a 1.a $5$ $22.834$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(3\) \(0\) \(12\) \(-18\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}+(4-\beta _{1}+\beta _{2}+\beta _{4})q^{4}+\cdots\)
387.4.a.g 387.a 1.a $5$ $22.834$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(3\) \(0\) \(3\) \(1\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}+(5+\beta _{2}+\beta _{4})q^{4}+(2\beta _{1}+\cdots)q^{5}+\cdots\)
387.4.a.h 387.a 1.a $6$ $22.834$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(-6\) \(0\) \(-43\) \(8\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{2}+(4-\beta _{1}+\beta _{3}-\beta _{4}+\cdots)q^{4}+\cdots\)
387.4.a.i 387.a 1.a $8$ $22.834$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(-1\) \(0\) \(-4\) \(52\) $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+(7+\beta _{2})q^{4}+\beta _{5}q^{5}+(6-2\beta _{1}+\cdots)q^{7}+\cdots\)
387.4.a.j 387.a 1.a $8$ $22.834$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(0\) \(0\) \(0\) \(-22\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(3+\beta _{2})q^{4}+(-\beta _{1}+\beta _{4}+\cdots)q^{5}+\cdots\)
387.4.a.k 387.a 1.a $10$ $22.834$ \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None \(0\) \(0\) \(0\) \(92\) $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(8+\beta _{2})q^{4}-\beta _{4}q^{5}+(9-\beta _{9})q^{7}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(387))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(387)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(43))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(129))\)\(^{\oplus 2}\)