Properties

Label 387.8.a
Level $387$
Weight $8$
Character orbit 387.a
Rep. character $\chi_{387}(1,\cdot)$
Character field $\Q$
Dimension $122$
Newform subspaces $8$
Sturm bound $352$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 387 = 3^{2} \cdot 43 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 387.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 8 \)
Sturm bound: \(352\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(387))\).

Total New Old
Modular forms 312 122 190
Cusp forms 304 122 182
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.
\(+\)\(+\)\(+\)\(26\)
\(+\)\(-\)\(-\)\(22\)
\(-\)\(+\)\(-\)\(36\)
\(-\)\(-\)\(+\)\(38\)
Plus space\(+\)\(64\)
Minus space\(-\)\(58\)

Trace form

\( 122q - 8q^{2} + 7692q^{4} + 530q^{5} - 1348q^{7} - 5136q^{8} + O(q^{10}) \) \( 122q - 8q^{2} + 7692q^{4} + 530q^{5} - 1348q^{7} - 5136q^{8} + 14510q^{10} + 1785q^{11} - 10349q^{13} - 14220q^{14} + 493820q^{16} + 32645q^{17} - 70634q^{19} - 41040q^{20} + 217930q^{22} - 8527q^{23} + 1704706q^{25} - 132414q^{26} - 336332q^{28} + 141316q^{29} + 252437q^{31} - 41436q^{32} - 1167746q^{34} + 691112q^{35} + 455050q^{37} + 603202q^{38} + 1115410q^{40} + 131969q^{41} - 159014q^{43} + 1336708q^{44} + 1201898q^{46} + 1184464q^{47} + 17217774q^{49} - 2810792q^{50} - 3005440q^{52} + 1107311q^{53} - 3234528q^{55} - 3040472q^{56} + 9979062q^{58} + 5447176q^{59} - 586404q^{61} - 13869234q^{62} + 44081316q^{64} - 10122152q^{65} - 5630593q^{67} - 8583026q^{68} - 6895612q^{70} + 5511942q^{71} + 7528880q^{73} + 4105134q^{74} - 35925172q^{76} + 4961408q^{77} - 9663292q^{79} + 19905464q^{80} - 17748978q^{82} + 12384849q^{83} + 24737748q^{85} + 3180280q^{86} + 38628952q^{88} - 16902450q^{89} + 78844q^{91} + 9966778q^{92} - 20403988q^{94} - 11817308q^{95} + 19702247q^{97} - 25281456q^{98} + O(q^{100}) \)

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(387))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 3 43
387.8.a.a \(10\) \(120.893\) \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None \(1\) \(0\) \(122\) \(-2052\) \(-\) \(+\) \(q+\beta _{1}q^{2}+(37+\beta _{1}+\beta _{2})q^{4}+(13-6\beta _{1}+\cdots)q^{5}+\cdots\)
387.8.a.b \(11\) \(120.893\) \(\mathbb{Q}[x]/(x^{11} - \cdots)\) None \(24\) \(0\) \(752\) \(-12\) \(-\) \(-\) \(q+(2+\beta _{1})q^{2}+(54+3\beta _{1}+\beta _{2})q^{4}+\cdots\)
387.8.a.c \(12\) \(120.893\) \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(7\) \(0\) \(766\) \(-1366\) \(-\) \(-\) \(q+(1-\beta _{1})q^{2}+(58+\beta _{2})q^{4}+(2^{6}-\beta _{1}+\cdots)q^{5}+\cdots\)
387.8.a.d \(13\) \(120.893\) \(\mathbb{Q}[x]/(x^{13} - \cdots)\) None \(-16\) \(0\) \(-998\) \(1360\) \(-\) \(+\) \(q+(-1-\beta _{1})q^{2}+(70+2\beta _{1}+\beta _{2})q^{4}+\cdots\)
387.8.a.e \(13\) \(120.893\) \(\mathbb{Q}[x]/(x^{13} - \cdots)\) None \(-9\) \(0\) \(266\) \(6\) \(-\) \(+\) \(q+(-1+\beta _{1})q^{2}+(73-2\beta _{1}+\beta _{2})q^{4}+\cdots\)
387.8.a.f \(15\) \(120.893\) \(\mathbb{Q}[x]/(x^{15} - \cdots)\) None \(-15\) \(0\) \(-378\) \(2064\) \(-\) \(-\) \(q+(-1+\beta _{1})q^{2}+(84-\beta _{1}+\beta _{2})q^{4}+\cdots\)
387.8.a.g \(22\) \(120.893\) None \(0\) \(0\) \(0\) \(-2046\) \(+\) \(-\)
387.8.a.h \(26\) \(120.893\) None \(0\) \(0\) \(0\) \(698\) \(+\) \(+\)

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_0(387))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_0(387)) \cong \) \(S_{8}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(43))\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(129))\)\(^{\oplus 2}\)