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Properties

Label 378.4.a
Level $378$
Weight $4$
Character orbit 378.a
Rep. character $\chi_{378}(1,\cdot)$
Character field $\Q$
Dimension $24$
Newform subspaces $18$
Sturm bound $288$
Trace bound $11$

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Defining parameters

Level: \( N \) \(=\) \( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 378.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 18 \)
Sturm bound: \(288\)
Trace bound: \(11\)
Distinguishing \(T_p\): \(5\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(378))\).

Total New Old
Modular forms 228 24 204
Cusp forms 204 24 180
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.

\(2\)\(3\)\(7\)FrickeDim
\(+\)\(+\)\(+\)$+$\(3\)
\(+\)\(+\)\(-\)$-$\(3\)
\(+\)\(-\)\(+\)$-$\(3\)
\(+\)\(-\)\(-\)$+$\(3\)
\(-\)\(+\)\(+\)$-$\(2\)
\(-\)\(+\)\(-\)$+$\(4\)
\(-\)\(-\)\(+\)$+$\(4\)
\(-\)\(-\)\(-\)$-$\(2\)
Plus space\(+\)\(14\)
Minus space\(-\)\(10\)

Trace form

\( 24 q + 96 q^{4} + O(q^{10}) \) \( 24 q + 96 q^{4} - 96 q^{10} + 144 q^{13} + 384 q^{16} - 48 q^{19} + 72 q^{22} + 1068 q^{25} + 456 q^{31} + 1200 q^{34} - 540 q^{37} - 384 q^{40} - 516 q^{43} - 240 q^{46} + 1176 q^{49} + 576 q^{52} - 96 q^{55} + 2616 q^{58} + 1080 q^{61} + 1536 q^{64} + 708 q^{67} - 504 q^{70} + 3768 q^{73} - 192 q^{76} + 768 q^{79} - 864 q^{82} - 1320 q^{85} + 288 q^{88} - 420 q^{91} - 336 q^{94} - 7368 q^{97} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(378))\) 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 7
378.4.a.a 378.a 1.a $1$ $22.303$ \(\Q\) None \(-2\) \(0\) \(-20\) \(-7\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{4}-20q^{5}-7q^{7}-8q^{8}+\cdots\)
378.4.a.b 378.a 1.a $1$ $22.303$ \(\Q\) None \(-2\) \(0\) \(1\) \(-7\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{4}+q^{5}-7q^{7}-8q^{8}+\cdots\)
378.4.a.c 378.a 1.a $1$ $22.303$ \(\Q\) None \(-2\) \(0\) \(7\) \(-7\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{4}+7q^{5}-7q^{7}-8q^{8}+\cdots\)
378.4.a.d 378.a 1.a $1$ $22.303$ \(\Q\) None \(-2\) \(0\) \(7\) \(-7\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{4}+7q^{5}-7q^{7}-8q^{8}+\cdots\)
378.4.a.e 378.a 1.a $1$ $22.303$ \(\Q\) None \(-2\) \(0\) \(9\) \(7\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{4}+9q^{5}+7q^{7}-8q^{8}+\cdots\)
378.4.a.f 378.a 1.a $1$ $22.303$ \(\Q\) None \(-2\) \(0\) \(15\) \(7\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{4}+15q^{5}+7q^{7}-8q^{8}+\cdots\)
378.4.a.g 378.a 1.a $1$ $22.303$ \(\Q\) None \(2\) \(0\) \(-15\) \(7\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}-15q^{5}+7q^{7}+8q^{8}+\cdots\)
378.4.a.h 378.a 1.a $1$ $22.303$ \(\Q\) None \(2\) \(0\) \(-9\) \(7\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}-9q^{5}+7q^{7}+8q^{8}+\cdots\)
378.4.a.i 378.a 1.a $1$ $22.303$ \(\Q\) None \(2\) \(0\) \(-7\) \(-7\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}-7q^{5}-7q^{7}+8q^{8}+\cdots\)
378.4.a.j 378.a 1.a $1$ $22.303$ \(\Q\) None \(2\) \(0\) \(-7\) \(-7\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}-7q^{5}-7q^{7}+8q^{8}+\cdots\)
378.4.a.k 378.a 1.a $1$ $22.303$ \(\Q\) None \(2\) \(0\) \(-1\) \(-7\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}-q^{5}-7q^{7}+8q^{8}+\cdots\)
378.4.a.l 378.a 1.a $1$ $22.303$ \(\Q\) None \(2\) \(0\) \(20\) \(-7\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}+20q^{5}-7q^{7}+8q^{8}+\cdots\)
378.4.a.m 378.a 1.a $2$ $22.303$ \(\Q(\sqrt{11}) \) None \(-4\) \(0\) \(-12\) \(14\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{4}+(-6+\beta )q^{5}+7q^{7}+\cdots\)
378.4.a.n 378.a 1.a $2$ $22.303$ \(\Q(\sqrt{23}) \) None \(-4\) \(0\) \(8\) \(-14\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{4}+(4+\beta )q^{5}-7q^{7}-8q^{8}+\cdots\)
378.4.a.o 378.a 1.a $2$ $22.303$ \(\Q(\sqrt{105}) \) None \(-4\) \(0\) \(9\) \(14\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{4}+(5+\beta )q^{5}+7q^{7}-8q^{8}+\cdots\)
378.4.a.p 378.a 1.a $2$ $22.303$ \(\Q(\sqrt{105}) \) None \(4\) \(0\) \(-9\) \(14\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}+(-5-\beta )q^{5}+7q^{7}+\cdots\)
378.4.a.q 378.a 1.a $2$ $22.303$ \(\Q(\sqrt{23}) \) None \(4\) \(0\) \(-8\) \(-14\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}+(-4+\beta )q^{5}-7q^{7}+\cdots\)
378.4.a.r 378.a 1.a $2$ $22.303$ \(\Q(\sqrt{11}) \) None \(4\) \(0\) \(12\) \(14\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}+(6+\beta )q^{5}+7q^{7}+8q^{8}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(378))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(378)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(18))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(27))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(42))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(54))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(63))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(126))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(189))\)\(^{\oplus 2}\)