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Properties

Label 396.4.a
Level $396$
Weight $4$
Character orbit 396.a
Rep. character $\chi_{396}(1,\cdot)$
Character field $\Q$
Dimension $13$
Newform subspaces $10$
Sturm bound $288$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 396 = 2^{2} \cdot 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 396.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 10 \)
Sturm bound: \(288\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(396))\).

Total New Old
Modular forms 228 13 215
Cusp forms 204 13 191
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.

\(2\)\(3\)\(11\)FrickeDim
\(-\)\(+\)\(+\)\(-\)\(3\)
\(-\)\(+\)\(-\)\(+\)\(3\)
\(-\)\(-\)\(+\)\(+\)\(4\)
\(-\)\(-\)\(-\)\(-\)\(3\)
Plus space\(+\)\(7\)
Minus space\(-\)\(6\)

Trace form

\( 13 q - 24 q^{5} - 8 q^{7} + O(q^{10}) \) \( 13 q - 24 q^{5} - 8 q^{7} - 11 q^{11} - 46 q^{13} - 18 q^{17} + 164 q^{19} + 189 q^{25} - 66 q^{29} - 220 q^{31} + 404 q^{37} + 546 q^{41} - 228 q^{43} + 348 q^{47} + 1101 q^{49} + 354 q^{53} + 198 q^{55} - 588 q^{59} - 1422 q^{61} + 396 q^{65} + 308 q^{67} + 216 q^{71} - 554 q^{73} - 220 q^{77} + 160 q^{79} + 36 q^{83} + 1080 q^{85} - 1656 q^{89} + 336 q^{91} + 168 q^{95} + 2016 q^{97} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(396))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 3 11
396.4.a.a 396.a 1.a $1$ $23.365$ \(\Q\) None 132.4.a.c \(0\) \(0\) \(-22\) \(-20\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-22q^{5}-20q^{7}-11q^{11}+22q^{13}+\cdots\)
396.4.a.b 396.a 1.a $1$ $23.365$ \(\Q\) None 396.4.a.b \(0\) \(0\) \(-12\) \(26\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-12q^{5}+26q^{7}-11q^{11}-34q^{13}+\cdots\)
396.4.a.c 396.a 1.a $1$ $23.365$ \(\Q\) None 132.4.a.d \(0\) \(0\) \(-10\) \(8\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-10q^{5}+8q^{7}+11q^{11}+18q^{13}+\cdots\)
396.4.a.d 396.a 1.a $1$ $23.365$ \(\Q\) None 132.4.a.b \(0\) \(0\) \(0\) \(2\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{7}+11q^{11}-88q^{13}+66q^{17}+\cdots\)
396.4.a.e 396.a 1.a $1$ $23.365$ \(\Q\) None 44.4.a.a \(0\) \(0\) \(7\) \(-26\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+7q^{5}-26q^{7}+11q^{11}+52q^{13}+\cdots\)
396.4.a.f 396.a 1.a $1$ $23.365$ \(\Q\) None 132.4.a.a \(0\) \(0\) \(12\) \(14\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+12q^{5}+14q^{7}-11q^{11}+56q^{13}+\cdots\)
396.4.a.g 396.a 1.a $1$ $23.365$ \(\Q\) None 396.4.a.b \(0\) \(0\) \(12\) \(26\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+12q^{5}+26q^{7}+11q^{11}-34q^{13}+\cdots\)
396.4.a.h 396.a 1.a $2$ $23.365$ \(\Q(\sqrt{31}) \) None 396.4.a.h \(0\) \(0\) \(-12\) \(-24\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(-6+\beta )q^{5}+(-12+\beta )q^{7}+11q^{11}+\cdots\)
396.4.a.i 396.a 1.a $2$ $23.365$ \(\Q(\sqrt{97}) \) None 44.4.a.b \(0\) \(0\) \(-11\) \(10\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(-5-\beta )q^{5}+(2+6\beta )q^{7}-11q^{11}+\cdots\)
396.4.a.j 396.a 1.a $2$ $23.365$ \(\Q(\sqrt{31}) \) None 396.4.a.h \(0\) \(0\) \(12\) \(-24\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(6+\beta )q^{5}+(-12-\beta )q^{7}-11q^{11}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(396))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(396)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 9}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(18))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(22))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(33))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(44))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(66))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(99))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(132))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(198))\)\(^{\oplus 2}\)