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Properties

Label 396.6.a
Level $396$
Weight $6$
Character orbit 396.a
Rep. character $\chi_{396}(1,\cdot)$
Character field $\Q$
Dimension $21$
Newform subspaces $10$
Sturm bound $432$
Trace bound $11$

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Defining parameters

Level: \( N \) \(=\) \( 396 = 2^{2} \cdot 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 396.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 10 \)
Sturm bound: \(432\)
Trace bound: \(11\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(396))\).

Total New Old
Modular forms 372 21 351
Cusp forms 348 21 327
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
\(-\)\(+\)\(+\)\(-\)\(4\)
\(-\)\(+\)\(-\)\(+\)\(4\)
\(-\)\(-\)\(+\)\(+\)\(6\)
\(-\)\(-\)\(-\)\(-\)\(7\)
Plus space\(+\)\(10\)
Minus space\(-\)\(11\)

Trace form

\( 21 q + 57 q^{5} - 102 q^{7} + O(q^{10}) \) \( 21 q + 57 q^{5} - 102 q^{7} + 121 q^{11} - 504 q^{13} - 210 q^{17} - 528 q^{19} - 6609 q^{23} + 17010 q^{25} - 6504 q^{29} + 4185 q^{31} - 26538 q^{35} - 19611 q^{37} + 20136 q^{41} + 20094 q^{43} - 17664 q^{47} + 39729 q^{49} - 13122 q^{53} - 6897 q^{55} + 62277 q^{59} + 88320 q^{61} + 11316 q^{65} + 807 q^{67} - 95619 q^{71} - 35616 q^{73} + 8954 q^{77} + 121482 q^{79} + 115386 q^{83} - 99342 q^{85} + 25983 q^{89} - 359928 q^{91} + 211128 q^{95} + 213207 q^{97} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(396))\) 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
396.6.a.a 396.a 1.a $1$ $63.512$ \(\Q\) None 132.6.a.d \(0\) \(0\) \(-42\) \(-208\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-42q^{5}-208q^{7}+11^{2}q^{11}-1054q^{13}+\cdots\)
396.6.a.b 396.a 1.a $1$ $63.512$ \(\Q\) None 132.6.a.b \(0\) \(0\) \(-22\) \(-52\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-22q^{5}-52q^{7}-11^{2}q^{11}+198q^{13}+\cdots\)
396.6.a.c 396.a 1.a $1$ $63.512$ \(\Q\) None 132.6.a.a \(0\) \(0\) \(36\) \(122\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+6^{2}q^{5}+122q^{7}-11^{2}q^{11}-208q^{13}+\cdots\)
396.6.a.d 396.a 1.a $1$ $63.512$ \(\Q\) None 132.6.a.c \(0\) \(0\) \(56\) \(58\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+56q^{5}+58q^{7}+11^{2}q^{11}+52q^{13}+\cdots\)
396.6.a.e 396.a 1.a $1$ $63.512$ \(\Q\) None 44.6.a.a \(0\) \(0\) \(79\) \(-50\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+79q^{5}-50q^{7}-11^{2}q^{11}-380q^{13}+\cdots\)
396.6.a.f 396.a 1.a $2$ $63.512$ \(\Q(\sqrt{31}) \) None 44.6.a.b \(0\) \(0\) \(22\) \(268\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(11+2\beta )q^{5}+(134+3\beta )q^{7}+11^{2}q^{11}+\cdots\)
396.6.a.g 396.a 1.a $3$ $63.512$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None 132.6.a.e \(0\) \(0\) \(-36\) \(-52\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(-12+\beta _{1})q^{5}+(-17-2\beta _{1}+\beta _{2})q^{7}+\cdots\)
396.6.a.h 396.a 1.a $3$ $63.512$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None 132.6.a.f \(0\) \(0\) \(-36\) \(-28\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(-12-\beta _{1})q^{5}+(-9+\beta _{1}-\beta _{2})q^{7}+\cdots\)
396.6.a.i 396.a 1.a $4$ $63.512$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None 396.6.a.i \(0\) \(0\) \(0\) \(-80\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{3}q^{5}+(-20+\beta _{1}+\beta _{3})q^{7}-11^{2}q^{11}+\cdots\)
396.6.a.j 396.a 1.a $4$ $63.512$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None 396.6.a.i \(0\) \(0\) \(0\) \(-80\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{3}q^{5}+(-20+\beta _{1}+\beta _{3})q^{7}+11^{2}q^{11}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(396))\) into lower level spaces

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