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Properties

Label 1296.2.s
Level $1296$
Weight $2$
Character orbit 1296.s
Rep. character $\chi_{1296}(431,\cdot)$
Character field $\Q(\zeta_{6})$
Dimension $48$
Newform subspaces $12$
Sturm bound $432$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 1296 = 2^{4} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1296.s (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 36 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 12 \)
Sturm bound: \(432\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(1296, [\chi])\).

Total New Old
Modular forms 504 48 456
Cusp forms 360 48 312
Eisenstein series 144 0 144

Trace form

\( 48 q + O(q^{10}) \) \( 48 q + 24 q^{25} + 24 q^{49} + 72 q^{73} + 36 q^{97} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(1296, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
1296.2.s.a 1296.s 36.h $2$ $10.349$ \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(-9\) $\mathrm{U}(1)[D_{6}]$ \(q+(-3-3\zeta_{6})q^{7}+5\zeta_{6}q^{13}+(-3+\cdots)q^{19}+\cdots\)
1296.2.s.b 1296.s 36.h $2$ $10.349$ \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(-6\) $\mathrm{U}(1)[D_{6}]$ \(q+(-2-2\zeta_{6})q^{7}+2\zeta_{6}q^{13}+(2-4\zeta_{6})q^{19}+\cdots\)
1296.2.s.c 1296.s 36.h $2$ $10.349$ \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(-3\) $\mathrm{U}(1)[D_{6}]$ \(q+(-1-\zeta_{6})q^{7}-7\zeta_{6}q^{13}+(-5+10\zeta_{6})q^{19}+\cdots\)
1296.2.s.d 1296.s 36.h $2$ $10.349$ \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(3\) $\mathrm{U}(1)[D_{6}]$ \(q+(1+\zeta_{6})q^{7}-7\zeta_{6}q^{13}+(5-10\zeta_{6})q^{19}+\cdots\)
1296.2.s.e 1296.s 36.h $2$ $10.349$ \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(6\) $\mathrm{U}(1)[D_{6}]$ \(q+(2+2\zeta_{6})q^{7}+2\zeta_{6}q^{13}+(-2+4\zeta_{6})q^{19}+\cdots\)
1296.2.s.f 1296.s 36.h $2$ $10.349$ \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(9\) $\mathrm{U}(1)[D_{6}]$ \(q+(3+3\zeta_{6})q^{7}+5\zeta_{6}q^{13}+(3-6\zeta_{6})q^{19}+\cdots\)
1296.2.s.g 1296.s 36.h $4$ $10.349$ \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(-6\) $\mathrm{SU}(2)[C_{6}]$ \(q+\zeta_{12}q^{5}+(-2+\zeta_{12}^{2})q^{7}+(\zeta_{12}+\cdots)q^{11}+\cdots\)
1296.2.s.h 1296.s 36.h $4$ $10.349$ \(\Q(\sqrt{-2}, \sqrt{-3})\) \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{6}]$ \(q+\beta _{1}q^{5}+(-4+4\beta _{2})q^{13}+\beta _{3}q^{17}+\cdots\)
1296.2.s.i 1296.s 36.h $4$ $10.349$ \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(6\) $\mathrm{SU}(2)[C_{6}]$ \(q+\zeta_{12}q^{5}+(2-\zeta_{12}^{2})q^{7}+(-\zeta_{12}+\cdots)q^{11}+\cdots\)
1296.2.s.j 1296.s 36.h $8$ $10.349$ \(\Q(\zeta_{24})\) None \(0\) \(0\) \(0\) \(-12\) $\mathrm{SU}(2)[C_{6}]$ \(q+\zeta_{24}^{6}q^{5}+(-1+\zeta_{24}-\zeta_{24}^{2})q^{7}+\cdots\)
1296.2.s.k 1296.s 36.h $8$ $10.349$ \(\Q(\zeta_{24})\) \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{6}]$ \(q+\zeta_{24}^{4}q^{5}+(2\zeta_{24}-\zeta_{24}^{2})q^{13}+(-\zeta_{24}^{5}+\cdots)q^{17}+\cdots\)
1296.2.s.l 1296.s 36.h $8$ $10.349$ \(\Q(\zeta_{24})\) None \(0\) \(0\) \(0\) \(12\) $\mathrm{SU}(2)[C_{6}]$ \(q+\zeta_{24}^{6}q^{5}+(1-\zeta_{24}+\zeta_{24}^{2})q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(1296, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(1296, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(36, [\chi])\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(72, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(108, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(144, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(216, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(432, [\chi])\)\(^{\oplus 2}\)