Properties

Label 1216.2.i
Level $1216$
Weight $2$
Character orbit 1216.i
Rep. character $\chi_{1216}(577,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $76$
Newform subspaces $18$
Sturm bound $320$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 1216 = 2^{6} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1216.i (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 19 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 18 \)
Sturm bound: \(320\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(3\), \(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(1216, [\chi])\).

Total New Old
Modular forms 344 84 260
Cusp forms 296 76 220
Eisenstein series 48 8 40

Trace form

\( 76 q + 2 q^{5} - 36 q^{9} + O(q^{10}) \) \( 76 q + 2 q^{5} - 36 q^{9} - 6 q^{13} - 2 q^{17} - 32 q^{25} + 2 q^{29} - 24 q^{33} + 24 q^{37} + 6 q^{41} - 24 q^{45} + 44 q^{49} + 2 q^{53} - 6 q^{57} - 38 q^{61} + 12 q^{65} + 60 q^{69} - 2 q^{73} + 16 q^{77} - 22 q^{81} + 22 q^{85} - 2 q^{89} - 2 q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(1216, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
1216.2.i.a 1216.i 19.c $2$ $9.710$ \(\Q(\sqrt{-3}) \) None 608.2.i.a \(0\) \(-3\) \(2\) \(4\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-3+3\zeta_{6})q^{3}+(2-2\zeta_{6})q^{5}+2q^{7}+\cdots\)
1216.2.i.b 1216.i 19.c $2$ $9.710$ \(\Q(\sqrt{-3}) \) None 152.2.i.b \(0\) \(-1\) \(-4\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1+\zeta_{6})q^{3}+(-4+4\zeta_{6})q^{5}+2\zeta_{6}q^{9}+\cdots\)
1216.2.i.c 1216.i 19.c $2$ $9.710$ \(\Q(\sqrt{-3}) \) None 76.2.e.a \(0\) \(-1\) \(-1\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1+\zeta_{6})q^{3}+(-1+\zeta_{6})q^{5}+2\zeta_{6}q^{9}+\cdots\)
1216.2.i.d 1216.i 19.c $2$ $9.710$ \(\Q(\sqrt{-3}) \) None 38.2.c.a \(0\) \(-1\) \(0\) \(8\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1+\zeta_{6})q^{3}+4q^{7}+2\zeta_{6}q^{9}+3q^{11}+\cdots\)
1216.2.i.e 1216.i 19.c $2$ $9.710$ \(\Q(\sqrt{-3}) \) None 152.2.i.a \(0\) \(-1\) \(3\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1+\zeta_{6})q^{3}+(3-3\zeta_{6})q^{5}+2\zeta_{6}q^{9}+\cdots\)
1216.2.i.f 1216.i 19.c $2$ $9.710$ \(\Q(\sqrt{-3}) \) None 152.2.i.b \(0\) \(1\) \(-4\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1-\zeta_{6})q^{3}+(-4+4\zeta_{6})q^{5}+2\zeta_{6}q^{9}+\cdots\)
1216.2.i.g 1216.i 19.c $2$ $9.710$ \(\Q(\sqrt{-3}) \) None 76.2.e.a \(0\) \(1\) \(-1\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1-\zeta_{6})q^{3}+(-1+\zeta_{6})q^{5}+2\zeta_{6}q^{9}+\cdots\)
1216.2.i.h 1216.i 19.c $2$ $9.710$ \(\Q(\sqrt{-3}) \) None 38.2.c.a \(0\) \(1\) \(0\) \(-8\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1-\zeta_{6})q^{3}-4q^{7}+2\zeta_{6}q^{9}-3q^{11}+\cdots\)
1216.2.i.i 1216.i 19.c $2$ $9.710$ \(\Q(\sqrt{-3}) \) None 152.2.i.a \(0\) \(1\) \(3\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1-\zeta_{6})q^{3}+(3-3\zeta_{6})q^{5}+2\zeta_{6}q^{9}+\cdots\)
1216.2.i.j 1216.i 19.c $2$ $9.710$ \(\Q(\sqrt{-3}) \) None 608.2.i.a \(0\) \(3\) \(2\) \(-4\) $\mathrm{SU}(2)[C_{3}]$ \(q+(3-3\zeta_{6})q^{3}+(2-2\zeta_{6})q^{5}-2q^{7}+\cdots\)
1216.2.i.k 1216.i 19.c $4$ $9.710$ \(\Q(\sqrt{-3}, \sqrt{7})\) None 38.2.c.b \(0\) \(0\) \(2\) \(-4\) $\mathrm{SU}(2)[C_{3}]$ \(q+\beta _{1}q^{3}+(1+\beta _{1}+\beta _{2})q^{5}+(-1-\beta _{3})q^{7}+\cdots\)
1216.2.i.l 1216.i 19.c $4$ $9.710$ \(\Q(\sqrt{-3}, \sqrt{7})\) None 38.2.c.b \(0\) \(0\) \(2\) \(4\) $\mathrm{SU}(2)[C_{3}]$ \(q+\beta _{1}q^{3}+(1-\beta _{1}+\beta _{2})q^{5}+(1-\beta _{3})q^{7}+\cdots\)
1216.2.i.m 1216.i 19.c $6$ $9.710$ 6.0.2696112.1 None 152.2.i.c \(0\) \(-1\) \(1\) \(4\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-\beta _{1}+\beta _{5})q^{3}+\beta _{1}q^{5}+(1+\beta _{2}+\cdots)q^{7}+\cdots\)
1216.2.i.n 1216.i 19.c $6$ $9.710$ 6.0.2696112.1 None 152.2.i.c \(0\) \(1\) \(1\) \(-4\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-\beta _{1}-\beta _{2}-\beta _{3}+\beta _{5})q^{3}+(-\beta _{1}+\cdots)q^{5}+\cdots\)
1216.2.i.o 1216.i 19.c $8$ $9.710$ 8.0.\(\cdots\).5 None 608.2.i.c \(0\) \(-4\) \(-2\) \(-8\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1-\beta _{2}+\beta _{4}-\beta _{6})q^{3}+(-1-\beta _{1}+\cdots)q^{5}+\cdots\)
1216.2.i.p 1216.i 19.c $8$ $9.710$ 8.0.\(\cdots\).1 None 608.2.i.d \(0\) \(0\) \(-2\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(\beta _{2}-\beta _{4})q^{3}-\beta _{1}q^{5}+(-2+2\beta _{1}+\cdots)q^{9}+\cdots\)
1216.2.i.q 1216.i 19.c $8$ $9.710$ 8.0.\(\cdots\).5 None 608.2.i.c \(0\) \(4\) \(-2\) \(8\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-\beta _{2}+\beta _{4})q^{3}+(-\beta _{1}-\beta _{2})q^{5}+\cdots\)
1216.2.i.r 1216.i 19.c $12$ $9.710$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None 608.2.i.f \(0\) \(0\) \(2\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(\beta _{5}-\beta _{8})q^{3}+\beta _{10}q^{5}+(\beta _{2}-\beta _{5}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(1216, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(1216, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(38, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(76, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(152, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(304, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(608, [\chi])\)\(^{\oplus 2}\)