Properties

Label 1216.2.a
Level $1216$
Weight $2$
Character orbit 1216.a
Rep. character $\chi_{1216}(1,\cdot)$
Character field $\Q$
Dimension $36$
Newform subspaces $24$
Sturm bound $320$
Trace bound $7$

Related objects

Downloads

Learn more

Defining parameters

Level: \( N \) \(=\) \( 1216 = 2^{6} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1216.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 24 \)
Sturm bound: \(320\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(3\), \(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1216))\).

Total New Old
Modular forms 172 36 136
Cusp forms 149 36 113
Eisenstein series 23 0 23

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)\(19\)FrickeDim
\(+\)\(+\)$+$\(8\)
\(+\)\(-\)$-$\(11\)
\(-\)\(+\)$-$\(10\)
\(-\)\(-\)$+$\(7\)
Plus space\(+\)\(15\)
Minus space\(-\)\(21\)

Trace form

\( 36 q + 36 q^{9} + O(q^{10}) \) \( 36 q + 36 q^{9} - 8 q^{17} + 28 q^{25} - 16 q^{29} - 16 q^{37} - 8 q^{41} + 48 q^{45} + 36 q^{49} + 32 q^{53} + 48 q^{69} + 8 q^{73} + 8 q^{77} + 4 q^{81} + 24 q^{85} - 24 q^{89} - 40 q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1216))\) 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 19
1216.2.a.a 1216.a 1.a $1$ $9.710$ \(\Q\) None \(0\) \(-3\) \(0\) \(-1\) $-$ $-$ $\mathrm{SU}(2)$ \(q-3q^{3}-q^{7}+6q^{9}-2q^{11}+q^{13}+\cdots\)
1216.2.a.b 1216.a 1.a $1$ $9.710$ \(\Q\) None \(0\) \(-2\) \(-3\) \(1\) $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{3}-3q^{5}+q^{7}+q^{9}+3q^{11}+\cdots\)
1216.2.a.c 1216.a 1.a $1$ $9.710$ \(\Q\) None \(0\) \(-2\) \(1\) \(-3\) $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{3}+q^{5}-3q^{7}+q^{9}-5q^{11}+\cdots\)
1216.2.a.d 1216.a 1.a $1$ $9.710$ \(\Q\) None \(0\) \(-2\) \(1\) \(3\) $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{3}+q^{5}+3q^{7}+q^{9}-3q^{11}+\cdots\)
1216.2.a.e 1216.a 1.a $1$ $9.710$ \(\Q\) None \(0\) \(-1\) \(0\) \(-1\) $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}-q^{7}-2q^{9}+6q^{11}-5q^{13}+\cdots\)
1216.2.a.f 1216.a 1.a $1$ $9.710$ \(\Q\) None \(0\) \(-1\) \(0\) \(3\) $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+3q^{7}-2q^{9}-2q^{11}-q^{13}+\cdots\)
1216.2.a.g 1216.a 1.a $1$ $9.710$ \(\Q\) None \(0\) \(-1\) \(4\) \(-3\) $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+4q^{5}-3q^{7}-2q^{9}+2q^{11}+\cdots\)
1216.2.a.h 1216.a 1.a $1$ $9.710$ \(\Q\) None \(0\) \(0\) \(-3\) \(-5\) $-$ $+$ $\mathrm{SU}(2)$ \(q-3q^{5}-5q^{7}-3q^{9}+5q^{11}+4q^{13}+\cdots\)
1216.2.a.i 1216.a 1.a $1$ $9.710$ \(\Q\) None \(0\) \(0\) \(-3\) \(5\) $-$ $-$ $\mathrm{SU}(2)$ \(q-3q^{5}+5q^{7}-3q^{9}-5q^{11}+4q^{13}+\cdots\)
1216.2.a.j 1216.a 1.a $1$ $9.710$ \(\Q\) None \(0\) \(0\) \(1\) \(-1\) $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{5}-q^{7}-3q^{9}-3q^{11}+4q^{13}+\cdots\)
1216.2.a.k 1216.a 1.a $1$ $9.710$ \(\Q\) None \(0\) \(0\) \(1\) \(1\) $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{5}+q^{7}-3q^{9}+3q^{11}+4q^{13}+\cdots\)
1216.2.a.l 1216.a 1.a $1$ $9.710$ \(\Q\) None \(0\) \(1\) \(0\) \(-3\) $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}-3q^{7}-2q^{9}+2q^{11}-q^{13}+\cdots\)
1216.2.a.m 1216.a 1.a $1$ $9.710$ \(\Q\) None \(0\) \(1\) \(0\) \(1\) $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+q^{7}-2q^{9}-6q^{11}-5q^{13}+\cdots\)
1216.2.a.n 1216.a 1.a $1$ $9.710$ \(\Q\) None \(0\) \(1\) \(4\) \(3\) $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+4q^{5}+3q^{7}-2q^{9}-2q^{11}+\cdots\)
1216.2.a.o 1216.a 1.a $1$ $9.710$ \(\Q\) None \(0\) \(2\) \(-3\) \(-1\) $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{3}-3q^{5}-q^{7}+q^{9}-3q^{11}+\cdots\)
1216.2.a.p 1216.a 1.a $1$ $9.710$ \(\Q\) None \(0\) \(2\) \(1\) \(-3\) $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{3}+q^{5}-3q^{7}+q^{9}+3q^{11}+\cdots\)
1216.2.a.q 1216.a 1.a $1$ $9.710$ \(\Q\) None \(0\) \(2\) \(1\) \(3\) $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{3}+q^{5}+3q^{7}+q^{9}+5q^{11}+\cdots\)
1216.2.a.r 1216.a 1.a $1$ $9.710$ \(\Q\) None \(0\) \(3\) \(0\) \(1\) $-$ $+$ $\mathrm{SU}(2)$ \(q+3q^{3}+q^{7}+6q^{9}+2q^{11}+q^{13}+\cdots\)
1216.2.a.s 1216.a 1.a $2$ $9.710$ \(\Q(\sqrt{17}) \) None \(0\) \(-1\) \(1\) \(6\) $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta q^{3}+(1-\beta )q^{5}+3q^{7}+(1+\beta )q^{9}+\cdots\)
1216.2.a.t 1216.a 1.a $2$ $9.710$ \(\Q(\sqrt{17}) \) None \(0\) \(1\) \(1\) \(-6\) $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{3}+(1-\beta )q^{5}-3q^{7}+(1+\beta )q^{9}+\cdots\)
1216.2.a.u 1216.a 1.a $3$ $9.710$ 3.3.961.1 None \(0\) \(-1\) \(-1\) \(4\) $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{3}+\beta _{2}q^{5}+(2-\beta _{1}+\beta _{2})q^{7}+\cdots\)
1216.2.a.v 1216.a 1.a $3$ $9.710$ 3.3.961.1 None \(0\) \(1\) \(-1\) \(-4\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{3}+\beta _{2}q^{5}+(-2+\beta _{1}-\beta _{2})q^{7}+\cdots\)
1216.2.a.w 1216.a 1.a $4$ $9.710$ 4.4.15317.1 None \(0\) \(-2\) \(-1\) \(-1\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(-1-\beta _{2})q^{3}+\beta _{1}q^{5}+(\beta _{2}-\beta _{3})q^{7}+\cdots\)
1216.2.a.x 1216.a 1.a $4$ $9.710$ 4.4.15317.1 None \(0\) \(2\) \(-1\) \(1\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(1+\beta _{2})q^{3}+(-\beta _{1}+\beta _{2})q^{5}-\beta _{3}q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1216))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1216)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(19))\)\(^{\oplus 7}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(38))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(64))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(76))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(152))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(304))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(608))\)\(^{\oplus 2}\)