Properties

Label 1216.4.a
Level $1216$
Weight $4$
Character orbit 1216.a
Rep. character $\chi_{1216}(1,\cdot)$
Character field $\Q$
Dimension $108$
Newform subspaces $34$
Sturm bound $640$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 1216 = 2^{6} \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1216.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 34 \)
Sturm bound: \(640\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(3\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(1216))\).

Total New Old
Modular forms 492 108 384
Cusp forms 468 108 360
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.

\(2\)\(19\)FrickeDim
\(+\)\(+\)$+$\(28\)
\(+\)\(-\)$-$\(25\)
\(-\)\(+\)$-$\(26\)
\(-\)\(-\)$+$\(29\)
Plus space\(+\)\(57\)
Minus space\(-\)\(51\)

Trace form

\( 108 q + 972 q^{9} + O(q^{10}) \) \( 108 q + 972 q^{9} + 104 q^{17} + 2612 q^{25} + 400 q^{29} + 16 q^{37} - 472 q^{41} - 1968 q^{45} + 5292 q^{49} - 1568 q^{53} - 1536 q^{65} + 1488 q^{69} - 296 q^{73} - 3656 q^{77} + 11212 q^{81} - 1176 q^{85} + 1784 q^{89} + 9216 q^{93} + 1160 q^{97} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(1216))\) 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 19
1216.4.a.a 1216.a 1.a $1$ $71.746$ \(\Q\) None 19.4.a.a \(0\) \(-5\) \(12\) \(-11\) $-$ $-$ $\mathrm{SU}(2)$ \(q-5q^{3}+12q^{5}-11q^{7}-2q^{9}-54q^{11}+\cdots\)
1216.4.a.b 1216.a 1.a $1$ $71.746$ \(\Q\) None 38.4.a.a \(0\) \(-2\) \(9\) \(31\) $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{3}+9q^{5}+31q^{7}-23q^{9}+57q^{11}+\cdots\)
1216.4.a.c 1216.a 1.a $1$ $71.746$ \(\Q\) None 608.4.a.a \(0\) \(-1\) \(8\) \(17\) $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}+8q^{5}+17q^{7}-26q^{9}-70q^{11}+\cdots\)
1216.4.a.d 1216.a 1.a $1$ $71.746$ \(\Q\) None 608.4.a.a \(0\) \(1\) \(8\) \(-17\) $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}+8q^{5}-17q^{7}-26q^{9}+70q^{11}+\cdots\)
1216.4.a.e 1216.a 1.a $1$ $71.746$ \(\Q\) None 38.4.a.a \(0\) \(2\) \(9\) \(-31\) $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{3}+9q^{5}-31q^{7}-23q^{9}-57q^{11}+\cdots\)
1216.4.a.f 1216.a 1.a $1$ $71.746$ \(\Q\) None 19.4.a.a \(0\) \(5\) \(12\) \(11\) $+$ $+$ $\mathrm{SU}(2)$ \(q+5q^{3}+12q^{5}+11q^{7}-2q^{9}+54q^{11}+\cdots\)
1216.4.a.g 1216.a 1.a $2$ $71.746$ \(\Q(\sqrt{73}) \) None 38.4.a.c \(0\) \(-9\) \(9\) \(-18\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(-4-\beta )q^{3}+(3+3\beta )q^{5}+(-7-4\beta )q^{7}+\cdots\)
1216.4.a.h 1216.a 1.a $2$ $71.746$ \(\Q(\sqrt{33}) \) None 76.4.a.a \(0\) \(-5\) \(5\) \(30\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(-2-\beta )q^{3}+(5-5\beta )q^{5}+(13+4\beta )q^{7}+\cdots\)
1216.4.a.i 1216.a 1.a $2$ $71.746$ \(\Q(\sqrt{93}) \) None 608.4.a.c \(0\) \(-2\) \(4\) \(-44\) $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}+(2+2\beta )q^{5}+(-22-\beta )q^{7}+\cdots\)
1216.4.a.j 1216.a 1.a $2$ $71.746$ \(\Q(\sqrt{177}) \) None 38.4.a.b \(0\) \(-1\) \(-10\) \(57\) $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta q^{3}+(-6+2\beta )q^{5}+(28+\beta )q^{7}+\cdots\)
1216.4.a.k 1216.a 1.a $2$ $71.746$ \(\Q(\sqrt{57}) \) None 152.4.a.a \(0\) \(-1\) \(5\) \(6\) $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta q^{3}+(3-\beta )q^{5}+(5-4\beta )q^{7}+(-13+\cdots)q^{9}+\cdots\)
1216.4.a.l 1216.a 1.a $2$ $71.746$ \(\Q(\sqrt{177}) \) None 38.4.a.b \(0\) \(1\) \(-10\) \(-57\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{3}+(-6+2\beta )q^{5}+(-28-\beta )q^{7}+\cdots\)
1216.4.a.m 1216.a 1.a $2$ $71.746$ \(\Q(\sqrt{57}) \) None 152.4.a.a \(0\) \(1\) \(5\) \(-6\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{3}+(3-\beta )q^{5}+(-5+4\beta )q^{7}+\cdots\)
1216.4.a.n 1216.a 1.a $2$ $71.746$ \(\Q(\sqrt{93}) \) None 608.4.a.c \(0\) \(2\) \(4\) \(44\) $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}+(2+2\beta )q^{5}+(22+\beta )q^{7}-26q^{9}+\cdots\)
1216.4.a.o 1216.a 1.a $2$ $71.746$ \(\Q(\sqrt{33}) \) None 76.4.a.a \(0\) \(5\) \(5\) \(-30\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(3-\beta )q^{3}+5\beta q^{5}+(-17+4\beta )q^{7}+\cdots\)
1216.4.a.p 1216.a 1.a $2$ $71.746$ \(\Q(\sqrt{73}) \) None 38.4.a.c \(0\) \(9\) \(9\) \(18\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(5-\beta )q^{3}+(6-3\beta )q^{5}+(11-4\beta )q^{7}+\cdots\)
1216.4.a.q 1216.a 1.a $3$ $71.746$ 3.3.3221.1 None 152.4.a.b \(0\) \(-5\) \(-2\) \(35\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-2-\beta _{2})q^{3}+(-1+\beta _{1}-2\beta _{2})q^{5}+\cdots\)
1216.4.a.r 1216.a 1.a $3$ $71.746$ 3.3.7057.1 None 152.4.a.c \(0\) \(-4\) \(-7\) \(7\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(-1-\beta _{2})q^{3}+(-2-2\beta _{1}-\beta _{2})q^{5}+\cdots\)
1216.4.a.s 1216.a 1.a $3$ $71.746$ 3.3.3144.1 None 19.4.a.b \(0\) \(-1\) \(-14\) \(-35\) $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{2}q^{3}+(-4+\beta _{1}-\beta _{2})q^{5}+(-11+\cdots)q^{7}+\cdots\)
1216.4.a.t 1216.a 1.a $3$ $71.746$ 3.3.35529.1 None 76.4.a.b \(0\) \(-1\) \(-9\) \(44\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(-\beta _{1}-\beta _{2})q^{3}+(-3-\beta _{2})q^{5}+(15+\cdots)q^{7}+\cdots\)
1216.4.a.u 1216.a 1.a $3$ $71.746$ 3.3.3144.1 None 19.4.a.b \(0\) \(1\) \(-14\) \(35\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{2}q^{3}+(-4+\beta _{1}-\beta _{2})q^{5}+(11+\cdots)q^{7}+\cdots\)
1216.4.a.v 1216.a 1.a $3$ $71.746$ 3.3.35529.1 None 76.4.a.b \(0\) \(1\) \(-9\) \(-44\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(\beta _{1}+\beta _{2})q^{3}+(-3-\beta _{2})q^{5}+(-15+\cdots)q^{7}+\cdots\)
1216.4.a.w 1216.a 1.a $3$ $71.746$ 3.3.7057.1 None 152.4.a.c \(0\) \(4\) \(-7\) \(-7\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(1+\beta _{2})q^{3}+(-2-2\beta _{1}-\beta _{2})q^{5}+\cdots\)
1216.4.a.x 1216.a 1.a $3$ $71.746$ 3.3.3221.1 None 152.4.a.b \(0\) \(5\) \(-2\) \(-35\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(2+\beta _{2})q^{3}+(-1+\beta _{1}-2\beta _{2})q^{5}+\cdots\)
1216.4.a.y 1216.a 1.a $5$ $71.746$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None 608.4.a.e \(0\) \(-6\) \(-5\) \(-7\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(-1-\beta _{2})q^{3}+(-1+\beta _{2}-\beta _{3})q^{5}+\cdots\)
1216.4.a.z 1216.a 1.a $5$ $71.746$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None 608.4.a.f \(0\) \(-3\) \(-27\) \(-20\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(-1-\beta _{3})q^{3}+(-5+\beta _{1})q^{5}+(-5+\cdots)q^{7}+\cdots\)
1216.4.a.ba 1216.a 1.a $5$ $71.746$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None 152.4.a.d \(0\) \(-2\) \(0\) \(22\) $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{2}q^{3}+(-\beta _{1}-\beta _{2})q^{5}+(4+\beta _{2}+\cdots)q^{7}+\cdots\)
1216.4.a.bb 1216.a 1.a $5$ $71.746$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None 152.4.a.d \(0\) \(2\) \(0\) \(-22\) $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{2}q^{3}+(-\beta _{1}-\beta _{2})q^{5}+(-4-\beta _{2}+\cdots)q^{7}+\cdots\)
1216.4.a.bc 1216.a 1.a $5$ $71.746$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None 608.4.a.f \(0\) \(3\) \(-27\) \(20\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(1+\beta _{3})q^{3}+(-5+\beta _{1})q^{5}+(5-\beta _{1}+\cdots)q^{7}+\cdots\)
1216.4.a.bd 1216.a 1.a $5$ $71.746$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None 608.4.a.e \(0\) \(6\) \(-5\) \(7\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(1+\beta _{2})q^{3}+(-1+\beta _{2}-\beta _{3})q^{5}+\cdots\)
1216.4.a.be 1216.a 1.a $7$ $71.746$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None 608.4.a.i \(0\) \(-9\) \(5\) \(28\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(-1-\beta _{1})q^{3}+(1-\beta _{5})q^{5}+(4+\beta _{3}+\cdots)q^{7}+\cdots\)
1216.4.a.bf 1216.a 1.a $7$ $71.746$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None 608.4.a.j \(0\) \(-3\) \(17\) \(42\) $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{3}+(2+\beta _{3})q^{5}+(6-\beta _{2}+\beta _{5}+\cdots)q^{7}+\cdots\)
1216.4.a.bg 1216.a 1.a $7$ $71.746$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None 608.4.a.j \(0\) \(3\) \(17\) \(-42\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{3}+(2+\beta _{3})q^{5}+(-6+\beta _{2}-\beta _{5}+\cdots)q^{7}+\cdots\)
1216.4.a.bh 1216.a 1.a $7$ $71.746$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None 608.4.a.i \(0\) \(9\) \(5\) \(-28\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(1+\beta _{1})q^{3}+(1-\beta _{5})q^{5}+(-4-\beta _{3}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(1216))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(1216)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(19))\)\(^{\oplus 7}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(38))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(64))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(76))\)\(^{\oplus 5}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(152))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(304))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(608))\)\(^{\oplus 2}\)