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

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

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(1216))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 19
1216.4.a.a \(1\) \(71.746\) \(\Q\) None \(0\) \(-5\) \(12\) \(-11\) \(-\) \(-\) \(q-5q^{3}+12q^{5}-11q^{7}-2q^{9}-54q^{11}+\cdots\)
1216.4.a.b \(1\) \(71.746\) \(\Q\) None \(0\) \(-2\) \(9\) \(31\) \(-\) \(-\) \(q-2q^{3}+9q^{5}+31q^{7}-23q^{9}+57q^{11}+\cdots\)
1216.4.a.c \(1\) \(71.746\) \(\Q\) None \(0\) \(-1\) \(8\) \(17\) \(+\) \(-\) \(q-q^{3}+8q^{5}+17q^{7}-26q^{9}-70q^{11}+\cdots\)
1216.4.a.d \(1\) \(71.746\) \(\Q\) None \(0\) \(1\) \(8\) \(-17\) \(+\) \(+\) \(q+q^{3}+8q^{5}-17q^{7}-26q^{9}+70q^{11}+\cdots\)
1216.4.a.e \(1\) \(71.746\) \(\Q\) None \(0\) \(2\) \(9\) \(-31\) \(+\) \(+\) \(q+2q^{3}+9q^{5}-31q^{7}-23q^{9}-57q^{11}+\cdots\)
1216.4.a.f \(1\) \(71.746\) \(\Q\) None \(0\) \(5\) \(12\) \(11\) \(+\) \(+\) \(q+5q^{3}+12q^{5}+11q^{7}-2q^{9}+54q^{11}+\cdots\)
1216.4.a.g \(2\) \(71.746\) \(\Q(\sqrt{73}) \) None \(0\) \(-9\) \(9\) \(-18\) \(+\) \(+\) \(q+(-4-\beta )q^{3}+(3+3\beta )q^{5}+(-7-4\beta )q^{7}+\cdots\)
1216.4.a.h \(2\) \(71.746\) \(\Q(\sqrt{33}) \) None \(0\) \(-5\) \(5\) \(30\) \(-\) \(+\) \(q+(-2-\beta )q^{3}+(5-5\beta )q^{5}+(13+4\beta )q^{7}+\cdots\)
1216.4.a.i \(2\) \(71.746\) \(\Q(\sqrt{93}) \) None \(0\) \(-2\) \(4\) \(-44\) \(+\) \(-\) \(q-q^{3}+(2+2\beta )q^{5}+(-22-\beta )q^{7}+\cdots\)
1216.4.a.j \(2\) \(71.746\) \(\Q(\sqrt{177}) \) None \(0\) \(-1\) \(-10\) \(57\) \(+\) \(-\) \(q-\beta q^{3}+(-6+2\beta )q^{5}+(28+\beta )q^{7}+\cdots\)
1216.4.a.k \(2\) \(71.746\) \(\Q(\sqrt{57}) \) None \(0\) \(-1\) \(5\) \(6\) \(-\) \(+\) \(q-\beta q^{3}+(3-\beta )q^{5}+(5-4\beta )q^{7}+(-13+\cdots)q^{9}+\cdots\)
1216.4.a.l \(2\) \(71.746\) \(\Q(\sqrt{177}) \) None \(0\) \(1\) \(-10\) \(-57\) \(-\) \(+\) \(q+\beta q^{3}+(-6+2\beta )q^{5}+(-28-\beta )q^{7}+\cdots\)
1216.4.a.m \(2\) \(71.746\) \(\Q(\sqrt{57}) \) None \(0\) \(1\) \(5\) \(-6\) \(+\) \(-\) \(q+\beta q^{3}+(3-\beta )q^{5}+(-5+4\beta )q^{7}+\cdots\)
1216.4.a.n \(2\) \(71.746\) \(\Q(\sqrt{93}) \) None \(0\) \(2\) \(4\) \(44\) \(+\) \(+\) \(q+q^{3}+(2+2\beta )q^{5}+(22+\beta )q^{7}-26q^{9}+\cdots\)
1216.4.a.o \(2\) \(71.746\) \(\Q(\sqrt{33}) \) None \(0\) \(5\) \(5\) \(-30\) \(+\) \(-\) \(q+(3-\beta )q^{3}+5\beta q^{5}+(-17+4\beta )q^{7}+\cdots\)
1216.4.a.p \(2\) \(71.746\) \(\Q(\sqrt{73}) \) None \(0\) \(9\) \(9\) \(18\) \(-\) \(-\) \(q+(5-\beta )q^{3}+(6-3\beta )q^{5}+(11-4\beta )q^{7}+\cdots\)
1216.4.a.q \(3\) \(71.746\) 3.3.3221.1 None \(0\) \(-5\) \(-2\) \(35\) \(-\) \(-\) \(q+(-2-\beta _{2})q^{3}+(-1+\beta _{1}-2\beta _{2})q^{5}+\cdots\)
1216.4.a.r \(3\) \(71.746\) 3.3.7057.1 None \(0\) \(-4\) \(-7\) \(7\) \(+\) \(-\) \(q+(-1-\beta _{2})q^{3}+(-2-2\beta _{1}-\beta _{2})q^{5}+\cdots\)
1216.4.a.s \(3\) \(71.746\) 3.3.3144.1 None \(0\) \(-1\) \(-14\) \(-35\) \(+\) \(-\) \(q-\beta _{2}q^{3}+(-4+\beta _{1}-\beta _{2})q^{5}+(-11+\cdots)q^{7}+\cdots\)
1216.4.a.t \(3\) \(71.746\) 3.3.35529.1 None \(0\) \(-1\) \(-9\) \(44\) \(+\) \(+\) \(q+(-\beta _{1}-\beta _{2})q^{3}+(-3-\beta _{2})q^{5}+(15+\cdots)q^{7}+\cdots\)
1216.4.a.u \(3\) \(71.746\) 3.3.3144.1 None \(0\) \(1\) \(-14\) \(35\) \(-\) \(+\) \(q+\beta _{2}q^{3}+(-4+\beta _{1}-\beta _{2})q^{5}+(11+\cdots)q^{7}+\cdots\)
1216.4.a.v \(3\) \(71.746\) 3.3.35529.1 None \(0\) \(1\) \(-9\) \(-44\) \(-\) \(-\) \(q+(\beta _{1}+\beta _{2})q^{3}+(-3-\beta _{2})q^{5}+(-15+\cdots)q^{7}+\cdots\)
1216.4.a.w \(3\) \(71.746\) 3.3.7057.1 None \(0\) \(4\) \(-7\) \(-7\) \(-\) \(+\) \(q+(1+\beta _{2})q^{3}+(-2-2\beta _{1}-\beta _{2})q^{5}+\cdots\)
1216.4.a.x \(3\) \(71.746\) 3.3.3221.1 None \(0\) \(5\) \(-2\) \(-35\) \(+\) \(+\) \(q+(2+\beta _{2})q^{3}+(-1+\beta _{1}-2\beta _{2})q^{5}+\cdots\)
1216.4.a.y \(5\) \(71.746\) \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(0\) \(-6\) \(-5\) \(-7\) \(+\) \(-\) \(q+(-1-\beta _{2})q^{3}+(-1+\beta _{2}-\beta _{3})q^{5}+\cdots\)
1216.4.a.z \(5\) \(71.746\) \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(0\) \(-3\) \(-27\) \(-20\) \(+\) \(+\) \(q+(-1-\beta _{3})q^{3}+(-5+\beta _{1})q^{5}+(-5+\cdots)q^{7}+\cdots\)
1216.4.a.ba \(5\) \(71.746\) \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(0\) \(-2\) \(0\) \(22\) \(+\) \(+\) \(q-\beta _{2}q^{3}+(-\beta _{1}-\beta _{2})q^{5}+(4+\beta _{2}+\cdots)q^{7}+\cdots\)
1216.4.a.bb \(5\) \(71.746\) \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(0\) \(2\) \(0\) \(-22\) \(-\) \(-\) \(q+\beta _{2}q^{3}+(-\beta _{1}-\beta _{2})q^{5}+(-4-\beta _{2}+\cdots)q^{7}+\cdots\)
1216.4.a.bc \(5\) \(71.746\) \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(0\) \(3\) \(-27\) \(20\) \(+\) \(-\) \(q+(1+\beta _{3})q^{3}+(-5+\beta _{1})q^{5}+(5-\beta _{1}+\cdots)q^{7}+\cdots\)
1216.4.a.bd \(5\) \(71.746\) \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(0\) \(6\) \(-5\) \(7\) \(+\) \(+\) \(q+(1+\beta _{2})q^{3}+(-1+\beta _{2}-\beta _{3})q^{5}+\cdots\)
1216.4.a.be \(7\) \(71.746\) \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(0\) \(-9\) \(5\) \(28\) \(-\) \(+\) \(q+(-1-\beta _{1})q^{3}+(1-\beta _{5})q^{5}+(4+\beta _{3}+\cdots)q^{7}+\cdots\)
1216.4.a.bf \(7\) \(71.746\) \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(0\) \(-3\) \(17\) \(42\) \(-\) \(-\) \(q-\beta _{1}q^{3}+(2+\beta _{3})q^{5}+(6-\beta _{2}+\beta _{5}+\cdots)q^{7}+\cdots\)
1216.4.a.bg \(7\) \(71.746\) \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(0\) \(3\) \(17\) \(-42\) \(-\) \(+\) \(q+\beta _{1}q^{3}+(2+\beta _{3})q^{5}+(-6+\beta _{2}-\beta _{5}+\cdots)q^{7}+\cdots\)
1216.4.a.bh \(7\) \(71.746\) \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(0\) \(9\) \(5\) \(-28\) \(-\) \(-\) \(q+(1+\beta _{1})q^{3}+(1-\beta _{5})q^{5}+(-4-\beta _{3}+\cdots)q^{7}+\cdots\)

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

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