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 $7$

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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

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

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1216))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 19
1216.2.a.a \(1\) \(9.710\) \(\Q\) None \(0\) \(-3\) \(0\) \(-1\) \(-\) \(-\) \(q-3q^{3}-q^{7}+6q^{9}-2q^{11}+q^{13}+\cdots\)
1216.2.a.b \(1\) \(9.710\) \(\Q\) None \(0\) \(-2\) \(-3\) \(1\) \(-\) \(-\) \(q-2q^{3}-3q^{5}+q^{7}+q^{9}+3q^{11}+\cdots\)
1216.2.a.c \(1\) \(9.710\) \(\Q\) None \(0\) \(-2\) \(1\) \(-3\) \(+\) \(-\) \(q-2q^{3}+q^{5}-3q^{7}+q^{9}-5q^{11}+\cdots\)
1216.2.a.d \(1\) \(9.710\) \(\Q\) None \(0\) \(-2\) \(1\) \(3\) \(-\) \(+\) \(q-2q^{3}+q^{5}+3q^{7}+q^{9}-3q^{11}+\cdots\)
1216.2.a.e \(1\) \(9.710\) \(\Q\) None \(0\) \(-1\) \(0\) \(-1\) \(+\) \(+\) \(q-q^{3}-q^{7}-2q^{9}+6q^{11}-5q^{13}+\cdots\)
1216.2.a.f \(1\) \(9.710\) \(\Q\) None \(0\) \(-1\) \(0\) \(3\) \(+\) \(+\) \(q-q^{3}+3q^{7}-2q^{9}-2q^{11}-q^{13}+\cdots\)
1216.2.a.g \(1\) \(9.710\) \(\Q\) None \(0\) \(-1\) \(4\) \(-3\) \(-\) \(+\) \(q-q^{3}+4q^{5}-3q^{7}-2q^{9}+2q^{11}+\cdots\)
1216.2.a.h \(1\) \(9.710\) \(\Q\) None \(0\) \(0\) \(-3\) \(-5\) \(-\) \(+\) \(q-3q^{5}-5q^{7}-3q^{9}+5q^{11}+4q^{13}+\cdots\)
1216.2.a.i \(1\) \(9.710\) \(\Q\) None \(0\) \(0\) \(-3\) \(5\) \(-\) \(-\) \(q-3q^{5}+5q^{7}-3q^{9}-5q^{11}+4q^{13}+\cdots\)
1216.2.a.j \(1\) \(9.710\) \(\Q\) None \(0\) \(0\) \(1\) \(-1\) \(+\) \(+\) \(q+q^{5}-q^{7}-3q^{9}-3q^{11}+4q^{13}+\cdots\)
1216.2.a.k \(1\) \(9.710\) \(\Q\) None \(0\) \(0\) \(1\) \(1\) \(+\) \(-\) \(q+q^{5}+q^{7}-3q^{9}+3q^{11}+4q^{13}+\cdots\)
1216.2.a.l \(1\) \(9.710\) \(\Q\) None \(0\) \(1\) \(0\) \(-3\) \(-\) \(-\) \(q+q^{3}-3q^{7}-2q^{9}+2q^{11}-q^{13}+\cdots\)
1216.2.a.m \(1\) \(9.710\) \(\Q\) None \(0\) \(1\) \(0\) \(1\) \(-\) \(-\) \(q+q^{3}+q^{7}-2q^{9}-6q^{11}-5q^{13}+\cdots\)
1216.2.a.n \(1\) \(9.710\) \(\Q\) None \(0\) \(1\) \(4\) \(3\) \(+\) \(-\) \(q+q^{3}+4q^{5}+3q^{7}-2q^{9}-2q^{11}+\cdots\)
1216.2.a.o \(1\) \(9.710\) \(\Q\) None \(0\) \(2\) \(-3\) \(-1\) \(+\) \(+\) \(q+2q^{3}-3q^{5}-q^{7}+q^{9}-3q^{11}+\cdots\)
1216.2.a.p \(1\) \(9.710\) \(\Q\) None \(0\) \(2\) \(1\) \(-3\) \(+\) \(-\) \(q+2q^{3}+q^{5}-3q^{7}+q^{9}+3q^{11}+\cdots\)
1216.2.a.q \(1\) \(9.710\) \(\Q\) None \(0\) \(2\) \(1\) \(3\) \(-\) \(+\) \(q+2q^{3}+q^{5}+3q^{7}+q^{9}+5q^{11}+\cdots\)
1216.2.a.r \(1\) \(9.710\) \(\Q\) None \(0\) \(3\) \(0\) \(1\) \(-\) \(+\) \(q+3q^{3}+q^{7}+6q^{9}+2q^{11}+q^{13}+\cdots\)
1216.2.a.s \(2\) \(9.710\) \(\Q(\sqrt{17}) \) None \(0\) \(-1\) \(1\) \(6\) \(-\) \(+\) \(q-\beta q^{3}+(1-\beta )q^{5}+3q^{7}+(1+\beta )q^{9}+\cdots\)
1216.2.a.t \(2\) \(9.710\) \(\Q(\sqrt{17}) \) None \(0\) \(1\) \(1\) \(-6\) \(-\) \(-\) \(q+\beta q^{3}+(1-\beta )q^{5}-3q^{7}+(1+\beta )q^{9}+\cdots\)
1216.2.a.u \(3\) \(9.710\) 3.3.961.1 None \(0\) \(-1\) \(-1\) \(4\) \(+\) \(-\) \(q-\beta _{1}q^{3}+\beta _{2}q^{5}+(2-\beta _{1}+\beta _{2})q^{7}+\cdots\)
1216.2.a.v \(3\) \(9.710\) 3.3.961.1 None \(0\) \(1\) \(-1\) \(-4\) \(-\) \(+\) \(q+\beta _{1}q^{3}+\beta _{2}q^{5}+(-2+\beta _{1}-\beta _{2})q^{7}+\cdots\)
1216.2.a.w \(4\) \(9.710\) 4.4.15317.1 None \(0\) \(-2\) \(-1\) \(-1\) \(+\) \(+\) \(q+(-1-\beta _{2})q^{3}+\beta _{1}q^{5}+(\beta _{2}-\beta _{3})q^{7}+\cdots\)
1216.2.a.x \(4\) \(9.710\) 4.4.15317.1 None \(0\) \(2\) \(-1\) \(1\) \(+\) \(-\) \(q+(1+\beta _{2})q^{3}+(-\beta _{1}+\beta _{2})q^{5}-\beta _{3}q^{7}+\cdots\)

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

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