Properties

Label 1216.3.e
Level $1216$
Weight $3$
Character orbit 1216.e
Rep. character $\chi_{1216}(1025,\cdot)$
Character field $\Q$
Dimension $78$
Newform subspaces $16$
Sturm bound $480$
Trace bound $17$

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Defining parameters

Level: \( N \) \(=\) \( 1216 = 2^{6} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1216.e (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 19 \)
Character field: \(\Q\)
Newform subspaces: \( 16 \)
Sturm bound: \(480\)
Trace bound: \(17\)
Distinguishing \(T_p\): \(3\), \(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(1216, [\chi])\).

Total New Old
Modular forms 332 82 250
Cusp forms 308 78 230
Eisenstein series 24 4 20

Trace form

\( 78q + 4q^{5} - 226q^{9} + O(q^{10}) \) \( 78q + 4q^{5} - 226q^{9} - 4q^{17} + 346q^{25} + 68q^{45} + 458q^{49} - 64q^{57} + 292q^{61} - 4q^{73} + 152q^{77} + 366q^{81} + 568q^{85} - 320q^{93} + O(q^{100}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(1216, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
1216.3.e.a \(1\) \(33.134\) \(\Q\) \(\Q(\sqrt{-19}) \) \(0\) \(0\) \(9\) \(-5\) \(q+9q^{5}-5q^{7}+9q^{9}-3q^{11}+15q^{17}+\cdots\)
1216.3.e.b \(1\) \(33.134\) \(\Q\) \(\Q(\sqrt{-19}) \) \(0\) \(0\) \(9\) \(5\) \(q+9q^{5}+5q^{7}+9q^{9}+3q^{11}+15q^{17}+\cdots\)
1216.3.e.c \(2\) \(33.134\) \(\Q(\sqrt{-2}) \) None \(0\) \(0\) \(-14\) \(-22\) \(q+\beta q^{3}-7q^{5}-11q^{7}-23q^{9}+3q^{11}+\cdots\)
1216.3.e.d \(2\) \(33.134\) \(\Q(\sqrt{-2}) \) None \(0\) \(0\) \(-14\) \(22\) \(q+\beta q^{3}-7q^{5}+11q^{7}-23q^{9}-3q^{11}+\cdots\)
1216.3.e.e \(2\) \(33.134\) \(\Q(\sqrt{57}) \) \(\Q(\sqrt{-19}) \) \(0\) \(0\) \(-9\) \(-5\) \(q+(-4-\beta )q^{5}+(-4+3\beta )q^{7}+9q^{9}+\cdots\)
1216.3.e.f \(2\) \(33.134\) \(\Q(\sqrt{57}) \) \(\Q(\sqrt{-19}) \) \(0\) \(0\) \(-9\) \(5\) \(q+(-4-\beta )q^{5}+(4-3\beta )q^{7}+9q^{9}+\cdots\)
1216.3.e.g \(2\) \(33.134\) \(\Q(\sqrt{-13}) \) None \(0\) \(0\) \(-8\) \(-10\) \(q+\beta q^{3}-4q^{5}-5q^{7}-4q^{9}+10q^{11}+\cdots\)
1216.3.e.h \(2\) \(33.134\) \(\Q(\sqrt{-13}) \) None \(0\) \(0\) \(-8\) \(10\) \(q+\beta q^{3}-4q^{5}+5q^{7}-4q^{9}-10q^{11}+\cdots\)
1216.3.e.i \(2\) \(33.134\) \(\Q(\sqrt{-2}) \) None \(0\) \(0\) \(2\) \(-10\) \(q+\beta q^{3}+q^{5}-5q^{7}+q^{9}+5q^{11}+\cdots\)
1216.3.e.j \(2\) \(33.134\) \(\Q(\sqrt{-2}) \) None \(0\) \(0\) \(2\) \(10\) \(q+\beta q^{3}+q^{5}+5q^{7}+q^{9}-5q^{11}+\cdots\)
1216.3.e.k \(2\) \(33.134\) \(\Q(\sqrt{-29}) \) None \(0\) \(0\) \(8\) \(-2\) \(q+\beta q^{3}+4q^{5}-q^{7}-20q^{9}-14q^{11}+\cdots\)
1216.3.e.l \(2\) \(33.134\) \(\Q(\sqrt{-29}) \) None \(0\) \(0\) \(8\) \(2\) \(q+\beta q^{3}+4q^{5}+q^{7}-20q^{9}+14q^{11}+\cdots\)
1216.3.e.m \(8\) \(33.134\) \(\mathbb{Q}[x]/(x^{8} + \cdots)\) None \(0\) \(0\) \(14\) \(-6\) \(q+\beta _{1}q^{3}+(2+\beta _{4})q^{5}+(-1+\beta _{3})q^{7}+\cdots\)
1216.3.e.n \(8\) \(33.134\) \(\mathbb{Q}[x]/(x^{8} + \cdots)\) None \(0\) \(0\) \(14\) \(6\) \(q+\beta _{1}q^{3}+(2+\beta _{4})q^{5}+(1-\beta _{3})q^{7}+\cdots\)
1216.3.e.o \(20\) \(33.134\) \(\mathbb{Q}[x]/(x^{20} + \cdots)\) None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{11}q^{3}+\beta _{2}q^{5}+\beta _{3}q^{7}+(-3+\beta _{5}+\cdots)q^{9}+\cdots\)
1216.3.e.p \(20\) \(33.134\) \(\mathbb{Q}[x]/(x^{20} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) \(q-\beta _{7}q^{3}-\beta _{6}q^{5}+\beta _{1}q^{7}+(-3+\beta _{4}+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(1216, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(1216, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(19, [\chi])\)\(^{\oplus 7}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(38, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(76, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(152, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(304, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(608, [\chi])\)\(^{\oplus 2}\)