Properties

Label 3136.2.f
Level $3136$
Weight $2$
Character orbit 3136.f
Rep. character $\chi_{3136}(3135,\cdot)$
Character field $\Q$
Dimension $76$
Newform subspaces $11$
Sturm bound $896$
Trace bound $53$

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Defining parameters

Level: \( N \) \(=\) \( 3136 = 2^{6} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3136.f (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 28 \)
Character field: \(\Q\)
Newform subspaces: \( 11 \)
Sturm bound: \(896\)
Trace bound: \(53\)
Distinguishing \(T_p\): \(3\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(3136, [\chi])\).

Total New Old
Modular forms 496 84 412
Cusp forms 400 76 324
Eisenstein series 96 8 88

Trace form

\( 76q + 72q^{9} + O(q^{10}) \) \( 76q + 72q^{9} - 56q^{25} + 24q^{29} - 4q^{37} - 20q^{53} + 4q^{57} - 16q^{65} + 92q^{81} + 76q^{85} + 36q^{93} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(3136, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
3136.2.f.a \(2\) \(25.041\) \(\Q(\sqrt{-3}) \) None \(0\) \(-2\) \(0\) \(0\) \(q-q^{3}-\zeta_{6}q^{5}-2q^{9}-\zeta_{6}q^{11}+\zeta_{6}q^{15}+\cdots\)
3136.2.f.b \(2\) \(25.041\) \(\Q(\sqrt{-3}) \) None \(0\) \(2\) \(0\) \(0\) \(q+q^{3}+\zeta_{6}q^{5}-2q^{9}-\zeta_{6}q^{11}+\zeta_{6}q^{15}+\cdots\)
3136.2.f.c \(4\) \(25.041\) 4.0.2048.2 \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(0\) \(0\) \(q-\beta _{1}q^{5}-3q^{9}+(\beta _{1}+\beta _{3})q^{13}+(\beta _{1}+\cdots)q^{17}+\cdots\)
3136.2.f.d \(4\) \(25.041\) 4.0.2048.2 \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(0\) \(0\) \(q+\beta _{2}q^{5}-3q^{9}+(\beta _{1}+\beta _{2})q^{13}+(-\beta _{1}+\cdots)q^{17}+\cdots\)
3136.2.f.e \(4\) \(25.041\) \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(0\) \(q-\zeta_{12}^{3}q^{3}-\zeta_{12}^{2}q^{5}+\zeta_{12}q^{11}+\cdots\)
3136.2.f.f \(4\) \(25.041\) \(\Q(\sqrt{-3}, \sqrt{7})\) None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{1}q^{3}-\beta _{2}q^{5}+4q^{9}-\beta _{3}q^{11}+\cdots\)
3136.2.f.g \(8\) \(25.041\) \(\Q(\zeta_{16})\) None \(0\) \(0\) \(0\) \(0\) \(q+\zeta_{16}^{7}q^{3}+(-\zeta_{16}^{2}+\zeta_{16}^{4})q^{5}+\cdots\)
3136.2.f.h \(8\) \(25.041\) 8.0.339738624.1 None \(0\) \(0\) \(0\) \(0\) \(q-\beta _{3}q^{3}+(-\beta _{2}+\beta _{4})q^{5}+(3-3\beta _{1}+\cdots)q^{9}+\cdots\)
3136.2.f.i \(8\) \(25.041\) 8.0.\(\cdots\).10 None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{4}q^{3}+(\beta _{1}-\beta _{2})q^{5}+(3+\beta _{3})q^{9}+\cdots\)
3136.2.f.j \(16\) \(25.041\) 16.0.\(\cdots\).2 None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{1}q^{3}+\beta _{13}q^{5}+(1-\beta _{2})q^{9}+(\beta _{4}+\cdots)q^{11}+\cdots\)
3136.2.f.k \(16\) \(25.041\) \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) \(q-\beta _{12}q^{3}+(\beta _{5}+\beta _{7})q^{5}+(2+\beta _{3})q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(3136, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(3136, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(28, [\chi])\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(112, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(196, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(224, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(448, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(784, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(1568, [\chi])\)\(^{\oplus 2}\)