Properties

Label 3136.2.b
Level $3136$
Weight $2$
Character orbit 3136.b
Rep. character $\chi_{3136}(1569,\cdot)$
Character field $\Q$
Dimension $82$
Newform subspaces $14$
Sturm bound $896$
Trace bound $17$

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Defining parameters

Level: \( N \) \(=\) \( 3136 = 2^{6} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3136.b (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 8 \)
Character field: \(\Q\)
Newform subspaces: \( 14 \)
Sturm bound: \(896\)
Trace bound: \(17\)
Distinguishing \(T_p\): \(3\), \(5\), \(17\), \(31\)

Dimensions

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

Total New Old
Modular forms 496 82 414
Cusp forms 400 82 318
Eisenstein series 96 0 96

Trace form

\( 82 q - 82 q^{9} + O(q^{10}) \) \( 82 q - 82 q^{9} - 12 q^{17} - 70 q^{25} + 24 q^{33} - 12 q^{41} - 40 q^{57} - 28 q^{73} + 154 q^{81} - 60 q^{89} - 44 q^{97} + O(q^{100}) \)

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

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
3136.2.b.a 3136.b 8.b $2$ $25.041$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+iq^{3}+2iq^{5}-q^{9}+iq^{11}+2iq^{13}+\cdots\)
3136.2.b.b 3136.b 8.b $2$ $25.041$ \(\Q(\sqrt{-1}) \) \(\Q(\sqrt{-2}) \) \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+iq^{3}-q^{9}+3iq^{11}+6q^{17}+iq^{19}+\cdots\)
3136.2.b.c 3136.b 8.b $2$ $25.041$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+iq^{3}-2iq^{5}-q^{9}+iq^{11}-2iq^{13}+\cdots\)
3136.2.b.d 3136.b 8.b $4$ $25.041$ \(\Q(\zeta_{8})\) \(\Q(\sqrt{-2}) \) \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q-\zeta_{8}^{2}q^{3}-5q^{9}-3\zeta_{8}q^{11}-\zeta_{8}^{3}q^{17}+\cdots\)
3136.2.b.e 3136.b 8.b $4$ $25.041$ \(\Q(\zeta_{8})\) \(\Q(\sqrt{-2}) \) \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+(\zeta_{8}-\zeta_{8}^{2})q^{3}+(-3+4\zeta_{8}^{3})q^{9}+\cdots\)
3136.2.b.f 3136.b 8.b $4$ $25.041$ \(\Q(\zeta_{8})\) \(\Q(\sqrt{-2}) \) \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+(\zeta_{8}-\zeta_{8}^{2})q^{3}+(-3+4\zeta_{8}^{3})q^{9}+\cdots\)
3136.2.b.g 3136.b 8.b $4$ $25.041$ \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\zeta_{12}q^{3}-\zeta_{12}q^{5}+(-1+\zeta_{12}^{2}+\cdots)q^{9}+\cdots\)
3136.2.b.h 3136.b 8.b $4$ $25.041$ \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\zeta_{12}q^{3}+\zeta_{12}q^{5}+(-1+\zeta_{12}^{2}+\cdots)q^{9}+\cdots\)
3136.2.b.i 3136.b 8.b $4$ $25.041$ \(\Q(i, \sqrt{7})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{1}q^{3}-\beta _{3}q^{5}+2q^{9}-\beta _{1}q^{11}+\cdots\)
3136.2.b.j 3136.b 8.b $4$ $25.041$ \(\Q(i, \sqrt{7})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{1}q^{3}+\beta _{3}q^{5}+2q^{9}+\beta _{1}q^{11}+\cdots\)
3136.2.b.k 3136.b 8.b $8$ $25.041$ \(\Q(\zeta_{24})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\zeta_{24}q^{3}+\zeta_{24}^{3}q^{5}+q^{9}-\zeta_{24}^{3}q^{13}+\cdots\)
3136.2.b.l 3136.b 8.b $12$ $25.041$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{10}q^{3}+\beta _{6}q^{5}+(-2+\beta _{4})q^{9}+\cdots\)
3136.2.b.m 3136.b 8.b $12$ $25.041$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{10}q^{3}-\beta _{6}q^{5}+(-2+\beta _{4})q^{9}+\cdots\)
3136.2.b.n 3136.b 8.b $16$ $25.041$ 16.0.\(\cdots\).1 None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{6}q^{3}+\beta _{12}q^{5}-\beta _{1}q^{9}+\beta _{4}q^{11}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(3136, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(56, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(64, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(224, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(392, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(448, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(1568, [\chi])\)\(^{\oplus 2}\)