Properties

Label 768.4.c
Level $768$
Weight $4$
Character orbit 768.c
Rep. character $\chi_{768}(767,\cdot)$
Character field $\Q$
Dimension $92$
Newform subspaces $22$
Sturm bound $512$
Trace bound $25$

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Defining parameters

Level: \( N \) \(=\) \( 768 = 2^{8} \cdot 3 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 768.c (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 12 \)
Character field: \(\Q\)
Newform subspaces: \( 22 \)
Sturm bound: \(512\)
Trace bound: \(25\)
Distinguishing \(T_p\): \(5\), \(7\), \(11\), \(13\), \(23\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(768, [\chi])\).

Total New Old
Modular forms 408 100 308
Cusp forms 360 92 268
Eisenstein series 48 8 40

Trace form

\( 92 q + 4 q^{9} + O(q^{10}) \) \( 92 q + 4 q^{9} - 1892 q^{25} + 104 q^{33} - 1900 q^{49} + 112 q^{57} + 872 q^{73} - 4 q^{81} + 3160 q^{97} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(768, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
768.4.c.a 768.c 12.b $2$ $45.313$ \(\Q(\sqrt{-2}) \) \(\Q(\sqrt{-2}) \) 384.4.f.b \(0\) \(-10\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+(-5-\beta )q^{3}+(23+10\beta )q^{9}-18q^{11}+\cdots\)
768.4.c.b 768.c 12.b $2$ $45.313$ \(\Q(\sqrt{-2}) \) None 384.4.f.c \(0\) \(-6\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(-3+3\beta )q^{3}+4\beta q^{5}-12\beta q^{7}+\cdots\)
768.4.c.c 768.c 12.b $2$ $45.313$ \(\Q(\sqrt{-2}) \) None 384.4.f.c \(0\) \(-6\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(-3-3\beta )q^{3}+4\beta q^{5}-12\beta q^{7}+\cdots\)
768.4.c.d 768.c 12.b $2$ $45.313$ \(\Q(\sqrt{-26}) \) None 384.4.f.e \(0\) \(-2\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(-1-\beta )q^{3}+2\beta q^{5}-2\beta q^{7}+(-5^{2}+\cdots)q^{9}+\cdots\)
768.4.c.e 768.c 12.b $2$ $45.313$ \(\Q(\sqrt{-26}) \) None 384.4.f.e \(0\) \(-2\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(-1+\beta )q^{3}+2\beta q^{5}-2\beta q^{7}+(-5^{2}+\cdots)q^{9}+\cdots\)
768.4.c.f 768.c 12.b $2$ $45.313$ \(\Q(\sqrt{-26}) \) None 384.4.f.e \(0\) \(2\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(1+\beta )q^{3}+2\beta q^{5}+2\beta q^{7}+(-5^{2}+\cdots)q^{9}+\cdots\)
768.4.c.g 768.c 12.b $2$ $45.313$ \(\Q(\sqrt{-26}) \) None 384.4.f.e \(0\) \(2\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(1-\beta )q^{3}+2\beta q^{5}+2\beta q^{7}+(-5^{2}+\cdots)q^{9}+\cdots\)
768.4.c.h 768.c 12.b $2$ $45.313$ \(\Q(\sqrt{-2}) \) None 384.4.f.c \(0\) \(6\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(3-3\beta )q^{3}+4\beta q^{5}+12\beta q^{7}+(-9+\cdots)q^{9}+\cdots\)
768.4.c.i 768.c 12.b $2$ $45.313$ \(\Q(\sqrt{-2}) \) None 384.4.f.c \(0\) \(6\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(3+3\beta )q^{3}+4\beta q^{5}+12\beta q^{7}+(-9+\cdots)q^{9}+\cdots\)
768.4.c.j 768.c 12.b $2$ $45.313$ \(\Q(\sqrt{-2}) \) \(\Q(\sqrt{-2}) \) 384.4.f.b \(0\) \(10\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+(5+\beta )q^{3}+(23+10\beta )q^{9}+18q^{11}+\cdots\)
768.4.c.k 768.c 12.b $4$ $45.313$ \(\Q(\zeta_{12})\) \(\Q(\sqrt{-3}) \) 192.4.f.b \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+\zeta_{12}^{2}q^{3}-5\zeta_{12}q^{7}-3^{3}q^{9}-3\zeta_{12}^{3}q^{13}+\cdots\)
768.4.c.l 768.c 12.b $4$ $45.313$ \(\Q(\zeta_{8})\) \(\Q(\sqrt{-2}) \) 24.4.f.a \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+(2\zeta_{8}-3\zeta_{8}^{2})q^{3}+(-23+5\zeta_{8}^{3})q^{9}+\cdots\)
768.4.c.m 768.c 12.b $4$ $45.313$ \(\Q(\sqrt{-6}, \sqrt{-14})\) None 384.4.f.g \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(\beta _{1}+\beta _{3})q^{3}+(-2\beta _{1}-\beta _{2})q^{5}+(3\beta _{1}+\cdots)q^{7}+\cdots\)
768.4.c.n 768.c 12.b $4$ $45.313$ \(\Q(\sqrt{-6}, \sqrt{-14})\) None 384.4.f.g \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{3}q^{3}+(-2\beta _{1}-\beta _{2})q^{5}+(3\beta _{1}-2\beta _{2}+\cdots)q^{7}+\cdots\)
768.4.c.o 768.c 12.b $4$ $45.313$ \(\Q(\sqrt{-6}, \sqrt{-14})\) None 384.4.f.g \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(-\beta _{1}-\beta _{3})q^{3}+(-2\beta _{1}-\beta _{2})q^{5}+\cdots\)
768.4.c.p 768.c 12.b $4$ $45.313$ \(\Q(\sqrt{-6}, \sqrt{-14})\) None 384.4.f.g \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{3}q^{3}+(-2\beta _{1}-\beta _{2})q^{5}+(-3\beta _{1}+\cdots)q^{7}+\cdots\)
768.4.c.q 768.c 12.b $4$ $45.313$ \(\Q(\sqrt{-2}, \sqrt{3})\) \(\Q(\sqrt{-6}) \) 384.4.f.a \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q-\beta _{3}q^{3}-7\beta _{1}q^{5}+\beta _{2}q^{7}+3^{3}q^{9}+\cdots\)
768.4.c.r 768.c 12.b $4$ $45.313$ \(\Q(\zeta_{12})\) \(\Q(\sqrt{-6}) \) 192.4.f.a \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+\zeta_{12}^{2}q^{3}-\zeta_{12}^{3}q^{5}+17\zeta_{12}q^{7}+\cdots\)
768.4.c.s 768.c 12.b $8$ $45.313$ 8.0.\(\cdots\).4 None 384.4.f.i \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(-\beta _{1}-\beta _{3})q^{3}+\beta _{4}q^{5}-\beta _{6}q^{7}+\cdots\)
768.4.c.t 768.c 12.b $8$ $45.313$ 8.0.1731891456.1 None 192.4.f.c \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{4}q^{3}+(3\beta _{2}+\beta _{4}+\beta _{6})q^{5}+\beta _{7}q^{7}+\cdots\)
768.4.c.u 768.c 12.b $8$ $45.313$ 8.0.1731891456.1 None 192.4.f.c \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{4}q^{3}+(3\beta _{2}+\beta _{4}+\beta _{6})q^{5}-\beta _{7}q^{7}+\cdots\)
768.4.c.v 768.c 12.b $16$ $45.313$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None 24.4.f.b \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{4}q^{3}+\beta _{9}q^{5}+\beta _{6}q^{7}+(6-3\beta _{7}+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(768, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(768, [\chi]) \simeq \) \(S_{4}^{\mathrm{new}}(12, [\chi])\)\(^{\oplus 7}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(24, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(48, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(96, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(192, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(384, [\chi])\)\(^{\oplus 2}\)