Properties

Label 704.6.a
Level $704$
Weight $6$
Character orbit 704.a
Rep. character $\chi_{704}(1,\cdot)$
Character field $\Q$
Dimension $100$
Newform subspaces $32$
Sturm bound $576$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 704 = 2^{6} \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 704.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 32 \)
Sturm bound: \(576\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(704))\).

Total New Old
Modular forms 492 100 392
Cusp forms 468 100 368
Eisenstein series 24 0 24

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)\(11\)FrickeDim
\(+\)\(+\)\(+\)\(24\)
\(+\)\(-\)\(-\)\(27\)
\(-\)\(+\)\(-\)\(26\)
\(-\)\(-\)\(+\)\(23\)
Plus space\(+\)\(47\)
Minus space\(-\)\(53\)

Trace form

\( 100 q + 8100 q^{9} + O(q^{10}) \) \( 100 q + 8100 q^{9} + 464 q^{13} - 808 q^{17} - 8496 q^{21} + 68732 q^{25} + 4304 q^{37} + 4952 q^{41} + 36024 q^{45} + 240100 q^{49} - 38856 q^{53} + 4144 q^{61} - 7680 q^{65} - 97896 q^{69} - 10072 q^{73} + 476740 q^{81} + 264800 q^{85} - 57592 q^{89} - 173544 q^{93} - 160808 q^{97} + O(q^{100}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(704))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 11
704.6.a.a 704.a 1.a $1$ $112.910$ \(\Q\) None 22.6.a.c \(0\) \(-29\) \(31\) \(230\) $-$ $-$ $\mathrm{SU}(2)$ \(q-29q^{3}+31q^{5}+230q^{7}+598q^{9}+\cdots\)
704.6.a.b 704.a 1.a $1$ $112.910$ \(\Q\) None 22.6.a.a \(0\) \(-21\) \(-81\) \(-98\) $-$ $-$ $\mathrm{SU}(2)$ \(q-21q^{3}-3^{4}q^{5}-98q^{7}+198q^{9}+\cdots\)
704.6.a.c 704.a 1.a $1$ $112.910$ \(\Q\) None 11.6.a.a \(0\) \(-15\) \(19\) \(-10\) $-$ $+$ $\mathrm{SU}(2)$ \(q-15q^{3}+19q^{5}-10q^{7}-18q^{9}+\cdots\)
704.6.a.d 704.a 1.a $1$ $112.910$ \(\Q\) None 44.6.a.a \(0\) \(-7\) \(79\) \(-50\) $+$ $+$ $\mathrm{SU}(2)$ \(q-7q^{3}+79q^{5}-50q^{7}-194q^{9}+\cdots\)
704.6.a.e 704.a 1.a $1$ $112.910$ \(\Q\) None 22.6.a.b \(0\) \(-1\) \(51\) \(-166\) $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}+51q^{5}-166q^{7}-242q^{9}+\cdots\)
704.6.a.f 704.a 1.a $1$ $112.910$ \(\Q\) None 22.6.a.b \(0\) \(1\) \(51\) \(166\) $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}+51q^{5}+166q^{7}-242q^{9}+\cdots\)
704.6.a.g 704.a 1.a $1$ $112.910$ \(\Q\) None 44.6.a.a \(0\) \(7\) \(79\) \(50\) $-$ $-$ $\mathrm{SU}(2)$ \(q+7q^{3}+79q^{5}+50q^{7}-194q^{9}+\cdots\)
704.6.a.h 704.a 1.a $1$ $112.910$ \(\Q\) None 11.6.a.a \(0\) \(15\) \(19\) \(10\) $+$ $-$ $\mathrm{SU}(2)$ \(q+15q^{3}+19q^{5}+10q^{7}-18q^{9}+\cdots\)
704.6.a.i 704.a 1.a $1$ $112.910$ \(\Q\) None 22.6.a.a \(0\) \(21\) \(-81\) \(98\) $+$ $+$ $\mathrm{SU}(2)$ \(q+21q^{3}-3^{4}q^{5}+98q^{7}+198q^{9}+\cdots\)
704.6.a.j 704.a 1.a $1$ $112.910$ \(\Q\) None 22.6.a.c \(0\) \(29\) \(31\) \(-230\) $+$ $+$ $\mathrm{SU}(2)$ \(q+29q^{3}+31q^{5}-230q^{7}+598q^{9}+\cdots\)
704.6.a.k 704.a 1.a $2$ $112.910$ \(\Q(\sqrt{793}) \) None 22.6.a.d \(0\) \(-29\) \(13\) \(-14\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(-14-\beta )q^{3}+(4+5\beta )q^{5}+(-4+\cdots)q^{7}+\cdots\)
704.6.a.l 704.a 1.a $2$ $112.910$ \(\Q(\sqrt{37}) \) None 88.6.a.a \(0\) \(-14\) \(-18\) \(-48\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-7-\beta )q^{3}-9q^{5}+(-24-11\beta )q^{7}+\cdots\)
704.6.a.m 704.a 1.a $2$ $112.910$ \(\Q(\sqrt{31}) \) None 44.6.a.b \(0\) \(-6\) \(22\) \(-268\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(-3+\beta )q^{3}+(11-2\beta )q^{5}+(-134+\cdots)q^{7}+\cdots\)
704.6.a.n 704.a 1.a $2$ $112.910$ \(\Q(\sqrt{31}) \) None 44.6.a.b \(0\) \(6\) \(22\) \(268\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(3+\beta )q^{3}+(11+2\beta )q^{5}+(134+3\beta )q^{7}+\cdots\)
704.6.a.o 704.a 1.a $2$ $112.910$ \(\Q(\sqrt{37}) \) None 88.6.a.a \(0\) \(14\) \(-18\) \(48\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(7+\beta )q^{3}-9q^{5}+(24+11\beta )q^{7}+\cdots\)
704.6.a.p 704.a 1.a $2$ $112.910$ \(\Q(\sqrt{793}) \) None 22.6.a.d \(0\) \(29\) \(13\) \(14\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(15-\beta )q^{3}+(9-5\beta )q^{5}+(10-6\beta )q^{7}+\cdots\)
704.6.a.q 704.a 1.a $3$ $112.910$ 3.3.54492.1 None 11.6.a.b \(0\) \(-34\) \(-24\) \(84\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(-11-\beta _{1})q^{3}+(-7-4\beta _{1}-\beta _{2})q^{5}+\cdots\)
704.6.a.r 704.a 1.a $3$ $112.910$ 3.3.1784453.1 None 88.6.a.b \(0\) \(-14\) \(-56\) \(112\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(-5+\beta _{2})q^{3}+(-19-\beta _{1}+\beta _{2})q^{5}+\cdots\)
704.6.a.s 704.a 1.a $3$ $112.910$ 3.3.1784453.1 None 88.6.a.b \(0\) \(14\) \(-56\) \(-112\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(5-\beta _{2})q^{3}+(-19-\beta _{1}+\beta _{2})q^{5}+\cdots\)
704.6.a.t 704.a 1.a $3$ $112.910$ 3.3.54492.1 None 11.6.a.b \(0\) \(34\) \(-24\) \(-84\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(11+\beta _{1})q^{3}+(-7-4\beta _{1}-\beta _{2})q^{5}+\cdots\)
704.6.a.u 704.a 1.a $4$ $112.910$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None 88.6.a.c \(0\) \(-13\) \(19\) \(-58\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(-3-\beta _{1})q^{3}+(5-2\beta _{1}+\beta _{3})q^{5}+\cdots\)
704.6.a.v 704.a 1.a $4$ $112.910$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None 88.6.a.d \(0\) \(-5\) \(-93\) \(94\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(-1-\beta _{1})q^{3}+(-24+2\beta _{1}+\beta _{3})q^{5}+\cdots\)
704.6.a.w 704.a 1.a $4$ $112.910$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None 88.6.a.d \(0\) \(5\) \(-93\) \(-94\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(1+\beta _{1})q^{3}+(-24+2\beta _{1}+\beta _{3})q^{5}+\cdots\)
704.6.a.x 704.a 1.a $4$ $112.910$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None 88.6.a.c \(0\) \(13\) \(19\) \(58\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(3+\beta _{1})q^{3}+(5-2\beta _{1}+\beta _{3})q^{5}+(13+\cdots)q^{7}+\cdots\)
704.6.a.y 704.a 1.a $6$ $112.910$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None 352.6.a.a \(0\) \(-27\) \(5\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-4-\beta _{1})q^{3}+(1+\beta _{5})q^{5}+(-2+\cdots)q^{7}+\cdots\)
704.6.a.z 704.a 1.a $6$ $112.910$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None 352.6.a.c \(0\) \(-9\) \(5\) \(196\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(-1-\beta _{2})q^{3}+(1-\beta _{1})q^{5}+(33-\beta _{3}+\cdots)q^{7}+\cdots\)
704.6.a.ba 704.a 1.a $6$ $112.910$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None 352.6.a.b \(0\) \(-9\) \(89\) \(-196\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{2})q^{3}+(15-\beta _{3})q^{5}+(-33+\cdots)q^{7}+\cdots\)
704.6.a.bb 704.a 1.a $6$ $112.910$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None 352.6.a.c \(0\) \(9\) \(5\) \(-196\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(1+\beta _{2})q^{3}+(1-\beta _{1})q^{5}+(-33+\beta _{3}+\cdots)q^{7}+\cdots\)
704.6.a.bc 704.a 1.a $6$ $112.910$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None 352.6.a.b \(0\) \(9\) \(89\) \(196\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(1-\beta _{2})q^{3}+(15-\beta _{3})q^{5}+(33+\beta _{1}+\cdots)q^{7}+\cdots\)
704.6.a.bd 704.a 1.a $6$ $112.910$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None 352.6.a.a \(0\) \(27\) \(5\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(4+\beta _{1})q^{3}+(1+\beta _{5})q^{5}+(2-3\beta _{1}+\cdots)q^{7}+\cdots\)
704.6.a.be 704.a 1.a $7$ $112.910$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None 352.6.a.g \(0\) \(-9\) \(-61\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(-1-\beta _{1})q^{3}+(-9+\beta _{1}+\beta _{2})q^{5}+\cdots\)
704.6.a.bf 704.a 1.a $7$ $112.910$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None 352.6.a.g \(0\) \(9\) \(-61\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(1+\beta _{1})q^{3}+(-9+\beta _{1}+\beta _{2})q^{5}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(704))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(704)) \simeq \) \(S_{6}^{\mathrm{new}}(\Gamma_0(4))\)\(^{\oplus 10}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 7}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(22))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(44))\)\(^{\oplus 5}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(64))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(88))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(176))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(352))\)\(^{\oplus 2}\)