Properties

Label 784.6.a
Level $784$
Weight $6$
Character orbit 784.a
Rep. character $\chi_{784}(1,\cdot)$
Character field $\Q$
Dimension $100$
Newform subspaces $41$
Sturm bound $672$
Trace bound $9$

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Defining parameters

Level: \( N \) \(=\) \( 784 = 2^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 784.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 41 \)
Sturm bound: \(672\)
Trace bound: \(9\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(784))\).

Total New Old
Modular forms 584 105 479
Cusp forms 536 100 436
Eisenstein series 48 5 43

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

\(2\)\(7\)FrickeDim.
\(+\)\(+\)\(+\)\(24\)
\(+\)\(-\)\(-\)\(27\)
\(-\)\(+\)\(-\)\(25\)
\(-\)\(-\)\(+\)\(24\)
Plus space\(+\)\(48\)
Minus space\(-\)\(52\)

Trace form

\( 100 q + 8 q^{3} - 18 q^{5} + 7674 q^{9} + O(q^{10}) \) \( 100 q + 8 q^{3} - 18 q^{5} + 7674 q^{9} - 62 q^{11} + 62 q^{13} + 1646 q^{15} - 602 q^{17} + 1520 q^{19} - 746 q^{23} + 55666 q^{25} + 2960 q^{27} + 8152 q^{29} - 8632 q^{31} + 5920 q^{33} - 618 q^{37} + 4292 q^{39} - 1298 q^{41} - 31720 q^{43} + 1574 q^{45} - 11832 q^{47} - 50402 q^{51} - 11058 q^{53} - 50424 q^{55} - 2854 q^{57} + 38968 q^{59} - 40650 q^{61} + 14492 q^{65} + 82042 q^{67} - 30608 q^{69} - 30920 q^{71} + 3542 q^{73} - 145960 q^{75} + 204710 q^{79} + 609412 q^{81} + 257712 q^{83} + 27438 q^{85} + 98560 q^{87} + 146550 q^{89} - 17002 q^{93} - 119342 q^{95} - 81130 q^{97} - 40644 q^{99} + O(q^{100}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(784))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 7
784.6.a.a 784.a 1.a $1$ $125.741$ \(\Q\) None \(0\) \(-19\) \(-19\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q-19q^{3}-19q^{5}+118q^{9}+559q^{11}+\cdots\)
784.6.a.b 784.a 1.a $1$ $125.741$ \(\Q\) None \(0\) \(-16\) \(-16\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q-2^{4}q^{3}-2^{4}q^{5}+13q^{9}+76q^{11}+\cdots\)
784.6.a.c 784.a 1.a $1$ $125.741$ \(\Q\) None \(0\) \(-14\) \(56\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q-14q^{3}+56q^{5}-47q^{9}-232q^{11}+\cdots\)
784.6.a.d 784.a 1.a $1$ $125.741$ \(\Q\) None \(0\) \(-12\) \(-54\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q-12q^{3}-54q^{5}-99q^{9}-540q^{11}+\cdots\)
784.6.a.e 784.a 1.a $1$ $125.741$ \(\Q\) None \(0\) \(-6\) \(-4\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q-6q^{3}-4q^{5}-207q^{9}+240q^{11}+\cdots\)
784.6.a.f 784.a 1.a $1$ $125.741$ \(\Q\) None \(0\) \(-2\) \(96\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{3}+96q^{5}-239q^{9}+720q^{11}+\cdots\)
784.6.a.g 784.a 1.a $1$ $125.741$ \(\Q\) \(\Q(\sqrt{-7}) \) \(0\) \(0\) \(0\) \(0\) $-$ $-$ $N(\mathrm{U}(1))$ \(q-3^{5}q^{9}+76q^{11}+4952q^{23}-5^{5}q^{25}+\cdots\)
784.6.a.h 784.a 1.a $1$ $125.741$ \(\Q\) None \(0\) \(8\) \(-10\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+8q^{3}-10q^{5}-179q^{9}+340q^{11}+\cdots\)
784.6.a.i 784.a 1.a $1$ $125.741$ \(\Q\) None \(0\) \(10\) \(-84\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+10q^{3}-84q^{5}-143q^{9}+336q^{11}+\cdots\)
784.6.a.j 784.a 1.a $1$ $125.741$ \(\Q\) None \(0\) \(16\) \(16\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+2^{4}q^{3}+2^{4}q^{5}+13q^{9}+76q^{11}+\cdots\)
784.6.a.k 784.a 1.a $1$ $125.741$ \(\Q\) None \(0\) \(19\) \(19\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+19q^{3}+19q^{5}+118q^{9}+559q^{11}+\cdots\)
784.6.a.l 784.a 1.a $1$ $125.741$ \(\Q\) None \(0\) \(20\) \(74\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+20q^{3}+74q^{5}+157q^{9}-124q^{11}+\cdots\)
784.6.a.m 784.a 1.a $1$ $125.741$ \(\Q\) None \(0\) \(26\) \(-16\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+26q^{3}-2^{4}q^{5}+433q^{9}-8q^{11}+\cdots\)
784.6.a.n 784.a 1.a $1$ $125.741$ \(\Q\) None \(0\) \(30\) \(-32\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+30q^{3}-2^{5}q^{5}+657q^{9}+624q^{11}+\cdots\)
784.6.a.o 784.a 1.a $2$ $125.741$ \(\Q(\sqrt{109}) \) None \(0\) \(-28\) \(42\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(-14-\beta )q^{3}+(21+6\beta )q^{5}+(62+\cdots)q^{9}+\cdots\)
784.6.a.p 784.a 1.a $2$ $125.741$ \(\Q(\sqrt{177}) \) None \(0\) \(-26\) \(62\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(-13-\beta )q^{3}+(31+5\beta )q^{5}+(103+\cdots)q^{9}+\cdots\)
784.6.a.q 784.a 1.a $2$ $125.741$ \(\Q(\sqrt{193}) \) None \(0\) \(-14\) \(-42\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(-7-\beta )q^{3}+(-21-5\beta )q^{5}+(-1+\cdots)q^{9}+\cdots\)
784.6.a.r 784.a 1.a $2$ $125.741$ \(\Q(\sqrt{130}) \) None \(0\) \(-14\) \(42\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-7+\beta )q^{3}+21q^{5}+(326-14\beta )q^{9}+\cdots\)
784.6.a.s 784.a 1.a $2$ $125.741$ \(\Q(\sqrt{79}) \) None \(0\) \(-14\) \(70\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(-7+\beta )q^{3}+(35+4\beta )q^{5}+(122+\cdots)q^{9}+\cdots\)
784.6.a.t 784.a 1.a $2$ $125.741$ \(\Q(\sqrt{37}) \) None \(0\) \(-8\) \(38\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-4-\beta )q^{3}+(19+10\beta )q^{5}+(-190+\cdots)q^{9}+\cdots\)
784.6.a.u 784.a 1.a $2$ $125.741$ \(\Q(\sqrt{345}) \) None \(0\) \(-6\) \(-82\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(-3-\beta )q^{3}+(-41+3\beta )q^{5}+(111+\cdots)q^{9}+\cdots\)
784.6.a.v 784.a 1.a $2$ $125.741$ \(\Q(\sqrt{57}) \) None \(0\) \(-6\) \(18\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-3-3\beta )q^{3}+(9-5\beta )q^{5}+(279+\cdots)q^{9}+\cdots\)
784.6.a.w 784.a 1.a $2$ $125.741$ \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(0\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+3\beta q^{3}-46\beta q^{5}-15^{2}q^{9}-274q^{11}+\cdots\)
784.6.a.x 784.a 1.a $2$ $125.741$ \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(0\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+6\beta q^{3}+7\beta q^{5}-171q^{9}-308q^{11}+\cdots\)
784.6.a.y 784.a 1.a $2$ $125.741$ \(\Q(\sqrt{46}) \) None \(0\) \(0\) \(0\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{3}+7\beta q^{5}-59q^{9}-476q^{11}+\cdots\)
784.6.a.z 784.a 1.a $2$ $125.741$ \(\Q(\sqrt{39}) \) None \(0\) \(0\) \(0\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{3}-3\beta q^{5}+381q^{9}+284q^{11}+\cdots\)
784.6.a.ba 784.a 1.a $2$ $125.741$ \(\Q(\sqrt{37}) \) None \(0\) \(8\) \(-38\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(4+\beta )q^{3}+(-19-10\beta )q^{5}+(-190+\cdots)q^{9}+\cdots\)
784.6.a.bb 784.a 1.a $2$ $125.741$ \(\Q(\sqrt{79}) \) None \(0\) \(14\) \(-70\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(7+\beta )q^{3}+(-35+4\beta )q^{5}+(122+\cdots)q^{9}+\cdots\)
784.6.a.bc 784.a 1.a $2$ $125.741$ \(\Q(\sqrt{130}) \) None \(0\) \(14\) \(-42\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(7+\beta )q^{3}-21q^{5}+(326+14\beta )q^{9}+\cdots\)
784.6.a.bd 784.a 1.a $2$ $125.741$ \(\Q(\sqrt{109}) \) None \(0\) \(28\) \(-42\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(14-\beta )q^{3}+(-21+6\beta )q^{5}+(62+\cdots)q^{9}+\cdots\)
784.6.a.be 784.a 1.a $4$ $125.741$ \(\Q(\sqrt{86}, \sqrt{134})\) None \(0\) \(0\) \(0\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{3}-\beta _{2}q^{5}+(-23+\beta _{3})q^{9}+\cdots\)
784.6.a.bf 784.a 1.a $4$ $125.741$ \(\Q(\sqrt{2}, \sqrt{113})\) None \(0\) \(0\) \(0\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(-\beta _{2}+\beta _{3})q^{3}+(-2\beta _{2}+4\beta _{3})q^{5}+\cdots\)
784.6.a.bg 784.a 1.a $4$ $125.741$ \(\Q(\sqrt{2}, \sqrt{793})\) None \(0\) \(0\) \(0\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(\beta _{1}+\beta _{3})q^{3}+(-4\beta _{1}-3\beta _{3})q^{5}+\cdots\)
784.6.a.bh 784.a 1.a $4$ $125.741$ 4.4.2732674592.1 None \(0\) \(0\) \(0\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{2}q^{3}+(\beta _{1}+2\beta _{2})q^{5}+(209-\beta _{3})q^{9}+\cdots\)
784.6.a.bi 784.a 1.a $4$ $125.741$ \(\Q(\sqrt{2}, \sqrt{1177})\) None \(0\) \(0\) \(0\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(\beta _{1}+\beta _{3})q^{3}+(8\beta _{1}+\beta _{3})q^{5}+(370+\cdots)q^{9}+\cdots\)
784.6.a.bj 784.a 1.a $5$ $125.741$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(0\) \(-13\) \(31\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(-3+\beta _{1})q^{3}+(6+\beta _{3})q^{5}+(47-4\beta _{1}+\cdots)q^{9}+\cdots\)
784.6.a.bk 784.a 1.a $5$ $125.741$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(0\) \(-5\) \(-81\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(-1-\beta _{1})q^{3}+(-2^{4}-\beta _{1}-\beta _{2}+\cdots)q^{5}+\cdots\)
784.6.a.bl 784.a 1.a $5$ $125.741$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(0\) \(5\) \(81\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(1+\beta _{1})q^{3}+(2^{4}+\beta _{1}+\beta _{2})q^{5}+(78+\cdots)q^{9}+\cdots\)
784.6.a.bm 784.a 1.a $5$ $125.741$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(0\) \(13\) \(-31\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(3-\beta _{1})q^{3}+(-6-\beta _{3})q^{5}+(47-4\beta _{1}+\cdots)q^{9}+\cdots\)
784.6.a.bn 784.a 1.a $6$ $125.741$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{2}q^{3}+(-\beta _{1}+\beta _{3})q^{5}+(-20+\beta _{4}+\cdots)q^{9}+\cdots\)
784.6.a.bo 784.a 1.a $8$ $125.741$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{2}q^{3}+(\beta _{1}+\beta _{4})q^{5}+(116+\beta _{3}+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(784))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(784)) \cong \) \(S_{6}^{\mathrm{new}}(\Gamma_0(4))\)\(^{\oplus 9}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 10}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(28))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 5}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(56))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(98))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(112))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(196))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(392))\)\(^{\oplus 2}\)