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Properties

Label 700.6.a
Level $700$
Weight $6$
Character orbit 700.a
Rep. character $\chi_{700}(1,\cdot)$
Character field $\Q$
Dimension $48$
Newform subspaces $16$
Sturm bound $720$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 700 = 2^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 700.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 16 \)
Sturm bound: \(720\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(700))\).

Total New Old
Modular forms 618 48 570
Cusp forms 582 48 534
Eisenstein series 36 0 36

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

\(2\)\(5\)\(7\)FrickeDim
\(-\)\(+\)\(+\)$-$\(11\)
\(-\)\(+\)\(-\)$+$\(11\)
\(-\)\(-\)\(+\)$+$\(13\)
\(-\)\(-\)\(-\)$-$\(13\)
Plus space\(+\)\(24\)
Minus space\(-\)\(24\)

Trace form

\( 48 q + 20 q^{3} + 3900 q^{9} + O(q^{10}) \) \( 48 q + 20 q^{3} + 3900 q^{9} + 152 q^{11} - 1660 q^{13} + 1480 q^{17} + 720 q^{19} + 1372 q^{21} - 9680 q^{23} + 2060 q^{27} + 11112 q^{29} + 17092 q^{31} - 16140 q^{33} + 6300 q^{37} - 38372 q^{39} + 34588 q^{41} - 8980 q^{43} + 2660 q^{47} + 115248 q^{49} - 57508 q^{51} - 2320 q^{53} - 28500 q^{57} + 44068 q^{59} - 33856 q^{61} + 9800 q^{63} + 19060 q^{67} + 204732 q^{69} + 4756 q^{71} + 101540 q^{73} - 11760 q^{77} - 69152 q^{79} + 140808 q^{81} + 56040 q^{83} + 196900 q^{87} - 118924 q^{89} + 59976 q^{91} - 27220 q^{93} - 18640 q^{97} - 305620 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(700))\) 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 5 7
700.6.a.a 700.a 1.a $1$ $112.269$ \(\Q\) None \(0\) \(-28\) \(0\) \(49\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-28q^{3}+7^{2}q^{7}+541q^{9}-577q^{11}+\cdots\)
700.6.a.b 700.a 1.a $1$ $112.269$ \(\Q\) None \(0\) \(-26\) \(0\) \(49\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-26q^{3}+7^{2}q^{7}+433q^{9}+8q^{11}+\cdots\)
700.6.a.c 700.a 1.a $1$ $112.269$ \(\Q\) None \(0\) \(-17\) \(0\) \(-49\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-17q^{3}-7^{2}q^{7}+46q^{9}-317q^{11}+\cdots\)
700.6.a.d 700.a 1.a $1$ $112.269$ \(\Q\) None \(0\) \(2\) \(0\) \(-49\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{3}-7^{2}q^{7}-239q^{9}-720q^{11}+\cdots\)
700.6.a.e 700.a 1.a $1$ $112.269$ \(\Q\) None \(0\) \(17\) \(0\) \(49\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+17q^{3}+7^{2}q^{7}+46q^{9}-317q^{11}+\cdots\)
700.6.a.f 700.a 1.a $1$ $112.269$ \(\Q\) None \(0\) \(28\) \(0\) \(-49\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+28q^{3}-7^{2}q^{7}+541q^{9}-577q^{11}+\cdots\)
700.6.a.g 700.a 1.a $2$ $112.269$ \(\Q(\sqrt{1009}) \) None \(0\) \(17\) \(0\) \(98\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(9-\beta )q^{3}+7^{2}q^{7}+(90-17\beta )q^{9}+\cdots\)
700.6.a.h 700.a 1.a $2$ $112.269$ \(\Q(\sqrt{1009}) \) None \(0\) \(23\) \(0\) \(-98\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(12-\beta )q^{3}-7^{2}q^{7}+(153-23\beta )q^{9}+\cdots\)
700.6.a.i 700.a 1.a $3$ $112.269$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(0\) \(-6\) \(0\) \(-147\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(-2-\beta _{1})q^{3}-7^{2}q^{7}+(94+5\beta _{1}+\cdots)q^{9}+\cdots\)
700.6.a.j 700.a 1.a $3$ $112.269$ 3.3.3101016.1 None \(0\) \(10\) \(0\) \(147\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(3+\beta _{1})q^{3}+7^{2}q^{7}+(-23-\beta _{1}+\cdots)q^{9}+\cdots\)
700.6.a.k 700.a 1.a $4$ $112.269$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(0\) \(-22\) \(0\) \(-196\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(-6+\beta _{1})q^{3}-7^{2}q^{7}+(-12-6\beta _{1}+\cdots)q^{9}+\cdots\)
700.6.a.l 700.a 1.a $4$ $112.269$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(0\) \(-1\) \(0\) \(196\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{3}+7^{2}q^{7}+(52-11\beta _{1}+\beta _{2}+\cdots)q^{9}+\cdots\)
700.6.a.m 700.a 1.a $4$ $112.269$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(0\) \(1\) \(0\) \(-196\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{3}-7^{2}q^{7}+(52-11\beta _{1}+\beta _{2}+\cdots)q^{9}+\cdots\)
700.6.a.n 700.a 1.a $4$ $112.269$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(0\) \(22\) \(0\) \(196\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(6-\beta _{1})q^{3}+7^{2}q^{7}+(-12-6\beta _{1}+\cdots)q^{9}+\cdots\)
700.6.a.o 700.a 1.a $8$ $112.269$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(0\) \(-13\) \(0\) \(-392\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(-2+\beta _{1})q^{3}-7^{2}q^{7}+(102-3\beta _{1}+\cdots)q^{9}+\cdots\)
700.6.a.p 700.a 1.a $8$ $112.269$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(0\) \(13\) \(0\) \(392\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(2-\beta _{1})q^{3}+7^{2}q^{7}+(102-3\beta _{1}+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(700))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(700)) \cong \) \(S_{6}^{\mathrm{new}}(\Gamma_0(4))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 12}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 9}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(25))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(28))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(70))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(100))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(140))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(175))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(350))\)\(^{\oplus 2}\)