Properties

Label 700.4.a
Level $700$
Weight $4$
Character orbit 700.a
Rep. character $\chi_{700}(1,\cdot)$
Character field $\Q$
Dimension $28$
Newform subspaces $20$
Sturm bound $480$
Trace bound $11$

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Defining parameters

Level: \( N \) \(=\) \( 700 = 2^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 700.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 20 \)
Sturm bound: \(480\)
Trace bound: \(11\)
Distinguishing \(T_p\): \(3\), \(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(700))\).

Total New Old
Modular forms 378 28 350
Cusp forms 342 28 314
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
\(-\)\(+\)\(+\)$-$\(7\)
\(-\)\(+\)\(-\)$+$\(7\)
\(-\)\(-\)\(+\)$+$\(7\)
\(-\)\(-\)\(-\)$-$\(7\)
Plus space\(+\)\(14\)
Minus space\(-\)\(14\)

Trace form

\( 28 q + 2 q^{3} + 228 q^{9} + O(q^{10}) \) \( 28 q + 2 q^{3} + 228 q^{9} - 88 q^{11} + 78 q^{13} - 20 q^{17} + 46 q^{19} - 98 q^{21} + 112 q^{23} - 100 q^{27} - 84 q^{29} - 220 q^{31} - 792 q^{33} + 92 q^{37} + 328 q^{39} + 160 q^{41} - 532 q^{43} + 644 q^{47} + 1372 q^{49} + 800 q^{51} + 536 q^{53} + 1212 q^{57} + 946 q^{59} + 86 q^{61} + 476 q^{63} - 856 q^{67} - 3384 q^{69} - 1472 q^{71} + 2104 q^{73} + 420 q^{77} + 1584 q^{79} - 1008 q^{81} - 306 q^{83} - 2444 q^{87} - 4252 q^{89} - 350 q^{91} + 1904 q^{93} - 1268 q^{97} - 3052 q^{99} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(700))\) 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
700.4.a.a 700.a 1.a $1$ $41.301$ \(\Q\) None \(0\) \(-9\) \(0\) \(7\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-9q^{3}+7q^{7}+54q^{9}+55q^{11}+69q^{13}+\cdots\)
700.4.a.b 700.a 1.a $1$ $41.301$ \(\Q\) None \(0\) \(-8\) \(0\) \(-7\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-8q^{3}-7q^{7}+37q^{9}+28q^{11}-82q^{13}+\cdots\)
700.4.a.c 700.a 1.a $1$ $41.301$ \(\Q\) None \(0\) \(-7\) \(0\) \(7\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-7q^{3}+7q^{7}+22q^{9}-7q^{11}-3q^{13}+\cdots\)
700.4.a.d 700.a 1.a $1$ $41.301$ \(\Q\) None \(0\) \(-5\) \(0\) \(-7\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-5q^{3}-7q^{7}-2q^{9}-65q^{11}+13q^{13}+\cdots\)
700.4.a.e 700.a 1.a $1$ $41.301$ \(\Q\) None \(0\) \(-4\) \(0\) \(-7\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-4q^{3}-7q^{7}-11q^{9}-12q^{11}+82q^{13}+\cdots\)
700.4.a.f 700.a 1.a $1$ $41.301$ \(\Q\) None \(0\) \(-1\) \(0\) \(7\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}+7q^{7}-26q^{9}-37q^{11}+38q^{13}+\cdots\)
700.4.a.g 700.a 1.a $1$ $41.301$ \(\Q\) None \(0\) \(-1\) \(0\) \(7\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}+7q^{7}-26q^{9}-7q^{11}+23q^{13}+\cdots\)
700.4.a.h 700.a 1.a $1$ $41.301$ \(\Q\) None \(0\) \(1\) \(0\) \(-7\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}-7q^{7}-26q^{9}-37q^{11}-38q^{13}+\cdots\)
700.4.a.i 700.a 1.a $1$ $41.301$ \(\Q\) None \(0\) \(4\) \(0\) \(7\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+4q^{3}+7q^{7}-11q^{9}+68q^{11}-22q^{13}+\cdots\)
700.4.a.j 700.a 1.a $1$ $41.301$ \(\Q\) None \(0\) \(5\) \(0\) \(-7\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+5q^{3}-7q^{7}-2q^{9}-15q^{11}+13q^{13}+\cdots\)
700.4.a.k 700.a 1.a $1$ $41.301$ \(\Q\) None \(0\) \(5\) \(0\) \(-7\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+5q^{3}-7q^{7}-2q^{9}+15q^{11}-17q^{13}+\cdots\)
700.4.a.l 700.a 1.a $1$ $41.301$ \(\Q\) None \(0\) \(5\) \(0\) \(7\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+5q^{3}+7q^{7}-2q^{9}-65q^{11}-13q^{13}+\cdots\)
700.4.a.m 700.a 1.a $1$ $41.301$ \(\Q\) None \(0\) \(7\) \(0\) \(-7\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+7q^{3}-7q^{7}+22q^{9}-7q^{11}+3q^{13}+\cdots\)
700.4.a.n 700.a 1.a $1$ $41.301$ \(\Q\) None \(0\) \(10\) \(0\) \(7\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+10q^{3}+7q^{7}+73q^{9}-40q^{11}+\cdots\)
700.4.a.o 700.a 1.a $2$ $41.301$ \(\Q(\sqrt{61}) \) None \(0\) \(-2\) \(0\) \(14\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(-1-\beta )q^{3}+7q^{7}+(35+2\beta )q^{9}+\cdots\)
700.4.a.p 700.a 1.a $2$ $41.301$ \(\Q(\sqrt{6}) \) None \(0\) \(0\) \(0\) \(-14\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{3}-7q^{7}-21q^{9}+(24+14\beta )q^{11}+\cdots\)
700.4.a.q 700.a 1.a $2$ $41.301$ \(\Q(\sqrt{6}) \) None \(0\) \(0\) \(0\) \(14\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{3}+7q^{7}-21q^{9}+(24-14\beta )q^{11}+\cdots\)
700.4.a.r 700.a 1.a $2$ $41.301$ \(\Q(\sqrt{61}) \) None \(0\) \(2\) \(0\) \(-14\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{3}-7q^{7}+(35+2\beta )q^{9}+(24+\cdots)q^{11}+\cdots\)
700.4.a.s 700.a 1.a $3$ $41.301$ 3.3.15629.1 None \(0\) \(-5\) \(0\) \(21\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(-2+\beta _{2})q^{3}+7q^{7}+(13-\beta _{1}-2\beta _{2})q^{9}+\cdots\)
700.4.a.t 700.a 1.a $3$ $41.301$ 3.3.15629.1 None \(0\) \(5\) \(0\) \(-21\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(2-\beta _{2})q^{3}-7q^{7}+(13-\beta _{1}-2\beta _{2})q^{9}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(700))\) into lower level spaces

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