Properties

Label 5700.2.a
Level $5700$
Weight $2$
Character orbit 5700.a
Rep. character $\chi_{5700}(1,\cdot)$
Character field $\Q$
Dimension $56$
Newform subspaces $30$
Sturm bound $2400$
Trace bound $17$

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Defining parameters

Level: \( N \) \(=\) \( 5700 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5700.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 30 \)
Sturm bound: \(2400\)
Trace bound: \(17\)
Distinguishing \(T_p\): \(7\), \(11\), \(13\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(5700))\).

Total New Old
Modular forms 1236 56 1180
Cusp forms 1165 56 1109
Eisenstein series 71 0 71

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

\(2\)\(3\)\(5\)\(19\)FrickeDim.
\(-\)\(+\)\(+\)\(+\)\(-\)\(8\)
\(-\)\(+\)\(+\)\(-\)\(+\)\(6\)
\(-\)\(+\)\(-\)\(+\)\(+\)\(7\)
\(-\)\(+\)\(-\)\(-\)\(-\)\(7\)
\(-\)\(-\)\(+\)\(+\)\(+\)\(5\)
\(-\)\(-\)\(+\)\(-\)\(-\)\(9\)
\(-\)\(-\)\(-\)\(+\)\(-\)\(9\)
\(-\)\(-\)\(-\)\(-\)\(+\)\(5\)
Plus space\(+\)\(23\)
Minus space\(-\)\(33\)

Trace form

\( 56 q - 2 q^{7} + 56 q^{9} + O(q^{10}) \) \( 56 q - 2 q^{7} + 56 q^{9} - 2 q^{11} - 8 q^{13} - 6 q^{17} - 2 q^{19} - 8 q^{23} + 36 q^{29} - 12 q^{31} - 8 q^{33} + 4 q^{37} - 8 q^{39} + 40 q^{41} + 6 q^{43} + 10 q^{47} + 54 q^{49} + 12 q^{51} - 36 q^{53} + 2 q^{57} + 24 q^{59} + 30 q^{61} - 2 q^{63} - 16 q^{67} + 4 q^{69} - 6 q^{73} - 18 q^{77} + 16 q^{79} + 56 q^{81} + 28 q^{89} + 28 q^{91} + 16 q^{93} - 24 q^{97} - 2 q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(5700))\) 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 3 5 19
5700.2.a.a 5700.a 1.a $1$ $45.515$ \(\Q\) None \(0\) \(-1\) \(0\) \(-2\) $-$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}-2q^{7}+q^{9}+4q^{13}-6q^{17}+\cdots\)
5700.2.a.b 5700.a 1.a $1$ $45.515$ \(\Q\) None \(0\) \(-1\) \(0\) \(-1\) $-$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}-q^{7}+q^{9}+2q^{17}-q^{19}+q^{21}+\cdots\)
5700.2.a.c 5700.a 1.a $1$ $45.515$ \(\Q\) None \(0\) \(-1\) \(0\) \(1\) $-$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}+q^{7}+q^{9}-4q^{11}-4q^{17}+\cdots\)
5700.2.a.d 5700.a 1.a $1$ $45.515$ \(\Q\) None \(0\) \(-1\) \(0\) \(1\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+q^{7}+q^{9}-4q^{11}+6q^{17}+\cdots\)
5700.2.a.e 5700.a 1.a $1$ $45.515$ \(\Q\) None \(0\) \(-1\) \(0\) \(1\) $-$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}+q^{7}+q^{9}+4q^{13}+q^{19}-q^{21}+\cdots\)
5700.2.a.f 5700.a 1.a $1$ $45.515$ \(\Q\) None \(0\) \(-1\) \(0\) \(2\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+2q^{7}+q^{9}+4q^{13}+2q^{17}+\cdots\)
5700.2.a.g 5700.a 1.a $1$ $45.515$ \(\Q\) None \(0\) \(-1\) \(0\) \(3\) $-$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+3q^{7}+q^{9}-2q^{11}+2q^{13}+\cdots\)
5700.2.a.h 5700.a 1.a $1$ $45.515$ \(\Q\) None \(0\) \(-1\) \(0\) \(4\) $-$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}+4q^{7}+q^{9}+2q^{11}-6q^{13}+\cdots\)
5700.2.a.i 5700.a 1.a $1$ $45.515$ \(\Q\) None \(0\) \(-1\) \(0\) \(5\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+5q^{7}+q^{9}+2q^{11}+2q^{13}+\cdots\)
5700.2.a.j 5700.a 1.a $1$ $45.515$ \(\Q\) None \(0\) \(1\) \(0\) \(-5\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}-5q^{7}+q^{9}+2q^{11}-2q^{13}+\cdots\)
5700.2.a.k 5700.a 1.a $1$ $45.515$ \(\Q\) None \(0\) \(1\) \(0\) \(-3\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}-3q^{7}+q^{9}-2q^{11}-2q^{13}+\cdots\)
5700.2.a.l 5700.a 1.a $1$ $45.515$ \(\Q\) None \(0\) \(1\) \(0\) \(-1\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}-q^{7}+q^{9}-5q^{11}+6q^{13}+\cdots\)
5700.2.a.m 5700.a 1.a $1$ $45.515$ \(\Q\) None \(0\) \(1\) \(0\) \(-1\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}-q^{7}+q^{9}-4q^{11}-6q^{17}+\cdots\)
5700.2.a.n 5700.a 1.a $1$ $45.515$ \(\Q\) None \(0\) \(1\) \(0\) \(-1\) $-$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}-q^{7}+q^{9}-4q^{11}+4q^{17}+\cdots\)
5700.2.a.o 5700.a 1.a $1$ $45.515$ \(\Q\) None \(0\) \(1\) \(0\) \(-1\) $-$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}-q^{7}+q^{9}-4q^{13}+q^{19}-q^{21}+\cdots\)
5700.2.a.p 5700.a 1.a $1$ $45.515$ \(\Q\) None \(0\) \(1\) \(0\) \(0\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}+q^{9}+2q^{11}-2q^{13}-6q^{17}+\cdots\)
5700.2.a.q 5700.a 1.a $1$ $45.515$ \(\Q\) None \(0\) \(1\) \(0\) \(1\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}+q^{7}+q^{9}-2q^{17}-q^{19}+q^{21}+\cdots\)
5700.2.a.r 5700.a 1.a $1$ $45.515$ \(\Q\) None \(0\) \(1\) \(0\) \(2\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+2q^{7}+q^{9}+4q^{11}+2q^{17}+\cdots\)
5700.2.a.s 5700.a 1.a $2$ $45.515$ \(\Q(\sqrt{10}) \) None \(0\) \(-2\) \(0\) \(-8\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}-4q^{7}+q^{9}+(1+\beta )q^{11}+2\beta q^{13}+\cdots\)
5700.2.a.t 5700.a 1.a $2$ $45.515$ \(\Q(\sqrt{33}) \) None \(0\) \(-2\) \(0\) \(-1\) $-$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}-\beta q^{7}+q^{9}+(-2+\beta )q^{11}+\cdots\)
5700.2.a.u 5700.a 1.a $2$ $45.515$ \(\Q(\sqrt{13}) \) None \(0\) \(2\) \(0\) \(-2\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}+(-1-\beta )q^{7}+q^{9}+(-1+\beta )q^{11}+\cdots\)
5700.2.a.v 5700.a 1.a $2$ $45.515$ \(\Q(\sqrt{10}) \) None \(0\) \(2\) \(0\) \(8\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}+4q^{7}+q^{9}+(1+\beta )q^{11}-2\beta q^{13}+\cdots\)
5700.2.a.w 5700.a 1.a $3$ $45.515$ 3.3.564.1 None \(0\) \(-3\) \(0\) \(-4\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+(-1+\beta _{2})q^{7}+q^{9}+(2+\beta _{1}+\cdots)q^{11}+\cdots\)
5700.2.a.x 5700.a 1.a $3$ $45.515$ 3.3.148.1 None \(0\) \(-3\) \(0\) \(0\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}-\beta _{2}q^{7}+q^{9}+(-1-\beta _{1})q^{11}+\cdots\)
5700.2.a.y 5700.a 1.a $3$ $45.515$ 3.3.148.1 None \(0\) \(3\) \(0\) \(0\) $-$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+\beta _{2}q^{7}+q^{9}+(-1-\beta _{1})q^{11}+\cdots\)
5700.2.a.z 5700.a 1.a $3$ $45.515$ 3.3.1524.1 None \(0\) \(3\) \(0\) \(0\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+\beta _{1}q^{7}+q^{9}+(-1-\beta _{2})q^{11}+\cdots\)
5700.2.a.ba 5700.a 1.a $4$ $45.515$ \(\Q(\sqrt{6}, \sqrt{22})\) None \(0\) \(-4\) \(0\) \(-2\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}+(-1-\beta _{2})q^{7}+q^{9}+(-\beta _{1}+\cdots)q^{11}+\cdots\)
5700.2.a.bb 5700.a 1.a $4$ $45.515$ \(\Q(\sqrt{6}, \sqrt{22})\) None \(0\) \(4\) \(0\) \(2\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+(1+\beta _{2})q^{7}+q^{9}+(\beta _{1}-\beta _{2}+\cdots)q^{11}+\cdots\)
5700.2.a.bc 5700.a 1.a $5$ $45.515$ 5.5.13090800.1 None \(0\) \(-5\) \(0\) \(0\) $-$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}-\beta _{1}q^{7}+q^{9}+(1-\beta _{3})q^{11}+\cdots\)
5700.2.a.bd 5700.a 1.a $5$ $45.515$ 5.5.13090800.1 None \(0\) \(5\) \(0\) \(0\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}+\beta _{1}q^{7}+q^{9}+(1-\beta _{3})q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(5700))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(5700)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(19))\)\(^{\oplus 18}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(38))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(57))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(76))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(95))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(100))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(114))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(150))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(190))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(228))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(285))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(300))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(380))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(475))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(570))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(950))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1140))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1425))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1900))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2850))\)\(^{\oplus 2}\)