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Properties

Label 3700.2.a
Level $3700$
Weight $2$
Character orbit 3700.a
Rep. character $\chi_{3700}(1,\cdot)$
Character field $\Q$
Dimension $57$
Newform subspaces $16$
Sturm bound $1140$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 3700 = 2^{2} \cdot 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3700.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 16 \)
Sturm bound: \(1140\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(3\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(3700))\).

Total New Old
Modular forms 588 57 531
Cusp forms 553 57 496
Eisenstein series 35 0 35

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

\(2\)\(5\)\(37\)FrickeDim
\(-\)\(+\)\(+\)\(-\)\(15\)
\(-\)\(+\)\(-\)\(+\)\(12\)
\(-\)\(-\)\(+\)\(+\)\(14\)
\(-\)\(-\)\(-\)\(-\)\(16\)
Plus space\(+\)\(26\)
Minus space\(-\)\(31\)

Trace form

\( 57 q - 2 q^{3} + 2 q^{7} + 55 q^{9} + O(q^{10}) \) \( 57 q - 2 q^{3} + 2 q^{7} + 55 q^{9} - 6 q^{11} - 4 q^{13} - 16 q^{17} - 8 q^{19} - 10 q^{21} + 6 q^{23} - 14 q^{27} - 2 q^{29} + 2 q^{31} - 10 q^{33} - q^{37} + 2 q^{39} + 4 q^{41} - 32 q^{43} - 14 q^{47} + 63 q^{49} - 30 q^{51} - 12 q^{53} + 2 q^{57} - 14 q^{59} + 2 q^{61} + 20 q^{63} + 16 q^{67} + 4 q^{69} + 6 q^{71} - 4 q^{73} + 14 q^{77} - 12 q^{79} + 57 q^{81} - 6 q^{83} + 8 q^{87} - 4 q^{89} - 22 q^{91} + 12 q^{93} - 2 q^{97} - 28 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(3700))\) 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 5 37
3700.2.a.a 3700.a 1.a $1$ $29.545$ \(\Q\) None 740.2.a.c \(0\) \(-3\) \(0\) \(3\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-3q^{3}+3q^{7}+6q^{9}+5q^{11}-2q^{13}+\cdots\)
3700.2.a.b 3700.a 1.a $1$ $29.545$ \(\Q\) None 3700.2.a.b \(0\) \(-2\) \(0\) \(2\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{3}+2q^{7}+q^{9}+6q^{11}-4q^{13}+\cdots\)
3700.2.a.c 3700.a 1.a $1$ $29.545$ \(\Q\) None 740.2.a.b \(0\) \(-1\) \(0\) \(1\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+q^{7}-2q^{9}-3q^{11}+4q^{13}+\cdots\)
3700.2.a.d 3700.a 1.a $1$ $29.545$ \(\Q\) None 740.2.a.a \(0\) \(1\) \(0\) \(-1\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}-q^{7}-2q^{9}-3q^{11}+6q^{13}+\cdots\)
3700.2.a.e 3700.a 1.a $1$ $29.545$ \(\Q\) None 148.2.a.a \(0\) \(1\) \(0\) \(3\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}+3q^{7}-2q^{9}+5q^{11}+6q^{17}+\cdots\)
3700.2.a.f 3700.a 1.a $1$ $29.545$ \(\Q\) None 3700.2.a.b \(0\) \(2\) \(0\) \(-2\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{3}-2q^{7}+q^{9}+6q^{11}+4q^{13}+\cdots\)
3700.2.a.g 3700.a 1.a $2$ $29.545$ \(\Q(\sqrt{17}) \) None 148.2.a.b \(0\) \(1\) \(0\) \(-1\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{3}-\beta q^{7}+(1+\beta )q^{9}+\beta q^{11}+\cdots\)
3700.2.a.h 3700.a 1.a $2$ $29.545$ \(\Q(\sqrt{3}) \) None 740.2.a.d \(0\) \(2\) \(0\) \(-6\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{3}+(-3+\beta )q^{7}+(1+2\beta )q^{9}+\cdots\)
3700.2.a.i 3700.a 1.a $3$ $29.545$ 3.3.148.1 None 740.2.a.e \(0\) \(0\) \(0\) \(-2\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{2}q^{3}+(-2\beta _{1}+\beta _{2})q^{7}+(-\beta _{1}+\cdots)q^{9}+\cdots\)
3700.2.a.j 3700.a 1.a $4$ $29.545$ 4.4.286164.1 None 740.2.a.f \(0\) \(-3\) \(0\) \(5\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{3}+(1+\beta _{1})q^{7}+(3+\beta _{2}+\cdots)q^{9}+\cdots\)
3700.2.a.k 3700.a 1.a $5$ $29.545$ 5.5.454057.1 None 3700.2.a.k \(0\) \(-1\) \(0\) \(0\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{3}+(\beta _{1}+\beta _{2})q^{7}+(\beta _{1}+\beta _{2})q^{9}+\cdots\)
3700.2.a.l 3700.a 1.a $5$ $29.545$ 5.5.454057.1 None 3700.2.a.k \(0\) \(1\) \(0\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{3}+(-\beta _{1}-\beta _{2})q^{7}+(\beta _{1}+\beta _{2}+\cdots)q^{9}+\cdots\)
3700.2.a.m 3700.a 1.a $6$ $29.545$ 6.6.17268201.1 None 3700.2.a.m \(0\) \(-1\) \(0\) \(2\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{2}q^{3}+\beta _{3}q^{7}+(1+\beta _{1}+\beta _{3}+\beta _{5})q^{9}+\cdots\)
3700.2.a.n 3700.a 1.a $6$ $29.545$ 6.6.17268201.1 None 3700.2.a.m \(0\) \(1\) \(0\) \(-2\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{2}q^{3}-\beta _{3}q^{7}+(1+\beta _{1}+\beta _{3}+\beta _{5})q^{9}+\cdots\)
3700.2.a.o 3700.a 1.a $9$ $29.545$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None 740.2.d.a \(0\) \(0\) \(0\) \(-4\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{5}q^{3}+\beta _{4}q^{7}+(\beta _{1}+\beta _{3}-\beta _{4})q^{9}+\cdots\)
3700.2.a.p 3700.a 1.a $9$ $29.545$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None 740.2.d.a \(0\) \(0\) \(0\) \(4\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{5}q^{3}-\beta _{4}q^{7}+(\beta _{1}+\beta _{3}-\beta _{4})q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(3700))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(3700)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(37))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(74))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(100))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(148))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(185))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(370))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(740))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(925))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1850))\)\(^{\oplus 2}\)