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Properties

Label 1740.2.a
Level $1740$
Weight $2$
Character orbit 1740.a
Rep. character $\chi_{1740}(1,\cdot)$
Character field $\Q$
Dimension $20$
Newform subspaces $14$
Sturm bound $720$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 1740 = 2^{2} \cdot 3 \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1740.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 14 \)
Sturm bound: \(720\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1740))\).

Total New Old
Modular forms 372 20 352
Cusp forms 349 20 329
Eisenstein series 23 0 23

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

\(2\)\(3\)\(5\)\(29\)FrickeDim
\(-\)\(+\)\(+\)\(+\)\(-\)\(2\)
\(-\)\(+\)\(+\)\(-\)\(+\)\(3\)
\(-\)\(+\)\(-\)\(+\)\(+\)\(3\)
\(-\)\(+\)\(-\)\(-\)\(-\)\(2\)
\(-\)\(-\)\(+\)\(+\)\(+\)\(1\)
\(-\)\(-\)\(+\)\(-\)\(-\)\(4\)
\(-\)\(-\)\(-\)\(+\)\(-\)\(4\)
\(-\)\(-\)\(-\)\(-\)\(+\)\(1\)
Plus space\(+\)\(8\)
Minus space\(-\)\(12\)

Trace form

\( 20 q + 20 q^{9} + O(q^{10}) \) \( 20 q + 20 q^{9} + 8 q^{13} + 8 q^{17} - 8 q^{19} + 8 q^{23} + 20 q^{25} + 8 q^{31} - 8 q^{35} + 8 q^{39} - 8 q^{41} + 8 q^{43} - 8 q^{47} + 28 q^{49} + 8 q^{51} + 8 q^{55} + 8 q^{57} - 24 q^{59} + 24 q^{61} + 8 q^{67} + 8 q^{69} - 32 q^{73} - 8 q^{77} + 16 q^{79} + 20 q^{81} + 32 q^{83} - 8 q^{85} - 24 q^{89} + 8 q^{91} + 24 q^{93} - 16 q^{95} + 40 q^{97} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1740))\) 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 3 5 29
1740.2.a.a 1740.a 1.a $1$ $13.894$ \(\Q\) None 1740.2.a.a \(0\) \(-1\) \(-1\) \(0\) $-$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}-q^{5}+q^{9}-6q^{11}+6q^{13}+\cdots\)
1740.2.a.b 1740.a 1.a $1$ $13.894$ \(\Q\) None 1740.2.a.b \(0\) \(-1\) \(1\) \(-2\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}+q^{5}-2q^{7}+q^{9}+3q^{11}-2q^{13}+\cdots\)
1740.2.a.c 1740.a 1.a $1$ $13.894$ \(\Q\) None 1740.2.a.c \(0\) \(-1\) \(1\) \(2\) $-$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+q^{5}+2q^{7}+q^{9}-4q^{11}+2q^{13}+\cdots\)
1740.2.a.d 1740.a 1.a $1$ $13.894$ \(\Q\) None 1740.2.a.d \(0\) \(-1\) \(1\) \(3\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}+q^{5}+3q^{7}+q^{9}+3q^{11}+3q^{13}+\cdots\)
1740.2.a.e 1740.a 1.a $1$ $13.894$ \(\Q\) None 1740.2.a.e \(0\) \(1\) \(-1\) \(-1\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}-q^{5}-q^{7}+q^{9}-3q^{11}-q^{13}+\cdots\)
1740.2.a.f 1740.a 1.a $1$ $13.894$ \(\Q\) None 1740.2.a.f \(0\) \(1\) \(1\) \(-5\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}+q^{5}-5q^{7}+q^{9}+5q^{11}-5q^{13}+\cdots\)
1740.2.a.g 1740.a 1.a $1$ $13.894$ \(\Q\) None 1740.2.a.g \(0\) \(1\) \(1\) \(-3\) $-$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+q^{5}-3q^{7}+q^{9}-3q^{11}+q^{13}+\cdots\)
1740.2.a.h 1740.a 1.a $1$ $13.894$ \(\Q\) None 1740.2.a.h \(0\) \(1\) \(1\) \(2\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}+q^{5}+2q^{7}+q^{9}+4q^{11}+2q^{13}+\cdots\)
1740.2.a.i 1740.a 1.a $2$ $13.894$ \(\Q(\sqrt{33}) \) None 1740.2.a.i \(0\) \(-2\) \(-2\) \(-1\) $-$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}-q^{5}-\beta q^{7}+q^{9}+(2+\beta )q^{11}+\cdots\)
1740.2.a.j 1740.a 1.a $2$ $13.894$ \(\Q(\sqrt{17}) \) None 1740.2.a.j \(0\) \(-2\) \(-2\) \(3\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}-q^{5}+(1+\beta )q^{7}+q^{9}+(-1+\cdots)q^{11}+\cdots\)
1740.2.a.k 1740.a 1.a $2$ $13.894$ \(\Q(\sqrt{17}) \) None 1740.2.a.k \(0\) \(-2\) \(2\) \(-5\) $-$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+q^{5}+(-2-\beta )q^{7}+q^{9}-\beta q^{11}+\cdots\)
1740.2.a.l 1740.a 1.a $2$ $13.894$ \(\Q(\sqrt{33}) \) None 1740.2.a.l \(0\) \(2\) \(-2\) \(1\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}-q^{5}+\beta q^{7}+q^{9}+(2-\beta )q^{11}+\cdots\)
1740.2.a.m 1740.a 1.a $2$ $13.894$ \(\Q(\sqrt{17}) \) None 1740.2.a.m \(0\) \(2\) \(-2\) \(2\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}-q^{5}+2\beta q^{7}+q^{9}+(-2+\beta )q^{11}+\cdots\)
1740.2.a.n 1740.a 1.a $2$ $13.894$ \(\Q(\sqrt{57}) \) None 1740.2.a.n \(0\) \(2\) \(2\) \(4\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}+q^{5}+2q^{7}+q^{9}+(-2+\beta )q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1740))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1740)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(29))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(58))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(60))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(87))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(116))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(145))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(174))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(290))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(348))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(435))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(580))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(870))\)\(^{\oplus 2}\)