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Properties

Label 1650.2.a
Level $1650$
Weight $2$
Character orbit 1650.a
Rep. character $\chi_{1650}(1,\cdot)$
Character field $\Q$
Dimension $32$
Newform subspaces $26$
Sturm bound $720$
Trace bound $13$

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Defining parameters

Level: \( N \) \(=\) \( 1650 = 2 \cdot 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1650.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 26 \)
Sturm bound: \(720\)
Trace bound: \(13\)
Distinguishing \(T_p\): \(7\), \(13\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1650))\).

Total New Old
Modular forms 384 32 352
Cusp forms 337 32 305
Eisenstein series 47 0 47

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

\(2\)\(3\)\(5\)\(11\)FrickeDim
\(+\)\(+\)\(+\)\(+\)$+$\(3\)
\(+\)\(+\)\(+\)\(-\)$-$\(1\)
\(+\)\(+\)\(-\)\(+\)$-$\(2\)
\(+\)\(+\)\(-\)\(-\)$+$\(2\)
\(+\)\(-\)\(+\)\(+\)$-$\(3\)
\(+\)\(-\)\(+\)\(-\)$+$\(2\)
\(+\)\(-\)\(-\)\(+\)$+$\(1\)
\(+\)\(-\)\(-\)\(-\)$-$\(3\)
\(-\)\(+\)\(+\)\(+\)$-$\(2\)
\(-\)\(+\)\(+\)\(-\)$+$\(1\)
\(-\)\(+\)\(-\)\(+\)$+$\(1\)
\(-\)\(+\)\(-\)\(-\)$-$\(3\)
\(-\)\(-\)\(+\)\(-\)$-$\(4\)
\(-\)\(-\)\(-\)\(+\)$-$\(4\)
Plus space\(+\)\(10\)
Minus space\(-\)\(22\)

Trace form

\( 32 q - 2 q^{2} + 2 q^{3} + 32 q^{4} + 4 q^{7} - 2 q^{8} + 32 q^{9} + O(q^{10}) \) \( 32 q - 2 q^{2} + 2 q^{3} + 32 q^{4} + 4 q^{7} - 2 q^{8} + 32 q^{9} + 2 q^{12} + 32 q^{16} + 4 q^{17} - 2 q^{18} + 28 q^{19} + 20 q^{21} + 2 q^{22} - 4 q^{23} + 32 q^{26} + 2 q^{27} + 4 q^{28} + 4 q^{29} + 8 q^{31} - 2 q^{32} + 2 q^{33} + 24 q^{34} + 32 q^{36} + 24 q^{37} - 4 q^{38} + 28 q^{39} + 28 q^{41} + 8 q^{42} - 4 q^{43} + 16 q^{46} + 28 q^{47} + 2 q^{48} + 48 q^{49} + 8 q^{51} - 12 q^{53} + 12 q^{57} - 24 q^{59} + 40 q^{61} + 16 q^{62} + 4 q^{63} + 32 q^{64} - 4 q^{66} + 40 q^{67} + 4 q^{68} + 4 q^{69} - 28 q^{71} - 2 q^{72} - 8 q^{73} + 4 q^{74} + 28 q^{76} + 8 q^{77} - 4 q^{78} - 20 q^{79} + 32 q^{81} + 8 q^{82} + 20 q^{84} + 12 q^{86} - 8 q^{87} + 2 q^{88} - 24 q^{89} - 8 q^{91} - 4 q^{92} + 8 q^{94} - 8 q^{97} - 2 q^{98} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1650))\) 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 11
1650.2.a.a 1650.a 1.a $1$ $13.175$ \(\Q\) None \(-1\) \(-1\) \(0\) \(-3\) $+$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+q^{6}-3q^{7}-q^{8}+\cdots\)
1650.2.a.b 1650.a 1.a $1$ $13.175$ \(\Q\) None \(-1\) \(-1\) \(0\) \(0\) $+$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+q^{6}-q^{8}+q^{9}+\cdots\)
1650.2.a.c 1650.a 1.a $1$ $13.175$ \(\Q\) None \(-1\) \(-1\) \(0\) \(2\) $+$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+q^{6}+2q^{7}-q^{8}+\cdots\)
1650.2.a.d 1650.a 1.a $1$ $13.175$ \(\Q\) None \(-1\) \(-1\) \(0\) \(2\) $+$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+q^{6}+2q^{7}-q^{8}+\cdots\)
1650.2.a.e 1650.a 1.a $1$ $13.175$ \(\Q\) None \(-1\) \(1\) \(0\) \(-4\) $+$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}-q^{6}-4q^{7}-q^{8}+\cdots\)
1650.2.a.f 1650.a 1.a $1$ $13.175$ \(\Q\) None \(-1\) \(1\) \(0\) \(-3\) $+$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}-q^{6}-3q^{7}-q^{8}+\cdots\)
1650.2.a.g 1650.a 1.a $1$ $13.175$ \(\Q\) None \(-1\) \(1\) \(0\) \(-1\) $+$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}-q^{6}-q^{7}-q^{8}+\cdots\)
1650.2.a.h 1650.a 1.a $1$ $13.175$ \(\Q\) None \(-1\) \(1\) \(0\) \(0\) $+$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}-q^{6}-q^{8}+q^{9}+\cdots\)
1650.2.a.i 1650.a 1.a $1$ $13.175$ \(\Q\) None \(-1\) \(1\) \(0\) \(2\) $+$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}-q^{6}+2q^{7}-q^{8}+\cdots\)
1650.2.a.j 1650.a 1.a $1$ $13.175$ \(\Q\) None \(-1\) \(1\) \(0\) \(2\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}-q^{6}+2q^{7}-q^{8}+\cdots\)
1650.2.a.k 1650.a 1.a $1$ $13.175$ \(\Q\) None \(-1\) \(1\) \(0\) \(4\) $+$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}-q^{6}+4q^{7}-q^{8}+\cdots\)
1650.2.a.l 1650.a 1.a $1$ $13.175$ \(\Q\) None \(1\) \(-1\) \(0\) \(-2\) $-$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}-q^{6}-2q^{7}+q^{8}+\cdots\)
1650.2.a.m 1650.a 1.a $1$ $13.175$ \(\Q\) None \(1\) \(-1\) \(0\) \(-2\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}-q^{6}-2q^{7}+q^{8}+\cdots\)
1650.2.a.n 1650.a 1.a $1$ $13.175$ \(\Q\) None \(1\) \(-1\) \(0\) \(-2\) $-$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}-q^{6}-2q^{7}+q^{8}+\cdots\)
1650.2.a.o 1650.a 1.a $1$ $13.175$ \(\Q\) None \(1\) \(-1\) \(0\) \(1\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}-q^{6}+q^{7}+q^{8}+\cdots\)
1650.2.a.p 1650.a 1.a $1$ $13.175$ \(\Q\) None \(1\) \(-1\) \(0\) \(3\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}-q^{6}+3q^{7}+q^{8}+\cdots\)
1650.2.a.q 1650.a 1.a $1$ $13.175$ \(\Q\) None \(1\) \(1\) \(0\) \(-2\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}+q^{6}-2q^{7}+q^{8}+\cdots\)
1650.2.a.r 1650.a 1.a $1$ $13.175$ \(\Q\) None \(1\) \(1\) \(0\) \(0\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}+q^{6}+q^{8}+q^{9}+\cdots\)
1650.2.a.s 1650.a 1.a $1$ $13.175$ \(\Q\) None \(1\) \(1\) \(0\) \(3\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}+q^{6}+3q^{7}+q^{8}+\cdots\)
1650.2.a.t 1650.a 1.a $1$ $13.175$ \(\Q\) None \(1\) \(1\) \(0\) \(4\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}+q^{6}+4q^{7}+q^{8}+\cdots\)
1650.2.a.u 1650.a 1.a $2$ $13.175$ \(\Q(\sqrt{2}) \) None \(-2\) \(-2\) \(0\) \(-4\) $+$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+q^{6}+(-2+\beta )q^{7}+\cdots\)
1650.2.a.v 1650.a 1.a $2$ $13.175$ \(\Q(\sqrt{73}) \) None \(-2\) \(-2\) \(0\) \(1\) $+$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+q^{6}+\beta q^{7}-q^{8}+\cdots\)
1650.2.a.w 1650.a 1.a $2$ $13.175$ \(\Q(\sqrt{10}) \) None \(-2\) \(2\) \(0\) \(4\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}-q^{6}+(2+\beta )q^{7}+\cdots\)
1650.2.a.x 1650.a 1.a $2$ $13.175$ \(\Q(\sqrt{10}) \) None \(2\) \(-2\) \(0\) \(-4\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}-q^{6}+(-2+\beta )q^{7}+\cdots\)
1650.2.a.y 1650.a 1.a $2$ $13.175$ \(\Q(\sqrt{73}) \) None \(2\) \(2\) \(0\) \(-1\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}+q^{6}-\beta q^{7}+q^{8}+\cdots\)
1650.2.a.z 1650.a 1.a $2$ $13.175$ \(\Q(\sqrt{2}) \) None \(2\) \(2\) \(0\) \(4\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}+q^{6}+(2+\beta )q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1650))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1650)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(22))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(33))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(55))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(66))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(110))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(150))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(165))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(275))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(330))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(550))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(825))\)\(^{\oplus 2}\)