Properties

Label 1550.2.a
Level $1550$
Weight $2$
Character orbit 1550.a
Rep. character $\chi_{1550}(1,\cdot)$
Character field $\Q$
Dimension $48$
Newform subspaces $19$
Sturm bound $480$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 1550 = 2 \cdot 5^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1550.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 19 \)
Sturm bound: \(480\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(3\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1550))\).

Total New Old
Modular forms 252 48 204
Cusp forms 229 48 181
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.

\(2\)\(5\)\(31\)FrickeDim
\(+\)\(+\)\(+\)\(+\)\(3\)
\(+\)\(+\)\(-\)\(-\)\(9\)
\(+\)\(-\)\(+\)\(-\)\(8\)
\(+\)\(-\)\(-\)\(+\)\(4\)
\(-\)\(+\)\(+\)\(-\)\(8\)
\(-\)\(+\)\(-\)\(+\)\(2\)
\(-\)\(-\)\(+\)\(+\)\(5\)
\(-\)\(-\)\(-\)\(-\)\(9\)
Plus space\(+\)\(14\)
Minus space\(-\)\(34\)

Trace form

\( 48 q - 2 q^{3} + 48 q^{4} + 2 q^{6} + 4 q^{7} + 60 q^{9} + O(q^{10}) \) \( 48 q - 2 q^{3} + 48 q^{4} + 2 q^{6} + 4 q^{7} + 60 q^{9} + 14 q^{11} - 2 q^{12} + 14 q^{13} + 12 q^{14} + 48 q^{16} + 12 q^{17} + 8 q^{18} + 4 q^{19} + 28 q^{21} - 6 q^{22} + 8 q^{23} + 2 q^{24} - 10 q^{26} - 8 q^{27} + 4 q^{28} - 18 q^{29} + 8 q^{33} - 24 q^{34} + 60 q^{36} - 6 q^{37} - 8 q^{38} + 28 q^{39} + 4 q^{41} - 4 q^{42} + 18 q^{43} + 14 q^{44} + 16 q^{46} - 36 q^{47} - 2 q^{48} + 88 q^{49} + 16 q^{51} + 14 q^{52} + 30 q^{53} - 28 q^{54} + 12 q^{56} + 16 q^{57} + 6 q^{58} - 20 q^{59} + 42 q^{61} - 4 q^{62} + 20 q^{63} + 48 q^{64} - 12 q^{66} + 16 q^{67} + 12 q^{68} + 8 q^{71} + 8 q^{72} + 24 q^{73} - 6 q^{74} + 4 q^{76} + 4 q^{77} + 12 q^{78} + 28 q^{79} + 144 q^{81} + 8 q^{82} - 38 q^{83} + 28 q^{84} - 22 q^{86} - 4 q^{87} - 6 q^{88} + 12 q^{89} + 76 q^{91} + 8 q^{92} - 2 q^{93} + 44 q^{94} + 2 q^{96} - 8 q^{98} - 2 q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1550))\) 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 31
1550.2.a.a 1550.a 1.a $1$ $12.377$ \(\Q\) None 310.2.a.b \(-1\) \(-2\) \(0\) \(0\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-2q^{3}+q^{4}+2q^{6}-q^{8}+q^{9}+\cdots\)
1550.2.a.b 1550.a 1.a $1$ $12.377$ \(\Q\) None 1550.2.a.b \(-1\) \(0\) \(0\) \(-5\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}-5q^{7}-q^{8}-3q^{9}+5q^{11}+\cdots\)
1550.2.a.c 1550.a 1.a $1$ $12.377$ \(\Q\) None 62.2.a.a \(-1\) \(0\) \(0\) \(0\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}-q^{8}-3q^{9}-2q^{13}+q^{16}+\cdots\)
1550.2.a.d 1550.a 1.a $1$ $12.377$ \(\Q\) None 1550.2.a.d \(-1\) \(1\) \(0\) \(0\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}-q^{6}-q^{8}-2q^{9}+\cdots\)
1550.2.a.e 1550.a 1.a $1$ $12.377$ \(\Q\) None 310.2.a.a \(-1\) \(2\) \(0\) \(4\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+2q^{3}+q^{4}-2q^{6}+4q^{7}-q^{8}+\cdots\)
1550.2.a.f 1550.a 1.a $1$ $12.377$ \(\Q\) None 1550.2.a.d \(1\) \(-1\) \(0\) \(0\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}-q^{6}+q^{8}-2q^{9}+\cdots\)
1550.2.a.g 1550.a 1.a $1$ $12.377$ \(\Q\) None 1550.2.a.b \(1\) \(0\) \(0\) \(5\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}+5q^{7}+q^{8}-3q^{9}+5q^{11}+\cdots\)
1550.2.a.h 1550.a 1.a $2$ $12.377$ \(\Q(\sqrt{3}) \) None 62.2.a.b \(2\) \(-2\) \(0\) \(-4\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+(-1+\beta )q^{3}+q^{4}+(-1+\beta )q^{6}+\cdots\)
1550.2.a.i 1550.a 1.a $2$ $12.377$ \(\Q(\sqrt{6}) \) None 310.2.a.d \(2\) \(0\) \(0\) \(4\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+\beta q^{3}+q^{4}+\beta q^{6}+2q^{7}+q^{8}+\cdots\)
1550.2.a.j 1550.a 1.a $2$ $12.377$ \(\Q(\sqrt{3}) \) None 310.2.a.c \(2\) \(2\) \(0\) \(0\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+(1+\beta )q^{3}+q^{4}+(1+\beta )q^{6}+\cdots\)
1550.2.a.k 1550.a 1.a $3$ $12.377$ 3.3.148.1 None 310.2.a.e \(-3\) \(-2\) \(0\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+(-1+\beta _{1})q^{3}+q^{4}+(1-\beta _{1}+\cdots)q^{6}+\cdots\)
1550.2.a.l 1550.a 1.a $3$ $12.377$ 3.3.564.1 None 1550.2.a.l \(-3\) \(-2\) \(0\) \(1\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+(-1-\beta _{2})q^{3}+q^{4}+(1+\beta _{2})q^{6}+\cdots\)
1550.2.a.m 1550.a 1.a $3$ $12.377$ 3.3.564.1 None 1550.2.a.l \(3\) \(2\) \(0\) \(-1\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+(1+\beta _{2})q^{3}+q^{4}+(1+\beta _{2})q^{6}+\cdots\)
1550.2.a.n 1550.a 1.a $4$ $12.377$ 4.4.11344.1 None 310.2.b.b \(-4\) \(-2\) \(0\) \(-6\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+(-1+\beta _{1})q^{3}+q^{4}+(1-\beta _{1}+\cdots)q^{6}+\cdots\)
1550.2.a.o 1550.a 1.a $4$ $12.377$ 4.4.6224.1 None 310.2.b.a \(-4\) \(4\) \(0\) \(6\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+(1+\beta _{1})q^{3}+q^{4}+(-1-\beta _{1}+\cdots)q^{6}+\cdots\)
1550.2.a.p 1550.a 1.a $4$ $12.377$ 4.4.6224.1 None 310.2.b.a \(4\) \(-4\) \(0\) \(-6\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+(-1-\beta _{1})q^{3}+q^{4}+(-1-\beta _{1}+\cdots)q^{6}+\cdots\)
1550.2.a.q 1550.a 1.a $4$ $12.377$ 4.4.11344.1 None 310.2.b.b \(4\) \(2\) \(0\) \(6\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+(1-\beta _{1})q^{3}+q^{4}+(1-\beta _{1})q^{6}+\cdots\)
1550.2.a.r 1550.a 1.a $5$ $12.377$ 5.5.12203472.1 None 1550.2.a.r \(-5\) \(-1\) \(0\) \(-4\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-\beta _{1}q^{3}+q^{4}+\beta _{1}q^{6}+(-1+\cdots)q^{7}+\cdots\)
1550.2.a.s 1550.a 1.a $5$ $12.377$ 5.5.12203472.1 None 1550.2.a.r \(5\) \(1\) \(0\) \(4\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+\beta _{1}q^{3}+q^{4}+\beta _{1}q^{6}+(1-\beta _{4})q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1550))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1550)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(31))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(62))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(155))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(310))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(775))\)\(^{\oplus 2}\)