Properties

Label 550.2.a
Level $550$
Weight $2$
Character orbit 550.a
Rep. character $\chi_{550}(1,\cdot)$
Character field $\Q$
Dimension $15$
Newform subspaces $14$
Sturm bound $180$
Trace bound $13$

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Defining parameters

Level: \( N \) \(=\) \( 550 = 2 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 550.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 14 \)
Sturm bound: \(180\)
Trace bound: \(13\)
Distinguishing \(T_p\): \(3\), \(7\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(550))\).

Total New Old
Modular forms 102 15 87
Cusp forms 79 15 64
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.

\(2\)\(5\)\(11\)FrickeDim.
\(+\)\(+\)\(+\)\(+\)\(1\)
\(+\)\(+\)\(-\)\(-\)\(2\)
\(+\)\(-\)\(+\)\(-\)\(2\)
\(+\)\(-\)\(-\)\(+\)\(2\)
\(-\)\(+\)\(+\)\(-\)\(3\)
\(-\)\(+\)\(-\)\(+\)\(1\)
\(-\)\(-\)\(+\)\(+\)\(1\)
\(-\)\(-\)\(-\)\(-\)\(3\)
Plus space\(+\)\(5\)
Minus space\(-\)\(10\)

Trace form

\( 15 q + q^{2} + 15 q^{4} + 4 q^{6} - 8 q^{7} + q^{8} + 19 q^{9} + O(q^{10}) \) \( 15 q + q^{2} + 15 q^{4} + 4 q^{6} - 8 q^{7} + q^{8} + 19 q^{9} + q^{11} - 2 q^{13} + 8 q^{14} + 15 q^{16} + 10 q^{17} + 13 q^{18} - 28 q^{19} - q^{22} + 12 q^{23} + 4 q^{24} - 10 q^{26} + 24 q^{27} - 8 q^{28} + 2 q^{29} - 12 q^{31} + q^{32} - 10 q^{34} + 19 q^{36} - 14 q^{37} - 4 q^{38} - 16 q^{39} + 6 q^{41} - 8 q^{42} - 12 q^{43} + q^{44} - 4 q^{47} - q^{49} + 24 q^{51} - 2 q^{52} - 14 q^{53} - 8 q^{54} + 8 q^{56} - 2 q^{58} + 36 q^{59} - 30 q^{61} - 8 q^{62} - 8 q^{63} + 15 q^{64} + 4 q^{66} - 40 q^{67} + 10 q^{68} + 8 q^{69} - 20 q^{71} + 13 q^{72} + 2 q^{73} - 14 q^{74} - 28 q^{76} - 8 q^{77} - 8 q^{78} - 8 q^{79} - 17 q^{81} + 10 q^{82} + 12 q^{83} + 32 q^{87} - q^{88} - 14 q^{89} - 24 q^{91} + 12 q^{92} + 16 q^{93} + 8 q^{94} + 4 q^{96} + 22 q^{97} + 25 q^{98} - 27 q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(550))\) 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 5 11
550.2.a.a 550.a 1.a $1$ $4.392$ \(\Q\) None \(-1\) \(-2\) \(0\) \(-4\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-2q^{3}+q^{4}+2q^{6}-4q^{7}-q^{8}+\cdots\)
550.2.a.b 550.a 1.a $1$ $4.392$ \(\Q\) None \(-1\) \(-2\) \(0\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-2q^{3}+q^{4}+2q^{6}-q^{8}+q^{9}+\cdots\)
550.2.a.c 550.a 1.a $1$ $4.392$ \(\Q\) None \(-1\) \(-2\) \(0\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-2q^{3}+q^{4}+2q^{6}-q^{8}+q^{9}+\cdots\)
550.2.a.d 550.a 1.a $1$ $4.392$ \(\Q\) None \(-1\) \(-1\) \(0\) \(1\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+q^{6}+q^{7}-q^{8}+\cdots\)
550.2.a.e 550.a 1.a $1$ $4.392$ \(\Q\) None \(-1\) \(1\) \(0\) \(-3\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}-q^{6}-3q^{7}-q^{8}+\cdots\)
550.2.a.f 550.a 1.a $1$ $4.392$ \(\Q\) None \(-1\) \(1\) \(0\) \(-3\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}-q^{6}-3q^{7}-q^{8}+\cdots\)
550.2.a.g 550.a 1.a $1$ $4.392$ \(\Q\) None \(-1\) \(3\) \(0\) \(1\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+3q^{3}+q^{4}-3q^{6}+q^{7}-q^{8}+\cdots\)
550.2.a.h 550.a 1.a $1$ $4.392$ \(\Q\) None \(1\) \(-3\) \(0\) \(-1\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-3q^{3}+q^{4}-3q^{6}-q^{7}+q^{8}+\cdots\)
550.2.a.i 550.a 1.a $1$ $4.392$ \(\Q\) None \(1\) \(-1\) \(0\) \(-5\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}-q^{6}-5q^{7}+q^{8}+\cdots\)
550.2.a.j 550.a 1.a $1$ $4.392$ \(\Q\) None \(1\) \(-1\) \(0\) \(3\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}-q^{6}+3q^{7}+q^{8}+\cdots\)
550.2.a.k 550.a 1.a $1$ $4.392$ \(\Q\) None \(1\) \(2\) \(0\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+2q^{3}+q^{4}+2q^{6}+q^{8}+q^{9}+\cdots\)
550.2.a.l 550.a 1.a $1$ $4.392$ \(\Q\) None \(1\) \(2\) \(0\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+2q^{3}+q^{4}+2q^{6}+q^{8}+q^{9}+\cdots\)
550.2.a.m 550.a 1.a $1$ $4.392$ \(\Q\) None \(1\) \(2\) \(0\) \(4\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+2q^{3}+q^{4}+2q^{6}+4q^{7}+q^{8}+\cdots\)
550.2.a.n 550.a 1.a $2$ $4.392$ \(\Q(\sqrt{33}) \) None \(2\) \(1\) \(0\) \(-1\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+\beta q^{3}+q^{4}+\beta q^{6}-\beta q^{7}+q^{8}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(550))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(550)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(55))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(110))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(275))\)\(^{\oplus 2}\)