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Properties

Label 850.2.a
Level $850$
Weight $2$
Character orbit 850.a
Rep. character $\chi_{850}(1,\cdot)$
Character field $\Q$
Dimension $24$
Newform subspaces $17$
Sturm bound $270$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 850 = 2 \cdot 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 850.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 17 \)
Sturm bound: \(270\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(3\), \(7\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(850))\).

Total New Old
Modular forms 146 24 122
Cusp forms 123 24 99
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.

\(2\)\(5\)\(17\)FrickeDim
\(+\)\(+\)\(+\)$+$\(4\)
\(+\)\(+\)\(-\)$-$\(2\)
\(+\)\(-\)\(+\)$-$\(3\)
\(+\)\(-\)\(-\)$+$\(3\)
\(-\)\(+\)\(+\)$-$\(5\)
\(-\)\(+\)\(-\)$+$\(1\)
\(-\)\(-\)\(+\)$+$\(1\)
\(-\)\(-\)\(-\)$-$\(5\)
Plus space\(+\)\(9\)
Minus space\(-\)\(15\)

Trace form

\( 24 q + 2 q^{3} + 24 q^{4} + 6 q^{6} - 4 q^{7} + 28 q^{9} + O(q^{10}) \) \( 24 q + 2 q^{3} + 24 q^{4} + 6 q^{6} - 4 q^{7} + 28 q^{9} + 6 q^{11} + 2 q^{12} - 4 q^{13} + 12 q^{14} + 24 q^{16} - 2 q^{17} + 4 q^{18} - 16 q^{19} - 16 q^{21} + 2 q^{22} + 16 q^{23} + 6 q^{24} - 4 q^{26} - 4 q^{27} - 4 q^{28} + 6 q^{29} - 12 q^{31} + 28 q^{33} + 2 q^{34} + 28 q^{36} + 6 q^{37} + 8 q^{38} - 12 q^{39} + 20 q^{41} + 16 q^{42} - 4 q^{43} + 6 q^{44} + 16 q^{46} + 2 q^{48} + 8 q^{49} + 2 q^{51} - 4 q^{52} + 28 q^{53} + 36 q^{54} + 12 q^{56} - 24 q^{57} + 2 q^{58} + 4 q^{59} - 22 q^{61} + 4 q^{62} - 20 q^{63} + 24 q^{64} + 4 q^{66} - 28 q^{67} - 2 q^{68} - 72 q^{69} - 80 q^{71} + 4 q^{72} - 40 q^{73} + 18 q^{74} - 16 q^{76} + 24 q^{77} + 20 q^{78} - 16 q^{79} + 20 q^{81} - 20 q^{82} - 12 q^{83} - 16 q^{84} + 4 q^{86} - 40 q^{87} + 2 q^{88} - 56 q^{89} - 56 q^{91} + 16 q^{92} - 16 q^{93} + 24 q^{94} + 6 q^{96} - 4 q^{97} - 40 q^{98} - 34 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(850))\) 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 17
850.2.a.a 850.a 1.a $1$ $6.787$ \(\Q\) None \(-1\) \(-3\) \(0\) \(-1\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-3q^{3}+q^{4}+3q^{6}-q^{7}-q^{8}+\cdots\)
850.2.a.b 850.a 1.a $1$ $6.787$ \(\Q\) None \(-1\) \(-1\) \(0\) \(-2\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+q^{6}-2q^{7}-q^{8}+\cdots\)
850.2.a.c 850.a 1.a $1$ $6.787$ \(\Q\) None \(-1\) \(1\) \(0\) \(-5\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}-q^{6}-5q^{7}-q^{8}+\cdots\)
850.2.a.d 850.a 1.a $1$ $6.787$ \(\Q\) None \(-1\) \(1\) \(0\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}-q^{6}-q^{8}-2q^{9}+\cdots\)
850.2.a.e 850.a 1.a $1$ $6.787$ \(\Q\) None \(-1\) \(2\) \(0\) \(4\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+2q^{3}+q^{4}-2q^{6}+4q^{7}-q^{8}+\cdots\)
850.2.a.f 850.a 1.a $1$ $6.787$ \(\Q\) None \(1\) \(-3\) \(0\) \(-2\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-3q^{3}+q^{4}-3q^{6}-2q^{7}+q^{8}+\cdots\)
850.2.a.g 850.a 1.a $1$ $6.787$ \(\Q\) None \(1\) \(-1\) \(0\) \(-2\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}-q^{6}-2q^{7}+q^{8}+\cdots\)
850.2.a.h 850.a 1.a $1$ $6.787$ \(\Q\) None \(1\) \(-1\) \(0\) \(0\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}-q^{6}+q^{8}-2q^{9}+\cdots\)
850.2.a.i 850.a 1.a $1$ $6.787$ \(\Q\) None \(1\) \(-1\) \(0\) \(5\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}-q^{6}+5q^{7}+q^{8}+\cdots\)
850.2.a.j 850.a 1.a $1$ $6.787$ \(\Q\) None \(1\) \(2\) \(0\) \(-2\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+2q^{3}+q^{4}+2q^{6}-2q^{7}+q^{8}+\cdots\)
850.2.a.k 850.a 1.a $1$ $6.787$ \(\Q\) None \(1\) \(2\) \(0\) \(2\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+2q^{3}+q^{4}+2q^{6}+2q^{7}+q^{8}+\cdots\)
850.2.a.l 850.a 1.a $1$ $6.787$ \(\Q\) None \(1\) \(3\) \(0\) \(1\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+3q^{3}+q^{4}+3q^{6}+q^{7}+q^{8}+\cdots\)
850.2.a.m 850.a 1.a $2$ $6.787$ \(\Q(\sqrt{2}) \) None \(-2\) \(-2\) \(0\) \(-2\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+(-1+\beta )q^{3}+q^{4}+(1-\beta )q^{6}+\cdots\)
850.2.a.n 850.a 1.a $2$ $6.787$ \(\Q(\sqrt{17}) \) None \(-2\) \(1\) \(0\) \(-2\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+\beta q^{3}+q^{4}-\beta q^{6}-2\beta q^{7}+\cdots\)
850.2.a.o 850.a 1.a $2$ $6.787$ \(\Q(\sqrt{2}) \) None \(2\) \(2\) \(0\) \(2\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+(1+\beta )q^{3}+q^{4}+(1+\beta )q^{6}+\cdots\)
850.2.a.p 850.a 1.a $3$ $6.787$ 3.3.568.1 None \(-3\) \(-1\) \(0\) \(0\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+\beta _{2}q^{3}+q^{4}-\beta _{2}q^{6}+(-\beta _{1}+\cdots)q^{7}+\cdots\)
850.2.a.q 850.a 1.a $3$ $6.787$ 3.3.568.1 None \(3\) \(1\) \(0\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-\beta _{2}q^{3}+q^{4}-\beta _{2}q^{6}+(\beta _{1}+\beta _{2})q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(850))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(850)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(17))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(34))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(85))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(170))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(425))\)\(^{\oplus 2}\)