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Properties

Label 1850.2.a
Level $1850$
Weight $2$
Character orbit 1850.a
Rep. character $\chi_{1850}(1,\cdot)$
Character field $\Q$
Dimension $57$
Newform subspaces $31$
Sturm bound $570$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 1850 = 2 \cdot 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1850.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 31 \)
Sturm bound: \(570\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(3\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1850))\).

Total New Old
Modular forms 296 57 239
Cusp forms 273 57 216
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.

\(2\)\(5\)\(37\)FrickeDim.
\(+\)\(+\)\(+\)\(+\)\(7\)
\(+\)\(+\)\(-\)\(-\)\(8\)
\(+\)\(-\)\(+\)\(-\)\(9\)
\(+\)\(-\)\(-\)\(+\)\(5\)
\(-\)\(+\)\(+\)\(-\)\(8\)
\(-\)\(+\)\(-\)\(+\)\(4\)
\(-\)\(-\)\(+\)\(+\)\(5\)
\(-\)\(-\)\(-\)\(-\)\(11\)
Plus space\(+\)\(21\)
Minus space\(-\)\(36\)

Trace form

\( 57 q - q^{2} - 2 q^{3} + 57 q^{4} + 4 q^{6} + 8 q^{7} - q^{8} + 67 q^{9} + O(q^{10}) \) \( 57 q - q^{2} - 2 q^{3} + 57 q^{4} + 4 q^{6} + 8 q^{7} - q^{8} + 67 q^{9} + 6 q^{11} - 2 q^{12} + 2 q^{13} + 4 q^{14} + 57 q^{16} + 2 q^{17} - 5 q^{18} + 4 q^{19} + 20 q^{21} - 8 q^{22} + 12 q^{23} + 4 q^{24} - 12 q^{26} + 4 q^{27} + 8 q^{28} + 22 q^{29} - 4 q^{31} - q^{32} + 32 q^{33} - 6 q^{34} + 67 q^{36} - q^{37} + 44 q^{39} - 8 q^{41} - 8 q^{42} + 16 q^{43} + 6 q^{44} + 2 q^{46} + 4 q^{47} - 2 q^{48} + 81 q^{49} - 12 q^{51} + 2 q^{52} - 18 q^{53} + 4 q^{54} + 4 q^{56} - 4 q^{57} - 4 q^{58} - 44 q^{59} + 14 q^{61} + 6 q^{62} + 68 q^{63} + 57 q^{64} - 36 q^{66} - 2 q^{67} + 2 q^{68} + 16 q^{69} - 5 q^{72} + 20 q^{73} + 5 q^{74} + 4 q^{76} - 28 q^{77} + 38 q^{78} + 60 q^{79} + 81 q^{81} - 2 q^{82} + 24 q^{83} + 20 q^{84} - 24 q^{86} + 20 q^{87} - 8 q^{88} + 38 q^{89} + 56 q^{91} + 12 q^{92} + 24 q^{93} + 24 q^{94} + 4 q^{96} - 14 q^{97} + 7 q^{98} - 40 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1850))\) 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 37
1850.2.a.a 1850.a 1.a $1$ $14.772$ \(\Q\) None \(-1\) \(-2\) \(0\) \(-4\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-2q^{3}+q^{4}+2q^{6}-4q^{7}-q^{8}+\cdots\)
1850.2.a.b 1850.a 1.a $1$ $14.772$ \(\Q\) None \(-1\) \(-2\) \(0\) \(0\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-2q^{3}+q^{4}+2q^{6}-q^{8}+q^{9}+\cdots\)
1850.2.a.c 1850.a 1.a $1$ $14.772$ \(\Q\) None \(-1\) \(-1\) \(0\) \(4\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+q^{6}+4q^{7}-q^{8}+\cdots\)
1850.2.a.d 1850.a 1.a $1$ $14.772$ \(\Q\) None \(-1\) \(0\) \(0\) \(-1\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}-q^{7}-q^{8}-3q^{9}-3q^{11}+\cdots\)
1850.2.a.e 1850.a 1.a $1$ $14.772$ \(\Q\) None \(-1\) \(0\) \(0\) \(2\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}+2q^{7}-q^{8}-3q^{9}-2q^{13}+\cdots\)
1850.2.a.f 1850.a 1.a $1$ $14.772$ \(\Q\) None \(-1\) \(2\) \(0\) \(-2\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+2q^{3}+q^{4}-2q^{6}-2q^{7}-q^{8}+\cdots\)
1850.2.a.g 1850.a 1.a $1$ $14.772$ \(\Q\) None \(-1\) \(3\) \(0\) \(0\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+3q^{3}+q^{4}-3q^{6}-q^{8}+6q^{9}+\cdots\)
1850.2.a.h 1850.a 1.a $1$ $14.772$ \(\Q\) None \(1\) \(-3\) \(0\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-3q^{3}+q^{4}-3q^{6}+q^{8}+6q^{9}+\cdots\)
1850.2.a.i 1850.a 1.a $1$ $14.772$ \(\Q\) None \(1\) \(-2\) \(0\) \(-1\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-2q^{3}+q^{4}-2q^{6}-q^{7}+q^{8}+\cdots\)
1850.2.a.j 1850.a 1.a $1$ $14.772$ \(\Q\) None \(1\) \(0\) \(0\) \(-2\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}-2q^{7}+q^{8}-3q^{9}+2q^{13}+\cdots\)
1850.2.a.k 1850.a 1.a $1$ $14.772$ \(\Q\) None \(1\) \(0\) \(0\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}+q^{8}-3q^{9}-4q^{11}-2q^{13}+\cdots\)
1850.2.a.l 1850.a 1.a $1$ $14.772$ \(\Q\) None \(1\) \(0\) \(0\) \(1\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}+q^{7}+q^{8}-3q^{9}-3q^{11}+\cdots\)
1850.2.a.m 1850.a 1.a $1$ $14.772$ \(\Q\) None \(1\) \(1\) \(0\) \(-4\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}+q^{6}-4q^{7}+q^{8}+\cdots\)
1850.2.a.n 1850.a 1.a $1$ $14.772$ \(\Q\) None \(1\) \(2\) \(0\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+2q^{3}+q^{4}+2q^{6}+q^{8}+q^{9}+\cdots\)
1850.2.a.o 1850.a 1.a $1$ $14.772$ \(\Q\) None \(1\) \(2\) \(0\) \(1\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+2q^{3}+q^{4}+2q^{6}+q^{7}+q^{8}+\cdots\)
1850.2.a.p 1850.a 1.a $1$ $14.772$ \(\Q\) None \(1\) \(2\) \(0\) \(4\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+2q^{3}+q^{4}+2q^{6}+4q^{7}+q^{8}+\cdots\)
1850.2.a.q 1850.a 1.a $2$ $14.772$ \(\Q(\sqrt{33}) \) None \(-2\) \(-4\) \(0\) \(3\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-2q^{3}+q^{4}+2q^{6}+(1+\beta )q^{7}+\cdots\)
1850.2.a.r 1850.a 1.a $2$ $14.772$ \(\Q(\sqrt{6}) \) None \(-2\) \(-2\) \(0\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+(-1+\beta )q^{3}+q^{4}+(1-\beta )q^{6}+\cdots\)
1850.2.a.s 1850.a 1.a $2$ $14.772$ \(\Q(\sqrt{6}) \) None \(-2\) \(0\) \(0\) \(-4\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+\beta q^{3}+q^{4}-\beta q^{6}+(-2-\beta )q^{7}+\cdots\)
1850.2.a.t 1850.a 1.a $2$ $14.772$ \(\Q(\sqrt{5}) \) None \(-2\) \(1\) \(0\) \(2\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+\beta q^{3}+q^{4}-\beta q^{6}+2\beta q^{7}+\cdots\)
1850.2.a.u 1850.a 1.a $2$ $14.772$ \(\Q(\sqrt{13}) \) None \(2\) \(-3\) \(0\) \(-2\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+(-1-\beta )q^{3}+q^{4}+(-1-\beta )q^{6}+\cdots\)
1850.2.a.v 1850.a 1.a $2$ $14.772$ \(\Q(\sqrt{6}) \) None \(2\) \(0\) \(0\) \(4\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+\beta q^{3}+q^{4}+\beta q^{6}+(2-\beta )q^{7}+\cdots\)
1850.2.a.w 1850.a 1.a $2$ $14.772$ \(\Q(\sqrt{6}) \) None \(2\) \(2\) \(0\) \(0\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+(1+\beta )q^{3}+q^{4}+(1+\beta )q^{6}+\cdots\)
1850.2.a.x 1850.a 1.a $2$ $14.772$ \(\Q(\sqrt{3}) \) None \(2\) \(2\) \(0\) \(6\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+(1+\beta )q^{3}+q^{4}+(1+\beta )q^{6}+\cdots\)
1850.2.a.y 1850.a 1.a $3$ $14.772$ 3.3.1524.1 None \(-3\) \(-1\) \(0\) \(-4\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-\beta _{1}q^{3}+q^{4}+\beta _{1}q^{6}+(-1+\cdots)q^{7}+\cdots\)
1850.2.a.z 1850.a 1.a $3$ $14.772$ 3.3.892.1 None \(-3\) \(0\) \(0\) \(1\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+\beta _{2}q^{3}+q^{4}-\beta _{2}q^{6}+\beta _{1}q^{7}+\cdots\)
1850.2.a.ba 1850.a 1.a $3$ $14.772$ 3.3.148.1 None \(-3\) \(3\) \(0\) \(4\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+(1+\beta _{2})q^{3}+q^{4}+(-1-\beta _{2})q^{6}+\cdots\)
1850.2.a.bb 1850.a 1.a $3$ $14.772$ 3.3.148.1 None \(3\) \(-3\) \(0\) \(-4\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+(-1-\beta _{2})q^{3}+q^{4}+(-1-\beta _{2})q^{6}+\cdots\)
1850.2.a.bc 1850.a 1.a $3$ $14.772$ 3.3.1524.1 None \(3\) \(1\) \(0\) \(4\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+\beta _{1}q^{3}+q^{4}+\beta _{1}q^{6}+(1+\beta _{1}+\cdots)q^{7}+\cdots\)
1850.2.a.bd 1850.a 1.a $5$ $14.772$ 5.5.1791440.1 None \(-5\) \(0\) \(0\) \(1\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+\beta _{1}q^{3}+q^{4}-\beta _{1}q^{6}+(\beta _{1}+\beta _{2}+\cdots)q^{7}+\cdots\)
1850.2.a.be 1850.a 1.a $5$ $14.772$ 5.5.1791440.1 None \(5\) \(0\) \(0\) \(-1\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-\beta _{1}q^{3}+q^{4}-\beta _{1}q^{6}+(-\beta _{1}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1850))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1850)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(37))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(74))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(185))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(370))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(925))\)\(^{\oplus 2}\)