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

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

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1850))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 5 37
1850.2.a.a \(1\) \(14.772\) \(\Q\) None \(-1\) \(-2\) \(0\) \(-4\) \(+\) \(-\) \(-\) \(q-q^{2}-2q^{3}+q^{4}+2q^{6}-4q^{7}-q^{8}+\cdots\)
1850.2.a.b \(1\) \(14.772\) \(\Q\) None \(-1\) \(-2\) \(0\) \(0\) \(+\) \(+\) \(+\) \(q-q^{2}-2q^{3}+q^{4}+2q^{6}-q^{8}+q^{9}+\cdots\)
1850.2.a.c \(1\) \(14.772\) \(\Q\) None \(-1\) \(-1\) \(0\) \(4\) \(+\) \(-\) \(+\) \(q-q^{2}-q^{3}+q^{4}+q^{6}+4q^{7}-q^{8}+\cdots\)
1850.2.a.d \(1\) \(14.772\) \(\Q\) None \(-1\) \(0\) \(0\) \(-1\) \(+\) \(-\) \(-\) \(q-q^{2}+q^{4}-q^{7}-q^{8}-3q^{9}-3q^{11}+\cdots\)
1850.2.a.e \(1\) \(14.772\) \(\Q\) None \(-1\) \(0\) \(0\) \(2\) \(+\) \(-\) \(-\) \(q-q^{2}+q^{4}+2q^{7}-q^{8}-3q^{9}-2q^{13}+\cdots\)
1850.2.a.f \(1\) \(14.772\) \(\Q\) None \(-1\) \(2\) \(0\) \(-2\) \(+\) \(+\) \(+\) \(q-q^{2}+2q^{3}+q^{4}-2q^{6}-2q^{7}-q^{8}+\cdots\)
1850.2.a.g \(1\) \(14.772\) \(\Q\) None \(-1\) \(3\) \(0\) \(0\) \(+\) \(-\) \(+\) \(q-q^{2}+3q^{3}+q^{4}-3q^{6}-q^{8}+6q^{9}+\cdots\)
1850.2.a.h \(1\) \(14.772\) \(\Q\) None \(1\) \(-3\) \(0\) \(0\) \(-\) \(+\) \(-\) \(q+q^{2}-3q^{3}+q^{4}-3q^{6}+q^{8}+6q^{9}+\cdots\)
1850.2.a.i \(1\) \(14.772\) \(\Q\) None \(1\) \(-2\) \(0\) \(-1\) \(-\) \(+\) \(-\) \(q+q^{2}-2q^{3}+q^{4}-2q^{6}-q^{7}+q^{8}+\cdots\)
1850.2.a.j \(1\) \(14.772\) \(\Q\) None \(1\) \(0\) \(0\) \(-2\) \(-\) \(-\) \(+\) \(q+q^{2}+q^{4}-2q^{7}+q^{8}-3q^{9}+2q^{13}+\cdots\)
1850.2.a.k \(1\) \(14.772\) \(\Q\) None \(1\) \(0\) \(0\) \(0\) \(-\) \(+\) \(-\) \(q+q^{2}+q^{4}+q^{8}-3q^{9}-4q^{11}-2q^{13}+\cdots\)
1850.2.a.l \(1\) \(14.772\) \(\Q\) None \(1\) \(0\) \(0\) \(1\) \(-\) \(-\) \(+\) \(q+q^{2}+q^{4}+q^{7}+q^{8}-3q^{9}-3q^{11}+\cdots\)
1850.2.a.m \(1\) \(14.772\) \(\Q\) None \(1\) \(1\) \(0\) \(-4\) \(-\) \(+\) \(-\) \(q+q^{2}+q^{3}+q^{4}+q^{6}-4q^{7}+q^{8}+\cdots\)
1850.2.a.n \(1\) \(14.772\) \(\Q\) None \(1\) \(2\) \(0\) \(0\) \(-\) \(-\) \(-\) \(q+q^{2}+2q^{3}+q^{4}+2q^{6}+q^{8}+q^{9}+\cdots\)
1850.2.a.o \(1\) \(14.772\) \(\Q\) None \(1\) \(2\) \(0\) \(1\) \(-\) \(+\) \(+\) \(q+q^{2}+2q^{3}+q^{4}+2q^{6}+q^{7}+q^{8}+\cdots\)
1850.2.a.p \(1\) \(14.772\) \(\Q\) None \(1\) \(2\) \(0\) \(4\) \(-\) \(+\) \(+\) \(q+q^{2}+2q^{3}+q^{4}+2q^{6}+4q^{7}+q^{8}+\cdots\)
1850.2.a.q \(2\) \(14.772\) \(\Q(\sqrt{33}) \) None \(-2\) \(-4\) \(0\) \(3\) \(+\) \(+\) \(+\) \(q-q^{2}-2q^{3}+q^{4}+2q^{6}+(1+\beta )q^{7}+\cdots\)
1850.2.a.r \(2\) \(14.772\) \(\Q(\sqrt{6}) \) None \(-2\) \(-2\) \(0\) \(0\) \(+\) \(-\) \(-\) \(q-q^{2}+(-1+\beta )q^{3}+q^{4}+(1-\beta )q^{6}+\cdots\)
1850.2.a.s \(2\) \(14.772\) \(\Q(\sqrt{6}) \) None \(-2\) \(0\) \(0\) \(-4\) \(+\) \(-\) \(+\) \(q-q^{2}+\beta q^{3}+q^{4}-\beta q^{6}+(-2-\beta )q^{7}+\cdots\)
1850.2.a.t \(2\) \(14.772\) \(\Q(\sqrt{5}) \) None \(-2\) \(1\) \(0\) \(2\) \(+\) \(+\) \(-\) \(q-q^{2}+\beta q^{3}+q^{4}-\beta q^{6}+2\beta q^{7}+\cdots\)
1850.2.a.u \(2\) \(14.772\) \(\Q(\sqrt{13}) \) None \(2\) \(-3\) \(0\) \(-2\) \(-\) \(+\) \(+\) \(q+q^{2}+(-1-\beta )q^{3}+q^{4}+(-1-\beta )q^{6}+\cdots\)
1850.2.a.v \(2\) \(14.772\) \(\Q(\sqrt{6}) \) None \(2\) \(0\) \(0\) \(4\) \(-\) \(-\) \(-\) \(q+q^{2}+\beta q^{3}+q^{4}+\beta q^{6}+(2-\beta )q^{7}+\cdots\)
1850.2.a.w \(2\) \(14.772\) \(\Q(\sqrt{6}) \) None \(2\) \(2\) \(0\) \(0\) \(-\) \(+\) \(+\) \(q+q^{2}+(1+\beta )q^{3}+q^{4}+(1+\beta )q^{6}+\cdots\)
1850.2.a.x \(2\) \(14.772\) \(\Q(\sqrt{3}) \) None \(2\) \(2\) \(0\) \(6\) \(-\) \(+\) \(+\) \(q+q^{2}+(1+\beta )q^{3}+q^{4}+(1+\beta )q^{6}+\cdots\)
1850.2.a.y \(3\) \(14.772\) 3.3.1524.1 None \(-3\) \(-1\) \(0\) \(-4\) \(+\) \(+\) \(+\) \(q-q^{2}-\beta _{1}q^{3}+q^{4}+\beta _{1}q^{6}+(-1+\cdots)q^{7}+\cdots\)
1850.2.a.z \(3\) \(14.772\) 3.3.892.1 None \(-3\) \(0\) \(0\) \(1\) \(+\) \(+\) \(-\) \(q-q^{2}+\beta _{2}q^{3}+q^{4}-\beta _{2}q^{6}+\beta _{1}q^{7}+\cdots\)
1850.2.a.ba \(3\) \(14.772\) 3.3.148.1 None \(-3\) \(3\) \(0\) \(4\) \(+\) \(+\) \(-\) \(q-q^{2}+(1+\beta _{2})q^{3}+q^{4}+(-1-\beta _{2})q^{6}+\cdots\)
1850.2.a.bb \(3\) \(14.772\) 3.3.148.1 None \(3\) \(-3\) \(0\) \(-4\) \(-\) \(-\) \(+\) \(q+q^{2}+(-1-\beta _{2})q^{3}+q^{4}+(-1-\beta _{2})q^{6}+\cdots\)
1850.2.a.bc \(3\) \(14.772\) 3.3.1524.1 None \(3\) \(1\) \(0\) \(4\) \(-\) \(-\) \(-\) \(q+q^{2}+\beta _{1}q^{3}+q^{4}+\beta _{1}q^{6}+(1+\beta _{1}+\cdots)q^{7}+\cdots\)
1850.2.a.bd \(5\) \(14.772\) 5.5.1791440.1 None \(-5\) \(0\) \(0\) \(1\) \(+\) \(-\) \(+\) \(q-q^{2}+\beta _{1}q^{3}+q^{4}-\beta _{1}q^{6}+(\beta _{1}+\beta _{2}+\cdots)q^{7}+\cdots\)
1850.2.a.be \(5\) \(14.772\) 5.5.1791440.1 None \(5\) \(0\) \(0\) \(-1\) \(-\) \(-\) \(-\) \(q+q^{2}-\beta _{1}q^{3}+q^{4}-\beta _{1}q^{6}+(-\beta _{1}+\cdots)q^{7}+\cdots\)

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

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