Properties

Label 2850.2.a
Level $2850$
Weight $2$
Character orbit 2850.a
Rep. character $\chi_{2850}(1,\cdot)$
Character field $\Q$
Dimension $56$
Newform subspaces $40$
Sturm bound $1200$
Trace bound $23$

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Defining parameters

Level: \( N \) \(=\) \( 2850 = 2 \cdot 3 \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2850.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 40 \)
Sturm bound: \(1200\)
Trace bound: \(23\)
Distinguishing \(T_p\): \(7\), \(11\), \(13\), \(23\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(2850))\).

Total New Old
Modular forms 624 56 568
Cusp forms 577 56 521
Eisenstein series 47 0 47

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)\(3\)\(5\)\(19\)FrickeDim.
\(+\)\(+\)\(+\)\(+\)\(+\)\(4\)
\(+\)\(+\)\(+\)\(-\)\(-\)\(4\)
\(+\)\(+\)\(-\)\(+\)\(-\)\(4\)
\(+\)\(+\)\(-\)\(-\)\(+\)\(2\)
\(+\)\(-\)\(+\)\(+\)\(-\)\(3\)
\(+\)\(-\)\(+\)\(-\)\(+\)\(4\)
\(+\)\(-\)\(-\)\(+\)\(+\)\(4\)
\(+\)\(-\)\(-\)\(-\)\(-\)\(4\)
\(-\)\(+\)\(+\)\(+\)\(-\)\(4\)
\(-\)\(+\)\(+\)\(-\)\(+\)\(2\)
\(-\)\(+\)\(-\)\(+\)\(+\)\(3\)
\(-\)\(+\)\(-\)\(-\)\(-\)\(5\)
\(-\)\(-\)\(+\)\(+\)\(+\)\(2\)
\(-\)\(-\)\(+\)\(-\)\(-\)\(5\)
\(-\)\(-\)\(-\)\(+\)\(-\)\(5\)
\(-\)\(-\)\(-\)\(-\)\(+\)\(1\)
Plus space\(+\)\(22\)
Minus space\(-\)\(34\)

Trace form

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

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(2850))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 3 5 19
2850.2.a.a \(1\) \(22.757\) \(\Q\) None \(-1\) \(-1\) \(0\) \(-4\) \(+\) \(+\) \(+\) \(+\) \(q-q^{2}-q^{3}+q^{4}+q^{6}-4q^{7}-q^{8}+\cdots\)
2850.2.a.b \(1\) \(22.757\) \(\Q\) None \(-1\) \(-1\) \(0\) \(-2\) \(+\) \(+\) \(+\) \(-\) \(q-q^{2}-q^{3}+q^{4}+q^{6}-2q^{7}-q^{8}+\cdots\)
2850.2.a.c \(1\) \(22.757\) \(\Q\) None \(-1\) \(-1\) \(0\) \(-2\) \(+\) \(+\) \(+\) \(-\) \(q-q^{2}-q^{3}+q^{4}+q^{6}-2q^{7}-q^{8}+\cdots\)
2850.2.a.d \(1\) \(22.757\) \(\Q\) None \(-1\) \(-1\) \(0\) \(2\) \(+\) \(+\) \(-\) \(+\) \(q-q^{2}-q^{3}+q^{4}+q^{6}+2q^{7}-q^{8}+\cdots\)
2850.2.a.e \(1\) \(22.757\) \(\Q\) None \(-1\) \(-1\) \(0\) \(2\) \(+\) \(+\) \(+\) \(+\) \(q-q^{2}-q^{3}+q^{4}+q^{6}+2q^{7}-q^{8}+\cdots\)
2850.2.a.f \(1\) \(22.757\) \(\Q\) None \(-1\) \(-1\) \(0\) \(4\) \(+\) \(+\) \(+\) \(-\) \(q-q^{2}-q^{3}+q^{4}+q^{6}+4q^{7}-q^{8}+\cdots\)
2850.2.a.g \(1\) \(22.757\) \(\Q\) None \(-1\) \(-1\) \(0\) \(4\) \(+\) \(+\) \(+\) \(-\) \(q-q^{2}-q^{3}+q^{4}+q^{6}+4q^{7}-q^{8}+\cdots\)
2850.2.a.h \(1\) \(22.757\) \(\Q\) None \(-1\) \(1\) \(0\) \(-4\) \(+\) \(-\) \(+\) \(-\) \(q-q^{2}+q^{3}+q^{4}-q^{6}-4q^{7}-q^{8}+\cdots\)
2850.2.a.i \(1\) \(22.757\) \(\Q\) None \(-1\) \(1\) \(0\) \(0\) \(+\) \(-\) \(-\) \(+\) \(q-q^{2}+q^{3}+q^{4}-q^{6}-q^{8}+q^{9}+\cdots\)
2850.2.a.j \(1\) \(22.757\) \(\Q\) None \(-1\) \(1\) \(0\) \(0\) \(+\) \(-\) \(+\) \(+\) \(q-q^{2}+q^{3}+q^{4}-q^{6}-q^{8}+q^{9}+\cdots\)
2850.2.a.k \(1\) \(22.757\) \(\Q\) None \(-1\) \(1\) \(0\) \(0\) \(+\) \(-\) \(-\) \(+\) \(q-q^{2}+q^{3}+q^{4}-q^{6}-q^{8}+q^{9}+\cdots\)
2850.2.a.l \(1\) \(22.757\) \(\Q\) None \(-1\) \(1\) \(0\) \(0\) \(+\) \(-\) \(+\) \(+\) \(q-q^{2}+q^{3}+q^{4}-q^{6}-q^{8}+q^{9}+\cdots\)
2850.2.a.m \(1\) \(22.757\) \(\Q\) None \(-1\) \(1\) \(0\) \(0\) \(+\) \(-\) \(+\) \(+\) \(q-q^{2}+q^{3}+q^{4}-q^{6}-q^{8}+q^{9}+\cdots\)
2850.2.a.n \(1\) \(22.757\) \(\Q\) None \(-1\) \(1\) \(0\) \(2\) \(+\) \(-\) \(+\) \(-\) \(q-q^{2}+q^{3}+q^{4}-q^{6}+2q^{7}-q^{8}+\cdots\)
2850.2.a.o \(1\) \(22.757\) \(\Q\) None \(-1\) \(1\) \(0\) \(2\) \(+\) \(-\) \(-\) \(-\) \(q-q^{2}+q^{3}+q^{4}-q^{6}+2q^{7}-q^{8}+\cdots\)
2850.2.a.p \(1\) \(22.757\) \(\Q\) None \(1\) \(-1\) \(0\) \(-4\) \(-\) \(+\) \(+\) \(+\) \(q+q^{2}-q^{3}+q^{4}-q^{6}-4q^{7}+q^{8}+\cdots\)
2850.2.a.q \(1\) \(22.757\) \(\Q\) None \(1\) \(-1\) \(0\) \(-2\) \(-\) \(+\) \(+\) \(-\) \(q+q^{2}-q^{3}+q^{4}-q^{6}-2q^{7}+q^{8}+\cdots\)
2850.2.a.r \(1\) \(22.757\) \(\Q\) None \(1\) \(-1\) \(0\) \(-2\) \(-\) \(+\) \(+\) \(-\) \(q+q^{2}-q^{3}+q^{4}-q^{6}-2q^{7}+q^{8}+\cdots\)
2850.2.a.s \(1\) \(22.757\) \(\Q\) None \(1\) \(-1\) \(0\) \(0\) \(-\) \(+\) \(-\) \(+\) \(q+q^{2}-q^{3}+q^{4}-q^{6}+q^{8}+q^{9}+\cdots\)
2850.2.a.t \(1\) \(22.757\) \(\Q\) None \(1\) \(-1\) \(0\) \(0\) \(-\) \(+\) \(-\) \(+\) \(q+q^{2}-q^{3}+q^{4}-q^{6}+q^{8}+q^{9}+\cdots\)
2850.2.a.u \(1\) \(22.757\) \(\Q\) None \(1\) \(-1\) \(0\) \(0\) \(-\) \(+\) \(-\) \(+\) \(q+q^{2}-q^{3}+q^{4}-q^{6}+q^{8}+q^{9}+\cdots\)
2850.2.a.v \(1\) \(22.757\) \(\Q\) None \(1\) \(-1\) \(0\) \(4\) \(-\) \(+\) \(+\) \(+\) \(q+q^{2}-q^{3}+q^{4}-q^{6}+4q^{7}+q^{8}+\cdots\)
2850.2.a.w \(1\) \(22.757\) \(\Q\) None \(1\) \(1\) \(0\) \(-4\) \(-\) \(-\) \(-\) \(-\) \(q+q^{2}+q^{3}+q^{4}+q^{6}-4q^{7}+q^{8}+\cdots\)
2850.2.a.x \(1\) \(22.757\) \(\Q\) None \(1\) \(1\) \(0\) \(-4\) \(-\) \(-\) \(+\) \(-\) \(q+q^{2}+q^{3}+q^{4}+q^{6}-4q^{7}+q^{8}+\cdots\)
2850.2.a.y \(1\) \(22.757\) \(\Q\) None \(1\) \(1\) \(0\) \(-2\) \(-\) \(-\) \(+\) \(+\) \(q+q^{2}+q^{3}+q^{4}+q^{6}-2q^{7}+q^{8}+\cdots\)
2850.2.a.z \(1\) \(22.757\) \(\Q\) None \(1\) \(1\) \(0\) \(-2\) \(-\) \(-\) \(+\) \(+\) \(q+q^{2}+q^{3}+q^{4}+q^{6}-2q^{7}+q^{8}+\cdots\)
2850.2.a.ba \(1\) \(22.757\) \(\Q\) None \(1\) \(1\) \(0\) \(2\) \(-\) \(-\) \(+\) \(-\) \(q+q^{2}+q^{3}+q^{4}+q^{6}+2q^{7}+q^{8}+\cdots\)
2850.2.a.bb \(1\) \(22.757\) \(\Q\) None \(1\) \(1\) \(0\) \(2\) \(-\) \(-\) \(+\) \(-\) \(q+q^{2}+q^{3}+q^{4}+q^{6}+2q^{7}+q^{8}+\cdots\)
2850.2.a.bc \(2\) \(22.757\) \(\Q(\sqrt{6}) \) None \(-2\) \(-2\) \(0\) \(-4\) \(+\) \(+\) \(-\) \(-\) \(q-q^{2}-q^{3}+q^{4}+q^{6}+(-2+\beta )q^{7}+\cdots\)
2850.2.a.bd \(2\) \(22.757\) \(\Q(\sqrt{3}) \) None \(-2\) \(-2\) \(0\) \(-2\) \(+\) \(+\) \(+\) \(+\) \(q-q^{2}-q^{3}+q^{4}+q^{6}+(-1+\beta )q^{7}+\cdots\)
2850.2.a.be \(2\) \(22.757\) \(\Q(\sqrt{3}) \) None \(-2\) \(2\) \(0\) \(-2\) \(+\) \(-\) \(+\) \(-\) \(q-q^{2}+q^{3}+q^{4}-q^{6}+(-1+\beta )q^{7}+\cdots\)
2850.2.a.bf \(2\) \(22.757\) \(\Q(\sqrt{10}) \) None \(-2\) \(2\) \(0\) \(0\) \(+\) \(-\) \(-\) \(+\) \(q-q^{2}+q^{3}+q^{4}-q^{6}+\beta q^{7}-q^{8}+\cdots\)
2850.2.a.bg \(2\) \(22.757\) \(\Q(\sqrt{10}) \) None \(2\) \(-2\) \(0\) \(0\) \(-\) \(+\) \(+\) \(+\) \(q+q^{2}-q^{3}+q^{4}-q^{6}+\beta q^{7}+q^{8}+\cdots\)
2850.2.a.bh \(2\) \(22.757\) \(\Q(\sqrt{3}) \) None \(2\) \(-2\) \(0\) \(2\) \(-\) \(+\) \(-\) \(-\) \(q+q^{2}-q^{3}+q^{4}-q^{6}+(1+\beta )q^{7}+\cdots\)
2850.2.a.bi \(2\) \(22.757\) \(\Q(\sqrt{3}) \) None \(2\) \(2\) \(0\) \(2\) \(-\) \(-\) \(-\) \(+\) \(q+q^{2}+q^{3}+q^{4}+q^{6}+(1+\beta )q^{7}+\cdots\)
2850.2.a.bj \(2\) \(22.757\) \(\Q(\sqrt{6}) \) None \(2\) \(2\) \(0\) \(4\) \(-\) \(-\) \(+\) \(-\) \(q+q^{2}+q^{3}+q^{4}+q^{6}+(2+\beta )q^{7}+\cdots\)
2850.2.a.bk \(3\) \(22.757\) 3.3.148.1 None \(-3\) \(-3\) \(0\) \(0\) \(+\) \(+\) \(-\) \(+\) \(q-q^{2}-q^{3}+q^{4}+q^{6}-\beta _{2}q^{7}-q^{8}+\cdots\)
2850.2.a.bl \(3\) \(22.757\) 3.3.148.1 None \(-3\) \(3\) \(0\) \(0\) \(+\) \(-\) \(-\) \(-\) \(q-q^{2}+q^{3}+q^{4}-q^{6}-\beta _{2}q^{7}-q^{8}+\cdots\)
2850.2.a.bm \(3\) \(22.757\) 3.3.148.1 None \(3\) \(-3\) \(0\) \(0\) \(-\) \(+\) \(-\) \(-\) \(q+q^{2}-q^{3}+q^{4}-q^{6}+\beta _{2}q^{7}+q^{8}+\cdots\)
2850.2.a.bn \(3\) \(22.757\) 3.3.148.1 None \(3\) \(3\) \(0\) \(0\) \(-\) \(-\) \(-\) \(+\) \(q+q^{2}+q^{3}+q^{4}+q^{6}+\beta _{2}q^{7}+q^{8}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(2850))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(2850)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(19))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(38))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(57))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(95))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(114))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(150))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(190))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(285))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(475))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(570))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(950))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1425))\)\(^{\oplus 2}\)