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

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

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(2850))\) 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 3 5 19
2850.2.a.a 2850.a 1.a $1$ $22.757$ \(\Q\) None \(-1\) \(-1\) \(0\) \(-4\) $+$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+q^{6}-4q^{7}-q^{8}+\cdots\)
2850.2.a.b 2850.a 1.a $1$ $22.757$ \(\Q\) None \(-1\) \(-1\) \(0\) \(-2\) $+$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+q^{6}-2q^{7}-q^{8}+\cdots\)
2850.2.a.c 2850.a 1.a $1$ $22.757$ \(\Q\) None \(-1\) \(-1\) \(0\) \(-2\) $+$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+q^{6}-2q^{7}-q^{8}+\cdots\)
2850.2.a.d 2850.a 1.a $1$ $22.757$ \(\Q\) None \(-1\) \(-1\) \(0\) \(2\) $+$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+q^{6}+2q^{7}-q^{8}+\cdots\)
2850.2.a.e 2850.a 1.a $1$ $22.757$ \(\Q\) None \(-1\) \(-1\) \(0\) \(2\) $+$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+q^{6}+2q^{7}-q^{8}+\cdots\)
2850.2.a.f 2850.a 1.a $1$ $22.757$ \(\Q\) None \(-1\) \(-1\) \(0\) \(4\) $+$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+q^{6}+4q^{7}-q^{8}+\cdots\)
2850.2.a.g 2850.a 1.a $1$ $22.757$ \(\Q\) None \(-1\) \(-1\) \(0\) \(4\) $+$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+q^{6}+4q^{7}-q^{8}+\cdots\)
2850.2.a.h 2850.a 1.a $1$ $22.757$ \(\Q\) None \(-1\) \(1\) \(0\) \(-4\) $+$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}-q^{6}-4q^{7}-q^{8}+\cdots\)
2850.2.a.i 2850.a 1.a $1$ $22.757$ \(\Q\) None \(-1\) \(1\) \(0\) \(0\) $+$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}-q^{6}-q^{8}+q^{9}+\cdots\)
2850.2.a.j 2850.a 1.a $1$ $22.757$ \(\Q\) None \(-1\) \(1\) \(0\) \(0\) $+$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}-q^{6}-q^{8}+q^{9}+\cdots\)
2850.2.a.k 2850.a 1.a $1$ $22.757$ \(\Q\) None \(-1\) \(1\) \(0\) \(0\) $+$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}-q^{6}-q^{8}+q^{9}+\cdots\)
2850.2.a.l 2850.a 1.a $1$ $22.757$ \(\Q\) None \(-1\) \(1\) \(0\) \(0\) $+$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}-q^{6}-q^{8}+q^{9}+\cdots\)
2850.2.a.m 2850.a 1.a $1$ $22.757$ \(\Q\) None \(-1\) \(1\) \(0\) \(0\) $+$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}-q^{6}-q^{8}+q^{9}+\cdots\)
2850.2.a.n 2850.a 1.a $1$ $22.757$ \(\Q\) None \(-1\) \(1\) \(0\) \(2\) $+$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}-q^{6}+2q^{7}-q^{8}+\cdots\)
2850.2.a.o 2850.a 1.a $1$ $22.757$ \(\Q\) None \(-1\) \(1\) \(0\) \(2\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}-q^{6}+2q^{7}-q^{8}+\cdots\)
2850.2.a.p 2850.a 1.a $1$ $22.757$ \(\Q\) None \(1\) \(-1\) \(0\) \(-4\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}-q^{6}-4q^{7}+q^{8}+\cdots\)
2850.2.a.q 2850.a 1.a $1$ $22.757$ \(\Q\) None \(1\) \(-1\) \(0\) \(-2\) $-$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}-q^{6}-2q^{7}+q^{8}+\cdots\)
2850.2.a.r 2850.a 1.a $1$ $22.757$ \(\Q\) None \(1\) \(-1\) \(0\) \(-2\) $-$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}-q^{6}-2q^{7}+q^{8}+\cdots\)
2850.2.a.s 2850.a 1.a $1$ $22.757$ \(\Q\) None \(1\) \(-1\) \(0\) \(0\) $-$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}-q^{6}+q^{8}+q^{9}+\cdots\)
2850.2.a.t 2850.a 1.a $1$ $22.757$ \(\Q\) None \(1\) \(-1\) \(0\) \(0\) $-$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}-q^{6}+q^{8}+q^{9}+\cdots\)
2850.2.a.u 2850.a 1.a $1$ $22.757$ \(\Q\) None \(1\) \(-1\) \(0\) \(0\) $-$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}-q^{6}+q^{8}+q^{9}+\cdots\)
2850.2.a.v 2850.a 1.a $1$ $22.757$ \(\Q\) None \(1\) \(-1\) \(0\) \(4\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}-q^{6}+4q^{7}+q^{8}+\cdots\)
2850.2.a.w 2850.a 1.a $1$ $22.757$ \(\Q\) None \(1\) \(1\) \(0\) \(-4\) $-$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}+q^{6}-4q^{7}+q^{8}+\cdots\)
2850.2.a.x 2850.a 1.a $1$ $22.757$ \(\Q\) None \(1\) \(1\) \(0\) \(-4\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}+q^{6}-4q^{7}+q^{8}+\cdots\)
2850.2.a.y 2850.a 1.a $1$ $22.757$ \(\Q\) None \(1\) \(1\) \(0\) \(-2\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}+q^{6}-2q^{7}+q^{8}+\cdots\)
2850.2.a.z 2850.a 1.a $1$ $22.757$ \(\Q\) None \(1\) \(1\) \(0\) \(-2\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}+q^{6}-2q^{7}+q^{8}+\cdots\)
2850.2.a.ba 2850.a 1.a $1$ $22.757$ \(\Q\) None \(1\) \(1\) \(0\) \(2\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}+q^{6}+2q^{7}+q^{8}+\cdots\)
2850.2.a.bb 2850.a 1.a $1$ $22.757$ \(\Q\) None \(1\) \(1\) \(0\) \(2\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}+q^{6}+2q^{7}+q^{8}+\cdots\)
2850.2.a.bc 2850.a 1.a $2$ $22.757$ \(\Q(\sqrt{6}) \) None \(-2\) \(-2\) \(0\) \(-4\) $+$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+q^{6}+(-2+\beta )q^{7}+\cdots\)
2850.2.a.bd 2850.a 1.a $2$ $22.757$ \(\Q(\sqrt{3}) \) None \(-2\) \(-2\) \(0\) \(-2\) $+$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+q^{6}+(-1+\beta )q^{7}+\cdots\)
2850.2.a.be 2850.a 1.a $2$ $22.757$ \(\Q(\sqrt{3}) \) None \(-2\) \(2\) \(0\) \(-2\) $+$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}-q^{6}+(-1+\beta )q^{7}+\cdots\)
2850.2.a.bf 2850.a 1.a $2$ $22.757$ \(\Q(\sqrt{10}) \) None \(-2\) \(2\) \(0\) \(0\) $+$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}-q^{6}+\beta q^{7}-q^{8}+\cdots\)
2850.2.a.bg 2850.a 1.a $2$ $22.757$ \(\Q(\sqrt{10}) \) None \(2\) \(-2\) \(0\) \(0\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}-q^{6}+\beta q^{7}+q^{8}+\cdots\)
2850.2.a.bh 2850.a 1.a $2$ $22.757$ \(\Q(\sqrt{3}) \) None \(2\) \(-2\) \(0\) \(2\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}-q^{6}+(1+\beta )q^{7}+\cdots\)
2850.2.a.bi 2850.a 1.a $2$ $22.757$ \(\Q(\sqrt{3}) \) None \(2\) \(2\) \(0\) \(2\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}+q^{6}+(1+\beta )q^{7}+\cdots\)
2850.2.a.bj 2850.a 1.a $2$ $22.757$ \(\Q(\sqrt{6}) \) None \(2\) \(2\) \(0\) \(4\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}+q^{6}+(2+\beta )q^{7}+\cdots\)
2850.2.a.bk 2850.a 1.a $3$ $22.757$ 3.3.148.1 None \(-3\) \(-3\) \(0\) \(0\) $+$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+q^{6}-\beta _{2}q^{7}-q^{8}+\cdots\)
2850.2.a.bl 2850.a 1.a $3$ $22.757$ 3.3.148.1 None \(-3\) \(3\) \(0\) \(0\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}-q^{6}-\beta _{2}q^{7}-q^{8}+\cdots\)
2850.2.a.bm 2850.a 1.a $3$ $22.757$ 3.3.148.1 None \(3\) \(-3\) \(0\) \(0\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}-q^{6}+\beta _{2}q^{7}+q^{8}+\cdots\)
2850.2.a.bn 2850.a 1.a $3$ $22.757$ 3.3.148.1 None \(3\) \(3\) \(0\) \(0\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}+q^{6}+\beta _{2}q^{7}+q^{8}+\cdots\)

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

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