Properties

Label 2800.2.a
Level $2800$
Weight $2$
Character orbit 2800.a
Rep. character $\chi_{2800}(1,\cdot)$
Character field $\Q$
Dimension $57$
Newform subspaces $44$
Sturm bound $960$
Trace bound $13$

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Defining parameters

Level: \( N \) \(=\) \( 2800 = 2^{4} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2800.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 44 \)
Sturm bound: \(960\)
Trace bound: \(13\)
Distinguishing \(T_p\): \(3\), \(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(2800))\).

Total New Old
Modular forms 516 57 459
Cusp forms 445 57 388
Eisenstein series 71 0 71

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

\(2\)\(5\)\(7\)FrickeDim.
\(+\)\(+\)\(+\)\(+\)\(7\)
\(+\)\(+\)\(-\)\(-\)\(7\)
\(+\)\(-\)\(+\)\(-\)\(7\)
\(+\)\(-\)\(-\)\(+\)\(7\)
\(-\)\(+\)\(+\)\(-\)\(8\)
\(-\)\(+\)\(-\)\(+\)\(5\)
\(-\)\(-\)\(+\)\(+\)\(7\)
\(-\)\(-\)\(-\)\(-\)\(9\)
Plus space\(+\)\(26\)
Minus space\(-\)\(31\)

Trace form

\( 57 q - q^{7} + 53 q^{9} + O(q^{10}) \) \( 57 q - q^{7} + 53 q^{9} - 8 q^{11} + 2 q^{13} - 6 q^{17} + 8 q^{19} - 8 q^{23} - 24 q^{27} + 6 q^{29} + 8 q^{31} + 14 q^{37} - 16 q^{39} + 2 q^{41} + 20 q^{43} + 57 q^{49} - 16 q^{51} + 6 q^{53} + 8 q^{57} - 6 q^{61} - 5 q^{63} - 28 q^{67} - 16 q^{69} + 12 q^{71} - 22 q^{73} + 4 q^{77} + 20 q^{79} + 57 q^{81} + 32 q^{83} - 8 q^{87} - 22 q^{89} - 6 q^{91} + 32 q^{93} - 22 q^{97} - 8 q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(2800))\) 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 7
2800.2.a.a 2800.a 1.a $1$ $22.358$ \(\Q\) None \(0\) \(-3\) \(0\) \(-1\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-3q^{3}-q^{7}+6q^{9}-3q^{11}+q^{13}+\cdots\)
2800.2.a.b 2800.a 1.a $1$ $22.358$ \(\Q\) None \(0\) \(-3\) \(0\) \(-1\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-3q^{3}-q^{7}+6q^{9}+5q^{11}+6q^{13}+\cdots\)
2800.2.a.c 2800.a 1.a $1$ $22.358$ \(\Q\) None \(0\) \(-3\) \(0\) \(1\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-3q^{3}+q^{7}+6q^{9}+5q^{11}+5q^{13}+\cdots\)
2800.2.a.d 2800.a 1.a $1$ $22.358$ \(\Q\) None \(0\) \(-2\) \(0\) \(-1\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{3}-q^{7}+q^{9}-5q^{11}+8q^{17}+\cdots\)
2800.2.a.e 2800.a 1.a $1$ $22.358$ \(\Q\) None \(0\) \(-2\) \(0\) \(1\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{3}+q^{7}+q^{9}-3q^{11}+4q^{13}+\cdots\)
2800.2.a.f 2800.a 1.a $1$ $22.358$ \(\Q\) None \(0\) \(-2\) \(0\) \(1\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{3}+q^{7}+q^{9}-q^{11}+4q^{13}+\cdots\)
2800.2.a.g 2800.a 1.a $1$ $22.358$ \(\Q\) None \(0\) \(-2\) \(0\) \(1\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{3}+q^{7}+q^{9}+4q^{13}-6q^{17}+\cdots\)
2800.2.a.h 2800.a 1.a $1$ $22.358$ \(\Q\) None \(0\) \(-1\) \(0\) \(-1\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}-q^{7}-2q^{9}-3q^{11}+2q^{13}+\cdots\)
2800.2.a.i 2800.a 1.a $1$ $22.358$ \(\Q\) None \(0\) \(-1\) \(0\) \(-1\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}-q^{7}-2q^{9}+5q^{11}-q^{13}+\cdots\)
2800.2.a.j 2800.a 1.a $1$ $22.358$ \(\Q\) None \(0\) \(-1\) \(0\) \(1\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}+q^{7}-2q^{9}+q^{11}-6q^{13}+\cdots\)
2800.2.a.k 2800.a 1.a $1$ $22.358$ \(\Q\) None \(0\) \(-1\) \(0\) \(1\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}+q^{7}-2q^{9}+q^{11}-q^{13}-3q^{17}+\cdots\)
2800.2.a.l 2800.a 1.a $1$ $22.358$ \(\Q\) None \(0\) \(-1\) \(0\) \(1\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}+q^{7}-2q^{9}+3q^{11}+q^{13}+\cdots\)
2800.2.a.m 2800.a 1.a $1$ $22.358$ \(\Q\) None \(0\) \(0\) \(0\) \(-1\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{7}-3q^{9}-4q^{11}+6q^{13}-2q^{17}+\cdots\)
2800.2.a.n 2800.a 1.a $1$ $22.358$ \(\Q\) None \(0\) \(0\) \(0\) \(-1\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{7}-3q^{9}-q^{11}-2q^{13}-4q^{17}+\cdots\)
2800.2.a.o 2800.a 1.a $1$ $22.358$ \(\Q\) None \(0\) \(0\) \(0\) \(-1\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{7}-3q^{9}+4q^{13}+4q^{17}-4q^{19}+\cdots\)
2800.2.a.p 2800.a 1.a $1$ $22.358$ \(\Q\) None \(0\) \(0\) \(0\) \(-1\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{7}-3q^{9}+4q^{11}-2q^{13}+6q^{17}+\cdots\)
2800.2.a.q 2800.a 1.a $1$ $22.358$ \(\Q\) None \(0\) \(0\) \(0\) \(-1\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{7}-3q^{9}+5q^{11}-6q^{13}+4q^{17}+\cdots\)
2800.2.a.r 2800.a 1.a $1$ $22.358$ \(\Q\) None \(0\) \(0\) \(0\) \(1\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{7}-3q^{9}-q^{11}+2q^{13}+4q^{17}+\cdots\)
2800.2.a.s 2800.a 1.a $1$ $22.358$ \(\Q\) None \(0\) \(0\) \(0\) \(1\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{7}-3q^{9}-4q^{13}-4q^{17}-4q^{19}+\cdots\)
2800.2.a.t 2800.a 1.a $1$ $22.358$ \(\Q\) None \(0\) \(0\) \(0\) \(1\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{7}-3q^{9}+5q^{11}+6q^{13}-4q^{17}+\cdots\)
2800.2.a.u 2800.a 1.a $1$ $22.358$ \(\Q\) None \(0\) \(1\) \(0\) \(-1\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}-q^{7}-2q^{9}+q^{11}+q^{13}+3q^{17}+\cdots\)
2800.2.a.v 2800.a 1.a $1$ $22.358$ \(\Q\) None \(0\) \(1\) \(0\) \(-1\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}-q^{7}-2q^{9}+q^{11}+6q^{13}+\cdots\)
2800.2.a.w 2800.a 1.a $1$ $22.358$ \(\Q\) None \(0\) \(1\) \(0\) \(-1\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}-q^{7}-2q^{9}+3q^{11}-q^{13}+\cdots\)
2800.2.a.x 2800.a 1.a $1$ $22.358$ \(\Q\) None \(0\) \(1\) \(0\) \(1\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+q^{7}-2q^{9}-3q^{11}-2q^{13}+\cdots\)
2800.2.a.y 2800.a 1.a $1$ $22.358$ \(\Q\) None \(0\) \(1\) \(0\) \(1\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+q^{7}-2q^{9}-3q^{11}+q^{13}+\cdots\)
2800.2.a.z 2800.a 1.a $1$ $22.358$ \(\Q\) None \(0\) \(1\) \(0\) \(1\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+q^{7}-2q^{9}+3q^{11}-5q^{13}+\cdots\)
2800.2.a.ba 2800.a 1.a $1$ $22.358$ \(\Q\) None \(0\) \(2\) \(0\) \(-1\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{3}-q^{7}+q^{9}-3q^{11}-4q^{13}+\cdots\)
2800.2.a.bb 2800.a 1.a $1$ $22.358$ \(\Q\) None \(0\) \(2\) \(0\) \(-1\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{3}-q^{7}+q^{9}-q^{11}-4q^{13}+\cdots\)
2800.2.a.bc 2800.a 1.a $1$ $22.358$ \(\Q\) None \(0\) \(2\) \(0\) \(1\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{3}+q^{7}+q^{9}-5q^{11}-8q^{17}+\cdots\)
2800.2.a.bd 2800.a 1.a $1$ $22.358$ \(\Q\) None \(0\) \(2\) \(0\) \(1\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{3}+q^{7}+q^{9}+2q^{17}+2q^{19}+\cdots\)
2800.2.a.be 2800.a 1.a $1$ $22.358$ \(\Q\) None \(0\) \(3\) \(0\) \(-1\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+3q^{3}-q^{7}+6q^{9}+5q^{11}+3q^{13}+\cdots\)
2800.2.a.bf 2800.a 1.a $1$ $22.358$ \(\Q\) None \(0\) \(3\) \(0\) \(1\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+3q^{3}+q^{7}+6q^{9}-3q^{11}-q^{13}+\cdots\)
2800.2.a.bg 2800.a 1.a $1$ $22.358$ \(\Q\) None \(0\) \(3\) \(0\) \(1\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+3q^{3}+q^{7}+6q^{9}+5q^{11}-6q^{13}+\cdots\)
2800.2.a.bh 2800.a 1.a $2$ $22.358$ \(\Q(\sqrt{5}) \) None \(0\) \(-2\) \(0\) \(2\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(-1-\beta )q^{3}+q^{7}+(3+2\beta )q^{9}+(-2+\cdots)q^{11}+\cdots\)
2800.2.a.bi 2800.a 1.a $2$ $22.358$ \(\Q(\sqrt{17}) \) None \(0\) \(-1\) \(0\) \(-2\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta q^{3}-q^{7}+(1+\beta )q^{9}-\beta q^{11}+(-2+\cdots)q^{13}+\cdots\)
2800.2.a.bj 2800.a 1.a $2$ $22.358$ \(\Q(\sqrt{17}) \) None \(0\) \(-1\) \(0\) \(-2\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta q^{3}-q^{7}+(1+\beta )q^{9}+(3-2\beta )q^{11}+\cdots\)
2800.2.a.bk 2800.a 1.a $2$ $22.358$ \(\Q(\sqrt{33}) \) None \(0\) \(-1\) \(0\) \(-2\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta q^{3}-q^{7}+(5+\beta )q^{9}+(-4+\beta )q^{11}+\cdots\)
2800.2.a.bl 2800.a 1.a $2$ $22.358$ \(\Q(\sqrt{6}) \) None \(0\) \(0\) \(0\) \(-2\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{3}-q^{7}+3q^{9}-2\beta q^{11}+(-2+\cdots)q^{13}+\cdots\)
2800.2.a.bm 2800.a 1.a $2$ $22.358$ \(\Q(\sqrt{6}) \) None \(0\) \(0\) \(0\) \(2\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{3}+q^{7}+3q^{9}+2\beta q^{11}+(2+\beta )q^{13}+\cdots\)
2800.2.a.bn 2800.a 1.a $2$ $22.358$ \(\Q(\sqrt{17}) \) None \(0\) \(1\) \(0\) \(2\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{3}+q^{7}+(1+\beta )q^{9}+\beta q^{11}+(-2+\cdots)q^{13}+\cdots\)
2800.2.a.bo 2800.a 1.a $2$ $22.358$ \(\Q(\sqrt{17}) \) None \(0\) \(1\) \(0\) \(2\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{3}+q^{7}+(1+\beta )q^{9}+(3-2\beta )q^{11}+\cdots\)
2800.2.a.bp 2800.a 1.a $2$ $22.358$ \(\Q(\sqrt{5}) \) None \(0\) \(2\) \(0\) \(-2\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{3}-q^{7}+(3+2\beta )q^{9}+(-2+\cdots)q^{11}+\cdots\)
2800.2.a.bq 2800.a 1.a $3$ $22.358$ 3.3.568.1 None \(0\) \(-1\) \(0\) \(3\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{3}+q^{7}+(1+2\beta _{1}+\beta _{2})q^{9}+\cdots\)
2800.2.a.br 2800.a 1.a $3$ $22.358$ 3.3.568.1 None \(0\) \(1\) \(0\) \(-3\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{3}-q^{7}+(1+2\beta _{1}+\beta _{2})q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(2800))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(2800)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(56))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(70))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(80))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(100))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(112))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(140))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(175))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(200))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(280))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(350))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(400))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(560))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(700))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1400))\)\(^{\oplus 2}\)