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

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

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(2800))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 5 7
2800.2.a.a \(1\) \(22.358\) \(\Q\) None \(0\) \(-3\) \(0\) \(-1\) \(-\) \(-\) \(+\) \(q-3q^{3}-q^{7}+6q^{9}-3q^{11}+q^{13}+\cdots\)
2800.2.a.b \(1\) \(22.358\) \(\Q\) None \(0\) \(-3\) \(0\) \(-1\) \(-\) \(+\) \(+\) \(q-3q^{3}-q^{7}+6q^{9}+5q^{11}+6q^{13}+\cdots\)
2800.2.a.c \(1\) \(22.358\) \(\Q\) None \(0\) \(-3\) \(0\) \(1\) \(+\) \(+\) \(-\) \(q-3q^{3}+q^{7}+6q^{9}+5q^{11}+5q^{13}+\cdots\)
2800.2.a.d \(1\) \(22.358\) \(\Q\) None \(0\) \(-2\) \(0\) \(-1\) \(+\) \(+\) \(+\) \(q-2q^{3}-q^{7}+q^{9}-5q^{11}+8q^{17}+\cdots\)
2800.2.a.e \(1\) \(22.358\) \(\Q\) None \(0\) \(-2\) \(0\) \(1\) \(-\) \(+\) \(-\) \(q-2q^{3}+q^{7}+q^{9}-3q^{11}+4q^{13}+\cdots\)
2800.2.a.f \(1\) \(22.358\) \(\Q\) None \(0\) \(-2\) \(0\) \(1\) \(+\) \(-\) \(-\) \(q-2q^{3}+q^{7}+q^{9}-q^{11}+4q^{13}+\cdots\)
2800.2.a.g \(1\) \(22.358\) \(\Q\) None \(0\) \(-2\) \(0\) \(1\) \(-\) \(+\) \(-\) \(q-2q^{3}+q^{7}+q^{9}+4q^{13}-6q^{17}+\cdots\)
2800.2.a.h \(1\) \(22.358\) \(\Q\) None \(0\) \(-1\) \(0\) \(-1\) \(-\) \(-\) \(+\) \(q-q^{3}-q^{7}-2q^{9}-3q^{11}+2q^{13}+\cdots\)
2800.2.a.i \(1\) \(22.358\) \(\Q\) None \(0\) \(-1\) \(0\) \(-1\) \(+\) \(+\) \(+\) \(q-q^{3}-q^{7}-2q^{9}+5q^{11}-q^{13}+\cdots\)
2800.2.a.j \(1\) \(22.358\) \(\Q\) None \(0\) \(-1\) \(0\) \(1\) \(+\) \(-\) \(-\) \(q-q^{3}+q^{7}-2q^{9}+q^{11}-6q^{13}+\cdots\)
2800.2.a.k \(1\) \(22.358\) \(\Q\) None \(0\) \(-1\) \(0\) \(1\) \(+\) \(-\) \(-\) \(q-q^{3}+q^{7}-2q^{9}+q^{11}-q^{13}-3q^{17}+\cdots\)
2800.2.a.l \(1\) \(22.358\) \(\Q\) None \(0\) \(-1\) \(0\) \(1\) \(-\) \(-\) \(-\) \(q-q^{3}+q^{7}-2q^{9}+3q^{11}+q^{13}+\cdots\)
2800.2.a.m \(1\) \(22.358\) \(\Q\) None \(0\) \(0\) \(0\) \(-1\) \(-\) \(+\) \(+\) \(q-q^{7}-3q^{9}-4q^{11}+6q^{13}-2q^{17}+\cdots\)
2800.2.a.n \(1\) \(22.358\) \(\Q\) None \(0\) \(0\) \(0\) \(-1\) \(+\) \(-\) \(+\) \(q-q^{7}-3q^{9}-q^{11}-2q^{13}-4q^{17}+\cdots\)
2800.2.a.o \(1\) \(22.358\) \(\Q\) None \(0\) \(0\) \(0\) \(-1\) \(-\) \(-\) \(+\) \(q-q^{7}-3q^{9}+4q^{13}+4q^{17}-4q^{19}+\cdots\)
2800.2.a.p \(1\) \(22.358\) \(\Q\) None \(0\) \(0\) \(0\) \(-1\) \(+\) \(+\) \(+\) \(q-q^{7}-3q^{9}+4q^{11}-2q^{13}+6q^{17}+\cdots\)
2800.2.a.q \(1\) \(22.358\) \(\Q\) None \(0\) \(0\) \(0\) \(-1\) \(-\) \(+\) \(+\) \(q-q^{7}-3q^{9}+5q^{11}-6q^{13}+4q^{17}+\cdots\)
2800.2.a.r \(1\) \(22.358\) \(\Q\) None \(0\) \(0\) \(0\) \(1\) \(+\) \(+\) \(-\) \(q+q^{7}-3q^{9}-q^{11}+2q^{13}+4q^{17}+\cdots\)
2800.2.a.s \(1\) \(22.358\) \(\Q\) None \(0\) \(0\) \(0\) \(1\) \(-\) \(-\) \(-\) \(q+q^{7}-3q^{9}-4q^{13}-4q^{17}-4q^{19}+\cdots\)
2800.2.a.t \(1\) \(22.358\) \(\Q\) None \(0\) \(0\) \(0\) \(1\) \(-\) \(-\) \(-\) \(q+q^{7}-3q^{9}+5q^{11}+6q^{13}-4q^{17}+\cdots\)
2800.2.a.u \(1\) \(22.358\) \(\Q\) None \(0\) \(1\) \(0\) \(-1\) \(+\) \(-\) \(+\) \(q+q^{3}-q^{7}-2q^{9}+q^{11}+q^{13}+3q^{17}+\cdots\)
2800.2.a.v \(1\) \(22.358\) \(\Q\) None \(0\) \(1\) \(0\) \(-1\) \(+\) \(+\) \(+\) \(q+q^{3}-q^{7}-2q^{9}+q^{11}+6q^{13}+\cdots\)
2800.2.a.w \(1\) \(22.358\) \(\Q\) None \(0\) \(1\) \(0\) \(-1\) \(-\) \(-\) \(+\) \(q+q^{3}-q^{7}-2q^{9}+3q^{11}-q^{13}+\cdots\)
2800.2.a.x \(1\) \(22.358\) \(\Q\) None \(0\) \(1\) \(0\) \(1\) \(-\) \(+\) \(-\) \(q+q^{3}+q^{7}-2q^{9}-3q^{11}-2q^{13}+\cdots\)
2800.2.a.y \(1\) \(22.358\) \(\Q\) None \(0\) \(1\) \(0\) \(1\) \(-\) \(+\) \(-\) \(q+q^{3}+q^{7}-2q^{9}-3q^{11}+q^{13}+\cdots\)
2800.2.a.z \(1\) \(22.358\) \(\Q\) None \(0\) \(1\) \(0\) \(1\) \(-\) \(+\) \(-\) \(q+q^{3}+q^{7}-2q^{9}+3q^{11}-5q^{13}+\cdots\)
2800.2.a.ba \(1\) \(22.358\) \(\Q\) None \(0\) \(2\) \(0\) \(-1\) \(-\) \(-\) \(+\) \(q+2q^{3}-q^{7}+q^{9}-3q^{11}-4q^{13}+\cdots\)
2800.2.a.bb \(1\) \(22.358\) \(\Q\) None \(0\) \(2\) \(0\) \(-1\) \(+\) \(+\) \(+\) \(q+2q^{3}-q^{7}+q^{9}-q^{11}-4q^{13}+\cdots\)
2800.2.a.bc \(1\) \(22.358\) \(\Q\) None \(0\) \(2\) \(0\) \(1\) \(+\) \(-\) \(-\) \(q+2q^{3}+q^{7}+q^{9}-5q^{11}-8q^{17}+\cdots\)
2800.2.a.bd \(1\) \(22.358\) \(\Q\) None \(0\) \(2\) \(0\) \(1\) \(+\) \(+\) \(-\) \(q+2q^{3}+q^{7}+q^{9}+2q^{17}+2q^{19}+\cdots\)
2800.2.a.be \(1\) \(22.358\) \(\Q\) None \(0\) \(3\) \(0\) \(-1\) \(-\) \(+\) \(+\) \(q+3q^{3}-q^{7}+6q^{9}+5q^{11}+3q^{13}+\cdots\)
2800.2.a.bf \(1\) \(22.358\) \(\Q\) None \(0\) \(3\) \(0\) \(1\) \(-\) \(-\) \(-\) \(q+3q^{3}+q^{7}+6q^{9}-3q^{11}-q^{13}+\cdots\)
2800.2.a.bg \(1\) \(22.358\) \(\Q\) None \(0\) \(3\) \(0\) \(1\) \(-\) \(-\) \(-\) \(q+3q^{3}+q^{7}+6q^{9}+5q^{11}-6q^{13}+\cdots\)
2800.2.a.bh \(2\) \(22.358\) \(\Q(\sqrt{5}) \) None \(0\) \(-2\) \(0\) \(2\) \(-\) \(-\) \(-\) \(q+(-1-\beta )q^{3}+q^{7}+(3+2\beta )q^{9}+(-2+\cdots)q^{11}+\cdots\)
2800.2.a.bi \(2\) \(22.358\) \(\Q(\sqrt{17}) \) None \(0\) \(-1\) \(0\) \(-2\) \(-\) \(+\) \(+\) \(q-\beta q^{3}-q^{7}+(1+\beta )q^{9}-\beta q^{11}+(-2+\cdots)q^{13}+\cdots\)
2800.2.a.bj \(2\) \(22.358\) \(\Q(\sqrt{17}) \) None \(0\) \(-1\) \(0\) \(-2\) \(+\) \(-\) \(+\) \(q-\beta q^{3}-q^{7}+(1+\beta )q^{9}+(3-2\beta )q^{11}+\cdots\)
2800.2.a.bk \(2\) \(22.358\) \(\Q(\sqrt{33}) \) None \(0\) \(-1\) \(0\) \(-2\) \(+\) \(+\) \(+\) \(q-\beta q^{3}-q^{7}+(5+\beta )q^{9}+(-4+\beta )q^{11}+\cdots\)
2800.2.a.bl \(2\) \(22.358\) \(\Q(\sqrt{6}) \) None \(0\) \(0\) \(0\) \(-2\) \(-\) \(-\) \(+\) \(q+\beta q^{3}-q^{7}+3q^{9}-2\beta q^{11}+(-2+\cdots)q^{13}+\cdots\)
2800.2.a.bm \(2\) \(22.358\) \(\Q(\sqrt{6}) \) None \(0\) \(0\) \(0\) \(2\) \(-\) \(-\) \(-\) \(q+\beta q^{3}+q^{7}+3q^{9}+2\beta q^{11}+(2+\beta )q^{13}+\cdots\)
2800.2.a.bn \(2\) \(22.358\) \(\Q(\sqrt{17}) \) None \(0\) \(1\) \(0\) \(2\) \(+\) \(+\) \(-\) \(q+\beta q^{3}+q^{7}+(1+\beta )q^{9}+\beta q^{11}+(-2+\cdots)q^{13}+\cdots\)
2800.2.a.bo \(2\) \(22.358\) \(\Q(\sqrt{17}) \) None \(0\) \(1\) \(0\) \(2\) \(+\) \(+\) \(-\) \(q+\beta q^{3}+q^{7}+(1+\beta )q^{9}+(3-2\beta )q^{11}+\cdots\)
2800.2.a.bp \(2\) \(22.358\) \(\Q(\sqrt{5}) \) None \(0\) \(2\) \(0\) \(-2\) \(-\) \(+\) \(+\) \(q+(1+\beta )q^{3}-q^{7}+(3+2\beta )q^{9}+(-2+\cdots)q^{11}+\cdots\)
2800.2.a.bq \(3\) \(22.358\) 3.3.568.1 None \(0\) \(-1\) \(0\) \(3\) \(+\) \(-\) \(-\) \(q-\beta _{1}q^{3}+q^{7}+(1+2\beta _{1}+\beta _{2})q^{9}+\cdots\)
2800.2.a.br \(3\) \(22.358\) 3.3.568.1 None \(0\) \(1\) \(0\) \(-3\) \(+\) \(-\) \(+\) \(q+\beta _{1}q^{3}-q^{7}+(1+2\beta _{1}+\beta _{2})q^{9}+\cdots\)

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

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