Properties

Label 5600.2.a
Level $5600$
Weight $2$
Character orbit 5600.a
Rep. character $\chi_{5600}(1,\cdot)$
Character field $\Q$
Dimension $114$
Newform subspaces $52$
Sturm bound $1920$
Trace bound $13$

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Defining parameters

Level: \( N \) \(=\) \( 5600 = 2^{5} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5600.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 52 \)
Sturm bound: \(1920\)
Trace bound: \(13\)
Distinguishing \(T_p\): \(3\), \(11\), \(13\), \(19\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(5600))\).

Total New Old
Modular forms 1008 114 894
Cusp forms 913 114 799
Eisenstein series 95 0 95

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.
\(+\)\(+\)\(+\)\(+\)\(14\)
\(+\)\(+\)\(-\)\(-\)\(16\)
\(+\)\(-\)\(+\)\(-\)\(16\)
\(+\)\(-\)\(-\)\(+\)\(12\)
\(-\)\(+\)\(+\)\(-\)\(13\)
\(-\)\(+\)\(-\)\(+\)\(11\)
\(-\)\(-\)\(+\)\(+\)\(14\)
\(-\)\(-\)\(-\)\(-\)\(18\)
Plus space\(+\)\(51\)
Minus space\(-\)\(63\)

Trace form

\( 114q + 122q^{9} + O(q^{10}) \) \( 114q + 122q^{9} - 4q^{13} + 20q^{17} - 4q^{29} - 16q^{33} - 20q^{37} - 12q^{41} + 114q^{49} + 44q^{53} - 32q^{57} - 20q^{61} + 48q^{69} + 36q^{73} + 98q^{81} + 36q^{89} + 32q^{93} + 52q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(5600))\) 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
5600.2.a.a \(1\) \(44.716\) \(\Q\) None \(0\) \(-3\) \(0\) \(-1\) \(-\) \(+\) \(+\) \(q-3q^{3}-q^{7}+6q^{9}+q^{11}+q^{13}+\cdots\)
5600.2.a.b \(1\) \(44.716\) \(\Q\) None \(0\) \(-3\) \(0\) \(-1\) \(+\) \(+\) \(+\) \(q-3q^{3}-q^{7}+6q^{9}+3q^{11}-q^{13}+\cdots\)
5600.2.a.c \(1\) \(44.716\) \(\Q\) None \(0\) \(-2\) \(0\) \(-1\) \(-\) \(+\) \(+\) \(q-2q^{3}-q^{7}+q^{9}+4q^{11}+4q^{13}+\cdots\)
5600.2.a.d \(1\) \(44.716\) \(\Q\) None \(0\) \(-1\) \(0\) \(1\) \(+\) \(-\) \(-\) \(q-q^{3}+q^{7}-2q^{9}-3q^{11}+2q^{13}+\cdots\)
5600.2.a.e \(1\) \(44.716\) \(\Q\) None \(0\) \(-1\) \(0\) \(1\) \(+\) \(+\) \(-\) \(q-q^{3}+q^{7}-2q^{9}-3q^{11}+7q^{13}+\cdots\)
5600.2.a.f \(1\) \(44.716\) \(\Q\) None \(0\) \(-1\) \(0\) \(1\) \(-\) \(+\) \(-\) \(q-q^{3}+q^{7}-2q^{9}-q^{11}-3q^{13}+\cdots\)
5600.2.a.g \(1\) \(44.716\) \(\Q\) None \(0\) \(-1\) \(0\) \(1\) \(-\) \(+\) \(-\) \(q-q^{3}+q^{7}-2q^{9}-q^{11}+q^{13}-q^{17}+\cdots\)
5600.2.a.h \(1\) \(44.716\) \(\Q\) None \(0\) \(-1\) \(0\) \(1\) \(+\) \(+\) \(-\) \(q-q^{3}+q^{7}-2q^{9}+3q^{11}-2q^{13}+\cdots\)
5600.2.a.i \(1\) \(44.716\) \(\Q\) None \(0\) \(-1\) \(0\) \(1\) \(-\) \(+\) \(-\) \(q-q^{3}+q^{7}-2q^{9}+5q^{11}-5q^{13}+\cdots\)
5600.2.a.j \(1\) \(44.716\) \(\Q\) None \(0\) \(0\) \(0\) \(-1\) \(+\) \(+\) \(+\) \(q-q^{7}-3q^{9}+2q^{13}-2q^{17}+8q^{19}+\cdots\)
5600.2.a.k \(1\) \(44.716\) \(\Q\) None \(0\) \(0\) \(0\) \(-1\) \(-\) \(+\) \(+\) \(q-q^{7}-3q^{9}+4q^{11}-2q^{13}-6q^{17}+\cdots\)
5600.2.a.l \(1\) \(44.716\) \(\Q\) None \(0\) \(0\) \(0\) \(1\) \(+\) \(+\) \(-\) \(q+q^{7}-3q^{9}-4q^{11}-2q^{13}-6q^{17}+\cdots\)
5600.2.a.m \(1\) \(44.716\) \(\Q\) None \(0\) \(0\) \(0\) \(1\) \(+\) \(+\) \(-\) \(q+q^{7}-3q^{9}+2q^{13}-2q^{17}-8q^{19}+\cdots\)
5600.2.a.n \(1\) \(44.716\) \(\Q\) None \(0\) \(1\) \(0\) \(-1\) \(-\) \(+\) \(+\) \(q+q^{3}-q^{7}-2q^{9}-5q^{11}-5q^{13}+\cdots\)
5600.2.a.o \(1\) \(44.716\) \(\Q\) None \(0\) \(1\) \(0\) \(-1\) \(-\) \(+\) \(+\) \(q+q^{3}-q^{7}-2q^{9}-3q^{11}-2q^{13}+\cdots\)
5600.2.a.p \(1\) \(44.716\) \(\Q\) None \(0\) \(1\) \(0\) \(-1\) \(+\) \(+\) \(+\) \(q+q^{3}-q^{7}-2q^{9}+q^{11}-3q^{13}+\cdots\)
5600.2.a.q \(1\) \(44.716\) \(\Q\) None \(0\) \(1\) \(0\) \(-1\) \(-\) \(+\) \(+\) \(q+q^{3}-q^{7}-2q^{9}+q^{11}+q^{13}-q^{17}+\cdots\)
5600.2.a.r \(1\) \(44.716\) \(\Q\) None \(0\) \(1\) \(0\) \(-1\) \(-\) \(-\) \(+\) \(q+q^{3}-q^{7}-2q^{9}+3q^{11}+2q^{13}+\cdots\)
5600.2.a.s \(1\) \(44.716\) \(\Q\) None \(0\) \(1\) \(0\) \(-1\) \(-\) \(+\) \(+\) \(q+q^{3}-q^{7}-2q^{9}+3q^{11}+7q^{13}+\cdots\)
5600.2.a.t \(1\) \(44.716\) \(\Q\) None \(0\) \(2\) \(0\) \(1\) \(-\) \(+\) \(-\) \(q+2q^{3}+q^{7}+q^{9}-4q^{11}+4q^{13}+\cdots\)
5600.2.a.u \(1\) \(44.716\) \(\Q\) None \(0\) \(3\) \(0\) \(1\) \(+\) \(+\) \(-\) \(q+3q^{3}+q^{7}+6q^{9}-3q^{11}-q^{13}+\cdots\)
5600.2.a.v \(1\) \(44.716\) \(\Q\) None \(0\) \(3\) \(0\) \(1\) \(+\) \(+\) \(-\) \(q+3q^{3}+q^{7}+6q^{9}-q^{11}+q^{13}+\cdots\)
5600.2.a.w \(2\) \(44.716\) \(\Q(\sqrt{5}) \) None \(0\) \(-2\) \(0\) \(-2\) \(-\) \(-\) \(+\) \(q-q^{3}-q^{7}-2q^{9}-\beta q^{11}+\beta q^{13}+\cdots\)
5600.2.a.x \(2\) \(44.716\) \(\Q(\sqrt{3}) \) None \(0\) \(-2\) \(0\) \(2\) \(+\) \(-\) \(-\) \(q+(-1+\beta )q^{3}+q^{7}+(1-2\beta )q^{9}+(-2+\cdots)q^{11}+\cdots\)
5600.2.a.y \(2\) \(44.716\) \(\Q(\sqrt{3}) \) None \(0\) \(-2\) \(0\) \(2\) \(-\) \(+\) \(-\) \(q+(-1+\beta )q^{3}+q^{7}+(1-2\beta )q^{9}+(2+\cdots)q^{11}+\cdots\)
5600.2.a.z \(2\) \(44.716\) \(\Q(\sqrt{5}) \) None \(0\) \(-2\) \(0\) \(2\) \(+\) \(+\) \(-\) \(q+(-1-\beta )q^{3}+q^{7}+(3+2\beta )q^{9}+(2+\cdots)q^{11}+\cdots\)
5600.2.a.ba \(2\) \(44.716\) \(\Q(\sqrt{17}) \) None \(0\) \(-1\) \(0\) \(2\) \(-\) \(+\) \(-\) \(q-\beta q^{3}+q^{7}+(1+\beta )q^{9}+\beta q^{11}+(-2+\cdots)q^{13}+\cdots\)
5600.2.a.bb \(2\) \(44.716\) \(\Q(\sqrt{17}) \) None \(0\) \(-1\) \(0\) \(2\) \(+\) \(+\) \(-\) \(q-\beta q^{3}+q^{7}+(1+\beta )q^{9}+(4-\beta )q^{11}+\cdots\)
5600.2.a.bc \(2\) \(44.716\) \(\Q(\sqrt{6}) \) None \(0\) \(0\) \(0\) \(-2\) \(-\) \(-\) \(+\) \(q+\beta q^{3}-q^{7}+3q^{9}+(-1-\beta )q^{11}+\cdots\)
5600.2.a.bd \(2\) \(44.716\) \(\Q(\sqrt{6}) \) None \(0\) \(0\) \(0\) \(-2\) \(-\) \(+\) \(+\) \(q+\beta q^{3}-q^{7}+3q^{9}+(1+\beta )q^{11}+(4+\cdots)q^{13}+\cdots\)
5600.2.a.be \(2\) \(44.716\) \(\Q(\sqrt{6}) \) None \(0\) \(0\) \(0\) \(2\) \(+\) \(+\) \(-\) \(q+\beta q^{3}+q^{7}+3q^{9}+(-1+\beta )q^{11}+\cdots\)
5600.2.a.bf \(2\) \(44.716\) \(\Q(\sqrt{6}) \) None \(0\) \(0\) \(0\) \(2\) \(+\) \(-\) \(-\) \(q+\beta q^{3}+q^{7}+3q^{9}+(1-\beta )q^{11}+(-4+\cdots)q^{13}+\cdots\)
5600.2.a.bg \(2\) \(44.716\) \(\Q(\sqrt{17}) \) None \(0\) \(1\) \(0\) \(-2\) \(+\) \(+\) \(+\) \(q+\beta q^{3}-q^{7}+(1+\beta )q^{9}+(-4+\beta )q^{11}+\cdots\)
5600.2.a.bh \(2\) \(44.716\) \(\Q(\sqrt{17}) \) None \(0\) \(1\) \(0\) \(-2\) \(+\) \(+\) \(+\) \(q+\beta q^{3}-q^{7}+(1+\beta )q^{9}-\beta q^{11}+(-2+\cdots)q^{13}+\cdots\)
5600.2.a.bi \(2\) \(44.716\) \(\Q(\sqrt{3}) \) None \(0\) \(2\) \(0\) \(-2\) \(-\) \(+\) \(+\) \(q+(1+\beta )q^{3}-q^{7}+(1+2\beta )q^{9}+(-2+\cdots)q^{11}+\cdots\)
5600.2.a.bj \(2\) \(44.716\) \(\Q(\sqrt{3}) \) None \(0\) \(2\) \(0\) \(-2\) \(+\) \(-\) \(+\) \(q+(1+\beta )q^{3}-q^{7}+(1+2\beta )q^{9}+(2-\beta )q^{11}+\cdots\)
5600.2.a.bk \(2\) \(44.716\) \(\Q(\sqrt{5}) \) None \(0\) \(2\) \(0\) \(-2\) \(-\) \(+\) \(+\) \(q+(1+\beta )q^{3}-q^{7}+(3+2\beta )q^{9}+(-2+\cdots)q^{11}+\cdots\)
5600.2.a.bl \(2\) \(44.716\) \(\Q(\sqrt{5}) \) None \(0\) \(2\) \(0\) \(2\) \(+\) \(-\) \(-\) \(q+q^{3}+q^{7}-2q^{9}+\beta q^{11}+\beta q^{13}+\cdots\)
5600.2.a.bm \(3\) \(44.716\) 3.3.1016.1 None \(0\) \(-1\) \(0\) \(-3\) \(+\) \(-\) \(+\) \(q-\beta _{1}q^{3}-q^{7}+(1+\beta _{1}+\beta _{2})q^{9}+(-3+\cdots)q^{11}+\cdots\)
5600.2.a.bn \(3\) \(44.716\) 3.3.1016.1 None \(0\) \(-1\) \(0\) \(-3\) \(+\) \(+\) \(+\) \(q-\beta _{1}q^{3}-q^{7}+(1+\beta _{1}+\beta _{2})q^{9}+(3+\cdots)q^{11}+\cdots\)
5600.2.a.bo \(3\) \(44.716\) 3.3.1016.1 None \(0\) \(1\) \(0\) \(3\) \(-\) \(+\) \(-\) \(q+\beta _{1}q^{3}+q^{7}+(1+\beta _{1}+\beta _{2})q^{9}+(-3+\cdots)q^{11}+\cdots\)
5600.2.a.bp \(3\) \(44.716\) 3.3.1016.1 None \(0\) \(1\) \(0\) \(3\) \(-\) \(-\) \(-\) \(q+\beta _{1}q^{3}+q^{7}+(1+\beta _{1}+\beta _{2})q^{9}+(3+\cdots)q^{11}+\cdots\)
5600.2.a.bq \(4\) \(44.716\) 4.4.48396.1 None \(0\) \(-2\) \(0\) \(-4\) \(+\) \(+\) \(+\) \(q-\beta _{1}q^{3}-q^{7}+(1+\beta _{1}+\beta _{2})q^{9}+(-1+\cdots)q^{11}+\cdots\)
5600.2.a.br \(4\) \(44.716\) 4.4.48396.1 None \(0\) \(-2\) \(0\) \(-4\) \(-\) \(-\) \(+\) \(q-\beta _{1}q^{3}-q^{7}+(1+\beta _{1}+\beta _{2})q^{9}+(1+\cdots)q^{11}+\cdots\)
5600.2.a.bs \(4\) \(44.716\) 4.4.48396.1 None \(0\) \(2\) \(0\) \(4\) \(-\) \(-\) \(-\) \(q+\beta _{1}q^{3}+q^{7}+(1+\beta _{1}+\beta _{2})q^{9}+(-1+\cdots)q^{11}+\cdots\)
5600.2.a.bt \(4\) \(44.716\) 4.4.48396.1 None \(0\) \(2\) \(0\) \(4\) \(+\) \(+\) \(-\) \(q+\beta _{1}q^{3}+q^{7}+(1+\beta _{1}+\beta _{2})q^{9}+(1+\cdots)q^{11}+\cdots\)
5600.2.a.bu \(5\) \(44.716\) 5.5.504568.1 None \(0\) \(-2\) \(0\) \(5\) \(+\) \(-\) \(-\) \(q-\beta _{2}q^{3}+q^{7}+(1+\beta _{2}+\beta _{3})q^{9}+(-1+\cdots)q^{11}+\cdots\)
5600.2.a.bv \(5\) \(44.716\) 5.5.504568.1 None \(0\) \(-2\) \(0\) \(5\) \(-\) \(-\) \(-\) \(q-\beta _{2}q^{3}+q^{7}+(1+\beta _{2}+\beta _{3})q^{9}+(1+\cdots)q^{11}+\cdots\)
5600.2.a.bw \(5\) \(44.716\) 5.5.504568.1 None \(0\) \(2\) \(0\) \(-5\) \(-\) \(-\) \(+\) \(q+\beta _{2}q^{3}-q^{7}+(1+\beta _{2}+\beta _{3})q^{9}+(-1+\cdots)q^{11}+\cdots\)
5600.2.a.bx \(5\) \(44.716\) 5.5.504568.1 None \(0\) \(2\) \(0\) \(-5\) \(+\) \(-\) \(+\) \(q+\beta _{2}q^{3}-q^{7}+(1+\beta _{2}+\beta _{3})q^{9}+(1+\cdots)q^{11}+\cdots\)
5600.2.a.by \(6\) \(44.716\) 6.6.29935424.1 None \(0\) \(-2\) \(0\) \(-6\) \(+\) \(-\) \(+\) \(q-\beta _{1}q^{3}-q^{7}+(1+\beta _{1}-\beta _{2})q^{9}+\beta _{4}q^{11}+\cdots\)
5600.2.a.bz \(6\) \(44.716\) 6.6.29935424.1 None \(0\) \(2\) \(0\) \(6\) \(-\) \(-\) \(-\) \(q+\beta _{1}q^{3}+q^{7}+(1+\beta _{1}-\beta _{2})q^{9}+\beta _{4}q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(5600))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(5600)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 15}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(56))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(70))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(80))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(100))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(112))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(140))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(160))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(175))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(200))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(224))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(280))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(350))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(400))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(560))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(700))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(800))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1120))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1400))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2800))\)\(^{\oplus 2}\)