Properties

Label 1600.2.a
Level 1600
Weight 2
Character orbit a
Rep. character \(\chi_{1600}(1,\cdot)\)
Character field \(\Q\)
Dimension 35
Newform subspaces 30
Sturm bound 480
Trace bound 13

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Defining parameters

Level: \( N \) = \( 1600 = 2^{6} \cdot 5^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 1600.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 30 \)
Sturm bound: \(480\)
Trace bound: \(13\)
Distinguishing \(T_p\): \(3\), \(7\), \(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1600))\).

Total New Old
Modular forms 276 41 235
Cusp forms 205 35 170
Eisenstein series 71 6 65

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

\(2\)\(5\)FrickeDim.
\(+\)\(+\)\(+\)\(8\)
\(+\)\(-\)\(-\)\(10\)
\(-\)\(+\)\(-\)\(9\)
\(-\)\(-\)\(+\)\(8\)
Plus space\(+\)\(16\)
Minus space\(-\)\(19\)

Trace form

\( 35q + 31q^{9} + O(q^{10}) \) \( 35q + 31q^{9} - 10q^{13} - 2q^{17} - 8q^{21} + 6q^{29} + 16q^{33} - 2q^{37} - 2q^{41} + 19q^{49} + 30q^{53} + 16q^{57} - 18q^{61} + 24q^{69} + 6q^{73} + 16q^{77} + 3q^{81} - 10q^{89} + 32q^{93} - 18q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1600))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 5
1600.2.a.a \(1\) \(12.776\) \(\Q\) None \(0\) \(-3\) \(0\) \(-2\) \(+\) \(+\) \(q-3q^{3}-2q^{7}+6q^{9}-q^{11}+4q^{13}+\cdots\)
1600.2.a.b \(1\) \(12.776\) \(\Q\) None \(0\) \(-3\) \(0\) \(-2\) \(-\) \(-\) \(q-3q^{3}-2q^{7}+6q^{9}+q^{11}-4q^{13}+\cdots\)
1600.2.a.c \(1\) \(12.776\) \(\Q\) None \(0\) \(-2\) \(0\) \(-2\) \(+\) \(+\) \(q-2q^{3}-2q^{7}+q^{9}+2q^{13}+6q^{17}+\cdots\)
1600.2.a.d \(1\) \(12.776\) \(\Q\) None \(0\) \(-2\) \(0\) \(2\) \(-\) \(-\) \(q-2q^{3}+2q^{7}+q^{9}-4q^{11}+4q^{13}+\cdots\)
1600.2.a.e \(1\) \(12.776\) \(\Q\) None \(0\) \(-2\) \(0\) \(2\) \(+\) \(+\) \(q-2q^{3}+2q^{7}+q^{9}+4q^{11}-6q^{13}+\cdots\)
1600.2.a.f \(1\) \(12.776\) \(\Q\) None \(0\) \(-2\) \(0\) \(2\) \(+\) \(-\) \(q-2q^{3}+2q^{7}+q^{9}+4q^{11}-4q^{13}+\cdots\)
1600.2.a.g \(1\) \(12.776\) \(\Q\) None \(0\) \(-1\) \(0\) \(-2\) \(+\) \(-\) \(q-q^{3}-2q^{7}-2q^{9}-5q^{11}+5q^{17}+\cdots\)
1600.2.a.h \(1\) \(12.776\) \(\Q\) None \(0\) \(-1\) \(0\) \(-2\) \(+\) \(+\) \(q-q^{3}-2q^{7}-2q^{9}+5q^{11}-5q^{17}+\cdots\)
1600.2.a.i \(1\) \(12.776\) \(\Q\) None \(0\) \(-1\) \(0\) \(2\) \(-\) \(+\) \(q-q^{3}+2q^{7}-2q^{9}-3q^{11}-4q^{13}+\cdots\)
1600.2.a.j \(1\) \(12.776\) \(\Q\) None \(0\) \(-1\) \(0\) \(2\) \(+\) \(-\) \(q-q^{3}+2q^{7}-2q^{9}+3q^{11}+4q^{13}+\cdots\)
1600.2.a.k \(1\) \(12.776\) \(\Q\) None \(0\) \(0\) \(0\) \(-4\) \(-\) \(+\) \(q-4q^{7}-3q^{9}+4q^{11}-2q^{13}-2q^{17}+\cdots\)
1600.2.a.l \(1\) \(12.776\) \(\Q\) \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(0\) \(0\) \(-\) \(-\) \(q-3q^{9}-4q^{13}+8q^{17}-10q^{29}+\cdots\)
1600.2.a.m \(1\) \(12.776\) \(\Q\) \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(0\) \(0\) \(-\) \(-\) \(q-3q^{9}+4q^{13}-8q^{17}-10q^{29}+\cdots\)
1600.2.a.n \(1\) \(12.776\) \(\Q\) \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(0\) \(0\) \(-\) \(+\) \(q-3q^{9}+6q^{13}-2q^{17}+10q^{29}+\cdots\)
1600.2.a.o \(1\) \(12.776\) \(\Q\) None \(0\) \(0\) \(0\) \(4\) \(+\) \(+\) \(q+4q^{7}-3q^{9}-4q^{11}-2q^{13}-2q^{17}+\cdots\)
1600.2.a.p \(1\) \(12.776\) \(\Q\) None \(0\) \(1\) \(0\) \(-2\) \(-\) \(-\) \(q+q^{3}-2q^{7}-2q^{9}-3q^{11}+4q^{13}+\cdots\)
1600.2.a.q \(1\) \(12.776\) \(\Q\) None \(0\) \(1\) \(0\) \(-2\) \(+\) \(+\) \(q+q^{3}-2q^{7}-2q^{9}+3q^{11}-4q^{13}+\cdots\)
1600.2.a.r \(1\) \(12.776\) \(\Q\) None \(0\) \(1\) \(0\) \(2\) \(+\) \(+\) \(q+q^{3}+2q^{7}-2q^{9}-5q^{11}-5q^{17}+\cdots\)
1600.2.a.s \(1\) \(12.776\) \(\Q\) None \(0\) \(1\) \(0\) \(2\) \(+\) \(-\) \(q+q^{3}+2q^{7}-2q^{9}+5q^{11}+5q^{17}+\cdots\)
1600.2.a.t \(1\) \(12.776\) \(\Q\) None \(0\) \(2\) \(0\) \(-2\) \(+\) \(+\) \(q+2q^{3}-2q^{7}+q^{9}-4q^{11}-6q^{13}+\cdots\)
1600.2.a.u \(1\) \(12.776\) \(\Q\) None \(0\) \(2\) \(0\) \(-2\) \(-\) \(-\) \(q+2q^{3}-2q^{7}+q^{9}-4q^{11}-4q^{13}+\cdots\)
1600.2.a.v \(1\) \(12.776\) \(\Q\) None \(0\) \(2\) \(0\) \(-2\) \(+\) \(-\) \(q+2q^{3}-2q^{7}+q^{9}+4q^{11}+4q^{13}+\cdots\)
1600.2.a.w \(1\) \(12.776\) \(\Q\) None \(0\) \(2\) \(0\) \(2\) \(-\) \(+\) \(q+2q^{3}+2q^{7}+q^{9}+2q^{13}+6q^{17}+\cdots\)
1600.2.a.x \(1\) \(12.776\) \(\Q\) None \(0\) \(3\) \(0\) \(2\) \(+\) \(-\) \(q+3q^{3}+2q^{7}+6q^{9}-q^{11}-4q^{13}+\cdots\)
1600.2.a.y \(1\) \(12.776\) \(\Q\) None \(0\) \(3\) \(0\) \(2\) \(-\) \(+\) \(q+3q^{3}+2q^{7}+6q^{9}+q^{11}+4q^{13}+\cdots\)
1600.2.a.z \(2\) \(12.776\) \(\Q(\sqrt{5}) \) \(\Q(\sqrt{-5}) \) \(0\) \(-2\) \(0\) \(6\) \(+\) \(-\) \(q+(-1-\beta )q^{3}+(3-\beta )q^{7}+(3+2\beta )q^{9}+\cdots\)
1600.2.a.ba \(2\) \(12.776\) \(\Q(\sqrt{5}) \) None \(0\) \(0\) \(0\) \(0\) \(-\) \(-\) \(q-\beta q^{3}+2\beta q^{7}+2q^{9}-\beta q^{11}-4q^{13}+\cdots\)
1600.2.a.bb \(2\) \(12.776\) \(\Q(\sqrt{5}) \) None \(0\) \(0\) \(0\) \(0\) \(-\) \(+\) \(q-\beta q^{3}+2\beta q^{7}+2q^{9}+\beta q^{11}+4q^{13}+\cdots\)
1600.2.a.bc \(2\) \(12.776\) \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(0\) \(0\) \(-\) \(+\) \(q+\beta q^{3}+\beta q^{7}+5q^{9}+2\beta q^{11}-2q^{13}+\cdots\)
1600.2.a.bd \(2\) \(12.776\) \(\Q(\sqrt{5}) \) \(\Q(\sqrt{-5}) \) \(0\) \(2\) \(0\) \(-6\) \(+\) \(-\) \(q+(1+\beta )q^{3}+(-3+\beta )q^{7}+(3+2\beta )q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1600))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1600)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(64))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(80))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(100))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(160))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(200))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(320))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(400))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(800))\)\(^{\oplus 2}\)