Properties

Label 10000.2.a
Level $10000$
Weight $2$
Character orbit 10000.a
Rep. character $\chi_{10000}(1,\cdot)$
Character field $\Q$
Dimension $232$
Newform subspaces $44$
Sturm bound $3000$
Trace bound $13$

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Defining parameters

Level: \( N \) \(=\) \( 10000 = 2^{4} \cdot 5^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 10000.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 44 \)
Sturm bound: \(3000\)
Trace bound: \(13\)
Distinguishing \(T_p\): \(3\), \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(10000))\).

Total New Old
Modular forms 1590 248 1342
Cusp forms 1411 232 1179
Eisenstein series 179 16 163

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.
\(+\)\(+\)\(+\)\(58\)
\(+\)\(-\)\(-\)\(62\)
\(-\)\(+\)\(-\)\(58\)
\(-\)\(-\)\(+\)\(54\)
Plus space\(+\)\(112\)
Minus space\(-\)\(120\)

Trace form

\( 232 q + 216 q^{9} + O(q^{10}) \) \( 232 q + 216 q^{9} - 4 q^{11} + 4 q^{21} - 4 q^{31} - 12 q^{39} + 4 q^{41} + 184 q^{49} + 16 q^{51} + 4 q^{61} + 12 q^{69} - 4 q^{71} - 60 q^{79} + 172 q^{81} - 64 q^{91} - 12 q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(10000))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 5
10000.2.a.a \(2\) \(79.850\) \(\Q(\sqrt{5}) \) None \(0\) \(-3\) \(0\) \(-6\) \(-\) \(+\) \(q+(-1-\beta )q^{3}-3q^{7}+(-1+3\beta )q^{9}+\cdots\)
10000.2.a.b \(2\) \(79.850\) \(\Q(\sqrt{5}) \) None \(0\) \(-3\) \(0\) \(-1\) \(-\) \(-\) \(q+(-1-\beta )q^{3}+(1-3\beta )q^{7}+(-1+3\beta )q^{9}+\cdots\)
10000.2.a.c \(2\) \(79.850\) \(\Q(\sqrt{5}) \) None \(0\) \(-2\) \(0\) \(1\) \(-\) \(+\) \(q-q^{3}+\beta q^{7}-2q^{9}+(4-2\beta )q^{11}+\cdots\)
10000.2.a.d \(2\) \(79.850\) \(\Q(\sqrt{5}) \) None \(0\) \(-2\) \(0\) \(1\) \(-\) \(+\) \(q-q^{3}+(1-\beta )q^{7}-2q^{9}+3q^{11}+(3+\cdots)q^{13}+\cdots\)
10000.2.a.e \(2\) \(79.850\) \(\Q(\sqrt{5}) \) None \(0\) \(-1\) \(0\) \(1\) \(-\) \(-\) \(q-\beta q^{3}+\beta q^{7}+(-2+\beta )q^{9}+(-2+\cdots)q^{11}+\cdots\)
10000.2.a.f \(2\) \(79.850\) \(\Q(\sqrt{5}) \) None \(0\) \(0\) \(0\) \(-3\) \(+\) \(+\) \(q+(1-2\beta )q^{3}+(-1-\beta )q^{7}+2q^{9}+\cdots\)
10000.2.a.g \(2\) \(79.850\) \(\Q(\sqrt{5}) \) None \(0\) \(0\) \(0\) \(-3\) \(+\) \(+\) \(q+(1-2\beta )q^{3}+(-2+\beta )q^{7}+2q^{9}+\cdots\)
10000.2.a.h \(2\) \(79.850\) \(\Q(\sqrt{5}) \) None \(0\) \(0\) \(0\) \(3\) \(+\) \(-\) \(q+(1-2\beta )q^{3}+(2-\beta )q^{7}+2q^{9}+(-5+\cdots)q^{11}+\cdots\)
10000.2.a.i \(2\) \(79.850\) \(\Q(\sqrt{5}) \) None \(0\) \(0\) \(0\) \(3\) \(+\) \(+\) \(q+(1-2\beta )q^{3}+(1+\beta )q^{7}+2q^{9}+(2+\cdots)q^{11}+\cdots\)
10000.2.a.j \(2\) \(79.850\) \(\Q(\sqrt{5}) \) None \(0\) \(1\) \(0\) \(-1\) \(-\) \(+\) \(q+\beta q^{3}-\beta q^{7}+(-2+\beta )q^{9}+(-2+\cdots)q^{11}+\cdots\)
10000.2.a.k \(2\) \(79.850\) \(\Q(\sqrt{5}) \) None \(0\) \(2\) \(0\) \(-1\) \(-\) \(-\) \(q+q^{3}-\beta q^{7}-2q^{9}+3q^{11}+(1-4\beta )q^{13}+\cdots\)
10000.2.a.l \(2\) \(79.850\) \(\Q(\sqrt{5}) \) None \(0\) \(2\) \(0\) \(-1\) \(-\) \(+\) \(q+q^{3}+(-1+\beta )q^{7}-2q^{9}+(2+2\beta )q^{11}+\cdots\)
10000.2.a.m \(2\) \(79.850\) \(\Q(\sqrt{5}) \) None \(0\) \(3\) \(0\) \(1\) \(-\) \(+\) \(q+(1+\beta )q^{3}+(-1+3\beta )q^{7}+(-1+3\beta )q^{9}+\cdots\)
10000.2.a.n \(2\) \(79.850\) \(\Q(\sqrt{5}) \) None \(0\) \(3\) \(0\) \(6\) \(-\) \(+\) \(q+(1+\beta )q^{3}+3q^{7}+(-1+3\beta )q^{9}+\cdots\)
10000.2.a.o \(4\) \(79.850\) \(\Q(\zeta_{20})^+\) None \(0\) \(-4\) \(0\) \(-8\) \(-\) \(-\) \(q+(-1+\beta _{1})q^{3}+(-2-\beta _{1}-\beta _{3})q^{7}+\cdots\)
10000.2.a.p \(4\) \(79.850\) 4.4.7625.1 None \(0\) \(-3\) \(0\) \(2\) \(+\) \(+\) \(q+(-1+\beta _{1})q^{3}-\beta _{2}q^{7}+(3-\beta _{1}+\beta _{3})q^{9}+\cdots\)
10000.2.a.q \(4\) \(79.850\) 4.4.7625.1 None \(0\) \(-2\) \(0\) \(-2\) \(+\) \(+\) \(q+\beta _{2}q^{3}+(-1-\beta _{3})q^{7}+(-2-\beta _{2}+\cdots)q^{9}+\cdots\)
10000.2.a.r \(4\) \(79.850\) 4.4.108625.1 None \(0\) \(-2\) \(0\) \(3\) \(+\) \(+\) \(q+(-1-\beta _{2})q^{3}+(\beta _{1}-\beta _{2})q^{7}+(-1+\cdots)q^{9}+\cdots\)
10000.2.a.s \(4\) \(79.850\) \(\Q(\zeta_{15})^+\) None \(0\) \(-1\) \(0\) \(-2\) \(-\) \(+\) \(q+(\beta _{2}+\beta _{3})q^{3}+(2\beta _{2}+2\beta _{3})q^{7}+(-1+\cdots)q^{9}+\cdots\)
10000.2.a.t \(4\) \(79.850\) 4.4.7625.1 None \(0\) \(-1\) \(0\) \(-2\) \(-\) \(+\) \(q-\beta _{1}q^{3}+(-1-\beta _{3})q^{7}+(2+\beta _{1}+\beta _{3})q^{9}+\cdots\)
10000.2.a.u \(4\) \(79.850\) 4.4.18625.1 None \(0\) \(-1\) \(0\) \(3\) \(-\) \(+\) \(q+(\beta _{1}-\beta _{2})q^{3}+(2-2\beta _{2}-\beta _{3})q^{7}+\cdots\)
10000.2.a.v \(4\) \(79.850\) 4.4.18625.1 None \(0\) \(1\) \(0\) \(-3\) \(-\) \(-\) \(q+(-\beta _{1}+\beta _{2})q^{3}+(-2+2\beta _{2}+\beta _{3})q^{7}+\cdots\)
10000.2.a.w \(4\) \(79.850\) \(\Q(\zeta_{15})^+\) None \(0\) \(1\) \(0\) \(2\) \(-\) \(-\) \(q+(-\beta _{2}-\beta _{3})q^{3}+(-2\beta _{2}-2\beta _{3})q^{7}+\cdots\)
10000.2.a.x \(4\) \(79.850\) 4.4.7625.1 None \(0\) \(1\) \(0\) \(2\) \(-\) \(+\) \(q+\beta _{1}q^{3}+(1+\beta _{3})q^{7}+(2+\beta _{1}+\beta _{3})q^{9}+\cdots\)
10000.2.a.y \(4\) \(79.850\) 4.4.108625.1 None \(0\) \(2\) \(0\) \(-3\) \(+\) \(-\) \(q+(1+\beta _{2})q^{3}+(-\beta _{1}+\beta _{2})q^{7}+(-1+\cdots)q^{9}+\cdots\)
10000.2.a.z \(4\) \(79.850\) 4.4.7625.1 None \(0\) \(2\) \(0\) \(2\) \(+\) \(+\) \(q-\beta _{2}q^{3}+(1+\beta _{3})q^{7}+(-2-\beta _{2})q^{9}+\cdots\)
10000.2.a.ba \(4\) \(79.850\) 4.4.7625.1 None \(0\) \(3\) \(0\) \(-2\) \(+\) \(-\) \(q+(1-\beta _{1})q^{3}+\beta _{2}q^{7}+(3-\beta _{1}+\beta _{3})q^{9}+\cdots\)
10000.2.a.bb \(4\) \(79.850\) \(\Q(\zeta_{20})^+\) None \(0\) \(4\) \(0\) \(8\) \(-\) \(-\) \(q+(1+\beta _{1})q^{3}+(2-\beta _{1}-\beta _{3})q^{7}+(1+\cdots)q^{9}+\cdots\)
10000.2.a.bc \(6\) \(79.850\) 6.6.103238125.1 None \(0\) \(-1\) \(0\) \(1\) \(-\) \(+\) \(q-\beta _{1}q^{3}+(1-\beta _{3}+\beta _{4})q^{7}+(2+\beta _{1}+\cdots)q^{9}+\cdots\)
10000.2.a.bd \(6\) \(79.850\) 6.6.103238125.1 None \(0\) \(1\) \(0\) \(-1\) \(-\) \(+\) \(q+\beta _{1}q^{3}+(-1+\beta _{3}-\beta _{4})q^{7}+(2+\beta _{1}+\cdots)q^{9}+\cdots\)
10000.2.a.be \(8\) \(79.850\) 8.8.6152203125.1 None \(0\) \(-5\) \(0\) \(-10\) \(-\) \(-\) \(q+(-1+\beta _{5})q^{3}+(-1+\beta _{1}+\beta _{3}-\beta _{5}+\cdots)q^{7}+\cdots\)
10000.2.a.bf \(8\) \(79.850\) 8.8.\(\cdots\).1 None \(0\) \(-5\) \(0\) \(0\) \(-\) \(+\) \(q+(-1+\beta _{1})q^{3}+(-\beta _{2}-\beta _{4}-\beta _{7})q^{7}+\cdots\)
10000.2.a.bg \(8\) \(79.850\) 8.8.3266578125.1 None \(0\) \(-3\) \(0\) \(-2\) \(+\) \(+\) \(q+(-1+\beta _{1}-\beta _{3})q^{3}+(\beta _{4}-\beta _{5})q^{7}+\cdots\)
10000.2.a.bh \(8\) \(79.850\) \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(0\) \(-2\) \(0\) \(-3\) \(+\) \(+\) \(q-\beta _{1}q^{3}+\beta _{2}q^{7}+(\beta _{1}-\beta _{2}+\beta _{3}-\beta _{4}+\cdots)q^{9}+\cdots\)
10000.2.a.bi \(8\) \(79.850\) 8.8.\(\cdots\).2 None \(0\) \(0\) \(0\) \(0\) \(-\) \(-\) \(q+\beta _{1}q^{3}+(-\beta _{1}-\beta _{6})q^{7}+\beta _{2}q^{9}+\cdots\)
10000.2.a.bj \(8\) \(79.850\) 8.8.\(\cdots\).2 None \(0\) \(0\) \(0\) \(0\) \(-\) \(-\) \(q+(\beta _{5}-\beta _{6})q^{3}+(-\beta _{1}-\beta _{6}+\beta _{7})q^{7}+\cdots\)
10000.2.a.bk \(8\) \(79.850\) \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(0\) \(2\) \(0\) \(3\) \(+\) \(+\) \(q+\beta _{1}q^{3}-\beta _{2}q^{7}+(\beta _{1}-\beta _{2}+\beta _{3}-\beta _{4}+\cdots)q^{9}+\cdots\)
10000.2.a.bl \(8\) \(79.850\) 8.8.3266578125.1 None \(0\) \(3\) \(0\) \(2\) \(+\) \(-\) \(q+(1-\beta _{1}+\beta _{3})q^{3}+(-\beta _{4}+\beta _{5})q^{7}+\cdots\)
10000.2.a.bm \(8\) \(79.850\) 8.8.\(\cdots\).1 None \(0\) \(5\) \(0\) \(0\) \(-\) \(-\) \(q+(1-\beta _{1})q^{3}+(\beta _{2}+\beta _{4}+\beta _{7})q^{7}+(2+\cdots)q^{9}+\cdots\)
10000.2.a.bn \(8\) \(79.850\) 8.8.6152203125.1 None \(0\) \(5\) \(0\) \(10\) \(-\) \(+\) \(q+(1-\beta _{5})q^{3}+(1-\beta _{1}-\beta _{3}+\beta _{5}-\beta _{7})q^{7}+\cdots\)
10000.2.a.bo \(12\) \(79.850\) \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(0\) \(-4\) \(0\) \(0\) \(+\) \(+\) \(q-\beta _{6}q^{3}+(1+\beta _{1}-\beta _{2}+\beta _{5})q^{7}+(2+\cdots)q^{9}+\cdots\)
10000.2.a.bp \(12\) \(79.850\) \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(0\) \(4\) \(0\) \(0\) \(+\) \(-\) \(q+\beta _{6}q^{3}+(-1-\beta _{1}+\beta _{2}-\beta _{5})q^{7}+\cdots\)
10000.2.a.bq \(16\) \(79.850\) \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(0\) \(-4\) \(0\) \(-8\) \(+\) \(-\) \(q-\beta _{1}q^{3}+(-1+\beta _{8})q^{7}+(1+\beta _{2})q^{9}+\cdots\)
10000.2.a.br \(16\) \(79.850\) \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(0\) \(4\) \(0\) \(8\) \(+\) \(-\) \(q+\beta _{1}q^{3}+(1-\beta _{8})q^{7}+(1+\beta _{2})q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(10000))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(10000)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(80))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(100))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(125))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(200))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(250))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(400))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(500))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(625))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1000))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1250))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2000))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2500))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(5000))\)\(^{\oplus 2}\)