Properties

Label 4026.2.a
Level 4026
Weight 2
Character orbit a
Rep. character \(\chi_{4026}(1,\cdot)\)
Character field \(\Q\)
Dimension 101
Newform subspaces 29
Sturm bound 1488
Trace bound 11

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Defining parameters

Level: \( N \) = \( 4026 = 2 \cdot 3 \cdot 11 \cdot 61 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4026.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 29 \)
Sturm bound: \(1488\)
Trace bound: \(11\)
Distinguishing \(T_p\): \(5\), \(7\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(4026))\).

Total New Old
Modular forms 752 101 651
Cusp forms 737 101 636
Eisenstein series 15 0 15

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

\(2\)\(3\)\(11\)\(61\)FrickeDim.
\(+\)\(+\)\(+\)\(+\)\(+\)\(7\)
\(+\)\(+\)\(+\)\(-\)\(-\)\(5\)
\(+\)\(+\)\(-\)\(+\)\(-\)\(7\)
\(+\)\(+\)\(-\)\(-\)\(+\)\(6\)
\(+\)\(-\)\(+\)\(+\)\(-\)\(8\)
\(+\)\(-\)\(+\)\(-\)\(+\)\(5\)
\(+\)\(-\)\(-\)\(+\)\(+\)\(3\)
\(+\)\(-\)\(-\)\(-\)\(-\)\(9\)
\(-\)\(+\)\(+\)\(+\)\(-\)\(6\)
\(-\)\(+\)\(+\)\(-\)\(+\)\(6\)
\(-\)\(+\)\(-\)\(+\)\(+\)\(5\)
\(-\)\(+\)\(-\)\(-\)\(-\)\(8\)
\(-\)\(-\)\(+\)\(+\)\(+\)\(4\)
\(-\)\(-\)\(+\)\(-\)\(-\)\(9\)
\(-\)\(-\)\(-\)\(+\)\(-\)\(10\)
\(-\)\(-\)\(-\)\(-\)\(+\)\(3\)
Plus space\(+\)\(39\)
Minus space\(-\)\(62\)

Trace form

\( 101q + q^{2} + q^{3} + 101q^{4} + 6q^{5} + q^{6} + 8q^{7} + q^{8} + 101q^{9} + O(q^{10}) \) \( 101q + q^{2} + q^{3} + 101q^{4} + 6q^{5} + q^{6} + 8q^{7} + q^{8} + 101q^{9} + 6q^{10} + q^{11} + q^{12} + 14q^{13} + 8q^{14} + 6q^{15} + 101q^{16} + 2q^{17} + q^{18} + 4q^{19} + 6q^{20} + 8q^{21} + q^{22} - 24q^{23} + q^{24} + 99q^{25} + 14q^{26} + q^{27} + 8q^{28} - 50q^{29} + 6q^{30} + q^{32} - 3q^{33} + 10q^{34} - 16q^{35} + 101q^{36} + 6q^{37} + 4q^{38} + 14q^{39} + 6q^{40} - 6q^{41} + 12q^{43} + q^{44} + 6q^{45} + 8q^{46} + 32q^{47} + q^{48} + 117q^{49} + 31q^{50} + 18q^{51} + 14q^{52} + 6q^{53} + q^{54} + 6q^{55} + 8q^{56} + 20q^{57} + 22q^{58} + 12q^{59} + 6q^{60} + q^{61} + 16q^{62} + 8q^{63} + 101q^{64} + 52q^{65} + q^{66} + 44q^{67} + 2q^{68} - 16q^{69} + 16q^{70} + 24q^{71} + q^{72} - 6q^{73} + 38q^{74} + 31q^{75} + 4q^{76} + 8q^{77} - 26q^{78} + 16q^{79} + 6q^{80} + 101q^{81} + 2q^{82} - 12q^{83} + 8q^{84} - 4q^{85} - 4q^{86} - 2q^{87} + q^{88} - 38q^{89} + 6q^{90} - 24q^{92} + 24q^{93} - 16q^{94} - 56q^{95} + q^{96} + 58q^{97} + 25q^{98} + q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(4026))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 3 11 61
4026.2.a.a \(1\) \(32.148\) \(\Q\) None \(-1\) \(-1\) \(1\) \(-2\) \(+\) \(+\) \(+\) \(-\) \(q-q^{2}-q^{3}+q^{4}+q^{5}+q^{6}-2q^{7}+\cdots\)
4026.2.a.b \(1\) \(32.148\) \(\Q\) None \(-1\) \(1\) \(3\) \(-4\) \(+\) \(-\) \(+\) \(-\) \(q-q^{2}+q^{3}+q^{4}+3q^{5}-q^{6}-4q^{7}+\cdots\)
4026.2.a.c \(1\) \(32.148\) \(\Q\) None \(-1\) \(1\) \(3\) \(-4\) \(+\) \(-\) \(-\) \(-\) \(q-q^{2}+q^{3}+q^{4}+3q^{5}-q^{6}-4q^{7}+\cdots\)
4026.2.a.d \(1\) \(32.148\) \(\Q\) None \(-1\) \(1\) \(4\) \(0\) \(+\) \(-\) \(-\) \(-\) \(q-q^{2}+q^{3}+q^{4}+4q^{5}-q^{6}-q^{8}+\cdots\)
4026.2.a.e \(1\) \(32.148\) \(\Q\) None \(-1\) \(1\) \(4\) \(4\) \(+\) \(-\) \(+\) \(+\) \(q-q^{2}+q^{3}+q^{4}+4q^{5}-q^{6}+4q^{7}+\cdots\)
4026.2.a.f \(1\) \(32.148\) \(\Q\) None \(1\) \(-1\) \(-1\) \(2\) \(-\) \(+\) \(+\) \(+\) \(q+q^{2}-q^{3}+q^{4}-q^{5}-q^{6}+2q^{7}+\cdots\)
4026.2.a.g \(1\) \(32.148\) \(\Q\) None \(1\) \(-1\) \(2\) \(4\) \(-\) \(+\) \(-\) \(-\) \(q+q^{2}-q^{3}+q^{4}+2q^{5}-q^{6}+4q^{7}+\cdots\)
4026.2.a.h \(1\) \(32.148\) \(\Q\) None \(1\) \(-1\) \(3\) \(2\) \(-\) \(+\) \(+\) \(+\) \(q+q^{2}-q^{3}+q^{4}+3q^{5}-q^{6}+2q^{7}+\cdots\)
4026.2.a.i \(1\) \(32.148\) \(\Q\) None \(1\) \(1\) \(0\) \(-4\) \(-\) \(-\) \(+\) \(-\) \(q+q^{2}+q^{3}+q^{4}+q^{6}-4q^{7}+q^{8}+\cdots\)
4026.2.a.j \(1\) \(32.148\) \(\Q\) None \(1\) \(1\) \(2\) \(0\) \(-\) \(-\) \(-\) \(+\) \(q+q^{2}+q^{3}+q^{4}+2q^{5}+q^{6}+q^{8}+\cdots\)
4026.2.a.k \(2\) \(32.148\) \(\Q(\sqrt{2}) \) None \(-2\) \(2\) \(-6\) \(4\) \(+\) \(-\) \(+\) \(+\) \(q-q^{2}+q^{3}+q^{4}+(-3+\beta )q^{5}-q^{6}+\cdots\)
4026.2.a.l \(2\) \(32.148\) \(\Q(\sqrt{17}) \) None \(2\) \(-2\) \(-3\) \(-2\) \(-\) \(+\) \(-\) \(-\) \(q+q^{2}-q^{3}+q^{4}+(-1-\beta )q^{5}-q^{6}+\cdots\)
4026.2.a.m \(2\) \(32.148\) \(\Q(\sqrt{5}) \) None \(2\) \(-2\) \(-2\) \(-1\) \(-\) \(+\) \(-\) \(+\) \(q+q^{2}-q^{3}+q^{4}-2\beta q^{5}-q^{6}+(-1+\cdots)q^{7}+\cdots\)
4026.2.a.n \(3\) \(32.148\) \(\Q(\zeta_{18})^+\) None \(-3\) \(3\) \(-3\) \(0\) \(+\) \(-\) \(-\) \(+\) \(q-q^{2}+q^{3}+q^{4}+(-1-\beta _{1})q^{5}-q^{6}+\cdots\)
4026.2.a.o \(3\) \(32.148\) 3.3.1129.1 None \(3\) \(-3\) \(-3\) \(-2\) \(-\) \(+\) \(-\) \(+\) \(q+q^{2}-q^{3}+q^{4}+(-1+\beta _{1})q^{5}-q^{6}+\cdots\)
4026.2.a.p \(3\) \(32.148\) \(\Q(\zeta_{14})^+\) None \(3\) \(3\) \(-5\) \(-6\) \(-\) \(-\) \(-\) \(-\) \(q+q^{2}+q^{3}+q^{4}+(-2-\beta _{2})q^{5}+q^{6}+\cdots\)
4026.2.a.q \(4\) \(32.148\) 4.4.6809.1 None \(-4\) \(-4\) \(0\) \(-2\) \(+\) \(+\) \(+\) \(-\) \(q-q^{2}-q^{3}+q^{4}+\beta _{1}q^{5}+q^{6}+(-1+\cdots)q^{7}+\cdots\)
4026.2.a.r \(4\) \(32.148\) 4.4.7537.1 None \(-4\) \(4\) \(1\) \(-4\) \(+\) \(-\) \(+\) \(-\) \(q-q^{2}+q^{3}+q^{4}-\beta _{2}q^{5}-q^{6}+(-1+\cdots)q^{7}+\cdots\)
4026.2.a.s \(4\) \(32.148\) 4.4.26825.1 None \(4\) \(-4\) \(3\) \(-2\) \(-\) \(+\) \(+\) \(+\) \(q+q^{2}-q^{3}+q^{4}+(1-\beta _{3})q^{5}-q^{6}+\cdots\)
4026.2.a.t \(4\) \(32.148\) 4.4.2777.1 None \(4\) \(4\) \(-4\) \(-6\) \(-\) \(-\) \(+\) \(+\) \(q+q^{2}+q^{3}+q^{4}+(-1-\beta _{2})q^{5}+q^{6}+\cdots\)
4026.2.a.u \(5\) \(32.148\) 5.5.9176805.1 None \(-5\) \(5\) \(-1\) \(-3\) \(+\) \(-\) \(+\) \(+\) \(q-q^{2}+q^{3}+q^{4}-\beta _{1}q^{5}-q^{6}+(-\beta _{1}+\cdots)q^{7}+\cdots\)
4026.2.a.v \(5\) \(32.148\) 5.5.11492689.1 None \(5\) \(-5\) \(7\) \(0\) \(-\) \(+\) \(-\) \(-\) \(q+q^{2}-q^{3}+q^{4}+(1+\beta _{1})q^{5}-q^{6}+\cdots\)
4026.2.a.w \(6\) \(32.148\) 6.6.30998405.1 None \(-6\) \(-6\) \(-1\) \(5\) \(+\) \(+\) \(-\) \(-\) \(q-q^{2}-q^{3}+q^{4}-\beta _{2}q^{5}+q^{6}+(1+\cdots)q^{7}+\cdots\)
4026.2.a.x \(6\) \(32.148\) 6.6.46101901.1 None \(6\) \(-6\) \(-6\) \(1\) \(-\) \(+\) \(+\) \(-\) \(q+q^{2}-q^{3}+q^{4}+(-1+\beta _{3})q^{5}-q^{6}+\cdots\)
4026.2.a.y \(7\) \(32.148\) \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(-7\) \(-7\) \(-2\) \(1\) \(+\) \(+\) \(+\) \(+\) \(q-q^{2}-q^{3}+q^{4}-\beta _{1}q^{5}+q^{6}+\beta _{6}q^{7}+\cdots\)
4026.2.a.z \(7\) \(32.148\) \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(-7\) \(-7\) \(2\) \(-4\) \(+\) \(+\) \(-\) \(+\) \(q-q^{2}-q^{3}+q^{4}-\beta _{2}q^{5}+q^{6}+(\beta _{4}+\cdots)q^{7}+\cdots\)
4026.2.a.ba \(7\) \(32.148\) \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(-7\) \(7\) \(-5\) \(9\) \(+\) \(-\) \(-\) \(-\) \(q-q^{2}+q^{3}+q^{4}+(-1+\beta _{1})q^{5}-q^{6}+\cdots\)
4026.2.a.bb \(8\) \(32.148\) \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(8\) \(8\) \(5\) \(13\) \(-\) \(-\) \(+\) \(-\) \(q+q^{2}+q^{3}+q^{4}+(1+\beta _{5})q^{5}+q^{6}+\cdots\)
4026.2.a.bc \(9\) \(32.148\) \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None \(9\) \(9\) \(8\) \(9\) \(-\) \(-\) \(-\) \(+\) \(q+q^{2}+q^{3}+q^{4}+(1-\beta _{1})q^{5}+q^{6}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(4026))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(4026)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(33))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(61))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(66))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(122))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(183))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(366))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(671))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1342))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2013))\)\(^{\oplus 2}\)