Properties

Label 4026.2.a
Level $4026$
Weight $2$
Character orbit 4026.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

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

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

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 3 11 61
4026.2.a.a 4026.a 1.a $1$ $32.148$ \(\Q\) None \(-1\) \(-1\) \(1\) \(-2\) $+$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+q^{5}+q^{6}-2q^{7}+\cdots\)
4026.2.a.b 4026.a 1.a $1$ $32.148$ \(\Q\) None \(-1\) \(1\) \(3\) \(-4\) $+$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}+3q^{5}-q^{6}-4q^{7}+\cdots\)
4026.2.a.c 4026.a 1.a $1$ $32.148$ \(\Q\) None \(-1\) \(1\) \(3\) \(-4\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}+3q^{5}-q^{6}-4q^{7}+\cdots\)
4026.2.a.d 4026.a 1.a $1$ $32.148$ \(\Q\) None \(-1\) \(1\) \(4\) \(0\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}+4q^{5}-q^{6}-q^{8}+\cdots\)
4026.2.a.e 4026.a 1.a $1$ $32.148$ \(\Q\) None \(-1\) \(1\) \(4\) \(4\) $+$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}+4q^{5}-q^{6}+4q^{7}+\cdots\)
4026.2.a.f 4026.a 1.a $1$ $32.148$ \(\Q\) None \(1\) \(-1\) \(-1\) \(2\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}-q^{5}-q^{6}+2q^{7}+\cdots\)
4026.2.a.g 4026.a 1.a $1$ $32.148$ \(\Q\) None \(1\) \(-1\) \(2\) \(4\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}+2q^{5}-q^{6}+4q^{7}+\cdots\)
4026.2.a.h 4026.a 1.a $1$ $32.148$ \(\Q\) None \(1\) \(-1\) \(3\) \(2\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}+3q^{5}-q^{6}+2q^{7}+\cdots\)
4026.2.a.i 4026.a 1.a $1$ $32.148$ \(\Q\) None \(1\) \(1\) \(0\) \(-4\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}+q^{6}-4q^{7}+q^{8}+\cdots\)
4026.2.a.j 4026.a 1.a $1$ $32.148$ \(\Q\) None \(1\) \(1\) \(2\) \(0\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}+2q^{5}+q^{6}+q^{8}+\cdots\)
4026.2.a.k 4026.a 1.a $2$ $32.148$ \(\Q(\sqrt{2}) \) None \(-2\) \(2\) \(-6\) \(4\) $+$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}+(-3+\beta )q^{5}-q^{6}+\cdots\)
4026.2.a.l 4026.a 1.a $2$ $32.148$ \(\Q(\sqrt{17}) \) None \(2\) \(-2\) \(-3\) \(-2\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}+(-1-\beta )q^{5}-q^{6}+\cdots\)
4026.2.a.m 4026.a 1.a $2$ $32.148$ \(\Q(\sqrt{5}) \) None \(2\) \(-2\) \(-2\) \(-1\) $-$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}-2\beta q^{5}-q^{6}+(-1+\cdots)q^{7}+\cdots\)
4026.2.a.n 4026.a 1.a $3$ $32.148$ \(\Q(\zeta_{18})^+\) None \(-3\) \(3\) \(-3\) \(0\) $+$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}+(-1-\beta _{1})q^{5}-q^{6}+\cdots\)
4026.2.a.o 4026.a 1.a $3$ $32.148$ 3.3.1129.1 None \(3\) \(-3\) \(-3\) \(-2\) $-$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}+(-1+\beta _{1})q^{5}-q^{6}+\cdots\)
4026.2.a.p 4026.a 1.a $3$ $32.148$ \(\Q(\zeta_{14})^+\) None \(3\) \(3\) \(-5\) \(-6\) $-$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}+(-2-\beta _{2})q^{5}+q^{6}+\cdots\)
4026.2.a.q 4026.a 1.a $4$ $32.148$ 4.4.6809.1 None \(-4\) \(-4\) \(0\) \(-2\) $+$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+\beta _{1}q^{5}+q^{6}+(-1+\cdots)q^{7}+\cdots\)
4026.2.a.r 4026.a 1.a $4$ $32.148$ 4.4.7537.1 None \(-4\) \(4\) \(1\) \(-4\) $+$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}-\beta _{2}q^{5}-q^{6}+(-1+\cdots)q^{7}+\cdots\)
4026.2.a.s 4026.a 1.a $4$ $32.148$ 4.4.26825.1 None \(4\) \(-4\) \(3\) \(-2\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}+(1-\beta _{3})q^{5}-q^{6}+\cdots\)
4026.2.a.t 4026.a 1.a $4$ $32.148$ 4.4.2777.1 None \(4\) \(4\) \(-4\) \(-6\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}+(-1-\beta _{2})q^{5}+q^{6}+\cdots\)
4026.2.a.u 4026.a 1.a $5$ $32.148$ 5.5.9176805.1 None \(-5\) \(5\) \(-1\) \(-3\) $+$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}-\beta _{1}q^{5}-q^{6}+(-\beta _{1}+\cdots)q^{7}+\cdots\)
4026.2.a.v 4026.a 1.a $5$ $32.148$ 5.5.11492689.1 None \(5\) \(-5\) \(7\) \(0\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}+(1+\beta _{1})q^{5}-q^{6}+\cdots\)
4026.2.a.w 4026.a 1.a $6$ $32.148$ 6.6.30998405.1 None \(-6\) \(-6\) \(-1\) \(5\) $+$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}-\beta _{2}q^{5}+q^{6}+(1+\cdots)q^{7}+\cdots\)
4026.2.a.x 4026.a 1.a $6$ $32.148$ 6.6.46101901.1 None \(6\) \(-6\) \(-6\) \(1\) $-$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}+(-1+\beta _{3})q^{5}-q^{6}+\cdots\)
4026.2.a.y 4026.a 1.a $7$ $32.148$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(-7\) \(-7\) \(-2\) \(1\) $+$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}-\beta _{1}q^{5}+q^{6}+\beta _{6}q^{7}+\cdots\)
4026.2.a.z 4026.a 1.a $7$ $32.148$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(-7\) \(-7\) \(2\) \(-4\) $+$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}-\beta _{2}q^{5}+q^{6}+(\beta _{4}+\cdots)q^{7}+\cdots\)
4026.2.a.ba 4026.a 1.a $7$ $32.148$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(-7\) \(7\) \(-5\) \(9\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}+(-1+\beta _{1})q^{5}-q^{6}+\cdots\)
4026.2.a.bb 4026.a 1.a $8$ $32.148$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(8\) \(8\) \(5\) \(13\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}+(1+\beta _{5})q^{5}+q^{6}+\cdots\)
4026.2.a.bc 4026.a 1.a $9$ $32.148$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None \(9\) \(9\) \(8\) \(9\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}+(1-\beta _{1})q^{5}+q^{6}+\cdots\)

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

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