Properties

Label 4012.2.a
Level 4012
Weight 2
Character orbit a
Rep. character \(\chi_{4012}(1,\cdot)\)
Character field \(\Q\)
Dimension 76
Newform subspaces 10
Sturm bound 1080
Trace bound 5

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Defining parameters

Level: \( N \) = \( 4012 = 2^{2} \cdot 17 \cdot 59 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4012.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 10 \)
Sturm bound: \(1080\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(3\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(4012))\).

Total New Old
Modular forms 546 76 470
Cusp forms 535 76 459
Eisenstein series 11 0 11

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

\(2\)\(17\)\(59\)FrickeDim.
\(-\)\(+\)\(+\)\(-\)\(19\)
\(-\)\(+\)\(-\)\(+\)\(17\)
\(-\)\(-\)\(+\)\(+\)\(19\)
\(-\)\(-\)\(-\)\(-\)\(21\)
Plus space\(+\)\(36\)
Minus space\(-\)\(40\)

Trace form

\( 76q + 4q^{5} - 4q^{7} + 80q^{9} + O(q^{10}) \) \( 76q + 4q^{5} - 4q^{7} + 80q^{9} - 8q^{13} - 4q^{15} + 4q^{17} - 16q^{19} + 16q^{21} + 8q^{23} + 64q^{25} - 12q^{27} + 8q^{29} + 8q^{31} + 16q^{33} - 12q^{35} - 24q^{37} - 28q^{39} + 8q^{41} - 8q^{43} + 76q^{45} - 4q^{47} + 80q^{49} + 4q^{51} + 4q^{53} + 8q^{55} + 20q^{57} + 12q^{61} + 20q^{63} + 12q^{65} - 16q^{67} + 8q^{69} + 52q^{71} + 8q^{75} - 4q^{77} + 20q^{79} + 60q^{81} + 16q^{83} + 16q^{87} + 24q^{89} - 28q^{91} - 16q^{93} - 60q^{95} - 28q^{97} - 4q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(4012))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 17 59
4012.2.a.a \(1\) \(32.036\) \(\Q\) None \(0\) \(-3\) \(-3\) \(-1\) \(-\) \(+\) \(+\) \(q-3q^{3}-3q^{5}-q^{7}+6q^{9}-2q^{11}+\cdots\)
4012.2.a.b \(1\) \(32.036\) \(\Q\) None \(0\) \(-3\) \(-1\) \(-5\) \(-\) \(-\) \(+\) \(q-3q^{3}-q^{5}-5q^{7}+6q^{9}-2q^{11}+\cdots\)
4012.2.a.c \(1\) \(32.036\) \(\Q\) None \(0\) \(-1\) \(-3\) \(1\) \(-\) \(+\) \(-\) \(q-q^{3}-3q^{5}+q^{7}-2q^{9}+2q^{11}+\cdots\)
4012.2.a.d \(2\) \(32.036\) \(\Q(\sqrt{5}) \) None \(0\) \(-2\) \(3\) \(-6\) \(-\) \(+\) \(-\) \(q-2\beta q^{3}+(2-\beta )q^{5}-3q^{7}+(1+4\beta )q^{9}+\cdots\)
4012.2.a.e \(2\) \(32.036\) \(\Q(\sqrt{5}) \) None \(0\) \(0\) \(-2\) \(0\) \(-\) \(+\) \(-\) \(q-\beta q^{3}-q^{5}+\beta q^{7}+2q^{9}-2\beta q^{11}+\cdots\)
4012.2.a.f \(3\) \(32.036\) 3.3.321.1 None \(0\) \(-3\) \(1\) \(3\) \(-\) \(-\) \(+\) \(q-q^{3}-\beta _{1}q^{5}+q^{7}-2q^{9}-2q^{11}+\cdots\)
4012.2.a.g \(12\) \(32.036\) \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(0\) \(-4\) \(3\) \(2\) \(-\) \(+\) \(-\) \(q-\beta _{1}q^{3}+\beta _{9}q^{5}+(-\beta _{2}+\beta _{8}-\beta _{9}+\cdots)q^{7}+\cdots\)
4012.2.a.h \(15\) \(32.036\) \(\mathbb{Q}[x]/(x^{15} - \cdots)\) None \(0\) \(-1\) \(1\) \(-11\) \(-\) \(-\) \(+\) \(q+\beta _{5}q^{3}-\beta _{8}q^{5}+(-1-\beta _{9})q^{7}+(1+\cdots)q^{9}+\cdots\)
4012.2.a.i \(18\) \(32.036\) \(\mathbb{Q}[x]/(x^{18} - \cdots)\) None \(0\) \(8\) \(4\) \(2\) \(-\) \(+\) \(+\) \(q+\beta _{1}q^{3}-\beta _{9}q^{5}-\beta _{8}q^{7}+(1+\beta _{1}+\cdots)q^{9}+\cdots\)
4012.2.a.j \(21\) \(32.036\) None \(0\) \(9\) \(1\) \(11\) \(-\) \(-\) \(-\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(4012))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(4012)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(17))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(34))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(59))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(68))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(118))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(236))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1003))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2006))\)\(^{\oplus 2}\)