Properties

Label 1045.2.a
Level $1045$
Weight $2$
Character orbit 1045.a
Rep. character $\chi_{1045}(1,\cdot)$
Character field $\Q$
Dimension $59$
Newform subspaces $11$
Sturm bound $240$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 1045 = 5 \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1045.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 11 \)
Sturm bound: \(240\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1045))\).

Total New Old
Modular forms 124 59 65
Cusp forms 117 59 58
Eisenstein series 7 0 7

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

\(5\)\(11\)\(19\)FrickeDim.
\(+\)\(+\)\(+\)\(+\)\(7\)
\(+\)\(+\)\(-\)\(-\)\(9\)
\(+\)\(-\)\(+\)\(-\)\(7\)
\(+\)\(-\)\(-\)\(+\)\(5\)
\(-\)\(+\)\(+\)\(-\)\(8\)
\(-\)\(+\)\(-\)\(+\)\(6\)
\(-\)\(-\)\(+\)\(+\)\(8\)
\(-\)\(-\)\(-\)\(-\)\(9\)
Plus space\(+\)\(26\)
Minus space\(-\)\(33\)

Trace form

\( 59 q - 3 q^{2} - 4 q^{3} + 53 q^{4} + 3 q^{5} + 20 q^{6} - 8 q^{7} + 9 q^{8} + 55 q^{9} + O(q^{10}) \) \( 59 q - 3 q^{2} - 4 q^{3} + 53 q^{4} + 3 q^{5} + 20 q^{6} - 8 q^{7} + 9 q^{8} + 55 q^{9} - 3 q^{10} - q^{11} + 36 q^{12} - 14 q^{13} + 32 q^{14} - 4 q^{15} + 37 q^{16} - 18 q^{17} + 25 q^{18} - q^{19} + 13 q^{20} - 3 q^{22} + 52 q^{24} + 59 q^{25} - 10 q^{26} + 8 q^{27} - 56 q^{28} + 2 q^{29} - 4 q^{30} - 24 q^{31} + 25 q^{32} - 4 q^{33} - 46 q^{34} + 17 q^{36} - 38 q^{37} + 9 q^{38} + 16 q^{39} - 15 q^{40} - 18 q^{41} - 16 q^{42} - 4 q^{43} + q^{44} + 23 q^{45} - 16 q^{46} + 32 q^{47} + 28 q^{48} + 43 q^{49} - 3 q^{50} + 40 q^{51} - 26 q^{52} - 38 q^{53} + 48 q^{54} + 7 q^{55} + 8 q^{56} - 4 q^{57} + 22 q^{58} + 44 q^{59} - 28 q^{60} - 22 q^{61} - 32 q^{62} + 45 q^{64} - 6 q^{65} - 12 q^{66} + 4 q^{67} - 62 q^{68} - 48 q^{69} + 8 q^{70} - 32 q^{71} + 5 q^{72} - 58 q^{73} - 50 q^{74} - 4 q^{75} - 7 q^{76} + 16 q^{77} - 80 q^{78} - 56 q^{79} + 13 q^{80} + 67 q^{81} - 14 q^{82} + 68 q^{83} - 104 q^{84} - 10 q^{85} - 52 q^{86} - 16 q^{87} - 15 q^{88} - 18 q^{89} - 39 q^{90} - 32 q^{91} - 24 q^{92} + 32 q^{93} + 72 q^{94} - q^{95} + 4 q^{96} - 42 q^{97} - 67 q^{98} - 13 q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1045))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 5 11 19
1045.2.a.a 1045.a 1.a $1$ $8.344$ \(\Q\) None \(-1\) \(-2\) \(-1\) \(-2\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-2q^{3}-q^{4}-q^{5}+2q^{6}-2q^{7}+\cdots\)
1045.2.a.b 1045.a 1.a $1$ $8.344$ \(\Q\) None \(1\) \(0\) \(1\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{4}+q^{5}-3q^{8}-3q^{9}+q^{10}+\cdots\)
1045.2.a.c 1045.a 1.a $2$ $8.344$ \(\Q(\sqrt{2}) \) None \(2\) \(4\) \(2\) \(0\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{2}+2q^{3}+(1+2\beta )q^{4}+q^{5}+\cdots\)
1045.2.a.d 1045.a 1.a $5$ $8.344$ \(\Q(\zeta_{22})^+\) None \(-3\) \(-7\) \(5\) \(-11\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1}-\beta _{4})q^{2}+(-2-\beta _{2}+\beta _{3}+\cdots)q^{3}+\cdots\)
1045.2.a.e 1045.a 1.a $5$ $8.344$ 5.5.36497.1 None \(-1\) \(-3\) \(-5\) \(3\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{4}q^{2}+(-1+\beta _{1})q^{3}+\beta _{3}q^{4}-q^{5}+\cdots\)
1045.2.a.f 1045.a 1.a $6$ $8.344$ 6.6.7281497.1 None \(-2\) \(-1\) \(-6\) \(5\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{4}q^{2}+\beta _{3}q^{3}+(\beta _{4}+\beta _{5})q^{4}-q^{5}+\cdots\)
1045.2.a.g 1045.a 1.a $6$ $8.344$ 6.6.131947641.1 None \(0\) \(3\) \(6\) \(5\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(1-\beta _{3})q^{3}+(2+\beta _{2}-\beta _{3}+\cdots)q^{4}+\cdots\)
1045.2.a.h 1045.a 1.a $7$ $8.344$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(1\) \(3\) \(-7\) \(-1\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+\beta _{5}q^{3}+(1+\beta _{2})q^{4}-q^{5}+\cdots\)
1045.2.a.i 1045.a 1.a $8$ $8.344$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(-6\) \(-7\) \(8\) \(-11\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{2}+(-1-\beta _{6})q^{3}+(2+\cdots)q^{4}+\cdots\)
1045.2.a.j 1045.a 1.a $9$ $8.344$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None \(3\) \(3\) \(-9\) \(-9\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+\beta _{5}q^{3}+(1-\beta _{1}-\beta _{8})q^{4}+\cdots\)
1045.2.a.k 1045.a 1.a $9$ $8.344$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None \(3\) \(3\) \(9\) \(13\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+\beta _{5}q^{3}+(1+\beta _{2})q^{4}+q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1045))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1045)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(19))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(55))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(95))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(209))\)\(^{\oplus 2}\)