Properties

Label 845.2.a
Level $845$
Weight $2$
Character orbit 845.a
Rep. character $\chi_{845}(1,\cdot)$
Character field $\Q$
Dimension $51$
Newform subspaces $15$
Sturm bound $182$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 845 = 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 845.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 15 \)
Sturm bound: \(182\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(845))\).

Total New Old
Modular forms 104 51 53
Cusp forms 77 51 26
Eisenstein series 27 0 27

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

\(5\)\(13\)FrickeDim.
\(+\)\(+\)\(+\)\(9\)
\(+\)\(-\)\(-\)\(16\)
\(-\)\(+\)\(-\)\(16\)
\(-\)\(-\)\(+\)\(10\)
Plus space\(+\)\(19\)
Minus space\(-\)\(32\)

Trace form

\( 51 q + 3 q^{2} + 53 q^{4} + q^{5} - 4 q^{7} + 3 q^{8} + 55 q^{9} + O(q^{10}) \) \( 51 q + 3 q^{2} + 53 q^{4} + q^{5} - 4 q^{7} + 3 q^{8} + 55 q^{9} + q^{10} + 4 q^{12} + 4 q^{14} + 61 q^{16} - 2 q^{17} + 11 q^{18} + 4 q^{19} - q^{20} - 4 q^{21} - 4 q^{24} + 51 q^{25} - 12 q^{27} - 28 q^{28} + 6 q^{29} - 8 q^{30} - 12 q^{31} - q^{32} + 20 q^{33} - 6 q^{34} - 4 q^{35} + 37 q^{36} + 10 q^{37} - 48 q^{38} + 9 q^{40} + 18 q^{41} - 16 q^{42} - 12 q^{43} - 4 q^{44} + 5 q^{45} - 8 q^{46} - 12 q^{47} - 12 q^{48} + 35 q^{49} + 3 q^{50} + 20 q^{51} - 2 q^{53} + 20 q^{54} - 12 q^{55} + 4 q^{57} - 2 q^{58} - 12 q^{59} + 12 q^{60} - 6 q^{61} - 8 q^{62} + 4 q^{63} + 49 q^{64} - 68 q^{66} + 16 q^{67} - 34 q^{68} - 20 q^{69} + 16 q^{70} - 16 q^{71} - 21 q^{72} + 14 q^{73} - 46 q^{74} - 12 q^{77} - 4 q^{79} - 17 q^{80} + 67 q^{81} - 18 q^{82} + 40 q^{83} + 28 q^{84} + 6 q^{85} - 36 q^{86} - 20 q^{87} + 4 q^{88} - 2 q^{89} - 23 q^{90} - 52 q^{92} - 60 q^{93} - 20 q^{94} - 12 q^{95} - 32 q^{96} + 2 q^{97} + 51 q^{98} + 20 q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(845))\) 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 13
845.2.a.a 845.a 1.a $1$ $6.747$ \(\Q\) None \(1\) \(-2\) \(1\) \(4\) $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-2q^{3}-q^{4}+q^{5}-2q^{6}+4q^{7}+\cdots\)
845.2.a.b 845.a 1.a $2$ $6.747$ \(\Q(\sqrt{5}) \) None \(-1\) \(0\) \(-2\) \(-4\) $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta q^{2}+(1-2\beta )q^{3}+(-1+\beta )q^{4}+\cdots\)
845.2.a.c 845.a 1.a $2$ $6.747$ \(\Q(\sqrt{13}) \) None \(-1\) \(2\) \(-2\) \(-2\) $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta q^{2}+q^{3}+(1+\beta )q^{4}-q^{5}-\beta q^{6}+\cdots\)
845.2.a.d 845.a 1.a $2$ $6.747$ \(\Q(\sqrt{3}) \) None \(0\) \(2\) \(2\) \(-4\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+(1+\beta )q^{3}+q^{4}+q^{5}+(3+\beta )q^{6}+\cdots\)
845.2.a.e 845.a 1.a $2$ $6.747$ \(\Q(\sqrt{5}) \) None \(1\) \(0\) \(2\) \(4\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+(1-2\beta )q^{3}+(-1+\beta )q^{4}+\cdots\)
845.2.a.f 845.a 1.a $2$ $6.747$ \(\Q(\sqrt{13}) \) None \(1\) \(2\) \(2\) \(2\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+q^{3}+(1+\beta )q^{4}+q^{5}+\beta q^{6}+\cdots\)
845.2.a.g 845.a 1.a $2$ $6.747$ \(\Q(\sqrt{2}) \) None \(2\) \(0\) \(-2\) \(-4\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{2}-\beta q^{3}+(1+2\beta )q^{4}-q^{5}+\cdots\)
845.2.a.h 845.a 1.a $3$ $6.747$ \(\Q(\zeta_{14})^+\) None \(-1\) \(-5\) \(3\) \(-5\) $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+(-1-\beta _{1}+\beta _{2})q^{3}+\beta _{2}q^{4}+\cdots\)
845.2.a.i 845.a 1.a $3$ $6.747$ 3.3.564.1 None \(-1\) \(-2\) \(3\) \(-2\) $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+(-1+\beta _{1})q^{3}+(2+\beta _{2})q^{4}+\cdots\)
845.2.a.j 845.a 1.a $3$ $6.747$ \(\Q(\zeta_{14})^+\) None \(1\) \(-5\) \(-3\) \(5\) $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(-1-\beta _{1}+\beta _{2})q^{3}+\beta _{2}q^{4}+\cdots\)
845.2.a.k 845.a 1.a $3$ $6.747$ 3.3.564.1 None \(1\) \(-2\) \(-3\) \(2\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(-1+\beta _{1})q^{3}+(2+\beta _{2})q^{4}+\cdots\)
845.2.a.l 845.a 1.a $4$ $6.747$ 4.4.4752.1 None \(-2\) \(-2\) \(4\) \(-10\) $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}-\beta _{3}q^{3}+(\beta _{1}+\beta _{2})q^{4}+q^{5}+\cdots\)
845.2.a.m 845.a 1.a $4$ $6.747$ 4.4.4752.1 None \(2\) \(-2\) \(-4\) \(10\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}-\beta _{3}q^{3}+(\beta _{1}+\beta _{2})q^{4}-q^{5}+\cdots\)
845.2.a.n 845.a 1.a $9$ $6.747$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None \(-3\) \(7\) \(-9\) \(-7\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(-\beta _{1}-\beta _{6})q^{2}+(1-\beta _{4})q^{3}+(2-\beta _{3}+\cdots)q^{4}+\cdots\)
845.2.a.o 845.a 1.a $9$ $6.747$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None \(3\) \(7\) \(9\) \(7\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(\beta _{1}+\beta _{6})q^{2}+(1-\beta _{4})q^{3}+(2-\beta _{3}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(845))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(845)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(65))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(169))\)\(^{\oplus 2}\)