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Properties

Label 1209.2.a
Level $1209$
Weight $2$
Character orbit 1209.a
Rep. character $\chi_{1209}(1,\cdot)$
Character field $\Q$
Dimension $59$
Newform subspaces $13$
Sturm bound $298$
Trace bound $4$

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Defining parameters

Level: \( N \) \(=\) \( 1209 = 3 \cdot 13 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1209.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 13 \)
Sturm bound: \(298\)
Trace bound: \(4\)
Distinguishing \(T_p\): \(2\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1209))\).

Total New Old
Modular forms 152 59 93
Cusp forms 145 59 86
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.

\(3\)\(13\)\(31\)FrickeDim
\(+\)\(+\)\(+\)$+$\(6\)
\(+\)\(+\)\(-\)$-$\(11\)
\(+\)\(-\)\(+\)$-$\(7\)
\(+\)\(-\)\(-\)$+$\(6\)
\(-\)\(+\)\(+\)$-$\(8\)
\(-\)\(+\)\(-\)$+$\(5\)
\(-\)\(-\)\(+\)$+$\(3\)
\(-\)\(-\)\(-\)$-$\(13\)
Plus space\(+\)\(20\)
Minus space\(-\)\(39\)

Trace form

\( 59 q + 5 q^{2} - q^{3} + 61 q^{4} + 2 q^{5} - 3 q^{6} + 8 q^{7} + 21 q^{8} + 59 q^{9} + O(q^{10}) \) \( 59 q + 5 q^{2} - q^{3} + 61 q^{4} + 2 q^{5} - 3 q^{6} + 8 q^{7} + 21 q^{8} + 59 q^{9} + 2 q^{10} + 12 q^{11} + q^{12} - q^{13} - 4 q^{14} - 6 q^{15} + 61 q^{16} - 2 q^{17} + 5 q^{18} + 4 q^{19} + 2 q^{20} - 8 q^{21} - 12 q^{22} + 32 q^{23} - 15 q^{24} + 37 q^{25} - 3 q^{26} - q^{27} + 44 q^{28} + 10 q^{29} + 22 q^{30} + 11 q^{31} + 17 q^{32} - 4 q^{33} - 6 q^{34} + 16 q^{35} + 61 q^{36} + 18 q^{37} - 48 q^{38} + 7 q^{39} - 10 q^{40} - 34 q^{41} - 8 q^{42} + 4 q^{43} + 44 q^{44} + 2 q^{45} - 8 q^{46} + 16 q^{47} + 17 q^{48} + 51 q^{49} - 25 q^{50} - 10 q^{51} + q^{52} + 2 q^{53} - 3 q^{54} + 8 q^{55} - 24 q^{56} - 4 q^{57} - 26 q^{58} - 36 q^{59} - 42 q^{60} - 22 q^{61} - 3 q^{62} + 8 q^{63} + 113 q^{64} - 6 q^{65} + 20 q^{66} + 28 q^{67} - 62 q^{68} + 8 q^{69} - 36 q^{70} + 16 q^{71} + 21 q^{72} - 50 q^{73} - 18 q^{74} + q^{75} - 8 q^{76} + 64 q^{77} + q^{78} + 16 q^{79} + 10 q^{80} + 59 q^{81} - 2 q^{82} + 12 q^{83} + 24 q^{84} - 4 q^{85} + 100 q^{86} - 14 q^{87} - 52 q^{88} - 42 q^{89} + 2 q^{90} + 128 q^{92} + 3 q^{93} - 72 q^{94} - 32 q^{95} - 63 q^{96} + 22 q^{97} - 63 q^{98} + 12 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1209))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 3 13 31
1209.2.a.a 1209.a 1.a $1$ $9.654$ \(\Q\) None \(-2\) \(-1\) \(2\) \(2\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}-q^{3}+2q^{4}+2q^{5}+2q^{6}+\cdots\)
1209.2.a.b 1209.a 1.a $1$ $9.654$ \(\Q\) None \(0\) \(1\) \(0\) \(-2\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}-2q^{4}-2q^{7}+q^{9}+q^{11}-2q^{12}+\cdots\)
1209.2.a.c 1209.a 1.a $2$ $9.654$ \(\Q(\sqrt{5}) \) None \(0\) \(-2\) \(1\) \(-3\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(1-2\beta )q^{2}-q^{3}+3q^{4}+(1-\beta )q^{5}+\cdots\)
1209.2.a.d 1209.a 1.a $2$ $9.654$ \(\Q(\sqrt{2}) \) None \(0\) \(2\) \(0\) \(-4\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+q^{3}-\beta q^{5}+\beta q^{6}-2q^{7}+\cdots\)
1209.2.a.e 1209.a 1.a $2$ $9.654$ \(\Q(\sqrt{5}) \) None \(2\) \(-2\) \(1\) \(1\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}-q^{4}-\beta q^{5}-q^{6}-\beta q^{7}+\cdots\)
1209.2.a.f 1209.a 1.a $2$ $9.654$ \(\Q(\sqrt{3}) \) None \(2\) \(-2\) \(-2\) \(4\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{2}-q^{3}+(2+2\beta )q^{4}+(-1+\cdots)q^{5}+\cdots\)
1209.2.a.g 1209.a 1.a $2$ $9.654$ \(\Q(\sqrt{21}) \) None \(2\) \(2\) \(-3\) \(3\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}-q^{4}+(-1-\beta )q^{5}+q^{6}+\cdots\)
1209.2.a.h 1209.a 1.a $5$ $9.654$ 5.5.135076.1 None \(-1\) \(5\) \(-4\) \(-10\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+q^{3}+\beta _{2}q^{4}+(-1+\beta _{1}+\cdots)q^{5}+\cdots\)
1209.2.a.i 1209.a 1.a $6$ $9.654$ 6.6.20329488.1 None \(-1\) \(-6\) \(-4\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}-q^{3}+(1+\beta _{2})q^{4}+(-1-\beta _{5})q^{5}+\cdots\)
1209.2.a.j 1209.a 1.a $6$ $9.654$ 6.6.41295872.1 None \(0\) \(-6\) \(0\) \(0\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}-q^{3}+\beta _{2}q^{4}-\beta _{4}q^{5}-\beta _{1}q^{6}+\cdots\)
1209.2.a.k 1209.a 1.a $8$ $9.654$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(2\) \(8\) \(2\) \(10\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+q^{3}+(1+\beta _{1}+\beta _{2}+\beta _{3}+\cdots)q^{4}+\cdots\)
1209.2.a.l 1209.a 1.a $11$ $9.654$ \(\mathbb{Q}[x]/(x^{11} - \cdots)\) None \(-2\) \(11\) \(3\) \(3\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+q^{3}+(2+\beta _{2})q^{4}-\beta _{5}q^{5}+\cdots\)
1209.2.a.m 1209.a 1.a $11$ $9.654$ \(\mathbb{Q}[x]/(x^{11} - \cdots)\) None \(3\) \(-11\) \(6\) \(4\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}-q^{3}+(1+\beta _{2})q^{4}+(1+\beta _{3}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1209))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1209)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(31))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(39))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(93))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(403))\)\(^{\oplus 2}\)