↑

Properties

Label 65.6.a
Level $65$
Weight $6$
Character orbit 65.a
Rep. character $\chi_{65}(1,\cdot)$
Character field $\Q$
Dimension $20$
Newform subspaces $5$
Sturm bound $42$
Trace bound $2$

Related objects

Downloads

Learn more

Defining parameters

Level: \( N \) \(=\) \( 65 = 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 65.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 5 \)
Sturm bound: \(42\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(65))\).

Total New Old
Modular forms 36 20 16
Cusp forms 32 20 12
Eisenstein series 4 0 4

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
\(+\)\(+\)\(+\)\(5\)
\(+\)\(-\)\(-\)\(6\)
\(-\)\(+\)\(-\)\(6\)
\(-\)\(-\)\(+\)\(3\)
Plus space\(+\)\(8\)
Minus space\(-\)\(12\)

Trace form

\( 20 q - 4 q^{2} + 44 q^{3} + 344 q^{4} - 50 q^{5} + 244 q^{6} - 196 q^{7} - 612 q^{8} + 2132 q^{9} + O(q^{10}) \) \( 20 q - 4 q^{2} + 44 q^{3} + 344 q^{4} - 50 q^{5} + 244 q^{6} - 196 q^{7} - 612 q^{8} + 2132 q^{9} + 100 q^{10} + 292 q^{11} - 68 q^{12} - 338 q^{13} - 232 q^{14} - 900 q^{15} + 488 q^{16} + 3140 q^{17} + 3840 q^{18} + 856 q^{19} - 2400 q^{20} - 9028 q^{21} + 10200 q^{22} + 744 q^{23} + 14632 q^{24} + 12500 q^{25} + 8228 q^{27} + 12348 q^{28} + 7384 q^{29} - 7600 q^{30} - 612 q^{31} - 34008 q^{32} - 28732 q^{33} - 27612 q^{34} + 3100 q^{35} + 36648 q^{36} + 28728 q^{37} - 41244 q^{38} + 15600 q^{40} + 44252 q^{41} - 56460 q^{42} - 5968 q^{43} - 81764 q^{44} + 5550 q^{45} - 5012 q^{46} - 57124 q^{47} + 85632 q^{48} + 76548 q^{49} - 2500 q^{50} - 82240 q^{51} - 16224 q^{52} + 40664 q^{53} - 10564 q^{54} + 1900 q^{55} - 50832 q^{56} - 71116 q^{57} - 154172 q^{58} - 28320 q^{59} - 76000 q^{60} + 60520 q^{61} + 172084 q^{62} + 123668 q^{63} - 97912 q^{64} - 16900 q^{65} + 53000 q^{66} - 120036 q^{67} - 2284 q^{68} + 34744 q^{69} + 47000 q^{70} - 67896 q^{71} + 163584 q^{72} - 91872 q^{73} + 85840 q^{74} + 27500 q^{75} + 224920 q^{76} + 71940 q^{77} + 73684 q^{78} + 74568 q^{79} - 89600 q^{80} + 242196 q^{81} - 178452 q^{82} + 117844 q^{83} - 352500 q^{84} + 101600 q^{85} + 7328 q^{86} + 50148 q^{87} - 133444 q^{88} - 210568 q^{89} + 103600 q^{90} + 37180 q^{91} + 58188 q^{92} + 165988 q^{93} + 207984 q^{94} + 75800 q^{95} + 155676 q^{96} + 388540 q^{97} + 30052 q^{98} + 325856 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(65))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 5 13
65.6.a.a 65.a 1.a $1$ $10.425$ \(\Q\) None 65.6.a.a \(5\) \(6\) \(-25\) \(-244\) $+$ $+$ $\mathrm{SU}(2)$ \(q+5q^{2}+6q^{3}-7q^{4}-5^{2}q^{5}+30q^{6}+\cdots\)
65.6.a.b 65.a 1.a $3$ $10.425$ 3.3.49857.1 None 65.6.a.b \(-2\) \(-16\) \(75\) \(-208\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{2}+(-5-\beta _{2})q^{3}+(-4+\cdots)q^{4}+\cdots\)
65.6.a.c 65.a 1.a $4$ $10.425$ 4.4.1878612.1 None 65.6.a.c \(-9\) \(-4\) \(-100\) \(-136\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(-2+\beta _{1}+\beta _{2})q^{2}+(-1-\beta _{2}-\beta _{3})q^{3}+\cdots\)
65.6.a.d 65.a 1.a $6$ $10.425$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None 65.6.a.d \(0\) \(38\) \(-150\) \(220\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(6+\beta _{1}+\beta _{2})q^{3}+(22+\beta _{2}+\cdots)q^{4}+\cdots\)
65.6.a.e 65.a 1.a $6$ $10.425$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None 65.6.a.e \(2\) \(20\) \(150\) \(172\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(3-\beta _{2})q^{3}+(23-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(65))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(65)) \simeq \) \(S_{6}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(13))\)\(^{\oplus 2}\)