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Properties

Label 180.6.a
Level $180$
Weight $6$
Character orbit 180.a
Rep. character $\chi_{180}(1,\cdot)$
Character field $\Q$
Dimension $9$
Newform subspaces $7$
Sturm bound $216$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 180 = 2^{2} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 180.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 7 \)
Sturm bound: \(216\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(180))\).

Total New Old
Modular forms 192 9 183
Cusp forms 168 9 159
Eisenstein series 24 0 24

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

\(2\)\(3\)\(5\)FrickeDim
\(-\)\(+\)\(+\)$-$\(2\)
\(-\)\(+\)\(-\)$+$\(2\)
\(-\)\(-\)\(+\)$+$\(2\)
\(-\)\(-\)\(-\)$-$\(3\)
Plus space\(+\)\(4\)
Minus space\(-\)\(5\)

Trace form

\( 9 q + 25 q^{5} - 102 q^{7} + O(q^{10}) \) \( 9 q + 25 q^{5} - 102 q^{7} + 816 q^{11} - 1182 q^{13} + 1494 q^{17} + 588 q^{19} - 3054 q^{23} + 5625 q^{25} - 1542 q^{29} + 2316 q^{31} - 550 q^{35} - 5022 q^{37} + 14034 q^{41} + 27150 q^{43} + 10254 q^{47} + 2229 q^{49} + 16374 q^{53} - 12000 q^{55} - 23148 q^{59} + 93186 q^{61} + 5450 q^{65} - 37782 q^{67} - 20868 q^{71} - 66222 q^{73} + 42240 q^{77} + 215784 q^{79} + 75426 q^{83} - 4650 q^{85} - 123474 q^{89} - 311628 q^{91} + 65900 q^{95} + 79698 q^{97} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(180))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 3 5
180.6.a.a 180.a 1.a $1$ $28.869$ \(\Q\) None \(0\) \(0\) \(-25\) \(-16\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-5^{2}q^{5}-2^{4}q^{7}+564q^{11}-370q^{13}+\cdots\)
180.6.a.b 180.a 1.a $1$ $28.869$ \(\Q\) None \(0\) \(0\) \(-25\) \(56\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-5^{2}q^{5}+56q^{7}-156q^{11}+350q^{13}+\cdots\)
180.6.a.c 180.a 1.a $1$ $28.869$ \(\Q\) None \(0\) \(0\) \(25\) \(-244\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+5^{2}q^{5}-244q^{7}+12^{2}q^{11}+50q^{13}+\cdots\)
180.6.a.d 180.a 1.a $1$ $28.869$ \(\Q\) None \(0\) \(0\) \(25\) \(44\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+5^{2}q^{5}+44q^{7}-6^{3}q^{11}+770q^{13}+\cdots\)
180.6.a.e 180.a 1.a $1$ $28.869$ \(\Q\) None \(0\) \(0\) \(25\) \(218\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+5^{2}q^{5}+218q^{7}+480q^{11}-622q^{13}+\cdots\)
180.6.a.f 180.a 1.a $2$ $28.869$ \(\Q(\sqrt{241}) \) None \(0\) \(0\) \(-50\) \(-80\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-5^{2}q^{5}+(-40-\beta )q^{7}+(120-5\beta )q^{11}+\cdots\)
180.6.a.g 180.a 1.a $2$ $28.869$ \(\Q(\sqrt{241}) \) None \(0\) \(0\) \(50\) \(-80\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+5^{2}q^{5}+(-40-\beta )q^{7}+(-120+5\beta )q^{11}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(180))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(180)) \cong \) \(S_{6}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 12}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(4))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 9}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(18))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(60))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(90))\)\(^{\oplus 2}\)