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Properties

Label 1980.2.a
Level $1980$
Weight $2$
Character orbit 1980.a
Rep. character $\chi_{1980}(1,\cdot)$
Character field $\Q$
Dimension $18$
Newform subspaces $12$
Sturm bound $864$
Trace bound $13$

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Defining parameters

Level: \( N \) \(=\) \( 1980 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1980.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 12 \)
Sturm bound: \(864\)
Trace bound: \(13\)
Distinguishing \(T_p\): \(7\), \(13\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1980))\).

Total New Old
Modular forms 456 18 438
Cusp forms 409 18 391
Eisenstein series 47 0 47

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

\(2\)\(3\)\(5\)\(11\)FrickeDim
\(-\)\(+\)\(+\)\(+\)\(-\)\(2\)
\(-\)\(+\)\(+\)\(-\)\(+\)\(2\)
\(-\)\(+\)\(-\)\(+\)\(+\)\(2\)
\(-\)\(+\)\(-\)\(-\)\(-\)\(2\)
\(-\)\(-\)\(+\)\(+\)\(+\)\(3\)
\(-\)\(-\)\(+\)\(-\)\(-\)\(3\)
\(-\)\(-\)\(-\)\(+\)\(-\)\(2\)
\(-\)\(-\)\(-\)\(-\)\(+\)\(2\)
Plus space\(+\)\(9\)
Minus space\(-\)\(9\)

Trace form

\( 18 q - 2 q^{5} - 4 q^{7} + O(q^{10}) \) \( 18 q - 2 q^{5} - 4 q^{7} - 4 q^{13} - 4 q^{17} - 8 q^{19} + 18 q^{25} - 12 q^{29} + 8 q^{31} - 4 q^{35} - 4 q^{37} + 4 q^{41} - 4 q^{43} - 32 q^{47} + 18 q^{49} + 4 q^{53} - 12 q^{61} - 4 q^{65} - 8 q^{67} + 24 q^{71} - 4 q^{73} - 12 q^{77} + 44 q^{83} - 4 q^{85} - 44 q^{89} - 16 q^{91} + 8 q^{95} - 12 q^{97} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1980))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 3 5 11
1980.2.a.a 1980.a 1.a $1$ $15.810$ \(\Q\) None 220.2.a.a \(0\) \(0\) \(-1\) \(-4\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{5}-4q^{7}+q^{11}-4q^{13}-4q^{19}+\cdots\)
1980.2.a.b 1980.a 1.a $1$ $15.810$ \(\Q\) None 220.2.a.b \(0\) \(0\) \(-1\) \(0\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{5}-q^{11}+4q^{17}-4q^{19}-6q^{23}+\cdots\)
1980.2.a.c 1980.a 1.a $1$ $15.810$ \(\Q\) None 660.2.a.c \(0\) \(0\) \(1\) \(-4\) $-$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{5}-4q^{7}+q^{11}-4q^{13}+6q^{17}+\cdots\)
1980.2.a.d 1980.a 1.a $1$ $15.810$ \(\Q\) None 660.2.a.a \(0\) \(0\) \(1\) \(-2\) $-$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{5}-2q^{7}+q^{11}+2q^{13}-8q^{17}+\cdots\)
1980.2.a.e 1980.a 1.a $1$ $15.810$ \(\Q\) None 660.2.a.b \(0\) \(0\) \(1\) \(0\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{5}-q^{11}-4q^{13}+2q^{17}+2q^{19}+\cdots\)
1980.2.a.f 1980.a 1.a $1$ $15.810$ \(\Q\) None 660.2.a.d \(0\) \(0\) \(1\) \(2\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{5}+2q^{7}-q^{11}+2q^{13}+2q^{19}+\cdots\)
1980.2.a.g 1980.a 1.a $2$ $15.810$ \(\Q(\sqrt{3}) \) None 1980.2.a.g \(0\) \(0\) \(-2\) \(-2\) $-$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{5}+(-1+\beta )q^{7}+q^{11}+(-1-3\beta )q^{13}+\cdots\)
1980.2.a.h 1980.a 1.a $2$ $15.810$ \(\Q(\sqrt{13}) \) None 660.2.a.e \(0\) \(0\) \(-2\) \(2\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{5}+(1+\beta )q^{7}-q^{11}+(-1-\beta )q^{13}+\cdots\)
1980.2.a.i 1980.a 1.a $2$ $15.810$ \(\Q(\sqrt{7}) \) None 1980.2.a.i \(0\) \(0\) \(-2\) \(2\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{5}+(1+\beta )q^{7}-q^{11}+(1+\beta )q^{13}+\cdots\)
1980.2.a.j 1980.a 1.a $2$ $15.810$ \(\Q(\sqrt{13}) \) None 660.2.a.f \(0\) \(0\) \(-2\) \(2\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{5}+(1+\beta )q^{7}+q^{11}+(3-\beta )q^{13}+\cdots\)
1980.2.a.k 1980.a 1.a $2$ $15.810$ \(\Q(\sqrt{3}) \) None 1980.2.a.g \(0\) \(0\) \(2\) \(-2\) $-$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{5}+(-1+\beta )q^{7}-q^{11}+(-1-3\beta )q^{13}+\cdots\)
1980.2.a.l 1980.a 1.a $2$ $15.810$ \(\Q(\sqrt{7}) \) None 1980.2.a.i \(0\) \(0\) \(2\) \(2\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{5}+(1+\beta )q^{7}+q^{11}+(1+\beta )q^{13}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1980))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1980)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 18}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(33))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(44))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(55))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(66))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(90))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(99))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(110))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(132))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(165))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(180))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(198))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(220))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(330))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(396))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(495))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(660))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(990))\)\(^{\oplus 2}\)