Properties

Label 441.2.a
Level $441$
Weight $2$
Character orbit 441.a
Rep. character $\chi_{441}(1,\cdot)$
Character field $\Q$
Dimension $14$
Newform subspaces $10$
Sturm bound $112$
Trace bound $13$

Related objects

Downloads

Learn more

Defining parameters

Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 441.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 10 \)
Sturm bound: \(112\)
Trace bound: \(13\)
Distinguishing \(T_p\): \(2\), \(5\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(441))\).

Total New Old
Modular forms 72 19 53
Cusp forms 41 14 27
Eisenstein series 31 5 26

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

\(3\)\(7\)FrickeDim.
\(+\)\(+\)\(+\)\(1\)
\(+\)\(-\)\(-\)\(5\)
\(-\)\(+\)\(-\)\(5\)
\(-\)\(-\)\(+\)\(3\)
Plus space\(+\)\(4\)
Minus space\(-\)\(10\)

Trace form

\( 14 q + 14 q^{4} - 2 q^{5} + 12 q^{8} + O(q^{10}) \) \( 14 q + 14 q^{4} - 2 q^{5} + 12 q^{8} + 10 q^{10} + 4 q^{11} - 2 q^{13} + 22 q^{16} - 6 q^{17} + 4 q^{19} + 2 q^{20} - 16 q^{22} - 10 q^{25} + 2 q^{26} + 8 q^{29} + 8 q^{31} + 4 q^{32} - 18 q^{34} + 4 q^{37} - 4 q^{38} - 6 q^{40} + 2 q^{41} + 20 q^{43} + 24 q^{44} - 20 q^{50} - 6 q^{52} - 12 q^{53} + 32 q^{55} - 36 q^{58} + 12 q^{59} + 22 q^{61} - 18 q^{64} - 24 q^{65} + 20 q^{67} + 6 q^{68} + 4 q^{71} - 22 q^{73} - 16 q^{74} + 12 q^{76} + 28 q^{79} + 2 q^{80} - 34 q^{82} - 12 q^{83} - 20 q^{85} - 44 q^{86} - 72 q^{88} - 14 q^{89} - 48 q^{92} - 24 q^{94} + 28 q^{95} - 46 q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(441))\) 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 7
441.2.a.a 441.a 1.a $1$ $3.521$ \(\Q\) None \(-2\) \(0\) \(-2\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+2q^{4}-2q^{5}+4q^{10}+2q^{11}+\cdots\)
441.2.a.b 441.a 1.a $1$ $3.521$ \(\Q\) None \(-2\) \(0\) \(2\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+2q^{4}+2q^{5}-4q^{10}+2q^{11}+\cdots\)
441.2.a.c 441.a 1.a $1$ $3.521$ \(\Q\) \(\Q(\sqrt{-7}) \) \(-1\) \(0\) \(0\) \(0\) $-$ $-$ $N(\mathrm{U}(1))$ \(q-q^{2}-q^{4}+3q^{8}-4q^{11}-q^{16}+\cdots\)
441.2.a.d 441.a 1.a $1$ $3.521$ \(\Q\) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(0\) $+$ $+$ $N(\mathrm{U}(1))$ \(q-2q^{4}-7q^{13}+4q^{16}-7q^{19}-5q^{25}+\cdots\)
441.2.a.e 441.a 1.a $1$ $3.521$ \(\Q\) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(0\) $+$ $-$ $N(\mathrm{U}(1))$ \(q-2q^{4}+7q^{13}+4q^{16}+7q^{19}-5q^{25}+\cdots\)
441.2.a.f 441.a 1.a $1$ $3.521$ \(\Q\) None \(1\) \(0\) \(-2\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{4}-2q^{5}-3q^{8}-2q^{10}+\cdots\)
441.2.a.g 441.a 1.a $2$ $3.521$ \(\Q(\sqrt{3}) \) None \(0\) \(0\) \(0\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+q^{4}+2\beta q^{5}-\beta q^{8}+6q^{10}+\cdots\)
441.2.a.h 441.a 1.a $2$ $3.521$ \(\Q(\sqrt{7}) \) \(\Q(\sqrt{-7}) \) \(0\) \(0\) \(0\) \(0\) $+$ $-$ $N(\mathrm{U}(1))$ \(q+\beta q^{2}+5q^{4}+3\beta q^{8}-2\beta q^{11}+11q^{16}+\cdots\)
441.2.a.i 441.a 1.a $2$ $3.521$ \(\Q(\sqrt{2}) \) None \(2\) \(0\) \(-4\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{2}+(1+2\beta )q^{4}+(-2+\beta )q^{5}+\cdots\)
441.2.a.j 441.a 1.a $2$ $3.521$ \(\Q(\sqrt{2}) \) None \(2\) \(0\) \(4\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{2}+(1+2\beta )q^{4}+(2-\beta )q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(441))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(441)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(63))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(147))\)\(^{\oplus 2}\)