Properties

Label 525.2.a
Level 525
Weight 2
Character orbit a
Rep. character \(\chi_{525}(1,\cdot)\)
Character field \(\Q\)
Dimension 20
Newforms 11
Sturm bound 160
Trace bound 11

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Defining parameters

Level: \( N \) = \( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 525.a (trivial)
Character field: \(\Q\)
Newforms: \( 11 \)
Sturm bound: \(160\)
Trace bound: \(11\)
Distinguishing \(T_p\): \(2\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(525))\).

Total New Old
Modular forms 92 20 72
Cusp forms 69 20 49
Eisenstein series 23 0 23

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

\(3\)\(5\)\(7\)FrickeDim.
\(+\)\(+\)\(+\)\(+\)\(3\)
\(+\)\(+\)\(-\)\(-\)\(1\)
\(+\)\(-\)\(+\)\(-\)\(3\)
\(+\)\(-\)\(-\)\(+\)\(3\)
\(-\)\(+\)\(+\)\(-\)\(4\)
\(-\)\(-\)\(+\)\(+\)\(1\)
\(-\)\(-\)\(-\)\(-\)\(5\)
Plus space\(+\)\(7\)
Minus space\(-\)\(13\)

Trace form

\(20q \) \(\mathstrut +\mathstrut 24q^{4} \) \(\mathstrut +\mathstrut 4q^{6} \) \(\mathstrut -\mathstrut 2q^{7} \) \(\mathstrut +\mathstrut 20q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(20q \) \(\mathstrut +\mathstrut 24q^{4} \) \(\mathstrut +\mathstrut 4q^{6} \) \(\mathstrut -\mathstrut 2q^{7} \) \(\mathstrut +\mathstrut 20q^{9} \) \(\mathstrut -\mathstrut 8q^{11} \) \(\mathstrut +\mathstrut 8q^{12} \) \(\mathstrut +\mathstrut 8q^{13} \) \(\mathstrut -\mathstrut 2q^{14} \) \(\mathstrut +\mathstrut 40q^{16} \) \(\mathstrut +\mathstrut 8q^{17} \) \(\mathstrut +\mathstrut 8q^{19} \) \(\mathstrut +\mathstrut 2q^{21} \) \(\mathstrut +\mathstrut 24q^{22} \) \(\mathstrut -\mathstrut 16q^{23} \) \(\mathstrut +\mathstrut 12q^{24} \) \(\mathstrut -\mathstrut 16q^{26} \) \(\mathstrut -\mathstrut 6q^{28} \) \(\mathstrut -\mathstrut 32q^{29} \) \(\mathstrut -\mathstrut 32q^{34} \) \(\mathstrut +\mathstrut 24q^{36} \) \(\mathstrut -\mathstrut 8q^{37} \) \(\mathstrut -\mathstrut 8q^{38} \) \(\mathstrut +\mathstrut 16q^{39} \) \(\mathstrut -\mathstrut 32q^{41} \) \(\mathstrut -\mathstrut 2q^{42} \) \(\mathstrut -\mathstrut 44q^{44} \) \(\mathstrut -\mathstrut 28q^{46} \) \(\mathstrut -\mathstrut 16q^{47} \) \(\mathstrut +\mathstrut 20q^{49} \) \(\mathstrut +\mathstrut 8q^{51} \) \(\mathstrut -\mathstrut 8q^{52} \) \(\mathstrut +\mathstrut 4q^{54} \) \(\mathstrut -\mathstrut 30q^{56} \) \(\mathstrut +\mathstrut 8q^{57} \) \(\mathstrut -\mathstrut 8q^{59} \) \(\mathstrut -\mathstrut 8q^{61} \) \(\mathstrut -\mathstrut 24q^{62} \) \(\mathstrut -\mathstrut 2q^{63} \) \(\mathstrut +\mathstrut 28q^{64} \) \(\mathstrut -\mathstrut 8q^{66} \) \(\mathstrut +\mathstrut 8q^{68} \) \(\mathstrut +\mathstrut 16q^{69} \) \(\mathstrut +\mathstrut 8q^{71} \) \(\mathstrut +\mathstrut 24q^{73} \) \(\mathstrut -\mathstrut 68q^{74} \) \(\mathstrut +\mathstrut 40q^{76} \) \(\mathstrut -\mathstrut 16q^{78} \) \(\mathstrut +\mathstrut 32q^{79} \) \(\mathstrut +\mathstrut 20q^{81} \) \(\mathstrut +\mathstrut 8q^{82} \) \(\mathstrut +\mathstrut 32q^{83} \) \(\mathstrut +\mathstrut 6q^{84} \) \(\mathstrut +\mathstrut 52q^{86} \) \(\mathstrut +\mathstrut 8q^{88} \) \(\mathstrut +\mathstrut 8q^{89} \) \(\mathstrut +\mathstrut 4q^{91} \) \(\mathstrut -\mathstrut 16q^{92} \) \(\mathstrut +\mathstrut 8q^{93} \) \(\mathstrut +\mathstrut 16q^{94} \) \(\mathstrut -\mathstrut 52q^{96} \) \(\mathstrut -\mathstrut 8q^{97} \) \(\mathstrut -\mathstrut 8q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(525))\) into irreducible Hecke orbits

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 3 5 7
525.2.a.a \(1\) \(4.192\) \(\Q\) None \(-1\) \(-1\) \(0\) \(-1\) \(+\) \(+\) \(+\) \(q-q^{2}-q^{3}-q^{4}+q^{6}-q^{7}+3q^{8}+\cdots\)
525.2.a.b \(1\) \(4.192\) \(\Q\) None \(-1\) \(1\) \(0\) \(-1\) \(-\) \(-\) \(+\) \(q-q^{2}+q^{3}-q^{4}-q^{6}-q^{7}+3q^{8}+\cdots\)
525.2.a.c \(1\) \(4.192\) \(\Q\) None \(1\) \(-1\) \(0\) \(1\) \(+\) \(-\) \(-\) \(q+q^{2}-q^{3}-q^{4}-q^{6}+q^{7}-3q^{8}+\cdots\)
525.2.a.d \(1\) \(4.192\) \(\Q\) None \(1\) \(-1\) \(0\) \(1\) \(+\) \(+\) \(-\) \(q+q^{2}-q^{3}-q^{4}-q^{6}+q^{7}-3q^{8}+\cdots\)
525.2.a.e \(2\) \(4.192\) \(\Q(\sqrt{5}) \) None \(-3\) \(-2\) \(0\) \(2\) \(+\) \(-\) \(-\) \(q+(-1-\beta )q^{2}-q^{3}+3\beta q^{4}+(1+\beta )q^{6}+\cdots\)
525.2.a.f \(2\) \(4.192\) \(\Q(\sqrt{13}) \) None \(-1\) \(-2\) \(0\) \(-2\) \(+\) \(+\) \(+\) \(q-\beta q^{2}-q^{3}+(1+\beta )q^{4}+\beta q^{6}-q^{7}+\cdots\)
525.2.a.g \(2\) \(4.192\) \(\Q(\sqrt{5}) \) None \(0\) \(2\) \(0\) \(-2\) \(-\) \(+\) \(+\) \(q-\beta q^{2}+q^{3}+3q^{4}-\beta q^{6}-q^{7}-\beta q^{8}+\cdots\)
525.2.a.h \(2\) \(4.192\) \(\Q(\sqrt{13}) \) None \(1\) \(2\) \(0\) \(2\) \(-\) \(-\) \(-\) \(q+\beta q^{2}+q^{3}+(1+\beta )q^{4}+\beta q^{6}+q^{7}+\cdots\)
525.2.a.i \(2\) \(4.192\) \(\Q(\sqrt{5}) \) None \(3\) \(2\) \(0\) \(-2\) \(-\) \(+\) \(+\) \(q+(1+\beta )q^{2}+q^{3}+3\beta q^{4}+(1+\beta )q^{6}+\cdots\)
525.2.a.j \(3\) \(4.192\) 3.3.148.1 None \(-1\) \(3\) \(0\) \(3\) \(-\) \(-\) \(-\) \(q-\beta _{1}q^{2}+q^{3}+(1+\beta _{1}+\beta _{2})q^{4}-\beta _{1}q^{6}+\cdots\)
525.2.a.k \(3\) \(4.192\) 3.3.148.1 None \(1\) \(-3\) \(0\) \(-3\) \(+\) \(-\) \(+\) \(q+\beta _{1}q^{2}-q^{3}+(1+\beta _{1}+\beta _{2})q^{4}-\beta _{1}q^{6}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(525))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(525)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(105))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(175))\)\(^{\oplus 2}\)