Properties

Label 575.2.a
Level 575
Weight 2
Character orbit a
Rep. character \(\chi_{575}(1,\cdot)\)
Character field \(\Q\)
Dimension 35
Newforms 12
Sturm bound 120
Trace bound 3

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Defining parameters

Level: \( N \) = \( 575 = 5^{2} \cdot 23 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 575.a (trivial)
Character field: \(\Q\)
Newforms: \( 12 \)
Sturm bound: \(120\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(2\), \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(575))\).

Total New Old
Modular forms 66 35 31
Cusp forms 55 35 20
Eisenstein series 11 0 11

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

\(5\)\(23\)FrickeDim.
\(+\)\(+\)\(+\)\(4\)
\(+\)\(-\)\(-\)\(13\)
\(-\)\(+\)\(-\)\(12\)
\(-\)\(-\)\(+\)\(6\)
Plus space\(+\)\(10\)
Minus space\(-\)\(25\)

Trace form

\(35q \) \(\mathstrut +\mathstrut 4q^{3} \) \(\mathstrut +\mathstrut 36q^{4} \) \(\mathstrut +\mathstrut 3q^{6} \) \(\mathstrut +\mathstrut 2q^{7} \) \(\mathstrut -\mathstrut 3q^{8} \) \(\mathstrut +\mathstrut 37q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(35q \) \(\mathstrut +\mathstrut 4q^{3} \) \(\mathstrut +\mathstrut 36q^{4} \) \(\mathstrut +\mathstrut 3q^{6} \) \(\mathstrut +\mathstrut 2q^{7} \) \(\mathstrut -\mathstrut 3q^{8} \) \(\mathstrut +\mathstrut 37q^{9} \) \(\mathstrut -\mathstrut 2q^{11} \) \(\mathstrut +\mathstrut 17q^{12} \) \(\mathstrut +\mathstrut 4q^{13} \) \(\mathstrut +\mathstrut 22q^{16} \) \(\mathstrut -\mathstrut 4q^{17} \) \(\mathstrut -\mathstrut q^{18} \) \(\mathstrut +\mathstrut 20q^{19} \) \(\mathstrut +\mathstrut 6q^{21} \) \(\mathstrut +\mathstrut 10q^{22} \) \(\mathstrut +\mathstrut 3q^{23} \) \(\mathstrut -\mathstrut 2q^{24} \) \(\mathstrut -\mathstrut 13q^{26} \) \(\mathstrut +\mathstrut 4q^{27} \) \(\mathstrut +\mathstrut 14q^{28} \) \(\mathstrut +\mathstrut 4q^{29} \) \(\mathstrut +\mathstrut 12q^{31} \) \(\mathstrut -\mathstrut 6q^{32} \) \(\mathstrut -\mathstrut 10q^{33} \) \(\mathstrut +\mathstrut 30q^{34} \) \(\mathstrut +\mathstrut 3q^{36} \) \(\mathstrut -\mathstrut 4q^{37} \) \(\mathstrut -\mathstrut 30q^{38} \) \(\mathstrut -\mathstrut 16q^{39} \) \(\mathstrut +\mathstrut 8q^{41} \) \(\mathstrut -\mathstrut 8q^{42} \) \(\mathstrut +\mathstrut 12q^{43} \) \(\mathstrut -\mathstrut 16q^{44} \) \(\mathstrut -\mathstrut 2q^{46} \) \(\mathstrut -\mathstrut 16q^{47} \) \(\mathstrut +\mathstrut 49q^{48} \) \(\mathstrut +\mathstrut 55q^{49} \) \(\mathstrut +\mathstrut 10q^{51} \) \(\mathstrut +\mathstrut 35q^{52} \) \(\mathstrut -\mathstrut 10q^{53} \) \(\mathstrut +\mathstrut 9q^{54} \) \(\mathstrut -\mathstrut 18q^{56} \) \(\mathstrut -\mathstrut 43q^{58} \) \(\mathstrut +\mathstrut 4q^{59} \) \(\mathstrut +\mathstrut 34q^{61} \) \(\mathstrut +\mathstrut 19q^{62} \) \(\mathstrut +\mathstrut 8q^{63} \) \(\mathstrut -\mathstrut 23q^{64} \) \(\mathstrut -\mathstrut 78q^{66} \) \(\mathstrut +\mathstrut 2q^{67} \) \(\mathstrut -\mathstrut 28q^{68} \) \(\mathstrut -\mathstrut 4q^{69} \) \(\mathstrut -\mathstrut 56q^{71} \) \(\mathstrut -\mathstrut 51q^{72} \) \(\mathstrut +\mathstrut 4q^{73} \) \(\mathstrut -\mathstrut 44q^{74} \) \(\mathstrut +\mathstrut 50q^{76} \) \(\mathstrut +\mathstrut 4q^{77} \) \(\mathstrut -\mathstrut 11q^{78} \) \(\mathstrut +\mathstrut 20q^{79} \) \(\mathstrut +\mathstrut 67q^{81} \) \(\mathstrut -\mathstrut 5q^{82} \) \(\mathstrut +\mathstrut 38q^{83} \) \(\mathstrut -\mathstrut 82q^{84} \) \(\mathstrut -\mathstrut 16q^{86} \) \(\mathstrut +\mathstrut 8q^{87} \) \(\mathstrut -\mathstrut 22q^{88} \) \(\mathstrut +\mathstrut 10q^{89} \) \(\mathstrut +\mathstrut 70q^{91} \) \(\mathstrut +\mathstrut 6q^{92} \) \(\mathstrut +\mathstrut 42q^{93} \) \(\mathstrut -\mathstrut 49q^{94} \) \(\mathstrut -\mathstrut 61q^{96} \) \(\mathstrut -\mathstrut 28q^{98} \) \(\mathstrut -\mathstrut 36q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(575))\) into irreducible Hecke orbits

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 5 23
575.2.a.a \(1\) \(4.591\) \(\Q\) None \(-2\) \(-2\) \(0\) \(-1\) \(-\) \(-\) \(q-2q^{2}-2q^{3}+2q^{4}+4q^{6}-q^{7}+\cdots\)
575.2.a.b \(1\) \(4.591\) \(\Q\) None \(-2\) \(0\) \(0\) \(-1\) \(+\) \(+\) \(q-2q^{2}+2q^{4}-q^{7}-3q^{9}+2q^{11}+\cdots\)
575.2.a.c \(1\) \(4.591\) \(\Q\) None \(-1\) \(0\) \(0\) \(1\) \(-\) \(-\) \(q-q^{2}-q^{4}+q^{7}+3q^{8}-3q^{9}-q^{11}+\cdots\)
575.2.a.d \(1\) \(4.591\) \(\Q\) None \(1\) \(0\) \(0\) \(-1\) \(+\) \(+\) \(q+q^{2}-q^{4}-q^{7}-3q^{8}-3q^{9}-q^{11}+\cdots\)
575.2.a.e \(1\) \(4.591\) \(\Q\) None \(2\) \(2\) \(0\) \(1\) \(-\) \(+\) \(q+2q^{2}+2q^{3}+2q^{4}+4q^{6}+q^{7}+\cdots\)
575.2.a.f \(2\) \(4.591\) \(\Q(\sqrt{5}) \) None \(1\) \(0\) \(0\) \(-2\) \(+\) \(+\) \(q+\beta q^{2}+(1-2\beta )q^{3}+(-1+\beta )q^{4}+\cdots\)
575.2.a.g \(2\) \(4.591\) \(\Q(\sqrt{5}) \) None \(3\) \(2\) \(0\) \(2\) \(+\) \(-\) \(q+(1+\beta )q^{2}+q^{3}+3\beta q^{4}+(1+\beta )q^{6}+\cdots\)
575.2.a.h \(4\) \(4.591\) 4.4.15317.1 None \(-2\) \(2\) \(0\) \(3\) \(+\) \(-\) \(q-\beta _{1}q^{2}+(1+\beta _{2})q^{3}+(1+\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
575.2.a.i \(4\) \(4.591\) 4.4.5744.1 None \(0\) \(-2\) \(0\) \(-6\) \(-\) \(-\) \(q-\beta _{1}q^{2}+(-1+\beta _{1}+\beta _{2})q^{3}+(\beta _{1}+\cdots)q^{4}+\cdots\)
575.2.a.j \(4\) \(4.591\) 4.4.5744.1 None \(0\) \(2\) \(0\) \(6\) \(-\) \(+\) \(q+\beta _{1}q^{2}+(1-\beta _{1}-\beta _{2})q^{3}+(\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
575.2.a.k \(7\) \(4.591\) \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(-1\) \(0\) \(0\) \(3\) \(-\) \(+\) \(q-\beta _{1}q^{2}+\beta _{5}q^{3}+(2+\beta _{2})q^{4}+(1-\beta _{1}+\cdots)q^{6}+\cdots\)
575.2.a.l \(7\) \(4.591\) \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(1\) \(0\) \(0\) \(-3\) \(+\) \(-\) \(q+\beta _{1}q^{2}-\beta _{5}q^{3}+(2+\beta _{2})q^{4}+(1-\beta _{1}+\cdots)q^{6}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(575))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(575)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(23))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(115))\)\(^{\oplus 2}\)