Properties

Label 575.2.a
Level $575$
Weight $2$
Character orbit 575.a
Rep. character $\chi_{575}(1,\cdot)$
Character field $\Q$
Dimension $35$
Newform subspaces $12$
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: \(\Q\)
Newform subspaces: \( 12 \)
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

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

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(575))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 5 23
575.2.a.a 575.a 1.a $1$ $4.591$ \(\Q\) None \(-2\) \(-2\) \(0\) \(-1\) $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}-2q^{3}+2q^{4}+4q^{6}-q^{7}+\cdots\)
575.2.a.b 575.a 1.a $1$ $4.591$ \(\Q\) None \(-2\) \(0\) \(0\) \(-1\) $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+2q^{4}-q^{7}-3q^{9}+2q^{11}+\cdots\)
575.2.a.c 575.a 1.a $1$ $4.591$ \(\Q\) None \(-1\) \(0\) \(0\) \(1\) $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{4}+q^{7}+3q^{8}-3q^{9}-q^{11}+\cdots\)
575.2.a.d 575.a 1.a $1$ $4.591$ \(\Q\) None \(1\) \(0\) \(0\) \(-1\) $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{4}-q^{7}-3q^{8}-3q^{9}-q^{11}+\cdots\)
575.2.a.e 575.a 1.a $1$ $4.591$ \(\Q\) None \(2\) \(2\) \(0\) \(1\) $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+2q^{3}+2q^{4}+4q^{6}+q^{7}+\cdots\)
575.2.a.f 575.a 1.a $2$ $4.591$ \(\Q(\sqrt{5}) \) None \(1\) \(0\) \(0\) \(-2\) $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+(1-2\beta )q^{3}+(-1+\beta )q^{4}+\cdots\)
575.2.a.g 575.a 1.a $2$ $4.591$ \(\Q(\sqrt{5}) \) None \(3\) \(2\) \(0\) \(2\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{2}+q^{3}+3\beta q^{4}+(1+\beta )q^{6}+\cdots\)
575.2.a.h 575.a 1.a $4$ $4.591$ 4.4.15317.1 None \(-2\) \(2\) \(0\) \(3\) $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+(1+\beta _{2})q^{3}+(1+\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
575.2.a.i 575.a 1.a $4$ $4.591$ 4.4.5744.1 None \(0\) \(-2\) \(0\) \(-6\) $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+(-1+\beta _{1}+\beta _{2})q^{3}+(\beta _{1}+\cdots)q^{4}+\cdots\)
575.2.a.j 575.a 1.a $4$ $4.591$ 4.4.5744.1 None \(0\) \(2\) \(0\) \(6\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(1-\beta _{1}-\beta _{2})q^{3}+(\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
575.2.a.k 575.a 1.a $7$ $4.591$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(-1\) \(0\) \(0\) \(3\) $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+\beta _{5}q^{3}+(2+\beta _{2})q^{4}+(1-\beta _{1}+\cdots)q^{6}+\cdots\)
575.2.a.l 575.a 1.a $7$ $4.591$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(1\) \(0\) \(0\) \(-3\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}-\beta _{5}q^{3}+(2+\beta _{2})q^{4}+(1-\beta _{1}+\cdots)q^{6}+\cdots\)

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

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