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Properties

Label 475.4.a
Level $475$
Weight $4$
Character orbit 475.a
Rep. character $\chi_{475}(1,\cdot)$
Character field $\Q$
Dimension $86$
Newform subspaces $15$
Sturm bound $200$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 475 = 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 475.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 15 \)
Sturm bound: \(200\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(2\), \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(475))\).

Total New Old
Modular forms 156 86 70
Cusp forms 144 86 58
Eisenstein series 12 0 12

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

\(5\)\(19\)FrickeDim
\(+\)\(+\)$+$\(21\)
\(+\)\(-\)$-$\(19\)
\(-\)\(+\)$-$\(21\)
\(-\)\(-\)$+$\(25\)
Plus space\(+\)\(46\)
Minus space\(-\)\(40\)

Trace form

\( 86 q - 4 q^{2} + 4 q^{3} + 358 q^{4} - 10 q^{6} - 4 q^{7} + 764 q^{9} + O(q^{10}) \) \( 86 q - 4 q^{2} + 4 q^{3} + 358 q^{4} - 10 q^{6} - 4 q^{7} + 764 q^{9} + 38 q^{11} + 56 q^{12} - 92 q^{13} + 36 q^{14} + 1338 q^{16} - 40 q^{17} - 36 q^{18} + 38 q^{19} - 240 q^{21} - 488 q^{22} - 94 q^{23} + 254 q^{24} + 114 q^{26} - 56 q^{27} + 606 q^{28} - 788 q^{29} + 44 q^{31} + 1096 q^{32} + 356 q^{33} - 208 q^{34} + 4088 q^{36} + 428 q^{37} - 114 q^{38} + 606 q^{39} - 96 q^{41} + 1246 q^{42} + 518 q^{43} - 1596 q^{44} + 328 q^{46} - 1922 q^{47} + 1996 q^{48} + 4410 q^{49} - 1212 q^{51} - 1144 q^{52} + 1180 q^{53} + 2182 q^{54} + 724 q^{56} - 114 q^{57} - 262 q^{58} + 1140 q^{59} + 1230 q^{61} + 1468 q^{62} - 66 q^{63} + 6366 q^{64} - 2056 q^{66} - 376 q^{67} + 58 q^{68} - 6212 q^{69} - 3360 q^{71} - 1876 q^{72} - 2020 q^{73} + 4540 q^{74} + 380 q^{76} - 262 q^{77} + 4444 q^{78} - 148 q^{79} + 7262 q^{81} - 3896 q^{82} + 20 q^{83} - 5152 q^{84} - 5560 q^{86} - 450 q^{87} - 2404 q^{88} + 2000 q^{89} - 2584 q^{91} - 2110 q^{92} - 5172 q^{93} + 712 q^{94} + 1122 q^{96} - 820 q^{97} + 2164 q^{98} - 2158 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(475))\) 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 19
475.4.a.a 475.a 1.a $1$ $28.026$ \(\Q\) None \(-5\) \(-4\) \(0\) \(32\) $+$ $-$ $\mathrm{SU}(2)$ \(q-5q^{2}-4q^{3}+17q^{4}+20q^{6}+2^{5}q^{7}+\cdots\)
475.4.a.b 475.a 1.a $1$ $28.026$ \(\Q\) None \(-3\) \(-7\) \(0\) \(-11\) $+$ $-$ $\mathrm{SU}(2)$ \(q-3q^{2}-7q^{3}+q^{4}+21q^{6}-11q^{7}+\cdots\)
475.4.a.c 475.a 1.a $1$ $28.026$ \(\Q\) None \(-3\) \(5\) \(0\) \(1\) $+$ $-$ $\mathrm{SU}(2)$ \(q-3q^{2}+5q^{3}+q^{4}-15q^{6}+q^{7}+\cdots\)
475.4.a.d 475.a 1.a $1$ $28.026$ \(\Q\) None \(0\) \(-4\) \(0\) \(22\) $+$ $-$ $\mathrm{SU}(2)$ \(q-4q^{3}-8q^{4}+22q^{7}-11q^{9}-12q^{11}+\cdots\)
475.4.a.e 475.a 1.a $1$ $28.026$ \(\Q\) None \(3\) \(5\) \(0\) \(-11\) $+$ $-$ $\mathrm{SU}(2)$ \(q+3q^{2}+5q^{3}+q^{4}+15q^{6}-11q^{7}+\cdots\)
475.4.a.f 475.a 1.a $3$ $28.026$ 3.3.3144.1 None \(-3\) \(-1\) \(0\) \(35\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(-1-\beta _{1}+\beta _{2})q^{2}+(\beta _{1}-2\beta _{2})q^{3}+\cdots\)
475.4.a.g 475.a 1.a $3$ $28.026$ 3.3.1304.1 None \(3\) \(11\) \(0\) \(5\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(1+\beta _{1})q^{2}+(4+\beta _{1}+\beta _{2})q^{3}+(1+\cdots)q^{4}+\cdots\)
475.4.a.h 475.a 1.a $5$ $28.026$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(3\) \(4\) \(0\) \(-72\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}+(\beta _{1}+\beta _{3})q^{3}+(4-\beta _{1}+\cdots)q^{4}+\cdots\)
475.4.a.i 475.a 1.a $6$ $28.026$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(1\) \(-5\) \(0\) \(-5\) $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(-1+\beta _{1}+\beta _{5})q^{3}+(4+\beta _{2}+\cdots)q^{4}+\cdots\)
475.4.a.j 475.a 1.a $9$ $28.026$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None \(-7\) \(-6\) \(0\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{2}+(-1-\beta _{4})q^{3}+(5+\cdots)q^{4}+\cdots\)
475.4.a.k 475.a 1.a $9$ $28.026$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None \(-5\) \(-6\) \(0\) \(-28\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{2}+(-\beta _{1}+\beta _{4})q^{3}+(5+\cdots)q^{4}+\cdots\)
475.4.a.l 475.a 1.a $9$ $28.026$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None \(5\) \(6\) \(0\) \(28\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}+(\beta _{1}-\beta _{4})q^{3}+(5-\beta _{1}+\cdots)q^{4}+\cdots\)
475.4.a.m 475.a 1.a $9$ $28.026$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None \(7\) \(6\) \(0\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}+(1+\beta _{4})q^{3}+(5-2\beta _{1}+\cdots)q^{4}+\cdots\)
475.4.a.n 475.a 1.a $12$ $28.026$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}-\beta _{5}q^{3}+(3+\beta _{2})q^{4}+(-5+\cdots)q^{6}+\cdots\)
475.4.a.o 475.a 1.a $16$ $28.026$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}-\beta _{11}q^{3}+(5+\beta _{2})q^{4}+(5+\cdots)q^{6}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(475))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(475)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(19))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(25))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(95))\)\(^{\oplus 2}\)