Properties

Label 475.2.a
Level $475$
Weight $2$
Character orbit 475.a
Rep. character $\chi_{475}(1,\cdot)$
Character field $\Q$
Dimension $28$
Newform subspaces $10$
Sturm bound $100$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 475 = 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 475.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 10 \)
Sturm bound: \(100\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(475))\).

Total New Old
Modular forms 56 28 28
Cusp forms 45 28 17
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.

\(5\)\(19\)FrickeDim
\(+\)\(+\)\(+\)\(6\)
\(+\)\(-\)\(-\)\(8\)
\(-\)\(+\)\(-\)\(9\)
\(-\)\(-\)\(+\)\(5\)
Plus space\(+\)\(11\)
Minus space\(-\)\(17\)

Trace form

\( 28 q + q^{2} - 2 q^{3} + 29 q^{4} - 3 q^{7} + 9 q^{8} + 16 q^{9} + O(q^{10}) \) \( 28 q + q^{2} - 2 q^{3} + 29 q^{4} - 3 q^{7} + 9 q^{8} + 16 q^{9} - 3 q^{11} - 6 q^{13} + 23 q^{16} - 3 q^{17} + 29 q^{18} - 2 q^{19} - 6 q^{21} + 12 q^{22} + 12 q^{23} + 8 q^{24} + 6 q^{26} - 8 q^{27} - 6 q^{28} - 16 q^{31} + 9 q^{32} - 6 q^{33} - 10 q^{34} - 27 q^{36} - 16 q^{37} + 3 q^{38} - 28 q^{39} + 4 q^{41} - 16 q^{42} + q^{43} - 2 q^{44} + 4 q^{46} + 15 q^{47} - 40 q^{48} - 13 q^{49} - 2 q^{51} - 30 q^{52} - 18 q^{53} - 52 q^{54} - 32 q^{56} + 2 q^{57} - 2 q^{58} + 2 q^{59} - 21 q^{61} - 48 q^{62} + 13 q^{63} + 79 q^{64} - 36 q^{66} + 20 q^{67} - 8 q^{68} + 18 q^{71} + 65 q^{72} - 23 q^{73} - 2 q^{74} - 5 q^{76} + 35 q^{77} - 24 q^{79} + 16 q^{81} + 2 q^{82} + 20 q^{83} - 56 q^{84} + 40 q^{86} + 76 q^{87} + 16 q^{88} + 2 q^{89} - 24 q^{91} + 20 q^{92} + 48 q^{93} + 24 q^{94} - 52 q^{96} - 58 q^{97} - 23 q^{98} + 9 q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(475))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 5 19
475.2.a.a 475.a 1.a $1$ $3.793$ \(\Q\) None 95.2.b.a \(-1\) \(0\) \(0\) \(2\) $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{4}+2q^{7}+3q^{8}-3q^{9}-4q^{11}+\cdots\)
475.2.a.b 475.a 1.a $1$ $3.793$ \(\Q\) None 19.2.a.a \(0\) \(2\) \(0\) \(1\) $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{3}-2q^{4}+q^{7}+q^{9}+3q^{11}+\cdots\)
475.2.a.c 475.a 1.a $1$ $3.793$ \(\Q\) None 95.2.b.a \(1\) \(0\) \(0\) \(-2\) $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{4}-2q^{7}-3q^{8}-3q^{9}-4q^{11}+\cdots\)
475.2.a.d 475.a 1.a $3$ $3.793$ \(\Q(\zeta_{14})^+\) None 475.2.a.d \(-4\) \(-2\) \(0\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(-1-\beta _{1})q^{2}+(-\beta _{1}+\beta _{2})q^{3}+(1+\cdots)q^{4}+\cdots\)
475.2.a.e 475.a 1.a $3$ $3.793$ 3.3.169.1 None 475.2.a.e \(-2\) \(-2\) \(0\) \(-4\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{2}+(-1-\beta _{2})q^{3}+(2+\cdots)q^{4}+\cdots\)
475.2.a.f 475.a 1.a $3$ $3.793$ 3.3.148.1 None 95.2.a.a \(-1\) \(-2\) \(0\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+(-1+\beta _{1}+\beta _{2})q^{3}+(\beta _{1}+\cdots)q^{4}+\cdots\)
475.2.a.g 475.a 1.a $3$ $3.793$ 3.3.169.1 None 475.2.a.e \(2\) \(2\) \(0\) \(4\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}+(1+\beta _{2})q^{3}+(2-\beta _{1}+\cdots)q^{4}+\cdots\)
475.2.a.h 475.a 1.a $3$ $3.793$ \(\Q(\zeta_{14})^+\) None 475.2.a.d \(4\) \(2\) \(0\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(1+\beta _{1})q^{2}+(\beta _{1}-\beta _{2})q^{3}+(1+2\beta _{1}+\cdots)q^{4}+\cdots\)
475.2.a.i 475.a 1.a $4$ $3.793$ 4.4.11344.1 None 95.2.a.b \(2\) \(-2\) \(0\) \(-4\) $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{2}q^{2}+\beta _{3}q^{3}+(1+\beta _{1}-\beta _{2})q^{4}+\cdots\)
475.2.a.j 475.a 1.a $6$ $3.793$ 6.6.66064384.1 None 95.2.b.b \(0\) \(0\) \(0\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{4}q^{2}+(-\beta _{1}-\beta _{5})q^{3}+(1-\beta _{3}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(475))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(475)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(19))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(95))\)\(^{\oplus 2}\)