Properties

Label 175.5.d
Level $175$
Weight $5$
Character orbit 175.d
Rep. character $\chi_{175}(76,\cdot)$
Character field $\Q$
Dimension $47$
Newform subspaces $9$
Sturm bound $100$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 175 = 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 175.d (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 7 \)
Character field: \(\Q\)
Newform subspaces: \( 9 \)
Sturm bound: \(100\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{5}(175, [\chi])\).

Total New Old
Modular forms 86 53 33
Cusp forms 74 47 27
Eisenstein series 12 6 6

Trace form

\( 47 q + 5 q^{2} + 341 q^{4} + q^{7} + 217 q^{8} - 1053 q^{9} + O(q^{10}) \) \( 47 q + 5 q^{2} + 341 q^{4} + q^{7} + 217 q^{8} - 1053 q^{9} + 294 q^{11} - 491 q^{14} + 1725 q^{16} - 815 q^{18} - 60 q^{21} - 2058 q^{22} + 1490 q^{23} - 679 q^{28} + 14 q^{29} + 7977 q^{32} - 6219 q^{36} - 4270 q^{37} - 8700 q^{39} - 5580 q^{42} - 3610 q^{43} + 12592 q^{44} - 6992 q^{46} + 4447 q^{49} - 13620 q^{51} - 6178 q^{53} + 2123 q^{56} + 12900 q^{57} + 17262 q^{58} + 341 q^{63} + 579 q^{64} + 19150 q^{67} + 12834 q^{71} - 36703 q^{72} - 3232 q^{74} - 16810 q^{77} - 20100 q^{78} - 16786 q^{79} + 30447 q^{81} + 14280 q^{84} - 34072 q^{86} + 882 q^{88} + 48700 q^{91} + 49242 q^{92} + 70980 q^{93} - 60115 q^{98} - 27126 q^{99} + O(q^{100}) \)

Decomposition of \(S_{5}^{\mathrm{new}}(175, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
175.5.d.a 175.d 7.b $1$ $18.090$ \(\Q\) \(\Q(\sqrt{-7}) \) \(-1\) \(0\) \(0\) \(-49\) $\mathrm{U}(1)[D_{2}]$ \(q-q^{2}-15q^{4}-7^{2}q^{7}+31q^{8}+3^{4}q^{9}+\cdots\)
175.5.d.b 175.d 7.b $2$ $18.090$ \(\Q(\sqrt{21}) \) \(\Q(\sqrt{-7}) \) \(-1\) \(0\) \(0\) \(98\) $\mathrm{U}(1)[D_{2}]$ \(q-\beta q^{2}+(31+\beta )q^{4}+7^{2}q^{7}+(-47+\cdots)q^{8}+\cdots\)
175.5.d.c 175.d 7.b $2$ $18.090$ \(\Q(\sqrt{-1}) \) \(\Q(\sqrt{-35}) \) \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+17iq^{3}-2^{4}q^{4}+7^{2}iq^{7}-208q^{9}+\cdots\)
175.5.d.d 175.d 7.b $2$ $18.090$ \(\Q(\sqrt{21}) \) \(\Q(\sqrt{-7}) \) \(1\) \(0\) \(0\) \(-98\) $\mathrm{U}(1)[D_{2}]$ \(q+\beta q^{2}+(31+\beta )q^{4}-7^{2}q^{7}+(47+2^{4}\beta )q^{8}+\cdots\)
175.5.d.e 175.d 7.b $4$ $18.090$ \(\Q(i, \sqrt{6})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{2}q^{2}-\beta _{1}q^{3}-10q^{4}+\beta _{3}q^{6}+\cdots\)
175.5.d.f 175.d 7.b $8$ $18.090$ \(\mathbb{Q}[x]/(x^{8} + \cdots)\) None \(-4\) \(0\) \(0\) \(128\) $\mathrm{SU}(2)[C_{2}]$ \(q+(-1-\beta _{2})q^{2}-\beta _{1}q^{3}+(1+\beta _{2}+\beta _{6}+\cdots)q^{4}+\cdots\)
175.5.d.g 175.d 7.b $8$ $18.090$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{4}q^{2}+3\beta _{3}q^{3}+(22+\beta _{1})q^{4}-3\beta _{2}q^{6}+\cdots\)
175.5.d.h 175.d 7.b $8$ $18.090$ \(\mathbb{Q}[x]/(x^{8} + \cdots)\) None \(4\) \(0\) \(0\) \(-128\) $\mathrm{SU}(2)[C_{2}]$ \(q+(1+\beta _{2})q^{2}-\beta _{1}q^{3}+(1+\beta _{2}+\beta _{6}+\cdots)q^{4}+\cdots\)
175.5.d.i 175.d 7.b $12$ $18.090$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(6\) \(0\) \(0\) \(50\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{1}q^{2}-\beta _{3}q^{3}+(10-\beta _{1}-\beta _{2})q^{4}+\cdots\)

Decomposition of \(S_{5}^{\mathrm{old}}(175, [\chi])\) into lower level spaces

\( S_{5}^{\mathrm{old}}(175, [\chi]) \cong \) \(S_{5}^{\mathrm{new}}(7, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(35, [\chi])\)\(^{\oplus 2}\)