Properties

Label 175.7.d
Level $175$
Weight $7$
Character orbit 175.d
Rep. character $\chi_{175}(76,\cdot)$
Character field $\Q$
Dimension $73$
Newform subspaces $9$
Sturm bound $140$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 175 = 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 7 \)
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: \(140\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{7}(175, [\chi])\).

Total New Old
Modular forms 126 79 47
Cusp forms 114 73 41
Eisenstein series 12 6 6

Trace form

\( 73 q - 3 q^{2} + 2197 q^{4} + 477 q^{7} - 1071 q^{8} - 14631 q^{9} + O(q^{10}) \) \( 73 q - 3 q^{2} + 2197 q^{4} + 477 q^{7} - 1071 q^{8} - 14631 q^{9} - 2358 q^{11} + 6869 q^{14} + 65805 q^{16} + 27313 q^{18} + 3876 q^{21} + 27166 q^{22} - 10422 q^{23} + 87361 q^{28} - 85790 q^{29} - 244047 q^{32} - 221475 q^{36} + 128802 q^{37} + 123432 q^{39} - 168540 q^{42} - 174462 q^{43} - 507496 q^{44} + 230816 q^{46} + 219173 q^{49} - 43608 q^{51} + 515586 q^{53} + 326251 q^{56} + 522720 q^{57} - 97946 q^{58} - 426547 q^{63} + 1160931 q^{64} - 1389038 q^{67} - 814158 q^{71} + 2991041 q^{72} + 3894664 q^{74} + 1087482 q^{77} + 2045340 q^{78} + 388330 q^{79} + 1671105 q^{81} - 928104 q^{84} - 260864 q^{86} + 1607450 q^{88} + 228476 q^{91} - 3382422 q^{92} - 1633200 q^{93} - 175227 q^{98} + 2894442 q^{99} + O(q^{100}) \)

Decomposition of \(S_{7}^{\mathrm{new}}(175, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
175.7.d.a 175.d 7.b $1$ $40.259$ \(\Q\) \(\Q(\sqrt{-7}) \) \(-9\) \(0\) \(0\) \(343\) $\mathrm{U}(1)[D_{2}]$ \(q-9q^{2}+17q^{4}+7^{3}q^{7}+423q^{8}+\cdots\)
175.7.d.b 175.d 7.b $2$ $40.259$ \(\Q(\sqrt{21}) \) \(\Q(\sqrt{-7}) \) \(-9\) \(0\) \(0\) \(-686\) $\mathrm{U}(1)[D_{2}]$ \(q+(-5-\beta )q^{2}+(92+9\beta )q^{4}-7^{3}q^{7}+\cdots\)
175.7.d.c 175.d 7.b $2$ $40.259$ \(\Q(\sqrt{-1}) \) \(\Q(\sqrt{-35}) \) \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q-26iq^{3}-2^{6}q^{4}-7^{3}iq^{7}+53q^{9}+\cdots\)
175.7.d.d 175.d 7.b $2$ $40.259$ \(\Q(\sqrt{21}) \) \(\Q(\sqrt{-7}) \) \(9\) \(0\) \(0\) \(686\) $\mathrm{U}(1)[D_{2}]$ \(q+(5+\beta )q^{2}+(92+9\beta )q^{4}+7^{3}q^{7}+\cdots\)
175.7.d.e 175.d 7.b $2$ $40.259$ \(\Q(\sqrt{-510}) \) None \(16\) \(0\) \(0\) \(-266\) $\mathrm{SU}(2)[C_{2}]$ \(q+8q^{2}-\beta q^{3}-8\beta q^{6}+(-133-7\beta )q^{7}+\cdots\)
175.7.d.f 175.d 7.b $14$ $40.259$ \(\mathbb{Q}[x]/(x^{14} + \cdots)\) None \(-4\) \(0\) \(0\) \(-526\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{2}q^{2}+\beta _{1}q^{3}+(20-2\beta _{2}-\beta _{3}+\cdots)q^{4}+\cdots\)
175.7.d.g 175.d 7.b $14$ $40.259$ \(\mathbb{Q}[x]/(x^{14} + \cdots)\) None \(4\) \(0\) \(0\) \(526\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{2}q^{2}+\beta _{1}q^{3}+(20-2\beta _{2}-\beta _{3}+\cdots)q^{4}+\cdots\)
175.7.d.h 175.d 7.b $16$ $40.259$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(-10\) \(0\) \(0\) \(400\) $\mathrm{SU}(2)[C_{2}]$ \(q+(-1+\beta _{1})q^{2}-\beta _{3}q^{3}+(38+\beta _{2}+\cdots)q^{4}+\cdots\)
175.7.d.i 175.d 7.b $20$ $40.259$ \(\mathbb{Q}[x]/(x^{20} + \cdots)\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{10}q^{2}-\beta _{11}q^{3}+(39-\beta _{1})q^{4}+\cdots\)

Decomposition of \(S_{7}^{\mathrm{old}}(175, [\chi])\) into lower level spaces

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