Properties

Label 175.10.a
Level $175$
Weight $10$
Character orbit 175.a
Rep. character $\chi_{175}(1,\cdot)$
Character field $\Q$
Dimension $85$
Newform subspaces $13$
Sturm bound $200$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 175 = 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 175.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 13 \)
Sturm bound: \(200\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{10}(\Gamma_0(175))\).

Total New Old
Modular forms 186 85 101
Cusp forms 174 85 89
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.

\(5\)\(7\)FrickeDim
\(+\)\(+\)\(+\)\(19\)
\(+\)\(-\)\(-\)\(22\)
\(-\)\(+\)\(-\)\(23\)
\(-\)\(-\)\(+\)\(21\)
Plus space\(+\)\(40\)
Minus space\(-\)\(45\)

Trace form

\( 85 q - 17 q^{2} + 294 q^{3} + 22273 q^{4} - 11386 q^{6} + 2401 q^{7} - 7905 q^{8} + 522233 q^{9} + O(q^{10}) \) \( 85 q - 17 q^{2} + 294 q^{3} + 22273 q^{4} - 11386 q^{6} + 2401 q^{7} - 7905 q^{8} + 522233 q^{9} - 54432 q^{11} + 480838 q^{12} - 48136 q^{13} - 88837 q^{14} + 5786121 q^{16} + 711878 q^{17} - 2153581 q^{18} - 640366 q^{19} - 408170 q^{21} + 2328236 q^{22} - 695176 q^{23} - 8763666 q^{24} - 3625100 q^{26} + 4282140 q^{27} + 5226977 q^{28} - 12919774 q^{29} - 356700 q^{31} + 2428023 q^{32} - 9834072 q^{33} + 10780618 q^{34} + 121310165 q^{36} - 34821542 q^{37} - 28892630 q^{38} + 53917784 q^{39} + 39695746 q^{41} - 8926918 q^{42} + 78756864 q^{43} + 10878162 q^{44} - 130524998 q^{46} + 87054668 q^{47} + 175660814 q^{48} + 490008085 q^{49} + 251405632 q^{51} + 98290868 q^{52} + 107791454 q^{53} - 350769592 q^{54} - 112316379 q^{56} + 128646980 q^{57} - 484167470 q^{58} - 517888282 q^{59} - 243558728 q^{61} + 529325916 q^{62} + 166900713 q^{63} + 1343015567 q^{64} + 1377678316 q^{66} - 526332 q^{67} - 511592234 q^{68} + 484724056 q^{69} - 774052540 q^{71} - 848550385 q^{72} + 317883234 q^{73} + 578853308 q^{74} + 340259166 q^{76} + 168242872 q^{77} - 370287692 q^{78} + 881409660 q^{79} + 1567163561 q^{81} - 1592705654 q^{82} + 989257114 q^{83} - 1017394938 q^{84} + 3261763934 q^{86} - 3212160700 q^{87} - 2388172120 q^{88} - 548591634 q^{89} + 326948972 q^{91} + 2656347768 q^{92} + 722379888 q^{93} + 8394540344 q^{94} + 3715282870 q^{96} + 2220934678 q^{97} - 98001617 q^{98} + 3534938932 q^{99} + O(q^{100}) \)

Decomposition of \(S_{10}^{\mathrm{new}}(\Gamma_0(175))\) 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 7
175.10.a.a 175.a 1.a $1$ $90.131$ \(\Q\) None 35.10.a.a \(-28\) \(116\) \(0\) \(-2401\) $+$ $+$ $\mathrm{SU}(2)$ \(q-28q^{2}+116q^{3}+272q^{4}-3248q^{6}+\cdots\)
175.10.a.b 175.a 1.a $2$ $90.131$ \(\Q(\sqrt{193}) \) None 7.10.a.a \(6\) \(86\) \(0\) \(4802\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(3+\beta )q^{2}+(43-11\beta )q^{3}+(-310+\cdots)q^{4}+\cdots\)
175.10.a.c 175.a 1.a $2$ $90.131$ \(\Q(\sqrt{2}) \) None 35.10.a.b \(24\) \(174\) \(0\) \(-4802\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(12+\beta )q^{2}+(87+54\beta )q^{3}+(-360+\cdots)q^{4}+\cdots\)
175.10.a.d 175.a 1.a $3$ $90.131$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None 7.10.a.b \(-21\) \(-84\) \(0\) \(-7203\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(-7+\beta _{2})q^{2}+(-28+\beta _{1}+\beta _{2})q^{3}+\cdots\)
175.10.a.e 175.a 1.a $4$ $90.131$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None 35.10.a.c \(19\) \(18\) \(0\) \(9604\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(5+\beta _{1})q^{2}+(4-\beta _{2})q^{3}+(435+9\beta _{1}+\cdots)q^{4}+\cdots\)
175.10.a.f 175.a 1.a $5$ $90.131$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None 35.10.a.d \(-2\) \(-140\) \(0\) \(-12005\) $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+(-28-\beta _{2})q^{3}+(168-4\beta _{1}+\cdots)q^{4}+\cdots\)
175.10.a.g 175.a 1.a $6$ $90.131$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None 35.10.a.e \(-15\) \(124\) \(0\) \(14406\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(-3+\beta _{1})q^{2}+(20+\beta _{1}-\beta _{2})q^{3}+\cdots\)
175.10.a.h 175.a 1.a $8$ $90.131$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None 175.10.a.h \(-27\) \(69\) \(0\) \(19208\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-3-\beta _{1})q^{2}+(8+\beta _{1}-\beta _{2})q^{3}+\cdots\)
175.10.a.i 175.a 1.a $8$ $90.131$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None 175.10.a.h \(27\) \(-69\) \(0\) \(-19208\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(3+\beta _{1})q^{2}+(-8-\beta _{1}+\beta _{2})q^{3}+\cdots\)
175.10.a.j 175.a 1.a $10$ $90.131$ \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None 175.10.a.j \(-22\) \(77\) \(0\) \(-24010\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(-2-\beta _{1})q^{2}+(8+\beta _{3})q^{3}+(237+\cdots)q^{4}+\cdots\)
175.10.a.k 175.a 1.a $10$ $90.131$ \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None 175.10.a.j \(22\) \(-77\) \(0\) \(24010\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(2+\beta _{1})q^{2}+(-8-\beta _{3})q^{3}+(237+\cdots)q^{4}+\cdots\)
175.10.a.l 175.a 1.a $13$ $90.131$ \(\mathbb{Q}[x]/(x^{13} - \cdots)\) None 35.10.b.a \(-32\) \(-158\) \(0\) \(31213\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-2-\beta _{1})q^{2}+(-12+\beta _{3})q^{3}+(208+\cdots)q^{4}+\cdots\)
175.10.a.m 175.a 1.a $13$ $90.131$ \(\mathbb{Q}[x]/(x^{13} - \cdots)\) None 35.10.b.a \(32\) \(158\) \(0\) \(-31213\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(2+\beta _{1})q^{2}+(12-\beta _{3})q^{3}+(208+5\beta _{1}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{10}^{\mathrm{old}}(\Gamma_0(175))\) into lower level spaces

\( S_{10}^{\mathrm{old}}(\Gamma_0(175)) \simeq \) \(S_{10}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 4}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 3}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(25))\)\(^{\oplus 2}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 2}\)