Properties

Label 225.10.a
Level $225$
Weight $10$
Character orbit 225.a
Rep. character $\chi_{225}(1,\cdot)$
Character field $\Q$
Dimension $70$
Newform subspaces $24$
Sturm bound $300$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 225.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 24 \)
Sturm bound: \(300\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(2\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{10}(\Gamma_0(225))\).

Total New Old
Modular forms 282 73 209
Cusp forms 258 70 188
Eisenstein series 24 3 21

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

\(3\)\(5\)FrickeDim
\(+\)\(+\)$+$\(13\)
\(+\)\(-\)$-$\(16\)
\(-\)\(+\)$-$\(20\)
\(-\)\(-\)$+$\(21\)
Plus space\(+\)\(34\)
Minus space\(-\)\(36\)

Trace form

\( 70 q + 32 q^{2} + 17922 q^{4} + 5638 q^{7} + 32940 q^{8} + O(q^{10}) \) \( 70 q + 32 q^{2} + 17922 q^{4} + 5638 q^{7} + 32940 q^{8} - 99738 q^{11} - 148516 q^{13} + 120072 q^{14} + 4164554 q^{16} + 315962 q^{17} + 1059982 q^{19} - 420944 q^{22} - 1192254 q^{23} + 3928908 q^{26} - 122144 q^{28} - 826002 q^{29} + 11154716 q^{31} + 24666772 q^{32} + 24557570 q^{34} - 18572232 q^{37} - 17323240 q^{38} - 25410504 q^{41} + 40819214 q^{43} - 228405234 q^{44} + 20818204 q^{46} + 125262422 q^{47} + 314859818 q^{49} - 196109592 q^{52} - 18769894 q^{53} - 27361620 q^{56} - 65147920 q^{58} - 391612404 q^{59} - 8832016 q^{61} + 38022984 q^{62} + 886502810 q^{64} + 564691858 q^{67} + 589124584 q^{68} - 465147732 q^{71} + 166031624 q^{73} + 176521572 q^{74} + 1142520042 q^{76} - 737802096 q^{77} + 82833184 q^{79} + 656687216 q^{82} + 1448251266 q^{83} - 2178393948 q^{86} - 654530880 q^{88} + 2699989944 q^{89} - 306057636 q^{91} - 3936201888 q^{92} - 2820398824 q^{94} - 1304850912 q^{97} + 6818382544 q^{98} + O(q^{100}) \)

Decomposition of \(S_{10}^{\mathrm{new}}(\Gamma_0(225))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 3 5
225.10.a.a 225.a 1.a $1$ $115.883$ \(\Q\) None \(-36\) \(0\) \(0\) \(4480\) $-$ $+$ $\mathrm{SU}(2)$ \(q-6^{2}q^{2}+28^{2}q^{4}+4480q^{7}-9792q^{8}+\cdots\)
225.10.a.b 225.a 1.a $1$ $115.883$ \(\Q\) None \(-8\) \(0\) \(0\) \(-4242\) $-$ $+$ $\mathrm{SU}(2)$ \(q-8q^{2}-448q^{4}-4242q^{7}+7680q^{8}+\cdots\)
225.10.a.c 225.a 1.a $1$ $115.883$ \(\Q\) None \(-4\) \(0\) \(0\) \(7680\) $-$ $+$ $\mathrm{SU}(2)$ \(q-4q^{2}-496q^{4}+7680q^{7}+4032q^{8}+\cdots\)
225.10.a.d 225.a 1.a $1$ $115.883$ \(\Q\) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(12580\) $+$ $+$ $N(\mathrm{U}(1))$ \(q-2^{9}q^{4}+12580q^{7}-118370q^{13}+\cdots\)
225.10.a.e 225.a 1.a $1$ $115.883$ \(\Q\) None \(18\) \(0\) \(0\) \(-9128\) $-$ $+$ $\mathrm{SU}(2)$ \(q+18q^{2}-188q^{4}-9128q^{7}-12600q^{8}+\cdots\)
225.10.a.f 225.a 1.a $1$ $115.883$ \(\Q\) None \(22\) \(0\) \(0\) \(5988\) $-$ $+$ $\mathrm{SU}(2)$ \(q+22q^{2}-28q^{4}+5988q^{7}-11880q^{8}+\cdots\)
225.10.a.g 225.a 1.a $2$ $115.883$ \(\Q(\sqrt{79}) \) None \(-36\) \(0\) \(0\) \(-3318\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(-18+\beta )q^{2}+(2^{7}-6^{2}\beta )q^{4}+(-1659+\cdots)q^{7}+\cdots\)
225.10.a.h 225.a 1.a $2$ $115.883$ \(\Q(\sqrt{1009}) \) None \(-10\) \(0\) \(0\) \(-1700\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(-5-\beta )q^{2}+(522+10\beta )q^{4}+(-850+\cdots)q^{7}+\cdots\)
225.10.a.i 225.a 1.a $2$ $115.883$ \(\Q(\sqrt{5}) \) \(\Q(\sqrt{-15}) \) \(0\) \(0\) \(0\) \(0\) $+$ $-$ $N(\mathrm{U}(1))$ \(q-19\beta q^{2}+1293q^{4}-14839\beta q^{8}+\cdots\)
225.10.a.j 225.a 1.a $2$ $115.883$ \(\Q(\sqrt{4729}) \) None \(19\) \(0\) \(0\) \(11872\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(10-\beta )q^{2}+(770-19\beta )q^{4}+(5964+\cdots)q^{7}+\cdots\)
225.10.a.k 225.a 1.a $2$ $115.883$ \(\Q(\sqrt{241}) \) None \(31\) \(0\) \(0\) \(-14112\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(15-\beta )q^{2}+(255-31\beta )q^{4}+(-6944+\cdots)q^{7}+\cdots\)
225.10.a.l 225.a 1.a $2$ $115.883$ \(\Q(\sqrt{79}) \) None \(36\) \(0\) \(0\) \(3318\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(18+\beta )q^{2}+(2^{7}+6^{2}\beta )q^{4}+(1659+\cdots)q^{7}+\cdots\)
225.10.a.m 225.a 1.a $3$ $115.883$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(-33\) \(0\) \(0\) \(5258\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-11-\beta _{1})q^{2}+(114+9\beta _{1}-\beta _{2})q^{4}+\cdots\)
225.10.a.n 225.a 1.a $3$ $115.883$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(-25\) \(0\) \(0\) \(-3890\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(-8+\beta _{1})q^{2}+(366-10\beta _{1}-\beta _{2})q^{4}+\cdots\)
225.10.a.o 225.a 1.a $3$ $115.883$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(25\) \(0\) \(0\) \(-3890\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(8-\beta _{1})q^{2}+(366-10\beta _{1}-\beta _{2})q^{4}+\cdots\)
225.10.a.p 225.a 1.a $3$ $115.883$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(33\) \(0\) \(0\) \(-5258\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(11+\beta _{1})q^{2}+(114+9\beta _{1}-\beta _{2})q^{4}+\cdots\)
225.10.a.q 225.a 1.a $4$ $115.883$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(-3\) \(0\) \(0\) \(-9834\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-1-\beta _{1})q^{2}+(149-2\beta _{1}-\beta _{2}+\cdots)q^{4}+\cdots\)
225.10.a.r 225.a 1.a $4$ $115.883$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(-2\) \(0\) \(0\) \(-13036\) $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(445-6\beta _{1}+\beta _{2})q^{4}+(-3237+\cdots)q^{7}+\cdots\)
225.10.a.s 225.a 1.a $4$ $115.883$ 4.4.49740556.1 None \(0\) \(0\) \(0\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+(342-\beta _{3})q^{4}+(133\beta _{1}-7^{2}\beta _{2}+\cdots)q^{7}+\cdots\)
225.10.a.t 225.a 1.a $4$ $115.883$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(2\) \(0\) \(0\) \(13036\) $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+(445-6\beta _{1}+\beta _{2})q^{4}+(3237+\cdots)q^{7}+\cdots\)
225.10.a.u 225.a 1.a $4$ $115.883$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(3\) \(0\) \(0\) \(9834\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(1+\beta _{1})q^{2}+(149-2\beta _{1}-\beta _{2})q^{4}+\cdots\)
225.10.a.v 225.a 1.a $6$ $115.883$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(0\) \(0\) \(0\) \(-9490\) $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(285+\beta _{2})q^{4}+(-1583+\cdots)q^{7}+\cdots\)
225.10.a.w 225.a 1.a $6$ $115.883$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(0\) \(0\) \(0\) \(9490\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(285+\beta _{2})q^{4}+(1583+5\beta _{2}+\cdots)q^{7}+\cdots\)
225.10.a.x 225.a 1.a $8$ $115.883$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{3}q^{2}+(18-\beta _{1})q^{4}+\beta _{2}q^{7}+(-220\beta _{3}+\cdots)q^{8}+\cdots\)

Decomposition of \(S_{10}^{\mathrm{old}}(\Gamma_0(225))\) into lower level spaces

\( S_{10}^{\mathrm{old}}(\Gamma_0(225)) \cong \) \(S_{10}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 6}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 6}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 3}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 4}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(25))\)\(^{\oplus 3}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 2}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 2}\)