Properties

Label 2025.2.a
Level $2025$
Weight $2$
Character orbit 2025.a
Rep. character $\chi_{2025}(1,\cdot)$
Character field $\Q$
Dimension $70$
Newform subspaces $26$
Sturm bound $540$
Trace bound $11$

Related objects

Downloads

Learn more

Defining parameters

Level: \( N \) \(=\) \( 2025 = 3^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2025.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 26 \)
Sturm bound: \(540\)
Trace bound: \(11\)
Distinguishing \(T_p\): \(2\), \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(2025))\).

Total New Old
Modular forms 306 82 224
Cusp forms 235 70 165
Eisenstein series 71 12 59

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.
\(+\)\(+\)\(+\)\(16\)
\(+\)\(-\)\(-\)\(20\)
\(-\)\(+\)\(-\)\(18\)
\(-\)\(-\)\(+\)\(16\)
Plus space\(+\)\(32\)
Minus space\(-\)\(38\)

Trace form

\( 70 q + 62 q^{4} + O(q^{10}) \) \( 70 q + 62 q^{4} + 6 q^{13} + 42 q^{16} + 12 q^{19} - 12 q^{28} + 8 q^{31} + 10 q^{34} + 18 q^{37} + 24 q^{43} - 52 q^{46} + 34 q^{49} + 54 q^{52} + 6 q^{58} + 2 q^{61} + 18 q^{64} - 36 q^{67} + 30 q^{73} + 88 q^{76} - 48 q^{79} + 72 q^{82} - 48 q^{88} - 16 q^{91} - 20 q^{94} + 12 q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(2025))\) 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
2025.2.a.a 2025.a 1.a $1$ $16.170$ \(\Q\) None \(-2\) \(0\) \(0\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+2q^{4}+5q^{11}-4q^{13}-4q^{16}+\cdots\)
2025.2.a.b 2025.a 1.a $1$ $16.170$ \(\Q\) None \(-1\) \(0\) \(0\) \(3\) $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{4}+3q^{7}+3q^{8}-2q^{11}+\cdots\)
2025.2.a.c 2025.a 1.a $1$ $16.170$ \(\Q\) None \(0\) \(0\) \(0\) \(-2\) $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{4}-2q^{7}-3q^{11}+4q^{13}+4q^{16}+\cdots\)
2025.2.a.d 2025.a 1.a $1$ $16.170$ \(\Q\) None \(0\) \(0\) \(0\) \(-2\) $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{4}-2q^{7}+3q^{11}+4q^{13}+4q^{16}+\cdots\)
2025.2.a.e 2025.a 1.a $1$ $16.170$ \(\Q\) None \(1\) \(0\) \(0\) \(3\) $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{4}+3q^{7}-3q^{8}+2q^{11}+\cdots\)
2025.2.a.f 2025.a 1.a $1$ $16.170$ \(\Q\) None \(2\) \(0\) \(0\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+2q^{4}-5q^{11}-4q^{13}-4q^{16}+\cdots\)
2025.2.a.g 2025.a 1.a $2$ $16.170$ \(\Q(\sqrt{3}) \) None \(-2\) \(0\) \(0\) \(6\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta )q^{2}+(2-2\beta )q^{4}+(3+\beta )q^{7}+\cdots\)
2025.2.a.h 2025.a 1.a $2$ $16.170$ \(\Q(\sqrt{13}) \) None \(-1\) \(0\) \(0\) \(-5\) $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta q^{2}+(1+\beta )q^{4}+(-2-\beta )q^{7}-3q^{8}+\cdots\)
2025.2.a.i 2025.a 1.a $2$ $16.170$ \(\Q(\sqrt{13}) \) None \(-1\) \(0\) \(0\) \(5\) $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta q^{2}+(1+\beta )q^{4}+(2+\beta )q^{7}-3q^{8}+\cdots\)
2025.2.a.j 2025.a 1.a $2$ $16.170$ \(\Q(\sqrt{3}) \) None \(0\) \(0\) \(0\) \(-4\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+q^{4}-2q^{7}-\beta q^{8}+2\beta q^{11}+\cdots\)
2025.2.a.k 2025.a 1.a $2$ $16.170$ \(\Q(\sqrt{13}) \) None \(1\) \(0\) \(0\) \(-5\) $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+(1+\beta )q^{4}+(-2-\beta )q^{7}+3q^{8}+\cdots\)
2025.2.a.l 2025.a 1.a $2$ $16.170$ \(\Q(\sqrt{13}) \) None \(1\) \(0\) \(0\) \(5\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+(1+\beta )q^{4}+(2+\beta )q^{7}+3q^{8}+\cdots\)
2025.2.a.m 2025.a 1.a $2$ $16.170$ \(\Q(\sqrt{3}) \) None \(2\) \(0\) \(0\) \(6\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{2}+(2+2\beta )q^{4}+(3-\beta )q^{7}+\cdots\)
2025.2.a.n 2025.a 1.a $3$ $16.170$ 3.3.564.1 None \(-1\) \(0\) \(0\) \(-5\) $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+(2+\beta _{2})q^{4}+(-2+\beta _{1})q^{7}+\cdots\)
2025.2.a.o 2025.a 1.a $3$ $16.170$ 3.3.564.1 None \(1\) \(0\) \(0\) \(-5\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(2+\beta _{2})q^{4}+(-2+\beta _{1})q^{7}+\cdots\)
2025.2.a.p 2025.a 1.a $4$ $16.170$ 4.4.11661.1 None \(-2\) \(0\) \(0\) \(-1\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{2}+(2-\beta _{1}+\beta _{2})q^{4}+\cdots\)
2025.2.a.q 2025.a 1.a $4$ $16.170$ 4.4.11661.1 None \(-2\) \(0\) \(0\) \(1\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{2}+(2-\beta _{1}+\beta _{2})q^{4}+\cdots\)
2025.2.a.r 2025.a 1.a $4$ $16.170$ \(\Q(\zeta_{24})^+\) None \(0\) \(0\) \(0\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+\beta _{2}q^{4}+(-\beta _{1}-2\beta _{3})q^{7}+\cdots\)
2025.2.a.s 2025.a 1.a $4$ $16.170$ \(\Q(\zeta_{24})^+\) None \(0\) \(0\) \(0\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{2}q^{2}+\beta _{3}q^{7}-2\beta _{2}q^{8}-\beta _{1}q^{11}+\cdots\)
2025.2.a.t 2025.a 1.a $4$ $16.170$ \(\Q(\zeta_{24})^+\) None \(0\) \(0\) \(0\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+\beta _{2}q^{4}+(\beta _{1}+2\beta _{3})q^{7}+\beta _{3}q^{8}+\cdots\)
2025.2.a.u 2025.a 1.a $4$ $16.170$ \(\Q(\sqrt{3}, \sqrt{7})\) None \(0\) \(0\) \(0\) \(-6\) $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{3}q^{2}-\beta _{2}q^{4}+(-1+\beta _{2})q^{7}+(\beta _{1}+\cdots)q^{8}+\cdots\)
2025.2.a.v 2025.a 1.a $4$ $16.170$ \(\Q(\sqrt{3}, \sqrt{7})\) None \(0\) \(0\) \(0\) \(6\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{3}q^{2}-\beta _{2}q^{4}+(1-\beta _{2})q^{7}+(\beta _{1}+\cdots)q^{8}+\cdots\)
2025.2.a.w 2025.a 1.a $4$ $16.170$ \(\Q(\sqrt{3}, \sqrt{11})\) None \(0\) \(0\) \(0\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(2+\beta _{2})q^{4}+(-\beta _{1}-\beta _{3})q^{7}+\cdots\)
2025.2.a.x 2025.a 1.a $4$ $16.170$ \(\Q(\sqrt{3}, \sqrt{11})\) None \(0\) \(0\) \(0\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(2+\beta _{2})q^{4}+(\beta _{1}+\beta _{3})q^{7}+\cdots\)
2025.2.a.y 2025.a 1.a $4$ $16.170$ 4.4.11661.1 None \(2\) \(0\) \(0\) \(-1\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}+(2-\beta _{1}+\beta _{2})q^{4}+(-1+\cdots)q^{7}+\cdots\)
2025.2.a.z 2025.a 1.a $4$ $16.170$ 4.4.11661.1 None \(2\) \(0\) \(0\) \(1\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}+(2-\beta _{1}+\beta _{2})q^{4}+(1+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(2025))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(2025)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(27))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(81))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(135))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(225))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(405))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(675))\)\(^{\oplus 2}\)