Properties

Label 2025.4.a
Level $2025$
Weight $4$
Character orbit 2025.a
Rep. character $\chi_{2025}(1,\cdot)$
Character field $\Q$
Dimension $222$
Newform subspaces $39$
Sturm bound $1080$
Trace bound $14$

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Defining parameters

Level: \( N \) \(=\) \( 2025 = 3^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2025.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 39 \)
Sturm bound: \(1080\)
Trace bound: \(14\)
Distinguishing \(T_p\): \(2\), \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(2025))\).

Total New Old
Modular forms 846 234 612
Cusp forms 774 222 552
Eisenstein series 72 12 60

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
\(+\)\(+\)\(+\)\(54\)
\(+\)\(-\)\(-\)\(56\)
\(-\)\(+\)\(-\)\(52\)
\(-\)\(-\)\(+\)\(60\)
Plus space\(+\)\(114\)
Minus space\(-\)\(108\)

Trace form

\( 222 q + 862 q^{4} - 14 q^{7} + O(q^{10}) \) \( 222 q + 862 q^{4} - 14 q^{7} + 40 q^{13} + 3434 q^{16} - 26 q^{19} - 42 q^{22} - 584 q^{28} - 78 q^{31} + 1018 q^{34} - 494 q^{37} + 4 q^{43} - 460 q^{46} + 9460 q^{49} - 80 q^{52} - 564 q^{58} + 1272 q^{61} + 11930 q^{64} + 1804 q^{67} + 676 q^{73} - 2650 q^{76} + 1462 q^{79} - 390 q^{82} - 1830 q^{88} - 1262 q^{91} - 188 q^{94} + 184 q^{97} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(2025))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 3 5
2025.4.a.a 2025.a 1.a $1$ $119.479$ \(\Q\) None 405.4.a.a \(-5\) \(0\) \(0\) \(-9\) $+$ $+$ $\mathrm{SU}(2)$ \(q-5q^{2}+17q^{4}-9q^{7}-45q^{8}+8q^{11}+\cdots\)
2025.4.a.b 2025.a 1.a $1$ $119.479$ \(\Q\) None 2025.4.a.b \(-1\) \(0\) \(0\) \(-15\) $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-7q^{4}-15q^{7}+15q^{8}+40q^{11}+\cdots\)
2025.4.a.c 2025.a 1.a $1$ $119.479$ \(\Q\) None 2025.4.a.b \(-1\) \(0\) \(0\) \(15\) $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-7q^{4}+15q^{7}+15q^{8}-40q^{11}+\cdots\)
2025.4.a.d 2025.a 1.a $1$ $119.479$ \(\Q\) None 2025.4.a.b \(1\) \(0\) \(0\) \(-15\) $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-7q^{4}-15q^{7}-15q^{8}-40q^{11}+\cdots\)
2025.4.a.e 2025.a 1.a $1$ $119.479$ \(\Q\) None 2025.4.a.b \(1\) \(0\) \(0\) \(15\) $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-7q^{4}+15q^{7}-15q^{8}+40q^{11}+\cdots\)
2025.4.a.f 2025.a 1.a $1$ $119.479$ \(\Q\) None 405.4.a.a \(5\) \(0\) \(0\) \(-9\) $+$ $+$ $\mathrm{SU}(2)$ \(q+5q^{2}+17q^{4}-9q^{7}+45q^{8}-8q^{11}+\cdots\)
2025.4.a.g 2025.a 1.a $2$ $119.479$ \(\Q(\sqrt{33}) \) None 9.4.c.a \(-3\) \(0\) \(0\) \(-7\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(-1-\beta )q^{2}+(1+3\beta )q^{4}+(-2-3\beta )q^{7}+\cdots\)
2025.4.a.h 2025.a 1.a $2$ $119.479$ \(\Q(\sqrt{57}) \) None 81.4.a.b \(-3\) \(0\) \(0\) \(-10\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(-1-\beta )q^{2}+(7+3\beta )q^{4}+(-8+6\beta )q^{7}+\cdots\)
2025.4.a.i 2025.a 1.a $2$ $119.479$ \(\Q(\sqrt{3}) \) None 405.4.a.c \(-2\) \(0\) \(0\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta )q^{2}+(-4-2\beta )q^{4}+4\beta q^{7}+\cdots\)
2025.4.a.j 2025.a 1.a $2$ $119.479$ \(\Q(\sqrt{33}) \) None 45.4.e.a \(-1\) \(0\) \(0\) \(-9\) $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta q^{2}+\beta q^{4}+(-5+\beta )q^{7}+(-8+\cdots)q^{8}+\cdots\)
2025.4.a.k 2025.a 1.a $2$ $119.479$ \(\Q(\sqrt{3}) \) None 81.4.a.c \(0\) \(0\) \(0\) \(44\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{2}-5q^{4}+22q^{7}-13\beta q^{8}-34\beta q^{11}+\cdots\)
2025.4.a.l 2025.a 1.a $2$ $119.479$ \(\Q(\sqrt{33}) \) None 45.4.e.a \(1\) \(0\) \(0\) \(-9\) $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+\beta q^{4}+(-5+\beta )q^{7}+(8-7\beta )q^{8}+\cdots\)
2025.4.a.m 2025.a 1.a $2$ $119.479$ \(\Q(\sqrt{3}) \) None 405.4.a.c \(2\) \(0\) \(0\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{2}+(-4+2\beta )q^{4}-4\beta q^{7}+\cdots\)
2025.4.a.n 2025.a 1.a $2$ $119.479$ \(\Q(\sqrt{33}) \) None 9.4.c.a \(3\) \(0\) \(0\) \(-7\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{2}+(1+3\beta )q^{4}+(-2-3\beta )q^{7}+\cdots\)
2025.4.a.o 2025.a 1.a $2$ $119.479$ \(\Q(\sqrt{57}) \) None 81.4.a.b \(3\) \(0\) \(0\) \(-10\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{2}+(7+3\beta )q^{4}+(-8+6\beta )q^{7}+\cdots\)
2025.4.a.p 2025.a 1.a $3$ $119.479$ 3.3.7032.1 None 405.4.a.g \(-1\) \(0\) \(0\) \(25\) $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+(2-\beta _{1}+\beta _{2})q^{4}+(10-5\beta _{1}+\cdots)q^{7}+\cdots\)
2025.4.a.q 2025.a 1.a $3$ $119.479$ 3.3.2292.1 None 45.4.e.b \(-1\) \(0\) \(0\) \(43\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{2}q^{2}+(3-\beta _{1}-\beta _{2})q^{4}+(14-2\beta _{1}+\cdots)q^{7}+\cdots\)
2025.4.a.r 2025.a 1.a $3$ $119.479$ 3.3.7032.1 None 405.4.a.g \(1\) \(0\) \(0\) \(25\) $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(2-\beta _{1}+\beta _{2})q^{4}+(10-5\beta _{1}+\cdots)q^{7}+\cdots\)
2025.4.a.s 2025.a 1.a $3$ $119.479$ 3.3.2292.1 None 45.4.e.b \(1\) \(0\) \(0\) \(43\) $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{2}q^{2}+(3-\beta _{1}-\beta _{2})q^{4}+(14-2\beta _{1}+\cdots)q^{7}+\cdots\)
2025.4.a.t 2025.a 1.a $4$ $119.479$ \(\Q(\sqrt{3}, \sqrt{17})\) None 405.4.b.a \(0\) \(0\) \(0\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{3}q^{2}+9q^{4}+4\beta _{2}q^{7}-\beta _{3}q^{8}+\cdots\)
2025.4.a.u 2025.a 1.a $5$ $119.479$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None 2025.4.a.u \(0\) \(0\) \(0\) \(-49\) $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+(7+\beta _{1}+\beta _{2})q^{4}+(-10+\cdots)q^{7}+\cdots\)
2025.4.a.v 2025.a 1.a $5$ $119.479$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None 2025.4.a.u \(0\) \(0\) \(0\) \(-49\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(7+\beta _{1}+\beta _{2})q^{4}+(-10+\cdots)q^{7}+\cdots\)
2025.4.a.w 2025.a 1.a $5$ $119.479$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None 2025.4.a.u \(0\) \(0\) \(0\) \(49\) $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+(7+\beta _{1}+\beta _{2})q^{4}+(10+\beta _{1}+\cdots)q^{7}+\cdots\)
2025.4.a.x 2025.a 1.a $5$ $119.479$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None 2025.4.a.u \(0\) \(0\) \(0\) \(49\) $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(7+\beta _{1}+\beta _{2})q^{4}+(10+\beta _{1}+\cdots)q^{7}+\cdots\)
2025.4.a.y 2025.a 1.a $6$ $119.479$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None 405.4.a.k \(-4\) \(0\) \(0\) \(-40\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{2}+(6-\beta _{1}+\beta _{3})q^{4}+\cdots\)
2025.4.a.z 2025.a 1.a $6$ $119.479$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None 405.4.a.k \(4\) \(0\) \(0\) \(-40\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}+(6-\beta _{1}+\beta _{3})q^{4}+(-7+\cdots)q^{7}+\cdots\)
2025.4.a.ba 2025.a 1.a $7$ $119.479$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None 45.4.e.c \(-2\) \(0\) \(0\) \(-22\) $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+(5+\beta _{2})q^{4}+(-3+\beta _{5})q^{7}+\cdots\)
2025.4.a.bb 2025.a 1.a $7$ $119.479$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None 45.4.e.c \(2\) \(0\) \(0\) \(-22\) $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(5+\beta _{2})q^{4}+(-3+\beta _{5})q^{7}+\cdots\)
2025.4.a.bc 2025.a 1.a $8$ $119.479$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None 405.4.b.b \(0\) \(0\) \(0\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(3+\beta _{2})q^{4}-\beta _{6}q^{7}+(2\beta _{1}+\cdots)q^{8}+\cdots\)
2025.4.a.bd 2025.a 1.a $8$ $119.479$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None 405.4.b.b \(0\) \(0\) \(0\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(3+\beta _{2})q^{4}+\beta _{6}q^{7}+(2\beta _{1}+\cdots)q^{8}+\cdots\)
2025.4.a.be 2025.a 1.a $12$ $119.479$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None 225.4.e.e \(-4\) \(0\) \(0\) \(-6\) $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+(4+\beta _{2})q^{4}+(-1+\beta _{6})q^{7}+\cdots\)
2025.4.a.bf 2025.a 1.a $12$ $119.479$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None 225.4.e.e \(-4\) \(0\) \(0\) \(6\) $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+(4+\beta _{2})q^{4}+(1-\beta _{6})q^{7}+\cdots\)
2025.4.a.bg 2025.a 1.a $12$ $119.479$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None 2025.4.a.bg \(0\) \(0\) \(0\) \(-44\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(3+\beta _{2})q^{4}+(-4+\beta _{7})q^{7}+\cdots\)
2025.4.a.bh 2025.a 1.a $12$ $119.479$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None 2025.4.a.bg \(0\) \(0\) \(0\) \(44\) $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(3+\beta _{2})q^{4}+(4-\beta _{7})q^{7}+\cdots\)
2025.4.a.bi 2025.a 1.a $12$ $119.479$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None 225.4.e.e \(4\) \(0\) \(0\) \(-6\) $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(4+\beta _{2})q^{4}+(-1+\beta _{6})q^{7}+\cdots\)
2025.4.a.bj 2025.a 1.a $12$ $119.479$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None 225.4.e.e \(4\) \(0\) \(0\) \(6\) $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(4+\beta _{2})q^{4}+(1-\beta _{6})q^{7}+\cdots\)
2025.4.a.bk 2025.a 1.a $16$ $119.479$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None 45.4.j.a \(0\) \(0\) \(0\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(3+\beta _{2})q^{4}+(-\beta _{1}+\beta _{11}+\cdots)q^{7}+\cdots\)
2025.4.a.bl 2025.a 1.a $16$ $119.479$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None 45.4.j.a \(0\) \(0\) \(0\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(3+\beta _{2})q^{4}+(\beta _{1}-\beta _{11})q^{7}+\cdots\)
2025.4.a.bm 2025.a 1.a $16$ $119.479$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None 405.4.b.d \(0\) \(0\) \(0\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{8}q^{2}+(4+\beta _{1})q^{4}-\beta _{4}q^{7}+(7\beta _{8}+\cdots)q^{8}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(2025))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(2025)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 10}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 9}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(25))\)\(^{\oplus 5}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(27))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(81))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(135))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(225))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(405))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(675))\)\(^{\oplus 2}\)