Properties

Label 675.4.a
Level $675$
Weight $4$
Character orbit 675.a
Rep. character $\chi_{675}(1,\cdot)$
Character field $\Q$
Dimension $76$
Newform subspaces $29$
Sturm bound $360$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 675 = 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 675.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 29 \)
Sturm bound: \(360\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(2\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(675))\).

Total New Old
Modular forms 288 76 212
Cusp forms 252 76 176
Eisenstein series 36 0 36

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.
\(+\)\(+\)\(+\)\(20\)
\(+\)\(-\)\(-\)\(19\)
\(-\)\(+\)\(-\)\(16\)
\(-\)\(-\)\(+\)\(21\)
Plus space\(+\)\(41\)
Minus space\(-\)\(35\)

Trace form

\( 76 q + 298 q^{4} - 40 q^{7} + O(q^{10}) \) \( 76 q + 298 q^{4} - 40 q^{7} + 2 q^{13} + 1030 q^{16} + 98 q^{19} + 294 q^{22} - 406 q^{28} - 46 q^{31} + 60 q^{34} + 1106 q^{37} + 980 q^{43} + 1956 q^{46} + 4500 q^{49} + 152 q^{52} + 1260 q^{58} - 2602 q^{61} + 610 q^{64} - 2722 q^{67} - 2632 q^{73} + 3476 q^{76} + 2114 q^{79} + 5040 q^{82} + 9294 q^{88} + 5002 q^{91} + 8712 q^{94} - 2644 q^{97} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(675))\) 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
675.4.a.a 675.a 1.a $1$ $39.826$ \(\Q\) None \(-3\) \(0\) \(0\) \(25\) $-$ $+$ $\mathrm{SU}(2)$ \(q-3q^{2}+q^{4}+5^{2}q^{7}+21q^{8}-15q^{11}+\cdots\)
675.4.a.b 675.a 1.a $1$ $39.826$ \(\Q\) None \(-2\) \(0\) \(0\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}-4q^{4}+24q^{8}-10q^{11}+80q^{13}+\cdots\)
675.4.a.c 675.a 1.a $1$ $39.826$ \(\Q\) None \(-1\) \(0\) \(0\) \(6\) $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-7q^{4}+6q^{7}+15q^{8}-47q^{11}+\cdots\)
675.4.a.d 675.a 1.a $1$ $39.826$ \(\Q\) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(-37\) $-$ $-$ $N(\mathrm{U}(1))$ \(q-8q^{4}-37q^{7}-70q^{13}+2^{6}q^{16}+\cdots\)
675.4.a.e 675.a 1.a $1$ $39.826$ \(\Q\) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(-17\) $-$ $+$ $N(\mathrm{U}(1))$ \(q-8q^{4}-17q^{7}+70q^{13}+2^{6}q^{16}+\cdots\)
675.4.a.f 675.a 1.a $1$ $39.826$ \(\Q\) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(17\) $+$ $-$ $N(\mathrm{U}(1))$ \(q-8q^{4}+17q^{7}-70q^{13}+2^{6}q^{16}+\cdots\)
675.4.a.g 675.a 1.a $1$ $39.826$ \(\Q\) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(37\) $+$ $+$ $N(\mathrm{U}(1))$ \(q-8q^{4}+37q^{7}+70q^{13}+2^{6}q^{16}+\cdots\)
675.4.a.h 675.a 1.a $1$ $39.826$ \(\Q\) None \(1\) \(0\) \(0\) \(6\) $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-7q^{4}+6q^{7}-15q^{8}+47q^{11}+\cdots\)
675.4.a.i 675.a 1.a $1$ $39.826$ \(\Q\) None \(2\) \(0\) \(0\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}-4q^{4}-24q^{8}+10q^{11}+80q^{13}+\cdots\)
675.4.a.j 675.a 1.a $1$ $39.826$ \(\Q\) None \(3\) \(0\) \(0\) \(25\) $+$ $+$ $\mathrm{SU}(2)$ \(q+3q^{2}+q^{4}+5^{2}q^{7}-21q^{8}+15q^{11}+\cdots\)
675.4.a.k 675.a 1.a $2$ $39.826$ \(\Q(\sqrt{5}) \) \(\Q(\sqrt{-15}) \) \(-9\) \(0\) \(0\) \(0\) $+$ $-$ $N(\mathrm{U}(1))$ \(q+(-4-\beta )q^{2}+(9+9\beta )q^{4}+(-13+\cdots)q^{8}+\cdots\)
675.4.a.l 675.a 1.a $2$ $39.826$ \(\Q(\sqrt{13}) \) None \(0\) \(0\) \(0\) \(-18\) $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta q^{2}+5q^{4}-9q^{7}+3\beta q^{8}-4\beta q^{11}+\cdots\)
675.4.a.m 675.a 1.a $2$ $39.826$ \(\Q(\sqrt{13}) \) None \(0\) \(0\) \(0\) \(18\) $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta q^{2}+5q^{4}+9q^{7}+3\beta q^{8}+4\beta q^{11}+\cdots\)
675.4.a.n 675.a 1.a $2$ $39.826$ \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(0\) \(-22\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+10q^{4}-11q^{7}+2\beta q^{8}-4\beta q^{11}+\cdots\)
675.4.a.o 675.a 1.a $2$ $39.826$ \(\Q(\sqrt{5}) \) \(\Q(\sqrt{-15}) \) \(9\) \(0\) \(0\) \(0\) $-$ $-$ $N(\mathrm{U}(1))$ \(q+(5-\beta )q^{2}+(18-9\beta )q^{4}+(59-46\beta )q^{8}+\cdots\)
675.4.a.p 675.a 1.a $3$ $39.826$ 3.3.1772.1 None \(-5\) \(0\) \(0\) \(4\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(-2+\beta _{1})q^{2}+(7-3\beta _{1}+\beta _{2})q^{4}+\cdots\)
675.4.a.q 675.a 1.a $3$ $39.826$ 3.3.5637.1 None \(-1\) \(0\) \(0\) \(-44\) $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+(7+\beta _{1}+\beta _{2})q^{4}+(-14+\cdots)q^{7}+\cdots\)
675.4.a.r 675.a 1.a $3$ $39.826$ 3.3.5637.1 None \(1\) \(0\) \(0\) \(-44\) $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(7+\beta _{1}+\beta _{2})q^{4}+(-14+\cdots)q^{7}+\cdots\)
675.4.a.s 675.a 1.a $3$ $39.826$ 3.3.1772.1 None \(5\) \(0\) \(0\) \(4\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(2-\beta _{1})q^{2}+(7-3\beta _{1}+\beta _{2})q^{4}+(2+\cdots)q^{7}+\cdots\)
675.4.a.t 675.a 1.a $4$ $39.826$ 4.4.467024.1 None \(-6\) \(0\) \(0\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-2-\beta _{3})q^{2}+3\beta _{3}q^{4}-\beta _{2}q^{7}+\cdots\)
675.4.a.u 675.a 1.a $4$ $39.826$ 4.4.183945.1 None \(-1\) \(0\) \(0\) \(-4\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(5+\beta _{2})q^{4}+(-1-3\beta _{1}+\cdots)q^{7}+\cdots\)
675.4.a.v 675.a 1.a $4$ $39.826$ 4.4.183945.1 None \(-1\) \(0\) \(0\) \(4\) $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(5+\beta _{2})q^{4}+(1+3\beta _{1}-\beta _{2}+\cdots)q^{7}+\cdots\)
675.4.a.w 675.a 1.a $4$ $39.826$ 4.4.3173728.1 None \(0\) \(0\) \(0\) \(-30\) $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(7+\beta _{2})q^{4}+(-7+\beta _{2})q^{7}+\cdots\)
675.4.a.x 675.a 1.a $4$ $39.826$ 4.4.3173728.1 None \(0\) \(0\) \(0\) \(30\) $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(7+\beta _{2})q^{4}+(7-\beta _{2})q^{7}+\cdots\)
675.4.a.y 675.a 1.a $4$ $39.826$ 4.4.183945.1 None \(1\) \(0\) \(0\) \(-4\) $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+(5+\beta _{2})q^{4}+(-1-3\beta _{1}+\cdots)q^{7}+\cdots\)
675.4.a.z 675.a 1.a $4$ $39.826$ 4.4.183945.1 None \(1\) \(0\) \(0\) \(4\) $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+(5+\beta _{2})q^{4}+(1+3\beta _{1}-\beta _{2}+\cdots)q^{7}+\cdots\)
675.4.a.ba 675.a 1.a $4$ $39.826$ 4.4.467024.1 None \(6\) \(0\) \(0\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(2+\beta _{3})q^{2}+3\beta _{3}q^{4}+\beta _{2}q^{7}+(-4+\cdots)q^{8}+\cdots\)
675.4.a.bb 675.a 1.a $6$ $39.826$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(0\) \(0\) \(0\) \(-36\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(5-\beta _{4})q^{4}+(-6+\beta _{2}+\beta _{4}+\cdots)q^{7}+\cdots\)
675.4.a.bc 675.a 1.a $6$ $39.826$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(0\) \(0\) \(0\) \(36\) $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(5-\beta _{4})q^{4}+(6-\beta _{2}-\beta _{4}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(675))\) into lower level spaces

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