Properties

Label 4761.2.a
Level $4761$
Weight $2$
Character orbit 4761.a
Rep. character $\chi_{4761}(1,\cdot)$
Character field $\Q$
Dimension $200$
Newform subspaces $50$
Sturm bound $1104$
Trace bound $13$

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Defining parameters

Level: \( N \) \(=\) \( 4761 = 3^{2} \cdot 23^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4761.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 50 \)
Sturm bound: \(1104\)
Trace bound: \(13\)
Distinguishing \(T_p\): \(2\), \(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(4761))\).

Total New Old
Modular forms 600 221 379
Cusp forms 505 200 305
Eisenstein series 95 21 74

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

\(3\)\(23\)FrickeTotalCuspEisenstein
AllNewOldAllNewOldAllNewOld
\(+\)\(+\)\(+\)\(144\)\(36\)\(108\)\(121\)\(36\)\(85\)\(23\)\(0\)\(23\)
\(+\)\(-\)\(-\)\(156\)\(48\)\(108\)\(132\)\(48\)\(84\)\(24\)\(0\)\(24\)
\(-\)\(+\)\(-\)\(156\)\(71\)\(85\)\(132\)\(61\)\(71\)\(24\)\(10\)\(14\)
\(-\)\(-\)\(+\)\(144\)\(66\)\(78\)\(120\)\(55\)\(65\)\(24\)\(11\)\(13\)
Plus space\(+\)\(288\)\(102\)\(186\)\(241\)\(91\)\(150\)\(47\)\(11\)\(36\)
Minus space\(-\)\(312\)\(119\)\(193\)\(264\)\(109\)\(155\)\(48\)\(10\)\(38\)

Trace form

\( 200 q + 190 q^{4} - 4 q^{5} + 6 q^{7} - 6 q^{8} + 4 q^{10} + 6 q^{11} - 8 q^{14} + 170 q^{16} - 10 q^{20} + 14 q^{22} + 162 q^{25} + 8 q^{26} + 22 q^{28} - 2 q^{29} + 2 q^{31} + 8 q^{32} - 18 q^{34} + 18 q^{35}+ \cdots + 2 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(4761))\) 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 23
4761.2.a.a 4761.a 1.a $1$ $38.017$ \(\Q\) None 1587.2.a.c \(-1\) \(0\) \(-4\) \(4\) $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{4}-4q^{5}+4q^{7}+3q^{8}+4q^{10}+\cdots\)
4761.2.a.b 4761.a 1.a $1$ $38.017$ \(\Q\) None 69.2.a.a \(-1\) \(0\) \(0\) \(2\) $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{4}+2q^{7}+3q^{8}+4q^{11}+\cdots\)
4761.2.a.c 4761.a 1.a $1$ $38.017$ \(\Q\) None 1587.2.a.c \(-1\) \(0\) \(4\) \(-4\) $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{4}+4q^{5}-4q^{7}+3q^{8}-4q^{10}+\cdots\)
4761.2.a.d 4761.a 1.a $1$ $38.017$ \(\Q\) \(\Q(\sqrt{-3}) \) 4761.2.a.d \(0\) \(0\) \(0\) \(-5\) $+$ $-$ $N(\mathrm{U}(1))$ \(q-2q^{4}-5q^{7}-7q^{13}+4q^{16}-8q^{19}+\cdots\)
4761.2.a.e 4761.a 1.a $1$ $38.017$ \(\Q\) \(\Q(\sqrt{-3}) \) 4761.2.a.d \(0\) \(0\) \(0\) \(5\) $+$ $-$ $N(\mathrm{U}(1))$ \(q-2q^{4}+5q^{7}-7q^{13}+4q^{16}+8q^{19}+\cdots\)
4761.2.a.f 4761.a 1.a $1$ $38.017$ \(\Q\) None 1587.2.a.a \(2\) \(0\) \(0\) \(-1\) $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+2q^{4}-q^{7}+4q^{11}-3q^{13}+\cdots\)
4761.2.a.g 4761.a 1.a $1$ $38.017$ \(\Q\) None 1587.2.a.a \(2\) \(0\) \(0\) \(1\) $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+2q^{4}+q^{7}-4q^{11}-3q^{13}+\cdots\)
4761.2.a.h 4761.a 1.a $2$ $38.017$ \(\Q(\sqrt{2}) \) None 4761.2.a.h \(-2\) \(0\) \(0\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{4}-\beta q^{5}+3q^{8}+\beta q^{10}+\cdots\)
4761.2.a.i 4761.a 1.a $2$ $38.017$ \(\Q(\sqrt{2}) \) None 529.2.a.d \(-2\) \(0\) \(-2\) \(4\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta )q^{2}+(1-2\beta )q^{4}+(-1-2\beta )q^{5}+\cdots\)
4761.2.a.j 4761.a 1.a $2$ $38.017$ \(\Q(\sqrt{2}) \) None 529.2.a.d \(-2\) \(0\) \(2\) \(-4\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta )q^{2}+(1-2\beta )q^{4}+(1+2\beta )q^{5}+\cdots\)
4761.2.a.k 4761.a 1.a $2$ $38.017$ \(\Q(\sqrt{2}) \) None 207.2.a.b \(-2\) \(0\) \(4\) \(4\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta )q^{2}+(1-2\beta )q^{4}+(2+\beta )q^{5}+\cdots\)
4761.2.a.l 4761.a 1.a $2$ $38.017$ \(\Q(\sqrt{3}) \) None 1587.2.a.k \(-2\) \(0\) \(-2\) \(-4\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta )q^{2}+(2-2\beta )q^{4}+(-1+\beta )q^{5}+\cdots\)
4761.2.a.m 4761.a 1.a $2$ $38.017$ \(\Q(\sqrt{3}) \) None 1587.2.a.k \(-2\) \(0\) \(2\) \(4\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta )q^{2}+(2-2\beta )q^{4}+(1-\beta )q^{5}+\cdots\)
4761.2.a.n 4761.a 1.a $2$ $38.017$ \(\Q(\sqrt{3}) \) None 1587.2.a.j \(0\) \(0\) \(0\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{4}-2\beta q^{5}-\beta q^{7}+2\beta q^{11}-5q^{13}+\cdots\)
4761.2.a.o 4761.a 1.a $2$ $38.017$ \(\Q(\sqrt{3}) \) \(\Q(\sqrt{-3}) \) 4761.2.a.o \(0\) \(0\) \(0\) \(0\) $+$ $-$ $N(\mathrm{U}(1))$ \(q-2q^{4}+\beta q^{7}+7q^{13}+4q^{16}+2\beta q^{19}+\cdots\)
4761.2.a.p 4761.a 1.a $2$ $38.017$ \(\Q(\sqrt{2}) \) None 1587.2.a.g \(0\) \(0\) \(-4\) \(2\) $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+(-2-\beta )q^{5}+(1-2\beta )q^{7}+\cdots\)
4761.2.a.q 4761.a 1.a $2$ $38.017$ \(\Q(\sqrt{2}) \) None 1587.2.a.g \(0\) \(0\) \(4\) \(-2\) $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+(2+\beta )q^{5}+(-1+2\beta )q^{7}+\cdots\)
4761.2.a.r 4761.a 1.a $2$ $38.017$ \(\Q(\sqrt{3}) \) None 4761.2.a.r \(0\) \(0\) \(-6\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+q^{4}-3q^{5}-2\beta q^{7}-\beta q^{8}+\cdots\)
4761.2.a.s 4761.a 1.a $2$ $38.017$ \(\Q(\sqrt{3}) \) None 529.2.a.b \(0\) \(0\) \(0\) \(-6\) $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+q^{4}+\beta q^{5}+(-3-\beta )q^{7}+\cdots\)
4761.2.a.t 4761.a 1.a $2$ $38.017$ \(\Q(\sqrt{3}) \) None 529.2.a.b \(0\) \(0\) \(0\) \(6\) $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+q^{4}-\beta q^{5}+(3+\beta )q^{7}-\beta q^{8}+\cdots\)
4761.2.a.u 4761.a 1.a $2$ $38.017$ \(\Q(\sqrt{3}) \) None 4761.2.a.r \(0\) \(0\) \(6\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+q^{4}+3q^{5}+2\beta q^{7}-\beta q^{8}+\cdots\)
4761.2.a.v 4761.a 1.a $2$ $38.017$ \(\Q(\sqrt{5}) \) None 69.2.a.b \(0\) \(0\) \(-2\) \(-2\) $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta q^{2}+3q^{4}+(-1-\beta )q^{5}+(-1+\cdots)q^{7}+\cdots\)
4761.2.a.w 4761.a 1.a $2$ $38.017$ \(\Q(\sqrt{5}) \) None 23.2.a.a \(1\) \(0\) \(-2\) \(-2\) $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+(-1+\beta )q^{4}-2\beta q^{5}+(-2+\cdots)q^{7}+\cdots\)
4761.2.a.x 4761.a 1.a $2$ $38.017$ \(\Q(\sqrt{2}) \) None 1587.2.a.f \(2\) \(0\) \(0\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{4}+2\beta q^{5}-3\beta q^{7}-3q^{8}+\cdots\)
4761.2.a.y 4761.a 1.a $2$ $38.017$ \(\Q(\sqrt{2}) \) None 4761.2.a.h \(2\) \(0\) \(0\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{4}+\beta q^{5}-3q^{8}+\beta q^{10}+\cdots\)
4761.2.a.z 4761.a 1.a $2$ $38.017$ \(\Q(\sqrt{2}) \) None 207.2.a.b \(2\) \(0\) \(-4\) \(4\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{2}+(1+2\beta )q^{4}+(-2+\beta )q^{5}+\cdots\)
4761.2.a.ba 4761.a 1.a $3$ $38.017$ 3.3.621.1 \(\Q(\sqrt{-23}) \) 529.2.a.f \(0\) \(0\) \(0\) \(0\) $-$ $-$ $N(\mathrm{U}(1))$ \(q-\beta _{1}q^{2}+(2+\beta _{1}+\beta _{2})q^{4}+(-3-2\beta _{1}+\cdots)q^{8}+\cdots\)
4761.2.a.bb 4761.a 1.a $4$ $38.017$ \(\Q(\zeta_{24})^+\) None 4761.2.a.bb \(-8\) \(0\) \(0\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+2q^{4}-2\beta _{1}q^{5}+(-2\beta _{1}-\beta _{3})q^{7}+\cdots\)
4761.2.a.bc 4761.a 1.a $4$ $38.017$ \(\Q(\zeta_{24})^+\) \(\Q(\sqrt{-3}) \) 4761.2.a.bc \(0\) \(0\) \(0\) \(0\) $+$ $+$ $N(\mathrm{U}(1))$ \(q-2q^{4}+(3\beta _{1}+2\beta _{3})q^{7}+\beta _{2}q^{13}+\cdots\)
4761.2.a.bd 4761.a 1.a $4$ $38.017$ \(\Q(\zeta_{24})^+\) None 1587.2.a.n \(0\) \(0\) \(0\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{2}q^{2}+q^{4}+(\beta _{1}+2\beta _{3})q^{5}+2\beta _{3}q^{7}+\cdots\)
4761.2.a.be 4761.a 1.a $4$ $38.017$ \(\Q(\zeta_{24})^+\) None 4761.2.a.be \(0\) \(0\) \(0\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{2}q^{2}+q^{4}+\beta _{1}q^{5}-\beta _{3}q^{7}-\beta _{2}q^{8}+\cdots\)
4761.2.a.bf 4761.a 1.a $4$ $38.017$ 4.4.268960.2 None 4761.2.a.bf \(0\) \(0\) \(0\) \(-2\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(3+\beta _{2})q^{4}-\beta _{1}q^{5}+\beta _{2}q^{7}+\cdots\)
4761.2.a.bg 4761.a 1.a $4$ $38.017$ 4.4.268960.2 None 4761.2.a.bf \(0\) \(0\) \(0\) \(2\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(3+\beta _{2})q^{4}+\beta _{1}q^{5}-\beta _{2}q^{7}+\cdots\)
4761.2.a.bh 4761.a 1.a $4$ $38.017$ \(\Q(\zeta_{24})^+\) None 4761.2.a.bh \(0\) \(0\) \(0\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+4q^{4}-\beta _{3}q^{5}-\beta _{2}q^{7}+2\beta _{1}q^{8}+\cdots\)
4761.2.a.bi 4761.a 1.a $4$ $38.017$ \(\Q(\sqrt{2}, \sqrt{13})\) None 529.2.a.h \(2\) \(0\) \(0\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{3}q^{2}+(1+\beta _{3})q^{4}-\beta _{1}q^{5}+(-\beta _{1}+\cdots)q^{7}+\cdots\)
4761.2.a.bj 4761.a 1.a $4$ $38.017$ \(\Q(\zeta_{24})^+\) None 529.2.a.g \(4\) \(0\) \(0\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{4}+\beta _{1}q^{5}+(\beta _{1}-\beta _{3})q^{7}+\cdots\)
4761.2.a.bk 4761.a 1.a $4$ $38.017$ \(\Q(\zeta_{24})^+\) None 1587.2.a.m \(4\) \(0\) \(0\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(1+\beta _{2})q^{2}+(2+2\beta _{2})q^{4}+(\beta _{1}+\beta _{3})q^{5}+\cdots\)
4761.2.a.bl 4761.a 1.a $4$ $38.017$ \(\Q(\zeta_{24})^+\) None 4761.2.a.bb \(8\) \(0\) \(0\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+2q^{4}-2\beta _{1}q^{5}+(2\beta _{1}+\beta _{3})q^{7}+\cdots\)
4761.2.a.bm 4761.a 1.a $5$ $38.017$ \(\Q(\zeta_{22})^+\) None 69.2.e.b \(-2\) \(0\) \(-7\) \(-3\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(\beta _{2}+\beta _{4})q^{2}+(2\beta _{1}+\beta _{3}-\beta _{4})q^{4}+\cdots\)
4761.2.a.bn 4761.a 1.a $5$ $38.017$ \(\Q(\zeta_{22})^+\) None 23.2.c.a \(-2\) \(0\) \(-7\) \(8\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(\beta _{2}+\beta _{4})q^{2}+(2\beta _{1}+\beta _{3}-\beta _{4})q^{4}+\cdots\)
4761.2.a.bo 4761.a 1.a $5$ $38.017$ \(\Q(\zeta_{22})^+\) None 23.2.c.a \(-2\) \(0\) \(7\) \(-8\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(\beta _{2}+\beta _{4})q^{2}+(2\beta _{1}+\beta _{3}-\beta _{4})q^{4}+\cdots\)
4761.2.a.bp 4761.a 1.a $5$ $38.017$ \(\Q(\zeta_{22})^+\) None 69.2.e.b \(-2\) \(0\) \(7\) \(3\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(\beta _{2}+\beta _{4})q^{2}+(2\beta _{1}+\beta _{3}-\beta _{4})q^{4}+\cdots\)
4761.2.a.bq 4761.a 1.a $5$ $38.017$ \(\Q(\zeta_{22})^+\) None 69.2.e.a \(2\) \(0\) \(-3\) \(4\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(\beta _{3}-\beta _{4})q^{2}+(1-\beta _{1}-\beta _{2}+\beta _{4})q^{4}+\cdots\)
4761.2.a.br 4761.a 1.a $5$ $38.017$ \(\Q(\zeta_{22})^+\) None 69.2.e.a \(2\) \(0\) \(3\) \(-4\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(\beta _{3}-\beta _{4})q^{2}+(1-\beta _{1}-\beta _{2}+\beta _{4})q^{4}+\cdots\)
4761.2.a.bs 4761.a 1.a $6$ $38.017$ 6.6.2803712.1 None 1587.2.a.s \(-2\) \(0\) \(0\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{2}q^{2}+(1-\beta _{2}+\beta _{4})q^{4}-\beta _{5}q^{5}+\cdots\)
4761.2.a.bt 4761.a 1.a $10$ $38.017$ 10.10.\(\cdots\).1 None 69.2.e.c \(2\) \(0\) \(-8\) \(19\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(1+\beta _{4}-\beta _{6}-\beta _{7})q^{4}+(-1+\cdots)q^{5}+\cdots\)
4761.2.a.bu 4761.a 1.a $10$ $38.017$ 10.10.\(\cdots\).1 None 69.2.e.c \(2\) \(0\) \(8\) \(-19\) $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(1+\beta _{4}-\beta _{6}-\beta _{7})q^{4}+(1+\cdots)q^{5}+\cdots\)
4761.2.a.bv 4761.a 1.a $12$ $38.017$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None 1587.2.a.v \(-4\) \(0\) \(0\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{2}q^{2}+(2-\beta _{7}+\beta _{9})q^{4}-\beta _{6}q^{5}+\cdots\)
4761.2.a.bw 4761.a 1.a $20$ $38.017$ \(\mathbb{Q}[x]/(x^{20} - \cdots)\) None 207.2.i.e \(0\) \(0\) \(0\) \(-18\) $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(1+\beta _{2}-\beta _{3})q^{4}+\beta _{19}q^{5}+\cdots\)
4761.2.a.bx 4761.a 1.a $20$ $38.017$ \(\mathbb{Q}[x]/(x^{20} - \cdots)\) None 207.2.i.e \(0\) \(0\) \(0\) \(18\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(1+\beta _{2}-\beta _{3})q^{4}-\beta _{19}q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(4761))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(4761)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(23))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(69))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(207))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(529))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1587))\)\(^{\oplus 2}\)