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Label Dim. \(A\) Field CM RM Traces Fricke sign $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
539.1.c.a \(1\) \(0.269\) \(\Q\) \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{-11}) \) \(\Q(\sqrt{77}) \) \(0\) \(0\) \(0\) \(0\) \(q+q^{4}-q^{9}+q^{11}+q^{16}-2q^{23}+\cdots\)
539.1.c.b \(2\) \(0.269\) \(\Q(\sqrt{2}) \) \(\Q(\sqrt{-11}) \) None \(0\) \(0\) \(0\) \(0\) \(q-\beta q^{3}+q^{4}+\beta q^{5}+q^{9}-q^{11}-\beta q^{12}+\cdots\)
539.1.h.a \(2\) \(0.269\) \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{-11}) \) \(\Q(\sqrt{77}) \) \(0\) \(0\) \(0\) \(0\) \(q+\zeta_{6}^{2}q^{4}+\zeta_{6}q^{9}+\zeta_{6}^{2}q^{11}-\zeta_{6}q^{16}+\cdots\)
539.1.h.b \(4\) \(0.269\) \(\Q(\sqrt{2}, \sqrt{-3})\) \(\Q(\sqrt{-11}) \) None \(0\) \(0\) \(0\) \(0\) \(q-\beta _{1}q^{3}+(-1-\beta _{2})q^{4}+(-\beta _{1}-\beta _{3})q^{5}+\cdots\)
539.1.l.a \(4\) \(0.269\) \(\Q(\zeta_{10})\) \(\Q(\sqrt{-7}) \) None \(0\) \(0\) \(0\) \(0\) \(q+(-\zeta_{10}+\zeta_{10}^{3})q^{2}+(-\zeta_{10}+\zeta_{10}^{2}+\cdots)q^{4}+\cdots\)
539.1.t.a \(8\) \(0.269\) \(\Q(\zeta_{15})\) \(\Q(\sqrt{-7}) \) None \(0\) \(0\) \(0\) \(0\) \(q+(-\zeta_{30}^{4}-\zeta_{30}^{13})q^{2}+(-\zeta_{30}^{2}+\cdots)q^{4}+\cdots\)
539.1.u.a \(8\) \(0.269\) \(\Q(\zeta_{15})\) \(\Q(\sqrt{-7}) \) None \(2\) \(0\) \(0\) \(0\) \(q+(\zeta_{30}^{8}-\zeta_{30}^{11})q^{2}+(-\zeta_{30}+\zeta_{30}^{4}+\cdots)q^{4}+\cdots\)
539.2.a.a \(1\) \(4.304\) \(\Q\) None None \(-2\) \(1\) \(-1\) \(0\) \(+\) \(q-2q^{2}+q^{3}+2q^{4}-q^{5}-2q^{6}-2q^{9}+\cdots\)
539.2.a.b \(1\) \(4.304\) \(\Q\) None None \(0\) \(-1\) \(-3\) \(0\) \(-\) \(q-q^{3}-2q^{4}-3q^{5}-2q^{9}-q^{11}+\cdots\)
539.2.a.c \(1\) \(4.304\) \(\Q\) None None \(0\) \(3\) \(1\) \(0\) \(-\) \(q+3q^{3}-2q^{4}+q^{5}+6q^{9}-q^{11}+\cdots\)
539.2.a.d \(1\) \(4.304\) \(\Q\) None None \(1\) \(-2\) \(2\) \(0\) \(+\) \(q+q^{2}-2q^{3}-q^{4}+2q^{5}-2q^{6}-3q^{8}+\cdots\)
539.2.a.e \(2\) \(4.304\) \(\Q(\sqrt{2}) \) None None \(-2\) \(0\) \(0\) \(0\) \(+\) \(q-q^{2}+\beta q^{3}-q^{4}-\beta q^{5}-\beta q^{6}+3q^{8}+\cdots\)
539.2.a.f \(2\) \(4.304\) \(\Q(\sqrt{5}) \) None None \(0\) \(-2\) \(4\) \(0\) \(-\) \(q-\beta q^{2}+(-1-\beta )q^{3}+3q^{4}+2q^{5}+\cdots\)
539.2.a.g \(3\) \(4.304\) \(\Q(\zeta_{18})^+\) None None \(0\) \(-3\) \(-6\) \(0\) \(+\) \(q+\beta _{1}q^{2}+(-1+\beta _{1}-\beta _{2})q^{3}+\beta _{2}q^{4}+\cdots\)
539.2.a.h \(3\) \(4.304\) 3.3.257.1 None None \(0\) \(-1\) \(-2\) \(0\) \(+\) \(q+\beta _{2}q^{2}-\beta _{1}q^{3}+(1+\beta _{1}-\beta _{2})q^{4}+\cdots\)
539.2.a.i \(3\) \(4.304\) 3.3.257.1 None None \(0\) \(1\) \(2\) \(0\) \(-\) \(q+\beta _{2}q^{2}+\beta _{1}q^{3}+(1+\beta _{1}-\beta _{2})q^{4}+\cdots\)
539.2.a.j \(3\) \(4.304\) \(\Q(\zeta_{18})^+\) None None \(0\) \(3\) \(6\) \(0\) \(-\) \(q+\beta _{1}q^{2}+(1-\beta _{1}+\beta _{2})q^{3}+\beta _{2}q^{4}+\cdots\)
539.2.a.k \(4\) \(4.304\) 4.4.9248.1 None None \(2\) \(0\) \(0\) \(0\) \(-\) \(q-\beta _{3}q^{2}+\beta _{2}q^{3}+(2-\beta _{3})q^{4}+(\beta _{1}+\cdots)q^{5}+\cdots\)
539.2.a.l \(10\) \(4.304\) \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None None \(2\) \(0\) \(0\) \(0\) \(-\) \(q+\beta _{5}q^{2}+\beta _{1}q^{3}+(2+\beta _{9})q^{4}+\beta _{7}q^{5}+\cdots\)
539.2.b.a \(8\) \(4.304\) 8.0.\(\cdots\).1 \(\Q(\sqrt{-11}) \) None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{1}q^{3}+2q^{4}+\beta _{5}q^{5}+(-3+\beta _{3}+\cdots)q^{9}+\cdots\)
539.2.b.b \(12\) \(4.304\) \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{6}q^{2}-\beta _{7}q^{3}+(-1+\beta _{2})q^{4}+(\beta _{5}+\cdots)q^{5}+\cdots\)
539.2.b.c \(16\) \(4.304\) \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{8}q^{2}-\beta _{7}q^{3}+(-2+\beta _{6})q^{4}-\beta _{4}q^{5}+\cdots\)
539.2.e.a \(2\) \(4.304\) \(\Q(\sqrt{-3}) \) None None \(-1\) \(-2\) \(2\) \(0\) \(q-\zeta_{6}q^{2}+(-2+2\zeta_{6})q^{3}+(1-\zeta_{6})q^{4}+\cdots\)
539.2.e.b \(2\) \(4.304\) \(\Q(\sqrt{-3}) \) None None \(-1\) \(2\) \(-2\) \(0\) \(q-\zeta_{6}q^{2}+(2-2\zeta_{6})q^{3}+(1-\zeta_{6})q^{4}+\cdots\)
539.2.e.c \(2\) \(4.304\) \(\Q(\sqrt{-3}) \) None None \(0\) \(-3\) \(-1\) \(0\) \(q+(-3+3\zeta_{6})q^{3}+(2-2\zeta_{6})q^{4}-\zeta_{6}q^{5}+\cdots\)
539.2.e.d \(2\) \(4.304\) \(\Q(\sqrt{-3}) \) None None \(0\) \(-1\) \(-3\) \(0\) \(q+(-1+\zeta_{6})q^{3}+(2-2\zeta_{6})q^{4}-3\zeta_{6}q^{5}+\cdots\)
539.2.e.e \(2\) \(4.304\) \(\Q(\sqrt{-3}) \) None None \(0\) \(1\) \(3\) \(0\) \(q+(1-\zeta_{6})q^{3}+(2-2\zeta_{6})q^{4}+3\zeta_{6}q^{5}+\cdots\)
539.2.e.f \(2\) \(4.304\) \(\Q(\sqrt{-3}) \) None None \(0\) \(3\) \(1\) \(0\) \(q+(3-3\zeta_{6})q^{3}+(2-2\zeta_{6})q^{4}+\zeta_{6}q^{5}+\cdots\)
539.2.e.g \(2\) \(4.304\) \(\Q(\sqrt{-3}) \) None None \(2\) \(-1\) \(1\) \(0\) \(q+2\zeta_{6}q^{2}+(-1+\zeta_{6})q^{3}+(-2+2\zeta_{6})q^{4}+\cdots\)
539.2.e.h \(2\) \(4.304\) \(\Q(\sqrt{-3}) \) None None \(2\) \(1\) \(-1\) \(0\) \(q+2\zeta_{6}q^{2}+(1-\zeta_{6})q^{3}+(-2+2\zeta_{6})q^{4}+\cdots\)
539.2.e.i \(4\) \(4.304\) \(\Q(\sqrt{-3}, \sqrt{5})\) None None \(0\) \(-2\) \(4\) \(0\) \(q-\beta _{2}q^{2}+(\beta _{1}-\beta _{2}-\beta _{3})q^{3}+3\beta _{1}q^{4}+\cdots\)
539.2.e.j \(4\) \(4.304\) \(\Q(\sqrt{-3}, \sqrt{5})\) None None \(0\) \(2\) \(-4\) \(0\) \(q-\beta _{2}q^{2}+(-\beta _{1}+\beta _{2}+\beta _{3})q^{3}+3\beta _{1}q^{4}+\cdots\)
539.2.e.k \(4\) \(4.304\) \(\Q(\sqrt{2}, \sqrt{-3})\) None None \(2\) \(0\) \(0\) \(0\) \(q-\beta _{2}q^{2}-\beta _{1}q^{3}+(1+\beta _{2})q^{4}+(-\beta _{1}+\cdots)q^{5}+\cdots\)
539.2.e.l \(6\) \(4.304\) 6.0.1783323.2 None None \(0\) \(-1\) \(-2\) \(0\) \(q+(-\beta _{3}+\beta _{5})q^{2}-\beta _{1}q^{3}+(-1-\beta _{1}+\cdots)q^{4}+\cdots\)
539.2.e.m \(6\) \(4.304\) \(\Q(\zeta_{18})\) None None \(0\) \(3\) \(6\) \(0\) \(q+(-\zeta_{18}^{2}+\zeta_{18}^{3}+\zeta_{18}^{4}-\zeta_{18}^{5})q^{2}+\cdots\)
539.2.e.n \(8\) \(4.304\) 8.0.6927565824.3 None None \(-2\) \(0\) \(0\) \(0\) \(q+(-1+\beta _{3}-\beta _{4}-\beta _{5})q^{2}+\beta _{2}q^{3}+\cdots\)
539.2.e.o \(20\) \(4.304\) \(\mathbb{Q}[x]/(x^{20} + \cdots)\) None None \(-2\) \(0\) \(0\) \(0\) \(q+\beta _{8}q^{2}+(-\beta _{1}+\beta _{10})q^{3}+(-2\beta _{2}+\cdots)q^{4}+\cdots\)
539.2.f.a \(4\) \(4.304\) \(\Q(\zeta_{10})\) None None \(2\) \(-1\) \(5\) \(0\) \(q+(\zeta_{10}-\zeta_{10}^{2})q^{2}+\zeta_{10}^{2}q^{3}+(-1+\cdots)q^{4}+\cdots\)
539.2.f.b \(4\) \(4.304\) \(\Q(\zeta_{10})\) None None \(2\) \(1\) \(-5\) \(0\) \(q+(\zeta_{10}-\zeta_{10}^{2})q^{2}-\zeta_{10}^{2}q^{3}+(-1+\cdots)q^{4}+\cdots\)
539.2.f.c \(8\) \(4.304\) 8.0.37515625.1 \(\Q(\sqrt{-7}) \) None \(-2\) \(0\) \(0\) \(0\) \(q+(-1+\beta _{1}+2\beta _{2}-\beta _{4}-\beta _{5}+\beta _{6}+\cdots)q^{2}+\cdots\)
539.2.f.d \(8\) \(4.304\) 8.0.159390625.1 None None \(-1\) \(4\) \(-3\) \(0\) \(q+\beta _{4}q^{2}+(\beta _{2}+\beta _{3})q^{3}+(1-\beta _{1}-\beta _{2}+\cdots)q^{4}+\cdots\)
539.2.f.e \(16\) \(4.304\) \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None None \(-3\) \(2\) \(5\) \(0\) \(q+(-\beta _{5}-\beta _{6})q^{2}+(-\beta _{9}+\beta _{11}-\beta _{13}+\cdots)q^{3}+\cdots\)
539.2.f.f \(16\) \(4.304\) \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None None \(-2\) \(0\) \(0\) \(0\) \(q+(-1-\beta _{2}+\beta _{5}-\beta _{7})q^{2}+(\beta _{11}-\beta _{12}+\cdots)q^{3}+\cdots\)
539.2.f.g \(20\) \(4.304\) \(\mathbb{Q}[x]/(x^{20} - \cdots)\) None None \(3\) \(-4\) \(4\) \(0\) \(q+\beta _{6}q^{2}+(-\beta _{1}-\beta _{7}-\beta _{18}+\beta _{19})q^{3}+\cdots\)
539.2.f.h \(20\) \(4.304\) \(\mathbb{Q}[x]/(x^{20} - \cdots)\) None None \(3\) \(4\) \(-4\) \(0\) \(q+\beta _{6}q^{2}+(\beta _{1}+\beta _{7}+\beta _{18}-\beta _{19})q^{3}+\cdots\)
539.2.f.i \(48\) \(4.304\) None None \(0\) \(0\) \(0\) \(0\)
539.2.i.a \(4\) \(4.304\) \(\Q(\sqrt{-3}, \sqrt{-7})\) \(\Q(\sqrt{-7}) \) None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{1}q^{2}+5\beta _{2}q^{4}+(3\beta _{1}-3\beta _{3})q^{8}+\cdots\)
539.2.i.b \(8\) \(4.304\) 8.0.3317760000.8 None None \(0\) \(0\) \(0\) \(0\) \(q+(\beta _{3}-\beta _{4})q^{2}+\beta _{5}q^{3}-\beta _{1}q^{5}+(\beta _{6}+\cdots)q^{6}+\cdots\)
539.2.i.c \(12\) \(4.304\) \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None None \(0\) \(6\) \(0\) \(0\) \(q+(\beta _{1}+\beta _{7})q^{2}+(1-\beta _{2}+\beta _{8}+\beta _{9}+\cdots)q^{3}+\cdots\)
539.2.i.d \(16\) \(4.304\) 16.0.\(\cdots\).7 \(\Q(\sqrt{-11}) \) None \(0\) \(0\) \(0\) \(0\) \(q+(\beta _{1}+\beta _{8})q^{3}-2\beta _{7}q^{4}+\beta _{5}q^{5}+(3+\cdots)q^{9}+\cdots\)
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