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Label Dim. \(A\) Field CM RM Traces Fricke sign $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
56.1.h.a \(1\) \(0.028\) \(\Q\) \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{-14}) \) \(\Q(\sqrt{2}) \) \(-1\) \(0\) \(0\) \(-1\) \(q-q^{2}+q^{4}-q^{7}-q^{8}-q^{9}+q^{14}+\cdots\)
56.2.a.a \(1\) \(0.447\) \(\Q\) None None \(0\) \(0\) \(2\) \(-1\) \(-\) \(q+2q^{5}-q^{7}-3q^{9}-4q^{11}+2q^{13}+\cdots\)
56.2.a.b \(1\) \(0.447\) \(\Q\) None None \(0\) \(2\) \(-4\) \(1\) \(-\) \(q+2q^{3}-4q^{5}+q^{7}+q^{9}-8q^{15}+\cdots\)
56.2.b.a \(2\) \(0.447\) \(\Q(\sqrt{-2}) \) None None \(0\) \(0\) \(0\) \(2\) \(q+\beta q^{2}+\beta q^{3}-2q^{4}-\beta q^{5}-2q^{6}+\cdots\)
56.2.b.b \(4\) \(0.447\) 4.0.2312.1 None None \(-1\) \(0\) \(0\) \(-4\) \(q-\beta _{1}q^{2}+(\beta _{1}-\beta _{2})q^{3}+\beta _{2}q^{4}+(-\beta _{1}+\cdots)q^{5}+\cdots\)
56.2.e.a \(2\) \(0.447\) \(\Q(\sqrt{-7}) \) \(\Q(\sqrt{-7}) \) None \(1\) \(0\) \(0\) \(0\) \(q+\beta q^{2}+(-2+\beta )q^{4}+(1-2\beta )q^{7}+\cdots\)
56.2.e.b \(4\) \(0.447\) \(\Q(i, \sqrt{6})\) None None \(-4\) \(0\) \(0\) \(0\) \(q+(-1+\beta _{1})q^{2}+\beta _{2}q^{3}-2\beta _{1}q^{4}+\cdots\)
56.2.i.a \(2\) \(0.447\) \(\Q(\sqrt{-3}) \) None None \(0\) \(-3\) \(1\) \(4\) \(q-3\zeta_{6}q^{3}+(1-\zeta_{6})q^{5}+(1+2\zeta_{6})q^{7}+\cdots\)
56.2.i.b \(2\) \(0.447\) \(\Q(\sqrt{-3}) \) None None \(0\) \(1\) \(1\) \(-4\) \(q+\zeta_{6}q^{3}+(1-\zeta_{6})q^{5}+(-3+2\zeta_{6})q^{7}+\cdots\)
56.2.m.a \(12\) \(0.447\) 12.0.\(\cdots\).2 None None \(0\) \(-6\) \(0\) \(0\) \(q+(\beta _{1}-\beta _{8})q^{2}+(-1-\beta _{10})q^{3}+(\beta _{2}+\cdots)q^{4}+\cdots\)
56.2.p.a \(12\) \(0.447\) 12.0.\(\cdots\).1 None None \(-2\) \(0\) \(0\) \(-4\) \(q+(-\beta _{1}+\beta _{5})q^{2}+\beta _{6}q^{3}-\beta _{3}q^{4}+\cdots\)
56.3.c.a \(4\) \(1.526\) 4.0.2048.2 None None \(0\) \(0\) \(0\) \(4\) \(q+\beta _{1}q^{3}+(\beta _{1}+\beta _{3})q^{5}+(1+\beta _{1}+\beta _{2}+\cdots)q^{7}+\cdots\)
56.3.g.a \(4\) \(1.526\) \(\Q(\sqrt{2}, \sqrt{-7})\) None None \(0\) \(8\) \(0\) \(0\) \(q+\beta _{1}q^{2}+(2-\beta _{2})q^{3}+(-3+\beta _{3})q^{4}+\cdots\)
56.3.g.b \(8\) \(1.526\) 8.0.\(\cdots\).3 None None \(1\) \(-8\) \(0\) \(0\) \(q+\beta _{5}q^{2}+(-1+\beta _{3}-\beta _{4})q^{3}+(1+\beta _{6}+\cdots)q^{4}+\cdots\)
56.3.h.a \(2\) \(1.526\) \(\Q(\sqrt{7}) \) \(\Q(\sqrt{-14}) \) None \(-4\) \(0\) \(0\) \(14\) \(q-2q^{2}+\beta q^{3}+4q^{4}-\beta q^{5}-2\beta q^{6}+\cdots\)
56.3.h.b \(2\) \(1.526\) \(\Q(\sqrt{-7}) \) \(\Q(\sqrt{-7}) \) None \(3\) \(0\) \(0\) \(14\) \(q+(1+\beta )q^{2}+(-1+3\beta )q^{4}+7q^{7}+\cdots\)
56.3.h.c \(2\) \(1.526\) \(\Q(\sqrt{2}) \) \(\Q(\sqrt{-14}) \) None \(4\) \(0\) \(0\) \(-14\) \(q+2q^{2}+\beta q^{3}+4q^{4}-3\beta q^{5}+2\beta q^{6}+\cdots\)
56.3.h.d \(8\) \(1.526\) 8.0.\(\cdots\).51 None None \(-4\) \(0\) \(0\) \(-16\) \(q+(-1-\beta _{2})q^{2}+\beta _{6}q^{3}+(-2-\beta _{1}+\cdots)q^{4}+\cdots\)
56.3.j.a \(28\) \(1.526\) None None \(-2\) \(0\) \(0\) \(-4\)
56.3.k.a \(2\) \(1.526\) \(\Q(\sqrt{-3}) \) None None \(-4\) \(-1\) \(-9\) \(-2\) \(q-2q^{2}-\zeta_{6}q^{3}+4q^{4}+(-3-3\zeta_{6})q^{5}+\cdots\)
56.3.k.b \(2\) \(1.526\) \(\Q(\sqrt{-3}) \) None None \(2\) \(-1\) \(9\) \(2\) \(q+2\zeta_{6}q^{2}-\zeta_{6}q^{3}+(-4+4\zeta_{6})q^{4}+\cdots\)
56.3.k.c \(12\) \(1.526\) \(\mathbb{Q}[x]/(x^{12} + \cdots)\) None None \(0\) \(6\) \(0\) \(0\) \(q-\beta _{7}q^{2}+(1+\beta _{1}+\beta _{4}+\beta _{8})q^{3}+(-1+\cdots)q^{4}+\cdots\)
56.3.k.d \(12\) \(1.526\) \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None None \(2\) \(-6\) \(0\) \(0\) \(q-\beta _{3}q^{2}+(\beta _{8}-\beta _{9}-\beta _{10})q^{3}+(\beta _{2}+\cdots)q^{4}+\cdots\)
56.3.o.a \(8\) \(1.526\) 8.0.\(\cdots\).2 None None \(0\) \(0\) \(0\) \(-4\) \(q-\beta _{5}q^{3}+\beta _{4}q^{5}+(1-3\beta _{1}-\beta _{2}+\beta _{3}+\cdots)q^{7}+\cdots\)
56.4.a.a \(1\) \(3.304\) \(\Q\) None None \(0\) \(-2\) \(-16\) \(-7\) \(-\) \(q-2q^{3}-2^{4}q^{5}-7q^{7}-23q^{9}+24q^{11}+\cdots\)
56.4.a.b \(1\) \(3.304\) \(\Q\) None None \(0\) \(6\) \(8\) \(-7\) \(+\) \(q+6q^{3}+8q^{5}-7q^{7}+9q^{9}+56q^{11}+\cdots\)
56.4.a.c \(2\) \(3.304\) \(\Q(\sqrt{57}) \) None None \(0\) \(-2\) \(22\) \(14\) \(+\) \(q+(-1-\beta )q^{3}+(11+\beta )q^{5}+7q^{7}+\cdots\)
56.4.b.a \(8\) \(3.304\) \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None None \(-2\) \(0\) \(0\) \(-56\) \(q-\beta _{1}q^{2}-\beta _{4}q^{3}+(2+\beta _{2})q^{4}+(\beta _{1}+\cdots)q^{5}+\cdots\)
56.4.b.b \(10\) \(3.304\) \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None None \(3\) \(0\) \(0\) \(70\) \(q+\beta _{1}q^{2}+\beta _{4}q^{3}+\beta _{2}q^{4}+(-\beta _{1}-\beta _{6}+\cdots)q^{5}+\cdots\)
56.4.e.a \(2\) \(3.304\) \(\Q(\sqrt{-7}) \) \(\Q(\sqrt{-7}) \) None \(-5\) \(0\) \(0\) \(0\) \(q+(-2-\beta )q^{2}+(2+5\beta )q^{4}+(-7+14\beta )q^{7}+\cdots\)
56.4.e.b \(4\) \(3.304\) \(\Q(\sqrt{-3}, \sqrt{-7})\) None None \(-4\) \(0\) \(0\) \(0\) \(q+(-1+\beta _{2})q^{2}+\beta _{3}q^{3}+(-6-2\beta _{2}+\cdots)q^{4}+\cdots\)
56.4.e.c \(16\) \(3.304\) \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None None \(8\) \(0\) \(0\) \(0\) \(q+(1+\beta _{2})q^{2}-\beta _{4}q^{3}-\beta _{8}q^{4}+\beta _{12}q^{5}+\cdots\)
56.4.i.a \(6\) \(3.304\) 6.0.11163123.4 None None \(0\) \(-1\) \(-13\) \(-20\) \(q+\beta _{3}q^{3}+(-5+5\beta _{1}-\beta _{3}-\beta _{4}-2\beta _{5})q^{5}+\cdots\)
56.4.i.b \(6\) \(3.304\) 6.0.11163123.4 None None \(0\) \(7\) \(3\) \(-4\) \(q+(2-2\beta _{1}-\beta _{3}+\beta _{5})q^{3}+(\beta _{1}+\beta _{4}+\cdots)q^{5}+\cdots\)
56.4.m.a \(44\) \(3.304\) None None \(-2\) \(-6\) \(0\) \(0\)
56.4.p.a \(44\) \(3.304\) None None \(0\) \(0\) \(0\) \(-4\)
56.5.c.a \(8\) \(5.789\) \(\mathbb{Q}[x]/(x^{8} + \cdots)\) None None \(0\) \(0\) \(0\) \(56\) \(q+\beta _{1}q^{3}-\beta _{2}q^{5}+(7-\beta _{5})q^{7}+(-31+\cdots)q^{9}+\cdots\)
56.5.g.a \(2\) \(5.789\) \(\Q(\sqrt{-7}) \) None None \(6\) \(-12\) \(0\) \(0\) \(q+(3+\beta )q^{2}-6q^{3}+(2+6\beta )q^{4}+4\beta q^{5}+\cdots\)
56.5.g.b \(22\) \(5.789\) None None \(-9\) \(12\) \(0\) \(0\)
56.5.h.a \(1\) \(5.789\) \(\Q\) \(\Q(\sqrt{-14}) \) None \(4\) \(-10\) \(22\) \(49\) \(q+4q^{2}-10q^{3}+2^{4}q^{4}+22q^{5}-40q^{6}+\cdots\)
56.5.h.b \(1\) \(5.789\) \(\Q\) \(\Q(\sqrt{-14}) \) None \(4\) \(10\) \(-22\) \(49\) \(q+4q^{2}+10q^{3}+2^{4}q^{4}-22q^{5}+40q^{6}+\cdots\)
56.5.h.c \(2\) \(5.789\) \(\Q(\sqrt{14}) \) \(\Q(\sqrt{-14}) \) None \(-8\) \(0\) \(0\) \(-98\) \(q-4q^{2}+\beta q^{3}+2^{4}q^{4}+3\beta q^{5}-4\beta q^{6}+\cdots\)
56.5.h.d \(2\) \(5.789\) \(\Q(\sqrt{-7}) \) \(\Q(\sqrt{-7}) \) None \(-1\) \(0\) \(0\) \(-98\) \(q-\beta q^{2}+(-2^{4}+\beta )q^{4}-7^{2}q^{7}+(2^{4}+\cdots)q^{8}+\cdots\)
56.5.h.e \(24\) \(5.789\) None None \(-4\) \(0\) \(0\) \(96\)
56.5.j.a \(60\) \(5.789\) None None \(2\) \(0\) \(0\) \(-4\)
56.5.k.a \(60\) \(5.789\) None None \(-4\) \(-2\) \(0\) \(0\)
56.5.o.a \(16\) \(5.789\) \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None None \(0\) \(0\) \(0\) \(-56\) \(q+(\beta _{2}-\beta _{6})q^{3}+\beta _{4}q^{5}+(-5-3\beta _{1}+\cdots)q^{7}+\cdots\)
56.6.a.a \(1\) \(8.981\) \(\Q\) None None \(0\) \(-6\) \(4\) \(49\) \(+\) \(q-6q^{3}+4q^{5}+7^{2}q^{7}-207q^{9}-240q^{11}+\cdots\)
56.6.a.b \(1\) \(8.981\) \(\Q\) None None \(0\) \(30\) \(32\) \(49\) \(-\) \(q+30q^{3}+2^{5}q^{5}+7^{2}q^{7}+657q^{9}+\cdots\)
56.6.a.c \(2\) \(8.981\) \(\Q(\sqrt{177}) \) None None \(0\) \(-26\) \(-62\) \(98\) \(-\) \(q+(-13-\beta )q^{3}+(-31-5\beta )q^{5}+7^{2}q^{7}+\cdots\)
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