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Label Dim. \(A\) Field CM RM Traces Fricke sign $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
684.1.h.a \(1\) \(0.341\) \(\Q\) \(\Q(\sqrt{-19}) \) None \(0\) \(0\) \(1\) \(-1\) \(q+q^{5}-q^{7}+q^{11}+q^{17}+q^{19}-2q^{23}+\cdots\)
684.1.h.b \(2\) \(0.341\) \(\Q(\sqrt{3}) \) \(\Q(\sqrt{-19}) \) None \(0\) \(0\) \(0\) \(2\) \(q-\beta q^{5}+q^{7}+\beta q^{11}+\beta q^{17}-q^{19}+\cdots\)
684.1.q.a \(8\) \(0.341\) \(\Q(\zeta_{24})\) None None \(0\) \(0\) \(0\) \(0\) \(q-\zeta_{24}^{5}q^{2}+\zeta_{24}^{10}q^{4}+(\zeta_{24}^{5}+\zeta_{24}^{11}+\cdots)q^{5}+\cdots\)
684.1.x.a \(4\) \(0.341\) \(\Q(\zeta_{12})\) None None \(-4\) \(0\) \(2\) \(0\) \(q-q^{2}-\zeta_{12}^{3}q^{3}+q^{4}+\zeta_{12}^{2}q^{5}+\cdots\)
684.1.bm.a \(4\) \(0.341\) \(\Q(\zeta_{12})\) None None \(2\) \(0\) \(-4\) \(0\) \(q+\zeta_{12}^{2}q^{2}+\zeta_{12}^{5}q^{3}+\zeta_{12}^{4}q^{4}+\cdots\)
684.1.bx.a \(6\) \(0.341\) \(\Q(\zeta_{18})\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(0\) \(0\) \(q+(\zeta_{18}^{5}+\zeta_{18}^{7})q^{7}+(\zeta_{18}^{3}+\zeta_{18}^{4}+\cdots)q^{13}+\cdots\)
684.2.a.a \(1\) \(5.462\) \(\Q\) None None \(0\) \(0\) \(-2\) \(0\) \(+\) \(q-2q^{5}-2q^{11}+2q^{13}-6q^{17}-q^{19}+\cdots\)
684.2.a.b \(1\) \(5.462\) \(\Q\) None None \(0\) \(0\) \(1\) \(-3\) \(+\) \(q+q^{5}-3q^{7}-5q^{11}-4q^{13}+3q^{17}+\cdots\)
684.2.a.c \(1\) \(5.462\) \(\Q\) None None \(0\) \(0\) \(3\) \(1\) \(-\) \(q+3q^{5}+q^{7}+5q^{11}-6q^{13}+5q^{17}+\cdots\)
684.2.a.d \(2\) \(5.462\) \(\Q(\sqrt{33}) \) None None \(0\) \(0\) \(-3\) \(1\) \(-\) \(q+(-1-\beta )q^{5}+(1-\beta )q^{7}+(1+\beta )q^{11}+\cdots\)
684.2.a.e \(2\) \(5.462\) \(\Q(\sqrt{7}) \) None None \(0\) \(0\) \(0\) \(6\) \(-\) \(q+\beta q^{5}+3q^{7}+\beta q^{11}+2q^{13}-3\beta q^{17}+\cdots\)
684.2.c.a \(4\) \(5.462\) \(\Q(\zeta_{8})\) None None \(0\) \(0\) \(0\) \(0\) \(q-\zeta_{8}^{2}q^{2}-2q^{4}-\zeta_{8}^{2}q^{5}+4\zeta_{8}q^{7}+\cdots\)
684.2.c.b \(32\) \(5.462\) None None \(0\) \(0\) \(0\) \(0\)
684.2.d.a \(8\) \(5.462\) 8.0.\(\cdots\).4 \(\Q(\sqrt{-19}) \) None \(0\) \(0\) \(0\) \(0\) \(q-\beta _{3}q^{5}-\beta _{2}q^{7}-\beta _{6}q^{11}+(\beta _{3}-\beta _{5}+\cdots)q^{17}+\cdots\)
684.2.f.a \(4\) \(5.462\) \(\Q(\sqrt{-2}, \sqrt{19})\) \(\Q(\sqrt{-57}) \) None \(0\) \(0\) \(0\) \(0\) \(q-\beta _{1}q^{2}-2q^{4}+2\beta _{1}q^{8}-\beta _{2}q^{11}+\cdots\)
684.2.f.b \(8\) \(5.462\) 8.0.\(\cdots\).1 None None \(0\) \(0\) \(4\) \(0\) \(q+\beta _{1}q^{2}+(-1+\beta _{2})q^{4}+(1-\beta _{2}+\beta _{6}+\cdots)q^{5}+\cdots\)
684.2.f.c \(10\) \(5.462\) 10.0.\(\cdots\).1 None None \(-2\) \(0\) \(0\) \(0\) \(q-\beta _{1}q^{2}+\beta _{2}q^{4}-\beta _{4}q^{5}-\beta _{5}q^{7}+\cdots\)
684.2.f.d \(10\) \(5.462\) 10.0.\(\cdots\).1 None None \(2\) \(0\) \(0\) \(0\) \(q+\beta _{1}q^{2}+\beta _{2}q^{4}-\beta _{4}q^{5}-\beta _{5}q^{7}+\cdots\)
684.2.f.e \(16\) \(5.462\) \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None None \(0\) \(0\) \(0\) \(0\) \(q-\beta _{3}q^{2}+(1+\beta _{6})q^{4}-\beta _{11}q^{5}+(\beta _{7}+\cdots)q^{7}+\cdots\)
684.2.i.a \(2\) \(5.462\) \(\Q(\sqrt{-3}) \) None None \(0\) \(0\) \(-3\) \(5\) \(q+(1-2\zeta_{6})q^{3}-3\zeta_{6}q^{5}+(5-5\zeta_{6})q^{7}+\cdots\)
684.2.i.b \(2\) \(5.462\) \(\Q(\sqrt{-3}) \) None None \(0\) \(0\) \(1\) \(-3\) \(q+(1-2\zeta_{6})q^{3}+\zeta_{6}q^{5}+(-3+3\zeta_{6})q^{7}+\cdots\)
684.2.i.c \(16\) \(5.462\) \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None None \(0\) \(-1\) \(2\) \(3\) \(q-\beta _{1}q^{3}-\beta _{13}q^{5}+(-\beta _{3}+\beta _{8}+\beta _{11}+\cdots)q^{7}+\cdots\)
684.2.i.d \(16\) \(5.462\) \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None None \(0\) \(-1\) \(6\) \(-5\) \(q+(\beta _{2}+\beta _{13})q^{3}+(\beta _{7}+\beta _{8})q^{5}+(-1+\cdots)q^{7}+\cdots\)
684.2.j.a \(40\) \(5.462\) None None \(0\) \(-2\) \(0\) \(-1\)
684.2.k.a \(2\) \(5.462\) \(\Q(\sqrt{-3}) \) None None \(0\) \(0\) \(-4\) \(-6\) \(q+(-4+4\zeta_{6})q^{5}-3q^{7}+4q^{11}-5\zeta_{6}q^{13}+\cdots\)
684.2.k.b \(2\) \(5.462\) \(\Q(\sqrt{-3}) \) None None \(0\) \(0\) \(-1\) \(0\) \(q+(-1+\zeta_{6})q^{5}+4q^{11}+\zeta_{6}q^{13}+\cdots\)
684.2.k.c \(2\) \(5.462\) \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(0\) \(-2\) \(q-q^{7}+7\zeta_{6}q^{13}+(5-2\zeta_{6})q^{19}+5\zeta_{6}q^{25}+\cdots\)
684.2.k.d \(2\) \(5.462\) \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(0\) \(10\) \(q+5q^{7}-5\zeta_{6}q^{13}+(3+2\zeta_{6})q^{19}+\cdots\)
684.2.k.e \(2\) \(5.462\) \(\Q(\sqrt{-3}) \) None None \(0\) \(0\) \(4\) \(-6\) \(q+(4-4\zeta_{6})q^{5}-3q^{7}-4q^{11}-5\zeta_{6}q^{13}+\cdots\)
684.2.k.f \(4\) \(5.462\) \(\Q(\sqrt{-2}, \sqrt{-3})\) None None \(0\) \(0\) \(0\) \(-4\) \(q-\beta _{2}q^{5}+(-1+\beta _{3})q^{7}-\beta _{3}q^{11}+\cdots\)
684.2.k.g \(4\) \(5.462\) \(\Q(\sqrt{-3}, \sqrt{7})\) None None \(0\) \(0\) \(2\) \(8\) \(q+(1+\beta _{1}+\beta _{2})q^{5}+(2-\beta _{3})q^{7}+(-3+\cdots)q^{11}+\cdots\)
684.2.l.a \(40\) \(5.462\) None None \(0\) \(1\) \(0\) \(-1\)
684.2.n.a \(2\) \(5.462\) \(\Q(\sqrt{-3}) \) None None \(0\) \(0\) \(0\) \(-1\) \(q+(1-2\zeta_{6})q^{3}+(-1+\zeta_{6})q^{7}-3q^{9}+\cdots\)
684.2.n.b \(38\) \(5.462\) None None \(0\) \(3\) \(0\) \(2\)
684.2.o.a \(232\) \(5.462\) None None \(-3\) \(0\) \(0\) \(0\)
684.2.r.a \(16\) \(5.462\) \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None None \(3\) \(0\) \(2\) \(0\) \(q-\beta _{12}q^{2}+(-1-\beta _{1}+\beta _{2}-2\beta _{3}+\cdots)q^{4}+\cdots\)
684.2.r.b \(20\) \(5.462\) \(\mathbb{Q}[x]/(x^{20} - \cdots)\) None None \(-1\) \(0\) \(0\) \(0\) \(q+(-\beta _{1}+\beta _{4})q^{2}+(-\beta _{2}-\beta _{9})q^{4}+\cdots\)
684.2.r.c \(20\) \(5.462\) \(\mathbb{Q}[x]/(x^{20} - \cdots)\) None None \(1\) \(0\) \(0\) \(0\) \(q+\beta _{16}q^{2}+(\beta _{14}-\beta _{17})q^{4}-\beta _{6}q^{5}+\cdots\)
684.2.r.d \(40\) \(5.462\) None None \(0\) \(0\) \(0\) \(0\)
684.2.u.a \(232\) \(5.462\) None None \(0\) \(0\) \(2\) \(0\)
684.2.w.a \(232\) \(5.462\) None None \(0\) \(0\) \(-4\) \(0\)
684.2.z.a \(8\) \(5.462\) 8.0.\(\cdots\).3 None None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{3}q^{2}-2\beta _{4}q^{4}-2\beta _{3}q^{5}+(-\beta _{5}+\cdots)q^{7}+\cdots\)
684.2.z.b \(72\) \(5.462\) None None \(0\) \(0\) \(0\) \(0\)
684.2.bb.a \(2\) \(5.462\) \(\Q(\sqrt{-3}) \) None None \(0\) \(-3\) \(0\) \(-1\) \(q+(-2+\zeta_{6})q^{3}-\zeta_{6}q^{7}+(3-3\zeta_{6})q^{9}+\cdots\)
684.2.bb.b \(38\) \(5.462\) None None \(0\) \(3\) \(0\) \(2\)
684.2.bd.a \(40\) \(5.462\) None None \(0\) \(0\) \(0\) \(-2\)
684.2.bg.a \(4\) \(5.462\) \(\Q(\zeta_{12})\) None None \(-2\) \(0\) \(-6\) \(-12\) \(q+(-\zeta_{12}-\zeta_{12}^{2}+\zeta_{12}^{3})q^{2}+(1-2\zeta_{12}^{2}+\cdots)q^{3}+\cdots\)
684.2.bg.b \(4\) \(5.462\) \(\Q(\zeta_{12})\) None None \(2\) \(0\) \(-6\) \(12\) \(q+(\zeta_{12}+\zeta_{12}^{2}-\zeta_{12}^{3})q^{2}+(-1+2\zeta_{12}^{2}+\cdots)q^{3}+\cdots\)
684.2.bg.c \(208\) \(5.462\) None None \(0\) \(0\) \(12\) \(0\)
684.2.bi.a \(232\) \(5.462\) None None \(0\) \(0\) \(-6\) \(0\)
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