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Label Dim. \(A\) Field CM RM Traces Fricke sign $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
420.1.o.a \(2\) \(0.210\) \(\Q(\sqrt{-1}) \) \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{-21}) \) \(\Q(\sqrt{105}) \) \(0\) \(0\) \(-2\) \(0\) \(q-iq^{2}-iq^{3}-q^{4}-q^{5}-q^{6}-iq^{7}+\cdots\)
420.1.o.b \(2\) \(0.210\) \(\Q(\sqrt{-1}) \) \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{-21}) \) \(\Q(\sqrt{105}) \) \(0\) \(0\) \(2\) \(0\) \(q+iq^{2}-iq^{3}-q^{4}+q^{5}+q^{6}-iq^{7}+\cdots\)
420.1.be.a \(4\) \(0.210\) \(\Q(\zeta_{12})\) \(\Q(\sqrt{-5}) \) None \(0\) \(0\) \(-2\) \(0\) \(q+\zeta_{12}q^{2}-\zeta_{12}q^{3}+\zeta_{12}^{2}q^{4}+\zeta_{12}^{4}q^{5}+\cdots\)
420.1.be.b \(4\) \(0.210\) \(\Q(\zeta_{12})\) \(\Q(\sqrt{-5}) \) None \(0\) \(0\) \(2\) \(0\) \(q+\zeta_{12}q^{2}-\zeta_{12}^{3}q^{3}+\zeta_{12}^{2}q^{4}-\zeta_{12}^{4}q^{5}+\cdots\)
420.2.a.a \(1\) \(3.354\) \(\Q\) None None \(0\) \(-1\) \(-1\) \(-1\) \(-\) \(q-q^{3}-q^{5}-q^{7}+q^{9}+2q^{11}+4q^{13}+\cdots\)
420.2.a.b \(1\) \(3.354\) \(\Q\) None None \(0\) \(-1\) \(1\) \(1\) \(-\) \(q-q^{3}+q^{5}+q^{7}+q^{9}-2q^{11}+4q^{13}+\cdots\)
420.2.a.c \(1\) \(3.354\) \(\Q\) None None \(0\) \(1\) \(-1\) \(1\) \(-\) \(q+q^{3}-q^{5}+q^{7}+q^{9}+6q^{11}-4q^{13}+\cdots\)
420.2.a.d \(1\) \(3.354\) \(\Q\) None None \(0\) \(1\) \(1\) \(-1\) \(-\) \(q+q^{3}+q^{5}-q^{7}+q^{9}+2q^{11}+4q^{13}+\cdots\)
420.2.c.a \(16\) \(3.354\) \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None None \(2\) \(-16\) \(0\) \(-4\) \(q+\beta _{1}q^{2}-q^{3}+\beta _{2}q^{4}+\beta _{4}q^{5}-\beta _{1}q^{6}+\cdots\)
420.2.c.b \(16\) \(3.354\) \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None None \(2\) \(16\) \(0\) \(4\) \(q+\beta _{1}q^{2}+q^{3}+\beta _{2}q^{4}-\beta _{4}q^{5}+\beta _{1}q^{6}+\cdots\)
420.2.d.a \(2\) \(3.354\) \(\Q(\sqrt{-3}) \) None None \(0\) \(-3\) \(-2\) \(4\) \(q+(-1-\zeta_{6})q^{3}-q^{5}+(1+2\zeta_{6})q^{7}+\cdots\)
420.2.d.b \(2\) \(3.354\) \(\Q(\sqrt{-3}) \) None None \(0\) \(3\) \(2\) \(4\) \(q+(1+\zeta_{6})q^{3}+q^{5}+(3-2\zeta_{6})q^{7}+3\zeta_{6}q^{9}+\cdots\)
420.2.d.c \(4\) \(3.354\) \(\Q(i, \sqrt{5})\) None None \(0\) \(-2\) \(4\) \(-6\) \(q+(-1+\beta _{3})q^{3}+q^{5}+(-1+\beta _{1}+\beta _{2}+\cdots)q^{7}+\cdots\)
420.2.d.d \(4\) \(3.354\) \(\Q(i, \sqrt{5})\) None None \(0\) \(2\) \(-4\) \(-6\) \(q+(1+\beta _{1})q^{3}-q^{5}+(-1+\beta _{1}+\beta _{2}+\cdots)q^{7}+\cdots\)
420.2.f.a \(8\) \(3.354\) 8.0.\(\cdots\).2 \(\Q(\sqrt{-35}) \) None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{1}q^{3}+\beta _{4}q^{5}+(-\beta _{1}-\beta _{6})q^{7}+\cdots\)
420.2.f.b \(8\) \(3.354\) 8.0.40960000.1 None None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{6}q^{3}-\beta _{2}q^{5}+(\beta _{5}+\beta _{6})q^{7}+(1+\cdots)q^{9}+\cdots\)
420.2.i.a \(48\) \(3.354\) None None \(0\) \(0\) \(0\) \(0\)
420.2.k.a \(2\) \(3.354\) \(\Q(\sqrt{-1}) \) None None \(0\) \(0\) \(-4\) \(0\) \(q+iq^{3}+(-2-i)q^{5}+iq^{7}-q^{9}+\cdots\)
420.2.k.b \(2\) \(3.354\) \(\Q(\sqrt{-1}) \) None None \(0\) \(0\) \(4\) \(0\) \(q+iq^{3}+(2-i)q^{5}+iq^{7}-q^{9}+4q^{11}+\cdots\)
420.2.l.a \(4\) \(3.354\) \(\Q(\zeta_{8})\) None None \(0\) \(-4\) \(0\) \(4\) \(q-\zeta_{8}^{2}q^{2}+(-1+\zeta_{8}^{2})q^{3}-2q^{4}+\cdots\)
420.2.l.b \(4\) \(3.354\) \(\Q(\zeta_{8})\) None None \(0\) \(4\) \(0\) \(-4\) \(q+\zeta_{8}^{2}q^{2}+(1-\zeta_{8}^{2})q^{3}-2q^{4}+(\zeta_{8}^{2}+\cdots)q^{5}+\cdots\)
420.2.l.c \(8\) \(3.354\) 8.0.386672896.3 None None \(0\) \(-2\) \(-8\) \(-8\) \(q-\beta _{3}q^{2}+\beta _{2}q^{3}+(1-\beta _{4}-\beta _{6})q^{4}+\cdots\)
420.2.l.d \(8\) \(3.354\) 8.0.386672896.3 None None \(0\) \(-2\) \(8\) \(-8\) \(q+\beta _{3}q^{2}-\beta _{6}q^{3}+(1-\beta _{4}-\beta _{6})q^{4}+\cdots\)
420.2.l.e \(8\) \(3.354\) 8.0.386672896.3 None None \(0\) \(2\) \(-8\) \(8\) \(q-\beta _{3}q^{2}+\beta _{6}q^{3}+(1-\beta _{4}-\beta _{6})q^{4}+\cdots\)
420.2.l.f \(8\) \(3.354\) 8.0.386672896.3 None None \(0\) \(2\) \(8\) \(8\) \(q+\beta _{3}q^{2}-\beta _{2}q^{3}+(1-\beta _{4}-\beta _{6})q^{4}+\cdots\)
420.2.l.g \(16\) \(3.354\) \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None None \(0\) \(0\) \(0\) \(-16\) \(q+\beta _{3}q^{2}+\beta _{4}q^{3}+\beta _{11}q^{4}+\beta _{5}q^{5}+\cdots\)
420.2.l.h \(16\) \(3.354\) \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None None \(0\) \(0\) \(0\) \(16\) \(q+\beta _{3}q^{2}+\beta _{6}q^{3}+\beta _{11}q^{4}+(-\beta _{4}+\cdots)q^{5}+\cdots\)
420.2.n.a \(24\) \(3.354\) None None \(0\) \(0\) \(0\) \(0\)
420.2.n.b \(24\) \(3.354\) None None \(0\) \(0\) \(0\) \(0\)
420.2.q.a \(2\) \(3.354\) \(\Q(\sqrt{-3}) \) None None \(0\) \(1\) \(-1\) \(5\) \(q+(1-\zeta_{6})q^{3}-\zeta_{6}q^{5}+(2+\zeta_{6})q^{7}+\cdots\)
420.2.q.b \(2\) \(3.354\) \(\Q(\sqrt{-3}) \) None None \(0\) \(1\) \(1\) \(-5\) \(q+(1-\zeta_{6})q^{3}+\zeta_{6}q^{5}+(-2-\zeta_{6})q^{7}+\cdots\)
420.2.q.c \(4\) \(3.354\) \(\Q(\sqrt{-3}, \sqrt{7})\) None None \(0\) \(-2\) \(-2\) \(0\) \(q+\beta _{2}q^{3}+(-1-\beta _{2})q^{5}+\beta _{1}q^{7}+(-1+\cdots)q^{9}+\cdots\)
420.2.q.d \(4\) \(3.354\) \(\Q(\sqrt{2}, \sqrt{-3})\) None None \(0\) \(-2\) \(2\) \(-2\) \(q+(-1-\beta _{1})q^{3}-\beta _{1}q^{5}+(\beta _{1}+\beta _{3})q^{7}+\cdots\)
420.2.s.a \(24\) \(3.354\) None None \(0\) \(-4\) \(0\) \(0\)
420.2.t.a \(4\) \(3.354\) \(\Q(\zeta_{8})\) None None \(4\) \(0\) \(-8\) \(0\) \(q+(1+\zeta_{8}^{2})q^{2}+\zeta_{8}q^{3}+2\zeta_{8}^{2}q^{4}+\cdots\)
420.2.t.b \(4\) \(3.354\) \(\Q(\zeta_{8})\) None None \(4\) \(0\) \(8\) \(0\) \(q+(1+\zeta_{8}^{2})q^{2}+\zeta_{8}q^{3}+2\zeta_{8}^{2}q^{4}+\cdots\)
420.2.t.c \(32\) \(3.354\) None None \(-4\) \(0\) \(-8\) \(0\)
420.2.t.d \(32\) \(3.354\) None None \(-4\) \(0\) \(8\) \(0\)
420.2.w.a \(8\) \(3.354\) 8.0.\(\cdots\).1 \(\Q(\sqrt{-21}) \) None \(-8\) \(0\) \(0\) \(0\) \(q+(-1+\beta _{2})q^{2}+\beta _{1}q^{3}-2\beta _{2}q^{4}+\cdots\)
420.2.w.b \(8\) \(3.354\) 8.0.\(\cdots\).1 \(\Q(\sqrt{-21}) \) None \(8\) \(0\) \(0\) \(0\) \(q+(1-\beta _{2})q^{2}+\beta _{1}q^{3}-2\beta _{2}q^{4}+(-\beta _{3}+\cdots)q^{5}+\cdots\)
420.2.w.c \(160\) \(3.354\) None None \(0\) \(0\) \(0\) \(0\)
420.2.x.a \(16\) \(3.354\) \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None None \(0\) \(0\) \(0\) \(0\) \(q-\beta _{7}q^{3}+(1+\beta _{1}+\beta _{2}-\beta _{3}+\beta _{5}+\cdots)q^{5}+\cdots\)
420.2.ba.a \(8\) \(3.354\) 8.0.3317760000.8 \(\Q(\sqrt{-5}) \) None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{5}q^{2}+(\beta _{5}+\beta _{7})q^{3}+2\beta _{3}q^{4}-\beta _{6}q^{5}+\cdots\)
420.2.ba.b \(8\) \(3.354\) 8.0.3317760000.8 \(\Q(\sqrt{-5}) \) None \(0\) \(0\) \(0\) \(0\) \(q+(-\beta _{4}-\beta _{5})q^{2}+(-\beta _{1}-\beta _{5}-\beta _{7})q^{3}+\cdots\)
420.2.ba.c \(160\) \(3.354\) None None \(0\) \(0\) \(0\) \(0\)
420.2.bb.a \(16\) \(3.354\) \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None None \(0\) \(0\) \(2\) \(0\) \(q-\beta _{6}q^{3}+(\beta _{1}+\beta _{14})q^{5}+(\beta _{2}-\beta _{3}+\cdots)q^{7}+\cdots\)
420.2.bf.a \(128\) \(3.354\) None None \(0\) \(0\) \(0\) \(0\)
420.2.bh.a \(10\) \(3.354\) 10.0.\(\cdots\).1 None None \(0\) \(-1\) \(-5\) \(-5\) \(q+\beta _{5}q^{3}+(-1-\beta _{7})q^{5}+(-1+\beta _{2}+\cdots)q^{7}+\cdots\)
420.2.bh.b \(10\) \(3.354\) 10.0.\(\cdots\).1 None None \(0\) \(1\) \(5\) \(-5\) \(q+\beta _{6}q^{3}+(1+\beta _{7})q^{5}+(-1+\beta _{2}+\beta _{3}+\cdots)q^{7}+\cdots\)
420.2.bi.a \(4\) \(3.354\) \(\Q(\zeta_{12})\) None None \(-4\) \(2\) \(0\) \(-2\) \(q+(-1+\zeta_{12}^{3})q^{2}+(1-\zeta_{12}^{2})q^{3}+\cdots\)
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