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Label Dim. \(A\) Field CM RM Traces Fricke sign $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
512.1.c.a \(2\) \(0.256\) \(\Q(\sqrt{-2}) \) \(\Q(\sqrt{-2}) \) None \(0\) \(0\) \(0\) \(0\) \(q-\beta q^{3}-q^{9}-\beta q^{11}+\beta q^{19}-q^{25}+\cdots\)
512.1.d.a \(2\) \(0.256\) \(\Q(\sqrt{2}) \) \(\Q(\sqrt{-2}) \) None \(0\) \(0\) \(0\) \(0\) \(q-\beta q^{3}+q^{9}+\beta q^{11}+\beta q^{19}+q^{25}+\cdots\)
512.1.f.a \(2\) \(0.256\) \(\Q(\sqrt{-1}) \) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(-2\) \(0\) \(q+(-1-i)q^{5}-iq^{9}+(1-i)q^{13}+\cdots\)
512.1.f.b \(2\) \(0.256\) \(\Q(\sqrt{-1}) \) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(2\) \(0\) \(q+(1+i)q^{5}-iq^{9}+(-1+i)q^{13}+\cdots\)
512.2.a.a \(2\) \(4.088\) \(\Q(\sqrt{2}) \) \(\Q(\sqrt{-2}) \) None \(0\) \(-4\) \(0\) \(0\) \(+\) \(q+(-2+\beta )q^{3}+(3-4\beta )q^{9}+(-2-3\beta )q^{11}+\cdots\)
512.2.a.b \(2\) \(4.088\) \(\Q(\sqrt{2}) \) None None \(0\) \(0\) \(-4\) \(0\) \(+\) \(q+\beta q^{3}-2q^{5}-2\beta q^{7}-q^{9}-3\beta q^{11}+\cdots\)
512.2.a.c \(2\) \(4.088\) \(\Q(\sqrt{2}) \) None None \(0\) \(0\) \(0\) \(-8\) \(+\) \(q+\beta q^{3}-2\beta q^{5}-4q^{7}-q^{9}+\beta q^{11}+\cdots\)
512.2.a.d \(2\) \(4.088\) \(\Q(\sqrt{2}) \) None None \(0\) \(0\) \(0\) \(8\) \(-\) \(q+\beta q^{3}+2\beta q^{5}+4q^{7}-q^{9}+\beta q^{11}+\cdots\)
512.2.a.e \(2\) \(4.088\) \(\Q(\sqrt{2}) \) None None \(0\) \(0\) \(4\) \(0\) \(-\) \(q+\beta q^{3}+2q^{5}+2\beta q^{7}-q^{9}-3\beta q^{11}+\cdots\)
512.2.a.f \(2\) \(4.088\) \(\Q(\sqrt{2}) \) \(\Q(\sqrt{-2}) \) None \(0\) \(4\) \(0\) \(0\) \(-\) \(q+(2+\beta )q^{3}+(3+4\beta )q^{9}+(2-3\beta )q^{11}+\cdots\)
512.2.a.g \(4\) \(4.088\) \(\Q(\zeta_{24})^+\) None None \(0\) \(0\) \(0\) \(0\) \(-\) \(q+\beta _{1}q^{3}-\beta _{2}q^{5}-\beta _{3}q^{7}+3q^{9}+\beta _{1}q^{11}+\cdots\)
512.2.b.a \(2\) \(4.088\) \(\Q(\sqrt{-2}) \) None None \(0\) \(0\) \(0\) \(-8\) \(q+\beta q^{3}-2\beta q^{5}-4q^{7}+q^{9}-\beta q^{11}+\cdots\)
512.2.b.b \(2\) \(4.088\) \(\Q(\sqrt{-2}) \) None None \(0\) \(0\) \(0\) \(8\) \(q+\beta q^{3}+2\beta q^{5}+4q^{7}+q^{9}-\beta q^{11}+\cdots\)
512.2.b.c \(4\) \(4.088\) \(\Q(\zeta_{8})\) \(\Q(\sqrt{-2}) \) None \(0\) \(0\) \(0\) \(0\) \(q+\zeta_{8}q^{3}+(-3+\zeta_{8}^{3})q^{9}-\zeta_{8}^{2}q^{11}+\cdots\)
512.2.b.d \(4\) \(4.088\) \(\Q(\sqrt{2}, \sqrt{-3})\) None None \(0\) \(0\) \(0\) \(0\) \(q-\beta _{1}q^{3}+\beta _{2}q^{5}+\beta _{3}q^{7}-3q^{9}+\beta _{1}q^{11}+\cdots\)
512.2.b.e \(4\) \(4.088\) \(\Q(\zeta_{8})\) None None \(0\) \(0\) \(0\) \(0\) \(q-\zeta_{8}^{2}q^{3}+\zeta_{8}q^{5}-\zeta_{8}^{3}q^{7}+q^{9}+\cdots\)
512.2.e.a \(2\) \(4.088\) \(\Q(\sqrt{-1}) \) None None \(0\) \(-4\) \(-2\) \(0\) \(q+(-2+2i)q^{3}+(-1-i)q^{5}+4iq^{7}+\cdots\)
512.2.e.b \(2\) \(4.088\) \(\Q(\sqrt{-1}) \) None None \(0\) \(-4\) \(2\) \(0\) \(q+(-2+2i)q^{3}+(1+i)q^{5}-4iq^{7}+\cdots\)
512.2.e.c \(2\) \(4.088\) \(\Q(\sqrt{-1}) \) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(-6\) \(0\) \(q+(-3-3i)q^{5}+3iq^{9}+(-1+i)q^{13}+\cdots\)
512.2.e.d \(2\) \(4.088\) \(\Q(\sqrt{-1}) \) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(-2\) \(0\) \(q+(-1-i)q^{5}+3iq^{9}+(5-5i)q^{13}+\cdots\)
512.2.e.e \(2\) \(4.088\) \(\Q(\sqrt{-1}) \) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(2\) \(0\) \(q+(1+i)q^{5}+3iq^{9}+(-5+5i)q^{13}+\cdots\)
512.2.e.f \(2\) \(4.088\) \(\Q(\sqrt{-1}) \) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(6\) \(0\) \(q+(3+3i)q^{5}+3iq^{9}+(1-i)q^{13}+\cdots\)
512.2.e.g \(2\) \(4.088\) \(\Q(\sqrt{-1}) \) None None \(0\) \(4\) \(-2\) \(0\) \(q+(2-2i)q^{3}+(-1-i)q^{5}-4iq^{7}+\cdots\)
512.2.e.h \(2\) \(4.088\) \(\Q(\sqrt{-1}) \) None None \(0\) \(4\) \(2\) \(0\) \(q+(2-2i)q^{3}+(1+i)q^{5}+4iq^{7}-5iq^{9}+\cdots\)
512.2.e.i \(8\) \(4.088\) \(\Q(\zeta_{16})\) None None \(0\) \(0\) \(-8\) \(0\) \(q-\zeta_{16}^{6}q^{3}+(-1+\zeta_{16}^{2}-\zeta_{16}^{4}+\cdots)q^{5}+\cdots\)
512.2.e.j \(8\) \(4.088\) \(\Q(\zeta_{16})\) None None \(0\) \(0\) \(8\) \(0\) \(q-\zeta_{16}^{6}q^{3}+(1-\zeta_{16}^{2}+\zeta_{16}^{4})q^{5}+\cdots\)
512.2.g.a \(4\) \(4.088\) \(\Q(\zeta_{8})\) None None \(0\) \(-4\) \(-8\) \(-4\) \(q+(-1-\zeta_{8}^{3})q^{3}+(-2-2\zeta_{8}+\zeta_{8}^{2}+\cdots)q^{5}+\cdots\)
512.2.g.b \(4\) \(4.088\) \(\Q(\zeta_{8})\) None None \(0\) \(-4\) \(8\) \(4\) \(q+(-1-\zeta_{8}^{3})q^{3}+(2+2\zeta_{8}-\zeta_{8}^{2}+\cdots)q^{5}+\cdots\)
512.2.g.c \(4\) \(4.088\) \(\Q(\zeta_{8})\) None None \(0\) \(4\) \(-8\) \(4\) \(q+(1+\zeta_{8}^{3})q^{3}+(-2-2\zeta_{8}+\zeta_{8}^{2}+\cdots)q^{5}+\cdots\)
512.2.g.d \(4\) \(4.088\) \(\Q(\zeta_{8})\) None None \(0\) \(4\) \(8\) \(-4\) \(q+(1+\zeta_{8}^{3})q^{3}+(2+2\zeta_{8}-\zeta_{8}^{2}+\zeta_{8}^{3})q^{5}+\cdots\)
512.2.g.e \(8\) \(4.088\) 8.0.18939904.2 None None \(0\) \(-4\) \(-8\) \(8\) \(q+(-1+\beta _{4}-\beta _{5})q^{3}+(-1-\beta _{6})q^{5}+\cdots\)
512.2.g.f \(8\) \(4.088\) 8.0.18939904.2 None None \(0\) \(-4\) \(8\) \(-8\) \(q+(-1+\beta _{4}-\beta _{5})q^{3}+(1+\beta _{6})q^{5}+\cdots\)
512.2.g.g \(8\) \(4.088\) 8.0.18939904.2 None None \(0\) \(4\) \(-8\) \(-8\) \(q+(1+\beta _{3}-\beta _{4}+\beta _{7})q^{3}+(-1-\beta _{4}+\cdots)q^{5}+\cdots\)
512.2.g.h \(8\) \(4.088\) 8.0.18939904.2 None None \(0\) \(4\) \(8\) \(8\) \(q+(1+\beta _{3}-\beta _{4}+\beta _{7})q^{3}+(1+\beta _{4})q^{5}+\cdots\)
512.2.i.a \(56\) \(4.088\) None None \(0\) \(-8\) \(8\) \(8\)
512.2.i.b \(56\) \(4.088\) None None \(0\) \(8\) \(8\) \(-8\)
512.2.k.a \(240\) \(4.088\) None None \(0\) \(16\) \(-16\) \(16\)
512.2.o.a \(4032\) \(4.088\) None None \(-64\) \(-64\) \(-64\) \(-64\)
512.3.c.a \(2\) \(13.951\) \(\Q(\sqrt{-2}) \) None None \(0\) \(0\) \(-16\) \(0\) \(q+\beta q^{3}-8q^{5}-8\beta q^{7}+7q^{9}+9\beta q^{11}+\cdots\)
512.3.c.b \(2\) \(13.951\) \(\Q(\sqrt{-2}) \) None None \(0\) \(0\) \(16\) \(0\) \(q+\beta q^{3}+8q^{5}+8\beta q^{7}+7q^{9}+9\beta q^{11}+\cdots\)
512.3.c.c \(4\) \(13.951\) \(\Q(\sqrt{-2}, \sqrt{-3})\) None None \(0\) \(0\) \(-16\) \(0\) \(q+(\beta _{1}+\beta _{2})q^{3}+(-4+\beta _{3})q^{5}+(2\beta _{1}+\cdots)q^{7}+\cdots\)
512.3.c.d \(4\) \(13.951\) \(\Q(\zeta_{8})\) \(\Q(\sqrt{-2}) \) None \(0\) \(0\) \(0\) \(0\) \(q+\zeta_{8}q^{3}+(-9+\zeta_{8}^{3})q^{9}+(-3\zeta_{8}+\cdots)q^{11}+\cdots\)
512.3.c.e \(4\) \(13.951\) \(\Q(\sqrt{-2}, \sqrt{-3})\) None None \(0\) \(0\) \(16\) \(0\) \(q+(\beta _{1}+\beta _{2})q^{3}+(4-\beta _{3})q^{5}+(-2\beta _{1}+\cdots)q^{7}+\cdots\)
512.3.c.f \(8\) \(13.951\) 8.0.40960000.1 None None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{1}q^{3}-\beta _{2}q^{5}+\beta _{7}q^{7}+(-5+\beta _{3}+\cdots)q^{9}+\cdots\)
512.3.c.g \(8\) \(13.951\) 8.0.18939904.2 None None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{5}q^{3}-\beta _{1}q^{5}-\beta _{4}q^{7}+(-1-\beta _{2}+\cdots)q^{9}+\cdots\)
512.3.d.a \(2\) \(13.951\) \(\Q(\sqrt{2}) \) \(\Q(\sqrt{-2}) \) None \(0\) \(-8\) \(0\) \(0\) \(q+(-4+\beta )q^{3}+(9-8\beta )q^{9}+(-12+\cdots)q^{11}+\cdots\)
512.3.d.b \(2\) \(13.951\) \(\Q(\sqrt{2}) \) \(\Q(\sqrt{-2}) \) None \(0\) \(8\) \(0\) \(0\) \(q+(4+\beta )q^{3}+(9+8\beta )q^{9}+(12+7\beta )q^{11}+\cdots\)
512.3.d.c \(4\) \(13.951\) \(\Q(\sqrt{-2}, \sqrt{-5})\) None None \(0\) \(-8\) \(0\) \(0\) \(q+(-2-\beta _{1})q^{3}+\beta _{3}q^{5}+(-\beta _{2}+2\beta _{3})q^{7}+\cdots\)
512.3.d.d \(4\) \(13.951\) \(\Q(\zeta_{8})\) None None \(0\) \(0\) \(0\) \(0\) \(q+\zeta_{8}^{2}q^{3}+\zeta_{8}q^{5}-\zeta_{8}^{3}q^{7}-7q^{9}+\cdots\)
512.3.d.e \(4\) \(13.951\) \(\Q(\sqrt{-2}, \sqrt{-5})\) None None \(0\) \(8\) \(0\) \(0\) \(q+(2-\beta _{1})q^{3}+\beta _{3}q^{5}+(-\beta _{2}-2\beta _{3})q^{7}+\cdots\)
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