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Label Dim. \(A\) Field CM RM Traces Fricke sign $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
1568.1.c.a \(4\) \(0.783\) 4.0.2048.2 \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) \(q-\beta _{1}q^{5}+q^{9}+\beta _{3}q^{13}-\beta _{3}q^{17}+\cdots\)
1568.1.d.a \(2\) \(0.783\) \(\Q(\sqrt{-1}) \) None None \(0\) \(0\) \(-2\) \(0\) \(q-iq^{3}-q^{5}-iq^{11}+iq^{15}-q^{17}+\cdots\)
1568.1.d.b \(2\) \(0.783\) \(\Q(\sqrt{-1}) \) None None \(0\) \(0\) \(2\) \(0\) \(q-iq^{3}+q^{5}+iq^{11}-iq^{15}+q^{17}+\cdots\)
1568.1.g.a \(1\) \(0.783\) \(\Q\) \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{-2}) \) \(\Q(\sqrt{14}) \) \(0\) \(0\) \(0\) \(0\) \(q-q^{9}+2q^{11}+q^{25}+2q^{43}-2q^{67}+\cdots\)
1568.1.g.b \(2\) \(0.783\) \(\Q(\sqrt{2}) \) \(\Q(\sqrt{-2}) \) None \(0\) \(0\) \(0\) \(0\) \(q-\beta q^{3}+q^{9}-\beta q^{17}+\beta q^{19}+q^{25}+\cdots\)
1568.1.n.a \(2\) \(0.783\) \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{-14}) \) \(\Q(\sqrt{2}) \) \(0\) \(0\) \(0\) \(0\) \(q-\zeta_{6}^{2}q^{9}-\zeta_{6}^{2}q^{23}+\zeta_{6}q^{25}+q^{71}+\cdots\)
1568.1.o.a \(2\) \(0.783\) \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{-2}) \) \(\Q(\sqrt{14}) \) \(0\) \(0\) \(0\) \(0\) \(q-\zeta_{6}^{2}q^{9}-\zeta_{6}q^{11}-\zeta_{6}q^{25}+q^{43}+\cdots\)
1568.1.o.b \(4\) \(0.783\) \(\Q(\sqrt{2}, \sqrt{-3})\) \(\Q(\sqrt{-2}) \) None \(0\) \(0\) \(0\) \(0\) \(q+(-\beta _{1}-\beta _{3})q^{3}+(-1-\beta _{2})q^{9}+(-\beta _{1}+\cdots)q^{17}+\cdots\)
1568.1.r.a \(4\) \(0.783\) \(\Q(\zeta_{12})\) None None \(0\) \(0\) \(2\) \(0\) \(q-\zeta_{12}q^{3}+\zeta_{12}^{2}q^{5}-\zeta_{12}q^{11}-\zeta_{12}^{3}q^{15}+\cdots\)
1568.1.s.a \(8\) \(0.783\) 8.0.339738624.1 \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) \(q+(\beta _{3}+\beta _{7})q^{5}+\beta _{4}q^{9}+(\beta _{1}-\beta _{5})q^{13}+\cdots\)
1568.1.x.a \(4\) \(0.783\) \(\Q(\zeta_{8})\) \(\Q(\sqrt{-7}) \) None \(0\) \(0\) \(0\) \(0\) \(q-\zeta_{8}^{3}q^{2}-\zeta_{8}^{2}q^{4}-\zeta_{8}q^{8}+\zeta_{8}q^{9}+\cdots\)
1568.1.bl.a \(8\) \(0.783\) \(\Q(\zeta_{24})\) \(\Q(\sqrt{-7}) \) None \(0\) \(0\) \(0\) \(0\) \(q-\zeta_{24}^{5}q^{2}+\zeta_{24}^{10}q^{4}+\zeta_{24}^{3}q^{8}+\cdots\)
1568.1.bo.a \(8\) \(0.783\) \(\Q(\zeta_{24})\) \(\Q(\sqrt{-7}) \) None \(0\) \(0\) \(0\) \(0\) \(q+\zeta_{24}^{7}q^{2}-\zeta_{24}^{2}q^{4}-\zeta_{24}^{9}q^{8}+\cdots\)
1568.2.a.a \(1\) \(12.521\) \(\Q\) None None \(0\) \(-2\) \(-2\) \(0\) \(-\) \(q-2q^{3}-2q^{5}+q^{9}+4q^{11}-6q^{13}+\cdots\)
1568.2.a.b \(1\) \(12.521\) \(\Q\) None None \(0\) \(-2\) \(0\) \(0\) \(-\) \(q-2q^{3}+q^{9}+4q^{11}+4q^{13}+2q^{17}+\cdots\)
1568.2.a.c \(1\) \(12.521\) \(\Q\) None None \(0\) \(-2\) \(2\) \(0\) \(+\) \(q-2q^{3}+2q^{5}+q^{9}-4q^{11}+6q^{13}+\cdots\)
1568.2.a.d \(1\) \(12.521\) \(\Q\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(-4\) \(0\) \(-\) \(q-4q^{5}-3q^{9}+4q^{13}-8q^{17}+11q^{25}+\cdots\)
1568.2.a.e \(1\) \(12.521\) \(\Q\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(2\) \(0\) \(+\) \(q+2q^{5}-3q^{9}-6q^{13}-2q^{17}-q^{25}+\cdots\)
1568.2.a.f \(1\) \(12.521\) \(\Q\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(4\) \(0\) \(-\) \(q+4q^{5}-3q^{9}-4q^{13}+8q^{17}+11q^{25}+\cdots\)
1568.2.a.g \(1\) \(12.521\) \(\Q\) None None \(0\) \(2\) \(-2\) \(0\) \(+\) \(q+2q^{3}-2q^{5}+q^{9}-4q^{11}-6q^{13}+\cdots\)
1568.2.a.h \(1\) \(12.521\) \(\Q\) None None \(0\) \(2\) \(0\) \(0\) \(-\) \(q+2q^{3}+q^{9}-4q^{11}+4q^{13}+2q^{17}+\cdots\)
1568.2.a.i \(1\) \(12.521\) \(\Q\) None None \(0\) \(2\) \(2\) \(0\) \(-\) \(q+2q^{3}+2q^{5}+q^{9}+4q^{11}+6q^{13}+\cdots\)
1568.2.a.j \(2\) \(12.521\) \(\Q(\sqrt{2}) \) None None \(0\) \(-2\) \(-2\) \(0\) \(+\) \(q+(-1+\beta )q^{3}+(-1+2\beta )q^{5}-2\beta q^{9}+\cdots\)
1568.2.a.k \(2\) \(12.521\) \(\Q(\sqrt{5}) \) None None \(0\) \(-2\) \(-2\) \(0\) \(+\) \(q+(-1-\beta )q^{3}+(-1+\beta )q^{5}+(3+2\beta )q^{9}+\cdots\)
1568.2.a.l \(2\) \(12.521\) \(\Q(\sqrt{2}) \) None None \(0\) \(-2\) \(2\) \(0\) \(+\) \(q+(-1+\beta )q^{3}+(1-2\beta )q^{5}-2\beta q^{9}+\cdots\)
1568.2.a.m \(2\) \(12.521\) \(\Q(\sqrt{7}) \) None None \(0\) \(0\) \(-6\) \(0\) \(+\) \(q+\beta q^{3}-3q^{5}+4q^{9}-\beta q^{11}-4q^{13}+\cdots\)
1568.2.a.n \(2\) \(12.521\) \(\Q(\sqrt{3}) \) None None \(0\) \(0\) \(-2\) \(0\) \(+\) \(q+\beta q^{3}-q^{5}-3\beta q^{11}-\beta q^{15}-5q^{17}+\cdots\)
1568.2.a.o \(2\) \(12.521\) \(\Q(\sqrt{2}) \) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) \(+\) \(q+\beta q^{5}-3q^{9}-\beta q^{13}-5\beta q^{17}-3q^{25}+\cdots\)
1568.2.a.p \(2\) \(12.521\) \(\Q(\sqrt{2}) \) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) \(-\) \(q+3\beta q^{5}-3q^{9}+5\beta q^{13}-3\beta q^{17}+\cdots\)
1568.2.a.q \(2\) \(12.521\) \(\Q(\sqrt{2}) \) None None \(0\) \(0\) \(0\) \(0\) \(+\) \(q+\beta q^{3}-q^{9}-2q^{11}-2\beta q^{13}-3\beta q^{17}+\cdots\)
1568.2.a.r \(2\) \(12.521\) \(\Q(\sqrt{2}) \) None None \(0\) \(0\) \(0\) \(0\) \(-\) \(q+\beta q^{3}-q^{9}+2q^{11}+2\beta q^{13}+3\beta q^{17}+\cdots\)
1568.2.a.s \(2\) \(12.521\) \(\Q(\sqrt{3}) \) None None \(0\) \(0\) \(2\) \(0\) \(-\) \(q+\beta q^{3}+q^{5}+3\beta q^{11}+\beta q^{15}+5q^{17}+\cdots\)
1568.2.a.t \(2\) \(12.521\) \(\Q(\sqrt{7}) \) None None \(0\) \(0\) \(6\) \(0\) \(-\) \(q+\beta q^{3}+3q^{5}+4q^{9}+\beta q^{11}+4q^{13}+\cdots\)
1568.2.a.u \(2\) \(12.521\) \(\Q(\sqrt{2}) \) None None \(0\) \(2\) \(-2\) \(0\) \(-\) \(q+(1+\beta )q^{3}+(-1-2\beta )q^{5}+2\beta q^{9}+\cdots\)
1568.2.a.v \(2\) \(12.521\) \(\Q(\sqrt{5}) \) None None \(0\) \(2\) \(-2\) \(0\) \(-\) \(q+(1+\beta )q^{3}+(-1+\beta )q^{5}+(3+2\beta )q^{9}+\cdots\)
1568.2.a.w \(2\) \(12.521\) \(\Q(\sqrt{2}) \) None None \(0\) \(2\) \(2\) \(0\) \(-\) \(q+(1+\beta )q^{3}+(1+2\beta )q^{5}+2\beta q^{9}+(-1+\cdots)q^{11}+\cdots\)
1568.2.a.x \(4\) \(12.521\) \(\Q(\sqrt{2}, \sqrt{5})\) None None \(0\) \(0\) \(0\) \(0\) \(-\) \(q-\beta _{2}q^{3}-2\beta _{1}q^{5}+7q^{9}-\beta _{3}q^{11}+\cdots\)
1568.2.b.a \(2\) \(12.521\) \(\Q(\sqrt{-2}) \) None None \(0\) \(0\) \(0\) \(0\) \(q+\beta q^{3}+\beta q^{5}+q^{9}-2\beta q^{11}+3\beta q^{13}+\cdots\)
1568.2.b.b \(2\) \(12.521\) \(\Q(\sqrt{-7}) \) \(\Q(\sqrt{-7}) \) None \(0\) \(0\) \(0\) \(0\) \(q+3q^{9}-\beta q^{11}-8q^{23}+5q^{25}-2\beta q^{29}+\cdots\)
1568.2.b.c \(4\) \(12.521\) 4.0.7168.1 \(\Q(\sqrt{-14}) \) None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{1}q^{3}+\beta _{2}q^{5}+(-3+\beta _{3})q^{9}+(-2\beta _{1}+\cdots)q^{13}+\cdots\)
1568.2.b.d \(4\) \(12.521\) 4.0.2312.1 None None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{1}q^{3}+\beta _{2}q^{5}+(-2+\beta _{3})q^{9}+(\beta _{1}+\cdots)q^{11}+\cdots\)
1568.2.b.e \(6\) \(12.521\) 6.0.1142512.1 None None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{3}q^{3}+(-\beta _{1}-\beta _{3})q^{5}+(-\beta _{2}+\beta _{5})q^{9}+\cdots\)
1568.2.b.f \(6\) \(12.521\) 6.0.1142512.1 None None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{3}q^{3}+(-\beta _{1}-\beta _{3})q^{5}+(-\beta _{2}+\beta _{5})q^{9}+\cdots\)
1568.2.b.g \(12\) \(12.521\) \(\mathbb{Q}[x]/(x^{12} + \cdots)\) None None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{9}q^{3}+\beta _{11}q^{5}+(-1-\beta _{4})q^{9}+\cdots\)
1568.2.e.a \(4\) \(12.521\) 4.0.2048.2 \(\Q(\sqrt{-2}) \) None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{1}q^{3}+(-7-\beta _{2})q^{9}+(2-3\beta _{2}+\cdots)q^{11}+\cdots\)
1568.2.e.b \(4\) \(12.521\) 4.0.2048.2 \(\Q(\sqrt{-2}) \) None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{1}q^{3}+(1+\beta _{2})q^{9}+(-2+3\beta _{2}+\cdots)q^{11}+\cdots\)
1568.2.e.c \(8\) \(12.521\) 8.0.339738624.1 None None \(0\) \(0\) \(0\) \(0\) \(q+(\beta _{2}-\beta _{4})q^{3}+\beta _{3}q^{5}+(-1+2\beta _{1}+\cdots)q^{9}+\cdots\)
1568.2.e.d \(8\) \(12.521\) 8.0.\(\cdots\).10 None None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{1}q^{3}+\beta _{6}q^{5}+(1-\beta _{3})q^{9}+3\beta _{3}q^{11}+\cdots\)
1568.2.e.e \(12\) \(12.521\) 12.0.\(\cdots\).2 None None \(0\) \(0\) \(0\) \(0\) \(q-\beta _{9}q^{3}+\beta _{3}q^{5}-\beta _{2}q^{9}+(-1+\beta _{1}+\cdots)q^{11}+\cdots\)
1568.2.f.a \(8\) \(12.521\) \(\Q(\zeta_{16})\) None None \(0\) \(0\) \(0\) \(0\) \(q+\zeta_{16}^{7}q^{3}+(-\zeta_{16}^{2}+\zeta_{16}^{4})q^{5}+\cdots\)
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