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Label Dim. \(A\) Field CM RM Traces Fricke sign $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
315.1.bg.a \(2\) \(0.157\) \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-35}) \) None \(0\) \(-2\) \(1\) \(1\) \(q-q^{3}+\zeta_{6}^{2}q^{4}-\zeta_{6}^{2}q^{5}+\zeta_{6}q^{7}+\cdots\)
315.1.bg.b \(2\) \(0.157\) \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-35}) \) None \(0\) \(2\) \(-1\) \(-1\) \(q+q^{3}+\zeta_{6}^{2}q^{4}+\zeta_{6}^{2}q^{5}-\zeta_{6}q^{7}+\cdots\)
315.2.a.a \(1\) \(2.515\) \(\Q\) None None \(-1\) \(0\) \(-1\) \(1\) \(+\) \(q-q^{2}-q^{4}-q^{5}+q^{7}+3q^{8}+q^{10}+\cdots\)
315.2.a.b \(1\) \(2.515\) \(\Q\) None None \(0\) \(0\) \(1\) \(1\) \(-\) \(q-2q^{4}+q^{5}+q^{7}+3q^{11}+5q^{13}+\cdots\)
315.2.a.c \(2\) \(2.515\) \(\Q(\sqrt{2}) \) None None \(-2\) \(0\) \(-2\) \(-2\) \(+\) \(q+(-1+\beta )q^{2}+(1-2\beta )q^{4}-q^{5}-q^{7}+\cdots\)
315.2.a.d \(2\) \(2.515\) \(\Q(\sqrt{5}) \) None None \(0\) \(0\) \(2\) \(2\) \(-\) \(q-\beta q^{2}+3q^{4}+q^{5}+q^{7}-\beta q^{8}-\beta q^{10}+\cdots\)
315.2.a.e \(2\) \(2.515\) \(\Q(\sqrt{17}) \) None None \(1\) \(0\) \(-2\) \(-2\) \(-\) \(q+\beta q^{2}+(2+\beta )q^{4}-q^{5}-q^{7}+(4+\beta )q^{8}+\cdots\)
315.2.a.f \(2\) \(2.515\) \(\Q(\sqrt{2}) \) None None \(2\) \(0\) \(2\) \(-2\) \(-\) \(q+(1+\beta )q^{2}+(1+2\beta )q^{4}+q^{5}-q^{7}+\cdots\)
315.2.b.a \(4\) \(2.515\) \(\Q(\sqrt{-2}, \sqrt{3})\) None None \(0\) \(0\) \(-4\) \(4\) \(q+\beta _{1}q^{2}+\beta _{2}q^{4}-q^{5}+(1+\beta _{1}+\beta _{3})q^{7}+\cdots\)
315.2.b.b \(4\) \(2.515\) \(\Q(\sqrt{-2}, \sqrt{3})\) None None \(0\) \(0\) \(4\) \(4\) \(q+\beta _{1}q^{2}+\beta _{2}q^{4}+q^{5}+(1-\beta _{1}-\beta _{3})q^{7}+\cdots\)
315.2.d.a \(2\) \(2.515\) \(\Q(\sqrt{-1}) \) None None \(0\) \(0\) \(4\) \(0\) \(q+2iq^{2}-2q^{4}+(2-i)q^{5}+iq^{7}+\cdots\)
315.2.d.b \(2\) \(2.515\) \(\Q(\sqrt{-1}) \) None None \(0\) \(0\) \(-4\) \(0\) \(q+iq^{2}+q^{4}+(-2+i)q^{5}+iq^{7}+\cdots\)
315.2.d.c \(2\) \(2.515\) \(\Q(\sqrt{-1}) \) None None \(0\) \(0\) \(-2\) \(0\) \(q+iq^{2}+q^{4}+(-1-2i)q^{5}-iq^{7}+\cdots\)
315.2.d.d \(2\) \(2.515\) \(\Q(\sqrt{-1}) \) None None \(0\) \(0\) \(4\) \(0\) \(q+iq^{2}+q^{4}+(2+i)q^{5}-iq^{7}+3iq^{8}+\cdots\)
315.2.d.e \(6\) \(2.515\) 6.0.350464.1 None None \(0\) \(0\) \(-2\) \(0\) \(q-\beta _{1}q^{2}+(-1+\beta _{3}+\beta _{5})q^{4}+(-\beta _{1}+\cdots)q^{5}+\cdots\)
315.2.g.a \(16\) \(2.515\) \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{3}q^{2}+(1+\beta _{14})q^{4}-\beta _{4}q^{5}-\beta _{12}q^{7}+\cdots\)
315.2.i.a \(2\) \(2.515\) \(\Q(\sqrt{-3}) \) None None \(-2\) \(-3\) \(1\) \(-1\) \(q+(-2+2\zeta_{6})q^{2}+(-1-\zeta_{6})q^{3}-2\zeta_{6}q^{4}+\cdots\)
315.2.i.b \(2\) \(2.515\) \(\Q(\sqrt{-3}) \) None None \(0\) \(3\) \(1\) \(-1\) \(q+(2-\zeta_{6})q^{3}+2\zeta_{6}q^{4}+\zeta_{6}q^{5}+(-1+\cdots)q^{7}+\cdots\)
315.2.i.c \(8\) \(2.515\) 8.0.142635249.1 None None \(1\) \(-1\) \(-4\) \(-4\) \(q+\beta _{7}q^{2}+(-\beta _{1}+\beta _{2}+\beta _{4}+\beta _{6})q^{3}+\cdots\)
315.2.i.d \(8\) \(2.515\) \(\Q(\zeta_{15})\) None None \(1\) \(3\) \(4\) \(4\) \(q+(1+\zeta_{15}-\zeta_{15}^{2}+\zeta_{15}^{3})q^{2}+(1+\cdots)q^{3}+\cdots\)
315.2.i.e \(12\) \(2.515\) \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None None \(3\) \(1\) \(6\) \(-6\) \(q+(1-\beta _{2}-\beta _{4})q^{2}+\beta _{9}q^{3}+(-\beta _{1}+\cdots)q^{4}+\cdots\)
315.2.i.f \(16\) \(2.515\) \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None None \(1\) \(1\) \(-8\) \(8\) \(q+(\beta _{2}+\beta _{13})q^{2}+\beta _{5}q^{3}+(-2-2\beta _{3}+\cdots)q^{4}+\cdots\)
315.2.j.a \(2\) \(2.515\) \(\Q(\sqrt{-3}) \) None None \(-2\) \(0\) \(1\) \(-1\) \(q-2\zeta_{6}q^{2}+(-2+2\zeta_{6})q^{4}+\zeta_{6}q^{5}+\cdots\)
315.2.j.b \(2\) \(2.515\) \(\Q(\sqrt{-3}) \) None None \(0\) \(0\) \(-1\) \(5\) \(q+(2-2\zeta_{6})q^{4}-\zeta_{6}q^{5}+(2+\zeta_{6})q^{7}+\cdots\)
315.2.j.c \(4\) \(2.515\) \(\Q(\zeta_{12})\) None None \(-2\) \(0\) \(2\) \(0\) \(q+(\zeta_{12}-\zeta_{12}^{2}+\zeta_{12}^{3})q^{2}+(-2+2\zeta_{12}+\cdots)q^{4}+\cdots\)
315.2.j.d \(4\) \(2.515\) \(\Q(\sqrt{2}, \sqrt{-3})\) None None \(0\) \(0\) \(-2\) \(-2\) \(q+\beta _{1}q^{2}+(-1-\beta _{2})q^{5}+(-1-\beta _{1}+\cdots)q^{7}+\cdots\)
315.2.j.e \(4\) \(2.515\) \(\Q(\sqrt{2}, \sqrt{-3})\) None None \(2\) \(0\) \(2\) \(2\) \(q+(1+\beta _{1}+\beta _{2})q^{2}+(2\beta _{1}+\beta _{2}+2\beta _{3})q^{4}+\cdots\)
315.2.j.f \(6\) \(2.515\) 6.0.4406832.1 None None \(-2\) \(0\) \(-3\) \(1\) \(q+(-\beta _{1}+\beta _{2}-\beta _{4})q^{2}+(-2+\beta _{1}+\cdots)q^{4}+\cdots\)
315.2.j.g \(6\) \(2.515\) 6.0.4406832.1 None None \(2\) \(0\) \(3\) \(1\) \(q+(\beta _{1}-\beta _{2}+\beta _{4})q^{2}+(-2+\beta _{1}+\beta _{3}+\cdots)q^{4}+\cdots\)
315.2.k.a \(4\) \(2.515\) \(\Q(\sqrt{-3}, \sqrt{13})\) None None \(1\) \(0\) \(-4\) \(-8\) \(q+\beta _{1}q^{2}+(1+2\beta _{2})q^{3}+(-1+\beta _{1}+\cdots)q^{4}+\cdots\)
315.2.k.b \(24\) \(2.515\) None None \(-1\) \(1\) \(-24\) \(7\)
315.2.k.c \(36\) \(2.515\) None None \(0\) \(-1\) \(36\) \(-1\)
315.2.l.a \(4\) \(2.515\) \(\Q(\sqrt{-3}, \sqrt{13})\) None None \(-2\) \(6\) \(2\) \(10\) \(q+(-1+\beta _{3})q^{2}+(1-\beta _{2})q^{3}+(2-\beta _{3})q^{4}+\cdots\)
315.2.l.b \(24\) \(2.515\) None None \(2\) \(-5\) \(12\) \(-11\)
315.2.l.c \(36\) \(2.515\) None None \(0\) \(-1\) \(-18\) \(-1\)
315.2.m.a \(12\) \(2.515\) \(\mathbb{Q}[x]/(x^{12} + \cdots)\) None None \(0\) \(0\) \(-4\) \(0\) \(q+\beta _{2}q^{2}+(\beta _{5}+\beta _{6}-\beta _{8})q^{4}+(-\beta _{3}+\cdots)q^{5}+\cdots\)
315.2.m.b \(12\) \(2.515\) \(\mathbb{Q}[x]/(x^{12} + \cdots)\) None None \(0\) \(0\) \(4\) \(0\) \(q+(\beta _{1}+\beta _{8}+\beta _{9})q^{2}+(-\beta _{5}-\beta _{6}+\beta _{8}+\cdots)q^{4}+\cdots\)
315.2.p.a \(4\) \(2.515\) \(\Q(i, \sqrt{10})\) None None \(0\) \(0\) \(-8\) \(4\) \(q+\beta _{1}q^{2}+3\beta _{2}q^{4}+(-2+\beta _{2})q^{5}+\cdots\)
315.2.p.b \(4\) \(2.515\) \(\Q(i, \sqrt{10})\) None None \(0\) \(0\) \(8\) \(4\) \(q+\beta _{1}q^{2}+3\beta _{2}q^{4}+(2-\beta _{2})q^{5}+(1+\cdots)q^{7}+\cdots\)
315.2.p.c \(4\) \(2.515\) \(\Q(i, \sqrt{10})\) None None \(4\) \(0\) \(0\) \(4\) \(q+(1+\beta _{2})q^{2}+\beta _{1}q^{5}+(1+\beta _{2}+\beta _{3})q^{7}+\cdots\)
315.2.p.d \(8\) \(2.515\) 8.0.40960000.1 None None \(0\) \(0\) \(0\) \(-8\) \(q+\beta _{1}q^{2}-\beta _{3}q^{4}+\beta _{5}q^{5}+(-1-\beta _{3}+\cdots)q^{7}+\cdots\)
315.2.p.e \(16\) \(2.515\) \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None None \(0\) \(0\) \(0\) \(-8\) \(q-\beta _{2}q^{2}+(-\beta _{6}-\beta _{7}+\beta _{13})q^{4}+(-\beta _{1}+\cdots)q^{5}+\cdots\)
315.2.r.a \(4\) \(2.515\) \(\Q(\zeta_{12})\) None None \(0\) \(0\) \(-2\) \(0\) \(q+\zeta_{12}^{3}q^{2}+(\zeta_{12}+\zeta_{12}^{3})q^{3}+q^{4}+\cdots\)
315.2.r.b \(84\) \(2.515\) None None \(0\) \(0\) \(3\) \(0\)
315.2.t.a \(2\) \(2.515\) \(\Q(\sqrt{-3}) \) None None \(0\) \(-3\) \(1\) \(4\) \(q+(1-2\zeta_{6})q^{2}+(-2+\zeta_{6})q^{3}-q^{4}+\cdots\)
315.2.t.b \(30\) \(2.515\) None None \(0\) \(4\) \(15\) \(-3\)
315.2.t.c \(32\) \(2.515\) None None \(0\) \(-1\) \(-16\) \(1\)
315.2.u.a \(88\) \(2.515\) None None \(0\) \(0\) \(-6\) \(0\)
315.2.z.a \(8\) \(2.515\) 8.0.\(\cdots\).2 \(\Q(\sqrt{-35}) \) None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{5}q^{3}+(2+2\beta _{4})q^{4}+(\beta _{1}-\beta _{6})q^{5}+\cdots\)
315.2.z.b \(80\) \(2.515\) None None \(0\) \(0\) \(0\) \(0\)
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