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Label Char Prim Dim $A$ Field CM Traces Fricke sign Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
15.2.a.a 15.a 1.a $1$ $0.120$ \(\Q\) None \(-1\) \(-1\) \(1\) \(0\) $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}-q^{4}+q^{5}+q^{6}+3q^{8}+\cdots\)
15.3.c.a 15.c 3.b $2$ $0.409$ \(\Q(\sqrt{-5}) \) None \(0\) \(-4\) \(0\) \(-12\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta q^{2}+(-2-\beta )q^{3}-q^{4}-\beta q^{5}+\cdots\)
15.3.d.a 15.d 15.d $1$ $0.409$ \(\Q\) \(\Q(\sqrt{-15}) \) \(-1\) \(3\) \(-5\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q-q^{2}+3q^{3}-3q^{4}-5q^{5}-3q^{6}+\cdots\)
15.3.d.b 15.d 15.d $1$ $0.409$ \(\Q\) \(\Q(\sqrt{-15}) \) \(1\) \(-3\) \(5\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+q^{2}-3q^{3}-3q^{4}+5q^{5}-3q^{6}+\cdots\)
15.3.f.a 15.f 5.c $4$ $0.409$ \(\Q(i, \sqrt{6})\) None \(-4\) \(0\) \(-4\) \(4\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-1+\beta _{1}-\beta _{2})q^{2}+\beta _{3}q^{3}+(-2\beta _{1}+\cdots)q^{4}+\cdots\)
15.4.a.a 15.a 1.a $1$ $0.885$ \(\Q\) None \(1\) \(3\) \(5\) \(-24\) $+$ $\mathrm{SU}(2)$ \(q+q^{2}+3q^{3}-7q^{4}+5q^{5}+3q^{6}+\cdots\)
15.4.a.b 15.a 1.a $1$ $0.885$ \(\Q\) None \(3\) \(-3\) \(-5\) \(20\) $+$ $\mathrm{SU}(2)$ \(q+3q^{2}-3q^{3}+q^{4}-5q^{5}-9q^{6}+\cdots\)
15.4.b.a 15.b 5.b $4$ $0.885$ \(\Q(i, \sqrt{41})\) None \(0\) \(0\) \(6\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{1}q^{2}+\beta _{2}q^{3}+(-5+\beta _{3})q^{4}+(2+\cdots)q^{5}+\cdots\)
15.4.e.a 15.e 15.e $8$ $0.885$ 8.0.\(\cdots\).8 None \(0\) \(-6\) \(0\) \(-16\) $\mathrm{SU}(2)[C_{4}]$ \(q-\beta _{3}q^{2}+(-1-\beta _{2}+\beta _{5})q^{3}+(\beta _{2}+\cdots)q^{4}+\cdots\)
15.5.c.a 15.c 3.b $6$ $1.551$ \(\mathbb{Q}[x]/(x^{6} + \cdots)\) None \(0\) \(8\) \(0\) \(76\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{2}+(1-\beta _{2})q^{3}+(-8+\beta _{2}+\beta _{4}+\cdots)q^{4}+\cdots\)
15.5.d.a 15.d 15.d $1$ $1.551$ \(\Q\) \(\Q(\sqrt{-15}) \) \(-7\) \(9\) \(25\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q-7q^{2}+9q^{3}+33q^{4}+5^{2}q^{5}-63q^{6}+\cdots\)
15.5.d.b 15.d 15.d $1$ $1.551$ \(\Q\) \(\Q(\sqrt{-15}) \) \(7\) \(-9\) \(-25\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+7q^{2}-9q^{3}+33q^{4}-5^{2}q^{5}-63q^{6}+\cdots\)
15.5.d.c 15.d 15.d $4$ $1.551$ \(\Q(\sqrt{10}, \sqrt{-26})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{2}q^{2}+(\beta _{1}+\beta _{2})q^{3}-6q^{4}+(2\beta _{2}+\cdots)q^{5}+\cdots\)
15.5.f.a 15.f 5.c $8$ $1.551$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(0\) \(0\) \(-84\) \(20\) $\mathrm{SU}(2)[C_{4}]$ \(q+\beta _{1}q^{2}+\beta _{5}q^{3}+(-\beta _{1}-12\beta _{2}-2\beta _{3}+\cdots)q^{4}+\cdots\)
15.6.a.a 15.a 1.a $1$ $2.406$ \(\Q\) None \(-2\) \(-9\) \(-25\) \(-132\) $+$ $\mathrm{SU}(2)$ \(q-2q^{2}-9q^{3}-28q^{4}-5^{2}q^{5}+18q^{6}+\cdots\)
15.6.a.b 15.a 1.a $1$ $2.406$ \(\Q\) None \(7\) \(9\) \(-25\) \(12\) $-$ $\mathrm{SU}(2)$ \(q+7q^{2}+9q^{3}+17q^{4}-5^{2}q^{5}+63q^{6}+\cdots\)
15.6.a.c 15.a 1.a $2$ $2.406$ \(\Q(\sqrt{409}) \) None \(-1\) \(-18\) \(50\) \(-112\) $-$ $\mathrm{SU}(2)$ \(q-\beta q^{2}-9q^{3}+(70+\beta )q^{4}+5^{2}q^{5}+\cdots\)
15.6.b.a 15.b 5.b $4$ $2.406$ \(\Q(i, \sqrt{89})\) None \(0\) \(0\) \(120\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{2}+(\beta _{1}+\beta _{2})q^{3}+(-11+\beta _{3})q^{4}+\cdots\)
15.6.e.a 15.e 15.e $16$ $2.406$ \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None \(0\) \(0\) \(0\) \(-80\) $\mathrm{SU}(2)[C_{4}]$ \(q-\beta _{7}q^{2}-\beta _{4}q^{3}+(-13\beta _{3}-\beta _{6})q^{4}+\cdots\)
15.7.c.a 15.c 3.b $8$ $3.451$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(0\) \(20\) \(0\) \(160\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{2}q^{2}+(2-\beta _{5})q^{3}+(-41-\beta _{3}+\cdots)q^{4}+\cdots\)
15.7.d.a 15.d 15.d $1$ $3.451$ \(\Q\) \(\Q(\sqrt{-15}) \) \(-11\) \(-27\) \(125\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q-11q^{2}-3^{3}q^{3}+57q^{4}+5^{3}q^{5}+\cdots\)
15.7.d.b 15.d 15.d $1$ $3.451$ \(\Q\) \(\Q(\sqrt{-15}) \) \(11\) \(27\) \(-125\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+11q^{2}+3^{3}q^{3}+57q^{4}-5^{3}q^{5}+\cdots\)
15.7.d.c 15.d 15.d $8$ $3.451$ \(\mathbb{Q}[x]/(x^{8} + \cdots)\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{5}q^{2}+(\beta _{1}+\beta _{3}+\beta _{5})q^{3}+(9+\beta _{2}+\cdots)q^{4}+\cdots\)
15.7.f.a 15.f 5.c $12$ $3.451$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(16\) \(0\) \(136\) \(-696\) $\mathrm{SU}(2)[C_{4}]$ \(q+(1+\beta _{1}-\beta _{4})q^{2}-\beta _{6}q^{3}+(18\beta _{1}+\cdots)q^{4}+\cdots\)
15.8.a.a 15.a 1.a $1$ $4.686$ \(\Q\) None \(-22\) \(27\) \(-125\) \(-420\) $-$ $\mathrm{SU}(2)$ \(q-22q^{2}+3^{3}q^{3}+356q^{4}-5^{3}q^{5}+\cdots\)
15.8.a.b 15.a 1.a $1$ $4.686$ \(\Q\) None \(-13\) \(-27\) \(-125\) \(1380\) $+$ $\mathrm{SU}(2)$ \(q-13q^{2}-3^{3}q^{3}+41q^{4}-5^{3}q^{5}+\cdots\)
15.8.a.c 15.a 1.a $2$ $4.686$ \(\Q(\sqrt{601}) \) None \(7\) \(54\) \(250\) \(1304\) $+$ $\mathrm{SU}(2)$ \(q+(4-\beta )q^{2}+3^{3}q^{3}+(38-7\beta )q^{4}+\cdots\)
15.8.b.a 15.b 5.b $8$ $4.686$ \(\mathbb{Q}[x]/(x^{8} + \cdots)\) None \(0\) \(0\) \(-444\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{1}q^{2}+\beta _{2}q^{3}+(-83-\beta _{4})q^{4}+\cdots\)
15.8.e.a 15.e 15.e $24$ $4.686$ None \(0\) \(24\) \(0\) \(1344\) $\mathrm{SU}(2)[C_{4}]$
15.9.c.a 15.c 3.b $10$ $6.111$ \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None \(0\) \(-112\) \(0\) \(7156\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{1}q^{2}+(-11-2\beta _{1}-\beta _{2})q^{3}+(-79+\cdots)q^{4}+\cdots\)
15.9.d.a 15.d 15.d $1$ $6.111$ \(\Q\) \(\Q(\sqrt{-15}) \) \(-17\) \(-81\) \(-625\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q-17q^{2}-3^{4}q^{3}+33q^{4}-5^{4}q^{5}+\cdots\)
15.9.d.b 15.d 15.d $1$ $6.111$ \(\Q\) \(\Q(\sqrt{-15}) \) \(17\) \(81\) \(625\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+17q^{2}+3^{4}q^{3}+33q^{4}+5^{4}q^{5}+\cdots\)
15.9.d.c 15.d 15.d $12$ $6.111$ \(\mathbb{Q}[x]/(x^{12} + \cdots)\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{2}q^{2}+(\beta _{2}-\beta _{3})q^{3}+(142-\beta _{1}+\cdots)q^{4}+\cdots\)
15.9.f.a 15.f 5.c $16$ $6.111$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(0\) \(0\) \(-444\) \(4540\) $\mathrm{SU}(2)[C_{4}]$ \(q+\beta _{2}q^{2}-\beta _{6}q^{3}+(158\beta _{1}-3\beta _{2}+3\beta _{3}+\cdots)q^{4}+\cdots\)
15.10.a.a 15.a 1.a $1$ $7.726$ \(\Q\) None \(-4\) \(81\) \(625\) \(-7680\) $+$ $\mathrm{SU}(2)$ \(q-4q^{2}+3^{4}q^{3}-496q^{4}+5^{4}q^{5}+\cdots\)
15.10.a.b 15.a 1.a $1$ $7.726$ \(\Q\) None \(22\) \(-81\) \(-625\) \(-5988\) $+$ $\mathrm{SU}(2)$ \(q+22q^{2}-3^{4}q^{3}-28q^{4}-5^{4}q^{5}+\cdots\)
15.10.a.c 15.a 1.a $2$ $7.726$ \(\Q(\sqrt{4729}) \) None \(19\) \(162\) \(-1250\) \(-11872\) $-$ $\mathrm{SU}(2)$ \(q+(10-\beta )q^{2}+3^{4}q^{3}+(770-19\beta )q^{4}+\cdots\)
15.10.a.d 15.a 1.a $2$ $7.726$ \(\Q(\sqrt{241}) \) None \(31\) \(-162\) \(1250\) \(14112\) $-$ $\mathrm{SU}(2)$ \(q+(15-\beta )q^{2}-3^{4}q^{3}+(255-31\beta )q^{4}+\cdots\)
15.10.b.a 15.b 5.b $8$ $7.726$ \(\mathbb{Q}[x]/(x^{8} + \cdots)\) None \(0\) \(0\) \(-690\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{3}q^{2}+\beta _{4}q^{3}+(-149-\beta _{1})q^{4}+\cdots\)
15.10.e.a 15.e 15.e $32$ $7.726$ None \(0\) \(-150\) \(0\) \(-9760\) $\mathrm{SU}(2)[C_{4}]$
15.11.c.a 15.c 3.b $14$ $9.530$ \(\mathbb{Q}[x]/(x^{14} + \cdots)\) None \(0\) \(44\) \(0\) \(-50548\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{2}q^{2}+(3+\beta _{2}-\beta _{3})q^{3}+(-629+\cdots)q^{4}+\cdots\)
15.11.d.a 15.d 15.d $1$ $9.530$ \(\Q\) \(\Q(\sqrt{-15}) \) \(-61\) \(243\) \(-3125\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q-61q^{2}+3^{5}q^{3}+2697q^{4}-5^{5}q^{5}+\cdots\)
15.11.d.b 15.d 15.d $1$ $9.530$ \(\Q\) \(\Q(\sqrt{-15}) \) \(61\) \(-243\) \(3125\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+61q^{2}-3^{5}q^{3}+2697q^{4}+5^{5}q^{5}+\cdots\)
15.11.d.c 15.d 15.d $16$ $9.530$ \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{6}q^{2}+(-\beta _{6}+\beta _{7})q^{3}+(137-\beta _{2}+\cdots)q^{4}+\cdots\)
15.11.f.a 15.f 5.c $20$ $9.530$ \(\mathbb{Q}[x]/(x^{20} - \cdots)\) None \(-64\) \(0\) \(10676\) \(10604\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-3-3\beta _{2}-\beta _{3})q^{2}+\beta _{6}q^{3}+(451\beta _{2}+\cdots)q^{4}+\cdots\)
15.12.a.a 15.a 1.a $1$ $11.525$ \(\Q\) None \(-56\) \(-243\) \(3125\) \(27984\) $-$ $\mathrm{SU}(2)$ \(q-56q^{2}-3^{5}q^{3}+1088q^{4}+5^{5}q^{5}+\cdots\)
15.12.a.b 15.a 1.a $2$ $11.525$ \(\Q(\sqrt{1609}) \) None \(-22\) \(486\) \(-6250\) \(-10864\) $-$ $\mathrm{SU}(2)$ \(q+(-11-\beta )q^{2}+3^{5}q^{3}+(-318+22\beta )q^{4}+\cdots\)
15.12.a.c 15.a 1.a $2$ $11.525$ \(\Q(\sqrt{1801}) \) None \(-13\) \(-486\) \(-6250\) \(7784\) $+$ $\mathrm{SU}(2)$ \(q+(-6-\beta )q^{2}-3^{5}q^{3}+(-1562+13\beta )q^{4}+\cdots\)
15.12.a.d 15.a 1.a $3$ $11.525$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(-1\) \(729\) \(9375\) \(-14608\) $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+3^{5}q^{3}+(1585+3\beta _{1}+\beta _{2})q^{4}+\cdots\)
15.12.b.a 15.b 5.b $12$ $11.525$ \(\mathbb{Q}[x]/(x^{12} + \cdots)\) None \(0\) \(0\) \(2556\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{3}q^{2}-\beta _{4}q^{3}+(-1359+\beta _{1})q^{4}+\cdots\)
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