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Label Char Prim Dim $A$ Field CM Minimal twist Traces Fricke sign Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
51.2.a.a 51.a 1.a $1$ $0.407$ \(\Q\) None 51.2.a.a \(0\) \(1\) \(3\) \(-4\) $-$ $\mathrm{SU}(2)$ \(q+q^{3}-2q^{4}+3q^{5}-4q^{7}+q^{9}-3q^{11}+\cdots\)
51.2.a.b 51.a 1.a $2$ $0.407$ \(\Q(\sqrt{17}) \) None 51.2.a.b \(-1\) \(-2\) \(3\) \(0\) $-$ $\mathrm{SU}(2)$ \(q-\beta q^{2}-q^{3}+(2+\beta )q^{4}+(1+\beta )q^{5}+\cdots\)
51.2.d.a 51.d 17.b $2$ $0.407$ \(\Q(\sqrt{-1}) \) None 51.2.d.a \(-4\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-2 q^{2}+i q^{3}+2 q^{4}+3 i q^{5}-2 i q^{6}+\cdots\)
51.2.d.b 51.d 17.b $2$ $0.407$ \(\Q(\sqrt{-1}) \) None 51.2.d.b \(2\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+q^{2}+i q^{3}-q^{4}+i q^{6}-4 i q^{7}+\cdots\)
51.2.e.a 51.e 17.c $8$ $0.407$ 8.0.836829184.2 None 51.2.e.a \(0\) \(0\) \(-4\) \(-4\) $\mathrm{SU}(2)[C_{4}]$ \(q+\beta _{1}q^{2}-\beta _{3}q^{3}+(-1-\beta _{4}-\beta _{5}+\cdots)q^{4}+\cdots\)
51.2.h.a 51.h 17.d $8$ $0.407$ \(\Q(\zeta_{16})\) None 51.2.h.a \(0\) \(0\) \(-8\) \(0\) $\mathrm{SU}(2)[C_{8}]$ \(q+(\zeta_{16}+\zeta_{16}^{3})q^{2}+\zeta_{16}^{7}q^{3}+(\zeta_{16}^{2}+\cdots)q^{4}+\cdots\)
51.2.i.a 51.i 51.i $32$ $0.407$ None 51.2.i.a \(0\) \(-8\) \(0\) \(-16\) $\mathrm{SU}(2)[C_{16}]$
51.3.b.a 51.b 3.b $10$ $1.390$ \(\mathbb{Q}[x]/(x^{10} + \cdots)\) None 51.3.b.a \(0\) \(2\) \(0\) \(-4\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{2}-\beta _{4}q^{3}+(-2+\beta _{2})q^{4}+\beta _{7}q^{5}+\cdots\)
51.3.c.a 51.c 51.c $1$ $1.390$ \(\Q\) \(\Q(\sqrt{-51}) \) 51.3.c.a \(0\) \(-3\) \(7\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q-3q^{3}+4q^{4}+7q^{5}+9q^{9}-5q^{11}+\cdots\)
51.3.c.b 51.c 51.c $1$ $1.390$ \(\Q\) \(\Q(\sqrt{-51}) \) 51.3.c.a \(0\) \(3\) \(-7\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+3q^{3}+4q^{4}-7q^{5}+9q^{9}+5q^{11}+\cdots\)
51.3.c.c 51.c 51.c $8$ $1.390$ \(\mathbb{Q}[x]/(x^{8} + \cdots)\) None 51.3.c.c \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{3}q^{2}+(-\beta _{1}+\beta _{7})q^{3}+(-4-\beta _{2}+\cdots)q^{4}+\cdots\)
51.3.f.a 51.f 51.f $20$ $1.390$ \(\mathbb{Q}[x]/(x^{20} + \cdots)\) None 51.3.f.a \(0\) \(-6\) \(0\) \(-4\) $\mathrm{SU}(2)[C_{4}]$ \(q-\beta _{1}q^{2}-\beta _{11}q^{3}+(1-\beta _{2})q^{4}-\beta _{9}q^{5}+\cdots\)
51.3.g.a 51.g 51.g $4$ $1.390$ \(\Q(\zeta_{8})\) None 51.3.g.a \(-4\) \(12\) \(8\) \(-4\) $\mathrm{SU}(2)[C_{8}]$ \(q+(-1-\zeta_{8}-\zeta_{8}^{2})q^{2}+3q^{3}+(2\zeta_{8}+\cdots)q^{4}+\cdots\)
51.3.g.b 51.g 51.g $4$ $1.390$ \(\Q(\zeta_{8})\) None 51.3.g.a \(4\) \(0\) \(-8\) \(-4\) $\mathrm{SU}(2)[C_{8}]$ \(q+(1+\zeta_{8}+\zeta_{8}^{2})q^{2}+3\zeta_{8}q^{3}+(2\zeta_{8}+\cdots)q^{4}+\cdots\)
51.3.g.c 51.g 51.g $32$ $1.390$ None 51.3.g.c \(0\) \(-16\) \(0\) \(0\) $\mathrm{SU}(2)[C_{8}]$
51.3.j.a 51.j 17.e $48$ $1.390$ None 51.3.j.a \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{16}]$
51.4.a.a 51.a 1.a $1$ $3.009$ \(\Q\) None 51.4.a.a \(-1\) \(-3\) \(16\) \(34\) $+$ $\mathrm{SU}(2)$ \(q-q^{2}-3q^{3}-7q^{4}+2^{4}q^{5}+3q^{6}+\cdots\)
51.4.a.b 51.a 1.a $1$ $3.009$ \(\Q\) None 51.4.a.b \(-1\) \(3\) \(-20\) \(-2\) $-$ $\mathrm{SU}(2)$ \(q-q^{2}+3q^{3}-7q^{4}-20q^{5}-3q^{6}+\cdots\)
51.4.a.c 51.a 1.a $1$ $3.009$ \(\Q\) None 51.4.a.c \(1\) \(-3\) \(-10\) \(-8\) $-$ $\mathrm{SU}(2)$ \(q+q^{2}-3q^{3}-7q^{4}-10q^{5}-3q^{6}+\cdots\)
51.4.a.d 51.a 1.a $2$ $3.009$ \(\Q(\sqrt{2}) \) None 51.4.a.d \(0\) \(-6\) \(6\) \(-8\) $+$ $\mathrm{SU}(2)$ \(q+\beta q^{2}-3q^{3}+10q^{4}+(3+4\beta )q^{5}+\cdots\)
51.4.a.e 51.a 1.a $3$ $3.009$ 3.3.5912.1 None 51.4.a.e \(5\) \(9\) \(8\) \(-8\) $+$ $\mathrm{SU}(2)$ \(q+(2-\beta _{1})q^{2}+3q^{3}+(5-2\beta _{1}+\beta _{2})q^{4}+\cdots\)
51.4.d.a 51.d 17.b $8$ $3.009$ \(\mathbb{Q}[x]/(x^{8} + \cdots)\) None 51.4.d.a \(-4\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(-1+\beta _{2})q^{2}-\beta _{4}q^{3}+(5-\beta _{2}+\beta _{3}+\cdots)q^{4}+\cdots\)
51.4.e.a 51.e 17.c $16$ $3.009$ \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None 51.4.e.a \(0\) \(0\) \(-32\) \(-8\) $\mathrm{SU}(2)[C_{4}]$ \(q+(\beta _{1}-\beta _{2})q^{2}+\beta _{4}q^{3}+(-4-\beta _{7}+\cdots)q^{4}+\cdots\)
51.4.h.a 51.h 17.d $40$ $3.009$ None 51.4.h.a \(0\) \(0\) \(32\) \(0\) $\mathrm{SU}(2)[C_{8}]$
51.4.i.a 51.i 51.i $128$ $3.009$ None 51.4.i.a \(0\) \(-8\) \(0\) \(-16\) $\mathrm{SU}(2)[C_{16}]$
51.5.b.a 51.b 3.b $22$ $5.272$ None 51.5.b.a \(0\) \(-10\) \(0\) \(-4\) $\mathrm{SU}(2)[C_{2}]$
51.5.c.a 51.c 51.c $1$ $5.272$ \(\Q\) \(\Q(\sqrt{-51}) \) 51.5.c.a \(0\) \(-9\) \(1\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q-9q^{3}+2^{4}q^{4}+q^{5}+3^{4}q^{9}+217q^{11}+\cdots\)
51.5.c.b 51.c 51.c $1$ $5.272$ \(\Q\) \(\Q(\sqrt{-51}) \) 51.5.c.a \(0\) \(9\) \(-1\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+9q^{3}+2^{4}q^{4}-q^{5}+3^{4}q^{9}-217q^{11}+\cdots\)
51.5.c.c 51.c 51.c $20$ $5.272$ \(\mathbb{Q}[x]/(x^{20} + \cdots)\) None 51.5.c.c \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{2}+\beta _{6}q^{3}+(-8-\beta _{4})q^{4}-\beta _{14}q^{5}+\cdots\)
51.5.f.a 51.f 51.f $44$ $5.272$ None 51.5.f.a \(0\) \(6\) \(0\) \(-4\) $\mathrm{SU}(2)[C_{4}]$
51.5.g.a 51.g 51.g $88$ $5.272$ None 51.5.g.a \(0\) \(-4\) \(0\) \(-8\) $\mathrm{SU}(2)[C_{8}]$
51.5.j.a 51.j 17.e $96$ $5.272$ None 51.5.j.a \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{16}]$
51.6.a.a 51.a 1.a $2$ $8.180$ \(\Q(\sqrt{145}) \) None 51.6.a.a \(-7\) \(18\) \(-113\) \(0\) $+$ $\mathrm{SU}(2)$ \(q+(-3-\beta )q^{2}+9q^{3}+(13+7\beta )q^{4}+\cdots\)
51.6.a.b 51.a 1.a $3$ $8.180$ 3.3.76361.1 None 51.6.a.b \(5\) \(-27\) \(-37\) \(-176\) $+$ $\mathrm{SU}(2)$ \(q+(2-\beta _{1})q^{2}-9q^{3}+(12+\beta _{1}+2\beta _{2})q^{4}+\cdots\)
51.6.a.c 51.a 1.a $4$ $8.180$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None 51.6.a.c \(5\) \(36\) \(146\) \(60\) $-$ $\mathrm{SU}(2)$ \(q+(1+\beta _{1})q^{2}+9q^{3}+(3^{3}+4\beta _{2}+\beta _{3})q^{4}+\cdots\)
51.6.a.d 51.a 1.a $5$ $8.180$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None 51.6.a.d \(1\) \(-45\) \(4\) \(-40\) $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}-9q^{3}+(24+\beta _{1}+2\beta _{2}+\beta _{4})q^{4}+\cdots\)
51.6.d.a 51.d 17.b $16$ $8.180$ \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None 51.6.d.a \(8\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{3}q^{2}+\beta _{9}q^{3}+(2^{4}-\beta _{2}-\beta _{3})q^{4}+\cdots\)
51.6.e.a 51.e 17.c $32$ $8.180$ None 51.6.e.a \(0\) \(0\) \(76\) \(236\) $\mathrm{SU}(2)[C_{4}]$
51.6.h.a 51.h 17.d $56$ $8.180$ None 51.6.h.a \(0\) \(0\) \(-88\) \(0\) $\mathrm{SU}(2)[C_{8}]$
51.6.i.a 51.i 51.i $224$ $8.180$ None 51.6.i.a \(0\) \(-8\) \(0\) \(-16\) $\mathrm{SU}(2)[C_{16}]$
51.7.b.a 51.b 3.b $32$ $11.733$ None 51.7.b.a \(0\) \(32\) \(0\) \(568\) $\mathrm{SU}(2)[C_{2}]$
51.7.c.a 51.c 51.c $1$ $11.733$ \(\Q\) \(\Q(\sqrt{-51}) \) 51.7.c.a \(0\) \(-27\) \(-182\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q-3^{3}q^{3}+2^{6}q^{4}-182q^{5}+3^{6}q^{9}+\cdots\)
51.7.c.b 51.c 51.c $1$ $11.733$ \(\Q\) \(\Q(\sqrt{-51}) \) 51.7.c.a \(0\) \(27\) \(182\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+3^{3}q^{3}+2^{6}q^{4}+182q^{5}+3^{6}q^{9}+\cdots\)
51.7.c.c 51.c 51.c $32$ $11.733$ None 51.7.c.c \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$
51.7.f.a 51.f 51.f $68$ $11.733$ None 51.7.f.a \(0\) \(18\) \(0\) \(-4\) $\mathrm{SU}(2)[C_{4}]$
51.7.g.a 51.g 51.g $136$ $11.733$ None 51.7.g.a \(0\) \(-4\) \(0\) \(-8\) $\mathrm{SU}(2)[C_{8}]$
51.7.j.a 51.j 17.e $144$ $11.733$ None 51.7.j.a \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{16}]$
51.8.a.a 51.a 1.a $2$ $15.932$ \(\Q(\sqrt{1177}) \) None 51.8.a.a \(7\) \(54\) \(320\) \(-1500\) $+$ $\mathrm{SU}(2)$ \(q+(4-\beta )q^{2}+3^{3}q^{3}+(182-7\beta )q^{4}+\cdots\)
51.8.a.b 51.a 1.a $3$ $15.932$ 3.3.1514860.1 None 51.8.a.b \(-7\) \(-81\) \(-440\) \(808\) $-$ $\mathrm{SU}(2)$ \(q+(-2-\beta _{1})q^{2}-3^{3}q^{3}+(78-2\beta _{1}+\cdots)q^{4}+\cdots\)
51.8.a.c 51.a 1.a $4$ $15.932$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None 51.8.a.c \(-15\) \(108\) \(-310\) \(-1036\) $-$ $\mathrm{SU}(2)$ \(q+(-4+\beta _{1})q^{2}+3^{3}q^{3}+(39-7\beta _{1}+\cdots)q^{4}+\cdots\)
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