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Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
7.3.b.a 7.b 7.b $1$ $0.191$ \(\Q\) \(\Q(\sqrt{-7}) \) \(-3\) \(0\) \(0\) \(-7\) $\mathrm{U}(1)[D_{2}]$ \(q-3q^{2}+5q^{4}-7q^{7}-3q^{8}+9q^{9}+\cdots\)
8.3.d.a 8.d 8.d $1$ $0.218$ \(\Q\) \(\Q(\sqrt{-2}) \) \(-2\) \(-2\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q-2q^{2}-2q^{3}+4q^{4}+4q^{6}-8q^{8}+\cdots\)
11.3.b.a 11.b 11.b $1$ $0.300$ \(\Q\) \(\Q(\sqrt{-11}) \) \(0\) \(-5\) \(-1\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q-5q^{3}+4q^{4}-q^{5}+2^{4}q^{9}-11q^{11}+\cdots\)
12.3.c.a 12.c 3.b $1$ $0.327$ \(\Q\) \(\Q(\sqrt{-3}) \) \(0\) \(-3\) \(0\) \(2\) $\mathrm{U}(1)[D_{2}]$ \(q-3q^{3}+2q^{7}+9q^{9}-22q^{13}+26q^{19}+\cdots\)
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 $1$ $0.409$ \(\Q\) \(\Q(\sqrt{-15}) \) \(1\) \(-3\) \(5\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+q^{2}-3q^{3}-3q^{4}+5q^{5}-3q^{6}+\cdots\)
16.3.c.a 16.c 4.b $1$ $0.436$ \(\Q\) \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(-6\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q-6q^{5}+9q^{9}+10q^{13}-30q^{17}+\cdots\)
19.3.b.a 19.b 19.b $1$ $0.518$ \(\Q\) \(\Q(\sqrt{-19}) \) \(0\) \(0\) \(-9\) \(-5\) $\mathrm{U}(1)[D_{2}]$ \(q+4q^{4}-9q^{5}-5q^{7}+9q^{9}+3q^{11}+\cdots\)
20.3.d.a 20.d 20.d $1$ $0.545$ \(\Q\) \(\Q(\sqrt{-5}) \) \(-2\) \(4\) \(-5\) \(-4\) $\mathrm{U}(1)[D_{2}]$ \(q-2q^{2}+4q^{3}+4q^{4}-5q^{5}-8q^{6}+\cdots\)
20.3.d.b 20.d 20.d $1$ $0.545$ \(\Q\) \(\Q(\sqrt{-5}) \) \(2\) \(-4\) \(-5\) \(4\) $\mathrm{U}(1)[D_{2}]$ \(q+2q^{2}-4q^{3}+4q^{4}-5q^{5}-8q^{6}+\cdots\)
20.3.d.c 20.d 20.d $2$ $0.545$ \(\Q(\sqrt{-1}) \) \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(6\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+iq^{2}-4q^{4}+(3-2i)q^{5}-4iq^{8}+\cdots\)
21.3.h.a 21.h 21.h $2$ $0.572$ \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) \(0\) \(3\) \(0\) \(-13\) $\mathrm{U}(1)[D_{6}]$ \(q+3\zeta_{6}q^{3}-4\zeta_{6}q^{4}+(-5-3\zeta_{6})q^{7}+\cdots\)
23.3.b.a 23.b 23.b $3$ $0.627$ 3.3.621.1 \(\Q(\sqrt{-23}) \) \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+(\beta _{1}+\beta _{2})q^{2}+(-2\beta _{1}-\beta _{2})q^{3}+(4+\cdots)q^{4}+\cdots\)
24.3.h.a 24.h 24.h $1$ $0.654$ \(\Q\) \(\Q(\sqrt{-6}) \) \(-2\) \(3\) \(2\) \(-10\) $\mathrm{U}(1)[D_{2}]$ \(q-2q^{2}+3q^{3}+4q^{4}+2q^{5}-6q^{6}+\cdots\)
24.3.h.b 24.h 24.h $1$ $0.654$ \(\Q\) \(\Q(\sqrt{-6}) \) \(2\) \(-3\) \(-2\) \(-10\) $\mathrm{U}(1)[D_{2}]$ \(q+2q^{2}-3q^{3}+4q^{4}-2q^{5}-6q^{6}+\cdots\)
27.3.b.a 27.b 3.b $1$ $0.736$ \(\Q\) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(-13\) $\mathrm{U}(1)[D_{2}]$ \(q+4q^{4}-13q^{7}-q^{13}+2^{4}q^{16}+11q^{19}+\cdots\)
31.3.b.b 31.b 31.b $3$ $0.845$ 3.3.837.1 \(\Q(\sqrt{-31}) \) \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+\beta _{2}q^{2}+(4+\beta _{1}-2\beta _{2})q^{4}+(-3\beta _{1}+\cdots)q^{5}+\cdots\)
32.3.d.a 32.d 8.d $1$ $0.872$ \(\Q\) \(\Q(\sqrt{-2}) \) \(0\) \(2\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+2q^{3}-5q^{9}-14q^{11}+2q^{17}+34q^{19}+\cdots\)
35.3.c.a 35.c 35.c $1$ $0.954$ \(\Q\) \(\Q(\sqrt{-35}) \) \(0\) \(-1\) \(5\) \(-7\) $\mathrm{U}(1)[D_{2}]$ \(q-q^{3}+4q^{4}+5q^{5}-7q^{7}-8q^{9}+\cdots\)
35.3.c.b 35.c 35.c $1$ $0.954$ \(\Q\) \(\Q(\sqrt{-35}) \) \(0\) \(1\) \(-5\) \(7\) $\mathrm{U}(1)[D_{2}]$ \(q+q^{3}+4q^{4}-5q^{5}+7q^{7}-8q^{9}+\cdots\)
36.3.d.a 36.d 4.b $1$ $0.981$ \(\Q\) \(\Q(\sqrt{-1}) \) \(-2\) \(0\) \(8\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q-2q^{2}+4q^{4}+8q^{5}-8q^{8}-2^{4}q^{10}+\cdots\)
36.3.d.b 36.d 4.b $1$ $0.981$ \(\Q\) \(\Q(\sqrt{-1}) \) \(2\) \(0\) \(-8\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+2q^{2}+4q^{4}-8q^{5}+8q^{8}-2^{4}q^{10}+\cdots\)
39.3.d.a 39.d 39.d $2$ $1.063$ \(\Q(\sqrt{13}) \) \(\Q(\sqrt{-39}) \) \(0\) \(-6\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q-\beta q^{2}-3q^{3}+9q^{4}+2\beta q^{5}+3\beta q^{6}+\cdots\)
39.3.d.b 39.d 39.d $2$ $1.063$ \(\Q(\sqrt{3}) \) \(\Q(\sqrt{-39}) \) \(0\) \(6\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+\beta q^{2}+3q^{3}-q^{4}-4\beta q^{5}+3\beta q^{6}+\cdots\)
39.3.h.a 39.h 39.h $2$ $1.063$ \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) \(0\) \(-3\) \(0\) \(9\) $\mathrm{U}(1)[D_{6}]$ \(q+(-3+3\zeta_{6})q^{3}+4\zeta_{6}q^{4}+(6-3\zeta_{6})q^{7}+\cdots\)
39.3.i.a 39.i 39.i $2$ $1.063$ \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) \(0\) \(3\) \(0\) \(13\) $\mathrm{U}(1)[D_{6}]$ \(q+3\zeta_{6}q^{3}+(-4+4\zeta_{6})q^{4}+(13-13\zeta_{6})q^{7}+\cdots\)
40.3.e.a 40.e 40.e $1$ $1.090$ \(\Q\) \(\Q(\sqrt{-10}) \) \(-2\) \(0\) \(5\) \(6\) $\mathrm{U}(1)[D_{2}]$ \(q-2q^{2}+4q^{4}+5q^{5}+6q^{7}-8q^{8}+\cdots\)
40.3.e.b 40.e 40.e $1$ $1.090$ \(\Q\) \(\Q(\sqrt{-10}) \) \(2\) \(0\) \(-5\) \(-6\) $\mathrm{U}(1)[D_{2}]$ \(q+2q^{2}+4q^{4}-5q^{5}-6q^{7}+8q^{8}+\cdots\)
43.3.b.a 43.b 43.b $1$ $1.172$ \(\Q\) \(\Q(\sqrt{-43}) \) \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+4q^{4}+9q^{9}-21q^{11}-17q^{13}+\cdots\)
44.3.d.a 44.d 11.b $2$ $1.199$ \(\Q(\sqrt{33}) \) \(\Q(\sqrt{-11}) \) \(0\) \(5\) \(1\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+(3-\beta )q^{3}+(-1+3\beta )q^{5}+(8-5\beta )q^{9}+\cdots\)
47.3.b.b 47.b 47.b $5$ $1.281$ 5.5.6903125.1 \(\Q(\sqrt{-47}) \) \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+\beta _{2}q^{2}+(-\beta _{1}-\beta _{4})q^{3}+(4+2\beta _{1}+\cdots)q^{4}+\cdots\)
48.3.e.a 48.e 3.b $1$ $1.308$ \(\Q\) \(\Q(\sqrt{-3}) \) \(0\) \(3\) \(0\) \(-2\) $\mathrm{U}(1)[D_{2}]$ \(q+3q^{3}-2q^{7}+9q^{9}-22q^{13}-26q^{19}+\cdots\)
49.3.d.a 49.d 7.d $2$ $1.335$ \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-7}) \) \(3\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{6}]$ \(q+3\zeta_{6}q^{2}+(-5+5\zeta_{6})q^{4}-3q^{8}+\cdots\)
51.3.c.a 51.c 51.c $1$ $1.390$ \(\Q\) \(\Q(\sqrt{-51}) \) \(0\) \(-3\) \(7\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q-3q^{3}+4q^{4}+7q^{5}+9q^{9}-5q^{11}+\cdots\)
51.3.c.b 51.c 51.c $1$ $1.390$ \(\Q\) \(\Q(\sqrt{-51}) \) \(0\) \(3\) \(-7\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+3q^{3}+4q^{4}-7q^{5}+9q^{9}+5q^{11}+\cdots\)
52.3.b.a 52.b 52.b $1$ $1.417$ \(\Q\) \(\Q(\sqrt{-13}) \) \(-2\) \(0\) \(0\) \(12\) $\mathrm{U}(1)[D_{2}]$ \(q-2q^{2}+4q^{4}+12q^{7}-8q^{8}+9q^{9}+\cdots\)
52.3.b.b 52.b 52.b $1$ $1.417$ \(\Q\) \(\Q(\sqrt{-13}) \) \(2\) \(0\) \(0\) \(-12\) $\mathrm{U}(1)[D_{2}]$ \(q+2q^{2}+4q^{4}-12q^{7}+8q^{8}+9q^{9}+\cdots\)
52.3.b.c 52.b 52.b $2$ $1.417$ \(\Q(\sqrt{-1}) \) \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+iq^{2}-4q^{4}+4iq^{5}-4iq^{8}+9q^{9}+\cdots\)
52.3.i.a 52.i 52.i $4$ $1.417$ \(\Q(\zeta_{12})\) \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{6}]$ \(q+\zeta_{12}q^{2}+4\zeta_{12}^{2}q^{4}+(3-6\zeta_{12}^{2}+\cdots)q^{5}+\cdots\)
55.3.d.a 55.d 55.d $2$ $1.499$ \(\Q(\sqrt{-11}) \) \(\Q(\sqrt{-11}) \) \(0\) \(0\) \(1\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+(1-2\beta )q^{3}-4q^{4}+(2-3\beta )q^{5}-2q^{9}+\cdots\)
55.3.d.b 55.d 55.d $2$ $1.499$ \(\Q(\sqrt{5}) \) \(\Q(\sqrt{-55}) \) \(0\) \(0\) \(10\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q-\beta q^{2}+q^{4}+5q^{5}+2\beta q^{7}+3\beta q^{8}+\cdots\)
55.3.d.c 55.d 55.d $2$ $1.499$ \(\Q(\sqrt{11}) \) \(\Q(\sqrt{-55}) \) \(0\) \(0\) \(-10\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+\beta q^{2}+7q^{4}-5q^{5}-4\beta q^{7}+3\beta q^{8}+\cdots\)
56.3.h.a 56.h 56.h $2$ $1.526$ \(\Q(\sqrt{7}) \) \(\Q(\sqrt{-14}) \) \(-4\) \(0\) \(0\) \(14\) $\mathrm{U}(1)[D_{2}]$ \(q-2q^{2}+\beta q^{3}+4q^{4}-\beta q^{5}-2\beta q^{6}+\cdots\)
56.3.h.b 56.h 56.h $2$ $1.526$ \(\Q(\sqrt{-7}) \) \(\Q(\sqrt{-7}) \) \(3\) \(0\) \(0\) \(14\) $\mathrm{U}(1)[D_{2}]$ \(q+(1+\beta )q^{2}+(-1+3\beta )q^{4}+7q^{7}+\cdots\)
56.3.h.c 56.h 56.h $2$ $1.526$ \(\Q(\sqrt{2}) \) \(\Q(\sqrt{-14}) \) \(4\) \(0\) \(0\) \(-14\) $\mathrm{U}(1)[D_{2}]$ \(q+2q^{2}+\beta q^{3}+4q^{4}-3\beta q^{5}+2\beta q^{6}+\cdots\)
57.3.l.a 57.l 57.l $6$ $1.553$ \(\Q(\zeta_{18})\) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{18}]$ \(q-3\zeta_{18}^{4}q^{3}+4\zeta_{18}^{2}q^{4}+(3\zeta_{18}+5\zeta_{18}^{2}+\cdots)q^{7}+\cdots\)
59.3.b.a 59.b 59.b $3$ $1.608$ 3.3.1593.1 \(\Q(\sqrt{-59}) \) \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+(\beta _{1}+\beta _{2})q^{3}+4q^{4}+(-\beta _{1}-2\beta _{2})q^{5}+\cdots\)
63.3.d.a 63.d 7.b $1$ $1.717$ \(\Q\) \(\Q(\sqrt{-7}) \) \(3\) \(0\) \(0\) \(-7\) $\mathrm{U}(1)[D_{2}]$ \(q+3q^{2}+5q^{4}-7q^{7}+3q^{8}+6q^{11}+\cdots\)
63.3.d.c 63.d 7.b $2$ $1.717$ \(\Q(\sqrt{7}) \) \(\Q(\sqrt{-7}) \) \(0\) \(0\) \(0\) \(14\) $\mathrm{U}(1)[D_{2}]$ \(q+\beta q^{2}+3q^{4}+7q^{7}-\beta q^{8}-8\beta q^{11}+\cdots\)
63.3.m.b 63.m 7.d $2$ $1.717$ \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(13\) $\mathrm{U}(1)[D_{6}]$ \(q+(4-4\zeta_{6})q^{4}+(8-3\zeta_{6})q^{7}+(-7+\cdots)q^{13}+\cdots\)
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