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Label Char Prim Dim $A$ Field CM RM Traces Fricke sign Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
225.1.g.a 225.g 5.c $2$ $0.112$ \(\Q(\sqrt{-1}) \) \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-15}) \) \(\Q(\sqrt{5}) \) \(0\) \(0\) \(0\) \(0\) \(q+iq^{4}-q^{16}-iq^{19}-q^{31}+iq^{49}+\cdots\)
225.2.a.a 225.a 1.a $1$ $1.797$ \(\Q\) None None \(-2\) \(0\) \(0\) \(-3\) $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+2q^{4}-3q^{7}-2q^{11}+q^{13}+\cdots\)
225.2.a.b 225.a 1.a $1$ $1.797$ \(\Q\) None None \(-1\) \(0\) \(0\) \(0\) $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{4}+3q^{8}+4q^{11}+2q^{13}+\cdots\)
225.2.a.c 225.a 1.a $1$ $1.797$ \(\Q\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(0\) \(-5\) $+$ $N(\mathrm{U}(1))$ \(q-2q^{4}-5q^{7}-5q^{13}+4q^{16}-q^{19}+\cdots\)
225.2.a.d 225.a 1.a $1$ $1.797$ \(\Q\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(0\) \(5\) $-$ $N(\mathrm{U}(1))$ \(q-2q^{4}+5q^{7}+5q^{13}+4q^{16}-q^{19}+\cdots\)
225.2.a.e 225.a 1.a $1$ $1.797$ \(\Q\) None None \(2\) \(0\) \(0\) \(3\) $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+2q^{4}+3q^{7}-2q^{11}-q^{13}+\cdots\)
225.2.a.f 225.a 1.a $2$ $1.797$ \(\Q(\sqrt{5}) \) \(\Q(\sqrt{-15}) \) None \(0\) \(0\) \(0\) \(0\) $-$ $N(\mathrm{U}(1))$ \(q-\beta q^{2}+3q^{4}-\beta q^{8}-q^{16}+2\beta q^{17}+\cdots\)
225.2.b.a 225.b 5.b $2$ $1.797$ \(\Q(\sqrt{-1}) \) None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+2iq^{2}-2q^{4}+3iq^{7}-2q^{11}+iq^{13}+\cdots\)
225.2.b.b 225.b 5.b $2$ $1.797$ \(\Q(\sqrt{-1}) \) None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+iq^{2}+q^{4}+3iq^{8}+4q^{11}+2iq^{13}+\cdots\)
225.2.b.c 225.b 5.b $2$ $1.797$ \(\Q(\sqrt{-1}) \) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+2q^{4}+iq^{7}-iq^{13}+4q^{16}+q^{19}+\cdots\)
225.2.e.a 225.e 9.c $2$ $1.797$ \(\Q(\sqrt{-3}) \) None None \(1\) \(3\) \(0\) \(-3\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1-\zeta_{6})q^{2}+(2-\zeta_{6})q^{3}+\zeta_{6}q^{4}+\cdots\)
225.2.e.b 225.e 9.c $6$ $1.797$ 6.0.954288.1 None None \(1\) \(-1\) \(0\) \(5\) $\mathrm{SU}(2)[C_{3}]$ \(q+(\beta _{1}-\beta _{2}+\beta _{4}+\beta _{5})q^{2}-\beta _{4}q^{3}+\cdots\)
225.2.e.c 225.e 9.c $8$ $1.797$ 8.0.1223810289.2 None None \(-2\) \(1\) \(0\) \(1\) $\mathrm{SU}(2)[C_{3}]$ \(q-\beta _{1}q^{2}+(-\beta _{4}+\beta _{7})q^{3}+(-1+\beta _{1}+\cdots)q^{4}+\cdots\)
225.2.e.d 225.e 9.c $8$ $1.797$ \(\Q(\zeta_{24})\) None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q-\zeta_{24}^{7}q^{2}+(\zeta_{24}^{6}-\zeta_{24}^{7})q^{3}-\zeta_{24}^{2}q^{4}+\cdots\)
225.2.e.e 225.e 9.c $8$ $1.797$ 8.0.1223810289.2 None None \(2\) \(-1\) \(0\) \(-1\) $\mathrm{SU}(2)[C_{3}]$ \(q+\beta _{1}q^{2}+(\beta _{4}-\beta _{7})q^{3}+(-1+\beta _{1}+\cdots)q^{4}+\cdots\)
225.2.f.a 225.f 15.e $4$ $1.797$ \(\Q(\zeta_{8})\) None None \(0\) \(0\) \(0\) \(8\) $\mathrm{SU}(2)[C_{4}]$ \(q+\zeta_{8}q^{2}-\zeta_{8}^{2}q^{4}+(2-2\zeta_{8}^{2})q^{7}+\cdots\)
225.2.f.b 225.f 15.e $8$ $1.797$ \(\Q(\zeta_{24})\) None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q-\zeta_{24}^{6}q^{2}-4\zeta_{24}^{2}q^{4}+\zeta_{24}q^{7}+\cdots\)
225.2.h.a 225.h 25.d $4$ $1.797$ \(\Q(\zeta_{10})\) None None \(1\) \(0\) \(-5\) \(0\) $\mathrm{SU}(2)[C_{5}]$ \(q+(1-\zeta_{10}+\zeta_{10}^{2}-\zeta_{10}^{3})q^{2}+\zeta_{10}^{3}q^{4}+\cdots\)
225.2.h.b 225.h 25.d $4$ $1.797$ \(\Q(\zeta_{10})\) None None \(2\) \(0\) \(5\) \(-2\) $\mathrm{SU}(2)[C_{5}]$ \(q+(\zeta_{10}-\zeta_{10}^{2})q^{2}+(-1+\zeta_{10}+\zeta_{10}^{3})q^{4}+\cdots\)
225.2.h.c 225.h 25.d $8$ $1.797$ 8.0.26265625.1 None None \(1\) \(0\) \(5\) \(4\) $\mathrm{SU}(2)[C_{5}]$ \(q+(\beta _{3}+\beta _{7})q^{2}+(-1+2\beta _{1}+2\beta _{3}+\cdots)q^{4}+\cdots\)
225.2.h.d 225.h 25.d $12$ $1.797$ \(\mathbb{Q}[x]/(x^{12} + \cdots)\) None None \(0\) \(0\) \(6\) \(-12\) $\mathrm{SU}(2)[C_{5}]$ \(q+\beta _{2}q^{2}+(-1+\beta _{1}-\beta _{5}-2\beta _{8}+\beta _{9}+\cdots)q^{4}+\cdots\)
225.2.h.e 225.h 25.d $16$ $1.797$ 16.0.\(\cdots\).3 None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{5}]$ \(q+(\beta _{3}-\beta _{10}+\beta _{12}-\beta _{14})q^{2}+(-\beta _{8}+\cdots)q^{4}+\cdots\)
225.2.k.a 225.k 45.j $4$ $1.797$ \(\Q(\zeta_{12})\) None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+\zeta_{12}q^{2}+(-\zeta_{12}-\zeta_{12}^{3})q^{3}-\zeta_{12}^{2}q^{4}+\cdots\)
225.2.k.b 225.k 45.j $12$ $1.797$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+(\beta _{1}+\beta _{6}-\beta _{7})q^{2}+\beta _{6}q^{3}+(2+2\beta _{8}+\cdots)q^{4}+\cdots\)
225.2.k.c 225.k 45.j $16$ $1.797$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+\beta _{1}q^{2}+(-\beta _{10}+\beta _{14})q^{3}+(1-\beta _{3}+\cdots)q^{4}+\cdots\)
225.2.m.a 225.m 25.e $8$ $1.797$ 8.0.58140625.2 None None \(5\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{10}]$ \(q+(1-\beta _{1})q^{2}+(1-2\beta _{1}-\beta _{2}+\beta _{3}+\cdots)q^{4}+\cdots\)
225.2.m.b 225.m 25.e $16$ $1.797$ \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{10}]$ \(q+(\beta _{1}-\beta _{3}+\beta _{4}-\beta _{10}+\beta _{15})q^{2}+\cdots\)
225.2.m.c 225.m 25.e $24$ $1.797$ None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{10}]$
225.2.p.a 225.p 45.l $16$ $1.797$ 16.0.\(\cdots\).9 None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{12}]$ \(q+(-\beta _{6}-\beta _{13})q^{2}+\beta _{1}q^{3}+(-\beta _{9}+\cdots)q^{4}+\cdots\)
225.2.p.b 225.p 45.l $16$ $1.797$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None None \(6\) \(6\) \(0\) \(2\) $\mathrm{SU}(2)[C_{12}]$ \(q+\beta _{5}q^{2}+(1-\beta _{9}+\beta _{11}+\beta _{12}+\beta _{13}+\cdots)q^{3}+\cdots\)
225.2.p.c 225.p 45.l $32$ $1.797$ None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{12}]$
225.2.q.a 225.q 225.q $224$ $1.797$ None None \(-3\) \(-8\) \(-8\) \(-8\) $\mathrm{SU}(2)[C_{15}]$
225.2.s.a 225.s 75.l $80$ $1.797$ None None \(0\) \(0\) \(0\) \(8\) $\mathrm{SU}(2)[C_{20}]$
225.2.u.a 225.u 225.u $224$ $1.797$ None None \(-5\) \(-10\) \(0\) \(0\) $\mathrm{SU}(2)[C_{30}]$
225.2.w.a 225.w 225.w $448$ $1.797$ None None \(-24\) \(-14\) \(-24\) \(-8\) $\mathrm{SU}(2)[C_{60}]$
225.3.c.a 225.c 3.b $2$ $6.131$ \(\Q(\sqrt{-2}) \) None None \(0\) \(0\) \(0\) \(-18\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta q^{2}+2q^{4}-9q^{7}+6\beta q^{8}+13\beta q^{11}+\cdots\)
225.3.c.b 225.c 3.b $2$ $6.131$ \(\Q(\sqrt{-2}) \) None None \(0\) \(0\) \(0\) \(18\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta q^{2}+2q^{4}+9q^{7}+6\beta q^{8}-13\beta q^{11}+\cdots\)
225.3.c.c 225.c 3.b $4$ $6.131$ \(\Q(\sqrt{-2}, \sqrt{-5})\) None None \(0\) \(0\) \(0\) \(-16\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{2}q^{2}+(-3-2\beta _{1})q^{4}+(-4-\beta _{1}+\cdots)q^{7}+\cdots\)
225.3.c.d 225.c 3.b $4$ $6.131$ \(\Q(\sqrt{-2}, \sqrt{-7})\) None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{2}q^{2}-3q^{4}-\beta _{1}q^{7}-\beta _{2}q^{8}-\beta _{3}q^{11}+\cdots\)
225.3.d.a 225.d 15.d $4$ $6.131$ \(\Q(\zeta_{8})\) None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\zeta_{8}^{3}q^{2}-2q^{4}+9\zeta_{8}q^{7}-6\zeta_{8}^{3}q^{8}+\cdots\)
225.3.d.b 225.d 15.d $8$ $6.131$ 8.0.40960000.1 None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{4}q^{2}+(3-\beta _{6})q^{4}+(2\beta _{1}-\beta _{3})q^{7}+\cdots\)
225.3.g.a 225.g 5.c $4$ $6.131$ \(\Q(i, \sqrt{6})\) None None \(-4\) \(0\) \(0\) \(-4\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-1+\beta _{1}-\beta _{2})q^{2}+(-2\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
225.3.g.b 225.g 5.c $4$ $6.131$ \(\Q(i, \sqrt{6})\) None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+2\beta _{1}q^{2}+8\beta _{2}q^{4}+5\beta _{1}q^{7}+8\beta _{3}q^{8}+\cdots\)
225.3.g.c 225.g 5.c $4$ $6.131$ \(\Q(i, \sqrt{30})\) \(\Q(\sqrt{-15}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{4}]$ \(q+\beta _{1}q^{2}+11\beta _{2}q^{4}+7\beta _{3}q^{8}-61q^{16}+\cdots\)
225.3.g.d 225.g 5.c $4$ $6.131$ \(\Q(i, \sqrt{6})\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{4}]$ \(q-4\beta _{2}q^{4}+\beta _{1}q^{7}-3\beta _{3}q^{13}-2^{4}q^{16}+\cdots\)
225.3.g.e 225.g 5.c $4$ $6.131$ \(\Q(i, \sqrt{6})\) None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+\beta _{1}q^{2}-\beta _{2}q^{4}+4\beta _{1}q^{7}-5\beta _{3}q^{8}+\cdots\)
225.3.g.f 225.g 5.c $4$ $6.131$ \(\Q(i, \sqrt{6})\) None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+\beta _{1}q^{2}-\beta _{2}q^{4}-6\beta _{1}q^{7}-5\beta _{3}q^{8}+\cdots\)
225.3.g.g 225.g 5.c $4$ $6.131$ \(\Q(i, \sqrt{10})\) None None \(0\) \(0\) \(0\) \(20\) $\mathrm{SU}(2)[C_{4}]$ \(q+\beta _{1}q^{2}+\beta _{2}q^{4}+(5+5\beta _{2})q^{7}-3\beta _{3}q^{8}+\cdots\)
225.3.i.a 225.i 45.h $4$ $6.131$ \(\Q(\zeta_{12})\) None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+(-\zeta_{12}-\zeta_{12}^{3})q^{2}+(-3\zeta_{12}+3\zeta_{12}^{3})q^{3}+\cdots\)
225.3.i.b 225.i 45.h $32$ $6.131$ None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$
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