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Label Char Prim Dim $A$ Field CM RM Traces Fricke sign Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
825.1.l.a 825.l 165.l $2$ $0.412$ \(\Q(\sqrt{-1}) \) \(\Q(\sqrt{-11}) \) None \(0\) \(0\) \(0\) \(0\) \(q+iq^{3}+iq^{4}-q^{9}+iq^{11}-q^{12}+\cdots\)
825.1.l.b 825.l 165.l $2$ $0.412$ \(\Q(\sqrt{-1}) \) \(\Q(\sqrt{-11}) \) None \(0\) \(2\) \(0\) \(0\) \(q+q^{3}+iq^{4}+q^{9}-iq^{11}+iq^{12}+\cdots\)
825.1.co.a 825.co 825.bo $8$ $0.412$ \(\Q(\zeta_{20})\) \(\Q(\sqrt{-11}) \) None \(0\) \(0\) \(0\) \(0\) \(q-\zeta_{20}^{3}q^{3}-\zeta_{20}q^{4}+\zeta_{20}^{5}q^{5}+\zeta_{20}^{6}q^{9}+\cdots\)
825.1.co.b 825.co 825.bo $8$ $0.412$ \(\Q(\zeta_{20})\) \(\Q(\sqrt{-11}) \) None \(0\) \(2\) \(0\) \(0\) \(q-\zeta_{20}^{4}q^{3}-\zeta_{20}q^{4}-\zeta_{20}^{5}q^{5}+\zeta_{20}^{8}q^{9}+\cdots\)
825.2.a.a 825.a 1.a $1$ $6.588$ \(\Q\) None None \(-1\) \(1\) \(0\) \(-4\) $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}-q^{4}-q^{6}-4q^{7}+3q^{8}+\cdots\)
825.2.a.b 825.a 1.a $1$ $6.588$ \(\Q\) None None \(0\) \(-1\) \(0\) \(1\) $+$ $\mathrm{SU}(2)$ \(q-q^{3}-2q^{4}+q^{7}+q^{9}-q^{11}+2q^{12}+\cdots\)
825.2.a.c 825.a 1.a $1$ $6.588$ \(\Q\) None None \(0\) \(1\) \(0\) \(-1\) $+$ $\mathrm{SU}(2)$ \(q+q^{3}-2q^{4}-q^{7}+q^{9}-q^{11}-2q^{12}+\cdots\)
825.2.a.d 825.a 1.a $2$ $6.588$ \(\Q(\sqrt{2}) \) None None \(-2\) \(-2\) \(0\) \(2\) $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta )q^{2}-q^{3}+(1-2\beta )q^{4}+(1+\cdots)q^{6}+\cdots\)
825.2.a.e 825.a 1.a $2$ $6.588$ \(\Q(\sqrt{3}) \) None None \(0\) \(-2\) \(0\) \(-4\) $+$ $\mathrm{SU}(2)$ \(q+\beta q^{2}-q^{3}+q^{4}-\beta q^{6}-2q^{7}-\beta q^{8}+\cdots\)
825.2.a.f 825.a 1.a $2$ $6.588$ \(\Q(\sqrt{2}) \) None None \(2\) \(2\) \(0\) \(-2\) $-$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{2}+q^{3}+(1+2\beta )q^{4}+(1+\beta )q^{6}+\cdots\)
825.2.a.g 825.a 1.a $2$ $6.588$ \(\Q(\sqrt{2}) \) None None \(2\) \(2\) \(0\) \(4\) $-$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{2}+q^{3}+(1+2\beta )q^{4}+(1+\beta )q^{6}+\cdots\)
825.2.a.h 825.a 1.a $3$ $6.588$ 3.3.148.1 None None \(-3\) \(3\) \(0\) \(-8\) $+$ $\mathrm{SU}(2)$ \(q+(-1-\beta _{2})q^{2}+q^{3}+(2-\beta _{1}+\beta _{2})q^{4}+\cdots\)
825.2.a.i 825.a 1.a $3$ $6.588$ 3.3.568.1 None None \(-2\) \(-3\) \(0\) \(3\) $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{2}-q^{3}+(3+\beta _{2})q^{4}+\cdots\)
825.2.a.j 825.a 1.a $3$ $6.588$ 3.3.148.1 None None \(-1\) \(-3\) \(0\) \(-4\) $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}-q^{3}+(\beta _{1}+\beta _{2})q^{4}+\beta _{1}q^{6}+\cdots\)
825.2.a.k 825.a 1.a $3$ $6.588$ 3.3.148.1 None None \(1\) \(-3\) \(0\) \(0\) $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}-q^{3}+(1+\beta _{1}+\beta _{2})q^{4}-\beta _{1}q^{6}+\cdots\)
825.2.a.l 825.a 1.a $3$ $6.588$ 3.3.148.1 None None \(1\) \(3\) \(0\) \(4\) $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+q^{3}+(\beta _{1}+\beta _{2})q^{4}+\beta _{1}q^{6}+\cdots\)
825.2.a.m 825.a 1.a $3$ $6.588$ 3.3.568.1 None None \(2\) \(3\) \(0\) \(-3\) $-$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}+q^{3}+(3+\beta _{2})q^{4}+(1+\cdots)q^{6}+\cdots\)
825.2.a.n 825.a 1.a $3$ $6.588$ 3.3.148.1 None None \(3\) \(-3\) \(0\) \(8\) $-$ $\mathrm{SU}(2)$ \(q+(1+\beta _{2})q^{2}-q^{3}+(2-\beta _{1}+\beta _{2})q^{4}+\cdots\)
825.2.c.a 825.c 5.b $2$ $6.588$ \(\Q(\sqrt{-1}) \) None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+iq^{2}+iq^{3}+q^{4}-q^{6}+4iq^{7}+\cdots\)
825.2.c.b 825.c 5.b $2$ $6.588$ \(\Q(\sqrt{-1}) \) None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-iq^{3}+2q^{4}-iq^{7}-q^{9}-q^{11}+\cdots\)
825.2.c.c 825.c 5.b $4$ $6.588$ \(\Q(\zeta_{12})\) None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\zeta_{12}^{2}q^{2}+\zeta_{12}q^{3}-q^{4}-\zeta_{12}^{3}q^{6}+\cdots\)
825.2.c.d 825.c 5.b $4$ $6.588$ \(\Q(\zeta_{8})\) None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(\zeta_{8}+\zeta_{8}^{2})q^{2}-\zeta_{8}q^{3}+(-1-2\zeta_{8}^{3})q^{4}+\cdots\)
825.2.c.e 825.c 5.b $4$ $6.588$ \(\Q(\zeta_{8})\) None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(\zeta_{8}+\zeta_{8}^{2})q^{2}-\zeta_{8}q^{3}+(-1-2\zeta_{8}^{3})q^{4}+\cdots\)
825.2.c.f 825.c 5.b $6$ $6.588$ 6.0.5161984.1 None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(\beta _{3}+\beta _{4})q^{2}-\beta _{3}q^{3}+(-3-\beta _{1}+\cdots)q^{4}+\cdots\)
825.2.c.g 825.c 5.b $6$ $6.588$ 6.0.350464.1 None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{1}q^{2}+\beta _{3}q^{3}+(-1+\beta _{2}+\beta _{4}+\cdots)q^{4}+\cdots\)
825.2.d.a 825.d 165.d $4$ $6.588$ \(\Q(i, \sqrt{11})\) \(\Q(\sqrt{-11}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+(-\beta _{1}+\beta _{2})q^{3}+2q^{4}+(2-\beta _{3})q^{9}+\cdots\)
825.2.d.b 825.d 165.d $8$ $6.588$ 8.0.619810816.2 None None \(0\) \(-2\) \(0\) \(16\) $\mathrm{SU}(2)[C_{2}]$ \(q+(\beta _{3}-\beta _{4}+\beta _{5}-\beta _{6})q^{2}+(-\beta _{4}+\beta _{5}+\cdots)q^{3}+\cdots\)
825.2.d.c 825.d 165.d $8$ $6.588$ 8.0.619810816.2 None None \(0\) \(-2\) \(0\) \(-16\) $\mathrm{SU}(2)[C_{2}]$ \(q+(\beta _{3}-\beta _{4}+\beta _{5}-\beta _{6})q^{2}+(-\beta _{3}+\beta _{6}+\cdots)q^{3}+\cdots\)
825.2.d.d 825.d 165.d $8$ $6.588$ 8.0.619810816.2 None None \(0\) \(2\) \(0\) \(-16\) $\mathrm{SU}(2)[C_{2}]$ \(q+(\beta _{3}-\beta _{4}+\beta _{5}-\beta _{6})q^{2}+(\beta _{3}-\beta _{6}+\cdots)q^{3}+\cdots\)
825.2.d.e 825.d 165.d $8$ $6.588$ 8.0.619810816.2 None None \(0\) \(2\) \(0\) \(16\) $\mathrm{SU}(2)[C_{2}]$ \(q+(\beta _{3}-\beta _{4}+\beta _{5}-\beta _{6})q^{2}+(\beta _{4}-\beta _{5}+\cdots)q^{3}+\cdots\)
825.2.d.f 825.d 165.d $32$ $6.588$ None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$
825.2.f.a 825.f 33.d $2$ $6.588$ \(\Q(\sqrt{-11}) \) \(\Q(\sqrt{-11}) \) None \(0\) \(-1\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q-\beta q^{3}-2q^{4}+(-3+\beta )q^{9}+(-1+\cdots)q^{11}+\cdots\)
825.2.f.b 825.f 33.d $4$ $6.588$ \(\Q(i, \sqrt{11})\) \(\Q(\sqrt{-11}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q-\beta _{1}q^{3}-2q^{4}+(3+\beta _{2})q^{9}+(1+2\beta _{2}+\cdots)q^{11}+\cdots\)
825.2.f.c 825.f 33.d $8$ $6.588$ 8.0.619810816.2 None None \(0\) \(2\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(\beta _{3}+\beta _{4})q^{2}+(-\beta _{3}-\beta _{5})q^{3}+(1+\cdots)q^{4}+\cdots\)
825.2.f.d 825.f 33.d $8$ $6.588$ 8.0.619810816.2 None None \(0\) \(2\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(-\beta _{3}-\beta _{4})q^{2}-\beta _{4}q^{3}+(1-\beta _{4}+\cdots)q^{4}+\cdots\)
825.2.f.e 825.f 33.d $16$ $6.588$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None None \(0\) \(-2\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{3}q^{2}+\beta _{4}q^{3}+(1+\beta _{1})q^{4}+\beta _{5}q^{6}+\cdots\)
825.2.f.f 825.f 33.d $16$ $6.588$ \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{6}q^{2}-\beta _{5}q^{3}+(2-\beta _{2})q^{4}+\beta _{14}q^{6}+\cdots\)
825.2.f.g 825.f 33.d $16$ $6.588$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None None \(0\) \(2\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{3}q^{2}-\beta _{4}q^{3}+(1+\beta _{1})q^{4}-\beta _{5}q^{6}+\cdots\)
825.2.j.a 825.j 55.e $8$ $6.588$ \(\Q(\zeta_{24})\) None None \(-4\) \(0\) \(0\) \(-8\) $\mathrm{SU}(2)[C_{4}]$ \(q-\zeta_{24}^{2}q^{2}-\zeta_{24}q^{3}+(-1+\zeta_{24}^{2}+\cdots)q^{4}+\cdots\)
825.2.j.b 825.j 55.e $8$ $6.588$ \(\Q(\zeta_{24})\) None None \(4\) \(0\) \(0\) \(8\) $\mathrm{SU}(2)[C_{4}]$ \(q+(1-\zeta_{24}^{2}+\zeta_{24}^{3})q^{2}-\zeta_{24}q^{3}+\cdots\)
825.2.j.c 825.j 55.e $24$ $6.588$ None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$
825.2.j.d 825.j 55.e $32$ $6.588$ None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$
825.2.k.a 825.k 15.e $4$ $6.588$ \(\Q(\zeta_{8})\) None None \(-4\) \(-4\) \(0\) \(-8\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-1+\zeta_{8}^{2}-\zeta_{8}^{3})q^{2}+(-1-\zeta_{8}^{2}+\cdots)q^{3}+\cdots\)
825.2.k.b 825.k 15.e $4$ $6.588$ \(\Q(\zeta_{8})\) None None \(-4\) \(-4\) \(0\) \(8\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-1+\zeta_{8}^{2}-\zeta_{8}^{3})q^{2}+(-1-\zeta_{8}^{2}+\cdots)q^{3}+\cdots\)
825.2.k.c 825.k 15.e $4$ $6.588$ \(\Q(\zeta_{8})\) None None \(-4\) \(0\) \(0\) \(8\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-1+\zeta_{8}^{2}-\zeta_{8}^{3})q^{2}+(-\zeta_{8}-\zeta_{8}^{2}+\cdots)q^{3}+\cdots\)
825.2.k.d 825.k 15.e $4$ $6.588$ \(\Q(i, \sqrt{6})\) None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+\beta _{1}q^{2}-\beta _{1}q^{3}+\beta _{2}q^{4}-3\beta _{2}q^{6}+\cdots\)
825.2.k.e 825.k 15.e $4$ $6.588$ \(\Q(i, \sqrt{6})\) None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+\beta _{1}q^{2}+\beta _{1}q^{3}+\beta _{2}q^{4}+3\beta _{2}q^{6}+\cdots\)
825.2.k.f 825.k 15.e $4$ $6.588$ \(\Q(\zeta_{8})\) None None \(4\) \(4\) \(0\) \(8\) $\mathrm{SU}(2)[C_{4}]$ \(q+(1-\zeta_{8}^{2}-\zeta_{8}^{3})q^{2}+(1-\zeta_{8}-\zeta_{8}^{3})q^{3}+\cdots\)
825.2.k.g 825.k 15.e $4$ $6.588$ \(\Q(\zeta_{8})\) None None \(4\) \(4\) \(0\) \(-8\) $\mathrm{SU}(2)[C_{4}]$ \(q+(1-\zeta_{8}^{2}-\zeta_{8}^{3})q^{2}+(1+\zeta_{8}^{2}-\zeta_{8}^{3})q^{3}+\cdots\)
825.2.k.h 825.k 15.e $4$ $6.588$ \(\Q(\zeta_{8})\) None None \(4\) \(4\) \(0\) \(8\) $\mathrm{SU}(2)[C_{4}]$ \(q+(1-\zeta_{8}^{2}-\zeta_{8}^{3})q^{2}+(1+\zeta_{8}^{2}+\zeta_{8}^{3})q^{3}+\cdots\)
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