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Label Char Prim Dim $A$ Field CM Traces Fricke sign Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
53.2.a.a 53.a 1.a $1$ $0.423$ \(\Q\) None \(-1\) \(-3\) \(0\) \(-4\) $+$ $\mathrm{SU}(2)$ \(q-q^{2}-3q^{3}-q^{4}+3q^{6}-4q^{7}+3q^{8}+\cdots\)
53.2.a.b 53.a 1.a $3$ $0.423$ 3.3.148.1 None \(-1\) \(3\) \(-2\) \(4\) $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+(1-\beta _{2})q^{3}+(\beta _{1}+\beta _{2})q^{4}+\cdots\)
53.2.b.a 53.b 53.b $4$ $0.423$ 4.0.7168.1 None \(0\) \(0\) \(0\) \(-8\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{2}-\beta _{3}q^{3}+(-1+\beta _{2})q^{4}+(-\beta _{1}+\cdots)q^{5}+\cdots\)
53.2.d.a 53.d 53.d $36$ $0.423$ None \(-8\) \(-9\) \(-5\) \(-5\) $\mathrm{SU}(2)[C_{13}]$
53.2.e.a 53.e 53.e $48$ $0.423$ None \(-13\) \(-13\) \(-13\) \(-5\) $\mathrm{SU}(2)[C_{26}]$
53.3.c.a 53.c 53.c $16$ $1.444$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(-6\) \(-4\) \(-10\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q-\beta _{1}q^{2}+\beta _{11}q^{3}+(-2\beta _{6}+\beta _{14}+\cdots)q^{4}+\cdots\)
53.3.f.a 53.f 53.f $192$ $1.444$ None \(-20\) \(-22\) \(-16\) \(-26\) $\mathrm{SU}(2)[C_{52}]$
53.4.a.a 53.a 1.a $1$ $3.127$ \(\Q\) None \(0\) \(1\) \(-18\) \(2\) $-$ $\mathrm{SU}(2)$ \(q+q^{3}-8q^{4}-18q^{5}+2q^{7}-26q^{9}+\cdots\)
53.4.a.b 53.a 1.a $4$ $3.127$ 4.4.222004.1 None \(-4\) \(-15\) \(-1\) \(-46\) $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{2})q^{2}+(-4-\beta _{2}+\beta _{3})q^{3}+\cdots\)
53.4.a.c 53.a 1.a $8$ $3.127$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(2\) \(16\) \(1\) \(68\) $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(2+\beta _{3})q^{3}+(6+\beta _{6}+\beta _{7})q^{4}+\cdots\)
53.4.b.a 53.b 53.b $12$ $3.127$ \(\mathbb{Q}[x]/(x^{12} + \cdots)\) None \(0\) \(0\) \(0\) \(24\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{2}-\beta _{6}q^{3}+(-5+\beta _{2})q^{4}+(-\beta _{1}+\cdots)q^{5}+\cdots\)
53.4.d.a 53.d 53.d $156$ $3.127$ None \(-11\) \(-15\) \(5\) \(-37\) $\mathrm{SU}(2)[C_{13}]$
53.4.e.a 53.e 53.e $144$ $3.127$ None \(-13\) \(-13\) \(-13\) \(-37\) $\mathrm{SU}(2)[C_{26}]$
53.5.c.a 53.c 53.c $34$ $5.479$ None \(4\) \(8\) \(64\) \(0\) $\mathrm{SU}(2)[C_{4}]$
53.5.f.a 53.f 53.f $408$ $5.479$ None \(-30\) \(-34\) \(-90\) \(-26\) $\mathrm{SU}(2)[C_{52}]$
53.6.a.a 53.a 1.a $9$ $8.500$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None \(-4\) \(-47\) \(-14\) \(-404\) $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+(-5-\beta _{3})q^{3}+(8+\beta _{1}+\beta _{3}+\cdots)q^{4}+\cdots\)
53.6.a.b 53.a 1.a $12$ $8.500$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(8\) \(43\) \(86\) \(380\) $-$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}+(4-\beta _{4})q^{3}+(19-\beta _{1}+\cdots)q^{4}+\cdots\)
53.6.b.a 53.b 53.b $22$ $8.500$ None \(0\) \(0\) \(0\) \(-24\) $\mathrm{SU}(2)[C_{2}]$
53.6.d.a 53.d 53.d $252$ $8.500$ None \(-17\) \(-9\) \(-85\) \(11\) $\mathrm{SU}(2)[C_{13}]$
53.6.e.a 53.e 53.e $264$ $8.500$ None \(-13\) \(-13\) \(-13\) \(11\) $\mathrm{SU}(2)[C_{26}]$
53.7.c.a 53.c 53.c $52$ $12.193$ None \(-4\) \(-34\) \(120\) \(0\) $\mathrm{SU}(2)[C_{4}]$
53.7.f.a 53.f 53.f $624$ $12.193$ None \(-22\) \(8\) \(-146\) \(-26\) $\mathrm{SU}(2)[C_{52}]$
53.8.a.a 53.a 1.a $14$ $16.556$ \(\mathbb{Q}[x]/(x^{14} - \cdots)\) None \(-16\) \(-149\) \(-374\) \(-3076\) $-$ $\mathrm{SU}(2)$ \(q+(-1-\beta _{1})q^{2}+(-11-\beta _{3})q^{3}+(55+\cdots)q^{4}+\cdots\)
53.8.a.b 53.a 1.a $17$ $16.556$ \(\mathbb{Q}[x]/(x^{17} - \cdots)\) None \(8\) \(121\) \(126\) \(2412\) $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(7-\beta _{4})q^{3}+(79+\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
53.8.b.a 53.b 53.b $30$ $16.556$ None \(0\) \(0\) \(0\) \(-664\) $\mathrm{SU}(2)[C_{2}]$
53.8.d.a 53.d 53.d $372$ $16.556$ None \(-5\) \(15\) \(235\) \(651\) $\mathrm{SU}(2)[C_{13}]$
53.8.e.a 53.e 53.e $360$ $16.556$ None \(-13\) \(-13\) \(-13\) \(651\) $\mathrm{SU}(2)[C_{26}]$
53.9.c.a 53.c 53.c $70$ $21.591$ None \(4\) \(68\) \(-1136\) \(0\) $\mathrm{SU}(2)[C_{4}]$
53.9.f.a 53.f 53.f $840$ $21.591$ None \(-30\) \(-94\) \(1110\) \(-26\) $\mathrm{SU}(2)[C_{52}]$
53.10.a.a 53.a 1.a $18$ $27.297$ \(\mathbb{Q}[x]/(x^{18} - \cdots)\) None \(-16\) \(-332\) \(-909\) \(-16732\) $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{2}+(-18-\beta _{1}+\beta _{3}+\cdots)q^{3}+\cdots\)
53.10.a.b 53.a 1.a $21$ $27.297$ None \(32\) \(478\) \(1591\) \(21684\) $-$ $\mathrm{SU}(2)$
53.10.b.a 53.b 53.b $40$ $27.297$ None \(0\) \(0\) \(0\) \(4952\) $\mathrm{SU}(2)[C_{2}]$
53.10.d.a 53.d 53.d $468$ $27.297$ None \(-29\) \(-159\) \(-695\) \(-4965\) $\mathrm{SU}(2)[C_{13}]$
53.10.e.a 53.e 53.e $480$ $27.297$ None \(-13\) \(-13\) \(-13\) \(-4965\) $\mathrm{SU}(2)[C_{26}]$
53.11.c.a 53.c 53.c $88$ $33.674$ None \(-12\) \(-64\) \(-1750\) \(0\) $\mathrm{SU}(2)[C_{4}]$
53.11.f.a 53.f 53.f $1056$ $33.674$ None \(-14\) \(38\) \(1724\) \(-26\) $\mathrm{SU}(2)[C_{52}]$
53.12.a.a 53.a 1.a $22$ $40.722$ None \(-64\) \(-1721\) \(-6844\) \(-116660\) $-$ $\mathrm{SU}(2)$
53.12.a.b 53.a 1.a $25$ $40.722$ None \(32\) \(709\) \(5656\) \(152252\) $+$ $\mathrm{SU}(2)$
53.12.b.a 53.b 53.b $48$ $40.722$ None \(0\) \(0\) \(0\) \(2104\) $\mathrm{SU}(2)[C_{2}]$
53.12.d.a 53.d 53.d $588$ $40.722$ None \(67\) \(495\) \(-8485\) \(-2117\) $\mathrm{SU}(2)[C_{13}]$
53.12.e.a 53.e 53.e $576$ $40.722$ None \(-13\) \(-13\) \(-13\) \(-2117\) $\mathrm{SU}(2)[C_{26}]$
53.14.a.a 53.a 1.a $27$ $56.832$ None \(-64\) \(-3047\) \(-31304\) \(-1140324\) $+$ $\mathrm{SU}(2)$
53.14.a.b 53.a 1.a $30$ $56.832$ None \(128\) \(4243\) \(31196\) \(742060\) $-$ $\mathrm{SU}(2)$
53.14.b.a 53.b 53.b $58$ $56.832$ None \(0\) \(0\) \(0\) \(-398264\) $\mathrm{SU}(2)[C_{2}]$
53.16.a.a 53.a 1.a $31$ $75.628$ None \(-256\) \(-8306\) \(-156539\) \(-7000492\) $-$ $\mathrm{SU}(2)$
53.16.a.b 53.a 1.a $34$ $75.628$ None \(128\) \(13564\) \(155961\) \(6176196\) $+$ $\mathrm{SU}(2)$
53.16.b.a 53.b 53.b $66$ $75.628$ None \(0\) \(0\) \(0\) \(4820616\) $\mathrm{SU}(2)[C_{2}]$
53.18.a.a 53.a 1.a $35$ $97.108$ None \(-256\) \(-30323\) \(-699174\) \(-66984244\) $+$ $\mathrm{SU}(2)$
53.18.a.b 53.a 1.a $38$ $97.108$ None \(512\) \(35287\) \(863326\) \(25252572\) $-$ $\mathrm{SU}(2)$
53.18.b.a 53.b 53.b $76$ $97.108$ None \(0\) \(0\) \(0\) \(-35279688\) $\mathrm{SU}(2)[C_{2}]$
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