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Label Char Prim Dim $A$ Field CM Traces Fricke sign Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
3.19.b.a 3.b 3.b $1$ $6.162$ \(\Q\) \(\Q(\sqrt{-3}) \) \(0\) \(-19683\) \(0\) \(77549186\) $\mathrm{U}(1)[D_{2}]$ \(q-3^{9}q^{3}+2^{18}q^{4}+77549186q^{7}+\cdots\)
4.17.b.a 4.b 4.b $1$ $6.493$ \(\Q\) \(\Q(\sqrt{-1}) \) \(256\) \(0\) \(329666\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+2^{8}q^{2}+2^{16}q^{4}+329666q^{5}+2^{24}q^{8}+\cdots\)
4.21.b.a 4.b 4.b $1$ $10.141$ \(\Q\) \(\Q(\sqrt{-1}) \) \(-1024\) \(0\) \(-19306574\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q-2^{10}q^{2}+2^{20}q^{4}-19306574q^{5}+\cdots\)
3.25.b.a 3.b 3.b $1$ $10.949$ \(\Q\) \(\Q(\sqrt{-3}) \) \(0\) \(531441\) \(0\) \(-4119710398\) $\mathrm{U}(1)[D_{2}]$ \(q+3^{12}q^{3}+2^{24}q^{4}-4119710398q^{7}+\cdots\)
7.17.b.a 7.b 7.b $1$ $11.363$ \(\Q\) \(\Q(\sqrt{-7}) \) \(449\) \(0\) \(0\) \(5764801\) $\mathrm{U}(1)[D_{2}]$ \(q+449q^{2}+136065q^{4}+7^{8}q^{7}+31667521q^{8}+\cdots\)
8.17.d.a 8.d 8.d $1$ $12.986$ \(\Q\) \(\Q(\sqrt{-2}) \) \(256\) \(-11966\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+2^{8}q^{2}-11966q^{3}+2^{16}q^{4}-3063296q^{6}+\cdots\)
7.19.b.a 7.b 7.b $1$ $14.377$ \(\Q\) \(\Q(\sqrt{-7}) \) \(-999\) \(0\) \(0\) \(-40353607\) $\mathrm{U}(1)[D_{2}]$ \(q-999q^{2}+735857q^{4}-7^{9}q^{7}-473239287q^{8}+\cdots\)
4.25.b.a 4.b 4.b $1$ $14.599$ \(\Q\) \(\Q(\sqrt{-1}) \) \(4096\) \(0\) \(64250786\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+2^{12}q^{2}+2^{24}q^{4}+64250786q^{5}+\cdots\)
8.19.d.a 8.d 8.d $1$ $16.431$ \(\Q\) \(\Q(\sqrt{-2}) \) \(-512\) \(-3266\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q-2^{9}q^{2}-3266q^{3}+2^{18}q^{4}+1672192q^{6}+\cdots\)
3.31.b.a 3.b 3.b $1$ $17.104$ \(\Q\) \(\Q(\sqrt{-3}) \) \(0\) \(-14348907\) \(0\) \(-79\!\cdots\!86\) $\mathrm{U}(1)[D_{2}]$ \(q-3^{15}q^{3}+2^{30}q^{4}-7945347009886q^{7}+\cdots\)
7.21.b.a 7.b 7.b $1$ $17.746$ \(\Q\) \(\Q(\sqrt{-7}) \) \(1201\) \(0\) \(0\) \(282475249\) $\mathrm{U}(1)[D_{2}]$ \(q+1201q^{2}+393825q^{4}+7^{10}q^{7}+\cdots\)
11.17.b.a 11.b 11.b $1$ $17.856$ \(\Q\) \(\Q(\sqrt{-11}) \) \(0\) \(-353\) \(543551\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q-353q^{3}+2^{16}q^{4}+543551q^{5}+\cdots\)
12.17.c.a 12.c 3.b $1$ $19.479$ \(\Q\) \(\Q(\sqrt{-3}) \) \(0\) \(6561\) \(0\) \(4743554\) $\mathrm{U}(1)[D_{2}]$ \(q+3^{8}q^{3}+4743554q^{7}+3^{16}q^{9}+\cdots\)
4.29.b.a 4.b 4.b $1$ $19.867$ \(\Q\) \(\Q(\sqrt{-1}) \) \(-16384\) \(0\) \(11167097746\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q-2^{14}q^{2}+2^{28}q^{4}+11167097746q^{5}+\cdots\)
8.21.d.a 8.d 8.d $1$ $20.281$ \(\Q\) \(\Q(\sqrt{-2}) \) \(1024\) \(114226\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+2^{10}q^{2}+114226q^{3}+2^{20}q^{4}+\cdots\)
7.23.b.a 7.b 7.b $1$ $21.470$ \(\Q\) \(\Q(\sqrt{-7}) \) \(393\) \(0\) \(0\) \(-1977326743\) $\mathrm{U}(1)[D_{2}]$ \(q+393q^{2}-4039855q^{4}-7^{11}q^{7}+\cdots\)
11.19.b.a 11.b 11.b $1$ $22.592$ \(\Q\) \(\Q(\sqrt{-11}) \) \(0\) \(-20870\) \(-3063526\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q-20870q^{3}+2^{18}q^{4}-3063526q^{5}+\cdots\)
15.17.d.a 15.d 15.d $1$ $24.349$ \(\Q\) \(\Q(\sqrt{-15}) \) \(-223\) \(6561\) \(390625\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q-223q^{2}+3^{8}q^{3}-15807q^{4}+5^{8}q^{5}+\cdots\)
15.17.d.b 15.d 15.d $1$ $24.349$ \(\Q\) \(\Q(\sqrt{-15}) \) \(223\) \(-6561\) \(-390625\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+223q^{2}-3^{8}q^{3}-15807q^{4}-5^{8}q^{5}+\cdots\)
8.23.d.a 8.d 8.d $1$ $24.537$ \(\Q\) \(\Q(\sqrt{-2}) \) \(-2048\) \(-199058\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q-2^{11}q^{2}-199058q^{3}+2^{22}q^{4}+\cdots\)
9.22.a.b 9.a 1.a $1$ $25.153$ \(\Q\) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(1123983020\) $+$ $N(\mathrm{U}(1))$ \(q-2^{21}q^{4}+1123983020q^{7}-370076825230q^{13}+\cdots\)
7.25.b.a 7.b 7.b $1$ $25.548$ \(\Q\) \(\Q(\sqrt{-7}) \) \(-5983\) \(0\) \(0\) \(13841287201\) $\mathrm{U}(1)[D_{2}]$ \(q-5983q^{2}+19019073q^{4}+7^{12}q^{7}+\cdots\)
11.21.b.a 11.b 11.b $1$ $27.887$ \(\Q\) \(\Q(\sqrt{-11}) \) \(0\) \(107527\) \(-10525249\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+107527q^{3}+2^{20}q^{4}-10525249q^{5}+\cdots\)
8.25.d.a 8.d 8.d $1$ $29.197$ \(\Q\) \(\Q(\sqrt{-2}) \) \(4096\) \(-629918\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+2^{12}q^{2}-629918q^{3}+2^{24}q^{4}+\cdots\)
7.27.b.a 7.b 7.b $1$ $29.981$ \(\Q\) \(\Q(\sqrt{-7}) \) \(16377\) \(0\) \(0\) \(-96889010407\) $\mathrm{U}(1)[D_{2}]$ \(q+16377q^{2}+201097265q^{4}-7^{13}q^{7}+\cdots\)
12.21.c.a 12.c 3.b $1$ $30.422$ \(\Q\) \(\Q(\sqrt{-3}) \) \(0\) \(59049\) \(0\) \(-77335774\) $\mathrm{U}(1)[D_{2}]$ \(q+3^{10}q^{3}-77335774q^{7}+3^{20}q^{9}+\cdots\)
15.19.d.a 15.d 15.d $1$ $30.808$ \(\Q\) \(\Q(\sqrt{-15}) \) \(-781\) \(19683\) \(-1953125\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q-781q^{2}+3^{9}q^{3}+347817q^{4}-5^{9}q^{5}+\cdots\)
15.19.d.b 15.d 15.d $1$ $30.808$ \(\Q\) \(\Q(\sqrt{-15}) \) \(781\) \(-19683\) \(1953125\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+781q^{2}-3^{9}q^{3}+347817q^{4}+5^{9}q^{5}+\cdots\)
19.17.b.a 19.b 19.b $1$ $30.842$ \(\Q\) \(\Q(\sqrt{-19}) \) \(0\) \(0\) \(-697729\) \(-11251873\) $\mathrm{U}(1)[D_{2}]$ \(q+2^{16}q^{4}-697729q^{5}-11251873q^{7}+\cdots\)
20.17.d.a 20.d 20.d $1$ $32.465$ \(\Q\) \(\Q(\sqrt{-5}) \) \(-256\) \(-11842\) \(390625\) \(7835518\) $\mathrm{U}(1)[D_{2}]$ \(q-2^{8}q^{2}-11842q^{3}+2^{16}q^{4}+5^{8}q^{5}+\cdots\)
20.17.d.b 20.d 20.d $1$ $32.465$ \(\Q\) \(\Q(\sqrt{-5}) \) \(256\) \(11842\) \(390625\) \(-7835518\) $\mathrm{U}(1)[D_{2}]$ \(q+2^{8}q^{2}+11842q^{3}+2^{16}q^{4}+5^{8}q^{5}+\cdots\)
16.19.c.a 16.c 4.b $1$ $32.862$ \(\Q\) \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(1844154\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+1844154q^{5}+3^{18}q^{9}-8483805110q^{13}+\cdots\)
11.23.b.a 11.b 11.b $1$ $33.738$ \(\Q\) \(\Q(\sqrt{-11}) \) \(0\) \(-349805\) \(87113399\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q-349805q^{3}+2^{22}q^{4}+87113399q^{5}+\cdots\)
21.17.h.a 21.h 21.h $2$ $34.088$ \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) \(0\) \(-6561\) \(0\) \(-11472481\) $\mathrm{U}(1)[D_{6}]$ \(q-3^{8}\zeta_{6}q^{3}-2^{16}\zeta_{6}q^{4}+(-5405345+\cdots)q^{7}+\cdots\)
8.27.d.a 8.d 8.d $1$ $34.263$ \(\Q\) \(\Q(\sqrt{-2}) \) \(-8192\) \(3051358\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q-2^{13}q^{2}+3051358q^{3}+2^{26}q^{4}+\cdots\)
7.29.b.a 7.b 7.b $1$ $34.768$ \(\Q\) \(\Q(\sqrt{-7}) \) \(-25199\) \(0\) \(0\) \(678223072849\) $\mathrm{U}(1)[D_{2}]$ \(q-25199q^{2}+366554145q^{4}+7^{14}q^{7}+\cdots\)
20.18.e.a 20.e 20.e $2$ $36.644$ \(\Q(\sqrt{-1}) \) \(\Q(\sqrt{-1}) \) \(-512\) \(0\) \(48956\) \(0\) $\mathrm{U}(1)[D_{4}]$ \(q+(-2^{8}-2^{8}i)q^{2}+2^{17}iq^{4}+(24478+\cdots)q^{5}+\cdots\)
12.23.c.a 12.c 3.b $1$ $36.805$ \(\Q\) \(\Q(\sqrt{-3}) \) \(0\) \(-177147\) \(0\) \(-3954581662\) $\mathrm{U}(1)[D_{2}]$ \(q-3^{11}q^{3}-3954581662q^{7}+3^{22}q^{9}+\cdots\)
23.17.b.a 23.b 23.b $3$ $37.335$ 3.3.621.1 \(\Q(\sqrt{-23}) \) \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+(48\beta _{1}-47\beta _{2})q^{2}+(-1075\beta _{1}+1359\beta _{2})q^{3}+\cdots\)
15.21.d.a 15.d 15.d $1$ $38.027$ \(\Q\) \(\Q(\sqrt{-15}) \) \(-1673\) \(-59049\) \(-9765625\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q-1673q^{2}-3^{10}q^{3}+1750353q^{4}+\cdots\)
15.21.d.b 15.d 15.d $1$ $38.027$ \(\Q\) \(\Q(\sqrt{-15}) \) \(1673\) \(59049\) \(9765625\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+1673q^{2}+3^{10}q^{3}+1750353q^{4}+\cdots\)
21.18.g.a 21.g 21.g $2$ $38.477$ \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) \(0\) \(19683\) \(0\) \(-2758181\) $\mathrm{U}(1)[D_{6}]$ \(q+(3^{8}+3^{8}\zeta_{6})q^{3}+(-2^{17}+2^{17}\zeta_{6})q^{4}+\cdots\)
24.17.h.a 24.h 24.h $1$ $38.958$ \(\Q\) \(\Q(\sqrt{-6}) \) \(-256\) \(-6561\) \(31294\) \(11491202\) $\mathrm{U}(1)[D_{2}]$ \(q-2^{8}q^{2}-3^{8}q^{3}+2^{16}q^{4}+31294q^{5}+\cdots\)
24.17.h.b 24.h 24.h $1$ $38.958$ \(\Q\) \(\Q(\sqrt{-6}) \) \(256\) \(6561\) \(-31294\) \(11491202\) $\mathrm{U}(1)[D_{2}]$ \(q+2^{8}q^{2}+3^{8}q^{3}+2^{16}q^{4}-31294q^{5}+\cdots\)
19.19.b.a 19.b 19.b $1$ $39.023$ \(\Q\) \(\Q(\sqrt{-19}) \) \(0\) \(0\) \(2373786\) \(11683330\) $\mathrm{U}(1)[D_{2}]$ \(q+2^{18}q^{4}+2373786q^{5}+11683330q^{7}+\cdots\)
8.29.d.a 8.d 8.d $1$ $39.735$ \(\Q\) \(\Q(\sqrt{-2}) \) \(16384\) \(-433454\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+2^{14}q^{2}-433454q^{3}+2^{28}q^{4}+\cdots\)
7.31.b.a 7.b 7.b $1$ $39.910$ \(\Q\) \(\Q(\sqrt{-7}) \) \(10089\) \(0\) \(0\) \(-47\!\cdots\!43\) $\mathrm{U}(1)[D_{2}]$ \(q+10089q^{2}-971953903q^{4}-7^{15}q^{7}+\cdots\)
11.25.b.a 11.b 11.b $1$ $40.146$ \(\Q\) \(\Q(\sqrt{-11}) \) \(0\) \(781282\) \(176017826\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+781282q^{3}+2^{24}q^{4}+176017826q^{5}+\cdots\)
20.19.d.a 20.d 20.d $1$ $41.077$ \(\Q\) \(\Q(\sqrt{-5}) \) \(-512\) \(11044\) \(-1953125\) \(-41066404\) $\mathrm{U}(1)[D_{2}]$ \(q-2^{9}q^{2}+11044q^{3}+2^{18}q^{4}-5^{9}q^{5}+\cdots\)
20.19.d.b 20.d 20.d $1$ $41.077$ \(\Q\) \(\Q(\sqrt{-5}) \) \(512\) \(-11044\) \(-1953125\) \(41066404\) $\mathrm{U}(1)[D_{2}]$ \(q+2^{9}q^{2}-11044q^{3}+2^{18}q^{4}-5^{9}q^{5}+\cdots\)
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