Newform search results

Level	Weight	Analytic conductor	\(Nk^2\)	Dim.
Bad \(p\)	Char.	Primitive character	Character order	Coefficient field
Self-twists	CM/RM discriminant	Inner twist count	Is self-dual	Analytic rank
Coefficient ring index	Coefficient ring gens.		Projective image	Projective image type
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Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Inner twists	Rank*	Traces				Fricke sign	Coefficient ring index	Sato-Tate	$q$-expansion
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
1.18.a.a	$1$	$18$	1.a	1.a	$1$	$1$	$1$	$1.832$	\(\Q\)	None	✓	$1$	$1$	\(-528\)	\(-4284\)	\(-1025850\)	\(3225992\)	$+$	$1$	$\mathrm{SU}(2)$	\(q-528q^{2}-4284q^{3}+147712q^{4}+\cdots\)
1.20.a.a	$1$	$20$	1.a	1.a	$1$	$1$	$1$	$2.288$	\(\Q\)	None	✓	$1$	$0$	\(456\)	\(50652\)	\(-2377410\)	\(-16917544\)	$+$	$1$	$\mathrm{SU}(2)$	\(q+456q^{2}+50652q^{3}-316352q^{4}+\cdots\)
1.22.a.a	$1$	$22$	1.a	1.a	$1$	$1$	$1$	$2.795$	\(\Q\)	None	✓	$1$	$1$	\(-288\)	\(-128844\)	\(21640950\)	\(-768078808\)	$+$	$1$	$\mathrm{SU}(2)$	\(q-288q^{2}-128844q^{3}-2014208q^{4}+\cdots\)
1.24.a.a	$1$	$24$	1.a	1.a	$1$	$2$	$2$	$3.352$	\(\Q(\sqrt{144169}) \)	None	✓	$1$	$0$	\(1080\)	\(339480\)	\(73069020\)	\(-1359184400\)	$+$	$2^{3}\cdot 3$	$\mathrm{SU}(2)$	\(q+(540-\beta )q^{2}+(169740+48\beta )q^{3}+\cdots\)
2.18.a.a	$2$	$18$	2.a	1.a	$1$	$1$	$1$	$3.664$	\(\Q\)	None	✓	$1$	$0$	\(256\)	\(6084\)	\(1255110\)	\(-22465912\)	$-$	$1$	$\mathrm{SU}(2)$	\(q+2^{8}q^{2}+78^{2}q^{3}+2^{16}q^{4}+1255110q^{5}+\cdots\)
1.26.a.a	$1$	$26$	1.a	1.a	$1$	$1$	$1$	$3.960$	\(\Q\)	None	✓	$1$	$1$	\(-48\)	\(-195804\)	\(-741989850\)	\(39080597192\)	$+$	$1$	$\mathrm{SU}(2)$	\(q-48q^{2}-195804q^{3}-33552128q^{4}+\cdots\)
2.20.a.a	$2$	$20$	2.a	1.a	$1$	$1$	$1$	$4.576$	\(\Q\)	None	✓	$1$	$0$	\(-512\)	\(-13092\)	\(6546750\)	\(96674264\)	$+$	$1$	$\mathrm{SU}(2)$	\(q-2^{9}q^{2}-13092q^{3}+2^{18}q^{4}+6546750q^{5}+\cdots\)
2.20.a.b	$2$	$20$	2.a	1.a	$1$	$1$	$1$	$4.576$	\(\Q\)	None	✓	$1$	$1$	\(512\)	\(-53028\)	\(-5556930\)	\(-44496424\)	$-$	$1$	$\mathrm{SU}(2)$	\(q+2^{9}q^{2}-53028q^{3}+2^{18}q^{4}-5556930q^{5}+\cdots\)
1.28.a.a	$1$	$28$	1.a	1.a	$1$	$2$	$2$	$4.619$	\(\Q(\sqrt{18209}) \)	None	✓	$1$	$0$	\(-8280\)	\(-1286280\)	\(5443587900\)	\(-175391963600\)	$+$	$2^{3}\cdot 3^{3}$	$\mathrm{SU}(2)$	\(q+(-4140-\beta )q^{2}+(-643140-192\beta )q^{3}+\cdots\)
3.17.b.a	$3$	$17$	3.b	3.b	$2$	$4$	$4$	$4.870$	\(\mathbb{Q}[x]/(x^{4} + \cdots)\)	None		$2$	$0$	\(0\)	\(-2052\)	\(0\)	\(-3141544\)		$2^{6}\cdot 3^{8}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}+(-513+\beta _{1}+\beta _{2})q^{3}+(-3116+\cdots)q^{4}+\cdots\)
1.30.a.a	$1$	$30$	1.a	1.a	$1$	$2$	$2$	$5.328$	\(\Q(\sqrt{51349}) \)	None	✓	$1$	$1$	\(8640\)	\(-4967640\)	\(-17477788500\)	\(-30\!\cdots\!00\)	$+$	$2^{6}\cdot 3$	$\mathrm{SU}(2)$	\(q+(4320-\beta )q^{2}+(-2483820+552\beta )q^{3}+\cdots\)
3.18.a.a	$3$	$18$	3.a	1.a	$1$	$1$	$1$	$5.497$	\(\Q\)	None	✓	$1$	$1$	\(204\)	\(-6561\)	\(-163554\)	\(-20846560\)	$+$	$1$	$\mathrm{SU}(2)$	\(q+204q^{2}-3^{8}q^{3}-89456q^{4}-163554q^{5}+\cdots\)
3.18.a.b	$3$	$18$	3.a	1.a	$1$	$2$	$2$	$5.497$	\(\Q(\sqrt{14569}) \)	None	✓	$1$	$0$	\(594\)	\(13122\)	\(382860\)	\(24471568\)	$-$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q+(297-\beta )q^{2}+3^{8}q^{3}+(88258-594\beta )q^{4}+\cdots\)
2.22.a.a	$2$	$22$	2.a	1.a	$1$	$1$	$1$	$5.590$	\(\Q\)	None	✓	$1$	$1$	\(-1024\)	\(71604\)	\(-28693770\)	\(-853202392\)	$+$	$1$	$\mathrm{SU}(2)$	\(q-2^{10}q^{2}+71604q^{3}+2^{20}q^{4}-28693770q^{5}+\cdots\)
2.22.a.b	$2$	$22$	2.a	1.a	$1$	$1$	$1$	$5.590$	\(\Q\)	None	✓	$1$	$0$	\(1024\)	\(59316\)	\(4975350\)	\(1427425832\)	$-$	$1$	$\mathrm{SU}(2)$	\(q+2^{10}q^{2}+59316q^{3}+2^{20}q^{4}+4975350q^{5}+\cdots\)
1.32.a.a	$1$	$32$	1.a	1.a	$1$	$2$	$2$	$6.088$	\(\mathbb{Q}[x]/(x^{2} - \cdots)\)	None	✓	$1$	$0$	\(39960\)	\(17363160\)	\(-19391218020\)	\(30\!\cdots\!00\)	$+$	$2^{3}\cdot 3$	$\mathrm{SU}(2)$	\(q+(19980-\beta )q^{2}+(8681580-432\beta )q^{3}+\cdots\)
3.19.b.a	$3$	$19$	3.b	3.b	$2$	$1$	$1$	$6.162$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓	$2$	$0$	\(0\)	\(-19683\)	\(0\)	\(77549186\)		$1$	$\mathrm{U}(1)[D_{2}]$	\(q-3^{9}q^{3}+2^{18}q^{4}+77549186q^{7}+\cdots\)
3.19.b.b	$3$	$19$	3.b	3.b	$2$	$4$	$4$	$6.162$	4.0.601940665.1	None		$2$	$0$	\(0\)	\(15876\)	\(0\)	\(-95744152\)		$2^{11}\cdot 3^{11}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{2}+(63^{2}+11\beta _{1}+\beta _{2})q^{3}+(-263384+\cdots)q^{4}+\cdots\)
4.17.b.a	$4$	$17$	4.b	4.b	$2$	$1$	$1$	$6.493$	\(\Q\)	\(\Q(\sqrt{-1}) \)	✓	$2$	$0$	\(256\)	\(0\)	\(329666\)	\(0\)		$1$	$\mathrm{U}(1)[D_{2}]$	\(q+2^{8}q^{2}+2^{16}q^{4}+329666q^{5}+2^{24}q^{8}+\cdots\)
4.17.b.b	$4$	$17$	4.b	4.b	$2$	$6$	$6$	$6.493$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None		$2$	$0$	\(-164\)	\(0\)	\(-506740\)	\(0\)		$2^{30}\cdot 3^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-3^{3}-\beta _{1})q^{2}+(1-3\beta _{1}+\beta _{2})q^{3}+\cdots\)
2.24.a.a	$2$	$24$	2.a	1.a	$1$	$1$	$1$	$6.704$	\(\Q\)	None	✓	$1$	$0$	\(-2048\)	\(-505908\)	\(-90135570\)	\(6872255096\)	$+$	$1$	$\mathrm{SU}(2)$	\(q-2^{11}q^{2}-505908q^{3}+2^{22}q^{4}+\cdots\)
3.20.a.a	$3$	$20$	3.a	1.a	$1$	$1$	$1$	$6.865$	\(\Q\)	None	✓	$1$	$1$	\(-1104\)	\(19683\)	\(3516270\)	\(-195590584\)	$-$	$1$	$\mathrm{SU}(2)$	\(q-1104q^{2}+3^{9}q^{3}+694528q^{4}+\cdots\)
3.20.a.b	$3$	$20$	3.a	1.a	$1$	$2$	$2$	$6.865$	\(\Q(\sqrt{87481}) \)	None	✓	$1$	$0$	\(702\)	\(-39366\)	\(6016140\)	\(113892064\)	$+$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q+(351-\beta )q^{2}-3^{9}q^{3}+(386242-702\beta )q^{4}+\cdots\)
4.18.a.a	$4$	$18$	4.a	1.a	$1$	$2$	$2$	$7.329$	\(\Q(\sqrt{9361}) \)	None	✓	$1$	$0$	\(0\)	\(-5880\)	\(604044\)	\(25350160\)	$-$	$2^{7}\cdot 3$	$\mathrm{SU}(2)$	\(q+(-2940-\beta )q^{3}+(302022+6^{2}\beta )q^{5}+\cdots\)
3.21.b.a	$3$	$21$	3.b	3.b	$2$	$6$	$6$	$7.605$	\(\mathbb{Q}[x]/(x^{6} + \cdots)\)	None		$2$	$0$	\(0\)	\(-4122\)	\(0\)	\(559607916\)		$2^{14}\cdot 3^{19}\cdot 5^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}+(-687+11\beta _{1}+\beta _{2})q^{3}+\cdots\)
2.26.a.a	$2$	$26$	2.a	1.a	$1$	$1$	$1$	$7.920$	\(\Q\)	None	✓	$1$	$1$	\(-4096\)	\(97956\)	\(341005350\)	\(-40882637368\)	$+$	$1$	$\mathrm{SU}(2)$	\(q-2^{12}q^{2}+97956q^{3}+2^{24}q^{4}+341005350q^{5}+\cdots\)
2.26.a.b	$2$	$26$	2.a	1.a	$1$	$2$	$2$	$7.920$	\(\Q(\sqrt{106705}) \)	None	✓	$1$	$0$	\(8192\)	\(379848\)	\(741953100\)	\(-376536944\)	$-$	$2^{7}\cdot 3\cdot 5^{2}$	$\mathrm{SU}(2)$	\(q+2^{12}q^{2}+(189924-\beta )q^{3}+2^{24}q^{4}+\cdots\)
5.17.c.a	$5$	$17$	5.c	5.c	$4$	$14$	$7$	$8.116$	\(\mathbb{Q}[x]/(x^{14} + \cdots)\)	None		$2$	$0$	\(-2\)	\(7908\)	\(192880\)	\(-386452\)		$2^{34}\cdot 3^{10}\cdot 5^{18}$	$\mathrm{SU}(2)[C_{4}]$	\(q-\beta _{2}q^{2}+(565-3\beta _{1}-565\beta _{4}-\beta _{6}+\cdots)q^{3}+\cdots\)
4.19.b.a	$4$	$19$	4.b	4.b	$2$	$8$	$8$	$8.215$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None		$2$	$0$	\(84\)	\(0\)	\(860880\)	\(0\)		$2^{56}\cdot 3^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(11-\beta _{1})q^{2}-\beta _{2}q^{3}+(44007-9\beta _{1}+\cdots)q^{4}+\cdots\)
3.22.a.a	$3$	$22$	3.a	1.a	$1$	$1$	$1$	$8.384$	\(\Q\)	None	✓	$1$	$1$	\(-2844\)	\(-59049\)	\(3109950\)	\(363303920\)	$+$	$1$	$\mathrm{SU}(2)$	\(q-2844q^{2}-3^{10}q^{3}+5991184q^{4}+\cdots\)
3.22.a.b	$3$	$22$	3.a	1.a	$1$	$1$	$1$	$8.384$	\(\Q\)	None	✓	$1$	$1$	\(1728\)	\(-59049\)	\(-41512770\)	\(538429808\)	$+$	$1$	$\mathrm{SU}(2)$	\(q+12^{3}q^{2}-3^{10}q^{3}+888832q^{4}+\cdots\)
3.22.a.c	$3$	$22$	3.a	1.a	$1$	$2$	$2$	$8.384$	\(\Q(\sqrt{649}) \)	None	✓	$1$	$0$	\(666\)	\(118098\)	\(996876\)	\(679896112\)	$-$	$2\cdot 3^{2}\cdot 7$	$\mathrm{SU}(2)$	\(q+(333-\beta )q^{2}+3^{10}q^{3}+(589618+\cdots)q^{4}+\cdots\)
4.20.a.a	$4$	$20$	4.a	1.a	$1$	$1$	$1$	$9.153$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-36\)	\(-196290\)	\(-35905576\)	$-$	$1$	$\mathrm{SU}(2)$	\(q-6^{2}q^{3}-196290q^{5}-35905576q^{7}+\cdots\)
5.18.a.a	$5$	$18$	5.a	1.a	$1$	$2$	$2$	$9.161$	\(\Q(\sqrt{39}) \)	None	✓	$1$	$1$	\(680\)	\(-10980\)	\(-781250\)	\(-22820700\)	$+$	$2^{2}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+(340+\beta )q^{2}+(-5490-52\beta )q^{3}+\cdots\)
5.18.a.b	$5$	$18$	5.a	1.a	$1$	$3$	$3$	$9.161$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓	$1$	$0$	\(118\)	\(15944\)	\(1171875\)	\(2139308\)	$-$	$2^{3}\cdot 5$	$\mathrm{SU}(2)$	\(q+(39-\beta _{1})q^{2}+(5317+8\beta _{1}+\beta _{2})q^{3}+\cdots\)
5.18.b.a	$5$	$18$	5.b	5.b	$2$	$8$	$8$	$9.161$	\(\mathbb{Q}[x]/(x^{8} + \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(379200\)	\(0\)		$2^{21}\cdot 3^{8}\cdot 5^{12}\cdot 11$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}-\beta _{2}q^{3}+(-72387+\beta _{3}+\cdots)q^{4}+\cdots\)
3.23.b.a	$3$	$23$	3.b	3.b	$2$	$6$	$6$	$9.201$	\(\mathbb{Q}[x]/(x^{6} + \cdots)\)	None		$2$	$0$	\(0\)	\(86670\)	\(0\)	\(-3447063060\)		$2^{21}\cdot 3^{22}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}+(14445-8\beta _{1}+\beta _{2})q^{3}+\cdots\)
2.28.a.a	$2$	$28$	2.a	1.a	$1$	$1$	$1$	$9.237$	\(\Q\)	None	✓	$1$	$0$	\(-8192\)	\(3984828\)	\(-2851889250\)	\(368721063704\)	$+$	$1$	$\mathrm{SU}(2)$	\(q-2^{13}q^{2}+3984828q^{3}+2^{26}q^{4}+\cdots\)
2.28.a.b	$2$	$28$	2.a	1.a	$1$	$1$	$1$	$9.237$	\(\Q\)	None	✓	$1$	$1$	\(8192\)	\(-1016388\)	\(-3341197410\)	\(-51021361384\)	$-$	$1$	$\mathrm{SU}(2)$	\(q+2^{13}q^{2}-1016388q^{3}+2^{26}q^{4}+\cdots\)
6.17.b.a	$6$	$17$	6.b	3.b	$2$	$6$	$6$	$9.739$	\(\mathbb{Q}[x]/(x^{6} + \cdots)\)	None		$2$	$0$	\(0\)	\(6006\)	\(0\)	\(-167892\)		$2^{27}\cdot 3^{13}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{2}+(1001+\beta _{1}-\beta _{2})q^{3}-2^{15}q^{4}+\cdots\)
3.24.a.a	$3$	$24$	3.a	1.a	$1$	$1$	$1$	$10.056$	\(\Q\)	None	✓	$1$	$1$	\(1128\)	\(177147\)	\(-48863730\)	\(-1723688680\)	$-$	$1$	$\mathrm{SU}(2)$	\(q+1128q^{2}+3^{11}q^{3}-7116224q^{4}+\cdots\)
3.24.a.b	$3$	$24$	3.a	1.a	$1$	$2$	$2$	$10.056$	\(\Q(\sqrt{530401}) \)	None	✓	$1$	$0$	\(-1242\)	\(-354294\)	\(-46808820\)	\(-211963904\)	$+$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q+(-621-\beta )q^{2}-3^{11}q^{3}+(-3229358+\cdots)q^{4}+\cdots\)
4.21.b.a	$4$	$21$	4.b	4.b	$2$	$1$	$1$	$10.141$	\(\Q\)	\(\Q(\sqrt{-1}) \)	✓	$2$	$0$	\(-1024\)	\(0\)	\(-19306574\)	\(0\)		$1$	$\mathrm{U}(1)[D_{2}]$	\(q-2^{10}q^{2}+2^{20}q^{4}-19306574q^{5}+\cdots\)
4.21.b.b	$4$	$21$	4.b	4.b	$2$	$8$	$8$	$10.141$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None		$2$	$0$	\(396\)	\(0\)	\(18568080\)	\(0\)		$2^{56}\cdot 3^{5}\cdot 5^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(7^{2}+\beta _{1})q^{2}+(6-12\beta _{1}-\beta _{2})q^{3}+\cdots\)
5.19.c.a	$5$	$19$	5.c	5.c	$4$	$16$	$8$	$10.269$	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)	None		$2$	$0$	\(510\)	\(-20130\)	\(3145170\)	\(78767350\)		$2^{39}\cdot 3^{12}\cdot 5^{25}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(2^{5}+\beta _{2}+2^{5}\beta _{4})q^{2}+(-1259-5\beta _{1}+\cdots)q^{3}+\cdots\)
2.30.a.a	$2$	$30$	2.a	1.a	$1$	$1$	$1$	$10.656$	\(\Q\)	None	✓	$1$	$1$	\(-16384\)	\(-2792556\)	\(6651856470\)	\(14\!\cdots\!48\)	$+$	$1$	$\mathrm{SU}(2)$	\(q-2^{14}q^{2}-2792556q^{3}+2^{28}q^{4}+\cdots\)
2.30.a.b	$2$	$30$	2.a	1.a	$1$	$1$	$1$	$10.656$	\(\Q\)	None	✓	$1$	$0$	\(16384\)	\(4782996\)	\(6065841750\)	\(904018883432\)	$-$	$1$	$\mathrm{SU}(2)$	\(q+2^{14}q^{2}+4782996q^{3}+2^{28}q^{4}+\cdots\)
3.25.b.a	$3$	$25$	3.b	3.b	$2$	$1$	$1$	$10.949$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓	$2$	$0$	\(0\)	\(531441\)	\(0\)	\(-4119710398\)		$1$	$\mathrm{U}(1)[D_{2}]$	\(q+3^{12}q^{3}+2^{24}q^{4}-4119710398q^{7}+\cdots\)
3.25.b.b	$3$	$25$	3.b	3.b	$2$	$6$	$6$	$10.949$	\(\mathbb{Q}[x]/(x^{6} + \cdots)\)	None		$2$	$0$	\(0\)	\(-616842\)	\(0\)	\(1988064876\)		$2^{13}\cdot 3^{24}\cdot 5^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}+(-102807+2^{4}\beta _{1}+\beta _{2}+\cdots)q^{3}+\cdots\)
6.18.a.a	$6$	$18$	6.a	1.a	$1$	$1$	$1$	$10.993$	\(\Q\)	None	✓	$1$	$1$	\(-256\)	\(-6561\)	\(645150\)	\(3974432\)	$+$	$1$	$\mathrm{SU}(2)$	\(q-2^{8}q^{2}-3^{8}q^{3}+2^{16}q^{4}+645150q^{5}+\cdots\)

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