LMFDB - Newform search results

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Level	Weight	Analytic conductor	$Nk^2$	Dim.

Bad $p$	Char.	Primitive character	Character order	Coefficient field

Self-twist	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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Results (1-50 of at least 1000)

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Label	Dim.	$A$	Field	CM	Traces				Fricke sign	$q$-expansion
Label	Dim.	$A$	Field	CM	$a_2$	$a_3$	$a_5$	$a_7$	Fricke sign	$q$-expansion
1.18.a.a	$1$	$1.832$	$\Q$	None	$-528$	$-4284$	$-1025850$	$3225992$	$+$	$q-528q^{2}-4284q^{3}+147712q^{4}+\cdots$
1.20.a.a	$1$	$2.288$	$\Q$	None	$456$	$50652$	$-2377410$	$-16917544$	$+$	$q+456q^{2}+50652q^{3}-316352q^{4}+\cdots$
1.22.a.a	$1$	$2.795$	$\Q$	None	$-288$	$-128844$	$21640950$	$-768078808$	$+$	$q-288q^{2}-128844q^{3}-2014208q^{4}+\cdots$
1.24.a.a	$2$	$3.352$	$\Q(\sqrt{144169}) $	None	$1080$	$339480$	$73069020$	$-1359184400$	$+$	$q+(540-\beta )q^{2}+(169740+48\beta )q^{3}+\cdots$
2.18.a.a	$1$	$3.664$	$\Q$	None	$256$	$6084$	$1255110$	$-22465912$	$-$	$q+2^{8}q^{2}+78^{2}q^{3}+2^{16}q^{4}+1255110q^{5}+\cdots$
1.26.a.a	$1$	$3.960$	$\Q$	None	$-48$	$-195804$	$-741989850$	$39080597192$	$+$	$q-48q^{2}-195804q^{3}-33552128q^{4}+\cdots$
2.20.a.a	$1$	$4.576$	$\Q$	None	$-512$	$-13092$	$6546750$	$96674264$	$+$	$q-2^{9}q^{2}-13092q^{3}+2^{18}q^{4}+6546750q^{5}+\cdots$
2.20.a.b	$1$	$4.576$	$\Q$	None	$512$	$-53028$	$-5556930$	$-44496424$	$-$	$q+2^{9}q^{2}-53028q^{3}+2^{18}q^{4}-5556930q^{5}+\cdots$
1.28.a.a	$2$	$4.619$	$\Q(\sqrt{18209}) $	None	$-8280$	$-1286280$	$5443587900$	$-175391963600$	$+$	$q+(-4140-\beta )q^{2}+(-643140-192\beta )q^{3}+\cdots$
3.17.b.a	$4$	$4.870$	$\mathbb{Q}[x]/(x^{4} + \cdots)$	None	$0$	$-2052$	$0$	$-3141544$		$q+\beta _{1}q^{2}+(-513+\beta _{1}+\beta _{2})q^{3}+(-3116+\cdots)q^{4}+\cdots$
1.30.a.a	$2$	$5.328$	$\Q(\sqrt{51349}) $	None	$8640$	$-4967640$	$-17477788500$	$-3\!\cdots\!00$	$+$	$q+(4320-\beta )q^{2}+(-2483820+552\beta )q^{3}+\cdots$
3.18.a.a	$1$	$5.497$	$\Q$	None	$204$	$-6561$	$-163554$	$-20846560$	$+$	$q+204q^{2}-3^{8}q^{3}-89456q^{4}-163554q^{5}+\cdots$
3.18.a.b	$2$	$5.497$	$\Q(\sqrt{14569}) $	None	$594$	$13122$	$382860$	$24471568$	$-$	$q+(297-\beta )q^{2}+3^{8}q^{3}+(88258-594\beta )q^{4}+\cdots$
2.22.a.a	$1$	$5.590$	$\Q$	None	$-1024$	$71604$	$-28693770$	$-853202392$	$+$	$q-2^{10}q^{2}+71604q^{3}+2^{20}q^{4}-28693770q^{5}+\cdots$
2.22.a.b	$1$	$5.590$	$\Q$	None	$1024$	$59316$	$4975350$	$1427425832$	$-$	$q+2^{10}q^{2}+59316q^{3}+2^{20}q^{4}+4975350q^{5}+\cdots$
1.32.a.a	$2$	$6.088$	$\mathbb{Q}[x]/(x^{2} - \cdots)$	None	$39960$	$17363160$	$-19391218020$	$30\!\cdots\!00$	$+$	$q+(19980-\beta )q^{2}+(8681580-432\beta )q^{3}+\cdots$
3.19.b.a	$1$	$6.162$	$\Q$	$\Q(\sqrt{-3}) $	$0$	$-19683$	$0$	$77549186$		$q-3^{9}q^{3}+2^{18}q^{4}+77549186q^{7}+\cdots$
3.19.b.b	$4$	$6.162$	4.0.601940665.1	None	$0$	$15876$	$0$	$-95744152$		$q-\beta _{1}q^{2}+(63^{2}+11\beta _{1}+\beta _{2})q^{3}+(-263384+\cdots)q^{4}+\cdots$
4.17.b.a	$1$	$6.493$	$\Q$	$\Q(\sqrt{-1}) $	$256$	$0$	$329666$	$0$		$q+2^{8}q^{2}+2^{16}q^{4}+329666q^{5}+2^{24}q^{8}+\cdots$
4.17.b.b	$6$	$6.493$	$\mathbb{Q}[x]/(x^{6} - \cdots)$	None	$-164$	$0$	$-506740$	$0$		$q+(-3^{3}-\beta _{1})q^{2}+(1-3\beta _{1}+\beta _{2})q^{3}+\cdots$
2.24.a.a	$1$	$6.704$	$\Q$	None	$-2048$	$-505908$	$-90135570$	$6872255096$	$+$	$q-2^{11}q^{2}-505908q^{3}+2^{22}q^{4}+\cdots$
3.20.a.a	$1$	$6.865$	$\Q$	None	$-1104$	$19683$	$3516270$	$-195590584$	$-$	$q-1104q^{2}+3^{9}q^{3}+694528q^{4}+\cdots$
3.20.a.b	$2$	$6.865$	$\Q(\sqrt{87481}) $	None	$702$	$-39366$	$6016140$	$113892064$	$+$	$q+(351-\beta )q^{2}-3^{9}q^{3}+(386242-702\beta )q^{4}+\cdots$
4.18.a.a	$2$	$7.329$	$\Q(\sqrt{9361}) $	None	$0$	$-5880$	$604044$	$25350160$	$-$	$q+(-2940-\beta )q^{3}+(302022+6^{2}\beta )q^{5}+\cdots$
3.21.b.a	$6$	$7.605$	$\mathbb{Q}[x]/(x^{6} + \cdots)$	None	$0$	$-4122$	$0$	$559607916$		$q+\beta _{1}q^{2}+(-687+11\beta _{1}+\beta _{2})q^{3}+\cdots$
2.26.a.a	$1$	$7.920$	$\Q$	None	$-4096$	$97956$	$341005350$	$-40882637368$	$+$	$q-2^{12}q^{2}+97956q^{3}+2^{24}q^{4}+341005350q^{5}+\cdots$
2.26.a.b	$2$	$7.920$	$\Q(\sqrt{106705}) $	None	$8192$	$379848$	$741953100$	$-376536944$	$-$	$q+2^{12}q^{2}+(189924-\beta )q^{3}+2^{24}q^{4}+\cdots$
5.17.c.a	$14$	$8.116$	$\mathbb{Q}[x]/(x^{14} + \cdots)$	None	$-2$	$7908$	$192880$	$-386452$		$q-\beta _{2}q^{2}+(565-3\beta _{1}-565\beta _{4}-\beta _{6}+\cdots)q^{3}+\cdots$
4.19.b.a	$8$	$8.215$	$\mathbb{Q}[x]/(x^{8} - \cdots)$	None	$84$	$0$	$860880$	$0$		$q+(11-\beta _{1})q^{2}-\beta _{2}q^{3}+(44007-9\beta _{1}+\cdots)q^{4}+\cdots$
3.22.a.a	$1$	$8.384$	$\Q$	None	$-2844$	$-59049$	$3109950$	$363303920$	$+$	$q-2844q^{2}-3^{10}q^{3}+5991184q^{4}+\cdots$
3.22.a.b	$1$	$8.384$	$\Q$	None	$1728$	$-59049$	$-41512770$	$538429808$	$+$	$q+12^{3}q^{2}-3^{10}q^{3}+888832q^{4}+\cdots$
3.22.a.c	$2$	$8.384$	$\Q(\sqrt{649}) $	None	$666$	$118098$	$996876$	$679896112$	$-$	$q+(333-\beta )q^{2}+3^{10}q^{3}+(589618+\cdots)q^{4}+\cdots$
4.20.a.a	$1$	$9.153$	$\Q$	None	$0$	$-36$	$-196290$	$-35905576$	$-$	$q-6^{2}q^{3}-196290q^{5}-35905576q^{7}+\cdots$
5.18.a.a	$2$	$9.161$	$\Q(\sqrt{39}) $	None	$680$	$-10980$	$-781250$	$-22820700$	$+$	$q+(340+\beta )q^{2}+(-5490-52\beta )q^{3}+\cdots$
5.18.a.b	$3$	$9.161$	$\mathbb{Q}[x]/(x^{3} - \cdots)$	None	$118$	$15944$	$1171875$	$2139308$	$-$	$q+(39-\beta _{1})q^{2}+(5317+8\beta _{1}+\beta _{2})q^{3}+\cdots$
5.18.b.a	$8$	$9.161$	$\mathbb{Q}[x]/(x^{8} + \cdots)$	None	$0$	$0$	$379200$	$0$		$q+\beta _{1}q^{2}-\beta _{2}q^{3}+(-72387+\beta _{3}+\cdots)q^{4}+\cdots$
3.23.b.a	$6$	$9.201$	$\mathbb{Q}[x]/(x^{6} + \cdots)$	None	$0$	$86670$	$0$	$-3447063060$		$q+\beta _{1}q^{2}+(14445-8\beta _{1}+\beta _{2})q^{3}+\cdots$
2.28.a.a	$1$	$9.237$	$\Q$	None	$-8192$	$3984828$	$-2851889250$	$368721063704$	$+$	$q-2^{13}q^{2}+3984828q^{3}+2^{26}q^{4}+\cdots$
2.28.a.b	$1$	$9.237$	$\Q$	None	$8192$	$-1016388$	$-3341197410$	$-51021361384$	$-$	$q+2^{13}q^{2}-1016388q^{3}+2^{26}q^{4}+\cdots$
6.17.b.a	$6$	$9.739$	$\mathbb{Q}[x]/(x^{6} + \cdots)$	None	$0$	$6006$	$0$	$-167892$		$q-\beta _{1}q^{2}+(1001+\beta _{1}-\beta _{2})q^{3}-2^{15}q^{4}+\cdots$
3.24.a.a	$1$	$10.056$	$\Q$	None	$1128$	$177147$	$-48863730$	$-1723688680$	$-$	$q+1128q^{2}+3^{11}q^{3}-7116224q^{4}+\cdots$
3.24.a.b	$2$	$10.056$	$\Q(\sqrt{530401}) $	None	$-1242$	$-354294$	$-46808820$	$-211963904$	$+$	$q+(-621-\beta )q^{2}-3^{11}q^{3}+(-3229358+\cdots)q^{4}+\cdots$
4.21.b.a	$1$	$10.141$	$\Q$	$\Q(\sqrt{-1}) $	$-1024$	$0$	$-19306574$	$0$		$q-2^{10}q^{2}+2^{20}q^{4}-19306574q^{5}+\cdots$
4.21.b.b	$8$	$10.141$	$\mathbb{Q}[x]/(x^{8} - \cdots)$	None	$396$	$0$	$18568080$	$0$		$q+(7^{2}+\beta _{1})q^{2}+(6-12\beta _{1}-\beta _{2})q^{3}+\cdots$
5.19.c.a	$16$	$10.269$	$\mathbb{Q}[x]/(x^{16} + \cdots)$	None	$510$	$-20130$	$3145170$	$78767350$		$q+(2^{5}+\beta _{2}+2^{5}\beta _{4})q^{2}+(-1259-5\beta _{1}+\cdots)q^{3}+\cdots$
2.30.a.a	$1$	$10.656$	$\Q$	None	$-16384$	$-2792556$	$6651856470$	$14\!\cdots\!48$	$+$	$q-2^{14}q^{2}-2792556q^{3}+2^{28}q^{4}+\cdots$
2.30.a.b	$1$	$10.656$	$\Q$	None	$16384$	$4782996$	$6065841750$	$904018883432$	$-$	$q+2^{14}q^{2}+4782996q^{3}+2^{28}q^{4}+\cdots$
3.25.b.a	$1$	$10.949$	$\Q$	$\Q(\sqrt{-3}) $	$0$	$531441$	$0$	$-4119710398$		$q+3^{12}q^{3}+2^{24}q^{4}-4119710398q^{7}+\cdots$
3.25.b.b	$6$	$10.949$	$\mathbb{Q}[x]/(x^{6} + \cdots)$	None	$0$	$-616842$	$0$	$1988064876$		$q+\beta _{1}q^{2}+(-102807+2^{4}\beta _{1}+\beta _{2}+\cdots)q^{3}+\cdots$
6.18.a.a	$1$	$10.993$	$\Q$	None	$-256$	$-6561$	$645150$	$3974432$	$+$	$q-2^{8}q^{2}-3^{8}q^{3}+2^{16}q^{4}+645150q^{5}+\cdots$

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Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Label	Dim.	\(A\)	Field	CM	Traces				Fricke sign	$q$-expansion
Label	Dim.	\(A\)	Field	CM	\(a_2\)	\(a_3\)	\(a_5\)	\(a_7\)	Fricke sign	$q$-expansion
1.18.a.a	\(1\)	\(1.832\)	\(\Q\)	None	\(-528\)	\(-4284\)	\(-1025850\)	\(3225992\)	\(+\)	\(q-528q^{2}-4284q^{3}+147712q^{4}+\cdots\)
1.20.a.a	\(1\)	\(2.288\)	\(\Q\)	None	\(456\)	\(50652\)	\(-2377410\)	\(-16917544\)	\(+\)	\(q+456q^{2}+50652q^{3}-316352q^{4}+\cdots\)
1.22.a.a	\(1\)	\(2.795\)	\(\Q\)	None	\(-288\)	\(-128844\)	\(21640950\)	\(-768078808\)	\(+\)	\(q-288q^{2}-128844q^{3}-2014208q^{4}+\cdots\)
1.24.a.a	\(2\)	\(3.352\)	\(\Q(\sqrt{144169}) \)	None	\(1080\)	\(339480\)	\(73069020\)	\(-1359184400\)	\(+\)	\(q+(540-\beta )q^{2}+(169740+48\beta )q^{3}+\cdots\)
2.18.a.a	\(1\)	\(3.664\)	\(\Q\)	None	\(256\)	\(6084\)	\(1255110\)	\(-22465912\)	\(-\)	\(q+2^{8}q^{2}+78^{2}q^{3}+2^{16}q^{4}+1255110q^{5}+\cdots\)
1.26.a.a	\(1\)	\(3.960\)	\(\Q\)	None	\(-48\)	\(-195804\)	\(-741989850\)	\(39080597192\)	\(+\)	\(q-48q^{2}-195804q^{3}-33552128q^{4}+\cdots\)
2.20.a.a	\(1\)	\(4.576\)	\(\Q\)	None	\(-512\)	\(-13092\)	\(6546750\)	\(96674264\)	\(+\)	\(q-2^{9}q^{2}-13092q^{3}+2^{18}q^{4}+6546750q^{5}+\cdots\)
2.20.a.b	\(1\)	\(4.576\)	\(\Q\)	None	\(512\)	\(-53028\)	\(-5556930\)	\(-44496424\)	\(-\)	\(q+2^{9}q^{2}-53028q^{3}+2^{18}q^{4}-5556930q^{5}+\cdots\)
1.28.a.a	\(2\)	\(4.619\)	\(\Q(\sqrt{18209}) \)	None	\(-8280\)	\(-1286280\)	\(5443587900\)	\(-175391963600\)	\(+\)	\(q+(-4140-\beta )q^{2}+(-643140-192\beta )q^{3}+\cdots\)
3.17.b.a	\(4\)	\(4.870\)	\(\mathbb{Q}[x]/(x^{4} + \cdots)\)	None	\(0\)	\(-2052\)	\(0\)	\(-3141544\)		\(q+\beta _{1}q^{2}+(-513+\beta _{1}+\beta _{2})q^{3}+(-3116+\cdots)q^{4}+\cdots\)
1.30.a.a	\(2\)	\(5.328\)	\(\Q(\sqrt{51349}) \)	None	\(8640\)	\(-4967640\)	\(-17477788500\)	\(-3\!\cdots\!00\)	\(+\)	\(q+(4320-\beta )q^{2}+(-2483820+552\beta )q^{3}+\cdots\)
3.18.a.a	\(1\)	\(5.497\)	\(\Q\)	None	\(204\)	\(-6561\)	\(-163554\)	\(-20846560\)	\(+\)	\(q+204q^{2}-3^{8}q^{3}-89456q^{4}-163554q^{5}+\cdots\)
3.18.a.b	\(2\)	\(5.497\)	\(\Q(\sqrt{14569}) \)	None	\(594\)	\(13122\)	\(382860\)	\(24471568\)	\(-\)	\(q+(297-\beta )q^{2}+3^{8}q^{3}+(88258-594\beta )q^{4}+\cdots\)
2.22.a.a	\(1\)	\(5.590\)	\(\Q\)	None	\(-1024\)	\(71604\)	\(-28693770\)	\(-853202392\)	\(+\)	\(q-2^{10}q^{2}+71604q^{3}+2^{20}q^{4}-28693770q^{5}+\cdots\)
2.22.a.b	\(1\)	\(5.590\)	\(\Q\)	None	\(1024\)	\(59316\)	\(4975350\)	\(1427425832\)	\(-\)	\(q+2^{10}q^{2}+59316q^{3}+2^{20}q^{4}+4975350q^{5}+\cdots\)
1.32.a.a	\(2\)	\(6.088\)	\(\mathbb{Q}[x]/(x^{2} - \cdots)\)	None	\(39960\)	\(17363160\)	\(-19391218020\)	\(30\!\cdots\!00\)	\(+\)	\(q+(19980-\beta )q^{2}+(8681580-432\beta )q^{3}+\cdots\)
3.19.b.a	\(1\)	\(6.162\)	\(\Q\)	\(\Q(\sqrt{-3}) \)	\(0\)	\(-19683\)	\(0\)	\(77549186\)		\(q-3^{9}q^{3}+2^{18}q^{4}+77549186q^{7}+\cdots\)
3.19.b.b	\(4\)	\(6.162\)	4.0.601940665.1	None	\(0\)	\(15876\)	\(0\)	\(-95744152\)		\(q-\beta _{1}q^{2}+(63^{2}+11\beta _{1}+\beta _{2})q^{3}+(-263384+\cdots)q^{4}+\cdots\)
4.17.b.a	\(1\)	\(6.493\)	\(\Q\)	\(\Q(\sqrt{-1}) \)	\(256\)	\(0\)	\(329666\)	\(0\)		\(q+2^{8}q^{2}+2^{16}q^{4}+329666q^{5}+2^{24}q^{8}+\cdots\)
4.17.b.b	\(6\)	\(6.493\)	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	\(-164\)	\(0\)	\(-506740\)	\(0\)		\(q+(-3^{3}-\beta _{1})q^{2}+(1-3\beta _{1}+\beta _{2})q^{3}+\cdots\)
2.24.a.a	\(1\)	\(6.704\)	\(\Q\)	None	\(-2048\)	\(-505908\)	\(-90135570\)	\(6872255096\)	\(+\)	\(q-2^{11}q^{2}-505908q^{3}+2^{22}q^{4}+\cdots\)
3.20.a.a	\(1\)	\(6.865\)	\(\Q\)	None	\(-1104\)	\(19683\)	\(3516270\)	\(-195590584\)	\(-\)	\(q-1104q^{2}+3^{9}q^{3}+694528q^{4}+\cdots\)
3.20.a.b	\(2\)	\(6.865\)	\(\Q(\sqrt{87481}) \)	None	\(702\)	\(-39366\)	\(6016140\)	\(113892064\)	\(+\)	\(q+(351-\beta )q^{2}-3^{9}q^{3}+(386242-702\beta )q^{4}+\cdots\)
4.18.a.a	\(2\)	\(7.329\)	\(\Q(\sqrt{9361}) \)	None	\(0\)	\(-5880\)	\(604044\)	\(25350160\)	\(-\)	\(q+(-2940-\beta )q^{3}+(302022+6^{2}\beta )q^{5}+\cdots\)
3.21.b.a	\(6\)	\(7.605\)	\(\mathbb{Q}[x]/(x^{6} + \cdots)\)	None	\(0\)	\(-4122\)	\(0\)	\(559607916\)		\(q+\beta _{1}q^{2}+(-687+11\beta _{1}+\beta _{2})q^{3}+\cdots\)
2.26.a.a	\(1\)	\(7.920\)	\(\Q\)	None	\(-4096\)	\(97956\)	\(341005350\)	\(-40882637368\)	\(+\)	\(q-2^{12}q^{2}+97956q^{3}+2^{24}q^{4}+341005350q^{5}+\cdots\)
2.26.a.b	\(2\)	\(7.920\)	\(\Q(\sqrt{106705}) \)	None	\(8192\)	\(379848\)	\(741953100\)	\(-376536944\)	\(-\)	\(q+2^{12}q^{2}+(189924-\beta )q^{3}+2^{24}q^{4}+\cdots\)
5.17.c.a	\(14\)	\(8.116\)	\(\mathbb{Q}[x]/(x^{14} + \cdots)\)	None	\(-2\)	\(7908\)	\(192880\)	\(-386452\)		\(q-\beta _{2}q^{2}+(565-3\beta _{1}-565\beta _{4}-\beta _{6}+\cdots)q^{3}+\cdots\)
4.19.b.a	\(8\)	\(8.215\)	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	\(84\)	\(0\)	\(860880\)	\(0\)		\(q+(11-\beta _{1})q^{2}-\beta _{2}q^{3}+(44007-9\beta _{1}+\cdots)q^{4}+\cdots\)
3.22.a.a	\(1\)	\(8.384\)	\(\Q\)	None	\(-2844\)	\(-59049\)	\(3109950\)	\(363303920\)	\(+\)	\(q-2844q^{2}-3^{10}q^{3}+5991184q^{4}+\cdots\)
3.22.a.b	\(1\)	\(8.384\)	\(\Q\)	None	\(1728\)	\(-59049\)	\(-41512770\)	\(538429808\)	\(+\)	\(q+12^{3}q^{2}-3^{10}q^{3}+888832q^{4}+\cdots\)
3.22.a.c	\(2\)	\(8.384\)	\(\Q(\sqrt{649}) \)	None	\(666\)	\(118098\)	\(996876\)	\(679896112\)	\(-\)	\(q+(333-\beta )q^{2}+3^{10}q^{3}+(589618+\cdots)q^{4}+\cdots\)
4.20.a.a	\(1\)	\(9.153\)	\(\Q\)	None	\(0\)	\(-36\)	\(-196290\)	\(-35905576\)	\(-\)	\(q-6^{2}q^{3}-196290q^{5}-35905576q^{7}+\cdots\)
5.18.a.a	\(2\)	\(9.161\)	\(\Q(\sqrt{39}) \)	None	\(680\)	\(-10980\)	\(-781250\)	\(-22820700\)	\(+\)	\(q+(340+\beta )q^{2}+(-5490-52\beta )q^{3}+\cdots\)
5.18.a.b	\(3\)	\(9.161\)	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	\(118\)	\(15944\)	\(1171875\)	\(2139308\)	\(-\)	\(q+(39-\beta _{1})q^{2}+(5317+8\beta _{1}+\beta _{2})q^{3}+\cdots\)
5.18.b.a	\(8\)	\(9.161\)	\(\mathbb{Q}[x]/(x^{8} + \cdots)\)	None	\(0\)	\(0\)	\(379200\)	\(0\)		\(q+\beta _{1}q^{2}-\beta _{2}q^{3}+(-72387+\beta _{3}+\cdots)q^{4}+\cdots\)
3.23.b.a	\(6\)	\(9.201\)	\(\mathbb{Q}[x]/(x^{6} + \cdots)\)	None	\(0\)	\(86670\)	\(0\)	\(-3447063060\)		\(q+\beta _{1}q^{2}+(14445-8\beta _{1}+\beta _{2})q^{3}+\cdots\)
2.28.a.a	\(1\)	\(9.237\)	\(\Q\)	None	\(-8192\)	\(3984828\)	\(-2851889250\)	\(368721063704\)	\(+\)	\(q-2^{13}q^{2}+3984828q^{3}+2^{26}q^{4}+\cdots\)
2.28.a.b	\(1\)	\(9.237\)	\(\Q\)	None	\(8192\)	\(-1016388\)	\(-3341197410\)	\(-51021361384\)	\(-\)	\(q+2^{13}q^{2}-1016388q^{3}+2^{26}q^{4}+\cdots\)
6.17.b.a	\(6\)	\(9.739\)	\(\mathbb{Q}[x]/(x^{6} + \cdots)\)	None	\(0\)	\(6006\)	\(0\)	\(-167892\)		\(q-\beta _{1}q^{2}+(1001+\beta _{1}-\beta _{2})q^{3}-2^{15}q^{4}+\cdots\)
3.24.a.a	\(1\)	\(10.056\)	\(\Q\)	None	\(1128\)	\(177147\)	\(-48863730\)	\(-1723688680\)	\(-\)	\(q+1128q^{2}+3^{11}q^{3}-7116224q^{4}+\cdots\)
3.24.a.b	\(2\)	\(10.056\)	\(\Q(\sqrt{530401}) \)	None	\(-1242\)	\(-354294\)	\(-46808820\)	\(-211963904\)	\(+\)	\(q+(-621-\beta )q^{2}-3^{11}q^{3}+(-3229358+\cdots)q^{4}+\cdots\)
4.21.b.a	\(1\)	\(10.141\)	\(\Q\)	\(\Q(\sqrt{-1}) \)	\(-1024\)	\(0\)	\(-19306574\)	\(0\)		\(q-2^{10}q^{2}+2^{20}q^{4}-19306574q^{5}+\cdots\)
4.21.b.b	\(8\)	\(10.141\)	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	\(396\)	\(0\)	\(18568080\)	\(0\)		\(q+(7^{2}+\beta _{1})q^{2}+(6-12\beta _{1}-\beta _{2})q^{3}+\cdots\)
5.19.c.a	\(16\)	\(10.269\)	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)	None	\(510\)	\(-20130\)	\(3145170\)	\(78767350\)		\(q+(2^{5}+\beta _{2}+2^{5}\beta _{4})q^{2}+(-1259-5\beta _{1}+\cdots)q^{3}+\cdots\)
2.30.a.a	\(1\)	\(10.656\)	\(\Q\)	None	\(-16384\)	\(-2792556\)	\(6651856470\)	\(14\!\cdots\!48\)	\(+\)	\(q-2^{14}q^{2}-2792556q^{3}+2^{28}q^{4}+\cdots\)
2.30.a.b	\(1\)	\(10.656\)	\(\Q\)	None	\(16384\)	\(4782996\)	\(6065841750\)	\(904018883432\)	\(-\)	\(q+2^{14}q^{2}+4782996q^{3}+2^{28}q^{4}+\cdots\)
3.25.b.a	\(1\)	\(10.949\)	\(\Q\)	\(\Q(\sqrt{-3}) \)	\(0\)	\(531441\)	\(0\)	\(-4119710398\)		\(q+3^{12}q^{3}+2^{24}q^{4}-4119710398q^{7}+\cdots\)
3.25.b.b	\(6\)	\(10.949\)	\(\mathbb{Q}[x]/(x^{6} + \cdots)\)	None	\(0\)	\(-616842\)	\(0\)	\(1988064876\)		\(q+\beta _{1}q^{2}+(-102807+2^{4}\beta _{1}+\beta _{2}+\cdots)q^{3}+\cdots\)
6.18.a.a	\(1\)	\(10.993\)	\(\Q\)	None	\(-256\)	\(-6561\)	\(645150\)	\(3974432\)	\(+\)	\(q-2^{8}q^{2}-3^{8}q^{3}+2^{16}q^{4}+645150q^{5}+\cdots\)

Introduction

L-functions

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This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.