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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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Results (1-50 of 140 matches)

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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
Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Inner twists	Rank*	$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$	Fricke sign	Coefficient ring index	Sato-Tate	$q$-expansion
38.2.a.a	$38$	$2$	38.a	1.a	$1$	$1$	$1$	$0.303$	$\Q$	None	✓	$1$	$0$	$-1$	$1$	$0$	$-1$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{3}+q^{4}-q^{6}-q^{7}-q^{8}+\cdots$
38.2.a.b	$38$	$2$	38.a	1.a	$1$	$1$	$1$	$0.303$	$\Q$	None	✓	$1$	$0$	$1$	$-1$	$-4$	$3$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}-q^{3}+q^{4}-4q^{5}-q^{6}+3q^{7}+\cdots$
38.2.c.a	$38$	$2$	38.c	19.c	$3$	$2$	$1$	$0.303$	$\Q(\sqrt{-3}) $	None		$2$	$0$	$1$	$-1$	$0$	$-8$		$1$	$\mathrm{SU}(2)[C_{3}]$	$q+(1-\zeta_{6})q^{2}+(-1+\zeta_{6})q^{3}-\zeta_{6}q^{4}+\cdots$
38.2.c.b	$38$	$2$	38.c	19.c	$3$	$4$	$2$	$0.303$	$\Q(\sqrt{-3}, \sqrt{7})$	None		$2$	$0$	$-2$	$0$	$-2$	$4$		$1$	$\mathrm{SU}(2)[C_{3}]$	$q+\beta _{2}q^{2}+(\beta _{1}+\beta _{3})q^{3}+(-1-\beta _{2}+\cdots)q^{4}+\cdots$
38.2.e.a	$38$	$2$	38.e	19.e	$9$	$6$	$1$	$0.303$	$\Q(\zeta_{18})$	None		$2$	$0$	$0$	$-3$	$0$	$-6$		$1$	$\mathrm{SU}(2)[C_{9}]$	$q-\zeta_{18}q^{2}+(-\zeta_{18}^{3}-\zeta_{18}^{5})q^{3}+\zeta_{18}^{2}q^{4}+\cdots$
38.3.b.a	$38$	$3$	38.b	19.b	$2$	$2$	$2$	$1.035$	$\Q(\sqrt{-2}) $	None		$2$	$0$	$0$	$0$	$-2$	$10$		$1$	$\mathrm{SU}(2)[C_{2}]$	$q+\beta q^{2}+2\beta q^{3}-2q^{4}-q^{5}-4q^{6}+\cdots$
38.3.d.a	$38$	$3$	38.d	19.d	$6$	$4$	$2$	$1.035$	$\Q(\sqrt{-2}, \sqrt{-3})$	None		$2$	$0$	$0$	$6$	$2$	$8$		$1$	$\mathrm{SU}(2)[C_{6}]$	$q+\beta _{1}q^{2}+(1-\beta _{1}+\beta _{2})q^{3}+2\beta _{2}q^{4}+\cdots$
38.3.f.a	$38$	$3$	38.f	19.f	$18$	$24$	$4$	$1.035$		None		$2$	$0$	$0$	$-6$	$0$	$-18$			$\mathrm{SU}(2)[C_{18}]$
38.4.a.a	$38$	$4$	38.a	1.a	$1$	$1$	$1$	$2.242$	$\Q$	None	✓	$1$	$1$	$-2$	$-2$	$-9$	$-31$	$-$	$1$	$\mathrm{SU}(2)$	$q-2q^{2}-2q^{3}+4q^{4}-9q^{5}+4q^{6}+\cdots$
38.4.a.b	$38$	$4$	38.a	1.a	$1$	$2$	$2$	$2.242$	$\Q(\sqrt{177}) $	None	✓	$1$	$0$	$-4$	$1$	$10$	$57$	$+$	$1$	$\mathrm{SU}(2)$	$q-2q^{2}+(1-\beta )q^{3}+4q^{4}+(4+2\beta )q^{5}+\cdots$
38.4.a.c	$38$	$4$	38.a	1.a	$1$	$2$	$2$	$2.242$	$\Q(\sqrt{73}) $	None	✓	$1$	$0$	$4$	$9$	$-9$	$-18$	$+$	$1$	$\mathrm{SU}(2)$	$q+2q^{2}+(5-\beta )q^{3}+4q^{4}+(-6+3\beta )q^{5}+\cdots$
38.4.c.a	$38$	$4$	38.c	19.c	$3$	$2$	$1$	$2.242$	$\Q(\sqrt{-3}) $	None		$2$	$0$	$-2$	$-5$	$-3$	$-64$		$1$	$\mathrm{SU}(2)[C_{3}]$	$q+(-2+2\zeta_{6})q^{2}+(-5+5\zeta_{6})q^{3}+\cdots$
38.4.c.b	$38$	$4$	38.c	19.c	$3$	$2$	$1$	$2.242$	$\Q(\sqrt{-3}) $	None		$2$	$0$	$-2$	$5$	$12$	$16$		$1$	$\mathrm{SU}(2)[C_{3}]$	$q+(-2+2\zeta_{6})q^{2}+(5-5\zeta_{6})q^{3}-4\zeta_{6}q^{4}+\cdots$
38.4.c.c	$38$	$4$	38.c	19.c	$3$	$6$	$3$	$2.242$	$\mathbb{Q}[x]/(x^{6} - \cdots)$	None		$2$	$0$	$6$	$-5$	$-1$	$52$		$3$	$\mathrm{SU}(2)[C_{3}]$	$q+(2-2\beta _{4})q^{2}+(-2-\beta _{1}-\beta _{2}+2\beta _{4}+\cdots)q^{3}+\cdots$
38.4.e.a	$38$	$4$	38.e	19.e	$9$	$12$	$2$	$2.242$	$\mathbb{Q}[x]/(x^{12} - \cdots)$	None		$2$	$0$	$0$	$-9$	$0$	$21$		$1$	$\mathrm{SU}(2)[C_{9}]$	$q-2\beta _{3}q^{2}+(-1+2\beta _{2}-\beta _{4}+2\beta _{6}+\cdots)q^{3}+\cdots$
38.4.e.b	$38$	$4$	38.e	19.e	$9$	$18$	$3$	$2.242$	$\mathbb{Q}[x]/(x^{18} + \cdots)$	None		$2$	$0$	$0$	$6$	$0$	$-33$		$3^{2}$	$\mathrm{SU}(2)[C_{9}]$	$q-2\beta _{8}q^{2}+(\beta _{3}+\beta _{4}+\beta _{6}+\beta _{9})q^{3}+\cdots$
38.5.b.a	$38$	$5$	38.b	19.b	$2$	$8$	$8$	$3.928$	$\mathbb{Q}[x]/(x^{8} + \cdots)$	None		$2$	$0$	$0$	$0$	$18$	$-162$		$2^{5}$	$\mathrm{SU}(2)[C_{2}]$	$q+\beta _{2}q^{2}+(\beta _{1}+\beta _{2})q^{3}-8q^{4}+(2-\beta _{6}+\cdots)q^{5}+\cdots$
38.5.d.a	$38$	$5$	38.d	19.d	$6$	$16$	$8$	$3.928$	$\mathbb{Q}[x]/(x^{16} - \cdots)$	None		$2$	$0$	$0$	$-12$	$-18$	$72$		$2^{10}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{6}]$	$q+(-\beta _{5}+\beta _{9})q^{2}+(-1+\beta _{3})q^{3}+8\beta _{7}q^{4}+\cdots$
38.5.f.a	$38$	$5$	38.f	19.f	$18$	$36$	$6$	$3.928$		None		$2$	$0$	$0$	$12$	$0$	$90$			$\mathrm{SU}(2)[C_{18}]$
38.6.a.a	$38$	$6$	38.a	1.a	$1$	$1$	$1$	$6.095$	$\Q$	None	✓	$1$	$1$	$-4$	$-6$	$31$	$-27$	$+$	$1$	$\mathrm{SU}(2)$	$q-4q^{2}-6q^{3}+2^{4}q^{4}+31q^{5}+24q^{6}+\cdots$
38.6.a.b	$38$	$6$	38.a	1.a	$1$	$1$	$1$	$6.095$	$\Q$	None	✓	$1$	$1$	$4$	$-14$	$-45$	$-121$	$+$	$1$	$\mathrm{SU}(2)$	$q+4q^{2}-14q^{3}+2^{4}q^{4}-45q^{5}-56q^{6}+\cdots$
38.6.a.c	$38$	$6$	38.a	1.a	$1$	$2$	$2$	$6.095$	$\Q(\sqrt{1441}) $	None	✓	$1$	$0$	$-8$	$3$	$-45$	$114$	$-$	$1$	$\mathrm{SU}(2)$	$q-4q^{2}+(2-\beta )q^{3}+2^{4}q^{4}+(-21+\cdots)q^{5}+\cdots$
38.6.a.d	$38$	$6$	38.a	1.a	$1$	$3$	$3$	$6.095$	$\mathbb{Q}[x]/(x^{3} - \cdots)$	None	✓	$1$	$0$	$12$	$13$	$81$	$228$	$-$	$1$	$\mathrm{SU}(2)$	$q+4q^{2}+(4+\beta _{1})q^{3}+2^{4}q^{4}+(3^{3}-\beta _{1}+\cdots)q^{5}+\cdots$
38.6.c.a	$38$	$6$	38.c	19.c	$3$	$6$	$3$	$6.095$	$\mathbb{Q}[x]/(x^{6} + \cdots)$	None		$2$	$0$	$12$	$-15$	$14$	$-624$		$2^{2}$	$\mathrm{SU}(2)[C_{3}]$	$q+(4-4\beta _{3})q^{2}+(-5+\beta _{1}+\beta _{2}+5\beta _{3}+\cdots)q^{3}+\cdots$
38.6.c.b	$38$	$6$	38.c	19.c	$3$	$8$	$4$	$6.095$	$\mathbb{Q}[x]/(x^{8} - \cdots)$	None		$2$	$0$	$-16$	$-14$	$-36$	$76$		$2^{2}\cdot 3$	$\mathrm{SU}(2)[C_{3}]$	$q+(-4+4\beta _{2})q^{2}+(-3+3\beta _{2}-\beta _{3}+\cdots)q^{3}+\cdots$
38.6.e.a	$38$	$6$	38.e	19.e	$9$	$24$	$4$	$6.095$		None		$2$	$0$	$0$	$18$	$0$	$438$			$\mathrm{SU}(2)[C_{9}]$
38.6.e.b	$38$	$6$	38.e	19.e	$9$	$30$	$5$	$6.095$		None		$2$	$0$	$0$	$15$	$0$	$-84$			$\mathrm{SU}(2)[C_{9}]$
38.7.b.a	$38$	$7$	38.b	19.b	$2$	$10$	$10$	$8.742$	$\mathbb{Q}[x]/(x^{10} + \cdots)$	None		$2$	$0$	$0$	$0$	$-112$	$-224$		$2^{11}$	$\mathrm{SU}(2)[C_{2}]$	$q-\beta _{6}q^{2}+(\beta _{5}+\beta _{6})q^{3}-2^{5}q^{4}+(-11+\cdots)q^{5}+\cdots$
38.7.d.a	$38$	$7$	38.d	19.d	$6$	$20$	$10$	$8.742$	$\mathbb{Q}[x]/(x^{20} + \cdots)$	None		$2$	$0$	$0$	$-30$	$112$	$-208$		$2^{20}\cdot 3^{4}\cdot 19^{2}$	$\mathrm{SU}(2)[C_{6}]$	$q-\beta _{8}q^{2}+(-2+\beta _{2}+\beta _{6})q^{3}+(2^{5}+\cdots)q^{4}+\cdots$
38.7.f.a	$38$	$7$	38.f	19.f	$18$	$60$	$10$	$8.742$		None		$2$	$0$	$0$	$30$	$0$	$432$			$\mathrm{SU}(2)[C_{18}]$
38.8.a.a	$38$	$8$	38.a	1.a	$1$	$1$	$1$	$11.871$	$\Q$	None	✓	$1$	$0$	$-8$	$77$	$440$	$951$	$+$	$1$	$\mathrm{SU}(2)$	$q-8q^{2}+77q^{3}+2^{6}q^{4}+440q^{5}+\cdots$
38.8.a.b	$38$	$8$	38.a	1.a	$1$	$2$	$2$	$11.871$	$\Q(\sqrt{2737}) $	None	✓	$1$	$0$	$-16$	$-61$	$175$	$-2592$	$+$	$1$	$\mathrm{SU}(2)$	$q-8q^{2}+(-30-\beta )q^{3}+2^{6}q^{4}+(95+\cdots)q^{5}+\cdots$
38.8.a.c	$38$	$8$	38.a	1.a	$1$	$2$	$2$	$11.871$	$\Q(\sqrt{17953}) $	None	✓	$1$	$1$	$-16$	$-11$	$-69$	$-348$	$-$	$1$	$\mathrm{SU}(2)$	$q-8q^{2}+(-5-\beta )q^{3}+2^{6}q^{4}+(-6^{2}+\cdots)q^{5}+\cdots$
38.8.a.d	$38$	$8$	38.a	1.a	$1$	$2$	$2$	$11.871$	$\Q(\sqrt{633}) $	None	✓	$1$	$1$	$16$	$-69$	$155$	$-2238$	$-$	$1$	$\mathrm{SU}(2)$	$q+8q^{2}+(-33-3\beta )q^{3}+2^{6}q^{4}+(62+\cdots)q^{5}+\cdots$
38.8.a.e	$38$	$8$	38.a	1.a	$1$	$4$	$4$	$11.871$	$\mathbb{Q}[x]/(x^{4} - \cdots)$	None	✓	$1$	$0$	$32$	$12$	$-279$	$2485$	$+$	$2\cdot 3$	$\mathrm{SU}(2)$	$q+8q^{2}+(3-\beta _{1})q^{3}+2^{6}q^{4}+(-70+\cdots)q^{5}+\cdots$
38.8.c.a	$38$	$8$	38.c	19.c	$3$	$12$	$6$	$11.871$	$\mathbb{Q}[x]/(x^{12} + \cdots)$	None		$2$	$0$	$-48$	$12$	$124$	$-1036$		$2^{6}\cdot 3^{3}\cdot 7^{2}$	$\mathrm{SU}(2)[C_{3}]$	$q+8\beta _{2}q^{2}+(\beta _{1}-2\beta _{2})q^{3}+(-2^{6}-2^{6}\beta _{2}+\cdots)q^{4}+\cdots$
38.8.c.b	$38$	$8$	38.c	19.c	$3$	$14$	$7$	$11.871$	$\mathbb{Q}[x]/(x^{14} - \cdots)$	None		$2$	$0$	$56$	$55$	$-126$	$1928$		$2^{8}\cdot 3^{6}$	$\mathrm{SU}(2)[C_{3}]$	$q+8\beta _{3}q^{2}+(\beta _{1}+\beta _{2}+8\beta _{3})q^{3}+(-2^{6}+\cdots)q^{4}+\cdots$
38.8.e.a	$38$	$8$	38.e	19.e	$9$	$30$	$5$	$11.871$		None		$2$	$0$	$0$	$45$	$0$	$1806$			$\mathrm{SU}(2)[C_{9}]$
38.8.e.b	$38$	$8$	38.e	19.e	$9$	$36$	$6$	$11.871$		None		$2$	$0$	$0$	$-84$	$0$	$-2988$			$\mathrm{SU}(2)[C_{9}]$
38.9.b.a	$38$	$9$	38.b	19.b	$2$	$12$	$12$	$15.480$	$\mathbb{Q}[x]/(x^{12} + \cdots)$	None		$2$	$0$	$0$	$0$	$558$	$-5422$		$2^{21}$	$\mathrm{SU}(2)[C_{2}]$	$q+\beta _{7}q^{2}+(\beta _{6}-\beta _{7})q^{3}-2^{7}q^{4}+(47+\cdots)q^{5}+\cdots$
38.9.d.a	$38$	$9$	38.d	19.d	$6$	$24$	$12$	$15.480$		None		$2$	$0$	$0$	$168$	$-558$	$11992$			$\mathrm{SU}(2)[C_{6}]$
38.9.f.a	$38$	$9$	38.f	19.f	$18$	$84$	$14$	$15.480$		None		$2$	$0$	$0$	$-168$	$0$	$-6570$			$\mathrm{SU}(2)[C_{18}]$
38.10.a.a	$38$	$10$	38.a	1.a	$1$	$1$	$1$	$19.571$	$\Q$	None	✓	$1$	$1$	$16$	$-119$	$-684$	$9149$	$+$	$1$	$\mathrm{SU}(2)$	$q+2^{4}q^{2}-119q^{3}+2^{8}q^{4}-684q^{5}+\cdots$
38.10.a.b	$38$	$10$	38.a	1.a	$1$	$1$	$1$	$19.571$	$\Q$	None	✓	$1$	$1$	$16$	$102$	$-1581$	$-4865$	$+$	$1$	$\mathrm{SU}(2)$	$q+2^{4}q^{2}+102q^{3}+2^{8}q^{4}-1581q^{5}+\cdots$
38.10.a.c	$38$	$10$	38.a	1.a	$1$	$3$	$3$	$19.571$	$\mathbb{Q}[x]/(x^{3} - \cdots)$	None	✓	$1$	$1$	$-48$	$3$	$486$	$-13317$	$+$	$2^{3}\cdot 3$	$\mathrm{SU}(2)$	$q-2^{4}q^{2}+(1+\beta _{1})q^{3}+2^{8}q^{4}+(162+\cdots)q^{5}+\cdots$
38.10.a.d	$38$	$10$	38.a	1.a	$1$	$4$	$4$	$19.571$	$\mathbb{Q}[x]/(x^{4} - \cdots)$	None	✓	$1$	$0$	$-64$	$84$	$-1395$	$12307$	$-$	$2\cdot 3$	$\mathrm{SU}(2)$	$q-2^{4}q^{2}+(21+\beta _{1})q^{3}+2^{8}q^{4}+(-350+\cdots)q^{5}+\cdots$
38.10.a.e	$38$	$10$	38.a	1.a	$1$	$4$	$4$	$19.571$	$\mathbb{Q}[x]/(x^{4} - \cdots)$	None	✓	$1$	$0$	$64$	$226$	$866$	$2670$	$-$	$2^{4}\cdot 3$	$\mathrm{SU}(2)$	$q+2^{4}q^{2}+(57+\beta _{2})q^{3}+2^{8}q^{4}+(218+\cdots)q^{5}+\cdots$
38.10.c.a	$38$	$10$	38.c	19.c	$3$	$14$	$7$	$19.571$	$\mathbb{Q}[x]/(x^{14} - \cdots)$	None		$2$	$0$	$112$	$165$	$909$	$3692$		$2^{10}\cdot 3^{6}\cdot 5^{2}$	$\mathrm{SU}(2)[C_{3}]$	$q+(2^{4}-2^{4}\beta _{2})q^{2}+(24-\beta _{1}-24\beta _{2}+\cdots)q^{3}+\cdots$
38.10.c.b	$38$	$10$	38.c	19.c	$3$	$16$	$8$	$19.571$	$\mathbb{Q}[x]/(x^{16} - \cdots)$	None		$2$	$0$	$-128$	$70$	$-341$	$3704$		$2^{12}\cdot 3^{5}\cdot 19^{2}$	$\mathrm{SU}(2)[C_{3}]$	$q+(-2^{4}+2^{4}\beta _{2})q^{2}+(9-\beta _{1}-9\beta _{2}+\cdots)q^{3}+\cdots$
38.10.e.a	$38$	$10$	38.e	19.e	$9$	$42$	$7$	$19.571$		None		$2$	$0$	$0$	$-252$	$0$	$14373$			$\mathrm{SU}(2)[C_{9}]$

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Elliptic curves over $\Q$
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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.