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 247 matches)

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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
78.2.a.a	$1$	$0.623$	$\Q$	None	$-1$	$-1$	$2$	$4$	$-$	$q-q^{2}-q^{3}+q^{4}+2q^{5}+q^{6}+4q^{7}+\cdots$
78.2.b.a	$2$	$0.623$	$\Q(\sqrt{-1}) $	None	$0$	$2$	$0$	$0$		$q+iq^{2}+q^{3}-q^{4}+2iq^{5}+iq^{6}+\cdots$
78.2.e.a	$2$	$0.623$	$\Q(\sqrt{-3}) $	None	$1$	$-1$	$6$	$-2$		$q+(1-\zeta_{6})q^{2}+(-1+\zeta_{6})q^{3}-\zeta_{6}q^{4}+\cdots$
78.2.e.b	$2$	$0.623$	$\Q(\sqrt{-3}) $	None	$1$	$1$	$-2$	$2$		$q+(1-\zeta_{6})q^{2}+(1-\zeta_{6})q^{3}-\zeta_{6}q^{4}+\cdots$
78.2.g.a	$12$	$0.623$	12.0.$\cdots$.52	None	$0$	$0$	$0$	$-12$		$q+\beta _{2}q^{2}-\beta _{4}q^{3}+\beta _{8}q^{4}+(-\beta _{1}-\beta _{11})q^{5}+\cdots$
78.2.i.a	$4$	$0.623$	$\Q(\zeta_{12})$	None	$0$	$-2$	$0$	$6$		$q+\zeta_{12}q^{2}+(-1+\zeta_{12}^{2})q^{3}+\zeta_{12}^{2}q^{4}+\cdots$
78.2.i.b	$4$	$0.623$	$\Q(\zeta_{12})$	None	$0$	$2$	$0$	$-6$		$q+\zeta_{12}q^{2}+(1-\zeta_{12}^{2})q^{3}+\zeta_{12}^{2}q^{4}+\cdots$
78.2.k.a	$16$	$0.623$	16.0.$\cdots$.9	None	$0$	$0$	$0$	$8$		$q+(-\beta _{7}-\beta _{15})q^{2}+(-\beta _{1}-\beta _{3}-\beta _{4}+\cdots)q^{3}+\cdots$
78.3.c.a	$8$	$2.125$	8.0.$\cdots$.2	None	$0$	$0$	$0$	$-8$		$q-\beta _{2}q^{2}+\beta _{4}q^{3}-2q^{4}+(\beta _{2}+\beta _{4}+\cdots)q^{5}+\cdots$
78.3.d.a	$4$	$2.125$	$\Q(\sqrt{2}, \sqrt{-5})$	None	$0$	$-8$	$0$	$0$		$q+\beta _{1}q^{2}+(-2-\beta _{3})q^{3}+2q^{4}+3\beta _{1}q^{5}+\cdots$
78.3.d.b	$4$	$2.125$	$\Q(\sqrt{2}, \sqrt{-5})$	None	$0$	$8$	$0$	$0$		$q+\beta _{1}q^{2}+(2+\beta _{3})q^{3}+2q^{4}-\beta _{1}q^{5}+\cdots$
78.3.f.a	$4$	$2.125$	$\Q(\zeta_{12})$	None	$4$	$0$	$12$	$-4$		$q+(1+\zeta_{12})q^{2}+\zeta_{12}^{3}q^{3}+2\zeta_{12}q^{4}+\cdots$
78.3.f.b	$8$	$2.125$	8.0.$\cdots$.1	None	$-8$	$0$	$0$	$4$		$q+(-1-\beta _{6})q^{2}+\beta _{1}q^{3}+2\beta _{6}q^{4}+\cdots$
78.3.h.a	$8$	$2.125$	8.0.4857532416.2	None	$0$	$-2$	$0$	$-12$		$q+\beta _{2}q^{2}+(-1-\beta _{1}+\beta _{3}-\beta _{4}-\beta _{5}+\cdots)q^{3}+\cdots$
78.3.h.b	$12$	$2.125$	$\mathbb{Q}[x]/(x^{12} - \cdots)$	None	$0$	$2$	$0$	$22$		$q+\beta _{7}q^{2}+(-\beta _{4}+\beta _{9})q^{3}+2\beta _{1}q^{4}+\cdots$
78.3.j.a	$20$	$2.125$	$\mathbb{Q}[x]/(x^{20} + \cdots)$	None	$0$	$0$	$0$	$18$		$q+\beta _{5}q^{2}-\beta _{1}q^{3}+(-2-2\beta _{6})q^{4}+\cdots$
78.3.l.a	$4$	$2.125$	$\Q(\zeta_{12})$	None	$-2$	$0$	$-12$	$-10$		$q+(-\zeta_{12}-\zeta_{12}^{2}+\zeta_{12}^{3})q^{2}+(\zeta_{12}+\cdots)q^{3}+\cdots$
78.3.l.b	$4$	$2.125$	$\Q(\zeta_{12})$	None	$2$	$0$	$6$	$20$		$q+(\zeta_{12}+\zeta_{12}^{2}-\zeta_{12}^{3})q^{2}+(-\zeta_{12}+\cdots)q^{3}+\cdots$
78.3.l.c	$8$	$2.125$	$\mathbb{Q}[x]/(x^{8} - \cdots)$	None	$4$	$0$	$-6$	$10$		$q+(-\beta _{4}-\beta _{5})q^{2}+(2\beta _{2}-\beta _{5})q^{3}+(2\beta _{2}+\cdots)q^{4}+\cdots$
78.4.a.a	$1$	$4.602$	$\Q$	None	$-2$	$-3$	$-16$	$28$	$+$	$q-2q^{2}-3q^{3}+4q^{4}-2^{4}q^{5}+6q^{6}+\cdots$
78.4.a.b	$1$	$4.602$	$\Q$	None	$-2$	$3$	$-16$	$-8$	$-$	$q-2q^{2}+3q^{3}+4q^{4}-2^{4}q^{5}-6q^{6}+\cdots$
78.4.a.c	$1$	$4.602$	$\Q$	None	$-2$	$3$	$10$	$-8$	$+$	$q-2q^{2}+3q^{3}+4q^{4}+10q^{5}-6q^{6}+\cdots$
78.4.a.d	$1$	$4.602$	$\Q$	None	$2$	$-3$	$-20$	$-32$	$-$	$q+2q^{2}-3q^{3}+4q^{4}-20q^{5}-6q^{6}+\cdots$
78.4.a.e	$1$	$4.602$	$\Q$	None	$2$	$-3$	$6$	$20$	$+$	$q+2q^{2}-3q^{3}+4q^{4}+6q^{5}-6q^{6}+\cdots$
78.4.a.f	$1$	$4.602$	$\Q$	None	$2$	$3$	$4$	$4$	$+$	$q+2q^{2}+3q^{3}+4q^{4}+4q^{5}+6q^{6}+\cdots$
78.4.b.a	$2$	$4.602$	$\Q(\sqrt{-1}) $	None	$0$	$-6$	$0$	$0$		$q+iq^{2}-3q^{3}-4q^{4}-4iq^{5}-3iq^{6}+\cdots$
78.4.b.b	$4$	$4.602$	$\Q(i, \sqrt{17})$	None	$0$	$12$	$0$	$0$		$q-\beta _{1}q^{2}+3q^{3}-4q^{4}+(4\beta _{1}-\beta _{2}+\cdots)q^{5}+\cdots$
78.4.e.a	$2$	$4.602$	$\Q(\sqrt{-3}) $	None	$-2$	$3$	$14$	$-16$		$q+(-2+2\zeta_{6})q^{2}+(3-3\zeta_{6})q^{3}-4\zeta_{6}q^{4}+\cdots$
78.4.e.b	$4$	$4.602$	$\Q(\sqrt{-3}, \sqrt{673})$	None	$-4$	$-6$	$26$	$-9$		$q+(-2+2\beta _{2})q^{2}+(-3+3\beta _{2})q^{3}+\cdots$
78.4.e.c	$4$	$4.602$	$\Q(\sqrt{-3}, \sqrt{61})$	None	$4$	$6$	$4$	$-18$		$q+(2-2\beta _{1})q^{2}+(3-3\beta _{1})q^{3}-4\beta _{1}q^{4}+\cdots$
78.4.e.d	$6$	$4.602$	$\mathbb{Q}[x]/(x^{6} + \cdots)$	None	$6$	$-9$	$-24$	$17$		$q+(2+2\beta _{2})q^{2}+(-3-3\beta _{2})q^{3}+4\beta _{2}q^{4}+\cdots$
78.4.g.a	$28$	$4.602$		None	$0$	$0$	$0$	$20$
78.4.i.a	$4$	$4.602$	$\Q(\zeta_{12})$	None	$0$	$-6$	$0$	$-42$		$q+2\zeta_{12}q^{2}+(-3+3\zeta_{12}^{2})q^{3}+4\zeta_{12}^{2}q^{4}+\cdots$
78.4.i.b	$8$	$4.602$	$\mathbb{Q}[x]/(x^{8} - \cdots)$	None	$0$	$12$	$0$	$24$		$q-\beta _{4}q^{2}+3\beta _{1}q^{3}+(4-4\beta _{1})q^{4}+(3+\cdots)q^{5}+\cdots$
78.4.k.a	$56$	$4.602$		None	$0$	$0$	$0$	$-92$
78.5.c.a	$16$	$8.063$	$\mathbb{Q}[x]/(x^{16} - \cdots)$	None	$0$	$12$	$0$	$80$		$q+\beta _{3}q^{2}+(1+\beta _{3}-\beta _{4})q^{3}-8q^{4}+\cdots$
78.5.d.a	$20$	$8.063$	$\mathbb{Q}[x]/(x^{20} - \cdots)$	None	$0$	$0$	$0$	$0$		$q+\beta _{2}q^{2}-\beta _{1}q^{3}+8q^{4}+(-2\beta _{2}-\beta _{6}+\cdots)q^{5}+\cdots$
78.5.f.a	$8$	$8.063$	$\mathbb{Q}[x]/(x^{8} - \cdots)$	None	$-16$	$0$	$-36$	$-8$		$q+(-2+2\beta _{2})q^{2}-3\beta _{1}q^{3}-8\beta _{2}q^{4}+\cdots$
78.5.f.b	$8$	$8.063$	8.0.$\cdots$.2	None	$16$	$0$	$60$	$88$		$q+(2+2\beta _{2})q^{2}+\beta _{3}q^{3}+8\beta _{2}q^{4}+(7+\cdots)q^{5}+\cdots$
78.5.h.a	$36$	$8.063$		None	$0$	$0$	$0$	$-90$
78.5.j.a	$36$	$8.063$		None	$0$	$0$	$0$	$126$
78.5.l.a	$8$	$8.063$	$\mathbb{Q}[x]/(x^{8} - \cdots)$	None	$-8$	$0$	$-72$	$118$		$q+(-2-2\beta _{1}+2\beta _{4})q^{2}+(-3\beta _{4}-6\beta _{5}+\cdots)q^{3}+\cdots$
78.5.l.b	$8$	$8.063$	8.0.$\cdots$.10	None	$8$	$0$	$-24$	$22$		$q+(2-2\beta _{1}+2\beta _{2})q^{2}+(3\beta _{2}+6\beta _{3}+\cdots)q^{3}+\cdots$
78.5.l.c	$12$	$8.063$	$\mathbb{Q}[x]/(x^{12} + \cdots)$	None	$-12$	$0$	$12$	$48$		$q+(-2+2\beta _{1}-2\beta _{3})q^{2}+(6\beta _{2}+3\beta _{3}+\cdots)q^{3}+\cdots$
78.5.l.d	$12$	$8.063$	$\mathbb{Q}[x]/(x^{12} - \cdots)$	None	$12$	$0$	$60$	$-48$		$q+(2+2\beta _{1}-2\beta _{2}-2\beta _{4})q^{2}+(-3\beta _{1}+\cdots)q^{3}+\cdots$
78.6.a.a	$1$	$12.510$	$\Q$	None	$-4$	$-9$	$4$	$-8$	$+$	$q-4q^{2}-9q^{3}+2^{4}q^{4}+4q^{5}+6^{2}q^{6}+\cdots$
78.6.a.b	$1$	$12.510$	$\Q$	None	$-4$	$9$	$24$	$-238$	$+$	$q-4q^{2}+9q^{3}+2^{4}q^{4}+24q^{5}-6^{2}q^{6}+\cdots$
78.6.a.c	$1$	$12.510$	$\Q$	None	$-4$	$9$	$76$	$100$	$-$	$q-4q^{2}+9q^{3}+2^{4}q^{4}+76q^{5}-6^{2}q^{6}+\cdots$
78.6.a.d	$1$	$12.510$	$\Q$	None	$4$	$-9$	$4$	$-146$	$+$	$q+4q^{2}-9q^{3}+2^{4}q^{4}+4q^{5}-6^{2}q^{6}+\cdots$
78.6.a.e	$1$	$12.510$	$\Q$	None	$4$	$-9$	$56$	$-68$	$-$	$q+4q^{2}-9q^{3}+2^{4}q^{4}+56q^{5}-6^{2}q^{6}+\cdots$

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Elliptic curves over $\Q$
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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
78.2.a.a	\(1\)	\(0.623\)	\(\Q\)	None	\(-1\)	\(-1\)	\(2\)	\(4\)	\(-\)	\(q-q^{2}-q^{3}+q^{4}+2q^{5}+q^{6}+4q^{7}+\cdots\)
78.2.b.a	\(2\)	\(0.623\)	\(\Q(\sqrt{-1}) \)	None	\(0\)	\(2\)	\(0\)	\(0\)		\(q+iq^{2}+q^{3}-q^{4}+2iq^{5}+iq^{6}+\cdots\)
78.2.e.a	\(2\)	\(0.623\)	\(\Q(\sqrt{-3}) \)	None	\(1\)	\(-1\)	\(6\)	\(-2\)		\(q+(1-\zeta_{6})q^{2}+(-1+\zeta_{6})q^{3}-\zeta_{6}q^{4}+\cdots\)
78.2.e.b	\(2\)	\(0.623\)	\(\Q(\sqrt{-3}) \)	None	\(1\)	\(1\)	\(-2\)	\(2\)		\(q+(1-\zeta_{6})q^{2}+(1-\zeta_{6})q^{3}-\zeta_{6}q^{4}+\cdots\)
78.2.g.a	\(12\)	\(0.623\)	12.0.\(\cdots\).52	None	\(0\)	\(0\)	\(0\)	\(-12\)		\(q+\beta _{2}q^{2}-\beta _{4}q^{3}+\beta _{8}q^{4}+(-\beta _{1}-\beta _{11})q^{5}+\cdots\)
78.2.i.a	\(4\)	\(0.623\)	\(\Q(\zeta_{12})\)	None	\(0\)	\(-2\)	\(0\)	\(6\)		\(q+\zeta_{12}q^{2}+(-1+\zeta_{12}^{2})q^{3}+\zeta_{12}^{2}q^{4}+\cdots\)
78.2.i.b	\(4\)	\(0.623\)	\(\Q(\zeta_{12})\)	None	\(0\)	\(2\)	\(0\)	\(-6\)		\(q+\zeta_{12}q^{2}+(1-\zeta_{12}^{2})q^{3}+\zeta_{12}^{2}q^{4}+\cdots\)
78.2.k.a	\(16\)	\(0.623\)	16.0.\(\cdots\).9	None	\(0\)	\(0\)	\(0\)	\(8\)		\(q+(-\beta _{7}-\beta _{15})q^{2}+(-\beta _{1}-\beta _{3}-\beta _{4}+\cdots)q^{3}+\cdots\)
78.3.c.a	\(8\)	\(2.125\)	8.0.\(\cdots\).2	None	\(0\)	\(0\)	\(0\)	\(-8\)		\(q-\beta _{2}q^{2}+\beta _{4}q^{3}-2q^{4}+(\beta _{2}+\beta _{4}+\cdots)q^{5}+\cdots\)
78.3.d.a	\(4\)	\(2.125\)	\(\Q(\sqrt{2}, \sqrt{-5})\)	None	\(0\)	\(-8\)	\(0\)	\(0\)		\(q+\beta _{1}q^{2}+(-2-\beta _{3})q^{3}+2q^{4}+3\beta _{1}q^{5}+\cdots\)
78.3.d.b	\(4\)	\(2.125\)	\(\Q(\sqrt{2}, \sqrt{-5})\)	None	\(0\)	\(8\)	\(0\)	\(0\)		\(q+\beta _{1}q^{2}+(2+\beta _{3})q^{3}+2q^{4}-\beta _{1}q^{5}+\cdots\)
78.3.f.a	\(4\)	\(2.125\)	\(\Q(\zeta_{12})\)	None	\(4\)	\(0\)	\(12\)	\(-4\)		\(q+(1+\zeta_{12})q^{2}+\zeta_{12}^{3}q^{3}+2\zeta_{12}q^{4}+\cdots\)
78.3.f.b	\(8\)	\(2.125\)	8.0.\(\cdots\).1	None	\(-8\)	\(0\)	\(0\)	\(4\)		\(q+(-1-\beta _{6})q^{2}+\beta _{1}q^{3}+2\beta _{6}q^{4}+\cdots\)
78.3.h.a	\(8\)	\(2.125\)	8.0.4857532416.2	None	\(0\)	\(-2\)	\(0\)	\(-12\)		\(q+\beta _{2}q^{2}+(-1-\beta _{1}+\beta _{3}-\beta _{4}-\beta _{5}+\cdots)q^{3}+\cdots\)
78.3.h.b	\(12\)	\(2.125\)	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	\(0\)	\(2\)	\(0\)	\(22\)		\(q+\beta _{7}q^{2}+(-\beta _{4}+\beta _{9})q^{3}+2\beta _{1}q^{4}+\cdots\)
78.3.j.a	\(20\)	\(2.125\)	\(\mathbb{Q}[x]/(x^{20} + \cdots)\)	None	\(0\)	\(0\)	\(0\)	\(18\)		\(q+\beta _{5}q^{2}-\beta _{1}q^{3}+(-2-2\beta _{6})q^{4}+\cdots\)
78.3.l.a	\(4\)	\(2.125\)	\(\Q(\zeta_{12})\)	None	\(-2\)	\(0\)	\(-12\)	\(-10\)		\(q+(-\zeta_{12}-\zeta_{12}^{2}+\zeta_{12}^{3})q^{2}+(\zeta_{12}+\cdots)q^{3}+\cdots\)
78.3.l.b	\(4\)	\(2.125\)	\(\Q(\zeta_{12})\)	None	\(2\)	\(0\)	\(6\)	\(20\)		\(q+(\zeta_{12}+\zeta_{12}^{2}-\zeta_{12}^{3})q^{2}+(-\zeta_{12}+\cdots)q^{3}+\cdots\)
78.3.l.c	\(8\)	\(2.125\)	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	\(4\)	\(0\)	\(-6\)	\(10\)		\(q+(-\beta _{4}-\beta _{5})q^{2}+(2\beta _{2}-\beta _{5})q^{3}+(2\beta _{2}+\cdots)q^{4}+\cdots\)
78.4.a.a	\(1\)	\(4.602\)	\(\Q\)	None	\(-2\)	\(-3\)	\(-16\)	\(28\)	\(+\)	\(q-2q^{2}-3q^{3}+4q^{4}-2^{4}q^{5}+6q^{6}+\cdots\)
78.4.a.b	\(1\)	\(4.602\)	\(\Q\)	None	\(-2\)	\(3\)	\(-16\)	\(-8\)	\(-\)	\(q-2q^{2}+3q^{3}+4q^{4}-2^{4}q^{5}-6q^{6}+\cdots\)
78.4.a.c	\(1\)	\(4.602\)	\(\Q\)	None	\(-2\)	\(3\)	\(10\)	\(-8\)	\(+\)	\(q-2q^{2}+3q^{3}+4q^{4}+10q^{5}-6q^{6}+\cdots\)
78.4.a.d	\(1\)	\(4.602\)	\(\Q\)	None	\(2\)	\(-3\)	\(-20\)	\(-32\)	\(-\)	\(q+2q^{2}-3q^{3}+4q^{4}-20q^{5}-6q^{6}+\cdots\)
78.4.a.e	\(1\)	\(4.602\)	\(\Q\)	None	\(2\)	\(-3\)	\(6\)	\(20\)	\(+\)	\(q+2q^{2}-3q^{3}+4q^{4}+6q^{5}-6q^{6}+\cdots\)
78.4.a.f	\(1\)	\(4.602\)	\(\Q\)	None	\(2\)	\(3\)	\(4\)	\(4\)	\(+\)	\(q+2q^{2}+3q^{3}+4q^{4}+4q^{5}+6q^{6}+\cdots\)
78.4.b.a	\(2\)	\(4.602\)	\(\Q(\sqrt{-1}) \)	None	\(0\)	\(-6\)	\(0\)	\(0\)		\(q+iq^{2}-3q^{3}-4q^{4}-4iq^{5}-3iq^{6}+\cdots\)
78.4.b.b	\(4\)	\(4.602\)	\(\Q(i, \sqrt{17})\)	None	\(0\)	\(12\)	\(0\)	\(0\)		\(q-\beta _{1}q^{2}+3q^{3}-4q^{4}+(4\beta _{1}-\beta _{2}+\cdots)q^{5}+\cdots\)
78.4.e.a	\(2\)	\(4.602\)	\(\Q(\sqrt{-3}) \)	None	\(-2\)	\(3\)	\(14\)	\(-16\)		\(q+(-2+2\zeta_{6})q^{2}+(3-3\zeta_{6})q^{3}-4\zeta_{6}q^{4}+\cdots\)
78.4.e.b	\(4\)	\(4.602\)	\(\Q(\sqrt{-3}, \sqrt{673})\)	None	\(-4\)	\(-6\)	\(26\)	\(-9\)		\(q+(-2+2\beta _{2})q^{2}+(-3+3\beta _{2})q^{3}+\cdots\)
78.4.e.c	\(4\)	\(4.602\)	\(\Q(\sqrt{-3}, \sqrt{61})\)	None	\(4\)	\(6\)	\(4\)	\(-18\)		\(q+(2-2\beta _{1})q^{2}+(3-3\beta _{1})q^{3}-4\beta _{1}q^{4}+\cdots\)
78.4.e.d	\(6\)	\(4.602\)	\(\mathbb{Q}[x]/(x^{6} + \cdots)\)	None	\(6\)	\(-9\)	\(-24\)	\(17\)		\(q+(2+2\beta _{2})q^{2}+(-3-3\beta _{2})q^{3}+4\beta _{2}q^{4}+\cdots\)
78.4.g.a	\(28\)	\(4.602\)		None	\(0\)	\(0\)	\(0\)	\(20\)
78.4.i.a	\(4\)	\(4.602\)	\(\Q(\zeta_{12})\)	None	\(0\)	\(-6\)	\(0\)	\(-42\)		\(q+2\zeta_{12}q^{2}+(-3+3\zeta_{12}^{2})q^{3}+4\zeta_{12}^{2}q^{4}+\cdots\)
78.4.i.b	\(8\)	\(4.602\)	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	\(0\)	\(12\)	\(0\)	\(24\)		\(q-\beta _{4}q^{2}+3\beta _{1}q^{3}+(4-4\beta _{1})q^{4}+(3+\cdots)q^{5}+\cdots\)
78.4.k.a	\(56\)	\(4.602\)		None	\(0\)	\(0\)	\(0\)	\(-92\)
78.5.c.a	\(16\)	\(8.063\)	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None	\(0\)	\(12\)	\(0\)	\(80\)		\(q+\beta _{3}q^{2}+(1+\beta _{3}-\beta _{4})q^{3}-8q^{4}+\cdots\)
78.5.d.a	\(20\)	\(8.063\)	\(\mathbb{Q}[x]/(x^{20} - \cdots)\)	None	\(0\)	\(0\)	\(0\)	\(0\)		\(q+\beta _{2}q^{2}-\beta _{1}q^{3}+8q^{4}+(-2\beta _{2}-\beta _{6}+\cdots)q^{5}+\cdots\)
78.5.f.a	\(8\)	\(8.063\)	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	\(-16\)	\(0\)	\(-36\)	\(-8\)		\(q+(-2+2\beta _{2})q^{2}-3\beta _{1}q^{3}-8\beta _{2}q^{4}+\cdots\)
78.5.f.b	\(8\)	\(8.063\)	8.0.\(\cdots\).2	None	\(16\)	\(0\)	\(60\)	\(88\)		\(q+(2+2\beta _{2})q^{2}+\beta _{3}q^{3}+8\beta _{2}q^{4}+(7+\cdots)q^{5}+\cdots\)
78.5.h.a	\(36\)	\(8.063\)		None	\(0\)	\(0\)	\(0\)	\(-90\)
78.5.j.a	\(36\)	\(8.063\)		None	\(0\)	\(0\)	\(0\)	\(126\)
78.5.l.a	\(8\)	\(8.063\)	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	\(-8\)	\(0\)	\(-72\)	\(118\)		\(q+(-2-2\beta _{1}+2\beta _{4})q^{2}+(-3\beta _{4}-6\beta _{5}+\cdots)q^{3}+\cdots\)
78.5.l.b	\(8\)	\(8.063\)	8.0.\(\cdots\).10	None	\(8\)	\(0\)	\(-24\)	\(22\)		\(q+(2-2\beta _{1}+2\beta _{2})q^{2}+(3\beta _{2}+6\beta _{3}+\cdots)q^{3}+\cdots\)
78.5.l.c	\(12\)	\(8.063\)	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)	None	\(-12\)	\(0\)	\(12\)	\(48\)		\(q+(-2+2\beta _{1}-2\beta _{3})q^{2}+(6\beta _{2}+3\beta _{3}+\cdots)q^{3}+\cdots\)
78.5.l.d	\(12\)	\(8.063\)	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	\(12\)	\(0\)	\(60\)	\(-48\)		\(q+(2+2\beta _{1}-2\beta _{2}-2\beta _{4})q^{2}+(-3\beta _{1}+\cdots)q^{3}+\cdots\)
78.6.a.a	\(1\)	\(12.510\)	\(\Q\)	None	\(-4\)	\(-9\)	\(4\)	\(-8\)	\(+\)	\(q-4q^{2}-9q^{3}+2^{4}q^{4}+4q^{5}+6^{2}q^{6}+\cdots\)
78.6.a.b	\(1\)	\(12.510\)	\(\Q\)	None	\(-4\)	\(9\)	\(24\)	\(-238\)	\(+\)	\(q-4q^{2}+9q^{3}+2^{4}q^{4}+24q^{5}-6^{2}q^{6}+\cdots\)
78.6.a.c	\(1\)	\(12.510\)	\(\Q\)	None	\(-4\)	\(9\)	\(76\)	\(100\)	\(-\)	\(q-4q^{2}+9q^{3}+2^{4}q^{4}+76q^{5}-6^{2}q^{6}+\cdots\)
78.6.a.d	\(1\)	\(12.510\)	\(\Q\)	None	\(4\)	\(-9\)	\(4\)	\(-146\)	\(+\)	\(q+4q^{2}-9q^{3}+2^{4}q^{4}+4q^{5}-6^{2}q^{6}+\cdots\)
78.6.a.e	\(1\)	\(12.510\)	\(\Q\)	None	\(4\)	\(-9\)	\(56\)	\(-68\)	\(-\)	\(q+4q^{2}-9q^{3}+2^{4}q^{4}+56q^{5}-6^{2}q^{6}+\cdots\)

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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.