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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Label	Dim.	$A$	Field	CM	RM	Traces				Fricke sign	$q$-expansion
Label	Dim.	$A$	Field	CM	RM	$a_2$	$a_3$	$a_5$	$a_7$	Fricke sign	$q$-expansion
1680.1.bb.a	$1$	$0.838$	$\Q$	$\Q(\sqrt{-5}) $, $\Q(\sqrt{-105}) $	$\Q(\sqrt{21}) $	$0$	$-1$	$-1$	$-1$		$q-q^{3}-q^{5}-q^{7}+q^{9}+q^{15}+q^{21}+\cdots$
1680.1.bb.b	$1$	$0.838$	$\Q$	$\Q(\sqrt{-5}) $, $\Q(\sqrt{-105}) $	$\Q(\sqrt{21}) $	$0$	$-1$	$1$	$1$		$q-q^{3}+q^{5}+q^{7}+q^{9}-q^{15}-q^{21}+\cdots$
1680.1.bb.c	$1$	$0.838$	$\Q$	$\Q(\sqrt{-5}) $, $\Q(\sqrt{-105}) $	$\Q(\sqrt{21}) $	$0$	$1$	$-1$	$1$		$q+q^{3}-q^{5}+q^{7}+q^{9}-q^{15}+q^{21}+\cdots$
1680.1.bb.d	$1$	$0.838$	$\Q$	$\Q(\sqrt{-5}) $, $\Q(\sqrt{-105}) $	$\Q(\sqrt{21}) $	$0$	$1$	$1$	$-1$		$q+q^{3}+q^{5}-q^{7}+q^{9}+q^{15}-q^{21}+\cdots$
1680.1.bb.e	$2$	$0.838$	$\Q(\sqrt{-1}) $	$\Q(\sqrt{-35}) $, $\Q(\sqrt{-21}) $	$\Q(\sqrt{15}) $	$0$	$0$	$0$	$0$		$q+iq^{3}-iq^{5}-iq^{7}-q^{9}-q^{11}+\cdots$
1680.1.bb.f	$2$	$0.838$	$\Q(\sqrt{-1}) $	$\Q(\sqrt{-35}) $, $\Q(\sqrt{-21}) $	$\Q(\sqrt{15}) $	$0$	$0$	$0$	$0$		$q-iq^{3}-iq^{5}+iq^{7}-q^{9}+q^{11}+\cdots$
1680.1.bb.g	$4$	$0.838$	$\Q(\zeta_{12})$	$\Q(\sqrt{-35}) $	None	$0$	$0$	$0$	$0$		$q+\zeta_{12}^{5}q^{3}-\zeta_{12}^{3}q^{5}+\zeta_{12}^{3}q^{7}+\cdots$
1680.1.bb.h	$4$	$0.838$	$\Q(\zeta_{12})$	$\Q(\sqrt{-35}) $	None	$0$	$0$	$0$	$0$		$q+\zeta_{12}q^{3}+\zeta_{12}^{3}q^{5}+\zeta_{12}^{3}q^{7}+\zeta_{12}^{2}q^{9}+\cdots$
1680.1.dp.a	$2$	$0.838$	$\Q(\sqrt{-3}) $	$\Q(\sqrt{-5}) $	None	$0$	$-2$	$-1$	$-1$		$q-q^{3}-\zeta_{6}q^{5}+\zeta_{6}^{2}q^{7}+q^{9}+\zeta_{6}q^{15}+\cdots$
1680.1.dp.b	$2$	$0.838$	$\Q(\sqrt{-3}) $	$\Q(\sqrt{-5}) $	None	$0$	$-1$	$1$	$-1$		$q-\zeta_{6}q^{3}+\zeta_{6}q^{5}+\zeta_{6}^{2}q^{7}+\zeta_{6}^{2}q^{9}+\cdots$
1680.1.dp.c	$2$	$0.838$	$\Q(\sqrt{-3}) $	$\Q(\sqrt{-5}) $	None	$0$	$1$	$1$	$1$		$q+\zeta_{6}q^{3}+\zeta_{6}q^{5}-\zeta_{6}^{2}q^{7}+\zeta_{6}^{2}q^{9}+\cdots$
1680.1.dp.d	$2$	$0.838$	$\Q(\sqrt{-3}) $	$\Q(\sqrt{-5}) $	None	$0$	$2$	$-1$	$1$		$q+q^{3}-\zeta_{6}q^{5}-\zeta_{6}^{2}q^{7}+q^{9}-\zeta_{6}q^{15}+\cdots$
1680.2.a.a	$1$	$13.415$	$\Q$	None	None	$0$	$-1$	$-1$	$-1$	$-$	$q-q^{3}-q^{5}-q^{7}+q^{9}-6q^{11}-4q^{13}+\cdots$
1680.2.a.b	$1$	$13.415$	$\Q$	None	None	$0$	$-1$	$-1$	$-1$	$-$	$q-q^{3}-q^{5}-q^{7}+q^{9}+2q^{13}+q^{15}+\cdots$
1680.2.a.c	$1$	$13.415$	$\Q$	None	None	$0$	$-1$	$-1$	$-1$	$+$	$q-q^{3}-q^{5}-q^{7}+q^{9}+2q^{13}+q^{15}+\cdots$
1680.2.a.d	$1$	$13.415$	$\Q$	None	None	$0$	$-1$	$-1$	$1$	$-$	$q-q^{3}-q^{5}+q^{7}+q^{9}+6q^{13}+q^{15}+\cdots$
1680.2.a.e	$1$	$13.415$	$\Q$	None	None	$0$	$-1$	$-1$	$1$	$-$	$q-q^{3}-q^{5}+q^{7}+q^{9}+4q^{11}-6q^{13}+\cdots$
1680.2.a.f	$1$	$13.415$	$\Q$	None	None	$0$	$-1$	$1$	$-1$	$+$	$q-q^{3}+q^{5}-q^{7}+q^{9}-6q^{13}-q^{15}+\cdots$
1680.2.a.g	$1$	$13.415$	$\Q$	None	None	$0$	$-1$	$1$	$-1$	$+$	$q-q^{3}+q^{5}-q^{7}+q^{9}+2q^{13}-q^{15}+\cdots$
1680.2.a.h	$1$	$13.415$	$\Q$	None	None	$0$	$-1$	$1$	$1$	$+$	$q-q^{3}+q^{5}+q^{7}+q^{9}-4q^{11}-2q^{13}+\cdots$
1680.2.a.i	$1$	$13.415$	$\Q$	None	None	$0$	$-1$	$1$	$1$	$-$	$q-q^{3}+q^{5}+q^{7}+q^{9}-2q^{11}+4q^{13}+\cdots$
1680.2.a.j	$1$	$13.415$	$\Q$	None	None	$0$	$-1$	$1$	$1$	$-$	$q-q^{3}+q^{5}+q^{7}+q^{9}+4q^{11}-2q^{13}+\cdots$
1680.2.a.k	$1$	$13.415$	$\Q$	None	None	$0$	$1$	$-1$	$-1$	$-$	$q+q^{3}-q^{5}-q^{7}+q^{9}-2q^{13}-q^{15}+\cdots$
1680.2.a.l	$1$	$13.415$	$\Q$	None	None	$0$	$1$	$-1$	$-1$	$-$	$q+q^{3}-q^{5}-q^{7}+q^{9}+4q^{11}+2q^{13}+\cdots$
1680.2.a.m	$1$	$13.415$	$\Q$	None	None	$0$	$1$	$-1$	$1$	$+$	$q+q^{3}-q^{5}+q^{7}+q^{9}-4q^{11}-2q^{13}+\cdots$
1680.2.a.n	$1$	$13.415$	$\Q$	None	None	$0$	$1$	$-1$	$1$	$-$	$q+q^{3}-q^{5}+q^{7}+q^{9}-2q^{11}+4q^{13}+\cdots$
1680.2.a.o	$1$	$13.415$	$\Q$	None	None	$0$	$1$	$-1$	$1$	$-$	$q+q^{3}-q^{5}+q^{7}+q^{9}+4q^{11}-2q^{13}+\cdots$
1680.2.a.p	$1$	$13.415$	$\Q$	None	None	$0$	$1$	$1$	$-1$	$+$	$q+q^{3}+q^{5}-q^{7}+q^{9}-4q^{11}-2q^{13}+\cdots$
1680.2.a.q	$1$	$13.415$	$\Q$	None	None	$0$	$1$	$1$	$-1$	$-$	$q+q^{3}+q^{5}-q^{7}+q^{9}-4q^{11}-2q^{13}+\cdots$
1680.2.a.r	$1$	$13.415$	$\Q$	None	None	$0$	$1$	$1$	$-1$	$-$	$q+q^{3}+q^{5}-q^{7}+q^{9}+2q^{11}+4q^{13}+\cdots$
1680.2.a.s	$1$	$13.415$	$\Q$	None	None	$0$	$1$	$1$	$1$	$-$	$q+q^{3}+q^{5}+q^{7}+q^{9}+2q^{13}+q^{15}+\cdots$
1680.2.a.t	$1$	$13.415$	$\Q$	None	None	$0$	$1$	$1$	$1$	$-$	$q+q^{3}+q^{5}+q^{7}+q^{9}+4q^{11}-2q^{13}+\cdots$
1680.2.a.u	$2$	$13.415$	$\Q(\sqrt{2}) $	None	None	$0$	$-2$	$2$	$-2$	$-$	$q-q^{3}+q^{5}-q^{7}+q^{9}+\beta q^{11}+2q^{13}+\cdots$
1680.2.a.v	$2$	$13.415$	$\Q(\sqrt{5}) $	None	None	$0$	$2$	$-2$	$-2$	$+$	$q+q^{3}-q^{5}-q^{7}+q^{9}+(-2-\beta )q^{11}+\cdots$
1680.2.d.a	$4$	$13.415$	$\Q(i, \sqrt{6})$	None	None	$0$	$-4$	$0$	$0$		$q-q^{3}+\beta _{1}q^{5}+(-\beta _{1}-\beta _{3})q^{7}+q^{9}+\cdots$
1680.2.d.b	$4$	$13.415$	$\Q(i, \sqrt{6})$	None	None	$0$	$4$	$0$	$0$		$q+q^{3}-\beta _{1}q^{5}+(-\beta _{1}-\beta _{3})q^{7}+q^{9}+\cdots$
1680.2.d.c	$12$	$13.415$	$\mathbb{Q}[x]/(x^{12} - \cdots)$	None	None	$0$	$-12$	$0$	$4$		$q-q^{3}-\beta _{1}q^{5}-\beta _{9}q^{7}+q^{9}-\beta _{7}q^{11}+\cdots$
1680.2.d.d	$12$	$13.415$	$\mathbb{Q}[x]/(x^{12} - \cdots)$	None	None	$0$	$12$	$0$	$-4$		$q+q^{3}+\beta _{1}q^{5}+\beta _{8}q^{7}+q^{9}-\beta _{7}q^{11}+\cdots$
1680.2.f.a	$2$	$13.415$	$\Q(\sqrt{-3}) $	None	None	$0$	$-3$	$2$	$-4$		$q+(-2+\zeta_{6})q^{3}+q^{5}+(-1-2\zeta_{6})q^{7}+\cdots$
1680.2.f.b	$2$	$13.415$	$\Q(\sqrt{-3}) $	None	None	$0$	$0$	$-2$	$-4$		$q+\zeta_{6}q^{3}-q^{5}+(-2-\zeta_{6})q^{7}-3q^{9}+\cdots$
1680.2.f.c	$2$	$13.415$	$\Q(\sqrt{-3}) $	None	None	$0$	$0$	$2$	$-4$		$q-\zeta_{6}q^{3}+q^{5}+(-2+\zeta_{6})q^{7}-3q^{9}+\cdots$
1680.2.f.d	$2$	$13.415$	$\Q(\sqrt{-3}) $	None	None	$0$	$3$	$-2$	$-4$		$q+(2-\zeta_{6})q^{3}-q^{5}+(-3+2\zeta_{6})q^{7}+\cdots$
1680.2.f.e	$4$	$13.415$	$\Q(i, \sqrt{5})$	None	None	$0$	$-2$	$-4$	$-6$		$q+(\beta _{1}+\beta _{2})q^{3}-q^{5}+(-1+\beta _{1}+2\beta _{2}+\cdots)q^{7}+\cdots$
1680.2.f.f	$4$	$13.415$	$\Q(i, \sqrt{5})$	None	None	$0$	$-2$	$-4$	$6$		$q+(-1-\beta _{1})q^{3}-q^{5}+(1-\beta _{1}-\beta _{2}+\cdots)q^{7}+\cdots$
1680.2.f.g	$4$	$13.415$	$\Q(\sqrt{-3}, \sqrt{-11})$	None	None	$0$	$-1$	$4$	$8$		$q-\beta _{1}q^{3}+q^{5}+(2+\beta _{2})q^{7}+(1+\beta _{2}+\cdots)q^{9}+\cdots$
1680.2.f.h	$4$	$13.415$	$\Q(\sqrt{-3}, \sqrt{-11})$	None	None	$0$	$1$	$-4$	$8$		$q+\beta _{1}q^{3}-q^{5}+(2-\beta _{2})q^{7}+(1+\beta _{2}+\cdots)q^{9}+\cdots$
1680.2.f.i	$4$	$13.415$	$\Q(i, \sqrt{5})$	None	None	$0$	$2$	$4$	$-6$		$q+(-\beta _{1}-\beta _{2})q^{3}+q^{5}+(-1-2\beta _{2}+\cdots)q^{7}+\cdots$
1680.2.f.j	$4$	$13.415$	$\Q(i, \sqrt{5})$	None	None	$0$	$2$	$4$	$6$		$q+(1+\beta _{1})q^{3}+q^{5}+(1+\beta _{2}+\beta _{3})q^{7}+\cdots$
1680.2.f.k	$16$	$13.415$	$\mathbb{Q}[x]/(x^{16} - \cdots)$	None	None	$0$	$0$	$-16$	$-2$		$q+\beta _{3}q^{3}-q^{5}-\beta _{9}q^{7}+\beta _{7}q^{9}+\beta _{8}q^{11}+\cdots$
1680.2.f.l	$16$	$13.415$	$\mathbb{Q}[x]/(x^{16} - \cdots)$	None	None	$0$	$0$	$16$	$-2$		$q-\beta _{3}q^{3}+q^{5}-\beta _{4}q^{7}+\beta _{7}q^{9}+\beta _{8}q^{11}+\cdots$

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$\zeta$ zeros

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

Label	Dim.	\(A\)	Field	CM	RM	Traces				Fricke sign	$q$-expansion
Label	Dim.	\(A\)	Field	CM	RM	\(a_2\)	\(a_3\)	\(a_5\)	\(a_7\)	Fricke sign	$q$-expansion
1680.1.bb.a	\(1\)	\(0.838\)	\(\Q\)	\(\Q(\sqrt{-5}) \), \(\Q(\sqrt{-105}) \)	\(\Q(\sqrt{21}) \)	\(0\)	\(-1\)	\(-1\)	\(-1\)		\(q-q^{3}-q^{5}-q^{7}+q^{9}+q^{15}+q^{21}+\cdots\)
1680.1.bb.b	\(1\)	\(0.838\)	\(\Q\)	\(\Q(\sqrt{-5}) \), \(\Q(\sqrt{-105}) \)	\(\Q(\sqrt{21}) \)	\(0\)	\(-1\)	\(1\)	\(1\)		\(q-q^{3}+q^{5}+q^{7}+q^{9}-q^{15}-q^{21}+\cdots\)
1680.1.bb.c	\(1\)	\(0.838\)	\(\Q\)	\(\Q(\sqrt{-5}) \), \(\Q(\sqrt{-105}) \)	\(\Q(\sqrt{21}) \)	\(0\)	\(1\)	\(-1\)	\(1\)		\(q+q^{3}-q^{5}+q^{7}+q^{9}-q^{15}+q^{21}+\cdots\)
1680.1.bb.d	\(1\)	\(0.838\)	\(\Q\)	\(\Q(\sqrt{-5}) \), \(\Q(\sqrt{-105}) \)	\(\Q(\sqrt{21}) \)	\(0\)	\(1\)	\(1\)	\(-1\)		\(q+q^{3}+q^{5}-q^{7}+q^{9}+q^{15}-q^{21}+\cdots\)
1680.1.bb.e	\(2\)	\(0.838\)	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-35}) \), \(\Q(\sqrt{-21}) \)	\(\Q(\sqrt{15}) \)	\(0\)	\(0\)	\(0\)	\(0\)		\(q+iq^{3}-iq^{5}-iq^{7}-q^{9}-q^{11}+\cdots\)
1680.1.bb.f	\(2\)	\(0.838\)	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-35}) \), \(\Q(\sqrt{-21}) \)	\(\Q(\sqrt{15}) \)	\(0\)	\(0\)	\(0\)	\(0\)		\(q-iq^{3}-iq^{5}+iq^{7}-q^{9}+q^{11}+\cdots\)
1680.1.bb.g	\(4\)	\(0.838\)	\(\Q(\zeta_{12})\)	\(\Q(\sqrt{-35}) \)	None	\(0\)	\(0\)	\(0\)	\(0\)		\(q+\zeta_{12}^{5}q^{3}-\zeta_{12}^{3}q^{5}+\zeta_{12}^{3}q^{7}+\cdots\)
1680.1.bb.h	\(4\)	\(0.838\)	\(\Q(\zeta_{12})\)	\(\Q(\sqrt{-35}) \)	None	\(0\)	\(0\)	\(0\)	\(0\)		\(q+\zeta_{12}q^{3}+\zeta_{12}^{3}q^{5}+\zeta_{12}^{3}q^{7}+\zeta_{12}^{2}q^{9}+\cdots\)
1680.1.dp.a	\(2\)	\(0.838\)	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-5}) \)	None	\(0\)	\(-2\)	\(-1\)	\(-1\)		\(q-q^{3}-\zeta_{6}q^{5}+\zeta_{6}^{2}q^{7}+q^{9}+\zeta_{6}q^{15}+\cdots\)
1680.1.dp.b	\(2\)	\(0.838\)	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-5}) \)	None	\(0\)	\(-1\)	\(1\)	\(-1\)		\(q-\zeta_{6}q^{3}+\zeta_{6}q^{5}+\zeta_{6}^{2}q^{7}+\zeta_{6}^{2}q^{9}+\cdots\)
1680.1.dp.c	\(2\)	\(0.838\)	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-5}) \)	None	\(0\)	\(1\)	\(1\)	\(1\)		\(q+\zeta_{6}q^{3}+\zeta_{6}q^{5}-\zeta_{6}^{2}q^{7}+\zeta_{6}^{2}q^{9}+\cdots\)
1680.1.dp.d	\(2\)	\(0.838\)	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-5}) \)	None	\(0\)	\(2\)	\(-1\)	\(1\)		\(q+q^{3}-\zeta_{6}q^{5}-\zeta_{6}^{2}q^{7}+q^{9}-\zeta_{6}q^{15}+\cdots\)
1680.2.a.a	\(1\)	\(13.415\)	\(\Q\)	None	None	\(0\)	\(-1\)	\(-1\)	\(-1\)	\(-\)	\(q-q^{3}-q^{5}-q^{7}+q^{9}-6q^{11}-4q^{13}+\cdots\)
1680.2.a.b	\(1\)	\(13.415\)	\(\Q\)	None	None	\(0\)	\(-1\)	\(-1\)	\(-1\)	\(-\)	\(q-q^{3}-q^{5}-q^{7}+q^{9}+2q^{13}+q^{15}+\cdots\)
1680.2.a.c	\(1\)	\(13.415\)	\(\Q\)	None	None	\(0\)	\(-1\)	\(-1\)	\(-1\)	\(+\)	\(q-q^{3}-q^{5}-q^{7}+q^{9}+2q^{13}+q^{15}+\cdots\)
1680.2.a.d	\(1\)	\(13.415\)	\(\Q\)	None	None	\(0\)	\(-1\)	\(-1\)	\(1\)	\(-\)	\(q-q^{3}-q^{5}+q^{7}+q^{9}+6q^{13}+q^{15}+\cdots\)
1680.2.a.e	\(1\)	\(13.415\)	\(\Q\)	None	None	\(0\)	\(-1\)	\(-1\)	\(1\)	\(-\)	\(q-q^{3}-q^{5}+q^{7}+q^{9}+4q^{11}-6q^{13}+\cdots\)
1680.2.a.f	\(1\)	\(13.415\)	\(\Q\)	None	None	\(0\)	\(-1\)	\(1\)	\(-1\)	\(+\)	\(q-q^{3}+q^{5}-q^{7}+q^{9}-6q^{13}-q^{15}+\cdots\)
1680.2.a.g	\(1\)	\(13.415\)	\(\Q\)	None	None	\(0\)	\(-1\)	\(1\)	\(-1\)	\(+\)	\(q-q^{3}+q^{5}-q^{7}+q^{9}+2q^{13}-q^{15}+\cdots\)
1680.2.a.h	\(1\)	\(13.415\)	\(\Q\)	None	None	\(0\)	\(-1\)	\(1\)	\(1\)	\(+\)	\(q-q^{3}+q^{5}+q^{7}+q^{9}-4q^{11}-2q^{13}+\cdots\)
1680.2.a.i	\(1\)	\(13.415\)	\(\Q\)	None	None	\(0\)	\(-1\)	\(1\)	\(1\)	\(-\)	\(q-q^{3}+q^{5}+q^{7}+q^{9}-2q^{11}+4q^{13}+\cdots\)
1680.2.a.j	\(1\)	\(13.415\)	\(\Q\)	None	None	\(0\)	\(-1\)	\(1\)	\(1\)	\(-\)	\(q-q^{3}+q^{5}+q^{7}+q^{9}+4q^{11}-2q^{13}+\cdots\)
1680.2.a.k	\(1\)	\(13.415\)	\(\Q\)	None	None	\(0\)	\(1\)	\(-1\)	\(-1\)	\(-\)	\(q+q^{3}-q^{5}-q^{7}+q^{9}-2q^{13}-q^{15}+\cdots\)
1680.2.a.l	\(1\)	\(13.415\)	\(\Q\)	None	None	\(0\)	\(1\)	\(-1\)	\(-1\)	\(-\)	\(q+q^{3}-q^{5}-q^{7}+q^{9}+4q^{11}+2q^{13}+\cdots\)
1680.2.a.m	\(1\)	\(13.415\)	\(\Q\)	None	None	\(0\)	\(1\)	\(-1\)	\(1\)	\(+\)	\(q+q^{3}-q^{5}+q^{7}+q^{9}-4q^{11}-2q^{13}+\cdots\)
1680.2.a.n	\(1\)	\(13.415\)	\(\Q\)	None	None	\(0\)	\(1\)	\(-1\)	\(1\)	\(-\)	\(q+q^{3}-q^{5}+q^{7}+q^{9}-2q^{11}+4q^{13}+\cdots\)
1680.2.a.o	\(1\)	\(13.415\)	\(\Q\)	None	None	\(0\)	\(1\)	\(-1\)	\(1\)	\(-\)	\(q+q^{3}-q^{5}+q^{7}+q^{9}+4q^{11}-2q^{13}+\cdots\)
1680.2.a.p	\(1\)	\(13.415\)	\(\Q\)	None	None	\(0\)	\(1\)	\(1\)	\(-1\)	\(+\)	\(q+q^{3}+q^{5}-q^{7}+q^{9}-4q^{11}-2q^{13}+\cdots\)
1680.2.a.q	\(1\)	\(13.415\)	\(\Q\)	None	None	\(0\)	\(1\)	\(1\)	\(-1\)	\(-\)	\(q+q^{3}+q^{5}-q^{7}+q^{9}-4q^{11}-2q^{13}+\cdots\)
1680.2.a.r	\(1\)	\(13.415\)	\(\Q\)	None	None	\(0\)	\(1\)	\(1\)	\(-1\)	\(-\)	\(q+q^{3}+q^{5}-q^{7}+q^{9}+2q^{11}+4q^{13}+\cdots\)
1680.2.a.s	\(1\)	\(13.415\)	\(\Q\)	None	None	\(0\)	\(1\)	\(1\)	\(1\)	\(-\)	\(q+q^{3}+q^{5}+q^{7}+q^{9}+2q^{13}+q^{15}+\cdots\)
1680.2.a.t	\(1\)	\(13.415\)	\(\Q\)	None	None	\(0\)	\(1\)	\(1\)	\(1\)	\(-\)	\(q+q^{3}+q^{5}+q^{7}+q^{9}+4q^{11}-2q^{13}+\cdots\)
1680.2.a.u	\(2\)	\(13.415\)	\(\Q(\sqrt{2}) \)	None	None	\(0\)	\(-2\)	\(2\)	\(-2\)	\(-\)	\(q-q^{3}+q^{5}-q^{7}+q^{9}+\beta q^{11}+2q^{13}+\cdots\)
1680.2.a.v	\(2\)	\(13.415\)	\(\Q(\sqrt{5}) \)	None	None	\(0\)	\(2\)	\(-2\)	\(-2\)	\(+\)	\(q+q^{3}-q^{5}-q^{7}+q^{9}+(-2-\beta )q^{11}+\cdots\)
1680.2.d.a	\(4\)	\(13.415\)	\(\Q(i, \sqrt{6})\)	None	None	\(0\)	\(-4\)	\(0\)	\(0\)		\(q-q^{3}+\beta _{1}q^{5}+(-\beta _{1}-\beta _{3})q^{7}+q^{9}+\cdots\)
1680.2.d.b	\(4\)	\(13.415\)	\(\Q(i, \sqrt{6})\)	None	None	\(0\)	\(4\)	\(0\)	\(0\)		\(q+q^{3}-\beta _{1}q^{5}+(-\beta _{1}-\beta _{3})q^{7}+q^{9}+\cdots\)
1680.2.d.c	\(12\)	\(13.415\)	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	None	\(0\)	\(-12\)	\(0\)	\(4\)		\(q-q^{3}-\beta _{1}q^{5}-\beta _{9}q^{7}+q^{9}-\beta _{7}q^{11}+\cdots\)
1680.2.d.d	\(12\)	\(13.415\)	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	None	\(0\)	\(12\)	\(0\)	\(-4\)		\(q+q^{3}+\beta _{1}q^{5}+\beta _{8}q^{7}+q^{9}-\beta _{7}q^{11}+\cdots\)
1680.2.f.a	\(2\)	\(13.415\)	\(\Q(\sqrt{-3}) \)	None	None	\(0\)	\(-3\)	\(2\)	\(-4\)		\(q+(-2+\zeta_{6})q^{3}+q^{5}+(-1-2\zeta_{6})q^{7}+\cdots\)
1680.2.f.b	\(2\)	\(13.415\)	\(\Q(\sqrt{-3}) \)	None	None	\(0\)	\(0\)	\(-2\)	\(-4\)		\(q+\zeta_{6}q^{3}-q^{5}+(-2-\zeta_{6})q^{7}-3q^{9}+\cdots\)
1680.2.f.c	\(2\)	\(13.415\)	\(\Q(\sqrt{-3}) \)	None	None	\(0\)	\(0\)	\(2\)	\(-4\)		\(q-\zeta_{6}q^{3}+q^{5}+(-2+\zeta_{6})q^{7}-3q^{9}+\cdots\)
1680.2.f.d	\(2\)	\(13.415\)	\(\Q(\sqrt{-3}) \)	None	None	\(0\)	\(3\)	\(-2\)	\(-4\)		\(q+(2-\zeta_{6})q^{3}-q^{5}+(-3+2\zeta_{6})q^{7}+\cdots\)
1680.2.f.e	\(4\)	\(13.415\)	\(\Q(i, \sqrt{5})\)	None	None	\(0\)	\(-2\)	\(-4\)	\(-6\)		\(q+(\beta _{1}+\beta _{2})q^{3}-q^{5}+(-1+\beta _{1}+2\beta _{2}+\cdots)q^{7}+\cdots\)
1680.2.f.f	\(4\)	\(13.415\)	\(\Q(i, \sqrt{5})\)	None	None	\(0\)	\(-2\)	\(-4\)	\(6\)		\(q+(-1-\beta _{1})q^{3}-q^{5}+(1-\beta _{1}-\beta _{2}+\cdots)q^{7}+\cdots\)
1680.2.f.g	\(4\)	\(13.415\)	\(\Q(\sqrt{-3}, \sqrt{-11})\)	None	None	\(0\)	\(-1\)	\(4\)	\(8\)		\(q-\beta _{1}q^{3}+q^{5}+(2+\beta _{2})q^{7}+(1+\beta _{2}+\cdots)q^{9}+\cdots\)
1680.2.f.h	\(4\)	\(13.415\)	\(\Q(\sqrt{-3}, \sqrt{-11})\)	None	None	\(0\)	\(1\)	\(-4\)	\(8\)		\(q+\beta _{1}q^{3}-q^{5}+(2-\beta _{2})q^{7}+(1+\beta _{2}+\cdots)q^{9}+\cdots\)
1680.2.f.i	\(4\)	\(13.415\)	\(\Q(i, \sqrt{5})\)	None	None	\(0\)	\(2\)	\(4\)	\(-6\)		\(q+(-\beta _{1}-\beta _{2})q^{3}+q^{5}+(-1-2\beta _{2}+\cdots)q^{7}+\cdots\)
1680.2.f.j	\(4\)	\(13.415\)	\(\Q(i, \sqrt{5})\)	None	None	\(0\)	\(2\)	\(4\)	\(6\)		\(q+(1+\beta _{1})q^{3}+q^{5}+(1+\beta _{2}+\beta _{3})q^{7}+\cdots\)
1680.2.f.k	\(16\)	\(13.415\)	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None	None	\(0\)	\(0\)	\(-16\)	\(-2\)		\(q+\beta _{3}q^{3}-q^{5}-\beta _{9}q^{7}+\beta _{7}q^{9}+\beta _{8}q^{11}+\cdots\)
1680.2.f.l	\(16\)	\(13.415\)	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None	None	\(0\)	\(0\)	\(16\)	\(-2\)		\(q-\beta _{3}q^{3}+q^{5}-\beta _{4}q^{7}+\beta _{7}q^{9}+\beta _{8}q^{11}+\cdots\)

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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.