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	Traces				Fricke sign	$q$-expansion
Label	Dim.	$A$	Field	CM	$a_2$	$a_3$	$a_5$	$a_7$	Fricke sign	$q$-expansion
6001.2.a.a	$113$	$47.918$		None	$-11$	$-11$	$-19$	$11$	$+$
6001.2.a.b	$114$	$47.918$		None	$-8$	$-23$	$-27$	$-53$	$+$
6001.2.a.c	$121$	$47.918$		None	$9$	$13$	$21$	$-13$	$-$
6001.2.a.d	$121$	$47.918$		None	$9$	$21$	$27$	$39$	$-$
6002.2.a.a	$47$	$47.926$		None	$47$	$-13$	$-14$	$-17$	$+$
6002.2.a.b	$56$	$47.926$		None	$-56$	$-11$	$0$	$-21$	$+$
6002.2.a.c	$69$	$47.926$		None	$-69$	$11$	$-2$	$23$	$-$
6002.2.a.d	$79$	$47.926$		None	$79$	$17$	$18$	$19$	$-$
6003.2.a.a	$1$	$47.934$	$\Q$	None	$0$	$0$	$-4$	$-4$	$+$	$q-2q^{4}-4q^{5}-4q^{7}-4q^{11}-5q^{13}+\cdots$
6003.2.a.b	$1$	$47.934$	$\Q$	None	$0$	$0$	$0$	$0$	$+$	$q-2q^{4}+4q^{11}+3q^{13}+4q^{16}-3q^{17}+\cdots$
6003.2.a.c	$1$	$47.934$	$\Q$	None	$1$	$0$	$-3$	$0$	$+$	$q+q^{2}-q^{4}-3q^{5}-3q^{8}-3q^{10}+\cdots$
6003.2.a.d	$2$	$47.934$	$\Q(\sqrt{17}) $	None	$-2$	$0$	$3$	$0$	$+$	$q-q^{2}-q^{4}+(1+\beta )q^{5}+3q^{8}+(-1+\cdots)q^{10}+\cdots$
6003.2.a.e	$2$	$47.934$	$\Q(\sqrt{6}) $	None	$0$	$0$	$0$	$-4$	$+$	$q-2q^{4}+\beta q^{5}+(-2+\beta )q^{7}+q^{13}+\cdots$
6003.2.a.f	$2$	$47.934$	$\Q(\sqrt{5}) $	None	$2$	$0$	$4$	$0$	$+$	$q+q^{2}-q^{4}+2q^{5}-3q^{8}+2q^{10}+\cdots$
6003.2.a.g	$4$	$47.934$	4.4.5744.1	None	$0$	$0$	$2$	$-2$	$-$	$q-2q^{4}+\beta _{1}q^{5}+(-1-\beta _{2}+\beta _{3})q^{7}+\cdots$
6003.2.a.h	$5$	$47.934$	5.5.312617.1	None	$2$	$0$	$3$	$-5$	$-$	$q+\beta _{3}q^{2}+(2-\beta _{1})q^{4}+(1-\beta _{1}-\beta _{2}+\cdots)q^{5}+\cdots$
6003.2.a.i	$7$	$47.934$	$\mathbb{Q}[x]/(x^{7} - \cdots)$	None	$1$	$0$	$5$	$-5$	$+$	$q+\beta _{3}q^{2}+(1+\beta _{4}+\beta _{5})q^{4}+(1-\beta _{3}+\cdots)q^{5}+\cdots$
6003.2.a.j	$7$	$47.934$	$\mathbb{Q}[x]/(x^{7} - \cdots)$	None	$3$	$0$	$3$	$-5$	$+$	$q+\beta _{1}q^{2}+(\beta _{1}+\beta _{2})q^{4}+\beta _{4}q^{5}+(-1+\cdots)q^{7}+\cdots$
6003.2.a.k	$10$	$47.934$	$\mathbb{Q}[x]/(x^{10} - \cdots)$	None	$-3$	$0$	$-6$	$3$	$+$	$q-\beta _{3}q^{2}+(2+\beta _{6})q^{4}+(-1-\beta _{4})q^{5}+\cdots$
6003.2.a.l	$10$	$47.934$	$\mathbb{Q}[x]/(x^{10} - \cdots)$	None	$3$	$0$	$10$	$1$	$-$	$q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(1+\beta _{7})q^{5}+\cdots$
6003.2.a.m	$11$	$47.934$	$\mathbb{Q}[x]/(x^{11} - \cdots)$	None	$-2$	$0$	$-2$	$3$	$+$	$q-\beta _{1}q^{2}+(2+\beta _{2})q^{4}+\beta _{4}q^{5}-\beta _{6}q^{7}+\cdots$
6003.2.a.n	$12$	$47.934$	$\mathbb{Q}[x]/(x^{12} - \cdots)$	None	$3$	$0$	$16$	$-7$	$-$	$q+\beta _{1}q^{2}+(1-\beta _{4}-\beta _{6}+\beta _{8})q^{4}+(1+\cdots)q^{5}+\cdots$
6003.2.a.o	$13$	$47.934$	$\mathbb{Q}[x]/(x^{13} - \cdots)$	None	$-4$	$0$	$-16$	$1$	$+$	$q-\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(-1+\beta _{8})q^{5}+\cdots$
6003.2.a.p	$14$	$47.934$	$\mathbb{Q}[x]/(x^{14} - \cdots)$	None	$2$	$0$	$3$	$-3$	$-$	$q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}-\beta _{11}q^{5}+(\beta _{3}+\cdots)q^{7}+\cdots$
6003.2.a.q	$16$	$47.934$	$\mathbb{Q}[x]/(x^{16} - \cdots)$	None	$-3$	$0$	$-16$	$1$	$+$	$q-\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(-1-\beta _{3})q^{5}+\cdots$
6003.2.a.r	$16$	$47.934$	$\mathbb{Q}[x]/(x^{16} - \cdots)$	None	$1$	$0$	$-3$	$13$	$-$	$q+\beta _{1}q^{2}+(2+\beta _{2})q^{4}+\beta _{8}q^{5}+(1-\beta _{10}+\cdots)q^{7}+\cdots$
6003.2.a.s	$20$	$47.934$	$\mathbb{Q}[x]/(x^{20} - \cdots)$	None	$-2$	$0$	$1$	$9$	$-$	$q-\beta _{1}q^{2}+(2+\beta _{2})q^{4}-\beta _{6}q^{5}-\beta _{14}q^{7}+\cdots$
6003.2.a.t	$22$	$47.934$		None	$-3$	$0$	$0$	$-6$	$+$
6003.2.a.u	$22$	$47.934$		None	$3$	$0$	$0$	$-6$	$+$
6003.2.a.v	$30$	$47.934$		None	$-1$	$0$	$0$	$10$	$-$
6003.2.a.w	$30$	$47.934$		None	$1$	$0$	$0$	$10$	$-$
6004.2.a.a	$1$	$47.942$	$\Q$	None	$0$	$-2$	$-1$	$-5$	$+$	$q-2q^{3}-q^{5}-5q^{7}+q^{9}+q^{11}-2q^{13}+\cdots$
6004.2.a.b	$1$	$47.942$	$\Q$	None	$0$	$0$	$1$	$-4$	$-$	$q+q^{5}-4q^{7}-3q^{9}+2q^{11}-4q^{13}+\cdots$
6004.2.a.c	$1$	$47.942$	$\Q$	None	$0$	$1$	$-3$	$-1$	$-$	$q+q^{3}-3q^{5}-q^{7}-2q^{9}+5q^{13}+\cdots$
6004.2.a.d	$8$	$47.942$	$\mathbb{Q}[x]/(x^{8} - \cdots)$	None	$0$	$0$	$4$	$1$	$-$	$q+\beta _{1}q^{5}-\beta _{4}q^{7}-3q^{9}+(-1+\beta _{1}+\cdots)q^{11}+\cdots$
6004.2.a.e	$24$	$47.942$		None	$0$	$1$	$9$	$2$	$-$
6004.2.a.f	$25$	$47.942$		None	$0$	$4$	$-8$	$2$	$+$
6004.2.a.g	$27$	$47.942$		None	$0$	$-4$	$-10$	$-8$	$+$
6004.2.a.h	$31$	$47.942$		None	$0$	$-4$	$10$	$11$	$-$
6005.2.a.a	$1$	$47.950$	$\Q$	None	$-2$	$-2$	$1$	$-2$	$-$	$q-2q^{2}-2q^{3}+2q^{4}+q^{5}+4q^{6}+\cdots$
6005.2.a.b	$1$	$47.950$	$\Q$	None	$-1$	$0$	$1$	$4$	$-$	$q-q^{2}-q^{4}+q^{5}+4q^{7}+3q^{8}-3q^{9}+\cdots$
6005.2.a.c	$4$	$47.950$	4.4.2777.1	None	$2$	$-2$	$-4$	$-7$	$+$	$q+(1-\beta _{3})q^{2}+(-1-\beta _{2}+\beta _{3})q^{3}+\cdots$
6005.2.a.d	$83$	$47.950$		None	$1$	$-4$	$-83$	$2$	$+$
6005.2.a.e	$88$	$47.950$		None	$-14$	$-34$	$88$	$-35$	$+$
6005.2.a.f	$111$	$47.950$		None	$20$	$40$	$111$	$39$	$-$
6005.2.a.g	$113$	$47.950$		None	$-3$	$6$	$-113$	$7$	$-$
6006.2.a.a	$1$	$47.958$	$\Q$	None	$-1$	$-1$	$-4$	$-1$	$+$	$q-q^{2}-q^{3}+q^{4}-4q^{5}+q^{6}-q^{7}+\cdots$
6006.2.a.b	$1$	$47.958$	$\Q$	None	$-1$	$-1$	$-4$	$-1$	$+$	$q-q^{2}-q^{3}+q^{4}-4q^{5}+q^{6}-q^{7}+\cdots$
6006.2.a.c	$1$	$47.958$	$\Q$	None	$-1$	$-1$	$-2$	$1$	$+$	$q-q^{2}-q^{3}+q^{4}-2q^{5}+q^{6}+q^{7}+\cdots$
6006.2.a.d	$1$	$47.958$	$\Q$	None	$-1$	$-1$	$-2$	$1$	$+$	$q-q^{2}-q^{3}+q^{4}-2q^{5}+q^{6}+q^{7}+\cdots$

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Overview	Random
Universe	Knowledge

Degree 1	Degree 2
Degree 3	Degree 4
$\zeta$ zeros

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
6001.2.a.a	\(113\)	\(47.918\)		None	\(-11\)	\(-11\)	\(-19\)	\(11\)	\(+\)
6001.2.a.b	\(114\)	\(47.918\)		None	\(-8\)	\(-23\)	\(-27\)	\(-53\)	\(+\)
6001.2.a.c	\(121\)	\(47.918\)		None	\(9\)	\(13\)	\(21\)	\(-13\)	\(-\)
6001.2.a.d	\(121\)	\(47.918\)		None	\(9\)	\(21\)	\(27\)	\(39\)	\(-\)
6002.2.a.a	\(47\)	\(47.926\)		None	\(47\)	\(-13\)	\(-14\)	\(-17\)	\(+\)
6002.2.a.b	\(56\)	\(47.926\)		None	\(-56\)	\(-11\)	\(0\)	\(-21\)	\(+\)
6002.2.a.c	\(69\)	\(47.926\)		None	\(-69\)	\(11\)	\(-2\)	\(23\)	\(-\)
6002.2.a.d	\(79\)	\(47.926\)		None	\(79\)	\(17\)	\(18\)	\(19\)	\(-\)
6003.2.a.a	\(1\)	\(47.934\)	\(\Q\)	None	\(0\)	\(0\)	\(-4\)	\(-4\)	\(+\)	\(q-2q^{4}-4q^{5}-4q^{7}-4q^{11}-5q^{13}+\cdots\)
6003.2.a.b	\(1\)	\(47.934\)	\(\Q\)	None	\(0\)	\(0\)	\(0\)	\(0\)	\(+\)	\(q-2q^{4}+4q^{11}+3q^{13}+4q^{16}-3q^{17}+\cdots\)
6003.2.a.c	\(1\)	\(47.934\)	\(\Q\)	None	\(1\)	\(0\)	\(-3\)	\(0\)	\(+\)	\(q+q^{2}-q^{4}-3q^{5}-3q^{8}-3q^{10}+\cdots\)
6003.2.a.d	\(2\)	\(47.934\)	\(\Q(\sqrt{17}) \)	None	\(-2\)	\(0\)	\(3\)	\(0\)	\(+\)	\(q-q^{2}-q^{4}+(1+\beta )q^{5}+3q^{8}+(-1+\cdots)q^{10}+\cdots\)
6003.2.a.e	\(2\)	\(47.934\)	\(\Q(\sqrt{6}) \)	None	\(0\)	\(0\)	\(0\)	\(-4\)	\(+\)	\(q-2q^{4}+\beta q^{5}+(-2+\beta )q^{7}+q^{13}+\cdots\)
6003.2.a.f	\(2\)	\(47.934\)	\(\Q(\sqrt{5}) \)	None	\(2\)	\(0\)	\(4\)	\(0\)	\(+\)	\(q+q^{2}-q^{4}+2q^{5}-3q^{8}+2q^{10}+\cdots\)
6003.2.a.g	\(4\)	\(47.934\)	4.4.5744.1	None	\(0\)	\(0\)	\(2\)	\(-2\)	\(-\)	\(q-2q^{4}+\beta _{1}q^{5}+(-1-\beta _{2}+\beta _{3})q^{7}+\cdots\)
6003.2.a.h	\(5\)	\(47.934\)	5.5.312617.1	None	\(2\)	\(0\)	\(3\)	\(-5\)	\(-\)	\(q+\beta _{3}q^{2}+(2-\beta _{1})q^{4}+(1-\beta _{1}-\beta _{2}+\cdots)q^{5}+\cdots\)
6003.2.a.i	\(7\)	\(47.934\)	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	\(1\)	\(0\)	\(5\)	\(-5\)	\(+\)	\(q+\beta _{3}q^{2}+(1+\beta _{4}+\beta _{5})q^{4}+(1-\beta _{3}+\cdots)q^{5}+\cdots\)
6003.2.a.j	\(7\)	\(47.934\)	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	\(3\)	\(0\)	\(3\)	\(-5\)	\(+\)	\(q+\beta _{1}q^{2}+(\beta _{1}+\beta _{2})q^{4}+\beta _{4}q^{5}+(-1+\cdots)q^{7}+\cdots\)
6003.2.a.k	\(10\)	\(47.934\)	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	\(-3\)	\(0\)	\(-6\)	\(3\)	\(+\)	\(q-\beta _{3}q^{2}+(2+\beta _{6})q^{4}+(-1-\beta _{4})q^{5}+\cdots\)
6003.2.a.l	\(10\)	\(47.934\)	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	\(3\)	\(0\)	\(10\)	\(1\)	\(-\)	\(q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(1+\beta _{7})q^{5}+\cdots\)
6003.2.a.m	\(11\)	\(47.934\)	\(\mathbb{Q}[x]/(x^{11} - \cdots)\)	None	\(-2\)	\(0\)	\(-2\)	\(3\)	\(+\)	\(q-\beta _{1}q^{2}+(2+\beta _{2})q^{4}+\beta _{4}q^{5}-\beta _{6}q^{7}+\cdots\)
6003.2.a.n	\(12\)	\(47.934\)	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	\(3\)	\(0\)	\(16\)	\(-7\)	\(-\)	\(q+\beta _{1}q^{2}+(1-\beta _{4}-\beta _{6}+\beta _{8})q^{4}+(1+\cdots)q^{5}+\cdots\)
6003.2.a.o	\(13\)	\(47.934\)	\(\mathbb{Q}[x]/(x^{13} - \cdots)\)	None	\(-4\)	\(0\)	\(-16\)	\(1\)	\(+\)	\(q-\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(-1+\beta _{8})q^{5}+\cdots\)
6003.2.a.p	\(14\)	\(47.934\)	\(\mathbb{Q}[x]/(x^{14} - \cdots)\)	None	\(2\)	\(0\)	\(3\)	\(-3\)	\(-\)	\(q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}-\beta _{11}q^{5}+(\beta _{3}+\cdots)q^{7}+\cdots\)
6003.2.a.q	\(16\)	\(47.934\)	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None	\(-3\)	\(0\)	\(-16\)	\(1\)	\(+\)	\(q-\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(-1-\beta _{3})q^{5}+\cdots\)
6003.2.a.r	\(16\)	\(47.934\)	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None	\(1\)	\(0\)	\(-3\)	\(13\)	\(-\)	\(q+\beta _{1}q^{2}+(2+\beta _{2})q^{4}+\beta _{8}q^{5}+(1-\beta _{10}+\cdots)q^{7}+\cdots\)
6003.2.a.s	\(20\)	\(47.934\)	\(\mathbb{Q}[x]/(x^{20} - \cdots)\)	None	\(-2\)	\(0\)	\(1\)	\(9\)	\(-\)	\(q-\beta _{1}q^{2}+(2+\beta _{2})q^{4}-\beta _{6}q^{5}-\beta _{14}q^{7}+\cdots\)
6003.2.a.t	\(22\)	\(47.934\)		None	\(-3\)	\(0\)	\(0\)	\(-6\)	\(+\)
6003.2.a.u	\(22\)	\(47.934\)		None	\(3\)	\(0\)	\(0\)	\(-6\)	\(+\)
6003.2.a.v	\(30\)	\(47.934\)		None	\(-1\)	\(0\)	\(0\)	\(10\)	\(-\)
6003.2.a.w	\(30\)	\(47.934\)		None	\(1\)	\(0\)	\(0\)	\(10\)	\(-\)
6004.2.a.a	\(1\)	\(47.942\)	\(\Q\)	None	\(0\)	\(-2\)	\(-1\)	\(-5\)	\(+\)	\(q-2q^{3}-q^{5}-5q^{7}+q^{9}+q^{11}-2q^{13}+\cdots\)
6004.2.a.b	\(1\)	\(47.942\)	\(\Q\)	None	\(0\)	\(0\)	\(1\)	\(-4\)	\(-\)	\(q+q^{5}-4q^{7}-3q^{9}+2q^{11}-4q^{13}+\cdots\)
6004.2.a.c	\(1\)	\(47.942\)	\(\Q\)	None	\(0\)	\(1\)	\(-3\)	\(-1\)	\(-\)	\(q+q^{3}-3q^{5}-q^{7}-2q^{9}+5q^{13}+\cdots\)
6004.2.a.d	\(8\)	\(47.942\)	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	\(0\)	\(0\)	\(4\)	\(1\)	\(-\)	\(q+\beta _{1}q^{5}-\beta _{4}q^{7}-3q^{9}+(-1+\beta _{1}+\cdots)q^{11}+\cdots\)
6004.2.a.e	\(24\)	\(47.942\)		None	\(0\)	\(1\)	\(9\)	\(2\)	\(-\)
6004.2.a.f	\(25\)	\(47.942\)		None	\(0\)	\(4\)	\(-8\)	\(2\)	\(+\)
6004.2.a.g	\(27\)	\(47.942\)		None	\(0\)	\(-4\)	\(-10\)	\(-8\)	\(+\)
6004.2.a.h	\(31\)	\(47.942\)		None	\(0\)	\(-4\)	\(10\)	\(11\)	\(-\)
6005.2.a.a	\(1\)	\(47.950\)	\(\Q\)	None	\(-2\)	\(-2\)	\(1\)	\(-2\)	\(-\)	\(q-2q^{2}-2q^{3}+2q^{4}+q^{5}+4q^{6}+\cdots\)
6005.2.a.b	\(1\)	\(47.950\)	\(\Q\)	None	\(-1\)	\(0\)	\(1\)	\(4\)	\(-\)	\(q-q^{2}-q^{4}+q^{5}+4q^{7}+3q^{8}-3q^{9}+\cdots\)
6005.2.a.c	\(4\)	\(47.950\)	4.4.2777.1	None	\(2\)	\(-2\)	\(-4\)	\(-7\)	\(+\)	\(q+(1-\beta _{3})q^{2}+(-1-\beta _{2}+\beta _{3})q^{3}+\cdots\)
6005.2.a.d	\(83\)	\(47.950\)		None	\(1\)	\(-4\)	\(-83\)	\(2\)	\(+\)
6005.2.a.e	\(88\)	\(47.950\)		None	\(-14\)	\(-34\)	\(88\)	\(-35\)	\(+\)
6005.2.a.f	\(111\)	\(47.950\)		None	\(20\)	\(40\)	\(111\)	\(39\)	\(-\)
6005.2.a.g	\(113\)	\(47.950\)		None	\(-3\)	\(6\)	\(-113\)	\(7\)	\(-\)
6006.2.a.a	\(1\)	\(47.958\)	\(\Q\)	None	\(-1\)	\(-1\)	\(-4\)	\(-1\)	\(+\)	\(q-q^{2}-q^{3}+q^{4}-4q^{5}+q^{6}-q^{7}+\cdots\)
6006.2.a.b	\(1\)	\(47.958\)	\(\Q\)	None	\(-1\)	\(-1\)	\(-4\)	\(-1\)	\(+\)	\(q-q^{2}-q^{3}+q^{4}-4q^{5}+q^{6}-q^{7}+\cdots\)
6006.2.a.c	\(1\)	\(47.958\)	\(\Q\)	None	\(-1\)	\(-1\)	\(-2\)	\(1\)	\(+\)	\(q-q^{2}-q^{3}+q^{4}-2q^{5}+q^{6}+q^{7}+\cdots\)
6006.2.a.d	\(1\)	\(47.958\)	\(\Q\)	None	\(-1\)	\(-1\)	\(-2\)	\(1\)	\(+\)	\(q-q^{2}-q^{3}+q^{4}-2q^{5}+q^{6}+q^{7}+\cdots\)

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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.