LMFDB - Newform search results

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Level	Weight	Analytic conductor	Analytic rank	Dim.

Bad $p$	Char.	Primitive character	Character order	Is maximal/largest

Coefficient field	Self-twists	CM/RM discriminant	Inner twist count	Is self-dual

Coefficient ring index	Coefficient ring gens.	Is twist minimal	Projective image

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Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Twist minimal	Largest	Maximal	Minimal twist	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	Twist minimal	Largest	Maximal	Minimal twist	Inner twists	Rank*	$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$	Fricke sign	Coefficient ring index	Sato-Tate	$q$-expansion
442.2.a.a	$442$	$2$	442.a	1.a	$1$	$1$	$1$	$3.529$	$\Q$	None	✓	✓	✓	✓	442.2.a.a	$1$	$0$	$-1$	$2$	$2$	$2$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+2q^{3}+q^{4}+2q^{5}-2q^{6}+2q^{7}+\cdots$
442.2.a.b	$442$	$2$	442.a	1.a	$1$	$1$	$1$	$3.529$	$\Q$	None	✓	✓	✓	✓	442.2.a.b	$1$	$1$	$1$	$0$	$-4$	$-2$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{4}-4q^{5}-2q^{7}+q^{8}-3q^{9}+\cdots$
442.2.a.c	$442$	$2$	442.a	1.a	$1$	$1$	$1$	$3.529$	$\Q$	None	✓	✓			442.2.a.c	$1$	$0$	$1$	$0$	$2$	$4$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{4}+2q^{5}+4q^{7}+q^{8}-3q^{9}+\cdots$
442.2.a.d	$442$	$2$	442.a	1.a	$1$	$1$	$1$	$3.529$	$\Q$	None	✓	✓			442.2.a.d	$1$	$0$	$1$	$2$	$-2$	$2$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+2q^{3}+q^{4}-2q^{5}+2q^{6}+2q^{7}+\cdots$
442.2.a.e	$442$	$2$	442.a	1.a	$1$	$1$	$1$	$3.529$	$\Q$	None	✓	✓			442.2.a.e	$1$	$0$	$1$	$2$	$4$	$-4$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+2q^{3}+q^{4}+4q^{5}+2q^{6}-4q^{7}+\cdots$
442.2.a.f	$442$	$2$	442.a	1.a	$1$	$2$	$2$	$3.529$	$\Q(\sqrt{2}) $	None	✓	✓	✓	✓	442.2.a.f	$1$	$1$	$-2$	$-4$	$0$	$0$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+(-2+\beta )q^{3}+q^{4}-\beta q^{5}+(2+\cdots)q^{6}+\cdots$
442.2.a.g	$442$	$2$	442.a	1.a	$1$	$2$	$2$	$3.529$	$\Q(\sqrt{5}) $	None	✓	✓	✓	✓	442.2.a.g	$1$	$0$	$-2$	$2$	$4$	$-2$	$-$	$2$	$\mathrm{SU}(2)$	$q-q^{2}+(1+\beta )q^{3}+q^{4}+2q^{5}+(-1+\cdots)q^{6}+\cdots$
442.2.a.h	$442$	$2$	442.a	1.a	$1$	$3$	$3$	$3.529$	3.3.148.1	None	✓	✓	✓	✓	442.2.a.h	$1$	$1$	$-3$	$-2$	$-4$	$0$	$+$	$2$	$\mathrm{SU}(2)$	$q-q^{2}+(-1+\beta _{1})q^{3}+q^{4}+(-1-\beta _{2})q^{5}+\cdots$
442.2.a.i	$442$	$2$	442.a	1.a	$1$	$3$	$3$	$3.529$	3.3.316.1	None	✓	✓	✓	✓	442.2.a.i	$1$	$0$	$3$	$2$	$4$	$0$	$-$	$2$	$\mathrm{SU}(2)$	$q+q^{2}+(1+\beta _{2})q^{3}+q^{4}+(1-\beta _{2})q^{5}+\cdots$
442.2.b.a	$442$	$2$	442.b	17.b	$2$	$2$	$2$	$3.529$	$\Q(\sqrt{-2}) $	None		✓			442.2.b.a	$2$	$0$	$-2$	$0$	$0$	$0$		$1$	$\mathrm{SU}(2)[C_{2}]$	$q-q^{2}+\beta q^{3}+q^{4}-\beta q^{5}-\beta q^{6}-q^{8}+\cdots$
442.2.b.b	$442$	$2$	442.b	17.b	$2$	$2$	$2$	$3.529$	$\Q(\sqrt{-1}) $	None		✓			442.2.b.b	$2$	$0$	$2$	$0$	$0$	$0$		$2$	$\mathrm{SU}(2)[C_{2}]$	$q+q^{2}+iq^{3}+q^{4}+2iq^{5}+iq^{6}+\cdots$
442.2.b.c	$442$	$2$	442.b	17.b	$2$	$4$	$4$	$3.529$	4.0.42632.1	None		✓			442.2.b.c	$2$	$0$	$4$	$0$	$0$	$0$		$1$	$\mathrm{SU}(2)[C_{2}]$	$q+q^{2}+\beta _{1}q^{3}+q^{4}-\beta _{1}q^{5}+\beta _{1}q^{6}+\cdots$
442.2.b.d	$442$	$2$	442.b	17.b	$2$	$4$	$4$	$3.529$	4.0.2048.2	None		✓			442.2.b.d	$2$	$0$	$4$	$0$	$0$	$0$		$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	$q+q^{2}+\beta _{2}q^{3}+q^{4}+\beta _{1}q^{5}+\beta _{2}q^{6}+\cdots$
442.2.b.e	$442$	$2$	442.b	17.b	$2$	$6$	$6$	$3.529$	6.0.7388168.1	None		✓	✓		442.2.b.e	$2$	$0$	$-6$	$0$	$0$	$0$		$2^{5}$	$\mathrm{SU}(2)[C_{2}]$	$q-q^{2}-\beta _{2}q^{3}+q^{4}-\beta _{2}q^{5}+\beta _{2}q^{6}+\cdots$
442.2.c.a	$442$	$2$	442.c	221.b	$2$	$4$	$4$	$3.529$	$\Q(\zeta_{8})$	None		✓			442.2.c.a	$4$	$0$	$0$	$0$	$0$	$0$		$2$	$\mathrm{SU}(2)[C_{2}]$	$q+\zeta_{8}q^{2}+\zeta_{8}^{2}q^{3}-q^{4}-\zeta_{8}^{3}q^{6}+\cdots$
442.2.c.b	$442$	$2$	442.c	221.b	$2$	$16$	$16$	$3.529$	$\mathbb{Q}[x]/(x^{16} + \cdots)$	None		✓	✓		442.2.c.b	$4$	$0$	$0$	$0$	$0$	$0$		$2^{7}$	$\mathrm{SU}(2)[C_{2}]$	$q+\beta _{1}q^{2}+\beta _{7}q^{3}-q^{4}+\beta _{6}q^{5}+\beta _{8}q^{6}+\cdots$
442.2.d.a	$442$	$2$	442.d	13.b	$2$	$4$	$4$	$3.529$	$\Q(\zeta_{12})$	None		✓			442.2.d.a	$2$	$0$	$0$	$4$	$0$	$0$		$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	$q+\zeta_{12}q^{2}+(1+\zeta_{12}^{3})q^{3}-q^{4}+2\zeta_{12}q^{5}+\cdots$
442.2.d.b	$442$	$2$	442.d	13.b	$2$	$6$	$6$	$3.529$	6.0.350464.1	None		✓			442.2.d.b	$2$	$0$	$0$	$0$	$0$	$0$		$2$	$\mathrm{SU}(2)[C_{2}]$	$q+\beta _{3}q^{2}+\beta _{1}q^{3}-q^{4}+(-\beta _{3}+\beta _{4}+\cdots)q^{5}+\cdots$
442.2.d.c	$442$	$2$	442.d	13.b	$2$	$6$	$6$	$3.529$	6.0.9144576.1	None		✓			442.2.d.c	$2$	$0$	$0$	$0$	$0$	$0$		$2$	$\mathrm{SU}(2)[C_{2}]$	$q+\beta _{2}q^{2}-\beta _{3}q^{3}-q^{4}+\beta _{5}q^{5}+\beta _{1}q^{6}+\cdots$
442.2.e.a	$442$	$2$	442.e	13.c	$3$	$2$	$1$	$3.529$	$\Q(\sqrt{-3}) $	None		✓			442.2.e.a	$2$	$0$	$-1$	$1$	$8$	$1$		$1$	$\mathrm{SU}(2)[C_{3}]$	$q+(-1+\zeta_{6})q^{2}+(1-\zeta_{6})q^{3}-\zeta_{6}q^{4}+\cdots$
442.2.e.b	$442$	$2$	442.e	13.c	$3$	$2$	$1$	$3.529$	$\Q(\sqrt{-3}) $	None		✓			442.2.e.b	$2$	$0$	$-1$	$3$	$-8$	$-1$		$1$	$\mathrm{SU}(2)[C_{3}]$	$q+(-1+\zeta_{6})q^{2}+(3-3\zeta_{6})q^{3}-\zeta_{6}q^{4}+\cdots$
442.2.e.c	$442$	$2$	442.e	13.c	$3$	$8$	$4$	$3.529$	8.0.1767277521.3	None		✓			442.2.e.c	$2$	$0$	$-4$	$-5$	$2$	$1$		$1$	$\mathrm{SU}(2)[C_{3}]$	$q+(-1+\beta _{6})q^{2}+(-1-\beta _{3}+\beta _{6}-\beta _{7})q^{3}+\cdots$
442.2.e.d	$442$	$2$	442.e	13.c	$3$	$8$	$4$	$3.529$	8.0.$\cdots$.1	None		✓			442.2.e.d	$2$	$0$	$-4$	$1$	$-6$	$5$		$1$	$\mathrm{SU}(2)[C_{3}]$	$q+(-1+\beta _{2})q^{2}+\beta _{1}q^{3}-\beta _{2}q^{4}+(-1+\cdots)q^{5}+\cdots$
442.2.e.e	$442$	$2$	442.e	13.c	$3$	$8$	$4$	$3.529$	8.0.6498455769.2	None		✓			442.2.e.e	$2$	$0$	$4$	$1$	$10$	$-1$		$1$	$\mathrm{SU}(2)[C_{3}]$	$q+\beta _{5}q^{2}+\beta _{1}q^{3}+(-1+\beta _{5})q^{4}+(1+\cdots)q^{5}+\cdots$
442.2.e.f	$442$	$2$	442.e	13.c	$3$	$12$	$6$	$3.529$	$\mathbb{Q}[x]/(x^{12} - \cdots)$	None		✓	✓		442.2.e.f	$2$	$0$	$6$	$-1$	$-6$	$3$		$1$	$\mathrm{SU}(2)[C_{3}]$	$q+(1-\beta _{6})q^{2}+\beta _{9}q^{3}-\beta _{6}q^{4}+(\beta _{2}+\cdots)q^{5}+\cdots$
442.2.h.a	$442$	$2$	442.h	17.c	$4$	$2$	$1$	$3.529$	$\Q(\sqrt{-1}) $	None		✓			442.2.h.a	$2$	$1$	$0$	$-4$	$-6$	$0$		$1$	$\mathrm{SU}(2)[C_{4}]$	$q+iq^{2}+(-2-2i)q^{3}-q^{4}+(-3+\cdots)q^{5}+\cdots$
442.2.h.b	$442$	$2$	442.h	17.c	$4$	$2$	$1$	$3.529$	$\Q(\sqrt{-1}) $	None		✓			442.2.h.b	$2$	$0$	$0$	$-2$	$-6$	$6$		$1$	$\mathrm{SU}(2)[C_{4}]$	$q-iq^{2}+(-1-i)q^{3}-q^{4}+(-3-3i)q^{5}+\cdots$
442.2.h.c	$442$	$2$	442.h	17.c	$4$	$2$	$1$	$3.529$	$\Q(\sqrt{-1}) $	None		✓			442.2.h.c	$2$	$1$	$0$	$0$	$-2$	$-4$		$1$	$\mathrm{SU}(2)[C_{4}]$	$q-iq^{2}-q^{4}+(-1-i)q^{5}+(-2+2i)q^{7}+\cdots$
442.2.h.d	$442$	$2$	442.h	17.c	$4$	$2$	$1$	$3.529$	$\Q(\sqrt{-1}) $	None		✓			442.2.h.d	$2$	$0$	$0$	$0$	$-2$	$0$		$1$	$\mathrm{SU}(2)[C_{4}]$	$q+iq^{2}-q^{4}+(-1-i)q^{5}-iq^{8}+\cdots$
442.2.h.e	$442$	$2$	442.h	17.c	$4$	$2$	$1$	$3.529$	$\Q(\sqrt{-1}) $	None		✓			442.2.h.e	$2$	$0$	$0$	$0$	$4$	$6$		$1$	$\mathrm{SU}(2)[C_{4}]$	$q+iq^{2}-q^{4}+(2+2i)q^{5}+(3-3i)q^{7}+\cdots$
442.2.h.f	$442$	$2$	442.h	17.c	$4$	$2$	$1$	$3.529$	$\Q(\sqrt{-1}) $	None		✓			442.2.h.f	$2$	$0$	$0$	$2$	$0$	$4$		$1$	$\mathrm{SU}(2)[C_{4}]$	$q-iq^{2}+(1+i)q^{3}-q^{4}+(1-i)q^{6}+\cdots$
442.2.h.g	$442$	$2$	442.h	17.c	$4$	$2$	$1$	$3.529$	$\Q(\sqrt{-1}) $	None		✓			442.2.h.g	$2$	$0$	$0$	$2$	$6$	$-2$		$1$	$\mathrm{SU}(2)[C_{4}]$	$q+iq^{2}+(1+i)q^{3}-q^{4}+(3+3i)q^{5}+\cdots$
442.2.h.h	$442$	$2$	442.h	17.c	$4$	$2$	$1$	$3.529$	$\Q(\sqrt{-1}) $	None		✓			442.2.h.h	$2$	$0$	$0$	$4$	$0$	$-6$		$1$	$\mathrm{SU}(2)[C_{4}]$	$q-iq^{2}+(2+2i)q^{3}-q^{4}+(2-2i)q^{6}+\cdots$
442.2.h.i	$442$	$2$	442.h	17.c	$4$	$2$	$1$	$3.529$	$\Q(\sqrt{-1}) $	None		✓			442.2.h.i	$2$	$0$	$0$	$4$	$2$	$0$		$1$	$\mathrm{SU}(2)[C_{4}]$	$q+iq^{2}+(2+2i)q^{3}-q^{4}+(1+i)q^{5}+\cdots$
442.2.h.j	$442$	$2$	442.h	17.c	$4$	$4$	$2$	$3.529$	$\Q(i, \sqrt{13})$	None		✓			442.2.h.j	$2$	$0$	$0$	$-2$	$-2$	$0$		$2$	$\mathrm{SU}(2)[C_{4}]$	$q-\beta _{1}q^{2}+(-1+\beta _{2})q^{3}-q^{4}+(-\beta _{1}+\cdots)q^{5}+\cdots$
442.2.h.k	$442$	$2$	442.h	17.c	$4$	$4$	$2$	$3.529$	$\Q(\zeta_{12})$	None		✓			442.2.h.k	$2$	$0$	$0$	$2$	$6$	$6$		$1$	$\mathrm{SU}(2)[C_{4}]$	$q-\zeta_{12}^{3}q^{2}+(1-\zeta_{12}-\zeta_{12}^{2}+\zeta_{12}^{3})q^{3}+\cdots$
442.2.h.l	$442$	$2$	442.h	17.c	$4$	$10$	$5$	$3.529$	$\mathbb{Q}[x]/(x^{10} - \cdots)$	None		✓	✓		442.2.h.l	$2$	$0$	$0$	$-2$	$-8$	$-2$		$2^{3}$	$\mathrm{SU}(2)[C_{4}]$	$q-\beta _{6}q^{2}-\beta _{1}q^{3}-q^{4}+(-1+\beta _{1}+\cdots)q^{5}+\cdots$
442.2.i.a	$442$	$2$	442.i	221.k	$4$	$2$	$1$	$3.529$	$\Q(\sqrt{-1}) $	None		✓			442.2.i.a	$2$	$1$	$-2$	$-2$	$-2$	$2$		$1$	$\mathrm{SU}(2)[C_{4}]$	$q-q^{2}+(-1-i)q^{3}+q^{4}+(-1+i)q^{5}+\cdots$
442.2.i.b	$442$	$2$	442.i	221.k	$4$	$2$	$1$	$3.529$	$\Q(\sqrt{-1}) $	None		✓			442.2.i.a	$2$	$0$	$2$	$-2$	$2$	$-2$		$1$	$\mathrm{SU}(2)[C_{4}]$	$q+q^{2}+(-1-i)q^{3}+q^{4}+(1-i)q^{5}+\cdots$
442.2.i.c	$442$	$2$	442.i	221.k	$4$	$18$	$9$	$3.529$	$\mathbb{Q}[x]/(x^{18} + \cdots)$	None		✓			442.2.i.c	$2$	$0$	$-18$	$2$	$-2$	$-4$		$2^{5}$	$\mathrm{SU}(2)[C_{4}]$	$q-q^{2}-\beta _{15}q^{3}+q^{4}-\beta _{11}q^{5}+\beta _{15}q^{6}+\cdots$
442.2.i.d	$442$	$2$	442.i	221.k	$4$	$18$	$9$	$3.529$	$\mathbb{Q}[x]/(x^{18} + \cdots)$	None		✓			442.2.i.c	$2$	$0$	$18$	$2$	$2$	$4$		$2^{5}$	$\mathrm{SU}(2)[C_{4}]$	$q+q^{2}-\beta _{15}q^{3}+q^{4}+\beta _{11}q^{5}-\beta _{15}q^{6}+\cdots$
442.2.l.a	$442$	$2$	442.l	13.e	$6$	$16$	$8$	$3.529$	$\mathbb{Q}[x]/(x^{16} - \cdots)$	None		✓			442.2.l.a	$2$	$0$	$0$	$2$	$0$	$0$		$1$	$\mathrm{SU}(2)[C_{6}]$	$q+\beta _{7}q^{2}+(-\beta _{2}-\beta _{13})q^{3}+\beta _{10}q^{4}+\cdots$
442.2.l.b	$442$	$2$	442.l	13.e	$6$	$24$	$12$	$3.529$		None		✓	✓		442.2.l.b	$2$	$0$	$0$	$-2$	$0$	$0$			$\mathrm{SU}(2)[C_{6}]$
442.2.m.a	$442$	$2$	442.m	221.l	$6$	$20$	$10$	$3.529$	$\mathbb{Q}[x]/(x^{20} - \cdots)$	None		✓			442.2.m.a	$4$	$0$	$-10$	$0$	$0$	$0$		$2^{2}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{6}]$	$q+(-1+\beta _{10})q^{2}+\beta _{1}q^{3}-\beta _{10}q^{4}+\cdots$
442.2.m.b	$442$	$2$	442.m	221.l	$6$	$24$	$12$	$3.529$		None		✓	✓		442.2.m.b	$4$	$0$	$12$	$0$	$0$	$0$			$\mathrm{SU}(2)[C_{6}]$
442.2.n.a	$442$	$2$	442.n	221.n	$6$	$40$	$20$	$3.529$		None		✓	✓	✓	442.2.n.a	$4$	$0$	$0$	$0$	$0$	$0$			$\mathrm{SU}(2)[C_{6}]$
442.2.p.a	$442$	$2$	442.p	17.d	$8$	$4$	$1$	$3.529$	$\Q(\zeta_{8})$	None		✓			442.2.p.a	$2$	$0$	$0$	$4$	$4$	$0$		$1$	$\mathrm{SU}(2)[C_{8}]$	$q-\zeta_{8}^{3}q^{2}+(1+\zeta_{8})q^{3}-\zeta_{8}^{2}q^{4}+(1+\cdots)q^{5}+\cdots$
442.2.p.b	$442$	$2$	442.p	17.d	$8$	$8$	$2$	$3.529$	8.0.110166016.2	None		✓			442.2.p.b	$2$	$0$	$0$	$-4$	$0$	$8$		$2$	$\mathrm{SU}(2)[C_{8}]$	$q+\beta _{3}q^{2}+(\beta _{1}-\beta _{4})q^{3}+\beta _{5}q^{4}+(\beta _{3}+\cdots)q^{5}+\cdots$
442.2.p.c	$442$	$2$	442.p	17.d	$8$	$12$	$3$	$3.529$	$\mathbb{Q}[x]/(x^{12} + \cdots)$	None		✓			442.2.p.c	$2$	$0$	$0$	$0$	$4$	$-8$		$2^{2}$	$\mathrm{SU}(2)[C_{8}]$	$q-\beta _{2}q^{2}+\beta _{8}q^{3}-\beta _{9}q^{4}+(-\beta _{2}+\beta _{3}+\cdots)q^{5}+\cdots$
442.2.p.d	$442$	$2$	442.p	17.d	$8$	$12$	$3$	$3.529$	$\mathbb{Q}[x]/(x^{12} + \cdots)$	None		✓			442.2.p.d	$2$	$0$	$0$	$0$	$4$	$-8$		$2^{2}$	$\mathrm{SU}(2)[C_{8}]$	$q+\beta _{6}q^{2}+\beta _{7}q^{3}+\beta _{9}q^{4}+(\beta _{6}+\beta _{9}+\cdots)q^{5}+\cdots$

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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.