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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Results (1-50 of 185 matches)

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Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	Image	CM	RM	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	Image	CM	RM	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
512.1.c.a	$512$	$1$	512.c	4.b	$2$	$2$	$2$	$0.256$	$\Q(\sqrt{-2}) $	$D_{4}$	$\Q(\sqrt{-2}) $	None		✓	✓	✓	512.1.c.a	$4$	$0$	$0$	$0$	$0$	$0$		$1$		$q-\beta q^{3}-q^{9}-\beta q^{11}+\beta q^{19}-q^{25}+\cdots$
512.1.d.a	$512$	$1$	512.d	8.d	$2$	$2$	$2$	$0.256$	$\Q(\sqrt{2}) $	$D_{4}$	$\Q(\sqrt{-2}) $	None	✓	✓	✓	✓	512.1.c.a	$4$	$0$	$0$	$0$	$0$	$0$		$1$		$q-\beta q^{3}+q^{9}+\beta q^{11}+\beta q^{19}+q^{25}+\cdots$
512.1.f.a	$512$	$1$	512.f	16.f	$4$	$2$	$1$	$0.256$	$\Q(\sqrt{-1}) $	$D_{4}$	$\Q(\sqrt{-1}) $	None		✓			512.1.f.a	$4$	$0$	$0$	$0$	$-2$	$0$		$1$		$q+(-i-1)q^{5}-i q^{9}+(-i+1)q^{13}+\cdots$
512.1.f.b	$512$	$1$	512.f	16.f	$4$	$2$	$1$	$0.256$	$\Q(\sqrt{-1}) $	$D_{4}$	$\Q(\sqrt{-1}) $	None		✓			512.1.f.a	$4$	$0$	$0$	$0$	$2$	$0$		$1$		$q+(i+1)q^{5}-i q^{9}+(i-1)q^{13}+\cdots$
512.2.a.a	$512$	$2$	512.a	1.a	$1$	$2$	$2$	$4.088$	$\Q(\sqrt{2}) $	$_{}$	$\Q(\sqrt{-2}) $	None	✓	✓			512.2.a.a	$2$	$1$	$0$	$-4$	$0$	$0$	$+$	$1$	$N(\mathrm{U}(1))$	$q+(-2+\beta )q^{3}+(3-4\beta )q^{9}+(-2-3\beta )q^{11}+\cdots$
512.2.a.b	$512$	$2$	512.a	1.a	$1$	$2$	$2$	$4.088$	$\Q(\sqrt{2}) $	$_{}$	None	None	✓	✓			512.2.a.b	$2$	$1$	$0$	$0$	$-4$	$0$	$+$	$1$	$\mathrm{SU}(2)$	$q+\beta q^{3}-2q^{5}-2\beta q^{7}-q^{9}-3\beta q^{11}+\cdots$
512.2.a.c	$512$	$2$	512.a	1.a	$1$	$2$	$2$	$4.088$	$\Q(\sqrt{2}) $	$_{}$	None	None	✓	✓			512.2.a.c	$2$	$1$	$0$	$0$	$0$	$-8$	$+$	$1$	$\mathrm{SU}(2)$	$q+\beta q^{3}-2\beta q^{5}-4q^{7}-q^{9}+\beta q^{11}+\cdots$
512.2.a.d	$512$	$2$	512.a	1.a	$1$	$2$	$2$	$4.088$	$\Q(\sqrt{2}) $	$_{}$	None	None	✓	✓			512.2.a.c	$2$	$0$	$0$	$0$	$0$	$8$	$-$	$1$	$\mathrm{SU}(2)$	$q+\beta q^{3}+2\beta q^{5}+4q^{7}-q^{9}+\beta q^{11}+\cdots$
512.2.a.e	$512$	$2$	512.a	1.a	$1$	$2$	$2$	$4.088$	$\Q(\sqrt{2}) $	$_{}$	None	None	✓	✓			512.2.a.b	$2$	$0$	$0$	$0$	$4$	$0$	$-$	$1$	$\mathrm{SU}(2)$	$q+\beta q^{3}+2q^{5}+2\beta q^{7}-q^{9}-3\beta q^{11}+\cdots$
512.2.a.f	$512$	$2$	512.a	1.a	$1$	$2$	$2$	$4.088$	$\Q(\sqrt{2}) $	$_{}$	$\Q(\sqrt{-2}) $	None	✓	✓			512.2.a.a	$2$	$0$	$0$	$4$	$0$	$0$	$-$	$1$	$N(\mathrm{U}(1))$	$q+(2+\beta )q^{3}+(3+4\beta )q^{9}+(2-3\beta )q^{11}+\cdots$
512.2.a.g	$512$	$2$	512.a	1.a	$1$	$4$	$4$	$4.088$	$\Q(\zeta_{24})^+$	$_{}$	None	None	✓	✓	✓		512.2.a.g	$4$	$0$	$0$	$0$	$0$	$0$	$-$	$2^{3}$	$\mathrm{SU}(2)$	$q+\beta _{1}q^{3}-\beta _{2}q^{5}-\beta _{3}q^{7}+3q^{9}+\beta _{1}q^{11}+\cdots$
512.2.b.a	$512$	$2$	512.b	8.b	$2$	$2$	$2$	$4.088$	$\Q(\sqrt{-2}) $	$_{}$	None	None		✓			512.2.a.c	$2$	$0$	$0$	$0$	$0$	$-8$		$1$	$\mathrm{SU}(2)[C_{2}]$	$q+\beta q^{3}-2\beta q^{5}-4q^{7}+q^{9}-\beta q^{11}+\cdots$
512.2.b.b	$512$	$2$	512.b	8.b	$2$	$2$	$2$	$4.088$	$\Q(\sqrt{-2}) $	$_{}$	None	None		✓			512.2.a.c	$2$	$0$	$0$	$0$	$0$	$8$		$1$	$\mathrm{SU}(2)[C_{2}]$	$q+\beta q^{3}+2\beta q^{5}+4q^{7}+q^{9}-\beta q^{11}+\cdots$
512.2.b.c	$512$	$2$	512.b	8.b	$2$	$4$	$4$	$4.088$	$\Q(\zeta_{8})$	$_{}$	$\Q(\sqrt{-2}) $	None		✓			512.2.a.a	$4$	$0$	$0$	$0$	$0$	$0$		$2^{6}$	$\mathrm{U}(1)[D_{2}]$	$q+\beta_1 q^{3}+(\beta_{3}-3)q^{9}-\beta_{2} q^{11}+\cdots$
512.2.b.d	$512$	$2$	512.b	8.b	$2$	$4$	$4$	$4.088$	$\Q(\sqrt{2}, \sqrt{-3})$	$_{}$	None	None		✓			512.2.a.g	$4$	$0$	$0$	$0$	$0$	$0$		$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	$q-\beta _{1}q^{3}+\beta _{2}q^{5}+\beta _{3}q^{7}-3q^{9}+\beta _{1}q^{11}+\cdots$
512.2.b.e	$512$	$2$	512.b	8.b	$2$	$4$	$4$	$4.088$	$\Q(\zeta_{8})$	$_{}$	None	None		✓			512.2.a.b	$4$	$0$	$0$	$0$	$0$	$0$		$2^{3}$	$\mathrm{SU}(2)[C_{2}]$	$q-\beta_{2} q^{3}+\beta_1 q^{5}-\beta_{3} q^{7}+q^{9}+\cdots$
512.2.e.a	$512$	$2$	512.e	16.e	$4$	$2$	$1$	$4.088$	$\Q(\sqrt{-1}) $	$_{}$	None	None		✓			512.2.e.a	$2$	$1$	$0$	$-4$	$-2$	$0$		$1$	$\mathrm{SU}(2)[C_{4}]$	$q+(2 i-2)q^{3}+(-i-1)q^{5}+4 i q^{7}+\cdots$
512.2.e.b	$512$	$2$	512.e	16.e	$4$	$2$	$1$	$4.088$	$\Q(\sqrt{-1}) $	$_{}$	None	None		✓			512.2.e.a	$2$	$0$	$0$	$-4$	$2$	$0$		$1$	$\mathrm{SU}(2)[C_{4}]$	$q+(2 i-2)q^{3}+(i+1)q^{5}-4 i q^{7}+\cdots$
512.2.e.c	$512$	$2$	512.e	16.e	$4$	$2$	$1$	$4.088$	$\Q(\sqrt{-1}) $	$_{}$	$\Q(\sqrt{-1}) $	None		✓			512.2.e.c	$4$	$1$	$0$	$0$	$-6$	$0$		$1$	$\mathrm{U}(1)[D_{4}]$	$q+(-3 i-3)q^{5}+3 i q^{9}+(i-1)q^{13}+\cdots$
512.2.e.d	$512$	$2$	512.e	16.e	$4$	$2$	$1$	$4.088$	$\Q(\sqrt{-1}) $	$_{}$	$\Q(\sqrt{-1}) $	None		✓			512.2.e.d	$4$	$0$	$0$	$0$	$-2$	$0$		$1$	$\mathrm{U}(1)[D_{4}]$	$q+(-i-1)q^{5}+3 i q^{9}+(-5 i+5)q^{13}+\cdots$
512.2.e.e	$512$	$2$	512.e	16.e	$4$	$2$	$1$	$4.088$	$\Q(\sqrt{-1}) $	$_{}$	$\Q(\sqrt{-1}) $	None		✓			512.2.e.d	$4$	$0$	$0$	$0$	$2$	$0$		$1$	$\mathrm{U}(1)[D_{4}]$	$q+(i+1)q^{5}+3 i q^{9}+(5 i-5)q^{13}+\cdots$
512.2.e.f	$512$	$2$	512.e	16.e	$4$	$2$	$1$	$4.088$	$\Q(\sqrt{-1}) $	$_{}$	$\Q(\sqrt{-1}) $	None		✓			512.2.e.c	$4$	$0$	$0$	$0$	$6$	$0$		$1$	$\mathrm{U}(1)[D_{4}]$	$q+(3 i+3)q^{5}+3 i q^{9}+(-i+1)q^{13}+\cdots$
512.2.e.g	$512$	$2$	512.e	16.e	$4$	$2$	$1$	$4.088$	$\Q(\sqrt{-1}) $	$_{}$	None	None		✓			512.2.e.a	$2$	$0$	$0$	$4$	$-2$	$0$		$1$	$\mathrm{SU}(2)[C_{4}]$	$q+(-2 i+2)q^{3}+(-i-1)q^{5}-4 i q^{7}+\cdots$
512.2.e.h	$512$	$2$	512.e	16.e	$4$	$2$	$1$	$4.088$	$\Q(\sqrt{-1}) $	$_{}$	None	None		✓			512.2.e.a	$2$	$0$	$0$	$4$	$2$	$0$		$1$	$\mathrm{SU}(2)[C_{4}]$	$q+(-2 i+2)q^{3}+(i+1)q^{5}+4 i q^{7}+\cdots$
512.2.e.i	$512$	$2$	512.e	16.e	$4$	$8$	$4$	$4.088$	$\Q(\zeta_{16})$	$_{}$	None	None		✓			512.2.e.i	$4$	$0$	$0$	$0$	$-8$	$0$		$2^{7}$	$\mathrm{SU}(2)[C_{4}]$	$q-\beta_{6} q^{3}+(-\beta_{4}+\beta_{2}-1)q^{5}-\beta_{3} q^{7}+\cdots$
512.2.e.j	$512$	$2$	512.e	16.e	$4$	$8$	$4$	$4.088$	$\Q(\zeta_{16})$	$_{}$	None	None		✓			512.2.e.i	$4$	$0$	$0$	$0$	$8$	$0$		$2^{7}$	$\mathrm{SU}(2)[C_{4}]$	$q-\beta_{6} q^{3}+(\beta_{4}-\beta_{2}+1)q^{5}+\beta_{3} q^{7}+\cdots$
512.2.g.a	$512$	$2$	512.g	32.g	$8$	$4$	$1$	$4.088$	$\Q(\zeta_{8})$	$_{}$	None	None					32.2.g.a	$2$	$1$	$0$	$-4$	$-8$	$-4$		$1$	$\mathrm{SU}(2)[C_{8}]$	$q+(-1-\zeta_{8}^{3})q^{3}+(-2-2\zeta_{8}+\zeta_{8}^{2}+\cdots)q^{5}+\cdots$
512.2.g.b	$512$	$2$	512.g	32.g	$8$	$4$	$1$	$4.088$	$\Q(\zeta_{8})$	$_{}$	None	None					32.2.g.a	$2$	$0$	$0$	$-4$	$8$	$4$		$1$	$\mathrm{SU}(2)[C_{8}]$	$q+(-1-\zeta_{8}^{3})q^{3}+(2+2\zeta_{8}-\zeta_{8}^{2}+\cdots)q^{5}+\cdots$
512.2.g.c	$512$	$2$	512.g	32.g	$8$	$4$	$1$	$4.088$	$\Q(\zeta_{8})$	$_{}$	None	None					32.2.g.a	$2$	$0$	$0$	$4$	$-8$	$4$		$1$	$\mathrm{SU}(2)[C_{8}]$	$q+(1+\zeta_{8}^{3})q^{3}+(-2-2\zeta_{8}+\zeta_{8}^{2}+\cdots)q^{5}+\cdots$
512.2.g.d	$512$	$2$	512.g	32.g	$8$	$4$	$1$	$4.088$	$\Q(\zeta_{8})$	$_{}$	None	None					32.2.g.a	$2$	$0$	$0$	$4$	$8$	$-4$		$1$	$\mathrm{SU}(2)[C_{8}]$	$q+(1+\zeta_{8}^{3})q^{3}+(2+2\zeta_{8}-\zeta_{8}^{2}+\zeta_{8}^{3})q^{5}+\cdots$
512.2.g.e	$512$	$2$	512.g	32.g	$8$	$8$	$2$	$4.088$	8.0.18939904.2	$_{}$	None	None					32.2.g.b	$2$	$0$	$0$	$-4$	$-8$	$8$		$2^{3}$	$\mathrm{SU}(2)[C_{8}]$	$q+(-1+\beta _{4}-\beta _{5})q^{3}+(-1-\beta _{6})q^{5}+\cdots$
512.2.g.f	$512$	$2$	512.g	32.g	$8$	$8$	$2$	$4.088$	8.0.18939904.2	$_{}$	None	None					32.2.g.b	$2$	$0$	$0$	$-4$	$8$	$-8$		$2^{3}$	$\mathrm{SU}(2)[C_{8}]$	$q+(-1+\beta _{4}-\beta _{5})q^{3}+(1+\beta _{6})q^{5}+\cdots$
512.2.g.g	$512$	$2$	512.g	32.g	$8$	$8$	$2$	$4.088$	8.0.18939904.2	$_{}$	None	None					32.2.g.b	$2$	$0$	$0$	$4$	$-8$	$-8$		$2^{3}$	$\mathrm{SU}(2)[C_{8}]$	$q+(1+\beta _{3}-\beta _{4}+\beta _{7})q^{3}+(-1-\beta _{4}+\cdots)q^{5}+\cdots$
512.2.g.h	$512$	$2$	512.g	32.g	$8$	$8$	$2$	$4.088$	8.0.18939904.2	$_{}$	None	None					32.2.g.b	$2$	$0$	$0$	$4$	$8$	$8$		$2^{3}$	$\mathrm{SU}(2)[C_{8}]$	$q+(1+\beta _{3}-\beta _{4}+\beta _{7})q^{3}+(1+\beta _{4})q^{5}+\cdots$
512.2.i.a	$512$	$2$	512.i	64.i	$16$	$56$	$7$	$4.088$		$_{}$	None	None					64.2.i.a	$2$	$0$	$0$	$-8$	$8$	$8$			$\mathrm{SU}(2)[C_{16}]$
512.2.i.b	$512$	$2$	512.i	64.i	$16$	$56$	$7$	$4.088$		$_{}$	None	None					64.2.i.a	$2$	$0$	$0$	$8$	$8$	$-8$			$\mathrm{SU}(2)[C_{16}]$
512.2.k.a	$512$	$2$	512.k	128.k	$32$	$240$	$15$	$4.088$		$_{}$	None	None			✓	✓	128.2.k.a	$2$	$0$	$0$	$16$	$-16$	$16$			$\mathrm{SU}(2)[C_{32}]$
512.2.o.a	$512$	$2$	512.o	512.o	$128$	$4032$	$63$	$4.088$		$_{}$	None	None		✓	✓	✓	512.2.o.a	$2$	$0$	$-64$	$-64$	$-64$	$-64$			$\mathrm{SU}(2)[C_{128}]$
512.3.c.a	$512$	$3$	512.c	4.b	$2$	$2$	$2$	$13.951$	$\Q(\sqrt{-2}) $	$_{}$	None	None		✓			512.3.c.a	$2$	$0$	$0$	$0$	$-16$	$0$		$1$	$\mathrm{SU}(2)[C_{2}]$	$q+\beta q^{3}-8q^{5}-8\beta q^{7}+7q^{9}+9\beta q^{11}+\cdots$
512.3.c.b	$512$	$3$	512.c	4.b	$2$	$2$	$2$	$13.951$	$\Q(\sqrt{-2}) $	$_{}$	None	None		✓			512.3.c.a	$2$	$0$	$0$	$0$	$16$	$0$		$1$	$\mathrm{SU}(2)[C_{2}]$	$q+\beta q^{3}+8q^{5}+8\beta q^{7}+7q^{9}+9\beta q^{11}+\cdots$
512.3.c.c	$512$	$3$	512.c	4.b	$2$	$4$	$4$	$13.951$	$\Q(\sqrt{-2}, \sqrt{-3})$	$_{}$	None	None		✓			512.3.c.c	$2$	$0$	$0$	$0$	$-16$	$0$		$2^{5}$	$\mathrm{SU}(2)[C_{2}]$	$q+(\beta _{1}+\beta _{2})q^{3}+(-4+\beta _{3})q^{5}+(2\beta _{1}+\cdots)q^{7}+\cdots$
512.3.c.d	$512$	$3$	512.c	4.b	$2$	$4$	$4$	$13.951$	$\Q(\zeta_{8})$	$_{}$	$\Q(\sqrt{-2}) $	None		✓			512.3.c.d	$4$	$0$	$0$	$0$	$0$	$0$		$2^{8}$	$\mathrm{U}(1)[D_{2}]$	$q+\beta_1 q^{3}+(\beta_{3}-9)q^{9}+(\beta_{2}-3\beta_1)q^{11}+\cdots$
512.3.c.e	$512$	$3$	512.c	4.b	$2$	$4$	$4$	$13.951$	$\Q(\sqrt{-2}, \sqrt{-3})$	$_{}$	None	None		✓			512.3.c.c	$2$	$0$	$0$	$0$	$16$	$0$		$2^{5}$	$\mathrm{SU}(2)[C_{2}]$	$q+(\beta _{1}+\beta _{2})q^{3}+(4-\beta _{3})q^{5}+(-2\beta _{1}+\cdots)q^{7}+\cdots$
512.3.c.f	$512$	$3$	512.c	4.b	$2$	$8$	$8$	$13.951$	8.0.40960000.1	$_{}$	None	None		✓			512.3.c.f	$4$	$0$	$0$	$0$	$0$	$0$		$2^{20}$	$\mathrm{SU}(2)[C_{2}]$	$q+\beta _{1}q^{3}-\beta _{2}q^{5}+\beta _{7}q^{7}+(-5+\beta _{3}+\cdots)q^{9}+\cdots$
512.3.c.g	$512$	$3$	512.c	4.b	$2$	$8$	$8$	$13.951$	8.0.18939904.2	$_{}$	None	None		✓			512.3.c.g	$4$	$0$	$0$	$0$	$0$	$0$		$2^{20}$	$\mathrm{SU}(2)[C_{2}]$	$q+\beta _{5}q^{3}-\beta _{1}q^{5}-\beta _{4}q^{7}+(-1-\beta _{2}+\cdots)q^{9}+\cdots$
512.3.d.a	$512$	$3$	512.d	8.d	$2$	$2$	$2$	$13.951$	$\Q(\sqrt{2}) $	$_{}$	$\Q(\sqrt{-2}) $	None	✓	✓			512.3.c.d	$2$	$0$	$0$	$-8$	$0$	$0$		$1$	$\mathrm{U}(1)[D_{2}]$	$q+(-4+\beta )q^{3}+(9-8\beta )q^{9}+(-12+\cdots)q^{11}+\cdots$
512.3.d.b	$512$	$3$	512.d	8.d	$2$	$2$	$2$	$13.951$	$\Q(\sqrt{2}) $	$_{}$	$\Q(\sqrt{-2}) $	None	✓	✓			512.3.c.d	$2$	$0$	$0$	$8$	$0$	$0$		$1$	$\mathrm{U}(1)[D_{2}]$	$q+(4+\beta )q^{3}+(9+8\beta )q^{9}+(12+7\beta )q^{11}+\cdots$
512.3.d.c	$512$	$3$	512.d	8.d	$2$	$4$	$4$	$13.951$	$\Q(\sqrt{-2}, \sqrt{-5})$	$_{}$	None	None		✓			512.3.c.f	$2$	$0$	$0$	$-8$	$0$	$0$		$2^{3}$	$\mathrm{SU}(2)[C_{2}]$	$q+(-2-\beta _{1})q^{3}+\beta _{3}q^{5}+(-\beta _{2}+2\beta _{3})q^{7}+\cdots$
512.3.d.d	$512$	$3$	512.d	8.d	$2$	$4$	$4$	$13.951$	$\Q(\zeta_{8})$	$_{}$	None	None		✓			512.3.c.a	$4$	$0$	$0$	$0$	$0$	$0$		$2^{7}$	$\mathrm{SU}(2)[C_{2}]$	$q+\beta_{2} q^{3}+\beta_1 q^{5}-\beta_{3} q^{7}-7 q^{9}+\cdots$
512.3.d.e	$512$	$3$	512.d	8.d	$2$	$4$	$4$	$13.951$	$\Q(\sqrt{-2}, \sqrt{-5})$	$_{}$	None	None		✓			512.3.c.f	$2$	$0$	$0$	$8$	$0$	$0$		$2^{3}$	$\mathrm{SU}(2)[C_{2}]$	$q+(2-\beta _{1})q^{3}+\beta _{3}q^{5}+(-\beta _{2}-2\beta _{3})q^{7}+\cdots$

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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.