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	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
900.1.c.a	$900$	$1$	900.c	4.b	$2$	$1$	$1$	$0.449$	$\Q$	$D_{2}$	$\Q(\sqrt{-1}) $, $\Q(\sqrt{-15}) $	$\Q(\sqrt{15}) $	✓				180.1.f.a	$4$	$0$	$-1$	$0$	$0$	$0$		$1$		$q-q^{2}+q^{4}-q^{8}+q^{16}+2q^{17}-q^{32}+\cdots$
900.1.c.b	$900$	$1$	900.c	4.b	$2$	$1$	$1$	$0.449$	$\Q$	$D_{2}$	$\Q(\sqrt{-1}) $, $\Q(\sqrt{-15}) $	$\Q(\sqrt{15}) $	✓				180.1.f.a	$4$	$0$	$1$	$0$	$0$	$0$		$1$		$q+q^{2}+q^{4}+q^{8}+q^{16}-2q^{17}+q^{32}+\cdots$
900.1.l.a	$900$	$1$	900.l	5.c	$4$	$4$	$2$	$0.449$	$\Q(i, \sqrt{6})$	$D_{6}$	$\Q(\sqrt{-3}) $	None		✓	✓	✓	900.1.l.a	$8$	$0$	$0$	$0$	$0$	$0$		$1$		$q-\beta _{1}q^{7}-\beta _{3}q^{13}-\beta _{2}q^{19}+q^{31}+\cdots$
900.1.m.a	$900$	$1$	900.m	60.l	$4$	$4$	$2$	$0.449$	$\Q(\zeta_{8})$	$D_{4}$	$\Q(\sqrt{-1}) $	None			✓	✓	180.1.m.a	$8$	$0$	$0$	$0$	$0$	$0$		$1$		$q+\zeta_{8}^{3}q^{2}-\zeta_{8}^{2}q^{4}+\zeta_{8}q^{8}+(1-\zeta_{8}^{2}+\cdots)q^{13}+\cdots$
900.1.t.a	$900$	$1$	900.t	36.f	$6$	$4$	$2$	$0.449$	$\Q(\zeta_{12})$	$D_{3}$	$\Q(\sqrt{-5}) $	None			✓	✓	180.1.p.a	$8$	$0$	$0$	$0$	$0$	$0$		$1$		$q+\zeta_{12}^{5}q^{2}-\zeta_{12}q^{3}-\zeta_{12}^{4}q^{4}+q^{6}+\cdots$
900.1.x.a	$900$	$1$	900.x	100.j	$10$	$4$	$1$	$0.449$	$\Q(\zeta_{10})$	$D_{5}$	$\Q(\sqrt{-1}) $	None			✓	✓	100.1.j.a	$4$	$0$	$1$	$0$	$1$	$0$		$1$		$q-\zeta_{10}^{4}q^{2}-\zeta_{10}^{3}q^{4}-\zeta_{10}^{2}q^{5}+\cdots$
900.1.bb.a	$900$	$1$	900.bb	100.h	$10$	$8$	$2$	$0.449$	$\Q(\zeta_{20})$	$D_{10}$	$\Q(\sqrt{-1}) $	None		✓	✓	✓	900.1.bb.a	$8$	$0$	$0$	$0$	$0$	$0$		$1$		$q-\zeta_{20}^{3}q^{2}+\zeta_{20}^{6}q^{4}+\zeta_{20}^{9}q^{5}+\cdots$
900.1.bd.a	$900$	$1$	900.bd	180.v	$12$	$8$	$2$	$0.449$	$\Q(\zeta_{24})$	$D_{6}$	$\Q(\sqrt{-5}) $	None		✓	✓	✓	900.1.bd.a	$16$	$0$	$0$	$0$	$0$	$0$		$1$		$q-\zeta_{24}^{11}q^{2}-\zeta_{24}q^{3}-\zeta_{24}^{10}q^{4}+\cdots$
900.1.bh.a	$900$	$1$	900.bh	300.u	$20$	$16$	$2$	$0.449$	$\Q(\zeta_{40})$	$D_{20}$	$\Q(\sqrt{-1}) $	None		✓	✓	✓	900.1.bh.a	$8$	$0$	$0$	$0$	$0$	$0$		$1$		$q-\zeta_{40}^{11}q^{2}-\zeta_{40}^{2}q^{4}+\zeta_{40}^{3}q^{5}+\cdots$
900.2.a.a	$900$	$2$	900.a	1.a	$1$	$1$	$1$	$7.187$	$\Q$	$_{}$	None	None	✓				60.2.d.a	$1$	$0$	$0$	$0$	$0$	$-4$	$-$	$1$	$\mathrm{SU}(2)$	$q-4q^{7}+4q^{11}+4q^{17}+4q^{23}+6q^{29}+\cdots$
900.2.a.b	$900$	$2$	900.a	1.a	$1$	$1$	$1$	$7.187$	$\Q$	$_{}$	None	None	✓				20.2.a.a	$1$	$1$	$0$	$0$	$0$	$-2$	$+$	$1$	$\mathrm{SU}(2)$	$q-2q^{7}-2q^{13}-6q^{17}-4q^{19}+6q^{23}+\cdots$
900.2.a.c	$900$	$2$	900.a	1.a	$1$	$1$	$1$	$7.187$	$\Q$	$_{}$	None	None	✓				300.2.a.b	$1$	$0$	$0$	$0$	$0$	$-1$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{7}-6q^{11}+5q^{13}+6q^{17}+5q^{19}+\cdots$
900.2.a.d	$900$	$2$	900.a	1.a	$1$	$1$	$1$	$7.187$	$\Q$	$_{}$	$\Q(\sqrt{-3}) $	None	✓	✓	✓	✓	900.2.a.d	$2$	$1$	$0$	$0$	$0$	$-1$	$+$	$1$	$N(\mathrm{U}(1))$	$q-q^{7}-7q^{13}-7q^{19}+11q^{31}-10q^{37}+\cdots$
900.2.a.e	$900$	$2$	900.a	1.a	$1$	$1$	$1$	$7.187$	$\Q$	$_{}$	None	None	✓				300.2.a.b	$1$	$1$	$0$	$0$	$0$	$1$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{7}-6q^{11}-5q^{13}-6q^{17}+5q^{19}+\cdots$
900.2.a.f	$900$	$2$	900.a	1.a	$1$	$1$	$1$	$7.187$	$\Q$	$_{}$	$\Q(\sqrt{-3}) $	None	✓	✓			900.2.a.d	$2$	$0$	$0$	$0$	$0$	$1$	$-$	$1$	$N(\mathrm{U}(1))$	$q+q^{7}+7q^{13}-7q^{19}+11q^{31}+10q^{37}+\cdots$
900.2.a.g	$900$	$2$	900.a	1.a	$1$	$1$	$1$	$7.187$	$\Q$	$_{}$	$\Q(\sqrt{-3}) $	None	✓				36.2.a.a	$2$	$0$	$0$	$0$	$0$	$4$	$-$	$1$	$N(\mathrm{U}(1))$	$q+4q^{7}-2q^{13}+8q^{19}-4q^{31}+10q^{37}+\cdots$
900.2.a.h	$900$	$2$	900.a	1.a	$1$	$1$	$1$	$7.187$	$\Q$	$_{}$	None	None	✓				60.2.d.a	$1$	$0$	$0$	$0$	$0$	$4$	$-$	$1$	$\mathrm{SU}(2)$	$q+4q^{7}+4q^{11}-4q^{17}-4q^{23}+6q^{29}+\cdots$
900.2.d.a	$900$	$2$	900.d	5.b	$2$	$2$	$2$	$7.187$	$\Q(\sqrt{-1}) $	$_{}$	None	None					300.2.a.b	$2$	$0$	$0$	$0$	$0$	$0$		$1$	$\mathrm{SU}(2)[C_{2}]$	$q+i q^{7}-6 q^{11}+5 i q^{13}-6 i q^{17}+\cdots$
900.2.d.b	$900$	$2$	900.d	5.b	$2$	$2$	$2$	$7.187$	$\Q(\sqrt{-1}) $	$_{}$	$\Q(\sqrt{-3}) $	None					36.2.a.a	$4$	$0$	$0$	$0$	$0$	$0$		$2$	$\mathrm{U}(1)[D_{2}]$	$q+2\beta q^{7}+\beta q^{13}-8 q^{19}-4 q^{31}+\cdots$
900.2.d.c	$900$	$2$	900.d	5.b	$2$	$2$	$2$	$7.187$	$\Q(\sqrt{-1}) $	$_{}$	None	None					20.2.a.a	$2$	$0$	$0$	$0$	$0$	$0$		$2$	$\mathrm{SU}(2)[C_{2}]$	$q+\beta q^{7}-\beta q^{13}+3\beta q^{17}+4 q^{19}+\cdots$
900.2.d.d	$900$	$2$	900.d	5.b	$2$	$2$	$2$	$7.187$	$\Q(\sqrt{-1}) $	$_{}$	$\Q(\sqrt{-3}) $	None		✓			900.2.a.d	$4$	$0$	$0$	$0$	$0$	$0$		$1$	$\mathrm{U}(1)[D_{2}]$	$q+i q^{7}-7 i q^{13}+7 q^{19}+11 q^{31}+\cdots$
900.2.e.a	$900$	$2$	900.e	12.b	$2$	$2$	$2$	$7.187$	$\Q(\sqrt{-2}) $	$_{}$	$\Q(\sqrt{-1}) $	None					180.2.h.a	$4$	$0$	$0$	$0$	$0$	$0$		$1$	$\mathrm{U}(1)[D_{2}]$	$q-\beta q^{2}-2q^{4}+2\beta q^{8}-6q^{13}+4q^{16}+\cdots$
900.2.e.b	$900$	$2$	900.e	12.b	$2$	$2$	$2$	$7.187$	$\Q(\sqrt{-2}) $	$_{}$	$\Q(\sqrt{-1}) $	None					36.2.b.a	$4$	$0$	$0$	$0$	$0$	$0$		$1$	$\mathrm{U}(1)[D_{2}]$	$q+\beta q^{2}-2q^{4}-2\beta q^{8}+4q^{13}+4q^{16}+\cdots$
900.2.e.c	$900$	$2$	900.e	12.b	$2$	$2$	$2$	$7.187$	$\Q(\sqrt{-2}) $	$_{}$	$\Q(\sqrt{-1}) $	None					180.2.h.a	$4$	$0$	$0$	$0$	$0$	$0$		$1$	$\mathrm{U}(1)[D_{2}]$	$q-\beta q^{2}-2q^{4}+2\beta q^{8}+6q^{13}+4q^{16}+\cdots$
900.2.e.d	$900$	$2$	900.e	12.b	$2$	$8$	$8$	$7.187$	8.0.18939904.2	$_{}$	None	None					180.2.e.a	$4$	$0$	$0$	$0$	$0$	$0$		$2^{3}$	$\mathrm{SU}(2)[C_{2}]$	$q+\beta _{5}q^{2}+(1-\beta _{3}-\beta _{4}-\beta _{7})q^{4}+(-2+\cdots)q^{7}+\cdots$
900.2.e.e	$900$	$2$	900.e	12.b	$2$	$8$	$8$	$7.187$	8.0.64000000.1	$_{}$	None	None		✓			900.2.e.e	$4$	$0$	$0$	$0$	$0$	$0$		$2^{6}$	$\mathrm{SU}(2)[C_{2}]$	$q+\beta _{4}q^{2}-\beta _{2}q^{4}+(-1+\beta _{1}-\beta _{2}+\cdots)q^{7}+\cdots$
900.2.e.f	$900$	$2$	900.e	12.b	$2$	$8$	$8$	$7.187$	$\Q(\sqrt{-2}, \sqrt{-3}, \sqrt{5})$	$_{}$	None	None					180.2.h.b	$8$	$0$	$0$	$0$	$0$	$0$		$2^{5}$	$\mathrm{SU}(2)[C_{2}]$	$q+\beta _{1}q^{2}+(1-\beta _{5})q^{4}+(\beta _{2}+\beta _{7})q^{7}+\cdots$
900.2.e.g	$900$	$2$	900.e	12.b	$2$	$8$	$8$	$7.187$	8.0.64000000.1	$_{}$	None	None		✓			900.2.e.e	$4$	$0$	$0$	$0$	$0$	$0$		$2^{6}$	$\mathrm{SU}(2)[C_{2}]$	$q-\beta _{1}q^{2}+\beta _{4}q^{4}+(-\beta _{4}+\beta _{7})q^{7}+\cdots$
900.2.h.a	$900$	$2$	900.h	60.h	$2$	$4$	$4$	$7.187$	$\Q(\zeta_{8})$	$_{}$	$\Q(\sqrt{-1}) $	None					36.2.b.a	$8$	$0$	$0$	$0$	$0$	$0$		$2^{2}$	$\mathrm{U}(1)[D_{2}]$	$q-\beta_{3} q^{2}+2 q^{4}-2\beta_{3} q^{8}+2\beta_1 q^{13}+\cdots$
900.2.h.b	$900$	$2$	900.h	60.h	$2$	$8$	$8$	$7.187$	8.0.18939904.2	$_{}$	None	None					180.2.e.a	$4$	$0$	$-4$	$0$	$0$	$0$		$2^{5}$	$\mathrm{SU}(2)[C_{2}]$	$q+(-1-\beta _{3})q^{2}+(\beta _{3}+\beta _{5})q^{4}+(-2\beta _{4}+\cdots)q^{7}+\cdots$
900.2.h.c	$900$	$2$	900.h	60.h	$2$	$8$	$8$	$7.187$	8.0.18939904.2	$_{}$	None	None					180.2.e.a	$4$	$0$	$4$	$0$	$0$	$0$		$2^{5}$	$\mathrm{SU}(2)[C_{2}]$	$q+(1+\beta _{3})q^{2}+(\beta _{3}+\beta _{5})q^{4}+(-2\beta _{4}+\cdots)q^{7}+\cdots$
900.2.h.d	$900$	$2$	900.h	60.h	$2$	$16$	$16$	$7.187$	$\Q(\zeta_{40})$	$_{}$	None	None		✓	✓		900.2.e.e	$8$	$0$	$0$	$0$	$0$	$0$		$2^{16}$	$\mathrm{SU}(2)[C_{2}]$	$q-\beta_{9} q^{2}+\beta_{3} q^{4}+(\beta_{10}+\beta_{5})q^{7}+\cdots$
900.2.i.a	$900$	$2$	900.i	9.c	$3$	$2$	$1$	$7.187$	$\Q(\sqrt{-3}) $	$_{}$	None	None					180.2.i.a	$2$	$0$	$0$	$-3$	$0$	$-1$		$1$	$\mathrm{SU}(2)[C_{3}]$	$q+(-2+\zeta_{6})q^{3}+(-1+\zeta_{6})q^{7}+(3+\cdots)q^{9}+\cdots$
900.2.i.b	$900$	$2$	900.i	9.c	$3$	$2$	$1$	$7.187$	$\Q(\sqrt{-3}) $	$_{}$	None	None					36.2.e.a	$2$	$1$	$0$	$0$	$0$	$-1$		$1$	$\mathrm{SU}(2)[C_{3}]$	$q+(1-2\zeta_{6})q^{3}+(-1+\zeta_{6})q^{7}-3q^{9}+\cdots$
900.2.i.c	$900$	$2$	900.i	9.c	$3$	$6$	$3$	$7.187$	6.0.954288.1	$_{}$	None	None					180.2.i.b	$2$	$0$	$0$	$1$	$0$	$3$		$3$	$\mathrm{SU}(2)[C_{3}]$	$q-\beta _{4}q^{3}+(-\beta _{1}-\beta _{2}-\beta _{3}-\beta _{5})q^{7}+\cdots$
900.2.i.d	$900$	$2$	900.i	9.c	$3$	$8$	$4$	$7.187$	8.0.142635249.1	$_{}$	None	None		✓			900.2.i.d	$2$	$0$	$0$	$-1$	$0$	$1$		$3^{3}$	$\mathrm{SU}(2)[C_{3}]$	$q+\beta _{3}q^{3}+(-\beta _{3}+\beta _{4}+\beta _{5}+\beta _{7})q^{7}+\cdots$
900.2.i.e	$900$	$2$	900.i	9.c	$3$	$8$	$4$	$7.187$	8.0.142635249.1	$_{}$	None	None		✓			900.2.i.d	$2$	$0$	$0$	$1$	$0$	$-1$		$3^{3}$	$\mathrm{SU}(2)[C_{3}]$	$q-\beta _{2}q^{3}+(2\beta _{2}+\beta _{5}-\beta _{6}-\beta _{7})q^{7}+\cdots$
900.2.i.f	$900$	$2$	900.i	9.c	$3$	$12$	$6$	$7.187$	$\mathbb{Q}[x]/(x^{12} + \cdots)$	$_{}$	None	None			✓		180.2.r.a	$4$	$0$	$0$	$0$	$0$	$0$		$1$	$\mathrm{SU}(2)[C_{3}]$	$q+\beta _{1}q^{3}+(\beta _{5}+\beta _{11})q^{7}+(-1+\beta _{2}+\cdots)q^{9}+\cdots$
900.2.j.a	$900$	$2$	900.j	15.e	$4$	$4$	$2$	$7.187$	$\Q(\zeta_{8})$	$_{}$	None	None					180.2.j.a	$4$	$0$	$0$	$0$	$0$	$-8$		$2$	$\mathrm{SU}(2)[C_{4}]$	$q+(2\beta_1-2)q^{7}+2\beta_{2} q^{11}+(-3\beta_1-3)q^{13}+\cdots$
900.2.j.b	$900$	$2$	900.j	15.e	$4$	$8$	$4$	$7.187$	$\Q(\zeta_{24})$	$_{}$	None	None		✓	✓		900.2.j.b	$8$	$0$	$0$	$0$	$0$	$0$		$2^{6}\cdot 3^{4}$	$\mathrm{SU}(2)[C_{4}]$	$q-\beta_{3} q^{7}-\beta_{5} q^{11}-\beta_1 q^{13}+\beta_{6} q^{17}+\cdots$
900.2.k.a	$900$	$2$	900.k	20.e	$4$	$2$	$1$	$7.187$	$\Q(\sqrt{-1}) $	$_{}$	$\Q(\sqrt{-5}) $	None					100.2.e.a	$4$	$1$	$-2$	$0$	$0$	$-6$		$1$	$\mathrm{U}(1)[D_{4}]$	$q+(i-1)q^{2}-2 i q^{4}+(3 i-3)q^{7}+\cdots$
900.2.k.b	$900$	$2$	900.k	20.e	$4$	$2$	$1$	$7.187$	$\Q(\sqrt{-1}) $	$_{}$	$\Q(\sqrt{-1}) $	None					180.2.k.a	$4$	$1$	$-2$	$0$	$0$	$0$		$1$	$\mathrm{U}(1)[D_{4}]$	$q+(-i-1)q^{2}+2 i q^{4}+(-2 i+2)q^{8}+\cdots$
900.2.k.c	$900$	$2$	900.k	20.e	$4$	$2$	$1$	$7.187$	$\Q(\sqrt{-1}) $	$_{}$	$\Q(\sqrt{-1}) $	None					20.2.e.a	$4$	$0$	$-2$	$0$	$0$	$0$		$1$	$\mathrm{U}(1)[D_{4}]$	$q+(-i-1)q^{2}+2 i q^{4}+(-2 i+2)q^{8}+\cdots$
900.2.k.d	$900$	$2$	900.k	20.e	$4$	$2$	$1$	$7.187$	$\Q(\sqrt{-1}) $	$_{}$	$\Q(\sqrt{-1}) $	None					180.2.k.a	$4$	$0$	$2$	$0$	$0$	$0$		$1$	$\mathrm{U}(1)[D_{4}]$	$q+(i+1)q^{2}+2 i q^{4}+(2 i-2)q^{8}+\cdots$
900.2.k.e	$900$	$2$	900.k	20.e	$4$	$2$	$1$	$7.187$	$\Q(\sqrt{-1}) $	$_{}$	$\Q(\sqrt{-5}) $	None					100.2.e.a	$4$	$0$	$2$	$0$	$0$	$6$		$1$	$\mathrm{U}(1)[D_{4}]$	$q+(-i+1)q^{2}-2 i q^{4}+(-3 i+3)q^{7}+\cdots$
900.2.k.f	$900$	$2$	900.k	20.e	$4$	$8$	$4$	$7.187$	$\Q(\zeta_{24})$	$_{}$	None	None		✓			900.2.k.f	$8$	$0$	$-4$	$0$	$0$	$0$		$2^{2}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{4}]$	$q+(\beta_{3}-\beta_1)q^{2}+(\beta_{4}+\beta_1)q^{4}+(-\beta_{7}+\beta_{6}+\cdots+\beta_{2})q^{7}+\cdots$
900.2.k.g	$900$	$2$	900.k	20.e	$4$	$8$	$4$	$7.187$	$\Q(\zeta_{24})$	$_{}$	None	None					300.2.j.a	$4$	$0$	$0$	$0$	$0$	$0$		$2^{2}$	$\mathrm{SU}(2)[C_{4}]$	$q-\beta_{4} q^{2}+(-\beta_{6}+\beta_{2}-2)q^{4}+\cdots$
900.2.k.h	$900$	$2$	900.k	20.e	$4$	$8$	$4$	$7.187$	$\Q(i, \sqrt{2}, \sqrt{7})$	$_{}$	None	None					300.2.j.b	$8$	$0$	$0$	$0$	$0$	$0$		$2^{2}$	$\mathrm{SU}(2)[C_{4}]$	$q+\beta _{1}q^{2}+(\beta _{4}+\beta _{6})q^{4}+(-\beta _{1}+\beta _{3}+\cdots)q^{7}+\cdots$
900.2.k.i	$900$	$2$	900.k	20.e	$4$	$8$	$4$	$7.187$	$\Q(i, \sqrt{6}, \sqrt{10})$	$_{}$	$\Q(\sqrt{-15}) $	None		✓			900.2.k.i	$16$	$0$	$0$	$0$	$0$	$0$		$2^{2}$	$\mathrm{U}(1)[D_{4}]$	$q+\beta _{5}q^{2}+(\beta _{2}-\beta _{4})q^{4}+(\beta _{1}-\beta _{3})q^{8}+\cdots$
900.2.k.j	$900$	$2$	900.k	20.e	$4$	$8$	$4$	$7.187$	$\Q(i, \sqrt{6}, \sqrt{10})$	$_{}$	None	None					100.2.e.d	$8$	$0$	$0$	$0$	$0$	$0$		$1$	$\mathrm{SU}(2)[C_{4}]$	$q-\beta _{5}q^{2}+(-\beta _{2}+\beta _{3})q^{4}+(\beta _{4}+2\beta _{6}+\cdots)q^{8}+\cdots$

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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.