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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Results (displaying matches 1-50 of 151) Next

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
12.3.c.a	$1$	$0.327$	$\Q$	$\Q(\sqrt{-3}) $	$0$	$-3$	$0$	$2$		$q-3q^{3}+2q^{7}+9q^{9}-22q^{13}+26q^{19}+\cdots$
12.3.d.a	$2$	$0.327$	$\Q(\sqrt{-3}) $	None	$-2$	$0$	$-4$	$0$		$q+(-1-\zeta_{6})q^{2}+\zeta_{6}q^{3}+(-2+2\zeta_{6})q^{4}+\cdots$
12.4.a.a	$1$	$0.708$	$\Q$	None	$0$	$3$	$-18$	$8$	$+$	$q+3q^{3}-18q^{5}+8q^{7}+9q^{9}+6^{2}q^{11}+\cdots$
12.4.b.a	$4$	$0.708$	$\Q(\sqrt{3}, \sqrt{-5})$	None	$0$	$0$	$0$	$0$		$q+\beta _{1}q^{2}+(-\beta _{1}-\beta _{3})q^{3}+(-2+\beta _{2}+\cdots)q^{4}+\cdots$
12.5.c.a	$1$	$1.240$	$\Q$	$\Q(\sqrt{-3}) $	$0$	$9$	$0$	$-94$		$q+9q^{3}-94q^{7}+3^{4}q^{9}+146q^{13}+\cdots$
12.5.d.a	$4$	$1.240$	$\Q(\sqrt{-3}, \sqrt{13})$	None	$6$	$0$	$24$	$0$		$q+(2-\beta _{1})q^{2}-\beta _{2}q^{3}+(-4-3\beta _{1}+\cdots)q^{4}+\cdots$
12.6.b.a	$8$	$1.925$	$\mathbb{Q}[x]/(x^{8} - \cdots)$	None	$0$	$0$	$0$	$0$		$q+\beta _{2}q^{2}-\beta _{1}q^{3}+(1-\beta _{3})q^{4}+(\beta _{2}+\cdots)q^{5}+\cdots$
12.7.c.a	$2$	$2.761$	$\Q(\sqrt{-5}) $	None	$0$	$-6$	$0$	$484$		$q+(-3+\beta )q^{3}+6\beta q^{5}+242q^{7}+(-711+\cdots)q^{9}+\cdots$
12.7.d.a	$6$	$2.761$	6.0.50898483.1	None	$-10$	$0$	$-44$	$0$		$q+(-2-\beta _{1})q^{2}-\beta _{2}q^{3}+(3^{3}+2\beta _{1}+\cdots)q^{4}+\cdots$
12.8.a.a	$1$	$3.749$	$\Q$	None	$0$	$-27$	$-378$	$-832$	$-$	$q-3^{3}q^{3}-378q^{5}-832q^{7}+3^{6}q^{9}+\cdots$
12.8.a.b	$1$	$3.749$	$\Q$	None	$0$	$27$	$270$	$1112$	$+$	$q+3^{3}q^{3}+270q^{5}+1112q^{7}+3^{6}q^{9}+\cdots$
12.8.b.a	$4$	$3.749$	$\Q(\sqrt{3}, \sqrt{-5})$	None	$0$	$0$	$0$	$0$		$q+\beta _{3}q^{2}+(-3\beta _{2}+3\beta _{3})q^{3}+(88+4\beta _{1}+\cdots)q^{4}+\cdots$
12.8.b.b	$8$	$3.749$	$\mathbb{Q}[x]/(x^{8} + \cdots)$	None	$0$	$0$	$0$	$0$		$q+\beta _{1}q^{2}+(-\beta _{1}+\beta _{2})q^{3}+(-41+\beta _{2}+\cdots)q^{4}+\cdots$
12.9.c.a	$1$	$4.889$	$\Q$	$\Q(\sqrt{-3}) $	$0$	$81$	$0$	$4034$		$q+3^{4}q^{3}+4034q^{7}+3^{8}q^{9}-35806q^{13}+\cdots$
12.9.c.b	$2$	$4.889$	$\Q(\sqrt{-110}) $	None	$0$	$-102$	$0$	$-6188$		$q+(-51+\beta )q^{3}-18\beta q^{5}-3094q^{7}+\cdots$
12.9.d.a	$8$	$4.889$	$\mathbb{Q}[x]/(x^{8} - \cdots)$	None	$6$	$0$	$-336$	$0$		$q+(1+\beta _{1})q^{2}+(-\beta _{1}-\beta _{3})q^{3}+(-6+\cdots)q^{4}+\cdots$
12.10.a.a	$1$	$6.180$	$\Q$	None	$0$	$-81$	$990$	$8576$	$-$	$q-3^{4}q^{3}+990q^{5}+8576q^{7}+3^{8}q^{9}+\cdots$
12.10.b.a	$16$	$6.180$	$\mathbb{Q}[x]/(x^{16} + \cdots)$	None	$0$	$0$	$0$	$0$		$q-\beta _{2}q^{2}-\beta _{4}q^{3}+(-21-\beta _{1})q^{4}+\cdots$
12.11.c.a	$1$	$7.624$	$\Q$	$\Q(\sqrt{-3}) $	$0$	$-243$	$0$	$22082$		$q-3^{5}q^{3}+22082q^{7}+3^{10}q^{9}+702218q^{13}+\cdots$
12.11.c.b	$2$	$7.624$	$\Q(\sqrt{-35}) $	None	$0$	$234$	$0$	$-20636$		$q+(117+\beta )q^{3}+6\beta q^{5}-10318q^{7}+\cdots$
12.11.d.a	$10$	$7.624$	$\mathbb{Q}[x]/(x^{10} - \cdots)$	None	$22$	$0$	$3116$	$0$		$q+(2-\beta _{2})q^{2}+\beta _{1}q^{3}+(-65-2\beta _{1}+\cdots)q^{4}+\cdots$
12.12.a.a	$1$	$9.220$	$\Q$	None	$0$	$-243$	$9990$	$-86128$	$-$	$q-3^{5}q^{3}+9990q^{5}-86128q^{7}+3^{10}q^{9}+\cdots$
12.12.a.b	$1$	$9.220$	$\Q$	None	$0$	$243$	$2862$	$9128$	$+$	$q+3^{5}q^{3}+2862q^{5}+9128q^{7}+3^{10}q^{9}+\cdots$
12.12.b.a	$20$	$9.220$	$\mathbb{Q}[x]/(x^{20} - \cdots)$	None	$0$	$0$	$0$	$0$		$q+\beta _{1}q^{2}-\beta _{3}q^{3}+(99+\beta _{2})q^{4}+(-5\beta _{1}+\cdots)q^{5}+\cdots$
12.13.c.a	$4$	$10.968$	$\mathbb{Q}[x]/(x^{4} + \cdots)$	None	$0$	$300$	$0$	$15800$		$q+(75+\beta _{1})q^{3}+(-3\beta _{1}+\beta _{2})q^{5}+(3950+\cdots)q^{7}+\cdots$
12.13.d.a	$12$	$10.968$	$\mathbb{Q}[x]/(x^{12} - \cdots)$	None	$-90$	$0$	$-10296$	$0$		$q+(-8+\beta _{2})q^{2}+(\beta _{1}+\beta _{2})q^{3}+(-386+\cdots)q^{4}+\cdots$
12.14.a.a	$1$	$12.868$	$\Q$	None	$0$	$-729$	$-14850$	$-62896$	$-$	$q-3^{6}q^{3}-14850q^{5}-62896q^{7}+\cdots$
12.14.a.b	$1$	$12.868$	$\Q$	None	$0$	$729$	$-24570$	$-173704$	$+$	$q+3^{6}q^{3}-24570q^{5}-173704q^{7}+\cdots$
12.14.b.a	$24$	$12.868$		None	$0$	$0$	$0$	$0$
12.15.c.a	$1$	$14.919$	$\Q$	$\Q(\sqrt{-3}) $	$0$	$-2187$	$0$	$-1389022$		$q-3^{7}q^{3}-1389022q^{7}+3^{14}q^{9}+\cdots$
12.15.c.b	$4$	$14.919$	$\mathbb{Q}[x]/(x^{4} + \cdots)$	None	$0$	$2148$	$0$	$1709288$		$q+(537-\beta _{1})q^{3}+(\beta _{1}+\beta _{2})q^{5}+(427322+\cdots)q^{7}+\cdots$
12.15.d.a	$14$	$14.919$	$\mathbb{Q}[x]/(x^{14} - \cdots)$	None	$182$	$0$	$-16124$	$0$		$q+(13-\beta _{1})q^{2}+\beta _{2}q^{3}+(665-13\beta _{1}+\cdots)q^{4}+\cdots$
12.16.a.a	$1$	$17.123$	$\Q$	None	$0$	$-2187$	$45702$	$1217888$	$-$	$q-3^{7}q^{3}+45702q^{5}+1217888q^{7}+\cdots$
12.16.a.b	$2$	$17.123$	$\Q(\sqrt{8017}) $	None	$0$	$4374$	$69660$	$2491504$	$+$	$q+3^{7}q^{3}+(34830-\beta )q^{5}+(1245752+\cdots)q^{7}+\cdots$
12.16.b.a	$28$	$17.123$		None	$0$	$0$	$0$	$0$
12.17.c.a	$1$	$19.479$	$\Q$	$\Q(\sqrt{-3}) $	$0$	$6561$	$0$	$4743554$		$q+3^{8}q^{3}+4743554q^{7}+3^{16}q^{9}+\cdots$
12.17.c.b	$4$	$19.479$	$\mathbb{Q}[x]/(x^{4} + \cdots)$	None	$0$	$-12420$	$0$	$-1050280$		$q+(-3105-\beta _{1})q^{3}+(-8\beta _{1}-\beta _{2}+\cdots)q^{5}+\cdots$
12.17.d.a	$16$	$19.479$	$\mathbb{Q}[x]/(x^{16} - \cdots)$	None	$-186$	$0$	$354144$	$0$		$q+(-12-\beta _{1})q^{2}+(\beta _{1}-\beta _{3})q^{3}+(8542+\cdots)q^{4}+\cdots$
12.18.a.a	$1$	$21.987$	$\Q$	None	$0$	$-6561$	$-1608930$	$-9417184$	$-$	$q-3^{8}q^{3}-1608930q^{5}-9417184q^{7}+\cdots$
12.18.a.b	$1$	$21.987$	$\Q$	None	$0$	$6561$	$130950$	$-14846776$	$+$	$q+3^{8}q^{3}+130950q^{5}-14846776q^{7}+\cdots$
12.18.b.a	$32$	$21.987$		None	$0$	$0$	$0$	$0$
12.19.c.a	$6$	$24.646$	$\mathbb{Q}[x]/(x^{6} + \cdots)$	None	$0$	$23934$	$0$	$11024364$		$q+(3989-\beta _{1})q^{3}+(-4\beta _{1}+\beta _{2})q^{5}+\cdots$
12.19.d.a	$18$	$24.646$	$\mathbb{Q}[x]/(x^{18} - \cdots)$	None	$-170$	$0$	$-1721764$	$0$		$q+(-9-\beta _{1})q^{2}+(-\beta _{1}+\beta _{2})q^{3}+(-24281+\cdots)q^{4}+\cdots$
12.20.a.a	$2$	$27.458$	$\Q(\sqrt{193153}) $	None	$0$	$-39366$	$624780$	$4667104$	$-$	$q-3^{9}q^{3}+(312390-\beta )q^{5}+(2333552+\cdots)q^{7}+\cdots$
12.20.a.b	$2$	$27.458$	$\mathbb{Q}[x]/(x^{2} - \cdots)$	None	$0$	$39366$	$-3267108$	$-27023984$	$+$	$q+3^{9}q^{3}+(-1633554-\beta )q^{5}+(-13511992+\cdots)q^{7}+\cdots$
12.20.b.a	$36$	$27.458$		None	$0$	$0$	$0$	$0$
12.21.c.a	$1$	$30.422$	$\Q$	$\Q(\sqrt{-3}) $	$0$	$59049$	$0$	$-77335774$		$q+3^{10}q^{3}-77335774q^{7}+3^{20}q^{9}+\cdots$
12.21.c.b	$6$	$30.422$	$\mathbb{Q}[x]/(x^{6} + \cdots)$	None	$0$	$-84378$	$0$	$-145040532$		$q+(-14063+\beta _{1})q^{3}+(3^{3}\beta _{1}-\beta _{2}+\cdots)q^{5}+\cdots$
12.21.d.a	$20$	$30.422$	$\mathbb{Q}[x]/(x^{20} - \cdots)$	None	$1254$	$0$	$1476984$	$0$		$q+(63-\beta _{1})q^{2}+(1-2\beta _{1}+\beta _{2})q^{3}+\cdots$
12.22.a.a	$1$	$33.537$	$\Q$	None	$0$	$59049$	$-11268090$	$281914136$	$+$	$q+3^{10}q^{3}-11268090q^{5}+281914136q^{7}+\cdots$

Introduction	Features
Universe	News

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
12.3.c.a	\(1\)	\(0.327\)	\(\Q\)	\(\Q(\sqrt{-3}) \)	\(0\)	\(-3\)	\(0\)	\(2\)		\(q-3q^{3}+2q^{7}+9q^{9}-22q^{13}+26q^{19}+\cdots\)
12.3.d.a	\(2\)	\(0.327\)	\(\Q(\sqrt{-3}) \)	None	\(-2\)	\(0\)	\(-4\)	\(0\)		\(q+(-1-\zeta_{6})q^{2}+\zeta_{6}q^{3}+(-2+2\zeta_{6})q^{4}+\cdots\)
12.4.a.a	\(1\)	\(0.708\)	\(\Q\)	None	\(0\)	\(3\)	\(-18\)	\(8\)	\(+\)	\(q+3q^{3}-18q^{5}+8q^{7}+9q^{9}+6^{2}q^{11}+\cdots\)
12.4.b.a	\(4\)	\(0.708\)	\(\Q(\sqrt{3}, \sqrt{-5})\)	None	\(0\)	\(0\)	\(0\)	\(0\)		\(q+\beta _{1}q^{2}+(-\beta _{1}-\beta _{3})q^{3}+(-2+\beta _{2}+\cdots)q^{4}+\cdots\)
12.5.c.a	\(1\)	\(1.240\)	\(\Q\)	\(\Q(\sqrt{-3}) \)	\(0\)	\(9\)	\(0\)	\(-94\)		\(q+9q^{3}-94q^{7}+3^{4}q^{9}+146q^{13}+\cdots\)
12.5.d.a	\(4\)	\(1.240\)	\(\Q(\sqrt{-3}, \sqrt{13})\)	None	\(6\)	\(0\)	\(24\)	\(0\)		\(q+(2-\beta _{1})q^{2}-\beta _{2}q^{3}+(-4-3\beta _{1}+\cdots)q^{4}+\cdots\)
12.6.b.a	\(8\)	\(1.925\)	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	\(0\)	\(0\)	\(0\)	\(0\)		\(q+\beta _{2}q^{2}-\beta _{1}q^{3}+(1-\beta _{3})q^{4}+(\beta _{2}+\cdots)q^{5}+\cdots\)
12.7.c.a	\(2\)	\(2.761\)	\(\Q(\sqrt{-5}) \)	None	\(0\)	\(-6\)	\(0\)	\(484\)		\(q+(-3+\beta )q^{3}+6\beta q^{5}+242q^{7}+(-711+\cdots)q^{9}+\cdots\)
12.7.d.a	\(6\)	\(2.761\)	6.0.50898483.1	None	\(-10\)	\(0\)	\(-44\)	\(0\)		\(q+(-2-\beta _{1})q^{2}-\beta _{2}q^{3}+(3^{3}+2\beta _{1}+\cdots)q^{4}+\cdots\)
12.8.a.a	\(1\)	\(3.749\)	\(\Q\)	None	\(0\)	\(-27\)	\(-378\)	\(-832\)	\(-\)	\(q-3^{3}q^{3}-378q^{5}-832q^{7}+3^{6}q^{9}+\cdots\)
12.8.a.b	\(1\)	\(3.749\)	\(\Q\)	None	\(0\)	\(27\)	\(270\)	\(1112\)	\(+\)	\(q+3^{3}q^{3}+270q^{5}+1112q^{7}+3^{6}q^{9}+\cdots\)
12.8.b.a	\(4\)	\(3.749\)	\(\Q(\sqrt{3}, \sqrt{-5})\)	None	\(0\)	\(0\)	\(0\)	\(0\)		\(q+\beta _{3}q^{2}+(-3\beta _{2}+3\beta _{3})q^{3}+(88+4\beta _{1}+\cdots)q^{4}+\cdots\)
12.8.b.b	\(8\)	\(3.749\)	\(\mathbb{Q}[x]/(x^{8} + \cdots)\)	None	\(0\)	\(0\)	\(0\)	\(0\)		\(q+\beta _{1}q^{2}+(-\beta _{1}+\beta _{2})q^{3}+(-41+\beta _{2}+\cdots)q^{4}+\cdots\)
12.9.c.a	\(1\)	\(4.889\)	\(\Q\)	\(\Q(\sqrt{-3}) \)	\(0\)	\(81\)	\(0\)	\(4034\)		\(q+3^{4}q^{3}+4034q^{7}+3^{8}q^{9}-35806q^{13}+\cdots\)
12.9.c.b	\(2\)	\(4.889\)	\(\Q(\sqrt{-110}) \)	None	\(0\)	\(-102\)	\(0\)	\(-6188\)		\(q+(-51+\beta )q^{3}-18\beta q^{5}-3094q^{7}+\cdots\)
12.9.d.a	\(8\)	\(4.889\)	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	\(6\)	\(0\)	\(-336\)	\(0\)		\(q+(1+\beta _{1})q^{2}+(-\beta _{1}-\beta _{3})q^{3}+(-6+\cdots)q^{4}+\cdots\)
12.10.a.a	\(1\)	\(6.180\)	\(\Q\)	None	\(0\)	\(-81\)	\(990\)	\(8576\)	\(-\)	\(q-3^{4}q^{3}+990q^{5}+8576q^{7}+3^{8}q^{9}+\cdots\)
12.10.b.a	\(16\)	\(6.180\)	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)	None	\(0\)	\(0\)	\(0\)	\(0\)		\(q-\beta _{2}q^{2}-\beta _{4}q^{3}+(-21-\beta _{1})q^{4}+\cdots\)
12.11.c.a	\(1\)	\(7.624\)	\(\Q\)	\(\Q(\sqrt{-3}) \)	\(0\)	\(-243\)	\(0\)	\(22082\)		\(q-3^{5}q^{3}+22082q^{7}+3^{10}q^{9}+702218q^{13}+\cdots\)
12.11.c.b	\(2\)	\(7.624\)	\(\Q(\sqrt{-35}) \)	None	\(0\)	\(234\)	\(0\)	\(-20636\)		\(q+(117+\beta )q^{3}+6\beta q^{5}-10318q^{7}+\cdots\)
12.11.d.a	\(10\)	\(7.624\)	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	\(22\)	\(0\)	\(3116\)	\(0\)		\(q+(2-\beta _{2})q^{2}+\beta _{1}q^{3}+(-65-2\beta _{1}+\cdots)q^{4}+\cdots\)
12.12.a.a	\(1\)	\(9.220\)	\(\Q\)	None	\(0\)	\(-243\)	\(9990\)	\(-86128\)	\(-\)	\(q-3^{5}q^{3}+9990q^{5}-86128q^{7}+3^{10}q^{9}+\cdots\)
12.12.a.b	\(1\)	\(9.220\)	\(\Q\)	None	\(0\)	\(243\)	\(2862\)	\(9128\)	\(+\)	\(q+3^{5}q^{3}+2862q^{5}+9128q^{7}+3^{10}q^{9}+\cdots\)
12.12.b.a	\(20\)	\(9.220\)	\(\mathbb{Q}[x]/(x^{20} - \cdots)\)	None	\(0\)	\(0\)	\(0\)	\(0\)		\(q+\beta _{1}q^{2}-\beta _{3}q^{3}+(99+\beta _{2})q^{4}+(-5\beta _{1}+\cdots)q^{5}+\cdots\)
12.13.c.a	\(4\)	\(10.968\)	\(\mathbb{Q}[x]/(x^{4} + \cdots)\)	None	\(0\)	\(300\)	\(0\)	\(15800\)		\(q+(75+\beta _{1})q^{3}+(-3\beta _{1}+\beta _{2})q^{5}+(3950+\cdots)q^{7}+\cdots\)
12.13.d.a	\(12\)	\(10.968\)	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	\(-90\)	\(0\)	\(-10296\)	\(0\)		\(q+(-8+\beta _{2})q^{2}+(\beta _{1}+\beta _{2})q^{3}+(-386+\cdots)q^{4}+\cdots\)
12.14.a.a	\(1\)	\(12.868\)	\(\Q\)	None	\(0\)	\(-729\)	\(-14850\)	\(-62896\)	\(-\)	\(q-3^{6}q^{3}-14850q^{5}-62896q^{7}+\cdots\)
12.14.a.b	\(1\)	\(12.868\)	\(\Q\)	None	\(0\)	\(729\)	\(-24570\)	\(-173704\)	\(+\)	\(q+3^{6}q^{3}-24570q^{5}-173704q^{7}+\cdots\)
12.14.b.a	\(24\)	\(12.868\)		None	\(0\)	\(0\)	\(0\)	\(0\)
12.15.c.a	\(1\)	\(14.919\)	\(\Q\)	\(\Q(\sqrt{-3}) \)	\(0\)	\(-2187\)	\(0\)	\(-1389022\)		\(q-3^{7}q^{3}-1389022q^{7}+3^{14}q^{9}+\cdots\)
12.15.c.b	\(4\)	\(14.919\)	\(\mathbb{Q}[x]/(x^{4} + \cdots)\)	None	\(0\)	\(2148\)	\(0\)	\(1709288\)		\(q+(537-\beta _{1})q^{3}+(\beta _{1}+\beta _{2})q^{5}+(427322+\cdots)q^{7}+\cdots\)
12.15.d.a	\(14\)	\(14.919\)	\(\mathbb{Q}[x]/(x^{14} - \cdots)\)	None	\(182\)	\(0\)	\(-16124\)	\(0\)		\(q+(13-\beta _{1})q^{2}+\beta _{2}q^{3}+(665-13\beta _{1}+\cdots)q^{4}+\cdots\)
12.16.a.a	\(1\)	\(17.123\)	\(\Q\)	None	\(0\)	\(-2187\)	\(45702\)	\(1217888\)	\(-\)	\(q-3^{7}q^{3}+45702q^{5}+1217888q^{7}+\cdots\)
12.16.a.b	\(2\)	\(17.123\)	\(\Q(\sqrt{8017}) \)	None	\(0\)	\(4374\)	\(69660\)	\(2491504\)	\(+\)	\(q+3^{7}q^{3}+(34830-\beta )q^{5}+(1245752+\cdots)q^{7}+\cdots\)
12.16.b.a	\(28\)	\(17.123\)		None	\(0\)	\(0\)	\(0\)	\(0\)
12.17.c.a	\(1\)	\(19.479\)	\(\Q\)	\(\Q(\sqrt{-3}) \)	\(0\)	\(6561\)	\(0\)	\(4743554\)		\(q+3^{8}q^{3}+4743554q^{7}+3^{16}q^{9}+\cdots\)
12.17.c.b	\(4\)	\(19.479\)	\(\mathbb{Q}[x]/(x^{4} + \cdots)\)	None	\(0\)	\(-12420\)	\(0\)	\(-1050280\)		\(q+(-3105-\beta _{1})q^{3}+(-8\beta _{1}-\beta _{2}+\cdots)q^{5}+\cdots\)
12.17.d.a	\(16\)	\(19.479\)	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None	\(-186\)	\(0\)	\(354144\)	\(0\)		\(q+(-12-\beta _{1})q^{2}+(\beta _{1}-\beta _{3})q^{3}+(8542+\cdots)q^{4}+\cdots\)
12.18.a.a	\(1\)	\(21.987\)	\(\Q\)	None	\(0\)	\(-6561\)	\(-1608930\)	\(-9417184\)	\(-\)	\(q-3^{8}q^{3}-1608930q^{5}-9417184q^{7}+\cdots\)
12.18.a.b	\(1\)	\(21.987\)	\(\Q\)	None	\(0\)	\(6561\)	\(130950\)	\(-14846776\)	\(+\)	\(q+3^{8}q^{3}+130950q^{5}-14846776q^{7}+\cdots\)
12.18.b.a	\(32\)	\(21.987\)		None	\(0\)	\(0\)	\(0\)	\(0\)
12.19.c.a	\(6\)	\(24.646\)	\(\mathbb{Q}[x]/(x^{6} + \cdots)\)	None	\(0\)	\(23934\)	\(0\)	\(11024364\)		\(q+(3989-\beta _{1})q^{3}+(-4\beta _{1}+\beta _{2})q^{5}+\cdots\)
12.19.d.a	\(18\)	\(24.646\)	\(\mathbb{Q}[x]/(x^{18} - \cdots)\)	None	\(-170\)	\(0\)	\(-1721764\)	\(0\)		\(q+(-9-\beta _{1})q^{2}+(-\beta _{1}+\beta _{2})q^{3}+(-24281+\cdots)q^{4}+\cdots\)
12.20.a.a	\(2\)	\(27.458\)	\(\Q(\sqrt{193153}) \)	None	\(0\)	\(-39366\)	\(624780\)	\(4667104\)	\(-\)	\(q-3^{9}q^{3}+(312390-\beta )q^{5}+(2333552+\cdots)q^{7}+\cdots\)
12.20.a.b	\(2\)	\(27.458\)	\(\mathbb{Q}[x]/(x^{2} - \cdots)\)	None	\(0\)	\(39366\)	\(-3267108\)	\(-27023984\)	\(+\)	\(q+3^{9}q^{3}+(-1633554-\beta )q^{5}+(-13511992+\cdots)q^{7}+\cdots\)
12.20.b.a	\(36\)	\(27.458\)		None	\(0\)	\(0\)	\(0\)	\(0\)
12.21.c.a	\(1\)	\(30.422\)	\(\Q\)	\(\Q(\sqrt{-3}) \)	\(0\)	\(59049\)	\(0\)	\(-77335774\)		\(q+3^{10}q^{3}-77335774q^{7}+3^{20}q^{9}+\cdots\)
12.21.c.b	\(6\)	\(30.422\)	\(\mathbb{Q}[x]/(x^{6} + \cdots)\)	None	\(0\)	\(-84378\)	\(0\)	\(-145040532\)		\(q+(-14063+\beta _{1})q^{3}+(3^{3}\beta _{1}-\beta _{2}+\cdots)q^{5}+\cdots\)
12.21.d.a	\(20\)	\(30.422\)	\(\mathbb{Q}[x]/(x^{20} - \cdots)\)	None	\(1254\)	\(0\)	\(1476984\)	\(0\)		\(q+(63-\beta _{1})q^{2}+(1-2\beta _{1}+\beta _{2})q^{3}+\cdots\)
12.22.a.a	\(1\)	\(33.537\)	\(\Q\)	None	\(0\)	\(59049\)	\(-11268090\)	\(281914136\)	\(+\)	\(q+3^{10}q^{3}-11268090q^{5}+281914136q^{7}+\cdots\)

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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.