Embedded newform 666.2.x.c.451.1

• Label: 666.2.x.c.451.1
• Level: $666$
• Weight: $2$
• Character: 666.451
• Analytic conductor: $5.318$
• Analytic rank: $0$
• Dimension: $6$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 666 = 2 \cdot 3^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	666.x (of order \(9\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.31803677462\)
Analytic rank:	\(0\)
Dimension:	\(6\)
Coefficient field:	\(\Q(\zeta_{18})\)
Defining polynomial:	\( x^{6} - x^{3} + 1 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 74)
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label			451.1
Root			\(-0.173648 + 0.984808i\) of defining polynomial
Character	\(\chi\)	\(=\)	666.451
Dual form			666.2.x.c.127.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.939693 + 0.342020i) q^{2} +(0.766044 - 0.642788i) q^{4} +(0.0209445 + 0.118782i) q^{5} +(0.233956 + 1.32683i) q^{7} +(-0.500000 + 0.866025i) q^{8} +(-0.0603074 - 0.104455i) q^{10} +(2.26604 - 3.92490i) q^{11} +(-0.592396 + 0.497079i) q^{13} +(-0.673648 - 1.16679i) q^{14} +(0.173648 - 0.984808i) q^{16} +(2.29813 + 1.92836i) q^{17} +(1.91875 + 0.698367i) q^{19} +(0.0923963 + 0.0775297i) q^{20} +(-0.786989 + 4.46324i) q^{22} +(0.0282185 + 0.0488759i) q^{23} +(4.68479 - 1.70513i) q^{25} +(0.386659 - 0.669713i) q^{26} +(1.03209 + 0.866025i) q^{28} +(-2.89053 + 5.00654i) q^{29} -3.34730 q^{31} +(0.173648 + 0.984808i) q^{32} +(-2.81908 - 1.02606i) q^{34} +(-0.152704 + 0.0555796i) q^{35} +(5.60607 - 2.36051i) q^{37} -2.04189 q^{38} +(-0.113341 - 0.0412527i) q^{40} +(5.47565 - 4.59462i) q^{41} +9.31315 q^{43} +(-0.786989 - 4.46324i) q^{44} +(-0.0432332 - 0.0362770i) q^{46} +(4.25877 + 7.37641i) q^{47} +(4.87211 - 1.77330i) q^{49} +(-3.81908 + 3.20459i) q^{50} +(-0.134285 + 0.761570i) q^{52} +(-0.482926 + 2.73881i) q^{53} +(0.513671 + 0.186961i) q^{55} +(-1.26604 - 0.460802i) q^{56} +(1.00387 - 5.69323i) q^{58} +(2.25624 - 12.7958i) q^{59} +(-8.82295 + 7.40333i) q^{61} +(3.14543 - 1.14484i) q^{62} +(-0.500000 - 0.866025i) q^{64} +(-0.0714517 - 0.0599551i) q^{65} +(0.889185 + 5.04282i) q^{67} +3.00000 q^{68} +(0.124485 - 0.104455i) q^{70} +(12.6236 + 4.59462i) q^{71} -8.71688 q^{73} +(-4.46064 + 4.13554i) q^{74} +(1.91875 - 0.698367i) q^{76} +(5.73783 + 2.08840i) q^{77} +(0.720285 + 4.08494i) q^{79} +0.120615 q^{80} +(-3.57398 + 6.19031i) q^{82} +(4.53596 + 3.80612i) q^{83} +(-0.180922 + 0.313366i) q^{85} +(-8.75150 + 3.18528i) q^{86} +(2.26604 + 3.92490i) q^{88} +(-1.32295 + 7.50281i) q^{89} +(-0.798133 - 0.669713i) q^{91} +(0.0530334 + 0.0193026i) q^{92} +(-6.52481 - 5.47497i) q^{94} +(-0.0427664 + 0.242540i) q^{95} +(5.12061 + 8.86916i) q^{97} +(-3.97178 + 3.33272i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	6 * q - 3 * q^5 + 6 * q^7 - 3 * q^8 - 6 * q^10 + 9 * q^11 - 3 * q^14 + 9 * q^19 - 3 * q^20 + 3 * q^22 + 15 * q^23 + 21 * q^25 + 9 * q^26 - 3 * q^28 - 18 * q^31 - 3 * q^35 + 9 * q^37 - 6 * q^38 + 6 * q^40 - 6 * q^41 + 12 * q^43 + 3 * q^44 + 15 * q^46 + 3 * q^47 - 6 * q^50 + 9 * q^52 + 18 * q^53 - 18 * q^55 - 3 * q^56 - 18 * q^58 + 6 * q^59 - 12 * q^61 + 3 * q^62 - 3 * q^64 - 3 * q^67 + 18 * q^68 - 12 * q^70 + 6 * q^71 - 36 * q^73 - 18 * q^74 + 9 * q^76 + 15 * q^77 + 30 * q^79 + 12 * q^80 - 6 * q^82 - 6 * q^83 - 18 * q^85 - 12 * q^86 + 9 * q^88 + 33 * q^89 + 9 * q^91 - 12 * q^92 - 12 * q^94 - 51 * q^95 + 42 * q^97 - 9 * q^98

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/666\mathbb{Z}\right)^\times\).

\(n\)	\(371\)	\(631\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{8}{9}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.

Currently showing only \(a_p\); display all \(a_n\)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.939693	+	0.342020i	−0.664463	+	0.241845i
							\(3\)		0			0
\(5\)	0.0209445	+	0.118782i	0.00936668	+	0.0531211i								0.989133	−	0.147024i	\(-0.0469696\pi\)
							−0.979766	+	0.200146i	\(0.935858\pi\)
\(7\)	0.233956	+	1.32683i	0.0884269	+	0.501494i	0.996564	+	0.0828217i	\(0.0263932\pi\)
							−0.908137	+	0.418672i	\(0.862496\pi\)
\(11\)	2.26604	−	3.92490i	0.683238	−	1.18340i	−0.290749	−	0.956799i	\(-0.593904\pi\)
							0.973987	−	0.226604i	\(-0.0727622\pi\)
\(13\)	−0.592396	+	0.497079i	−0.164301	+	0.137865i	−0.721231	−	0.692694i	\(-0.756424\pi\)
							0.556930	+	0.830559i	\(0.311979\pi\)
\(17\)	2.29813	+	1.92836i	0.557379	+	0.467697i	0.877431	−	0.479703i	\(-0.159256\pi\)
							−0.320051	+	0.947400i	\(0.603700\pi\)
\(19\)	1.91875	+	0.698367i	0.440191	+	0.160216i	0.552602	−	0.833445i	\(-0.313635\pi\)
							−0.112411	+	0.993662i	\(0.535857\pi\)
\(23\)	0.0282185	+	0.0488759i	0.00588396	+	0.0101913i	0.868952	−	0.494896i	\(-0.164794\pi\)
							−0.863068	+	0.505087i	\(0.831460\pi\)
\(29\)	−2.89053	+	5.00654i	−0.536758	+	0.929692i	0.462318	+	0.886714i	\(0.347018\pi\)
							−0.999076	+	0.0429778i	\(0.986316\pi\)
\(31\)	−3.34730			−0.601192			−0.300596	−	0.953752i	\(-0.597186\pi\)
							−0.300596	+	0.953752i	\(0.597186\pi\)
\(37\)	5.60607	−	2.36051i	0.921632	−	0.388066i
							\(41\)	5.47565	−	4.59462i	0.855153	−	0.717559i	−0.105765	−	0.994391i	\(-0.533729\pi\)
0.960918	+	0.276832i	\(0.0892847\pi\)
\(43\)	9.31315			1.42024			0.710121	−	0.704080i	\(-0.248640\pi\)
							0.710121	+	0.704080i	\(0.248640\pi\)
\(47\)	4.25877	+	7.37641i	0.621206	+	1.07596i	0.989262	+	0.146156i	\(0.0466901\pi\)
							−0.368056	+	0.929804i	\(0.619977\pi\)
\(53\)	−0.482926	+	2.73881i	−0.0663350	+	0.376204i	0.933509	+	0.358553i	\(0.116730\pi\)
							−0.999844	+	0.0176510i	\(0.994381\pi\)
\(59\)	2.25624	−	12.7958i	0.293738	−	1.66587i	−0.378550	−	0.925581i	\(-0.623577\pi\)
							0.672288	−	0.740290i	\(-0.265312\pi\)
\(61\)	−8.82295	+	7.40333i	−1.12966	+	0.947900i	−0.999052	−	0.0435341i	\(-0.986138\pi\)
							−0.130611	+	0.991434i	\(0.541694\pi\)
\(67\)	0.889185	+	5.04282i	0.108631	+	0.616079i	0.989708	+	0.143104i	\(0.0457085\pi\)
							−0.881076	+	0.472974i	\(0.843180\pi\)
\(71\)	12.6236	+	4.59462i	1.49815	+	0.545281i	0.955580	−	0.294732i	\(-0.0952304\pi\)
							0.542567	+	0.840013i	\(0.317453\pi\)
\(73\)	−8.71688			−1.02023			−0.510117	−	0.860105i	\(-0.670398\pi\)
							−0.510117	+	0.860105i	\(0.670398\pi\)
\(79\)	0.720285	+	4.08494i	0.0810384	+	0.459592i	0.998141	+	0.0609412i	\(0.0194102\pi\)
							−0.917103	+	0.398650i	\(0.869479\pi\)
\(83\)	4.53596	+	3.80612i	0.497886	+	0.417776i	0.856843	−	0.515578i	\(-0.172423\pi\)
							−0.358956	+	0.933354i	\(0.616867\pi\)
\(89\)	−1.32295	+	7.50281i	−0.140232	+	0.795297i	0.830840	+	0.556511i	\(0.187860\pi\)
							−0.971072	+	0.238785i	\(0.923251\pi\)
\(97\)	5.12061	+	8.86916i	0.519920	+	0.900527i	0.999732	+	0.0231560i	\(0.00737146\pi\)
							−0.479812	+	0.877371i	\(0.659295\pi\)

Currently showing only \(a_p\); display all \(a_n\)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	666.2.x.c.451.1		6
3.2	odd	2		74.2.f.a.7.1	✓	6
12.11	even	2		592.2.bc.b.81.1		6
37.16	even	9	inner	666.2.x.c.127.1		6
111.41	odd	18		2738.2.a.p.1.2		3
111.53	odd	18		74.2.f.a.53.1	yes	6
111.107	odd	18		2738.2.a.m.1.2		3
444.275	even	18		592.2.bc.b.497.1		6

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
74.2.f.a.7.1	✓	6	3.2	odd	2
74.2.f.a.53.1	yes	6	111.53	odd	18
592.2.bc.b.81.1		6	12.11	even	2
592.2.bc.b.497.1		6	444.275	even	18
666.2.x.c.127.1		6	37.16	even	9	inner
666.2.x.c.451.1		6	1.1	even	1	trivial
2738.2.a.m.1.2		3	111.107	odd	18
2738.2.a.p.1.2		3	111.41	odd	18