Embedded newform 666.2.f.j.433.2

• Label: 666.2.f.j.433.2
• Level: $666$
• Weight: $2$
• Character: 666.433
• 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.f (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.31803677462\)
Analytic rank:	\(0\)
Dimension:	\(6\)
Relative dimension:	\(3\) over \(\Q(\zeta_{3})\)
Coefficient field:	6.0.4406832.1
Defining polynomial:	\( x^{6} - x^{5} + 6x^{4} + 7x^{3} + 24x^{2} + 5x + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2^{2} \)
Twist minimal:	no (minimal twist has level 74)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			433.2
Root			\(-0.827721 + 1.43366i\) of defining polynomial
Character	\(\chi\)	\(=\)	666.433
Dual form			666.2.f.j.343.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.500000 + 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(0.629755 - 1.09077i) q^{5} +(-1.52569 + 2.64257i) q^{7} -1.00000 q^{8} +1.25951 q^{10} -5.31088 q^{11} +(-1.00000 + 1.73205i) q^{13} -3.05137 q^{14} +(-0.500000 - 0.866025i) q^{16} +(2.55137 + 4.41911i) q^{17} +(-2.39593 + 4.14988i) q^{19} +(0.629755 + 1.09077i) q^{20} +(-2.65544 - 4.59936i) q^{22} -1.74049 q^{23} +(1.70682 + 2.95629i) q^{25} -2.00000 q^{26} +(-1.52569 - 2.64257i) q^{28} +4.05137 q^{29} -0.791864 q^{31} +(0.500000 - 0.866025i) q^{32} +(-2.55137 + 4.41911i) q^{34} +(1.92162 + 3.32834i) q^{35} +(-1.37024 - 5.92642i) q^{37} -4.79186 q^{38} +(-0.629755 + 1.09077i) q^{40} +(-0.104068 + 0.180251i) q^{41} -4.36226 q^{43} +(2.65544 - 4.59936i) q^{44} +(-0.870245 - 1.50731i) q^{46} +3.20814 q^{47} +(-1.15544 - 2.00128i) q^{49} +(-1.70682 + 2.95629i) q^{50} +(-1.00000 - 1.73205i) q^{52} +(5.65544 + 9.79551i) q^{53} +(-3.34456 + 5.79294i) q^{55} +(1.52569 - 2.64257i) q^{56} +(2.02569 + 3.50859i) q^{58} +(-6.36226 - 11.0198i) q^{59} +(-3.02569 + 5.24064i) q^{61} +(-0.395932 - 0.685774i) q^{62} +1.00000 q^{64} +(1.25951 + 2.18154i) q^{65} +(4.65544 - 8.06346i) q^{67} -5.10275 q^{68} +(-1.92162 + 3.32834i) q^{70} +(-3.52569 + 6.10667i) q^{71} -3.31088 q^{73} +(4.44731 - 4.14988i) q^{74} +(-2.39593 - 4.14988i) q^{76} +(8.10275 - 14.0344i) q^{77} +(5.70682 - 9.88450i) q^{79} -1.25951 q^{80} -0.208136 q^{82} +(-1.78520 - 3.09205i) q^{83} +6.42697 q^{85} +(-2.18113 - 3.77783i) q^{86} +5.31088 q^{88} +(1.63642 + 2.83436i) q^{89} +(-3.05137 - 5.28514i) q^{91} +(0.870245 - 1.50731i) q^{92} +(1.60407 + 2.77833i) q^{94} +(3.01770 + 5.22681i) q^{95} +2.20814 q^{97} +(1.15544 - 2.00128i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	6 * q + 3 * q^2 - 3 * q^4 + q^5 - 6 * q^8 + 2 * q^10 - 8 * q^11 - 6 * q^13 - 3 * q^16 - 3 * q^17 - 8 * q^19 + q^20 - 4 * q^22 - 16 * q^23 - 20 * q^25 - 12 * q^26 + 6 * q^29 + 8 * q^31 + 3 * q^32 + 3 * q^34 - 4 * q^35 - 11 * q^37 - 16 * q^38 - q^40 - 7 * q^41 + 16 * q^43 + 4 * q^44 - 8 * q^46 + 32 * q^47 + 5 * q^49 + 20 * q^50 - 6 * q^52 + 22 * q^53 - 32 * q^55 + 3 * q^58 + 4 * q^59 - 9 * q^61 + 4 * q^62 + 6 * q^64 + 2 * q^65 + 16 * q^67 + 6 * q^68 + 4 * q^70 - 12 * q^71 + 4 * q^73 + 2 * q^74 - 8 * q^76 + 12 * q^77 + 4 * q^79 - 2 * q^80 - 14 * q^82 + 4 * q^83 - 18 * q^85 + 8 * q^86 + 8 * q^88 + 9 * q^89 + 8 * q^92 + 16 * q^94 - 36 * q^95 + 26 * q^97 - 5 * q^98

Character values

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

\(n\)	\(371\)	\(631\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{1}{3}\right)\)

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 \(a_n\)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.500000	+	0.866025i	0.353553	+	0.612372i
							\(3\)		0			0
\(5\)	0.629755	−	1.09077i	0.281635	−	0.487806i								−0.690153	−	0.723664i	\(-0.742457\pi\)
							0.971788	+	0.235858i	\(0.0757899\pi\)
\(7\)	−1.52569	+	2.64257i	−0.576656	+	0.998797i	0.419204	+	0.907892i	\(0.362309\pi\)
							−0.995860	+	0.0909046i	\(0.971024\pi\)
\(11\)	−5.31088			−1.60129			−0.800646	−	0.599138i	\(-0.795510\pi\)
							−0.800646	+	0.599138i	\(0.795510\pi\)
\(13\)	−1.00000	+	1.73205i	−0.277350	+	0.480384i	−0.970725	−	0.240192i	\(-0.922790\pi\)
							0.693375	+	0.720577i	\(0.256123\pi\)
\(17\)	2.55137	+	4.41911i	0.618799	+	1.07179i	0.989705	+	0.143121i	\(0.0457139\pi\)
							−0.370906	+	0.928670i	\(0.620953\pi\)
\(19\)	−2.39593	+	4.14988i	−0.549664	+	0.952047i	0.448633	+	0.893716i	\(0.351911\pi\)
							−0.998297	+	0.0583306i	\(0.981422\pi\)
\(23\)	−1.74049			−0.362917			−0.181459	−	0.983399i	\(-0.558082\pi\)
							−0.181459	+	0.983399i	\(0.558082\pi\)
\(29\)	4.05137			0.752321			0.376161	−	0.926554i	\(-0.377244\pi\)
							0.376161	+	0.926554i	\(0.377244\pi\)
\(31\)	−0.791864			−0.142223			−0.0711115	−	0.997468i	\(-0.522655\pi\)
							−0.0711115	+	0.997468i	\(0.522655\pi\)
\(37\)	−1.37024	−	5.92642i	−0.225267	−	0.974297i
							\(41\)	−0.104068	+	0.180251i	−0.0162527	+	0.0281505i	−0.874037	−	0.485859i	\(-0.838507\pi\)
0.857785	+	0.514009i	\(0.171840\pi\)
\(43\)	−4.36226			−0.665238			−0.332619	−	0.943061i	\(-0.607932\pi\)
							−0.332619	+	0.943061i	\(0.607932\pi\)
\(47\)	3.20814			0.467955			0.233977	−	0.972242i	\(-0.424826\pi\)
							0.233977	+	0.972242i	\(0.424826\pi\)
\(53\)	5.65544	+	9.79551i	0.776835	+	1.34552i	0.933757	+	0.357906i	\(0.116509\pi\)
							−0.156923	+	0.987611i	\(0.550157\pi\)
\(59\)	−6.36226	−	11.0198i	−0.828296	−	1.43465i	−0.899374	−	0.437179i	\(-0.855978\pi\)
							0.0710788	−	0.997471i	\(-0.477356\pi\)
\(61\)	−3.02569	+	5.24064i	−0.387400	+	0.670996i	−0.992099	−	0.125458i	\(-0.959960\pi\)
							0.604699	+	0.796454i	\(0.293293\pi\)
\(67\)	4.65544	−	8.06346i	0.568753	−	0.985109i	−0.427937	−	0.903809i	\(-0.640759\pi\)
							0.996690	−	0.0813001i	\(-0.0259072\pi\)
\(71\)	−3.52569	+	6.10667i	−0.418422	+	0.724728i	−0.995781	−	0.0917622i	\(-0.970750\pi\)
							0.577359	+	0.816491i	\(0.304083\pi\)
\(73\)	−3.31088			−0.387510			−0.193755	−	0.981050i	\(-0.562067\pi\)
							−0.193755	+	0.981050i	\(0.562067\pi\)
\(79\)	5.70682	−	9.88450i	0.642067	−	1.11209i	−0.342904	−	0.939371i	\(-0.611410\pi\)
							0.984971	−	0.172722i	\(-0.0552563\pi\)
\(83\)	−1.78520	−	3.09205i	−0.195951	−	0.339397i	0.751261	−	0.660005i	\(-0.229446\pi\)
							−0.947212	+	0.320608i	\(0.896113\pi\)
\(89\)	1.63642	+	2.83436i	0.173460	+	0.300442i	0.939627	−	0.342199i	\(-0.111172\pi\)
							−0.766167	+	0.642641i	\(0.777838\pi\)
\(97\)	2.20814			0.224202			0.112101	−	0.993697i	\(-0.464242\pi\)
							0.112101	+	0.993697i	\(0.464242\pi\)

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.f.j.433.2		6
3.2	odd	2		74.2.c.c.63.3	yes	6
12.11	even	2		592.2.i.e.433.1		6
37.10	even	3	inner	666.2.f.j.343.2		6
111.11	odd	6		2738.2.a.n.1.1		3
111.26	odd	6		2738.2.a.o.1.1		3
111.47	odd	6		74.2.c.c.47.3	✓	6
444.47	even	6		592.2.i.e.417.1		6

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
74.2.c.c.47.3	✓	6	111.47	odd	6
74.2.c.c.63.3	yes	6	3.2	odd	2
592.2.i.e.417.1		6	444.47	even	6
592.2.i.e.433.1		6	12.11	even	2
666.2.f.j.343.2		6	37.10	even	3	inner
666.2.f.j.433.2		6	1.1	even	1	trivial
2738.2.a.n.1.1		3	111.11	odd	6
2738.2.a.o.1.1		3	111.26	odd	6