Embedded newform 336.2.bl.a.271.1

• Label: 336.2.bl.a.271.1
• Level: $336$
• Weight: $2$
• Character: 336.271
• Analytic conductor: $2.683$
• Analytic rank: $1$
• Dimension: $2$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	336.bl (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.68297350792\)
Analytic rank:	\(1\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\zeta_{6})\)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			271.1
Root			\(0.500000 - 0.866025i\) of defining polynomial
Character	\(\chi\)	\(=\)	336.271
Dual form			336.2.bl.a.31.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.500000 - 0.866025i) q^{3} +(-3.00000 - 1.73205i) q^{5} +(0.500000 + 2.59808i) q^{7} +(-0.500000 + 0.866025i) q^{9} +(-3.00000 + 1.73205i) q^{11} -5.19615i q^{13} +3.46410i q^{15} +(-6.00000 + 3.46410i) q^{17} +(-3.50000 + 6.06218i) q^{19} +(2.00000 - 1.73205i) q^{21} +(3.50000 + 6.06218i) q^{25} +1.00000 q^{27} +(-2.50000 - 4.33013i) q^{31} +(3.00000 + 1.73205i) q^{33} +(3.00000 - 8.66025i) q^{35} +(-0.500000 + 0.866025i) q^{37} +(-4.50000 + 2.59808i) q^{39} -10.3923i q^{41} -1.73205i q^{43} +(3.00000 - 1.73205i) q^{45} +(3.00000 - 5.19615i) q^{47} +(-6.50000 + 2.59808i) q^{49} +(6.00000 + 3.46410i) q^{51} +12.0000 q^{55} +7.00000 q^{57} +(-2.50000 - 0.866025i) q^{63} +(-9.00000 + 15.5885i) q^{65} +(1.50000 - 0.866025i) q^{67} +3.46410i q^{71} +(7.50000 - 4.33013i) q^{73} +(3.50000 - 6.06218i) q^{75} +(-6.00000 - 6.92820i) q^{77} +(-13.5000 - 7.79423i) q^{79} +(-0.500000 - 0.866025i) q^{81} -6.00000 q^{83} +24.0000 q^{85} +(6.00000 + 3.46410i) q^{89} +(13.5000 - 2.59808i) q^{91} +(-2.50000 + 4.33013i) q^{93} +(21.0000 - 12.1244i) q^{95} +6.92820i q^{97} -3.46410i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q - q^3 - 6 * q^5 + q^7 - q^9 - 6 * q^11 - 12 * q^17 - 7 * q^19 + 4 * q^21 + 7 * q^25 + 2 * q^27 - 5 * q^31 + 6 * q^33 + 6 * q^35 - q^37 - 9 * q^39 + 6 * q^45 + 6 * q^47 - 13 * q^49 + 12 * q^51 + 24 * q^55 + 14 * q^57 - 5 * q^63 - 18 * q^65 + 3 * q^67 + 15 * q^73 + 7 * q^75 - 12 * q^77 - 27 * q^79 - q^81 - 12 * q^83 + 48 * q^85 + 12 * q^89 + 27 * q^91 - 5 * q^93 + 42 * q^95

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/336\mathbb{Z}\right)^\times\).

\(n\)	\(85\)	\(113\)	\(127\)	\(241\)
\(\chi(n)\)	\(1\)	\(1\)	\(-1\)	\(e\left(\frac{5}{6}\right)\)

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			0
							\(3\)	−0.500000	−	0.866025i	−0.288675	−	0.500000i
\(5\)	−3.00000	−	1.73205i	−1.34164	−	0.774597i								−0.354593	−	0.935021i	\(-0.615380\pi\)
							−0.987048	+	0.160424i	\(0.948714\pi\)
\(7\)	0.500000	+	2.59808i	0.188982	+	0.981981i
							\(11\)	−3.00000	+	1.73205i	−0.904534	+	0.522233i	−0.878668	−	0.477432i	\(-0.841568\pi\)
−0.0258656	+	0.999665i	\(0.508234\pi\)
\(13\)		−	5.19615i		−	1.44115i	−0.693375	−	0.720577i	\(-0.743877\pi\)
							0.693375	−	0.720577i	\(-0.256123\pi\)
\(17\)	−6.00000	+	3.46410i	−1.45521	+	0.840168i	−0.998770	−	0.0495842i	\(-0.984210\pi\)
							−0.456444	+	0.889752i	\(0.650877\pi\)
\(19\)	−3.50000	+	6.06218i	−0.802955	+	1.39076i	0.114708	+	0.993399i	\(0.463407\pi\)
							−0.917663	+	0.397360i	\(0.869927\pi\)
\(23\)		0			0		0.500000	−	0.866025i	\(-0.333333\pi\)
							−0.500000	+	0.866025i	\(0.666667\pi\)
\(29\)		0			0			−	1.00000i	\(-0.5\pi\)
									1.00000i	\(0.5\pi\)
\(31\)	−2.50000	−	4.33013i	−0.449013	−	0.777714i	0.549309	−	0.835619i	\(-0.314891\pi\)
							−0.998322	+	0.0579057i	\(0.981558\pi\)
\(37\)	−0.500000	+	0.866025i	−0.0821995	+	0.142374i	−0.904194	−	0.427121i	\(-0.859528\pi\)
							0.821995	+	0.569495i	\(0.192861\pi\)
\(41\)		−	10.3923i		−	1.62301i	−0.584349	−	0.811503i	\(-0.698650\pi\)
							0.584349	−	0.811503i	\(-0.301350\pi\)
\(43\)		−	1.73205i		−	0.264135i	−0.991241	−	0.132068i	\(-0.957838\pi\)
							0.991241	−	0.132068i	\(-0.0421616\pi\)
\(47\)	3.00000	−	5.19615i	0.437595	−	0.757937i	−0.559908	−	0.828554i	\(-0.689164\pi\)
							0.997503	+	0.0706177i	\(0.0224970\pi\)
\(53\)		0			0		0.866025	−	0.500000i	\(-0.166667\pi\)
							−0.866025	+	0.500000i	\(0.833333\pi\)
\(59\)		0			0		0.866025	−	0.500000i	\(-0.166667\pi\)
							−0.866025	+	0.500000i	\(0.833333\pi\)
\(61\)		0			0		0.500000	−	0.866025i	\(-0.333333\pi\)
							−0.500000	+	0.866025i	\(0.666667\pi\)
\(67\)	1.50000	−	0.866025i	0.183254	−	0.105802i	−0.405567	−	0.914066i	\(-0.632926\pi\)
							0.588821	+	0.808264i	\(0.299592\pi\)
\(71\)			3.46410i			0.411113i	0.978645	+	0.205557i	\(0.0659005\pi\)
							−0.978645	+	0.205557i	\(0.934100\pi\)
\(73\)	7.50000	−	4.33013i	0.877809	−	0.506803i	0.00787336	−	0.999969i	\(-0.497494\pi\)
							0.869935	+	0.493166i	\(0.164160\pi\)
\(79\)	−13.5000	−	7.79423i	−1.51887	−	0.876919i	−0.999753	−	0.0222151i	\(-0.992928\pi\)
							−0.519115	−	0.854704i	\(-0.673739\pi\)
\(83\)	−6.00000			−0.658586			−0.329293	−	0.944228i	\(-0.606810\pi\)
							−0.329293	+	0.944228i	\(0.606810\pi\)
\(89\)	6.00000	+	3.46410i	0.635999	+	0.367194i	0.783072	−	0.621932i	\(-0.213652\pi\)
							−0.147073	+	0.989126i	\(0.546985\pi\)
\(97\)			6.92820i			0.703452i	0.936103	+	0.351726i	\(0.114405\pi\)
							−0.936103	+	0.351726i	\(0.885595\pi\)

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	336.2.bl.a.271.1	yes	2
3.2	odd	2		1008.2.cs.n.271.1		2
4.3	odd	2		336.2.bl.e.271.1	yes	2
7.2	even	3		2352.2.b.h.1567.2		2
7.3	odd	6		336.2.bl.e.31.1	yes	2
7.4	even	3		2352.2.bl.f.31.1		2
7.5	odd	6		2352.2.b.a.1567.1		2
7.6	odd	2		2352.2.bl.l.607.1		2
8.3	odd	2		1344.2.bl.d.1279.1		2
8.5	even	2		1344.2.bl.h.1279.1		2
12.11	even	2		1008.2.cs.m.271.1		2
21.2	odd	6		7056.2.b.l.1567.1		2
21.5	even	6		7056.2.b.a.1567.2		2
21.17	even	6		1008.2.cs.m.703.1		2
28.3	even	6	inner	336.2.bl.a.31.1	✓	2
28.11	odd	6		2352.2.bl.l.31.1		2
28.19	even	6		2352.2.b.h.1567.1		2
28.23	odd	6		2352.2.b.a.1567.2		2
28.27	even	2		2352.2.bl.f.607.1		2
56.3	even	6		1344.2.bl.h.703.1		2
56.45	odd	6		1344.2.bl.d.703.1		2
84.23	even	6		7056.2.b.a.1567.1		2
84.47	odd	6		7056.2.b.l.1567.2		2
84.59	odd	6		1008.2.cs.n.703.1		2

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
336.2.bl.a.31.1	✓	2	28.3	even	6	inner
336.2.bl.a.271.1	yes	2	1.1	even	1	trivial
336.2.bl.e.31.1	yes	2	7.3	odd	6
336.2.bl.e.271.1	yes	2	4.3	odd	2
1008.2.cs.m.271.1		2	12.11	even	2
1008.2.cs.m.703.1		2	21.17	even	6
1008.2.cs.n.271.1		2	3.2	odd	2
1008.2.cs.n.703.1		2	84.59	odd	6
1344.2.bl.d.703.1		2	56.45	odd	6
1344.2.bl.d.1279.1		2	8.3	odd	2
1344.2.bl.h.703.1		2	56.3	even	6
1344.2.bl.h.1279.1		2	8.5	even	2
2352.2.b.a.1567.1		2	7.5	odd	6
2352.2.b.a.1567.2		2	28.23	odd	6
2352.2.b.h.1567.1		2	28.19	even	6
2352.2.b.h.1567.2		2	7.2	even	3
2352.2.bl.f.31.1		2	7.4	even	3
2352.2.bl.f.607.1		2	28.27	even	2
2352.2.bl.l.31.1		2	28.11	odd	6
2352.2.bl.l.607.1		2	7.6	odd	2
7056.2.b.a.1567.1		2	84.23	even	6
7056.2.b.a.1567.2		2	21.5	even	6
7056.2.b.l.1567.1		2	21.2	odd	6
7056.2.b.l.1567.2		2	84.47	odd	6