Embedded newform 1680.2.bg.d.1201.1

• Label: 1680.2.bg.d.1201.1
• Level: $1680$
• Weight: $2$
• Character: 1680.1201
• Analytic conductor: $13.415$
• Analytic rank: $0$
• Dimension: $2$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1680.bg (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(13.4148675396\)
Analytic rank:	\(0\)
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:	no (minimal twist has level 210)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			1201.1
Root			\(0.500000 + 0.866025i\) of defining polynomial
Character	\(\chi\)	\(=\)	1680.1201
Dual form			1680.2.bg.d.961.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.500000 + 0.866025i) q^{3} +(-0.500000 - 0.866025i) q^{5} +(2.00000 + 1.73205i) q^{7} +(-0.500000 - 0.866025i) q^{9} +(-2.50000 + 4.33013i) q^{11} -5.00000 q^{13} +1.00000 q^{15} +(2.00000 - 3.46410i) q^{17} +(-3.50000 - 6.06218i) q^{19} +(-2.50000 + 0.866025i) q^{21} +(0.500000 + 0.866025i) q^{23} +(-0.500000 + 0.866025i) q^{25} +1.00000 q^{27} +(-1.00000 + 1.73205i) q^{31} +(-2.50000 - 4.33013i) q^{33} +(0.500000 - 2.59808i) q^{35} +(-0.500000 - 0.866025i) q^{37} +(2.50000 - 4.33013i) q^{39} +5.00000 q^{41} -12.0000 q^{43} +(-0.500000 + 0.866025i) q^{45} +(-5.50000 - 9.52628i) q^{47} +(1.00000 + 6.92820i) q^{49} +(2.00000 + 3.46410i) q^{51} +(4.50000 - 7.79423i) q^{53} +5.00000 q^{55} +7.00000 q^{57} +(2.00000 - 3.46410i) q^{59} +(-2.00000 - 3.46410i) q^{61} +(0.500000 - 2.59808i) q^{63} +(2.50000 + 4.33013i) q^{65} +(-6.00000 + 10.3923i) q^{67} -1.00000 q^{69} -2.00000 q^{71} +(-5.00000 + 8.66025i) q^{73} +(-0.500000 - 0.866025i) q^{75} +(-12.5000 + 4.33013i) q^{77} +(-6.00000 - 10.3923i) q^{79} +(-0.500000 + 0.866025i) q^{81} +12.0000 q^{83} -4.00000 q^{85} +(-7.00000 - 12.1244i) q^{89} +(-10.0000 - 8.66025i) q^{91} +(-1.00000 - 1.73205i) q^{93} +(-3.50000 + 6.06218i) q^{95} -8.00000 q^{97} +5.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q - q^3 - q^5 + 4 * q^7 - q^9 - 5 * q^11 - 10 * q^13 + 2 * q^15 + 4 * q^17 - 7 * q^19 - 5 * q^21 + q^23 - q^25 + 2 * q^27 - 2 * q^31 - 5 * q^33 + q^35 - q^37 + 5 * q^39 + 10 * q^41 - 24 * q^43 - q^45 - 11 * q^47 + 2 * q^49 + 4 * q^51 + 9 * q^53 + 10 * q^55 + 14 * q^57 + 4 * q^59 - 4 * q^61 + q^63 + 5 * q^65 - 12 * q^67 - 2 * q^69 - 4 * q^71 - 10 * q^73 - q^75 - 25 * q^77 - 12 * q^79 - q^81 + 24 * q^83 - 8 * q^85 - 14 * q^89 - 20 * q^91 - 2 * q^93 - 7 * q^95 - 16 * q^97 + 10 * q^99

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1680\mathbb{Z}\right)^\times\).

\(n\)	\(241\)	\(337\)	\(421\)	\(1121\)	\(1471\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(1\)	\(1\)	\(1\)	\(1\)

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
\(5\)	−0.500000	−	0.866025i	−0.223607	−	0.387298i
							\(7\)	2.00000	+	1.73205i	0.755929	+	0.654654i
\(11\)	−2.50000	+	4.33013i	−0.753778	+	1.30558i								0.192201	+	0.981356i	\(0.438437\pi\)
							−0.945979	+	0.324227i	\(0.894896\pi\)
\(13\)	−5.00000			−1.38675			−0.693375	−	0.720577i	\(-0.743877\pi\)
							−0.693375	+	0.720577i	\(0.743877\pi\)
\(17\)	2.00000	−	3.46410i	0.485071	−	0.840168i	−0.514782	−	0.857321i	\(-0.672127\pi\)
							0.999853	+	0.0171533i	\(0.00546033\pi\)
\(19\)	−3.50000	−	6.06218i	−0.802955	−	1.39076i	−0.917663	−	0.397360i	\(-0.869927\pi\)
							0.114708	−	0.993399i	\(-0.463407\pi\)
\(23\)	0.500000	+	0.866025i	0.104257	+	0.180579i	0.913434	−	0.406986i	\(-0.133420\pi\)
							−0.809177	+	0.587565i	\(0.800087\pi\)
\(29\)		0			0			−	1.00000i	\(-0.5\pi\)
									1.00000i	\(0.5\pi\)
\(31\)	−1.00000	+	1.73205i	−0.179605	+	0.311086i	−0.941745	−	0.336327i	\(-0.890815\pi\)
							0.762140	+	0.647412i	\(0.224149\pi\)
\(37\)	−0.500000	−	0.866025i	−0.0821995	−	0.142374i	0.821995	−	0.569495i	\(-0.192861\pi\)
							−0.904194	+	0.427121i	\(0.859528\pi\)
\(41\)	5.00000			0.780869			0.390434	−	0.920631i	\(-0.372325\pi\)
							0.390434	+	0.920631i	\(0.372325\pi\)
\(43\)	−12.0000			−1.82998			−0.914991	−	0.403473i	\(-0.867803\pi\)
							−0.914991	+	0.403473i	\(0.867803\pi\)
\(47\)	−5.50000	−	9.52628i	−0.802257	−	1.38955i	−0.918127	−	0.396286i	\(-0.870299\pi\)
							0.115870	−	0.993264i	\(-0.463035\pi\)
\(53\)	4.50000	−	7.79423i	0.618123	−	1.07062i	−0.371706	−	0.928351i	\(-0.621227\pi\)
							0.989828	−	0.142269i	\(-0.0454398\pi\)
\(59\)	2.00000	−	3.46410i	0.260378	−	0.450988i	−0.705965	−	0.708247i	\(-0.749486\pi\)
							0.966342	+	0.257260i	\(0.0828195\pi\)
\(61\)	−2.00000	−	3.46410i	−0.256074	−	0.443533i	0.709113	−	0.705095i	\(-0.249096\pi\)
							−0.965187	+	0.261562i	\(0.915762\pi\)
\(67\)	−6.00000	+	10.3923i	−0.733017	+	1.26962i	0.222571	+	0.974916i	\(0.428555\pi\)
							−0.955588	+	0.294706i	\(0.904778\pi\)
\(71\)	−2.00000			−0.237356			−0.118678	−	0.992933i	\(-0.537866\pi\)
							−0.118678	+	0.992933i	\(0.537866\pi\)
\(73\)	−5.00000	+	8.66025i	−0.585206	+	1.01361i	0.409644	+	0.912245i	\(0.365653\pi\)
							−0.994850	+	0.101361i	\(0.967680\pi\)
\(79\)	−6.00000	−	10.3923i	−0.675053	−	1.16923i	−0.976453	−	0.215728i	\(-0.930788\pi\)
							0.301401	−	0.953498i	\(-0.402546\pi\)
\(83\)	12.0000			1.31717			0.658586	−	0.752506i	\(-0.271155\pi\)
							0.658586	+	0.752506i	\(0.271155\pi\)
\(89\)	−7.00000	−	12.1244i	−0.741999	−	1.28518i	−0.951584	−	0.307389i	\(-0.900545\pi\)
							0.209585	−	0.977790i	\(-0.432789\pi\)
\(97\)	−8.00000			−0.812277			−0.406138	−	0.913812i	\(-0.633125\pi\)
							−0.406138	+	0.913812i	\(0.633125\pi\)

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	1680.2.bg.d.1201.1		2
4.3	odd	2		210.2.i.b.151.1	yes	2
7.2	even	3	inner	1680.2.bg.d.961.1		2
12.11	even	2		630.2.k.g.361.1		2
20.3	even	4		1050.2.o.g.949.1		4
20.7	even	4		1050.2.o.g.949.2		4
20.19	odd	2		1050.2.i.p.151.1		2
28.3	even	6		1470.2.a.o.1.1		1
28.11	odd	6		1470.2.a.l.1.1		1
28.19	even	6		1470.2.i.e.961.1		2
28.23	odd	6		210.2.i.b.121.1	✓	2
28.27	even	2		1470.2.i.e.361.1		2
84.11	even	6		4410.2.a.j.1.1		1
84.23	even	6		630.2.k.g.541.1		2
84.59	odd	6		4410.2.a.u.1.1		1
140.23	even	12		1050.2.o.g.499.2		4
140.39	odd	6		7350.2.a.u.1.1		1
140.59	even	6		7350.2.a.a.1.1		1
140.79	odd	6		1050.2.i.p.751.1		2
140.107	even	12		1050.2.o.g.499.1		4

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.2.i.b.121.1	✓	2	28.23	odd	6
210.2.i.b.151.1	yes	2	4.3	odd	2
630.2.k.g.361.1		2	12.11	even	2
630.2.k.g.541.1		2	84.23	even	6
1050.2.i.p.151.1		2	20.19	odd	2
1050.2.i.p.751.1		2	140.79	odd	6
1050.2.o.g.499.1		4	140.107	even	12
1050.2.o.g.499.2		4	140.23	even	12
1050.2.o.g.949.1		4	20.3	even	4
1050.2.o.g.949.2		4	20.7	even	4
1470.2.a.l.1.1		1	28.11	odd	6
1470.2.a.o.1.1		1	28.3	even	6
1470.2.i.e.361.1		2	28.27	even	2
1470.2.i.e.961.1		2	28.19	even	6
1680.2.bg.d.961.1		2	7.2	even	3	inner
1680.2.bg.d.1201.1		2	1.1	even	1	trivial
4410.2.a.j.1.1		1	84.11	even	6
4410.2.a.u.1.1		1	84.59	odd	6
7350.2.a.a.1.1		1	140.59	even	6
7350.2.a.u.1.1		1	140.39	odd	6