Embedded newform 671.2.dc.a.6.3

• Label: 671.2.dc.a.6.3
• Level: $671$
• Weight: $2$
• Character: 671.6
• Analytic conductor: $5.358$
• Analytic rank: $0$
• Dimension: $960$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 671 = 11 \cdot 61 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	671.dc (of order \(60\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.35796197563\)
Analytic rank:	\(0\)
Dimension:	\(960\)
Relative dimension:	\(60\) over \(\Q(\zeta_{60})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{60}]$

Embedding invariants

Embedding label			6.3
Character	\(\chi\)	\(=\)	671.6
Dual form			671.2.dc.a.112.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.57499 - 0.134950i) q^{2} +(-3.10048 + 1.00741i) q^{3} +(4.62333 + 0.485932i) q^{4} +(2.47030 + 1.42623i) q^{5} +(8.11965 - 2.17565i) q^{6} +(-0.119782 - 0.00627753i) q^{7} +(-6.74591 - 1.06845i) q^{8} +(6.17104 - 4.48352i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 960 q - 20 q^{2} - 20 q^{3} - 8 q^{4} - 18 q^{5} - 20 q^{6} - 20 q^{7} - 20 q^{8} + 220 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/671\mathbb{Z}\right)^\times\).

\(n\)	\(123\)	\(551\)
\(\chi(n)\)	\(e\left(\frac{9}{10}\right)\)	\(e\left(\frac{7}{60}\right)\)

Coefficient data

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

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.57499	−	0.134950i	−1.82079	−	0.0954238i	−0.888889	−	0.458123i	\(-0.848522\pi\)
							−0.931906	+	0.362699i	\(0.881855\pi\)
\(3\)	−3.10048	+	1.00741i	−1.79006	+	0.581626i	−0.999526	−	0.0307909i	\(-0.990197\pi\)
							−0.790535	+	0.612417i	\(0.790197\pi\)
\(5\)	2.47030	+	1.42623i	1.10475	+	0.637830i	0.937466	−	0.348078i	\(-0.113166\pi\)
							0.167288	+	0.985908i	\(0.446499\pi\)
\(7\)	−0.119782	−	0.00627753i	−0.0452735	−	0.00237268i	0.0296858	−	0.999559i	\(-0.490549\pi\)
							−0.0749592	+	0.997187i	\(0.523883\pi\)
\(11\)	2.99989	−	1.41444i	0.904502	−	0.426469i
							\(13\)	0.913391	−	4.29716i	0.253329	−	1.19182i	−0.649006	−	0.760783i	\(-0.724815\pi\)
0.902335	−	0.431036i	\(-0.141852\pi\)
\(17\)	1.17298	−	4.37761i	0.284489	−	1.06173i	−0.664723	−	0.747090i	\(-0.731451\pi\)
							0.949212	−	0.314637i	\(-0.101883\pi\)
\(19\)	−0.403085	+	0.447671i	−0.0924741	+	0.102703i	−0.787606	−	0.616180i	\(-0.788679\pi\)
							0.695131	+	0.718883i	\(0.255346\pi\)
\(23\)	−2.54284	+	4.99061i	−0.530219	+	1.04061i	0.458198	+	0.888850i	\(0.348495\pi\)
							−0.988417	+	0.151763i	\(0.951505\pi\)
\(29\)	−0.186989	−	0.487123i	−0.0347230	−	0.0904564i	0.915123	−	0.403175i	\(-0.132094\pi\)
							−0.949846	+	0.312719i	\(0.898760\pi\)
\(31\)	−3.64961	−	4.50690i	−0.655490	−	0.809463i	0.335692	−	0.941972i	\(-0.391030\pi\)
							−0.991182	+	0.132509i	\(0.957697\pi\)
\(37\)	−4.67001	−	9.16541i	−0.767745	−	1.50678i	−0.859575	−	0.511010i	\(-0.829272\pi\)
							0.0918298	−	0.995775i	\(-0.470728\pi\)
\(41\)	−9.34446			−1.45936			−0.729680	−	0.683789i	\(-0.760331\pi\)
							−0.729680	+	0.683789i	\(0.760331\pi\)
\(43\)	0.463799	−	8.84981i	0.0707287	−	1.34958i	−0.700786	−	0.713372i	\(-0.747167\pi\)
							0.771514	−	0.636212i	\(-0.219500\pi\)
\(47\)	−0.366840	+	3.49025i	−0.0535090	+	0.509105i	0.934639	+	0.355599i	\(0.115723\pi\)
							−0.988148	+	0.153506i	\(0.950944\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	671.2.dc.a.6.3	yes	960
11.2	odd	10		671.2.cu.a.189.3	✓	960
61.51	odd	60		671.2.cu.a.600.3	yes	960
671.112	even	60	inner	671.2.dc.a.112.3	yes	960

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
671.2.cu.a.189.3	✓	960	11.2	odd	10
671.2.cu.a.600.3	yes	960	61.51	odd	60
671.2.dc.a.6.3	yes	960	1.1	even	1	trivial
671.2.dc.a.112.3	yes	960	671.112	even	60	inner