Embedded newform 671.2.dc.a.6.8

• Label: 671.2.dc.a.6.8
• Level: $671$
• Weight: $2$
• Character: 671.6
• Analytic conductor: $5.358$
• Analytic rank: $0$
• Dimension: $960$
• 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			6.8
Character	\(\chi\)	\(=\)	671.6
Dual form			671.2.dc.a.112.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.20560 - 0.115591i) q^{2} +(1.11197 - 0.361302i) q^{3} +(2.86227 + 0.300837i) q^{4} +(1.76874 + 1.02118i) q^{5} +(-2.49433 + 0.668355i) q^{6} +(1.41323 + 0.0740642i) q^{7} +(-1.91538 - 0.303367i) q^{8} +(-1.32110 + 0.959838i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 960 q - 20 q^{2} - 20 q^{3} - 8 q^{4} - 18 q^{5} - 20 q^{6} - 20 q^{7} - 20 q^{8} + 220 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.20560	−	0.115591i	−1.55960	−	0.0817349i	−0.747104	−	0.664708i	\(-0.768556\pi\)
							−0.812492	+	0.582973i	\(0.801889\pi\)
\(3\)	1.11197	−	0.361302i	0.641999	−	0.208598i	0.0301158	−	0.999546i	\(-0.490412\pi\)
							0.611883	+	0.790948i	\(0.290412\pi\)
\(5\)	1.76874	+	1.02118i	0.791005	+	0.456687i	0.840316	−	0.542097i	\(-0.182369\pi\)
							−0.0493113	+	0.998783i	\(0.515703\pi\)
\(7\)	1.41323	+	0.0740642i	0.534150	+	0.0279936i	0.317505	−	0.948257i	\(-0.397155\pi\)
							0.216645	+	0.976250i	\(0.430488\pi\)
\(11\)	−0.224428	−	3.30902i	−0.0676677	−	0.997708i
							\(13\)	1.19881	−	5.63998i	0.332491	−	1.56425i	−0.421167	−	0.906983i	\(-0.638380\pi\)
0.753659	−	0.657266i	\(-0.228287\pi\)
\(17\)	−0.387718	+	1.44698i	−0.0940355	+	0.350945i	−0.996871	−	0.0790399i	\(-0.974815\pi\)
							0.902836	+	0.429985i	\(0.141481\pi\)
\(19\)	4.17429	−	4.63602i	0.957647	−	1.06357i	−0.0402779	−	0.999189i	\(-0.512824\pi\)
							0.997925	−	0.0643864i	\(-0.0205090\pi\)
\(23\)	1.20866	−	2.37212i	0.252022	−	0.494622i	−0.729985	−	0.683463i	\(-0.760473\pi\)
							0.982008	+	0.188841i	\(0.0604731\pi\)
\(29\)	−0.559813	−	1.45836i	−0.103955	−	0.270811i	0.871731	−	0.489984i	\(-0.162997\pi\)
							−0.975686	+	0.219173i	\(0.929664\pi\)
\(31\)	3.01785	+	3.72673i	0.542021	+	0.669341i	0.972403	−	0.233306i	\(-0.0749545\pi\)
							−0.430382	+	0.902647i	\(0.641621\pi\)
\(37\)	0.519056	+	1.01871i	0.0853324	+	0.167474i	0.929729	−	0.368244i	\(-0.120041\pi\)
							−0.844397	+	0.535718i	\(0.820041\pi\)
\(41\)	1.43095			0.223477			0.111738	−	0.993738i	\(-0.464358\pi\)
							0.111738	+	0.993738i	\(0.464358\pi\)
\(43\)	0.0266024	−	0.507604i	0.00405683	−	0.0774089i	−0.995851	−	0.0910011i	\(-0.970993\pi\)
							0.999908	+	0.0135922i	\(0.00432665\pi\)
\(47\)	−0.257548	+	2.45041i	−0.0375672	+	0.357428i	0.959549	+	0.281541i	\(0.0908456\pi\)
							−0.997116	+	0.0758872i	\(0.975821\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	671.2.dc.a.6.8	yes	960
11.2	odd	10		671.2.cu.a.189.8	✓	960
61.51	odd	60		671.2.cu.a.600.8	yes	960
671.112	even	60	inner	671.2.dc.a.112.8	yes	960

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
671.2.cu.a.189.8	✓	960	11.2	odd	10
671.2.cu.a.600.8	yes	960	61.51	odd	60
671.2.dc.a.6.8	yes	960	1.1	even	1	trivial
671.2.dc.a.112.8	yes	960	671.112	even	60	inner