Embedded newform 671.2.dc.a.6.13

• Label: 671.2.dc.a.6.13
• 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			6.13
Character	\(\chi\)	\(=\)	671.6
Dual form			671.2.dc.a.112.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.93406 - 0.101360i) q^{2} +(-1.88163 + 0.611379i) q^{3} +(1.74127 + 0.183015i) q^{4} +(-2.84234 - 1.64103i) q^{5} +(3.70115 - 0.991721i) q^{6} +(-0.892121 - 0.0467541i) q^{7} +(0.476568 + 0.0754810i) q^{8} +(0.739695 - 0.537420i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 960 q - 20 q^{2} - 20 q^{3} - 8 q^{4} - 18 q^{5} - 20 q^{6} - 20 q^{7} - 20 q^{8} + 220 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.93406	−	0.101360i	−1.36759	−	0.0716722i	−0.645651	−	0.763632i	\(-0.723414\pi\)
							−0.721936	+	0.691960i	\(0.756747\pi\)
\(3\)	−1.88163	+	0.611379i	−1.08636	+	0.352980i	−0.796839	−	0.604192i	\(-0.793496\pi\)
							−0.289521	+	0.957172i	\(0.593496\pi\)
\(5\)	−2.84234	−	1.64103i	−1.27113	−	0.733890i	−0.295933	−	0.955209i	\(-0.595630\pi\)
							−0.975201	+	0.221319i	\(0.928964\pi\)
\(7\)	−0.892121	−	0.0467541i	−0.337190	−	0.0176714i	−0.117010	−	0.993131i	\(-0.537331\pi\)
							−0.220180	+	0.975459i	\(0.570664\pi\)
\(11\)	−0.840446	−	3.20837i	−0.253404	−	0.967361i
							\(13\)	0.634752	−	2.98627i	0.176048	−	0.828243i	−0.798134	−	0.602480i	\(-0.794179\pi\)
0.974182	−	0.225763i	\(-0.0724874\pi\)
\(17\)	1.27803	−	4.76967i	0.309968	−	1.15681i	−0.618617	−	0.785693i	\(-0.712307\pi\)
							0.928584	−	0.371122i	\(-0.121027\pi\)
\(19\)	0.804961	−	0.893999i	0.184671	−	0.205098i	−0.643700	−	0.765278i	\(-0.722602\pi\)
							0.828371	+	0.560180i	\(0.189268\pi\)
\(23\)	0.404129	−	0.793147i	0.0842666	−	0.165383i	−0.845028	−	0.534722i	\(-0.820416\pi\)
							0.929294	+	0.369340i	\(0.120416\pi\)
\(29\)	−2.46029	−	6.40927i	−0.456864	−	1.19017i	−0.947769	−	0.318959i	\(-0.896667\pi\)
							0.490904	−	0.871214i	\(-0.336666\pi\)
\(31\)	0.245589	+	0.303277i	0.0441091	+	0.0544702i	0.798723	−	0.601699i	\(-0.205509\pi\)
							−0.754614	+	0.656169i	\(0.772176\pi\)
\(37\)	2.84473	+	5.58309i	0.467670	+	0.917854i	0.997562	+	0.0697889i	\(0.0222326\pi\)
							−0.529892	+	0.848065i	\(0.677767\pi\)
\(41\)	−9.25267			−1.44502			−0.722512	−	0.691358i	\(-0.757013\pi\)
							−0.722512	+	0.691358i	\(0.757013\pi\)
\(43\)	0.0740858	−	1.41364i	0.0112980	−	0.215578i	−0.987146	−	0.159819i	\(-0.948909\pi\)
							0.998444	−	0.0557589i	\(-0.0177578\pi\)
\(47\)	−0.552870	+	5.26021i	−0.0806444	+	0.767280i	0.877226	+	0.480078i	\(0.159392\pi\)
							−0.957870	+	0.287202i	\(0.907275\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	671.2.dc.a.6.13	yes	960
11.2	odd	10		671.2.cu.a.189.13	✓	960
61.51	odd	60		671.2.cu.a.600.13	yes	960
671.112	even	60	inner	671.2.dc.a.112.13	yes	960

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
671.2.cu.a.189.13	✓	960	11.2	odd	10
671.2.cu.a.600.13	yes	960	61.51	odd	60
671.2.dc.a.6.13	yes	960	1.1	even	1	trivial
671.2.dc.a.112.13	yes	960	671.112	even	60	inner