Embedded newform 671.2.dc.a.6.11

• Label: 671.2.dc.a.6.11
• 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			6.11
Character	\(\chi\)	\(=\)	671.6
Dual form			671.2.dc.a.112.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.98489 - 0.104024i) q^{2} +(2.95525 - 0.960219i) q^{3} +(1.93993 + 0.203895i) q^{4} +(-0.684600 - 0.395254i) q^{5} +(-5.96574 + 1.59851i) q^{6} +(3.99020 + 0.209117i) q^{7} +(0.0969418 + 0.0153541i) q^{8} +(5.38443 - 3.91201i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 960 q - 20 q^{2} - 20 q^{3} - 8 q^{4} - 18 q^{5} - 20 q^{6} - 20 q^{7} - 20 q^{8} + 220 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.98489	−	0.104024i	−1.40353	−	0.0735559i	−0.664531	−	0.747261i	\(-0.731368\pi\)
							−0.739000	+	0.673705i	\(0.764702\pi\)
\(3\)	2.95525	−	0.960219i	1.70621	−	0.554382i	0.716518	−	0.697569i	\(-0.245735\pi\)
							0.989696	+	0.143186i	\(0.0457347\pi\)
\(5\)	−0.684600	−	0.395254i	−0.306162	−	0.176763i	0.339046	−	0.940770i	\(-0.389896\pi\)
							−0.645208	+	0.764007i	\(0.723229\pi\)
\(7\)	3.99020	+	0.209117i	1.50815	+	0.0790390i	0.788388	−	0.615178i	\(-0.210916\pi\)
							0.719766	+	0.694217i	\(0.244249\pi\)
\(11\)	3.17335	+	0.964282i	0.956801	+	0.290742i
							\(13\)	0.688144	−	3.23746i	0.190857	−	0.897911i	−0.773601	−	0.633673i	\(-0.781546\pi\)
0.964458	−	0.264238i	\(-0.0851203\pi\)
\(17\)	−1.18827	+	4.43467i	−0.288197	+	1.07557i	0.658274	+	0.752778i	\(0.271287\pi\)
							−0.946471	+	0.322788i	\(0.895380\pi\)
\(19\)	−2.86702	+	3.18415i	−0.657739	+	0.730493i	−0.976063	−	0.217489i	\(-0.930213\pi\)
							0.318324	+	0.947982i	\(0.396880\pi\)
\(23\)	−1.50598	+	2.95566i	−0.314019	+	0.616297i	−0.993034	−	0.117827i	\(-0.962407\pi\)
							0.679015	+	0.734124i	\(0.262407\pi\)
\(29\)	3.52993	+	9.19579i	0.655492	+	1.70762i	0.705350	+	0.708860i	\(0.250790\pi\)
							−0.0498573	+	0.998756i	\(0.515877\pi\)
\(31\)	−1.99143	−	2.45921i	−0.357671	−	0.441687i	0.566271	−	0.824219i	\(-0.308386\pi\)
							−0.923942	+	0.382532i	\(0.875052\pi\)
\(37\)	−0.264743	−	0.519588i	−0.0435235	−	0.0854197i	0.868221	−	0.496178i	\(-0.165264\pi\)
							−0.911744	+	0.410759i	\(0.865264\pi\)
\(41\)	−5.52963			−0.863582			−0.431791	−	0.901974i	\(-0.642118\pi\)
							−0.431791	+	0.901974i	\(0.642118\pi\)
\(43\)	0.108079	−	2.06227i	0.0164819	−	0.314494i	−0.977931	−	0.208930i	\(-0.933002\pi\)
							0.994413	−	0.105564i	\(-0.0336647\pi\)
\(47\)	−0.752215	+	7.15685i	−0.109722	+	1.04393i	0.791676	+	0.610942i	\(0.209209\pi\)
							−0.901398	+	0.432992i	\(0.857458\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	671.2.dc.a.6.11	yes	960
11.2	odd	10		671.2.cu.a.189.11	✓	960
61.51	odd	60		671.2.cu.a.600.11	yes	960
671.112	even	60	inner	671.2.dc.a.112.11	yes	960

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
671.2.cu.a.189.11	✓	960	11.2	odd	10
671.2.cu.a.600.11	yes	960	61.51	odd	60
671.2.dc.a.6.11	yes	960	1.1	even	1	trivial
671.2.dc.a.112.11	yes	960	671.112	even	60	inner