Embedded newform 671.2.cu.a.600.5

• Label: 671.2.cu.a.600.5
• Level: $671$
• Weight: $2$
• Character: 671.600
• 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.cu (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			600.5
Character	\(\chi\)	\(=\)	671.600
Dual form			671.2.cu.a.189.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.93657 + 1.56820i) q^{2} -0.862016i q^{3} +(0.875213 - 4.11755i) q^{4} +(-1.74379 - 0.183280i) q^{5} +(1.35181 + 1.66935i) q^{6} +(-0.263647 + 0.983944i) q^{7} +(2.49965 + 4.90584i) q^{8} +2.25693 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 960 q - 20 q^{2} - 8 q^{4} - 8 q^{5} + 20 q^{6} - 20 q^{7} - 920 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(123\)	\(551\)
\(\chi(n)\)	\(e\left(\frac{9}{10}\right)\)	\(e\left(\frac{53}{60}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.93657	+	1.56820i	−1.36936	+	1.10889i	−0.386959	+	0.922097i	\(0.626475\pi\)
							−0.982400	+	0.186788i	\(0.940192\pi\)
\(3\)		−	0.862016i		−	0.497685i	−0.968544	−	0.248843i	\(-0.919950\pi\)
							0.968544	−	0.248843i	\(-0.0800502\pi\)
\(5\)	−1.74379	−	0.183280i	−0.779849	−	0.0819654i	−0.293765	−	0.955878i	\(-0.594908\pi\)
							−0.486084	+	0.873912i	\(0.661575\pi\)
\(7\)	−0.263647	+	0.983944i	−0.0996492	+	0.371896i	−0.997683	−	0.0680300i	\(-0.978329\pi\)
							0.898034	+	0.439926i	\(0.144995\pi\)
\(11\)	3.31586	+	0.0714210i	0.999768	+	0.0215342i
							\(13\)	−4.56604	−	4.11128i	−1.26639	−	1.14026i	−0.983460	−	0.181125i	\(-0.942026\pi\)
−0.282933	−	0.959140i	\(-0.591307\pi\)
\(17\)	0.891877	−	2.32342i	0.216312	−	0.563512i	−0.781892	−	0.623413i	\(-0.785745\pi\)
							0.998204	+	0.0599017i	\(0.0190787\pi\)
\(19\)	−3.51388	+	6.08622i	−0.806140	+	1.39628i	0.109379	+	0.994000i	\(0.465114\pi\)
							−0.915519	+	0.402275i	\(0.868219\pi\)
\(23\)	2.66459	+	5.22955i	0.555605	+	1.09044i	0.982522	+	0.186146i	\(0.0595997\pi\)
							−0.426917	+	0.904291i	\(0.640400\pi\)
\(29\)	0.0577875	+	0.0467954i	0.0107309	+	0.00868969i	0.634671	−	0.772783i	\(-0.281136\pi\)
							−0.623940	+	0.781472i	\(0.714469\pi\)
\(31\)	0.437661	−	8.35108i	0.0786063	−	1.49990i	−0.620119	−	0.784508i	\(-0.712916\pi\)
							0.698725	−	0.715390i	\(-0.253751\pi\)
\(37\)	−6.51756	−	6.51756i	−1.07148	−	1.07148i	−0.997240	−	0.0742395i	\(-0.976347\pi\)
							−0.0742395	−	0.997240i	\(-0.523653\pi\)
\(41\)	−0.910740	−	2.80297i	−0.142234	−	0.437750i	0.854411	−	0.519597i	\(-0.173918\pi\)
							−0.996645	+	0.0818472i	\(0.973918\pi\)
\(43\)	0.0709303	+	1.35343i	0.0108168	+	0.206396i	0.998691	+	0.0511424i	\(0.0162862\pi\)
							−0.987875	+	0.155254i	\(0.950380\pi\)
\(47\)	6.18564	−	2.75402i	0.902268	−	0.401716i	0.0974522	−	0.995240i	\(-0.468931\pi\)
							0.804816	+	0.593525i	\(0.202264\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	671.2.cu.a.600.5	yes	960
11.2	odd	10		671.2.dc.a.112.5	yes	960
61.6	odd	60		671.2.dc.a.6.5	yes	960
671.189	even	60	inner	671.2.cu.a.189.5	✓	960

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
671.2.cu.a.189.5	✓	960	671.189	even	60	inner
671.2.cu.a.600.5	yes	960	1.1	even	1	trivial
671.2.dc.a.6.5	yes	960	61.6	odd	60
671.2.dc.a.112.5	yes	960	11.2	odd	10