Embedded newform 671.2.dc.a.6.19

• Label: 671.2.dc.a.6.19
• 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			6.19
Character	\(\chi\)	\(=\)	671.6
Dual form			671.2.dc.a.112.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.29101 - 0.0676589i) q^{2} +(-1.97523 + 0.641792i) q^{3} +(-0.326917 - 0.0343603i) q^{4} +(2.16259 + 1.24857i) q^{5} +(2.59347 - 0.694917i) q^{6} +(-3.89004 - 0.203868i) q^{7} +(2.97346 + 0.470949i) q^{8} +(1.06260 - 0.772023i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 960 q - 20 q^{2} - 20 q^{3} - 8 q^{4} - 18 q^{5} - 20 q^{6} - 20 q^{7} - 20 q^{8} + 220 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.29101	−	0.0676589i	−0.912881	−	0.0478421i	−0.409891	−	0.912134i	\(-0.634433\pi\)
							−0.502990	+	0.864292i	\(0.667767\pi\)
\(3\)	−1.97523	+	0.641792i	−1.14040	+	0.370539i	−0.817520	−	0.575900i	\(-0.804652\pi\)
							−0.322882	+	0.946439i	\(0.604652\pi\)
\(5\)	2.16259	+	1.24857i	0.967138	+	0.558378i	0.898363	−	0.439255i	\(-0.144757\pi\)
							0.0687757	+	0.997632i	\(0.478091\pi\)
\(7\)	−3.89004	−	0.203868i	−1.47030	−	0.0770550i	−0.699727	−	0.714410i	\(-0.746695\pi\)
							−0.770570	+	0.637355i	\(0.780028\pi\)
\(11\)	−2.02813	−	2.62425i	−0.611505	−	0.791241i
							\(13\)	−0.102086	+	0.480277i	−0.0283136	+	0.133205i	−0.990034	−	0.140827i	\(-0.955024\pi\)
0.961721	+	0.274032i	\(0.0883573\pi\)
\(17\)	1.02982	−	3.84334i	0.249768	−	0.932146i	−0.721159	−	0.692770i	\(-0.756390\pi\)
							0.970927	−	0.239377i	\(-0.0769431\pi\)
\(19\)	−4.65808	+	5.17333i	−1.06864	+	1.18684i	−0.0869761	+	0.996210i	\(0.527720\pi\)
							−0.981662	+	0.190632i	\(0.938946\pi\)
\(23\)	−1.70852	+	3.35316i	−0.356251	+	0.699181i	−0.997686	−	0.0679960i	\(-0.978339\pi\)
							0.641435	+	0.767178i	\(0.278339\pi\)
\(29\)	−1.38263	−	3.60189i	−0.256749	−	0.668853i	0.743251	−	0.669012i	\(-0.233283\pi\)
							−1.00000		0.000159009i	\(0.999949\pi\)
\(31\)	3.60094	+	4.44679i	0.646747	+	0.798666i	0.990074	−	0.140550i	\(-0.0448870\pi\)
							−0.343326	+	0.939216i	\(0.611554\pi\)
\(37\)	0.281368	+	0.552216i	0.0462566	+	0.0907837i	0.912978	−	0.408009i	\(-0.133777\pi\)
							−0.866721	+	0.498793i	\(0.833777\pi\)
\(41\)	12.6565			1.97661			0.988305	−	0.152489i	\(-0.0487287\pi\)
							0.988305	+	0.152489i	\(0.0487287\pi\)
\(43\)	0.171180	−	3.26630i	0.0261047	−	0.498107i	−0.954022	−	0.299736i	\(-0.903102\pi\)
							0.980127	−	0.198371i	\(-0.0635651\pi\)
\(47\)	1.17532	−	11.1824i	0.171437	−	1.63112i	−0.483436	−	0.875380i	\(-0.660611\pi\)
							0.654873	−	0.755739i	\(-0.272722\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	671.2.dc.a.6.19	yes	960
11.2	odd	10		671.2.cu.a.189.19	✓	960
61.51	odd	60		671.2.cu.a.600.19	yes	960
671.112	even	60	inner	671.2.dc.a.112.19	yes	960

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
671.2.cu.a.189.19	✓	960	11.2	odd	10
671.2.cu.a.600.19	yes	960	61.51	odd	60
671.2.dc.a.6.19	yes	960	1.1	even	1	trivial
671.2.dc.a.112.19	yes	960	671.112	even	60	inner