Embedded newform 671.2.ck.a.14.16

• Label: 671.2.ck.a.14.16
• Level: $671$
• Weight: $2$
• Character: 671.14
• Analytic conductor: $5.358$
• Analytic rank: $0$
• Dimension: $480$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 671 = 11 \cdot 61 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	671.ck (of order \(30\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.35796197563\)
Analytic rank:	\(0\)
Dimension:	\(480\)
Relative dimension:	\(60\) over \(\Q(\zeta_{30})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{30}]$

Embedding invariants

Embedding label			14.16
Character	\(\chi\)	\(=\)	671.14
Dual form			671.2.ck.a.48.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.616002 + 1.38356i) q^{2} +(-0.492021 - 1.51429i) q^{3} +(-0.196528 - 0.218267i) q^{4} +(-0.0864920 + 0.822917i) q^{5} +(2.39820 + 0.252061i) q^{6} +(0.972024 + 4.57301i) q^{7} +(-2.45770 + 0.798555i) q^{8} +(0.376076 - 0.273235i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 480 q - 9 q^{2} - 12 q^{3} - 61 q^{4} - 5 q^{5} + 3 q^{6} - 18 q^{7} - 124 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(123\)	\(551\)
\(\chi(n)\)	\(e\left(\frac{4}{5}\right)\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.616002	+	1.38356i	−0.435579	+	0.978327i	0.553757	+	0.832678i	\(0.313194\pi\)
							−0.989336	+	0.145649i	\(0.953473\pi\)
\(3\)	−0.492021	−	1.51429i	−0.284069	−	0.874273i	−0.986676	−	0.162696i	\(-0.947981\pi\)
							0.702608	−	0.711577i	\(-0.252019\pi\)
\(5\)	−0.0864920	+	0.822917i	−0.0386804	+	0.368019i	0.958010	+	0.286733i	\(0.0925694\pi\)
							−0.996691	+	0.0812861i	\(0.974097\pi\)
\(7\)	0.972024	+	4.57301i	0.367390	+	1.72844i	0.641844	+	0.766836i	\(0.278170\pi\)
							−0.274453	+	0.961600i	\(0.588497\pi\)
\(11\)	−2.24454	+	2.44173i	−0.676755	+	0.736208i
							\(13\)	−0.170737	−	0.0760168i	−0.0473538	−	0.0210833i	0.382923	−	0.923780i	\(-0.374917\pi\)
−0.430277	+	0.902697i	\(0.641584\pi\)
\(17\)	−2.60508	−	5.85112i	−0.631826	−	1.41910i	−0.891327	−	0.453361i	\(-0.850225\pi\)
							0.259501	−	0.965743i	\(-0.416442\pi\)
\(19\)	1.31620	+	0.279767i	0.301957	+	0.0641828i	0.356398	−	0.934334i	\(-0.384005\pi\)
							−0.0544418	+	0.998517i	\(0.517338\pi\)
\(23\)			5.33212i			1.11182i	0.831241	+	0.555912i	\(0.187631\pi\)
							−0.831241	+	0.555912i	\(0.812369\pi\)
\(29\)	−3.40559	+	3.06641i	−0.632402	+	0.569417i	−0.921739	−	0.387810i	\(-0.873232\pi\)
							0.289337	+	0.957227i	\(0.406565\pi\)
\(31\)	−8.18103	+	0.859861i	−1.46936	+	0.154436i	−0.805131	−	0.593097i	\(-0.797905\pi\)
							−0.664226	+	0.747532i	\(0.731239\pi\)
\(37\)	−4.89304	−	1.58984i	−0.804410	−	0.261369i	−0.122182	−	0.992508i	\(-0.538989\pi\)
							−0.682228	+	0.731139i	\(0.738989\pi\)
\(41\)	3.72497	+	11.4643i	0.581743	+	1.79042i	0.611973	+	0.790878i	\(0.290376\pi\)
							−0.0302303	+	0.999543i	\(0.509624\pi\)
\(43\)	3.43986	−	1.98600i	0.524573	−	0.302862i	−0.214231	−	0.976783i	\(-0.568724\pi\)
							0.738804	+	0.673921i	\(0.235391\pi\)
\(47\)	−5.81270	−	1.23553i	−0.847870	−	0.180220i	−0.236565	−	0.971616i	\(-0.576022\pi\)
							−0.611305	+	0.791395i	\(0.709355\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	671.2.ck.a.14.16	✓	480
11.4	even	5	inner	671.2.ck.a.136.16	yes	480
61.48	even	6	inner	671.2.ck.a.597.16	yes	480
671.48	even	30	inner	671.2.ck.a.48.16	yes	480

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
671.2.ck.a.14.16	✓	480	1.1	even	1	trivial
671.2.ck.a.48.16	yes	480	671.48	even	30	inner
671.2.ck.a.136.16	yes	480	11.4	even	5	inner
671.2.ck.a.597.16	yes	480	61.48	even	6	inner