Embedded newform 671.2.bt.a.24.11

• Label: 671.2.bt.a.24.11
• Level: $671$
• Weight: $2$
• Character: 671.24
• Analytic conductor: $5.358$
• Analytic rank: $0$
• Dimension: $480$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 671 = 11 \cdot 61 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	671.bt (of order \(20\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.35796197563\)
Analytic rank:	\(0\)
Dimension:	\(480\)
Relative dimension:	\(60\) over \(\Q(\zeta_{20})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{20}]$

Embedding invariants

Embedding label			24.11
Character	\(\chi\)	\(=\)	671.24
Dual form			671.2.bt.a.28.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.48741 - 1.48741i) q^{2} +(0.466480 - 0.642054i) q^{3} +2.42479i q^{4} +(1.24388 - 1.71206i) q^{5} +(-1.64885 + 0.261152i) q^{6} +(3.16101 + 0.500655i) q^{7} +(0.631841 - 0.631841i) q^{8} +(0.732421 + 2.25416i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 480 q - 10 q^{2} - 10 q^{3} - 10 q^{5} - 30 q^{6} - 20 q^{7} + 10 q^{8} + 110 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(123\)	\(551\)
\(\chi(n)\)	\(e\left(\frac{1}{10}\right)\)	\(e\left(\frac{3}{20}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.48741	−	1.48741i	−1.05176	−	1.05176i	−0.998585	−	0.0531742i	\(-0.983066\pi\)
							−0.0531742	−	0.998585i	\(-0.516934\pi\)
\(3\)	0.466480	−	0.642054i	0.269322	−	0.370690i	−0.652839	−	0.757497i	\(-0.726422\pi\)
							0.922161	+	0.386807i	\(0.126422\pi\)
\(5\)	1.24388	−	1.71206i	0.556281	−	0.765655i	−0.434567	−	0.900640i	\(-0.643098\pi\)
							0.990848	+	0.134984i	\(0.0430984\pi\)
\(7\)	3.16101	+	0.500655i	1.19475	+	0.189230i	0.721954	−	0.691941i	\(-0.243244\pi\)
							0.472797	+	0.881171i	\(0.343244\pi\)
\(11\)	3.07689	+	1.23805i	0.927717	+	0.373286i
							\(13\)	−3.28017	−	1.06579i	−0.909756	−	0.295598i	−0.183498	−	0.983020i	\(-0.558742\pi\)
−0.726258	+	0.687422i	\(0.758742\pi\)
\(17\)	2.52739	−	0.400300i	0.612983	−	0.0970870i	0.157780	−	0.987474i	\(-0.449566\pi\)
							0.455204	+	0.890387i	\(0.349566\pi\)
\(19\)	6.76048	+	4.91178i	1.55096	+	1.12684i	0.942962	+	0.332900i	\(0.108027\pi\)
							0.607998	+	0.793938i	\(0.291973\pi\)
\(23\)	−0.330502	+	2.08670i	−0.0689143	+	0.435108i	0.928973	+	0.370146i	\(0.120692\pi\)
							−0.997888	+	0.0649616i	\(0.979308\pi\)
\(29\)	−4.58974	+	0.726943i	−0.852293	+	0.134990i	−0.567271	−	0.823531i	\(-0.692001\pi\)
							−0.285022	+	0.958521i	\(0.592001\pi\)
\(31\)	0.866320	+	5.46973i	0.155596	+	0.982393i	0.934684	+	0.355479i	\(0.115682\pi\)
							−0.779089	+	0.626914i	\(0.784318\pi\)
\(37\)	7.31505	+	1.15859i	1.20259	+	0.190471i	0.725398	−	0.688330i	\(-0.241656\pi\)
							0.477189	+	0.878801i	\(0.341656\pi\)
\(41\)	−1.69331	−	5.21147i	−0.264450	−	0.813894i	−0.991820	−	0.127648i	\(-0.959257\pi\)
							0.727369	−	0.686246i	\(-0.240743\pi\)
\(43\)	−9.10774	+	1.44252i	−1.38892	+	0.219983i	−0.805695	−	0.592330i	\(-0.798208\pi\)
							−0.583222	+	0.812313i	\(0.698208\pi\)
\(47\)	−5.08499	+	3.69446i	−0.741723	+	0.538893i	−0.893250	−	0.449560i	\(-0.851581\pi\)
							0.151527	+	0.988453i	\(0.451581\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	671.2.bt.a.24.11	✓	480
11.6	odd	10		671.2.by.a.512.11	yes	480
61.28	odd	20		671.2.by.a.211.11	yes	480
671.28	even	20	inner	671.2.bt.a.28.11	yes	480

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
671.2.bt.a.24.11	✓	480	1.1	even	1	trivial
671.2.bt.a.28.11	yes	480	671.28	even	20	inner
671.2.by.a.211.11	yes	480	61.28	odd	20
671.2.by.a.512.11	yes	480	11.6	odd	10