Embedded newform 882.2.bk.a.131.49

• Label: 882.2.bk.a.131.49
• Level: $882$
• Weight: $2$
• Character: 882.131
• Analytic conductor: $7.043$
• Analytic rank: $0$
• Dimension: $672$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	882.bk (of order \(42\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.04280545828\)
Analytic rank:	\(0\)
Dimension:	\(672\)
Relative dimension:	\(56\) over \(\Q(\zeta_{42})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{42}]$

Embedding invariants

Embedding label			131.49
Character	\(\chi\)	\(=\)	882.131
Dual form			882.2.bk.a.101.49

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.781831 + 0.623490i) q^{2} +(1.06398 - 1.36673i) q^{3} +(0.222521 + 0.974928i) q^{4} +(-0.278820 - 3.72059i) q^{5} +(1.68399 - 0.405170i) q^{6} +(2.25843 - 1.37822i) q^{7} +(-0.433884 + 0.900969i) q^{8} +(-0.735891 - 2.90834i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 672 q + 112 q^{4} + 14 q^{6} - 2 q^{7} + 10 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/882\mathbb{Z}\right)^\times\).

\(n\)	\(199\)	\(785\)
\(\chi(n)\)	\(e\left(\frac{41}{42}\right)\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.781831	+	0.623490i	0.552838	+	0.440874i
							\(3\)	1.06398	−	1.36673i	0.614289	−	0.789081i
\(5\)	−0.278820	−	3.72059i	−0.124692	−	1.66390i								−0.612142	−	0.790748i	\(-0.709692\pi\)
							0.487450	−	0.873151i	\(-0.337927\pi\)
\(7\)	2.25843	−	1.37822i	0.853608	−	0.520917i
							\(11\)	2.85142	−	1.11910i	0.859734	−	0.337421i	0.105813	−	0.994386i	\(-0.466255\pi\)
0.753921	+	0.656965i	\(0.228160\pi\)
\(13\)	−3.80994	+	1.49529i	−1.05669	+	0.414719i	−0.829136	−	0.559047i	\(-0.811167\pi\)
							−0.227551	+	0.973766i	\(0.573072\pi\)
\(17\)	−5.72964	+	1.76736i	−1.38964	+	0.428647i	−0.897171	−	0.441682i	\(-0.854382\pi\)
							−0.492469	+	0.870330i	\(0.663906\pi\)
\(19\)	−0.0199420	−	0.0115135i	−0.00457501	−	0.00264138i	0.497711	−	0.867343i	\(-0.334174\pi\)
							−0.502286	+	0.864702i	\(0.667507\pi\)
\(23\)	4.13399	+	4.45538i	0.861997	+	0.929012i	0.998096	−	0.0616835i	\(-0.0196469\pi\)
							−0.136099	+	0.990695i	\(0.543456\pi\)
\(29\)	−0.857859	−	2.78111i	−0.159300	−	0.516439i	0.840373	−	0.542009i	\(-0.182336\pi\)
							−0.999673	+	0.0255698i	\(0.991860\pi\)
\(31\)			6.03606i			1.08411i	0.840344	+	0.542054i	\(0.182353\pi\)
							−0.840344	+	0.542054i	\(0.817647\pi\)
\(37\)	3.39094	+	3.14633i	0.557467	+	0.517254i	0.907818	−	0.419365i	\(-0.137747\pi\)
							−0.350351	+	0.936619i	\(0.613938\pi\)
\(41\)	6.46596	−	4.40842i	1.00981	−	0.688479i	0.0590045	−	0.998258i	\(-0.481207\pi\)
							0.950809	+	0.309779i	\(0.100255\pi\)
\(43\)	7.27786	+	4.96196i	1.10986	+	0.756692i	0.972285	−	0.233799i	\(-0.0751157\pi\)
							0.137578	+	0.990491i	\(0.456068\pi\)
\(47\)	4.70757	−	5.90310i	0.686669	−	0.861056i	−0.309280	−	0.950971i	\(-0.600088\pi\)
							0.995949	+	0.0899151i	\(0.0286596\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	882.2.bk.a.131.49	yes	672
9.2	odd	6		882.2.bc.a.425.23	✓	672
49.3	odd	42		882.2.bc.a.689.23	yes	672
441.101	even	42	inner	882.2.bk.a.101.49	yes	672

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
882.2.bc.a.425.23	✓	672	9.2	odd	6
882.2.bc.a.689.23	yes	672	49.3	odd	42
882.2.bk.a.101.49	yes	672	441.101	even	42	inner
882.2.bk.a.131.49	yes	672	1.1	even	1	trivial