Embedded newform 882.2.bk.a.131.21

• Label: 882.2.bk.a.131.21
• 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			131.21
Character	\(\chi\)	\(=\)	882.131
Dual form			882.2.bk.a.101.21

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.781831 - 0.623490i) q^{2} +(1.23540 + 1.21400i) q^{3} +(0.222521 + 0.974928i) q^{4} +(0.00765300 + 0.102122i) q^{5} +(-0.208957 - 1.71940i) q^{6} +(0.299732 - 2.62872i) q^{7} +(0.433884 - 0.900969i) q^{8} +(0.0524141 + 2.99954i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 672 q + 112 q^{4} + 14 q^{6} - 2 q^{7} + 10 q^{9}+O(q^{10}) \)

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
							\(3\)	1.23540	+	1.21400i	0.713257	+	0.700903i
\(5\)	0.00765300	+	0.102122i	0.00342253	+	0.0456705i								0.998654	−	0.0518753i	\(-0.0165198\pi\)
							−0.995231	+	0.0975457i	\(0.968901\pi\)
\(7\)	0.299732	−	2.62872i	0.113288	−	0.993562i
							\(11\)	−5.12342	+	2.01080i	−1.54477	+	0.606278i	−0.976299	−	0.216425i	\(-0.930560\pi\)
−0.568471	+	0.822703i	\(0.692465\pi\)
\(13\)	4.20806	−	1.65154i	1.16710	−	0.458055i	0.298911	−	0.954281i	\(-0.403376\pi\)
							0.868193	+	0.496226i	\(0.165281\pi\)
\(17\)	6.92787	−	2.13697i	1.68026	−	0.518290i	0.699684	−	0.714452i	\(-0.253324\pi\)
							0.980572	+	0.196162i	\(0.0628478\pi\)
\(19\)	0.0178077	+	0.0102813i	0.00408536	+	0.00235868i	0.502041	−	0.864844i	\(-0.332583\pi\)
							−0.497956	+	0.867202i	\(0.665916\pi\)
\(23\)	5.44611	+	5.86951i	1.13559	+	1.22388i	0.971400	+	0.237449i	\(0.0763112\pi\)
							0.164192	+	0.986428i	\(0.447498\pi\)
\(29\)	0.735019	+	2.38287i	0.136490	+	0.442488i	0.997852	−	0.0655017i	\(-0.0208648\pi\)
							−0.861363	+	0.507990i	\(0.830389\pi\)
\(31\)		−	1.90003i		−	0.341256i	−0.985336	−	0.170628i	\(-0.945420\pi\)
							0.985336	−	0.170628i	\(-0.0545796\pi\)
\(37\)	−1.09678	−	1.01766i	−0.180309	−	0.167302i	0.584855	−	0.811138i	\(-0.301152\pi\)
							−0.765164	+	0.643836i	\(0.777342\pi\)
\(41\)	4.75262	−	3.24028i	0.742235	−	0.506047i	−0.132180	−	0.991226i	\(-0.542198\pi\)
							0.874415	+	0.485179i	\(0.161245\pi\)
\(43\)	1.56856	+	1.06943i	0.239203	+	0.163086i	0.676989	−	0.735993i	\(-0.263285\pi\)
							−0.437785	+	0.899080i	\(0.644237\pi\)
\(47\)	−6.30561	+	7.90698i	−0.919768	+	1.15335i	0.0680415	+	0.997682i	\(0.478325\pi\)
							−0.987809	+	0.155670i	\(0.950246\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	882.2.bk.a.131.21	yes	672
9.2	odd	6		882.2.bc.a.425.48	✓	672
49.3	odd	42		882.2.bc.a.689.48	yes	672
441.101	even	42	inner	882.2.bk.a.101.21	yes	672

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
882.2.bc.a.425.48	✓	672	9.2	odd	6
882.2.bc.a.689.48	yes	672	49.3	odd	42
882.2.bk.a.101.21	yes	672	441.101	even	42	inner
882.2.bk.a.131.21	yes	672	1.1	even	1	trivial