Embedded newform 882.2.bc.a.563.18

• Label: 882.2.bc.a.563.18
• Level: $882$
• Weight: $2$
• Character: 882.563
• 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.bc (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			563.18
Character	\(\chi\)	\(=\)	882.563
Dual form			882.2.bc.a.47.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.680173 + 0.733052i) q^{2} +(0.579418 + 1.63226i) q^{3} +(-0.0747301 - 0.997204i) q^{4} +(1.91351 - 2.39947i) q^{5} +(-1.59064 - 0.685476i) q^{6} +(-2.49051 + 0.892955i) q^{7} +(0.781831 + 0.623490i) q^{8} +(-2.32855 + 1.89152i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 672 q - 56 q^{4} + 14 q^{6} - 2 q^{7} - 26 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(199\)	\(785\)
\(\chi(n)\)	\(e\left(\frac{37}{42}\right)\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.680173	+	0.733052i	−0.480955	+	0.518346i
							\(3\)	0.579418	+	1.63226i	0.334527	+	0.942386i
\(5\)	1.91351	−	2.39947i	0.855748	−	1.07307i								−0.140798	−	0.990038i	\(-0.544967\pi\)
							0.996547	−	0.0830360i	\(-0.0264617\pi\)
\(7\)	−2.49051	+	0.892955i	−0.941324	+	0.337505i
							\(11\)	−4.59943	−	1.04979i	−1.38678	−	0.316523i	−0.536964	−	0.843605i	\(-0.680429\pi\)
−0.849815	+	0.527082i	\(0.823286\pi\)
\(13\)	−0.750189	+	0.808512i	−0.208065	+	0.224241i	−0.828459	−	0.560049i	\(-0.810782\pi\)
							0.620394	+	0.784290i	\(0.286973\pi\)
\(17\)	−0.0297019	+	0.396344i	−0.00720377	+	0.0961276i	−0.999647	−	0.0265794i	\(-0.991539\pi\)
							0.992443	+	0.122707i	\(0.0391576\pi\)
\(19\)	−3.33432	+	1.92507i	−0.764946	+	0.441642i	−0.831069	−	0.556170i	\(-0.812270\pi\)
							0.0661230	+	0.997811i	\(0.478937\pi\)
\(23\)	0.262978	−	0.546080i	0.0548348	−	0.113866i	−0.871751	−	0.489949i	\(-0.837015\pi\)
							0.926586	+	0.376084i	\(0.122730\pi\)
\(29\)	−4.66868	+	6.84770i	−0.866953	+	1.27159i	0.0941959	+	0.995554i	\(0.469972\pi\)
							−0.961148	+	0.276032i	\(0.910980\pi\)
\(31\)	−6.87537	+	3.96950i	−1.23485	+	0.712943i	−0.968038	−	0.250805i	\(-0.919305\pi\)
							−0.266815	+	0.963748i	\(0.585971\pi\)
\(37\)	−4.16629	−	2.84053i	−0.684934	−	0.466980i	0.170185	−	0.985412i	\(-0.445564\pi\)
							−0.855119	+	0.518432i	\(0.826516\pi\)
\(41\)	2.59064	+	6.60085i	0.404590	+	1.03088i	0.977439	+	0.211218i	\(0.0677431\pi\)
							−0.572848	+	0.819661i	\(0.694162\pi\)
\(43\)	0.522414	−	1.33109i	0.0796675	−	0.202989i	−0.885490	−	0.464658i	\(-0.846177\pi\)
							0.965158	+	0.261668i	\(0.0842726\pi\)
\(47\)	7.35616	+	6.82552i	1.07301	+	0.995604i	0.999999	−	0.00172717i	\(-0.000549774\pi\)
							0.0730076	+	0.997331i	\(0.476740\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	882.2.bc.a.563.18	yes	672
9.2	odd	6		882.2.bk.a.857.12	yes	672
49.47	odd	42		882.2.bk.a.635.12	yes	672
441.47	even	42	inner	882.2.bc.a.47.18	✓	672

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
882.2.bc.a.47.18	✓	672	441.47	even	42	inner
882.2.bc.a.563.18	yes	672	1.1	even	1	trivial
882.2.bk.a.635.12	yes	672	49.47	odd	42
882.2.bk.a.857.12	yes	672	9.2	odd	6