Embedded newform 441.2.z.a.142.50

• Label: 441.2.z.a.142.50
• Level: $441$
• Weight: $2$
• Character: 441.142
• Analytic conductor: $3.521$
• Analytic rank: $0$
• Dimension: $648$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 441 = 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	441.z (of order \(21\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.52140272914\)
Analytic rank:	\(0\)
Dimension:	\(648\)
Relative dimension:	\(54\) over \(\Q(\zeta_{21})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{21}]$

Embedding invariants

Embedding label			142.50
Character	\(\chi\)	\(=\)	441.142
Dual form			441.2.z.a.382.50

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.880400 - 2.24322i) q^{2} +(-0.697542 + 1.58538i) q^{3} +(-2.79084 - 2.58952i) q^{4} +(-0.193333 + 0.0931042i) q^{5} +(2.94225 + 2.96051i) q^{6} +(-2.55672 - 0.680587i) q^{7} +(-3.92360 + 1.88951i) q^{8} +(-2.02687 - 2.21174i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 8 q^{2} - 13 q^{3} + 44 q^{4} + 3 q^{5} + 2 q^{6} - 7 q^{7} - 16 q^{8} - 39 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/441\mathbb{Z}\right)^\times\).

\(n\)	\(199\)	\(344\)
\(\chi(n)\)	\(e\left(\frac{4}{21}\right)\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.880400	−	2.24322i	0.622537	−	1.58620i	−0.176092	−	0.984374i	\(-0.556346\pi\)
							0.798629	−	0.601824i	\(-0.205559\pi\)
\(3\)	−0.697542	+	1.58538i	−0.402726	+	0.915321i
							\(5\)	−0.193333	+	0.0931042i	−0.0864611	+	0.0416375i	−0.476615	−	0.879112i	\(-0.658136\pi\)
0.390153	+	0.920750i	\(0.372422\pi\)
\(7\)	−2.55672	−	0.680587i	−0.966348	−	0.257238i
							\(11\)	−2.29711	−	2.88048i	−0.692605	−	0.868499i	0.303842	−	0.952722i	\(-0.401730\pi\)
−0.996447	+	0.0842236i	\(0.973159\pi\)
\(13\)	1.02958	−	2.62332i	0.285554	−	0.727579i	−0.714068	−	0.700076i	\(-0.753149\pi\)
							0.999622	−	0.0275027i	\(-0.00875548\pi\)
\(17\)	−4.84728	+	4.49762i	−1.17564	+	1.09083i	−0.181425	+	0.983405i	\(0.558071\pi\)
							−0.994213	+	0.107428i	\(0.965739\pi\)
\(19\)	−2.18628	+	3.78675i	−0.501567	+	0.868740i	0.498431	+	0.866929i	\(0.333910\pi\)
							−0.999998	+	0.00181047i	\(0.999424\pi\)
\(23\)	1.32859	−	5.82093i	0.277030	−	1.21375i	−0.624497	−	0.781027i	\(-0.714696\pi\)
							0.901528	−	0.432722i	\(-0.142447\pi\)
\(29\)	−4.39860	−	1.35679i	−0.816800	−	0.251949i	−0.141921	−	0.989878i	\(-0.545328\pi\)
							−0.674879	+	0.737929i	\(0.735804\pi\)
\(31\)	3.77434	−	6.53735i	0.677892	−	1.17414i	−0.297723	−	0.954652i	\(-0.596227\pi\)
							0.975615	−	0.219491i	\(-0.0704396\pi\)
\(37\)	4.89040	+	1.50849i	0.803978	+	0.247994i	0.669394	−	0.742908i	\(-0.266554\pi\)
							0.134584	+	0.990902i	\(0.457030\pi\)
\(41\)	2.47625	+	1.68828i	0.386725	+	0.263665i	0.741039	−	0.671462i	\(-0.234333\pi\)
							−0.354314	+	0.935126i	\(0.615286\pi\)
\(43\)	−2.32210	+	1.58318i	−0.354117	+	0.241433i	−0.727292	−	0.686328i	\(-0.759222\pi\)
							0.373175	+	0.927761i	\(0.378269\pi\)
\(47\)	4.35634	−	11.0998i	0.635437	−	1.61907i	−0.141826	−	0.989892i	\(-0.545297\pi\)
							0.777263	−	0.629175i	\(-0.216607\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.z.a.142.50	yes	648
9.4	even	3		441.2.y.a.436.50	yes	648
49.39	even	21		441.2.y.a.88.50	✓	648
441.382	even	21	inner	441.2.z.a.382.50	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.y.a.88.50	✓	648	49.39	even	21
441.2.y.a.436.50	yes	648	9.4	even	3
441.2.z.a.142.50	yes	648	1.1	even	1	trivial
441.2.z.a.382.50	yes	648	441.382	even	21	inner