Embedded newform 441.2.z.a.4.10

• Label: 441.2.z.a.4.10
• Level: $441$
• Weight: $2$
• Character: 441.4
• 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			4.10
Character	\(\chi\)	\(=\)	441.4
Dual form			441.2.z.a.331.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.152487 + 2.03479i) q^{2} +(-0.743876 - 1.56418i) q^{3} +(-2.13947 - 0.322473i) q^{4} +(0.723985 - 3.17199i) q^{5} +(3.29621 - 1.27512i) q^{6} +(-2.58955 - 0.542443i) q^{7} +(0.0742997 - 0.325528i) q^{8} +(-1.89330 + 2.32711i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 8 q^{2} - 13 q^{3} + 44 q^{4} + 3 q^{5} + 2 q^{6} - 7 q^{7} - 16 q^{8} - 39 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(199\)	\(344\)
\(\chi(n)\)	\(e\left(\frac{5}{21}\right)\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.152487	+	2.03479i	−0.107824	+	1.43882i	0.636644	+	0.771158i	\(0.280322\pi\)
							−0.744468	+	0.667658i	\(0.767297\pi\)
\(3\)	−0.743876	−	1.56418i	−0.429477	−	0.903078i
							\(5\)	0.723985	−	3.17199i	0.323776	−	1.41856i	−0.506999	−	0.861947i	\(-0.669245\pi\)
0.830775	−	0.556609i	\(-0.187898\pi\)
\(7\)	−2.58955	−	0.542443i	−0.978757	−	0.205024i
							\(11\)	−2.58062	+	1.24276i	−0.778086	+	0.374707i	−0.780392	−	0.625291i	\(-0.784980\pi\)
0.00230572	+	0.999997i	\(0.499266\pi\)
\(13\)	−0.295982	+	3.94961i	−0.0820906	+	1.09542i	0.792278	+	0.610161i	\(0.208895\pi\)
							−0.874368	+	0.485263i	\(0.838724\pi\)
\(17\)	−0.938284	+	0.141424i	−0.227567	+	0.0343003i	−0.261836	−	0.965112i	\(-0.584328\pi\)
							0.0342686	+	0.999413i	\(0.489090\pi\)
\(19\)	−2.87174	−	4.97400i	−0.658822	−	1.14111i	−0.980921	−	0.194408i	\(-0.937722\pi\)
							0.322098	−	0.946706i	\(-0.395612\pi\)
\(23\)	−4.06608	+	5.09870i	−0.847836	+	1.06315i	0.149395	+	0.988778i	\(0.452268\pi\)
							−0.997230	+	0.0743747i	\(0.976304\pi\)
\(29\)	−2.93158	+	7.46955i	−0.544381	+	1.38706i	0.348730	+	0.937223i	\(0.386613\pi\)
							−0.893111	+	0.449837i	\(0.851482\pi\)
\(31\)	−2.87652	−	4.98228i	−0.516639	−	0.894844i	−0.999813	−	0.0193203i	\(-0.993850\pi\)
							0.483175	−	0.875524i	\(-0.339484\pi\)
\(37\)	2.19264	−	5.58675i	0.360468	−	0.918456i	−0.629479	−	0.777017i	\(-0.716732\pi\)
							0.989947	−	0.141439i	\(-0.0451730\pi\)
\(41\)	2.02726	+	1.88102i	0.316604	+	0.293766i	0.822426	−	0.568872i	\(-0.192620\pi\)
							−0.505822	+	0.862638i	\(0.668811\pi\)
\(43\)	7.14555	−	6.63010i	1.08969	−	1.01108i	0.0897998	−	0.995960i	\(-0.471377\pi\)
							0.999886	−	0.0151205i	\(-0.00481320\pi\)
\(47\)	0.690837	−	9.21858i	0.100769	−	1.34467i	−0.686081	−	0.727525i	\(-0.740671\pi\)
							0.786850	−	0.617144i	\(-0.211710\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.z.a.4.10	yes	648
9.7	even	3		441.2.y.a.151.45	✓	648
49.37	even	21		441.2.y.a.184.45	yes	648
441.331	even	21	inner	441.2.z.a.331.10	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.y.a.151.45	✓	648	9.7	even	3
441.2.y.a.184.45	yes	648	49.37	even	21
441.2.z.a.4.10	yes	648	1.1	even	1	trivial
441.2.z.a.331.10	yes	648	441.331	even	21	inner