Embedded newform 441.2.z.a.4.2

• Label: 441.2.z.a.4.2
• 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			4.2
Character	\(\chi\)	\(=\)	441.4
Dual form			441.2.z.a.331.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.193400 + 2.58074i) q^{2} +(1.69682 + 0.347567i) q^{3} +(-4.64515 - 0.700143i) q^{4} +(0.742208 - 3.25183i) q^{5} +(-1.22514 + 4.31183i) q^{6} +(2.07445 - 1.64215i) q^{7} +(1.55350 - 6.80632i) q^{8} +(2.75839 + 1.17952i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 8 q^{2} - 13 q^{3} + 44 q^{4} + 3 q^{5} + 2 q^{6} - 7 q^{7} - 16 q^{8} - 39 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.193400	+	2.58074i	−0.136754	+	1.82486i	0.333977	+	0.942581i	\(0.391609\pi\)
							−0.470731	+	0.882277i	\(0.656010\pi\)
\(3\)	1.69682	+	0.347567i	0.979659	+	0.200668i
							\(5\)	0.742208	−	3.25183i	0.331926	−	1.45426i	−0.483472	−	0.875360i	\(-0.660624\pi\)
0.815398	−	0.578901i	\(-0.196518\pi\)
\(7\)	2.07445	−	1.64215i	0.784068	−	0.620675i
							\(11\)	3.28065	−	1.57988i	0.989154	−	0.476352i	0.131910	−	0.991262i	\(-0.457889\pi\)
0.857244	+	0.514910i	\(0.172175\pi\)
\(13\)	0.138244	−	1.84473i	0.0383419	−	0.511637i	−0.944661	−	0.328048i	\(-0.893609\pi\)
							0.983003	−	0.183589i	\(-0.0587716\pi\)
\(17\)	−4.12084	+	0.621116i	−0.999450	+	0.150643i	−0.628334	−	0.777944i	\(-0.716263\pi\)
							−0.371116	+	0.928586i	\(0.621025\pi\)
\(19\)	−1.33027	−	2.30410i	−0.305185	−	0.528596i	0.672117	−	0.740445i	\(-0.265385\pi\)
							−0.977303	+	0.211848i	\(0.932052\pi\)
\(23\)	−2.74847	+	3.44647i	−0.573095	+	0.718638i	−0.980918	−	0.194423i	\(-0.937717\pi\)
							0.407823	+	0.913061i	\(0.366288\pi\)
\(29\)	−3.53951	+	9.01853i	−0.657271	+	1.67470i	0.0780243	+	0.996951i	\(0.475139\pi\)
							−0.735295	+	0.677747i	\(0.762956\pi\)
\(31\)	−2.49657	−	4.32419i	−0.448398	−	0.776648i	0.549884	−	0.835241i	\(-0.314672\pi\)
							−0.998282	+	0.0585931i	\(0.981339\pi\)
\(37\)	−1.13960	+	2.90366i	−0.187350	+	0.477359i	−0.993429	−	0.114448i	\(-0.963490\pi\)
							0.806080	+	0.591807i	\(0.201585\pi\)
\(41\)	8.14022	+	7.55302i	1.27129	+	1.17958i	0.974437	+	0.224661i	\(0.0721275\pi\)
							0.296853	+	0.954923i	\(0.404063\pi\)
\(43\)	−0.329987	+	0.306183i	−0.0503225	+	0.0466924i	−0.704933	−	0.709274i	\(-0.749023\pi\)
							0.654611	+	0.755966i	\(0.272833\pi\)
\(47\)	−0.605395	+	8.07843i	−0.0883059	+	1.17836i	0.760386	+	0.649472i	\(0.225010\pi\)
							−0.848691	+	0.528888i	\(0.822609\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.z.a.4.2	yes	648
9.7	even	3		441.2.y.a.151.53	✓	648
49.37	even	21		441.2.y.a.184.53	yes	648
441.331	even	21	inner	441.2.z.a.331.2	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.y.a.151.53	✓	648	9.7	even	3
441.2.y.a.184.53	yes	648	49.37	even	21
441.2.z.a.4.2	yes	648	1.1	even	1	trivial
441.2.z.a.331.2	yes	648	441.331	even	21	inner