Embedded newform 441.2.ba.a.22.5

• Label: 441.2.ba.a.22.5
• Level: $441$
• Weight: $2$
• Character: 441.22
• Analytic conductor: $3.521$
• Analytic rank: $0$
• Dimension: $648$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 441 = 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	441.ba (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			22.5
Character	\(\chi\)	\(=\)	441.22
Dual form			441.2.ba.a.421.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.11562 - 1.44241i) q^{2} +(-0.314751 + 1.70321i) q^{3} +(1.66464 + 4.24143i) q^{4} +(-1.01272 - 0.939663i) q^{5} +(3.12262 - 3.14936i) q^{6} +(1.62237 - 2.08995i) q^{7} +(1.45657 - 6.38167i) q^{8} +(-2.80186 - 1.07218i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 5 q^{2} - 10 q^{3} + 47 q^{4} - 9 q^{5} - 34 q^{6} - 7 q^{7} - 28 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(199\)	\(344\)
\(\chi(n)\)	\(e\left(\frac{4}{7}\right)\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.11562	−	1.44241i	−1.49597	−	1.01994i	−0.987281	−	0.158987i	\(-0.949177\pi\)
							−0.508691	−	0.860949i	\(-0.669870\pi\)
\(3\)	−0.314751	+	1.70321i	−0.181722	+	0.983350i
							\(5\)	−1.01272	−	0.939663i	−0.452900	−	0.420230i	0.420455	−	0.907313i	\(-0.361870\pi\)
−0.873356	+	0.487083i	\(0.838061\pi\)
\(7\)	1.62237	−	2.08995i	0.613199	−	0.789929i
							\(11\)	−0.137403	−	0.0936794i	−0.0414284	−	0.0282454i	0.542429	−	0.840102i	\(-0.317505\pi\)
−0.583857	+	0.811856i	\(0.698457\pi\)
\(13\)	−0.294157	+	3.92525i	−0.0815844	+	1.08867i	0.794741	+	0.606949i	\(0.207607\pi\)
							−0.876325	+	0.481720i	\(0.840012\pi\)
\(17\)	−0.253332	−	0.317668i	−0.0614420	−	0.0770458i	0.750161	−	0.661255i	\(-0.229976\pi\)
							−0.811603	+	0.584210i	\(0.801405\pi\)
\(19\)	−0.482092			−0.110599			−0.0552997	−	0.998470i	\(-0.517611\pi\)
							−0.0552997	+	0.998470i	\(0.517611\pi\)
\(23\)	0.0836313	+	0.213089i	0.0174383	+	0.0444322i	0.939328	−	0.343021i	\(-0.111450\pi\)
							−0.921889	+	0.387453i	\(0.873355\pi\)
\(29\)	3.55251	−	9.05164i	0.659684	−	1.68085i	−0.0704251	−	0.997517i	\(-0.522436\pi\)
							0.730109	−	0.683331i	\(-0.239469\pi\)
\(31\)	4.26561	−	7.38825i	0.766126	−	1.32697i	−0.173524	−	0.984830i	\(-0.555515\pi\)
							0.939649	−	0.342139i	\(-0.111151\pi\)
\(37\)	4.16051	+	5.21711i	0.683983	+	0.857688i	0.995714	−	0.0924825i	\(-0.0294802\pi\)
							−0.311731	+	0.950170i	\(0.600909\pi\)
\(41\)	8.45536	+	7.84543i	1.32051	+	1.22525i	0.955761	+	0.294143i	\(0.0950341\pi\)
							0.364744	+	0.931108i	\(0.381156\pi\)
\(43\)	1.88320	−	1.74735i	0.287185	−	0.266468i	−0.523404	−	0.852085i	\(-0.675338\pi\)
							0.810588	+	0.585616i	\(0.199147\pi\)
\(47\)	−3.45064	−	2.35260i	−0.503327	−	0.343163i	0.284890	−	0.958560i	\(-0.408043\pi\)
							−0.788217	+	0.615398i	\(0.788995\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.ba.a.22.5	✓	648
9.7	even	3	inner	441.2.ba.a.169.5	yes	648
49.29	even	7	inner	441.2.ba.a.274.5	yes	648
441.421	even	21	inner	441.2.ba.a.421.5	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.ba.a.22.5	✓	648	1.1	even	1	trivial
441.2.ba.a.169.5	yes	648	9.7	even	3	inner
441.2.ba.a.274.5	yes	648	49.29	even	7	inner
441.2.ba.a.421.5	yes	648	441.421	even	21	inner