Embedded newform 441.2.z.a.142.32

• Label: 441.2.z.a.142.32
• 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			142.32
Character	\(\chi\)	\(=\)	441.142
Dual form			441.2.z.a.382.32

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.187748 - 0.478375i) q^{2} +(0.742299 - 1.56493i) q^{3} +(1.27251 + 1.18072i) q^{4} +(-0.268987 + 0.129537i) q^{5} +(-0.609256 - 0.648909i) q^{6} +(-0.691339 + 2.55383i) q^{7} +(1.72975 - 0.833004i) q^{8} +(-1.89799 - 2.32328i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 8 q^{2} - 13 q^{3} + 44 q^{4} + 3 q^{5} + 2 q^{6} - 7 q^{7} - 16 q^{8} - 39 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.187748	−	0.478375i	0.132758	−	0.338262i	−0.849001	−	0.528391i	\(-0.822795\pi\)
							0.981759	+	0.190129i	\(0.0608907\pi\)
\(3\)	0.742299	−	1.56493i	0.428566	−	0.903510i
							\(5\)	−0.268987	+	0.129537i	−0.120295	+	0.0579309i	−0.493063	−	0.869994i	\(-0.664123\pi\)
0.372768	+	0.927924i	\(0.378409\pi\)
\(7\)	−0.691339	+	2.55383i	−0.261301	+	0.965257i
							\(11\)	2.99540	+	3.75611i	0.903146	+	1.13251i	0.990661	+	0.136346i	\(0.0435358\pi\)
−0.0875157	+	0.996163i	\(0.527893\pi\)
\(13\)	2.29352	−	5.84380i	0.636108	−	1.62078i	−0.139980	−	0.990154i	\(-0.544704\pi\)
							0.776089	−	0.630624i	\(-0.217201\pi\)
\(17\)	5.90240	−	5.47662i	1.43154	−	1.32828i	0.568952	−	0.822371i	\(-0.307349\pi\)
							0.862589	−	0.505905i	\(-0.168841\pi\)
\(19\)	−2.00716	+	3.47650i	−0.460474	+	0.797564i	−0.998985	−	0.0450546i	\(-0.985654\pi\)
							0.538511	+	0.842619i	\(0.318987\pi\)
\(23\)	1.15465	−	5.05886i	0.240762	−	1.05485i	−0.699564	−	0.714570i	\(-0.746623\pi\)
							0.940326	−	0.340275i	\(-0.110520\pi\)
\(29\)	−5.43072	−	1.67515i	−1.00846	−	0.311068i	−0.253850	−	0.967244i	\(-0.581697\pi\)
							−0.754609	+	0.656175i	\(0.772173\pi\)
\(31\)	−2.70214	+	4.68025i	−0.485319	+	0.840597i	−0.999858	−	0.0168702i	\(-0.994630\pi\)
							0.514539	+	0.857467i	\(0.327963\pi\)
\(37\)	−1.43299	−	0.442020i	−0.235583	−	0.0726676i	0.174716	−	0.984619i	\(-0.444099\pi\)
							−0.410299	+	0.911951i	\(0.634576\pi\)
\(41\)	0.475844	+	0.324425i	0.0743144	+	0.0506667i	0.599907	−	0.800070i	\(-0.295204\pi\)
							−0.525593	+	0.850736i	\(0.676156\pi\)
\(43\)	−3.57858	+	2.43983i	−0.545728	+	0.372071i	−0.804578	−	0.593847i	\(-0.797609\pi\)
							0.258851	+	0.965917i	\(0.416656\pi\)
\(47\)	−0.940942	+	2.39748i	−0.137250	+	0.349709i	−0.982929	−	0.183984i	\(-0.941101\pi\)
							0.845679	+	0.533692i	\(0.179196\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.z.a.142.32	yes	648
9.4	even	3		441.2.y.a.436.32	yes	648
49.39	even	21		441.2.y.a.88.32	✓	648
441.382	even	21	inner	441.2.z.a.382.32	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.y.a.88.32	✓	648	49.39	even	21
441.2.y.a.436.32	yes	648	9.4	even	3
441.2.z.a.142.32	yes	648	1.1	even	1	trivial
441.2.z.a.382.32	yes	648	441.382	even	21	inner