Embedded newform 441.2.z.a.4.12

• Label: 441.2.z.a.4.12
• 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			4.12
Character	\(\chi\)	\(=\)	441.4
Dual form			441.2.z.a.331.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.129339 + 1.72591i) q^{2} +(-1.02279 + 1.39782i) q^{3} +(-0.984361 - 0.148369i) q^{4} +(-0.845939 + 3.70630i) q^{5} +(-2.28022 - 1.94603i) q^{6} +(-1.47863 + 2.19400i) q^{7} +(-0.386868 + 1.69498i) q^{8} +(-0.907804 - 2.85935i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 8 q^{2} - 13 q^{3} + 44 q^{4} + 3 q^{5} + 2 q^{6} - 7 q^{7} - 16 q^{8} - 39 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.129339	+	1.72591i	−0.0914563	+	1.22040i	0.742987	+	0.669306i	\(0.233408\pi\)
							−0.834443	+	0.551094i	\(0.814211\pi\)
\(3\)	−1.02279	+	1.39782i	−0.590508	+	0.807032i
							\(5\)	−0.845939	+	3.70630i	−0.378315	+	1.65751i	0.324312	+	0.945950i	\(0.394867\pi\)
−0.702628	+	0.711558i	\(0.747990\pi\)
\(7\)	−1.47863	+	2.19400i	−0.558871	+	0.829255i
							\(11\)	4.18953	−	2.01757i	1.26319	−	0.608320i	0.322173	−	0.946681i	\(-0.395587\pi\)
0.941016	+	0.338361i	\(0.109872\pi\)
\(13\)	−0.125001	+	1.66802i	−0.0346689	+	0.462625i	0.952671	+	0.304004i	\(0.0983237\pi\)
							−0.987340	+	0.158621i	\(0.949295\pi\)
\(17\)	7.93952	−	1.19669i	1.92562	−	0.290240i	0.928841	−	0.370478i	\(-0.120806\pi\)
							0.996776	+	0.0802376i	\(0.0255679\pi\)
\(19\)	2.46469	+	4.26896i	0.565438	+	0.979367i	0.997009	+	0.0772883i	\(0.0246262\pi\)
							−0.431571	+	0.902079i	\(0.642040\pi\)
\(23\)	1.08815	−	1.36450i	0.226896	−	0.284518i	−0.655332	−	0.755341i	\(-0.727471\pi\)
							0.882228	+	0.470823i	\(0.156043\pi\)
\(29\)	0.129427	−	0.329775i	0.0240340	−	0.0612377i	−0.918353	−	0.395762i	\(-0.870480\pi\)
							0.942387	+	0.334524i	\(0.108576\pi\)
\(31\)	−3.92318	−	6.79515i	−0.704624	−	1.22045i	−0.966827	−	0.255432i	\(-0.917782\pi\)
							0.262202	−	0.965013i	\(-0.415551\pi\)
\(37\)	−0.872456	+	2.22298i	−0.143431	+	0.365456i	−0.984479	−	0.175505i	\(-0.943844\pi\)
							0.841048	+	0.540961i	\(0.181939\pi\)
\(41\)	2.90224	+	2.69289i	0.453254	+	0.420558i	0.873479	−	0.486862i	\(-0.161858\pi\)
							−0.420225	+	0.907420i	\(0.638049\pi\)
\(43\)	7.14693	−	6.63138i	1.08990	−	1.01128i	0.0900138	−	0.995941i	\(-0.471309\pi\)
							0.999882	−	0.0153354i	\(-0.00488159\pi\)
\(47\)	0.0477675	−	0.637413i	0.00696760	−	0.0929762i	−0.992636	−	0.121137i	\(-0.961346\pi\)
							0.999603	+	0.0281607i	\(0.00896503\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.z.a.4.12	yes	648
9.7	even	3		441.2.y.a.151.43	✓	648
49.37	even	21		441.2.y.a.184.43	yes	648
441.331	even	21	inner	441.2.z.a.331.12	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.y.a.151.43	✓	648	9.7	even	3
441.2.y.a.184.43	yes	648	49.37	even	21
441.2.z.a.4.12	yes	648	1.1	even	1	trivial
441.2.z.a.331.12	yes	648	441.331	even	21	inner