Embedded newform 441.2.z.a.4.8

• Label: 441.2.z.a.4.8
• 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			4.8
Character	\(\chi\)	\(=\)	441.4
Dual form			441.2.z.a.331.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.163627 + 2.18345i) q^{2} +(-1.55326 - 0.766401i) q^{3} +(-2.76302 - 0.416458i) q^{4} +(-0.523133 + 2.29200i) q^{5} +(1.92755 - 3.26607i) q^{6} +(1.85748 + 1.88409i) q^{7} +(0.386968 - 1.69542i) q^{8} +(1.82526 + 2.38085i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 8 q^{2} - 13 q^{3} + 44 q^{4} + 3 q^{5} + 2 q^{6} - 7 q^{7} - 16 q^{8} - 39 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.163627	+	2.18345i	−0.115702	+	1.54393i	0.573405	+	0.819272i	\(0.305622\pi\)
							−0.689107	+	0.724660i	\(0.741997\pi\)
\(3\)	−1.55326	−	0.766401i	−0.896778	−	0.442482i
							\(5\)	−0.523133	+	2.29200i	−0.233952	+	1.02501i	0.712374	+	0.701800i	\(0.247620\pi\)
−0.946327	+	0.323212i	\(0.895237\pi\)
\(7\)	1.85748	+	1.88409i	0.702060	+	0.712118i
							\(11\)	−1.90979	+	0.919705i	−0.575822	+	0.277301i	−0.699045	−	0.715078i	\(-0.746391\pi\)
0.123223	+	0.992379i	\(0.460677\pi\)
\(13\)	0.0368951	−	0.492330i	0.0102328	−	0.136548i	−0.989747	−	0.142829i	\(-0.954380\pi\)
							0.999980	+	0.00628080i	\(0.00199925\pi\)
\(17\)	−1.06254	+	0.160152i	−0.257704	+	0.0388426i	−0.276623	−	0.960978i	\(-0.589215\pi\)
							0.0189196	+	0.999821i	\(0.493977\pi\)
\(19\)	−1.55406	−	2.69170i	−0.356525	−	0.617519i	0.630853	−	0.775903i	\(-0.282705\pi\)
							−0.987378	+	0.158383i	\(0.949372\pi\)
\(23\)	−2.83621	+	3.55649i	−0.591390	+	0.741580i	−0.984008	−	0.178122i	\(-0.942998\pi\)
							0.392618	+	0.919702i	\(0.371569\pi\)
\(29\)	−0.0595642	+	0.151767i	−0.0110608	+	0.0281825i	−0.936297	−	0.351208i	\(-0.885771\pi\)
							0.925237	+	0.379390i	\(0.123866\pi\)
\(31\)	−3.17443	−	5.49827i	−0.570144	−	0.987518i	−0.996551	−	0.0829867i	\(-0.973554\pi\)
							0.426407	−	0.904532i	\(-0.359779\pi\)
\(37\)	2.77018	−	7.05831i	0.455415	−	1.16038i	−0.500238	−	0.865888i	\(-0.666754\pi\)
							0.955654	−	0.294492i	\(-0.0951504\pi\)
\(41\)	−1.14510	−	1.06250i	−0.178835	−	0.165934i	0.585676	−	0.810545i	\(-0.300829\pi\)
							−0.764511	+	0.644611i	\(0.777019\pi\)
\(43\)	−7.34554	+	6.81567i	−1.12018	+	1.03938i	−0.121278	+	0.992619i	\(0.538699\pi\)
							−0.998906	+	0.0467606i	\(0.985110\pi\)
\(47\)	−0.341640	+	4.55887i	−0.0498333	+	0.664979i	0.915225	+	0.402943i	\(0.132013\pi\)
							−0.965058	+	0.262036i	\(0.915606\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.z.a.4.8	yes	648
9.7	even	3		441.2.y.a.151.47	✓	648
49.37	even	21		441.2.y.a.184.47	yes	648
441.331	even	21	inner	441.2.z.a.331.8	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.y.a.151.47	✓	648	9.7	even	3
441.2.y.a.184.47	yes	648	49.37	even	21
441.2.z.a.4.8	yes	648	1.1	even	1	trivial
441.2.z.a.331.8	yes	648	441.331	even	21	inner