Embedded newform 441.2.z.a.331.8

• Label: 441.2.z.a.331.8
• Level: $441$
• Weight: $2$
• Character: 441.331
• 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			331.8
Character	\(\chi\)	\(=\)	441.331
Dual form			441.2.z.a.4.8

$q$-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 \(\left(\mathbb{Z}/441\mathbb{Z}\right)^\times\).

\(n\)	\(199\)	\(344\)
\(\chi(n)\)	\(e\left(\frac{16}{21}\right)\)	\(e\left(\frac{2}{3}\right)\)

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)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.y.a.151.47	✓	648	49.4	even	21
441.2.y.a.184.47	yes	648	9.4	even	3
441.2.z.a.4.8	yes	648	441.4	even	21	inner
441.2.z.a.331.8	yes	648	1.1	even	1	trivial