Embedded newform 441.2.z.a.331.11

• Label: 441.2.z.a.331.11
• 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			331.11
Character	\(\chi\)	\(=\)	441.331
Dual form			441.2.z.a.4.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.132809 - 1.77221i) q^{2} +(1.41906 + 0.993117i) q^{3} +(-1.14543 + 0.172646i) q^{4} +(-0.788881 - 3.45631i) q^{5} +(1.57155 - 2.64676i) q^{6} +(2.57823 + 0.593925i) q^{7} +(-0.332830 - 1.45822i) q^{8} +(1.02744 + 2.81858i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 8 q^{2} - 13 q^{3} + 44 q^{4} + 3 q^{5} + 2 q^{6} - 7 q^{7} - 16 q^{8} - 39 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.132809	−	1.77221i	−0.0939101	−	1.25314i	−0.822728	−	0.568435i	\(-0.807549\pi\)
							0.728818	−	0.684708i	\(-0.240070\pi\)
\(3\)	1.41906	+	0.993117i	0.819292	+	0.573377i
							\(5\)	−0.788881	−	3.45631i	−0.352798	−	1.54571i	−0.770688	−	0.637213i	\(-0.780087\pi\)
0.417889	−	0.908498i	\(-0.362770\pi\)
\(7\)	2.57823	+	0.593925i	0.974478	+	0.224483i
							\(11\)	1.33579	+	0.643285i	0.402757	+	0.193958i	0.624285	−	0.781196i	\(-0.285390\pi\)
−0.221528	+	0.975154i	\(0.571105\pi\)
\(13\)	−0.172992	−	2.30842i	−0.0479794	−	0.640241i	−0.968415	−	0.249346i	\(-0.919785\pi\)
							0.920435	−	0.390895i	\(-0.127835\pi\)
\(17\)	0.421630	+	0.0635504i	0.102260	+	0.0154132i	0.199973	−	0.979801i	\(-0.435915\pi\)
							−0.0977128	+	0.995215i	\(0.531153\pi\)
\(19\)	−2.91097	+	5.04194i	−0.667821	+	1.15670i	0.310691	+	0.950511i	\(0.399440\pi\)
							−0.978512	+	0.206189i	\(0.933894\pi\)
\(23\)	1.15315	+	1.44601i	0.240449	+	0.301513i	0.887383	−	0.461032i	\(-0.152521\pi\)
							−0.646935	+	0.762546i	\(0.723949\pi\)
\(29\)	−3.05029	−	7.77201i	−0.566424	−	1.44323i	−0.871261	−	0.490821i	\(-0.836697\pi\)
							0.304836	−	0.952405i	\(-0.401398\pi\)
\(31\)	3.82249	−	6.62074i	0.686539	−	1.18912i	−0.286412	−	0.958107i	\(-0.592462\pi\)
							0.972951	−	0.231014i	\(-0.0742042\pi\)
\(37\)	−1.11492	−	2.84076i	−0.183291	−	0.467018i	0.809464	−	0.587169i	\(-0.199758\pi\)
							−0.992756	+	0.120150i	\(0.961662\pi\)
\(41\)	−1.79590	+	1.66635i	−0.280473	+	0.260241i	−0.807847	−	0.589392i	\(-0.799367\pi\)
							0.527374	+	0.849633i	\(0.323177\pi\)
\(43\)	8.01694	+	7.43863i	1.22257	+	1.13438i	0.986692	+	0.162598i	\(0.0519875\pi\)
							0.235879	+	0.971782i	\(0.424203\pi\)
\(47\)	−0.218989	−	2.92221i	−0.0319429	−	0.426248i	−0.990134	−	0.140121i	\(-0.955251\pi\)
							0.958192	−	0.286127i	\(-0.0923681\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.z.a.331.11	yes	648
9.4	even	3		441.2.y.a.184.44	yes	648
49.4	even	21		441.2.y.a.151.44	✓	648
441.4	even	21	inner	441.2.z.a.4.11	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.y.a.151.44	✓	648	49.4	even	21
441.2.y.a.184.44	yes	648	9.4	even	3
441.2.z.a.4.11	yes	648	441.4	even	21	inner
441.2.z.a.331.11	yes	648	1.1	even	1	trivial