Embedded newform 441.2.y.a.25.4

• Label: 441.2.y.a.25.4
• Level: $441$
• Weight: $2$
• Character: 441.25
• 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.y (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			25.4
Character	\(\chi\)	\(=\)	441.25
Dual form			441.2.y.a.247.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.533217 + 2.33617i) q^{2} +(1.50562 + 0.856211i) q^{3} +(-3.37146 - 1.62361i) q^{4} +(0.810456 + 2.06501i) q^{5} +(-2.80308 + 3.06086i) q^{6} +(1.78727 - 1.95082i) q^{7} +(2.60267 - 3.26364i) q^{8} +(1.53381 + 2.57826i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 5 q^{2} - 13 q^{3} - 109 q^{4} - 12 q^{5} + 2 q^{6} - 7 q^{7} - 16 q^{8} - 3 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/441\mathbb{Z}\right)^\times\).

\(n\)	\(199\)	\(344\)
\(\chi(n)\)	\(e\left(\frac{8}{21}\right)\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.533217	+	2.33617i	−0.377041	+	1.65193i	0.329428	+	0.944181i	\(0.393144\pi\)
							−0.706469	+	0.707744i	\(0.749713\pi\)
\(3\)	1.50562	+	0.856211i	0.869272	+	0.494333i
							\(5\)	0.810456	+	2.06501i	0.362447	+	0.923500i	0.989501	+	0.144525i	\(0.0461653\pi\)
−0.627054	+	0.778976i	\(0.715740\pi\)
\(7\)	1.78727	−	1.95082i	0.675523	−	0.737339i
							\(11\)	−3.87240	+	1.19448i	−1.16757	+	0.360148i	−0.817139	−	0.576440i	\(-0.804441\pi\)
−0.350432	+	0.936588i	\(0.613965\pi\)
\(13\)	−0.720250	+	0.222168i	−0.199761	+	0.0616182i	−0.393021	−	0.919529i	\(-0.628570\pi\)
							0.193260	+	0.981148i	\(0.438094\pi\)
\(17\)	−0.536710	+	7.16190i	−0.130171	+	1.73702i	0.424924	+	0.905229i	\(0.360301\pi\)
							−0.555095	+	0.831787i	\(0.687318\pi\)
\(19\)	−0.480533	−	0.832307i	−0.110242	−	0.190944i	0.805626	−	0.592425i	\(-0.201829\pi\)
							−0.915868	+	0.401480i	\(0.868496\pi\)
\(23\)	−0.508812	+	0.346902i	−0.106095	+	0.0723340i	−0.615205	−	0.788367i	\(-0.710927\pi\)
							0.509111	+	0.860701i	\(0.329974\pi\)
\(29\)	0.599840	−	8.00430i	0.111387	−	1.48636i	−0.609210	−	0.793009i	\(-0.708514\pi\)
							0.720598	−	0.693353i	\(-0.243867\pi\)
\(31\)	0.283781			0.0509686			0.0254843	−	0.999675i	\(-0.491887\pi\)
							0.0254843	+	0.999675i	\(0.491887\pi\)
\(37\)	9.54258	+	6.50602i	1.56879	+	1.06958i	0.961391	+	0.275186i	\(0.0887395\pi\)
							0.607399	+	0.794397i	\(0.292213\pi\)
\(41\)	3.34735	+	0.504532i	0.522769	+	0.0787947i	0.405125	−	0.914261i	\(-0.367228\pi\)
							0.117643	+	0.993056i	\(0.462466\pi\)
\(43\)	−5.57380	+	0.840115i	−0.849997	+	0.128116i	−0.559572	−	0.828782i	\(-0.689035\pi\)
							−0.290424	+	0.956898i	\(0.593796\pi\)
\(47\)	0.605043	−	2.65087i	0.0882547	−	0.386669i	−0.911439	−	0.411436i	\(-0.865027\pi\)
							0.999693	+	0.0247672i	\(0.00788445\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.y.a.25.4	✓	648
9.4	even	3		441.2.z.a.319.4	yes	648
49.2	even	21		441.2.z.a.394.4	yes	648
441.247	even	21	inner	441.2.y.a.247.4	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.y.a.25.4	✓	648	1.1	even	1	trivial
441.2.y.a.247.4	yes	648	441.247	even	21	inner
441.2.z.a.319.4	yes	648	9.4	even	3
441.2.z.a.394.4	yes	648	49.2	even	21