Embedded newform 441.2.bn.a.5.18

• Label: 441.2.bn.a.5.18
• Level: $441$
• Weight: $2$
• Character: 441.5
• 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.bn (of order \(42\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.52140272914\)
Analytic rank:	\(0\)
Dimension:	\(648\)
Relative dimension:	\(54\) over \(\Q(\zeta_{42})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{42}]$

Embedding invariants

Embedding label			5.18
Character	\(\chi\)	\(=\)	441.5
Dual form			441.2.bn.a.353.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.920093 + 0.733749i) q^{2} +(-1.68844 + 0.386212i) q^{3} +(-0.136860 + 0.599621i) q^{4} +(1.62477 + 1.10775i) q^{5} +(1.27014 - 1.59424i) q^{6} +(1.62874 - 2.08500i) q^{7} +(-1.33527 - 2.77272i) q^{8} +(2.70168 - 1.30419i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 21 q^{2} - 11 q^{3} + 99 q^{4} - 18 q^{5} - 34 q^{6} - 5 q^{7} - 23 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(199\)	\(344\)
\(\chi(n)\)	\(e\left(\frac{29}{42}\right)\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.920093	+	0.733749i	−0.650604	+	0.518839i	−0.892260	−	0.451523i	\(-0.850881\pi\)
							0.241656	+	0.970362i	\(0.422309\pi\)
\(3\)	−1.68844	+	0.386212i	−0.974823	+	0.222979i
							\(5\)	1.62477	+	1.10775i	0.726620	+	0.495401i	0.869249	−	0.494375i	\(-0.164603\pi\)
−0.142629	+	0.989776i	\(0.545556\pi\)
\(7\)	1.62874	−	2.08500i	0.615604	−	0.788055i
							\(11\)	−0.122952	−	0.815733i	−0.0370714	−	0.245953i	0.962618	−	0.270862i	\(-0.0873086\pi\)
−0.999690	+	0.0249088i	\(0.992070\pi\)
\(13\)	−0.961378	−	6.37833i	−0.266638	−	1.76903i	−0.575383	−	0.817884i	\(-0.695147\pi\)
							0.308745	−	0.951145i	\(-0.400091\pi\)
\(17\)	−0.0185444	+	0.0172067i	−0.00449767	+	0.00417323i	−0.682418	−	0.730962i	\(-0.739072\pi\)
							0.677921	+	0.735135i	\(0.262881\pi\)
\(19\)	4.79472	+	2.76823i	1.09998	+	0.635076i	0.936217	−	0.351423i	\(-0.114302\pi\)
							0.163767	+	0.986499i	\(0.447635\pi\)
\(23\)	−1.23754	−	4.01202i	−0.258046	−	0.836564i	−0.988185	−	0.153266i	\(-0.951021\pi\)
							0.730139	−	0.683299i	\(-0.239455\pi\)
\(29\)	4.79359	+	5.16626i	0.890147	+	0.959350i	0.999399	−	0.0346642i	\(-0.0110362\pi\)
							−0.109252	+	0.994014i	\(0.534846\pi\)
\(31\)		−	8.65610i		−	1.55468i	−0.629079	−	0.777341i	\(-0.716568\pi\)
							0.629079	−	0.777341i	\(-0.283432\pi\)
\(37\)	7.29110	+	2.24901i	1.19865	+	0.369734i	0.828900	−	0.559397i	\(-0.188967\pi\)
							0.369749	+	0.929132i	\(0.379444\pi\)
\(41\)	−0.728162	+	9.71665i	−0.113720	+	1.51749i	0.590230	+	0.807235i	\(0.299037\pi\)
							−0.703949	+	0.710250i	\(0.748582\pi\)
\(43\)	−0.181649	−	2.42394i	−0.0277012	−	0.369647i	−0.993777	−	0.111388i	\(-0.964470\pi\)
							0.966076	−	0.258259i	\(-0.0831488\pi\)
\(47\)	−1.10538	−	1.38611i	−0.161237	−	0.202184i	0.694650	−	0.719348i	\(-0.255559\pi\)
							−0.855887	+	0.517164i	\(0.826988\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.bn.a.5.18	yes	648
9.2	odd	6		441.2.bd.a.299.18	yes	648
49.10	odd	42		441.2.bd.a.59.18	✓	648
441.353	even	42	inner	441.2.bn.a.353.18	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.bd.a.59.18	✓	648	49.10	odd	42
441.2.bd.a.299.18	yes	648	9.2	odd	6
441.2.bn.a.5.18	yes	648	1.1	even	1	trivial
441.2.bn.a.353.18	yes	648	441.353	even	42	inner