Embedded newform 441.2.bn.a.5.1

• Label: 441.2.bn.a.5.1
• 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			5.1
Character	\(\chi\)	\(=\)	441.5
Dual form			441.2.bn.a.353.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.16353 + 1.72536i) q^{2} +(0.581018 - 1.63169i) q^{3} +(1.25896 - 5.51585i) q^{4} +(2.91211 + 1.98544i) q^{5} +(1.55820 + 4.53267i) q^{6} +(-2.07232 + 1.64484i) q^{7} +(4.39168 + 9.11943i) q^{8} +(-2.32484 - 1.89608i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 21 q^{2} - 11 q^{3} + 99 q^{4} - 18 q^{5} - 34 q^{6} - 5 q^{7} - 23 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.16353	+	1.72536i	−1.52984	+	1.22001i	−0.635953	+	0.771728i	\(0.719393\pi\)
							−0.893892	+	0.448283i	\(0.852036\pi\)
\(3\)	0.581018	−	1.63169i	0.335451	−	0.942058i
							\(5\)	2.91211	+	1.98544i	1.30234	+	0.887917i	0.997910	−	0.0646186i	\(-0.0205831\pi\)
0.304426	+	0.952536i	\(0.401535\pi\)
\(7\)	−2.07232	+	1.64484i	−0.783263	+	0.621691i
							\(11\)	0.128415	+	0.851980i	0.0387187	+	0.256882i	0.999813	−	0.0193387i	\(-0.00615609\pi\)
−0.961094	+	0.276220i	\(0.910918\pi\)
\(13\)	0.596855	+	3.95988i	0.165538	+	1.09827i	0.905105	+	0.425189i	\(0.139792\pi\)
							−0.739567	+	0.673083i	\(0.764970\pi\)
\(17\)	−1.51858	+	1.40904i	−0.368311	+	0.341742i	−0.842515	−	0.538672i	\(-0.818926\pi\)
							0.474205	+	0.880415i	\(0.342736\pi\)
\(19\)	1.72443	+	0.995602i	0.395612	+	0.228407i	0.684589	−	0.728929i	\(-0.259982\pi\)
							−0.288977	+	0.957336i	\(0.593315\pi\)
\(23\)	0.379891	+	1.23158i	0.0792127	+	0.256801i	0.986513	−	0.163681i	\(-0.0523369\pi\)
							−0.907301	+	0.420483i	\(0.861861\pi\)
\(29\)	6.03902	+	6.50852i	1.12142	+	1.20860i	0.975662	+	0.219281i	\(0.0703712\pi\)
							0.145757	+	0.989320i	\(0.453438\pi\)
\(31\)		−	2.77807i		−	0.498956i	−0.968380	−	0.249478i	\(-0.919741\pi\)
							0.968380	−	0.249478i	\(-0.0802590\pi\)
\(37\)	1.25748	+	0.387883i	0.206729	+	0.0637675i	0.396390	−	0.918082i	\(-0.370263\pi\)
							−0.189661	+	0.981850i	\(0.560739\pi\)
\(41\)	0.381381	−	5.08918i	0.0595618	−	0.794797i	−0.884817	−	0.465938i	\(-0.845717\pi\)
							0.944379	−	0.328859i	\(-0.106664\pi\)
\(43\)	−0.323102	−	4.31150i	−0.0492726	−	0.657497i	−0.966093	−	0.258196i	\(-0.916872\pi\)
							0.916820	−	0.399301i	\(-0.130747\pi\)
\(47\)	2.93124	+	3.67566i	0.427566	+	0.536150i	0.948219	−	0.317618i	\(-0.102883\pi\)
							−0.520653	+	0.853768i	\(0.674311\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.bn.a.5.1	yes	648
9.2	odd	6		441.2.bd.a.299.1	yes	648
49.10	odd	42		441.2.bd.a.59.1	✓	648
441.353	even	42	inner	441.2.bn.a.353.1	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.bd.a.59.1	✓	648	49.10	odd	42
441.2.bd.a.299.1	yes	648	9.2	odd	6
441.2.bn.a.5.1	yes	648	1.1	even	1	trivial
441.2.bn.a.353.1	yes	648	441.353	even	42	inner