Embedded newform 441.2.bd.a.122.10

• Label: 441.2.bd.a.122.10
• Level: $441$
• Weight: $2$
• Character: 441.122
• 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.bd (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			122.10
Character	\(\chi\)	\(=\)	441.122
Dual form			441.2.bd.a.47.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.29221 + 1.39267i) q^{2} +(-1.37046 + 1.05916i) q^{3} +(-0.120265 - 1.60482i) q^{4} +(0.149465 - 0.187424i) q^{5} +(0.295858 - 3.27726i) q^{6} +(2.64547 + 0.0386573i) q^{7} +(-0.580288 - 0.462764i) q^{8} +(0.756341 - 2.90309i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 18 q^{2} - 11 q^{3} - 60 q^{4} - 15 q^{5} - 22 q^{6} - 5 q^{7} + 13 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(199\)	\(344\)
\(\chi(n)\)	\(e\left(\frac{37}{42}\right)\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.29221	+	1.39267i	−0.913728	+	0.984765i	−0.999934	−	0.0115124i	\(-0.996335\pi\)
							0.0862053	+	0.996277i	\(0.472526\pi\)
\(3\)	−1.37046	+	1.05916i	−0.791238	+	0.611509i
							\(5\)	0.149465	−	0.187424i	0.0668429	−	0.0838184i	−0.747286	−	0.664503i	\(-0.768643\pi\)
0.814129	+	0.580684i	\(0.197215\pi\)
\(7\)	2.64547	+	0.0386573i	0.999893	+	0.0146111i
							\(11\)	1.56749	+	0.357768i	0.472615	+	0.107871i	0.452193	−	0.891920i	\(-0.350642\pi\)
0.0204218	+	0.999791i	\(0.493499\pi\)
\(13\)	1.25783	−	1.35562i	0.348860	−	0.375982i	−0.534107	−	0.845417i	\(-0.679352\pi\)
							0.882967	+	0.469435i	\(0.155542\pi\)
\(17\)	0.408731	−	5.45414i	0.0991318	−	1.32282i	−0.696725	−	0.717338i	\(-0.745360\pi\)
							0.795857	−	0.605484i	\(-0.207021\pi\)
\(19\)	5.94328	−	3.43136i	1.36348	−	0.787207i	0.373397	−	0.927672i	\(-0.378193\pi\)
							0.990086	+	0.140465i	\(0.0448596\pi\)
\(23\)	2.21068	−	4.59052i	0.460958	−	0.957189i	−0.532864	−	0.846201i	\(-0.678884\pi\)
							0.993822	−	0.110988i	\(-0.0354015\pi\)
\(29\)	−4.55891	+	6.68669i	−0.846568	+	1.24169i	0.121898	+	0.992543i	\(0.461102\pi\)
							−0.968466	+	0.249144i	\(0.919851\pi\)
\(31\)	−4.61998	+	2.66735i	−0.829774	+	0.479070i	−0.853775	−	0.520642i	\(-0.825693\pi\)
							0.0240016	+	0.999712i	\(0.492359\pi\)
\(37\)	−5.46391	−	3.72523i	−0.898261	−	0.612424i	0.0236948	−	0.999719i	\(-0.492457\pi\)
							−0.921956	+	0.387295i	\(0.873409\pi\)
\(41\)	1.82330	+	4.64570i	0.284752	+	0.725536i	0.999654	+	0.0263114i	\(0.00837614\pi\)
							−0.714902	+	0.699225i	\(0.753529\pi\)
\(43\)	3.52974	−	8.99363i	0.538280	−	1.37152i	−0.360354	−	0.932816i	\(-0.617344\pi\)
							0.898634	−	0.438699i	\(-0.144561\pi\)
\(47\)	−3.59668	−	3.33723i	−0.524630	−	0.486785i	0.372819	−	0.927904i	\(-0.378391\pi\)
							−0.897449	+	0.441119i	\(0.854582\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.bd.a.122.10	yes	648
9.2	odd	6		441.2.bn.a.416.10	yes	648
49.47	odd	42		441.2.bn.a.194.10	yes	648
441.47	even	42	inner	441.2.bd.a.47.10	✓	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.bd.a.47.10	✓	648	441.47	even	42	inner
441.2.bd.a.122.10	yes	648	1.1	even	1	trivial
441.2.bn.a.194.10	yes	648	49.47	odd	42
441.2.bn.a.416.10	yes	648	9.2	odd	6