Embedded newform 441.2.bn.a.320.13

• Label: 441.2.bn.a.320.13
• Level: $441$
• Weight: $2$
• Character: 441.320
• 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			320.13
Character	\(\chi\)	\(=\)	441.320
Dual form			441.2.bn.a.164.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.714955 - 1.48462i) q^{2} +(0.502561 - 1.65754i) q^{3} +(-0.445953 + 0.559208i) q^{4} +(3.74883 - 1.15636i) q^{5} +(-2.82012 + 0.438953i) q^{6} +(0.974463 + 2.45976i) q^{7} +(-2.06393 - 0.471079i) q^{8} +(-2.49487 - 1.66603i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 21 q^{2} - 11 q^{3} + 99 q^{4} - 18 q^{5} - 34 q^{6} - 5 q^{7} - 23 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(199\)	\(344\)
\(\chi(n)\)	\(e\left(\frac{17}{42}\right)\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.714955	−	1.48462i	−0.505549	−	1.04978i	−0.985054	−	0.172245i	\(-0.944898\pi\)
							0.479505	−	0.877539i	\(-0.340816\pi\)
\(3\)	0.502561	−	1.65754i	0.290154	−	0.956980i
							\(5\)	3.74883	−	1.15636i	1.67653	−	0.517140i	0.696764	−	0.717301i	\(-0.254623\pi\)
0.979763	+	0.200161i	\(0.0641465\pi\)
\(7\)	0.974463	+	2.45976i	0.368312	+	0.929702i
							\(11\)	0.613086	+	0.0459444i	0.184852	+	0.0138528i	0.166835	−	0.985985i	\(-0.446645\pi\)
0.0180178	+	0.999838i	\(0.494264\pi\)
\(13\)	−0.250153	−	0.0187464i	−0.0693800	−	0.00519931i	0.0399949	−	0.999200i	\(-0.487266\pi\)
							−0.109375	+	0.994001i	\(0.534885\pi\)
\(17\)	2.15951	−	5.50235i	0.523759	−	1.33452i	−0.387191	−	0.921999i	\(-0.626555\pi\)
							0.910950	−	0.412516i	\(-0.135350\pi\)
\(19\)	−4.00625	−	2.31301i	−0.919097	−	0.530641i	−0.0357503	−	0.999361i	\(-0.511382\pi\)
							−0.883347	+	0.468720i	\(0.844715\pi\)
\(23\)	−0.747730	+	4.96087i	−0.155913	+	1.03441i	0.764827	+	0.644235i	\(0.222824\pi\)
							−0.920740	+	0.390177i	\(0.872414\pi\)
\(29\)	4.93871	+	1.93830i	0.917096	+	0.359934i	0.776488	−	0.630132i	\(-0.216999\pi\)
							0.140608	+	0.990065i	\(0.455094\pi\)
\(31\)			1.59878i			0.287150i	0.989639	+	0.143575i	\(0.0458599\pi\)
							−0.989639	+	0.143575i	\(0.954140\pi\)
\(37\)	−5.30653	+	0.799831i	−0.872388	+	0.131491i	−0.569952	−	0.821678i	\(-0.693038\pi\)
							−0.302436	+	0.953170i	\(0.597800\pi\)
\(41\)	−3.87577	−	3.59619i	−0.605293	−	0.561630i	0.316852	−	0.948475i	\(-0.397374\pi\)
							−0.922145	+	0.386845i	\(0.873565\pi\)
\(43\)	−4.22389	+	3.91920i	−0.644137	+	0.597672i	−0.933078	−	0.359675i	\(-0.882887\pi\)
							0.288941	+	0.957347i	\(0.406697\pi\)
\(47\)	9.83900	−	4.73822i	1.43517	−	0.691140i	0.455216	−	0.890381i	\(-0.349562\pi\)
							0.979950	+	0.199241i	\(0.0638478\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.bn.a.320.13	yes	648
9.2	odd	6		441.2.bd.a.173.42	✓	648
49.17	odd	42		441.2.bd.a.311.42	yes	648
441.164	even	42	inner	441.2.bn.a.164.13	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.bd.a.173.42	✓	648	9.2	odd	6
441.2.bd.a.311.42	yes	648	49.17	odd	42
441.2.bn.a.164.13	yes	648	441.164	even	42	inner
441.2.bn.a.320.13	yes	648	1.1	even	1	trivial