Embedded newform 441.2.bn.a.5.7

• Label: 441.2.bn.a.5.7
• 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			5.7
Character	\(\chi\)	\(=\)	441.5
Dual form			441.2.bn.a.353.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.70698 + 1.36127i) q^{2} +(0.921293 - 1.46670i) q^{3} +(0.615683 - 2.69748i) q^{4} +(-1.79313 - 1.22254i) q^{5} +(0.423952 + 3.75777i) q^{6} +(2.16539 + 1.52022i) q^{7} +(0.726444 + 1.50848i) q^{8} +(-1.30244 - 2.70253i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 21 q^{2} - 11 q^{3} + 99 q^{4} - 18 q^{5} - 34 q^{6} - 5 q^{7} - 23 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.70698	+	1.36127i	−1.20702	+	0.962565i	−0.999876	−	0.0157452i	\(-0.994988\pi\)
							−0.207143	+	0.978311i	\(0.566416\pi\)
\(3\)	0.921293	−	1.46670i	0.531909	−	0.846802i
							\(5\)	−1.79313	−	1.22254i	−0.801913	−	0.546735i	0.0915877	−	0.995797i	\(-0.470806\pi\)
−0.893501	+	0.449062i	\(0.851758\pi\)
\(7\)	2.16539	+	1.52022i	0.818441	+	0.574591i
							\(11\)	0.0434937	+	0.288562i	0.0131138	+	0.0870046i	0.994430	−	0.105396i	\(-0.0336109\pi\)
−0.981317	+	0.192400i	\(0.938373\pi\)
\(13\)	0.0974010	+	0.646214i	0.0270142	+	0.179227i	0.998269	−	0.0588153i	\(-0.0187323\pi\)
							−0.971255	+	0.238043i	\(0.923494\pi\)
\(17\)	1.81171	−	1.68102i	0.439403	−	0.407707i	−0.429204	−	0.903208i	\(-0.641206\pi\)
							0.868608	+	0.495501i	\(0.165015\pi\)
\(19\)	−2.44871	−	1.41376i	−0.561772	−	0.324339i	0.192084	−	0.981378i	\(-0.438475\pi\)
							−0.753857	+	0.657039i	\(0.771809\pi\)
\(23\)	−1.98007	−	6.41923i	−0.412873	−	1.33850i	−0.888509	−	0.458859i	\(-0.848258\pi\)
							0.475636	−	0.879642i	\(-0.342218\pi\)
\(29\)	−2.76486	−	2.97981i	−0.513422	−	0.553337i	0.422126	−	0.906537i	\(-0.361284\pi\)
							−0.935548	+	0.353200i	\(0.885094\pi\)
\(31\)		−	6.20148i		−	1.11382i	−0.830573	−	0.556910i	\(-0.811987\pi\)
							0.830573	−	0.556910i	\(-0.188013\pi\)
\(37\)	−1.99374	−	0.614988i	−0.327769	−	0.101103i	0.126503	−	0.991966i	\(-0.459625\pi\)
							−0.454273	+	0.890863i	\(0.650101\pi\)
\(41\)	0.108151	−	1.44317i	0.0168903	−	0.225385i	−0.982371	−	0.186944i	\(-0.940142\pi\)
							0.999261	−	0.0384413i	\(-0.0122393\pi\)
\(43\)	−0.767123	−	10.2365i	−0.116985	−	1.56106i	−0.679129	−	0.734019i	\(-0.737642\pi\)
							0.562144	−	0.827039i	\(-0.309977\pi\)
\(47\)	2.84186	+	3.56358i	0.414528	+	0.519802i	0.944632	−	0.328131i	\(-0.106419\pi\)
							−0.530104	+	0.847932i	\(0.677847\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.bn.a.5.7	yes	648
9.2	odd	6		441.2.bd.a.299.7	yes	648
49.10	odd	42		441.2.bd.a.59.7	✓	648
441.353	even	42	inner	441.2.bn.a.353.7	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.bd.a.59.7	✓	648	49.10	odd	42
441.2.bd.a.299.7	yes	648	9.2	odd	6
441.2.bn.a.5.7	yes	648	1.1	even	1	trivial
441.2.bn.a.353.7	yes	648	441.353	even	42	inner