Embedded newform 441.2.bd.a.248.37

• Label: 441.2.bd.a.248.37
• Level: $441$
• Weight: $2$
• Character: 441.248
• 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			248.37
Character	\(\chi\)	\(=\)	441.248
Dual form			441.2.bd.a.425.37

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.191689 + 1.27177i) q^{2} +(-0.850569 - 1.50882i) q^{3} +(0.330488 - 0.101942i) q^{4} +(3.27747 - 1.57835i) q^{5} +(1.75583 - 1.37095i) q^{6} +(-2.62368 - 0.341068i) q^{7} +(1.30907 + 2.71830i) q^{8} +(-1.55306 + 2.56671i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 18 q^{2} - 11 q^{3} - 60 q^{4} - 15 q^{5} - 22 q^{6} - 5 q^{7} + 13 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(199\)	\(344\)
\(\chi(n)\)	\(e\left(\frac{1}{42}\right)\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.191689	+	1.27177i	0.135544	+	0.899278i	0.948460	+	0.316897i	\(0.102641\pi\)
							−0.812916	+	0.582381i	\(0.802121\pi\)
\(3\)	−0.850569	−	1.50882i	−0.491076	−	0.871116i
							\(5\)	3.27747	−	1.57835i	1.46573	−	0.705859i	0.480485	−	0.877003i	\(-0.340461\pi\)
0.985246	+	0.171145i	\(0.0547465\pi\)
\(7\)	−2.62368	−	0.341068i	−0.991656	−	0.128912i
							\(11\)	3.32774	−	2.65379i	1.00335	−	0.800147i	0.0234705	−	0.999725i	\(-0.492528\pi\)
0.979882	+	0.199578i	\(0.0639570\pi\)
\(13\)	0.102425	+	0.679544i	0.0284075	+	0.188472i	0.998534	−	0.0541281i	\(-0.0172379\pi\)
							−0.970126	+	0.242600i	\(0.922000\pi\)
\(17\)	−2.80364	−	0.864808i	−0.679982	−	0.209747i	−0.0645237	−	0.997916i	\(-0.520553\pi\)
							−0.615458	+	0.788170i	\(0.711029\pi\)
\(19\)	5.82672	−	3.36406i	1.33674	−	0.771768i	0.350418	−	0.936593i	\(-0.386040\pi\)
							0.986323	+	0.164826i	\(0.0527062\pi\)
\(23\)	−4.37893	−	0.999463i	−0.913071	−	0.208403i	−0.259920	−	0.965630i	\(-0.583696\pi\)
							−0.653151	+	0.757228i	\(0.726553\pi\)
\(29\)	−2.07655	−	2.23799i	−0.385607	−	0.415585i	0.510181	−	0.860067i	\(-0.329578\pi\)
							−0.895788	+	0.444482i	\(0.853388\pi\)
\(31\)	6.92768	−	3.99970i	1.24425	−	0.718367i	0.274292	−	0.961647i	\(-0.411557\pi\)
							0.969956	+	0.243280i	\(0.0782233\pi\)
\(37\)	−5.70667	+	5.29502i	−0.938171	+	0.870495i	−0.991765	−	0.128068i	\(-0.959123\pi\)
							0.0535948	+	0.998563i	\(0.482932\pi\)
\(41\)	−0.642076	+	8.56791i	−0.100275	+	1.33808i	0.689321	+	0.724456i	\(0.257909\pi\)
							−0.789596	+	0.613627i	\(0.789710\pi\)
\(43\)	0.157605	+	2.10309i	0.0240345	+	0.320718i	0.996243	+	0.0866015i	\(0.0276007\pi\)
							−0.972209	+	0.234117i	\(0.924780\pi\)
\(47\)	−1.96142	+	0.295637i	−0.286103	+	0.0431231i	−0.290526	−	0.956867i	\(-0.593830\pi\)
							0.00442302	+	0.999990i	\(0.498592\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.bd.a.248.37	✓	648
9.2	odd	6		441.2.bn.a.101.18	yes	648
49.33	odd	42		441.2.bn.a.131.18	yes	648
441.425	even	42	inner	441.2.bd.a.425.37	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.bd.a.248.37	✓	648	1.1	even	1	trivial
441.2.bd.a.425.37	yes	648	441.425	even	42	inner
441.2.bn.a.101.18	yes	648	9.2	odd	6
441.2.bn.a.131.18	yes	648	49.33	odd	42