Embedded newform 441.2.bn.a.101.28

• Label: 441.2.bn.a.101.28
• Level: $441$
• Weight: $2$
• Character: 441.101
• 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			101.28
Character	\(\chi\)	\(=\)	441.101
Dual form			441.2.bn.a.131.28

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.117772 - 0.0939203i) q^{2} +(1.43809 + 0.965344i) q^{3} +(-0.439993 + 1.92773i) q^{4} +(-0.0359157 + 0.479261i) q^{5} +(0.260033 - 0.0213753i) q^{6} +(2.50465 - 0.852469i) q^{7} +(0.259952 + 0.539796i) q^{8} +(1.13622 + 2.77651i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 21 q^{2} - 11 q^{3} + 99 q^{4} - 18 q^{5} - 34 q^{6} - 5 q^{7} - 23 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.117772	−	0.0939203i	0.0832776	−	0.0664117i	−0.580966	−	0.813928i	\(-0.697325\pi\)
							0.664244	+	0.747516i	\(0.268754\pi\)
\(3\)	1.43809	+	0.965344i	0.830283	+	0.557342i
							\(5\)	−0.0359157	+	0.479261i	−0.0160620	+	0.214332i	0.983398	+	0.181463i	\(0.0580834\pi\)
−0.999460	+	0.0328687i	\(0.989536\pi\)
\(7\)	2.50465	−	0.852469i	0.946671	−	0.322203i
							\(11\)	0.562715	+	0.220849i	0.169665	+	0.0665886i	0.448653	−	0.893706i	\(-0.351904\pi\)
−0.278988	+	0.960294i	\(0.589999\pi\)
\(13\)	−1.45022	−	0.569169i	−0.402218	−	0.157859i	0.155604	−	0.987820i	\(-0.450268\pi\)
							−0.557822	+	0.829961i	\(0.688363\pi\)
\(17\)	−2.66621	−	0.822416i	−0.646650	−	0.199465i	−0.0459552	−	0.998944i	\(-0.514633\pi\)
							−0.600695	+	0.799478i	\(0.705109\pi\)
\(19\)	4.20074	−	2.42530i	0.963716	−	0.556402i	0.0664011	−	0.997793i	\(-0.478848\pi\)
							0.897315	+	0.441391i	\(0.145515\pi\)
\(23\)	−5.94720	+	6.40956i	−1.24008	+	1.33649i	−0.318680	+	0.947862i	\(0.603240\pi\)
							−0.921397	+	0.388623i	\(0.872951\pi\)
\(29\)	1.63599	−	5.30375i	0.303796	−	0.984882i	−0.667017	−	0.745043i	\(-0.732429\pi\)
							0.970813	−	0.239840i	\(-0.0770948\pi\)
\(31\)		−	1.32601i		−	0.238159i	−0.992885	−	0.119080i	\(-0.962006\pi\)
							0.992885	−	0.119080i	\(-0.0379944\pi\)
\(37\)	−3.16513	+	2.93681i	−0.520344	+	0.482809i	−0.896064	−	0.443924i	\(-0.853586\pi\)
							0.375720	+	0.926733i	\(0.377395\pi\)
\(41\)	−2.94495	−	2.00784i	−0.459925	−	0.313571i	0.311099	−	0.950377i	\(-0.399303\pi\)
							−0.771024	+	0.636806i	\(0.780255\pi\)
\(43\)	2.44885	−	1.66960i	0.373446	−	0.254611i	−0.362024	−	0.932169i	\(-0.617914\pi\)
							0.735470	+	0.677558i	\(0.236962\pi\)
\(47\)	0.0590258	+	0.0740160i	0.00860980	+	0.0107963i	0.786118	−	0.618077i	\(-0.212088\pi\)
							−0.777508	+	0.628873i	\(0.783516\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.bn.a.101.28	yes	648
9.5	odd	6		441.2.bd.a.248.27	✓	648
49.33	odd	42		441.2.bd.a.425.27	yes	648
441.131	even	42	inner	441.2.bn.a.131.28	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.bd.a.248.27	✓	648	9.5	odd	6
441.2.bd.a.425.27	yes	648	49.33	odd	42
441.2.bn.a.101.28	yes	648	1.1	even	1	trivial
441.2.bn.a.131.28	yes	648	441.131	even	42	inner