Embedded newform 441.2.bd.a.59.2

• Label: 441.2.bd.a.59.2
• Level: $441$
• Weight: $2$
• Character: 441.59
• 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			59.2
Character	\(\chi\)	\(=\)	441.59
Dual form			441.2.bd.a.299.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.56687 + 1.00742i) q^{2} +(1.42091 - 0.990465i) q^{3} +(4.10783 - 3.81151i) q^{4} +(-1.27532 - 0.614163i) q^{5} +(-2.64947 + 3.97385i) q^{6} +(-2.60240 + 0.477010i) q^{7} +(-4.31162 + 8.95316i) q^{8} +(1.03796 - 2.81472i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 18 q^{2} - 11 q^{3} - 60 q^{4} - 15 q^{5} - 22 q^{6} - 5 q^{7} + 13 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(199\)	\(344\)
\(\chi(n)\)	\(e\left(\frac{13}{42}\right)\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.56687	+	1.00742i	−1.81505	+	0.712356i	−0.826216	+	0.563353i	\(0.809511\pi\)
							−0.988837	+	0.149003i	\(0.952394\pi\)
\(3\)	1.42091	−	0.990465i	0.820362	−	0.571845i
							\(5\)	−1.27532	−	0.614163i	−0.570342	−	0.274662i	0.126406	−	0.991979i	\(-0.459656\pi\)
−0.696747	+	0.717317i	\(0.745370\pi\)
\(7\)	−2.60240	+	0.477010i	−0.983613	+	0.180293i
							\(11\)	−0.259260	−	0.206753i	−0.0781699	−	0.0623384i	0.583625	−	0.812023i	\(-0.301634\pi\)
−0.661795	+	0.749685i	\(0.730205\pi\)
\(13\)	0.427864	−	0.167924i	0.118668	−	0.0465738i	−0.305264	−	0.952268i	\(-0.598745\pi\)
							0.423932	+	0.905694i	\(0.360649\pi\)
\(17\)	−1.77059	−	1.64287i	−0.429432	−	0.398455i	0.435625	−	0.900128i	\(-0.356527\pi\)
							−0.865057	+	0.501674i	\(0.832718\pi\)
\(19\)	4.18608	−	2.41683i	0.960352	−	0.554460i	0.0640709	−	0.997945i	\(-0.479592\pi\)
							0.896281	+	0.443486i	\(0.146258\pi\)
\(23\)	−6.69104	+	1.52719i	−1.39518	+	0.318440i	−0.853038	−	0.521848i	\(-0.825243\pi\)
							−0.542139	+	0.840289i	\(0.682385\pi\)
\(29\)	0.620247	+	2.01079i	0.115177	+	0.373395i	0.994746	−	0.102377i	\(-0.0326449\pi\)
							−0.879569	+	0.475772i	\(0.842169\pi\)
\(31\)	−7.17604	+	4.14309i	−1.28885	+	0.744121i	−0.978450	−	0.206484i	\(-0.933798\pi\)
							−0.310405	+	0.950605i	\(0.600465\pi\)
\(37\)	−8.11144	+	2.50205i	−1.33351	+	0.411334i	−0.877900	−	0.478844i	\(-0.841056\pi\)
							−0.455613	+	0.890178i	\(0.650580\pi\)
\(41\)	1.10936	−	0.756346i	0.173252	−	0.118121i	−0.473589	−	0.880746i	\(-0.657042\pi\)
							0.646842	+	0.762624i	\(0.276089\pi\)
\(43\)	−5.18324	−	3.53387i	−0.790436	−	0.538910i	0.0994772	−	0.995040i	\(-0.468283\pi\)
							−0.889913	+	0.456130i	\(0.849235\pi\)
\(47\)	−3.16950	−	8.07576i	−0.462320	−	1.17797i	−0.952008	−	0.306073i	\(-0.900985\pi\)
							0.489688	−	0.871898i	\(-0.337111\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.bd.a.59.2	✓	648
9.2	odd	6		441.2.bn.a.353.2	yes	648
49.5	odd	42		441.2.bn.a.5.2	yes	648
441.299	even	42	inner	441.2.bd.a.299.2	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.bd.a.59.2	✓	648	1.1	even	1	trivial
441.2.bd.a.299.2	yes	648	441.299	even	42	inner
441.2.bn.a.5.2	yes	648	49.5	odd	42
441.2.bn.a.353.2	yes	648	9.2	odd	6