Embedded newform 441.2.bn.a.353.14

• Label: 441.2.bn.a.353.14
• Level: $441$
• Weight: $2$
• Character: 441.353
• 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			353.14
Character	\(\chi\)	\(=\)	441.353
Dual form			441.2.bn.a.5.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.33541 - 1.06496i) q^{2} +(1.50168 + 0.863111i) q^{3} +(0.204153 + 0.894452i) q^{4} +(0.337281 - 0.229955i) q^{5} +(-1.08619 - 2.75183i) q^{6} +(-2.64519 - 0.0547315i) q^{7} +(-0.802272 + 1.66593i) q^{8} +(1.51008 + 2.59223i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 21 q^{2} - 11 q^{3} + 99 q^{4} - 18 q^{5} - 34 q^{6} - 5 q^{7} - 23 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(199\)	\(344\)
\(\chi(n)\)	\(e\left(\frac{13}{42}\right)\)	\(e\left(\frac{1}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.33541	−	1.06496i	−0.944279	−	0.753037i	0.0248247	−	0.999692i	\(-0.492097\pi\)
							−0.969103	+	0.246655i	\(0.920669\pi\)
\(3\)	1.50168	+	0.863111i	0.866995	+	0.498317i
							\(5\)	0.337281	−	0.229955i	0.150837	−	0.102839i	−0.485550	−	0.874209i	\(-0.661381\pi\)
0.636387	+	0.771370i	\(0.280428\pi\)
\(7\)	−2.64519	−	0.0547315i	−0.999786	−	0.0206866i
							\(11\)	−0.812681	+	5.39178i	−0.245032	+	1.62568i	0.440577	+	0.897715i	\(0.354774\pi\)
−0.685610	+	0.727969i	\(0.740464\pi\)
\(13\)	−0.625877	+	4.15242i	−0.173587	+	1.15168i	0.717070	+	0.697001i	\(0.245483\pi\)
							−0.890658	+	0.454675i	\(0.849756\pi\)
\(17\)	−2.55461	−	2.37033i	−0.619585	−	0.574891i	0.306657	−	0.951820i	\(-0.400789\pi\)
							−0.926242	+	0.376929i	\(0.876980\pi\)
\(19\)	5.35673	−	3.09271i	1.22892	−	0.709516i	0.262114	−	0.965037i	\(-0.415580\pi\)
							0.966804	+	0.255521i	\(0.0822470\pi\)
\(23\)	−1.61560	+	5.23765i	−0.336876	+	1.09213i	0.615740	+	0.787949i	\(0.288857\pi\)
							−0.952616	+	0.304176i	\(0.901619\pi\)
\(29\)	−2.64071	+	2.84600i	−0.490367	+	0.528490i	−0.928952	−	0.370200i	\(-0.879289\pi\)
							0.438585	+	0.898690i	\(0.355480\pi\)
\(31\)		−	8.33578i		−	1.49715i	−0.663050	−	0.748575i	\(-0.730738\pi\)
							0.663050	−	0.748575i	\(-0.269262\pi\)
\(37\)	7.68811	−	2.37147i	1.26392	−	0.389867i	0.410867	−	0.911695i	\(-0.365226\pi\)
							0.853050	+	0.521829i	\(0.174750\pi\)
\(41\)	0.165105	+	2.20317i	0.0257851	+	0.344078i	0.995144	+	0.0984293i	\(0.0313818\pi\)
							−0.969359	+	0.245648i	\(0.920999\pi\)
\(43\)	0.0259394	−	0.346138i	0.00395572	−	0.0527855i	−0.994877	−	0.101097i	\(-0.967765\pi\)
							0.998832	+	0.0483112i	\(0.0153839\pi\)
\(47\)	−2.08232	+	2.61115i	−0.303738	+	0.380876i	−0.910153	−	0.414273i	\(-0.864036\pi\)
							0.606414	+	0.795149i	\(0.292607\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.bn.a.353.14	yes	648
9.5	odd	6		441.2.bd.a.59.14	✓	648
49.5	odd	42		441.2.bd.a.299.14	yes	648
441.5	even	42	inner	441.2.bn.a.5.14	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.bd.a.59.14	✓	648	9.5	odd	6
441.2.bd.a.299.14	yes	648	49.5	odd	42
441.2.bn.a.5.14	yes	648	441.5	even	42	inner
441.2.bn.a.353.14	yes	648	1.1	even	1	trivial