Embedded newform 441.2.y.a.88.38

• Label: 441.2.y.a.88.38
• Level: $441$
• Weight: $2$
• Character: 441.88
• 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.y (of order \(21\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.52140272914\)
Analytic rank:	\(0\)
Dimension:	\(648\)
Relative dimension:	\(54\) over \(\Q(\zeta_{21})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{21}]$

Embedding invariants

Embedding label			88.38
Character	\(\chi\)	\(=\)	441.88
Dual form			441.2.y.a.436.38

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.774857 - 0.971640i) q^{2} +(0.403927 - 1.68429i) q^{3} +(0.101361 + 0.444092i) q^{4} +(0.508183 - 0.346474i) q^{5} +(-1.32354 - 1.69756i) q^{6} +(2.64043 - 0.167778i) q^{7} +(2.74944 + 1.32406i) q^{8} +(-2.67369 - 1.36066i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 5 q^{2} - 13 q^{3} - 109 q^{4} - 12 q^{5} + 2 q^{6} - 7 q^{7} - 16 q^{8} - 3 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(199\)	\(344\)
\(\chi(n)\)	\(e\left(\frac{17}{21}\right)\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.774857	−	0.971640i	0.547907	−	0.687053i	−0.428364	−	0.903606i	\(-0.640910\pi\)
							0.976271	+	0.216553i	\(0.0694814\pi\)
\(3\)	0.403927	−	1.68429i	0.233207	−	0.972427i
							\(5\)	0.508183	−	0.346474i	0.227267	−	0.154948i	−0.444334	−	0.895861i	\(-0.646560\pi\)
0.671600	+	0.740914i	\(0.265607\pi\)
\(7\)	2.64043	−	0.167778i	0.997987	−	0.0634140i
							\(11\)	−0.621135	−	0.0936210i	−0.187279	−	0.0282278i	0.0547329	−	0.998501i	\(-0.482569\pi\)
−0.242012	+	0.970273i	\(0.577807\pi\)
\(13\)	1.83066	+	0.275928i	0.507734	+	0.0765286i	0.397914	−	0.917423i	\(-0.369734\pi\)
							0.109820	+	0.993951i	\(0.464972\pi\)
\(17\)	−0.396881	−	0.368251i	−0.0962577	−	0.0893141i	0.630576	−	0.776127i	\(-0.282819\pi\)
							−0.726834	+	0.686813i	\(0.759009\pi\)
\(19\)	−0.957228	−	1.65797i	−0.219603	−	0.380364i	0.735083	−	0.677977i	\(-0.237143\pi\)
							−0.954687	+	0.297613i	\(0.903810\pi\)
\(23\)	−5.98466	−	1.84602i	−1.24789	−	0.384922i	−0.400697	−	0.916211i	\(-0.631232\pi\)
							−0.847191	+	0.531288i	\(0.821708\pi\)
\(29\)	−2.61206	−	2.42364i	−0.485048	−	0.450059i	0.399346	−	0.916800i	\(-0.369237\pi\)
							−0.884394	+	0.466742i	\(0.845428\pi\)
\(31\)	5.31250			0.954153			0.477076	−	0.878862i	\(-0.341696\pi\)
							0.477076	+	0.878862i	\(0.341696\pi\)
\(37\)	−4.58247	+	1.41351i	−0.753354	+	0.232379i	−0.647566	−	0.762010i	\(-0.724213\pi\)
							−0.105788	+	0.994389i	\(0.533737\pi\)
\(41\)	−0.0555415	−	0.741149i	−0.00867412	−	0.115748i	0.991186	−	0.132478i	\(-0.0422934\pi\)
							−0.999860	+	0.0167301i	\(0.994674\pi\)
\(43\)	−0.181404	+	2.42066i	−0.0276638	+	0.369148i	0.966142	+	0.258013i	\(0.0830676\pi\)
							−0.993805	+	0.111135i	\(0.964551\pi\)
\(47\)	−0.554848	+	0.695757i	−0.0809329	+	0.101487i	−0.820649	−	0.571432i	\(-0.806388\pi\)
							0.739717	+	0.672919i	\(0.234960\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.y.a.88.38	✓	648
9.4	even	3		441.2.z.a.382.38	yes	648
49.44	even	21		441.2.z.a.142.38	yes	648
441.436	even	21	inner	441.2.y.a.436.38	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.y.a.88.38	✓	648	1.1	even	1	trivial
441.2.y.a.436.38	yes	648	441.436	even	21	inner
441.2.z.a.142.38	yes	648	49.44	even	21
441.2.z.a.382.38	yes	648	9.4	even	3