Embedded newform 441.2.y.a.121.36

• Label: 441.2.y.a.121.36
• Level: $441$
• Weight: $2$
• Character: 441.121
• 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			121.36
Character	\(\chi\)	\(=\)	441.121
Dual form			441.2.y.a.277.36

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.893702 - 0.430384i) q^{2} +(0.804260 + 1.53400i) q^{3} +(-0.633506 + 0.794392i) q^{4} +(-1.72952 + 0.533487i) q^{5} +(1.37898 + 1.02480i) q^{6} +(-0.957626 - 2.46636i) q^{7} +(-0.665725 + 2.91673i) q^{8} +(-1.70633 + 2.46748i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 5 q^{2} - 13 q^{3} - 109 q^{4} - 12 q^{5} + 2 q^{6} - 7 q^{7} - 16 q^{8} - 3 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(199\)	\(344\)
\(\chi(n)\)	\(e\left(\frac{19}{21}\right)\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.893702	−	0.430384i	0.631943	−	0.304328i	−0.0903550	−	0.995910i	\(-0.528800\pi\)
							0.722298	+	0.691582i	\(0.243086\pi\)
\(3\)	0.804260	+	1.53400i	0.464340	+	0.885657i
							\(5\)	−1.72952	+	0.533487i	−0.773466	+	0.238583i	−0.656270	−	0.754526i	\(-0.727867\pi\)
−0.117196	+	0.993109i	\(0.537391\pi\)
\(7\)	−0.957626	−	2.46636i	−0.361948	−	0.932198i
							\(11\)	−0.0695797	+	0.928476i	−0.0209791	+	0.279946i	0.976842	+	0.213963i	\(0.0686373\pi\)
−0.997821	+	0.0659828i	\(0.978982\pi\)
\(13\)	−0.415671	+	5.54674i	−0.115286	+	1.53839i	0.576986	+	0.816754i	\(0.304229\pi\)
							−0.692273	+	0.721636i	\(0.743390\pi\)
\(17\)	−1.33842	+	3.41025i	−0.324615	+	0.827106i	0.671659	+	0.740861i	\(0.265582\pi\)
							−0.996274	+	0.0862455i	\(0.972513\pi\)
\(19\)	2.79980	−	4.84939i	0.642317	−	1.11253i	−0.342597	−	0.939483i	\(-0.611306\pi\)
							0.984914	−	0.173044i	\(-0.0553602\pi\)
\(23\)	5.31826	+	0.801599i	1.10893	+	0.167145i	0.677874	−	0.735178i	\(-0.262901\pi\)
							0.431061	+	0.902323i	\(0.358140\pi\)
\(29\)	0.146472	−	0.373204i	0.0271991	−	0.0693022i	−0.916629	−	0.399739i	\(-0.869101\pi\)
							0.943828	+	0.330437i	\(0.107196\pi\)
\(31\)	−5.66247			−1.01701			−0.508505	−	0.861059i	\(-0.669802\pi\)
							−0.508505	+	0.861059i	\(0.669802\pi\)
\(37\)	−0.0638757	+	0.00962771i	−0.0105011	+	0.00158279i	−0.154291	−	0.988025i	\(-0.549309\pi\)
							0.143790	+	0.989608i	\(0.454071\pi\)
\(41\)	7.06791	+	6.55807i	1.10382	+	1.02420i	0.999560	+	0.0296479i	\(0.00943860\pi\)
							0.104262	+	0.994550i	\(0.466752\pi\)
\(43\)	9.16914	−	8.50772i	1.39828	−	1.29742i	0.494657	−	0.869088i	\(-0.335294\pi\)
							0.903624	−	0.428327i	\(-0.140897\pi\)
\(47\)	3.33654	−	1.60679i	0.486684	−	0.234375i	−0.174417	−	0.984672i	\(-0.555804\pi\)
							0.661101	+	0.750297i	\(0.270090\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.y.a.121.36	✓	648
9.7	even	3		441.2.z.a.268.19	yes	648
49.32	even	21		441.2.z.a.130.19	yes	648
441.277	even	21	inner	441.2.y.a.277.36	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.y.a.121.36	✓	648	1.1	even	1	trivial
441.2.y.a.277.36	yes	648	441.277	even	21	inner
441.2.z.a.130.19	yes	648	49.32	even	21
441.2.z.a.268.19	yes	648	9.7	even	3