Embedded newform 441.2.y.a.277.36

• Label: 441.2.y.a.277.36
• Level: $441$
• Weight: $2$
• Character: 441.277
• 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			277.36
Character	\(\chi\)	\(=\)	441.277
Dual form			441.2.y.a.121.36

$q$-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 \(\left(\mathbb{Z}/441\mathbb{Z}\right)^\times\).

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

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)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.y.a.121.36	✓	648	441.121	even	21	inner
441.2.y.a.277.36	yes	648	1.1	even	1	trivial
441.2.z.a.130.19	yes	648	9.4	even	3
441.2.z.a.268.19	yes	648	49.23	even	21