Embedded newform 441.2.y.a.25.3

• Label: 441.2.y.a.25.3
• Level: $441$
• Weight: $2$
• Character: 441.25
• 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			25.3
Character	\(\chi\)	\(=\)	441.25
Dual form			441.2.y.a.247.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.596104 + 2.61170i) q^{2} +(0.768590 - 1.55218i) q^{3} +(-4.66370 - 2.24592i) q^{4} +(1.23942 + 3.15800i) q^{5} +(3.59567 + 2.93259i) q^{6} +(1.25551 + 2.32888i) q^{7} +(5.30523 - 6.65255i) q^{8} +(-1.81854 - 2.38598i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 5 q^{2} - 13 q^{3} - 109 q^{4} - 12 q^{5} + 2 q^{6} - 7 q^{7} - 16 q^{8} - 3 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(199\)	\(344\)
\(\chi(n)\)	\(e\left(\frac{8}{21}\right)\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.596104	+	2.61170i	−0.421509	+	1.84675i	0.102075	+	0.994777i	\(0.467452\pi\)
							−0.523584	+	0.851974i	\(0.675405\pi\)
\(3\)	0.768590	−	1.55218i	0.443746	−	0.896153i
							\(5\)	1.23942	+	3.15800i	0.554287	+	1.41230i	0.883673	+	0.468105i	\(0.155063\pi\)
−0.329386	+	0.944195i	\(0.606842\pi\)
\(7\)	1.25551	+	2.32888i	0.474537	+	0.880236i
							\(11\)	3.65272	−	1.12671i	1.10134	−	0.339717i	0.309777	−	0.950809i	\(-0.399746\pi\)
0.791559	+	0.611092i	\(0.209270\pi\)
\(13\)	−2.70452	+	0.834234i	−0.750099	+	0.231375i	−0.646153	−	0.763208i	\(-0.723623\pi\)
							−0.103946	+	0.994583i	\(0.533147\pi\)
\(17\)	−0.176704	+	2.35795i	−0.0428569	+	0.571886i	0.933900	+	0.357535i	\(0.116383\pi\)
							−0.976757	+	0.214351i	\(0.931236\pi\)
\(19\)	3.40536	+	5.89826i	0.781243	+	1.35315i	0.931218	+	0.364463i	\(0.118748\pi\)
							−0.149974	+	0.988690i	\(0.547919\pi\)
\(23\)	−2.66652	+	1.81800i	−0.556007	+	0.379079i	−0.808479	−	0.588525i	\(-0.799709\pi\)
							0.252472	+	0.967604i	\(0.418757\pi\)
\(29\)	0.106019	−	1.41472i	0.0196872	−	0.262708i	−0.978671	−	0.205433i	\(-0.934140\pi\)
							0.998358	−	0.0572750i	\(-0.0182412\pi\)
\(31\)	−4.55809			−0.818657			−0.409329	−	0.912387i	\(-0.634237\pi\)
							−0.409329	+	0.912387i	\(0.634237\pi\)
\(37\)	−7.32425	−	4.99359i	−1.20410	−	0.820941i	−0.216247	−	0.976339i	\(-0.569382\pi\)
							−0.987852	+	0.155398i	\(0.950334\pi\)
\(41\)	4.97115	+	0.749280i	0.776363	+	0.117018i	0.525259	−	0.850942i	\(-0.323968\pi\)
							0.251104	+	0.967960i	\(0.419206\pi\)
\(43\)	4.65361	−	0.701418i	0.709668	−	0.106965i	0.215718	−	0.976456i	\(-0.430791\pi\)
							0.493950	+	0.869490i	\(0.335553\pi\)
\(47\)	1.53831	−	6.73976i	0.224385	−	0.983095i	−0.729749	−	0.683715i	\(-0.760363\pi\)
							0.954134	−	0.299380i	\(-0.0967798\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.y.a.25.3	✓	648
9.4	even	3		441.2.z.a.319.3	yes	648
49.2	even	21		441.2.z.a.394.3	yes	648
441.247	even	21	inner	441.2.y.a.247.3	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.y.a.25.3	✓	648	1.1	even	1	trivial
441.2.y.a.247.3	yes	648	441.247	even	21	inner
441.2.z.a.319.3	yes	648	9.4	even	3
441.2.z.a.394.3	yes	648	49.2	even	21