Embedded newform 441.2.z.a.130.35

• Label: 441.2.z.a.130.35
• Level: $441$
• Weight: $2$
• Character: 441.130
• 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.z (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			130.35
Character	\(\chi\)	\(=\)	441.130
Dual form			441.2.z.a.268.35

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.821420 - 0.560034i) q^{2} +(-1.08036 - 1.35382i) q^{3} +(-0.369590 + 0.941701i) q^{4} +(-0.0503266 - 0.220495i) q^{5} +(-1.64561 - 0.507013i) q^{6} +(1.99876 + 1.73348i) q^{7} +(0.666242 + 2.91900i) q^{8} +(-0.665643 + 2.92522i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 8 q^{2} - 13 q^{3} + 44 q^{4} + 3 q^{5} + 2 q^{6} - 7 q^{7} - 16 q^{8} - 39 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(199\)	\(344\)
\(\chi(n)\)	\(e\left(\frac{2}{21}\right)\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.821420	−	0.560034i	0.580831	−	0.396004i	−0.236960	−	0.971520i	\(-0.576151\pi\)
							0.817791	+	0.575515i	\(0.195199\pi\)
\(3\)	−1.08036	−	1.35382i	−0.623746	−	0.781627i
							\(5\)	−0.0503266	−	0.220495i	−0.0225067	−	0.0986085i	0.962427	−	0.271540i	\(-0.0875331\pi\)
−0.984934	+	0.172932i	\(0.944676\pi\)
\(7\)	1.99876	+	1.73348i	0.755459	+	0.655195i
							\(11\)	2.39852	+	1.15506i	0.723180	+	0.348265i	0.758998	−	0.651093i	\(-0.225689\pi\)
−0.0358179	+	0.999358i	\(0.511404\pi\)
\(13\)	2.29139	−	1.56224i	0.635518	−	0.433289i	−0.202236	−	0.979337i	\(-0.564821\pi\)
							0.837754	+	0.546048i	\(0.183868\pi\)
\(17\)	1.07034	+	2.72719i	0.259596	+	0.661440i	0.999942	−	0.0107864i	\(-0.00343349\pi\)
							−0.740346	+	0.672226i	\(0.765338\pi\)
\(19\)	−3.31409	−	5.74017i	−0.760305	−	1.31689i	−0.942694	−	0.333660i	\(-0.891716\pi\)
							0.182389	−	0.983226i	\(-0.441617\pi\)
\(23\)	3.25875	+	4.08634i	0.679496	+	0.852061i	0.995308	−	0.0967608i	\(-0.0308482\pi\)
							−0.315812	+	0.948822i	\(0.602277\pi\)
\(29\)	−0.700759	−	0.105622i	−0.130128	−	0.0196136i	0.0836554	−	0.996495i	\(-0.473340\pi\)
							−0.213783	+	0.976881i	\(0.568579\pi\)
\(31\)	1.46177	+	2.53186i	0.262541	+	0.454735i	0.966917	−	0.255093i	\(-0.0821061\pi\)
							−0.704375	+	0.709828i	\(0.748773\pi\)
\(37\)	−8.10899	−	1.22223i	−1.33311	−	0.200934i	−0.556464	−	0.830872i	\(-0.687842\pi\)
							−0.776646	+	0.629938i	\(0.783080\pi\)
\(41\)	11.2533	+	3.47117i	1.75746	+	0.542106i	0.994537	−	0.104381i	\(-0.0332860\pi\)
							0.762926	+	0.646486i	\(0.223762\pi\)
\(43\)	1.58948	−	0.490290i	0.242394	−	0.0747685i	−0.171178	−	0.985240i	\(-0.554757\pi\)
							0.413572	+	0.910472i	\(0.364281\pi\)
\(47\)	−6.76399	+	4.61161i	−0.986629	+	0.672672i	−0.945208	−	0.326469i	\(-0.894141\pi\)
							−0.0414213	+	0.999142i	\(0.513189\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.z.a.130.35	yes	648
9.7	even	3		441.2.y.a.277.20	yes	648
49.23	even	21		441.2.y.a.121.20	✓	648
441.268	even	21	inner	441.2.z.a.268.35	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.y.a.121.20	✓	648	49.23	even	21
441.2.y.a.277.20	yes	648	9.7	even	3
441.2.z.a.130.35	yes	648	1.1	even	1	trivial
441.2.z.a.268.35	yes	648	441.268	even	21	inner