Embedded newform 441.2.z.a.268.35

• Label: 441.2.z.a.268.35
• Level: $441$
• Weight: $2$
• Character: 441.268
• 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			268.35
Character	\(\chi\)	\(=\)	441.268
Dual form			441.2.z.a.130.35

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

\(n\)	\(199\)	\(344\)
\(\chi(n)\)	\(e\left(\frac{19}{21}\right)\)	\(e\left(\frac{2}{3}\right)\)

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)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.y.a.121.20	✓	648	9.4	even	3
441.2.y.a.277.20	yes	648	49.32	even	21
441.2.z.a.130.35	yes	648	441.130	even	21	inner
441.2.z.a.268.35	yes	648	1.1	even	1	trivial