Embedded newform 441.2.z.a.268.40

• Label: 441.2.z.a.268.40
• 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			268.40
Character	\(\chi\)	\(=\)	441.268
Dual form			441.2.z.a.130.40

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.19380 + 0.813921i) q^{2} +(1.71916 - 0.210914i) q^{3} +(0.0320159 + 0.0815752i) q^{4} +(-0.000507118 + 0.00222183i) q^{5} +(2.22401 + 1.14747i) q^{6} +(0.314524 - 2.62699i) q^{7} +(0.614850 - 2.69383i) q^{8} +(2.91103 - 0.725191i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 8 q^{2} - 13 q^{3} + 44 q^{4} + 3 q^{5} + 2 q^{6} - 7 q^{7} - 16 q^{8} - 39 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.19380	+	0.813921i	0.844146	+	0.575529i	0.906347	−	0.422535i	\(-0.138859\pi\)
							−0.0622007	+	0.998064i	\(0.519812\pi\)
\(3\)	1.71916	−	0.210914i	0.992558	−	0.121771i
							\(5\)	−0.000507118		0.00222183i	−0.000226790		0.000993632i	−0.975041	−	0.222024i	\(-0.928734\pi\)
0.974814	+	0.223018i	\(0.0715908\pi\)
\(7\)	0.314524	−	2.62699i	0.118879	−	0.992909i
							\(11\)	−2.93257	+	1.41225i	−0.884203	+	0.425810i	−0.820158	−	0.572137i	\(-0.806114\pi\)
−0.0640452	+	0.997947i	\(0.520400\pi\)
\(13\)	1.97136	+	1.34405i	0.546758	+	0.372773i	0.804970	−	0.593315i	\(-0.202181\pi\)
							−0.258213	+	0.966088i	\(0.583134\pi\)
\(17\)	0.0598202	−	0.152419i	0.0145085	−	0.0369672i	−0.923436	−	0.383753i	\(-0.874632\pi\)
							0.937944	+	0.346785i	\(0.112727\pi\)
\(19\)	−1.97578	+	3.42215i	−0.453275	+	0.785095i	−0.998587	−	0.0531381i	\(-0.983078\pi\)
							0.545313	+	0.838233i	\(0.316411\pi\)
\(23\)	−4.65452	+	5.83659i	−0.970535	+	1.21701i	0.00562997	+	0.999984i	\(0.498208\pi\)
							−0.976165	+	0.217029i	\(0.930364\pi\)
\(29\)	−3.91333	+	0.589839i	−0.726687	+	0.109530i	−0.501959	−	0.864891i	\(-0.667387\pi\)
							−0.224727	+	0.974422i	\(0.572149\pi\)
\(31\)	−2.86923	+	4.96966i	−0.515330	+	0.892577i	0.484512	+	0.874785i	\(0.338997\pi\)
							−0.999842	+	0.0177927i	\(0.994336\pi\)
\(37\)	5.85601	−	0.882652i	0.962723	−	0.145107i	0.351171	−	0.936311i	\(-0.385784\pi\)
							0.611552	+	0.791204i	\(0.290546\pi\)
\(41\)	2.43244	−	0.750308i	0.379883	−	0.117178i	−0.0989320	−	0.995094i	\(-0.531543\pi\)
							0.478815	+	0.877916i	\(0.341066\pi\)
\(43\)	−2.71093	−	0.836211i	−0.413413	−	0.127521i	0.0810700	−	0.996708i	\(-0.474166\pi\)
							−0.494483	+	0.869187i	\(0.664642\pi\)
\(47\)	0.462999	+	0.315667i	0.0675353	+	0.0460448i	0.596617	−	0.802526i	\(-0.296511\pi\)
							−0.529082	+	0.848571i	\(0.677464\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.z.a.268.40	yes	648
9.4	even	3		441.2.y.a.121.15	✓	648
49.32	even	21		441.2.y.a.277.15	yes	648
441.130	even	21	inner	441.2.z.a.130.40	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.y.a.121.15	✓	648	9.4	even	3
441.2.y.a.277.15	yes	648	49.32	even	21
441.2.z.a.130.40	yes	648	441.130	even	21	inner
441.2.z.a.268.40	yes	648	1.1	even	1	trivial