Embedded newform 441.2.z.a.331.18

• Label: 441.2.z.a.331.18
• Level: $441$
• Weight: $2$
• Character: 441.331
• 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			331.18
Character	\(\chi\)	\(=\)	441.331
Dual form			441.2.z.a.4.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.0832951 - 1.11150i) q^{2} +(-0.959106 - 1.44226i) q^{3} +(0.749175 - 0.112920i) q^{4} +(0.757564 + 3.31911i) q^{5} +(-1.52318 + 1.18618i) q^{6} +(-1.18524 + 2.36542i) q^{7} +(-0.683962 - 2.99663i) q^{8} +(-1.16023 + 2.76656i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 8 q^{2} - 13 q^{3} + 44 q^{4} + 3 q^{5} + 2 q^{6} - 7 q^{7} - 16 q^{8} - 39 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(199\)	\(344\)
\(\chi(n)\)	\(e\left(\frac{16}{21}\right)\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.0832951	−	1.11150i	−0.0588986	−	0.785947i	−0.945958	−	0.324291i	\(-0.894874\pi\)
							0.887059	−	0.461656i	\(-0.152745\pi\)
\(3\)	−0.959106	−	1.44226i	−0.553740	−	0.832690i
							\(5\)	0.757564	+	3.31911i	0.338793	+	1.48435i	0.801584	+	0.597882i	\(0.203991\pi\)
−0.462791	+	0.886467i	\(0.653152\pi\)
\(7\)	−1.18524	+	2.36542i	−0.447979	+	0.894044i
							\(11\)	3.12006	+	1.50254i	0.940733	+	0.453033i	0.840428	−	0.541924i	\(-0.182304\pi\)
0.100305	+	0.994957i	\(0.468018\pi\)
\(13\)	0.357646	+	4.77245i	0.0991932	+	1.32364i	0.795525	+	0.605921i	\(0.207195\pi\)
							−0.696332	+	0.717720i	\(0.745186\pi\)
\(17\)	3.97020	+	0.598412i	0.962915	+	0.145136i	0.611640	−	0.791136i	\(-0.290510\pi\)
							0.351275	+	0.936272i	\(0.385748\pi\)
\(19\)	−2.15962	+	3.74057i	−0.495450	+	0.858145i	−0.999986	−	0.00524540i	\(-0.998330\pi\)
							0.504536	+	0.863391i	\(0.331664\pi\)
\(23\)	−4.61972	−	5.79294i	−0.963278	−	1.20791i	−0.978124	−	0.208022i	\(-0.933297\pi\)
							0.0148462	−	0.999890i	\(-0.495274\pi\)
\(29\)	0.595483	+	1.51727i	0.110578	+	0.281749i	0.975442	−	0.220258i	\(-0.0706901\pi\)
							−0.864863	+	0.502008i	\(0.832595\pi\)
\(31\)	2.25870	−	3.91218i	0.405674	−	0.702648i	−0.588726	−	0.808333i	\(-0.700370\pi\)
							0.994400	+	0.105685i	\(0.0337035\pi\)
\(37\)	2.43477	+	6.20370i	0.400274	+	1.01988i	0.978909	+	0.204296i	\(0.0654904\pi\)
							−0.578635	+	0.815587i	\(0.696414\pi\)
\(41\)	−1.11546	+	1.03499i	−0.174205	+	0.161639i	−0.762457	−	0.647039i	\(-0.776007\pi\)
							0.588252	+	0.808678i	\(0.299817\pi\)
\(43\)	−1.06162	−	0.985037i	−0.161895	−	0.150217i	0.595066	−	0.803677i	\(-0.297126\pi\)
							−0.756961	+	0.653460i	\(0.773317\pi\)
\(47\)	−0.818159	−	10.9176i	−0.119341	−	1.59249i	−0.660010	−	0.751257i	\(-0.729448\pi\)
							0.540670	−	0.841235i	\(-0.318171\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.z.a.331.18	yes	648
9.4	even	3		441.2.y.a.184.37	yes	648
49.4	even	21		441.2.y.a.151.37	✓	648
441.4	even	21	inner	441.2.z.a.4.18	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.y.a.151.37	✓	648	49.4	even	21
441.2.y.a.184.37	yes	648	9.4	even	3
441.2.z.a.4.18	yes	648	441.4	even	21	inner
441.2.z.a.331.18	yes	648	1.1	even	1	trivial