Embedded newform 441.2.y.a.151.44

• Label: 441.2.y.a.151.44
• Level: $441$
• Weight: $2$
• Character: 441.151
• 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			151.44
Character	\(\chi\)	\(=\)	441.151
Dual form			441.2.y.a.184.44

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.60118 - 0.771090i) q^{2} +(-0.884294 + 1.48930i) q^{3} +(0.722233 - 0.905652i) q^{4} +(-2.59882 - 2.41135i) q^{5} +(-0.267532 + 3.06652i) q^{6} +(-1.80347 - 1.93585i) q^{7} +(-0.332830 + 1.45822i) q^{8} +(-1.43605 - 2.63396i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 5 q^{2} - 13 q^{3} - 109 q^{4} - 12 q^{5} + 2 q^{6} - 7 q^{7} - 16 q^{8} - 3 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(199\)	\(344\)
\(\chi(n)\)	\(e\left(\frac{5}{21}\right)\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.60118	−	0.771090i	1.13221	−	0.545243i	0.228565	−	0.973529i	\(-0.426597\pi\)
							0.903643	+	0.428286i	\(0.140882\pi\)
\(3\)	−0.884294	+	1.48930i	−0.510548	+	0.859850i
							\(5\)	−2.59882	−	2.41135i	−1.16223	−	1.07839i	−0.995727	−	0.0923460i	\(-0.970563\pi\)
−0.166499	−	0.986042i	\(-0.553246\pi\)
\(7\)	−1.80347	−	1.93585i	−0.681647	−	0.731682i
							\(11\)	−1.22500	−	0.835189i	−0.369351	−	0.251819i	0.364393	−	0.931245i	\(-0.381276\pi\)
−0.733744	+	0.679426i	\(0.762229\pi\)
\(13\)	−1.91266	−	1.30403i	−0.530475	−	0.361672i	0.268268	−	0.963344i	\(-0.413549\pi\)
							−0.798743	+	0.601673i	\(0.794501\pi\)
\(17\)	0.421630	−	0.0635504i	0.102260	−	0.0154132i	−0.0977128	−	0.995215i	\(-0.531153\pi\)
							0.199973	+	0.979801i	\(0.435915\pi\)
\(19\)	−2.91097	−	5.04194i	−0.667821	−	1.15670i	−0.978512	−	0.206189i	\(-0.933894\pi\)
							0.310691	−	0.950511i	\(-0.399440\pi\)
\(23\)	0.675703	+	1.72166i	0.140894	+	0.358991i	0.983851	−	0.178989i	\(-0.0572827\pi\)
							−0.842957	+	0.537980i	\(0.819187\pi\)
\(29\)	8.25590	−	1.24438i	1.53308	−	0.231075i	0.672389	−	0.740198i	\(-0.265268\pi\)
							0.860694	+	0.509123i	\(0.170030\pi\)
\(31\)	−7.64497			−1.37308			−0.686539	−	0.727093i	\(-0.740871\pi\)
							−0.686539	+	0.727093i	\(0.740871\pi\)
\(37\)	−1.11492	+	2.84076i	−0.183291	+	0.467018i	−0.992756	−	0.120150i	\(-0.961662\pi\)
							0.809464	+	0.587169i	\(0.199758\pi\)
\(41\)	2.34106	−	0.722120i	0.365612	−	0.112776i	−0.106505	−	0.994312i	\(-0.533966\pi\)
							0.472117	+	0.881536i	\(0.343490\pi\)
\(43\)	−10.4505	−	3.22356i	−1.59369	−	0.491588i	−0.634160	−	0.773201i	\(-0.718654\pi\)
							−0.959528	+	0.281614i	\(0.909130\pi\)
\(47\)	2.64020	−	1.27145i	0.385113	−	0.185461i	−0.231302	−	0.972882i	\(-0.574299\pi\)
							0.616415	+	0.787421i	\(0.288584\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.y.a.151.44	✓	648
9.4	even	3		441.2.z.a.4.11	yes	648
49.37	even	21		441.2.z.a.331.11	yes	648
441.184	even	21	inner	441.2.y.a.184.44	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.y.a.151.44	✓	648	1.1	even	1	trivial
441.2.y.a.184.44	yes	648	441.184	even	21	inner
441.2.z.a.4.11	yes	648	9.4	even	3
441.2.z.a.331.11	yes	648	49.37	even	21