Embedded newform 441.2.z.a.382.38

• Label: 441.2.z.a.382.38
• Level: $441$
• Weight: $2$
• Character: 441.382
• 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			382.38
Character	\(\chi\)	\(=\)	441.382
Dual form			441.2.z.a.142.38

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.454036 + 1.15687i) q^{2} +(1.71544 + 0.239337i) q^{3} +(0.333914 - 0.309827i) q^{4} +(-0.554147 - 0.266863i) q^{5} +(0.501990 + 2.09320i) q^{6} +(-1.17491 + 2.37056i) q^{7} +(2.74944 + 1.32406i) q^{8} +(2.88544 + 0.821133i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 8 q^{2} - 13 q^{3} + 44 q^{4} + 3 q^{5} + 2 q^{6} - 7 q^{7} - 16 q^{8} - 39 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(199\)	\(344\)
\(\chi(n)\)	\(e\left(\frac{17}{21}\right)\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.454036	+	1.15687i	0.321052	+	0.818028i	0.996728	+	0.0808287i	\(0.0257567\pi\)
							−0.675676	+	0.737199i	\(0.736148\pi\)
\(3\)	1.71544	+	0.239337i	0.990407	+	0.138181i
							\(5\)	−0.554147	−	0.266863i	−0.247822	−	0.119345i	0.305850	−	0.952080i	\(-0.401059\pi\)
−0.553672	+	0.832735i	\(0.686774\pi\)
\(7\)	−1.17491	+	2.37056i	−0.444076	+	0.895989i
							\(11\)	0.391646	−	0.491108i	0.118086	−	0.148075i	−0.719275	−	0.694725i	\(-0.755526\pi\)
0.837361	+	0.546651i	\(0.184097\pi\)
\(13\)	−0.676370	−	1.72336i	−0.187591	−	0.477975i	0.805877	−	0.592083i	\(-0.201694\pi\)
							−0.993468	+	0.114108i	\(0.963599\pi\)
\(17\)	−0.396881	−	0.368251i	−0.0962577	−	0.0893141i	0.630576	−	0.776127i	\(-0.282819\pi\)
							−0.726834	+	0.686813i	\(0.759009\pi\)
\(19\)	−0.957228	−	1.65797i	−0.219603	−	0.380364i	0.735083	−	0.677977i	\(-0.237143\pi\)
							−0.954687	+	0.297613i	\(0.903810\pi\)
\(23\)	1.39363	+	6.10588i	0.290591	+	1.27316i	0.883705	+	0.468045i	\(0.155041\pi\)
							−0.593113	+	0.805119i	\(0.702101\pi\)
\(29\)	3.40496	−	1.05029i	0.632286	−	0.195034i	0.0379866	−	0.999278i	\(-0.487906\pi\)
							0.594299	+	0.804244i	\(0.297429\pi\)
\(31\)	−2.65625	−	4.60076i	−0.477076	−	0.826321i	0.522578	−	0.852591i	\(-0.324970\pi\)
							−0.999655	+	0.0262705i	\(0.991637\pi\)
\(37\)	−4.58247	+	1.41351i	−0.753354	+	0.232379i	−0.647566	−	0.762010i	\(-0.724213\pi\)
							−0.105788	+	0.994389i	\(0.533737\pi\)
\(41\)	−0.614083	+	0.418675i	−0.0959037	+	0.0653861i	−0.610322	−	0.792153i	\(-0.708960\pi\)
							0.514419	+	0.857539i	\(0.328008\pi\)
\(43\)	−2.00565	−	1.36743i	−0.305859	−	0.208531i	0.400657	−	0.916228i	\(-0.368782\pi\)
							−0.706516	+	0.707697i	\(0.749734\pi\)
\(47\)	−0.325120	−	0.828391i	−0.0474236	−	0.120833i	0.905199	−	0.424987i	\(-0.139721\pi\)
							−0.952623	+	0.304154i	\(0.901626\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.z.a.382.38	yes	648
9.7	even	3		441.2.y.a.88.38	✓	648
49.44	even	21		441.2.y.a.436.38	yes	648
441.142	even	21	inner	441.2.z.a.142.38	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.y.a.88.38	✓	648	9.7	even	3
441.2.y.a.436.38	yes	648	49.44	even	21
441.2.z.a.142.38	yes	648	441.142	even	21	inner
441.2.z.a.382.38	yes	648	1.1	even	1	trivial