Embedded newform 441.2.z.a.142.14

• Label: 441.2.z.a.142.14
• Level: $441$
• Weight: $2$
• Character: 441.142
• 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			142.14
Character	\(\chi\)	\(=\)	441.142
Dual form			441.2.z.a.382.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.643114 + 1.63863i) q^{2} +(-1.73201 - 0.0118877i) q^{3} +(-0.805400 - 0.747302i) q^{4} +(0.303403 - 0.146111i) q^{5} +(1.13336 - 2.83047i) q^{6} +(0.198573 + 2.63829i) q^{7} +(-1.42946 + 0.688390i) q^{8} +(2.99972 + 0.0411794i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 8 q^{2} - 13 q^{3} + 44 q^{4} + 3 q^{5} + 2 q^{6} - 7 q^{7} - 16 q^{8} - 39 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(199\)	\(344\)
\(\chi(n)\)	\(e\left(\frac{4}{21}\right)\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.643114	+	1.63863i	−0.454750	+	1.15868i	0.501246	+	0.865305i	\(0.332875\pi\)
							−0.955996	+	0.293380i	\(0.905220\pi\)
\(3\)	−1.73201	−	0.0118877i	−0.999976	−	0.00686339i
							\(5\)	0.303403	−	0.146111i	0.135686	−	0.0653429i	−0.364809	−	0.931082i	\(-0.618866\pi\)
0.500495	+	0.865740i	\(0.333151\pi\)
\(7\)	0.198573	+	2.63829i	0.0750534	+	0.997180i
							\(11\)	−0.238584	−	0.299175i	−0.0719359	−	0.0902048i	0.744562	−	0.667554i	\(-0.232659\pi\)
−0.816497	+	0.577349i	\(0.804087\pi\)
\(13\)	−1.09482	+	2.78955i	−0.303648	+	0.773683i	0.694863	+	0.719142i	\(0.255465\pi\)
							−0.998512	+	0.0545413i	\(0.982630\pi\)
\(17\)	−4.16184	+	3.86162i	−1.00939	+	0.936582i	−0.997991	−	0.0633534i	\(-0.979820\pi\)
							−0.0114038	+	0.999935i	\(0.503630\pi\)
\(19\)	−0.105464	+	0.182669i	−0.0241951	+	0.0419072i	−0.877869	−	0.478900i	\(-0.841036\pi\)
							0.853674	+	0.520807i	\(0.174369\pi\)
\(23\)	1.89055	−	8.28303i	0.394206	−	1.72713i	−0.255378	−	0.966841i	\(-0.582200\pi\)
							0.649584	−	0.760290i	\(-0.274943\pi\)
\(29\)	−8.58689	−	2.64871i	−1.59455	−	0.491852i	−0.634790	−	0.772685i	\(-0.718913\pi\)
							−0.959757	+	0.280833i	\(0.909389\pi\)
\(31\)	1.34111	−	2.32288i	0.240871	−	0.417201i	−0.720092	−	0.693879i	\(-0.755900\pi\)
							0.960963	+	0.276678i	\(0.0892336\pi\)
\(37\)	6.63269	+	2.04591i	1.09041	+	0.336346i	0.787275	−	0.616602i	\(-0.211491\pi\)
							0.303132	+	0.952948i	\(0.401968\pi\)
\(41\)	−7.86782	−	5.36419i	−1.22875	−	0.837746i	−0.237711	−	0.971336i	\(-0.576397\pi\)
							−0.991037	+	0.133590i	\(0.957349\pi\)
\(43\)	6.35750	−	4.33447i	0.969510	−	0.661000i	0.0285760	−	0.999592i	\(-0.490903\pi\)
							0.940934	+	0.338591i	\(0.109950\pi\)
\(47\)	−4.13503	+	10.5359i	−0.603156	+	1.53682i	0.223849	+	0.974624i	\(0.428138\pi\)
							−0.827006	+	0.562194i	\(0.809958\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.z.a.142.14	yes	648
9.4	even	3		441.2.y.a.436.14	yes	648
49.39	even	21		441.2.y.a.88.14	✓	648
441.382	even	21	inner	441.2.z.a.382.14	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.y.a.88.14	✓	648	49.39	even	21
441.2.y.a.436.14	yes	648	9.4	even	3
441.2.z.a.142.14	yes	648	1.1	even	1	trivial
441.2.z.a.382.14	yes	648	441.382	even	21	inner