Embedded newform 441.2.y.a.121.38

• Label: 441.2.y.a.121.38
• Level: $441$
• Weight: $2$
• Character: 441.121
• 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			121.38
Character	\(\chi\)	\(=\)	441.121
Dual form			441.2.y.a.277.38

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.919527 - 0.442821i) q^{2} +(1.70959 - 0.278009i) q^{3} +(-0.597540 + 0.749292i) q^{4} +(3.10620 - 0.958137i) q^{5} +(1.44891 - 1.01268i) q^{6} +(-2.22035 - 1.43877i) q^{7} +(-0.671861 + 2.94362i) q^{8} +(2.84542 - 0.950565i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 5 q^{2} - 13 q^{3} - 109 q^{4} - 12 q^{5} + 2 q^{6} - 7 q^{7} - 16 q^{8} - 3 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(199\)	\(344\)
\(\chi(n)\)	\(e\left(\frac{19}{21}\right)\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.919527	−	0.442821i	0.650204	−	0.313122i	−0.0795508	−	0.996831i	\(-0.525349\pi\)
							0.729755	+	0.683709i	\(0.239634\pi\)
\(3\)	1.70959	−	0.278009i	0.987034	−	0.160509i
							\(5\)	3.10620	−	0.958137i	1.38914	−	0.428492i	0.492135	−	0.870519i	\(-0.336217\pi\)
0.897002	+	0.442027i	\(0.145740\pi\)
\(7\)	−2.22035	−	1.43877i	−0.839213	−	0.543803i
							\(11\)	0.141022	−	1.88181i	0.0425198	−	0.567387i	−0.934739	−	0.355336i	\(-0.884367\pi\)
0.977259	−	0.212051i	\(-0.0680144\pi\)
\(13\)	0.173309	−	2.31265i	0.0480673	−	0.641413i	−0.920192	−	0.391467i	\(-0.871968\pi\)
							0.968260	−	0.249947i	\(-0.0804131\pi\)
\(17\)	−1.50942	+	3.84594i	−0.366088	+	0.932776i	0.622566	+	0.782567i	\(0.286090\pi\)
							−0.988654	+	0.150209i	\(0.952005\pi\)
\(19\)	−1.82246	+	3.15659i	−0.418100	+	0.724171i	−0.995748	−	0.0921149i	\(-0.970637\pi\)
							0.577648	+	0.816286i	\(0.303971\pi\)
\(23\)	5.97698	+	0.900884i	1.24629	+	0.187847i	0.738867	−	0.673852i	\(-0.235361\pi\)
							0.507419	+	0.861699i	\(0.330599\pi\)
\(29\)	−2.29539	+	5.84856i	−0.426243	+	1.08605i	0.542954	+	0.839763i	\(0.317306\pi\)
							−0.969197	+	0.246287i	\(0.920789\pi\)
\(31\)	−9.28793			−1.66816			−0.834081	−	0.551642i	\(-0.814001\pi\)
							−0.834081	+	0.551642i	\(0.814001\pi\)
\(37\)	0.627734	−	0.0946157i	0.103199	−	0.0155547i	−0.0972398	−	0.995261i	\(-0.531001\pi\)
							0.200439	+	0.979706i	\(0.435763\pi\)
\(41\)	−3.50284	−	3.25016i	−0.547052	−	0.507590i	0.357523	−	0.933904i	\(-0.383621\pi\)
							−0.904575	+	0.426314i	\(0.859812\pi\)
\(43\)	2.01305	−	1.86784i	0.306987	−	0.284843i	−0.511601	−	0.859223i	\(-0.670947\pi\)
							0.818589	+	0.574380i	\(0.194757\pi\)
\(47\)	−2.77756	+	1.33760i	−0.405149	+	0.195110i	−0.625349	−	0.780345i	\(-0.715043\pi\)
							0.220200	+	0.975455i	\(0.429329\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.y.a.121.38	✓	648
9.7	even	3		441.2.z.a.268.17	yes	648
49.32	even	21		441.2.z.a.130.17	yes	648
441.277	even	21	inner	441.2.y.a.277.38	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.y.a.121.38	✓	648	1.1	even	1	trivial
441.2.y.a.277.38	yes	648	441.277	even	21	inner
441.2.z.a.130.17	yes	648	49.32	even	21
441.2.z.a.268.17	yes	648	9.7	even	3