Embedded newform 441.2.y.a.25.10

• Label: 441.2.y.a.25.10
• Level: $441$
• Weight: $2$
• Character: 441.25
• 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			25.10
Character	\(\chi\)	\(=\)	441.25
Dual form			441.2.y.a.247.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.464927 + 2.03698i) q^{2} +(0.999043 - 1.41489i) q^{3} +(-2.13119 - 1.02633i) q^{4} +(-0.231038 - 0.588676i) q^{5} +(2.41762 + 2.69285i) q^{6} +(1.57123 - 2.12867i) q^{7} +(0.476059 - 0.596959i) q^{8} +(-1.00382 - 2.82707i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 5 q^{2} - 13 q^{3} - 109 q^{4} - 12 q^{5} + 2 q^{6} - 7 q^{7} - 16 q^{8} - 3 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(199\)	\(344\)
\(\chi(n)\)	\(e\left(\frac{8}{21}\right)\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.464927	+	2.03698i	−0.328753	+	1.44036i	0.492757	+	0.870167i	\(0.335989\pi\)
							−0.821510	+	0.570195i	\(0.806868\pi\)
\(3\)	0.999043	−	1.41489i	0.576798	−	0.816887i
							\(5\)	−0.231038	−	0.588676i	−0.103324	−	0.263264i	0.869858	−	0.493302i	\(-0.164210\pi\)
−0.973182	+	0.230038i	\(0.926115\pi\)
\(7\)	1.57123	−	2.12867i	0.593870	−	0.804561i
							\(11\)	−2.00237	+	0.617650i	−0.603738	+	0.186229i	−0.581530	−	0.813525i	\(-0.697546\pi\)
−0.0222080	+	0.999753i	\(0.507070\pi\)
\(13\)	4.68226	−	1.44429i	1.29862	−	0.400573i	0.433058	−	0.901366i	\(-0.357435\pi\)
							0.865567	+	0.500794i	\(0.166958\pi\)
\(17\)	0.472737	−	6.30824i	0.114656	−	1.52997i	−0.582369	−	0.812925i	\(-0.697874\pi\)
							0.697024	−	0.717047i	\(-0.254507\pi\)
\(19\)	2.25973	+	3.91396i	0.518417	+	0.897924i	0.999771	+	0.0213978i	\(0.00681166\pi\)
							−0.481354	+	0.876526i	\(0.659855\pi\)
\(23\)	0.541499	−	0.369188i	0.112910	−	0.0769810i	−0.505548	−	0.862798i	\(-0.668710\pi\)
							0.618459	+	0.785817i	\(0.287757\pi\)
\(29\)	−0.320163	+	4.27227i	−0.0594527	+	0.793341i	0.885188	+	0.465234i	\(0.154030\pi\)
							−0.944640	+	0.328107i	\(0.893589\pi\)
\(31\)	4.75896			0.854734			0.427367	−	0.904078i	\(-0.359441\pi\)
							0.427367	+	0.904078i	\(0.359441\pi\)
\(37\)	1.12082	+	0.764164i	0.184262	+	0.125628i	0.651938	−	0.758272i	\(-0.273956\pi\)
							−0.467676	+	0.883900i	\(0.654909\pi\)
\(41\)	4.59705	+	0.692893i	0.717938	+	0.108212i	0.497844	−	0.867267i	\(-0.334125\pi\)
							0.220095	+	0.975479i	\(0.429363\pi\)
\(43\)	−8.88826	+	1.33969i	−1.35545	+	0.204301i	−0.786254	−	0.617903i	\(-0.787982\pi\)
							−0.569193	+	0.822204i	\(0.692744\pi\)
\(47\)	−0.0669222	+	0.293206i	−0.00976161	+	0.0427684i	−0.979574	−	0.201083i	\(-0.935554\pi\)
							0.969813	+	0.243852i	\(0.0784110\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.y.a.25.10	✓	648
9.4	even	3		441.2.z.a.319.10	yes	648
49.2	even	21		441.2.z.a.394.10	yes	648
441.247	even	21	inner	441.2.y.a.247.10	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.y.a.25.10	✓	648	1.1	even	1	trivial
441.2.y.a.247.10	yes	648	441.247	even	21	inner
441.2.z.a.319.10	yes	648	9.4	even	3
441.2.z.a.394.10	yes	648	49.2	even	21