Embedded newform 245.2.x.a.108.21

• Label: 245.2.x.a.108.21
• Level: $245$
• Weight: $2$
• Character: 245.108
• Analytic conductor: $1.956$
• Analytic rank: $0$
• Dimension: $624$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 245 = 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	245.x (of order \(84\), degree \(24\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.95633484952\)
Analytic rank:	\(0\)
Dimension:	\(624\)
Relative dimension:	\(26\) over \(\Q(\zeta_{84})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{84}]$

Embedding invariants

Embedding label			108.21
Character	\(\chi\)	\(=\)	245.108
Dual form			245.2.x.a.152.21

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.28575 - 0.948928i) q^{2} +(-0.110744 + 0.585295i) q^{3} +(0.163183 - 0.529026i) q^{4} +(0.775003 + 2.09747i) q^{5} +(0.413014 + 0.857632i) q^{6} +(-0.606720 - 2.57525i) q^{7} +(0.763381 + 2.18162i) q^{8} +(2.46232 + 0.966387i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 624 q - 26 q^{2} - 22 q^{3} - 28 q^{5} - 28 q^{6} - 18 q^{7} - 24 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/245\mathbb{Z}\right)^\times\).

\(n\)	\(101\)	\(197\)
\(\chi(n)\)	\(e\left(\frac{13}{42}\right)\)	\(e\left(\frac{3}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.28575	−	0.948928i	0.909164	−	0.670993i	−0.0353817	−	0.999374i	\(-0.511265\pi\)
							0.944546	+	0.328380i	\(0.106503\pi\)
\(3\)	−0.110744	+	0.585295i	−0.0639380	+	0.337920i	−0.999899	−	0.0141822i	\(-0.995486\pi\)
							0.935961	+	0.352102i	\(0.114533\pi\)
\(5\)	0.775003	+	2.09747i	0.346592	+	0.938016i
							\(7\)	−0.606720	−	2.57525i	−0.229319	−	0.973351i
\(11\)	1.01666	+	2.59041i	0.306535	+	0.781038i								0.998265	+	0.0588816i	\(0.0187535\pi\)
							−0.691730	+	0.722156i	\(0.743151\pi\)
\(13\)	−0.754220	−	6.69388i	−0.209183	−	1.85655i	−0.459232	−	0.888316i	\(-0.651875\pi\)
							0.250049	−	0.968233i	\(-0.419553\pi\)
\(17\)	0.0996795	+	2.66399i	0.0241758	+	0.646113i	0.957729	+	0.287673i	\(0.0928816\pi\)
							−0.933553	+	0.358440i	\(0.883309\pi\)
\(19\)	−2.69893	−	4.67469i	−0.619178	−	1.07245i	−0.989636	−	0.143598i	\(-0.954133\pi\)
							0.370458	−	0.928849i	\(-0.379201\pi\)
\(23\)	−5.61066	−	0.209936i	−1.16990	−	0.0437747i	−0.554636	−	0.832093i	\(-0.687142\pi\)
							−0.615267	+	0.788318i	\(0.710952\pi\)
\(29\)	1.09900	+	0.250839i	0.204079	+	0.0465797i	0.323338	−	0.946283i	\(-0.395195\pi\)
							−0.119259	+	0.992863i	\(0.538052\pi\)
\(31\)	−3.55443	−	2.05215i	−0.638395	−	0.368577i	0.145601	−	0.989343i	\(-0.453488\pi\)
							−0.783996	+	0.620766i	\(0.786822\pi\)
\(37\)	3.58107	−	6.77572i	0.588725	−	1.11392i	−0.392263	−	0.919853i	\(-0.628308\pi\)
							0.980988	−	0.194069i	\(-0.0621685\pi\)
\(41\)	2.43308	−	5.05234i	0.379983	−	0.789043i	−0.620007	−	0.784596i	\(-0.712870\pi\)
							0.999990	−	0.00444673i	\(-0.00141544\pi\)
\(43\)	−0.871829	−	0.305066i	−0.132953	−	0.0465221i	0.262983	−	0.964801i	\(-0.415294\pi\)
							−0.395935	+	0.918278i	\(0.629580\pi\)
\(47\)	−2.24234	−	3.03826i	−0.327079	−	0.443176i	0.609944	−	0.792445i	\(-0.291192\pi\)
							−0.937023	+	0.349269i	\(0.886430\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	245.2.x.a.108.21	yes	624
5.2	odd	4	inner	245.2.x.a.157.21	yes	624
49.5	odd	42	inner	245.2.x.a.103.21	✓	624
245.152	even	84	inner	245.2.x.a.152.21	yes	624

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
245.2.x.a.103.21	✓	624	49.5	odd	42	inner
245.2.x.a.108.21	yes	624	1.1	even	1	trivial
245.2.x.a.152.21	yes	624	245.152	even	84	inner
245.2.x.a.157.21	yes	624	5.2	odd	4	inner