Embedded newform 245.2.x.a.103.21

• Label: 245.2.x.a.103.21
• Level: $245$
• Weight: $2$
• Character: 245.103
• 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			103.21
Character	\(\chi\)	\(=\)	245.103
Dual form			245.2.x.a.157.21

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.948928 - 1.28575i) q^{2} +(0.585295 - 0.110744i) q^{3} +(-0.163183 - 0.529026i) q^{4} +(0.453736 + 2.18955i) q^{5} +(0.413014 - 0.857632i) q^{6} +(2.57525 + 0.606720i) q^{7} +(2.18162 + 0.763381i) q^{8} +(-2.46232 + 0.966387i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 624 q - 26 q^{2} - 22 q^{3} - 28 q^{5} - 28 q^{6} - 18 q^{7} - 24 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(101\)	\(197\)
\(\chi(n)\)	\(e\left(\frac{29}{42}\right)\)	\(e\left(\frac{3}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.948928	−	1.28575i	0.670993	−	0.909164i	−0.328380	−	0.944546i	\(-0.606503\pi\)
							0.999374	+	0.0353817i	\(0.0112647\pi\)
\(3\)	0.585295	−	0.110744i	0.337920	−	0.0639380i	−0.0141822	−	0.999899i	\(-0.504514\pi\)
							0.352102	+	0.935961i	\(0.385467\pi\)
\(5\)	0.453736	+	2.18955i	0.202917	+	0.979196i
							\(7\)	2.57525	+	0.606720i	0.973351	+	0.229319i
\(11\)	1.01666	−	2.59041i	0.306535	−	0.781038i								−0.691730	−	0.722156i	\(-0.743151\pi\)
							0.998265	−	0.0588816i	\(-0.0187535\pi\)
\(13\)	−6.69388	−	0.754220i	−1.85655	−	0.209183i	−0.888316	−	0.459232i	\(-0.848125\pi\)
							−0.968233	+	0.250049i	\(0.919553\pi\)
\(17\)	−2.66399	−	0.0996795i	−0.646113	−	0.0241758i	−0.287673	−	0.957729i	\(-0.592882\pi\)
							−0.358440	+	0.933553i	\(0.616691\pi\)
\(19\)	2.69893	−	4.67469i	0.619178	−	1.07245i	−0.370458	−	0.928849i	\(-0.620799\pi\)
							0.989636	−	0.143598i	\(-0.0458673\pi\)
\(23\)	−0.209936	−	5.61066i	−0.0437747	−	1.16990i	−0.832093	−	0.554636i	\(-0.812858\pi\)
							0.788318	−	0.615267i	\(-0.210952\pi\)
\(29\)	−1.09900	+	0.250839i	−0.204079	+	0.0465797i	−0.323338	−	0.946283i	\(-0.604805\pi\)
							0.119259	+	0.992863i	\(0.461948\pi\)
\(31\)	−3.55443	+	2.05215i	−0.638395	+	0.368577i	−0.783996	−	0.620766i	\(-0.786822\pi\)
							0.145601	+	0.989343i	\(0.453488\pi\)
\(37\)	6.77572	−	3.58107i	1.11392	−	0.588725i	0.194069	−	0.980988i	\(-0.437831\pi\)
							0.919853	+	0.392263i	\(0.128308\pi\)
\(41\)	2.43308	+	5.05234i	0.379983	+	0.789043i	0.999990	+	0.00444673i	\(0.00141544\pi\)
							−0.620007	+	0.784596i	\(0.712870\pi\)
\(43\)	−0.305066	−	0.871829i	−0.0465221	−	0.132953i	0.918278	−	0.395935i	\(-0.129580\pi\)
							−0.964801	+	0.262983i	\(0.915294\pi\)
\(47\)	3.03826	+	2.24234i	0.443176	+	0.327079i	0.792445	−	0.609944i	\(-0.208808\pi\)
							−0.349269	+	0.937023i	\(0.613570\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
245.2.x.a.103.21	✓	624	1.1	even	1	trivial
245.2.x.a.108.21	yes	624	49.10	odd	42	inner
245.2.x.a.152.21	yes	624	5.2	odd	4	inner
245.2.x.a.157.21	yes	624	245.157	even	84	inner