Embedded newform 245.2.x.a.103.14

• Label: 245.2.x.a.103.14
• 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			103.14
Character	\(\chi\)	\(=\)	245.103
Dual form			245.2.x.a.157.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.0769155 + 0.104217i) q^{2} +(-1.66116 + 0.314309i) q^{3} +(0.584565 + 1.89511i) q^{4} +(-1.50931 - 1.64985i) q^{5} +(0.0950130 - 0.197296i) q^{6} +(2.27914 + 1.34370i) q^{7} +(-0.486981 - 0.170402i) q^{8} +(-0.131947 + 0.0517854i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 624 q - 26 q^{2} - 22 q^{3} - 28 q^{5} - 28 q^{6} - 18 q^{7} - 24 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.0769155	+	0.104217i	−0.0543875	+	0.0736924i	−0.830947	−	0.556351i	\(-0.812201\pi\)
							0.776560	+	0.630043i	\(0.216963\pi\)
\(3\)	−1.66116	+	0.314309i	−0.959073	+	0.181466i	−0.641804	−	0.766869i	\(-0.721814\pi\)
							−0.317270	+	0.948335i	\(0.602766\pi\)
\(5\)	−1.50931	−	1.64985i	−0.674982	−	0.737834i
							\(7\)	2.27914	+	1.34370i	0.861434	+	0.507869i
\(11\)	−1.23682	+	3.15138i	−0.372916	+	0.950176i								0.614053	+	0.789265i	\(0.289538\pi\)
							−0.986969	+	0.160911i	\(0.948557\pi\)
\(13\)	−6.14213	−	0.692052i	−1.70352	−	0.191941i	−0.794091	−	0.607799i	\(-0.792053\pi\)
							−0.909430	+	0.415858i	\(0.863481\pi\)
\(17\)	−0.450188	−	0.0168448i	−0.109187	−	0.00408547i	−0.0172580	−	0.999851i	\(-0.505494\pi\)
							−0.0919287	+	0.995766i	\(0.529303\pi\)
\(19\)	−3.38636	+	5.86535i	−0.776884	+	1.34560i	0.156846	+	0.987623i	\(0.449868\pi\)
							−0.933730	+	0.357979i	\(0.883466\pi\)
\(23\)	−0.143129	−	3.82520i	−0.0298444	−	0.797610i	−0.931195	−	0.364520i	\(-0.881233\pi\)
							0.901351	−	0.433089i	\(-0.142577\pi\)
\(29\)	3.87280	−	0.883941i	0.719161	−	0.164144i	0.152748	−	0.988265i	\(-0.451188\pi\)
							0.566413	+	0.824122i	\(0.308331\pi\)
\(31\)	4.62475	−	2.67010i	0.830629	−	0.479564i	−0.0234390	−	0.999725i	\(-0.507462\pi\)
							0.854068	+	0.520161i	\(0.174128\pi\)
\(37\)	−1.20552	+	0.637134i	−0.198186	+	0.104744i	−0.563417	−	0.826173i	\(-0.690514\pi\)
							0.365232	+	0.930917i	\(0.380990\pi\)
\(41\)	5.04961	+	10.4856i	0.788617	+	1.63758i	0.770245	+	0.637748i	\(0.220134\pi\)
							0.0183720	+	0.999831i	\(0.494152\pi\)
\(43\)	2.89531	+	8.27433i	0.441531	+	1.26182i	0.922810	+	0.385256i	\(0.125887\pi\)
							−0.481279	+	0.876568i	\(0.659828\pi\)
\(47\)	−4.33222	−	3.19732i	−0.631919	−	0.466378i	0.230159	−	0.973153i	\(-0.426075\pi\)
							−0.862078	+	0.506775i	\(0.830837\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	245.2.x.a.103.14	✓	624
5.2	odd	4	inner	245.2.x.a.152.14	yes	624
49.10	odd	42	inner	245.2.x.a.108.14	yes	624
245.157	even	84	inner	245.2.x.a.157.14	yes	624

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
245.2.x.a.103.14	✓	624	1.1	even	1	trivial
245.2.x.a.108.14	yes	624	49.10	odd	42	inner
245.2.x.a.152.14	yes	624	5.2	odd	4	inner
245.2.x.a.157.14	yes	624	245.157	even	84	inner