Embedded newform 245.2.x.a.108.8

• Label: 245.2.x.a.108.8
• 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			108.8
Character	\(\chi\)	\(=\)	245.108
Dual form			245.2.x.a.152.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.06718 + 0.787615i) q^{2} +(-0.505129 + 2.66967i) q^{3} +(-0.0709737 + 0.230091i) q^{4} +(-0.576678 - 2.16043i) q^{5} +(-1.56361 - 3.24687i) q^{6} +(-2.33061 - 1.25230i) q^{7} +(-0.981614 - 2.80529i) q^{8} +(-4.07936 - 1.60103i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 624 q - 26 q^{2} - 22 q^{3} - 28 q^{5} - 28 q^{6} - 18 q^{7} - 24 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.06718	+	0.787615i	−0.754611	+	0.556928i	−0.901786	−	0.432182i	\(-0.857744\pi\)
							0.147176	+	0.989110i	\(0.452982\pi\)
\(3\)	−0.505129	+	2.66967i	−0.291636	+	1.54133i	0.463697	+	0.885994i	\(0.346523\pi\)
							−0.755333	+	0.655341i	\(0.772525\pi\)
\(5\)	−0.576678	−	2.16043i	−0.257898	−	0.966172i
							\(7\)	−2.33061	−	1.25230i	−0.880888	−	0.473324i
\(11\)	−0.869792	−	2.21619i	−0.262252	−	0.668207i								0.737724	−	0.675102i	\(-0.235901\pi\)
							−0.999976	+	0.00689489i	\(0.997805\pi\)
\(13\)	0.458349	+	4.06796i	0.127123	+	1.12825i	0.881625	+	0.471951i	\(0.156450\pi\)
							−0.754501	+	0.656299i	\(0.772121\pi\)
\(17\)	0.0973853	+	2.60268i	0.0236194	+	0.631242i	0.959958	+	0.280144i	\(0.0903821\pi\)
							−0.936339	+	0.351098i	\(0.885808\pi\)
\(19\)	−2.06051	−	3.56891i	−0.472714	−	0.818764i	0.526799	−	0.849990i	\(-0.323392\pi\)
							−0.999512	+	0.0312261i	\(0.990059\pi\)
\(23\)	−3.64174	−	0.136264i	−0.759356	−	0.0284131i	−0.345091	−	0.938569i	\(-0.612152\pi\)
							−0.414265	+	0.910156i	\(0.635961\pi\)
\(29\)	−0.312373	−	0.0712970i	−0.0580062	−	0.0132395i	0.193419	−	0.981116i	\(-0.438042\pi\)
							−0.251426	+	0.967877i	\(0.580899\pi\)
\(31\)	−5.82339	−	3.36213i	−1.04591	−	0.603857i	−0.124409	−	0.992231i	\(-0.539704\pi\)
							−0.921502	+	0.388374i	\(0.873037\pi\)
\(37\)	−2.64950	+	5.01310i	−0.435575	+	0.824148i	−0.999999	−	0.00133693i	\(-0.999574\pi\)
							0.564424	+	0.825485i	\(0.309098\pi\)
\(41\)	4.47810	−	9.29886i	0.699361	−	1.45224i	−0.183701	−	0.982982i	\(-0.558808\pi\)
							0.883062	−	0.469256i	\(-0.155478\pi\)
\(43\)	−9.83454	−	3.44126i	−1.49975	−	0.524787i	−0.549293	−	0.835630i	\(-0.685103\pi\)
							−0.950461	+	0.310843i	\(0.899389\pi\)
\(47\)	3.32377	+	4.50355i	0.484822	+	0.656910i	0.976914	−	0.213633i	\(-0.0685297\pi\)
							−0.492092	+	0.870543i	\(0.663768\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	245.2.x.a.108.8	yes	624
5.2	odd	4	inner	245.2.x.a.157.8	yes	624
49.5	odd	42	inner	245.2.x.a.103.8	✓	624
245.152	even	84	inner	245.2.x.a.152.8	yes	624

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
245.2.x.a.103.8	✓	624	49.5	odd	42	inner
245.2.x.a.108.8	yes	624	1.1	even	1	trivial
245.2.x.a.152.8	yes	624	245.152	even	84	inner
245.2.x.a.157.8	yes	624	5.2	odd	4	inner