Embedded newform 245.2.x.a.103.8

• Label: 245.2.x.a.103.8
• 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			103.8
Character	\(\chi\)	\(=\)	245.103
Dual form			245.2.x.a.157.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.787615 + 1.06718i) q^{2} +(2.66967 - 0.505129i) q^{3} +(0.0709737 + 0.230091i) q^{4} +(-0.248242 - 2.22225i) q^{5} +(-1.56361 + 3.24687i) q^{6} +(1.25230 + 2.33061i) q^{7} +(-2.80529 - 0.981614i) q^{8} +(4.07936 - 1.60103i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 624 q - 26 q^{2} - 22 q^{3} - 28 q^{5} - 28 q^{6} - 18 q^{7} - 24 q^{8}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.787615	+	1.06718i	−0.556928	+	0.754611i	−0.989110	−	0.147176i	\(-0.952982\pi\)
							0.432182	+	0.901786i	\(0.357744\pi\)
\(3\)	2.66967	−	0.505129i	1.54133	−	0.291636i	0.655341	−	0.755333i	\(-0.272525\pi\)
							0.885994	+	0.463697i	\(0.153477\pi\)
\(5\)	−0.248242	−	2.22225i	−0.111017	−	0.993818i
							\(7\)	1.25230	+	2.33061i	0.473324	+	0.880888i
\(11\)	−0.869792	+	2.21619i	−0.262252	+	0.668207i								−0.999976	−	0.00689489i	\(-0.997805\pi\)
							0.737724	+	0.675102i	\(0.235901\pi\)
\(13\)	4.06796	+	0.458349i	1.12825	+	0.127123i	0.656299	−	0.754501i	\(-0.272121\pi\)
							0.471951	+	0.881625i	\(0.343550\pi\)
\(17\)	−2.60268	−	0.0973853i	−0.631242	−	0.0236194i	−0.280144	−	0.959958i	\(-0.590382\pi\)
							−0.351098	+	0.936339i	\(0.614192\pi\)
\(19\)	2.06051	−	3.56891i	0.472714	−	0.818764i	−0.526799	−	0.849990i	\(-0.676608\pi\)
							0.999512	+	0.0312261i	\(0.00994120\pi\)
\(23\)	−0.136264	−	3.64174i	−0.0284131	−	0.759356i	−0.938569	−	0.345091i	\(-0.887848\pi\)
							0.910156	−	0.414265i	\(-0.135961\pi\)
\(29\)	0.312373	−	0.0712970i	0.0580062	−	0.0132395i	−0.193419	−	0.981116i	\(-0.561958\pi\)
							0.251426	+	0.967877i	\(0.419101\pi\)
\(31\)	−5.82339	+	3.36213i	−1.04591	+	0.603857i	−0.921502	−	0.388374i	\(-0.873037\pi\)
							−0.124409	+	0.992231i	\(0.539704\pi\)
\(37\)	−5.01310	+	2.64950i	−0.824148	+	0.435575i	−0.825485	−	0.564424i	\(-0.809098\pi\)
							0.00133693	+	0.999999i	\(0.499574\pi\)
\(41\)	4.47810	+	9.29886i	0.699361	+	1.45224i	0.883062	+	0.469256i	\(0.155478\pi\)
							−0.183701	+	0.982982i	\(0.558808\pi\)
\(43\)	−3.44126	−	9.83454i	−0.524787	−	1.49975i	−0.835630	−	0.549293i	\(-0.814897\pi\)
							0.310843	−	0.950461i	\(-0.399389\pi\)
\(47\)	−4.50355	−	3.32377i	−0.656910	−	0.484822i	0.213633	−	0.976914i	\(-0.431470\pi\)
							−0.870543	+	0.492092i	\(0.836232\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
245.2.x.a.103.8	✓	624	1.1	even	1	trivial
245.2.x.a.108.8	yes	624	49.10	odd	42	inner
245.2.x.a.152.8	yes	624	5.2	odd	4	inner
245.2.x.a.157.8	yes	624	245.157	even	84	inner