Embedded newform 245.2.x.a.103.19

• Label: 245.2.x.a.103.19
• 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			103.19
Character	\(\chi\)	\(=\)	245.103
Dual form			245.2.x.a.157.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.796498 - 1.07922i) q^{2} +(2.57654 - 0.487508i) q^{3} +(0.0592118 + 0.191960i) q^{4} +(-1.63558 - 1.52475i) q^{5} +(1.52608 - 3.16894i) q^{6} +(-1.87091 + 1.87075i) q^{7} +(2.78641 + 0.975007i) q^{8} +(3.60827 - 1.41614i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 624 q - 26 q^{2} - 22 q^{3} - 28 q^{5} - 28 q^{6} - 18 q^{7} - 24 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.796498	−	1.07922i	0.563209	−	0.763121i	−0.426766	−	0.904362i	\(-0.640347\pi\)
							0.989975	+	0.141241i	\(0.0451092\pi\)
\(3\)	2.57654	−	0.487508i	1.48757	−	0.281463i	0.622301	−	0.782778i	\(-0.286198\pi\)
							0.865265	+	0.501315i	\(0.167150\pi\)
\(5\)	−1.63558	−	1.52475i	−0.731455	−	0.681889i
							\(7\)	−1.87091	+	1.87075i	−0.707136	+	0.707078i
\(11\)	1.32624	−	3.37920i	0.399876	−	1.01887i								−0.579167	−	0.815209i	\(-0.696622\pi\)
							0.979042	−	0.203657i	\(-0.0652828\pi\)
\(13\)	−3.93188	−	0.443017i	−1.09051	−	0.122871i	−0.451650	−	0.892195i	\(-0.649164\pi\)
							−0.638858	+	0.769325i	\(0.720593\pi\)
\(17\)	3.16567	+	0.118451i	0.767789	+	0.0287286i	0.418416	−	0.908255i	\(-0.362585\pi\)
							0.349372	+	0.936984i	\(0.386395\pi\)
\(19\)	−3.27569	+	5.67367i	−0.751496	+	1.30163i	0.195602	+	0.980683i	\(0.437334\pi\)
							−0.947098	+	0.320946i	\(0.895999\pi\)
\(23\)	0.0875098	+	2.33875i	0.0182471	+	0.487663i	0.978220	+	0.207570i	\(0.0665556\pi\)
							−0.959973	+	0.280093i	\(0.909635\pi\)
\(29\)	−8.14532	+	1.85912i	−1.51255	+	0.345229i	−0.896701	−	0.442637i	\(-0.854043\pi\)
							−0.615846	+	0.787866i	\(0.711186\pi\)
\(31\)	−1.63334	+	0.943010i	−0.293357	+	0.169370i	−0.639455	−	0.768829i	\(-0.720840\pi\)
							0.346098	+	0.938198i	\(0.387507\pi\)
\(37\)	8.33924	−	4.40742i	1.37096	−	0.724575i	0.390375	−	0.920656i	\(-0.372345\pi\)
							0.980588	+	0.196081i	\(0.0628215\pi\)
\(41\)	0.959744	+	1.99293i	0.149887	+	0.311243i	0.962371	−	0.271741i	\(-0.0875993\pi\)
							−0.812484	+	0.582984i	\(0.801885\pi\)
\(43\)	−3.38697	−	9.67939i	−0.516507	−	1.47609i	−0.846644	−	0.532160i	\(-0.821380\pi\)
							0.330136	−	0.943933i	\(-0.392905\pi\)
\(47\)	−5.98363	−	4.41612i	−0.872803	−	0.644158i	0.0624851	−	0.998046i	\(-0.480097\pi\)
							−0.935288	+	0.353888i	\(0.884859\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	245.2.x.a.103.19	✓	624
5.2	odd	4	inner	245.2.x.a.152.19	yes	624
49.10	odd	42	inner	245.2.x.a.108.19	yes	624
245.157	even	84	inner	245.2.x.a.157.19	yes	624

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
245.2.x.a.103.19	✓	624	1.1	even	1	trivial
245.2.x.a.108.19	yes	624	49.10	odd	42	inner
245.2.x.a.152.19	yes	624	5.2	odd	4	inner
245.2.x.a.157.19	yes	624	245.157	even	84	inner