Embedded newform 245.2.x.a.17.7

• Label: 245.2.x.a.17.7
• Level: $245$
• Weight: $2$
• Character: 245.17
• 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			17.7
Character	\(\chi\)	\(=\)	245.17
Dual form			245.2.x.a.173.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.959983 + 1.11552i) q^{2} +(1.54513 + 2.92353i) q^{3} +(-0.0247339 - 0.164099i) q^{4} +(-2.03817 - 0.919712i) q^{5} +(-4.74455 - 1.08291i) q^{6} +(0.161542 - 2.64082i) q^{7} +(-2.28548 - 1.43606i) q^{8} +(-4.46963 + 6.55575i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 624 q - 26 q^{2} - 22 q^{3} - 28 q^{5} - 28 q^{6} - 18 q^{7} - 24 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(101\)	\(197\)
\(\chi(n)\)	\(e\left(\frac{25}{42}\right)\)	\(e\left(\frac{1}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.959983	+	1.11552i	−0.678810	+	0.788792i	−0.986686	−	0.162635i	\(-0.948001\pi\)
							0.307876	+	0.951426i	\(0.400382\pi\)
\(3\)	1.54513	+	2.92353i	0.892081	+	1.68790i	0.709495	+	0.704710i	\(0.248923\pi\)
							0.182586	+	0.983190i	\(0.441553\pi\)
\(5\)	−2.03817	−	0.919712i	−0.911497	−	0.411308i
							\(7\)	0.161542	−	2.64082i	0.0610571	−	0.998134i
\(11\)	−2.72820	+	1.86005i	−0.822582	+	0.560827i								−0.899858	−	0.436183i	\(-0.856330\pi\)
							0.0772760	+	0.997010i	\(0.475378\pi\)
\(13\)	0.818278	+	2.33851i	0.226950	+	0.648585i	0.999900	+	0.0141186i	\(0.00449424\pi\)
							−0.772951	+	0.634466i	\(0.781220\pi\)
\(17\)	0.613278	+	0.267570i	0.148742	+	0.0648953i	0.472944	−	0.881092i	\(-0.343191\pi\)
							−0.324203	+	0.945988i	\(0.605096\pi\)
\(19\)	2.66038	+	4.60792i	0.610334	+	1.05713i	0.991184	+	0.132493i	\(0.0422981\pi\)
							−0.380850	+	0.924637i	\(0.624369\pi\)
\(23\)	0.449805	+	1.03096i	0.0937909	+	0.214971i	0.957133	−	0.289647i	\(-0.0935380\pi\)
							−0.863343	+	0.504618i	\(0.831633\pi\)
\(29\)	6.41683	+	5.11725i	1.19157	+	0.950249i	0.999515	−	0.0311340i	\(-0.00991185\pi\)
							0.192060	+	0.981383i	\(0.438483\pi\)
\(31\)	1.62430	+	0.937790i	0.291733	+	0.168432i	0.638723	−	0.769437i	\(-0.279463\pi\)
							−0.346990	+	0.937869i	\(0.612796\pi\)
\(37\)	−1.46121	−	1.97987i	−0.240221	−	0.325488i	0.667856	−	0.744290i	\(-0.267212\pi\)
							−0.908078	+	0.418802i	\(0.862450\pi\)
\(41\)	1.69549	−	0.386984i	0.264791	−	0.0604368i	−0.0880642	−	0.996115i	\(-0.528068\pi\)
							0.352855	+	0.935678i	\(0.385211\pi\)
\(43\)	0.734804	+	1.16943i	0.112057	+	0.178337i	0.898044	−	0.439905i	\(-0.144988\pi\)
							−0.785988	+	0.618242i	\(0.787845\pi\)
\(47\)	−1.74250	−	1.49954i	−0.254170	−	0.218731i	0.515969	−	0.856607i	\(-0.327432\pi\)
							−0.770139	+	0.637876i	\(0.779813\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	245.2.x.a.17.7	✓	624
5.3	odd	4	inner	245.2.x.a.213.20	yes	624
49.26	odd	42	inner	245.2.x.a.222.20	yes	624
245.173	even	84	inner	245.2.x.a.173.7	yes	624

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
245.2.x.a.17.7	✓	624	1.1	even	1	trivial
245.2.x.a.173.7	yes	624	245.173	even	84	inner
245.2.x.a.213.20	yes	624	5.3	odd	4	inner
245.2.x.a.222.20	yes	624	49.26	odd	42	inner